Location and space-economy: a general theory relating to industrial location, market areas, land use, trade, and urban structure

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Location and Space-Economy
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Location

and

Space-Economy

A General Theory Relating to Industrial Location,

Market Areas, Land Use, Trade,

and Urban Structure

by

WALTER ISARD

Professor of Economics
University of Pennsylvania

PUBLISHED JOINTLY BY

The Technology Press of
Massachusetts Institute of Technology

AND

John Wiley & Sons, Inc., New York
Chapman & Hall, Ltd., London

Copyright © 1956

by

Tlie Massachusetts Institute of Technology

All Rights Reserved

This book or any part thereof must
not be reproduced in any form without
the written permission of the publisher.

Library of Congress Catalog Card Number: 56-11026
Printed in the United States of America

To
Alvin H. Hansen

and
Abbott P. Usher

Digitized by the Internet Archive

in 2010 with funding from

Lyrasis IVIembers and Sloan Foundation

http://www.archive.org/details/locationspaceecoOOisar

Preface

Over the historical record, the process of social development and
economic growth has been for the most part cumulative though at
times seemingly erratic. On occasion it has been recessive. Some of its
stages have been forced to start at their origins more than once.
Moreover, these stages have started independently at diverse places
on the earth at different times. Together with the manifold cultural,
political, social, and economic forces which have evolved this fact has
led, over the centuries, to marked regional differentiations and to tre-
mendous disparities in the welfare of the peoples of the several regions
of the world.

Such a record of man's adaptation to, and interaction with, his
physical environment suggests that a comprehensive theory of society
or economy should embrace both time and space dimensions. It should
be able to unravel the dynamic interplay of forces not only currently
but also over the long past. Its propositions should be testable against
the background of historic development in the several regions of the
world and concomitantly should offer explanation of this development.
Its hypotheses should encompass the influence of past events and inter-
mingling of forces upon existing economic and social structure and
conditions. Ideally, its conceptual framework should enable one to
anticipate the course of future development, given certain premises
and judgments.

Unhappily, the state of the social science disciplines leaves very

viii PREFACE

much to be desired in the way of achieving such a comprehensive
theory. The dynamic frameworks of these disciplines, frameworks
which are designed to catch the effect of what some characterize as the
"time variable," are rather crude and even at times naive. Still more
so are their spatial and regional frameworks.

It is the basic objective of this volume to improve the spatial and
regional frameworks of the social science disciplines, particularly of
economics, through the development of a more adequate general theory
of location and space-economy. As a matter of course, we must work
within the setting of dynamic techniques and frameworks currently
available. Since these are inadequate, a chief limitation of this volume
is its inability to cope satisfactorily with the development process over
time.

Another limitation is imposed by the inherent weaknesses of general
theories. General theories have a widespread reputation for being
sterile in terms of concrete social problems and related policy recom-
mendations. As one critic has put it, we "are all too apt, seizing upon
the conditions of a particular period of time and extent of space and
applying the simplifying processes of selection and emphasis to which
economic theorizing seems all too prone, to come forth with a system
of universalized 'laws,' a system which is perhaps not implausible at
its time and place of origin, but which, as it becomes increasingly
concerned about the consistency of its internal logic, turns in upon
itself, and grows more and more remote from reality." i

Admittedly the general theory of location and space-economy evolved
in this volume is of little direct utility for handling specific problems
of reality. That it should be of immediate relevance was not the
author's intention. But as one proceeds to the materials to be con-
tained in a second volume, ^ where less general and more useful theo-
retical frameworks are developed, he may perceive how the general
principles and constructs derived in this volume are of use in furnish-
ing penetrating insights into the operation of economic processes in the
real world. Also, they facilitate the choice of those structural relations
subject to empirical estimation which are the more significant ones for
analytic purposes. At least the author has found the general statement
of conditions of equilibrium extremely helpful as a background against
which various location and spatial doctrines could be contrasted and

1 John H. Williams, "An Economist's Confessions," The American Economic
Review, XLII, March 1952, pp. 6-7. The words in itahcs have been inserted by
the author.

Significant, however, is Professor WiUiams' wilUngness to attribute some signif-
icance to Keynes' remark that without theory we are "lost in the woods" (ibid.,
p. 23).

2 For some details on the contents of this second volume, see pp. x-xi.

PREFACE ix

subsequently ordered and evaluated according to content. This eased
the process of digestion, enabling the author to achieve for himself an
improved and less confusing reformulation of ideas. It also provided
a sharper conception of the manifold spatial processes at play and the
manner in which individual ones are related to each other and to the
array as a whole.

Still another serious limitation of the analysis is to be noted. A
presentation of conditions of equilibrium in a theoretical system may
seem to imply a tendency toward the attainment of a state of equilib-
rium in the real world. But in a full historic sense, actual economic
life never does realize a state of equilibrium. There are always changes
impinging upon the economy. The process of adjustment is constantly
in operation. Witness, for example, the adaptation of population to
environment. There has never been a complete adjustment which
might be said to characterize an optimum or equilibrium spatial dis-
tribution of population.

As Usher has neatly put it,

"Classical and neo-classical theory rest upon a concept of equilibrium that
becomes a source of serious difficulty in historical analysis. It is implied
that disturbances of the socio-economic equilibrium are small in magnitude
and quickly corrected by adaptive changes. Such disturbances do exist,
and market processes have developed that deal with some measure of
adequacy with these minor disturbances of the equilibrium. But these
are not the only disturbances that occur in the socio-economic world.
The world economy is beset by other disturbances, whose magnitude is
of such an order that adjustments require several generations. . ." 3

"The most dangerous of all transfers of the equilibrium concept appears
when the ideal of stability is represented as a characteristic of long-run
conditions. All the data of history show that the empirical phenomena
are dynamic, and that for the world economy as a whole secular change
is positive; in some periods, the rates of growth are not large; in partic-
ular regions contraction may occur over substantial periods of time.
Whether it be growth, or cumulative change that may be classed as
progress, secular change cannot be described as a condition of equi-
librium." 4

Despite the disrupting effects of technological advance and other
dynamic phenomena and the consequent failure to attain equilibrium
in the secular sense, ^ there is still value in equilibrium analysis. It is

3 Abbott P. Usher, "The Pattern of the World Economy," unpublished essay,
pp. 10-11.

"* Ibid., p. 17. These statements were written in the 1930's. and should be modi-
fied somewhat in view of subsequent developments in equilibrium analysis.

5 Usher's thesis is of broader content. Its more extensive ramifications are worth
noting here. In Usher's words:

"The broad elements of differentiation among societies in their various his-
torical worlds rest on differences in the maturity of settlement of the various

X PREFACE

thought pertinent and worthwhile by some who conceive of the socio-
economic system as a body tending toward a moving equilibrium and
by others who find in equilibrium analysis categories of reference with
which the extent of disequilibrium can be measured. Most important,
equilibrium analysis is valuable because it enables one to grasp better
the laws of change and the workings of a system. In doing so, it
necessarily casts light upon the long-run interaction of diverse forces
and can yield valuable insights for historical trend projection in any
concrete situation. ^

As the reader will quickly perceive, the first and third chapters of
this volume were designed for a book which would encompass a broader
field than is actually covered. The initial plan was to devote a con-
siderable part of this volume to regional analysis. However, as the
structure of the manuscript took form, it became advisable, in view
of the quantity of materials to be developed, to present the materials
in two volumes. During the next years therefore a companion volume
will be prepared which will treat the principles of regional science and
general regional theory. (The impatient reader may glean some of
the contents of this second volume from the articles cited below.'')
Together these two volumes will form a unit, to which the first and
third chapters of this first completed volume relate.

regions, differences in culture, differences in technical knowledge, differences
in the material resources of the region. Much theoretical discussion presumes
that the first three classes of differences will be progressively equalized by the
diffusion of population, culture, and technical skills. Some even presume that
differentiation of material resources becomes less and less important as tech-
nical knowledge progresses. . ."

"There is nothing in the historical record to warrant the presumption that the
elements of difference have diminished significantly in intensity. Changes of
great importance have occurred, but it would not be difficult to defend the
proposition that the world of effective contacts at the present time is more
highly differentiated than the world as known to the Romans of the Augustan
age, or the world of the third century a.d." (Ibid., p. 4.)

". . . it must be evident that the discovery of means of utilizing new ma^-
terials merely sets new limits to the scarcities that dominate our social life.
The unbalanced and unequal distribution of material resources must always
leave us with a world pattern of unequal and unbalanced distribution of the
resources of primary importance to social life. Regional differentiation will
remain important however much the specific patterns of distribution may be
changed by new discoveries and new technologies." (Ibid., pp 7-8)

6 For example, concentration upon substitution among transport inputs along
the lines of classical and neoclassical equilibrium analysis to be discussed in a
subsequent chapter yields a framework for partially explaining not only the
current locational pattern of the iron and steel industry but also the changing
historical pattern over the last two centuries.

7 The author's writings, which contain materials to be incorporated and further

PREFACE XI

The writing of this volume was begun some ten years ago. Conse-
quently, certain materials may seem a bit dated. In particular, Chap.
2, which was largely completed by 1947, points up an imbalance in
the Anglo-Saxon literature which during recent years has been some-
what corrected. Nonetheless, the earlier statements have not been
qualified or tempered. The sharp criticism and emphasis of these
statements are retained in order not to lose whatever potency and vigor
the argument as first developed may have contained.

In the light of this partial correction of the imbalance in the Anglo-
Saxon literature, I have changed my terminology, for which I must
apologize. In previous writings the movement of a unit weight of a
particular commodity over a unit of distance was defined as a distance
input. Such a movement could also have been defined as a transport
input. This latter term would have been more in keeping with the
existing usage of words. However, given the neglect by economic
theorists of the distance variable and role of space, it was decided in
these early writings to use the term distance inputs. This was judged
desirable in order to make as explicit as possible the significance of
transport costs and the distance factor in shaping economic phenomena.

expanded in this volume, are in chronological order of development: "Inter-
regional and Regional Input-Output Analysis: A Model of a Space-Economy,"
Review of Economics and Statistics, Vol. 33 (November 1951), pp. 318-328; "Re-
gional and National Product Projections and Their Interrelations" (with G.
Freutel), Long-Range Economic Projection, National Bureau of Economic Re-
search, Studies in Income and Wealth, Vol. 16, Princeton University Press,
Princeton, 1954, pp. 427-471 ; "Some Empirical Results and Problems of Regional
Input-Output Analysis," in Leontief et al., Studies in the Structure of the Ameri-
can Economy, Oxford University Press, New York, 1953, pp. 116-181; "Some
Emerging Concepts and Techniques for Regional Analysis," Zeitschrift fur die
Gesamte Staatswissenschaft, Vol. 109 (1953), pp. 240-250; "Regional Commodity
Balances and Interregional Commodity Flows," Papers and Proceedings of the
American Economic Association, Vol. 43 (May 1953), pp. 167-180; "The Impact
of Steel upon the Greater New York-Philadelphia Industrial Region: A Study
in Agglomeration Projection" (with R. E. Kuenne), Review of Economics and
Statistics, Vol. 35 (November 1953), pp. 289-301; "Location Theory and Trade
Theory: Short-run Analysis," Quarterly Journal of Economics, Vol. 68 (May
1954), pp. 305-320; "Economic Structural Interrelations of Metropohtan Regions"
(with R. Kavesh), American Journal of Sociology, Vol. 60 (September 1954), pp.
152-162 ; "Industrial Complex Analysis and Regional Development with Particular
Reference to Puerto Rico" (with T. Vietorisz), Abstract in Papers and Proceed-
ings of the Regional Science Association, Vol. 1 (1955); "The Value of the
Regional Approach in Economic Analysis," Regional Income, National Bureau
of Economic Research, Studies in Income and Wealth, Vol. 21, Princeton Univer-
sity Press, Princeton, 1956; "Regional Science, The Concept of Region, and
Regional Structure," Papers and Proceedings of the Regional Science Association,
Vol. 2 (1956), pp. 13-26; and an extensive manuscript on tools and techniques
of regional analysis to be published by Resources for the Future, Inc.

xii PREFACE

Now that economists are more aware of the space axis and the spatial
aspects of their subject matter, it seems appropriate to employ the
more customary terminology of trans-port inputs, particularly in order
to facilitate interdisciplinary exchange of knowledge, ideas, and tech-
niques of analysis in the broad field of regional and area studies.

I am indebted to a host of individuals, each of whom has contributed
in one way or other to the writing and preparation of this volume.
The major influence of one of the two outstanding economists and
personalities to whom this book is dedicated is immediately apparent
in the first few sentences of the Preface. Professor Abbott P. Usher
has been a constant source of inspiration. His ever-present encourage-
ment and guidance and his vast fund of knowledge, which was always
generously made available, have been invaluable in the development
of ideas.

An equally strong influence stems from the teachings and writings
of Professor Alvin H. Hansen. The tremendous stimulation derived
from discussions with Professor Hansen, especially in the formative
stages of this study, cleared the way of obstacles which often beset an
attempt at general theorizing. His persistent urgings to make bold
and creative attacks upon problems have added immeasurably to the
contents of the volume.

My wife has given freely of her time and patience in the preparation
of this volume. She has assisted not only in the presentation of ideas
but also in their logical derivation.

Many others — former teachers, graduate students, and associates —
have been helpful in diverse ways. At the risk of failing to mention
all who deserve such mention, I gratefully acknowledge the help of
Joseph Airov, Martin J. Beckmann, John F. Bell, Edward H.
Chamberlin, Edgar Dunn, Guy Freutel, Gottfried von Haberler,
Seymour E. Harris, Edgar M. Hoover, Robert Kavesh, John Kimber,
Robert E. Kuenne, Sven Laursen, Wassily W. Leontief, Fritz Machlup,
Russell Mack, Leon Moses, C. Reinold Noyes, Merton J. Peck, Win-
field Riefler, Paul A. Samuelson, Eugene Schooler, Benjamin Stevens,
Edward L. Ullman, and Thomas Vietorisz.

Gerald A. P. Carrothers has rendered invaluable service in the con-
struction of the figures; and Richard Pfister, in the development of
the index. I am indebted to Alexia Hanitsch and Gabrielle Fuchs for
competent secretarial assistance.

The Harvard University Press, the Yale University Press, Richard
D. Irwin, Inc., the Addison-Wesley Publishing Company, Inc., and
the editors of the Quarterly Journal of Economics, Econometrica, and
Metroeconomica have kindly granted permission to use previously

PREFACE xiii

published materials in direct or amended form. Chapters 2, 4, 9, and
parts of Chap. 5 are largely drawn from articles appearing in the
November 1949, May 1951, February 1954, and August 1951 issues,
respectively, of the Quarterly Journal of Economics. Some materials
from these articles also appear in other chapters of this book. Chapter
10 is a splicing of two articles, one of which appeared in Econometrica,
Vol. 20, No. 3, July 1952, and the other in Metroeconomica, Vol. 5,
No. 1, April 1953.

I am grateful to the Social Science Research Council for a post-
doctoral fellowship which enabled me to initiate the development of
this general theory. The writing of the final chapters, the construction
of the index, and the completion of this book has been facilitated by
a grant from Resources for the Future, Inc.

Walter Isard
Cambridge, Massachusetts
September, 1956

Contents

chapter page

1 • Introduction: Posing the Location and Regional Problem ... 1

1 • Some Basic Development Processes 1

2 • Some Fundamental Questions 9

3 • Possible Theoretical Approaches 15

2 • Some General Theories of Location and Space-Economy .... 24

1 • The Anglo-Saxon Bias 24

2 • Some Early Attempts at General Theory 27

3 • Predohl's Conception 31

4 • Weigmann's Formulation 37

5 • Palander's Criticisms and Losch's General System 42

6 • Ohlin's View of Trade and Location Theory 50

7 • Closing Remarks 53

3 • Some Empirical Regularities of the Space-Economy 55

4 • Transport Inputs and Related Spatial Concepts 77

1 • General Introductory Remarks 77

2 • Transport Inputs Contrasted with Capital Inputs 81

3 • Transport Rate: The Price of a Transport Input 86

4 • Transport Inputs and the Classification of Factors 89

5 • The Locational Equilibrium of the Firm: Transport — Orientation 91

1 • Some Definitional and Classificational Remarks 91

2 • Transport-oriented Equilibrium under SimpHfied Conditions 95

3 • Transport-oriented Equilibrium with Realistic Rate Structures 104

4 • Transport-oriented Equilibrium Further Extended 113

Appendix to Chapter 5 — Transport Inputs and Some Formulations of
the Transport-orientation Problem 119

xvi CONTENTS

chapter page

6 • The Locational Equilibrium of the Firm: Labor and Other

Orientation 126

1 • Introductory Remarks 126

2 • Labor Orientation 127

3 • Some Other Forms of Orientation 131

4 • A Re-examination of the Substitution Framework for Spatial

Analysis 138
Appendix to Chapter 6 — The Labor Coefficient and a Related Ratio 141

7 • Market and Supply Area Analysis and Competitive Locational

Equilibrium 143

1 • Market Area Analysis 143

2 • Supply (Purchasing) Area Analysis 154

3 • Some Remarks on Spatial Pricing Systems and Competitive

Locational Equihbrium 158

4 • Concluding Remarks 169

8 • Agglomeration Analysis and Agricultural Location Theory . . . 172

1 • Economies of Scale 173

2 • Localization Economies 176

3 • Urbanization Economies 182

'4 • Agricultural Location Theory 188

Appendix to Chapter 8 — Some Theoretical Notes on Urban
Land-use 200

9 • Some Basic Interrelations of Location and Trade Theory .... 207

1 • Prehminary Remarks 207

2 • A Fusion of Opportunity Cost Doctrine and Transport-

orientation 208

3 • The Effects of a Change in the Distance Variable upon Trade,

Industrial Location, and Geographic SpeciaHzation 215

4 • Some Conclusions 219

10 • Aspects of General Location Theory : A Mathematical Formulation 221

1 • Weberian Theory Restated and Generalized 222

2 • Inclusion of Market and Supply Areas as Variables 231

3 • The Analysis Extended to the Case of Many Producers 235

4 • Losch Market Area Analysis Encompassed 239

5 • Agriculture Location Theory Embraced and GeneraUzed 243

6 • Concluding Remarks 251

11 • Partial Graphic Synthesis and Summary 254

List of Figures

FIGURE PAGE

1 • A hypothetical region. 4

2 • Communities of 2500 or more inhabitants, ranked in decreasing order

of population size. U.S.A. 1790-1930. 56

3 • Frequency distribution of cities, at shortest distance intervals, by size

classes, Iowa 1930. 59

4 • Railway express. Movement by weight (less than carload lots) between

13 arbitrary cities in the U.S.A., May 1939. 61

5 • Telephone messages. Number of messages interchanged between 311

arbitrary pairs of cities in the U.S.A., 1940. 62

6 • Bus passengers. Movement of persons by highway bus between 29 arbi-

trary cities in the U.S.A. during intervals in 1933 and 1934. 63

7 • Family migration. Number of families (plus 100) moving varying dis-

tances within or between separated areas in Cleveland, 1933-1935. 64

8 • Contours of equal population potential for the United States, 1940. 67

9 • Population per square mile and dollar value of selected sustenance

activities per 1/100 sq mile of hinterland, by distance outward from the
nearest metropolis: for 67 metropoHtan communities in the U.S.A.,
1939-1940. 69

10 • U.S.A. Class I railroad shipments. Tonnage of all commodities, by

distance shipped (25-mile zones), 1949. Bar chart. 71

11 • U.S.A. Class I railroad shipments. Tonnage of all commodities, by

distance shipped (100-mile zones), 1949. Bar chart. 72

12 • U.SA. Class I railroad shipments. Tonnage of all commodities, by

distance shipped (25-mile zones), 1949. Point chart. 73

xvii

xviii LIST OF FIGURES

FIGURE PAGE

13 • World ocean-going freight. Tonnage, by distance shipped (500-mile

zones), 1925. 74

14 • World ocean-going freight. Tonnage, by distance shipped (2000-mile

zones), 1925. 75

15 • A locational line. 96

16 • A transformation line for the line case. 97

17 • A locational triangle. 98

18 • A transformation line for the triangle case. 98

19 • A four-sided locational polygon. 99

20 • A transformation line for a four-sided polygon. 100

21 • Locational equilibrium: discontinuous transformation line. 102

22 • Locational equilibrium: reahstic rate structures. 106

23 • Locational equilibrium: break in transport system. Ill

24 • Shift of transformation curve and equilibrium site with change in

weights. 114

25 • An outlay-substitution line in a case of labor orientation. 129

26 • A revenue-outlay substitution line. 134

27 • Margin hnes: two competitors. 149

28 • The division of a spatial market: two competitors. 152

29 • The division of a market among several supply points. 156

30 • A case of agglomeration from economies of scale. 173

31 • A case of indeterminacy in location. 174

32 • Non-intersecting critical isodapanes: no agglomeration. 177

33 • Intersecting critical isodapanes: agglomeration. 178

34 • Economies of scale in power generation with urban size. 184

35 • Hypothetical economies of scale with urban size. 187

36 • Price and cost curves of an agricultural enterprise. 191

37 • Input proportions, scale, and equilibrium for an agricultural enterprise. 192

38 • Rent functions for different agricultural land uses. 195

39 • Variation of sales with distance from urban core. 200

40 • Variation of cost and profit with volume of sales. 202

41 • Rent functions for different urban land uses. 203

42 • A locational triangle. 223

43 • Change in market boundary pattern with shift of production points in a

square matrix. 241

44 • The Launhardt-Palander construction. ' 256

45 • The effects of a weight change. 258

46 • Division of a market region between two sources of each of two raw

materials. 261

47 • Spatial production patterns: two sources of each of two raw materials,

one labor location. 262

48 • Spatial production patterns: scale economies introduced. 266

LIST OF FIGURES xix

FIGURE PAGE

49 • Spatial production patterns: localization and scale economies introduced. 268

50 • Spatial production patterns: urbanization, localization, and scale econo-

mies introduced. 269

51 • A simple Losch system of nets of market areas. 270

52 • A modified Losch system consistent with resulting population distribu-

tion. 272

53 • An agricultural land-use pattern. 277

54 • An urban land-use pattern. 279

55 • A commodity flow pattern: intranational trade. 282

56 • A commodity flow pattern: international trade. 284

57 • A commodity flow pattern with modified geographic position of trading

nations. 285

Chapter

Introduction:

Posing the Location

and Regional Problem

In this chapter, we wish to paint broadly the location and regional
problem, especially for the reader unacquainted with the literature and
with little training sympathetic to and appreciative of the spatial
nature of social phenomena. We shall cast the discussion against a
background which traces the evolution of an area and shall raise certain
key questions. In a final section, we shall consider several possible
approaches to the analysis of the location and regional problem, i

1. Some Basic Development Processes

Broadly speaking, economic evolution stems from the action of tech-
nologic man upon the elements of his physical environment. On the
whole, these elements are passive, most of them changing imperceptibly
over human time. However, certain changes in environmental features,
such as soil erosion or silt agglomeration at mouths of rivers, do in
our time accumulate to a critical point and then provoke wholesale
economic and social adjustments. These relatively few instances are
the exceptions to the statement that the dynamic force in economic
development lies in the activities of man almost to the point of exclu-
sion. His reaction with his environment, his constant modification
of the restraints and scarcities which it imposes, and his incessant

1 The reader is reminded that this chapter serves as an introduction not only to
this volume but also to a future volume on the principles of regional science and
general regional theory.

2 LOCATION AND SPACE-ECONOMY

construction of techniques which revalue resources and cause certain
natural features to be less restrictive and others more spell economic
and social change and progress.

Hence, it is not inappropriate to begin with a framework in which
natural resources, physical configuration, and the matrix of techno-
logical conditions are given. We may imagine an area at the start
isolated from other areas because of the friction of physical distance.
Upon this area of varying topography and uneven resource content
settlement takes place.

One or several individuals or family units may be presumed to begin
the occupation. The selection of a site for initial habitation and cul-
tivation of crops will depend on a host of factors. These include the
existing vegetation and the difficulties of clearing, transport resources,
climate, topography, type of soil and nature of drainage, the available
tools and techniques, defense considerations and the cultural inherit-
ance of the individuals and family units which in any given instance
sharply defines existing knowledge and organizational experience and,
thus, the horizon of possibilities. But also, in at least some respects,
the selection may be arbitraiy and indeterminate in terms of any
ex ante rational framework. This selection may be subject to the
whims and fancies of the first inhabitants and perhaps to a particular
series of historical events. And, because of the play of the economic
"irrational," the first site of settlement may be later abandoned for
another not bristling with so many hazards and rigors.

Once a fairly stable adjustment with the environment is attained,
the process of development is more subject to predictive analysis
though capricious elements remain in the picture exerting influence to
varying degrees. As more individuals and family units come to inhabit
the area, presumably they will settle in close vicinity to the first. The
"gregarious instinct," as early social psychologists were prone to term
it, or, more accurately, previously acquired behavior patterns would
tend to foster nucleation. It is not to be denied that where individuals
have diverse cultural backgrounds there may be clash and dispersal
rather than agglomeration, or that idiosyncratic elements of individual
personalities may dominate those which are socially and culturally
determined and thereby induce an unpredictable spatial pattern of
settlement, though in all likelihood a more unstable one. However,
economic forces or, more specifically, increasing returns from co-
operation in combating the elements when population numbers are
small, operate strongly to encourage nucleation. In a sense these
economic forces are already imbedded in an existent culture, having
previously conditioned the emergence of particular culture traits and

POSING THE LOCATION AND REGIONAL PROBLEM 3

complexes and previously influenced the dominant interests and focal
values and attitudes to which a given cultural pattern is oriented.

In any case, sooner or later a population cluster does precipitate.
What is its internal structure? What principles govern its spatial
configuration and, in particular, the spatial configuration of its
economic activities?

For a society which engages predominantly in agricultural activities,
the conceptual framework developed by von Thiinen and his followers
is illuminating. In a uniformly fertile plain of considerable extent,
which is undifferentiated in its physical features and isolated from
the rest of the world and which contains a single population cluster
at some distance from its periphery, cultivation of diverse crops and
production of other farm commodities will tend to take place in
concentric zones around the cluster as center. To each zone there
will correspond a particular agricultural product or combination of
products. The demand for the various products by the given popula-
tion, the effort involved in transporting a unit of each of the several
products over any unit of distance, the intensity and associated cost
at which a unit of area can yield each product or combination of
products, and the resulting prices or barter ratios are among the
various factors determining in which zone each product will be
produced. The relaxation of the uniformity assumptions and the
introduction of realities such as differentiation in ■ soil, climate, and
topography and a finite number of transport routes in general
irregularly placed engender serious distortions of the concentric
pattern. An enclave of land devoted to grazing may appear in a
wheat-growing zone simply because the topography of the enclave
precludes any other activity. In an area stretching along a transport
route, cultivation of land may be much more intense than in an area
closer to the population cluster but untapped by transport media
and may yield entirely different crops. In short, any physical
semblance of zonal arrangement may be completely absent. However,
in terms of time-cost distance and in terms of other concepts which
would give explicit recognition to areal differentiation with respect to
significant variables, the concentric zonal arrangement would remain
undisturbed, as will be indicated in a later chapter.

This conception of competitive equilibrium in land use is at least
partially refutable, however, because of its static nature. As already
mentioned, society is in a constant state of disequilibrium, continually
striving toward a condition of perfect adjustment but just as per-
sistently being jarred off its course by forces of change. A population
nucleus and its associated hinterland are no less a dynamic organism.

4 LOCATION AND SPACE-ECONOMY

The nucleus typically grows in size from an initial small compact
mass reflecting the centripetal drive of increasing returns to a larger
and larger but less and less compressed body, at times even sprawling
seemingly chaotic and without coherence. The centrifugal effects of
diminishing returns from increasing intensity in the use of land and
the mounting diseconomies and congestion from multiplying numbers
become manifest in the growing extent of the spatial spread. At the
peripheries of each of the zones of cultivation, the process entails
transition from one type of land use to another. But does the transi-
tion take place smoothly and orderly or abruptly and haphazardly?
Unquestionably, cultural values and institutions condition the nature

E

X

H D

Fig. 1. A hypothetical region.

of the transition. But just as certainly economic forces are at work
pressing forward the succession of uses to which a given piece of
land is subject and intensifying its exploitation. But how? Unfor-
tunately, little is known and can be said about this dynamic process.

Imagine before the nucleus and its hinterland have reached massive
proportions that a new item appears in the food basket of the
inhabitants of the city which we shall designate as point A (Fig. 1).
Fish are found to abound in a stream some days' journey from A.
Owing to previously acquired inclinations or to initially favourable
reactions and a non-resisting set of social attitudes and institutions,
this new commodity is cumulatively accepted by the populace. A small
colony of settlers is established at point B contiguous to the stream.
(See Fig. 1.) Since, in terms of the best known transportation means.

POSING THE LOCATION AND REGIONAL PROBLEM 5

B is located at a distance of several days' journey from A, still smaller
colonies are established at C and D, primarily to facilitate the process
of transportation and exchange between A and B through servicing
the porters and their carriage animals.

The phenomenon of geographic specialization among separated areas
unevenly endowed with resources appears in full flush. Geographic
specialization per se is not a new phenomenon as it has already been
present on a local level. Because of economies of scale, individuals, of
necessity residing at different points in city A, have come to specialize
in particular activities. The specialization in the cultivation of land
differently situated with respect to the center of A has already been
alluded to. In both these forms of local specialization, transport
considerations are of fundamental importance. Likewise, the critical
significance of transport cannot be denied for geographic specialization
among separated areas. For imagine that the intervening distance
between A and B were to be gradually lengthened. Transport costs
would gradually mount and reach a point where they would become
intolerable. Trade and specialization between A and B would gradually
diminish and ultimately cease. Or, imagine the opposite. A and B are
gradually brought into apposition. Interareal exchange and specializa-
tion become more pronounced as the resistance of intervening distance
declines.

The critical significance of transport considerations is indirect as
well as direct. The process of zone formation is not unique to A.
Only a fraction of the food requirements of the populace of B will, in
general, be procured in exchange for fish. Transport costs on certain
commodities may be prohibitive and may compel the cultivation or
production of these commodities in the immediate hinterland of B. A
zonal spatial design, distorted by peculiarities of topography, soil
characteristics, and a host of other factors, emerges around B as the
focus. Clearly, the number of commodities for which transport costs
will be judged by consumers at B as prohibitive, the number of persons
engaged in fishing, the width of the zones, and the intensity of land
use about B are interrelated and dependent, too, upon distance from
A. The greater the intervening distance, in general, the smaller the
trade between A and B, the fewer who ply at fishing, and the higher
the degree of self-sufficiency at B. But the effect upon the spatial
pattern of cultivation at B is not subject to easy perception. It is
evident, too, that because the magnitude of the exchange of agricultural
products for fish is related to intervening distance, the amount of
agricultural product required of A's hinterland and hence its zonal
pattern and associated intensity of land use will also be affected by

6 LOCATION AND SPACE-ECONOMY

intervening distance. This distance will also affect the amount of
servicing activities provided by C and D, and consequently the
character of their hinterlands, which will be of more limited extent
than those of A and B.

As trade develops pure barter arrangements may be presumed to be
displaced by the introduction of a medium of exchange and money
prices. We do not purport to treat at length here or elsewhere the
culture complex embodying price and monetary mechanisms, broadly
conceived. However, like all sets of associated culture traits, it
comprises behavior forms which have evolved consciously or uncon-
sciously as part of an effort to achieve directly or indirectly a more
effective adjustment to a given environment. And as do all culture
traits and sets of interrelated culture traits, price and monetary
mechanisms condition the changes in cultural forms, complexes, and
patterns which transpire as a society evolves. It is invalid, then, to
take the position that price and monetary phenomena are merely
surface manifestations and reflections of the more nearly basic and
underlying relations and interaictions of man with his physical
environment. Price systems and monetary institutions are in modern
society an indispensable set of cultural tools, which are inextricably
interwoven into the fabric of man's culture and which strongly shape
the evolving organizational form and nature of man's economic and
social activities. It is equally invalid to assign a secondary role to
the geographic distribution of resources, topography and spatial
position, and other characteristics of environment as some trade and
economic theorists are inclined to do. It is all too clear from the
above remarks that trade, relative and absolute spatial position, and
the geographic pattern of resources are fundamentally interrelated.
Our basic position, which we shall reiterate again and again, is that
location and trade are as the two sides of the same coin. The forces
determining one simultaneously determine the other. To understand
and anticipate the interaction of these forces, a knowledge of resources,
position, topography, and other environmental characteristics and a
knowledge of price, exchange control and monetary mechanisms, and
other cultural institutions and behavior complexes and patterns are
each indispensable.

Into this relatively simple frame of general interdependence, another
element of change may be injected. Iron ore is discovered. Previously
acquired experience or successful experimentation with iron products
leads to the exploitation of the iron ore deposits. Deposits, qualita-
tively and quantitatively adequate, are found to exist at points E, F,
and G. Since charcoal derived from timber is required for smelting

POSING THE LOCATION AND REGIONAL PROBLEM 7

the ore, the use of coal being infeasible for one reason or another, and
since timber may be assumed to be ubiquitous in this early stage of
development, iron will tend to be produced at E and G. Working of
ore close to markets, scattered over an area according to the scatter
of markets, is characteristic at this stage, provided local timber
resources and supplies of water and water power suffice. This pattern
minimizes total transport costs on raw materials and finished products.
Necessarily, new transport routes emerge, interconnecting E and A,
and G and B.

The iron manufacture sparks an incipient industrialization. The
character of this industrialization is markedly different from that
observable in modern industrialized societies. The agricultural
stratum serves . as the base to which the structure of industry is
oriented. The agricultural stratum provides the market for industrial
products, the raw materials where they are agricultural, and the labor
for the factories. This labor frequently is available only seasonally
or secondarily as female labor and thereby immobile. Additionally,
the agricultural stratum furnishes the food and drink for the worker,
which frequently exceeds the weight of the raw materials used by
the worker combined with the weight of the finished product attributed
to his effort. This condition dictates orientation to points of food
supply. Hence, the industrial pattern becomes tied to and reflects the
underlying agricultural stratum, as do the patterns of tertiary and
secondary activities for related reasons.

Deviations, however, do crop up as when indispensable but immobile
and highly localized mineral resources must be tapped. More impor-
tant in fostering deviations, and in nurturing a cumulatively mounting
emancipation of the industrial stratum from the agricultural, is the
increasing productivity stemming from improvements in technology.
Technological advance revalues resources not only by broadening the
horizon of materials subject to transformation and by multiplying
production possibilities but also by altering the strategy of diverse
spatial patterns of activities, of mineral deposits, and of transportation
routes through changing rates of output. As the laborer works up
greater quantities of raw materials which yield larger amounts of
finished products, while simultaneously his consumption of food and
drink at most rises at a considerably smaller rate, the pull of the
agricultural stratum is attenuated and in time loses its dominance.
Relocation at sites of mineral reserves and at new critical nodes takes
place. Industry severs its geographic bonds to agriculture and,
concomitantly, partially binds agriculture.

As an instance, posit with the passage of decades the discovery of

8 LOCATION AND SPACE-ECONOMY

an excellent bed of coking coal in the vicinity of F and the acquisition
of crude blast furnace techniques. Compared to the weight of food
and drink of the worker and of iron ore smelted per weight unit of
manufactured iron, the weight of coal required is so prodigious as to
dictate location of blast furnaces at the coal site. Major industrializa-
tion ensues at F oriented to coal resources and iron manufacture. A
host of iron and steel fabricating activities, of heavy fuel-consuming
processes, of ancillary by-product, input-product, and service func-
tions, as well as the allied subsidiary operations indirectly induced
or required to support the flow generated by the basic activities
develops in juxtaposition, each activity in its peculiar way attuned
to the major transport and agglomeration economies at F. The rise
of F signifies drastic changes in the pattern of transport routes and
commodity movements, and the emerging steam-steel complex makes
feasible more modern transport facilities. A superior type of facility,
more suited to large-scale movement of heavy industrial goods, may
be presumed to be constructed to connect F with the old established
transport route AB at H. The crossroads position of H, its centrality
as well as the probable imbalance of traffic and thus unutilized trans-
port capacity in certain directions, hastens the advent and growth of
new enterprises at H.

Realignment of agriculture is necessitated. Zone formation proceeds
about F and H while the configurations of cultivation around the old
foci repattern themselves to the changing geographic structure of
industrial and household demands. More commercialization of agri-
culture and greater exchange of the products of the farm for those of
the factory take place. Dependence of agriculture upon industry
mounts, and the associated problems of agricultural prices, output,
and income emerge and grow in complexity.

The rise of F and H spells relative and perhaps even absolute
decline of E and G, and C and D. At these fading cities, workers are
displaced at their iron-manufacture and transport-servicing trades.
Owing to established modes of behavior and institutions and the
occupational and geographic immobilities attached thereto, the workers
of these cities form a pool of cheap, downgraded labor upon which
parasitic industry may draw. Elsewhere, socially sanctioned agricul-
tural practices deeply rooted in folkways and community structure may
resist change. A resulting inadequate income or material content of
living may impel elements of the agricultural population to offer their
services at bargain rates, thereby attracting industry that has no
strong inclination to locate at any particular site. Additionally, at
F and H where heavy basic industry, mining, and transport can

POSING THE LOCATION AND REGIONAL PROBLEM 9

utilize effectively the labor of the chief breadwinner only, the reserve
of surplus women and child labor nurtures the growth of otherwise
footloose industry. This growth of footloose activities at F and H,
however, is not independent of the growth of such activities as well
as basic industry and agriculture elsewhere. It takes place within
a system of activities interdependent spatially and industrially, and
only against such a system as background can it be fully understood.

2. Some Fundamental Questions

This general descriptive analysis of the evolutionary course of an
hypothetical area could be carried forward and developed extensively.
To proceed thus, however, would not point up as forcefully as we should
like: (1) certain fundamental location and regional problems; and
(2) the need for the general type of theoretical analysis with which
this book is concerned. We now wish to focus upon these problems
and this need by posing a number of key questions which, at the
same time, maintain the emphasis on the spatial order.

For example, as the area industrializes, at what point does it abandon
the phase of isolation and commence trading with the outside world?
Which sites develop as major and minor ports? In what commodities
will the area trade? If our hypothetical area has developed late
relative to other regions of the world, it may at the beginning of trade
export agricultural and mineral commodities in exchange for manu-
factures. If so, does its general composition of imports and exports
change toward export of manufactures and import of raw materials,
food, and specialized machinery and equipment as newer areas are
brought into the orbit of world trade? How does the transition from
one import-export pattern to another proceed? If, because of the
conjunction of various natural factors, cultural forces and historical
incidents, P emerges as a dominant port, what types of industry
develop at P and how much of each?

With further economic development, a diversified transport network
evolves. Internal waterway and highway projects are pushed to
completion; pipelines are laid. But what is an optimum mix and
spatial pattern of the diverse transport facilities? In what ways is
such a mix or pattern conditioned by the character of industrial traffic
generated and the nature of intercommunity and intracommunity
population movement? Should any government subsidy be forth-
coming? If so, how should it be distributed among the several
facilities and the several areas and between current and capital
expense?
As industrialization proceeds, problems of relocation arise. In the

10 LOCATION AND SPACE-ECONOMY

initial stages of heavy manufacturing, textiles may have been drawn
to F because of the location there of excellent coal deposits and of a
reserve of women and child labor available at relatively low rates.
With time a heavy concentration of textiles may have developed at F,
reflecting, partially, the attraction of a pool of skilled labor and,
partially, historical inertia. But, as the force of innovation presses
the spatial structure of the economy and its parts into ever new
configurations, the strategy of existing textile sites alters. The reserve
of low-cost labor at these textile sites may have become depleted ; what
was once low-cost labor may have been converted into high-cost labor
owing to the continued use of obsolete equipment and processes. Else-
where, cheap and abundant labor incident to "cultural lag" and
creeping industrialization and urbanization may make available
savings which warrant geographic shift despite heavy costs of reloca-
tion. Resisting, however, is the force of historical inertia, a force
intimately associated with the cultural variable. A critical point may
exist at which this force is overcome. But exactly how does this point
vary with the institutional environment? Furthermore, if relocation
is expedient, what would be an optimum localization pattern of
textiles? What distribution among regions? How large the factory?
To what extent can a textile factory serve as a focal point for
employment in an esthetically designed "new town"? How do creative
minds generate ideas which modify accepted ideals and values which
in turn form the scaffolding within which decisions are made by
consumers, and political, social, and business leaders? How do these
creative minds condition the manner in which the spatial structure
yields to stresses and strains?

The problem of adjustment to change in the relative strength of the
several location factors may be raised again in connection with iron
and steel location. Foreign ores become accessible while concomitantly
improvement in smelting and rolling techniques and utilization of
a swelling supply of scrap reduce the quantity of coal spent per ton
of steel. Should a major integrated iron and steel works be established
contiguous to the port P? Or should the manufacture of steel at P be
limited to the operation of scrap furnaces only? At what point will
the pull of the market and scrap supplies at A be suflEicient to draw
some steel capacity to A? Does the relative decline of steel at F also
imply absolute decline? Furthermore, how do competitive ethics and
practices and institutional pricing arrangements, whether f.o.b., basing
point, or some other, affect the process of locational adaptation?
Finally, do impending new techniques, continuous casting for example,
spell complete decentralization for the far future?

POSING THE LOCATION AND REGIONAL PROBLEM 11

Intimately associated with the historic shiftings of iron and steel
location are the dynamic processes of the urban-metropolitan com-
plexes at P, F, H, and A. Clearly, the magnitude of each complex is
linked to its content of such basic economic activity as iron and steel.
Spatial realignment of such activity is tantamount to reshuffling of
the ranks of cities. But exactly how does employment in any particu-
lar basic activity or combination of basic activities generate employ-
ment in non-basic activities? Why does the ratio of non-basic to basic
vary from city to city, and in what way is it related to the input
structure and gravitational impulse of a particular basic activity?
Can a meaningful dynamic structure of an urban-metropolitan complex
be constructed and depicted in terms of interactivity relations?

Additionally, spatial physiognomy of the urban-metropolitan region
is a primary concern. Intensity of land use is a function of, among
other factors, distance from the core. At the core the pyramiding of
activities attains a maximum maximorum. With movement away
along radials in all directions the intensity of land use diminishes, but
differentially in the various directions. Moreover, the rate at which
this intensity falls in any direction changes with distance. In addition,
after a point along many of the radials, the intensity reaches a relative
minimum, reverses its trend by creeping upward, and attains a relative
maximum, only to decline again and perhaps to repeat this undulatory
performance. Along a few of the radials, generally the strategic ones
along which the rate of decline from the core is among the least for
all radials, relative maxima for the entire metropolitan region are
realized. These latter represent subfoci from which, for some distance
at least, intensity falls off in all directions. The strategic radials also
exercise a degree of dominance to the extent that, in directions
perpendicular to their course, intensity decreases.

In short, an urban-metropolitan region comes to comprise an
hierarchy of strategic nodal sites, classifiable by order and degree of
dominance. This multinucleated body is, viewed from another angle,
a network of transport interconnections and hence of interstitial areas
each subject to hierarchical order. What is the nature of the ecological
process that gives rise to this dynamic organism? How do (1) the
cost relations of the numerous economic activities, (2) the spatial
and product preferences of consumers, of familial and various asso-
ciational units, and (3) the friction of distance interact? How does
progress in the state of transport technology, which in turn is condi-
tioned by the preferences and ideals of these units, impel change and
rearrangement of centers and population nuclei? On a more concrete
level, what determines the use to which any given piece of land is put?

12 LOCATION AND SPACE-ECONOMY

Which types of retail, wholesale, cultural, governmental and adminis-
trative, industrial, and service activities tend to appear in the core
and in each of the several types of subcenters and satellite cities in
the various phases of urban-metropolitan growth? In what form does
specialization among the several metropolitan regions emerge? Are
there forces which pervade the intrametropolitan and intermetropolitan
structures which yield stability in the size distribution of sites and
cities as a whole, even though there is constant reshuffling of the ranks
of individual members of any meaningfully defined population of sites
and cities? Do such forces deny the concept of an optimum-size city?

For our hypothetical region these questions become more crucial
with time. Impending innovations, such as aircraft and atomic energy
with destructive as well as constructive potential, compel a recasting
of national and regional values. The inherited cultural ideals and
productivity sanctioning and extolling the cosmopolitan life in peace-
time must be weighed against the suddenly increased risks, whether
calculated or imagined, of vulnerability when a state of war exists or
portends. Policy questions concerning urban decentralization arise.
Intelligent answers to these questions, however, require foresight on
the probable "normal" effects of these innovations. To such effects
we first turn.

Environmental barriers deflect the path of growth of any community,
metropolitan area, or regional unit. Physical features, however, have
critical value as barriers only with respect to a given or assumed state
of technology, and in particular of transport technology. With the
advent of aircraft, what aspects of topography lose significance as
obstacles to the movement of people, ideas, and goods? Which
acquire new importance? Will any new trade routes and major
realignment of trade patterns develop? Will the strategy of sites for
trading activities be altered, with a consequent relative decline of P?
Will the emphasis on air traffic in international trade provoke the rise
of new centers more suitably geared to such traffic? To what extent
will aircraft find widespread use in industry and foster increased
productivity and new location patterns? How will metropolitan
structures be affected both indirectly through increased effective
demand for a myriad of service activities made possible by greater
productivity and directly through increased population mobility
inherent in the use of family aircraft? What should be the position of
aircraft in a diversified transport system, and to what degree and in
what manner should air transport be subsidized?

The harnessing of atomic energy poses somewhat similar questions.
Will nuclear power be competitive with conventionally produced

POSING THE LOCATION AND REGIONAL PROBLEM 13

power? If not, will a given society deem justifiable the allocation of
a disproportionate share of costs to the military to encourage active
participation by private enterprise in the production of nuclear power
and in the atomic energy industry in general, and hence to build up
and maintain skills and capacity for the production and effective
utilization of fissionable materials? Does competitive or cheap nuclear
power augur for our hypothetical area the emancipation of heavy
industry from coal belts, the dissolution of industrial concentrations,
and a wide and more even spread of economic activities? Or will
such nuclear power alter the significance of merely one of many
location factors and in doing so modify location patterns to a small
degree only, even to the extent of fostering increased industrial
concentration rather than diffusion or dispersion? Will a host of new
processes and industries develop concomitantly to yield indirect effects
of revolutionary proportions? Which, if any, of the existing industrial
sectors, including the public utilities, are likely to become obsolete
in whole or in part? What may be the impact of nuclear power upon
other regions of the world deficient in fuel and power resources, upon
trading relations with these regions, and thus indirectly upon the
structure of our hypothetical space-economy?

Tentative answers to these and many other questions are required
for an intelligent approach to urban-metropolitan and industrial de-
centralization. A program of decentralization must be geared to the
anticipated future geographic, economic, and social organization of
metropolitan regions, not just to current demands. Decentralization
policy is partially "deviational" policy, a policy designed to accelerate
certain tendencies and strengthen the effectiveness of certain forces,
not all necessarily centrifugal, in accordance with certain values judged
to be in the interest of the commonweal. It therefore involves insight
into processes indicated in the above paragraphs, which are subject
to a fair amount of objective analysis. It also involves evaluation of
the impact of innovation upon the psychology of the individual and
social groups and requires some estimate of how imagined or impend-
ing military applications of such innovations as aircraft and atomic
energy ultimately affect individual and group space preferences. These
questions, like a host of others of similar stamp, are subject to
considerably less objective analysis.

Identification of meaningful sectors of the metropolitan region which
should be dissected from the urban mass and located elsewhere,
whether in the periphery or in more distant districts, is not unlike
some of the age-old problems of regional resource use and conservation.
A meaningful sector is not necessarily, and perhaps only infrequently.

14 LOCATION AND SPACE-ECONOMY

a wedge of activities contiguous in space. It is a complex of activities
where association leads to definite agglomeration economies, but where,
subject to certain restraints, presence of these activities within a
Greater Metropolitan Region in any of many possible patterns of
scatter and concentration may more often than not satisfy the spatial
associational requirement. Hence, contiguity in physical space is only
one of several constraining factors each of which is critical in the
determination of some of the several combinations of activities which
comprise meaningful sectors for decentralization purposes. Other
restraints bear upon such matters as the volume and time-pattern of
demand for transportation and diverse utility services, the structure
of labor requirements by occupation and other characteristics, and
the interrelations of activities in terms of stages of production, by-
product use, and input-output functions. Yet, given a knowledge of
the types of restraints operative and even detailed information about
the nature of some, how evaluate the net interaction of different groups
of them under different sets of circumstances in order to identify and
effectively to carve out sectors from the metropolitan region?

Correlatively, how achieve a satisfactory geographic balance in our
hypothetical region if the pattern of activity is adjudged too con-
centrated and vulnerable in certain sections, for example, along the
industrial band which stretches from A to H and H to Fl Put other-
wise, how plan the long-run utilization and conservation of resources
in the entire region, weighing the economic desirability of each de-
velopment design against its military, political, and other virtues and
limitations. Guiding new growth along certain channels and in specific
parts of the region is one expedient for achieving a redistribution of
activity. Some costly relocation of other activities may be justifiable.
More significant, an entire set of activities located within an industrial
band may be subject to decentralization as an integral unit, when
decentralization of the activities individually would be, from an
economic standpoint, highly irrational because: (1) they feed inputs
into each other and utilize each other's by-products, (2) together they
can maintain the high quality transport service which each requires,
(3) together their labor forces form a substantial pool of diversified
skills, the existence of which is a sine qua non for each, (4) together
their combined demand for diverse urban and professional services is
of such a magnitude as to insure efficient performance at relatively
low cost, and (5) their combined labor force constitutes a market for
consumer products of such a scale as to restrict the cost of living and
hence money wages to moderate, if not low, bounds. To ascertain each
of these sets of activities where the force of historical inertia binds

POSING THE LOCATION AND REGIONAL PROBLEM 15

location to original or old sites of development, where relocation in
the small is precluded but where relocation in the large can be eco-
nomically sanctioned is, to reiterate, a difficult problem. To attack
this problem, and, more generally, to fonxiulate effective governmental
policy for decentralization and for the long-run planning of resource
development to achieve an optimum geographic distribution of activi-
ties, requires full utilization of all the existing skills of social scientists.
And, more important, it requires continued and, if possible, accelerated
progress in the social sciences in the fashioning of new tools and tech-
niques for analyzing the interdependence of the various sectors of the
space-economy, and hence the net effects of certain specified changes.

3. Possible Theoretical Approaches 2

Having posed a host of key questions, w^e must confess to an inability
to provide even partial answers. Our intention, as stated earlier, is
to p6int up the need for the general type of theoretical and empirical
analysis with which the remainder of this book is concerned. What
follow^s is a modest attempt to depict and understand some of the
basic spatial interrelations which underlie the location of economic
activities and regional development.

Various approaches were possible in undertaking this assignment.
Several are recorded in the event that future students of location and
regional development may find' them of some use. One obvious
approach toward a more general theory would involve amassing new
and reinterpreting existing historical material — to a very limited extent
along the lines of Roscher, Schaffle, Ritschl, Weber, and Englander —
to a very significant extent along the paths explored by Dean and
Usher. (The von Thiinen framew^ork for agricultural location would
be of critical value here for understanding initial stages of develop-
ment.) Unquestionably, historical generalizations, more comprehensive
and at the same time more incisive than those we now possess, can
be achieved in the study of the past and current spatial and regional
structure of the world economy and its various sectors.

A second approach might begin with the pure abstractions of Losch.
These are presented at some length in the following chapter. Cannot
one further pursue these abstractions with profit as Losch does to
some extent? Imagine one relaxes the assumption of uniform popula-
tion scatter, an assumption which is inconsistent with the derived

2 This section is written primarily for the research worker. It assumes a knowl-
edge of the works and approaches of the several individuals cited. The contribu-
tions of these individuals are at best only partially recorded in the following
chapters.

16 LOCATION AND SPACE-ECONOMY

results which yield cities and towns of diverse sizes. Further, one
can introduce geographic inequalities in endowment of resources of
all kinds, differences in consumers' space preferences, a finite number
of transport routes reflecting high fixed costs and economies of scale,
and a number of other restraints which permit a closer correspondence
to actual conditions. Could not a more meaningful system of nets
and hierarchy of sites be depicted? Since the character, magnitude,
and direction of trade among all sites are functionally related to the
nature of the postulates underlying one's abstractions, would not a
more realistic trade pattern be yielded? Furthermore, the Loschian
framework, which has most relevance for market-oriented processes
and service functions which use ubiquitous raw materials or none at
all, may be extended to incorporate agricultural activities. About
each population cluster, whatever its size, zones of cultivation
tend to develop a la Thiinen though within a complex web of
interrelations about which some remarks will be made in the following
paragraphs.

Another approach might rather build upon Thiinen's model with
extensions at later stages to encompass the significant elements of
Loschian and Weberian theory. Imagine two isolated city regions,
each region comprising the central city and its surrounding hinterland
devoted to agriculture and forestry. With the development of transport
technology we may suppose that resistance imposed on movement by
topographic obstacles and by the sheer friction of distance is gradually
reduced. This change, as well as growth in population numbers, will
induce an increase in the consumption of various agricultural and
forest products and perhaps also in the types of products available.
A new equilibrium arrangement of expanded zones will, after a sufficient
lapse of time, be established for each city region although, as we have
already indicated, little is known about the dynamic process of
transition from one equilibrium pattern to another, a process which is
intimately linked with cultural values and institutional modes of
behavior. In time, and under the assumption that resources are uni-
formly and equally distributed among the city regions, the hinterlands
of these two isolated city regions may have expanded to such a degree
that their outer boundaries become tangent or coincide along a certain
stretch. Nonetheless, the hinterlands do not overlap and still can
be rigorously defined. However, the symmetrical concentric zonal
arrangement about each city is disturbed although in a systematic,
identifiable manner. The introduction of a third, a fourth, and finally
an nth city region merely adds to the complexity of the resulting
design. It does not entail any overlapping of hinterlands or any less

POSING THE LOCATION AND REGIONAL PROBLEM 17

rigor in their demarcation using standard Thiinen principles, to be
discussed in a later chapter.

The schema of a space-economy composed of many city-regions with
some hinterlands bordering on each other for limited stretches, and
without interregional trade, is subject to considerable improvement.
Recognition of relations which Ohlin and Losch have emphasized —
namely, the force of concentration of production imbedded in economies
of scale and the force of scatter of production engendered by the
cost of transportation — immediately lays the basis for trade. Return-
ing for the moment to the fiction of two city-regions, we may recognize
that not all commodities produced are agricultural and forest, that
some commodities are services to be performed at central places, and
that others are associated with market-oriented production processes
which currently we may postulate use only ubiquitous raw materials.
Each of the commodities belonging in these latter two sets may be
classified as to the size of market area most economical for its produc-
tion. Some may be produced and distributed most efficiently when
one factory serves both city-regions. These commodities may be
classified as supraregional and national. Others may be produced and
distributed most efficiently when two factories operate, one located in
each city, each serving consumers in its corresponding city-region.
These may be classified as regional. Still others may be most efficiently
produced and distributed when they are produced in many factories,
symmetrically located within the combined area of the city-regions.
The commodities these factories produce may be considered sub-
regional; and since their sites of production to a large extent lie outside
the core of the two main cities, they give rise to new central function
sites. As these sites come to represent population nuclei, effective
demand for agricultural products gives rise to zone formations at
each of them.

In sum, superimposing a Loschian framework upon a Thiinen
arrangement leads to an hierarchical pattern of sites within each city-
region and generates interregional as well as increased intraregional
trade. Confining attention to only two city-regions raises the basic
question: Which of the two serves as the site for producing supra-
regional and national commodities, or does a new intermediate city
emerge as the site for producing such commodities? Historical evidence
would seem to favor the hypothesis that one of the two existing cities
usurp national commodity production. This tends to spell relative
decline for the other. The latter tends, as a consequence, to expand
its output of agricultural products (and hence its hinterland) in order
to provide the wherewithal to purchase supplies of national com-

18 LOCATION AND SPACE-ECONOMY

modities from the former. This condition in turn entails relative
contraction of the agricultural supply hinterland of the former and
specialization among the city-regions. However, it should be borne
in mind that there may be decided transport advantage to an inter-
mediate location and that, where there are a sufficient number of
national commodities, a major shift to an intermediate location may
take place. This shift may involve either a simultaneous relocation
of many producers or a gradual transition whereby a one-by-one
sequence of relocations is experienced.

To pursue this interesting two city-region schema in greater detail
is outside the scope of this book. More relevant, but still beyond
our defined bounds, would be a multicity-region construct, resulting
again from both population growth and advance in transport tech-
nology. Here, more determinacy is likely to be present. The city-
region central to all is, from the transport standpoint, the ideal location
for national commodity production. It tends to specialize in these
commodities while on balance the remaining city-regions tend to
specialize in agriculture; necessarily this specialization involves zonal
adaptations. At the same time, there may be subnational commodities
which are also supraregional. The existence of such commodities
generates a still gTeater hierarchical arrangement of city-regions.

Hitherto, the postulate that resources are uniformly and equally
distributed among city-regions has been maintained. Let this postu-
late be relaxed; let us return to the simple two city-region schema
where both regions are initially isolated from the rest of the world
and from one another. In time, as there is advance in transport
technology and increase in population numbers, trade may well take
place before the hinterlands have expanded so as to be contiguous
with one another over some stretch. There may be partial or even
complete specialization in the production of various agricultural prod-
ucts, each city-region supplying to the other crops for whose produc-
tion it is most suited by reason of its resources. Upon this improved
Thiinen arrangement, which should also be expanded to embrace many
city-regions, a Loschian framework may be superimposed to inject
into the analysis service activities and ubiquitous raw-material-using,
market-oriented production processes. Once again, a hierarchical
arrangement of cities and towns evolves, with consequent major altera-
tions in zonal patterns. A more realistic portrayal of a space-economy
results.

However, the analysis of the space-economy can be further improved.
Relaxing the assumption of uniform and equal resource distribution
allows one to take into account basic production processes (for example.

POSING THE LOCATION AND REGIONAL PROBLEM 19

steel, aluminum, and chemicals) which use raw materials which are
localized in varying degrees (for example, coal, ores, and hydropower).
Here the Weberian type of analysis can be added to the derived
Thiinen-Loschian model. Consideration of transport costs (as they
are related to weight loss, relative weights, relative positions, and
rate structure) , of labor costs and other costs, and of agglomeration
and deglomeration economies is required. New sites of production
and cities may emerge, to be added to the Thiinen-Loschian hierarchy;
and of the existing cities, some may grow while others decline. The
spatial pattern of agricultural production and of service and market-
oriented activities based on ubiquitous raw materials will con-
comitantly undergo change, particularly where strong local multiplier
effects are associated with basic industry (reflecting partially agglom-
eration effects) .3

In short, the development of a combined Thiinen-Loschian-Weberian
framework along channels suggested above and elsewhere in this book
represents another approach toward achieving a general theory of
location and space-economy and toward a more thorough understand-
ing of the interrelations of the space-economy. •*

A fourth approach might proceed through entirely different channels.
Imagine there exists only one non-agricultural, non-service production

3 To be specific, imagine that in the Thiinen-Loschian framework a steel plant,
using locaUzed ore and coal, is introduced. Since steel production on the whole
tends to be transport-oriented, the steel plant may or may not be located in an
existing city. If the point of minimum transport cost for steel production does
not coincide with or lie close to an existing city, a new city tends to arise especially
since steel production is an agglomeration-inducing type of activity. It is easily
seen how service and agricultural activities in city-regions close by are directty
affected, and more remote city-regions are indirectly affected. Likewise, that there
are important direct and indirect repercussions when the optimal transport point
for steel production coincides with an existing city is easily perceived.

4 The above approach takes into account a steady advance in the state of trans-
port technology and a gradual reduction in resistances to and costs of movement.
A variant of this approach would start at the other extreme. Transport costs are
assumed zero. A corresponding spatial pattern of economic activities is derived.
By degrees, transport costs are permitted to rise, approaching more and more the
levels of reality. For a while, at least, inequalities of human and natural resources
become more and more critical. The previous spatial pattern related to consumer
and social group space preferences increasingly takes on an economic hue. An
hierarchy of sites, of trade routes, and of flow phenomena in general develops, as
well as zone formations and industrial agglomerations at strategic mineral and
nodal sites. Ultimately, as transport costs rise to exorbitant levels, the hierarchy
disappears, as does interregional trade, and the hypothetical isolated city-regions
of Thiinen are attained.

In this Thiinen-Loschian- Weberian approach and its variant, the effects of tariff
barriers, exchange rate and balance of payments considerations, central banking

20 LOCATION AND SPACE-ECONOMY

process which involves only a single stage. Its location will be
associated with a population cluster. Either it is a market-oriented
operation and, hence, drawn to an agricultural and related service
population nucleus, or it is an operation which is tied to a raw
material source and which, hence, binds a population cluster composed
of the families of its labor force and of the labor force of allied
service activities. Associated, too, with the latter population cluster
will be a farming population, but one which does not need to be in
the immediate environs owing to the possibility of engaging in long-
distance (interregional) trade.

Consider an advance in technology which permits the production
of the identical product at lower cost, but by a two-stage process.
Both stages may be carried on at a market site or at a raw material
site. But also the production process may be split geographically,
the earlier stage being linked to the raw material site, the later stage
to the market. Another type of long-distance (interregional) flow
phenomenon may be observed, namely, the movement of semifinished
manufactures in exchange for finished products. History records that
in time such a flow phenomenon appears as a region develops. But
what are the conditions that the process be split geographically and,
hence, that such a flow phenomenon exists? Are these conditions
similar to the conditions for long-distance (interregional) trade in
finished products? Cannot the conditions for both these types of
flow phenomenon be expressed in terms of common concepts and
principles?

Permit the economy to become still more complex. Technological
advance compels, through making possible significant cost savings,
the shift to a three-stage, to a four-stage, and finally to an n-stage
production process. Again, what are the conditions that the process
be split geographically and among more than two sites when more
than one raw material is utilized? The statement of these conditions
should involve merely a further generalization of a common set of
principles.

Pursue the generalization further. Allow the production by a single-
stage process of another industrial commodity. AVhat determines the
location of such production? In addition to the factors which were
relevant for the first industrial commodity, linkage relations between
the two industrial commodities must be considered. Location at a

and monetary institutions, obstacles to immigration and labor mobility, and vari-
ous other political and social frictions and policies should be investigated to obtain
a still better understanding of the space-economy of reality and in particular of
international trade.

POSING THE LOCATION AND REGIONAL PROBLEM 21

site of one of the raw materials utilized in the manufacture of the
first industrial commodity, at any intermediate point at which a
stage in such manufacture may have become situated as well as at
the market, at a site of a raw material consumed in the production
of the second industrial commodity, or at other relevant intermediate
points must be investigated. The appearance of agglomeration
economies may alter the spatial configuration simply derived by super-
imposing upon the location pattern of the first, the location pattern
of the second industrial commodity which would have obtained in
the absence of production of the first. As already intimated, additive
procedures are insufficient. But how evaluate the effect of agglomera-
tion? What are the principles which relate these forces to others?

In response to successive technological advances, the production
of the second industrial commodity is performed in two stages, three
stages, and finally n stages. Should not the identical principles apply
to the spatial affinities of one or more stages of one production process
as to one or more stages of the other? Further introduce more in-
dustrial commodities each produced in one or more stages. Can these
principles be as extensively generalized? Moreover, since these prin-
ciples relate the location of any one stage of a production process to
the location of the next stage or the consumer (industrial or house-
hold), should not these same principles simultaneously govern flow
phenomena in the postulated model and, hence, all trade?

Why consider industrial commodities singly? Slicing the economy
by individual industrial commodities represents an extreme of indus-
trial disaggregation. For certain purposes a more aggregative analysis
in terms of groups may be more useful. For example, a somewhat
less disaggregated classification such as a fine Leontief-type break-
down may be more expedient for purposes of operation. At the other
extreme, the highly aggregative Colin Clark classification of primary,
secondary, and tertiary, or a fourfold classification of heavy industry,
light industry, agriculture, and trade and services may be more
appropriate in other instances. Could not the previously derived
principles be restated to be valid for groups, the nature of the re-
formulation being dependent upon the degree and manner of aggrega-
tion? Could these principles be extended, though at best only partially,
to encompass the effects upon the spatial industrial pattern of different
types of pricing policies and institutional business ties, different geo-
graphic patterns of income distribution, different kinds of space
preferences of individual household consumers and diverse types of
aggregates of household consumers, and like factors?

The framework may even be extended to consider interrelations of

22 LOCATION AND SPACE-ECONOMY

large regions. We have alluded to long-distance trade as interregional.
Recognizing that an hierarchy of regions exists, we may conceive at
the start of this long-distance (interregional) trade as taking place
among the smaller regions of one large major region, but not as
occurring beyond the bounds of a large major region. Thus, it is
possible to have a construct of several large major regions, each
initially isolated from the other, but each experiencing its own variety
of internal development which is subject, however, to explanation
in terms of a common set of general principles.

Of the several large regions, consider two neighboring ones. One
may possess, relative to existing technology, abundant mineral resources
and niggardly amounts of agricultural resources. In the other, the
opposite situation may obtain. (It is clear that, ceteris paribus, in
one region the character of development will be more industrial in
the other more agricultural.) Let there be progress in transport tech-
niques such that the geographic barriers to commodity movement are
reduced sufficiently to permit intercourse. The pattern of geographic
specialization in each large region changes. Industry shifts from one
region to the other with concomitant increases in productivity, gross
output, and degree of specialization. Trade internal to each large
region likewise changes, and an hierarchical order of trade relations
develops. As further transport advances permit the introduction of
a third, fourth, and finally an nth large region into the circle of inter-
relations, further changes ensue, each functionally dependent upon the
state of transport technology and level of transport costs. To under-
stand the resulting spatial phenomena, notwithstanding their com-
plexity, could not the same common set of general principles, somewhat
adapted and extended here and there, be invoked? Does not the
introduction of hierarchical relations merely add complexity to the
statement of these principles, and not basic change?

With the raising of these questions the discussion of the fourth type
of approach to a general theory of location and space-economy is
concluded. The recently developed approaches of Vining, who places
emphasis on the search for distributional stability characteristics
among spatial flow phenomena, and of Koopmans and others, whose
primary attack is upon the transportation problem, are other fruitful
ones. Several other approaches associated with the older location
literature will be discussed at length in the following chapter. Still
another approach, a variant of the fourth, might start with individual
persons rather than with individual production processes. The spatial
preference of each person, both as a consuming unit and as an income-
earning unit, would be considered. Aggregating individuals to form

POSING THE LOCATION AND REGIONAL PROBLEM 23

meaningful social and economic groups would introduce group space
preferences, which in turn would affect the geographic association of
employment opportunities, and thus industrial location and trade.
Optimal spatial patterns from the standpoint of a group are not
necessarily consistent with an optimal position for each individual.
Conflicts may arise which involve some of the hitherto unsolved prob-
lems of welfare economics. In this manner the variant of the fourth
approach might proceed.

Before this chapter is brought to a close, a few words might be
said about the author's approach. In a real sense it is eclectic, drawing
upon various elements of the works of others. It aims at developing
in this volume principles for a general theory through reducing to
common simple terms the basic elements of the various location
theories, beginning first with Weberian dogma. It does not purport
to present in detail and evaluate various location theories nor to weave
the fabric of a complete and realistic space-economy. Rather, with
a full recognition of the deficiencies and non-operational character of
our general location principles, it seeks to bring the separate location
theories into one general doctrine, to the extent possible; and to fuse
the resulting doctrine, where this can be done, with existing production,
price, and trade theory. This is sought in order to develop a superior
set of tools and conceptual framework for the theoretical and empirical
type of regional analysis which will be attempted in a second volume
and which, together with the general location analysis, aims at increas-
ing the understanding of the structure and changing character of
spatial phenomena.

Chapter

Some General Theories
of Location and Space-Economy'

1. The Anglo-Saxon Bias

The difficulties of the problem depend chiefly on variations in the area
of space, and the period of time over which the market in question ex-
tends; the influence of time being more fundamental than that of space. 2

Thus spoke Marshall, in line with Anglo-Saxon tradition, and in
the half-century to follow Anglo-Saxon economists were to hearken to
his cry. Theoreticians of today are chiefly preoccupied with intro-
ducing the time element in full into their analyses, and the literature
abounds with models of a dynamic nature. Yet who can deny the
spatial aspect of economic development: that all economic processes
exist in space, as well as over time? Realistically, both time and
space must be vital considerations in any theory of economy. Unfor-
tunately, however, aside from those of the monopolistic competition

1 As already indicated in the Preface, the basic material for this chapter was
written in 1947 and published in 1949. At that time we emphasized an imbalance
of the Anglo-Saxon literature. Since then this imbalance has been partially cor-
rected, in particular in the works of Enke, Samuelson, Koopmans, Beckmann, and
Fox which will be cited in a later chapter. Nevertheless, the statements are left
unqualified in order to retain the forcefulness of the argument as first developed.

It should be noted that this chapter is not intended as a survey of all location
theory but only of general location theory up to 1947. It therefore fails to treat
many of the contributions to partial location theory of such individuals as Thiinen,
Launhardt, Englander, Palander, and Hoover.

2 Alfred Marshall, Principles of Economics, 8th ed., London, 1936, Book V,
Chap. XV, Sect. 1.

24

SOME GENERAL THEORIES OF LOCATION 25

school of thought, particularly Chamberlin,^ the architects of our
finest theoretical structures have intensified the prejudice exhibited
by Marshall. They continue to abstract from the element of space,
and in so doing they are approaching a position of great imbalance. ^

Let us consider, as an example, modern general equilibrium theory.
The latest contributors have concentrated their efforts on attacking the
problem of time to the exclusion of that of space. Hicks, ^ Mosak,^
Lange,''' and Samuelson,^ to name a few, have all treated an economy
in which all factors and producers, commodities and consumers
are, in effect, congregated at one point. Hicks, to be sure, begins
by formulating the problem in a manner pregnant with spatial
implications:

It turns out, on investigation, that most of the problems of several
variables, with which economic theory has to concern itself, are problems
of interrelations of markets. Thus, the more complex problems of inter-
national trade involve the interrelations of the markets for imports and
exports with the capital market. . . .

. . . The method of General Equilibrium, which these writers (Walras,
Pareto, and Wicksell) elaborated, was especially designed to exhibit the
economic system as a whole, in the form of a complex pattern of inter-
relations of markets. Our work is bound to be in their tradition, and to
be a continuation of theirs. ^

But actually he confines himself to a wonderland of no spatial dimen-

3 E. H. Chamberlin, The Theory of Monopolistic Competition, Cambridge,
Mass., 1933; and Chamberlin 's doctoral dissertation deposited under the same
title in the Harvard University library, 1927. In his doctoral dissertation Cham-
berlin treats the space factor somewhat more thoroughly and more as an integral
part of his theory. See also S. Enke, "Space and Value," Quarterly Journal oj
Economics, Vol. LVI (August 1942), pp. 627-37.

4 Outside of the field of monopolistic competition there have been scattered
treatments by Anglo-Saxon theorists of certain aspects of space as an economic
factor. For example, F. A. Fetter has treated space in "The Economic Law of
Market Areas," Quarterly Journal of Economics, Vol. XXXVIII (May 1924), p.
525; treatises on international trade have attributed some importance to spatial
resistances; rent theorists have been forced to recognize, however inadequately,
the existence of space in the separation of immobile natural resources and markets ;
and so forth. But in these latter instances only passing attention has been given
to this vital consideration.

5 J. R. Hicks, Value and Capital, Oxford, 1939.

6 Jacob L. Mosak, General Equilibrium Theory in International Trade, Cowles
Commission Monograph No. 7, Bloomington, Ind., 1944.

"^ Oscar Lange, Price Flexibility and Employment, Cowles Commission Mono-
graph No. 8, Bloomington, Ind., 1944.

8 Paul A. Samuelson, Foundations of Economic Analysis, Cambridge, Mass.,
1947.

9 Hicks, op. cit., p. 2. The words in parentheses are added.

26 LOCATION AND SPACE-ECONOMY

sions. Apparently he assumes markets to be perfect, one price ruling
throughout each of them. Or, otherwise expressed, transport costs
and other costs involved in movement within a "market" are assumed
to be zero. In this sense the factor of space is repudiated, everything
within the economy is in effect compressed to a point, and all spatial
resistance disappears. ^^

The approach to unreality which is inherent in such a treatment is
best illustrated by Mosak's work, Gerieral Equilibrium Theory in
International Trade, which is excellent in other respects. Although
* Mosak expands Hicks's analysis to embrace an international economy,
spatially speaking he is still dwelling within a dimensionless habitat.
His study of the effects of international exchange, of unilateral pay-
ments, and of impediments to international trade, n can be interpreted
as treating an anomalous field: a one-point ivorld, which somehow or
other is conceived as divided into n -parts, representing n nations,
between which trade and trade barriers exist. ^"^

We may now consider the relations of general equilibrium theory
to the general theory of location and space-economy envisaged in this
book. We conceive the general theory of location and space-economy
to be one which comprehends the economy in its totality. Not only

10 The explanation may partly lie in Hicks's rejection of monopolistic competi-
tion theory generally in favor of perfect competition on the ground that the
former introduces elements of indeterminacj^, whereas his preference is for deter-
minate solutions (ojo. cit., pp. 83-85). It is clear that Hicks fails to reveal an
appreciation of the spatial aspect of monopolistic competition theory and of the
extent to which determinate solutions are obtained in monopolistic competition
analysis through its consideration of the space factor.

Professor Schumpeter has pointed out to me in conversation that one might
maintain that transport cost is implicitly contained in production cost, and that
the Hicksian analysis is thus sufficiently comprehensive. My point is this: pro-
duction theory, having gone bej'ond the mere statement that the producer maxi-
mizes his profits (in which statement all production costs are implicitly treated),
from a methodological standpoint cannot justifiably treat certain production costs
expUcitly and other important ones implicitly in order to avoid the obstacles to
anatysis which the latter present. For a balanced treatment, the particular effects
of transport and spatial costs in separating producers from each other must be
considered. They are too vital to be sidestepped through impHcit treatment, as
Hicks and others may be interpreted as having done.

11 Transport costs are not explicitly treated. The analysis of their effects, it is
maintained, follows similar lines to that of the effects of import and export taxes
(Mosak, op. cit., pp. 64r-65).

12 At this stage in the development of theory, it is as unjustifiable and inade-
quate to lump transport costs into one category along with all trade resistances
in the theory of trade as it is to treat them as implicit production costs in the
theory of production.

Although these indictments are levelled specifically at Mosak, they apply to
others as well.

SOME GENERAL THEORIES OF LOCATION 27

are the mutual relations and interdependence of all economic elements,
both in the aggregate and atomistically, of fundamental importance;
but the spatial as well as the temporal (dynamic) character of the
interrelated economic processes must enter the picture. Seen in this
perspective, Hicksian general equilibrium analysis is but a very special
case of a general theory of location and space-economy which con-
cerns itself with the local distribution of factors and resources as well
as with local variations in prices and, thus, with the immobilities and
spatial inelasticities of factors and goods.

In the sections to follow, the contributions of several authors who
have pioneered in this field will be restated and critically evaluated.
It will not be surprising to find that these authors have come under
the influence of German thought, i^ The classical school and their
followers were too prone to overlook the local differences within the
English economy. England's dominant international position and
the dynamic aspects of her industrial development fm'ther helped to
cloud their vision. It was in international trade theory that the spatial
structure of the domestic economy was most explicitly assumed away
or relegated to the background. This step facilitated a macroscopic
process analysis (though quite elementary) of international trade w^iich
seemed so urgent to the classical school.

On the other hand, the reaction of German thought to classical
teachings, which precipitated the rise of the German historical school,
ploughed the ground for contributions in the field of "Raumwdrtschaft."
In the study of the stages of economic development, the spatial
structure of economic processes w^as necessarily a primary concern.
And, wdth the impress of the Lausanne school of thought upon German
economics, it was almost inevitable that attempts w^ould be made at
a fusion of space with general equilibrium analysis.

2. Some Early Atteivipts at General Theory

The first attempt to construct a general location theorj^ is to be
attributed to Alfred Weber in his Chap. VII, "Manufacturing Industry
Within the Economic System." i4 it is true that the father of location
theorists, von Thiinen, who was far in advance of his time, did progress

13 In his various writings Chamberlin, who has not come under German influence,
does treat spatial position explicitly, but only as one of the leading manifestations
of the broader category of "product differentiation." From his analysis emerges
explicitly the need for applying the techniques of monopolistic competition in
handling the space-econom}' of reality. However, his works cannot be classified
as general location theory.

!■* i/ber den Standort der Industnen, Tubingen. 1909; English translation with
introduction and notes by Carl J. Friedrich, Alfred Weber's Theory of the Loca-
tion oj Industiies, Chicago, 1929.

28 LOCATION AND SPACE-ECONOMY

somewhat toward a general locational analysis. It may have been
that his interests and experiences in the operation of his estate 'Gut
Tellow' served to restrict the generality of his abstract thinking.
Nonetheless, the seeds for developing the basic methodology in analysis
of specific as well as general location problems can be found in Thiinen's
work. 15 The science of economics has suffered from the relative
neglect of his methods during the nineteenth and early twentieth
centuries.

Launhardt, the other major predecessor of Weber, also failed to
achieve sufficient generality in his analysis. In fact Launhardt's studies
of industrial location and market areas ^^ treated a narrower set of
circumstances than were encompassed in Thiinen's isolated state.

Weber's attempt at general locational analysis was undoubtedly
greatly influenced by the writings of Roscher and Schaffle.i'^ Weber
pursued an essentially evolutionary approach. He tried to develop
the general basis upon which any given historical system orients
itself or, in other words, a theory of the transformation of locational
structures.

His method is to inquire into the forces that come into operation

The material in the rest of Weber's book does not concern general location. It
deals with what is usually conceived of as Weberian location theory, namely, an
industrial location theory under the special conditions that: (1) the location and
the size of the places of consumption are fixed; (2) the location of the material
deposits is given; (3) the geographic cost pattern of labor is given, and at any one
point labor is unlimited in supply at constant cost.

Weber's other important contribution ("Industrielle Standortslehre : Allgemeine
und kapitahstische Theorie des Standortes," Grundriss der Sozialokonomik, Part
VI, 2nd rev. ed., Tiibingen, 1923) merely touches the field of general location
theory.

15 Johann Heinrich von Thiinen, Der isolierte Staat in Beziehung auf Land-
wirtschajt und N ationalokonomie , Hamburg, 1826. See also the interesting article
by Bertil Ohlin, "Some Aspects of the Theory of Rent : von Thiinen vs. Ricardo,"
Economics, Sociology and the Modern World: Essays in Honor of T. N. Carver,
Cambridge, Mass., 1935.

16 See, in particular, "Die Bestimmung des zweckmassigsten Standortes einer
gewerbhchen Anlage," Zeitschrijt des Vereins deutscher Ingenieure, Vol. XXVI,
No. 3, Berlin, 1882, and Mathematische Begrundung der Volkswirischaftslehre,
Leipzig, 1885, Part III.

1'^ Wilhelm Roscher, "Studien liber die Naturgesetze, welche den zweckmassigen
Standort der Industriezweige bestimmen," Ansichten der Volkswirtschaft aus dem
geschichtlichen Standpunkte, 3rd ed., 1878; A. Schaffie, Das gesellschajtliche Sys-
tem der menschlichen Wirtschajt, 3rd ed., Tiibingen, 1873. Both were of the
German historical school and were primarily concerned with discovering whether
or not there were any natural laws or regularities in the evolving locational struc-
tures of economies. Their contribution rests in their collection of historical facts
and in their presentation of an abundance of conflicting ideas.

SOME GENERAL THEORIES OF LOCATION 29

when a people occupy an undeveloped country and establish an isolated
economic system. At first an agricultural stratum forms to produce
the necessary means of subsistence. As indicated in the preceding
chapter, the settled area with its agricultural population serves then
as the geographical foundation for all other strata. It determines in
the first instance the loci (places) of consumption for the second
stratum, namely, the primary industrial stratum, which produces for
the agricultural stratum. In turn, the primary industrial stratum
serves as the loci of consumption for the third stratum, namely, the
secondary industrial stratum. This third stratum actually consists of
numerous substrata, each of which is oriented to and is smaller than
the preceding one, the first substratum being the only one directly
oriented to the primary industrial stratum. These three strata form
the core of the economic system. The mass of local tradesmen and
functionaries, engaged in the process of circulation and in performing
personal services, strengthens proportionally the different parts of this
system.

A fourth stratum, the central organizing stratum, is essentially
independent of any of the three preceding ones. It consists of officials
and businessmen with general organizing and managing functions,
members of the liberal professions, and persons living off accumulated
wealth. Their pattern of locations within the economic system, if
not arbitrary, is determined not by economic forces but by others.
A fifth stratum, the central dependent stratum, is formed and tied to
the central organizing stratum in the same way as is the secondary
industrial to the primary industrial stratum.

The locational structures of these five strata are interrelated with
forces playing back and forth among them. For example, though the
agricultural stratum appears on the scene first, the formation of cities
incident to industrial development induces rearrangements of the
agricultural structure to conform more closely to the pattern of con-
centric zones as conceived by von Thiinen.

This is as far as Weber goes. Despite the later writings of
Englander,i8 which elaborate and develop the evolutionary approach

18 Oskar Englander, "Kritisches und Positives zu einer allgemeinen reinen Lehre
vom Standort," Zeitschrift fur Volkswirtschaft und Sozialpolitik, Neue Folge,
Vol. V, Nos. 7-9 (1926). With Englander the problem is to investigate first the
spatial form of primary production, i.e., of a land and forest economy, where all
households are self-sufficient. Next, specialized products are assumed to be culti-
vated on land of particular quality, and the consequent changes in the spatial
structure of the economy are observed. In turn, agricultural industries, mining,
manufacturing, and other economic elements and complicating factors are suc-
essively introduced and the resulting spatial realignments of economic activities

30 LOCATION AND SPACE-ECONOMY

in other ways, and of Ritschl/^ too, no essential advance in this
technique of general analysis is made. The technique is currently
inadequate; it does not present any general, heuristic principle by
means of which one can order the spatial complexities involved in
the total location of economic activities. It merely records the inter-
relations of the various strata and some of the reactions of one stratum
upon another. For any given stratum, or combination of strata, it
fails to get at the rule or rules governing structure and provides no
common denominator in terms of which all the forces stemming from
the various interrelations can be expressed and evaluated and by means
of which a net effect could perhaps be deduced. ^o This is the task of
a general theory of location and space-economy.

Nonetheless, the evolutionary approach is very useful. It not only
furnishes a convenient and meaningful breakdown for studying
historical sequences of locational structures and for classifying his-
torical facts but also will be very suggestive for pursuing dynamic
analysis, once an improved general static theory has been achieved. ^i

noted. Ultimately, according to Englander, an approximation to the picture of
a modern economy is realized.

19 Hans Ritschl, "Reine und historische Dynamik des Standortes der Erzeu-
gungszweige," Schmollers Jahrbuch, Vol. LI (1927), pp. 813-70. Ritschl, recog-
nizing that the location picture is historically relative, follows Weber's classification
of strata and traces in detail their development during the periods of village, city,
territorial, national, and world economy.

Also see R. G. Hawtrey, The Economic Problem, New York, 1925, Chaps. VII
and IX, in which to some extent he adopts the approach of Englander and Ritschl.

20 This is clearly indicated in Ritschl's work. After describing in detail, in a
section on pure dynamics, the various possible effects of changes in ten or more
major locational elements, he is unable to find a method for combining their
effects. (Op. cit., pp. 853-56.)

21 In the order of treatment of subject matter, the work of E. M. Hoover, Loca-
tion Theory and the Shoe and Leather Industries, Cambridge, Mass., 1937, some-
what resembles the evolutionary approach. But Hoover's analysis is definitely
partial, though in a broad setting. Through carefully drawing up a set of assump-
tions and relaxing them one by one, he is able to proceed from an analysis of
extractive industries to a treatment of manufacturing, first under simple condi-
tions and then under more complex ones. He emphasizes the major specific forces
at work and does not pay too much attention to general interrelations, especially
when they can be stated only in broad terms. In this way he is able to synthesize
the various theoretical contributions of his predecessors that are of practical value
and, by employing illustrative empirical material, is able to stick close to reahty.
From the standpoint of balance and sound judgment Hoover's writings are the
best. See also his Location of Economic Activity, New York, 1948.

Other major works in English on location theory are by A. P. Usher {A Dynamic
Analysis oj the Location of Economic Activity, unpubUshed) and by W. H. Dean,
Jr. {The Theory of the Geographic Location of Economic Activities, doctoral

SOME GENERAL THEORIES OF LOCATION 31

3. Predohl's Conception

Shortly after the appearance of Weber's book, Bortkiewicz^s and
Schumpeter2 3 recognized the need of a general equilibrium analysis
to supplement partial locational theories. 2 4 Considerably later
Englander2 5 came to appreciate in full the implications of a general
theory of location. The pure theory of location, according to
Englander, is the general theory of "local conditionality" within an
economy. Any given entrepreneur, in choosing the site at which to
produce or render services, considers the various supply prices existing
in the various localities for the inputs that he might possibly employ.
At the same time he considers the various prices which might be
obtained in the various localities for his product or services. When
finally he does locate at a site, he influences in turn the prices of various
inputs and outputs. Through being so interrelated, the pattern of
local price differences and the location of economic activities are simul-
taneously determined by a general theory of "local conditionality." 2 6

dissertation, Harvard University, 1938: Selections published by Edward Brothers,
Inc., Ann Arbor, Mich., 1938). Usher and Dean, too, follow a partial approach
both in their static analyses and in their dynamics where they rely upon extensive
use of historical matei'ial. Their interests are in "the broader aspects of the
developing geographic patterns of population density" and in "the relations of
these patterns to localized resources and to the significance that regional resources
possess under the technological conditions of each historical period." (Usher,
op. cit., p. 2.) They study: (1) topography as a vital and partially independent
factor in the pattern of settlement; (2) the impact of transport innovation par-
ticularly as it relates to the accessibility of resource deposits to the primary
regions of the world; (3) nodality and industrial agglomeration; (4) externally-
conditioned labor and the controlling role of energy resources; (5) the historical
patterns of urban settlements; and (6) other related subjects. These studies are
in a sense interwoven, but at best only loosely. The result is a set of highly valu-
able partial analyses, and not a finely spun general framework.

^^ Deutsche Literaturzeitung, Vol. XXXI (1910), pp. 1717-24.

^^ Jahrhuch fiir Gesetzgebung, Verwaltung und Volkswirtschajt, Vol. XXXIV,
No. 3 (1910), pp. 444-47.

24 V. Furlan ["Die Standortsprobleme in der Volks- und Weltwirtschaftslehre,"
Weltwirtschajtliches Archiv, Vol. II (1913), pp. 1-34] makes a somewhat abortive
attempt at general locational analysis. The complicated interrelations of various
economic factors as well as the "spatial transformation of goods" are fully recog-
nized; but the contributions to knowledge are essentially along the lines of devel-
oping overly simplified models of markets, domestic and international, more
specifically of determining points of collection and distribution of goods and of
export and import, and the related paths of commerce.

25 0?). cit.

26 Further: (1) by classifying raw materials and factors of production, whether
mobile or immobile, as place-free (available everywhere under the same condi-

32 LOCATION AND SPACE-ECONOMY

Somewhat earlier (1925) than Englander's publications there ap-
peared an article, "Das Standortsproblem in der Wirtschaftstheorie," ^^
by Andreas Predohl, which utilized a principle by means of which a
general equilibrium approach could be systematically applied to
location analysis. This was none other than the familiar substitution
principle, already well established in general equilibrium theory.

Although Predohl did visualize new horizons in the extended use of
this principle, he unfortunately tried to remain within the scope of
traditional thought. He purported to deduce a general location theory
as a special case of the existing general economic theory, as a logical
and inherent element of it. The general economic theory to which he
alluded was the theory of interdependent prices and quantities, of
general equilibrium as expounded successively by Walras, Pareto, and
Cassel. He wished to investigate how far the location problem is a
price problem; location theory, a price theory. In other words, to
what extent does the local distribution of production lie inside the
economic relationship of interdependent prices? ^8

Predohl contends that the problem of the local distribution of
economic activity is synonymous with the problem of the distribu-
tion of determined groups (bundles) of productive factors (he groups
productive factors under the categories of land, labor, and capital)
since every economic activity uses a grouping of factors. The distribu-
tion of determined groups of productive factors in turn is a special
case of the distribution of productive factors in general. ^ 9 To Predohl,

tions), conditionally place-bound (available at all or some places under unequal
conditions), and unconditionally place-bound (present at one site), and (2) by con-
ceiving immobile goods as goods of infinite weight which enter into production
with infinite weight-loss, Englander brings together the specific location theories of
industry and agriculture within the confines of his pure location theory, not as
distinct compartments, but as internally related sectors.

Elsewhere, too, Englander has attacked a broad range of location problems,
but only through elucidation of simplified, isolated cases. See his Theone des
Giiterverkehrs und der Frachtsdtze, Jena 1924, and "Standort" in Handworterbuch
der Staatswissenschajten, 4th rev. ed., Jena 1926, Vol. VII.

27 Weltwirtschaftliches Archiv, Vol. XXI (1925), pp. 294-331. A briefer article
in English, "The Theory of Location in its Relation to General Economics," ap-
peared in the Journal of Political Economy, Vol. XXXVI (1928), pp. 371-90. A
much more recent statement in German which was available only after the fol-
lowing paragraphs were written is contained in his book, Aussenwirtschajt : Welt-
ivirtschaft, Handelspolitik und Wdhrungspolitik, Gottingen, 1949. Also see
Predohl's reply to Englander's criticism, "Zur Frage einer allgemein Standorts-
theorie," Zeitschrijt filr Volkswirtschajt und Sozialpolitik, Vol. V, Nos. 10-12
(1927), pp. 756-63.

28 "Das Standortsprobleme . . ." Op. cit., pp. 295-97.

29 As will be shown later, this statement is very weak, if not untenable.

SOME GENERAL THEORIES OF LOCATION 33

general interdependence theory explains the distribution of productive
factors in general by means of the principle of substitution. Therefore
general location theory is deducible from the application of the
principle of substitution to the employment of the several groups of
productive factors. ^^

Predohl in his reasoning overestimates the scope of Walrasian-
Casselian general equilibrium analysis. On the whole he seems to be
under the impression that this analysis implicitly embraces the space
element in its entirety. However, as indicated previously, modern as
well as earlier general equilibrium analyses, with minor exceptions,
concern a one-point world. The element of transport cost is generally
abstracted; factors and products possess perfect mobility. In essence ,
there is no spatial distribution of factors; the relevant problem is the
distribution of factors among the various types of production. In
reality, then, the situation is the reverse of what Predohl has con-
ceived although several times he appears to realize the truth of the
matter. As we have pointed out, Walrasian-Casselian general equi-
librium analysis is but a special case of a general location theory. ^i

Nevertheless, the tools shaped by general equilibrium theory are
useful, as Predohl discovered. Starting with the familiar case of
Thiinen's isolated state, ^^ Predohl assumes all locations fixed except
that of one enterprise. A shift of this enterprise toward the periphery
implies that capital and labor outlays (including transport outlays)
are substituted for land-use outlays. The reverse takes place in a
shift toward the central consumption point. Application of the prin-
ciple of substitution will yield the site of minimum cost so far as these
two all-inclusive groups of expenditures are concerned. However,
within these two all-inclusive groups, there are other substitution
points. For example, within the former group, there is a substitution
point between transport outlays and local capital and labor outlays
(such as is involved in determining whether or not to process a product

30 Op. cit., pp. 299-303.

31 However, see Predohl's reaction to these statements in his recent article
"Von der Standortslehre zur Raumwirtschaftslehre," Jahrbuch fur Sozialwissen-
schaft, Band 2, Heft 1, pp. 97-102.

32 The features of Thiinen's familiar model are : a uniform plain with equal
fertiUty and possibilities for agricultural production at all points, at the center
of which hes a city possessing potential transport facilities of similar character in
all directions (i.e., transport costs proportional to weight and distance). Produc-
tion aUgns itself around the city in rings in accordance with the price and transport
cost of each particular product cultivated. Predohl adopts at the start an ex-
panded version of Thiinen where all conditions for all production, whether agricul-
tural or industrial, are uniform throughout the plain (ibid., p. 299).

34 LOCATION AND SPACE-ECONOMY

in order to reduce its weight or bulk) ; and within the category of
transport outlay, there may be a substitution point involved in
allocating a given portion between transporting a raw material lying
at the periphery and transporting a raw material lying near the con-
sumption center. In this manner innumerable interdependent points
of substitution arise which determine the location of any individual
enterprise. This proposition, states Predohl, can be extended by means
of general equilibrium analysis to cover the location of all economic
activities. 3 3

Inequalities in local resource patterns, land, labor, capital, and
transport do not invalidate the operation of the substitution principle.
They present various technical possibilities for production which are
different from those that would exist in Thiinen's homogeneous plain;
but essentially these new production possibilities, like the old, can
be expressed in terms of economic values and, thus, fall within the
scope of substitution operations. Similarly, economic values can be
imputed to various historical-political forces, though here many more
difficulties and arbitrary elements creep in. Recognizing these various
limitations (for example, in accounting for the locus of consumption
of the rentier classes) , Predohl does, however, maintain that the
locationally relevant substitution points, thus logically deduced, are
applicable in general. ^ 4

It is to be expected that Predohl in this first attempt at substitution
analysis would be unable to resolve all the difficulties that beset his
path. His argument is particularly weak when he becomes specific
and illustrates substitution operations — a step which he avoids as much
as possible. For example, he utilizes a vague concept, namely, a
land-use unit, and speaks of rent outlays at different sites as being
proportional to the quantities of land-use units at those sites. Land
more distant from a city and yielding less rent therefore relates to
fewer technical units of land use than does land less distant which
yields greater rent. Englander easily demonstrated that this proposi-
tion is false: that two pieces of land unequally distant from a city can

33 "Der Standort der Produktion bzw. Produktionsstufe ist also bestimmt durch
ein System von Substitutionspimkten, das derart gegliedert ist, dass die Gruppen
einer iibergeordneten Kombination untergeordnete Kombination in sich enthalten.
Ubertragen wir diese Losung auf samtliche Produktionen, dann konnen wir unter
Zuspitzung eines allgemeinen Casselschen Satzes auf unser besonderes Problem
sagen: Wenn das Preisverhaltnis in dieser Weise fur jeden einzelnen Betrieb die
standortlich relevanten Substitutionspunkte bestimmt, sind offenbar durch dasselbe
fUr die gesamte Gesellschaft die zu verwendenden Mengen in Verhaltnis zueinan-
der, mithin die Standorte bestimmt." Ihid., pp. 306-7.

34 /bid., pp. 308-11.

SOME GENERAL THEORIES OF LOCATION 35

be of the exact same quality and be utilized to the same degree, and
yet yield different rents. ^^

Further, Predohl tends to convert all spatial and quality differences
into differences in quantities of use units. Immobile labor, situated at
diverse places and of different qualities, can be converted into amounts
of labor-use units and thus made comparable. And so with all types
of resources. 3 6 In this way all geographic differences in land, labor,
and capital can be summed up into use units of land, labor, and
capital at any given point. This reasoning lies behind Predohl's
argument that the distribution among various economic activities of
determined groups of productive factors (each group at any point of
time having a unique spatial position) is a special case of the distribu-
tion among various economic activities of productive factors in general
in terms of a one-point society.

It is not necessary to carry the argument to such an extreme, if
not untenable, position. It is appropriate now to suggest certain
revisions and extensions to be developed in subsequent chapters in
order to strengthen the basis for a widespread use of the substitution
principle in location analysis.

First one ought to distinguish between two types of substitutions:
(1) that between transport inputs; and (2) that between outlays,
between revenues, and between outlays and revenues. If there is any
sense at all to location economics, it is because there are certain
regularities in the variations of costs and prices over space. These
regularities arise primarily because transport cost is some function of
distance. If this were not so, if transport costs were completely
irregular and their changes unpredictable — for example, if transport
costs on a certain item were positive for a distance of 100 miles
and negative for a distance of 101 miles — there would be little sense
in searching for a general economics of plant locations. The spatial
pattern of industrial concentrations, of consuming centers, and of the
production of raw materials would be quite arbitrarj^ from the economic
standpoint.

Since it is the distance factor that is the heart of locational analysis,
there is every reason to speak of transport inputs (a concept to be
defined later) , wherein distance and weight are the two basic factors,
and of transport rates as prices of these inputs. Location theorists
unfortunately have shied away from such a concept. However, it

35 "Kritisches imd Positives . . . ," op. cit., pp. 499-500. See Predohl's weak
reply ("Zur Frage . . . ," op. cit., pp. 758-60). Also see his later and stronger reply
("Von der Standortslehre . . . ," op. cit., pp. 100-101).

36 "The Theory of Location . . . ," op. cit., pp. 380-81.

36 LOCATION AND SPACE-ECONOMY

brings into bold relief the basic aspects of spatial analysis without
the necessity of tagging each unit of land, labor, and capital with a
set of absolute spatial co-ordinates or of converting them into common
units, if, indeed, this can be done. The problem of production becomes
a problem of choosing the right combination of the various types of
capital, 'labor, land, and transport inputs. In the case of transport-
oriented industries and transport- oriented sets of economic activities,
how the essential location analysis reduces to a consideration of
substitution between transport inputs will be demonstrated.

The selection of the correct substitution points between transport
inputs is easy to visualize although in practice it may be difficult to
effect because of the complicated nature of transport rate structures.
However, selection of the correct substitution point between a transport
input and a labor input, or between the two groups, transport inputs
and labor inputs, cannot be so satisfactorily handled. That here the
choice of the optimum location requires an outright comparison of
outlays on the various kinds of labor, or of total labor outlays and
total transport outlays, or of total labor outlays and total interest
outlays, and so forth will be indicated. Substitution analysis in terms
of outlays and revenues must supplement substitution analysis in
terms of transport inputs in order to achieve a proper locational
methodology.

By this approach, Predohl's original conception can be made more
digestible and broadened into a general equilibrium theory of space-
economy, which includes as special cases various types of location
theories as well as actual modern general equilibrium theory. ^'^

37 It is interesting to note that Predohl and others have rightly pointed out
that Weber's industrial location theory is chiefly based on technical empirical
knowledge. Transport costs are reduced to weight and distance, i.e., to technical
factors; varying raw material prices and other elements are reduced in similar
fashion. Technical concepts such as locational weight, material index, coefficient
of labor, Formkoeffizient, and others are the critical measures. The point of trans-
port orientation is merely the point of minimum transportation in terms of ton-
kilometers. Essentially Weber abstracts from most economic interrelations and
reactions. Only under severe limitations is Weberian doctrine generally applicable.

However these criticisms do not detract from the merits of Weber's contribu-
tion. Formal theory, in and of itself, is highly unsatisfactory, too general and,
accordingly, too sterile. As Predohl emphasizes, it needs to be supplemented by
concrete information; abstract and vague values must be replaced by exact,
quantitative data. In other words, supplementary explanations are required, even
if they are obtained in such a manner as to limit their general validity. It is in
fulfilling this need that Weber's work is of great significance. If the general theory
of location constructed upon the principle of substitution is to be of pragmatic
value, to it must be added empirical location theory and statistical investigation
which seeks out regular movements in major economic variables, even though

SOME GENERAL THEORIES OF LOCATION 37

4. Weigmann's Formulations

In this section a summary of the contributions of Hans Weigmann,3 8
which have received but slight attention in the literature is presented.
Weigmann's writings on general location theory are very difficult to
comprehend, because of both his vague style and the complexity of the
basic concepts. These concepts do not lend themselves to a general
synthesis as do those of other contributors. Nonetheless, they seem
to disclose some of the more promising channels of exploration for
further theoretical development.

Weigmann attempts to formulate the foundations for a realistic
economic theory which embraces the spatial structure of economic
processes, the spatial extent and bonds of markets, and the spatial
interrelations of all economic quantities.

The first principle that Weigmann establishes is that a theory of
space-economy embraces a theory of limited competition. Actually all
factors and goods, regardless of setting, face immobilities of varying
extent in all directions; and, in accordance with the nature of the
obstacles to movement, whether they be economic, social, political, or
cultural, markets are restricted in scope. The competition which any
good or factor can ofTer to other goods and factors at different locations
is incomplete. The existence of physical space implies immobility,

this means eliminating the numerous special factors which affect each individual
situation. Weber's theory of industrial location is just such a supplementary,
empirical theory (it excludes economic details which he considers relatively unim-
portant, and thus in great part hypothesizes that the set of technical substitution
points approximately parallels the set of economic substitution points). But
ultimately all such empirical technical functional observations must be translated
into economic terms.

Indeed, it is only by utilizing chiefly the Weberian approach with supplementary
economic data that I have found it meaningful to analyze the locational structure
of the iron and steel industry. See my "Some Locational Factors in the Iron
and Steel Industry since the Early Nineteenth Century," Journal of Political
Economy, Vol. LVI (June 1948) ; and (with W. Capron) "The Future Locational
Pattern of Iron and Steel Production in the United States," Journal of Political
Economy, Vol. LVII (April 1949). Also see E. Niederhauser, "Die Standorts-
theorie Alfred Webers," Staatswissenschaftliche Studien, Vol. XIV (Weinfelden,
1944).

38 "Ideen zu einer Theorie der Raumwirtschaft," WeUwirtschaftliches Archiv,
Vol. XXXIV (1931), pp. 1-40; and "Standortstheorie und Raumwirtschaft" in Joh.
Heinr. von Thilnen zuvi 150 Geburtstag, ed. by W. Seedorf and H. Jurgen, Rostock,
Carl Hinstorffs, 1933, pp. 137-57. To trace the development of Weigmann's
thought the reader is also referred to the following of his works : Kritischer Beitrag
zur Theorie des internalionalen Handels, G. Fischer, Jena, 1926, and Politische
Raumsordnung, Hanseatische Verlagsanstalt, Hamburg, 1935.

38 LOCATION AND SPACE-ECONOMY

limited competition, and spatial inelasticity (or negative spatial
elasticity). Thus the generally accepted principle of pure competition
is not applicable to the analysis of spatial economic processes. ^^

A second basic principle concerns the question of form. In place of
customary linear causal analysis, Weigmann favors the approach
of general equilibrium theory in the employment of Gestalt analysis.
He observes the space-economy as a whole in its full array of spatial
markets. In that sense he aims at presenting a realistic functional
picture of the "form-full" of economic life, wherein the various
elements are weighted in accord with their importance. Having
adopted this methodology, he confronts the primary problem of
determining the basic form (Grundgestalt) of economic phenomena,
i.e., the Gestalt core. This basic form should then provide an
heuristic principle to help master and order systematically the "form
wealth" of real economic life, or in other words, the countless spatial
forms of moving economic processes. ^o

At this point Weigmann differentiates between statics (mobility or
competition as potential energy) and dynamics. Since he purports
to describe the space-economy in its realistic setting, he is compelled to
complicate his problem manifoldly by introducing the time element and
by assigning time co-ordinates to his various markets and processes.
Weigmann poses the perplexing problem of dynamics as follows: to
choose that time period which would yield in the resulting spatial array
of markets a competition field (a broad market area in time and space)
which could be valid as the basic form.^i He resolves the problem by
formulating a concept quite difficult to comprehend, the concept of
"relative maximum." It states that as an increasing amount of physical
space (therefore spatial resistance) is to be overcome in movement
by an economic object, the time period necessary for such movement
increases until it reaches a maximum — a maximum in the sense that
given still more time a further spatial movement would be improbable
because of the overpowering force of the countless obstacles. There,

39 "Ideen zu . . . ," op. cit., pp. 6-9. These points are also developed by Cham-
berlin in his doctoral dissertation of 1927, op. cit., especially pp. 105-09, 167-84;
and in The Theory oj Monopolistic Competition, 1933, passim. In the latter, a
portion of the earlier analysis devoted to two-dimensional space was simplified
to one dimension and removed from Chapter 5 to Appendix C.

40 /bid., pp. 9-12.

•*i ". . . welches Konkurrenzgebeit ist essentiell im Sinne des Gestaltganzen,
wenn behebiger Absteckung der Zeitgrenzen eine Fiille raumhche variierender
Flachen entsteht? Oder anders ausgedriickt: welcher Zeitraum ist zu wahlen,
damit ein mit diesem Zeitmass gegebenes Konkurrenzfeld als Grundgestalt
gelten kann?" Ihid., p. 14.

SOME GENERAL THEORIES OF LOCATION 39

where the time period reaches its maximum, competition ends and the
competition field becomes bounded. In other words, the force of com-
petition does not have the power to span a distance greater than the
radius (or axis) of its field, irrespective of the time factor for all
practical purposes. This principle contains the definition of basic
form. The basic form is depicted as that unit of space (corresponding
to a market region or competition field) of the relatively greatest time-
weight, hence, of the relatively greatest stability and permanence. ^ 2

Having exposed the tremendous magnitude of the task of formulat-
ing a theory of space-economy, Weigmann stops for breath. How to
locate the basic form? How to represent as an empirical Gestalt unit
the multitude of interlaced, mutually related individual markets,
market strata, and market densities? From here on our author can
only offer fruitful suggestions and preliminary observations for con-
quering the manifold difficulties which appear. First we have the
classification of markets according to structure. Each individual
commodity market including its labor, capital, and land orientation
possesses a particular structure which offers a certain resistance to
change. Some change frequently, others slowly. Some are active,
others highly inactive. By definition those markets of a relatively
permanent nature, of persistent inactivity, are grouped together as the
essential ones, as the basic form; their combined structure determines
the basic structure of the Gestalt whole, of the space-economy under
question. On the other hand, the rapidly changing markets are con-
sidered as accidental or secondary; their movements are charac-
terized as minor modifications of the Gestalt form, and these
movements are to a certain extent conditioned by the already
determined basic form, by the core of markets of greatest continuity.
Fundamental organic change of the Gestalt picture of the space-
economy, therefore, implies only change within this relatively immu-
table core of persistent markets. ^^

42 ". . . und fiihren angesichts des Vorhandenseins eines relativen Maximums
zeitkostender Bewegung innerhalb jedes Gestaltganzen zu einem Bilde sich biin-
delnder und iiberschneidender Konkurrenzfelder als den akzidentiellen und peri-
pheren Erscheinungsformen einer zentralen Grundgestalt. Der Begriff des
relativen Maximums besagt dabei folgendes: Die Konkurrenz wird gradweise
beschrankt und als dort aufhorend bedacht, wo die zahlreichen hemmenden
Faktoren den sukzessiv steigenden Zeitaufwand der Bewegung zu einem Maximum
hinfuhren, bei dem unter den konkret gegebenen tjmstanden die weitere Bewegung
unwahrscheinlich wird oder auf lange Sicht nicht mehr die Kraft besitzt, mit
Ansicht auf anhaltenden Erfolg Spannungsunterschiede aufzugleichen. Die Grund-
gestalt ist also die Raumeinheit des relative grossten Zeitgewichtes; die Dauer
des Bestandes gibt ihr den Charakter der essentiellen Form." Ibid., pp. 14-15.

43 /bid., pp. 16-19.

40 LOCATION AND SPACE-ECONOMY

Our task is further illuminated by reference to the structures of the
specific markets for land, labor, and capital goods. Weigmann
maintains that the markets for the productive factors of labor and
land are primary constituencies of the basic form. Movements in all
commodity markets course back to these two, whether directly or
indirectly, through semifinished products and various stages of produc-
tion. And in these markets for labor and land, which offer great
resistance to change, are focused the facts of scarcity within the
economy.

The land market is portrayed as a spatially-connected area of
supplied land services. Actually each individual piece of land is
distinct and immobilized by nature, so that its supply area has no
spatial extent. But for practical purposes Weigmann conceives of a
Gestalt whole (space-economy) already in existence. This whole
exerts an hypothetical aggregate demand which, in turn, defines the
boundaries of the land market, the peripheral area being considered
as marginal land. The supply of land in general is not perfectly
inelastic, spatially speaking. Change in the land market ensues (1)
from additions or subtractions at the fringe to the land under cultiva-
tion, i.e., an expansion or contraction of the space base of the economy,
and (2) from variations in the intensity of use and in the methods of
cultivation and organization of each individual land unit. ^ 4

The size and nature of the hypothetical demand mentioned above is
obviously related to the labor market. The labor market, in contrast
to the market for land, is much less rigid and invariable; for that
reason the conception and description of it are theoretically and
empirically much more difficult. There are many forms of labor
immobility and inelasticity. Weigmann makes a beginning at analysis
by explaining one, namely migi^ation mobility. To delimit the labor
market accordingly, one must recognize the various tiijae stages of
migration (e.g., seasonal, cyclical, and secular) and their spatial
forms. In line with familiar Weberian technique, the long-run labor
base is presented as a continually moving, organic process whereby
labor, step-by-step through varying intervals of time, gradually
moves from farms or rural communities to giant metropolitan centers
via town and urban clusters of increasing size.^s This ever structural

^^Ibid., pp. 20-23.

45 Weber ("Industrielle Standortslehre . . . ," op. cit., pp. 74-84) distinguishes
two stages of modem capitalistic development: (1) bound (gebundene) capitalism
and (2) free capitalism. In the former, which characterizes the sixteenth to the
eighteenth centuries, labor is historically fixed, locationally immobile. In the
latter, which characterizes the present times, labor becomes mobile, released from

SOME GENERAL THEORIES OF LOCATION 41

movement within the labor base is designated as one of the essential
dynamic aspects of modern space-economies.'* ^

With respect to markets for capital goods, Weigmann offers a few
suggestions. First we must distinguish capital in substance from
capital in title. The former obviously has far more limited mobility.
Second, capital goods (a concept which in its broadest formulation
includes all commodities) must be classified according to the extent
to which they become bound up in production. At one end of the
scale would be "combination-free" capital goods; at the other end
would be capital goods permanently tied to a given production
combination. The spatial elasticities and markets of the several
divisions of capital goods would vary accordingly. Unfortunately,
present day terminology regarding capital (e.g., fixed and circulating)
is unsuited for depicting its spatial elasticity; nor has theory recog-
nized the influence of spatial elasticity upon the various other elasticity
forms of capital.

The formulation of the problem of a theory of space-economy thus
is more comprehensive than that of traditional location theory. The
latter has chiefly treated capital as a "combination free" factor in its
long-run agglomerative setting but has given little attention to the
mobility of existing equipment, to the short-run adaptability of capital
goods. In fact, for an empirical theory, there are even strong grounds
for considering the mobility of a given combination of various produc-
tive factors as a whole, rather than of their constituent parts — a phase
of the problem which location theory has rarely posed. ^^ Furthermore,
location theory has frequently been of restricted scope in that it has
often sought the ideal location of a given firm, others assumed in
equilibrium.

Thus Weigmann sketches his picture of the space-economy as a
rhythmic-moving Gestalt whole with a core composed of the markets
for land, labor, and capital goods and of numerous other markets
superimposed upon these, overlapping and irregularly intersecting
each other and at times extending into other space-economies.^s His

its historical bonds. The economies of concentration and large-scale organization
can come into operation and can offer incentives for huge masses of labor to
agglomerate at given points. On the other hand, these forces are offset by the
community attachments (home feeling) of the individual laborer, by his lack of
perspective and initiative, and by the consequent increase in rent at the points
of agglomeration. The net result is the step-by-step migration already mentioned.

46"Ideen zu . . . ," op. cit., pp. 23-27.

^Ubid., pp. 27-32.

48 In the Thiinen Festschrift (op. cit.) Weigmann commences with existing loca-
tion theory (the Englander and Predohl versions) and approaches a theory of

42 LOCATION AND SPACE-ECONOMY

presentation lacks clarity and frequently one is forced to construe an
imaginary model in order to follow the argument. Nevertheless, one
obtains penetrating insight into the subtle spatial relations of economic
life and is given an original as well as a challenging view of the
immense magnitude of the assignment.

5. Palander's Criticisms and Losch's General System

In Sect. 3 a development of the framework of a general (static)
theory of location and space-economy in terms of the substitution
principle has been suggested. However, it has been customary in
general equilibrium analysis to present the relations of a given one-
point economic order by means of a system of mathematical equations.
Should a solution for this system of equations exist, the merit of the
presentation is generally regarded as considerably enhanced. Can a
solvable system of equations be evolved for a space-economy?

Tord Palander, in the first major work on location theory to originate
outside of Germany, addressed himself to this question. ^ 9 He con-
sidered insuperable the difficulties encountered by the general approach
in representing or even closely approximating reality.

First, states Palander, writing in 1935, the Walras-Pareto-Cassel
general equilibrium theory in its present form is meaningful for a
locational analysis only of an economic district wherein transport
costs are zero, capital and labor perfectly mobile, and technical
conditions of production uniform throughout — in other words, where
the district in question can be compressed into a point market. To be
sure, he continues, a somewhat closer approximation to reality can
be obtained by withdrawing one by one the simplifying assumptions
given above. For example, there might be introduced into the simpli-
fied model the following series of complications: freight costs on
product based on distance and weight, transport costs for mobile pro-
duction factors, equal real wages throughout the district, consumption
as dependent upon location choice, and so forth. Even so, contends
Palander, this procedure would not take us far, for in respects other
than the neglect of local differences in demand and supply of factors
and commodities, the deviations of a general equilibrium theory from

space-economy ("total localization") in part through the extension of Predohl's
substitution principle to include "quantity elasticity." Quantity elasticity is
synonymous with a broad definition of elasticity of supply, one that embraces,
among others, spatial elasticity. In this way Weigmann brings out the logical
bond of location theory and general price theory.

'^^ Beitrdge zur Standortstheorie, Almqvist & Wiksells Boktryckeri-A.-B.,
Uppsala, 1935, Chaps. X and XI.

SOME GENERAL THEORIES OF LOCATION 43

reality are severe. Interdependence theory has as an underlying
premise the principle of pure competition. Yet, in no sense at all,
can the traditional interpretation of this premise hold when we intro-
duce space and thus transport costs into the analysis. If the various
places in a region under consideration are treated as different markets
(corresponding in this way to the varying local prices resulting from
transport costs between these places) , then the necessary condition
of a large number of buyers and sellers for each commodity and factor
at each market, cannot be fulfilled. If the region itself is viewed as
one market, one could interpret the different prices ruling for a given
commodity at the various places within the region (1) as signifying
non-uniformity of product, or better yet, (2) as signifying a uniform
product in a persistently imperfect market where individuals are in
monopoly situations in accordance with the advantages of their
respective positions. Neither case could be regarded as pure
competition. 5 0

If one now discards the premise of pure competition, he must
necessarily forsake certain supplementary simple principles which
have served as scaffolding for general equilibrium theory, namely, that
the price of a commodity equals average cost (the latter including a
normal profit) and that the price of a factor equals the value of its
marginal product, ^i

Further censure of general equilibrium analysis follows from its
limitation to static conditions, a widely recognized limitation which
does not need to be discussed here.^^ Palander insists on the necessity
of depicting the economic development process. His conscience thus
compels him to forego Walrasian economics in favor of the Launhardt-
, Weber tradition. His energies are confined to analyzing the economic
starting point, the adaptations of enterprise during a time period, the
movement of factors during the same period, and the concomitant
changes of technique, institutions, and consumer base. ^ 3

August Losch, however, has not accepted these views. In his
monumental work, Die rdumliche Ordnung der Wirtschaft,^^ he goes

50/6id., pp. 273-77.
51- Ibid., pp. 277-78.

52 However, see the work of Samuelson which relates the comparative statical
behavior of a general equilibrium system to its dynamical stability properties
{op. cit., Part II).

53 Palander, op. cit., pp. 278-85.

54 G. Fischer, Jena, 1st ed. 1940, 2nd ed. 1944. All page references are to the
second edition. Part of the material of this book is available in English in the
article "The Nature of Economic Regions," Southern Economic Journal, Vol. V
(July 1938), pp. 71-78, and in a review article by W. F. Stolper, American Eco-

44 LOCATION AND SPACE-ECONOMY

beyond partial analysis and the mere recognition of the complex spatial
interrelations of economic factors. He presents succinctly, through a
set of elementary equations, a highly simplified static model of a space-
economy operating under conditions of monopolistic competition. To
appreciate fully this model, one must understand Losch's concept of the
market, by means of which space is introduced into the problem and
which represents his other major contribution to location theory.
What is the market area? How is it bounded? These questions are
fundamental.

Losch postulates the following: a broad, homogeneous plain with
uniform transport features in all directions and with an even scatter
of industrial raw materials in sufficient quantity for production; a
uniform distribution of agricultural population with a uniform set of
tastes and preferences, each homestead at the start being self-
sufiicient; technical knowledge disseminated throughout the plain and
production opportunities available to all. In all other respects, too,
extra-economic forces are excluded. If in this situation an individual
finds it profitable (owing to the economies of large-scale production as
opposed to the handicap of transport cost) to produce a commodity
over and above the needs of his homestead, his market area would
assume a circular form. However, if one farmer finds it profitable
to produce over and above his needs, so will others, and the force of
competition, by eliminating all excess profits, not only will contract
the market area of the original producer but also will transform the
circular shape of the market area into a hexagon. The hexagon is
the ideal economic form of market area, it is maintained. Firstly, a
net of hexagonal market forms will exhaust (completely cover) any
area under consideration, whereas circular ones will leave empty
unutilized corners, as is readily seen from a graphic presentation.
Secondly, of all the regular polygons (hexagon, square, and triangle)
which will exhaust a given area, the hexagon deviates least from the
circle form and, in consequence, minimizes the transport expenditures
in supplying a given demand or, expressed differently, maximizes
the demand of the population of a given area.^s

For each commodity, then, the plain is dissected into a honeycomb

nomic Review, Vol. XXXIII (September 1943), pp. 626-36. Also see Losch's
article, "Beitrage zur Standortstheorie," Schmollers Jahrbuch, Vol. LXII (1938),
pp. 329-35.

Since the writing of this and other chapters, Losch's basic work has become
available in an English translation: The Economics of Location, Yale University-
Press, New Haven, 1954.

55 Die rdumliche Ordnung . . . , op. cit., pp. 70-78.

SOME GENERAL THEORIES OF LOCATION 45

(a net of hexagons) of market areas. Losch next groups these
honeycombs according to the size of their respective market units.
And, in a manner consistent with the estabUshed criterion of minimum
transport effort, he orders the resulting nets about a common, central
production point to obtain his system of nets. ^ 6

We are now in a position to reproduce Losch's attempt at a general
equilibrium scheme. At the start Losch attacks the problem of the
location of the production of industrial goods alone. The same
hypotheses which were basic to the above determination of market
areas are retained for the general analysis. Table I presents the
symbols of spatial arrangement. The position in the plain of each
production place of each commodity is designated by a set of x, y
co-ordinates; the boundary of the market area of each production place
is described by a set of equations, each equation being represented by
a corresponding Greek symbol in Table I.

Losch puts forth the following, as either given or imknown:

A. Given:

^m ^ /'"(tt) individual demand for good m
TTg™ = (f)"^{Dq) the factory price 1 of the good m at place g as a

I function of the total demand
j^m = x^iDq) the average produc- 1 Dg'" = ^(J"", x^y^, a^i^^

tion cost J ■ ■ ■ ^|^'^ ^1 ^9™ ■ ■ ■)•

S>^ = Dg™(7rg™ — /c/O the profit on product m at place g

0- = rural population per square kilometer
a^ = population of the city Pq"^
r = freight rate
m = number of products
G = total surface area

B, To be sought:

Number of

unknowns

L

TT,-

= factory price of the good m at
location Pq^

n

2.

Qm

= market area of the location Pq""
in square kilometers

n

3.

g-

= the number of towns which pro-
duce good m

m

4.

X^:

. 2/r

= co-ordinates of the location Pq"^

2n

5.

«r

,/3r--

• • €q"^ = equations of the boundaries of
the market area of Pg^

N

Total: ^n + m + N

Corresponding to the list of unknowns, Losch presents in Table II
a set of equilibrium conditions. The first condition for equilibrium

56 /bid., pp. 79-90.

46

LOCATION AND SPACE-ECONOMY

Table I. Symbols of Spatial Arrangement

Prod-

Production Places

Market Boundaries

uct

Num-

Num-

Abbreviations of

Num-

ber

Position

ber

their equations

ber

1

Pi

^{xi'yi');}H'---Va'

a

al^/3ll•••ell;a2^/32l••

A

2

Vi

Hxihj,'^);P2'---pb'

b

«l^^l^•••r?l2;a2^/32^••

■ B

ii"'(xi™yi'"); ^2"

(total)
A+B + ---

m (total) n (total) N

= a + b + --- + q

which must be fulfilled is that each producer occupy a spatial position
which maximizes his profits; as a result he will not find it desirable
to change his location either in the x or y direction.

Second, the whole plain under consideration must be exhausted by the
various market areas for any particular good. Third, no abnormal
profit may exist; the cost of each commodity produced at any factory
must equal its factory price. Fourth, the changes in average price
and average cost ensuing from an infinitesimal change in the size of
any producer's market area must be equal. This follows from the
assumption of free entry into any fine of production and from
the negatively sloping demand curve confronting each producer. In
other words, a Chamberlinian tangency solution results which guaran-
tees that the size of each producer's market area must be the minimum
economically possible. This condition together with the second and
third insures a maximum number of independent producers.

The fifth condition requires that any consumer on any boundary
line be indifferent as to the possible production sources from which he
can obtain a given commodity at the same minimum delivered price.

Since, in toto, the number of fulfilling equations of Table II equals
the number of unknowns, the system of spatial economy is determinate ;
the unknowns can be derived. ^"^

^'^ Ibid., pp. 63-68. Tables I and II and the lists of given and unknown condi-
tions are for the most part literal translations.

SOME GENERAL THEORIES OF LOCATION

47

^

T3

vli;

II

3

+

^~

o

Cb"

Xi

^

o

II

+

w

o

5S5

s. s

Q

H

1

+

g

X

*&

8^

fl

>

S

II

'^

Cb

'o

o

+

'^

f^

a

^

II

Q

g

1

s

OS

5^
+

^^i

S^

^

Cb*

o

s

c^ ^

^

-0-

^

^

f^

^

G

c«

73

a:>

i=l

rS

3

O

o

6

' ^

m

M

X!

1

2

a

a

5:;

a

s

3 c3

o

is

c3

Xi

§

1

X '+-'

'^

cS

03

11

I

o

2

^

l-H

gJ90npojd
JO

48 LOCATION AND SPACE-ECONOMY

In a manner analogous to the above, states Losch, the location of
the production of agricultural goods can be analyzed. In a similar
manner, too, the reverse propositions, which concern the conditions for
the best location of industrial and agricultural places in their capacity
as consumption centers, can be attacked. Unfortunately, however,
the optimal location for production does not necessarily coincide
with the optimal location for consumption; and Losch fails to develop
the necessary additional sets of equations.

This is the way in which Losch spins his web of general equilibrium..
Although his approach minimizes the elements of interdependence and
does not comprehend the space-economy as a whole but as consisting
of several major sectors, and, although it has other severe limitations, ^^
we have here for the first time an attempt to encompass general spatial
relations in a set of equations. And through eschewing the assumption
of pure competition and postulating monopolistic competition in its
stead, Losch goes far toward meeting Palander's objections to spatial
general equilibrium analysis.

One need not, however, proceed, as does Losch, in deriving a set of
equations. Losch assigns a set of spatial co-ordinates to each producer
and consumer. This step permits, in a sense, a geographic description
of a space-economy. But his presentation would become exceedingly
complex if one were to relax the simple uniformity assumptions which
are basic to his model — if one were to allow inequality in raw material,
labor, and capital resources, an uneven and discontinuous distribution

58 Some of the more significant criticisms of Losch's scheme may be noted :

1. Losch's model is not a true general equilibrium system since his commodity
and factor markets are not interrelated via utility and production functions in the
complex manner which tj^pifies a Walrasian system. E.g., the model is based on
the assumption that the price of a commodity is a simple function of its demand,
an assumption which is open to serious objection.

2. A simple coimt of equations and unknowns does not necessarily prove the
existence of an equilibrium.

3. His treatment of boundarj^ equations as single unknowns is also subject to
major criticism. A conception of a continuous field of price gradients would be
much superior.

4. It may be claimed that Losch's model has both too many unknowns and too
many equations. If conditions are assumed which lead to the formation of a
honeycomb of regular hexagonal market areas for any given commodity, then
once the location of one producer of that commodity is fixed and once the equa-
tion of one of his six boundary lines is known, the location of all other producers
of that commodity and the equation of all other boundary lines are known and
determined.

5. His system is built implicitly upon an hexagonal net of market areas whose
derivation and construction involve inconsistencies some of which will be noted
later in this book.

SOME GENERAL THEORIES OF LOCATION 49

of population, and all other types of local differences. To introduce
inequality in the spatial pattern of inputs alone is a very difficult task.

From a functional standpoint — one that is relevant to the incessant
struggle within the economy as a whole, as well as within its various
parts, to obtain the correct set of substitution points with respect to
inputs, outputs, outlays, and revenues — Losch's model is anemic. It is
much more meaningful to design a set of equations depicting general
equilibrium in terms of input-output relations and price-cost relations,
including therein transport inputs (and if possible local price-cost
variations) in order to give explicit recognition to the factor of space.
This latter model, constructed without reference to absolute spatial
co-ordinates, would be much more able to cope with further possible
theoretical developments, for example, with the structural, dynamic
developments visualized by Weigmann; although, to be sure, any
spatial description of the order of Losch is desirable if it does not
impose restrictions upon the basic operations of the model. ^ 9

With respect to input-output relations, the Leontief technique, within
the severe limits to substitution imposed by its assumptions, offers a
powerful tool of analysis. ^^ It will be seen in a second volume that
one can give an increasing amount of play to spatial substitution
operations (1) through rearranging the activities included in the
structural matrix and bill of goods sector in order to incorporate
locational shifts of basic industries (and the associated local multiplier
effects) resulting from substitution between transport inputs and
between various outlays (including transport outlays) and revenues;
(2) through detailing interindustry flow tables by decomposing the
nation into regions and establishing an hierarchy of regions; and (3)
through introducing resource limitations and other non-linearities by
employing an iterative approach, by changing relevant coefficients, and
by other devices. Such a modified model can thus reflect, to a large
extent, the interactivity relationships of the space-economy.

Apropos of price-cost relations, it cannot be too strongly emphasized
that the theories of space-economy and of monopolistic competition
(broadly conceived^^) are inextricably bound together. The note-

59 For other relevant criticism of Losch, see W. F. Stolper, op. cit. and Hans
Ritschl, "Aufgabe und Methode der Standortslehre," WeltwirtschaftUches Archiv,
Vol. LIII (1941), pp. 115-25.

60 w, "w. Leontief, The Structure of American Economy, ] 919-1939, New York,
1951; and W. W. Leontief et al.. Studies in the Structure of the American
Economy, New York, 1953.

61 To include oligopoly with or without product differentiation. See E. H.
Chamberlin, "Some Final Comments," The Review of Economics and Statistics,
Vol. XXXI (May 1949), pp. 123-4.

50 LOCATION AND SPACE-ECONOMY

worthy contribution of Chamberlin in developing techniques for spatial
analysis has not been treated specifically since it has been largely
digested by Losch and Palander. Progress along Chamberlinian lines,
however, is a sine qua non for developing further the theory of the
space-economy in its welfare aspects^s which, however, are beyond
the scope of this book. Triffin has already built upon Chamberlin's
structure, setting the monopolistic competition techniques in a general
equilibrium framework. 6 3 Triffin's interdependence analysis, in many
places explicitly cloaked in substitution terms, is not unlike Predohl's
substitution technique (although, to be sure, Triffin hardly thinks in
terms of spatial or location relations). In this sense, then, a gen-
eralized theory of monopolistic competition, broadly defined and
including the physical production (input- output) problem in its spatial
setting, can be conceived as synonymous with our general theory of
location and space-economy.

6. Ohlin's View of Trade and Location Theory

One final matter should be discussed, namely, the interrelation of
trade theory and the general theory of location and space-economy.
In 1911 Weber pointed out that classical trade theory ignored entirely
the transport cost involved in traversing space. ^ 4 He particularly
criticized the classicists for overlooking the large portion of inter-
nationally distributed industry which is transport-oriented and which
seeks the minimum transport cost point with respect to raw materials
and markets, and for attributing to international division of labor and
capital the international distribution of transport-oriented industry.

Furlan, Englander, Ritschl, Weigmann, and others have stressed

62 For most situations of the space-economy, it is quite meaningless to apply
the norms of pure competition. Also see Chamberlin, Theory of Monopolistic
Competition, Cambridge, Mass., 3rd ed., 1938, pp. 208-13.

63 R. Triffin, Monopolistic Competition and General Equilibrium Theory, Cam-
bridge, Mass., 1940. Although the monopolistic competition of Chamberlin seems
to be more than a particular equilibrium theory (certainly it is at least a quite
broad particular equilibrium theory for it embraces the problems both of indi-
vidual equilibrium and of equilibrium for an elastically defined group), nonetheless
it is not a general equilibrium theory in the full meaning of the term (see Triffin,
pp. 8-9, 54, 67, and elsewhere). Triffin's contribution consists of extending the
scope of monopolistic competition to encompass the complex net of competitive
interrelationships throughout the entire economic collectivity. In doing this,
Triffin discards the concepts both of an industry, and of a group of firms. Rather,
he emphasizes the individual firm (or more strictly, the maximizing unit) and
the various coefficients of interdependence between any given firm and each of
all the other fiims in the economy, both with respect to factors and to products.

6-1 "Die Standortslehre und die Handelspolitik," Archiv fur Sozialwissenschaft
und Sozialpolitik, Vol. XXXII (May 1911), pp. 667-88.

SOME GENERAL THEORIES OF LOCATION 51

the interrelation of trade and location theories, but not until the
appearance of Ohlin's Interregional and International Trade^^ do we
have a serious attempt to integrate the two. As one of his objectives
Ohlin purports:

"to demonstrate that the theory of international trade is only part of a
general localization theory, wherein the space aspects of pricing are taken
into full account, and to frame certain fundamentals of such a theory as
a background for a theory of international trade, wherein the influence of
local differences in the supply of factors of production and transportation
costs within each country is duly considered." 6 6

Ohlin plants his objectives within the framework of a mutual-
interdependence theory of pricing, the latter to be expanded to enfold
the multitude of markets and local price variations which ensue
from the varying spatial immobilities and indivisibilities of goods and
factors. Thus his general localization theory would determine simul-
taneously prices, markets, location of industry, commerce and
agriculture, spatial distribution of factors and commodities, and other
economic magnitudes.

It would seem logical that Ohlin should first develop a general
localization theory. Then, by focusing upon certain forms of immo-
bilities of factors and goods (consideration of other relations set aside
for the time being) he could develop at length his theory of inter-
regional and international trade as a special case. Unfortunately,
Ohlin adopts an entirely different procedure and as a consequence has
to employ a nearly unique casuistry. Parts I and II of his book are
devoted to the theory of interregional trade and a simplified version
of international trade, respectively. These parts, however, are
constructed upon an unrealistic set of hypotheses. The region is
defined as that area within which there is perfect mobility of factors.
Between regions factors are considered perfectly immobile. And all
impediments to movement of commodities are assumed away. In
Part III he attempts to approach reality through the successive intro-
duction of the following: (1) interregional costs of transfer of
commodities; (2) interregional factor movements; (3) interior costs
of transfer and factor movement; (4) local differences in labor and
capital supply. The inclusion of the last two items represents an effort
to subject the theory of interregional trade to a broadening process
and thus convert it into a general localization theory.

65 Cambridge, Mass., 1933. Some of his other relevant works are: Handelns
teon, Stockholm, 1924, and "Some Aspects of the Theory of Rent: von Thiinen
vs. Ricardo," op. cit.

66 Op. cit., p. vii.

52 LOCATION AND SPACE-ECONOMY

Ohlin at most achieves a weak (and only verbal) general localization
theory. He does not attain the total systematic analysis which
characterizes his interregional trade theory. His treatment of loca-
tional forces (Chaps. X-XII, inclusive) is quite sketchy and flimsy.
At the start, a modified version of Thiinen's isolated state is applied
to industrial production within a district, "the frontiers of which are
not described." Ohlin imagines at first that his district possesses
"uniform transport features" throughout its area and that within it
the factors of labor and capital are perfectly mobile. At the center lies
a strategic natural resource, perhaps coal or iron ore deposits. The
surrounding zones of cultivation of various agricultural products, rent
of land, and prices of commodities can be determined only through a
mutual-interdependence system. Next, the general approach on the
whole is abandoned in favor of a step-by-step analysis commencing
in typical Weberian style. The localization of manufacturing, of raw
material production, of consumers' markets, local differences in trans-
port resources and facilities, economies of large-scale operation and
concentration, local differences in capital and labor supply are
successively considered. In the end, however, Ohlin returns to a
general interdependence setting in depicting the relations of the various
economic forces.

This singular approach has turned out to be misleading to many.
One sympathetic critic maintains that Ohlin does not successfully
bridge the gap between interregional trade theory and general localiza-
tion theory, and thus does not achieve a unified theory ; for the district
whose total localization is supposed to be explained by a general theory
does not necessarily have to possess the same mobility characteristics
as the region, which is the unit of study for interregional trade. ^^ To
the extent that the exposition of the total analysis for the district is
deficient (and of this there is no question), the gap is not bridged, but
the district itself can be conceived as boundless or, more realistically,
as synonymous with the world (which Ohlin does not explicitly do).
A satisfactory, exhaustive, total analysis for the district would then
describe all economic relations within the world and explain all manner
of trade. One could then deduce interregional trade analysis (no

67Tord Palander, op. cit., pp. 266-67. Elsewhere (pp. 262-64) Palander sum-
marizes Ohlin's earlier study, Handelns teori. Here an attempt is made to extend
interregional trade to a theory of interlocal trade (thus accounting for local differ-
ences in factor supply) through subdividing the region (within which perfect
mobility of factors reigns) until the subregions become identical with the locali-
ties themselves. Obviously this technique is inadeciuate; it assumes away the
location problem, for at the start the basis of interregional trade is presumed to
be the different relative ' scarcities of productive factors among regions.

SOME GENERAL THEORIES OF LOCATION 53

matter on what basis the region and subregions are delineated) by
singhng out from among the complex of relations those of relevance. ^8
However, it is not necessary at all to view trade theory as narrowly
as Ohlin does. It is true that international trade theory historically
and as it exists in such standard works as Viner's^^ and Haberler's'^'o
does tend to correspond to Ohlin's conception of it. It is still subject
to Weber's criticism: it does not incorporate transport-oriented
industry into its analytical framework and is thus inadequate for
determining policy. Nonetheless, one can view trade theory and the
general theory of location and space-economy as synonymous. For
(1) location cannot be explained unless at the same time trade is
accounted for and (2) trade cannot be explained without the simul-
taneous determination of locations. Once we recognize this it is futile
to argue whether trade theory is or is not a special form of general
location theory. ''^i As we shall see later, an improved location-trade
doctrine can be achieved through synthesis of the better elements of
existing trade and location theories.

7. Closing Remarks

In summary, the general theory of location and space-economy is
conceived as embracing the total spatial array of economic activities,
with attention paid to the geographic distribution of inputs and
outputs and the geographic variations in prices and costs. Modern
general equilibrium theory is a special case of this theory, in which
transport costs are taken as zero and all inputs and outputs are viewed
as perfectly mobile ; international trade theory, as narrowly conceived
by Ohlin, is also a special case of this theory. One proceeds from the
latter to the former by assuming a given locational structure of
economic activities, by erecting appropriate barriers within the world
economy to correspond to the boundaries of nations, and so forth.

68Losch ("Beitrage zur Standortstheorie," op. cit., p. 331) has also charged
Ohlin with lacking a clear answer to the location problem within his regions, and
with a failure to perceive labor distribution as a result of economic activity be-
tween men, not between regions. The latter accusation falls, however, with the
definition of the district as the world, or as an area greater in extent than, and
inclusive of, the region.

On the other hand, one may perhaps with some justification raise objections
to the liberal and generous interpretation given herein to Ohlin's reasoning.

^^ Studies in the Theory oj International Trade, New York, 1937.

70 The Theory oj International Trade, London, 1936.

■^i Thus Viner's cynical remark about Ohlin's dictum that the theory of inter-
national trade is nothing but international location theory is really unnecessary
and indicates either Viner's confusion or his failure to appreciate the scope of
location theory (op. cit., p. 468 note).

54 LOCATION AND SPACE-ECONOMY

However, it is important to bear in mind that the distinction between
trade theory and the general theory of location and space-economy
is one of definition only. Trade theory can be broadly conceived as
synonymous with the general theory of location and space-economy.
And in a sense, too, because of the monopoly elements which are almost
invariably present in spatial relations, a broadly defined general theory
of monopolistic competition can be conceived as identical with the
general theory of location and space-economy.

The substitution principle provides one of the best analytical tools
for developing this general theory. However, Predohl's use of this
tool must be modified and extended to embrace various substitution
relations between transport inputs, and between various types of
outlays and revenues. The formulation of these relations in terms
of a system of mathematical equations ought first to embrace the
concept of transport inputs and later, if possible, Losch's sets of
spatial co-ordinates.

The evolutionary approach of Weber and others, and especially the
writings of Weigmann, who conceives the space-economy as a
rhythmic-moving Gestalt whole with a basic structural core of land
and labor markets, should be very helpful in suggesting lines along
which this general theory may be nurtured to embody dynamic
relations.

Chapte

r3

Some Empirical Regularities
of the Space-Economy

Before the development of appropriate concepts is undertaken, it
is pertinent to examine some of the currently available empirical
material and observations on the space-economy. This should
establish whether or not there are in fact significant regularities
associated with variation in the distance factor. If there are, these
empirical regularities should furnish a valuable background against
which concepts and techniques may be silhouetted. Concomitantly
they should afford insights into the ways in which new concepts and
techniques should be fashioned. A comprehensive canvass and
processing of available material is not intended, since this is beyond
the scope of the book. Rather we wish to benefit from what a prelim-
inary and cursory examination reveals.

Historically, the empirical rank-size rule for cities, noted by
Auerbach, Lotka, Gibrat, Singer, i and others, has spurred on the search
for related empirical regularities over space. The rank-size rule which
is claimed to have widespread validity is given by the equation:

(1) r-P'^ = K

1 F. Auerbach, "Das Gesetz der Bevolkerungskonzentration," Petermanns Mit-
teilungen, Vol. 59 (February 1913), pp. 74-76, and Chart 14; A. J. Lotka, Elements
of Physical Biology, Baltimore, 1925, pp. 306-7; R. Gibrat, Les Inegalites Eco-
nomiques, Paris, 1931, pp. 250-52, 280; H. W. Singer, "The 'Courbe des Popula-
tions.' A Parallel to Pareto's Law," Economic Journal, Vol. XLVI (June 1936),
pp. 254r-63.

55

56

LOCATION AND SPACE-ECONOMY

where q and K are constants for the given group of cities, r stands for
the rank of a particular city in population, and P its population. For
example, according to the 1940 census data on metropolitan districts
for the United States, in which instance q is approximately equal to

8000

1930
1920

\VV^ ^1890

1880

•1870
1860

50 100
Rank

Fig. 2. Communities of 2500 or more inhabitants, ranked in decreasing order
of population size. U. S. A. 1790-1930. (Source, G. K. Zipf, Human Behavior
and the Principle of Least Ejfort, Addison-Wesley, Cambridge, Mass., 1949.)

unity, the population of the New York metropolitan area was
11,690,520. This roughly sets the value for K since the New York area
ranks first. And in line with the rank-size rule the Boston metropolitan
area which ranked fifth (r = 5) had a population of 2,350,514, approxi-
mately one-fifth of K. In logarithmic terms the above equation is:

log r = —q log P + C

which, just as does the equation representing Pareto's "law" of income
distribution, yields a straight line on double logarithmic paper.

EMPIRICAL REGULARITIES OF SPACE-ECONOMY 57

The best presentation of the empirical findings on rank and size
of cities is given by Zipf.2 In Fig. 2 he has plotted the relevant
decennial data for the United States over the period 1790-1930,
logarithmic scales being used on both the horizontal and vertical axes.^
The closest approximation to a linear distribution, as is implied by
Eq. 1, seems to be reached in year 1930. The distributions for earlier
years seem to diverge increasingly from a linear form. It should also
be noted that the distributions for metropolitan districts of the United
States for years 1940* and 1950^ each show roughly as close an
approximation to a straight line as do the distributions in Fig. 2
since 1900.

How much validity and universality should be attributed to this
rank-size rule is, at this stage, a matter of individual opinion and
judgment. 6 However, it cannot be denied that, to a limited extent at
least, there is some basis for the formulation of hypotheses and
additional exploration. If further research corroborates the belief that
these and other distributions exhibit a statistical regularity over
various time periods and diverse parts of the world, expectation that
there are regularities associated with the distance variable would not
be unjustified. For, according to one possible line of reasoning, modern
cities have become increasingly centers of numerous market-oriented
activities, each activity tending to have a defined sales area. Since
the size of a city is positively associated with the number of activities
which locate within it and since economies of scale and other factors

2 G. K. Zipf, National Unity and Disunity, The Principia Press, Bloomington,
Ind., 1941; and Human Behavior and the Principle of Least Effort, Addison-
Wesley Press, Cambridge, Mass., 1949, Chaps. 9 and 10.

3 The encircled points are estimates of Zipf. For full particulars on the process-
ing of the data, see National Unity and Disunity, pp. 41-43. It should be observed
that early census data are particularly faulty.

4 G. K. Zipf, Human Behavior . . . , op. cit., p. 375.

5 See Figs. 1 and 2 in Rutledge Vining, "A Description of Certain Spatial Aspects
of an Economic System," Economic Development and Cidtural Change, Vol. Ill
(January 1955), pp. 147-195.

6 See the interesting statistical analysis in G. R. Allen, "The 'Courbe des Popula-
tions': A Further Analysis," Bulletin of the Oxford University Institute of Statis-
tics, Vol. 16 (May and June 1954). This analysis lends considerable support to
the rank-size rule.

Additionally, Zipf has plotted relevant rank-size data on cities for Canada,
1881-1931, for Germany, 1875-1939, for France, 1886-1936, for India, 1911 and 1921,
and for other areas. In general, he concludes that the data conform to a rec-
tilinear pattern (logarithmic scales). However, as in the case of the United
States in 1840 and of the Austro-Hungarian empire in 1910, the conformity is not
always good; this fact is interpreted by Zipf as an indication of some inherent
poUtical, economic, or social instability within the system. (Zipf, Human
Behavior . . . , op. cit., Chap. 10.)

58 LOCATION AND SPACE-ECONOMY

preclude the presence of each activity in each city, cities of different
sizes emerge. Further, one can expect the longer (and a larger volume
of) population and commodity flows to be generally associated with
the larger cities which have been fortunate in usurping those functions
wherein economies of scale are marked and with which the larger
market areas are linked. Still more, economies of scale have varying
significance for different commodities and activities; when they are
most dominant, the commodity produced (or service rendered) tends
to be national, being supplied to all market points from one location.
Hence, that city which captures the largest amount of these "national
market area activities" and which concomitantly engages in all other
activities whose market areas are of lesser geographic scope tends to
be largest in size. It tends to be a terminal point of the longest
average population and commodity flows and of the largest volume of
such flows, ceteris paribus. And that city which captures the next
largest amount of these national activities, while at the same time
being a center of all "non-national" activities, would tend to be second
in size and to rank second in average length (and volume) of popula-
tion and commodity flows, ceteris paribus. And in like fashion the
third, fourth . . . nth largest city would tend to rank third, fourth . . .
nth. in average length (and volume) of flows, ceteris paribus, the
progression of activities from those with national markets to those with
major regional markets to those with minor regional markets ... to
those with only local markets being duly taken into account.'^ As a
consequence, a statistically regular hierarchy of average length and
volume of flows emerges. Thus regularity of flows over distance
and regularity in the spatial patterning of cities can come to be
associated with a statistically regular hierarchy of cities, ceteris
paribus.^

The ceteris paribus clause, however, excludes so many differentials
(such as in the geographic distribution of mineral resources) that the
above argument not only is open to serious qualification but even may
be subject to major restatement. Nonetheless, there is some basis, as

'^For development of this classification of markets see A. Losch, Die rdum-
liche Ordnung der Wirtschajt, 2nd ed., G. Fischer, Jena, 1944, pp. 70-79, 307-16;
W. Christaller, Die zentralen Orte in Silddeutschland, G. Fischer, Jena, 1935;
E. Ullman, "A Theory of Location for Cities," American Journal of Sociology,
Vol. XLVI (May 1941), pp. 853-64; and W. Isard, "Some Empirical Results and
Problems of Regional Input-Output Analysis," in W. Leontief, et al., Studies in
the Structure of the Amencan Economy, New York, 1953, pp. 148-81.

8 In this regard also see E. M. Hoover, "The Concept of a System of Cities:
A Comment on Rutledge Vining's Paper," Economic Development and Cultural
Change, Vol. Ill (January 1955), pp. 196-98.

EMPIRICAL REGULARITIES OF SPACE-ECONOMY

59

sapp JO jaqranj^

60 LOCATION AND SPACE-ECONOMY

Losch would maintain, for searching for statistical regularities of
commodity and population flows over distance and regularities in the
spatial patterning of cities — the more so if one attributes, as Zipf
does, statistical regularity to the rank-size findings on cities.

Among others, Christaller and Losch have studied the spatial
patterning of cities, recognizing the numerous resource inequalities
which tend to distort regularities inherent in the resistance of distance,
per se. Figure 3 is one of the better illustrations of spatial regularity
in city patterning. The data which it presents are consistent with
the reasoning in the above two paragraphs and with the central place
theories of Christaller and Losch. They indicate that as one proceeds
from smaller to larger class sizes of cities, the distance separating cities
of like class size increases, although with considerable variation about
any average. ^

Zipf, as have many others, has intuitively associated city size with
market area complex. He has searched for simple rectilinear, and
presumably stable, interactions over distance, although the logic
connecting his statistical findings on the one hand and his Forces of
Unification and Diversification and principle of least effort on the
other is not at all clear. In Figs. 4 to 7 some of his findings are
presented.

Figure 4 refers to railway-express shipments (less than carload lots).
Since the data are recorded for shipments between pairs of cities with
different populations and since there is a presumption that tonnage of
shipment will vary directly with the number of originating and
terminating units (individuals), Zipf employs a Pi'P2 element.

9 Christaller's study of settlements in South Germany (op. cit.), most of which
provide services (or central functions) for the surrounding population, also indi-
cates that settlements of a typical size tend to be spaced regularly. According to
his interpretation of the data, not only do the populations of the several sizes of
tj^pical settlements tend to bear a regular relation to each other but also the
distances separating any pair of settlements of hke size tend to increase by the
Vs as one proceeds from a settlement of a given size to the settlement of next
higher size. Thus market hamlets are found to be spaced roughly 7 kilometers
apart, township centers 12 kilometers, county seats 21 kilometers, district cities
36 kilometers, small state capitals 62 kilometers, provincial head cities 108 kilo-
meters, and regional capital cities 186 kilometers. For a full theoretical explana-
tion of Christaller's findings, see Losch, op. cit., pp. 70-97; and for an objective
evaluation, see Ullman, op. cit.

Losch has accumulated additional empirical evidence on settlement patterns as
well as evidence on a host of other significant relationships involving the distance
factor, such as the spatial distributions of various non-agricultural activities, the
sizes and shapes of market areas, the variations of prices, wages, and interest with
distance from strategic geographic points of reference or over space or both.

EMPIRICAL REGULARITIES OF SPACE-ECONOMY 61

(Pi and P2 represent the populations of any given pair of cities.)
Accordingly, weight of shipments between any pair of cities and the
corresponding P1P2/D factor for the given pair of cities (times 1/10'^)
are plotted in Fig. 4, where D represents distance. If one accepts

10,000

•

/

oj 1»000

-

'O

• . v^ •

j3

3

• *x

0

• yf

\ .yfii^ ••

0

mX*

^ 100

-

^

• /*•

M

x**<*

s

.•

y*^

0

r^ • • •

^

y

•

10
1

/

1

•

1 1

1

10

100

P1P2

1,000

1

107

10,000

Fig. 4. Railway express. Movement by weight (less than carload lots) between
13 arbitrary cities in the U. S. A., May 1939. (Source: G. K. Zipf, Human
Behavior and the Principle of Least Effort, Addison-Wesley, Cambridge,

1949.)

Zipf 's Pi • P2 element, then the linear tendency of the data portrayed
by Fig. 4 indicates a definite inverse relationship between tonnage of
railway express and distance, as is indeed demonstrated by other
data. 10

Figures 5 and 6 refer respectively to telephone messages and bus-
passenger movements. Again Zipf introduces, for the same reason as
in Fig. 4, a Pi • P2 element. If one accepts this element, the charts
indicate that the numbers of both telephone messages and bus pas-

' Zipf, Human Behavior

op. cit., p. 402, Fig. 9-20(a).

62

LOCATION AND SPACE-ECONOMY

sengers definitely fall off in linear fashion as the distance between any
pair of cities increases. ^

•

•

100,000

—

10,000

-

•

t 0 •.

f

1

• «

•

• •

• • • X

s

,

•'.;>. . /

s

.• .

-. , X

^ 1,000

o
u

p

100

X > •

• * X

« i». •*:

» •

••» r-.

•

• ,•

* .••

• ,. •

10

1

1 1

1

10

100

1,000

10,000

P1P9

i^^_l

D

107

Fig. 5. Telephone messages. Number of messages interchanged between 311

arbitrary pairs of cities in the U. S. A., 1940. (The hne has a slope of 1.00.)

(Source: G. K. Zipf, Huvian Behavior and the Principle oj Least Effort,

Addison-Wesley, Cambridge, Mass., 1949.)

11 That number of messages falls off with distance is directly depicted in Zipf,
ibid., p. 402, Fig. 9-20 (b).

Among others, Zipf presents the following interesting distributions (although
at times their rectilinearity is questionable) : on railway passengers, airway pas-
sengers, and the P1P2/D factor; on number of different news items in The
Chicago Tribune, number of obituaries in The New York Times, average cir-
culation per day of The New York Times, and the P1P2/D factor; on charge
accounts of Jordan Marsh Co., Boston, in 96 cities and a, PID factor; on length
and number of one-way trips for both trucks and passenger cars; and on number
of marriage licenses issued and distance separating applicants. For full details

EMPIRICAL REGULARITIES OF SPACE-ECONOMY 63

Figure 7 refers to some limited data processed by Stouffer on internal
migration within Cleveland. ^^ These data as they are chartered by

1,000,000

100,000

o

en

a

^ 1,000

o

Xi

100

10

• • • ••

0.5 1

10

100

1,000

1

107

10,000 100,000

Fig. 6. Bus passengers. Movement of persons by highway bus between 29

arbitrary cities in the U. S. A. during intervals in 1933 and 1934. (The line has

a slope of 1.25.) (Source: G. K. Zipf, Human Behavior and the Principle oj

Least Effort, Addison-Wesley, Cambridge, Mass., 1949.)

Zipf clearly suggest that the number of families moving between
separated areas varies inversely with distance, and in general the data

on the nature of his samples and on the particular points of time to which they
refer see Zipf, Human Behavior . . . , op. cit., pp. 386-414.

12 S. A. Stouffer, "Intervening Opportunities: A Theory Relating Mobility and
Distance," American Sociological Review, Vol. 5 (December 1940), pp. 845-67,

64

LOCATION AND SPACE-ECONOMY

10,000

5,000

2,000

1,000

500

200

100-

J I Mill

LiT]

5 10 20

Distance (thousands of feet)

50

Fig. 7. Family migration. Number of families (plus 100) moving varjdng

distances within or between separated areas in Cleveland, 1933-1935. (Source:

G. K. Zipf, Human Behavior and the Principle of Least Effort, Addison-Wesley,

Cambridge, Mass. 1949. Adapted from the data of S. A. Stouffer.)

are consistent with the reasoning and limited empirical findings of
Ravenstein^^ and others. ^^

13 E. G. Ravenstein, "The Laws of Migration," Journal of the Royal Statistical
Society, Vol. 48 (June 1885), pp. 167-227, and Vol. 52 (June 1889), pp. 241-301.

Ravenstein was perhaps first to classify migrants in terms of distance spanned
and to show comprehensively with statistical material that the extent of migration
into a given center of absoi-ption from any given point, in general, varies inversely
with the distance separating the two.

Stouffer has attempted to generalize the relationship between migration and
distance. His hypothesis "assumes that there is no necessary relationship between
mobility and distance. ... It proposes that the number of persons going a given
distance is directly proportional to the number of opportunities at that distance

EMPIRICAL REGULARITIES OF SPACE-ECONOMY 65

Stewart has also been inspired to search for empirical regularities
associated with the distance variable in connection with his study
of social physics. 15 Following Newtonian physics, Stewart has for-
mulated new concepts to observe vital aspects of the space-economy.
Stewart advances the thesis: (1) that the demographic (gravitational)
force F of attraction between two groups of Ni and A^2 average
Americans separated by r distance is given hy F = Ni N^/r^, where
F acts along the line joining the two groups ;i6 (2) that accordingly
their demographic energy by virtue of this force field is given by
E = GNi N2/r, where G is a constant; (3) that the potential which

and inversely -proportional to the number oj intervening opportunities" {op. cit.,
p. 846).

Mathematically Ay/ As = aAx/xAs, where Ay equals the number of persons
moving from an origin to a circular band of width As; x equals the number of
intervening opportunities, i.e., the cumulated number of opportunities between
the origin and distance s; and Ax equals the number of opportunities within the
band of width As. Data on residential mobility in Cleveland, on net interstate
migration for the United States in 1930, and on intercounty migration in Sweden
between 1921 and 1930 tend to substantiate this hypothesis, an hypothesis which
conceivably can be further generalized to cover to some extent movement of
commodities as well as persons. See also M. L. Bright and D. S. Thomas, "Inter-
state Migration and Intervening Opportunities," American Sociological Review,
Vol. 6 (December 1941), pp. 773-83; and E. C. Isbell, "Internal Migration in
Sweden and Intervening Opportunities," A7nerican Sociological Review, Vol. 9
(December 1944), pp. 627-39.

14 See Stuart C. Dodd, "The Interactance Hypothesis: A Gravity Model Fitting
Physical Masses and Human Groups," American Sociological Review, Vol. 15
(April 1950), pp. 245-256. For a very interesting set of empirical tests of the Zipf
and Stouffer hypotheses as they relate to migration, refer to T. R. Anderson,
"Intermetropolitan Migration," American Sociological Review, Vol. 20 (June
1955), pp. 287-91.

1^ J. Q. Stewart, "Empirical Mathematical Rules Concerning the Distribution
and Equilibrium of Population," Geographical Review, Vol. XXXVII (July 1947),
pp. 461-85; "Demographic Gravitation: Evidence and Applications," Sociometry,
Vol. XI (February-May 1948), pp. 31-58; "Potential of Population and its Rela-
tionship to Marketing," Theory in Marketing, ed. by R. Cox and W. Alderson,
Chicago, 1950, pp. 19-40; "The Development of Social Physics," American Journal
of Physics, Vol. 18 (May 1950), pp. 239-53; and other studies cited in these four
articles.

1^ Stewart reduces the gravitational constant to unity by a suitable choice of
other units. The molecular mass of the "average American" is taken as unity,
thus permitting this simple formulation. Later, Stewart relaxes this assumption.
The reader is referred to any standard college physics textbook for explanation
of the concepts and equations used in this paragraph.

It should be noted that Stewart's concept of demographic force is translatable
into Reilly's law of retail gravitation (W. J. Reilly, "Methods for the Study of
Retail Relationships," University oj Texas Bulletin, No. 2944, November 1929).

66 LOCATION AND SPACE-ECONOMY

the group of A^i individuals produces at the point where the second
group is located is given by V2 = GNi/r; and (4) that the potential
at any point produced by the entire population of any given terrain is

given by y = / - DdS, where D is the density of population over

the infinitesimal element of area dS, the integration being extended
to all areas of the plane where D is not zero. The potential at any
point, according to Stewart, may also be taken as an inverted measure
of the proximity of the point to people in general, i'''

Stewart has computed population potentials for various areas of the
world for different periods of time. 1 ^ Since population is reported not
for infinitesimal elements of area but rather for comparatively large
units of area, only approximations to potentials can be achieved. In
Fig. 8 are depicted equipotential contour lines for the United States
in 1940.19 It is extremely interesting to observe that east of the
Sierras there is, in all directions, a continuous fall in potential with
increase in distance from New York City, the major peak of the
country, except that all other cities are local peaks on the general
downhill slope. 20

Working with approximate averages of potential for rural areas in
a sequence of 28 states from Texas to Maine^i and using double log

1'^ "Demographic Gravitation . . .," op. cit., pp. 32-36.

18 See "Empirical Mathematical Rules . . . ," op. cit., pp. 476-79; "Potential of
Population . . . ," op. cit., p. 22; and "The Development of Social Physics," op. cit.,
p. 250.

19 "Potential of Population . . .," op. cit., p. 22. In this article Stewart does not
indicate the number of areas in which he divided the United States. Obviously
the larger the number of areas, and hence control points, the more precise the
computed potentials, and the more likely that local peaks corresponding to cities
will appear. See "Empirical Mathematical Rules . . . ," op. cit., pp. 473-82, for a
discussion of some of the problems in computing potentials.

In the construction of Fig. 8, Stewart weighted population in the Deep South
by a factor of 0.8; in the main sequence of 28 states from Maine to Texas by
unity; and in the Far West by 2. For his reasoning on this step, see "Potential
of Population . . . ," op. cit., pp. 29-30. However, it should be noted that if
population is unweighted, a similar contour map results except that the potentials
in the Far West are of considerably smaller value. See his "unweighted" map in
"Concerning Social Physics," Scientific American, Vol. 178 (May 1948), p. 22.

20 This statement, of course, might require quahfication if a finer-grained map
were constructed. Nonetheless, the resulting configuration of contours and its
relation to the distance variable would still be impressive. And it is very likely
that all of the contour lines in non-urban parts would close around New York City.

According to Stewart, the major structure of United States' potentials has not
altered much since 1840 when New York City was already the principal peak.

21 Observations are confined to these 28 states because these states exhibit a con-

EMPIRICAL REGULARITIES OF SPACE-ECONOMY

67

68 LOCATION AND SPACE-ECONOMY

paper, Stewart claims to obtain fairly good linear relations by plotting
the data on potential for these areas against the data on each of the
following: (1) density of rural population; (2) density of rural non-
farm population; (3) rents of rural non-farm dwelling units; (4) value
of farmland per acre; (5) rural road mileage per square mile; and
(6) railroad mileage per square mile. In each case, as potential rises
from area to area (the potential, in essence, measures the influence
upon any particular area of people at a distance) , each of these items
tends to increase. Other still more interesting associations which
Stewart notes and judges to be linear (logarithmic scales) are those:
(1) between potential and number of wage-earners in manufacturing
for 253 rural counties ;22 (2) between demographic energy (a concept
related to distance) and income for various states (a) in the main
sequence of 28 states, (b) in 11 western states, and (c) in 9 southern
states; and (3) between potential produced at New York City by the
populations of various branch Federal Reserve districts and the daily
flow of bank checks for October 1948, into New York City from these
districts when grouped according to the above three classes of states. 2 3
Human ecologists — McKenzie,24 Hawley,25 Bogue,26 and others —
have also closely studied spatial phenomena and the impact of distance
upon the interrelations of human beings in adapting to environment.
Of these, Bogue has most explicitly considered the distance variable
within the framework of metropolitan regional analysis. Figure 9
depicts some of his summary findings in a forceful manner. As Bogue
has neatly stated:

"On the average, as the distance from the metropolis increases, the number
of persons per square mile of land decreases. With increasing distance,
each square mile of land area supports steadily decreasing average amounts
of retail trade, services, wholesale trade, and manufacturing activities.
This finding is noteworthy for the following reasons: first, it is a statement
of a set of conditions which applies to the entire land area of the United
States. . . .

siderable degree of statistical homogeneity, a homogeneity which has already been
noted to be different from that of the 11 western states and from that of the 9
southern states.

22 Stewart uses counties (in the main sequence of 28 states) which contain zero
or only relatively small urban populations in order to avoid the local distortion
of potentials that is produced by the influence of nearby cities.

23 "Demographic Gravitation . . . ," op. cit., pp. 39-51 ; and "Potential of Popu-
lation . . . ," op. cit., pp. 27-30

24 R. D. McKenzie, The Metropolitan Community, New York, 1933.

25 A. H. Hawley, Human Ecology, New York, 1950.

26 D. J. Bogue, The Structure of the Metropolitan Community, University of
Michigan, Ann Arbor, 1949. "

EMPIRICAL REGULARITIES OF SPACE-ECONOMY 69

10 25 50 100 200 500

Distance from the nearest metropolis (miles)

Fig. 9. Population per square mile and dollar value of selected sustenance

activities per 1/100 sq mile of hinterland, by distance outward from the nearest

metropolis: for 67 metropoUtan communities in the U. S. A., 1939-1940.

70 LOCATION AND SPACE-ECONOMY

"Second, the distance pattern encompasses the entire area which has been
called the 'hinterland.' The effect of distance from the metropolis does not
cease to exist at the suburb, but continues throughout all distances. . . .

"Third, the distance patterns for the suburban and for the most distant
zones are shown here to be only different aspects of the same phenom-
enon. There is a fairly constant rate of change between relative decreases
in land occupancy and relative increases in distance."2 7

It should also be noted that in general these regularities with distance
hold for each size class of metropolitan region, though with different
rates of change. Likewise, on a per capita basis, retail sales, receipts
from services, wholesale sales, and value added by manufacture each
manifests a fairly regular pattern with distance from the nearest
metropolis, for all metropolitan regions, for metropolitan regions
broken down by size classes, and for various types of sectors of
metropolitan regions. In addition Bogue finds many other related
regularities with distance. 2 8

To complete this brief survey of empirical material, we wish to
re-examine more carefully data on tonnage of commodity flows over
distance, both on an intranational and international basis. Fortu-
nately, data for the United States have recently become available,
recording for each of 261 I.C.C. commodity classes and for each of
five consolidated groups of these classes state-to-state Class I railroad
shipments as well as the average short-line haul per ton (in miles) for
commodities in each class. These data are probably the most compre-
hensive commodity-flow data currently available. ^ 9

As is to be expected, the friction of distance is of different signifi-
cance for the diverse commodities. Certain flows, such as those of
cement, are extremely sensitive to the distance variable; others, such
as those of oranges and grapefruit, extremely insensitive. For our
purposes, it is sufficient to chart the data on total commodity flows.
Doing this tends to average out the particular sets of resource, market,
transport rate, and other relations peculiar to any given commodity
flow and to isolate more effectively the general impact of the distance
variable.

2''' Bogue, op. cit., p. 31.

28 Also, see the supporting material in C. Clark "Urban Population Densities,"
Journal of the Royal Statistical Society, Vol. CXIV, Part IV (1951), pp. 490-96,

-9 The I.C.C. data are derived from a continuous representative 1 per cent
sample of the carload waybill terminations of Class I railroads only. For our
purposes the data can be used with confidence. Errors from omissions because
of the disclosure rule, from rebilling, from double billing of rail- water-rail move-
ments, etc. are minor. For full details on shortcomings of the sample see the
introductory note of Interstate Commerce Commission, Carload Waybill Analyses,
1949, Washington, D.C., 1950-1951.

EMPIRICAL REGULARITIES OF SPACE-ECONOMY 71

In Figs. 10 and 11 are plotted the data on tonnage of total com-
modities moving over various distances, by 25-mile zones and 100-mile
zones respectively. ^0 The impact of the distance variable is unques-

400 600 800 1200

Distance (miles)

1600

2000

Fig. 10.

U. S. A. Class I railroad shipments. Tonnage of all commodities, by-
distance shipped (25-mile zones), 1949. Bar chart.

tioned, whether the data are recorded by 25-mile zones, or by the
larger 100-mile zones in order to smooth out some of the irregular-

30 The assistance of Mr. Merton J. Peck in preparing the data for Figs. 10-14
is gratefully acknowledged.

The reader is also referred to the excellent set of charts, independently derived,
in Vining op. cit., and in R. Vining, "Delimitation of Economic Areas: Statistical
Conceptions in the Study of the Spatial Structure of an Economic System,"
Journal of the American Statistical Association, Vol. 48 (March 1953), pp. 44-64;
and to Edward L. UUman, Maps of State-to-State Freight Movement for 13 States
of the United States in 1948, mimeographed.

72

LOCATION AND SPACE-ECONOMY

ities.31 Also, to examine whether or not any linear relation is present,
using double log paper, the data for the mile zones lying between 75
and 1400 miles are graphed as a set of points in Fig. 12.32 A straight

3500 f-

3000

2500

2000

1500

1000

500

800

1200 1600 2000
Distance (miles)

2400 2800 3200

Fig. 11.

U. S. A. Class I railroad shipments. Tonnage of all commodities, by-
distance shipped (100-mile zones), 1949. Bar chart.

31 Also it is interesting to note that each of the four significant I.C.C. major
groups of commodities — products of forests, products of agriculture, products of
mines, and manufactures and miscellaneous — show shipments falling off fairly
regularly with distance, though less regularly than the aggregate for all commodi-
ties. Shipments of products of mines fall off most precipitously; those of products
of agriculture least. Shipments of the fifth, and by far the least significant of the
I.C.C. major groups, namely, animals and products, evidence no real tendency to
fall off with distance.

32 In Fig. 12 the tonnage data for the extreme zones are not plotted. For the
zones falling between 0 and 75 miles they have little significance since rail transport
over relatively short distances is infeasible for so many commodities. For zones
falling beyond 1400 miles, the tonnage of shipments is very small and the data not
only become less reliable but also reflect shipments of commodities under special
conditions. However, the reader may still care to plot the data for the extreme
zones (see Fig. 10).

EMPIRICAL REGULARITIES OF SPACE-ECONOMY 73

line has been drawn in freehand. A priori, the data seem to conform
well to a linear pattern.

On an international level, there is, unfortunately, a paucity of com-
prehensive physical shipment data. However, one set of reliable and

3000
2000

1000

500

200

o 100

50
40
30

20

10 k

J L

70 100

200 300 400 500
Distance (miles)

1000 1500

Fig. 12. U. S. A. Class I railroad shipments. Tonnage of all commodities, by
distance shipped (25-miIe zones), 1949. Point chart.

relevant data has been developed during the 1920's by the German
National Bureau of Statistics. ^^ The world is divided into 23 areas
and the total flow of goods via water in tons from any given area to
itself and each of the other 22 is indicated for years 1913, 1924, and

33 "Der GUterverkehr der Weltschiffahrt," Vierteljahrshefte zur Statistik des
Deutschen Reichs, Erganzungsheft zu Heft 1928, I, vom Statistichen Reichsamt,
Berlin, 1928. For sources and adequacy of data, see the first page of this mono-
graph.

74

LOCATION AKD SPACE-ECONOMY

1925. Upon the data for 1925, Figs. 13 and 14 are based; in these the
distance between any two areas is taken as the distance between the
two ports, each of which was the most important port in its area in
1925.3^

50

40

Q 30

G

i 20

10

"^L^^-T-

0 2,500 5,000 7,500 10,000 12,500
Distance (miles)

Fig. 13. World ocean-going freight. Tonnage, by distance shipped (500-mile
zones), 1925.

34 The 23 areas and the corresponding ports of origin and destination are:
Baltic countries and Scandinavia (Stockholm) ; Germanj^ (Hamburg) ; Great
Britain and Ireland (London); Holland and Belgium (AntAverp); France (Cher-
bourg, and for trade through the ^Mediterranean, Marseilles) ; Spain and Portugal
(Lisbon) ; Italy (Genoa) ; Eastern Mediterranean countries and Turkey (Alex-
andria) ; countries on the Black Sea (Odessa) ; North Africa excluding Egj-pt
(Algiers) : West Africa (Lagos) ; South Africa (Durban) ; East Africa (Zanzibar) ;
Arabia, Persia, and British India (Bombay) ; Indo-China, the East Indies, and
the Philippines (Singapore) ; Eastern Asia including Asiatic Russia (Yokohama) ;
Austraha, New Zealand, and Oceania (Sydney) ; West Coast of Canada and the
United States (San Francisco) ; East Coast of Canada and the United States
(New York) ; Mexico, Central America, and West Indies (Havana) ; Brazil and

140

120

ai 100

CM 80

o

g 60

S 40

20

EMPIRICAL REGULARITIES OF SPACE-ECONOMY 75

Once again the significance of the distance variable is demon-
strated, ^s This is so whether one observes the configuration by the
500-mile zones of Fig. 13 or
the more regular configura-
tion by 2000-mile zones of
Fig. 14. The latter, based
upon more aggregative data,
tends to conceal the signifi-
cance for international trade
of the uneven world pat-
tern of resources and of the
particular cultural and polit-
ical institutions which have
evolved in the individual
nations participating in in-
ternational trade. 3 6

In conclusion, it must be
recognized that significant
regularities are associated
with variation in the distance
factor. However, considera-
ble caution and circumspec-
tion must be exercised in
attributing significance to particular sets of data and related findings
which have been presented here and elsewhere. In a number of cases

Northern South America (Rio de Janeiro) ; remainder of the East Coast of South
America (Buenos Aires); and West Coast of South America (Antofagasta).

In selecting the chief port in each area and in deriving the distances between
ports the author had to use his best judgment in several instances. Other indi-
viduals may have made different choices. Also, inaccuracy creeps in because not
all traffic moves by the shortest navigable routes as is assumed here. But adjust-
ment for these factors would affect Figs. 13 and 14 only in a minor way.

A more serious shortcoming is that international trade within any given area
was excluded because we had no knowledge of the distances over which such
trade moved. As a result the volume of shipment over distances falling within
the smaller distance zones is considerably underestimated in Figs. 13 and 14.

Shortest navigable distances between ports in nautical miles were obtained or
estimated from U.S. Navy Department, Hydrographic Office, Table of Distances
Between Ports, Washington, D.C., 1942.

35 Also see Losch, op. cit., pp. 305-07.

36 The manner in which the data are aggregated, of course, influences the
smoothness with which tonnage falls off with distance when the aggregated data
are charted. When the data are aggregated by 1500-mile zones, tonnage for the
second zone is considerably less than for the third. However, by 2500- and 3000-
mile zones, the "falling-off-effect" is regular as it is by 2000-mile zones.

0 2 4 6 8 10 12 14
Distance (thousands of miles)

Fig. 14. World ocean-going freight. Ton-
nage, by distance shipped (2000-mile zones),
1925.

76 LOCATION AND SPACE-ECONOMY

the data are inadequate; and the statistical processing and techniques
are suspect or deficient or both. Nonetheless, after an over-all view
of the empirical material, it is undeniable that the friction of distance
manifests itself in a number of important ways and markedly condi-
tions the structure and functioning of critical sectors of the social
system. The impact of transport, both direct and indirect, is crucial.
In at least certain major analyses of the economy the spatial frame-
work cannot be ignored.

Chapter

4

Transport Inputs and
Related Spatial Concepts

1. General Introductory Remarks

Having presented some of the more important statistical findings on
spatial relations and having established that there are significant regu-
larities associated with variation in the distance factor, we are now
in a position to develop appropriate concepts for theoretical analysis.
This is not to deny the usefulness of the approach of the astron-
omer Stewart who advocates the methods of Tycho Brahe and Kepler,
namely, accumulating extensive observations and condensing the volu-
minous data into concise empirical rules before 'formulating a general
theory. Such has been the point of departure, in business cycle analy-
sis, of the National Bureau of Economic Research. Nonetheless, it
is also valuable at this stage, given the extensive material already
amassed, to formulate operational and functional concepts for a general
theory of space-economy which could facilitate systematic and large-
scale observation and possibly abet the extraction of more empirical
rules from the data.i

As before we must recognize the obvious fact that economic activity
takes place in a time-space continuum. In general, to minimize effort

1 For a thorough discussion of tlie methodological issues involved see T. C.
Koopmans, "Measurement without Theory," Review of Economic Statistics, Vol.
XXIX (August 1947), pp. 161-72; and R. Vining and T. C. Koopmans, "Method-
ological Issues in Quantitative Economics," idem, Vol. XXXI (May 1949), pp.
77-94.

77

78 LOCATION AND SPACE-ECONOMY

or factor services in producing a given social output or to maximize
social output with a given amount of effort and factor services, is not
to choose a path of action with respect to the time axis alone, or to the
space axis alone, but rather with respect to both axes.

Aside from personal preferences and acquired interpersonal behavior
forms, there are, as mentioned above, at least two major sets of eco-
nomic factors which press society into a spatial framework: one, de-
glomeration forces, including the operation of the law of diminishing
returns; the other, inequality of resource endowment. ^ Viewed in
terms of a young developing economy, the transition and growth of a
population nucleation from a village into a town, in general, involves
spatial extension of the agricultural hinterland. Obviously, despite the
greater transport effort or cost that may be incurred in marketing
the output of the new, more distant land brought under cultivation, the
net return from the application of intramarginal doses of capital and
labor to that new land is usually greater than that from the application
of additional doses to old, less distant land. And obviously too, spatial
bonds must be established with other geographic points if this nuclea-
tion is to consume mineral or other products not available locally. In
terms of the existing economy of the United States, it is inconceivable
for these reasons that the entire population be herded about New York
City, and perhaps a few additional focal points, ^ notwithstanding the
sharpening of the New York peak implicit in Stewart's * application of
physical analogues to demographic study. ^

2 Space can also be introduced through assumption as it is by Losch {Die rdum-
liche Ordnung der Wirtschaft, Jena, 1944, Chaps. 8 and 9). All economic
resources as well as completely self-sufficient homesteads can be postulated to
be uniformly distributed over any given plain. Spatial relations, however, do
not become important until speciaHzation and trade ensue.

3 Even herding of population is inconsistent with a spaceless one-point economy.

4 "Empirical Mathematical Rules . ... ," op. cit., pp. 480-81. Diminishing returns
and inequality of resources in a sense set in motion centrifugal forces which
balance the tendency toward centripetal shift along the lines of force in a field
toward the peak of potential.

5 Our interpretation of the factors at play seems to be at variance with Zipf's.
Zipf offers a theoretical explanation of the spatial arrangement of the economy
as well as of other processes of human ecology in terms of two major forces: the
Force of Diversification and the Force of Unification. The Force of Diversification
reflects the economy of moving "the population to the immediate sources of raw
materials in order to save the work of transporting the materials to the per-
sons . . ." and operates "to split the population into a larger n number of small,
widely scattered and largely autarchical communities . . ." The Force of Unifica-
tion, on the other hand, reflects the economy of saving the work of transporting
finished products to each consumer and "operates in the opposite direction of
moving the materials to the population, with the result that all production and

TRANSPORT INPUTS, RELATED SPATIAL CONCEPTS 79

It is proposed to express some of the complex spatial relations of an
economy in terms of a simple common concept of transport inputs.
We define a transport input as the movement of a unit of weight over
a unit distance; e.g., we may speak of pound-miles, ton-kilometers, etc.^
In an indirect sense, transport inputs correspond to the exertions of
effort and other factor services required to overcome resistance encoun-
tered in movement through space where friction is present. In a space-
economy we obviously wish to minimize these, ceteris paribus.

It is imperative to think in terms of some such concept as transport
inputs if one is to comprehend fully the significance of space in actu-
ality. One cannot ignore transport cost and merely concentrate upon

consumption will take place in one big city where the entire population . . . will
live" (op. cit., p. 352). Only in terms of the functioning of both forces can the
actual location of population be understood, according to Zipf.

Though in its present form there may be considerable value to Zipf's theoretical
framework for a general study of human behavior, it needs to be substantially
revised and extended for an analysis of the space-economy. Though his frame-
work rightly points out the essential factor of minimizing transport effort in terms
of probable distributions of raw materials in general, it should in addition encom-
pass the vital consideration of agglomeration (and deglomeration) economies
which consist of the economies (and diseconomies) of scale within the plant, the
economies (and diseconomies) of localization, and urbanization economies (and
diseconomies). On the one hand, agglomeration economies partially nulUfy the
Force of Diversification acting toward scattered production and autarchical com-
munities when a small "diversity of raw materials is used, with an increased likeli-
hood of finding them in a restricted area" (op. cit., p. 365). On the other hand,
deglomeration economies prevent the economy from virtually collapsing to a
point as impUed by the Force of Unification when the "diversity of needed raw
materials increases, with the decreasing probability of finding them in a single
spot" {op. cit., p. 365).

Also, Zipf's framework tends to minimize the major roles played by certain
industries in our economy — such as iron and steel, aluminum, and glass — whose
chief raw materials are highly localized in a relatively few places. The location
forces operating on these industries in terms of both factor immobilities and
transport costs on raw materials and products tend to lead to large population
nuclei at places other than Zipf's least-work center which would be the point
at which the sum of all "least-work distances to every person on the terrain" is
at a minimum. Nor need all innovations which increase the diversity of materials
increase the Force of Unification as Zipf maintains. For, by revaluing the attri-
butes and resources of certain areas, as atomic energy has recently done, a major
innovation can introduce more "dispersion" from a least-work center.

6 Just as we frequently use the general term, man-hours, when we speak of
social aggregates and specify man-hours of particular types of labor — skilled, un-
skilled, etc. — when we speak of the production process of a particular firm, so with
transport inputs we shall speak of ton-miles when we think in terms of aggregates
and specify ton-miles of particular commodities when we deal with an individual
line of production.

80 LOCATION AND SPACE-ECONOMY

the labor, raw material, and other costs which compose transport cost;
nor can one ignore transport inputs and merely concentrate upon the
labor, capital, and other inputs which, as we shall presently show, com-
pose transport inputs if he is to understand the full array of dynamic
spatial phenomena.'''

(To avoid confusion with earlier writings of mine, it should be noted
again that the term transport inputs is being substituted for the term
distance inputs which has been previously used. As mentioned in the
preface, distance inputs is inferior to transport inputs as a term when
considered with respect to the everyday usage of words; and also dis-
tance inputs is misleading to non-economists who use the term transport
inputs with a correct spatial perspective. It is felt that the non-spatial
bias of traditional economic theory has by now been sufficiently over-
come to justify the employment in this book of the superior term,
transport inputs.)

■^ A highly simplified example may be illuminating at this point. Assume that
a society's iron manufacture has advanced to the stage where it requires 300 lb
of coal, 200 lb of ore, and 10 hours of direct labor to produce 100 lb of iron (we
ignore other raw material and factor requirements). Transportation of coal and
ore is required if production is at the market site. A; of coal and the finished
product if production is at the ore site, B; and of ore and the finished product
if production is at the coal site, C. Allow 20 labor-hours to transport the necessary
coal and ore to produce 100 lb of iron at A, 10 labor-hours to transport the neces-
sary coal per 100 lb of iron to B and 100 lb of finished product from B to A, and
7 labor-hours to transport the necessary ore per 100 lb of iron to C and 100 lb of
finished product from C to A. (We postulate that the services of land and capital
goods required in the above transportation are negligible.) Therefore we have:

For production at A : 30/i + 2r + 3c -> i

For production at B: 20/i + 2r + 3c -> i

For production at C : 17h + 2r + 3c -> i
where h = 1 labor-hour, r = 100 lb of ore, c = 100 lb of coal, and i = 100 lb of
iron delivered at A.

This formulation, following traditional Unas, points out the need for minimizing
the amount of labor inputs. But it conceals the real issue which is to minimize
the effort at overcoming spatial resistances, in a sense, to economize on distances
to be traversed with appropriate weighting for the type and amount of traffic in
any direction. The presentation of alternatives should be:

For production at A: 20d + lOh + 2r + 3c -> i

For production at B: lOd + lOh + 2r + 3c -> r

For production at C: 7d + lOh + 2r + Sc ^ i
where d = a transport input so defined as to require 1 hour of effort.

This presentation does emphasize exphcitly that it is transport inputs or effort
at overcoming space resistance which is to be minimized. It does not leave implicit
the real problem which, if left implicit, becomes quickly concealed as soon as we
treat a modern economy with complex stages of production.

TRANSPORT INPUTS, RELATED SPATIAL CONCEPTS 81

2. Transport Inputs Contrasted with Capital Inputs

Before certain operational uses of transport inputs are demonstrated,
this and related concepts should be developed more fully.

It is instructive to contrast transport inputs and capital ^ inputs.
Neither can be considered an ultimate factor of production. Both in
a sense are derived even if the analyst (a la Knight) must go back to
the beginnings of time to justify this view.^ As capital goods, and thus
services of capital goods, transport inputs stem from direct labor inputs
with or without direct land inputs (as, for example, the services of land
underlying the roadbed upon which a railway is constructed) , with or
without the services of capital goods (as, for example, that of rail
equipment), and with or without other transport inputs (as, for
example, the transport inputs required in bringing coal to the loco-
motive) . Ultimately, they can be traced back to direct labor and land
inputs only.

The same motive lies behind decisions respecting the use of capital
inputs and the use of transport inputs. The motive in the traditional
sense is to maximize profit. Methods requiring the use of capital
goods or the increased use of capital goods are adopted when they are
found to be more productive, given any initial amount of labor-hours
and other inputs to be expended. Or these methods may enable the
production of goods otherwise unobtainable and at an expenditure of
effort which society is willing to make. Likewise with transport
inputs. When in a simple economy a farmer with a given amount of
capital and other resources chooses to apply his efforts at cultivating
new land on the periphery of the hinterland of a growing town rather
than at cultivating intensively a more limited quantity of old land near
the town, in general he anticipates reaping greater returns despite the
fact that he applies less of his available labor to cultivation and more
to marketing his harvest. In effect he substitutes transport inputs
(indirect labor inputs) for direct labor inputs. He finds it profitable^o

s In using the word capital, we are thinking in real terms and chiefly of capital
goods. Controversy over the definition of capital is not desired. The argument
that follows holds, except perhaps for minor revisions, regardless of the particular
definition of real capital adopted.

9 And even if he must contrast capital with unskilled labor and virgin soil.
Among others, see F. Knight, "The Theory of Investment Once More : Mr. Bould-
ing and the Austrians," Quarterly Journal of Economics, Vol. 50 (November 1935),
pp. 45-50; and K. Wicksell, Lectures on Political Economy, New York, 1934,
Vol. I, Part II, especially pp. 145-46, 149-51, 185-86.

10 It should be emphasized that just as every investment for a longer period
(made possible by the accumulation of additional capital) will not necessarily

82 LOCATION AND SPACE-ECONOMY

to do so in the same way that in using a plough that he has built, he
finds it profitable to substitute services of capital goods (indirect labor
inputs) for direct labor inputs. ^ Also, when the United States public
over the years has chosen to consume coffee and at an increasing rate
rather than to consume more of domestically produced commodities,
there has resulted an increase in the spatial extent of the United States
economy. This resembles the increase in the time extent of production
which took place when society decided to mine and refine uranium ore
(as well as produce the requisite equipment) partly with labor drawn
from unmechanized agriculture. 12

Many economists think of methods using the services of capital
goods as roundabout methods which increase the time extent of
production, or to be more precise, the time period of investment.!^ We
need not judge the validity of such reasoning. i* If one accepts it, one
can draw a parallel with respect to the use of transport inputs.
Methods which use transport inputs are also roundabout, and they
tend to increase the spatial extent of production.

Further, if one adheres to Hayek's theory of capital and discards
the concept of an average investment period, to account for a greater
use of capital "it is sufficient to say that the investment period of some

yield a larger product, so every increase in the use of transport inputs and in the
spatial extent of society will not necessarily be desirable. Only those spatial
lengthenings of production which are profitable will be adopted. Compare F. von
Hayek, The Pure Theory of Capital, London, 1941, p. 60.

11 For a discussion of relations between direct and indirect labor which is
designed to elucidate certain capital aspects of production but which to a large
extent is also applicable to spatial aspects, see 0. Lange, "The Place of Interest
in the Theory of Production," Review of Economic Studies, Vol. Ill (June 1936),
pp. 159-70.

12 In the sense that both capital and transport inputs are derived, society with-
draws certain resources from immediate direct application in order to exploit,
or exploit more efficiently, other potential resources. And these potential re-
sources, as they are drawn into the production process, need not be immediately
consumed, but may be employed to exploit still other potential resources and
thus to increase further the time and space extent of production. In this way
the process of capital and spatial growth of the economy can be cumulative.

13 As Hayek has pointed out, the concept of a single or average period of pro-
duction may be not only a confusing but also a meaningless abstraction. One
must think in terms of periods for which particular factors are invested. "The
Mythology of Capital," Quarterly Journal of Economics, Vol. 50 (February 1936),
pp. 199-205.

i^J. B. Clark, Knight, Nurkse, Smithies, and others have attacked it. For
bibhography on this controversy, see N. Kaldor, "Annual Survey of Economic
Theory: The Recent Controversy on the Theory of Capital," Econometrica, Vol.
5 (July 1937), and Readings in the Theory of Income Distribution, Blakiston,
Philadelphia, 1946, pp. 694-99.

TRANSPORT INPUTS, RELATED SPATIAL CONCEPTS 83

factors has been lengthened while those of all others have remained
unchanged; or that the investment periods of a greater quantity of
factors have been lengthened than the quantity of factors whose
investment periods have been shortened by an equal amount; or that
the investment period of a given quantity of factors has been
lengthened by more than the investment period of another equal
amount has been shortened. "i^ In parallel fashion, we need not speak
of an average spatial extent of production, which concept if meaningful
could pose serious problems in measurement. But when more transport
inputs are utilized, and profitably so, we can assume that the spatial
extent of production in general is increased: (1) that the spatial dimen-
sion of some production lines is lengthened (as, for example, through
the extension of marketing and purchasing areas) ; (2) that the spatial
dimensions of certain production lines are lengthened while those of
others are shortened by an equal amount, but that the former group
is of greater quantitative significance; or (3) that of two equally
important groups of production lines, the increase in the spatial
dimension of those lengthened is greater than the decrease of those
shortened. 16

Connected with the roundaboutness of capitalistic production one
frequently finds the concept of time preference over which has raged a
controversy that I do not wish to discuss. i''' However, I do wish to

15 "The Mythology of Capital," op. cit., p. 206. Also see F. Machlup, "Professor
Knight and the 'Period of Production,' " Journal of Political Economy, Vol. 43
(October 1935), pp. 584-93.

16 To correspond to Knight's contention that increase in the use of capital does
not necessarily entail an increase in the period of production [see F. Knight,
"Professor Hayek and the Theory of Investment," Economic Journal, Vol. 45
(March 1935), pp. 79-81], it is difficult to visualize how a profitable absorption
of additional transport inputs might shorten the spatial extent of production
in general.

I am incHned to reject Knight's views that all capital is normally conceptually
perpetual, that its replacement has to be taken for granted as a technological
detail, that, in consequence there is no production process of determinate
length other than zero or "all history," and that, in the only sense of timing in
terms of which economic analysis is possible, production and consumption are
simultaneous. (Hayek, "Mythology of Capital," op. cit., p. 202.) It certainly
would be meaningless to use the Knightian emphasis for the development of
space concepts. It would negate the very existence of a space-economy. There
could be no space separating production and consumption. And so forth.

1'^ For various points of view, see, among others, E. von Bohm-Bawerk, The
Positive Theory of Capital, 1891, Book V; I. Fisher, The Theory of Interest, New
York, 1939, especially Chap. XX; F. Knight, "Professor Fisher's Interest Theory:
A Case in Point," Journal of Political Economy, Vol. 39 (April 1931), pp. 176-212;
and F. von Hayek, Pure Theory of Capital, London, 1941, Chaps. 17 and 18, and
Appendix I.

84 LOCATION AND SPACE-ECONOMY

emphasize that, if one thinks in terms of time preference, there is
strong justification for thinking in terms of space preference. ^^
Psychologists and sociologists, whether speaking of a gregarious
instinct or of acquired behavior patterns or of both, have emphasized
the social nature of man and his propensity to associate with groups
of various sorts. i^ One can reason that such a propensity, acquired or
instinctive, is a manifestation of a positive space preference. In the
extreme, unreal case, where there are ubiquitous resources, no diminish-
ing returns on land, and no congestion problem that sets in motion
dispersive forces, people would aggregate in one or many herds of dif-
ferent sizes — a phenomenon which is biologically valid. ^o To induce
them to separate, there would have to be an incentive. In the real
world the incentive for non-herd existence, economically speaking, is
greater productivity obtainable through (1) capitalizing deglomeration
economies (such as postponing the operation of the laws of diminishing
returns) and (2) exploiting the uneven geographic distribution of
resources. This incentive (it can be maintained) is analogous to that
which induces people to defer present consumption of commodities
for the possession of a greater amount at a later date.

It should be stressed that not all individuals need have a positive
space preference. There are hermits. They exhibit negative space
preference, being willing in general to accept a lower productivity (a
lower standard of living) in order to be spatially apart from society.
They resemble those well-to-do individuals who fear and exaggerate
the insecurities of the future, who have a negative rate of time
preference, and who would be willing to accept if necessary a negative
interest rate.

Less extreme is the introvert whose need for social contact is not
intense. He possesses a mild space preference. He is easily induced
to lead a fairly isolated life, though in actuality he may not do so.
Not so with the extrovert. His need for social interchange is acute;
his space preference markedly high. He parallels the sailor, child,

18 For this expression I am indebted to Dr. E. M. Hoover.

19 Refer, among others, to W. McDougall, An Introduction to Social Psychology,
Boston, 1926, pp. 87-90, 175-78, 303-8, 456-60; L. L. Bernard, Instinct, A Study in
Social Psychology, New York, 1924, especially pp. 357-59, 369-72; K. Young, Source
Book for Social Psychology, New York, 1927, Chaps. I-IV; and W. F. Ogburn
and M. F. Nimkoff, Sociology, Cambridge, Mass., 1940, Part IV.

20 See W. Trotter, Instincts of the Herd in Peace and War, London, 1916; and
W. C. Allee, The Social Life of Animals, New York, 1938. Small herds (cities),
and not necessarily one huge herd (city), may be sufficient to satisfy the social
needs of man.

TRANSPORT INPUTS, RELATED SPATIAL CONCEPTS 85

savage, and spendthrift whose positive rate of time preference is
likewise towering. 21

On balance it does seem that, despite the many serious qualifications
one must make in generalizing about such psychological principles as
time preference and space preference and despite the recent centrifugal
tendencies in population movement (which we will discuss briefly
below) , individuals in society, in general, do manifest a positive space
preference just as they seem to manifest a positive time preference.

Associated with time preference has been the procedure of discount-
ing over time. The present value of a future product is equal to its
expected future price discounted by the prevailing rate of interest.
The return to an input is equated to its discounted marginal produc-
tivity. And so forth. The usefulness of this procedure is obvious.
But there is also a discounting over space, which enables one to
compare values of two or more goods, yields, or inputs spatially
separated and differently distant from any particular geographic
point of reference. The rate of discount in space is of course the
transport rate. Though economists have never spoken explicitly of
spatial discounting, nonetheless they have performed the operation. 2 2
In doing so, they have most frequently been considering a one-point
market served by a surface producing area (as we find in farming).
In such a case the farther the site of production from the market the
more the market price is discounted to yield the net price on the output
of that site. However, far more complex situations, involving all
types of possible purchasing areas and selling areas for the relevant
inputs and outputs, have been handled by location analysts. 2 3 The

21 The reader who is interested in constructing other parallels may refer to the
various possible types of time preference cited by Fisher (op. cit.) and Bohm-
Bawerk (op. cit.).

Just as we can have different rates of time preference for various commodities
(e.g. see Hayek, The Pure Theory of Capital, op. cit., pp. 241-42) we can conceive
of different space preferences with respect to various social activities. And cer-
tainly the state of technology, the geographic environment, and the cultural milieu
affect the nature of one's space preference. Compare the age of the automobile
with the age of the horsecar.

22 As an instance, Alfred Marshall states : "If in any industry, whether agricul-
tural or not, two producers have equal facilities in all respects, except that one
has a more convenient situation than the other, and can buy or sell in the same
markets with less cost of carnage, the differential advantage which his situation
gives him is the aggregate of the excess charges for cost of carriage to which his
rival is put. And we may suppose that other advantages of situation, such for
instance as the near access to a labour market specially adapted to his trade, can
be translated in like manner into money values." (Principles of Economics, Book
V, Chap. XI, Sect. 1.)

23 We cite a few works: O. Englander, Theorie des GUterverkehrs und der

86 LOCATION AND SPACE-ECONOMY

explicit use of spatial discounting can clear the ground for a more
functional analysis of the factor of situs in economic activities.

3. Transport Rate: The Price of a Transport Input

In speaking of an input, one also thinks in terms of the price of, or
the return to, that input. What determines its reward? What
determines interest, rent, wages, profits, or in Knightian terms the
annual rate of return in perpetuity? What is the return or price cor-
responding to a transport input? The last question can be partly
answered with a simple supply and demand approach conventionally
used to answer the first two questions.

If, from the standpoint of society, we think of a transport input as
equivalent to the movement of a ton of any commodity over 1 mile
and if, for the moment, we put aside the complicated transport struc-
tures of reality, then the price of a transport input is the transport
rate. From the standpoint of suppliers, at higher and higher trans-
port rates, there will be a tendency for more and more transport inputs
to be furnished. More and more direct labor and land services and
services of capital and capital goods will flow into the area of transport
inputs. The supply curve for transport inputs is positively inclined
(where transport rate is measured along the vertical axis and quantity
along the horizontal). On the other hand, the demand curve for
transport inputs, as can be expected, is negatively inclined. It may
be claimed that such a demand curve reflects the marginal productivity
corresponding to various quantities of transport inputs. It would then
be anticipated, given a state of technology, tastes, and resources, that
as the spatial extent of production is continually lengthened through
the application of more and more transport inputs, ^-i the additional
product associated with each successive lengthening, after a point, tends
to fall off. 2 5

Frachtsdtze, Jena, 1924; T. Palander, Beitrdge zur Standortstheorie, Upsala, 1935,
especially Chaps. VII and XII; E. M. Hoover, Location Theory and the Shoe
and Leather Industries, Cambridge, Mass., 1937; and A. Losch, op. cit.

As Professor Haberler has suggested, transport costs may be compared with
storage costs: the former, in moving from one point in space to another; the
latter, from one point in time to another in the future.

24 Of course, for many lines of production the spatial extent, like the time
period of investment, increases by only large jumps.

25 It seems trivial to argue whether or not physical output is to be attributed
to the use of transport inputs and whether or not a marginal physical product
can be assigned to the use of an additional dose of transport inputs. The farmer
who does use an additional dose of transport inputs, when he finds it profitable
to shift his farming operations to a location somewhat more distant from his

TRANSPORT INPUTS, RELATED SPATIAL CONCEPTS 87

It is instructive to examine somewhat more thoroughly the effect of
a change in the price of a transport input. Suppose an advance in
the state of transport technology pushes the supply curve of transport
inputs to the right and results in a lower price. From the viewpoint of
industrial production there will be both a scale and a substitution
effect. Historically we find that reduced transport rates have tended
(1) to transform a scattered, ubiquitous pattern of production into an
increasingly concentrated one, and (2) to effect progressive differentia-
tion and selection between sites with superior and inferior resources
and trade routes. ^^ The resulting increase in geographic specialization
and in the spatial extent of production in general is, in essence, a
substitution of transport inputs for various other inputs (particularly
for those inputs at inferior sites) as well as a substitution of inputs
in general at the favored sites for inputs in general at the disfavored
sites. As to the scale effect, the tremendous increases in output
engendered by the cheapening of transport inputs are too well known
to require discussion.

Also with respect to consumer behavior, there are scale and
substitution effects. With a fall in the time and money cost of popu-
lation movement (as realized with the development of the street and
electric railway, the automobile, bus, and aircraft), a person in
general can maintain a given level of social contact (or space prefer-
ence) and at the same time consume more of other products. He can,

market point, does realize an addition to total product, ceteris paribus. Also, in
the case where a firm decides to exploit a deposit of richer ore which, however,
is more removed from the point of smelting, there corresponds to the increase
in transport inputs (whether large or small) an increase in physical product, ceteris
paribus.

Some may contend that in both cases the addition to total product is not the
result of the additional use of transport inputs per se but rather a result of a more
efficient combination of land, labor, and capital services. Others may insist that
just as a (marginal) productivity is attributed to capital, even though capital
inputs (e.g. the services of capital equipment) are not productive per se but
merely allow the services of labor to be more productive, so should a (marginal)
productivity be assigned to transport inputs.

From our standpoint, the significant point is the association, whether one inter-
prets it causally or merely statistically, of greater physical product with increased
use of transport inputs, ceteris paribus. This explains in part the demand for
transport inputs and, for obvious reasons, a demand schedule which indicates
that, at a lower and lower price for transport inputs, there will be a tendency for
more and more of transport inputs to be purchased.

26 See W. H. Dean, Jr., The Theory of the Geographic Location of Economic
Activities, Ann Arbor, Mich., 1938, especially Chap. 3; E. M. Hoover, op. cit.,
Chap. 3; and H. Ritschl, "Reine und historische Dynamik des Standortes der
Erzeugungszweige," Schmollers Jahrbuch, Vol. 51 (1927), pp. 813-70.

88 LOCATION AKD SPACE-ECONOMY

at the given level of social contact, enjoy more of the amenities of life
that come from living in a less congested area away from the compact
urban mass. This scale effect which obviously requires a greater
consumption of transport inputs partly accounts for the process of
dispersion of urban populations^ and the settlement of peripheral
metropolitan areas that has taken place during the last half-century. 2 8
But this development is also partly due to the operation of the
substitution effect. Consumption of transport inputs has been substi-
tuted for the consumption of other commodities and services.
Expenditures on travel, whether intraurban or other, appear to have
absorbed an increasing proportion of the consumer's budget.

Hitherto we have spoken of a single transport rate as the price of
a transport input. However, in modern society there is a multitude
of rates which vary with length of haul, nature of haul, type of
commodity, degree of competition, character of topography, etc.
Similarly, although we speak of a single prevailing interest rate as the
price of capital, there is a multitude of interest rates varying according
to the nature of the risk, length of the loan, type of region, etc. 2 9 It
is to be expected that in the operation of a complex, institutionalized
society there will be all types of transport rates and discriminations
in the application of these rates. But this does not invalidate
thinking in general terms of the transport rate as a hypothetical,
representative one, one that reflects the general movement of the
multitude of actual transport rates. The basic analysis is essentially
unaffected by such a fiction, ^o

^■'^ All income, classes have, in general, been affected. At one extreme the lowest
income groups have gradually moved out of the worst slums into somewhat better
districts, generally somewhat younger, less congested, and farther from the core
of the city. At the other extreme the highest income groups, who though on
higher planes of living do not necessarily possess different space preferences, in
general have vacated by degrees their existing residences to construct new ones
more removed, although some individuals have moved closer to the core. For
further material see H. Hoyt, The Structure and Growth of Residential Neighbor-
hoods in American Cities (Federal Housing Administration), Washington, 1939;
and U.S. National Resources Committee, Our Cities, Washington, 1937.

28 Also, the mobility and flexibility of rural population has been increased.
Living apart from one's fellow creatures, as is involved in many rural occupa-
tions, may require sacrifice of social contact for increased productivity. Reduced
time and money cost of movement allows greater social intercourse for a given
productivity. It may also induce the individual with a given space preference to
exploit resources which hitherto lay idle because of the social isolation involved.

29 Refer to Fisher, op. cit., Chap. IX.

30 It should be borne in mind that the determination of the transport rate and
spatial extent of production is not independent of the interest rate (and vice
versa). Since an increase in the potential availabihty of transport inputs in a

TRANSPORT INPUTS, RELATED SPATIAL CONCEPTS 89

4. Transport Inputs and the Classification of Factors

A few words should be said about how transport inputs fit into the
various classifications of productive agents or the conceptual classifica-
tions presented as substitutes. One might venture the hypothesis that
historically, had there been a certain social class which owned all
transport facilities and performed all transport services, the Classicals
might well have thought of transport as a fourth factor of production
and have been more conscious of distance and the spatial aspect of
production. Such was not the case, and in any event a classification
based on socio-economic groups could not have much meaning today.

Perhaps the most salient feature of a transport input is its momen-
tary character. A transport input is realized at a given time from
the performance of various services. There can be no stock of trans-
port inputs. There can only be a stock of services which can be used
to yield transport inputs. A particular individual engaged solely in
transporting goods represents a stock of potential labor services. It
would be wrong to conceive of him as also a stock of potential transport
inputs since in the future his services need not be employed at rendering
transport inputs.

Or take a piece of equipment, a locomotive. It should not be
considered a stock of transport inputs. Rather it represents a stock of
services of a particular capital good, which services when combined
with labor and other services simultaneously yield transport inputs.
This, too, is merely another way of saying that a transport input is an
indirect input.

From the standpoint of orienting transport inputs within the frame-
work of other types of inputs, it seems best to utilize the approach of
Walras,^! which has been well developed by Knight. ^^ There the
fundamental dichotomy is between resources or capital (as broadly
conceived by Knight) and services. There is nothing of a resource or
capital nature in the concept of a transport input. Rather it is in the
nature of a service and has the same time dimension as the service of

modern society involves an increase in capital investment in transport equipment
and facilities, the conditions under which capital is available do influence the
nature of the supply curve of transport inputs and, hence, the price of a transport
input and the structure of the space-economy. In this way, too, land values and
differential rents from superior situations are affected by the interest rate.

31 L. Walras, Elements d'economie politique pure, Lausanne, 1926, pp. 175-84.

32 F. Knight, "The Ricardian Theory of Production and Distribution," Canadian
Journal oj Economics and Political Science, Vol. I (May 1935), pp. 3-25; and
"Capital and Interest," Encyclopaedia Britannica, Vol. IV, 1946, pp. 799-801. Also
see the approach of I. Fisher {op. oil., Chap. I).

90 LOCATION AND SPACE-ECONOMY

a given person, piece of land, or capital good, though to be sure there
are stocks which correspond to these latter services. It flows indirectly
from given resources and capital and competes with and substitutes
for all other types of inputs (services) in the production process.

Nonetheless, one can utilize the Marshallian approach, if one follows
a functional analysis. In a production process there are requirements
for labor at a given place, capital at a given place, land services,
organizing ability, and finally transport inputs, i.e., the composite of
services needed to move raw materials, equipment, labor, and finished
product to the appropriate places. Expressed differently, the transport
function (defined in the broadest sense possible) can be singled out
as a vital aspect of production, as vital perhaps as the functions of
labor, capital, land, and the entrepreneur. The inputs corresponding
to the transport function we have called transport inputs, and by
paying attention to this function and its associated inputs we are
able to describe the spatial aspects of the economy. But one need not
necessarily think of the transport function as another factor of produc-
tion. The important thing is to recognize the role that transport inputs
do play in production and consumption processes.

Chapter

The Locational

Equilibrium of the Firm:

Transport — Orientation

1. Some Definitional and Classificational Remarks

The theoretical, conceptual, and empirical materials of the previous
chapters form a background against which we shall re-examine,
restate in part, and attempt to extend existing location theories. In
doing so we shall have as a prime objective the synthesizing of partial
location theories into a more general theory which yields any given
specific location theory capable of being spelled out in detail when
the appropriate set of conditions are postulated.

In this chapter we shall confine ourselves to the locational
equilibrium of the firmi when the problem of transport-orientation
obtains. The utility of the concept of transport inputs in the determi-
nation of the firm's geographical position will be demonstrated. Also,
this concept will enable us to fuse much of traditional Weberian
locational doctrine and modern production theory. At the same time
certain difficulties which have confronted location theorists for a long
time will be resolved.

1 For simplicity's sake we shall speak of a firm as consisting of one or more
plants operating at one and only one site and of a producer as managing one and
only one firm. There is no logical difiiculty in extending the analysis to enter-
prises which operate plants spatially separated, whether these plants correspond
to the same or different stages of production. However, one must then consider
transport inputs within the enterprise and consequently the analysis becomes
more complex.

91

92 LOCATION AND SPACE-ECONOMY

However, it is advisable, first, clearly to define the problem of this
chapter and to stress the various levels of abstraction at which
locational analysis is possible. This is necessary in order to avoid
criticism similar to some which has been directed at Weber but which
has failed to appreciate that the interpretation and significance of the
Weberian doctrine are different for each of these levels. 2 It is possible,
of course, to establish different classifications of levels at which inquiry
can be conducted. Although the reader may prefer to adopt another
classification, it is satisfactory for our purposes to set up the following
one which distinguishes among at least four levels of inquiry :

1. For the small, individual producer who has a negligible influence
upon prices (with the exception of the price of his own product) , the
locus of consumption, the supply costs and sources of factors, transport
rates, agglomeration economies, and other locational variables;

2. For the small or large producer who does influence these variables ;

3. For an industry as a whole or for a group of producers who form
a meaningful aggregate for analysis because they are homogeneous
with respect to certain characteristics, or because, though hetero-
geneous, they complement each other ;3

4. For a regional or world economy (where general analysis should
account for the determination of values for all possible location
variables) .

In this chapter we shall confine ourselves chiefly to the first level.
However, it should be emphasized that the accepted dualism in
location theory — viz., a Thiinen type of analysis for the agricultural
sphere, a Weberian schema for the industrial sector — ^^and the opposition
to incorporating these two models into one general framework totters
once we recognize these levels of inquiry. The Thiinen school confines
itself to an aggregative analysis. Its problem is the distribution of
agricultural production over a given region. It assumes away any
problems of location for the individual producer by assigning to him

2 A systematic presentation and refutation (in many respects, valid) of these
criticisms is found in E. Niederhauser, "Die Standortstheorie Alfred Webers,"
Staatswissenschaftliche Studien, Vol. XIV (Weinfelden, 1944).

3 Chamberlin has pointed out the limitations of the group concept for analyz-
ing substitution effects among the products of individual firms {The Theory of
Monopolistic Competition, Cambridge, Mass., 1933, especially pp. 103-4) and has
particularly criticized the industry in this respect. See his "Product Heterogeneity
and Public Pohcy," Papers and Proceedings of the American Economic Associa-
tion, Vol. XL (May 1950), pp. 85-92; and "Monopolistic Competition Revisited,"
Economica, November 1951. Our industry or group of firms, however, may be
conceived in terms of similar techniques of production, or inputs, or in terms of
a set of external economies achieved by agglomeration of similar or dissimilar
lines of production.

LOCATION EQUILIBRIUM: TRANSPORT— ORIENTATION 93

a fixed location, an infinite immobility. The Weberian school, on the
other hand, is primarily concerned with the locational problem of an
individual firm which produces a given product. True, the Weberian
doctrine frequently shifts to aggregative analysis when it considers
agglomeration economies, the various economic strata of society, and
the like. But in this type of aggregative analysis the Weberian school
explicitly avoids the problem of the efficient spatial distribution, both
qualitative and quantitative, of the various types of industrial produc-
tion over a given region. Thus, the Thiinen and Weberian schools have
carved out for themselves separate, non-overlapping areas of inquiry.
In real life, of course, this clear-cut line of demarcation in locational
decisions disappears. No agricultural producer is perfectly immobile;
he quite frequently does consider changing location. Accordingly, the
Thiinen scheme is insufficient for explaining such an agriculturalist's
decision. Analysis on the individual level is also required. On the
other hand, the Weberian dogma is grossly inadequate for the over-all
regional type of industrial planning which has been undertaken in the
last decade or two by international, national, and regional authorities.
The Thiinen methodology can be of great service here. The task ahead
is thus to conduct analysis at each level of inquiry and ultimately to
fuse the results into one comprehensive framework. ^ This task I can
hope to accomplish only partially in this and subsequent chapters.

It is also desirable at this point to consider the categories of
commodities (embracing factor services) which have grown out
of Weber's doctrines. Commodities have been classified according to
mobility, dispensability, geographic occurrence, and weight loss. We
frequently encounter in the literature^ commodities described in terms
of the first three of these characteristics and thus falling into one of the
following categories:

1. Indispensable, single-source, immobile commodities

2. Indispensable, single-source, mobile commodities

3. Indispensable, many-source, immobile commodities

4. Indispensable, many-source, mobile commodities

5. Dispensable, single-source, immobile commodities

6. Dispensable, single-source, mobile commodities

7. Dispensable, many-source, immobile commodities

8. Dispensable, many-source, mobile commodities

This classification may be useful for certain purposes. From our

4 Compare 0. Englander, "Kritisches und Positives zu einer allgemeinen reinen
Lehre vom Standort," Zeitschrift filr Volkswirtschaft und Sozialpolitik, Neue
Folge, Vol. V (1926), pp. 475-79.

5 See, for example, Dean (Selections) op. cit., pp. 8-12.

94 LOCATION AND SPACE-ECONOMY

viewpoint, however, these various categories can be reduced to a series
of relations which involve substitution, both in the large and small.
Category 8 is the most general. Here, three explicit types of substitu-
tion possibilities exist: (1) substitution between transport inputs and
between various outlays and revenues associated with the use of any
of several different commodities or combinations of commodities in the
production process, (2) substitution associated with the use of any
of several sources of any one commodity, (3) substitution associated
with the various places to which a commodity can be transported.
The fourth characteristic which Weber underscored, namely, weight
loss or the degree to which the weight of a good does enter into the
weight of the finished product, also lends itself to a substitution
analysis which emphasizes the desirability of various places as the site
of production according to transport expense.

Categories 1 to 7 can be viewed as special cases of category 8, each
limiting in some respect the range of substitution. Thus, when a
commodity is technically indispensable for a given production process
(though from the social-aggregative standpoint, no commodity is
indispensable), substitution between the transport inputs and outlays
associated with the location of the given commodity and those
associated with the location of a potential substitute commodity is
non-existent. When only one deposit or locality exists as a source of a
commodity (as is rare from a world-economic standpoint), then the
substitution problem connected with diverse sources disappears.
Finally, when a commodity is perfectly immobile, no substitution
problem arises in connection with production at places other than
sources (or points of consumption) of this commodity. Although, to
maintain a formal, complete substitution framework we may, as
Englander has done, eschew from our analysis the attribute of mobility
by considering immobile commodities to be goods of infinite weight
entailing infinite weight loss in production (or consumption). ^

No matter what scheme of classification is selected, the whole
production process, as Predohl indicated, may be conceived as a
complex substitution problem in space, involving such spatial substi-
tutions in the large and small as implied by the above classification

6 A strategic and rare labor skill available at only one locality might in the
short run be an instance of a service faUing in category 1. Category 2 would
embrace those raw materials which Weber considers in the early part of his book ;
however, as soon as he introduces replacement deposits, the goods which he
treats come to fall in category 4. Categories 3 and 4 have been treated by
Palander and Hoover in their supply and market area analyses, and to some extent
at least, categories 5 to 8 in their more generalized analyses.

LOCATION EQUILIBRIUM: TRANSPORT— ORIENTATION 95

as well as those substitutions ordinarily conceived in the production
theory (and consumption theory) of a one-point economy. '''

2. Transport-oriented Equilibrium Under Simplified Conditions

We commence the analysis of the locational equilibrium of the firm
under the simplifying assumptions that: (1) its productive activities
do not affect the locus of consumption, transport rates, prices of raw
materials, labor and other factors and products, and agglomeration
economies and other locational variables; and (2) its actions do not
provoke retaliatory measures by other producers.

Assume a point C whereat are concentrated all consumers of the
product of an individual firm. Also let point Mi be the only source
of a raw material indispensable for the production of the good. Other

■^ Palander's criticisms of Predohl's principle of substitution are in the main
unwarranted — or at least are unjustified in view of modern developments in pro-
duction theory. (See Tord Palander, Beitrage zur Standortstheorie, Uppsala, 1935,
pp. 254-61.) Palander has first of all underestimated the fruitfulness of decom-
posing the whole production problem into a set of substitution problems between
the various possible pairs of spatially-defined inputs and outputs — as well as into
a set of substitution problems between various pairs of groups of these spatially-
defined inputs and outputs. Clearly, Predohl had in mind the substitution prob-
lems between groups of commodities (commodities as defined in our broad sense)
and those between possible subgroups when he concludes : "Der Standort der Pro-
duktion bzw. Produktionsstufe ist also bestimmt durch ein System von Substi-
tutionspunkten, das derart gegliedert ist, dass die Gruppen einer ijbergeordneten
Kombination untergeordnete Kombination in sich enthalten" ("Das Standorts-
problem in der Wirtschaftstheorie," Weltwirtschaftliches Archiv, Band XXI, 1925,
pp. 306-7). Such substitution analysis between groups, subgroups, and pairs of
commodities has been developed by Hicks and others and has been generally
considered to be feasible and of value.

Secondly, Palander's point that such problems as scale of plant are excluded
from substitution analysis is also no longer valid. For, through admitting dis-
continuities in the technical transformation function, as can be done, variations
in scale can easily be treated, being viewed as sudden large and jumpy increases
in plant, equipment, and the like. This fact, too, overrides Palander's objection
to substitution analysis for its restriction to cases of continuous variation and
its failure to treat such important discontinuous variations as are involved in
shifts to labor locations (a problem which will be fully discussed at a later point).

Thirdly, there are not two distinct substitution problems as Palander maintains,
one between the various factors where scale, location, and prices are given, and
the other where technique (the proportion of factors) and scale are fixed, but
where production is free to adjust to the spatial variations in the prices of factors
and products. Fundamentally decisions in both of these categories are inter-
related and are contained within the over-all substitution problem of the indi-
vidual firm. Further, on a simple two-dimensional diagram, changes in prices
and other locational variables can be treated together with changes in the propor-
tion of factors.

96 LOCATION AND SPACE-ECONOMY

productive factors are taken to be ubiquitous, available everywhere
in the correct amounts and at the same price. ^ If the raw material

at Ml were immobile, such as ore de-

• — -• posits, then the productive activity

1 (mining) would be at Mi, price and
Fig. 15. A locational line. profit conditions permitting. Here,

however, we assume that the indispen-
sable raw material is mobile, and, further, that a straight line rail-
way connects points Mi and C. See Fig. 15. Where will the firm
locate?

Before we attempt to answer this question it is wise to make clear
our use of the terms transformation function and transformation line.
We conceive the transformation function to embrace the numerous
technical substitution relations between any pair of outputs, any
input and any output, and any pair of inputs. As indicated in the
preceding chapter, transport inputs are viewed as any other set of
inputs in the transformation-production process. They substitute for
other inputs and products.

In the rest of this section and in Sect. 3, we reformulate the Weberian
transport orientation doctrine. In this doctrine, the weights of various
raw materials and the market demand are assumed to be constant.
Therefore, variation in the transport input variables reduces to
variation in the distances over which the raw materials and finished
product must move. Hence, a transformation relation between any
two transport inputs reduces to a consistent set of variations in two
distance variables. In what follows we shall view the relevant
consistent sets of variation in two distance variables as a transforma-
tion line between these two distance variables, although the trans-
formation line rigorously speaking has reference to variation in the
corresponding transport inputs. The reader, however, need not accept
this procedure. He can deny transformation relations between distance
variables, and proceed, as in Sect. 4, to state the Weberian dogma in
terms of transformation relations between the variable transport inputs.
The basic analysis and conclusions, however, remain unchanged.

Returning to the problem of Fig. 15, we observe that in our simple
case we have two distance variables, (1) distance from point C and
(2) distance from point Mi. When we plot these two variables on
Fig. 16 we obtain a straight transformation line with a slope of —1.
Of course it is possible to select a location involving unnecessary

8 This implies that none of the inputs and outputs of the transformation func-
tion changes as we move production from site to site except those which will be
associated below with the distance variables.

Distance from C

Fig. 16. A transformation line for the line

case.

LOCATION EQUILIBRIUM: TRANSPORT— ORIENTATION 97

transportation, i.e., a point not on line MiC, or in other words, a set
of distances from points C and Mj lying above and to the right of line
VW in Fig. 16. But since we
assume that the producer min-
imizes costs, he will not select
a location involving unneces-
sary distance, just as he
will not employ unnecessary
labor.9

Let us complicate our case.
Production now requires a
second raw material present
at only one source, M2. If
this good is both indispensable
and immobile, the site of pro-
duction, if production is at all
feasible, will coincide with this
source. Where this second
good is mobile, for each pos-
sible (realistic) distance from

M2, there exists a transformation line between the variables, distance
from C and distance from Mi ; and for each possible (realistic) dis-
tance from Ml there exists a transformation line between the varia-
bles, distance from C and distance from M2; and finally, for each
possible (realistic) distance from C there exists a transformation line
between the variables, distance from M^ and distance from M2. The
exact nature of all the transformation lines will, of course, depend
upon the relative positions of Mj, Ma, and C. Take an example where
the distances between C and M2, C and Mi, and M2 and Mi are 8, 5,
and 7 units respectively (Fig. 17) . For any value, let us say 3 units,
of the variable distance from C, we obtain a transformation line rep-
resenting the different possible sets of the variables, distance from Mi
and distance from AI2, given by arc TS, the locus of points constructed
with a radius of 3 units from point C. The transformation line turns
out to be convex to the origin Q (Fig. 18) . Obviously this transforma-
tion line contains no sets of variables represented by points outside
the triangle CM1M2 of Fig. 17. That would be covering unnecessary
distance.

9 However, as we shall see later, when discriminatory transport rates nullify
the distance principle and cause the cost between two termini to be less than that
between an intermediate point and one of the termini, it is quite possible for an
entrepreneur to choose a location involving unnecessary distance in the transfor-
mation sense.

98

LOCATION AND SPACE-ECONOMY

Our problem becomes still more complicated when we introduce
additional indispensable but mobile raw materials, each obtainable

Fig. 17. A locational triangle.

at an only source. Let M3 represent the sole source of a third indis-
pensable but mobile raw material. Let M 3 be 7 units distant from C

S
T

\

£

\

g

\

0

\

\

*M

V

01

\

«

^^^

-

^--•T

4^

X

•iM

Q

— 1 1 1 1 1 1

1 1

Q

Distance from M2

Fig. 18. A transformation line for the triangle case.

and 2 units from Mo. See Fig. 19. Here we must pose the substitution
question somewhat differently. Ordinarily we would hold constant all
inputs and outputs but two and observe the substitution relations

LOCATION EQUILIBRIUM: TRANSPORT— ORIENTATION 99

between these two. Here, if we assign fixed values to the variables,
distance from Mi and distance from M2, say RM^ and RM2, respec-
tively, and values which also permit a location at point R within the
polygon of Fig. 19, then necessarily the values of the other two
variables, distance from C and distance from M3 , are determined, being
RC and RM^, respectively. No substitution problem arises. The
situation would resemble one emerging under the ordinary non-spatial
conception of the production problem, where we examine the substitu-

FiG. 19. A four-sided locational polygon.

tion relations between two factors, each for technical reasons bearing
a fixed relationship to another factor in the given basket of commodi-
ties. There would be no possibility of substitution.

However, a real substitution problem does exist in the case of our
polygon of Fig. 19. The entrepreneur is concerned with minimizing
transport expense. He will move to a new position if, for example, the
shorter distance from C lessens his transport expense by more than
the amount which the increased distance from M3 adds to his transport
expense, the summed expenses of transporting the fixed quantities of
raw materials from M^ and Mo remaining constant. Hence, given
the sum to be expended on transport to consumption place and from
all raw material sources but two, what are the technical substitution
relations between the distance variables from these two points? Thus
in this formulation all distances are variables, although the values of
all but the relevant two are restrained by a total cost condition and
although all distances are subject to the obvious condition that they

100

LOCATION AND SPACE-ECONOMY

be measured to a common point, the production point. If, for example,
as in Fig. 19, we assume that transport cost is proportional to
distance and that equal weights of raw materials from sources Mi and
M2 are required^^^ (simple hypotheses which we later discard), we can

L^-

Q

Distance from M3

Fig. 20. A transformation line for a four-sided polygon.

indicate the locus of possible sites by an elliptical curve at any point
on which the sum of distance from M^ and distance from M 2 , and thus
the amount expended on transport from M^ and M2, is constant.
From this curve we obtain the transformation line of Fig. 20, which
represents the possible sets of values for the variables, distance from

10 Obviously in reality the weights of raw materials to be transported from
various sources are of different magnitude, and the transport tariffs vary from
raw material to raw material as well as for longer and shorter distances. But,
given any set of real information, we can construct theoretically and the entre-
preneur can calculate empirically a curve of constant transport costs to or from
all points but two.

LOCATION EQUILIBRIUM: TRANSPORT— ORIENTATION 101

C and distance from M3.11 Similarly, we can obtain a transformation
line for these same two variables if a different total amount is to be
expended on transporting the fixed quantities of raw materials from
Ml and M2 respectively. And likewise we can obtain transformation
lines for any two of the possible distance variables, given the sum
to be spent on transport of given amounts of raw materials and product
from other sources or to the consumption place. This same procedure
can be extended to the cases where we have 4, 5, 6, . . ., n raw materials
and consumption places. 12

Thus far we have tacitly assumed that transport facilities of uniform
cost character radiate in all directions from all points and thus cover
the entire plane under consideration. We drop this assumption, and
for the time being adopt a less abstract one. Within any locational
polygon transport facilities of uniform cost character are taken to
connect a finite number of points with all or some of the corners of
that polygon. For simplicity we assume that: (1) on arc TS of Fig. 17
only points T, J, H, and S can be considered as possible production
sites because of the limited transport facilities; and (2) these points
are connected by straight transport lines to each corner of the triangle.
We have at once injected discontinuity into our spatial model. Given
the parametric value of 3 units for the variable distance from C, the
corresponding transformation line for the two variables, distance from
Ml and distance from M2, degenerates into a series of points which
can be connected by straight lines with different slopes and which we
shall continue to call a transformation line. See Fig. 21.

Paralleling an established procedure in production analysis, we now
desire information about the prices or costs of every pair of variables.

11 It should be noted that beyond both points K and L the transformation Hne
of Fig. 20 turns back on itself; both variables increase as we move respectively to
the right of K' and to the left of L' on the elHptical curve of Fig. 19. Theoretically
such movement involves unnecessary distance and would not happen ; realistically,
as we shall see, this can occur.

12 It should be reiterated that the substitutions are subject to a spatial restraint.
The values assigned to the distances of the production site from the several raw
material sources and from the consumption place (s) must be spatially (geometri-
cally) consistent. A change in one of the two distance variables in general not
only involves a change in the second distance variable under consideration but
almost invariably necessitates changes in the parametric values of at least some
of the other distances. Here then we cannot speak of a "fixed basket" of goods.
And in order to obtain a meaningful substitution relation between the two variable
distances in question we need to nullify the effects of changes in the other dis-
tances by imposing a total cost restraint upon the transport of all items over
these other distances.

102

LOCATION AKD SPACE-ECONOMY

With these prices and costs we can derive the price-ratio lines (a la
Hicks) which together with the technical relations depicted by the
various transformation lines allow us to determine a partial locational
equilibrium position.

Q G E B

Distance from M2

Fig. 21. Locational equilibrium: discontinuous transformation line.

Before we can identify the costs associated with the two variables,
distance from Mi and distance from Mo, we must know: (1) the
weight of each of the raw materials to be transported; and (2) the
transport rate(s). For the locational problem depicted in Fig. 17,
assume that 1 ton of the first raw material must be transported from
source M^ , that 1 ton of the second raw material must be transported
from source M2, and that the transport rate on these commodities is
the same and is proportional to distance. As a consequence, the trans-
port charge per unit of distance which is associated with each distance
variable is identical. It then follows that the various price-ratio lines
for these two distance variables must be straight and must have a
slope of —1. Two such lines are EF and BD of Fig. 21.

The transformation line of Fig. 21 and the derived price-ratio lines
yield point J as a partial equilibrium position. This follows since J
is that realistic point on the transformation line TS which lies on the
lowest possible price-ratio (iso-outlay) line. This lowest line is EF.

LOCATION EQUILIBRIUM: TRANSPORT— ORIENTATION 103

Note that at J the following conditions are satisfied: (1) to the right
of J the price-ratio line is steeper than the transformation line; and
(2) to the left of J the same price-ratio line is less steep than the
transformation line.^^

However, what if we allow the distance from C to vary? Taking as
fixed the distance from M2 consistent with location at point J, we can
construct a transformation line for the variables, distance from Mi
and distance from C. And, knowing the transport rate structure,
which for the present we shall take to be the simple one described
above, we can construct price-ratio lines and determine the partial
equilibrium position for these two variables. Presumably, this new
(and better) partial equilibrium position will be consistent for a value
of the variable distance from C different from that assumed in the
preceding paragraph and in Fig. 21. As a consequence, the transforma-
tion line between the variables, distance from M-^ and distance from
M2, changes, and therefore it may be necessary to find a new partial
equilibrium position with respect to these two variables. And so this
process continues. We finally reach a "full" equilibrium position
when the three partial equilibrium positions with respect to (1)
distance from C and distance from Afi, (2) distance from C and dis-
tance from M2, and (3) distance from Mj and distance from M2
coincide. Here there will be no tendency to alter any of the values

13 When the variables, distance from M\ and distance from M2 (or more strictly
speaking the corresponding two transport input variables), are considered as com-
modities r and s (and as commodities are expressed as negative quantities since
they correspond to inputs) and when the graphical solution is approached from
an origin from which positive quantities of these commodities are measured, the
stated inequalities have to be reversed. And in mathematical terms, we have: (1)
to the left of the partial equilibrium point, -prlps > — Ay^/ Ayr] and (2) to the
right of the partial equilibrium point, pr/ps < — Ays/ Ayr, where Ayr and Ays
are finite changes in the quantities of the commodities r and s when one moves
from the partial equilibrium point to a point representing the next possible site
of production in the relevant direction along arc TS, and pr and ps are respectively
the transport charges per unit distance on the raw materials from Mi and M2
required for the production of a unit of output.

In this situation, where a transformation line consists of only a finite number
of points, twin partial solutions may be possible for any pair of distance variables.
This possibility will occur if the slope of the price-ratio line is the same as that
of the segment connecting two consecutive points of the transformation line, each
of which lies on the lowest of the price-ratio lines which course through points
of the transformation line. A certain degree of indeterminacy is thus introduced
into the over-all solution. But this indeterminacy is not a major consideration in
view of the analysis to come. The reader can easily restate the graphic and
mathematical conditions for a twin solution.

Also see Samuelson, op. cit., pp. 70-74 for a treatment of discontinuities in the
production function.

104 LOCATION AND SPACE-ECONOMY

of the variables. Each partial equilibrium position must satisfy the
two conditions stated above. ^^

Although we have dealt with only a triangle of raw material sources
and consumption place, this procedure is applicable to any polygon
of such sources and place. In the case of a four- or more-sided polygon,
we do not hold constant all distance variables but two, but only the
total transport expenditure upon these other distance variables. We
then find the partial equilibrium position with respect to these two
variables. When there are n distance variables, there are of course
%n(n —1) partial equilibrium points. At a full equilibrium position
all of these coincide.

3. Transport-oriented Equilibrium with
Realistic Rate Structures

Having described locational equilibrium under extremely simple
conditions, we commence now to introduce various real complexities,
especially those dealing with transport rate structure and costs.

We abandon the postulate that equal weights of raw materials or
product are transported from the various sources and to the consump-
tion place. Our price-ratio lines need no longer cut the horizontal
and vertical axes symmetrically ; they can cut these axes at any angle
depending upon the relative weights of the raw materials moved.
Suppose, for example, that the production of II/2 tons of final product
requires 1 ton of the first raw material (from M^) and 2 tons of second
raw material (from M2). Retaining a pure weight and distance
basis for computing the transport cost schedule, we obtain a new
series of price-ratio lines with different slope. Of this new series line
GL in Fig. 21 is the relevant one for the circumstances represented
there. Point H rather than point J becomes the partial equilibrium
position for the two variables, distance from Mi and distance from M2.
Similarly, the other partial equilibrium points and the final equilibrium
position will be altered by the change in relative weights. ^^

i"* Except for qualifications relating to twin solutions noted in footnote 13.

We also have assumed in our simple case that the dynamic postulates implied
by our iterative procedure do not lead to an oscillatory situation.

15 Save in one respect the problems of this paragraph resemble Weber's : fixed
points indicate the consumption place and raw material sources, unequal weights
are transported from the several sources to the production site and from there to
the consumption place, and transport tariffs are based solely on weight and dis-
tance. His analysis, though later quahfied (op. cit., pp. 82-83 and elsewhere)
assumes a uniform transport system completely flexible, i.e., uniform transport
facilities radiating from all points in all directions. His is a case of continuous
variation of all distance variables. Ours, thus far, is one of simple discontinuous
variation.

LOCATION EQUILIBRIUM: TRANSPORT— ORIENTATION 105

Of still greater moment is the relaxation of our assumption that
transport rates are proportional to distance, an assumption which is
valid only for areas where primitive transport mechanisms still operate.
In industrialized areas modern transport media require large overhead
expenditures and incur many costs and offer many services (especially
terminal) which are unrelated to the distance covered in a given
shipment. Typically, tariff per distance unit or zone is steep for the
first zone and falls abruptly from the first to the second zone and
considerably less abruptly between each succeeding zone (or set of
zones) and the one after. Tariff structures are graduated, rates being
less than proportional to distance. ^^

One can best demonstrate the significance of typical modern rate
structures for the spatial equilibrium of the firm by constructing
appropriate price-ratio lines or iso-outlay lines. We utilize the 1945
standard maximum first class rates on freight shipments prescribed
by the I.C.C. for railroads operating in the Eastern Territory. ^"^ On
the basis of 2 short tons of raw material from source M2 and 1 ton
from source Mi per ton product, we have constructed, in Fig. 22,
price-ratio or iso-outlay lines corresponding to outlays of $24.00,
$26.40, and $30.00.18

Several characteristics of these iso-outlay lines are important to
note. First, in effect, they are not lines but a series of rectangles and
squares which have been blacked in. These rectangles and squares
border each other or are connected by dashed lines. This particular
form of iso-outlay line results from the zonal character of the rate
structure. For example, the rate for a shipment of a given weight
is the same for all distances 40 miles or less but greater than 35 miles.
Hence, if we consider two shipments of different goods and weights,
we find that total cost of these shipments will not vary for any
combination of distances for these two shipments which can be
represented by a point lying within a square (such as square A of
Fig. 22) which is bounded on two sides by two 40-mile lines and on

16 For further details, see E. M. Hoover, The Location of Economic Activity,
New York, 1948, Chaps. 2-4.

'^'^ As published in I.C.C. Docket 15879 Appendix E and as given in The Freight
Traffic Red Book, New York, 1945, pp. 1194^95 in the column Appendix E under
Eastern Class Rates.

18 Along each axis of Fig. 22 we measure mileage from the respective source
and also cost of transporting over the different distances the amounts of raw ma-
terials required per ton of output. The $30.00 line, for example, shows the various
combinations of values for distance from Mi and distance from M2 which would
occasion a total transport outlay on the raw materials from Mi and M2 of $30.00
per ton of product.

106

LOCATION AND SPACE-ECONOMY

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LOCATION EQUILIBRIUM: TRANSPORT— ORIENTATION 107

the other two sides by two lines approaching the limit of 35 miles.
Thus the producer may have some leeway in choosing a rational
location and may select a site which compels him to traverse "unneces-
sary distance" (in terms of the minimum quantities hypothesized in
the transformation function) without increasing his costs. ^ 9

Second, because the rate structure is graduated, the iso-outlay lines
tend to be convex to origin Q, as in Fig. 22. This fact has significant
implications. Frequently, price-ratio or iso-outlay lines are taken
to be straight or concave to the origin. One then obtains a unique
stable solution that involves quantities of both inputs, when, as is
usually the case, the transformation curve is convex to the origin from
which positive quantities of inputs are measured. If both the trans-
formation line and iso-outlay line are irregularly convex to the origin,
particularly if the latter is more convex than the former, as it may
well be with modern rate structures, then the equilibrium point is
likely to be an "end" point, that is, a realistic point on one of the ends
of the transformation line and one which also may correspond to a
corner of the locational polygon. The equilibrium point always is an
end point in the case where the locational polygon collapses to a line,
e.g., where there is only one raw material used in the production

process

20

Further, tl\e likelihood of an end point solution is considerably
increased by the fact that modern rate structures call for a relatively
large increment in shipping charges from the zero (i.e., not shipping
at all) to the first zone and relatively small increment for any other
two successive zones. For example, on the horizontal scale of Fig. 22
transport cost for 1 ton of the raw material from M^ rises $6.00 if one
decides to ship 1 mile instead of zero miles, but rises only 20 cents if
one decides to ship 6 miles instead of 5. As a result price-ratio or
iso-outlay lines have "tails" on both ends. Our $30.00 and $26.40 lines
have respectively the vertical stretches LM and BG as tails on the
left.21 Their tails on the right (horizontal stretches) are not shown

19 Such a minor degree of indeterminacy also appears with respect to the quan-
tities of other factor inputs employed when these inputs are sold in lot quantities
which are not divisible, as where the services of a machine are rented by the
month. The analysis is not seriously qualified, even though the typical mathe-
matical solution imphes unique amounts of inputs.

-^ In this case, the transformation line is a straight line or consists of a number
of realistic points lying on a straight line. Compare Dean, op. cit., pp. 17-18. Also
see the Appendix to this chapter for some discussion of the simple line case.

21 In the case of our $30.00 line, a decision not to ship the raw material from
Ml at all instead of to ship it to a point within 5 miles permits a saving of $6.00
or allows the movement of the raw material from source M2 over an additional
90 miles without increasing transport cost. Thus, the tail LM.

108 LOCATION AND SPACE-ECONOMY

in Fig. 22, above, since they extend beyond the limits of the diagram.

In Fig. 22 we have constructed the transformation line, BEFDC.
Point B corresponds to the som-ce of Mi ; it represents a zero value
for the variable distance from M^ . At point D the usual equilibrium
conditions are satisfied since to the left of D the arithmetic slope of the
iso-outlay line ($30.00) is less than that of the transformation line,
and, to the right of D, the arithmetic slope of the former exceeds that
of the latter. However, D is only a relative minimum transport cost
point. Only in contrast to the points in its own neighborhood does it
represent the most desirable combination of values for the distance
variables. End point B, lying on the tail GB, is still more desirable
for it falls on the $26.40 iso-outlay line not on the $30.00 one. In the
given situation and for this pair of values for the distance variables,
B represents the position of stable equilibrium.

Thus, our technique clearly demonstrates the strategy of locating
at corners of the locational polygon, given modern transport rate
structures. 2 2 It is consistent with the emphasis that Palander,
Hoover, and others^s have given to such location and the minor
importance they have attached to locations intermediate between raw
material sources and market centers. It implies that the individual
producer must not be content with an equilibrium position arrived at
by the usual substitution operations conceived in production theory,
but must compare the spatial equilibrium so obtained with each
possible relative minimum transport cost position which corresponds
to a zero value for one of the distance variables. 2-±

We pause to consider a significant aspect of our technique. One
of the most devastating shortcomings of the Weberian model has been
its inability to encompass realistic transport rate structures less than

22 In situations where neither one raw material nor the market is dominant (in
Weber's sense) so that locational polygons are meaningful constructs, it should
not be inferred that relative minimum points will never exist at corners of these
polygons unless there is a large initial increment followed by relatively small
increments in the rate structure. Palander (op. cit., pp. 314-16) has demonstrated
that relative minima may occur at corners of polygons when a tariff structure
mildly graduated from beginning to end is in vogue. However, the advantage of
corner location is generally not so pronounced with such a structure.

23 Palander, op. cit., pp. 198-99, 330-33; Hoover, op. cit., pp. 52-57; and B. Ohlin,
Interregional and International Trade, Cambridge, Mass., 1933, pp. 185-202.

24 Making this comparison is not so difficult as might appear. When any distance
variable is assigned a zero value, the values of the other distance variables are
uniquely determined. Thus the producer need only calculate the total transport
cost for each comer of his locational polygon and for whatever relative minimum
cost points (in the usual case, only one) which may be determined through spatial
substitution within the locational polygon.

LOCATION EQUILIBRIUM: TRANSPORT— ORIENTATION 109

proportional to distance. Weber proposed to take account of such
rate structures by using fictitious distances. Distances should not be
stated in their geographic length, but in proportion to the decreasing
rate scale. In general the longer the distance, the more it should be
shortened for geometrical analysis. ^ 5 Bortkiewicz early showed that
such a procedure is inconsistent with the construction of a locational
polygon. 2 6 For, how can we know how much to shorten the distance
of any corner of the locational polygon from the given site of
production and thus be able to calculate the relative distances between
the various corners of the locational polygon until the actual location of
the production site is determined; whilst on the other hand the very
location of the production site is dependent upon the relative distances
between the various corners of the locational polygon?

It is just this Weberian dilemma that our present technique cuts
through. We need not speak of fictitious distances nor are we bound
to a geometrical technique applicable only to situations where rates
are proportional to distance, ^'i' Further, our technique brings out the
critical importance of terminal and loading charges which the Weberian
analysis essentially sidesteps.

25 We quote Weber: ". . . Die sinkenden Staffelungen der Siitze mit wachsender
Entfernung machen dabei keine Schwierigkeiten .... Man hat sich zu sagen, dass
kartenmassige Entfernungen bei Vorhandensein solcher Staffelungen nicht mit
ihrer tatsachlichen Lange in die Rechnung einzustellen sind, sondern unter Reduk-
tion derselben entsprechend den sinkenden Staffeln. Wird also, wie beim deu-
tschen Stiickguttarif der allgemeinen Klasse, fiir 50 Kilometer ein Satz von 11
Pfg., fiir die nachsten 150 km einer von 10, fiir weitere 100 km von 9 Pfg. usw.
berechnet, so ist dabei eine Strecke von 100 km nicht mit ihrer vollen Lange,
sondern mit 50 + (50 — 50/11) km, also mit 95,4 km einzustellen . . ." {o-p. cit.,
pp. 43-44).

26 L. Von Bortkiewicz, "Eine geometrische Fundierung der Lehre vom Standort
der Industrien," Archiv fiir Sozialwissenschaft und Sozialpolitik, Vol. XXX (1910),
pp. 769-71. On this point Niederhauser's defense of Weber {op. cit., pp. 173-75)
is not convincing.

^"^ In one sense Weber should not be criticized too severely for failing to realize
the inconsistency in his geometrical construction. At the time of his writing the
German rate structure (although not the French, British, United States, and Bel-
gian, among others) was proportional to distance, save for small consignments
and a few bulky goods, and was quite uniform (refer to M. Colson, Transports
et Tarijs, 3rd ed., 1907) ; and Weber, as he frequently stated, was preoccupied
with conditions of the German economy. On the other hand, Weber was explicitly
attempting to erect a pure theoretical model applicable to all times and all regions.

We should mention that when the transport rate structure is not proportional
to weight, it is perfectly admissible, though we do not do so, to make adjustments
as Weber does {op. cit., pp. 44-46). A rate per ton-mile greater than normal for
a given good, whatever the reason, implies an "ideal" weight greater than actual;
a rate below normal implies an "ideal" weight less than actual.

110 LOCATION AND SPACE-ECONOMY

Returning to the main thread of the argument, we find it also possible
to account for the effects of breaks in transport routes, which charac-
teristically occur at transport junctions where the direction of
movement has to be changed, where the shipment has to be unloaded
and reloaded onto another transport medium or system, or where
another scale of transport charges becomes effective, and so forth.
To pass over such breaks entails sudden large increments in transport
cost, whether due to switching, to loading, or to other charges. Industry
often locates at such breaks in order to avoid these large increments.
This is borne out by Fig. 23. Here we assume a break in a given
transport route 100 miles from each of the two sources M^ and M2.
Along each axis is measured the cost of transporting over various
distances the amount of raw material from the respective source
which is required to produce 1 ton of finished product. If there were
no break in the transport route we would have a cost scale along each
axis similar to that on Fig. 22. But a break does exist, and to pass
over it involves an added cost of 50 cents per ton of raw material
from M2 and 60 cents per ton of raw material from Mi . Accordingly
we have added another scale (the outer one) of transport charges for
various distances along each axis after the hundred-mile mark. At
point F, the graphic position of the break, the raw materials can be
assembled from both sources Mj and ilf 2 without either one's bearing
the added expense of passing over the break. The total transport cost
is $33.60, and we have constructed the iso-outlay line of $33.60.^8
We have also inscribed a transformation line GFJEH. If no break
existed, the equilibrium point would be G where the transformation
line would meet the tail end of the iso-outlay line of $32.80. But
because there is a break, which necessitates additional transport

28 In constructing this iso-outlay line we have assumed that to reach points
other than the break itself one of the raw materials must pass over the break.
Obviously the other must travel less than 100 miles in this case since the line
goes through the break itself and since we have taken it to be regular in its course.

For the case where both materials traverse distances greater than 100 miles to
any place where they are assembled, we need to know for each potential site
whether both pass over the break or, if only one does so, which one. However,
complications arise as soon as we introduce alternative routes from raw material
sources to any potential site, or to alternative transshipment points varying in
distance from each source, or to both. Then the iso-outlay Hues tend to be highly
irregular and tend to criss-cross one another. We can avoid considerable irregu-
larity by excluding from the analysis certain potential sites which are known to
be inferior for any number of reasons — such perhaps as those involving unneces-
sary distance. Or, alternatively, we can derive the best equilibrium position for
each transshipment point or route or both and then compare these sites and select
the optimum one.

LOCATION EQUILIBRIUM: TRANSPORT— ORIENTATION 111

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112 LOCATION AND SPACE-ECONOMY

expense, G lies on the tail end of an iso-outlay line of $33.80. Point F,
which satisfies the equilibrium conditions when both distance variables
have positive values, lies on the iso-outlay line of $33.60. It is the
preferred position.

There are numerous other deviations from a pure weight and
distance rate structure. Most of these can be encompassed by a set
of iso-outlay lines. When different ton-mile rates apply to different
commodities— whether because one is more bulky than the other, more
valuable, more fragile or perishable, more difficult to handle, closer
to the stage of finished product, better able to bear a high transport
charge, etc. — one can account for these differences in rate structure
by adjusting the transport scale on the appropriate axis. Different
rates are frequently set for different directions of movement because
one is uphill and the other downhill, because one encounters more
severe topographical obstacles, because one bears a greater volume of
traffic, because greater speeds are attainable in one direction, etc. By
changing the transport cost scales on the respective axes one can
incorporate these rate differences into the analysis so long as the
direction of movement of each raw material and finished product
remains the same for all realistic points on the transformation line.
If this condition does not hold, then for the movement of any particular
good it is necessary to use several different transport cost scales along
the axis relevant for such movement. And, accordingly, in constructing
any iso-outlay line on which a given realistic point on the transforma-
tion line lies, we must use, for each raw material and finished product,
that transport cost scale which is relevant for the direction in which
the good moves to reach the realistic site under consideration.

Again, if different types of transport facilities are used in any one
journey or if alternative types of transport facilities are available for
reaching the various possible production sites, we must construct
several transport cost scales along the relevant axis or axes and use
the appropriate one in deriving the iso-outlay line upon which a given
realistic point on the transformation line lies. ^ 9 It is also necessary
to adjust the transport cost scales when various types of import and
export duties, or special levies or transport expenses, are incurred
at different points along a given route. Where there are alternative
routes with various special charges and duties along them, alterna-
tive transport cost scales must be set up.^o

29 See Palander, o-p. cit., pp. 333-58, for a full discussion of substitution possi-
bilities between various types of transport media and between cheaper and dearer
facilities.

30 It should also be mentioned that eccentricities in transport rate structures

LOCATION EQUILIBRIUM: TRANSPORT— ORIENTATION 113

4. Transport-oriented Equilibrium Further Extended

The analysis of the preceding sections has emphasized distance as a
variable. This initial emphasis has been pm^sued partly to counteract
the traditional bias toward consideration of spatial relations. For a
complete analysis of transport-orientation, however, change in more
variables than merely distance must be examined. Clearly, the
amounts of the raw materials used may vary with the location of
the plant, particularly if alternative sources of different quality are
exploitable. Also, in any true transport-orientation problem the
variations from site to site in other costs such as labor and power
must not be assumed away. They must be explicitly introduced as
possible deviational forces, even though they turn out to be dominated
by variations in transport outlay. To comprehend better the inter-
action of these variables, a transport-orientation framework less
restrictive than the one already developed (which has reference to
variations in distances alone) is desired. Further, a more general
framework is imperative, as will be seen below, in order to establish
connections among the several types of location theories and to uncover
principles common to all. Moreover, such a framework would facilitate
the fusion of location theory and production theory.

We now propose to employ the concept of transport inputs as defined
in Chap. 4. To repeat: a transport input represents the movement of
a unit weight over a unit distance. It may be expressed in such terms
as a hundredweight-kilometer and a ton-mile. We therefore encounter
transport inputs in the shipment of any raw material to the production
site and in the shipment of the finished product from the production

which run counter to transport cost as a monotonically increasing function of dis-
tance are reflected in eccentricities of iso-outlay Hues. For example, the histori-
cally important practice in the United States of levying a smaller total charge for
movement between two nodal termini served by two or more competing railways
than for movement between one of these termini and an intermediate point or
between two intermediate points on one of the alternative routes causes the iso-
outlay lines to criss-cross. If, for instance, the raw material from M2 could be
moved from its source, a terminal, to another terminal 210 miles distant on a
special rate, then point K in Fig. 22, which may represent a combination of
values of 210 miles for distance from M2 and 100 miles for distance from Mi,
would he on a lower iso-outlay line than point /. Obviously such eccentricity
enhances the attraction of terminal sites, particularly when these sites are raw
material sources or consumption places. As a consequence, a commodity may
travel an unnecessary distance, that is, to a site lying outside the locational poly-
gon. Similarly this occurs when back hauls and roundabout hauls take place
because of the rigidities of the transport net, especially in water and water-rail
transport.

114

LOCATION AND SPACE-ECONOMY

site to the market. Since the distance variable as well as the weight
variable is encompassed by the concept of transport inputs, all the
relations among distance variables discussed above may be translated
into relations among transport inputs. To demonstrate this, reconsider

Transport inputs on raw material M2

Fig. 24. Shift of transformation curve and equilibrium site with change in

weights.

. 17 we substitute for the three
distance from M2, and distance
(1) transport inputs on the raw

the minimum transport cost solution for the locational triangle of
Fig. 17.

In the problem associated with Fi§
distance variables (distance from M^ ,
from C) three new variables, namely:
material from source M^ ; (2) transport inputs on the raw material
from source Af 2 ; and (3) transport inputs on the product to the market.
Relations among these new variables can be presented in terms of
meaningful transformation lines. For the situation where we consider
as a possible production site each point along arc TS of Fig. 17, we
obtain a transformation line between transport inputs on the raw
material from M^ (which we shall henceforth designate transport
inputs on raw material Mj ) and transport inputs on the raw material
from Mo (which we shall designate transport inputs on raw
material M2). This transformation line, SHJT, plotted in Fig. 24

LOCATION EQUILIBRIUM: TRANSPORT— ORIENTATION 115

resembles the transformation line of Fig. 18. The only difference
between the two charts thus far is that in one case distance variables,
in the other transport inputs, are measured along the axes.

It should be noted that the transformation line is drawn as a
continuous curve. Although in reality it should be a finite number of
points, as in Fig. 21, we present it as a curve in order to facilitate
the analysis and the synthesis of transport-orientation theory and
production theory. The latter in its more familiar form utilizes curves
rather than a finite number of points and continuous production
functions rather than discontinuous ones.

If transport rates are proportional to distance and if they are the
same for both raw materials, then the relevant set of price-ratio lines
are straight lines which cut both axes symmetrically. The prices
involved are the rates per ton-mile transportation of the two raw
materials. In this case the prices are different from those in the
situation depicted in Fig. 21. There each price was the cost of
moving 1 mile the weight (tonnage) of a given raw material required
per weight unit (1 ton) of the product. Hence, when the weight
requirements of raw materials were altered, so were the slopes of the
price-ratio lines, the transformation line expressed in terms of
the distance variables remaining unchanged. Here in the situation
depicted in Fig. 24, the prices are rates on the movement of a weight
unit over a unit of distance (e.g. rates per ton-mile) . When weight re-
quirements of raw materials change, the price-ratio lines remain the
same. The requirements of transport inputs, however, change. A new
transformation line between transport inputs becomes relevant. For
example, suppose we assume as previously that the production of 1^/^
tons of finished product requires 1 ton of the raw material from Mi
and 2 tons of the raw material from Mo . These new weight relations
yield, for the locational figure of Fig. 17, a transformation line
S'H'J'T' between transport inputs on raw material M-i and transport
inputs on raw material M2. See Fig. 24. The point S" on the dashed
transformation line refers to the identical geographic location as does
S; iy as does H'; J as J'; and T as T\ However, although the
transformation line rather than the price-ratio line shifts in this new
formulation, the equilibrium point of location is unaffected. Before
the change in weight relations, both the old and the new formulation
of the problem yielded a location at or in the vicinity of J in Fig. 17. ^1

31 When the transformation hne is a series of points, as in Fig. 21, the partial
equilibrium point is at J. When the transformation line is assumed continuous,
as in Fig. 24, the partial equilibrium point is somewhat to the left of J in the
direction of H at the point K where the transformation line is tangent to a
price-ratio line.

116 LOCATION AND SPACE-ECONOMY

After the change in weight relations, both formulations yield a location
at H in Fig. 17.

As indicated, the new formulation of the problem in terms of
transport inputs is superior to the first one in several respects.
However, in at least one basic respect it is inferior to a formulation
in terms of distance variables. In the first formulation, shifts of the
transformation line can take place only when the distance variable
from C assumes another value or when new sources of raw materials
are utilized with a consequent alteration of the locational triangle.
In the new formulation, shifts of the transformation line can take
place as well when weight relations change. Hence, in the new formu-
lation it becomes impossible to identify the cause of a shift from mere
observation of the transformation lines and to know whether or not
a change of location (geographic position) is entailed. The new
formulation hence loses the spatial perspective which the first formu-
lation permits. (Since the new formulation is much more in line
with orthodox economic analysis, this is consistent with the weak
spatial perspective of orthodox economic thinking.) To reiterate, it is
for this reason that the less elegant formulation in terms of distance
variables has been presented first for emphasis.

Returning to the problem of partial locational equilibrium which
is achieved when the geographic point of production lies within the
locational triangle and corresponds to the point on the transformation
line which lies on the lowest price-ratio line (and hence incurs least
total transport cost) , we note two necessary conditions. One, the first
order condition, is that the geographic point of production corre-
spond to a point of tangency between a price-ratio line and the trans-
formation line. This signifies, in general, that the rate of substitution
at the margin, or the marginal rate of substitution, between any two
transport inputs be equal to the reciprocal of their prices (the cor-
responding transport rates). Two, the second order {or stability)
condition, is that at the point of tangency the transformation line
be more convex to the origin than the price-ratio line. Since in Fig.
24 the transformation lines SHJT and S'H'J'T are convex throughout,
whereas the price-ratio lines are straight, this stability condition is
fulfilled at any point which may prove to be a point of tangency.
The convexity of the transformation lines to the origin signifies a
diminishing marginal rate of substitution between the two transport
inputs. It should be noted that at both points K and H', these two
necessary conditions are satisfied. Since throughout its course each
transformation line is convex to the origin, no other points of tangency
occur, and K and H' each represents the partial locational equilibrium
position for its particular situation.

LOCATION EQUILIBRIUM: TRANSPORT— ORIENTATION 117

As before, a "full" equilibrium position is attained when the three
partial equilibrium positions with respect to the three pairs of variables
[here: (1) transport inputs on product and transport inputs on raw ma-
terial Ml ; (2) transport inputs on product and transport inputs on raw
material M2 ; and (3) transport inputs on raw material Mi and trans-
port inputs on raw material M2 ] coincide.

Just as we have converted the locational triangle problem of Figs.
19 and 20 into a problem of substitution among transport inputs, so
we can convert the other locational problems depicted in the previous
sections. The set of possible sites of production in the locational line
case (see Figs. 15 and 16) can be presented in terms of a transforma-
tion line for two transport input variables rather than for two distance
variables. In the case of the four-sided polygon (see Figs. 19 and
20), transformation lines for pairs of transport inputs substitute for
transformation lines for pairs of distance variables. Along the axes
of a diagram similar to Fig. 20, we measure transport inputs rather
than distances. Again, in the identification of the path of substitu-
tion between any pair of transport input variables, the same restraint
obtains as in the formulation with distance variables: the sum of
transport costs associated with all other variables (here, transport
inputs) must remain constant. Once again, the over-all location
equilibrium involves the positional coincidence of more than one
partial equilibrium between pairs of variables. ^ 2

When we inject realistic rate structures and discontinuities in the
transformation function, we can (though we need not) construct
for a locational triangle problem figures somewhat comparable to
Figs. 22 and 23. The scales along the axes, however, would be
different. Scale b of the figure corresponding to Fig. 22 and scale c
of the figure corresponding to Fig. 23 would measure transport inputs
on raw material M2 in ton-mile units, and would be numerically
twice as large as the existing scale. Scale d of the figure correspond-
ing to Fig. 22 and scale / of the figure corresponding to Fig. 23
would measure transport inputs on raw material Mi in ton-mile
units and would be numerically the same as the existing scale. The
other scales would be transport cost scales as in Figs. 22 and 23,
respectively, but would be designated somewhat differently. The
transformation lines would correspond respectively to the transforma-
tion lines of Figs. 22 and 23, but would of course refer to transport
inputs rather than distance variables. The equilibrium analysis and
statement of equilibrium conditions would essentially parallel that
presented for Figs. 22 and 23.

32 See Chap. 10 for a full statement.

118 LOCATION AND SPACE-ECONOMY

Before this chapter is brought to a close stress should be placed on
this important point. The two necessary (the first- and second-order)
conditions, as stated above for a partial locational equilibrium, when
generalized to consider for any transport-orientation problem sub-
stitution among any pair of transport inputs, are no different from
those formulated by Allen, Hicks, and other production theorists
when the problem of substitution between two inputs or factors of
production is posed. ^^ At an equilibrium point, in Hicks' words,
the "price-ratio between any two factors [inputs] must equal their
marginal rate of substitution," and for ''the substitution of one factor
(input) for another, 'diminishing marginal rate of substitution' "
must hold. 3 4

Even when the transformation line is taken as a finite number of
points, the necessary conditions for transport-oriented equilibrium
are essentially the same as those stated by Samuelson for a firm given
a discontinuous production function. ^ 5 Further, when realistic rate
(transport price) structures are introduced, the necessary conditions
again resemble those which can be derived from Samuelson's statements.

Thus, it is along these lines that by using the concept of transport
inputs^e we are able to fuse much traditional (Weberian) doctrine

33 R. G. D. Allen, Mathematical Analysis for Economists, New York, 1939,
Chaps. XIV and XIX; and J. R. Hicks, Value and Capital, Oxford, 1939, Chap. VI.
34 /bid., pp. 86-87. Italics are mine.

35 Samuelson, op. cit., Chaps. Ill and IV.

36 In order to illustrate the general applicability of the concept of transport
inputs, let us refer to a transport-oriented production process such as iron and
steel. The United States Steel Corporation recently constructed 1.8 million tons
of steel capacity immediately below Trenton for serving the Eastern seaboard
market centering around New York City with steel produced from Venezuelan
ore. Of the suitable waterfront sites, this was probably the closest to New York
City. In choosing a site at Trenton rather than one farther from the market
but closer to coal, the corporation substituted transport inputs on coal from the
coal source for transport inputs on product to the market. Considering, first,
the possible combinations of quantities of these two transport inputs, given the
quantity of transport inputs on ore from Cerro Bolivar, Venezuela, which for
practical purposes remains constant for all Middle Atlantic seaboard points, and
second, considering the quantities of coal and scrap that might be used per ton
steel, the ton-mile transport charges on these items, and as a consequence the
significantly higher price of a transport input on product to the market than
of a transport input on coal from the coal source — one can easily portray the
Trenton site as the point of locational equilibrium in the given situation. In
making this statement we also consider variation in the quantity of transport
inputs on ore from Cerro Bolivar, for the given requirement of ore per ton steel.
A meaningful larger quantity of this input would involve an inland location
which, in view of the significant cost of transshipment to rail and of the geography
of ore, coal, and market sites, would not be an over-all locational equilibrium

LOCATION EQUILIBRIUM: TRANSPORT— ORIENTATION 119

on transport-orientation and production theory for the firm, thereby
extending both.^'^ Restatement of the problem of transport-orienta-
tion in terms of substitutions among transport inputs automatically
establishes a point of connection with other substitutions among the
diverse inputs (and outputs) of a firm and, hence, allows improve-
ment in any statement on transport-orientation. At the same time,
inclusion of transport inputs in the transformation function of the
firm adds a spatial dimension to production theory and allows this
theory to embrace the situation of transport-orientation. ^^

Appendix to Chapter 5

Transport Inputs and Some Formulations of the
Transport-orientation Problem

In this chapter, the general transport-orientation problem has been essen-
tially restated. In Chap. 10 the transport-orientation problem is presented
in more rigorous mathematical terms. Nevertheless, it may be helpful to
some readers to translate some of the more famihar formulations of the
transport-orientation problem into substitution relations among transport
inputs in a way which is simple and direct.

We may begin with the line case already alluded to at the beginning of
Sect. 2 of this chapter. Suppose 1 ton of raw material from source M-^ is
required to produce 1 ton of product which is consumed at point C. Let C
be connected by a straight line railway to M-^^. See Fig. 15. It then follows
that the various combinations of the variables, transport inputs on raw
material M-^ and transport inputs on product, corresponding to the innumer-
able efficient locations possible along line CM-^^ are given by a transformation
line with a slope of — 1. Such a transformation line is given by Fig. 16 when
we appropriately measure along the axes transport inputs rather than
distances.

If we now posit that transport rates on both product and raw material are
identical and proportional to weight and distance, we obtain a set of price-
ratio lines whose slopes are also — 1. When we superimpose such a set of
lines upon Fig. 16, we find that the transformation line and one and only

point when translated into the relevant sets of transformation and iso-outlay
lines. Neither would a location in the Southern Atlantic seaboard, which would
involve a smaller quantity of this input. See, in this connection, W. Isard and
J. Cumberland, "New England as a Possible Location for an Integrated Iron and
Steel Works," Economic Geography, Vol. 26 (October 1950), pp. 245-59.

37 This fusion is more rigorously demonstrated in Chap. 10.

38 It is perhaps unnecessary to reiterate the point in the final paragraph of
Chap. 4, namely, that the extension of production theory to include transport
inputs as another set of inputs does not commit a person to the acceptance of the
transport function as another factor of production, if he is not inclined to do so.

120 LOCATION AND SPACE-ECONOMY

one of the set of price-ratio lines coincide completely. This fact signifies that
each of the innumerable possible locations along line CM^ incurs the same
total transport cost. The locational equilibrium problem is therefore indeter-
minate, as Weber noted. Any point on the transformation line corresponds
to as good a site of production as any other.

Determinacy is immediately introduced if an ubiquity which enters into the
weight of the product plus a pure material from source M^ are required for
production. In this instance, the relevant transformation line will fall off
less rapidly than the transformation line of Fig. 16. When price-ratio lines
with slope of —1 are superimposed upon this new transformation line, the
point of the transformation line which lies on the lowest price-ratio line will
correspond to an end-point solution, specifically to location at the market, as
Weber indicated.

Determinacy can also be introduced if we assume that the raw material is
weight-losing and that, as a consequence, more than a ton of raw material is
required per ton product. In this case the new transformation line falls off
more rapidly than the transformation line of Fig. 16. When price-ratio lines
of slope — 1 are superimposed, that combination of transport inputs on raw
material and finished product which corresponds to location at M-^ lies on the
lowest price-ratio line.

Most of the indeterminacy of the first case is eliminated if graduated
(Staff el) transport tariff structures obtain. For the price-ratio lines are no
longer straight fines with slope of — 1, but curved lines as indicated in the
discussion pertaining to Fig. 22. As a consequence, those two combinations
of the two transport input variables which correspond to location at the
market and location at the raw material source lie on the same price-ratio
line; all other combinations fie on higher price-ratio lines.

Essentially, ubiquities which enter into the final product increase transport
inputs on the finished product for any location away from the market; and
the occurrence of weight-loss in the use of a localized raw material increases
transport inputs on the raw material for any location away from the source.
Further, the appfication of different transport rates to the movement of
finished product and raw material can be incorporated into the problem
either: (1) indirectly by the use of "ideal" weights, in which case price- ratio
hues are unaffected and the relative values of transport input variables change ;
or (2) directly by changing the appropriate transport scales along the axes,
in which case the slopes of the price-ratio lines change, and the values of the
transport input variables remain unaffected. With these considerations in
mind, it is easily perceived how the various correct propositions which have
been advanced for the line case by Weber and others are translatable into
transformation lines with respect to transport inputs and price-ratio lines.

We pass on to the use of weight triangles for the solution of the transport-
orientation problem when a locational triangle is given. Dean has expounded
this type of solution most precisely, and we shall have reference to his
formulation.

Let us consider first the case where the weight triangle does not exist.
This would obtain, for example, if Weber's material index were less than unity
(and his locational weight, less than two). In this situation the ideal weight
of the product exceeds the combined ideal weights of locahzed raw materials,
the weight of the product being the dominant weight. When transport inputs
are based on ideal weights, the price of every type of transport input is the

LOCATION EQUILIBRIUM: TRANSPORT— ORIENTATION 121

same regardless of which raw material or finished product is being moved;
and, accordingly, the problem is to minimize total transport inputs. Hence,
if the material index is less than unitj^ and if we consider a location away
from the locus of consumption, it is always feasible to shift toward the locus
of consumption, until that locus is reached. For, with any such shift, the
total of transport inputs diminishes, since any increase in transport inputs on
localized raw materials is always less than the decrease of transport inputs on
the finished product. In efi"ect such a shift corresponds to one or more
movements along one or more transformation lines which involves one or
more substitutions of fewer transport inputs of one type for more transport
inputs of another type. This process entails a concomitant movement to lower
price-ratio lines, whose slopes by definition are — 1.

When transport inputs are more meaningfully based on actual weights rather
than ideal weights, again it follows that if the material index is less than
unity, it is not economic to locate at a site other than the locus of consumption.
For, with any shift from a location not the locus of consumption toward the
locus of consumption, any increase in the costs of transport inputs on localized
raw materials (because of the increase in the amounts of such transport
inputs) will always be smaller than the decrease in the costs of transport
inputs on finished product (because of the decrease in the amount of such
transport inputs). Once more such a shift involves substitutions among
transport inputs which correspond to movements along transformation lines
on to lower price-ratio lines whose slopes are ordinarily different from — 1.

The weight triangle does not exist when Dean's generalized index yields
a value less than unity. [In Dean's generalized index test the denominator is
either: (1) the largest localized raw material or group of spatially localized
raw materials, or (2) the product, whichever is the larger by weight.] If, for
example, the weight of a locaUzed raw material is dominant (the generalized
index will then be less than unity) and if the site of production is not at
the site of this locahzed raw material, it is always feasible to shift toward
this site. With such a shift, either: (1) total transport inputs on localized
raw materials and finished product will decrease, when these transport inputs
are based on "ideal" weights; or (2) the decrease in the costs of transport
inputs on the localized raw material (because of the decrease in the amount of
such transport inputs) will be greater than any increase in the costs of
transport inputs on other localized raw materials and finished product
(because of an increase in the amount of such transport inputs) .

Let us now turn to the general case where the weight triangle does exist.
Here, the generalized index test gives values always greater than unity.
Also, Weber's material index yields a value greater than unity. However, it
does not follow that location cannot be at an end point (a corner of the
locational triangle), and for that matter at a source of a pure material. The
problem is one of the equilibrium of forces in which relative weights and
relative distances are the basic factors.

The w^eight triangle is a geometric device used to obtain the point at which
the several locational forces are in equilibrium. Its counterpart in the physical
world is a mechanical model such as Varignon's which was designed to demon-
strate the parallelogram of forces, and the use of which Pick describes in an
appendix to Weber's book. But what are the locational fo7'ces in a transport-
orientation problem?

If we base transport inputs on ideal weights, as Weber and Dean tend to

122 LOCATION AND SPACE-ECONOMY

do, the three locational forces in a locational triangle are the ideal weights
of the finished product and the two raw materials. They pull against each
other in a maimer wliich minimizes total transport inputs at the point of
equilibrium when that point lies within or on the locational triangle. When
the point of equilibrium lies outside the locational triangle, transport inputs
are minimized when location is at that corner of the locational triangle whose
exterior angle is less than the corresponding angle of the weight triangle.
This holds whether the corner corresponds to a pure or a weight-losing raw
material.

If we base transport inputs on actual weight, as is done in the mathematical
statement in Chap. 10, the three locational forces in a locational triangle are
the three costs involved in moving one at a time the respective three actual
weights a unit of distance. These three forces, each of which is expressed in
terms of transport costs per unit of distance, interact to determine a point
of equilibrium. If such a point lies within or on the locational triangle, it
corresponds to a point of minimum total transport cost. If it hes outsicle the
locational triangle, the corner of the locational triangle whose exterior angle
is less than its corresponding angle in the weight triangle is the point of
minimum total transport cost.

Again, one easily perceives that any shift from a location which is not a
point of equilibrium within or on a locational triangle toward such a point,
or toward the appropriate corner of the triangle when the point of equilibrium
hes outside the triangle, entails substitutions among transport inputs which
involve movements along transformation lines on to lower price-ratio lines.

In essence, the weight triangle solution shortcuts the process of determining
the site of locational equilibrium. Likewise does the somewhat similar geo-
metric construction which is based upon the use of the "pole principle,"
which Launhardt first sketched and Palander later amplified. However, in
doing so, both the weight triangle and the pole construction fail to point up
explicitly the interaction of the basic economic forces, namely, the several
transport costs per unit distance. More satisfactory in this regard is the
Varignon mechanical model (which can be employed for locational polygons
of more than three sides), even though it is oriented to "ideal" weights rather
than "economic" weights.

A third approach to the solution of the transport-orientation problem rehes
upon the use of isodapanes.3 9 An isodapane, as used by Palander and Hoover,
is a locus of points at each of which the location of the production process
would incur the same over-all (combined) transport costs in the movement
of both raw materials and finished product. Typically, to derive a set of
isodapanes, in which each isodapane refers to a different value for total
(combined) transport costs, one constructs isovectors (a la Palander) or
isotims (a la Hoover) about each raw material source and finished product.
An isovector is a line which, in the case of a localized raw material, connects
points to which the required quantity of localized raw material may be shipped
from a point source at the same transport cost. Hence, around each raw

39 The isodapane technique is more flexible than the weight triangle method.
It can encompass graduated tariff structures and transport nets composed of dif-
ferent media, avoid adjustment to an "ideal" weight basis, and relate to loca-
tional polygons of more than three comers. It is however a much more cumber-
some technique.

LOCATION EQUILIBRIUM: TRANSPORT— ORIENTATION 123

material source, a set of isovectors may be drawn; to each isovector we may
assign a transport cost which generally rises with distance from the raw
material source. (See Figs. 48 and 49, Palander, op. cit., and Fig. 15, Hoover,
op. cit.) In the case of a finished product, an isovector is a line wliich
connects points from which the finished product may be shipped to a specified
market point at the same transport cost. Here, too, a set of isovectors may
be drawn, where each represents a different level of transport cost which
generally rises with distance from the market point.

Once a set of isovectors is constructed about each raw material source and
market point relevant in a locational problem, isodapanes may be drawn by
connecting all those points wliich as a location for production would incur
the same sum of the several transport costs on the raw materials and finished
product. The procedure of employing maps of isovectors to derive isodapanes
is discussed by Palander in coimection with liis Fig. 52, and Hoover in
connection with his Fig. 15.

Essentially, isodapanes are contour lines of a total transport cost surface.
Such a surface is precisely treated in Chap. 10, and is briefly described in
Pick's Appendix (Weber, op. cit., pp. 244-5). Once the contour lines are
mapped, the identification of the minimum transport cost point, the point of
locational equihbrium in the transport-orientation problem, is a matter of
course. See, for example, the situation depicted by Palander, op. cit., in Figs.
53 to 68, and Hoover, op. cit., in Figs. 15 and 20.

As already intimated, both the process of substitution among transport
inputs, as discussed in this chapter, and isodapanes refer to the identical
transport cost surface. Our substitution process refers to a path of movement
along the transport cost surface in a direction toward the trough point of the
set of isodapanes (the minimum point of the surface). An isodapane, in
contrast, refers to a path of movement around a trough point. As such our
substitution process and an isodapane represent merely two different paths
of journey along the same surface.

That (1) the process of substitution among transport inputs and (2) the
movement along an isodapane and from isodapane to isodapane can involve
the same basic considerations is clearly demonstrated by reference to Palan-
der's method of constructing isodapanes when three or more commodities are
to be moved. Imagine a locational triangle such as Fig. 17. About each raw
material source and market Palander advises as a first step the construction
of a set of isovectors. As a second step, any two of the three sets of isovectors
are used to derive a subset of isodapanes (partial isodapanes) which refer
to the sum of only two transport costs. As a third step, the subset of
isodapanes are combined with the third set of isovectors to obtain the desired
set of isodapanes which refer to the sum of three transport costs.

For the moment consider ( 1 ) the subset of isodapanes derived in the second
step and (2) the third set of isovectors, which, let us posit, centers around
the market point. Suppose we select that isovector which corresponds to
three distance units from C in Fig. 17 and thus to arc TS. Suppose also we
select any one isodapane from the subset of isodapanes which intersect this
isovector. As we proceed along this isovector in one direction we ^vill be
moving on to subset isodapanes of greater and greater values; and in the
other direction to subset isodapanes of lower and lower values. Such move-
ment necessarily involves a substitution between transport inputs on the raw
material from M-^ and transport inputs on the raw material from Mg, since

124 LOCATION AND SPACE-ECONOMY

distances from M-^^ and ikfg change while weights remain constant. A move-
ment in the former direction is identical with a movement along the trans-
formation hne ST of Fig. 24 to higher and higher price-ratio (iso-outlay)
lines; and in the latter direction, to lower and lower price-ratio hnes. This
is so because we are holding constant transport inputs on finished product to
C, and, hence, the transport cost on finished product to C (since the transport
rate is given) . The desirability of moving along any isovector until the subset
isodapane of lowest value is reached (which wiU be at a point of tangency)
is clear. This is equivalent to substituting between transport inputs and
moving along the transformation line ST of Fig. 24 until point K is reached.
Of all points on the transformation line, K lies on the lowest price-ratio line
and is also a point of tangency. K represents a partial locational equilibrium
point.

We proceed further. Having identified the point which would be K in
Fig. 17 and which is the point of tangency of the isovector of the previous
paragraph with the lowest value isodapane of the subset isodapanes having
a point in common with the isovector, let us select that isovector of the set
of isovectors centering around raw material source M2 which passes through
what would be point K in Fig. 17. Also, construct another subset of isoda-
panes based upon the sum of transport costs on raw material from M^ and
finished product to C. Once again, we move along the isovector until we
reach that isodapane of the new subset which has the lowest value; it will
be tangent to the isovector. Once again, we are substituting between trans-
port inputs — this time between transport inputs on the raw material from
i¥i and on the finished product to C. Or, to put it otherwise, we are moving
along a transformation line on to the lowest price-ratio line. We reach
another point of partial locational equilibrium, partial since we hold fixed the
value for the variable, transport inputs from Mg-

Next we (1) select either that isovector centering around C or that isovector
centering around M^ which passes through this new partial equihbrium point
and (2) construct still another subset of relevant isodapanes. As before
we move along the isovector to lower and lower subset isodapanes, etc., etc.

We can continue this procedure until we reach that point where it is no
longer possible to move on to any lower subset isodapane along any of the
three isovectors which can be constructed through this point. That is, it is
no longer economic to substitute between any pair of transport inputs. In
essence, we have a coincidence of three partial locational equilibrium points.
We are at the point around which all the isodapanes center. We are at the
trough of the transport cost surface.

Thus, it is clear that on a transport cost surface (1) economic movement
from isodapane to isodapane and (2) economic movement which corresponds
to substitution between any pair of transport inputs in our transformation
sense differ in direction only. They aim at the same goal. And, in fact,
when the former movement is restricted to paths along isovectors on to subset
isodapanes of lower and lower value, they are identical and translatable one
into the other.4 0

40 In the above discussion we have implicitly assumed that one and only one
minimum point exists. Actually, for most problems, more than one exists. How-
ever, this does not qualify our basic analysis. Following Palander and Hoover,
the reader can easily reword the above statements to cover cases with several
minimum points.

LOCATION EQUILIBRIUM: TRANSPORT— ORIENTATION 125

In this appendix an attempt has been made to translate some of the better
formulations of the transport-orientation problem into substitution relations
among transport inputs. It has not been the intention to treat and translate
each formulation comprehensively. The discussion has not covered all the
refinements of the several doctrines which are presented in Launhardt, Weber,
Palander, Dean, Losch, Hoover, and elsewhere; rather, it has sought to relate
and restate only the basic threads. From here the reader can easily proceed
further.

Chapter

The Locational Equilibrium

of the Firm:
Labor and Other Orientation

1. Introductory Remarks

In the previous chapter, we have examined the conditions for the
locational equilibrium of transport- oriented processes, postulating
that among sites differentials (except those arising from different
transport costs on raw materials and finished product) either do not
exist or are insignificant. Each productive factor and service, other
than a transported raw material or finished product, was considered
to be available everywhere in correct amounts and at the same price.
When a raw material was present at several sources, it was taken to be
adequately available at the same price for all sources. When more
than one market point for a finished product existed, at each the
revenue potential or the ruling price on the finished product was
posited to be identical.

We now relax some of these assumptions and introduce differentials
in factor costs and revenue potentials, i To incorporate such differen-
tials into general locational analysis for the firm, it is necessary to
think in terms of substitution between outlays, between revenues,
and between outlays and revenues. It is insufficient here to speak
of substitution between the commodities encompassed by our trans-

1 We consider differentials in revenue potential since a firm may consider several
production locations which may serve directly, or be gateway points to, different
markets where different prices and demand elasticities for the firm's product (s)
obtain.

126

LOCATION EQUILIBRIUM: LABOR ORIENTATION 127

formation function, for it is the variations from site to site in the
prices of these inputs and outputs which, along with other forces,
influence location. For example, Weber discusses the phenomenon
of labor orientation where a firm does not locate at the transport
optimum point representing the best combination of transport inputs
but rather at a cheap labor point. When these two points are not
identical, the firm thereby consumes more transport inputs and in-
creases its transport outlays while it simultaneously holds constant
(or even increases) its labor inputs but reduces its labor outlays,
ceteris paribus. We do not have substitution between transport
inputs and labor inputs but rather between transport outlays and
labor outlays.

2. Labor Orientation

It is possible to develop conceptual schemes to treat substitutions
between outlays, between revenues, and between outlays and revenues.
Suppose within the typical Weberian framework we allow first in-
equalities in labor resources among sites and consequent differentials
in labor costs. To every realistic point on the transformation line for
a pair of transport input variables, we can assign not only a necessary-
transport outlay as given by the iso-outlay line which passes through
it but also a labor outlay. Thus, if we take the realistic points
G, F, J, E, and H on the transformation line in Fig. 23 and assume
that labor outlay per ton of product is $20.00 at each of the sites
represented by these points, except the cheap labor site^ represented

2 The term cheap labor is employed here in a broad sense. A site where cheap
money wages are paid to labor is not necessarily a cheap labor site if the labor
is inefficient; on the other hand, a site where high money wages are paid can be
a cheap labor site if the efficiency of labor more than counterbalances the high
money wages. The fundamental concept is the wage per labor service of a given
quality or per efficiency unit.

Cheap labor areas or sites arise from a number of circumstances. Often rela-
tively low wage payments are found to be characteristic of a surplus agricultural
region. Such payments reflect the relatively small transport cost for food and
drink consumed by the laborer and his family. A relatively low wage payment
permits a satisfactory content of living. However, if one considers food and drink
for laborers as raw materials in the production process, which conceptually is a
consistent procedure, then surplus agricultural regions, which contain good assem-
bly points for these raw materials, need not be locationally classified as cheap
labor areas.

As Ohlin, Hoover, and others have noted, differences in wages which are attrib-
utable to differences in transport costs, and, therefore, total costs of the same
basket of consumer goods (budget materials) at different places may be classified
as "equalizing" differences. These contrast with "real" differences. Since equaliz-
ing differences arise from transport cost differentials, they represent part of the

128 LOCATION AND SPACE-ECONOMY

by J where it is only $16.00/^ we can depict the respective transport
and labor outlays incurred at these sites by corresponding points
in Fig. 25. In this figure, labor outlays and transport outlays are
measured along the vertical and horizontal axes, respectively. As
in Fig. 23, point F is taken to be the optimum transport point.

Also in Fig. 25, we have plotted the points L, M, N, and R, which
represent other cheap labor sites. These additional positions do not
have corresponding realistic points on the transformation line of
Fig. 23. But, in the light of all possible variations in all transport
inputs (transport inputs on raw material ilfi, on raw material M2,
and on finished product), such additional positions correspond to
realistic points on transformation lines when the quantity of transport
inputs on the finished product to C is different from that assumed
in Fig. 23. Other realistic points which may or may not be cheap
labor sites can also be plotted, but plotting them is not necessary
to elucidate the argument.

When two or more points incur the same labor outlay, we consider
only the one which involves the least transport outlay; and when,
in Fig. 25, we connect these points in order according to transport
outlay, we obtain the line FJLMNR, which may be called an "outlay-
substitution" line. It presents the meaningful substitution possibilities

transport orientation problem and should be treated as such. Their inclusion in
the transport orientation problem is, however, more complex than is often realized.

Major industrial centers based upon coal mining, iron and steel manufacture,
and other primary economic activities which engage the chief breadwinners of
families frequently have supplies of secondary labor available, i.e., labor which
is surplus to the primary industries yet immobile because the location of the
chief breadwinner determines the location of the secondary labor supplied by the
family. Like any other surplus commodity, this labor can usually be purchased
at bargain rates. Parasitic industry is attracted by it. But it is important to
recognize that this kind of attraction is fundamentally conditioned by a site's
attraction for dominant industries; it can be explained only in terms of the total
situation. Basically, such parasitic industry is not a case of labor orientation.
Differentials in regional development of secondary industry must be stated in
terms of differentials in regional development of different primary industries.

Cheap labor sites are most often attributable to cultural factors such as are
found in poverty stricken regions where long-run immobility of labor obtains.
Frequently, the laborer may be willing to forego leisure and to sell his services at
a low price in order to supplement his other income and thus obtain the purchas-
ing power requisite for minimum subsistence living.

For full discussion of these and related points, refer to Dean (Selections), op. cit.,
pp. 22-30; Ohlin, op. cit., pp. 212-20; and Hoover, op. cit., pp. 60-74.

3 We repeat : Weber assumes that the wage levels of labor at the several loca-
tions are fixed and that the labor supply available at each location is unlimited.
(Friedrich, o-p. cit., p. 101.)

LOCATION EQUILIBRIUM: LABOR ORIENTATION 129

between transport outlays and labor outlays, just as the transformation
line does for two inputs. We also construct a new set of iso-outlay
(transport plus labor outlay) lines which are straight and obviously
have a negative slope of unity when the same scale is used along
both axes. TU and CD are two such lines representing (on transport
and labor) combined outlays of $56.00 and $50.00 respectively.

Q 30

Fig. 25. An outlay-substitution line in a case of labor orientation.

40 50 W U 60

Transport outlay ($)

Of all points on the outlay-substitution line, J lies on the lowest
iso-outlay line. It indicates location at a cheap labor site rather
than at F, the minimum transport cost point, and, hence, a substitution
of transport outlays for labor outlays if location were initially at F.

If the reader cares to, he may state formal conditions for equilibrium
with respect to these two outlays. These conditions might almost
parallel those stated in the preceding chapter. With Q as origin
(Fig. 25) , to the left of the equilibrium point when it is not an end
point, the outlay-substitution line between transport outlay and
labor outlay must be steeper than the relevant iso-outlay line; and,
to the right of the equilibrium point when it is not an end point, either

130 LOCATION AND SPACE-ECONOMY

the iso-outlay line must be steeper than the outlay-substitution line
or else the outlay-substitution line must have a positive slope. ^

Incidentally, the approach illustrated in Fig. 25 does away with the
unnecessarily complex Weberian technique for determining whether
or not an operation will be labor oriented and, if so, to which labor
site. According to Weber, one must construct the critical isodapane^
for each cheap labor site, consider only those sites which fall within
their critical isodapanes, and select that site which in terms of "ideal"
distance lies farthest from its critical isodapane. Using the above
graphic technique, one needs to observe whether or not any points
representing cheap labor sites lie on lower iso-outlay lines than does

4 Along certain stretches, the slope of an outlay-substitution line may well be
positive. For, in moving from one cheap labor site to another, from left to right
along the outlay-substitution line of Fig. 25, not only will transport outlay rise
but also labor outlay will increase if at the second cheap labor site labor is some-
what more dear. In contrast, when dealing with transformation lines for two
transport inputs (or distance variables) where only a finite number of points can
be considered as location sites, one would encounter positive slopes only when
"unnecessary distance" is traversed, e.g., where a firm locates outside the locational
polygon. This possibility has been logically excluded from our analysis except
for the case where there is a particular type of irregularity in the transport rate
structure.

Also, the above conditions, as in the previous chapter, may be satisfied by more
than one point. It becomes necessary to select the best from among such points.
For example, in Fig. 25 both J and N meet the requirements stated above. Obvi-
ously J, lying on the lowest iso-outlay line, is the more desirable site of the two.

Again it is worthy to note the extent of discontinuity involved in reality.
Usually the possibihties of substitution do not permit a gradual geographic shift
through sites in turn necessitating less and less labor outlay and more and more
transport outlay until the optimum combination is reached. Rather, as Weber
states, the problem is one of an alternative attraction (Alternativattraktion) not
an approaching attraction (Annaherungsattraktion). It does no good to move
from the optimum transport point to a site nearer the cheap labor location. In
such a shift, labor outlays are not usually significantly affected. To derive the
benefits of cheap labor, one has to migrate to the cheap labor site itself. Spatially
speaking, there is no continuity in this migration. It involves in the typical case
a discrete geographic jump.

5 In contrast to Palander and Hoover, Weber uses the isodapane concept in a
marginal sense rather than in the total sense as defined in the Appendix to Chap. 5.
Thus according to Weber, whom we follow in this and subsequent chapters, an
isodapane is a curve connecting points representing locations involving the same
increases of transport cost over the cost incurred at the transport optimum point.
The critical isodapane for any cheap labor site is the one which represents an addi-
tional transport outlay equivalent to the saving in labor outlay at the cheap labor
site. If the site lies anywhere within the area bounded by the critical isodapane,
it becomes eligible for attracting the production process under consideration
(Weber, op. cit., pp. 102-04).

LOCATION EQUILIBRIUM: LABOR ORIENTATION 131

the point representing the transport optimum site. If none does,
there will be no deviation to a labor location. If more than one do,
then the firm will shift to a cheap labor site and to that one which
lies on the lowest iso-outlay line. In Fig. 23 such a site is J.^

The above presentation of the labor orientation problem, however,
is inadequate. For, assuming still that all other factors are ubiquitous
and available everywhere at the same cost, it is reasonable to expect
another type of adjustment, specifically, a tendency for substitution
at the cheap labor site of cheap labor for transport inputs, provided
production coefficients and other technical relations are not fixed. This
might take the form of reduction in bulk of the product, in weight, in
perishability, and so forth. And, if the cheap labor site's advantage
lay in the exploitability of a special class of labor, let us say un-
skilled labor, we might face the additional substitution problem of
skilled labor vs. unskilled labor.'' This is the type of substitution
problem within a substitution problem that Predohl had in mind.
The adjustment, however, would not affect the conditions to be met
by an equilibrium point. Cheap labor points would lie on lower
iso-outlay lines above and to the left of their respective positions on
Fig. 25. This adjustment, of course, might involve a change of the
equilibrium site.

3. Some Other Forms of Orientation

In the above section we have relaxed the assumption that the costs
of an efficiency unit of labor at all locations were alike. Differentials
in labor costs were introduced, while, by assumption, differentials in
other costs (except transport) were precluded. We now proceed to
preclude differentials in labor cost, while allowing differentials in
transport costs and in just one other cost item, let us say power.

It is clear that in this new situation a procedure paralleling that
of the preceding section may be utilized. To every realistic point on
the transformation line for a pair of transport inputs, we can assign
not only a necessary transport outlay as given by the iso-outlay line
which passes through it but also a power outlay. For each point, the

^ Also, the above graphic approach provides a simpler framework when adjust-
ments need to be made for transport savings from the use of replacement deposits
in Weber's sense. However, logically the use of a replacement deposit signifies a
shift from the supply area of one source of the raw material in question to the
supply area of a second source. Such use is therefore more appropriately consid-
ered as a phase of supply area analysis to be treated below.

'^ Since for the present all other factors are considered ubiquitous and available
everywhere at the same cost, there would also tend to be a substitution of cheap
labor for these factors at cheap labor sites.

132 LOCATION AND SPACE-ECONOMY

associated transport outlay and power outlay can be plotted on a graph
along whose horizontal and vertical axes transport and power outlays
are measured respectively. Also, for every realistic point on other
transformation lines for any pair of transport inputs, both transport
and power outlays can be derived and plotted on the same graph.
When we view the entire set of points depicted on this graph and,
of two or more points incurring the same power outlay, consider only
the one which involves the least transport outlay, and when we con-
nect these points in order according to transport outlay, we obtain
another type of outlay-substitution line. This type presents the
meaningful substitution possibilities between transport and power
outlays, just as the type of outlay-substitution line in Fig. 25 presents
the meaningful substitution possibilities between transport and labor
outlays. We can also construct a new set of iso-outlay (transport plus
power outlay) lines, which, like the iso-outlay lines in Fig. 25, are
straight and have a negative slope of unity when the same scale is
used along both axes.

Given the iso-outlay lines and the outlay-substitution line, it would
be easy to identify in our assumed situation the optimum location.
This location would correspond to that point on the outlay-substitu-
tion line lying on the lowest of the iso-outlay lines. Formal condi-
tions of locational equilibrium could be stated. Allowance could be
made for the substitution at cheap power sites of power inputs for
transport inputs and other inputs, technological and economic condi-
tions permitting. This substitution would be in addition to the sub-
stitution associated with locational shift which takes place along the
outlay-substitution line.

In this way, the significance which variation in power cost has for
locational equilibrium can be evaluated under our given set of assump-
tions. When the point on the outlay-substitution line which lies on
the lowest iso-outlay line (after adjustment has been made for sub-
stitution among inputs) corresponds to a cheap power point, we
have the familiar case of power orientation. According to Weber's
terminology, the cheap power point lies within the critical isodapane;
or, according to Dean's terminology, there are deviational economies,
and the largest possible deviational economies, in shifting to this
particular cheap power point in our assumed situation. ^

In similar fashion, it is possible to consider formally the variation

8 Paralleling the concept of labor coefficient and the ratio of labor savings to
additional transport outlays (which are discussed in the Appendix to this chapter),
one can construct a power coefficient and the ratio of power cost savings to addi-
tional transport outlays and use them as we use the labor coefficient and corre-
sponding ratio.

LOCATION EQUILIBRIUM: LABOR ORIENTATION 133

in the costs of other input items resulting from diverse inequalities
in mineral and human resources and from differences in cultural and
political institutions. Along with variation in transport outlays, the
variation of the interest rate among sites and regions might be con-
sidered in isolation. We could construct outlay-substitution lines
(referring to the variation at different sites in outlays on interest
and on transport) and relevant iso-outlay lines to determine which
(if any) cheap interest point might deviate the location of production
from the minimum transport cost point. In this problem, at least
implicit consideration would have to be given to the alternatives in
the use of different types of venture and other capital and to the
entire set of input substitution points associated with the width
and depth of capital structure.

Or we might consider in isolation variation in transport outlays
and tax outlays ; or in transport outlays and rent outlays ; or in trans-
port outlays and in general production outlays where differences in
general production outlays obtain because of differences among sites
in climatic and other environmental features, in union restrictions
and diverse social burdens, in agglomeration and industrial and
population density, and so forth.

Another differential that may confront the individual producer may
be in the price of raw materials at different sources, whether owing
to differences in fertility, richness of ore, processing costs, etc., or to
imposed competitive differences. As before, assuming all other factor
costs geographically uniform, one can measure transport outlay along
one axis and outlay (at the source) on raw material A along the other.
One can then plot the respective outlays for any possible site. How-
ever, only those sites are relevant which involve, for each possible
source of the raw material under consideration, the least transport
outlay for assembling all the raw materials required in production
and carrying the finished product to the market. There will be a
deviation from the site involving the least aggregate transport outlay
of all (i.e., the best of the several relative minimum points with
respect to transport cost) to an optimum position with respect to the
cheaper raw material source if the saving from the lower price of raw
material more than counterbalances the increased outlay on transport
occasioned by the shift. The new site would lie on an iso-outlay
(transport plus raw material A outlay) line lower than the first.
Usually the shift would involve a discrete spatial jump, except in the
case of agricultural raw materials or the like for which there are
large supply areas. And similarly with sources of other raw materials,
each in turn.

Again one might consider differentials in price received per unit

134

LOCATION AND SPACE-ECONOMY

of product at various consumption places for the case where a firm
confronts a geographic pattern of market prices upon which it has
httle, if any, infiuence, and where the differential aspects of this pattern
are unrelated to the given firm's transport outlays. In Fig. 26, one

Transport outlay

Fig. 26. A revenue-outlay substitution line.

measures total transport outlay on inputs and output along the
horizontal axis and revenue from sale of product along the vertical,
all other outlays and revenues being held constant. Here, one is
concerned with iso-revenue-less-outlay lines, i.e., constant product
revenue less transport outlay lines, with the result that of two such
lines a position on the one to the left is more desirable. These
straight iso-revenue-less-outlay lines, unlike those involving two kinds
of outlays, have positive slopes. In Fig. 26, point S is that site which
incurs the least transport outlay. However, positions R, N, and M,
being "higher-price" markets or being gateway points to higher-price
consumption places, all offer larger revenue from sale of the product
than does point S. In the case of point R, the additional revenue
more than counterbalances the additional transport outlay required
for assembling raw materials at R and for shipping the product to the
corresponding consumption place. R then is a preferred position

LOCATION EQUILIBRIUM: LABOR ORIENTATION 135

lying on a higher iso-revenue-less-outlay line than does S, N, or M.^
If more than one output is produced, we can construct a similar
chart for each.io

This sort of partial analysis, however, where all costs and revenues
but two are held constant, is inadequate and of little pragmatic value.
Weber recognized this fact when, speaking of shifts to labor locations,
he took into consideration the economies to be gained by replacement
of material deposits, that is, by utilization of new raw material sources
closer to labor locations rather than those sources which were more
favorably situated in relation to the market to be served and the
optimum transport point. Without doubt, if a plant does find it
expedient to shift to a cheap labor site, a cheap power site, or a site
better able to serve a higher revenue consumption place, other out-
lays and revenues besides transport outlay do change in the usual
case. We thus require a substitution analysis which can treat differen-
tials in all types of costs and revenues at one time, not just differentials
in two types alone.

An extension of the analytical technique to meet this requirement
can be forced along familiar lines. When one attacks the problem of
choosing correct quantities of physical inputs and outputs, he may first
consider the substitution point relating any two commodities, the
basket of all other commodities being given. Then in turn he may
derive the substitution point for a second pair, a third pair, and so on.
During these latter operations, if the derived quantity of any com-
modity is inconsistent with the quantity previously assigned to the
same commodity within the fixed basket of goods when the substitution
relations between a first pair of goods was being considered, then one
changes the composition of the basket to make it consistent with the
new quantity. In turn, this change is likelj^ to lead to a new sub-
stitution point for the first two commodities and a change in the
quantities desired. And this change in the first pair of values is likely
to alter the substitution relations between other pairs of goods, and
different quantities of these other goods might then be desirable. This
process continues until finally a consistent set of substitution points

9 Note also that the formal partial equilibrium conditions can be stated. They
are met in this situation. Viewed with point Q as origin, to the left of point R
the iso-revenue-less-outlay Hne has a slope less steep than that of the revenue-
outlay substitution hne (SRNM) ; and to the right of point R the revenue-outlay
substitution line has a negative slope. R would still be an equilibrium point if
to its right the slope of the revenue-outlay substitution line were positive so
long as it were less than the slope of the iso-revenue-less-outlay line.

10 The substitution analysis centering upon the relation between the location
of a firm and its scale of operations is considered in Chap. 8.

136 LOCATION AND SPACE-ECONOMY

is obtained, wherein the equilibrium conditions are satisfied for every
meaningful pair of commodities.

At the start one may be inclined to approach the problem in the same
fashion when differentials in several types of costs and revenues exist
between sites. One takes two outlays which are different at several
sites that in other respects possess similar price and cost structures
and finds the best combination of these outlays, as in Fig. 25 above.
Similarly with another pair of outlays until all possible pairs are
considered. However, one finds that different sets of sites pertain to
different pairs of outlays. This finding means that the above pro-
cedure, though logical for certain equilibrium analyses, overlooks one
important aspect of the location problem. Sites are often unlike
with respect to more than tvv^o outlays. Where all sites differ in at
least three outlays, the above process is inappropriate. But by lump-
ing together two or more outlays, so that two or more sites are alike
except with respect to two groups of outlays, one can still utiHze this
analytical technique.

It is important to recognize that this is essentially no different from
the procedure where two or more commodities are used in fixed pro-
portions. If the consumption of commodity j, belonging to a group
of commodities used in fixed proportions, is increased to replace some
of commodity m which is not included in the group, we cannot
construct a transformation line between commodities j and m, the
basket of all other commodities being fixed. For, as the quantity of ;
varies, so vary the quantities of those commodities used in fixed
proportion to commodity ;. As a consequence, the basket of all com-
modities other than ; and m cannot be given. The only method of
attack is to consider as a whole the commodities used in fixed pro-
portions and to inquire into the substitutability of this group for
commodity m, the basket of other commodities being given.

When one lumps together two or more outlays to derive an outlay-
substitution line between any two groups of outlays, he is using this
same technique. Here he has no such explicit constraints on the
variation in quantities of commodities as those given by the trans-
formation function. He has only a geographic pattern of resources
and facilities resulting in price-cost differentials. But this does not
preclude using an identical procedure. He can construct outlay-
substitution, revenue-outlay substitution, and revenue-substitution lines
between all possible pairs of outlays and revenues or groups of outlays
and revenues. If all sites manifest price differentials with respect
to only three outlays — transport, labor, and interest — one can con-
struct an outlay-substitution line, measuring transport outlay along
one axis and labor plus interest outlay along the other. The relevant

LOCATION EQUILIBRIUM: LABOR ORIENTATION 137

price-ratio lines would be iso-outlay (transport plus labor plus in-
terest outlay) lines. 11 If sites exhibit differences among four items —
whether they be outlays or revenues — measuring along the first axis
either one or the aggregate of two items and along the second axis
the aggregate of respectively three or two items, one can derive a
substitution line from plotting the values associated with each site to
contrast with appropriately derived iso-lines. And in similar fashion
one can treat the situation where differences among sites exists among
five or more items. 12

If there are differentials among all sites with respect to each cost
and revenue, then, the above procedure being followed, the problem of
optimum location would reduce to a simple substitution relation be-
tween two groups of outlays and revenues. At this juncture, however,
cloaking the analysis in terms of substitution would be of little value.
A forthright comparison of total costs at each site will achieve the
desired result more readily. Likewise, the site of production being
given, the substitution technique degenerates when all inputs and
outputs fall into two groups, the commodities within each group being
used in fixed proportions. In this situation there is only one sub-
stitution relation, namely, between the two groups of commodities.
In reality, however, the various constraints to the transformation
function usually leave room for many substitution relations between
the different commodities. Likewise, potential sites for a production
process do not manifest major price differentials in many categories
of costs and revenues. The fewer constraints to the transforma-
tion function and the fewer major price differentials among all sites,
the more pertinent is substitution analysis (except for the extreme
case).

11 And a la Weber, he might be incHned to compute "labor-interest coefficients."
However, he would soon discover extreme difficulties in the use of such a
coefficient.

12 An alternative to this procedure is to divide sites into groups, each group
containing sites which differ with respect to two items only (when sites in general
differ in at least three items), or with respect to three items only (when sites in
general differ in at least four items), or with respect to four items only (when sites
in general differ in at least five items), and so forth. For each group of sites the
optimum one can be obtained through the above substitution analysis where along
each axis one, or the aggregate of two or more items, is measured. The total
revenue and cost situation for each of the resulting "partial" optimum sites can
be directly compared to yield the over-all optimum location. Or the over-all opti-
mum location may be derived by a division of all the differential revenue and
cost items into two groups, and by a plot for each of the "partial" optimum
points of the aggregate of each group (upon a chart along whose axes the two
aggregates are measured) in order to yield a substitution hne to contrast with
an appropriately derived iso-line.

138 LOCATION AND SPACE-ECONOMY

4. A Re-examination of the Substitution Framework for
Spatial Analysis

It is appropriate at this juncture to evaluate the substitution ap-
proach developed in this and the preceding chapter. As already inti-
mated, it may be contended that at least to some degree the substitution
framework has been pushed to undesirable length. Such a view stems
from a consideration of the spatial setting within which various types
of costs and revenues vary.

For certain purposes it is convenient to classify location factors
into three groups. These groups, though generally valid for our
purposes, overlap to some degree and cannot be precisely delineated.
In the first group may be included transport costs and certain other
transfer costs. The distinguishing feature of these transport and trans-
fer costs is that they vary regularly with distance from any given
point of reference, usually increasing in step-by-step fashion as dis-
tance increases. Hence, given a relevant set of reference points,
whether they be raw material, service, nodal, or market points, we
find systematic variation of these costs over space. The structure of
transport rates and the tariff structures for other transfer costs which
are a function of distance being given, the systematic variation of
these costs over space becomes predictable.

This is not to deny that exceptions to this systematic variation
exist. Such exceptions in the transport rate structure have already
been alluded to in the previous chapter, and the necessary techniques
to incorporate these irregularities into the main body of rates, without
jeopardizing the system of this body, have already been indicated. In
similar ways, exceptions in tariff structures for other transfer costs
which are a function of distance can be handled without destroying
the general systematic variation of these costs over space.

A second group of location factors comprises the several costs
associated with labor, power, water, taxes, insurance, interest (as
payment for the services of capital) , climate, topography, social and
political milieu, and a number of other items. The geographic cost
pattern of many of these items may be said to be relatively stable.
However, in contrast to the first group, it cannot be said that the
costs of any of these items generally vary systematically with dis-
tance from any given reference point. Rather, they tend to vary
haphazardly, independently of direction and distance, i^ Yor example,

13 On the surface, there appear to be exceptions to this statement. Power rates,
for example, may rise regularly with distance from a generating station. Once
beyond the feasible transmission range, however, the variation of power rates is

LOCATION EQUILIBRIUM: LABOR ORIENTATION 139

cheap labor points generally occur around any given reference
point in an unpredictable fashion. There is no reason to anticipate
that, given any set of spatial co-ordinates, cheap labor points will
be some function of distance and direction from that defined position.
Hence, in this sense, analysis of labor costs, as well as the other
costs which fall in this second group, seeks in vain for any meaningful
general spatial framework.

A third group of location factors comprises the diverse elements
which give rise to agglomeration and deglomeration economies. In-
cluded in agglomeration economies are: (1) economies of scale; (2)
localization economies; and (3) urbanization economies. i* Degiom-
erative forces embrace chiefly: (1) diseconomies within a firm as its
scale of operation becomes too large; (2) the rise of rents and costs
of urban services as increase in the intensity of land use and population
settlement leads to congestion; and (3) the rise in the cost of food
supply as the increase in the size of population settlement compels
resort to surplus agricultural areas farther and farther afield. ^^

It is clear from an analysis of agglomerative and deglomerative
forces that their operation is independent of geographic position.
Their associated economies and diseconomies are functionally de-
pendent upon the magnitude of activities. These economies and

not subject to regularity, except insofar as the variation reflects differences in the
transport cost of fuel. Essentially, where power transmission or transportation
of fuel is possible, a regularity may ensue; where power transmission and fuel
transportation are infeasible, irregularity is characteristic. It is thus transport
cost which imparts to the geographic variation of power rates any regularity
which it possesses, both intraregionally and interregionally. There is no inherent
regularity in the geographic distribution of energy and power resources.

Likewise with labor costs. To some degree, regularity in the variation of wage
rates over space may be said to have existed historically and to persist in current
times. As already intimated, such regularity is for the most part related to
differences in transport cost in obtaining the goods in the laborer's market basket.
For a given content of living, the variation in wages resulting from these differ-
ences in transport cost have already been referred to as "equalizing" differences
in money wages. They are to be distinguished from "real" differences in money
wages which do not exhibit any spatial regularity. Again, it is the transport
element which imparts a regularity to the spatial cost pattern of an input, which
otherwise does not possess any such regularity. Compare Hoover, Location
Theory and the Shoe and Leather Industries, Cambridge, Mass., 1937, Chap IV.

!•* See Chap. 8 for full discussion of them.

15 To the extent that agglomerative and deglomerative forces are associated with
the increase and decrease of transport cost, or of any other cost item falling in
the second group, to the same extent these categories of location forces, as already
suggested, overlap. However, such an overlap does not seriously interfere with
our argument.

140 LOCATION AND SPACE-ECONOMY

diseconomies obtain regardless of the locality at which any given
magnitude and situational interaction of activities occur (though, to
be sure, elements of the physical setting, such as topography and
bedrock conditions can influence to some extent the intensity with
which agglomerative and deglomerative forces operate). These forces
are, for the most part, spatially passive. They are adaptive, and
they materialize at localities where other considerations either dictate
or hinder location. Their geographic pattern is a derived one, and it
reflects the regularities and irregularities of spatial pattern associated
with other location factors.

Hence, we are led to conclude from our general consideration of
these three groups of location forces that only the transport factor
and other transfer factors whose costs are functionally related to
distance impart regularity to the spatial setting of activities. Sub-
stitution analysis among various transport inputs, of the type dis-
cussed in Chap. 5, is vital for understanding the spatial configuration
of economic activities and its inherent order and for unearthing the
pervasive impact of the friction of distance. On the other hand,
substitution analysis which is more elaborate than the sort sketched
in the preceding sections and which particularly treats at one time
several factors that either haphazardly distort or intensify the sys-
tematic spatial arrangement imposed by the transport factor, may be
judged to be less significant. Such analysis possesses less value at this
stage where we are concerned with the development of equilibrium
analysis for the firm as an integral part of a general theory of space-
economy which is independent of any particular cultural, institutional,
or geographic frame of reference. ^^

16 For example, it may be maintained by some that, where a particular process
is not transport-oriented, i.e., where transport cost differentials among sites are
not the dominant ones, it is best in ascertaining the maximum profit location
merely to recognize that substitutions do take place among outlays and revenues
and to focus attention upon the more important substitutions. To be specific, the
indirect shift of textile capacity from New England to the cheap labor point of
Puerto Rico involved a substitution of transport outlays for labor outlays. The
cheap labor point, however, was in no way generally related to a distance factor.
It could have existed elsewhere. This arbitrariness, it may be contended, weakens
any attempt at stating general formal equihbrium conditions regarding substitu-
tion in analyzing the spatial equilibrium of this case. This effect is even greater,
it may be held, when tax outlays and other outlays are introduced into the picture.

LOCATION EQUILIBRIUM: LABOR ORIENTATION 141

Appendix to Chapter 6
The Labor Coefficient and a Related Ratio

Since Weber's concept of labor coefficient has considerable significance for
location analysis, it is useful to present briefly some of the connections
between this concept and a related ratio directly obtainable from substitu-
tion between transport and labor outlays.

In considering the feasibility of the shift of production processes from their
transport optimal points to cheap labor locations, Weber emphasizes two
industrial characteristics: (1) locational weight (based upon "ideal" instead
of actual weights) and (2) index of labor costs (i.e., average labor costs per
ton product). Also, he considers of primary significance three environmental
conditions: (1) geographic position of locational figures and labor locations,
(2) transport rates, and (3) actual percentage of compression of the labor
cost index. If we combine locational weight, geographic position of locational
figures and labor locations, and transport rates, we obtain the set of addi-
tional transport outlays associated with any pattern of shift from the transport
optimal site. [These outlays may be comprehensively depicted by a set of
isodapanes. Obviously the smaller (greater) the locational weight and the
lower (higher) the transport rate, the farther apart (the closer) do consecu-
tive isodapanes lie and the greater (less) is the likelihood of deviation to a
labor location. Unless the locational figure for a production process is
symmetric and unit weights apply to all corners, there will not be an equal
tendency to deviate in all directions. Isodapanes will not be circular; and,
ceteris paribus, they will be pulled toward the corners with heaviest weight.
For possible deviations over short distances, this distortion of circular form
may be significant. As the distance of possible deviation increases, the
isodapanes tend to approximate more and more a set of concentric circles,
and the distorting effect of any asymmetric locational figure and pattern of
weights can be increasingly neglected.]

Likewise, if we combine the index of labor costs and actual percentages
of compression, we obtain the set of savings in labor outlays associated with
existing labor locations. [Obviously the higher (lower) the index of labor
cost and the percentage of compression, the greater (smaller) the hkelihood
of deviation to a labor location, ceteris paribus.]

To measure quantitatively the extent to which different industries may be
deviated to labor locations, Weber develops the concept of labor coefficient.
The labor coefficient of an industry is defined as the ratio of its labor cost
per ton product to its locational weight, or labor cost per locational ton. If
only deviations over significant distances are considered so that the distorting
effects upon isodapanes of any particular locational figure and its set of forces
can be ignored, then, according to Weber, the higher the labor coefficient the
more likely that an industry will be labor oriented, given a fixed transport
rate structure proportional to weight and distance and after due allowance is
made for savings from replacement deposits which tend to become increasingly
significant with increase of deviational distance. Weber's labor coefficient is
also useful in establishing priorities for different types of industries which
might be induced to locate at a cheap labor location. Ceteris paribus, the
higher the coefficient, the higher the economic priority.

142 LOCATION AND SPACE-ECONOMY

If we consider a specific labor location and if we multiply the numerator
of the labor coefficient for an industry by the relevant percentage of compres-
sion and the denominator by the transport rate and distance between the labor
location and the transport optimal point, we obtain a ratio of outlays — of labor
savings per locational ton to additional transport expense per locational ton.
So long as this ratio is greater than unity for any given industrial process
and environmental situation, the labor location lies within the critical isoda-
pane and attracts the industry in question. It pays to substitute transport
outlays for labor outlays. The greater the ratio, the greater the savings
achieved in shifting to a labor location.

Compared to the labor coefficient from which it is derived, the ratio per
locational ton of labor savings to additional transport outlays has the virtue
of being able to indicate directly whether or not a shift to a labor location is
feasible. Further it can yield more directly answers to questions like these:
By how much must labor costs be compressed to attract a specific industrial
process to a given location, ceteris -paribus'! By how much must the transport
rate be cut to allow an industrial process to be attracted to a cheap labor
location, ceteris paribus f To which of several labor locations will an industrial
process be attracted? In contrast, the labor coefficient has the major advan-
tage of having more general applicability. It indicates relative tendency of
various industrial processes to shift to labor locations and is generaUy inde-
pendent of the percentage of compression, transport rate, or distance relations
in any particular situation. (In still other contexts, another ratio might be
useful, namely, one which is derived through multiplying the numerator of
the labor coefficient by a relevant percentage of compression and the denomi-
nator by the transport rate. This ratio is independent of spatial situation.)

Chapter

7

Market and Supply

Area Analysis and Competitive

Locational Equilibrium

1. Market Area Analysis

Hitherto we have treated the firm as serving for the most part a
one-point market. We now relax this simplifying postulate and con-
sider the market as an area. Further, with the use of the concept
of transport inputs it is easily shown how production analysis for
a one-point consumption place may be viewed as a special case of
production analysis for a market area.i

In Chap. 2 we have briefly sketched Losch's conception of the

1 Incidentally, for a long time, location theorists treated separately the problems
of production for a one-point consumption place and production for a market
area. Launhardt, who presented the first significant treatment of industrial loca-
tion theory, distinguished between the partial problem of determining the site
of production within or at the corners of a locational polygon, where the corners
represented raw material sources and a one-point consumption place ["Die
Bestimmung des zweckmassigsten Standortes einer gewerbhchen Anlage," Zeit-
schrift des Vereines Deutscher Ingenieure, Bd. 26 (March 1882)], and the partial
problem of supplying a consuming area from a given point of production {Malhe-
matische Begrundung der Volkswirthschaftslehre, Leipzig, 1885, Part III). Al-
though he handled both problems comprehensively for his time, he made no
attempt to put them together. Weber, in his analysis, treated only the first of
these problems. Englander, perhaps the first to recognize that these two prob-
lems are fundamentally one and the same (in his caustic criticism of Weber,
"Kritisches und Positives . . .," op. cit.), nevertheless did not adequately synthe-
size them in this and his other works. The later writings of Palander, Hoover,
and Losch are much more satisfactoiy in this respect.

143

144 LOCATION AND SPACE-ECONOMY

space-economy built upon the elements of a market area, a net of
market areas, and a system of nets of market areas. We nevertheless
need to start over again in order to investigate thoroughly the concept
of market area and to relate it to the theoretical structure thus far
developed. 2 In Chaps. 10 and 11 we shall attempt an integration of
market area analysis with supply area analysis to be discussed in the
next section and with other location doctrine.

Imagine that a producer secures each of his raw materials and inputs
at the site of his factory (hence at zero transport cost) but serves a
spatial array of consumers. If consumers come to one and only one
particular site and make their purchases there, or arrange trans-
portation from that site on the item purchased, then to the producer
of this item, that site is the market. For that individual producer,
whom for the present we take to be an isolated monopolist in the
Chamberlinian sense, ^ our spatial equilibrium analysis need not be
extended. From the standpoint of society, however, when consumers
are actively responsible for the transportation of the item, another
set of transport inputs may be involved. If the consumers are other
producers farther along in the stage of manufacture, then this trans-
portation appears as transport inputs on raw material from a point
source in these producers' calculations ; and again no extension of our
analysis is required. On the other hand, if consumers are households,
we are not able thus far to account for the transport inputs for
which they are actively responsible. * However, once we make the

2 In this section we shall cover only the more important theoretical aspects of
market area analysis. For supplementary details, graphic illustrations, and more
extensive discussions refer among others to W. Launhardt, Mathematische Be-
grundung . . . , op. cit.; F. A. Fetter, "The Economic Law of Market Areas,"
Quarterly Journal of Economics, Vol. 38 (May 1924), p. 525; Tord Palander,
Beitrdge zur Standortstheone, Uppsala: Almqvist and Wiksells, 1935, Chap. IX;
Edgar M. Hoover, Jr., Location Theory and the Shoe and Leather Industries,
Cambridge, Mass., Harvard University Press, 1937, Chaps. 2, 3, 5, and 6; CD.
and W. P. Hyson, "The Economic Law of Market Areas," Quarterly Journal of
Economics, Vol. LXIV (May 1950), pp. 319-27; and Melvin L. Greenhut, "Inte-
grating the Leading Theories of Plant Location" and "The Size and Shape of
the Market Area of a Firm," Southern Economic Journal, Vol. 18 (April 1952),
pp. 526-38 and Vol. 19 (July 1952), pp. 37-50, respectively.

3 See E. Chamberlin, The Theory of Monopolistic Competition, Cambridge,
Mass., 1938, p. 74; and "Monopolistic Competition Revisited," Economica, No-
vember 1951, pp. 351-54.

4 To do so would take us into the realms of sociology and social psychology.
For, to explain the spatial distribution of household consumers around focal
points — for example, the population spread around any given metropolitan core —
requires knowledge of the process by which tastes are molded and, in particular,
understanding of the space preferences of consumers. Hiunan ecology promises

MARKET AND SUPPLY AREA ANALYSIS 145

assumption, usually considered legitimate for economic analysis, that
tastes and space preferences are known, and thus demand schedules
and the spatial pattern of population about any given pattern of
focal points are known, we necessarily "explain" the total quantity
of transport inputs for which household consumers are actively
responsible.

Consider situations, where, in contrast, the producer is actively
responsible for transporting his products to the places of use, where
each consumer is charged a price equal to a quoted price at a focal
point plus transport cost to his place, and where the producer arranges
the distribution of his product from that focal point. If the con-
sumer is an industrial producer, he has in effect made the decision
to contract for the transport inputs involved in delivery from the focal
point to the place of use by having chosen to locate where he is rather
than at a site closer to the focal point. He incurs the expense of
these transport inputs indirectly by paying a higher delivered price.
From society's standpoint, we may still consider the transport cost
on the product paid by the first producer as outlays by the second
producer on transport inputs required to obtain one of his raw
materials. Once again our analysis needs no extension. If the con-
sumer is a household, and we take its tastes, space preference, and
thus demand schedules as given, we necessarily "explain" the trans-
port inputs involved in the delivery of the product from the focal
point. The consumer's willingness to pay the delivered price signifies
his willingness to incur the costs of these transport inputs. When
the household is unwilling to incur these costs and when the industrial
consumer finds it expedient to avoid transport inputs in obtaining
the product which he uses as a raw material, the market for the
producer of this product reduces to a point. Viewed in this narrow
framework, production for a one-point market is thus a special case
of production for a market area, continuous or discontinuous.

It is fruitful to spell out these general statements, particularly
as they link up with existing market area analysis. If the producer
does not encounter competition from other producers in serving con-
sumers, to all of whom he quotes an identical factory price, his
market area takes the familiar shape of a circle where consumers of like
tastes and income are uniformly scattered over a plain of even
topography (provided they are willing to incur the costs of transport
inputs). Unevenness of consumer spread, inequalities in effective
purchasing power and differences of tastes among consumers, irregu-

eventually to provide such an understanding. (See Bogue, op. cit.; McKenzie,
op. cit.; and A. Hawley, Human Ecology, New York, 1950.)

146 LOCATION AND SPACE-ECONOMY

larities in geographic feature, economies of scale in transport, and a
host of other factors distort the "natural" circular regularity. How-
ever, the essential condition that the market boundary be a locus
of consumers who are just willing to pay for the first unit of product
a price which is equal to the factory price plus transport cost to the
point of consumption still obtains. ^

Introduction of a competitor producing the identical commodity
alters the condition in the area in which competition is in force.
Where both producers set the same factory price, effective for all
consumers, and where the freight rate is invariant with direction,
being a function of weight and distance only, the boundary separating
the consumers served by each producer is the perpendicular bisector of
the straight line joining the two producers. (For example, see the
boundary line ZV of Fig. 28 which separates the markets of the
two producers at A and B.) Only this perpendicular bisector yields
a locus of points of equal delivered price. In the districts where
competition is absent, the previous condition for the determination of
the boundary line for each producer still obtains.

In the equally familiar case where one producer establishes
identically for all consumers a factory price lower than the other, the
locus of points of equal delivered price in the area of competition
becomes an hyperbola. (For example, see the boundary line which
separates the market areas tributary to production points L and Mj
in Fig. 45 in Chap. 11.) In the more unusual circumstance where
both producers charge one and the same factory price but where the
product of one producer bears a higher transport rate than the product
of the other, the market of the former ultimately becomes enclosed
by the market of the latter (provided, of course, that the area of
effective consumption extends far enough in the geographic hinter-
land of the former) ; the market of the latter is limited in its outer
extents only by the area of effective consumption. Finally, where
inequality in factory prices as well as in transport rates obtains, the
market area of the producer upon whose product the higher transport
rate applies ultimately becomes contained by the market area of the

5 Since consumers are not typically distributed so that one is at every con-
ceivable point on a plain, the market boundary must necessarily cut through
certain stretches of the plain where it does not pass through possible points of
consumption. In these stretches its course is somewhat indeterminate, being
restricted only by the condition that it enclose those consumers who are more
than willing to pay for the first unit of product the factory price plus transport
cost and that it exclude those consumers who are unwilling to do so. Because of
differences of incomes and tastes and other factors, enclaves of "excluded" con-
sumers may come to exist within a producer's general market territory.

MARKET AND SUPPLY AREA ANALYSIS 147

other producer, again provided that the area of effective consumption
extends far enough in the geographic hinterland of the former. ^

At this point we pause to consider how the market area analysis thus
far developed can be formulated in a substitution framework. It has
already been noted that the market area of the firm which is not con-
fronted by competitors assumes a circular form under "uniformity"
conditions with respect to terrain, the geographic scatter, income and
tastes of consumers, and other factors. Viewed from the firm's stand-
point, for any given size of output it is always profitable, when the set
of consumers it initially serves does not approximate a circular terri-
tory, to substitute transport inputs in one direction for transport inputs
in another direction and to continue doing so until an approximation
to a circular market area is attained. The substitution is effected
simply by the curtailment of sales to the most distant consumers and
the extension of sales to new, less distant consumers (transport rates
being a simple function of weight and distance and invariant with
direction) . Viewed from society's standpoint, any given size of output
can be distributed with less average cost to consumers, and hence
with less effort devoted to transportation, when under the above speci-
fied situation a non-circular market territory is transformed into a
circular one through the substitution of transport inputs in one direc-
tion for transport inputs in another direction. Or alternatively (and
again under our simple assumptions) , given a fixed amount of labor
and resources devoted to both production and transportation, the path
toward maximum social output and consumption of goods involves the
reshaping of a non-circular market area into a circular one through the
substitution of transport inputs in one direction for transport inputs in
another direction.

The introduction of a competitor, as already indicated, establishes
another type of boundary line, a locus of points of equal delivered price.
A boundary consisting of any other set of connected points implies, in
our simplified model, a social inefficiency. If such a boundary exists,
it becomes economic for certain consumers to shift their buying from
one producer to another. In effect society is thereby substituting trans-

6 Where pi and P2 are the two factory prices and ri and ra the respective freight
rates on the product, the competitive boundary is defined by the equation:
Pi + nsi = P2 + ^2S2 where si and S2 are the respective distances from the
two factory locations to any given point on the boundary line. This equation can
be expressed in the form: si — hs2 = ±k, where h = r2/ri; +k = (p2 — Pi) /ri
when (p2 — Pi) / n '> 0 ; and —k= (p2 — Pi)/ri when (p2 — Px)lr\ < 0. This
equation describes a family of indifference curves which the Hysons have called
hypercircles and which comprise half the family of Descartes' ovals. See C. D.
and W. P. Hyson, op. cit.

148 LOCATION AND SPACE-ECONOMY

port inputs on the product of one producer for transport inputs on the
product of the second producer and should continue to do so until a
boundary comprising points of equal delivered price is attained. At
the same time society is also substituting production outlays by one
producer for production outlays by the second producer.

Where conditions of constant cost prevail, the marginal rate of sub-
stitution of production outlays by one producer for production outlays
by the second producer is also a constant. On the other hand, the
transfer of consumers from one producer's market to that of the second,
as the dividing boundary line is gradually shifted, entails a changing
marginal rate of substitution between transport inputs of the two pro-
ducers in serving consumers on the common boundary line.''' It is
through this changing marginal rate of substitution between transport
inputs of the two producers that we reach the partial equilibrium situa-
tion defined by a locus of points where the difference in transport
outlays of the two producers equals the difference in their production
outlays. When conditions of constant cost do not prevail, it is through
the changing marginal rates of substitution both between transport
inputs and between production outlays that we attain the desired
locus of points.

Hoover has aptly portrayed the competitive situation for two pro-
ducers with the use of graphs.^ In Fig. 27 let a producer be located at
A. Consumers are posited to be arrayed along line AB. If the pro-
ducer at A were to supply the needs of consumers at A only, his mar-
ginal costs would be AK. However, other consumers not at A may wish
to purchase from the producer at A. If A's market area extends as far as
L, his marginal costs, owing to economies of scale, fall to AJ, assuming
that the factory price he charges is equal to marginal costs. ^ Adding
transport cost to the consumers along the stretch AL yields a delivered
price line JG whose vertical height at any point is the delivered price
to the corresponding consumer along stretch AL. JG is also a transport
gradient line since it indicates how transport cost on a unit of product
rises as distance from A increases. i<>

"^ It is implicitly assumed that each producer charges the identical factory price
to all consumers and that his factory price equals his unit production cost.

8 Hoover, op. cit., Chap. II, particularly Fig. 7.

9 If the factory price is equal to average unit cost, the factory price would be
higher, sales smaller, and marginal costs higher; also, the market area would be
more Umited if the consumer at L were just willing to pay a delivered price equal
to AH{= GL).

10 Hoover, op. cit., pp. 8-11. Where irregularities in rate structure exist, these
will be reflected in the transport gradient line.

The transport gradient line of Fig. 27 is constructed to portray a rate structure

MARKET AND SUPPLY AREA ANALYSIS

149

If A's market area is now enlarged to reach as far as M, sales under
marginal cost pricing correspond to that output at which marginal costs
fall to AE. At this output marginal costs are at a minimum, and the
derived delivered price (transport gradient) line is accordingly at its
lowest level (where factory price is taken to equal marginal costs). ^

/s

\

f

/

^v /
u

^ ^^" *'

L M NX R

Fig. 27. Margin lines: two competitors.

B

For all other outputs, marginal costs (factory price) are higher and
the delivered price line higher. For example, if A's market is enlarged
to AN ^ marginal costs rise to AD, and the transport gradient line starts
from point D and necessarily parallels EF at a higher level. The
delivered price to the consumer at A^ is AD plus transport cost, or
in toto CN.

If points G, F, and C are connected with other points, each of which
by its vertical height represents for a given size market area the deliv-
ered price to the consumer on the boundary line, the curve KGFCS is

proportional to weight and distance. If the rate structure is graduated and less
than proportional to distance, the transport gradient from J would rise continu-
ously but curve downward.

11 Obviously, under a different pricing system, sales would be of a different mag-
nitude, and marginal costs would not be at a minimum when the edge of the
producer's market is at M.

150 LOCATION AND SPACE-ECONOMY

obtained; Hoover has designated this a margin line. It indicates how
delivered price at the edge of the market varies with the geographic
extent of the market. It is evident that the margin line changes in
form as the basis of pricing at the factory changes. ^^

It is fruitful to point out that, in accordance with his market area
theory, Losch would maintain that the stretch of the margin line from
K through G, F, and C to P corresponds to his natural market area
when competition is absent and when the margin line is based upon
factory prices which are equated to average unit costs. i^ Within this
stretch, the economies of scale (including the advantages and disad-
vantages of specialization) outweigh the diseconomy of transport cost.
In contrast, beyond point P, the economies of scale (which owing to
rising marginal cost are less than KQ) fail to match transport costs
(which exceed KQ). Consumers beyond R either produce for them-
selves or purchase from another producer, i^

We may now introduce a second producer situated at B. If he con-
fronts the same set of conditions as A — similar cost curves, types and
spatial spread of consumers, transport rates — and if he, too, pursues
an average cost pricing system (as is postulated for A in the preceding
paragraph), his margin line for serving consumers along the stretch
AB will be identical to ^'s. His margin line (TUV) intersects A's at

12 Conceivably, the producer's marginal cost and average cost curves could be
superimposed upon Fig. 27 if the pricing system and consumer demand curves
were stipulated beforehand. Points on the horizontal axis would not only indicate
distance from A but also quantity of output which would be purchased were the
edge of the market at each point.

It is clear that if the pricing sj^stem is changed, the quantity purchased in each
size of market area would also change; and correspondingly the shape and form
of any marginal or average cost curve which might be superimposed on Fig. 27
would change. This would be so even though, in orthodox fashion, such a mar-
ginal or average cost curve is taken to be independent of price and a function of
output only. Likewise, the margin line would shift since, for each size of market
area, total output and hence marginal or average cost and the transport gradient
starting point along the vertical axis would change (provided, of course, we retain
the postulate that factory price is in some way related to costs).

If the margin Hne of Fig. 27 were based on a factory price equal to average
cost rather than marginal cost, it would lie above the line KGFCS in Fig. 27 in
the initial stretch and would remain above it so long as marginal costs were below
average costs. It would intersect this line were the edge of the market extended
sufficiently to call forth an output at which marginal cost equals average cost. At
this output the extent of the market would be identical under both a marginal cost
and average cost pricing system.

13 Though the margin line of Fig. 27 was constructed upon a marginal cost
pricing system, we assume here and in the following discussion that it was con-
structed upon an average cost pricing system.

14 Losch, op. cit., pp. 71-74.

MARKET AND SUPPLY AREA ANALYSIS 151

W, which corresponds to point X on the straight line ABA^ It may
then be said that at X the delivered prices from the two producers are
identical. Consumers to the left of X purchase from A because they
bear less transport cost in the delivery of the product from A than
from B. For a similar reason consumers to the right of X purchase
from B.

Had B been located farther to the right so that the two margin lines
would have intersected at a point to the right of P (rather than to the
left), A and B would be non-competitive. i^ As already indicated,
beyond P consumers in the Losch scheme find it preferable to produce
for themselves at a cost of AK rather than to purchase from A at a
delivered price exceeding AK. Likewise, if any of these consumers
are at the same time to the left of the new position of point V, they
find it preferable to produce for themselves rather than to purchase
from B. Hence, we would have had a situation (temporary in the
Loschian scheme) where each producer was a monopolist within his own
natural market area and where each unserved consumer lying between
these two market areas produced for his own needs, i''

We return to the situation depicted by Fig. 27 and to our substitution
framework. As soon as B becomes effective as a producer, consumers
along XR shift their purchasing from A to B. In doing so, they are
substituting transport inputs on the product from B for transport inputs
on the product from A. They are also substituting production outlays
by B for production outlays by A. In this case, because of the sym-
metry assumptions, they are substituting lower production outlays
by B for higher production outlays by A. In the more general situa-
tion, the marginal production outlays by B might be either smaller or
greater than those by A. However, if they are greater, the transport
inputs on product from A must exceed those on product from B to
these consumers by a still greater amount if these consumers are to
shift their purchasing allegiance.

In this analysis the transition from a market area which is a single
straight line (as when in Losch's scheme the y co-ordinate of every
consumer's position takes the value of zero) to one which comprises
any number of straight lines radiating in all directions from each of
any set of focal points (where the x and y co-ordinates of consumers
may take all values) is easily effected. Imagine A and B as two focal

15 Given our assumptions, X is necessarily the mid-point of AB.

16 Diagrammatically, V would be to the left of P (rather than to the right of P
as in Fig. 27) and X would be to the left of R (rather than to the right of R in
Fig. 27).

1''^ In Fig. 27, AR would be A's market area; and beyond R to the edge of B's
market area, consumers would be self-sufficient.

152

LOCATION AND SPACE-ECONOMY

points lying on the a;-axis. Radiating from each is a straight line at
an angle a (less than 90°) from A and at an angle 180° — a from B.
See Fig. 28. Again let Losch's uniformity assumptions be adopted. If
as a logical consequence A and B serve the same number and kinds of
consumers in their respective market areas except for consumers along

Fig. 28. The division of a spatial market: two competitors.

lines AW and BW, it also follows that B serves all the consumers
along line BW and that A serves all the consumers along line AW,
where AW = BW. For, if A were to serve a consumer on line BW (but
not at W) , A's delivery price to this consumer would be greater than
J5's.is Society would deem it desirable for this consumer to shift his
allegiance from A to B, and thereby for him to substitute transport
inputs on the product from B for transport inputs on the product from
A and marginal production outlays at B for marginal production
outlays at A, even though the latter might be somewhat smaller.

IS Except perhaps in rare instances where the marginal cost curve behaves in
unusual fashion.

MARKET AND SUPPLY AREA ANALYSIS 153

For all values of a defined by the competitive stretch ZV (see Fig.
28) the same can be demonstrated. In short, the competitive boundary
line between A and B is, as already indicated earlier in this chapter,
a perpendicular bisector of the straight line connecting A and B and is
a locus of points equidistant from A and B.'^^ If we follow Losch
and permit complete freedom of entry and exit for many producers and
impose the restraint that all consumers be served by at least one
producer, we obtain boundary lines which must divide large regions
into identical and regularly shaped market areas in order to be con-
sistent with our uniformity assumptions. Only equilateral triangles,
squares, and regular hexagons satisfy these requirements. And Losch
has demonstrated algebraically that the division of any large region
into regular hexagonal market areas is more efficient than a division
into either equilateral triangular or square market areas. ^o Put in
another way, it is efficient for society to substitute among various
sets of transport inputs and marginal production outlays in order to
proceed to a pattern of regular hexagons from any other pattern of
regularly or irregularly shaped market areas. 21

Hence the Losch scheme of a net of regularly shaped hexagons is a
logical outgrowth of the simple Fetter-Launhardt approach when ad-
ditional assumptions and restraints are introduced. As a consequence,
too, it can be described in terms of simple substitution relations.

When it is additionally assumed that the production, sales, and
price of any commodity are independent of the production, sales,
and price of any other, and when it is recognized that differences
among commodities in applicable transport rates and economies of
scale will cause different sizes of regular hexagons to characterize the
market areas of various commodities, a system of nets of hexagonal
market areas similar to Losch's becomes a logical derivation. (For
example, see Fig. 51 in Chap. 11.) However, it does not seem fruitful
for one to pursue analysis on a multicommodity basis at this level
of extreme simplification. He would need to relax the assumption
of the independence of the production, sales, and prices of the several
commodities. In addition to economies of scale he would need to
recognize other economies of agglomeration, whether they be localiza-
tion, urbanization, or other forms of juxtaposition economies. He would
need to eliminate among others the inconsistency between the set
of uniformity assumptions, particularly with reference to population

19 For additional details see Hoover, Location Theory . . . , o-p. cit., Chaps. 2
and 3.

20 Losch, op. cit., pp. 76-78.

21 For a more rigorous demonstration, see Chap. 10, Sect. 4.

154 LOCATION AND SPACE-ECONOMY

distribution, and the hierarchical pattern of concentrations of economic
activities which results from the superimposition upon one another
of nets of hexagonal market areas. This is especially so when they
are ordered around a common core as Losch is inclined to do. If one
were to overcome these several obstacles, the Losch multicommodity
framework embodied in a system of nets of market areas would be
a logical point of departure for the pursuit of regional analysis. Since
the assumptions of the Losch framework have most relevance to service
activities (where the pull of raw materials tends to be minor) and
least relevance to basic industry oriented wholly or partially to
localized raw materials, the Losch approach is most pertinent for the
study of highly urbanized regions in which service activities dominate
the economic structure.

2. Supply (Purchasing) Area Analysis 2 2

The previous section posited that raw materials were essentially
ubiquitous, available at every potential factory site at the same cost.
As already intimated, this assumption cannot be tolerated for the
case of wholly or partially resource-oriented industry. We must con-
sider the theoretical significance of quantitative and qualitative in-
equalities in the spatial distribution of raw material deposits.

It is clear that when there are many scattered factories (manu-
facturers) to be served by relatively few sources of a raw material,
the analysis of the preceding section applies. Each of the many
factories (manufacturers) can be viewed as a consumer of the raw
material, and each of the raw material sources as a production point.
The problem is to define boundary lines which delineate the market
area composed of industrial consumers to be served by each pro-
ducing raw material source, where different extraction or production
costs may obtain at the several raw material sources. 2 3 For example,
in Fig. 46 of Chap. 11 assumed conditions lead to an hyperbolic
boundary line dividing the market areas of M^ and M^', two sources
of a first raw material, and to a straight line boundary line separating
the market areas of M2 and ilf 2'; 'two sources of a second raw material.

22 The term supply area or purchasing area is used here to indicate the geo-
graphic area from which a raw material is furnished to a producer. It normally
consists of many sites which produce the raw material. It does not refer to the
geographic area of consumers whose purchases of a given commodity are sup-
plied by a specified factory. This latter area has already been designated a
market area.

23 This is the case for which Hoover initially develops his margin line concept.
Points A and B of Fig. 27 are sites of an extractive activity. Line AB represents
a locus of possible consumers, industrial or household. (Hoover, op. cit., pp. 11-16.)

MARKET AND SUPPLY AREA ANALYSIS 155

The case is different when each industrial consumer must procure
his raw material requirements from many sources. Here, one may
assert, we have market area analysis in reverse. And as with market
areas, analysis of supply areas is facilitated when we proceed from
the simple to the more complex situations.

Imagine an industrial consumer who does not confront competition
in procuring his raw material from many potential sources of supply.
If these sources are uniformly scattered in a plain, if each can yield
the raw material at the same constant unit cost, and if no single source
can furnish the full amount of the raw material demanded at a price
which is equal to unit cost, the industrial consumer's supply area will
tend to be circular. ^ 4 Since the delivered price of the marginal unit of
raw material procured rises as the radius of the supply area increases,
the supply area is limited by the condition that the delivered price
be consistent from a profit standpoint with the price at which the
marginal unit of output can be marketed. ^ 5

Even in the more realistic case where increasing costs are encountered
in supplying the raw material, the industrial consumer's supply area
tend-s to be circular. The delivered price to the industrial consumer
of the marginal unit from each source of raw material supply utilized
will be the same. The difference in marginal costs for any two
sources will equal the difference in the unit transport costs borne
by these two sources. And, consequently, intensity in the utilization
of any source of raw material will fall off with increasing distance
from the point of industrial consumption. These latter conditions
are also fulfilled when the industrial consumer can be served by only
a relatively few sources. ^ 6 Needless to say, if these conditions are

24 The supply area reduces to a point when each source can furnish the full
amount of raw material demanded.

25 The industrial consumer procures the maximum possible quantity of raw
material from any given source before purchasing from another, more distant
source. Hence, each source lying within the circular supply area tends to be fully
exploited and to generate locational rent.

26 When there are only a relatively few sources each of which operates under
different cost conditions, we can graphically depict the situation with a modified
Hoover-type diagram. In Fig. 29 point T is a site of industrial consumption.
Points A, B, and C are raw material sources which need not be along a straight
line from T but which are, respectively, AT, BT, and CT distance from T. At
raw material source A, curve aa is a traditional supply curve representing the dif-
ferent quantities of the raw material which would be forthcoming at A at different
net prices, where net price is measured along a vertical hne passing through A
parallel to TL and where quantity is measured along AT. At raw material source
B, the corresponding supply curve is bb, where net pi'ice is measured along a ver-
tical line passing through B parallel to TL and where quantity is measured along
line BT. At raw material source C, the corresponding supply curve is cc.

156

LOCATION AND SPACE-ECONOMY

Since transport costs are incurred in moving the raw material to point T, the
price at T necessary to ehcit the production of any quantity of raw material at
A, B, or C must exceed the corresponding price at A, B, or C by an amount equal
to the cost of shipping a unit from A, B, or C. In the above figure the dash-dot
transport gradient lines are constructed to indicate transport costs on a unit of
raw material from sources A, B, and C to T. Hence, to the industrial consumer
at T, the supply curve aa appears as a' a'; the price at a' a' associated with any
given quantity of raw material exceeds the corresponding price at aa by an amount
equal to transport costs which are represented by the vertical rise in the corre-

A B C T

Fig. 29. The division of a market among several supply points.

sponding dash-dot transport gradient line. Likewise, to the industrial consumer
at T, bb appears as b'b' and cc as c'c'. Since for any price established at T the
several quantities available from the three raw material sources are additive, we
obtain at T the aggregate supply curve b'def. If the industrial consumer's derived
demand curve for the raw material is DD, the equilibrium price and aggregate
quantity are TS and SR, respectively. A, B, and C furnish respectively SE, SF,
and SG quantities of the raw material {SR = SE + SF + SG).

With this diagram it can be easily demonstrated how shifts of demand, different
transport rates, increases and decreases in transport tariffs, and changes in other
elements affect the quantities produced at, and commodity flows stemming from,
each raw material source.

This type of diagram can also be employed when one raw material source or
producer, say at T, serves several industrial or household consumers, say Sit A, B,
and C. At A, B, and C one constructs the respective demand cui-ves for the raw
material or commodity produced at T. Each of these demand curves can be
transformed into an effective demand curve at T through depressing its vertical
level by an amount equivalent to the cost of shipping the raw material or com-
modity from T to the corresponding point of consumption. In this case, the
dash-dot transport gradient lines are downward sloping. At point T one can

MARKET AND SUPPLY AREA ANALYSIS 157

not met (or if circularity of supply area is not realized in a con-
tinuous potential supply area),^^ both society and the industrial con-
sumer will find it desirable to substitute among the various transport
inputs and among the various marginal production outlays associated
with the several or many sources until these conditions are fulfilled
(or circularity attained).

Where the industrial consumer confronts competition from other
industrial consumers in purchasing his raw materials, the boundary
lines separating the supply areas of the several industrial consumers
tend to be straight lines, given the Losch type of uniformity assump-
tions. In contrast, when transport rates on raw material are differen-
tiated according to the point of termination, or when prices paid
at these points of termination by the several industrial producers
differ, or both, boundary lines take the general form of hypercircles,
which in special instances degenerate into circles, hyperbolas, and, as
already discussed, straight lines. ^ 8

If these types of boundary lines do not obtain in the situations
alluded to, once again it becomes economically feasible for society
to shift raw material sources from the supply hinterland of one in-
dustrial consumer to the supply hinterland of another, and thereby
to substitute among the relevant transport inputs and among the
relevant marginal production outlays. It is not necessary to develop
this point in detail and with the use of figures. The reasoning and
graphic analysis of the preceding section, appropriately refashioned,
apply here.

construct an aggregate demand curve by adding together horizontalwise the several
reduced demand curves. One can also construct a supply curve and proceed in
traditional fashion to determine the equilibrium price and quantity and the
apportionment of the total equilibrium quantity among the several consumers.
Again, one can easily demonstrate how a shift of the supply curve, different
transport rates, and changes in tariff structure can affect the realized pattern of
consumption and commodity flows from T.

In the situation where both consumers and producers are at different distances
from a point T to which all output is transported for sale and from which deUv-
eries to consumers are made, we can construct both the aggregate demand curve
and the aggregate supply curve which obtain at T. Proceeding as above, we can
determine market price at T, net price to each producer, delivered price to each
consumer, the distribution of output among consumers, the allocation of sales
among producers, and commodity flows to and from T.

2'^ Where a continuous potential supply area encompasses subsections in which
extraction costs differ, the circularity restriction must be relaxed. Also, where
different transport rates apply in different directions, circularity must be expressed
in terms of economic distance and not physical distance.

28 Hyson, op. cit.

158 LOCATION AND SPACE-ECONOMY

In a pure abstract sense one could even proceed to the analysis of
systems of supply areas in a way analogous to the manner in which
Losch molds systems of market areas. But in reality, physical
space restrictions essentially preclude this. It is possible to associate
with any particular point a consumer who purchases a number of
products and accordingly is served by a number of producers. How-
ever, for the most part it is not physically possible to associate with
any particular point the production of more than one raw material
or type of raw material mix. This fact raises the vital problem
of competition in the use of land and logically leads to the Thiinen
type of agricultural location theory and to land rent analysis, which
will be discussed later. Furthermore, many raw materials such as
coal and bauxite are highly localized. In these cases the Weberian
framework, as already discussed in Chap. 5 and as extended in later
chapters, is particularly applicable rather than a Loschian system of

3. Some Remarks on Spatial Pricing Systems and
Competitive Locational Equilibrium

Hitherto we have avoided discussion of spatial pricing systems and
competition. For the most part we have assumed that either (1)
the firm establishes a price at the factory on the basis of cost or
some other objectively given consideration; or (2) the firm has a
negligible influence upon a ruling market price and accepts it as a
datum. This latter situation obtains where a small firm and a large
number of competitors are concentrated at one point, which is mostly
a theoretical possibility, or where competitors' markets are concen-
trated at a point, as is the case with groups of farmers. In any case,
our firm has not been concerned with the reactions of competitors.

Losch has demonstrated that for the more usual case in the space-
economy of reality the above assumptions are invalid. Even in the
pure situation, where there are many independent producers with
complete freedom of entry and exit, with complete knowledge on
technology and markets, and with equal access to inputs, each
firm has at least from the standpoint of classical economics a limited
control over the market price it realizes. Because of the spatial
spread of consumers and producers characteristic of reality, the
demand curve for the individual firm's product, save for exceptional
instances, is not a horizontal line of infinite elasticity at the price de-
termined by the intersection of the demand and supply curves for
the industry, as it is under the traditional pure competition. Rather,
the friction of distance imparts to each producer a limited monopolistic

MARKET AND SUPPLY AREA ANALYSIS 159

position with regard to consumers closer to him transportation-wise
than to other producers. The demand curve for the firm's product,
as Losch has neatly portrayed, ^ 9 has a negative slope. We should
therefore pursue locational analysis on the second of the four levels
discussed in the early part of Chap. 5.

If reactions by competitors can be ignored, we may view the firm
as weighing, for each possible location, the net revenue potentials
from different possible sets of prices and outputs on the one hand and
the related sets of outlays on transport and other inputs on the other.
For each possible location the firm selects that price (or an array of
prices in the case of price discrimination) which, in view of its esti-
mate of the aggregate demand curve for its product (or of each in-
dividual consumer's demand curve), yields maximum net revenue.
Since the firm is in a position to control prices at least to a limited
extent and does tend to associate different prices with different outputs,
its relevant price-ratio and iso-revenue-less-outlay lines are no longer
straight as in Fig. 24 of Chap. 5 and Fig. 26 of Chap. 6. Rather
they become convex and concave curves or sets of points. ^o None-
theless, the firm's calculations can be set forth and expressed in terms
of equilibrium conditions.

Having determined for each possible location the maximum net
revenue, the firm selects among locations. In choosing one rather
than another, the firm substitutes among potential revenues, outlays,
and transport inputs. The conditions of locational equilibrium, which
are satisfied by the location yielding the greatest net revenue of all,
can be stated in formal terms (if the reader so desires) with the
use of transformation lines, outlay-substitution lines, revenue-outlay
substitution lines, price-ratio lines, iso-outlay lines, iso-revenue lines,
and iso-revenue-less-outlay lines, ^i

29 Losch, op. cit., pp. 70-74. Losch calculates for a given factory price the
amount which will be taken by each consumer along a straight line from the fac-
tory. At a lower (higher) price each consumer will take more (less), and the
extent of the firm's market will be lengthened (shortened) to include more (fewer)
consumers, whether or not at the expense of competitors. Hence, the aggregate
demand curve for the firm's product with reference to consumers along the line
is negatively sloping. Deriving the demand curve with reference to consumers
along every other line which can radiate from the factory site and summing yields
the total demand curve for the firm's product in the entire spatial market area.
This, too, will be negatively sloping, whether or not we follow Losch in postulating
an even spatial spread of consumers of like tastes and income, a homogeneous
transport network, absence of competition, etc.

30 Each pair of consecutive points may be connected by a straight line. The
resulting straight lines would have a different slope.

31 It is implicitly assumed that the firm's operations have only a neghgible effect

160 LOCATION AND SPACE-ECONOMY

Once we admit the essential reality of a firm's ability to influence
prices, we are logically compelled to recognize that price changes by
one firm frequently provoke retaliatory measures by other firms. A
firm then, in seeking to maximize its profits and to locate at the most
desirable site, in many instances must take cognizance of the possible
reactions of other firms, and must select its location, set its price,
and determine its output after considering not only the direct conse-
quences but also the possible indirect repercussions from the reactions
of other firms. Unfortunately, our theoretical schemata for under-
standing the complex oligopolistic situations of reality, despite recent
major advances scored by the theory of games and related studies,
are rather inadequate, particularly as they pertain to locational
equilibrium. We still have not developed what might be considered
a pattern of rational behavior on the part of the firm in response to
possible reactions of other firms, their indirect repercussions, and
the many uncertainties which becloud the problem. Nonetheless, it
is of value to have in mind some of the more important methodology
and achievements of those who have struggled with this problem in
its locational aspects. This allows a more comprehensive perspective of
the space-economy of reality and a finer appreciation of the magnitude
of the assignment.

We have already noted several times that Launhardt presented the
earliest systematic treatment of the division of a market area among
competing firms. However, in his analysis, the spatial co-ordinates of
the firm were given; only variations in output and prices were con-
sidered. Fetter's attention likewise centered on firms having fixed
locations; and Englander did not explicitly analyze the combined
effects of variations in both price and location in expounding his
doctrine of the "local conditionality of demand and supply. "^^
Hotelling was the first to allow price and location to vary simul-
taneously. He used a simple case primarily designed to elucidate

upon the local and regional income stream. Where regional income is influenced
significantly by a firm's location, as in the case of the Fairless Steelworks and the
Delaware River economy, then forces are generated which in turn affect price
and output of the firm, as well as the prices of various other inputs such as labor,
power, and transport. These indirect and cumulative effects are, at least in the
traditional economic sense, beyond the pale of individual firm analysis and are
judiciously relegated to the categoiy of regional analysis. They clearly point up
the close interrelation between firm and regional analysis.

32 "Kritisches und Positives . . . "op. cit. A general, though incomplete, mathe-
matical formulation of the problem of price policy for two or more spatially fixed
competitors is given by E. Schneider, "Observations on a Theory of Space Econ-
omy," Econometrica, Vol. 3 (1935), pp. 79-105.

MARKET AND SUPPLY AREA ANALYSIS 161

the imperfect competitive conditions of reality. ^^ Palander has justly
christened the following as the Laimhardt-Hotelling problem.

Hotelling's simplification pictures two firms each producing a given
quantity of a commodity under constant cost conditions and each com-
peting for a market stretched along a line of length I. Corresponding
to each unit length of line, one unit quantity of commodity is con-
sumed during each unit of time. The demand from each unit of the
line is infinitely inelastic. Customers' purchases are influenced only
by price — factory price plus transport cost; customers are indifferent
to all other circumstances surrounding the act of selling. Firms A
and B are located at distances a and b from their respective ends of
the line and at distances x and y respectively from a point at which
the delivered prices from A and B are the same. First, Hotelling
determines the set of equilibrium factory prices for A and B in terms of
a, b, I, and c (the cost of transport per unit quantity, per unit dis-
tance) where costs of production are zero, assuming that neither
producer attempts to drive the other out of the market by undercutting.
He comes to consider the case, among others, where producer A is
spatially immobilized and producer B is free to locate wherever he
wishes. B proceeds on the assumption that A will not change his
price in reaction to B's price or location, that is, that A trades
autonomously. B therefore finds it profitable to locate as close to A
as other conditions permit, on the side of A facing the more extensive
market. 3 4 This solution of Hotelling's is also borne out by Zeuthen's
graphic analysis. ^ 5

Both Hotelling and Zeuthen consider another type of price policy.
Instead of sharing the market under the best possible terms, com-
petitors can aim to eliminate one another by undercutting. This
course of action becomes the more likely the closer the two com-
petitors are to each other. ^^ 5 realizes maximum profits, given ^'s
price, when he sets his price so that his own delivered price will be

33 H. Hotelling, "Stability in Competition," Economic Journal, Vol. 39 (March
1929), pp. 41-57.

34 If B were approaching A on the short side of the market stretch, a position
as close as possible to A on the short side would not represent a maximum nor a
stable location. B would find it profitable to skip over A, as it were, and locate
as close as possible on the other side.

35 F. Zeuthen, "Theoretical Remarks on Price Policy : Hotelling's Case with
Variations," Quarterly Journal of Economics, Vol. 47, pp. 231-53.

36 Where the two firms come to be and remain fixed at locations infinitesimally
close, we have a problem similar to the traditional one treated by Cournot, Amo-
roso, Edgeworth, and others, i.e., of two firms producing like products and located
at a one-point market center.

162 LOCATION AND SPACE-ECONOMY

just below ^'s delivered price at all points in A's hinterland when
the quantity he offers corresponds to the total market demand. 3''

Where both firms are free to move, each adopting a policy of sharing
the market, and where also the same given factory price is charged
by each, we have a stable solution. Both firms locate at the center
of the market line or, more rigorously, one at the center, the other
adjacent to the first on either side. ^ 8 For, if either firm were to serve
a shorter stretch (that is, one noticeably less than one-half of the
market) with factory prices given and equal and the other assump-
tions still valid, that firm could improve its situation by skipping
over to the other side of its competitor, and thus usurping the larger
market stretch. Where three or more firms serve the market line,
each spatially mobile, then under the above simplified postulates
Chamberlin reasons that there will not be a tendency for concentra-
tion at the market center, as Hotelling implies. Rather there will be
a dispersion, the dispersion being least when sellers are grouped in pairs:

Taking the length of the line as unity, the general conclusion for n sellers
is that the space between the last sellers at either end and the ends of the

line would never exceed 1/n (if the number of sellers is odd it can never

1 \ v

exceed ) , and that the space between any two sellers can never

n + 1 /

exceed 2/n, this limit being reached only in the extreme case where

sellers are grouped by twos.^^

Anyone entering the market or any firm considering a new site,
operating under the assumption that the positions of all of its com-
petitors are fixed, will locate either adjacent to that firm which
services the longest market stretch on one of its sides or, in certain
likely circumstances, at a point on the longest stretch between any
two consecutively placed competitors.

Palander, eschewing certain of Hotelling's, Zeuthen's, and Chamber-
lin's assumptions, penetrates still further this problem of spatial com-
petition. First, he shows that Hotelling's basic solution for two firms
trading autonomously is not generally valid. A stable equilibrium is
reached only where the two firms are far apart and have small hinter-
lands. Otherwise, a constant fluctuation of price takes place within

37 Apropos the applicability of conclusions relevant for a market of fixed linear
extent to a market of unlimited areal character, refer to Losch (op. cit., 1st ed.,
1940, pp. 12-14).

38 See E. Chamberlin, The Theory of Monopolistic Competition, Cambridge,
Mass., 1938, 3rd ed., Appendix C.

39 Ibid., p. 209.

MARKET AND SUPPLY AREA ANALYSIS 163

limits, and price may even be cut to the level of costs (constant)
if the firms lie very close to each other. The solution for two firms
where one is free to locate is thus inherently unstable. ^^

Second, Palander generalizes the problem to apply when firm B, in
selecting a new location, can choose (1) to eliminate its competitor
from the market by undercutting; (2) to share the intermediate market
with its competitor; or (3) by raising its price, to confine its sales to
consumers in its own hinterland, that is, to adopt a policy of hinterland
defense. A¥here B trades autonomously, as A does, then B adopts either
(1) a policy of eliminating A from the market in which case he locates
right next to A and slightly undercuts A's price; or (2) a policy of
hinterland defense, in which case he may locate adjacent to .A, or as
far away as the mid-point on the largest market stretch — this depend-
ing upon A^s price and location. ^i In these circumstances, B would
never find it advantageous to share the intermediate market with A.
Where B trades ''superpolitisch," i.e., is aware of A's path of reaction
to his (B's) choice of location and price, he will pursue a different but
determinate course. He will locate relatively far away from A. Where
both A and B trade superpolitisch, there results an express tendency
for deglomeration.

Lerner and Singer, ^ 2 somewhat more realistic, set an upper limit to
the price which each buyer is willing to pay for one unit of the com-
modity (that is, limit the stretch of infinite inelasticity on the demand
curve of each consumer) ;4 3 and they have suggested analytic methods
for the case where a demand with some elasticity is postulated for each
consumer. As a consequence their conclusions at times differ signifi-
cantly from those of the above authors. To a certain extent, too, they
examine more comprehensively the range of conditions, patterns of

40 Palander, op. cit.. Chap. IX, especially p. 248. Furthermore, Palander finds
a logical inconsistency at the base of Hotelling's agglomeration tendency. Hotel-
ling postulates that firm B trades autonomously with respect to price (assumes
that .4's price will not change) yet, in selecting the most desirable location, some-
how comes to possess knowledge of the final equilibrium price that should result
from mutual price adaptations by A and B (ibid., p. 251).

"^1 When A's price is relatively high and his position relatively near the center
of the market, B will tend to eliminate A from the market. When ^'s price is
relatively low and he lies relatively near one of the ends of the market line, B
will tend to locate at some distance from A and set a price so high that he will
just be able to retain the consumers lying in his own hinterland. (Ibid., pp.
391-93.)

42 A. P. Lerner and H. W. Singer, "Some Notes on Duopoly and Spatial Com-
petition," Journal of Political Economy, Vol. 45 (1937), pp. 145-86.

43 See also in this connection, A. Robinson, "A Problem in the Theory of Indus-
trial Location," Economic Journal, Vol. 51 (June-Sept. 1941), pp. 270-75.

164 LOCATION AND SPACE-ECONOMY

reaction, and possible solutions that are inherent within the problem.
For example, they consider an alternative to the short-sighted policy
where two mobile competitors undercut each other, being primarily
concerned with short-run gains. Each producer assumes that his
competitor will not respond to the sort of encroachment where the
former shifts location — so long as his encroachment is limited to one-
half of his competitor's customers — but that his competitor will, on the
other hand, reciprocate undercutting of price with undercutting. This
assumption yields the result that A and B may be located adjacent to
one another or at various distances apart, depending upon transport
cost, size of market, and the upper limit to price.**

Smithies* 5 has gone farther and discarded the postulate of an inelas-
tic demand for each consumer. He examines the case where identical
linear demand functions exist at every point on the linear market and
accordingly arrives at solutions different from those deduced by others.
For example, the solution to the full competition case where both firms
are mobile and each assumes, in any move, that the other's price and
location are given, is not that one which follows Hotelling's line of
reasoning. It does not conclude with the two firms adjacent — a solu-
tion which others have suggested when special conditions obtain. In-
stead, the two firms remain apart although less than half the linear
market separates them. This is so because each producer is compelled
to weigh the gains from further encroachment upon his competitor as
he moves closer to the market center against the loss of consumer
patronage in his own hinterland from greater freight charges.* ^

■*^ In addition, Lemer and Singer have treated the question of spatial price dis-
crimination in some detail, a question which Hotelling, Zeuthen, and Palander
had already recognized to different extents. Hoover, too, has made a number of
astute observations on the problem (E. M. Hoover, Jr., "Spatial Price Discrimina-
tion," The Review of Economic Studies, Vol. IV, No. 3, pp. 182-91). Others, such
as A. Smithies ["Monopolistic Price Policy in a Spatial Market," Econometrica,
Vol. 9 (1941), pp. 63-73], E. Schneider, in several articles, and H. Moller ["Grund-
lagen einer Theorie der regionalen Preisdifferenzierung," Weltwirtschajtliches
Archiv, Bd. 58 (1943), pp. 335-90] have been concerned with the question either
in a monopolistic or oligopolistic setting.

Price discrimination, which tends to be most expedient within spatial markets
where distance and other geographic obstacles enable the producer to deal sepa-
rately with the various sectors of his market, offers to the individual firm oppor-
tunities for additional profits. Obviously as one proceeds from a monopolistic to
an oligopolistic situation, these opportunities become limited, the more so as the
number of competitors increases and their locations draw nearer to that of the
given firm.

45 A. Smithies, "Optimum Location in Spatial Competition," Journal of Political
Economy, Vol. 44 (June 1941), pp. 423-39.

46 Smithies has also considered a more generalized set of conjectural hypotheses

MARKET AND SUPPLY AREA ANALYSIS 165

Ackley4 7 examines still more realistic market conditions in spatial
competition. He analyzes a number of cases where there is a discon-
tinuous distribution of customers with different demand functions, that
is, cases of spatially discrete demand where the quantity sold by either
seller is a discontinuous function of his own and his rival's prices and
locations. He shows clearly that no precise generalized solutions
emerge, even when rigid assumptions are made as to competitor's be-
havior. The solution of each specific discrete case needs to be worked
out anew under various assumptions regarding competitor's reactions.
Often the very type of market discontinuity conditions the type of
assumption sellers make as to their competitor's conduct. In striving
for maximum profits the sellers do not necessarily confront any less
determinate or more unstable situations than where a continuous
spatial market exists.'* ^

Finally, we should mention a generalized approach being developed
in connection with game theory, an analytical advance which has
received its initial and principal stimulus from the work of von
Neumann and Morgenstern.^s Game theory pertains to situations

for each competitor. In addition to the full competition case cited, he examines
the cases where (1) "Each competitor in making an adjustment assumes that his
rival will set a price equal to his own and will adopt a location symmetrical with
his own" {ibid., p. 427), and (2) "each competitor assumes that his rival will have
the same price reactions as above but will keep his location unchanged" {ibid.,
p. 427). The case where each firm assumes that his competitor will not react, if
he (the competitor) is cut out of the market entirely by his rival, is discarded as
fantastic.

In addition, Smithies investigates somewhat the effect upon the final equilibrium
relationship of changes in marginal cost of one or both firms.

47 G. Ackley, "Spatial Competition in a Discontinuous Market," Quarterly
Journal of Economics, Vol. 56 (February 1942), pp. 212-30.

48 Mbller {op. cit.), following H. von Stackelberg's approach, discusses at length
the problem of stability of equilibrium under various regional competitive and
price-setting situations.

49 J. von Neumann and 0. Morgenstern, Theory of Games and Economic Be-
havior, Princeton, 1944. Also refer to the excellent and somewhat complementary
expository reviews: J. Marschak, "Neumann's and Morgenstern 's New Approach
to Static Economics," The Journal of Political Economy, Vol. LIV (April 1946),
pp. 97-115; L. Hurwicz, "The Theory of Economic Behavior," The American
Economic Review, Vol. XXXV (December 1945), pp. 909-925; and C. Kaysen,
"A Revolution in Economic Theory," Review of Economic Studies, Vol. XIV,
No. 35 (1946-47), pp. 1-15. A recent survey of game theory is contained in R.
Duncan Luce and Howard Raiffa, A Survey of the Theory of Games, Behavioral
Models Project, Columbia University, 1954, hectographed. Also see John Nash,
"Two-Person Cooperative Games" and J. P. Mayberry, J. F. Nash, and M. Shubik,
"A Comparison of Treatments of a Duopoly Situation," Econometrica, Vol. 21
(January 1953), pp. 12&-40 and 141-54, respectively.

166 LOCATION AND SPACE-ECONOMY

of interest conflict; it has relevance to the above locational equilibrium
problem where a firm, either individually or in collusion with others,
competes with other firms or coalitions of firms in serving a given
consumer market.

Basic to game theory in its current form are certain postulates. It
is posited that the variables within a given situation are well specified
and that the values which they may take and the possible outcomes of
the situation can be precisely characterized. (In our spatial equi-
librium problem the behavior of competitors need not be postulated as
invariant, or as varying within certain limits, but can be considered
an unrestricted variable.) Individuals are assumed to be completely
informed about the physical characteristics of the given situation and
to be "able to perform all the statistical, mathematical, etc. operations
which this knowledge makes possible."50 Further, they are able, either
directly or indirectly, to assign to each possible outcome a numerical
utility, for all practical purposes a money value, which in coalition
activity must be transferable.

Assuming that each individual (producer) desires to maximize utility
(gains), von Neumann and Morgenstern define rational behavior of an
individual as the choice of that strategy which permits him the best
of all possible minima, that is, the maximum of the minima. This
follows since he knows that his competitors (viewed as a coalition if
we wish to consider the individual competitor alone) in attempting to
maximize their gains will minimize his own. On the other hand, the
coalition of competitors, knowing that its rival will tend to maximize
his gains, will pursue a strategy which permits its rival the least of all
maxima, that is, the minimum of the maxima. When both parties
pursue their respective policies simultaneously, it is conceivable that
a relatively simple stable solution will be arrived at; it would be a
"saddle-point" solution where the maximum of the minima coincides
with the minimum of the maxima. In the usual case a saddle-point
will not exist. However, the authors have abstractly demonstrated
that when each party pursues a course of "mixed strategies," that is,
chooses several strategies and assigns definite probabilities to each,
then a solution will always exist. The competitive struggle between
an individual firm and a coalition of rivals can thus be resolved. By
a similar reasoning process the individual competitor may find it profit-
able to ally himself with others and be part of a coalition.

50 Neumann and Morgenstern, op. cit., p. 30. This assumption does not specify
"perfect information" on the part of all competitors. The rules of the game "may
explicitly prescribe that certain participants should not possess certain pieces of
information" (p. 30 note).

MARKET AND SUPPLY AREA ANALYSIS 167

In addition to confronting the conceptual complications of collusive
action and the tremendous problem of empirical verification in a situa-
tion where the variety and complexity of solutions are overwhelming,
game theorists do labor under some very unrealistic assumptions. It
is difficult to accept the assigning of complex probabilities to various
courses of action as characteristic of man's behavior in the competitive
struggle. It is perhaps even more difficult to accept the assumption
of complete knowledge in a very involved situation when experience
teaches us that any human being is far more restricted in his per-
ceptions. Moreover, empirical studies do show a great variation in
individual abilities which runs counter to von Neumann's and Mor-
genstern's condition that all rivals are equally capable of drawing
inferences from given amounts of information. ^i (Nor does the usual
businessman concede the point that his competitors are his equal.
Rather, it is an everyday observation that businessmen strive to outwit
their competitors, being convinced of their own superiority.) Lastly,
in a spatial competition setting, von Neumann and Morgenstern's ap-
proach would overemphasize the dependence of any individual's choice
of strategy upon his competitors' reactions. In a social exchange econ-
omy, geographic separation of rivals acts as insulation from reactions
and in many instances simplifies the problem of maximization.

Despite these difficulties and shortcomings, the contributions in the
area of game theory represent an initial major achievement. As game
theory develops it will undoubtedly cast considerable light upon a host
of basic problems as well as the locational equilibrium problem. ^ 2

51 However, J. Marschak {op. cit.), after presenting some simple illustrations,
concludes : "it seems to us that properly stated differences in degrees of knowledge
or intelligence of individual players can also be regarded as rules of the game"
(p. 106).

52 Allied to the locational equilibrium problem of this section are the contribu-
tions of S. Enke ["Equilibrium among Spatially Separated Markets: Solution by
Electric Analogue," Econometrica, Vol. 19 (January 1951), pp. 40-47], P. Samuel-
son ["Spatial Price Equilibrium and Linear Programming," American Economic
Review, Vol. XLII (June 1952), pp. 283-303], M. Beckmann ["A Continuous
Model of Transportation," Econometrica, Vol. 20 (October 1952), pp. 643-660, and
"The Partial Equilibrium of a Continuous Spatial Market," Weltwirischaftliches
Archiv, Bd. 71 (1953), Heft 1, pp. 73-89], K. A. Fox ["A Spatial Equilibrium
Model of the Livestock-Feed Economy in the United States," Econometrica, Vol.
21 (October 1953), pp. 547-566], and others on spatial price equilibrium, especially
as related to transportation flow patterns.

In one of its simplest forms Enke poses the problem as follows :
"There are three regions trading a homogeneous good. Each region consti-
tutes a single and distinct market. The regions of each possible pair of regions
are separated — but not isolated — by a transportation cost per physical unit
which is independent of volume. There are no legal restrictions to limit the

168 LOCATION AND SPACE-ECONOMY

actions of the profit-seeking traders in each region. For each region the
functions which relate local production and local use to local price are known,
and consequently the magnitude of the difference which will be exported or
imported at each local price is also known. Given these trade functions and
transportation costs, we wish to ascertain:

(1) the net price in each region,

(2) the quantity of exports or imports for each region,

(3) which regions export, import, or do neither,

(4) the aggregate trade in the commodity,

(5) the volume and direction of trade between each possible pair of regions
..." {o-p. cit., p. 41).

Viewed in this way the problem is essentially a transportation problem and not a
basic location problem. Enke demonstrates how an electric analog can be
employed to derive a solution to this problem.

As Samuelson has shown, the Enke problem contains within it the following
Koopmans-Hitchcock minimum transport cost problem : "A specified total number
of (empty or ballast) ships is to be sent out from each of a number of ports. They
are to be allocated among a number of other receiving ports, with the total sent in
to each such port being specified. If we are given the unit costs of shipment
between every two ports, how can we minimize the total costs of the program?"
[Samuelson, op. cit., p. 284. For full discussion of this problem see T. C. Koop-
mans, "Optimum Utilization of the Transportation System," Econometrica, Vol.
17, Supplement (July 1949), pp. 136-146; and T. C. Koopmans and S. Reiter, "A
Model of Transportation," Chap. XIV in Activity Analysis of Production and
Allocation, ed. by T. C. Koopmans, John Wiley & Sons, Inc., 1951.] It should
also be noted that the Enke problem contains within it a bit of the location prob-
lem. For the Enke problem determines the scale of output in each given region
(the Koopmans-Hitchcock problem implicitly assumes that the scale is given).

Enke has not confined himself to only three regions. His analog solution is
proposed as applicable to a problem embracing many regions. Samuelson has
also probed the many-region problem, and Beckmann has gone even further and
considered the case of "continuous geographical intensity distributions of produc-
tion," i.e., where every infinitesimally small area in an economy which can consist
of many regions both produces and consumes a commodity.

If excess supply functions could be derived for each infinitesimally small area
of the world and if the Samuelson-Beckmann formulation could be considered
relevant and adequate and could yield a quantitative solution, then the location
problem would be solved. Corresponding to each infinitesimally small area, there
would be a unique scale of output (zero or positive amount of production), such
as Enke obtains for each region in his more limited model. We would have our
geographic distribution of production. Theoretically, both the location and
transportation patterns would have been derived simultaneously.

In practice, however, the Samuelson-Beckmann formulation ignores a number
of basic locational forces, as Beckmann fully recognizes, and more important is
not now able, and is not likely in the future to be able, to yield a quantitative
solution for every infinitesimally small area. It is at this juncture that location
theory makes its contribution. For location theory seeks principles to narrow
down, and greatly narrow down, the number of points to be considered as potential
locations for the production of any given commodity. Once a relatively small
number of production points or regions are isolated, the Enke-Samuelson-Beck-

MARKET AND SUPPLY AREA ANALYSIS 169

4. Concluding Remarks

In bringing this chapter to a close we should fully appreciate the
progress which still needs to be achieved to understand rational be-
havior for the individual firm, even under simplified cost assumptions.
When more realistic cost conditions are introduced, when geographic
mobility of the firm is permitted not only along a line as in most of the
discussion of the preceding section but also within a geographic area,
and when the uneven areal distribution of consumer demand is recog-
nized and different pricing policies are allowed, still greater progress
is required.

From the standpoint of enabling one to reach precise results, the
market and supply area analyses discussed in the first two sections of
this chapter are more satisfactory than the locational equilibrium
analysis following Hotelling's approach, even when the latter is sup-
plemented by game theory. Market and supply area analyses achieve
these more precise results through postulating a relatively simple prob-
lem and through abstracting from, among other factors, competitors'
reactions, pricing policy as a variable, and for the most part locational
mobility.

In contrast, the locational equilibrium approach stemming from
Hotelling's work is much more sophisticated in its consideration of
reactions and mobility. However, this approach yields results only
within a very restrictive framework. It almost completely ignores the
cost side of the picture and the inequalities in the spatial distribution
of natural and human resources. Apart from the fact that entrepre-
neurial ability and organization and scale of output may vary from
firm to firm and thus cause each firm to face a different cost situation,
which in turn affects each one's competitive policy, it is generally true
that production and distribution costs will not be the same for all sites.
Access to raw material sources and power facilities, transport relations
with consumers, availability of skilled and unskilled labor, labor organ-
ization, external economies from association with other industries, taxes
and other social burdens, political conditions, relevant geographic fea-
tures such as bed-rock conditions for power plants and soil for farming,
capital supply, markets for by-products, opportunities for waste dis-

mann formulation may offer a more efficient approach to the determination of the
resulting geographic flows of commodities (e.g., see Fox, op. dt.).

However, in a second volume on principles of regional science we hope to be
able to demonstrate how the activity analysis approach can further regional theory
and thereby our understanding of industrial location.

170 LOCATION AND SPACE-ECONOMY

posal, etc., do vary from site to site and give rise to significant cost
differentials. Herein lies probably the most serious weakness of the
Hotelling approach.

Though the Hotelling approach does not yield as precise results as
do market and supply area analyses, it can nonetheless be cloaked in
a formal substitution framework for most specific situations where
adequate assumptions are made about the behavior of such variables
as competitors' reactions, price policies, and cost functions. For exam-
ple, consider the first of Hotelling's problems discussed above. In
terms of outlay-substitution lines, revenue-outlay substitution lines,
and iso-revenue-less-outlay lines which relate to transport outlays and
commodity revenue, in moving toward A's location, B substitutes trans-
port outlays in one direction for transport outlays in another direction.
(This corresponds to a shift along an outlay-substitution line.) B is
at the same time proceeding along a revenue-outlay substitution line
on to iso-revenue-less-outlay lines of higher and higher order (see Fig.
26, Chap. 6). When B finally comes as close as possible to A, on the
side of A facing the more extensive market, his corresponding position
on the revenue-outlay substitution line rests on an iso-revenue-less-
outlay line higher in order than any other iso-revenue-less-outlay line
with which the substitution line has a point in common. This point
corresponds to maximum profits and stability, given Hotelling's
assumptions. 5 3

As already indicated game theory promises to furnish additional
insights into the Hotelling locational equilibrium (interdependence)
problem. When this is achieved, there still remains the task of inte-
grating the Hotelling approach with the type of market and supply
area analyses presented in the earlier sections of this chapter and with
the Weberian doctrine reformulated in the preceding and later chap-
es Or consider a more complex situation in which both processing costs and
transport costs on raw materials are variables. Along the horizontal axis of a
relevant graph (once again see Fig. 26, Chap. 6), we would measure transport plus
processing outlays. Along the vertical axis we would measure product revenue.
B would then shift his location until he reached a site which would correspond
to that point on his revenue-outlay substitution line which lies on the highest
iso-revenue-less-outlay line. However, in this case the revenue-outlay substitu-
tion Hne would in turn be associated not only with a substitution line between
transport outlays on product in one direction and transport outlays on product in
the opposite direction but also with a second substitution line between transport
outlays on product and transport outlays on raw material and a third substitution
line between total transport outlays and processing outlays.

For an interesting set of graphs which illuminates this problem see Greenhut,
"Integrating the Leading Theories . . ." op. cit.

MARKET AND SUPPLY AREA ANALYSIS 171

ters.54 Even allowing for progress on both these scores, we must still
grapple with other variables which basically condition industrial loca-
tion and regional development. Several of these are introduced into
the analysis in the succeeding chapter.

54 Greenhut, in his several articles, op. cit., has explored and probed considerably
into this problem.

Chapter

8

Agglomeration Analysis and
Agricultural Location Theory

In his classic work on location theory, Alfred Weber emphasizes
three basic location forces. ^ Two of these, transport cost differentials
and labor cost differentials, interplay to determine the regional dis-
tribution of industries. We have already treated these two forces
among others in the preceding chapters. The third general location
factor, agglomeration (deglomeration) economies and diseconomies,
acts, according to Weber, to concentrate or disperse industries within
any given region. In our discussion thus far we have touched inciden-
tally upon this third locational factor. We now probe deeper.

Following Ohlin, Hoover has neatly classified agglomeration (de-
glomeration) factors as follows:

(a) Large-scale economies within a firm, consequent upon the enlarge-
ment of the firm's scale of production at one point.

(b) Localization economies for all firms in a single industry at a
single location, consequent upon the enlargement of the total output of
that industry at that location.

(c) Urbanization economies for all firms in all industries at a single
location, consequent upon the enlargement of the total economic size
(population, income, output, or wealth) of that location, for all industries
taken together.

Bearing this classification in mind, we can now examine how agglom-
eration theory can be reformulated in order to facilitate a more satis-

1 C. J. Friedrich, Alfred Weber's Theory of Location of Industries, University
of Chicago Press, Chicago, 1929.

172

AGGLOMERATION AND AGRICULTURAL LOCATION 173

factory integration with the previous substitution analysis and with
orthodox production theory.

1. Economies of Scale

As the first step let us investigate the influence of large-scale econo-
mies upon the location of production. Let us reconsider the case to
which Fig. 27 of the preceding chapter pertains. Two firms, A and B,
are competing for the market along the straight line connecting their

Fig. 30. A case of agglomeration from economies of scale.

factory sites. Their respective margin lines are KGWP and TUWV.
The delivered price of each to the consumer at X, when X is the mar-
ginal consumer, is the same. A comes to serve consumers lying along
the stretch AX; B serves those along the stretch XB.^

In this case the marginal costs of both firms are rising sufficiently
rapidly to make feasible a division of the market. Suppose, however,
that significant economies of scale extend over a much larger range of
production and that the margin lines of the two producers are as in
Fig. 30. ^'s margin line always lies below B's; no matter which con-
sumer is designated as marginal, A can deliver to him at a lower price
than B. A usurps the entire market and can do so because production
economies realized with increase of his output more than balance the
mounting transport cost disadvantage as more distant consumers are
served.

2 A la Hoover, it is assumed that a producer always serves first the nearer of
any two consumers.

174

LOCATION AND SPACE-ECONOMY

The case of Fig. 30 warrants concentration of production at A.^
When contrasted with Fig. 27 it neatly illustrates the impulse toward
agglomeration which stems from economies of scale. In another con-
nection, namely, in determining for each commodity the appropriate
size of an hexagonal market area, Losch has also developed this point. ^

Although the concentration of production at A (rather than a divi-
sion between A and B) entails an increase of over-all transport outlays,
it permits a still greater decrease in over-all production outlays. In
terms of our substitution framework, transport outlays are substituted
for production outlays. If we were to visualize one parent company
controlling the two subsidiary firms A and B, the parent company,
by concentrating production at A, would be shifting along an outlay-
substitution line and proceeding on to lower iso-outlay (transport plus
production outlay) lines. Viewed from society's standpoint, such con-
centration of production is desirable; it allows the production of any
given output at lower cost and thus releases resources for other pro-
duction and use.

The solution, however, does not necessarily rest with concentration

3 However, in other types of situations there may not be a clear-cut indication
of the location at which production should concentrate. Take, for example, the
case represented by Fig. 31. A and B have identical cost curves and confront
identical transport rate structures. (Consumers are uniformly distributed along

K

Fig. 31. A case of indeterminacy in location.

AB.) Economies of scale dictate concentration of production in one plant, but
such concentration is as feasible at 5 as at ^. In the traditional sense the solution
to this problem is indeterminate; and as Hoover has pointed out (op. cit., pp.
98-99), the location at which production is undertaken first is the location at which
production is more likely to concentrate. (The alternative of a division of the
market between the two sites involves instability.)
4 Losch, o-p. cit., pp. 70-85.

AGGLOMERATION AND AGRICULTURAL LOCATION 175

of production at A. There may be other sites lying between A and B
which would not labor at such high production costs as B must. Since
they would occasion less transport outlays than A in serving the total
array of consumers (except in extreme instances) , they must be con-
sidered as potential locations. If any of these intermediate sites can
produce over the relevant range of output at lower costs than A, obvi-
ously production should shift from A. If none can, then comparison
must be made between savings on transport outlays engendered by
any given shift from A and the corresponding increase in production
outlays. If a shift were found to be desirable, the producer (parent
company) would be proceeding along a revenue-outlay substitution line
(as well as along several outlay-substitution lines) on to a higher iso-
revenue-less-outlay line.^

The significant generalization which emerges from consideration of
cases of this sort is that in any location decision the scale of output is
one of several basic, interdependent variables. As scale varies, so may
the substitution points between any pair of transport outlays, between
any two sets of outlays, between outlays and revenues, and so forth. ^

5 Hoover's neat discussion {op. cit., pp. 99-104) of the factors governing the
location of marketing and other intermediary estabhshments presents another
variation on this theme. He focuses upon three basic elements: (1) costs of
transport (transfer) from the factory; (2) costs of operation of the intermediary
establishment; and (3) costs of transport (transfer) to the consumer. His diagram
illustrates how production (operation) outlays substitute for transport outlays as
one shifts the intermediary establishment to successive transport junction points
along the path from the factory to the consumer. At the same time it illustrates
the concomitant substitution of transport outlays (inputs) on the unprocessed
commodity for transport outlays (inputs) on the finished commodity.

6 In the case depicted by Fig. 30, we considered two firms, a given linear dis-
tribution of consumers, and a fixed pattern of raw material supply. The relation
between scale and location, however, can be illustrated in any number of ways.
To take another example suppose we consider a single firm to whom the market
and raw materials supply are variables, as well as scale. For a predetermined rate
of output, an optimum site may be identified with which is associated an equi-
librium relation between, let us say, transport outlays to the west and transport
outlays to the east. For a larger scale of output the resources from a different
source of a vital input may be required — a source, say, which lies further eastward
— because the other source is taxed to capacity, or because a step-up in output at
the original source involves such steeply rising costs that it becomes feasible to
commence exploiting a more distant source. Or, perhaps for a larger scale of
output, a larger supply area of raw materials is required, one more easily expanded
in the east because of certain natural conditions. Or, perhaps the larger market
area which the enterprise contemplates serving extends much farther to the east
than to the west of the original site. An obvious conclusion follows. Total trans-
port charges will increase, and if the site of production remains unchanged, most
of this increase will be accounted for by greater transport outlays to the east.

176 LOCATION AND SPACE-ECONOMY

This point is widely accepted and recognized, and reflects the fact that
the phase of agglomeration theory which treats economies of scale is
already embraced by existing production theory. It is easily incor-
porated into substitutional location analysis.

2. Localization Economies

A more controversial issue in agglomeration theory revolves around
the influence of localization economies. Weber raised this question
early. Without clearly distinguishing among the three different types
of agglomeration factors already noted, he asks under what conditions
and where several units of production will agglomerate. He provides
precise answers to these questions. Several individual units of produc-
tion will agglomerate when (in relation to any assumed unit of agglom-
eration): (1) their critical isodapanes''' intersect and (2) together

However, there will be a tendency for transport outlays to the west to be sub-
stituted for transport outlays to the east, that is, for the firm to shift its location
eastward in order to lower total transport outlays.

It should be observed that such a shift of site can involve a considerable spatial
jump. For example, until a certain size of output is reached, a dominant raw
material may be supplied by a single source so that the site of production is at
that source. With a larger output it may become feasible, as we have seen, to
utilize a second source of the raw material and thus to locate at an intermediate
site between the two sources. Or it may be that with the larger output the raw
material loses its dominance and production becomes market-oriented or is most
suitably located within the locational polygon of raw material sources and market
points. Or it is conceivable that up to a certain output production is essentially
transport-oriented, i.e., carried on at the minimum transport cost point. But with
a larger output a cheap labor location, or a cheap power location, etc., becomes
effective in attracting production to itself. Again all these conditions can be
formally presented in terms of substitution and transformation lines which embrace
all scales of output in order to point up the interrelation of scale and location.

It is also apparent that the question of the most efficient size of output to a
large extent depends on the manner in which sources of raw materials and other
inputs are exploitable and markets for finished goods available both in terms of
their spatial distribution and their quantitative importance. And in turn the
scale variable influences to a major degree the specific sources of raw materials
utilized and the specific markets served.

7 In this connection the critical isodapane for any unit of production is that
locus of points for each of which transport costs in assembhng the raw materials
and shipping the finished product exceed the corresponding transport costs asso-
ciated with the optimal transport point by a constant amount. This amount is
equal to the economies of agglomeration that would be realized by association
with the assumed unit of agglomeration.

See Weber for extensive discussion of the critical isodapane and of its depend-
ence upon locational weight, transport rates, the function of economy of agglom-
eration, and other variables.

AGGLOMERATION AND AGRICULTURAL LOCATION 177

they attain within the common segment the requisite quantity of
production.

Suppose three units of production, Pj, P2, and P3, each transport-
oriented, are located as in Fig. 32. Around each are drawn its locational

Fig. 32. Non-intersecting critical isodapanes: no agglomeration.

figure and critical isodapane. The critical isodapanes do not intersect.
Agglomeration is infeasible. In contrast stands the situation depicted
by Fig. 33 where these same three units are assumed initially to lie
closer to one another. Here, their critical isodapanes, the heavy
undashed circles, do intersect. (For the present, ignore the dashed
circles). A la Weber, agglomeration will take place at a site within
the common segment which is shaded.

Weber's determination of the center of agglomeration is as precise
as his statement of conditions under which agglomeration will occur.
The center of agglomeration "will be located at that one of the several

178

LOCATION AKD SPACE-ECONOMY

possible points of agglomeration which has the lowest transportation
costs in relation to the total agglomerated output.''^ This point is
derived by means of a locational figure and analysis of the equilibrium
of forces in much the same way as is the optimal transport point

Fig. 33. Intersecting critical isodapanes: agglomeration.

for any given unit of production. However, in the derivation of this
point, Weber permits the use of new sources of raw material supplies
(replacement deposits) for each unit of production.

Weber gives a precise answer also to the question of the size of the
unit of agglomeration to which each unit of production will be at-
tracted. Each unit of production will select that unit of agglomeration
whose center lies most distant from the relevant critical isodapane
of the given unit of production.

Weber's analysis is not unsophisticated. He does consider for each
unit of production a function of economy of agglomeration which
varies with the size of agglomeration. He admits exceptions to his

8 Friedrich, o-p. cit., p. 138.

AGGLOMERATION AND AGRICULTURAL LOCATION 179

conditions under which agglomeration will be precipitated. ^ He
emphasizes labor locations as centers of agglomerations, where both
cheap labor and agglomeration economies are obtainable, and intro-
duces various realities into his analysis. i<^ Nonetheless it must be
said that Weber's schema has limited application, especially in under-
standing the forces which determine the site at which agglomeration
obtains in actuality.

Imagine an entrepreneur who controls three units of production and
who confronts the location problem, de novo. Considering the locational
polygon of raw material sources and markets relevant for each unit
and assuming that economies of scale are not operative, he could
locate each unit at its optimal transport point. Or, he could locate
the three units adjacent to each other at a center of agglomeration,
thereby achieving localization economies but only by incurring larger
transportation costs. This is one type of situation to which Weber's
schema has most application. In this type of situation, each unit
of production may be visualized as substituting transport outlays for
production outlays of one sort or another when it shifts to the center
of agglomeration. And in this sense, that phase of agglomeration
theory which concerns localization economies could be integrated into
our substitutional framework just as we have integrated that phase of
agglomeration theory which concerns economies of scale. ^

^ For example, the critical isodapane of a given unit of production may not
quite reach the common segment formed by the intersection of the critical
isodapanes of other units. Nevertheless, if the given unit's production is neces-
sary for the group to attain the requisite total of production and if other units
would enjoy sizable economies from agglomeration, the given unit of production
can be induced to shift to the potential center of agglomeration by some form
of subsidy or side payment.

10 As with the analyses of other location factors, it is not our intention to present
here these more sophisticated aspects of Weber's analysis nor to study the
agglomeration factor in full. Rather, we touch upon it to the extent necessary
to integrate it with our general location analysis and other existing theories.
The reader is referred to other studies for more comprehensive and detailed
treatments of the agglomeration and deglomeration variables. He can easily
graft these more extensive treatments onto the analysis presented in this chapter.

11 Palander contends that where the several units of production are controlled
by one firm, the localization economies problem disappears. According to
Palander, the firm confronts a scale problem; it must determine the amounts of
production to engage in at various locations. In any case, however one views
the problem, the substitutional framework applies.

The reader should also bear in mind that as soon as Weber considers centers
of agglomeration where several units of production locate, his assumptions of
fixed raw material prices, transport rates, wages, and other costs are less valid
than when only one unit of production is associated with a given site.

180 LOCATION AND SPACE-ECONOMY

However, as Engliinder and Palander have rightly indicated in their
sharp criticism of Weber's agglomeration theory, this type of situation
is not widely characteristic of reality. Societal development is an
historical process. At any given point of time there exists an in-
herited physical structural framework. Plants have already been
erected and are producing. To relocate these plants involves oppor-
tunity costs since one would forego the use of facilities forced into
obsolescence. Critics of Weber have therefore emphasized the ad-
vantages of existing production points as centers of agglomeration,
whether they reflect labor or any other form of orientation. As new
units of production come into existence, they will tend to gain localiza-
tion economies by agglomerating around established production points.
Thereby they frequently strengthen the gravitational pull of these
points. From this standpoint, the evolutionary framework becomes
critical as a locational factor; and any pure substitutional theory
which is not linked to specific regional structure is of severely limited
significance.

Moreover, even if the opportunity costs of relocation could be
ignored and plants were completely mobile, the problem is not as
simple as Weber depicted. In shifting to a center of agglomeration, it
is to the advantage of each unit of production to deviate as little as
possible from its optimal transport site. At the same time, the
managers of these units of production differ in bargaining ability.
Therefore, it is to be expected that the center of agglomeration will
not be at the over-all minimum transport cost point of a new over-all
locational polygon; rather, it will tend to lie within the common
segment closer to the firms with greater bargaining ability. It could
even lie at a point outside the common segment if an appropriate set
of side payments were made to firms who could not otherwise be
induced to agglomerate. And, if costs of relocation are reintroduced
into the problem, the center of agglomeration could lie at the site of an
already existing production point. Since this would eliminate one
group of relocation costs, in many situations each unit of production
could be made better off through an appropriate set of side payments
than if all were to shift to Weber's over-all transport optimal point.

Clearly, game theory strikes at the heart of this latter type of
situation. The several participants are the several units of pro-
duction. Whether they be new units with whom no relocation costs
are associated or existing units confronted with relocation costs, they
interact engaging in various forms of collusive action. The bargain-
ing which ensues is complicated not only because of the innumerable
coalitions which are possible but also because of the different scales of

AGGLOMERATION AND AGRICULTURAL LOCATION 181

agglomeration which are potentially feasible for each unit of produc-
tion. 12 As with the Hotelling and similar locational equilibrium
problems examined in the previous chapter, progress in this phase of
agglomeration theory and its incorporation into existing forms of
analysis must await further development of game theory.

From an entirely different standpoint, however, Weber's agglomera-
tion theory may be justly defended. Suppose a new area is to be
opened for development by a governmental planning authority. Tech-
nological and other factors dictate, for any given commodity, the range
of feasible scales for the units of production. Should these units

12 To spell out somewhat more the way in which game theory pertains to this
phase of agglomeration theory, imagine there are three units of production
(parties) placed as in Fig. 33. Their critical isodapanes intersect with respect
to two sizes of agglomeration. (We already oversimplify the problem by con-
sidering only two sizes.) The critical isodapanes relevant for the smaller unit
of agglomeration are the dashed circles; those relevant for the larger unit are the
undashed circles. Any two parties could agglomerate to form the smaller unit
of agglomeration. The third party would consequently gain nothing. It is there-
fore to his advantage to encourage the formation of the larger unit of agglomera-
tion in which he could participate and from which he could reap gain. Leaving
aside the determination of which party is the third party, we encounter the
problem of identifying types of collusive actions which might develop. Whom
will the third party approach to form a coalition? To make an effective approach
he must offer a gain to the co-operating (second) party which will be greater than
what the latter obtains in the smaller unit of agglomeration. The third party may
offer a side payment. Or he may propose to agglomerate at a site closer to the
second party's initial location (optimal transport point) ; this proposal may, or
may not, be contingent upon the participation of the remaining (first) party.
Or the second party may be strong enough to force agglomeration at his own
optimal transport point, provided the first and third parties reap gain either
directly or indirectly through side payments. However, the first party cannot
be presumed to be an inactive participant. His power, like the power of any of
the other two parties, rests in the fact that without his co-operation the additional
gains of the larger unit of agglomeraion are not possible. He too has bargaining
power and can be presumed to exercise it.

Costs of relocation complicate the problem still more by altering the proba-
bilities of diverse moves. They significantly affect the range of collusive action.
Furthermore, the problem as presented is not a constant-sum game. As Weber
demonstrates, there is a center, the over-all optimal transport point, at which
agglomeration can proceed with a minimum addition to the sum of the transporta-
tion costs of all parties. Any deviation from this point reduces the "surplus" or
"net gain" to be apportioned among the participants. In certain situations it may
therefore be useful to introduce a fourth participant, a dummy, in order to
convert the problem into a constant-sum or zero-sum game. This entails further
complexities, as well as does any variation from the symmetrical situation
presented, such as with respect to initial geographic positions, size of output of
each unit of production, ability to relocate as measured by opportunity costs,
and so forth.

182 LOCATION AND SPACE-ECONOMY

be agglomerated to realize localization economies, or should they be
spatially disconnected in order to reduce transportation costs? From
this social welfare approach, irrationalities and differences among
managers in bargaining ability do not enter the problem. Nor do
inherited physical structures. The localization economies achievable
at Weber's over-all transport optimal point (and not at any other
point) must be compared with the additional transport outlays occa-
sioned by agglomeration at this point. Moreover, this social welfare
approach implicit in Weber, though not generally realistic, provides
a useful guidepost; in certain contexts it can indicate directions in
which existing structure should be transformed in order to approach
optimum resource utilization. Hence, from these standpoints, too,
the Weberian agglomeration theory is relevant, i^ and likewise the
substitutional locational framework within which it fits.

3. Urbanization Economies

The third phase of agglomeration theory, which concerns urbaniza-
tion economies, is in as unsatisfactory a state as that phase which
treats localization economies. This is to be expected to some extent
since the analysis of urbanization economies can be said: (1) to re-
semble, or (2) partially to evolve from, or (3) even to contain, accord-
ing to some persons, the analysis of localization economies, i-*

In the previous section we did not specify types of localization
economies (and diseconomies) which arise, such as those associated
with access to a larger pool of skilled labor, with fuller use of spe-
cialized and auxiliary industrial and repair facilities, with large-lot
buying and selling through common brokers and jobbers. We at-
tacked the problem as if these economies were known and furnished
us in the form of a function of economy of agglomeration (a la
Weber). In the study of urbanization economies we face all these
in a broader context as well as, among others: (1) economies which
stem from a higher level of use of the general apparata of an urban
structure (such as transportation facilities, gas and water mains, and
the like) and from a finer articulation of economic activities (daily,

13 Critics of Weber's agglomeration theory are too often inclined to forget that
in his Vber den Standort der Industrien Weber seeks a "pure" theory such as one
which is relevant for social planning rather than a "realistic" theory wherein
institutional forces are duly considered.

14 The discerning reader may have already concluded that in several respects
there is also only a fine line of distinction between localization economies and
economies of scale. He may have observed for example that the figures used in
connection with the discussion of economies of scale are also relevant, with
appropriate changes in initial premises, for inquiry into localization economies.

AGGLOMERATION AND AGRICULTURAL LOCATION 183

seasonally, and interindustrially) ; and (2) diseconomies engendered
by rises in the cost of living and money wages, in the costs of local
materials produced under conditions of diminishing returns, in time-
cost and other costs of transportation, and in land values and rents.
Consequently, we can theoretically employ the approach linked to
critical isodapanes, as developed in the preceding section, for the
investigation of the impact of at least some of these urbanization
economies and diseconomies. Here, the units of production need not
engage in the same type of activity.

In this general sense little can be added to the existing state of
analysis. It can be presumed that Weber's approach would have still
less application here than in the treatment of localization economies.
Cities evolve over time. They are much less subject to relocation
than are individual units of production. To put it another way, the
accumulated fixed investments of an urban mass in conjunction with
its vested social institutions entail major geographic immobilities and
rigidities and, for the most part, tend to preclude urban relocation.
Cities attract or repel units of production in accordance with the
urbanization (for the most part, external) economies or diseconomies
relevant to each unit of production. In this sense, one concludes
that agglomeration analysis, particularly that of the substitutional
variety, has little to say beyond the obvious; units are attracted to
or repelled from cities according to a simple comparison of advantages
and disadvantages generated by these cities.

However, it is instructive to pursue a tangential extension of the
above discussion. Once more we pose the following problem. A new
area is to be developed. Given a full knowledge of existing tech-
nology and likely changes in this technology, of the human and
natural resources of the area, and of other relevant relations and
materials, how should one plan a net of cities for this area? What
is the optimum spatial distribution and hierarchy of cities of different
sizes? Within each urban-metropolitan region what is the best spatial
distribution of different types of satellite cities and centers? Within
each city what is the most desirable constellation of community and
neighborhood sites of various sorts? In what ways should the inten-
sities of land use and traffic generation be controlled to be consistent
with an optimum structure of cities?

Closely allied with the above theoretical questions is a more
practical one. Given a network of cities and corresponding patterns
of land use, along what channels should changes in the structure of
this network and these patterns be fostered in order to attain a situa-
tion closer to optimum? Since cities are conglomerations of economic

184

LOCATION AND SPACE-ECONOMY

activities, in what directions should the joint geographic distribution
of economic activities be reshuffled when flexibility in the structure
exists?

At this point, only a meager beginning can be made at answers to
these basic questions, which have already been posed several times,
either implicitly or explicitly. To start exploratory analysis, let there
be given the economies of scale associated with every utility and
service which a city provides and with every other activity or service

1,000
350

10,000
3,750

100,000
40,000

Population
Kw capacity

Fig. 34. Economies of scale in power generation with urban size.

subject to urbanization economies or diseconomies. Let us focus first
upon the economies and diseconomies associated with the provision of
electric power. To do so we have constructed Fig. 34. Along the
vertical axis we measure net economies (economies less diseconomies)
in power generation. Along the horizontal axis we measure both scale
of output and the corresponding size of city which can absorb any scale
of output. We posit more or less representative cities for each of
which an approximate level of power output can be identified. (In
reality, of course, the power consumption of any city is a function
of many characteristics; and consequently the kilowatt-hour require-
ments of any two cities of the same size can differ widely.) We also
postulate that fuel of stated BTU quality is available to all sizes of
cities at a fixed price or at prices which are a function of quantity
of purchase but not of geographic position ;i5 and that for each scale of
output the most efficient power plant design is in effect. Excess capac-
ity beyond that necessitated by peak load requirement (which is

15 Hence, all cities are taken to be equally distant from a fuel source.

AGGLOMERATION AND AGRICULTURAL LOCATION 185

taken to exceed average load by a uniform percentage in all cities) does
not exist. 16

Given these assumptions, Fig. 34 indicates annual net economies
in power generation which would be realized if any given size of
population together with its associated industrial and commercial
activities were concentrated in one city rather than in cities of 1000
population each.^''' The solid curve refers to a situation in which
the cities of 1000 each would be too distant from one another to permit
service by one or a relatively few power stations. The dashed curve
refers to a situation in which the cities of 1000 each would be
sufficiently close to permit some integration in power production and
service. 18 Obviously, a set of curves can be constructed to depict
different degrees of integration which might be feasible.

For every other service and commodity whose production or cost
reflects urbanization economies and diseconomies, we can construct a
similar set of net economy curves when appropriate assumptions are
stipulated. One set would reflect in the early stage of each curve
the general economies which arise from access to larger and more
diversified pools of skilled labor, and in a later stage the diseconomies
in the use of labor which stem from internal congestion and inefficiency
(the journey-to-work problem), increases in money wages, and other
factors as the urban mass multiplies. Another set would reflect
economies and diseconomies in the operation of an urban transportation
system (including streets, rail and truck terminals, parking facilities
as well as bus, subway, street and electric railway, and other transit
media) or of a subset of the transportation system if an identifiable
subset of the transportation system can be meaningfully isolated for
study. In reality the size and character of urban transportation
systems vary with the spatial distribution of population, of economic,
cultural, and other activities within the city, with patterns of group
behavior and social organization, with topography and other geo-
graphic features, with the state of technology, and with many other
factors. However, we postulate that it is possible to associate at

18 Already the reader may seriously object to the set of postulates adopted
whereby cities are standardized in terms of power consumption, price of fuel,
and other factors.

1'^ In Fig. 34 we depict increasing per capita consumption of power in all uses
with increase in size of city in order to reflect the effect upon power consumption
of the lower power rates which larger cities tend to charge because of lower unit
power costs. The kilowatt capacity scale has specific reference to the solid curve.

18 Both curves fall off slightly after a certain size of city is reached because of
slowly mounting diseconomies in the co-ordination and management of larger and
larger power systems.

186 LOCATION AND SPACE-ECONOMY

least approximately a size and character of transportation system
with each size of city, and that the difficult problem of defining
a unit of transportation service can be surmounted. ^^ Each curve
in the resulting set of net economy curves would rise to a maximum and
then fall significantly as deglomerative forces, such as congestion and
co-ordination problems, grow in relative importance.

Still other sets of curves would depict economies and diseconomies in
the performance of various municipal functions: in the provision of
fire and police protection; in the administration and operation of an
educational system; in the construction and maintenance of gas, water,
and sanitation facilities; in the organization and supply of recreational
facilities and services; and in other activities. Here, too, as population
numbers increase, as congestion multiplies, as rents, land values, and
the costs of food supply rise, diseconomies mount in relative importance.
Imagine that somehow or other it is possible a priori to identify for
any large region, either already settled or about to be settled, that
curve in each set of net economy curves which tends to be most
relevant or representative for the situation under study. These curves
(one from each set) could be plotted as in Fig. 35, where only four of
them have been constructed. It is tempting to sum all these curves
and to interpret the resulting total curve as an over-all index of
economy or function of economy which defines the over-all urbaniza-
tion economies and diseconomies associated with cities of different
sizes. Unfortunately, this is not justifiable. There are many logical
objections to this procedure.

To reiterate a point already mentioned, the standardization of cities
is subject to serious criticism. There are no standard cities. Each
is unique. Furthermore, the selection of a relevant or representative
curve from any given set of net economy curves presents logical
difficulties of a somewhat similar nature. It is unnecessary to spell
out these fairly obvious points.

Additionally, there are a number of other objections of which at
least two should be explicitly mentioned. One involves the problem
of weighting. Summing the individual curves of Fig. 35 to derive
the total curve depicted thereon implies that the sets of economies are
of equal importance. Yet it is clearly evident that the relative im-
portance of any set of economies depends upon many characteristics
of a given situation; among others, industrial composition, income,

19 In effect, we standardize cities in terms of: (1) the ratio of industrial activity
to commercial and service activities; (2) industrial mix (heavy and light) and
composition of commercial and service activities; (3) land-use patterns; (4)
journey-to-work and commodity flow configurations; and many other relations.

AGGLOMERATION AND AGRICULTURAL LOCATION 187

culture and social organization, consumption patterns, and geographic
setting. In a situation where cities are, or are likely to be, charac-
terized by a heavy proportion of electro-process activities within
their basic industry sectors, greater significance should be attached
to the power economy curves, ceteris paribus. In another situation

Total economies

Transportation economies
__^JV— Labor economies

1,000 10,000 100,000 1,000,000 ^

Population

Fig. 35. Hypothetical economies of scale with urban size.

\

\

where general urban settlement is severely restricted by mountainous
slopes, ocean, and other barriers, the transportation economy curves
assume greater relative importance, ceteris paribus. Thus, some form
of weighting must be introduced into the problem in a valid analytical
manner.

Another objection, perhaps the most serious of all, stems from the
neglect of interdependence among the sets of net economy curves
(as well as interurban and urban-rural interrelations) . Economies in
urban transit are directly related to economies in power generation
since considerable power is consumed in street and electric railways

188 LOCATION AND SPACE-ECONOMY

and other transit facilities. In turn, economies in power generation
are directly related to economies in the operation of port and rail
facilities since a significant fraction of the delivered price of fuel
may consist of transportation charges. Economies and diseconomies
in the educational system, in the provision of fire and police protec-
tion, and in various other municipal services are also directly in-
fluenced by the character and efficiency of the transportation system.
Likewise, labor economies and diseconomies are related to a number
of other sets of economies and diseconomies. In short, it is erroneous
to consider the various economies and diseconomies associated with
the numerous urban activities as simply additive. Rather, they are
multiplicative in a complex fashion.

The above considerations are sufficient in themselves to invalidate
the use, even in an approximative fashion, of a simple total curve
or index of economies and diseconomies in the functioning of cities
of various sizes. The construction of Fig. 35 does not answer the basic
questions posed earlier in this section; nor does it cast significant
light upon the third phase of agglomeration theory, which is con-
cerned with urbanization economies and diseconomies. We are still
thrown back upon the simple statement that, with respect to each
firm, there are attracting and repelling forces for location in rural
areas and in cities of different sizes. When the potential savings of
a location shift override the additional costs involved, the firm will
shift. In doing so it will be substituting one set of outlays and
revenues for another set. (And, as already indicated, in the presenta-
tion of this problem the critical isodapane techniques, a la Weber, are
helpful only to a limited extent.)

Nonetheless, it seems worthwhile to explore further the above
approach in the analysis of urbanization economies and diseconomies.
In a future volume, which will place chief emphasis upon regional
analysis, it will be shown how within urban-metropolitan regions some
of the complex interdependence of the sets of net economy curves can
be understood and partially identified in quantitative terms. Yet, this
interdependence problem, together with the probleras of weighting
and of introducing flexibility and non-standardization into the ap-
proach, is of tremendous scope and requires a large amount of
additional research.

4. Agricultural Location Theory

As indicated at the beginning of Chap. 5, there is a traditional
dualism in location theory — viz., a Thiinen type of analysis for the
agricultural sphere and a Weberian scheme for the industrial. His-

AGGLOMERATION AND AGRICULTURAL LOCATION 189

torically, the former has confined itself to an aggregative analysis; the
latter primarily to individual firm analysis. Since these two schools
have been concerned with two different levels of inquiry, it might
appear, as it has to some in the past, that their doctrines are non-
integrable. It might seem illogical to discuss agricultural location
theory in a section of a chapter where the preceding sections have
centered primarily around the individual firm. However, we contend:

(1) that there is a smooth transition from individual firm analysis in
the Weberian sense to the analysis of the individual farm enterprise;

(2) that the analysis of the individual farm enterprise must investigate
the bonds which link the agricultural firm to the aggregate of agricul-
tural activities; and (3) that casting the locational analysis of the
individual farm enterprise against the background of locational forces
which interplay on the aggregative level in agriculture can throw at
least some light upon the way location forces which influence the
individual industrial enterprise are interlocked with those governing the
spatial distribution of industry on a more aggregative level. 20

The previous discussion has emphasized the distance variable.
Transport inputs and transport outlays have been given a focal position
in the theoretical scaffolding. Yet the internal spatial dimensions of
the firm, of industrial agglomerations, or of urban aggregates have
been ignored. On the surface, at least, to ignore them was justifiable
in individual firm analysis where it was implicitly assumed that the
firm was associated with a factory-type operation whose requirements
of land were relatively small. Differentials associated with the cost
of land services were taken to be minor as they typically are for most
industrial factories. In contrast this neglect is not justifiable for
industrial agglomerations, for cities, and for firms whose internal
spatial dimensions are relatively large. These latter firms are well
typified by agricultural enterprises.

In agricultural location theory, rent differentials have always
occupied a central position. Since the internal spatial extent of the
typical agricultural enterprise is relatively large, differences in the
price of land services associated with different spatial positions (as
well as resource content) are a major location influence. However,
to the agricultural enterprise, the rent differential plays the same
role as the labor cost differential to an enterprise in an industry which
is labor intensive, or the power cost differential to an enterprise in
an industry which is power intensive. In the case of the agricultural

20 The imperative need to attack the analysis of industrial agglomeration from
a more aggregative standpoint has already been pointed up in the preceding
sections.

190 LOCATION AND SPACE-ECONOMY

enterprise, it is critical to investigate the substitution relation between
rent outlays and transport outlays; in the case of textiles, it is im-
portant to examine the substitution relation between labor outlays and
transport outlays; in the ease of the aluminum industry, between
power outlays and transport outlays. In this sense, location theory
for agricultural firms is no different from that for industrial firms.
Comparison of cost differentials and investigation of substitution re-
lations among the several outlays are basic to both. The only signifi-
cant difference rests, it may be contended, in the fact that the small
character of the many agricultural enterprises cultivating most major
crops coupled with the concentration of markets at particular points
in general permits a deeper locational analysis of the agricultural
sphere than of the industrial. We therefore proceed to the discussion of
agricultural location theory as a logical extension of the general
analytical framework evolved in the preceding pages. 21

Consider the operator of an agricultural enterprise. Among others,
his decisions regarding the location of his enterprise (in terms of dis-
tance from the market), product (s) to be cultivated, factor proportions
to be used, and intensity of production are interrelated. Some of these
interrelations can be neatly demonstrated with the use of a set of graphs
developed by Dunn. 2 2

In Fig. 36 are presented a set of price and cost curves for the indi-
vidual farm operator. OE represents the price at the market for the
particular product he cultivates. OD represents the net farm price, i.e.,
market price less the cost to transport the unit of product to the market
which is indicated by ED. The solid curves, AC and MC, are respec-
tively his average and marginal cost curves; they exclude rent pay-
ments or are relevant when the price of land is taken to be zero.

If the price of land were zero, the operator would extend output to
the level, 01 1, at which level marginal costs equal net farm price
(marginal revenue) and where, ex definitione, gross marginal costs
(including cost of transport to the market) equal market price (gross

21 In this section, we shall treat only some of the basic elements of agricultural
location theory. For a full statement, see Edgar S. Dunn, The Location 0} Agri-
cultural Production, University of Florida Press, Gainesville, 1954; August Losch,
Die rdumliche Ordnung der Wirtschaft, G. Fischer, Jena, 1944, pp. 24-44; E. M.
Hoover, Jr., Location Theory and the Shoe and Leather Industries, Harvard Uni-
versity Press, Cambridge, Mass., 1937, Chap. 2; E. T. Benedict, et at., Theodor
Brinkman's Economics of the Farm Business, University of California Press,
Berkeley, 1935; F. Aeroboe, Allgemeine landwirtschajtsliche Betriebslehre, P.
Parey, Berlin, 1923, and J. H. von Thiinen, Der isolierte Staat in Beziehung auj
Landwirtschajt und Nationalokonomie, Hempel and Parey, Berlin, 1895.

22 Dunn, op. cit.. Chap. HI.

AGGLOMERATION AND AGRICULTURAL LOCATION 191

marginal revenue). Total surplus is represented by the area of the
rectangle, ABCD.

To supplement Fig. 36 one can construct Fig. 37 which focuses upon
other relations of this situation. Along the vertical and horizontal

'Market price

D

^marKct price j^ "^^ ^ ^^^„

Net farm price / /^' ,mC

L

I,

Output

Fig. 36. Price and cost curves of an agricultural enterprise.

axes we measure respectively amounts of land inputs and amounts of

a composite set of other inputs. Each of the curves /]

/.

repre-

sents an iso-product curve, ^ 3 and, in order, they relate to decreasing
levels of output. The dashed lines 0»Si, 0S)2 . . . OSn are scale lines. 2*

23 Each iso-product curve depicts the various combinations of the amounts of
land inputs and of the composite set of other inputs required to produce that
given level of output corresponding to the iso-product curve.

24 Each scale line is a locus of points indicating the combinations of the
amounts of land inputs and of the composite set of other inputs which involve
the least total outlay for the production of each level of output, for a given
specific price-ratio oj inputs.

192

LOCATION AND SPACE-ECONOMY

Since rent, the price of land services, is taken to be zero, the Hicksian
price-ratio (the iso-outlay or iso-cost) lines are vertical, such as line
VX. Given the level of output, OIi, to be produced (as determined
by Fig. 36), the equilibrium amounts of land and other inputs are given

^-Sn

100 200

600 700

300 400 500
Other inputs

Fig. 37. Input proportions, scale, and equilibrium for an agricultural enterprise.

by the point of tangency, P, of the iso-product curve Zj (which corre-
sponds to an output of 01 1) with the price-ratio line VX.

Dunn has discussed the inconsistency of the above derivation. The
surplus, ABCD (Fig. 36), when divided by the 30 units of land utilized
(as given by point P, Fig. 37) yields rent per unit of land. This rent
or price for the use of the services of land may be, in certain circum-
stances, an opportunity cost or, in other circumstances, an actual pay-
ment by the farm operator who leases the land in full competition

AGGLOMERATION AND AGRICULTURAL LOCATION 193

with other potential users. In any case, a price for the use of land
services exists, and vertical price-ratio lines like VX are incorrect. As
a first approximation, the rent of each of the 30 units of land under
use may be taken to be ABCD/30. This would yield a set of price-
ratio lines whose slopes are identical to that of line B^Ai. Accordingly,
U would be the new point of tangency of the Ii iso-product curve with
the relevant price-ratio line of the new set. It would be economic to
reduce land inputs and increase other inputs as indicated by the shift
along curve Ji from P to U, and thereby to substitute outlays on other
inputs for rent outlays.

However, if the quantities of land inputs and other inputs given by
point U were to be employed to produce 01^ of output, Dunn has
shown that the resulting rent per unit of land is greater than that
assumed by the price-ratio line BiA^. As a consequence, a new set
of price-ratio lines, with less steep slope, becomes relevant. Another
substitution of outlays on other inputs for rent outlays is justified.
This, in turn, generates a still greater rent per unit of land and makes
relevant another slope for the price-ratio lines. In Dunn's words: "a
series of successive substitution adjustments is made with the tangency
point moving down the iso-product line from [P to Q in Fig. 37] . Each
of these successive adjustments, however, yields a rapidly decreasing
increment to the rent per acre. When the point Q is reached, the rent
per acre cannot be increased any further by increasing the ratio of
the 'other' input factors to land. "^ 5

A second basic set of substitutions occurs simultaneously with these
adjustments. As rent is explicitly acknowledged as a cost, the marginal
and average cost curves rise. (Land inputs are a variable.) In Fig. 36,
the intersection of the new dashed marginal cost curve M'C^ in which
the rent is included as a cost, with the net farm price line suggests OI2
as the equilibrium output, rather than 01 1. This would correspond
to a shift from Q to R along scale line OS2 in Fig. 37. However, the
change of scale increases rent per unit land (when the enterprise is
operating under decreasing returns to scale), decreases the steepness
of the relevant set of price-ratio lines, and indicates a series of substi-
tutions which lead to point T on the I^ iso-product line. Once again
contraction of scale is warranted. Ultimately, according to Dunn, the
various scale contractions and the substitution of outlays on other
inputs for outlays on land inputs lead to point W where neither scale
adjustments nor outlay substitutions are prompted. 2 6 Corresponding

25 Dunn, op. cit., p. 30. The letters and number in brackets are mine.

26 The reader should bear in mind that the step-by-step set of adjustments
depicted in Fig. 37 is for illustrative purposes only. Both scale adjustments and

194 LOCATION AND SPACE-ECONOMY

to point W are the dot-dash marginal and average cost curves [M'^C"
and A''C"0 of Fig. 36 ; the average cost curve inclusive of rent is tangent
at its minimum point to the net farm price line. Given the spatial
position of the land to be used and the crop to he cultivated, point W
represents a stable equilibrium position for the agricultural enterprise.

Thus far, only part of the total problem confronting the agricultural
operator has been presented. Another part concerns his location.
Abstracting from changes in the resource content of the land, in posi-
tion relative to fixed transport facilities, in the tax, legal, and other
socio-economic and political frameworks which condition land use,^'''
we can depict the impact of a shift of location upon the operation of
an agricultural enterprise with the use of Fig. 36. In terms of economic
distance, let the agricultural enterprise shift closer to the market. Ac-
cordingly, its net farm price line rises, the transport cost to the market
being diminished. ^s A higher net farm price line would obviously
permit a greater rent per unit of land if inputs and output and all other
prices are constant. However, since the ratio of the price of a land
input to the price of a composite of other inputs rises, the slope of the
relevant price-ratio lines becomes less steep. A series of substitution
adjustments leading to still greater rent per unit of land and to a higher
ratio of other inputs to land inputs is warranted ; and consequently the
intensity with which a unit of land is cultivated increases.

Allow the agricultural enterprise to shift its location to a third site
still closer to the market and in succession to a fourth and a fifth. It
becomes evident that the closer the enterprise to the market the higher
the rent per unit of land and the greater the intensity at which it is
economically feasible to cultivate a unit of land ; and, the more distant
the enterprise from the market the less the rent which the operator
can afford to pay for a unit of land and the less the intensity with

substitutions can and do take place simultaneously. Many different kinds of paths
leading from the hypothetical point P to the equilibrium point W are possible.

27 AH these elements influence the firm's operations through their effect upon
its cost curves. Thus, if the resource content of the land deteriorates and taxes
rise as a firm shifts from one location to another, we can expect the relevant
average cost curve for the second location to course at a higher level than that
pertinent for the first location, ceteris paribus. Since the variation in the
influence of these elements does not exhibit a spatial regularity, a study of their
effects lies outside the scope of this volume, although for any particular situation
their pi'esence must be fully recognized. The reader interested in their effects
is referred, among others', to Brinkmann, op. cit.; Hoover, op. cit., pp. 22-33; and
Dunn, op. cit., Chap. 5.

28 The extent of this diminution would be given by the typical transport
gradient line over the relevant stretch.

AGGLOMERATION AND AGRICULTURAL LOCATION 195

which he cultivates it. If the rent potential per unit of land were
plotted for each possible location of the agricultural enterprise along
a line extending from the market, a rent function would be described
such as is depicted by curve A A on Fig. 38.^9

Distance from market

Fig. 38. Rent functions for different agricultural land uses.

It should be emphasized that any shift of the agricultural enterprise
involves the general substitution between rent outlays and all other
outlays combined. Of still greater relevance are certain subsidiary

29 Our rent function is identical with Hoover's "rent surface" and with Dunn's
"industry rent function." At times this rent function has been depicted as a
straight line (e.g. Losch, op. cit. pp. 25-40; and Dunn, op. cit., Chap. 2) when it
has been either implicitly or explicitly assumed that intensity of cultivation is
invariant with distance from the market. In these instances, the rent function is
given as: R = E{p — a) — Ejk where R is rent per unit of land, k (an inde-
pendent variable) is distance from the market, and E, p, a, and / (all constants
or parameters) are, respectively, yield per unit of land, market price, average
production cost, and transport rate (Dunn, op. cit., p. 7).

Because of its emphasis upon the firm as the core unit our formulation of the
problem precludes the use of a straight line rent function. A straight hne rent
function, although helpful for pedagogical purposes, is logically invalid.

196 LOCATION AND SPACE-ECONOMY

substitutions. Viewed from the standpoint of transport orientation
theory, the enterprise finds its transport optimal (minimum transport
cost) point at the market. As the enterprise moves away from the
market, it substitutes transport outlays for rent outlays, ceteris paribus;
and concomitantly, because the price of land becomes more moderate,
it substitutes rent outlays for other (excluding transport) outlays,
ceteris paribus.^^ This substitution represents a deviation from the
transport optimal point. This deviation is of the same order as the
deviation which occurs when, following traditional Weberian dogma,
a firm shifts from its transport optimal point to a cheap labor location.
Among others, this latter shift involves two basic substitutions: (1) a
substitution of transport outlays for labor outlays, ceteris paribus; and
(2) a substitution of labor outlays for other (excluding transport) out-
lays since the price of labor is less at the cheap labor location. The
only basic distinction between these two shifts, it can be maintained,
is that on the one hand the shift of the agricultural enterprise can be
a somewhat continuous one whilst on the other hand the shift of the
industrial firm to a cheap labor location typically involves a discrete
jump.

It should be borne in mind that any one farmer or agricultural
operator need not approach the location decision in the comprehensive
fashion depicted above. He need not be fully aware of the breadth
of the problem. His vision may be narrowly circumscribed. Yet he
behaves as if he were fully aware of the entire problem. This result
obtains because the relative freedom of entry into and exit from agri-
cultural production tends to force the farmer into efficient paths or to
weed him out if he persistently deviates from these paths, ^i Essen-
tially there corresponds to every site an appropriate set of character-
istics relating to size of enterprise, intensity of land use, and ratios
of factor inputs. This appropriate set yields, for each site, the maxi-
mum rent, all of which accrues to the landowner. The competitive
process wipes away any surplus profit which might befall an operator
at any site, and at the same time ensures "normal returns" or "normal
profits."32 Thus, theoretically, the farm operator is indifferent to the

30 A move in the opposite direction, let us say from no-rent land (re : the given
commodity) toward the market, involves a substitution of rent outlays for
transportation outlays, ceteris paribus, and a substitution of other (excluding
transport) outlays for rent outlays.

31 Unless, being a landowner, he can pursue inefficient practices at the expense
of economically justifiable rent.

32 If the operator makes an abnormal profit after allowance for unusual
entrepreneurial ability, etc., by implication his rent payment to the landowner
is too low. Accordingly other individuals will be attracted to the use of the

AGGLOMERATION AND AGRICULTURAL LOCATION 197

position at which he is located, provided, of course, he is within the
rent-yielding hinterland. Hence, observing the adjustments of the
individual agricultural entrepreneur at each possible distance from
the market yields, for the given crop, a theoretically valid rent func-
tion of the general order of the curve AA on Fig. 38.

As already intimated, there is another basic substitution path along
which an individual farm operator can proceed. He can elect to pro-
duce a different crop. With respect to this second crop we can derive
for each meaningful location the relevant net farm price line, ratios
of factor inputs, scale of output, and the rent potential which through
the process of competitive bidding would accrue to the landowner.
As with the first crop, a rent function results, let us say curve BB of
Fig. 38. Again, when production is restricted to the second crop, the
agricultural operator is theoretically indifferent to location within the
rent-yielding hinterland since at all locations his average unit cost
curve (inclusive of rent payments) would be tangent at its minimum
point to the effective net farm price line. Of course, to avoid losses
and to attain a position corresponding to this tangency point, the
operator is compelled to adhere to the appropriate set of substitution
points, which set varies with distance from the market.

When the possibility of producing each crop is permitted, speciali-
zation in land use may ensue. Crop A should be cultivated at those
sites, if any, at which the ordinate of the rent function curve AA exceeds
the ordinate of the rent function curve BB. And similarly, crop B
should be produced at those sites, if any, at which its cultivation
yields greater rent per acre land. If the agricultural operator were
to persist in the cultivation of either crop A or crop B at any location
when conditions dictate otherwise, his land would be bid away from
him by potential operators who channel their activity in accord with
the practice which yields maximum rent per unit of land at that loca-
tion. Thereby, in shifting from one crop to another, he (or the ultimate
operator of the land) substitutes among various revenues and among
various outlays, as generally developed in the individual firm discus-
sion of Chap. 6.

given land and continuously bid up rent until the surplus profits associated with
its use are eliminated. In contrast, if the operator contracts for the use of land
at a price greater than its rent potential, his average unit costs (inclusive of
adequate payments for entrepreneurial abihty, privately-owned capital, etc.) will
exceed his net farm price. He will tend to leave the industry. The land which he
cultivated will tend to remain idle until the landowner reduces rent to the point
where the operator's average unit cost curve can be tangent with his net farm
price hne. At all sites, zero surplus tends to obtain after appropriate payment
is made for all inputs.

198 LOCATION AND SPACE-ECONOMY

Likewise, we can derive a rent function for a third crop, a fourth
crop . . . and for an nth crop. Each of these can be graphically depicted
in Fig. 38.3 3 Land use at any location is governed by the rent function
which has the highest value at that location. For example, in Fig. 38,
the stretch of land OZ should be devoted to the production of crop A,
ZY to crop B, YX to crop C, and XW to crop D.^^ It is clear that
the individual farmer at any one location is forced to cultivate that
crop which yields greatest rent to the landowner, so long as relative
freedom of entry, exit, and contract is granted. Thus, when we con-
centrate upon the individual farm's adjustments, we obtain the typical
set of rent functions traditionally central to agricultural location
theory. Further, when we rotate lines OZ, OY, OX, and OW around
point 0 as the center, we obtain the familiar Thiinen rings (concentric
circles).

In the above manner, individual firm analysis moves in the direction
of aggregative equilibrium analysis for the industry. The analysis of
the problem on the firm level involving resources devoted to a single
use is extended to encompass the problem involving competition of
different uses for the same land. It is, of course, a necessary condition
that the supply and demand for each crop be equated at the given
market price. If not, relative market prices must change, rent functions
shift, appropriate firm adjustments occur, and a new pattern of land
use be established consistent with this necessary condition. 3 5 Thus
conditions relating to the agricultural industry as a whole — conditions
which the Thiinen school has emphasized — are of equal importance as
those governing individual firm adjustments in deriving an equilibrium
land use pattern. It should be pointed out that the resulting pattern is
logically more precise and different from the approximative pattern
which the Thiinen school derives when it abstracts from individual
firm adjustments.

Where several concentrated markets, rather than one, exist for agri-
cultural products, the problem becomes more complex graphically but
remains the same conceptually. For each crop as many rent functions
may be identified as there are markets, each rent function being linked
to the consumption of the crop at one and only one market. Each rent
function throughout its course reflects appropriate sets of individual
firm adjustments. As before, any given unit of land becomes geared
to that use which yields maximum rent returns within the entire range

33 Curves CC and DD represent rent functions for two of these crops.

34 For more extensive discussion of boundary and related conditions see Chap.
10 and Dunn, op. cit., Chaps. 2 and 3 and Appendix A.

35 For a fuller treatment which considers the income factor as well, see Dunn,
op. cit., pp. 16-24.

AGGLOMERATION AND AGRICULTURAL LOCATION 199

of markets and crops. ^^ Furthermore, these principles governing land
use apply with equal force whether we consider for each unit of land
the production of single crops or any number of products in combina-
tion. In reality products in combination (farming systems) tend to
characterize the operations of agricultural enterprises. However, the
above analysis remains unchanged since a crop can be easily redefined
as a set of products in combination. ^^

Hence we observe how the agricultural farm location problem can be
viewed as an integral part of the general individual firm location prob-
lem. In the case of industrial firms, outlays on land services may
frequently be less critical locationally than outlays on transport, labor,
power, and other inputs. In the case of the agricultural enterprise,
some of these latter outlays (excluding transport) may be the less
significant ones; and typically, outlays on land are major. Despite
these contrasts on the concrete level, in both cases the location problem
of the individual unit involves the identical basic substitution process;
the individual unit whether agricultural or non-agricultural must, ex-
plicitly or implicitly, substitute among outlays, among revenues, and
among outlays and revenues.

Additionally, because of the relative freedom of entry and exit and
because of the many small units which typically produce a given crop,
the analysis of the adjustments of the individual farm enterprise within
the full array of feasible locations, crops, and systems of crops, leads
to fruitful conclusions on a more aggregative level. When combined
with total conditions and restraints bearing upon supply, demand, price,
income, and related variables within the agricultural industry as a
whole, this analysis leads to over-all equilibrium land-use patterns
within the industry and to a more precise determination of the familiar
Thiinen spatial configurations. However, to specify these total con-
ditions and restraints requires the study of forces governing the inter-
relations of various urban-metropolitan markets, their connections
with regional hinterlands, and their respective sizes. These forces
operating on the more aggregative regional level will be analyzed in a
future volume.

36 This multimarket framework is dealt with at greater length in Chap 10. Also
see Dunn, op. cit., pp. 57-63.

Likewise, differentials in input prices among sites at different distances from
the market, an aspect of the agricultural location problem with which Brinkmann
and Losch among others have been concerned, can also be easily encompassed
by the substitution framework. See Chap. 10. Additionally, other phases of the
agricultural location problem can be incorporated into the above framework
(see Dunn, op. cit., Chaps. 5 and 6).

37 For further discussion, see Dunn, op. cit., pp. 46-52.

200

LOCATION AND SPACE-ECONOMY

Appendix to Chapter 8
Some Theoretical Notes on Urban Land-use

Traditionally, the analysis of urban land-use patterns has fallen outside
the realm of location theory. Yet, in many respects, urban land-use theory
is a logical extension of agricultural location theory. In this appendix we
shall dwell upon some of the interconnections of these two forms of theories.
We must leave to the reader and subsequent writers the task of compre-
hensively stating these interconnections and of relating the literature in these
two fields.

Arbitrarily select a unit of urban land. To what use should it be put?
The price each potential user is willing to bid is dependent upon many factors
such as: (1) effective distance from the core; (2) accessibility of the site to

T K

Effective distance

Fig. 39. Variation of sales with distance from urban core.

potential customers; (3) number of competitors, their locations, and the
intensity with wliich they vie for sales; and (4) proximity to land devoted
to an individual use or a set of uses which are complementary in terms of
both attracting potential customers and cutting costs, whether they be pro-
duction, service, advertising, or other.

Couple with the selected unit of land a particular use, let us say, use A.
The relations governing the decision of a businessman to bid for the unit of
land and devote it to such use may be depicted by a set of graphs. In
Fig. 39 we measure along the vertical and horizontal axes, respectively, dollar
volume of sales^s and effective distance from the core. (Effective distance

38 An alternative would be to measure physical volume of sales along the vertical
axis. Since retail and commercial activities tend to dominate urban economic

AGGLOMERATION AND AGRICULTURAL LOCATION 201

is not synonymous with physical distance. Rather it is physical distance
adjusted in the time-cost dimensions. As a consequence, equal physical dis-
tances from the core along the several routes feeding into the core, whether
directly or indirectly via other routes, correspond to different effective
distances.) When the values for advertising outlays, price mark-up, quality
of product or service, and other relevant variables are set, the businessman
presumably estimates dollar volume of sales. This volume of sales and the
effective distance separating the core and the selected unit of land may be
represented, let us say, by point R, in Fig. 39.

The businessman interested in producing the commodity or service corre-
sponding to use A also considers other sites as potential locations. Given the
same set of values for advertising outlays, price mark-up, quality of product,
and other relevant factors, he anticipates for each possible site a dollar volume
of sales. When the effective distance and dollar volume of sales corresponding
to each site are plotted on Fig. 39, we obtain curve W . In general, it is to
be expected that dollar volume of sales falls off with effective distance from
the core although at times it rises to secondary peaks only to decline again. 3 9

However, the potential land user can consider other sets of parametric
values for quality of service, advertising outlays, price mark-up, and other
relevant variables. There are, in effect, an infinite number of sets possible.
For each possible set the potential land user can derive a curve similar to VV.
We have added to Fig. 39 curves WW and ZZ which represent two of the
infinite number possible. 4 o

In addition to Fig. 39, we construct Fig. 40. Figure 40 depicts marginal
cost and average cost curves in whose construction rent payments are
excluded. These curves relate to that quahty of product or service and that

activities and since the price of a retail or service activity may often be syn-
onymous with a price mark-up based upon dollar values, we have chosen to speak
in terms of dollar volume of sales and price mark-ups. The reader may do
otherwise without affecting the validity of the argument.

39 Where the urban structural pattern is set and relatively inflexible, these
secondary peaks can be readily ascertained. They appear at those effective
distances which separate secondary and satellite commercial and shopping centers
from the core.

In contrast, where the urban structural pattern is extremely fluid and largely
to be determined, it is difBcult to identify the effective distances from the core
at which secondary peaks occur. Since secondary peaks reflect the juxtaposition of
complementary activities and since in this latter setting the spatial configurations
of all economic activities including transportation patterns are interdependent,
only a general equihbrium (simultaneous equations) approach, which takes into
account all aspects of complementarity and competition, yields the effective
distances at which secondary peaks occur.

Nonetheless, if only for purposes of exposition, we have inserted secondary
peaks in Fig. 39.

40 Rigorously speaking, the problem involves n-variables, n-dimensions, and
(to — 1) spaces. However, the reader may care to Hmit the framework of refer-
ence of Fig. 39 to three variables: dollar volume of sales, effective distance, and
price mark-up. In this case curves WW, VV, and ZZ refer to three different values
for the variable, price mark-up, curve ZZ being associated with the highest of the
three and curve WW with the lowest.

202

LOCATION AND SPACE-ECONOMY

amount of advertising outlays associated with curve VV of Fig. 39. Also,
they reflect variation of costs with sales volume only. Just as we assumed
that agricultural cost functions (exclusive of rent) vary insignificantly with
distance from the market, we postulate that the cost functions relating to
land use A are invariant with distance from the core. 4 1 As with Fig. 36, we

$0.20
$0.15 f
$0.10

$0.05 -

o

p \ ^

J

MC

/

AC

/

. \

M

>

/

'. \

Dollar volume of sales

Fig. 40. Variation of cost and profit with volume of sales.

measure along the vertical axis not only costs in dollars, but also price
(price mark-up).

Associated with the curve VV of Fig. 39 is a price mark-up, let us say
15 cents. This price mark-up is represented by the price line PP in Fig. 40.
Therefore, for any given volume of sales, the spread between the price line
and the ayerage cost curve is profits per dollar of sales (exclusive of rent
payments). Multiplying this spread by volimie of sales yields total profits.
Since we have posited freedom of entry, exit, and contract in the long run,
this total profit in general will accrue to the landowner as rent through the
process of competitive bidding.

Returning to Fig. 39, we find that the businessman estimates his volume
of sales at OS if the arbitrarily selected unit of land (at effective distance
OT from the core) is devoted to use A. Corresponding to this voltime of

41 As already mentioned, in reality fertility of soil, topography, labor costs, and
other factors do vary in a specific regional situation with distance from a central
market and thus affect agricultural cost functions. Similarly, topography, accessi-
bility to labor force, and other factors do vary in a specific urban setting with dis-
tance from the core and do cause cost functions (excluding rent) associated with
any given land use to vary with the position of the land employed. However,
these features of the broad physical and social environment do not vary in a
regular fashion with distance from any pertinent focal point but rather vary hap-
hazardly. Hence, their variation cannot be incorporated into our analysis which
aims at generality and which abstracts from specific environmental situations.

AGGLOMERATION AND AGRICULTURAL LOCATION 203

sales in Fig. 40 is the spread (profit per dollar of sales) MN and, therefore,
a rent (total profits) which is represented by the area of the rectangle PNML.
Accordingly, we can plot in Fig. 41 point F which indicates that a total rent
OJ {= PNML) is yielded by the arbitrarily selected unit of land (at OT
distance from the core) if devoted to use A. In Fig. 41, we measure effective
distance along the horizontal axis and rent along the vertical.

c

>

_^

-

F

O D

E

A gB

Effective distance

Fig. 41. Rent functions for different urban land uses.

For every other unit of land, the businessman, who has already estimated
volume of sales corresponding to any effective distance from the core (as
given by curve VV, Fig. 39), can determine profits per dollar sales and total
profits (rent) in accordance with Fig. 40. We can therefore plot in Fig. 41
the rent corresponding to each effective distance separating any relevant unit
of land from the core. We obtain the dashed curve EE. It depicts rent
generally declining with increase in effective distance though rising to sec-
ondary peaks at satelhte centers where sales volume also attains secondary
peaks.

Curve EE might be construed as the rent function relating to use A, much
as curve AA in Fig. 38 was construed. Except in a very limited sense, this
interpretation would be incorrect. In considering the employment of different
units of land in use A, the businessman would not confine his potential opera-
tions to the values for price mark-up, quality of product or service, advertising
outlays, and other variables which underlie curve VV of Fig. 39. He would
find that total profits could be enhanced if one or more of these values were
varied at each potential location. Simultaneously, of course, his cost functions
would be altered.

For example, the businessman might realize greater profits at effective

204 LOCATION AND SPACE-ECONOMY

distance OK (Fig. 39) if he operated under conditions specified by curve ZZ
rather than by curve VV. His profit could be greater even though his volume
of sales would not be HK, but only JK, since the cost curves associated with
curve ZZ would be significantly lower than those in Fig. 40.4 2

Hence, at this and most locations the rent wliich could potentially accrue
to the landowner would exceed that indicated by curve EE (Fig. 41) ; at no
location would rent be less. We therefore let curve AA be the locus of
points, each of which depicts at the given effective distance the maximum
rent obtainable from the variation of all factors under control of the business-
man. It is the relevant rent function for use 4.43

In similar fashion we can derive for each possible use — retail, commercial,
wholesale, industrial, residential, cultural, etc. — a rent function which would
depict the maximum rent potential at each possible site. For purposes of
illustration, we have added two of these functions, BB and CC, to Fig. 41.
As with Fig. 38, at each effective distance, that use whose rent function has
the highest ordinate wins out in the competition. For example, if there were
only three possible uses of urban land and if at each effective distance the
rent potential in any use were completely independent of the uses to which
other land was put, the units of land along stretch OD would be channelled
into use A, along stretch DG into use B, and beyond G into use C. In reality,
at any effective distance the rent potential of land in a given use may be
significantly affected by the uses, whether competitive or complementary, to
which adjacent and other units of land in the urban area are put. Competi-
tion coupled with limited volume of sales is likely to depress markedly the
rent potential of any given unit of land when decisions are or have been
made to devote other units to the same use. 4 4 Furthermore, innumerable
rent functions obtain, each of which corresponds to a unique use to which
urban land can be put. Thus, aside from general residential and industrial
land, only a relatively few sites of the many possible ones are allocated by the
market mechanism to any particular use. Such sites may comprise clusters

42 For related discussion on the problem of choosing for any given location the
correct set of values for price mark-up, advertising outlays, and quaUty of product,
the reader is referred to, among others: Kenneth E. Boulding, Economic Analysis,
Harper & Brothers, New York, rev. ed. 1948, Chaps. 31 and 32, especially pp.
717-727; Edward Chamberhn, The Theory a) Monopolistic Competition, Harvard
University Press, Cambridge, Mass., 1938, Chaps. V and VI; L. Abbott, "Vertical
Equilibrium Under Pure Quahty Competition," American Economic Review, Vol.
XVIII (December 1953), pp. 826-845; and R. Dorfman and P. 0. Steiner, "Opti-
mal Advertising and Optimal Quahty," American Economic Review, Vol. XLIV
(December 1954), pp. 826-836.

43 If there were plotted in Fig. 41 an infinite number of curves, each corre-
sponding to the relevant conditions depicted by one of the infinite number of
curves which we can draw in Fig. 39, we would find that at each effective distance
curve AA passes through the maximum of the infinite number of values of the
ordinate given by these curves at that effective distance.

4'i This is not to deny that the rent potential of certain units of land in a given
use may be greatly enhanced when the use (whether like or unlike) of adjacent
land results in increasing the attractive power (dollar volume of sales) of the
several juxtaposed units as a whole.

AGGLOMERATION AND AGRICULTURAL LOCATION 205

of adjacent units, units spatially distributed throughout the urban area in
more or less regular fashion, or both. 4 5

In the above fashion, it is thus possible to proceed with the derivation of
optimal land-use patterns for different urban settings. Only a skeletal frame-
work has been traced in this appendix. Yet it suffices to disclose the rather
obvious intercomiections of agricultural location theory and urban land-use
theory. In each, rent functions (surfaces) guide the allocating hand of the
market. For both, relations to transportation facilities and systems are
critical in the definition of effective economic distance; and changes in the
transportation grid provoke in each a dynamic pattern of adjustments whose
outline is difficult to unravel ex post, let alone anticipate. In agricultural loca-
tion theory, transport outlays are explicitly considered; they depress net farm
price. In urban land-use theory, transport outlays enter into the picture just
as forcefully, though only implicitly. Transport outlays in terms of both time
and cost are borne by the consumer but strategically condition revenue poten-
tials of the business firm through affecting the accessibility of his location to
customers and, hence, his volume of sales.

Both agricultural location and land-use theory must be concerned with
complementarity and competitive relations. On the farm, complementarity
revolves around the full utilization of the farmer's time, equipment, land,
and other indivisible units, and it customarily results in the cultivation of a
system of crops rather than a single crop. In the urban setting, comple-
mentarity in terms of configuration of uses spatially juxtaposed critically
affects both costs and accessibility. ^ 6 Finally, technology, the legal frame-
work, and other institutional and cultural factors in their full dynamic setting
establish constantly changing limits to which a given unit of land, whether
rural or urban, may be put. In many cases, these hmits are rational so far
as the commonweal is concerned; in other instances, irrational. Where
rational, they too add to the similarities of and interconnections between
agricultural location theory and urban land-use theory.

In bringing this appendix to a close, we wish to state that the urban
land-use problem can be presented in terms of substitution analysis and as an
integral part of general location theory, much as agricultural location theory
has been. In essence the businessman substitutes among various outlays and

45 The reader is referred to Hoover, op. cit., Figs. 11 and 12, for suggestive
graphic illustrations.

To avoid complicated diagrams we have presented the analysis in terms of
effective distance along only one line radiating from core point 0. A more com-
prehensive presentation would have considered units of land along all possible
lines radiating in different directions both directly and indirectly from point 0.
Such a presentation (in which, to reiterate, equal physical distances would corre-
spond to different effective distances) would have yielded rent surfaces for each
of the innumerable uses. However, the analysis would not have been altered and
would have led to similar general results.

46 The complementarity problem is as difficult to attack as the agglomeration
problem. In many respects, these two problems are two sides of the same coin.
Both require an analytical approach which looks at each individual activity not
in vacuum but as an element of a complex of activities. Such a complex approach
will be explored in a future volume.

206 LOCATION AND SPACE-ECONOMY

revenues when he selects both the commodity (product or service) to be
produced and his location. He may substitute rent outlays for advertising
outlays or for outlays to alter the quahty of his commodity when he considers
shifting his location to any site closer to the core; or, if he maintains quality
and advertising outlays, he incurs additional rent outlays to acquire addi-
tional revenue potentials. In weighing the several commodities wliich he
might produce, once again he substitutes among the outlays and revenues
associated with the several commodities, much as the farmer does in selecting
the particular set of crops to be cultivated. And so forth. Thus, although
the typical businessman may not attack his problem in such a comprehensive
fashion, tracing out in a substitution framework what his logical reactions
would be allows us to arrive at optimal patterns of land use.

It must constantly be borne in mind, however, that the businessman
operates within a setting of restraints. Certain of these restraints are im-
posed by the features of his physical environment, such as topography and
existing structures. Certain are associated with social and economic conditions
which relate to such factors as total demand, total income, tastes, and cultural
patterns, whose treatment falls within the scope of a volume on regional
analysis. These restraints are of as great importance in shaping land-use
patterns as are the businessman's own decisions. Since these restraints differ
from urban area to urban area, they in turn induce logical patterns of land
use which differ from area to area. They furnish a partial justification for
the kaleidoscopic variety of reality.

Chapter

.9

Some Basic Interrelations
of Location and Trade Theory'

1. Preliminary Remarks

Heretofore, we have treated trade rather incidentally. We have
focused upon the location of the individual industrial or agricultural
enterprise (or group of enterprises) with respect to markets, whether
one-point or areal. We have implicitly posited that once locations
are determined, the associated flows of commodities, both as inputs
and products, are likewise determined. This postulate is consistent
with the statement in Chap. 2 that "(1) location cannot be explained
without at the same time accounting for trade and (2) trade cannot
be explained without the simultaneous determination of locations."

Although trade and location are as the two sides of the same coin,
it does not follow that the general location theory developed thus far
is adequate to explain all forms of trade. First, the general theory of
location developed in this volume does not consider, except in minor
fashion, the aggregate demand and income side of the picture, par-
ticularly as they relate to regions and to interregional trade. These
aspects of both the location and trade problem are to be considered
in another volume. ^

1 The contents of this chapter are largely drawn from a manuscript written
jointly with Merton J. Peck.

- The reader will find some of these aggregative demand and income aspects
treated in W. Isard, "Location Theory and Trade Theory: Short-Run Analysis,"
Quarterly Journal of Economics, Vol. LXVIII (May 1954), pp. 305-320.

207

208 LOCATION AND SPACE-ECONOMY

Second, trade may be international as well as intranational. An
international setting introduces into our conceptual framework certain
additional basic factors. Nonetheless some of these factors can be
incorporated into our analysis, as we shall now attempt to demonstrate
in this chapter.

In Chap. 2 we have already noted that Weber criticized classical
trade theory for ignoring the significant amount of industry which
is transport- oriented, and whose geographic distribution, internation-
ally speaking, is governed primarily by considerations of transport
cost of raw materials, fuel, and finished product. This criticism was
undoubtedly a major source of inspiration for Ohlin's attempt "to
demonstrate that the theory of international trade is only a part of
a general localization theory."^ This attempt has already been dis-
cussed in Chap. 2. Other location theorists — Furlan, Englander,
Weigmann, Predohl, and particularly Losch^ — have reiterated Weber's
contention. But despite Ohlin's excellent formulation of the problem,
Williams' classic criticism of the mobility and immobility premises
of trade theory, ^ and Losch's major contribution in visualizing and
portraying the spatial structure of an economic system, the presenta-
tion of the basic interrelations which should logically obtain between
location and trade theories, and which should lead to improvements
in both, is still wanting. As one of the objectives of this chapter we
hope to cast additional light on these basic interrelations.

2. A Fusion of Opportunity Cost Doctrine and
Transport-orientation

The empirical materials presented in Chap. 3 testify to the signifi-
cance of the friction of distance both in interregional and in inter-

3 Bertil Ohlin, Interregional and International Trade, Harvard University Press,
Cambridge, Mass., 1933, p. vii.

4 V. Furlan, "Die Standortsprobleme in der Volks- und Weltwirtschaftslehre,"
Weltwirtschaftliches Archiv, Vol. II (1913), pp. 1-34; O. Englander, "Kritisches
und Positives zu einer allgemeinen reinen Lehre vom Standort," Zeitschrijt fur
Volkswirtschaft und Sozialpolitik, Vol. V (Neue Folge), Nos. 7-9 (1926) ; H. Weig-
mann, "Ideen zu einer Theorie der Raumwirtschaft," Weltwirtschaftliches Archiv,
Vol. XXXIV (1931), pp. 1-40; and "Standortstheorie und Raumwirtschaft" in
Johann Heinrich von Thunen zum 150 Geburtstag, ed. W. Seedorf and H. Jurgen,
Rostock, 1933, pp. 137-57; A. Predohl, "Aussenwirtschaft," Grundriss der Sozial-
wissenschaft, Bd. 17 (Gottingen, 1949) ; A. Losch, Die rdumliche Ordnung der Wirt-
schaft, Jena, 1944, Part III.

5 J. H. Williams, "The Theory of International Trade Reconsidered," Economic
Journal, Vol. XXXIX (June 1929), pp. 195-209.

INTERRELATIONS OF LOCATION, TRADE THEORY 209

national trade. The "falling off with distance" effect is pronounced. ^
Yet how incorporate this effect in both location and trade theory?

In the previous chapters, especially in Chap. 5, the distance variable
has already been partly incorporated into location theory via the con-
cept of transport input. Fuller treatment of this variable by location
analysis lies in that direction which would involve the development
of gravity models and models of other types.''' (These models are to be
discussed in a subsequent volume.) In contrast, in international trade
theory the distance variable has hardly been explicitly recognized. This
situation reflects the fact that the various international trade doctrines
have been pushed along certain channels to extreme refinement but
left in a primitive stage of development in other channels. Those
aspects of trade theory which have remained undeveloped are the very
ones which would involve the explicit treatment of the distance variable
and which would thereby contribute to the fusion of trade theory
and location theory. Hence an attack must be made on these aspects.

A major obstacle to such an attack is the disagreement among trade
theorists as to what is "good and relevant" trade theory. Into
such a controversy we do not wish to enter, particularly since we feel
that, for the most part, whatever the trade theory considered, im-
provement can be effected when attention is paid to the spatial aspects
of the economy and when the techniques and concepts of location
theory are embraced. For our purposes, however, it seems sufficient
to proceed with a very crude classification of trade doctrines, namely,
those which are concerned with or emphasize long-run effects and ad-
justments, and those which attribute greater, though not exclusive,
significance to short-run repercussions and forces. Since the extension
of the latter set of trade doctrines to incorporate more explicitly the
distance variable revolves around the appropriate development of
gravity models, modified interregional input-output schemes and
activity analysis, which are to be discussed in a subsequent volume,
we shall confine ourselves in this chapter to the extension of the first
type of trade doctrine. ^ This type is best exemplified by Graham's
work. 9

6 Also, see W. Beckerman, "Distance and the Pattern of Intra-European Trade,"
Review of Economics and Statistics, forthcoming.

■^ See W. Isard and G. Freutel, "Regional and National Product Projections and
Their Interrelations," Long-Range Economic Projection, Studies in Income and
Wealth, Vol. XVI, Princeton University Press, Princeton, N.J., 1954, pp. 434-439.

8 For some preliminary extension of the latter set of trade doctrines the reader
is referred to W. Isard, op. cit.

9 Frank D. Graham, The Theory of International Values, Princeton University
Press, Princeton, N.J., 1940.

210 LOCATION AND SPACE-ECONOMY

Graham's approach has strong appeal to location theorists since he
adopts a multicountry, multicommodity approach (one which until
recently has not been common among trade theorists) and since he
emphasizes supply and cost conditions. lo However, Graham ignores
the very heart of location analysis, namely, that sector of any national
or international economy for which transport costs are the primary
location factor and which is characterized as transport-oriented. As
pointed out above, if there is any significance to location analysis, it
lies in the fact that transport costs vary systematically with distance
and thus provide an underlying stratum for systematic analysis.

Though transport-orientation analysis should be included in trade
doctrine in order to facilitate the understanding of flows of commodities,
it must be admitted that existing location theory does not readily
adapt itself to being so included since it treats costs only in terms of
a given currency. We wish now to demonstrate that, by extending the
analytic framework of transport-orientation to consider not costs in
any particular currency (as dollars or sterling) but rather opportunity
costs, it is possible to incorporate transport-orientation into existing
trade theory. We illustrate by an extended, though simple, example.

Assume three countries. A, B, and C, each possessing, as with
Graham, 12 productive units. After trade, two finished goods, steel and
textiles, are consumed by each. To produce 1 weight unit of steel
requires 2 weight units of ore and 4 weight units of coal and, in addi-
tion, shipping for the finished steel and for one or both raw materials.
If production is at A, coal which is assumed to exist in B alone will need
to be shipped as well as finished steel. If production is at 5, ore which
is assumed to exist in A alone will need to be shipped as well as steel.
If production is at C which possesses neither coal nor ore, both raw
materials and steel will need to be shipped. To simplify computations,
shipping requirements on textiles are assumed to be negligible, n

10 By and large, the traditional location theory of the Launhardt, Weber,
Palander, and Hoover type has posed the problem of finding the point of minimum
cost for assembling raw materials, processing them, and distributing the finished
product to the market point or area. For the most part, demand has been taken
as given, or its variation as of minor consequence for determining the optimum
plant location. Even agricultural location theory of the Thijnen tj^pe takes prices
and hence demand at the city market as set. The problem is essentially to con-
sider the variation in transport and production costs associated with the various
possible patterns of zones in the cultivation of several crops and to select that
pattern which maximizes rent for each unit of land.

11 The reader may wish to consider not only shipping requirements on textiles,
but also the raw material and other factor requirements in both textile and steel
production, as one must in reality. To do so compHcates manyfold the computa-
tions and does not affect the essential nature of the conclusions to be derived.

INTERRELATIONS OF LOCATION, TRADE THEORY 211

In each country let a productive unit, when devoted to the pro-
duction of a given commodity alone, produce under conditions of
pure competition and constant cost, the quantities (in standard weight
units) of each commodity as listed in Table I. Shipping is necessarily
expressed in terms not of weight units but of transport inputs where
a transport input (say a ton-mile) is defined as the movement of a
weight unit (a ton) over a unit of distance (a mile) . Also, although
a productive unit in each country can produce 8 units of steel when
the ore and coal are at hand, the ore, coal, and transportation of the
ore, coal, and finished product must be purchased. Hence, a productive
unit in countries A, B, and C produces respectively 8 — Xa, 8 — Xb,
and S — Xc units of steel where Xa, Xb, and Xc correspond to the
amounts of steel (or their equivalents in terms of textiles or shipping)

Table I. Amount of Each Commodity a Productive

Unit in Each Country Can Produce when Devoted

to the Production of One Commodity Alone

Commodity

Country

A

B

C

Ore

Coal

Textiles

Shipping (in transport inputs)

30

0

5

2500

0

20

4

600

0

0

2

600

Steel S - Xa 8 - Xb 8 - Xc

exchanged for the coal, ore, and associated shipping required if a par-
ticular country were to produce and deliver the steel. The figures of
Table I, once the values of the variables Xa, Xb, and Xc are determined
for any given situation, express for each country the opportunity costs
in the use of a productive unit for the production of any commodity.
In determining the values of Xa, Xb, and Xc we must construct a
table on shipping requirements of steel in terms of transport inputs.
For this to be done, however, the distances separating countries must
be specified. For the moment, posit that each country is 100 distance
units from each of the other two, so that their geographic position is

The two activities, textile and steel production, are purposefully chosen. The
former typifies an industry usually treated by trade theorists and is one in the
location of which labor and other immobile local (national) resources are domi-
nant factors while transport cost is, at most, minor. The latter represents a
transport-oriented industry, in the location of which labor and other immobile
resources have been considered incidental (W. Isard, "Some Locational Factors
in the Iron and Steel Industry since the Early Nineteenth Century," Journal of
Political Economy, Vol. LVI (June 1948), pp. 203-217).

212

LOCATION AND SPACE-ECONOMY

as the corners of an equilateral triangle. See the figure at the upper
right hand corner of Table II. Immediately Table II can be filled in.
Since in serving any market B has less shipping expense (incurs fewer
transport inputs) than either A or C, and A less than C, and since the

Table II.

Transport Input Requirements Per Weight Unit of
Steel, Given Relative Position: b

100

Country
to Wliich
Delivered

On Coal

On Ore

On Steel

Total

A

If production at A
If production at B
If production at C

400

0

400

0
200
200

0

100
100

400
300
700

B

If production at A
If production at B
If production at C

400

0

400

0
200
200

100

0

100

500
200
700

C

If production at A
If production at B
If production at C

400

0

400

0
200
200

100

100

0

500
300
600

mine prices of ore and coal wherever steel may be produced will be
identical for the three countries, 12 it follows that:

S-Xb>8-Xa>S-Xc.

To derive the values of X^, Xb, and Xc as well as the patterns of
production and trade, demand conditions must also be specified since
otherwise exchange ratios cannot be obtained. To simplify the prob-
lem assume, within the range of variation of real income considered
below, that each country desires to consume twice as much textiles as
steel. 1^ It then follows that:

12 The mine prices of ore and coal in terms of finished steel will, of course,
depend upon which country or countries produce steel.

13 It is traditional to posit that variation of real income, within definite Hmits,
does not affect the simplified expenditure (demand) pattern usually assumed.
See, for example, Graham, op. cit., Chap. V, or L. Metzler, "Graham's Theory of
International Values," American Economic Review, Vol. XL (June 1950), pp.
304-13.

INTERRELATIONS OF LOCATION, TRADE THEORY 213

(1) A produces all the ore required; also she furnishes all the ship-
ping (since, given demand conditions, her 12 productive units are not
completely utilized in both ore production and in shipping and since
she has greater advantage compared to B and C in shipping than in
any other activity except ore production) ; and in addition she engages
in some textile production (since she has greater comparative ad-
vantage in textiles than in steel or coal).

(2) B produces all the coal required and some textiles (since even
if B were to produce all the steel required in addition to the coal, she
would have left over some productive units, given the demand condi-
tions, and since relative to A she has less disadvantage in the produc-
tion of textiles than in shipping or ore production) .

(3) C produces some textiles (since even if C were to produce all
the steel required, she would have unutilized some productive units,
and, since relative to A or B, she has less disadvantage in the produc-
tion of textiles than in shipping or ore production or coal production) .

Since each country produces textiles, exchange values are, as with
Graham, 1 unit of textiles for 6 of ore, for 500 of shipping, i^ for 5
of coal and for (8 — Zb)/4 or (8 — Xc)/2 of steel. However, it is not
clear whether B or C will produce steel, even though B has absolute
advantage over C in steel production and is the point of minimum
transport cost for serving each steel market. The answer hinges upon
whether B's productivity in steel (8 — Xb) is more than twice C's
(S — Xc). Simple calculation shows that this is the case.i^ As a

14 Traditional location theoiy assumes a given transport rate structure which
applies to the movement of a commodity whether or not the commodity repre-
sents return cargo on a ship which otherwise might make the trip empty. This
procedure assumes certain monopolistic elements and inefficiencies in rate-making
which to some extent at least exist in reality.

Logically, the entire pattern of commodity movements and requirements for
transport sei-vices in all directions should be considered in setting rates, as, for
example, Koopmans has done (T. C. Koopmans and S. Reiter, "A Model of
Transportation," in Activity Analysis of Production and Allocation, ed. by T. C.
Koopmans, New York, 1951, Chap. XIV). To do this here, however, would require
an extension of location theory in a direction which is beyond the scope of this
chapter. The rate of 500 transport inputs for 1 unit of textiles is taken as fixed.

15 From Table II it is seen that in producing and delivering steel to any market
C's absolute disadvantage relative to B is least with respect to the market in C,
the absolute disadvantage in this case being measured by 300 transport inputs.
Consider then the question of productivity in producing and delivering steel to
C. To produce and deliver 8 units of steel to C, B requires: (1) 16 units of ore
for which, according to the above exchange ratios, she must pay A 2.7 units of
textiles (or its equivalent) ; (2) 32 units of coal for which she must pay her coal
producers 6.4 units of textiles; and (3) 2400 transport inputs for which she must

214 LOCATION AND SPACE-ECONOMY

consequence, B produces steel and C textiles since C has comparative
advantage in textiles alone.

It is not necessary to show here how one derives the equilibrium
situation consonant with the several postulates on demand, ^^ oppor-
tunity costs, and relative position of countries, i '^ Suffice it to indicate
that country C consumes 5.7 units of steel and twice as much textiles;
B, 11.9 imits of steel, and twice as much textiles; and A, 14.2 units of
steel and twice as much textiles. C exports to B 12.7 units of textiles
in exchange for 5.7 units of steel. B exports to A 14.2 units of steel in
exchange for 63.5 units of ore, 4.4 of textiles, and 8331 transport inputs.
(These data together with data on outputs and raw materials and
shipping consumption are recorded in Table IV.) In addition, the
exchange ratio of textiles for steel at the point of consumption is 2.2
to unity in A and C, and 2.0 to unity in BA^

pay A 4.8 units of textiles. Thus, to obtain the ore, coal, and shipping, she must
pay out 13.9 units of textiles (or its equivalent) which requires the employment of
3.5 of her productive units since, according to Table I, each of her productive units
can produce 4 units of textiles. In addition another productive unit is engaged
in the actual manufacture of 8 units of steel. All told, B must devote 4.5 of her
productive units to the task of producing and delivering 8 units of steel to C. Her
productivity in steel relative to the market in C is 1.8.

To produce and deliver 8 units of steel to C, C requires: (1) 16 units of ore for
which she must pay A 2.7 units of textiles; (2) 32 units of coal for which she
must pay B 6.4 units of textiles; and (3) 4800 transport inputs for which she must
pay A 9.6 units of textiles. Thus she must pay out 187 units of textiles (or its
equivalent) which, according to Table I, requires the employment of 9.3 of her
productive units. In addition, one more productive unit is engaged in the actual
manufacture of steel. All told, C must apply 10.3 of her productive units to the
job of producing 8 units of steel for her own consumption. C's productivity in
steel relative to the market in C is 0.8. Since C's productivity in this case is less
than one-half of B's (1.8), B has comparative advantage in producing steel.

If B has comparative advantage over C in producing and delivering steel to the
market in C, for which market B has least absolute advantage over C, it follows
that B has comparative advantage over C in producing and delivering steel to
the markets in A and B, for which markets B's absolute advantage is greater.
Hence, B produces all the steel, and C produces onty textiles. (The above numer-
ical results and those to follow are rounded to the first decimal point.)

16 The reader should bear in mind that the simplifying assumption concerning
demand is not basic to the argument. Any other assumption is equally suitable,
provided it does not unduly complicate computations.

'^'^ The reader can refer to Graham, op. cit., pp. 76-82, for a demonstration of
how a final, stable exchange situation may be derived.

It should also be noted that the problem can be set up in a linear programming
form. This has been done by John S. Chipman, who has verified in this maimer
some of the results obtained.

18 The ratio is smaller in B since transport input requirements are smaller in
dehvering to the market in B. See Table II.

INTERRELATIONS OF LOCATION, TRADE THEORY 215

The above example illustrates how the basic core of location theory,
namely, transport-orientation, can be fused with the opportunity cost
doctrine of trade theory to yield a superior understanding of the
simultaneous determination of the location of economic activities and
of trade. Steel production is concentrated at B. This result is ob-
tained and represents a case of transport-orientation, whether we em-
ploy the traditional (intranational) location approach which minimizes
the cost of transport inputs or the superior approach which considers
opportunity cost as well. From the standpoint of trade theory we have
introduced explicitly the distance factor (in the concept of transport
inputs) and shown how the opportunity cost formulation can be easily
extended to embrace industries which are typically transport-oriented
intranationally. The extended opportunity cost formulation and the
improved formulation of transport-orientation are, of course, one and
the same.

3. The Effects of a Change in the Distance Variable Upon
Trade, Industrial Location, and Geographic Specialization

We now consider the implications of a change in distance relations.
Are trade and geographic specialization among nations and regions
significantly dependent upon their relative position, upon the absolute
distances separating any pair?i9 To answer this basic question, alter
the relative positions of the three countries so that they are as a
straight line, 200 distance units separating A and B, with C at the
midpoint, 100 distance units from both A and B. See the line diagram
at the upper right hand corner of Table III. Transport input require-
ments are simultaneously changed and are recorded in Table III. Note
that once again B has an absolute advantage over all countries in pro-
ducing steel for all markets because (1) the prices of coal, ore, and
transport inputs and the unit requirements of coal and ore are the same
for all countries as potential producers, and (2) the transport input re-

19 Again the reader is reminded that provided distances are not so great as to
make transport cost prohibitive for certain commodities and thus stifle trade in
these commodities, trade theory (aside from Ohhn's contribution) has usually
presumed that variation in distances and hence transport costs will cause variation
in the divergences of exchange values in the several countries but will not affect
the commodities produced and traded by each countrj'. See, for example, R. F.
Harrod, International Economics, London, 1947, p. 20; Haberler, op. cit., pp. 140-
42; Viner, op. cit., pp. 314-18 and 467-70; and even Graham, op. cit., pp. 139^6,
who perhaps treats transport cost more realistically than any of the above. It
should be noted that at one point Viner writes: "In fact, differences in freight
costs may create a comparative advantage which in the absence of freight costs
would not exist at all" (p. 470 note). However, this pregnant statement is left
undeveloped.

216

LOCATION AND SPACE-ECONOMY

quirements in serving each market are least for B. Thus, B, according
to orthodox location doctrine, is the optimal transport point. It should
be the point of steel production for all markets (as it was in the
triangular situation) provided the net deviating force of cheap sites of

Table III. Transport Input Requirements per Weight Unit of

Steel, Given Relative Position:

100

100

Country
to Which
Delivered

On Coal

On Ore

On Steel

Total

A

If production at A
If production at B
If production at C

800

0

400

0
400
200

0
200
100

800
600
700

B

If production at A
If production at B
If production at C

800

0

400

0

400
200

200

0

100

1,000
400
700

C

If production at A
If production at B
If production at C

800

0

400

0
400
200

100

100

0

900
500
600

labor and other inputs does not offset the transport minimizing force.
The resulting stable exchange situation, which does testify to the
partial dominance of the deviating force of cheap labor at C, is pre-
sented in Table IV, along with the old exchange situation of the
triangular setup. 20 The new situation strongly contrasts with the old.
C now produces steel for both itself and A. There are new geographic
flows of ore, transport inputs, and textiles from A to C, of coal from
B to C, of steel from C to A, and of textiles from B to A. The old flows
of steel from B to A, of textiles from A to B, and of steel from B to C
have been eliminated. The only flow which has not been subject to
major change is that of textiles from C to B. In sum, changing the
distance variable as we did almost completely revamps the geographic
flow of commodities and the structure of trade. In contrast, the

20 As in the old situation, we quickly perceive in the new that: (1) A should
produce all the ore, all the shipping, and some textiles; (2) B should produce all
the coal and some textiles; and (3) C should produce some textiles. Again, the
critical question is: should B, C, or both produce steel? If he cares to, the reader
can make the necessary computations to derive an answer to this question in the
same manner as was done in an earlier footnote for the triangle situation (the
following footnotes also indicate another computational approach). The results
are: {I) B produces steel for its own market; (2) C produces steel for both its
own and A's market.

INTERRELATIONS OF LOCATION, TRADE THEORY 217

resulting changes in real income (consumption of the finished goods,
textiles, and steel) are relatively minor.21

Thus we see how, once trade theory is extended to embrace the
realistic situations of commodities sensitive to differentials in trans-
port cost, the distance variable can have a major influence on trade
as it does in fact. With such an extension the criticisms of Weber are
met, and the vague intuitive formulation of Ohlin can be concretely
expressed. Although one example (to which we limit ourselves in this
chapter) is inadequate to point up in full the implications of such an
extension, it develops the required procedure and the significance of
the distance factor.

At the same time this extension represents an extension of location
theory. As already noted in the line position case, B by orthodox
location doctrine is the optimal transport point for steel production.
However, it develops that B produces steel for its own consumption
only and not for A and C. This result obtains because B has compara-
tive advantage over C in producing steel for B's internal market (costs
being expressed in terms of textiles) but comparative disadvantage
(though still absolute advantage) in producing steel for A and C.^^
The required restatement of location doctrine is immediately apparent:
costs must be expressed as opportunity costs. Thus, a transport-
oriented industry must be defined as one in which the differential
advantage of the optimal transport point completely offsets the net
differential advantage of any other site where costs are expressed in
terms of a commodity produced in common by two or more nationn.
And a labor-oriented industry must be defined as one in which the
differential advantage of a cheap labor point completely offsets the
net differential advantage of any other site, again where costs are
expressed in terms of a commodity produced in common. In the tri-
angle case, B was the optimal transport point, and in terms of each
market its transport cost advantage more than offset the labor cost
advantage of C. In the line case B is still the optimal transport point,
but, in serving the markets at A and C (but not at 5) , its transport

21 The exchange values also are subject to some change. One unit of textiles
still exchanges for 6 of ore, for 500 of shipping, and for 5 of coal. However, in
A, 1 unit of textile exchanges for 0.359 units of steel as against 0.448 of steel in
the triangular situation; in B, for 0.411 as against 0.492; and in C, for 0.387 as
against 0.448.

22 In terms of the market at B, 1 productive unit in B (whose opportunity cost
is 4 units of textiles) can produce 1.64 units of steel; and 1 in C (whose oppor-
tunity cost is 2 units of textiles), 0.72 units of steel. In terms of the markets at
A and C, 1 productive unit in B can produce respectively 1.41 and 1.52 units of
steel; and in C, 0.72 and 0.77 units of steel.

218

LOCATION AND SPACE-ECONOMY

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INTERRELATIONS OF LOCATION, TRADE THEORY 219

cost advantage is more than offset by C's labor cost advantage. 2 3
Hence the steel industry in the straight line case is partially transport-
oriented and partially labor-oriented. Incidentally, expressing costs
as opportunity costs brings out the significant conclusion that industries
which have traditionally been considered transport-oriented within
a national framework may, because the international set of cost
differentials diverge considerably from the national, be oriented other-
wise within an international framework.

4. Some Conclusions

For long-run analysis it is shown how, with the use of the concept
of transport inputs, transport-orientation and the opportunity cost
doctrine can be fused to yield a superior set of tools. It becomes
possible to trace the impact of change in the distance variable upon
the industrial structure (geographic specialization) of each country,
the composition of trade, exchange values, and the magnitude and
characteristics of other significant elements associated with the simul-
taneous determination of location of economic activities and com-
modity flows. At the same time this fusion represents, on the one hand,
an extension of location theory since the traditional cost approach of
location theory is reformulated in terms of opportunity costs, and, on
the other hand, an extension of long-run trade theory since commod-
ities sensitive to differentials in transport costs are introduced into
the traditional comparative cost framework. By this achievement
the way is paved for each type of theory to take over many of the
sound doctrines of the other. Practically all location doctrine, such
as labor orientation, power orientation, agglomeration, etc., can now be
adapted and employed in trade theory; and, similarly, all developments
in long-run trade theory can be adapted and employed in location

-3 In the triangle case, to produce and deliver 8 units of steel to C requires :
(1) shipping which costs 4.8 units of textiles if production is at B and 9.6 units
of textiles if at C ; (2) 1 productive unit which costs 4 units of textiles at B and
2 at C; and (3) ore and coal whose prices are identical for both B and C. Thus,
B has a differential advantage of 4.8 units of textiles with regard to transport and
a differential disadvantage of 2 units of textiles with regard to labor and other
resources contained in a productive unit. The transport differential is the greater;
hence, the industry is transport-oriented at B.

In the straight line case, shipping costs 8 units of textiles if production is at B
and 9.6 units if at C; 1 productive unit is still required whose cost is 4 at 5 and
2 at C. The differential in the cost of the productive unit is now greater. As a
result, in serving the market at C the industry becomes oriented (presumably
labor-oriented) to the cheap site of the productive unit, namely C.

220 LOCATION AKD SPACE-ECONOMY

theory. Actually to perform this adaptation, however, would require
more extended treatment than is possible in this volume. ^ 4

24 The reader is again reminded that other relations of location theory and
trade theory, particularly the short-run aspects of trade theory, will be discussed
in a second volume. Certain of these have been pointed up in W. Isard, "Location
Theory and Trade Theory . . .," op. cit.

Also the reader is referred to Gottfried von Haberler's "A Survey of Inter-
national Trade Theory," Special Papers in International Economics, No. 1,
September 1955, Princeton University Press, Princeton, New Jersey, for a brief,
but penetrating, statement on the directions in which one should approach the goal
of fusing trade and location theory.

Chapter

10

Aspects of General Location

Theory: A Mathematical

Formulation

The previous chapters, particularly Chaps. 5 to 8, have treated several
types and elements of location theory in a manner which emphasized
their interrelations and interconnections. The substitution framework
was advanced as a means of pointing up these interrelations and inter-
connections better. In this chapter we shall, in one sense, repeat the
argument, though more explicitly and cogently through the use of
mathematical symbols. Yet, more important, we shall try to probe
deeper and indicate more effectively how the substitution framework
coupled with the concept of transport inputs permits at least a partial
fusion of the several location dogmas not only with each other but also
with much of production theory. We hope (1) to demonstrate more
satisfactorily the pervasiveness of a general principle of an optimum
space-economy,^ and (2) to signify how, when supported by appropriate

^ The space-economy will be assumed to operate imder a set of conditions cus-
tomarily postulated by economists. To the same extent that the postulates are
unacceptable to the reader, the ensuing analysis wiU be devoid of significance. For
a dissent from the profit maximization (or cost minimization) principle tj^pical of
human ecologists and economists of similar inclinations, see Kenneth E. Boulding,
A Reconstruction of Economics, John Wiley & Sons, Inc., New York, 1950, Chaps.
1 and 2.

Also, it is recognized that the term optimum can be interpreted in several different
ways. The analysis that follows will be concerned with an optimum space-economy,
primarily from a location standpoint with emphasis on transportation costs. Further,
we approach the problem only in terms of an economy with given transport facilities
and rate structure. Equally important, however, is an approach which attempts to

221

222 LOCATION AND SPACE-ECONOMY

side conditions and other hypothesized relations, this principle implies
various existing location theories, and therefore must be considered a
core element of a general theory of location.

1. Weberian Theory Restated and Generalized

Imagine a general spatial transformation function as

(1) (j){yi, yo, • ■ • , Vk) niASA, niBSB, ■ ■ ■ , mLSL] Xk+i, Xk+2, ■ ■ ■ , Xn) =0

where the variables yi, Vz, • ■ • , Vk represent quantities of various inputs
other than transport inputs; ulaSa, msSB, • • • , miSL represent quanti-
ties of various transport inputs, and Xk+i, Xk+2, • ■ • , Xn represent
quantities of various outputs. In this formulation, niA, rriB, • • • , rriL
represent the weights of various raw materials and finished products
subject to shipment, and Sa, Sb, • ■ ■ , sl represent the distances the
respective raw materials and products are moved. By definition, miSi
represents transport inputs (say, ton-miles of transportation) involved
in the shipment of the raw material I from its source (s) to the site of
production, or the product / from this site to the consumption point (s).
Transport inputs are explicitly set apart in this function, for a study
of their variation is basic to an understanding of the operation of the
space-economy. To facilitate this study, we shall start the analysis at a
very simple level and assume the Weberian problem of transport
orientation.2 Given the locational triangle IJC in Fig. 42 where points /
and J each represent a unique source of a raw material and C the mar-
ket point, find in this enclosed area the location for the point of pro-
duction that minimizes total transportation costs per unit of output.
This problem is subject to the postulate that prices and the required
amounts per unit of output of all inputs (including raw materials)
except transport inputs are invariant with the site of production. The

determine the transport network and rate structure simultaneously with the location
of economic activities. Excellent analyses of an optimum transportation system,
given the geographic distribution of economic activities, are to be found in T.C.
Koopmans, "Optimum Utilization of the Transportation System," Econometrica,
Vol. 17 (July 1949), Supplement, pp. 136-46; and T. C. Koopmans and S. Reiter,
"A Model of Transportation" and G. B. Dantzig, "Apphcation of the Simplex
Method to a Transportation Problem," in Activity Analysis of Production and Alloca-
tion (ed. by T. C. Koopmans), John Wiley & Sons, Inc., New York, 1951, Chaps.
XIV and XXIII, respectively. Also, see M. Beckmann, "A Continuous Model of
Transportation," Econometrica, Vol. 20 (October 1952), pp. 643-60.

It seems very likely that the analysis developed in this chapter will also prove
valid for an optimum space-economy where the character of the transport rate
structure is also viewed as variable and to be determined.

^ Alfred Weber's Theory of Location of Industries, ed. by C. J. Friedrich, University
of Chicago Press, Chicago, 1929, Chap. III.

LOCATION THEORY: MATHEMATICAL FORMULATION 223

only variables are the three transport inputs miSi, mjSj, and mcSc;
since mi and mj are fixed and mc is taken as unity, transport inputs vary
simply because distances vary.

Since total revenue and costs on all inputs (except transport inputs)
are thus fixed, the firm's customary problem of maximizing profits

V

Piyi - P2y2

PkVk — TaMaSa — VBiriBSB

(2)

— rLiriLSL + Pk+lXk+l + Pk+2Xk+2 + • • • + PnXn,

Fig. 42. A locational triangle.

where pi, p2, • • • , Pn are prices, and va, Tb, • • • , tl are transport rates,
is reduced to the problem of minimizing transport costs,

(3) K = rtmisi + rjirijSj + rcmcSc-

Either in maximizing profits or minimizing transport costs, the firm is
restrained by the spatial transformation function of Eq. 1 which in the
latter case becomes simply

(4)

f(si, sj, sc) = 0,

and implies that the firm must choose quantities of transport inputs con-
sistent with measuring the three distances Si, sj, and sc from a com-
mon point P within the locational triangle (see Fig. 42).

A necessary and sufficient condition that within the locational tri-
angle P be a stationary point of K is that

(5)

dK = d{rimisi) -{- dirjmjSj) -f- d(rcmcSc)

224 LOCATION AND SPACE-ECONOMY

or that

ri d{mjSj)

(6)

rj d{misi)

ri _ d(mcSc)

re dimiSi)

rj d{mcSc)

(mcSc) = const.
(mjSj) = const.
(miSi) = const.

re d{mjsj)

Equations (6) obtain since

(7) diriMiSi) = rid{miSi) {i = /, J, C),

for, by definition, ri is fixed. The right-hand terms in Eqs. 6 represent
the marginal rate of substitution between the respective transport
inputs. Of Eqs. 6, any two conditions are necessary and sufficient,
since with Eq. 4 any two imply the third. ^ We thus have three equa-
tions to determine the three unknowns.

For the stationary value to be a minimum, i.e., a point of minimum
transport cost, it is sufficient that the second derivative be positive along
any arbitrary straight line through P.^ Since along any such line

(8) d^'^^ {i = I,J,C),

where u is the arc length on an arbitrary line along which the derivative
is evaluated and where the equality cannot hold for all values of i in
this Weberian problem of fixed Wi and ri,

The transport cost surface is everywhere convex downward and of
necessity there is only one minimum point.

In terms of the space-economy, Eqs. 6 state the important condition
that at the point of minimum transport cost, the marginal rate of substi-
tution between any two transport inputs, the other held constant, must
equal the reciprocal of the ratio of their prices, namely, the corresponding
transport rates.

^ For, since all directional derivatives of the transport cost function are con-
tinuous, the vanishing of the derivative of the function in two directions at point
P (in our case, let us say along the two arcs where sc and sj in turn are constant)
implies that the derivative vanishes in all directions (for example, along the arc
where s/ is constant).

■* If such is the case, it follows that in a neighborhood of P, total cost increases
with any arbitrary movement away from P. Therefore P must be a minimum point.

LOCATION THEORY: MATHEMATICAL FORMULATION 225

The above condition, it must be emphasized, holds when the mini-
mum transport cost point is not at one of the vertices of the triangle,
i.e., when

{rimiY < (vj-mj cos dtj + rkruk cos diu)^

(10) + irjUij sin dij + Vkmic sin Oik)^

{i^j^ k;i,j,k = I,J,C),

where dij and dik are angles cut off by an arbitrary straight line through
i and directions i toj and i to k, respectively.^

In words, if we conceive the problem as one of the equilibrium of
forces,^ the magnitude of the resultant of the forces (locational pulls)
acting from any two corners on the third corner is greater than the
magnitude of the force (locational pull) acting from the third. ^

The fact that any one of the inequalities of Eqs. 10 does not hold is
necessary and sufficient that the point corresponding to i be the minimum
transport cost point. In a movement away from i the marginal rate of
substitution between transport inputs associated with i and transport
inputs associated with any other corner y is then given by:

(11) '-1 ^ ''('"'■^'■)

d{miSi)

rrikSk = const.

Economically speaking, this means that any small movement away from
the transport optimum point along a path for which transport inputs to

^ In Fig. 42 we illustrate for i = J, j = C, and k = I.

^ See Alfred Weber's Theory of Location of Industry, op. cit., pp. 227-32.

' When any one of the inequaUties of Eqs. 10 does not hold, and this can be true of
only one of the three which compose the set, three types of cases may be dis-
tinguished (1) when viin > mp-j + rrihrk. This corresponds to the Weberian cate-
gory where one raw material or the product is dominant; (2) when one of the
inequalities of Eqs. 10 becomes an equaUty. Here, in the direction of the resultant
force, the slope of the transport cost surface vanishes as point i is reached from within
the triangle. However, in all other directions, the directional derivative f^ 0
as i is approached; (3) when one of the inequalities of Eqs. 10 is reversed but type
(1) does not obtain. Here, as in type (1), all directional derivatives 5^ 0 as t is
approached.

As Dean has neatly pointed out, in cases of type (1) Weber's emphasis on weight
loss, purity, and addition of ubiquities is correctly placed; however, in cases of
types (2) and (3) "the only pertinent concerns are relative gross weight and rela-
tive distance." (W. H. Dean, Jr., The Theory of the Geographic Location of Economic
Activities, Selections from the doctoral dissertation, Edward Brothers, Inc., Ann
Arbor, Mich., 1938, p. 19.) Tliis is clear from inequaUties of Eqs. 10. Weber and
others who have employed his concepts have "... seriously overestimated the deter-
minate influence upon location of weight-losing materials, when they are not
dominant, and underestimated the attractiveness of pure materials, which are
never dominant" (ibid.).

226 LOCATION AND SPACE-ECONOMY

or from k is constant involves savings of transport cost on one set of
transport inputs which are smaller than (or in an extreme case, equal
to)^ the additional outlays on the other set of transport inputs.

We now extend the problem and consider the shipment of many raw
materials to a production point and of product (s) to many consuming
points (i' variable distances).

The transport cost equation with f terms

(12) K = TAtnASA + rsniBSB + • • • + rimLSL
is subject to (i" — 2) restraints

(13) 0^(s^, SB, • • • , Sl) = 0 (t = 1, ■ • • , i- - 2),

since the determination of the values for any two of the distance vari-
ables necessarily determines the values of all the others. There are
only two independent variables.

As in the case of thi^ee distance variables, a necessary and sufficient
condition that a point P, not coinciding with any raw material or con-
sumption site, be a stationary point is that

Ti d{mjsj)

Tj diniiSi)

L

XI ricnikSk = const.

■.=A

Of these equations, any two imply the rest.® These two together with
the f — 2 restraints determine the f unknowns.

For the stationary point to be a minimum it is sufficient that the
second derivative be positive along any arbitrary straight line through
P, which is the case as shown above.

However, the above conditions exist only when P does not coincide
with any raw material or consumption site. The sufficient and neces-
sary conditions that P does coincide with any such site are exactly
analogous to those stated above for three distance variables except
that there are (i' — 1) terms in each of the parentheses of inequalities
of Eqs. 10. In a movement away from corner i the marginal rate of

^ This extreme case occurs when one of the inequaUties of Eqs. 10 becomes an
equaUty and when the movement is at the same time along the path of the result-
ant acting on corner i.

^ Except in the extreme case where the curve (defined by 2Za=4 rkmuSk = const.;
j 9^ k 7^ i) along which the first of the derivatives is taken is tangent at P to the
other curve along which the second of the derivatives is taken. In such a case the
two equations are not independent, and a second independent equation must be
introduced.

LOCATION THEORY: MATHEMATICAL FORMULATION 227

substitution between any two transport inputs is then given by:

^ dinijSj)

(15) Tj d{miSi)

L

^ rkVikSk = const.

U^k^i){i,j = A,--- ,L).

In words, Eqs. 14 state that at the point of minimum transport cost, the
marginal rate of substitution between any two transport inputs, total
cost on all other transport inputs being held constant, must equal the
reciprocal of the ratio of their prices, namely, the corresponding transport
rates. This statement applies, however, only to a minimum transport
cost point which does not coincide with a raw material or consumption
site. Such correspondence will exist when the locational pull at any site
exceeds (or in an extreme case, just equals) the combined locational
pulls of other sites. Then, according to Eqs. 15, any small movement
away from such a site along any path for which total costs on all trans-
port inputs but two are constant may involve a saving on expenditures
for one of these transport inputs, but a saving which is less than (or, in
an extreme case, just equal to) the additional expenditures on the second
of these transport inputs. ^°

Equations 14 have some interesting implications. If the P transport
inputs are arbitrarily divided into any two groups, ^^ say, for example,
transport inputs on raw materials and transport inputs on products,
and if for each group a set of constant total cost curves (generalized
isotims) is constructed, the point of minimum transport cost P is a
point of tangency between two of these constant total cost curves, one
from each set. For, employing vector notation, at P

F L

(16) grad K = grad ^ miViSi + grad XI f^^j^jSj = 0

i=A j=G

and by hypothesis each constant total cost curve in groups one and
two is of the form

F

(17) E niiViSi = Ci

i=A
L

(18) XI mjTjSj = C2
i = G

^^ See Chap. 5 for some parallel geometric presentation.

^^ It is relevant to consider this phase of the problem since in reality there are
marked differentiations in transport rate structures. For example, on the whole
raw materials bear a significantly lower rate than finished products; movement in
one direction may entail rates markedly different from those for movement in a
second direction; shipments over one major type of transport facility usually incur
charges different from those for shipments over a second major type. We shall
merely sketch how analysis of substitution between groups of transport inputs may
proceed.

228 LOCATION AND SPACE-ECONOMY

respectively, where Ci and C2 are constants. Since at each point of

F

any curve defined by Eq. 17, grad ^3 miTiSi has a direction normal

L

to the curve, and likewise for grad ^ mjTjSj at each point of each curve

j = G

defined by Eq. 18, these two gradients can be in opposite directions at
a point (as required by Eq. 16) only when the respective curves passing
through that point are tangent to each other. In terms of the familiar
Edgeworth-type box diagram, ^^ where the two families of curves may
be for illustrative purposes alone taken to represent Eqs. 17 and 18, it
is always possible to reduce total costs on each of the two groups of
transport inputs by shifting location from a point not a point of tangency
to some point which is a point of tangency. However, in contrast to
the Edgeworth solution Eq. 16 states more. Of all the possible points
of tangency there is one which is the best, namely, the one at which the
two gradients are not only opposite in direction, but also equal in
magnitude.

For policy purposes, a more generally useful approach in treating
groups of transport inputs is to replace actual weights by ideal weights
a la Weber. A finished product with a representative transport rate
may be taken as the standard for all finished products. The weight
of any given finished product can be adjusted upward or downward
according as its rate is greater or smaller than that for the standard
finished product. One thousand tons of product X moving at a rate
three-quarters that of the standard product would have an ideal weight
of 750 tons. Likewise, ideal weights of raw materials can be derived
with respect to a standard raw material.

It then follows from Eq. 12 that since the rates are the same for all
transport inputs of a group, at P

L

(19) '-^--^^

dT.

mrsi

where Wi* and rrij* are ideal weights.

Under this circumstance, at the optimum location, the marginal
rate of substitution between the two groups of transport inputs must
be equal to the reciprocal of the ratio of the corresponding transport
rates.

Hitherto, transport rates proportional to distance and a space-

^^ For example, see W. F. Stolper and P. A. Samuelson "Protection and Real
Wages," Review of Economic Studies, Vol. IX (November 1941), p. 67.

LOCATION THEORY: MATHEMATICAL FORMULATION 229

economy which is continuous transportationwise have been postulated.
Needless to say, the real world differs considerably from such a fiction
and accordingly it is important to explore needed modifications of the
above analysis. We wish to make a few preliminary comments here.

Typically, with modern transport media, there is an initial terminal
and loading charge incurred by any shipment, invariant with distance
of the shipment, and in addition a line charge where the rate per ton-mile
tends to be a decreasing function of distance. Consider first the effect
of the line charge alone.

Letting, Vi = Vi (si), from Eq. 12 we have:

L

(20) dK = J2 '^iiri + Sir/)dsi

where r^ = -— •
asi

The necessary and sufficient condition that P not coinciding with any
raw material or market site be a stationary point of K is that the
following relations hold at P:^^

Ti + Sir/ dimjSj)

(zlj

^i + SjTj' d{miSi)

L

^ TknikSk = const.

k=A

i,j = A,--- ,L.

Here, in contrast with the situation where transport rates are propor-
tional to distance there need not be one and only one stationary point
and a stationary point which is necessarily a minimum. For along any
arbitrary straight line through P,

L

(22) d^K = X^ mdin + Sir/)(d^Si) + (2r/ + s^r/'){dsiy].

i = A

The first term in the brackets is necessarily non-negative, whereas the
second term can be both positive and negative, though usually nega-
tive.^* Thus d^K can be both positive and negative. For point P to be
a transport optimum point, the usual second-order quadratic form

^^ Provided the two relations are independent. See footnote 9 of this Chapter.

^^ The first term is non-negative since d'^Si is non-negative, and since to,- (ri + Si r/)
is the partial derivative of total transport cost (K) with respect to Sj and thus
necessarily positive.

The expression (2ri' +Siri") measures the rate of change of slope of nii ri Sj
as Si and r, alone vary. In modern rate structures r/ tends to be negative. Also,
Ti" is typically positive but not sufficiently so that s,- r/' > — 2ri'; i. e., miViSiis
typically concave downward as a function of Sj.

230 LOCATION AND SPACE-ECONOMY

conditions for a minimum must be satisfied; if more than one stationary
point satisfy these conditions, then, obviously, by direct calculation, of
these points P must incur least total transport cost.^^

In economic terms, the first-order conditions of Eqs. 21 state that the
marginal rate of substitution between any two transport inputs must
equal the reciprocal of the ratios of their modified transport rates, each
modified transport rate being equal to the actual transport rate for
distance Si adjusted for the saving or added expense per ton-mile result-
ing from the change in the rate that would ensue from a small change in
the distance variable Si.

When, in addition to a line charge, there is an initial terminal and
loading charge, a minimum must exist at each raw material and market
site. An infinitesimal movement in any direction from such a site
involves a significant initial charge which, in reality, far exceeds the
transport cost savings that may be realized by diminishing other
transport inputs. Also, for the same reason, a minimum will exist at
each point corresponding to a break in the transportation network,
where an additional transshipment, loading, or other charge is levied.
However, the conditions that P not coinciding with any raw material
or market site or break be a minimum point remain unaltered. ^^

Thus, terminal and loading charges, transshipment expenses at breaks,
special transit privileges at particular junction points, and other trans-
port rate abnormalities introduce discontinuities into the transport cost
surface. Pictorially the entire transport cost surface is raised by the
sum of all the initial terminal and loading charges, transshipment
expenses, and the like except at each raw material, market, junction, or
special privilege site where the surface is punctured. Each punched out
point is raised not by the sum of these charges and expenses but by the
sum less the charge or expense encountered in movement from the
corresponding site. As a consequence, the space-economy tends to
comprise a hierarchical set of focal points of different degrees of domi-

^^ The full statement of second-order conditions may be found, for example, in
P. A. Samuelson, op. cit., Mathematical Appendix A.

It should be noted, however, that in a specific location problem one can avoid
the cumbersome quadratic form conditions with complicated side relations by
calculating d'^K directly with d~Si and dsi expressed as trigonometric functions of
angles formed at P by the lines from P to Pi and the line along which d'^K is
evaluated.

A necessary and sufficient condition that P coinciding with a raw rnaterial or
market site Pi be a minimum point is that the inequalities of Eqs. 10 extended to
embrace f variables is not valid where ry is replaced by (ry + Sj r/), j = A, ■ ■ ■, L.

^^ However, it is unlikely with modern rate structures that P will be an absolute
minimum point. For full elaboration of this point see Chap. 5.

LOCATION THEORY: MATHEMATICAL FORMULATION 231

2. Inclusion of Market and Supply Areas as Variables

In the previous section the market points to be served are stipulated
beforehand. To the extent that there are many, and particularly if
there is an infinite number in an area of approximately continuous
density, the above location analysis may be said in a sense to embrace
market area theory. But in a major respect such a statement would be
invalid. Market area analysis has as its essential core the problem of
demarcating boundaries and consumers to be served. The problems of
determining transport relations and sites of production are also vital,
but only in a framework where the area itself is a variable. ^^ The
analysis hitherto developed posits a fixed market area and is thereby
inapplicable.

However, it is not difficult to extend the analysis to encompass the
market area (and later, the supply area) as a variable. The initial step
is to state the condition of indifference that defines the market boundary,
which we take to be:

F

(23) r*s* + X) biriSi = T,

i=A

where r* represents transport rate (with regard to the unit of product)
to the boundary line, being invariant with direction;^^ s* represents
radius of circle defining boundary line; A, • ■ ■ , F denote the various raw
materials required; hi represents a constant coefficient indicating the
number of units of raw material i used per unit of product; ri represents
transport rate on a unit of raw material i; and T represents the difference
between the maximum price po the consumer is willing to pay and the
unit costs of production t (excluding transport costs) which are held
constant throughout this section of the analysis. Equation 23 states that
at the market boundary the sum of the transport costs on the unit
product and on the raw materials required to yield the unit product is

^^ See, for example, W. Launhardt, Mathematische Begriindung der Volkswirtschafts-
lehre, Leipzig, 1885, Part III; F. A. Fetter, "The Economic Law of Market Areas,"
Quarterly Journal of Economics, Vol. XXXVIII (May 1924), p. 525; O. Englander,
Theorie des Guterverkehrs und der Frachtsdtze, Jena, 1924; T. Palander, Beitrdge zur
Standortstheorie, Uppsala, 1935, Chaps. IX and XII; E. Schneider, "Bemerkungen
zu einer Theorie der Raumwirtschaft," Econometrica, Vol. Ill (January 1935),
pp. 79-89; E.Hoover, Location Theory and the Shoe and Leather Industries, Cambridge,
Mass., 1937, Chaps. Ill and V; A. Losch, Die rdumliche Ordnung der Wirtschaft,
Jena, 1944, Part II; and C. D. Hyson and W. P. Hyson, "The Economic Law of
Market Areas," Quarterly Journal of Economics, Vol. LXIV (May 1950), pp. 319-327.

^^ When r* is not invariant with direction, the resulting market area is non-
circular. The ensuing analysis, however, is not altered save that visual concep-
tion becomes more difficult.

232 LOCATION AND SPACE-ECONOMY

just equal to the difference between unit costs of production and the
maximum price the consumer is wilUng to pay.

For the moment it is useful to consider the simplified case where each
consumer purchases one and only one unit of product, for which he is
wilHng to pay a maximum price po, {po = T + ir), but for which he
actually pays a delivered cost price (i.e., t plus transport costs on raw
materials and the unit of product he purchases). The resulting total
(consumer or social) surplus is:^^

F f*

(24) Tm - K =^ Tm - Y. miUSi - j rs#(i

i = A d

where m represents number of units produced (consumed), r represents
transport rate (with regard to the unit of product), s represents distance
from P to the consumer, -^{s) represents the quantity consumed inside
circle with radius s and center at P, and the Stieltjes integral is evaluated
over an area with P as center and s* as radius.
Since, by definition,

(25) m = r#(s)
we rewrite Eq. 24:

F

(26) Tm - K = Tm - X) ^i^iSi - nirs

i=A

where rs represents the average unit cost of transporting the product
from P to all consumers.

To maximize surplus,^° we set:

F

(27) d{Tm - K) = d{Tm) - ^ rid{miSi) - d{mTs) = 0

i=A

which with Eq. 23 is subject to (/ — 2) restraints expressing as before
the fact that only two of the Si can be independent. Equation 27 implies:

(28) (i^/c^^■)(^,i = ^, •••,P)

Ti

d{mjSj)

F

rj

d{miSi)

Tm -

X) fkiUkSk — mrs = const.

k=A

r

d(mjsj)

F

rj

d{ms)

Tm -

12 rkmkSk = const.

^^ See below in text and footnote 25 for some discussion of the concept of surplus.

^^ The problem is not to minimize total transport cost subject to Eq. 23 and
the (/ — 2) restraints. For in such a problem P would tend to be a center in a
market area containing as sparse a population as the restraints permit. Clearly,
this is not optimum from a social standpoint.

LOCATION THEORY: MATHEMATICAL FORMULATION 233

Of Eqs. 28 any two independent ones imply all the rest.^^ These two
in addition to Eq. 23 and the (/ — 2) restraints on the variation of the
Si determine the / unknown distances and s*. Equations 28, however,
are only necessary and not sufficient conditions for a maximum point
except where the density of consumption is constant throughout the
region, 22 and perhaps certain other special cases. It is easily seen, for
example, that variation in consumer density over a region may lead to
relative maxima at points tending to be central with respect to districts
of heavy density and to relative minima in sparsely populated districts
in between. As before, second-order conditions can be stated to dis-
tinguish between stationary points, and the best of the maxima can only
be determined by direct computation.

Equations 28 state that at the point of maximum surplus the marginal
rate of substitution between any two transport inputs (transport inputs
on the product being equal to the sum of transport inputs involved in
delivering each individual unit from P) is equal to the reciprocal of the
ratio of their transport rates, the difference between Tm and total costs
on all other transport inputs held constant.^^ Also, it logically follows
that the point of maximum surplus is the point of minimum total
transport cost for serving the market area defined with P as center. ^^

^^ Equations 27 and 28 and others to follow which are based upon Eq. 24 and
others which employ the Stieltjes integral are valid only when the Stieltjes integral

Irs dip (s) is differentiable in the relevant region. The Stieltjes integral rather

than the Riemann is employed since the Riemann is a special case of the Stieltjes
and since the Riemann cannot be used as the Stieltjes can for cases where discrete
consumption points exist at finite distances from the boundary line within a market
(or supply) area.

^^ When such is the case, the point of minimum transport cost on raw materials
is the point of maximum surplus P. Any movement away from P increases average
cost (K/m) and decreases m, and thereby decreases surplus {Tm — K). Thus
P is a maximum.

^^ Any movement away from P resulting from the substitution of a transport
input on one raw material for a transport input on a second raw material may
involve a shrinking or expansion of the circular market area and change in m as
well as change in the other distance variables. The market area itself can of course
encompass raw material sites.

^* Suppose the m consumers, contained in the circle with P as center and s*
as radius, could be served with lower total transport costs from P', necessarily
not forming a circular area around P'. P' would then yield greater surplus. But
since a circular area with center at a production site will, by our equations, al-
ways yield greater surplus than a noncircular area around the same point, a cir-
cular area around P' must then yield still greater surplus than that around P.
But this contradicts the fact that P is the point of maximum surplus. Thus P
must be the point of minimum transport cost for the m consumers.

It also follows that at P average transport cost (K/m) for the given market is

234 LOCATION AND SPACE-ECONOMY

The simplification that each consumer purchase one and only one unit
can now be relaxed. Without specifying the nature of each consumer's
demand function, we can conceive of: (1) a firm, located at one site
only, levying a fixed profit a per unit; or (2) each consumer, except
when he is on the boundary, obtaining per unit product purchased a
surplus jS measured, let us say, by the difference between some given
price and the lower delivered price (tt plus costs of transportation) ; or
(3) society attributing a value y (in addition to the delivered price the
consumer pays) to the consumption of each unit.^^ In each of these
cases and a multitude of others that the reader may wish to construct,
the form of the necessary conditions for equilibrium is not altered though
their content is.

In the first case we maximize

(29) am = j {a -\- p - it) #(s) - K = (a -\- p - 7r)w - K

where p represents the delivered price (excluding the profit charge) to
consumers at s, the distance from P, and p represents an average
delivered price (excluding profit charge) over all units sold. In effect,
Eq. 29 resembles Eq. 26 except that (a -\- p — t) has been substituted
for T. The form of the necessary conditions for a maximum in this
problem resemble those expressed in Eqs. 27 and 28 for the previous
problem, save that (a + ^ — tt) is always substituted for T.^^ How-
ever, the path along which substitution of transport inputs on raw
material i for transport inputs on raw material j can take place is totally
different. In such a substitution, the market area tends to shift, expand,
or contract; each consumer tends to alter the number of units he pur-

at a minimum which, since

d(-) =d Y. binsi + d{rs) =0
\m/ i=A

yields the relations,

n d{bjSj) p (■ ^]- ^ i)(i ■
Tj d{hiSi) Yl h^kSk +rs = const.

= A,--

■,F).

These relations which are implied by, but do not imply, Eqs. 28 may facilitate test-
ing the stationary character of points.

^^ We fully appreciate the unreality of these conceptions. However, since eco-
nomics has not yet reached the stage where the welfare of a group of consumers
can be quantitativelj'^ evaluated, and since the validity of the relations emphasized
in this paper is independent of the nature of any welfare function, these simple
conceptions suffice for the immediate purpose.

^® Also, a must be subtracted from T in Eq. 23.

LOCATION THEORY: MATHEMATICAL FORMULATION 235

chases; and thus p and m tend to change. To determine a path of
substitution becomes more difficult.

In the case of (2) or (3) above, we merely substitute ;3 or 7 for a in
all the equations. The form of the necessary equihbrium conditions
remains the same, although again the content differs.^^

Heretofore, we have posited that each raw material originates at a
single point. The analysis can now be extended to embrace raw material
supply areas, each composed of any number of originating sites. To do
so with respect to any raw material i requires the substitution of
y TiSi d\l/i{si) or miTiSi for the term miTiSi in the above equations where
\l/i{Si) is the quantity of raw material i supplied within a circle of radius Si
and P as center. With the introduction of a supply (or purchasing) area
for any raw material the problem is changed in a way exactly analogous
to the way it was changed with the introduction of a market area. The
reader will find, if he cares to reformulate the problem mathematically,
that the fundamental form of the substitution relations remains un-
changed, though these relations bear upon different paths and have
different content. ^^

3. The Analysis Extended to the Case of Many Producers

Having generalized the analysis to embrace a market area, variation
in consumption patterns over space, and a supply area for each raw
material where such areas overlap so that a point may be both a market
site and a site at which several raw materials originate, we now proceed
to allow more than one production site. In doing this we could maintain
the postulate of constant unit production cost (excluding transporta-
tion). We would then obtain a statement implying geographic patterns
such as those depicted in Palander and Hoover"^ where, for example,
we may have at one and the same time production: (1) at raw material
sites, each serving a district of consumers, (2) at each point along a
closed elliptical-shaped curve, each point serving the consumers in the

^^ Where no raw materials are required in production, Eqs. 23 and 27 become
r*s* = T and d{Tm — K) = d(Tm) — d{mrs) = 0. The radius s* is thus invar-
iant with production site. However, if we wish to express equihbrium condi-
tions in terms of transport inputs and not in terms of the vanishing of the partial de-
rivatives in the x and y directions, we can divide transport inputs on the product
into three or more subdivisions of transport inputs, each corresponding to a set
of consumers asymmetrically located. The analysis would proceed as above where
there are several transport inputs to consider.

^^ Of course, substitution can now involve change in one raw material supply
area vis-a-vis change in another raw material supply area, or in the market area,
and so forth.

2^ T. Palander, op. ait., Chap. VI; and E. M. Hoover, op. cit., pp. 53-55. Also
see Fig. 44 below and the accompanying discussion in Chap. 11.

236 LOCATION AND SPACE-ECONOMY

hinterland along its pole line only, and (3) at each consumption site
contained in the elliptical-shaped curve, each such site meeting its ovm
needs only.

However, we shall not dwell upon this unrealistic situation which
can be considered a special case of the more general type which allows
variation in the unit cost of production (as Losch does). A region may
be conceived as divided into several market areas, each served from a
production site and bounded on all sides. As before, any boundary hne,
not a boundary hne between two producers, is defined by a condition
corresponding to Eq. 23, namely,

F

(30) a, + r^*s,* + X bi^iSi, = po, (m = 1, • • • , 7?),

where a^ represents the marginal production costs (excluding trans-
portation) at site P^, and Sj> represents the distance between P^ and the
site of raw material i.^'^

Our problem is to maximize, let us say, social surplus:

7?n ^ m{y -\- p — t) — K
(31)

F V V _

= w(7 + p-7f)— X)Z) ^'''ii^^iSi^ - J2 m^rs^,

where p represents the average delivered price on all units produced
(consumed) ; if represents the average unit production costs on all units;
mt> represents the total weight of raw material i used by producer n;
m^ represents the total units (weight) of product consumed in market

area served by producer ix; and rs^ = — / rs^ #m(sm) represents the

average transport cost per unit of product in shipping the product from
P„ to customers in the corresponding market area.

^•^ We have defined the boundary in terms of marginal production cost plus
unit transport cost rather than average unit production cost plus unit transport
cost. This ensures an optimum spatial arrangement for society as a whole, but
involves a net loss for each producer when he is producing on the faUing section
of his average cost curve and when dehvered price is based on marginal cost at
P^. The reader may substitute ir^ (average unit production cost) for a^ in Eqs. 30
and in subsequent equations which define boundary conditions among producers.
This, however, would not be consistent with an optimum space-economy, though
of course it would be consistent with an optimum space-economy subject to the
restraint that dehvered prices be based on average vmit production cost.

Ttiis problem, which hes outside the scope of this book, has been treated at length
in the hterature on welfare economics. Refer, for example, to A. Bergson, "Sociahst
Economics," in A Survey of Conteviporary Economics, ed. by H. S. EUis, Blakiston,
Philadelphia, 1948, pp. 424-28, and the literature cited therein.

LOCATION THEORY: MATHEMATICAL FORMULATION 237

First, it should be noted that since there is an infinity of market sites
(consumers) in our wholly or partially continuous areas, one may treat
an infinity of variables, the distance from any consumer to a correspond-
ing producer being a variable. However, it is immediately possible to
reduce the infinity of variables to a finite number through considering
each market area as a whole and introducing boundary conditions
between the market areas of any two producers. Imagine that any two
producers shift their common boundary within any small element of
area without affecting the market areas and outputs of other producers.
Let the first obtain dnip new sales from this element while the second
lose dmp (= —dm^) sales. If ym is a maximum, such a shift should not
reduce total costs when through some pricing arrangement the intensity
pattern of consumption of all other elements in the two market areas is
held unchanged, as it can be. For, if this were not so, the resulting
decrease in total costs would make possible an increase in m and thus
in ym. Therefore, with such a shift, for d(ym) = 0, we must have:

diirpnip) + d(rpnipSp) + d[7np ^ biriSipj
(32)

= —diiTpmy) — diryniySv) — dim:, ^ biriSipj

where Wp and Wy represent the average unit production costs (excluding
transportation) of producers at sites Pp and Py, respectively. Since

diiTpmp) = (Tp dmp,
(33)

(dwymy) = (Tydmy = —ay dmp,
and since

d{mpSp) = Sp^ dnip,
(34)

d(mySy) = Sy^ dmy = —Sy^ dmp,

where Sp^ and Sy^ represent the distances from Pp and Py, respectively,
to any point on their common boundary line, Eq. 32 becomes, after
cancelling dmp,

F F

(Tp + TpSp^ + X) biViSip = 0-, + TySy^ + X) biViSiy
(35) *=^ '■=-'

{p 9^ v){p,V = I,' • • ,7]).

Equations 35 furnish the boundary (indifference) conditions dividing
a market domain between any two producers, each boundary repre-
senting a locus of points of equal deUvered prices.

238 LOCATION AND SPACE-ECONOMY

The problem is now reduced to one involving a finite number of vari-
ables, namely, to that of maximizing ym (where the r? market areas are
defined by Eqs. 30 and 35), subject to i7(/ — 2) restraints on the variation
of the distances, which express for each producer the fact that in choosing
his production site only two of his / distance variables can be inde-
pendent. Thus ym can be considered as a function of independent
coordinates in 2j? dimensional space.

In this new framework, setting d(y7n) = 0, we obtain:

(36) '■■= y^'^^'l

evaluated along the path

F V V
k=A /x=l M=l

= const.

Up 9^ kfx 9^ iv).

Of Eqs. 36,^^ each of which holds in 2?/ — 1 independent directions,
and yields 2?? — 1 independent equations, any two together holding in
2t? and only 2?? independent directions are required to provide necessary
and sufficient conditions for a stationary point. These provide the
equations for determining the 2?? independent unknowns. Again,
complex second-order conditions are required to distinguish among
maxima and other stationary points, and only direct computation will
yield the best of the maxima points.

Economically speaking, Eqs. 36 state that in a small variation
of any production site from its corresponding position in a geographic
pattern of production sites which yields maximum surplus, the mar-
ginal rate of substitution of one transport input for another must be
equal to the reciprocal of the ratio of their transport rates, social surplus
plus total revenue less total production costs and less total cost on all
other transport inputs being held constant. It should be emphasized
that variation of any production site tends to entail variation in all
production sites, as well as market areas, and so forth.

In order to avoid further complications in detailing the above rela-
tions, it has been postulated that each raw material originated from one
fixed site only. However, just as we have treated many producers serv-
ing a spatially extended market, we can treat many producers procur-
ing their supplies of each raw material from a spatially extended supply
area. New unknown boundary equations are introduced, but so are new

^^ Analogous to the second of the Eqs. 28 the ratios in Eqs. 36 should be viewed
as involving transport inputs on product [e.g. d{m^s^) vis-a-vis d(mi^iy) or vis-a-vis
d(mySy)] as weU as transport inputs on raw materials.

LOCATION THEORY: MATHEMATICAL FORMULATION 239

conditions to determine them. The reader can easily develop analysis
in this direction.

One further salient point must be noted. The derived boundary
(isotant) Eqs. 35 contain market area theory, developed by Launhardt,
Fetter, Palander, Hyson, and others.^^ j^ the usual case, only two
producers, each at a particular site, are considered, and transport costs
on raw materials are neglected or assumed to be zero or already ac-
counted for in the price which each producer charges. Where marginal
cost is the basis for determining the price at the factory^^ then Eqs. 35
are relevant and they will yield (1) straight line boundaries, when
dp = a^ and fp = Ty, (2) hyperbolic boundaries when ap 9^ ay and Vp = r^,
(3) circular boundaries when cxp = ay and fp 7^ r^, and (4) Descartes
ovals or hypercircles as boundaries when ap 9^ ay and Vp 9^ r^.^*

4. LoscH Market Area Analysis Encompassed

In this section we ^^dsh to demonstrate that the principle and general
analysis thus far developed logically embraces the Losch system of
market areas.

In Chap. 2 we have sketched Losch's elaborately developed theory
of market areas based upon empirical evidence and deductive reasoning.
Assuming raw materials equally and adequately present at any site,
population uniformly distributed and having like tastes and of Uke
stamp, full technical knowledge and other resources available to every-
one, transportation possible in all directions at a fixed rate, and complete
freedom of entry and exit for producers, Losch has demonstrated how
a regular hexagonal net of market areas will evolve. ^^ It can be shown
that such a net of market areas, given Losch's restraining postulates, is
implied by the general relations in Eqs. 35 and 36.

First, the assumption of raw materials equally and adequately present
at every site reduces transport inputs on raw materials to zero and

^^ See footnote 17.

^^ Where another method of pricing is employed, the process of maximizing
social surplus ym wall be constrained by such a method and will jdeld boundary
equations similar to Eqs. 35 except that the factory price charged the peripheral
consumer by each producer will substitute for marginal cost. The boundaries
^'ielded by these new equations /will still be of the same tj^pe as those derived b}'
traditional market area analysis.

^^ Where the transport rate is a function of distance, then the hyperbolic, cir-
cular, and hj-i^ercircular boundaries become distorted and need to be described
bj' more complex functions.

^^ Losch, op- cii-1 Part III. Also contrast vidth the empirical materials in; W.
Christaller, Die zentralen Orte in Siiddeutschland, Jena, 1935; and E. Ullman, "A
Theory of Location of Cities," American Journal of Sociology, Vol. XL VI (May 1941),
pp. 853-64.

240 LOCATION AND SPACE-ECONOMY

eliminates transport inputs on any raw material as a variable. Equa-
tions 36 can now be simply stated:

d{mpSp)

m(y -\- p — tt) — ^ mf,rs^ = const.

(37) 1

d{mvSy)

p 9^ fJi 9^ V

p, V = 1, ' ■ • , n

where r^ = Vp = r, a constant rate per unit product; or the marginal
rate of substitution between any two transport inputs on product from
any two production points is unity along a path where social surplus plus
total revenue less total production costs less total costs on all other
transport inputs is held constant. Boundary Eqs. 35 become

(38) (Tp + rsp(^ = (T, + rs.o

p,v = 1, • • • ,n

Losch's assumption of free entry and exit of producers ensures that
there will be no profits, namely, that price will equal average unit
production cost for any producer. Furthermore, in Losch's schema the
delivered price is equal to average unit production cost (factory price)
plus transport cost (the consumer is responsible for the transportation
of the product he purchases). Hence, the boundary (indifference) line
between the market areas of any two producers, which is a locus of
points of equal delivered price, is given by:

(39) TTp + rSpO - TT, + rs,o

Consider for the moment all situations, whether optimum or not,
where Eqs. 38 are satisfied. If the Loschian schema necessitating
the conditions of Eq. 39 is to be simultaneously satisfied, then, by
subtraction:

(40) TTp — O-p = TTy — O-y

p,V= 1, ■ • • ,71

This states that the difference between average and marginal costs is
the same for all producers. Such can be the case, when each producer
confronts the same regular demand and U-shaped average cost functions,
if, and only if, as Losch posits, the outputs of all producers are of the
same size; and thus where each produces at a point where the demand
curve is tangent to the average cost curve. Furthermore, with a uniform
distribution of consumers and producers of the same size, Eqs. 38 as
derived from the general problem ensure a straight fine boundary
between any two producers which is a perpendicular bisector of the

LOCATION THEORY: MATHEMATICAL FORMULATION 241

straight line joining the two. Of the various regular^^ geometric shapes
which possess such boundary hues, only the equilateral triangle, square,
and regular hexagon can exhaust any given domain, as required by Losch.
Equations 38 do not constitute sufficient conditions for an optimum
areal distribution. A distribution, under Losch's restraint that all
producers be of the same size, must also satisfy the relations in Eqs. 37
to ensure necessary (though still not sufficient) conditions for a maxi-
mum. We do find that the distribution of a domain into squares verti-

ri

1^-

U.

n

Fig. 43. Change in market boundary pattern with shift of production points
in a square matrix.

cally and horizontally aligned as in Fig. 43 (unbroken lines) does,
because of pattern symmetry, satisfy Eqs. 37 as well as boundary
Eqs. 38 and 39 and Losch's other postulates. However, we can easily
demonstrate that this distribution is not a maximum one.

Let all producers and their associated consumers in the second
row of Fig. 43 shift their positions equally to the right by a small
arbitrary amount so that the dashed lines become the boundary lines
separating the market areas of producers in the second row. Since
any consumer is no more distant from his respective producer in the

^® Competition compresses irregular shapes into regvdar areas.

242 LOCATION AND SPACE-ECONOMY

new situation than in the old, m and m^ are unaffected and hkewise
a^^, TTy,, and the profit of each producer. It also follows that since con-
sumers of like tastes and stamp are uniformly distributed throughout
the area, the same variables are unaffected if no consumers are shifted
when all producers in the second row and their mutual boundary lines
are shifted as above. ^^ But this new situation does not fulfill Eqs. 38
and 39. These equations require that the boundary separating any two
producers be a perpendicular bisector of the line connecting the two,
and require that, given the new (shifted) pattern of production sites,
the market area of any producer in the second row take on an irregular
hexagonal shape as indicated for the producer at P;, in Fig. 43. Therefore
the new pattern of production sites Avith irregular hexagonal market
areas for the second row producers involves a greater m and a greater
social welfare than the same pattern of production sites with rigid
square market areas and thus than the old pattern of production sites
with square market areas vertically and horizontally aligned. Hence
this latter square pattern is not a maximum one. Likewise with the
pattern where a domain is divided into equilateral triangles.

A distribution into regular hexagons, too, satisfies Eqs. 37 because of
pattern symmetry. However, unlike the square and equilateral triangle
distributions, it does represent a maximum, given producers of like size.
Any substitution of transport inputs on the product of one producer for
that on the product of another which entails small shifts of a number of
production sites leads to six-sided polygons which are not regular hex-
agons^* (as can be easily verified by the reader). Since a regular
hexagon is more efficient (requires less transport inputs) in serving any
area of given size than any other six-sided polygon, the regular hexagonal
pattern corresponds to a minimum transport cost, or maximum surplus
arrangement. Thus, given Losch's restricting postulates, the regular
hexagonal, and only the regular hexagonal, pattern is implied by the
derived general location principle and analysis.^^

^^ Producers in the second row will, it is true, serve different consumers; and
some consumers will become worse off, while others better off. However, overall
consumption and social welfare will not have been changed by the shift.

^^ This contrasts with the situation where in certain directions small shifts in
production sites transform a square or equilateral triangle pattern into a pattern
of more-sided polj^gons which is more efficient than the initial pattern.

^^ However, the Losch hexagonal pattern is not in general consistent with an
optimum space-economy if Losch's restraining postulates are relaxed.

Imagine average unit production cost falls in such a way with increase in output
(engendered by extending the radius of a circular market area) that the sum of
average unit production cost and average transport cost of product to consumers
contained in any size market area yields a cost function which decreases only
slightly as the size of the circular market area increases. Also, for simphcity,
imagine each consumer buys one and only one unit. The market domain then can

LOCATION THEORY: MATHEMATICAL FORMULATION 243

5. Agriculture Location Theory Embraced and Generalized

Another major branch of location theory, stemming from the work of
von Thtinen, developed by Aereboe and Brinkmann and most recently
by Dunn,'*o has as its object the explanation of the geographic pattern
of agricultural activities. Immediately, it is seen that, since the
von Thtinen problem concerns itself with the formation of zones each
devoted to the cultivation of a particular crop or combination of crops,
the general location statement hitherto developed must be extended to
treat more than one commodity if it is to encompass agricultural
location theory.

As in the previous analysis, we shall abstract from price changes,
prices being determined and fixed beforehand. However, this position
is much less tenable in the case of agricultural location theory. Since
several commodities vie for the purchasing potential of the city market
and since significant changes in the outputs of the various agricultural
commodities may be involved in spatial (zonal) shifts, it is only by price
changes, direct and indirect, and their resultant effects upon the several
outputs that appropriate and complete adjustments can be made to
locational shifts. To account for repercussions through price changes,
as a truly general equilibrium system would, is, however, beyond the
scope of this chapter.^ ^ Since agricultural location theorists have tra-
ditionally omitted price changes from their formulations of concrete

be overlain with non-overlapping circles of different sizes, tangent to one another,
and which at the same time satisfy boundary conditions in Eqs. 38. Any unfilled
gap between any three circles, each tangent to the other two, can be partially filled
with a circle of still smaller size, even to the extreme where the circles themselves
become infinitesimal in area. For practical purposes, however, we can allow slight
distortions of the circles well before such an extreme is reached. Since a circle can
more efficiently serve a given size area (consumers) than a regular hexagon (see
Losch, op.cit., pp. 76-78) and since, in this example, the distortions of circular form
are minor and the variation in total average cost (including transport) associated
with the various sizes of circles is but slight, each size circle is more efficient than
any regular hexagon which may be derived as optimum size. Hence, a pattern of
circles, slightly distorted to exhaust the area, serves the area more efficiently than
any pattern of regular hexagons.

^•^ J. H. von Thiinen, Der isolierte Staat in Beziehung auf Landwirtschaft und
Nationalokonomie, Hempel and Parey, Berlin, 1895; F. Aereboe, Allgemeine land-
wirtschaftsliche Betriebslehre, P. Parey, Berlin, 1923; E. T. Benedict, H. Stippler
and M. R. Benedict, Theodor Brinkmann' s Economics of the Farm Business, University
of California Press, Berkeley, 1935; and Edgar S. Dunn, Jr., The Location of
Agricultural Production, University of Florida Press, Gainesville, 1954.

*^ The reader may refer to the non-operational general equihbrium statements,
involving the simultaneous determination of price and spatial structure, in Losch,
op. cit., 1st ed., 1940, pp. 57-63, and Dunn, op. cit.. Chap. 2.

244 LOCATION AND SPACE-ECONOMY

equilibrium conditions, there is no inconsistency in demonstrating how
their analysis is implied by our general analysis.

We shall proceed from the simple to the more complex. Imagine a
single city, surrounded by land of uniform quality, consuming commodi-
ties k -\- 1, ■ ■ ■ , n whose prices {pk+i, • • • , Pn) are set. Our problem
is to determine the location and the quantities {nik+i, • • • , w„) of the
agricultural commodities which will be produced, given the freight
rate (rk+i, • • • , r„) and cost function for each commodity, 7rA;+i, • • • ,
TTn, and (Tk+i, • • • , an representing their average unit and marginal
costs, respectively.

Discarding the assumptions of constant yield of a given crop per acre,
regardless of distance from the city, and of constant average unit cost —
assumptions which have characterized the algebraic statements of Brink-
mann, Losch, and Dunn^^ — ^^g have for the given commodity a unique
marginal rent function, which traces out the amount of rent which
would be yielded by each unit circumferential band of land as we pro-
ceed radially outward. Each marginal rent function declines continu-
ously since local price (net of transport cost) falls off with distance
from the city market and since, too, as a result intensity per unit of land
falls off given uniform fertility of land and hence a production function
invariant with distance.*^ However, since only one commodity (or one
combination of commodities that is fixed proportionally and thus can
be viewed as a single commodity) can be cultivated on any given piece
of land, it is necessary to think in terms of stretches of land devoted to
the cultivation of one and only one commodity.

Consider total rent for society

(41)

R = J2 \£^r(Pr - TT,) dW - £ TrVrS dJv]

under the simplification that no transportation costs are incurred on
raw materials, labor, and other inputs'*'* and where the Riemann integral

^^ As previously noted, Dunn has pushed the analysis on to the individual firm
level and has shown the inconsistency of these assumptions for firm analysis.
Thereby, he has been able to sketch graphically the approximate character of the
necessary modifications of analysis on the industry level. A revision of his mathe-
matical statement, however, was not made.

''^ Under the usual assumption that the farmer is not incurring loss and, thus,
is operating at a point on the rising section of his marginal cost curve.

^^ The simpUfication is made merely to facilitate presentation. As will be shown
later, another term representing transport costs on raw materials and on labor
can be brought into the brackets of Eq. 41; the analytical technique remains
unaffected.

It should be noted that von Thiinen and Brinkmann would insist upon two
additional terms, a negative one for transport costs on industrially produced goods

modity, Tr representing the intensity of production of r associated with
the element of area dW^^ Setting dR = 0, we obtain

LOCATION THEORY: MATHEMATICAL FORMULATION 245

/ is taken over the area devoted to the production of the rth com-

le intensity of p
Setting dR =

d f Tjs dW

(42) ^< = -4

' dJT.sdW

evaluated along the path

X; f r^iPr - Trr)dW - E f "^rVrS dW = COUSt.

ihj = k + I,- • • ,n).

Immediately, it is seen that these relations imply a concentric circular
zone (in the extreme case approaching a line) pattern of cultivation.
First, zones, whatever their shape, must be contiguous to each other and
to the city market. Otherwise, there would be empty spaces which
would then permit a shifting of some zone closer to the city market
thereby reducing transport costs on one commodity without affecting
transport costs on any other commodity. But this is inconsistent with
Eqs. 42. Second, since the transport rate is independent of direction,
the contiguous zones must also be concentric and circular because
Eqs. 42 also imply pattern symmetry.'*^

Of Eqs. 42, only n-k are independent, and they determine the n- k
variable boundaries separating the zones in which the n-k commodities

used as inputs and shipped from the city market, and a positive one for transport
costs of the agriculturally produced inputs from any element of area to the city
market. The latter positive term represents the differential price advantage any
element of area has over the city market in procuring agricultural raw materials
(including food for labor), since the difference between the price at the city market
and the local price at the element of area for any such raw material is the cost of
shipping that raw material from the element of area to the market.

■^^ We employ the Riemann integral throughout this section since intensity of
production on any element of area (farm) is a critical issue in agricultural loca-
tion theory. The use of the Stieltjes integral would conceal the intensity variable.
Also, since agricultural production tends to assume a continuous character, the
Stieltjes integral would in any case tend to reduce to a Riemann integral. However,
in the latter part of this section where a continuous market area is considered, the
use of the Stieltjes integral would have definite advantage in an extension of the
analysis to include cities as discrete consumption points viithin the market area,
provided the integral is differentiable in the relevant region.

^® Thus the analysis in its more vital aspects can be reduced to the problem of
examining relations along any straight line through the point representing the
city market.

246 LOCATION AND SPACE-ECONOMY

are produced. These equations are only necessary conditions for maxi-
mum rent. Elaborate second-order conditions are required to dis-
tinguish a maximum from other stationary points. With marginal rent
functions for all commodities plotted on a single graph (marginal rent
measured along the ordinate), the absolute maximum obtains when the
commodity produced on any given unit of land corresponds to the mar-
ginal rent function having the highest ordinate for that unit of land.*^

Economically speaking, Eqs. 42 state that for maximum rent the
marginal rate of substitution between any two transport inputs, each on
a particular crop, must equal the reciprocal of the ratio of their transport
rates, the sum of rent on all other commodities plus the total difference
between sales value and production costs for the two crops being held
constant. It must be borne in mind that the path along which the sum
is held constant may involve shifts of the zonal boundaries of other
crops as well as of the tAvo explicitly considered and may call for changes
in the intensity of cultivation and in unit production costs of other crops
as well as of these two. Thus the path may be quite complex.

Despite this complexity, Eqs. 42 imply relatively simple boundary
determining equations. Imagine that only two boundaries shift, namely,
those Sp and s,, distances from the city, representing respectively the
outer boundaries of the zones producing crops p and m- Further, let at
least one other zone intervene, p being nearer the city. From Eqs. 42
where i = p and j = (j. we w^ould have with a small change of Sp and
corresponding change of Sp,:

(43) VpVpSp^dSp = -r^VpS^^dSp

where 27rs ds substitutes for dW and 27r is cancelled. Also from the
restraint of Eqs. 42 defining the path, w^e have, after cancelling 27r,^^

Vpipp — Trp)Sp dSp — Tp+i(pp+i — Tp+i)Sp dSp + T^ipp, — Tp.)s^ ds,,
(44)

- T^+i(p^+i — 7r^+i)s^ dSf, + rp+iTp+iSp^ dSp + r^+iT^+is^^ ^^^ ^ q

where Tp+i and Xp+i are evaluated at Sp, and F^+i and tt^+i at s^. Simi-
larly two equations of like form are obtained when in Eqs. 42 we let

^^ Given any position other than the absolute maximum already described, there
is always at least one way in which a substitution of transport inputs on one commod-
ity for transport inputs on another can increase rent. This may involve a shift
of some element of area from the cultivation of one crop to the other, the formation
of new, or deletion of old, zones, and a corresponding increase or decrease in the
number of boundary variables.

^* In Eq. 44 are contained all the elements which change in the restraint governing
the path, namely, the sales revenue and production costs associated with crops
p, p + 1, /i, and ju + 1 and transport costs associated with crops p + 1 and /x + 1.
Since the path defines a constant sum, the total of these changes must be zero.

LOCATION THEORY: MATHEMATICAL FORMULATION 247

i = p and ^ = /x + 1, and where as before distances Sp and s^ only are
varied. These latter two equations, with Eqs. 43 and 44, yield alge-
braically:*^

(45) r^s^r^ - r^+is^r^+i = T^{j)^ - tt^) - r^+i(p^+i - x;,+i).

Equations of the type of Eqs. 45 represent another way of expressing
necessary conditions for maximum rent.^° Each such equation states
that with a small shift of the boundary line between any two zones, the
change in over-all transport costs for the two crops is equal to the change
in the sum of the difference between sales value and production costs
for each of the two crops. Or, if in each of Eqs. 45 the first term on the
left side is carried over to the right and the last term on the right to the
left, the necessary conditions for maximum rent are that with any
infinitesimal shift of the boundary between two zones, marginal rent on
the one crop must equal marginal rent on the other. In graphic terms,
the marginal rent functions of the two crops must intersect or be tangent
at the boundary line.^^

These boundary conditions, implied by Eqs. 42, are identical with
those obtained by Losch and Dunn,^^ except that Vi and tt^ are con-
stants in Losch's and Dunn's schema whereas they are variables in our
formulation. Treating intensity and unit production costs as variables
not only is realistic but has the distinct advantage, as Dunn noted, of
facilitating the unification of theory for the industry and for the indi-
vidual farm unit.^^ Postulating Vi and in as constants throughout a
zone precludes any such thing as a firm adjustment to a lower or higher
local (net) price for product and rent payment for land and, thus, in
essence is inconsistent with customary firm analysis. However, with Vi
and TTi as variables, we derive that, since a farmer will produce on the
rising section of his marginal cost curve up to the point where his
marginal costs {(n) just equals the local (net) price (pi — riSi), dai/dsi =
— Vi; and therefore, dVi/dsi = —ridVi/dai. As dVi/dai is positive on
the rising section of the marginal cost curve, dVi/dsi is negative. Thus
intensity of cultivation falls off in any zone from land unit to land unit,
or farm unit to farm unit with increase of distance from the market. ^^

*^ By eliminating the three unknowns dsp, ds^, and Sp.

^^ These equations are also obtainable when more than two boundaries are shifted.

^^ This condition also characterizes stationary points other than the absolute
maximum, including relative maxima when each crop is restricted to production
within one zone onl3^

^^ Losch, op. cit., pp. 28-32; Dunn, op. cit., Chap. 2.

^^ Though, of course, when there is no definite bunching of firms with respect
to similarity of product or output mix, there may not be any justification or mean^
ing in the distinction between firm and industry.

^* However, because of differences in production cost functions, transport rates,

248 LOCATION AND SPACE-ECONOMY

And thus we have an adjustment to the fall in local price appropriate
to both firm and industry analysis.

When within any zone it is desirable to consider each firm by itself,
because outputs or output combinations for various firms are too
heterogeneous or for other reasons, each firm can be considered an
industry. More boundary variables are introduced but so are more
equations to determine the boundaries. Boundary Eq. 45 comes to
hold between firms whether or not they produce a homogeneous output
as well as between groups of firms (industries), each corresponding to a
zone. Thus firm and industry analysis are mutually consistent in this
spatial framework, and in the extreme case, may be considered as one
and the same thing.

Hitherto, we have excluded consideration of transport costs on raw
materials. Such can easily be introduced explicitly into the analysis by
inserting into the brackets of Eq. 41 the term

■J T.TrhirriSidW,

where Si is the distance of the source of raw material i from the element
dW. With this term inserted, marginal rates of substitution between
transport inputs on raw materials, and between transport inputs on a
raw material and transport inputs on a product, are obtainable from the
required revision of Eqs. 42. It should be noted that in general the
zonal boundaries will no longer be concentric circles because pattern
symmetry will have been lost.^^

If, however, raw material i is supplied not from a single point source
but from an area, Si becomes a variable for any given element, dW^
representing the distance from dW of the element of the raw material
supply area, dU, which furnishes the i required by dW. As a result
additional equations of the order of Eqs. 42 result which relate, for
example, transport inputs on raw material i used by crop r,^^

X

TiSirdU,
u

and market prices, intensity per land unit, however measured, need not fall ofif
from zone to zone or from a farm unit in one zone to a contiguous farm unit in the
next zone.

^^ Hence, it becomes necessary to think in terms of a boundary defined by an
equation rather than by a radial distance alone. Analysis along a straight line is
in general no longer valid.

Tins is, of course, equivalent to I TrbirSi dW.

to / Trb

LOCATION THEORY: MATHEMATICAL FORMULATION 249

and transport inputs on raw material i used by crop r + 1,

riS,v+i dU.

I

+1

These determine the boundary Hnes cutting off the portion of the supply
area of raw material i which serves each crop.^^

A still more general framework, encompassing more than one city
market, can be handled. Consider first a relatively small number of city
markets close enough to compete for the potential crops of at least some
land. Here an industry (agricultural activity) and its corresponding
zone of cultivation must be defined not only in terms of the crop pro-
duced but also the city market served. The number of industries is
thus multiplied but so also is the number of Eqs. 42 to determine
boundary lines. ^^

Where the number of city markets becomes large and attains the
approximate continuity of a market area, we reach the most general
type of framework, that is, many markets each consuming many
products, many producers of any given product, many raw materials
used by each producer, and many sources (a supply area) of each raw
material. An alternative approach to this most general framework
follows from the analysis of Sect. 3. There we considered many pro-
ducers of a single commodity demanded by a population spread over an
area and requiring many raw materials, each furnished from a supply
area. By introducing many commodities in this framework we reach
the same problem as when we inject the market area into the Thiinen
type of framework. It seems, however, more desirable to generalize
from the Thiinen type of framework. In this framework prices are
explicitly assumed as given and we avoid the problem, which thus far
has not been satisfactorily handled, of relating social surpluses, or
consumer utilities, or satisfactions derivable from several commodities.^®

^^ Also, dR = 0 implies equations guaranteeing the appropriate pairing off
of elements of area in any portion with the elements in the crop zone which the
portion serves.

^^ In the case of only two cities, in which the ruling market prices are different,
there will tend to be two hinterlands, one corresponding to each city. In each
hinterland, zonal pattern symmetry will exist with respect to the cultivation of
any given crop when no raw materials are employed. However, on the two sides
of any smaU stretch of the border separating the two hinterlands, in general different
types of crops wiU be cultivated at different intensities per unit of land, though
marginal rent is the same.

^^ In essence, the Thiinen framework assumes away this subjective evaluation
problem. It treats only objective, measurable elements and, on the basis of these
latter alone, yields a spatial pattern of economic activity and land use.

250 LOCATION AND SPACE-ECONOMY

This problem would confront us if we were to proceed from the analysis
of Sect. 3.

Generalizing the Thiinen analysis w^e have for total rent:

(46) , -.

- J VHrTrSr ^^ - J ^ ^rhirfiSi dW ,

where Tht represents the intensity of the effective demand for product
T in the element dH of the consumption market area at the price pr,
Pr being fixed for each element but varying from element to element;
TTr represents the average production cost of r at the element dW, where
an element dW may serve more or less than one element dH; and Sr
represents the distance from the element dH of the element dW produc-
ing the T which is demanded at element dH.

Setting dR = 0 yields again equations of the order of Eqs. 42, though
more complex. These, nonetheless, can be easily set down by the reader.
A whole new set of substitution relations, however, is embraced. For
example, there may be substitution between transport inputs on the

product T going to city A, I TrSri dW,^^ and transport inputs on the

product T going to city B, I TrSrB dW, or on the product t -\- 1 going

to city B, etc. More generally, the consumption market area may be
divided into any two or more meaningful parts and transport inputs
on product r going to any a part can be considered vis-a-vis transport
inputs on any other product going to any area or meaningful part of an
area, or on any raw material going to any production area or meaningful
part of a production area of any good,^i and so forth. In this way, within
the hypothesized framework of the space-economy, one can make the
most comprehensive statements concerning equilibrium conditions and

JThtSt dH where the integral /
H Jh

^'^ Equivalent to / TffrST dH where the integral / is evaluated over the area of

city A.

^^ As a consequence, if attention is paid to a given crop alone without regard
to destination or market, multiple zones of cultivation will appear, each in gen-
eral asymmetrical with respect to any selected focal point.

Where a given good is produced at a finite number of plants, as in the analj'sis
of Sect. 3, the cultivation zone or production area of that good has in effect been
reduced to a finite number of points, the intensity of cultivation or production
at other points being zero. Any one point may still be a potential site of production
of all other goods, a potential source of all raw materials, and a potential market
for all goods.

LOCATION THEORY: MATHEMATICAL FORMULATION 251

substitution relations among transport inputs. This the reader can
easily do by setting down equations of the order of Eqs. 42 that, however,
correspond to total rent as defined in Eq. 46.

6. Concluding Remarks

We have treated what may be called a continuous space-economy,
continuous transportationwise and continuous to some extent at least
with respect to market and supply areas. This stands in strong contrast
to the highly discontinuous realistic one. We have noted how modern
rate structures in effect yield a transport cost surface punctured at
terminal sites, the transport cost for each such point being significantly
below that for points contiguous in space. The fact that in reality
transport routes radiate from one point not in all directions but in only
a few, particularly in modern industrial societies with a relatively small
number of fixed trunk routes, means that the transport cost surface is
punctured linearly, too. Each locus of points corresponding to an exist-
ing transport route lies significantly below any locus of points through
which no transport route courses. The presence of junction and trans-
shipment points, of loading and unloading facilities, of various auxiliary
transport services, and of the whole complex of agglomeration econ-
omies, as well as of varying topography, back-haul and other abnormal
rates, special transit privileges, and other factors, imposes still larger
discontinuities and distortions upon the transport cost surface derived
from simplifying assumptions. Hence we discover in the space-economy
of reality hierarchies and different degrees of dominance in sets of focal
points and channels of movement. ^^ To a large extent substitution
among transport inputs is not in the small but rather in the large, en-
tailing geographic shifts over substantial distance from one focal point to
another. To this extent the preceding analysis is not directly relevant.
However, the extent of discontinuity must not be exaggerated. The
demarcation of many agricultural and raw material supply areas and
many industrial market areas can be handled in terms of substitution in
the small. Likewise with many industrial location problems. It should
be kept in mind that the extent to which spatial continuity and dis-
continuity exist is yet to be determined or even approximately estimated
by empirical study.

^^ For an excellent pictorial presentation, refer to "Interregional Highways,"
House Document 379, 78th Congress, 2nd session, Wasliington 1944, Fig. 22. For
discussion, see A. Hawley, Human Ecology; A Theory of Community Structure,
New York, 1950; D. J. Bogue, The Structure of the Metropolitan Community, Ann
Arbor, 1949; and W. Isard and G. Freutel, "Regional and National Product Projec-
tions and their Interrelations" in Long-Range Economic Projection, National Bureau
of Economic Research, Studies in Income and Wealth, Vol. 16, Princeton University
Press, Princeton, 1954, pp. 427-471.

252 LOCATION AND SPACE-ECONOMY

To analyze the space-economy in terms of geographic shifts in the
large, from focal point to focal point, is highly desirable and will be
partially attempted in a future volume. In this chapter we have tried to
demonstrate the usefulness of the concept of transport inputs in deriving
conditions for the efficient operation of a space-economy. A basic
principle — in a sense an intuitively obvious one — has emerged, namely,
that the marginal rate of substitution between any two transport inputs
or groups of transport inputs, however the transport inputs or groups
of transport inputs may be defined, must equal the reciprocal of the
ratio of their transport rates, social surplus (however defined) less
transport costs on all other transport inputs being held constant. This
principle impHes a large part of existing location theory. Weberian
transport orientation is embodied in such a principle, and with this
principle we are better able to take transport orientation out of the nar-
row geometric framework in which it has hitherto been confined and
determine the optimal transport point for the more generafized case when
many raw material and market points are involved. The principle, too,
encompasses all market (and purchasing) area theory, implying the
customary boundary conditions which separate the market areas of
producers (and hinterlands of focal points). Likewise, Losch's spatial
designs are embraced as well as the von Thiinen type of agricultural
location theory. Both the Losch and the von Thiinen types of theory
can be generalized to harbor a much broader range and a more reaHstic
set of situations.

More important, this general principle fuses the separate partial
location theories. It thus serves as a basic core of a general location
theory from which, for the most part, existing location theories are
derived as special cases of the most general situation, embracing many
market points, each consuming many commodities, each of which is
produced by many producers, each of whom uses many raw materials
and inputs, each of which is furnished by a supply area.^^

Perhaps most important is that this principle allows existing location
theory to be stated in a form comparable to that of production theory.
By incorporating transport inputs into the transformation function, and
thereby yielding a spatial transformation function, we can extend exist-
ing production theory so that to a large extent it embodies the location
factor explicitly. At the same time location theory can now consider
change in a number of parameters. For example, the relation between

^^ For example, one can derive Weber's theory of transport orientation by making
assumptions such as the existence of a single market point, consuming one unit of
one commodit}^, produced by a single producer, using two raw materials, each
obtainable at a single source.

LOCATION THEORY: MATHEMATICAL FORMULATION 253

economies of scale and the number and geographic distribution of plants
can be examined through substitution between transport inputs and all
other inputs as a whole; or the relation between the spatial extent and
capital intensity (time extent) of production, through substitution
between transport inputs and capital inputs. Thus a more compre-
hensive framework emerges for both types of theory.

Chapter

11

Partial Graphic Synthesis
and Summary

We must now take stock. What has been accomplished? In attack-
ing this question we shall, where the use of illustrations makes it
possible, attempt a finer integration of the materials than has thus far
been achieved. However, in order to keep repetition at a minimum
we shall be brief. We shall assume that the reader has thoroughly
digested previous materials.

In Chap. 2 we have envisaged the general theory of location and
space-economy as embracing, within a temporal (dynamic) framework,
the total spatial array of economic activities and their interrelations,
both aggregatively and atomistically, with attention paid to the geo-
graphic distribution of inputs and outputs and to the variations over
space in prices and costs. This theory eschews the narrow framework
of Marshallian doctrine. In many ways it includes modern general
equilibrium theory as but a special case; in other respects it becomes
synonymous with a broadly defined trade theory and a broadly defined
theory of monopolistic competition.

The empirical materials of Chap. 3 strongly testify to the fact that
there are significant regularities associated with the distance variable
and that in many important respects there is a basic structure to a
space-economy. To understand this structure and to analyze the cur-
rent and anticipate the future functioning of a space-economy, we
adopt a substitution approach as initially suggested by Predohl. The
substitution approach, covering substitution in the large as well as
substitution in the small, is familiar to economists and needs no further
comment.

254

PARTIAL GRAPHIC SYNTHESIS AND SUMMARY 255

Coupled with the substitution approach is the concept of transport
inputs. A transport input is defined as the movement of a unit weight
over a unit distance. In the growth of a simple nucleus of population,
transport inputs early become basic elements. They permit the in-
creased productivity which accrues from (1) postponing and mitigating
diseconomies from excessive agglomeration and the forces of diminish-
ing returns and (2) exploiting the unequal distribution of natural
resources. These same basic forces operate in our modern world-
economy with its complex hierarchy of cities and spatial distributions
of population.

As with capital inputs, transport inputs can be thought of as
derived and as indicating roundaboutness in the production process.
Corresponding to transport inputs there are such concepts as spatial
extent of production, space discount, and space preference. The
transport rate is the price of a transport input. The determination of
this price may be accounted for by a conventional demand and supply
analysis for transport inputs. A fall in the price of a transport input
induces a spatial lengthening of production and may be associated (as
is historically the case) with both a scale and substitution effect.

In Chap. 5 we couple the substitution approach and the concept
of transport inputs in order to restate and reformulate transport-
orientation doctrine. The problem of finding the transport optimal
point reduces to a problem of finding the correct substitution points
between pairs of transport inputs. This is so whether one treats
simplified transport rate structures or the complicated ones of reality,
whether one assumes uniform transport facilities radiating in all direc-
tions from all points or the discontinuous and heterogeneous network
of reality, whether one analyzes a process using one localized raw
material or many, and whether one considers one market point or
many. It is demonstrated how the various geometric, graphic, and
physical solutions propounded are translatable into substitution points
among transport inputs. The most rigorous presentation of the
solution to the transport-orientation problem in terms of substitutions
among transport inputs is found in Sect. 1 of Chap. 10. The formula-
tion of the transport-orientation problem as well as other problems of
location in terms of substitution among transport inputs, where trans-
port inputs are viewed as any other set of inputs embraced by a
transformation function, has the decided advantage of permitting at
least a partial fusion of production theory and location theory.

Much of the argument of Chap. 5 centers around the locational
triangle, as has historically been the case with transport-orientation
analysis. However, the argument does embrace more than one market

256

LOCATION AND SPACE-ECONOMY

point and a single source of each of two raw materials, n market
points and raw materials are considered, and in Sect. 1 of Chap. 10
this extension of the argument is most rigorously presented. Yet
another useful way of presenting the problem of transport-orientation
when no weight is dominant and when there exist many market points,
or in essence an area of consumers, has been developed by Launhardt
and Palander. Their graphic presentation can be extended to portray
in a forceful manner the interrelations and influences of the various
location factors.

In constructing his basic diagram which we largely depict in Fig. 44,
Palander, following Launhardt, in effect postulates (1) the absence of

Fig. 44. The Launhardt-Palander construction.

the various agglomeration economies and of geographic variations in
the prices of various inputs and outputs except those resulting from
transport cost on the product and on the two raw materials con-
sidered, and (2) uniform transport facilities radiating in all directions
from all points, i Given ilfi as the only source of the first raw
material and M2 as the only source of the second raw material, at
what points should production occur to serve consumers? Starting
with consumer at C, we construct the locational triangle CMiMo and
the corresponding weight triangle OM1M2 erected upon the side M1M2
of the locational triangle. 0 is ori'fe of Launhardt's poles. We circum-
scribe a circle around the weight triangle and connect the pole 0 with
the point of consumption C by a straight line. P, the point of inter-
section of the pole line OC and the circumscribed circle, is the desired
location, the transport optimal point, for serving the consumer at C.
1 Palander, op. cit., pp. 143-146.

PARTIAL GRAPHIC SYNTHESIS AND SUMMARY 257

Take another consumer at Ci. We could construct a second loca-
tional triangle CiMiMo- Its weight triangle, too, would be OMiM^-
We find that the relevant pole line OCi coincides with pole line OC.
Since the point of intersection with the unchanged circumscribed circle
remains the same, P is the logical production point to serve not only
C but also Cj. Likewise, it can be demonstrated that P is the optimal
transport point for all consumers along the pole line OCi from P to Ci
and beyond.

Take still another consumer at Co- We could construct the loca-
tional triangle C2M1M2. Its weight triangle erected upon side M1M2
would as before be OM-^Mo, since the relevant weights have not
changed. The corresponding circumscribed circle therefore remains
the same. Thus, to derive the production point for serving C2, we
need not construct locational triangle C2M1M2. We simply connect
the pole 0 and C2 with a straight line and locate P2, the point of
intersection of the pole line OC2 with the circumscribed circle.

Take a fourth consumer at C3. Connecting C3 with the pole 0
yields pole line OC3 with M2, the source of the second raw material,
as the point of intersection and thus the logical production point for
C3. For C4, M2 is the transport optimal point, too. This is so even
though the pole line OC4 does not intersect the circumscribed circle
at M2 but rather at a point outside the locational triangle C4M1M2.
At M2 the external angle of the locational triangle is less than the
corresponding angle of the weight triangle. This fact indicates pro-
duction at M2.

As with C3 and C4, we derive that consumption points C5 and Cq
should be served by a producer at a raw material source, this time Mi ,
the source of the first raw material. When we consider the consumer
at C7, once more we have the situation where the pole line OC7 does
not intersect the circumscribed circle at a point within the locational
triangle C7M1M2. Angle conditions yield C7 as the logical point of
production for the consumer at Cj.

Smce it can be demonstrated that the breakdown of situations for
the consumer field lying below a straight line coursing through Mi
and M2 is an exact reflection of the breakdown of situations above
the line, we can generalize. For pU consumers in the horizontally
dashed area, including the boundary lines, production should be at M2.
For all consumers in the vertically dashed area, production should be
at Ml . For all consumers along any given pole line, production should
be at the intersection of that pole line and the relevant circumscribed
circle. For each point of consumption in the dotted area, production
should be located at that very point. Thus we obtain an infinite

258

LOCATION AND SPACE-ECONOMY

number of logical production points contained within and lying on the
two relevant circular arcs of Fig. 44. In this way the graphic presenta-
tion of transport-orientation when a weight triangle generally exists
is extended to embrace an area of consumers, whether or not consumers
are actually spread continuously throughout the area. 2

Fig. 45. The effects of a weight change.

Despite its unreality, this derivation of Launhardt and Palander
can yield fruitful insights, as Palander has demonstrated. For ex-
ample, consider the effect of technological change. Such change may
result in the use of a new raw material source, or the use of a new
process of production utilizing the same or different raw materials, or
some other change in the transformation function. Suppose we take
a situation where the efl&ciency in the use of a material, say coal, is
increased, ceteris paribus. Coal is located at M^ in Fig. 45 drawn
largely from Palander. ^ Before the technological change, the spatial
production and market patterns are given by the heavy lines. (For
the moment ignore the unmarked area around L.) The new spatial
patterns consequent to the reduction in the weight of coal per unit
product are given by the dashed lines. The shaded areas indicate
changes. The area tributary to M^ (the source of coal) contracts,
that to M2 increases. The area in which production is market-
oriented also expands. Simultaneously, all the production points de-

2 When a weight triangle does not exist, i.e., when the weight of a raw material
or the product is dominant, production for all consumers whether they are con-
centrated at a point or scattered takes place at a raw material source (when a
raw material is dominant) or always at the market (when the product is dominant) .

3 Palander, op. cit., p. 153.

PARTIAL GRAPHIC SYNTHESIS AND SUMMARY 259

pendent on markets along a pole line shift position, e.g., P3 loses its
market along a pole line and P3' comes to serve consumers along such
a line.

It could have been the case that before the technological change coal
was dominant. The entire area under consideration would have been
tributary to Mi (vertically dashed). With technological change coal
loses its dominance, and this Palander-type diagram suggests a theo-
retical location pattern that might emerge. Or, given the situation
as initially depicted, the technological change might have completely
eliminated the use of the raw material at Mi. The resulting new
pattern would not have a vertically dashed area.

In this way and others the Launhardt-Palander construction can
give insights into locational shifts. It particularly points up some
of the dynamic locational implications of changing weight relations
whether the changes are due to technological advance or to other forces.
It also can be employed theoretically to anticipate some of the
locational effects of the exploitation of a new raw material source*
or market potential.

In Chap. 6 we treat the case of labor and other similar types of
orientation. The shift of a production process from the transport
optimal point to a cheap labor point lying within the critical isodapane
of the relevant locational figure is depicted as the substitution of trans-
port outlays for labor outlays. In addition to permitting both a more
direct attack upon the problem when several cheap labor points exist
and a more comprehensive presentation of the interrelations of labor
outlays and other types of production outlays as well as of the different
kinds of labor outlays, the substitution approach allows a closer tie
with production theory in general.

Somewhat similar statements relate to orientation to a cheap power
site, to a low tax site, to a low rent site, or to any site at which a
significant saving in a given type of production outlay (or increase
in revenue) is obtainable. Substitution between transport outlay
(which tends to vary systematically with distance from a given
reference point) and any other outlay or revenue (which tends to vary
haphazardly with distance from a reference point), whether or not
depicted in terms of outlay-substitution, iso-outlay, and other lines
of like character, is a fruitful alternative to the critical isodapane
technique. It can be extended, though with decreasing returns, to
embrace meaningful groups of outlays (and revenues).

Incorporation of labor or a similar type of orientation into the
Launhardt-Palander construction is easily achieved. In Fig. 45 let

4 In this connection, see Palander, op. cit., pp. 157-162.

260 LOCATION AND SPACE-ECONOMY

L be a cheap labor location. For each of the innumerable points in the
market region, we construct a locational triangle as before. Around
each locational triangle, we construct the critical isodapane with
reference to point L. We group together those locational triangles
within whose critical isodapanes point L falls. The market points
corresponding to these locational triangles together comprise the
consumer market served by the cheap labor location. They are indi-
cated in Fig. 45 by the unmarked area centering around L. The
boundary line between the market area tributary to L and that to M-^
is a locus of market points corresponding to the locational triangles
whose critical isodapanes course through point L.

Clearly, if the labor cost advantage of L increases, the consumer
market tributary to L expands. At one extreme the total area becomes
tributary to L. At the other extreme, as the labor cost advantage of L
diminishes, the consumer market served by L is entirely regained
by Ml.

In like manner, we can insert other cheap labor sites, cheap power
sites, cheap tax sites, etc. into Fig. 45 and determine the consumer
points, if any, which might be served by these sites. In this way, the
transport-orientation problem already extended to embrace many
market points, is converted into a more generalized location problem
which considers the pull of sites possessing advantages with reference
to factors other than transport and relative spatial position.

Viewed from a different angle, the location of production at L plays
up another phase of location theory, namely, market area analysis.
The market area variable does not crop up when we consider the typical
Weberian problem where the market is concentrated at a point. Point
L either is or is not the best site at which to produce to serve that point.
When many market points exist, the identification of those points to be
served by production at L becomes a problem of defining a boundary
line cutting across a market region. In Fig. 45 we need not adopt the
cumbersome procedure of constructing for each market point a loca-
tional triangle and its critical isodapane to determine points to be
served by L. Rather we can view L and Mj as two points, each
producing at constant though different unit costs and competing for
the market in Mj's hinterland. This is a customary market area
problem.

In Chap. 7, we treat market and supply area analysis. Once again
it is demonstrated that all such analysis can be embraced by a general
substitution framework involving substitution among transport inputs
and among outlays and revenues. This can be shown whether we
consider a single isolated monopolist, a set of competing firms pro-

PARTIAL GRAPHIC SYNTHESIS AND SUMMARY 261

ducing at constant or variable unit costs, a single consumption market,
or several competing markets offering the same or different prices for
a commodity. The substitution relations among various types of
transport inputs in the analysis of market and supply areas are
particularly pointed up in Sect. 2 of Chap. 10.

As already indicated, the introduction of a cheap labor point into
the Launhardt-Palander construction can simultaneously introduce
a market area problem. It is instructive, however, to inject the market
area problem into this construction in another way.

::L L-

:;i;:::::;4;Mi:::::::::::::i:::

Fig.

46. Division of a market region between two sources of each of two raw
materials.

Suppose we have a situation where, owing to the addition of
ubiquities, the weight of product becomes dominant. Production
becomes market-oriented, except for a small area tributary to L. The
dotted area bounded by the two circular arcs in Fig. 45 expands to
include almost the entire region. See Fig. 46. Into this situation we
now allow a second deposit of each raw material, which we designate
Ml' and M2 respectively. Since we still postulate that the transport
rate structure is proportional to weight and distance and since we take
the price (unit cost) of the first raw material to be less at ilf 1 than at
Ml' we obtain a hyperbola as a boundary line (the locus of equal
delivered prices) . It delineates the market area of industrial consumers
tributary to source Mi and that tributary to Mi'. Additionally, since
we take the price of the second raw material to be the same at both
M2 and M2', we obtain a straight line boundary which marks off
the market areas of industrial consumers served by each of these two
sources. Except for the small district of household consumers oriented

262

LOCATION AND SPACE-ECONOMY

to production at L, we demarcate altogether four districts of industrial
consumers, each employing the same combination of raw materials, but
each procuring them from a different combination of sources. Since
production is market-oriented, there is spatially coincident with each
district of industrial consumers a district of household consumers. It
is hardly necessary to repeat that, if an industrial consumer procures

Fig. 47. Spatial production patterns: two sources of each of two raw materials,
one labor location.

any raw material from a source other than indicated, he will find it
profitable to switch his allegiance. In doing so, he will effect the
socially desirable substitutions between transport inputs on raw
material from one source and the other and between production outlays
at one source and the other.

Figure 46 presents a rather simple and conventional case of market
delineation. It is fruitful to investigate a more complicated situation.
Suppose we eliminate the ubiquities required in production and postu-
late that equal weights of the two raw materials and finished product
are pertinent. Further, let us follow Palander and add to Fig. 45, as
he has done, an additional source of the first raw material at M^'; and

PARTIAL GRAPHIC SYNTHESIS AND SUMMARY 263

of the second at M2'. Accordingly, we derive Fig. 47 which is largely
taken from Palander. Once again, aside from the district of household
consumers served from the cheap labor point L, we obtain four
groupings of industrial consumers, whose respective districts of house-
hold consumers are indicated by double-weight solid lines. ^ This
time, however, production in each of these four districts need not be
market-oriented.

In district a, households are served by producers who utilize raw
materials from sources M^ and ^2'.^ Production takes place at the
raw material sources, at market points, and at points where relevant
pole lines intersect with an arc of a circle circumscribed about the
relevant weight triangle. At the lower right, district a is partly bounded
by district c. In district c producers at raw material sources, market
points, and intersection points procure their raw materials from Mi'
and M2. District a is also partly bounded by district d. In district d
all producers are market-oriented; they obtain raw materials from Mi'
and M2'. Because this pair of raw material sources is separated by a
greater distance than any other relevant pair, its competitive ability
is not so great.''' The district of household consumers which this pair
of sources can indirectly serve with economy is the most restricted of
the four. It contrasts with the pair of sources, M^' and M2, which
being the closest of any relevant pair serves indirectly the largest
district of household consumers.

Finally at the upper right, district a is bounded by district b. Since
M2 and M2' are equally distant from Mj and since they supply the
second raw material at the same price, it is indifferent whether produc-
tion at Ml is based upon either of these sources of the second raw
material. Therefore, the subdistrict served by Mi can be part of either
a or b.

Figure 47 neatly illustrates how a market region of household
consumers can be indirectly, via industrial producers, assigned to the
market areas of competing raw material sources. For example, in
district c none of the ultimate consumers procures his raw materials
directly from Mi' and M2. Rather, household consumers are served
by industrial producers who directly consume the raw materials from
Ml' and M2 and who at times may be located at the point of ultimate

5 These lines are loci of equal delivered prices to household consumers.

6 Sources Mi and ilf 2' also furnish the raw materials for the production of the
goods consumed by households in the unmarked area tributary to L.

■^ Palander's diagram and discussion suggest that he postulates for the first raw
material the same price at each of its two sources, and likewise for the second
raw material.

264 LOCATION AND SPACE-ECONOMY

consumption. Thus, boundary lines which pertain here to the division
of a market region of ultimate consumers are more complex than those
discussed in the first two sections of Chap. 7 and illustrated in Fig. 46.
This is so because a complex transport-orientation problem as well as
a market area problem is involved. Boundary lines come to be defined
by substitution points which have reference to transport inputs on the
finished product, transport inputs on the first raw material, transport
inputs on the second raw material, and, if a cheap labor location
exists, labor outlays and transport outlays.

It is to be noted that when we consider the market of industrial
producers who are supplied with raw materials we find it to be
discontinuous because of the transport-orientation problem which is
involved. Industrial production occurs only on the four pairs of
circular arcs, or within the areas contained by them, and at L. The
sole section at which there is effective competition between alternative
sources of a raw material, of the sort illustrated by Fig. 46, is along
the straight line from M-^ to A. Along this stretch, industrial producers
are indifferent as to the source of their second raw material. ^

As is widely recognized, the sharpness of the boundary lines presented
in Fig. 47 as well as in preceding figures is much exaggerated.
Producers who compete for the household market do not behave
according to the criteria which have been implicitly assumed. They
generally do not establish at the factory a single price based on unit
cost and applicable to all customers. They typically are able to
influence price, to discriminate among consumers, to induce consumers
by advertising, price cuts, or other means to shift their allegiance
from a competitor. Producers relocate at times, take cognizance of
each other's reactions, form coalitions, set prices and quotas. All these
types of monopolistic and oligopolistic behavior tend to invalidate
the simple, clear-cut boundary lines customarily depicted. At best
boundary lines are blurred and tend to degenerate into overlapping
zones.

Even if we were to allow zonal types of boundaries, cross-hauling, and
market interpenetration in the above diagrams, we must recognize that
the geographic patterns of production which they would depict have
limited validity. When we treated, in Sect. 3 of Chap. 7, the simple
case of locational equilibrium along a line where pricing policy and firm

8 When the raw material sources are pulled farther apart, even this competitive
stretch disappears. See Palander, o-p. cit., Figs. 27 and 28.

It should also be noted that along stretch BD of Fig. 47, the two producers
located at M\' and M2' compete for household consumers in a way consistent
with the simpler framework discussed in Chap. 7.

PARTIAL GRAPHIC SYNTHESIS AND SUMMARY 265

location were variables, as they are in reality, we noted that many
types of location patterns were possible, depending upon one's set of
initial assumptions. Even the application of the powerful tools of game
theory does not, at the present time, cast additional light on the elusive
problem of rational behavior for a group of firms in terms of their
pricing and location policies. Thus we must contantly bear in mind
that Fig. 47 and similar diagrams pertain to a situation which abstracts
from interest conflicts, undercutting and retaliation, advertising
strategies, collusive action, market encroachment, and similar
phenomena characteristic of firm behavior.

In addition to exaggerating the sharpness of boundary lines and the
detenninateness of locational equilibrium, we greatly overstate in
Fig. 47 the number of producers. We derive this unrealistic result
because Fig. 47 is a theoretical construction which abstracts from a
number of forces, especially economies of scale both in production and
transportation.

In Chap. 8 we treat economies of scale as a subset of agglomeration
factors, localization economies and urbanization economies being the
other two subsets. We recognize that these three subsets are not always
clearly distinguishable from one another. Section 1 of Chap. 8 demon-
strates how the economies of scale factor can be frequently embodied
in a substitution of transport outlays for production outlays in gen-
eral. The achievement of these economies of scale can be visualized
in many instances as a movement along an outlay- substitution line
on to a lower iso-outlay line.

The impact of economies of scale can be easily portrayed. In Fig.
47 the smallest scale of output is associated with those producers
who are market-oriented. Granted significant economies of scale, we
have postulated that output in each of the three largest market-oriented
production areas will be concentrated at a single (central) point.
These points are designated as I^, h, and /o in Fig. 48. ^ We also
assume that the smallest market-oriented production area, the d dis-
trict, does not have a demand sufficient to justify a production point
within its bounds when economies of scale exist.

In Fig. 47, the production points serving consumers along one and
only one pole line also operate at a small scale. In a setting in
which there are significant scale economies, we postulate that along
any one arc production will be concentrated at a single point. Thus

9 These points as well as others which are identified are only roughly located.
We do not attempt in this and the following figures to determine a set of produc-
tion points and market areas consistent with a given scale economy function.
These figures are for illustrative purposes only.

266

LOCATION AND SPACE-ECONOMY

we account for points Pi, P2, P3, P^, and P5 of Fig. 48. 10 Finally,
we posit that the scale of output at each of the raw material points
and the cheap labor point in Fig. 47 is large enough to warrant the
retention of each as a production point in Fig. 48.

Fig. 48. Spatial production patterns: scale economies introduced.

Thus Fig. 48 depicts a situation reflecting the impact of the scale
variable. It portrays a much more realistic production pattern
than that of Fig. 47. Because only a relatively few production points
are justified, each production point comes to serve a market area.
The cases of production for a single market point or a single pole
line are eliminated. Market boundary lines, of the type illustrated in
Fig. 46, become significant once again, as they are in blurred form
in reality. However, since there are many competing production sites
surrounding any given producing location, the boundary line determin-
ing the market area served by this location is a connected series of
different types of boundary stretches, where each stretch pertains

10 We do not allow for any production points on the circular arcs enclosing dis-
trict d and on the lower circular arc of district b. There is no pole line market
for the former and too small a market to justify the latter.

PARTIAL GRAPHIC SYT^THESIS AND SUMMARY 267

to the competition between the given location and one other producing
site. 11

As the next step in our graphic presentation, imagine that the sources
of the two raw materials are multiplied many times so that for all
practical purposes the two raw materials become ubiquities, each
available everywhere at the same price. Further, take consumers
of like taste and stamp to be uniformly distributed, and adopt Losch's
various other assumptions and conditions pertaining to his market area
analysis, which we have noted several times. One can easily visualize
how the pattern of production sites takes on a uniform character such
as to yield the logically derived pattern of hexagonal market areas. 12
Thus, from this angle, the Losch derivation can be considered as a
special case of the Launhardt-Palander construction into which the
factor of economies of scale has been injected. In his derivation
Losch has pointed up the conflicting pulls of the scale variable and
the transport outlay variable; in essence, the basic substitution re-
lation between transport outlays and production outlays. As with the
Launhardt-Palander construction, the Losch derivation is implied by
the principles governing substitution among transport inputs once
Losch's set of assumptions is admitted. This is demonstrated in Sect.
4 of Chap. 10.

We proceed with the summary discussion of Chap. 8. Section 2
of this chapter treats localization economies, a second subset of ag-
glomeration economies. In contrast to scale economies which are in-
ternal to a firm, localization economies are external to a firm. They
are contingent upon the spatial juxtaposition of several firms of like
character. They are reflected, for example, in lower cost service inputs
when such juxtaposition permits the more efficient use of an auxiliary
repair facility.

The realization of localization economies involves a physical move
and additional transport outlays by at least one firm. At least one
firm must and will find it profitable to substitute transport outlays
for production outlays in general. Exactly which firm or firms will
relocate and exactly which points will prove to be the points of ag-
glomeration are questions which revolve around a complex interplay
of historical and institutional forces relating to decision making and

11 The interested reader may construct these boundary lines for himself. He is
reminded that the market areas of the raw material sources and the cheap labor
site are greater than the corresponding ones of Fig. 47. Production sites which
effectively compete with the raw material sources and L are farther removed from
them in Fig. 48 than in Fig. 47.

12 See Fig. 51 below for an illustration of several nets of hexagonal market areas.

268

LOCATION AND SPACE-ECONOMY

rational behavior by the firm. We have as yet been unable to un-
ravel the concrete manifestations of this interplay. The clearest picture
of the degree and pattern of localization and of specific substitutions
emerges when we abstract from these forces and approach from a social
welfare standpoint the problem of industrial planning for a completely
undeveloped region.

Fig. 49. Spatial production patterns : localization and scale economies introduced.

For illustrative purposes, we posit as only one of many possible
situations that depicted by Fig. 49. (For the moment ignore the
small circles which are not filled in.) In this situation, firms Ii, P2,
and Pi have relocated around M2' to realize localization economies;
P3 and I2 around Mj ; P4 , P5 , and I^ around M2 ; and none around
Ml' and L.

Into the locational pattern of Fig. 49, we can introduce the forces
associated with urbanization economies, the third subset of agglomera-
tion economies. To do so compels us to expand into a multicommodity
framework since, as already noted, urbanization economies refer to
those savings in production outlays which are realizable when firms
producing a variety of commodities agglomerate around a point. As
we have discussed in Sect. 3 of Chap. 8, urbanization economies, like

PARTIAL GRAPHIC SYNTHESIS AND SUMMARY 269

localization economies, reflect a complex interplay of historical and
institutional forces. The factors governing the specific localities at
which different degrees of urbanization economies become obtainable
are beyond the pale of our current analytic frameworks. We can only
make the simple statement that for many firms the advantages of

•v^

■ 00 +

Fig. 50. Spatial production patterns: urbanization, localization, and scale
economies introduced.

locating at an urban center outweigh the advantages of a non-urban
location. The decision to settle in an urban area thereby involves
substitutions among various outlays and revenues.

We portray the impact of urbanization economies with the use of
Fig. 50. In Fig. 49 we have already noted the fairly concentrated
geographic pattern of production to which localization economies
lead when only a single commodity is considered. Suppose in Fig. 49
we also depict, for the given region or nation, the geographic pattern
of firms producing a second commodity. These firms are represented
by the small circles which are not filled in. Their pattern also
reflects localization economies. A geographic pattern of firms pro-
ducing a third commodity could be marked in Fig. 49. Likewise,
for a fourth, fifth . . . and nth commodity. To avoid confusion these
have not been presented in Fig. 49.

In Fig. 50 urbanization economies act to bring together the firms
represented by the small black and white circles which would other-
wise be separated as in Fig. 49. In some instances the firms pro-
ducing the second commodity shift to a center of production of the
first commodity ; in other instances the firms producing the first com-

270

LOCATION AND SPACE-ECONOMY

modity shift to a center of the second. To these sets of locations
on Fig. 50 we have added sets of locations of firms producing a third,
fourth, and fifth commodity, represented respectively by small black
squares, crosses, and white triangles. In the absence of urbanization

Fig. 51. A simple Losch system of nets of market areas. (Source: Adapted

from A. Losch, The Economics of Location, Yale University Press, New Haven,

Conn., 1954, Fig. 28.)

economies, many of the firms producing the third and fourth com-
modities would be situated differently. Not so, however, with the
firms producing the fifth commodity, whose locations are indicated
by white triangles. They are not led to relocate because of the
pull of urbanization economies. Moreover, they are not very sensitive
to localization economies. They retain a fairly dispersed pattern.

A second, less satisfactory way of graphically depicting the impact
of urbanization economies is to follow Losch. In Fig. 51, we repro-

PARTIAL GRAPHIC SYNTHESIS AND SUMMARY 271

duce one of Losch's inconsistent diagrams. This diagram involves
the superimposition of several nets of hexagonal market areas, i^ As-
sociated with each net is a set of commodities resembling each other
only in that they have market areas of the same size and are produced
at each of the same set of production points. The several nets of
market areas are arranged with at least one production center in
common and so that, according to Losch: (1) the greatest number
of locations coincide; (2) local effective demand is at a maximum;
(3) the sum of the shortest distances between industrial locations
is at a minimum; and, as a consequence, (4) shipments and total
length of transport lines are at a minimum. To this diagram we have
added Losch's twelve major radial transport routes; and we have
indicated his six sectors rich in number of production sites (shaded)
and his six sectors poor in number of production sites (unshaded) .

Perhaps the most serious deficiency of this Loschian construction is
that it yields different sizes of concentrations of industrial activity and
thus jobs at various production centers, and yet it postulates uniform
distribution of consuming population. It is beyond the scope of this
volume to modify the Loschian argument in order to eliminate this
inconsistency. However, it is clear that the Loschian diagram would
need to exhibit for each commodity greater concentrations of market
areas and producers about the central city (the common production
center) in order to square with the central city's high level of industrial
activity and large laboring population. We construct Fig. 52 merely
to suggest such greater concentrations. In Fig. 52 at the lower right
we have indicated a second, though less important center, at which
production activity is concentrated. Also, below and to the right of
this second center is a set of zones which is to be ignored for the present.

Because Losch's construction implies a relatively high density of
laborers and thus population at the core, the size of a market area in
square kilometers necessary to generate sufficient demand for a com-
modity to justify production is much smaller at the core than at a great
distance from the core. Further, at a great distance from the core,
market areas must be much larger because not only are production
sites and industrial population fewer in number but also, as a logical
consequence of differential industrial population, agricultural activity
is less intensive and agricultural population more sparse than in the
immediate hinterland of the central city. Thus, we obtain a pattern of
distorted hexagons (if we insist on maintaining the hexagonal form)
which in general decrease in size as we approach the central city from

13 Specifically, the figure covers only four sizes of market areas, the four smallest
of the th-^oretical ones derived by Losch.

272

LOCATION AND SPACE-ECONOMY

\ /

y A

/

/ \-

T^Mi

R? i

^

<^

il

rWCV .

1 i

"\

---J.,

Fig. 52. A modified Losch system consistent with resulting
population distribution.

PARTIAL GRAPHIC SYNTHESIS AND SUMMARY 273

any direction. In fact at the central city the hexagonal market areas
are so small for certain commodities that they reduce to points when
we attempt to depict them on a small scale figure. On Fig. 52 we por-
tray this reduction to points for two sets of commodities, where each
set would have a market area of different size in Losch's scheme, i^
(To avoid confusion we present in Fig. 52 hypothetical market areas
for only three sets of commodities. The reader may superimpose
others.)

In Fig. 52 we have indicated a secondary center at which production
sites and thus industrial population are concentrated. Once again, the
hexagonal market areas decrease in size as we approach the core.
Moreover, as Losch recognized, economic forces lead to the develop-
ment around each core of sectors alternately rich and poor in number
of production sites. This is depicted with respect to both centers.
Further, the number of production sites tends to be greater in any
sector as the concentrations of production sites (cities) which the
transport route coursing through the sector interconnects grow larger
in magnitude. 15 This relation is shown in Fig. 52 by the heavier con-
centrations of production sites, and generally smaller hexagonal market
areas, along the transport route which interconnects the two centers
indicated. Actually, we should have presented in Fig. 52 secondary
centers (smaller in size than the first) along each transport route and
a hierarchical array of satellite centers, as is a logical consequence
of the Loschian argument and as occurs in reality. In order to avoid a
complicated and visually meaningless diagram, we have not done so.^^

Thus Fig. 52 suggests the impact of urbanization economies upon
the spatial pattern of production sites when the Losch uniformity

^■^ With reference to these two sets of commodities we attempted to adhere to
the second and fourth smallest of Losch's theoretical market areas. Because of
specifications for constructing Fig. 52 which are cited below, we were not able to
do so in any satisfactory manner.

1^ Unlike Losch, we locate major transport routes through the heart of city-rich
and city-poor sectors rather than at their boundaries in order to catch more fully
the significant scale (urbanization) economies in the use of modern transport
media.

16 For examples of patterns of secondary and satellite centers which we have
in mind, the reader is referred to the map : United States, Population Distribution,
Urban and Rural, 1950, U.S. Bureau of Census, Washington, D.C., 1953; to popu-
lation dot and land use maps for the area around such cities as Indianapolis; to
Robert E. Dickinson, City, Region and Regionalism, Kegan Paul, London, 1947,
especially Figs. 2, 5, 24, 32, and 48; and an expansion of a figure by M. J. Proud-
foot in Amos H. Hawley, Human Ecology, Ronald Press, New York, 1950, p. 271.
In addition, the reader may gain some impression of hierarchical arrangement
from Fig. 53 below.

274 LOCATION AND SPACE-ECONOM\

assumptions are admitted, except for modification with respect to
population distribution. The resulting 'pattern is at best only one of
many which can be evolved.^'^ Moreover, because of the underlying
uniformity assumptions, this pattern incorporates unrealities in many
important characteristics.

A second basic limitation of the Loschian argument is that it pertains
to situations where raw materials are not required (as in service activi-
ties) or are ubiquitous and everywhere available at the same costs.
The argument therefore excludes that production (whether market- or
material-oriented) where material sources exert significant locational
pulls. Yet this is the very type of production which Weber has treated
extensively and for which the Launhardt-Palander construction was
designed. It therefore seems generally more valid to envisage an urban
economy as consisting of a concentration of firms using localized raw
materials 18 (such as the concentration at the lower left of Fig. 50)
upon which concentration is superimposed a modified Loschian diagram
similar to Fig. 52 to account for industrial, commercial, and service
activities utilizing ubiquitous raw materials or none at all. Hence-
forth, we have in mind this type of structure when we speak generally
of urban economies, i^ We shall outline it in some detail in connection

1'' In planning the construction of Fig. 52, 1 instructed my draftsman, Mr. Gerald
A. P. Carrothers, to (1) retain the Loschian deduction that each producer of any
given commodity operates at approximately the same cost so that the boundary
separating the markets of any pair of neighboring producers is a perpendicular
bisector of the line connecting the two; (2) adhere to hexagonal market areas in
so far as possible in order to deviate as little as possible from the distinguishing
characteristic of the Loschian derivation; (3) depict hexagonal market areas which
increase in size with distance from the core in any direction; and (4) construct the
hexagonal market areas so that, along any circle drawn with the core as center, the
size of the market areas in general tends to decrease as we approach the transport
axis of a city-rich sector and increase as we approach the transport axis of a city-
poor sector. In the time made available, Mr. Carrothers was not able to adhere
strictly to hexagonal forms. The extreme difficulty met in working with hexag-
onal forms only and, as a consequence, the need to reshuffle constantly the sites
of production in order to meet this specification (for this reason the contrast
between the size of market areas in the city-poor and city-rich sectors is not as
sharp as we initially planned), strongly suggest that the hexagon is a pure concept
much as is perfect competition. The hexagon loses much of its significance as a
spatial form once agglomeration forces are admitted and, as a logical outcome,
inequaUties in population distribution recognized. In general, non-hexagonal
forms are more consistent with the full interplay of location forces.

18 Raw materials are conceived broadly so as to include semifabricated and
fabricated products which are subject to further processing as well as minerals
and other substances in their crude form.

19 This is not to deny that urban areas may have as their basic activities func-
tions and services and even industries which do not utilize localized raw materials.

PARTIAL GRAPHIC SYNTHESIS AND SUMMARY 275

with Fig. 54. But first let us proceed with our summary discussion
of Chap. 8.

Hitherto, the discussion and figures of this chapter have referred to
location factors as they govern interregional and intraregional distri-
bution of basic industry and service activities and the urbanization
process. We must now introduce those location forces determining
the pattern of agricultural land use.

In Sect. 4 of Chap. 8 the interaction of these latter forces was dis-
cussed. The various adjustments of the agricultural farm enterprise
were examined in detail. At any given location, the farm enterprise
must select the correct proportion of factor inputs and scale of opera-
tions for the production of any given crop or commodity mix; this
involves, in addition to others, the basic substitution point between
land inputs and other inputs, between rent outlays and outlays on
inputs other than land. In selecting a particular site for farming, the
enterprise again substitutes between rent outlays and all other outlays
combined. However, this latter substitution decomposes into a subset
of substitutions: one between rent outlays and transport outlays as
the farm enterprise considers locations at different distances from the
market ; and another between rent outlays and the sum of other outlays
(excluding transport) since the price of a land input decreases with
distance from the market and therefore leads to different factor pro-
portions at sites at different distances from the market.

With respect to these types of substitutions, the general location
analysis for the individual farm enterprise is identical with the general
location analysis for the industrial firm. The farm enterprise inten-
sively uses land inputs; therefore, to it, cost differentials among sites
on land inputs are critical. The industrial firm which is labor-oriented
or power-oriented uses labor or power inputs intensively; therefore,
to it, cost differentials among sites on labor inputs or power inputs are
critical. This firm confronts the same types of substitution relations
as identified for the farm enterprise in the previous paragraph, save
that labor outlays or power outlays take the place of rent outlays.

In this regard the traditional dualism of an industrial location theory
and an agricultural location theory, separate and for the most part
unrelated, loses most of its significance. Both the location of the
industrial firm and that of the farm enterprise can be treated in the
same general analytic framework.

When we view the location problem of the farm enterprise still more

For example, medical, educational, and governmental activities oriented to national
markets can serve as basic activities.

276 LOCATION AND SPACE-ECONOMY

comprehensively within a setting where market prices are given, or
income and demand functions specified, and when we confront the
enterprise with the problem of choosing a crop (commodity-mix) to
be produced, we are able to derive a set of rent functions. We can do
this by noting the substitution adjustments of the enterprise at all
locations with respect to each crop. These rent functions, as they inter-
sect, permit the identification of the familiar Thiinen rings, distorted
of course by any restraints we may wish to impose which relate to
resource content of land, physical barriers, legal, political, and social
institutions, etc. When we formulate mathematically the substitution
relations governing agricultural land use, as in Sect. 5 of Chap. 10,
we are able to harbor a still more embracive set of situations. This
set can involve the use of many raw materials, each furnished
from a single source or supply area, and a framework of many
markets.

A general graphic representation of equilibrium agricultural land-
use patterns, as yielded by the analysis, is attempted in Fig. 53. Here
we consider a hierarchy of several urban areas and their agricultural
hinterlands. A considerable distance intervenes between each pair of
cities. Once again to avoid a confusing diagram, we ignore satellite
type centers which may exist between any pair and which the reader
should bear in mind.

About each city we have drawn Thiinen rings. These rings are not
concentric circles. Rather they are distorted bands. In part they
reflect the impact of lower transport rates along major transport
routes, which routes comprise the net suggested by Losch and indicated
in Fig. 51.2 0 This net embodies those urbanization economies which
stem from scale economies in transportation.

Many other forces distort the symmetry of the Thiinen pattern about
each city and of the boundary lines which separate the agricultural
hinterlands tributary to each city.^i To the right in Fig. 53 we have
indicated an area of marshland which precludes any agricultural activ-
ity. To the left we have indicated an area containing soil particularly
suited, and therefore devoted, to the production of a crop which ordi-
narily would not be cultivated so close to a city.

Because we have depicted the impact of only a few disturbing forces,
Fig. 53 greatly exaggerates the symmetry of agricultural land-use pat-

20 Again, we revolve the net so that the transport routes course through the
middle of city-rich and city-poor sectors.

21 The boundary lines are yielded by supply area analysis wherein any one com-
modity (or combination) of many possible ones may be yielded by a given unit
of land. Note that one city's agricultural hinterland is entirely enclosed.

PARTIAL GRAPHIC SYNTHESIS AND SUMMARY 277

a\'

II 1 1 1 1
II 1 1 II
II 1 1 II
1 1 1 1 1 1

u-^^

^

1 1 1 ' ii^

1 r

*%!&

Mi
1 1 1 1 II
1 1 1 1 II

i|ii:ii|!|ii
;ii!i|ii!{ii

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 [ T ;■■;

I 1 1 1 1 1 1 1 1 iji

II 1 1 1 1 1 1 II
II 1 1 1 1 1 1 1 1

1

/C

//

%- ' i' ' ' ' ' '

i;
r

/^

\

^:0' \
\

/
- /

\

\
\

/
\/

\

Fig. 53. An agricultural land-use pattern.

terns. 2 2 The pattern of reality is much more complex. Yet whether
we consider the complex pattern of reality or the oversimplified pat-
tern of Fig. 53, the relevant substitution framework reflecting the
interplay of competitive forces is the same.

It should be borne in mind that the grid of agricultural bands of
Fig. 53 does not conflict with the modified Loschian market areas of

-- For this reason, we have not attempted to construct Fig. 53 to be rigorouslj'
consistent with our theoretical framework. This figure was designed to be visually
suggestive.

278 LOCATION AND SPACE-ECONOMY

Fig. 52. Within a city the market areas cover urban residents. As
distance from the city increases, the market areas cover increasingly
farming populations. The isolated dots at considerable distances from
the city indicate small centers of retail and service activities oriented
to farming population. To portray this consonance in the use of land
we have superimposed upon the market areas at the lower right of
Fig. 52 a set of bands indicating an hypothetical agricultural land-use
pattern.

We are now in a position to probe somewhat more deeply into the
impact of urbanization economies and to examine the pattern of urban
land use which results from superimposing upon a concentration of
heavy, localized raw material using industries (such as depicted at
the lower left of Fig. 50) a modified Loschian system of nets of market
areas (such as depicted in Fig. 52) . For the more customary situation,
the logic of this procedure has been stated.

As one of many suggestive patterns of urban land use, we present
Fig. 54 from which boundary lines separating market areas of producers
are omitted. Figure 54 is constructed on a larger scale than is Fig. 52.
The central city is outlined in greater detail. Apart from the apparent
effect upon size resulting from the use of a larger scale, the dimensions
of the central city (when compared with those of Fig. 52) are signifi-
cantly greater because of the city's expanded industrial base. The
city's total economy now comprises those activities using localized raw
materials as well as those using ubiquitous raw materials or none
at all. Because of the addition of operations utilizing localized raw
materials, the population and the income stream of the city have
mounted, and in turn these have stimulated the expansion within the
city of those activities using ubiquitous raw materials or none at all.
These latter are for the most part commercial and service type activi-
ties. The locations of firms engaging in commercial and service type
activities are represented by small dots. ^ 3 The density of these firms
is so great at and around the core of the city that the core and its
immediate environs appear as almost a solid black mass.

Figure 54 portrays four industrial districts (light grey shading)
greatly differing in size. Apart from those firms which manufacture
miscellaneous items or use ubiquitous raw materials and which are
indicated by a double dagger sign (|), we have concentrated in one of

23 The specific spatial pattern as well as the magnitude of these activities is
clearly a function of, among other variables, the amount of manufacture which is
based on localized raw materials.

It should also be noted that in Fig. 54 dots are no longer used to represent in
addition the industrial firms utilizing ubiquitous raw materials, as was the case
in Fig. 52.

PARTIAL GRAPHIC SYNTHESIS AND SUMMARY 279

Fig. 54. An urban land-use pattern.

the several industrial districts all producers of any given commodity.
By so doing, we retain a pattern consistent with the localization econo-
mies hypothesized in connection with Figs. 49 and 50. Thus the four
firms in the lower left agglomeration of Fig. 50, which produce the first
commodity associated with Fig. 50 and which are represented by a
black circle (•), are placed together in the industrial district at the
lower center of Fig. 54. The five firms producing the fourth commodity
represented by cross signs (+) are located in the same industrial dis-
trict. The four firms producing the third commodity represented by
black squares (■) and the three producing the second commodity

280 LOCATION AND SPACE-ECONOMY

represented by white circles (O) are put together in a second indus-
trial district. Lastly, the three producing the fifth commodity and rep-
resented by white triangles (^) are concentrated in a third industrial
district. Together with the firms which manufacture miscellaneous
items or use ubiquitous raw materials and which are present in all four
industrial districts, these firms compose the industrial sector of the city.

In addition to designating areas of industrial activity and of com-
mercial and service activity, Fig. 54 identifies a third set of areas
allocated to residential use (shown by crosshatched shading, where
varying density of shading indicates varying intensity of use), and
a fourth set of areas devoted to parks and recreational activities
(shrub-like shading). 2*

The land-use design of Fig. 54 represents one of many possible brews
of (1) intuition, (2) logic and analytic principles relating to the inter-
action of general forces governing land use, and (3) facts. It is not
a rigorous theoretical derivation. The development of a body of ab-
stract thought on urban land use has not proceeded sufficiently far to
allow a firmer statement on optimum urban land-use patterns. One
important avenue along which such development might proceed would
involve a comprehensive investigation of the interconnections of urban
land-use theory and agricultural location theory. As indicated in the
Appendix of Chap. 8, there are basic similarities between these two.
As in agricultural location theory, urban land-use analysis must place
central emphasis upon rent outlays (land values) . Further, along the
methodological lines of agricultural location theory, rent functions can
be derived and used as allocators of urban land. To both the urban
land user and the farmer, transport relations are critical. To the former
these relations are in terms of effective distance from the core and
accessibility to potential customers, although the user of any particular
urban site does not customarily incur the major part of the transport
outlays (both real and implicit) connected with the sale of product or
service. As in the rural setting, we observe that in an urban economy
complementary and competitive relations in terms of configuration of
uses spatially juxtaposed critically affect unit production costs and
accessibility. Moreover, technology, physical and cultural environ-
ment, legal institutions, and other factors serve to impose restraints
as well as distortions upon otherwise rational land-use patterns.

2-1 In Fig. 54 we do not purport to indicate all types of land use. Only four major
categories are presented. As a consequence, we must define these categories very
broadly so that they embrace other types of uses. For example, we include within
the land area assigned to commercial and service activities that land required for
governmental and institutional functions of a similar character; and within the
land for residential purposes that land set aside for elementary school functions.

PARTIAL GRAPHIC SYNTHESIS AND SUMMARY 281

Despite these basic similarities, there are important forces influenc-
ing the array of urban land uses which do not have a correspondingly
strong counterpart in the rural setting. As examples, we refer to those
shaping journey -to-work phenomena, the pattern of shopping trips
and social contact, advertising outlays, and quality competition. None-
theless, we find a general substitution framework of relevance in ap-
proaching urban land-use problems, let alone in attacking the entire
range of land-use problems wherein the competition between agricul-
tural uses and industrial, commercial, and residential uses is encom-
passed as well.

To complete the graphic presentation of this chapter we should
depict the flows of commodities and people connected with the various
locations, land uses, and cities which have been examined. For the
most part our graphic conceptions have referred to static structural
situations. They have not underscored the kinetic characteristics: the
constant stream of raw material and commodity shipments, inter-
regional and intraregional ; the journey -to-work patterns; trips to
shopping centers and points of social contact; intercity movement of
people; various interconnections via communications media; etc. To
illustrate all these diverse flow phenomena is beyond the scope of this
summary chapter. We shall have reference to only one type, namely,
that associated with the movement of selected raw materials and
finished product. We leave to the reader the task of superimposing
upon the patterns already depicted the kaleidoscopic variety of realistic
flows. 2 5

, In Chap. 9, we partially analyze the interrelations of trade (com-
modity flows) and industrial location. Our particular concern is with
the possibility of improving by reformulation both international trade
doctrine and location theory, and thereby to obtain a superior set of
analytical tools.

With examples of simple sets of conditions, we demonstrate how
the distance variable, and thus industries locationally sensitive to
transport cost differentials, can be incorporated into trade doctrine.
This step involves the explicit consideration of transport inputs and
consequently meets certain harsh criticisms levelled at traditional trade
theory. Concomitantly, we achieve an extension of location theory,

25 For illustrative materials on flows, the reader is referred to: E. L. Ullman,
"Die wirtschaftliche Verflechtung verschiedener Regionen. der USA betrachtet am
Giiteraustausch Connecticuts, lowas und Washington^ mit den anderen Staaten,"
Die Erde, 1955, Heft 2; "Interregional Highways," House Document 379, 78th
Congress, 2nd Session, Washington, 1944; and Gerald W. Breese, The Daytime
Population oj the Central Business District of Chicago, University of Chicago
Press, Chicago, 1949.

282

LOCATION AND SPACE-ECONOMY

namely, the restatement of transport orientation doctrine in terms of
opportunity cost. We find, for example, that traditional Weberian
doctrine might suggest in an international trade setting complete
transport-orientation when an optimum solution would involve only
partial transport-orientation. These reformulations of both long-run

Region
A

Fig. 55. A commodity flow pattern: intranational trade.

opportunity cost doctrine and transport-orientation dogma are in
essence one and the same. They lead to a partial fusion of trade and
location theories and pave the way for a more comprehensive inte-
gration.

Our examples clearly demonstrate how a change in the distance
variable can completely revamp the geographic flow of commodities
and the composition of trade and influence in a major way the indus-
trial structure of nations and international location patterns. When
graphically presented, these examples also point up basic interrelations
with urban economic structure and land-use patterns.

To begin, we assume within a nation three regions A, B, and C with
their central cities as terminating and originating points of commodity
flows. We take the central city and surrounding area of Fig. 54 to
be our region B. Of region B we reproduce in Fig. 55 and subsequent
figures only its lower central industrial district. Except for relative

PARTIAL GRAPHIC SYNTHESIS AND SUMMARY 283

position regions A and C are not indicated in Fig. 55. As with nations
A, B, and C of Table II of Chap. 9, regions A and C together with
region B are geographically situated as the corners of an equilateral
triangle. We also postulate initially the same resource endowments,
demand conditions, and production functions as are assumed for nations
A, 5, and C in Chap. 9.

Since it is clear from Table I of Chap. 9 that region C has an abso-
lute disadvantage in the production of each commodity and since
among regions within a nation long-run mobility of productive factors,
especially labor and capital, may be posited, long-run equilibrium
may be assumed to entail the shift of productive factors from C to A.^^
Trade between A and C would be non-existent. Trade between A and
B would be approximately as depicted in Fig. 55 with B producing all
steel and coal, and A all ore, textiles, and shipping. The producers of
steel may be taken to be represented by the black circular marks {*) .^'^
B's imports from and exports to A are indicated by arrows whose
widths approximately represent values.

We now introduce national boundaries, in essence change the para-
metric value of the political variable. We assume regions A, B, and C
are three nations as in Chap. 9. Since long-run immobility of factors
is characteristic of the international setting, we can no longer postulate
that factors will shift from A to C. Rather, the productive units remain
at C, and C engages in those activities in which she has least compara-
tive disadvantage. The tables and discussion of Chap. 9 spell out the
long-run equilibrium position.

Figure 56 depicts this second situation as it bears upon region B.
The change from an intranational to an international setting alters
the magnitudes of the flows between B and A and introduces new
flows between B and C. Also, we observe that the industrial structure
of B changes somewhat. B engages in a new activity, namely, textiles.
The firm producing textiles is indicated by a star sign (*) .

In Chap. 9 a change in another basic variable, namely, the distance

2*3 This assumption oversimplifies any real situation. C may possess other
resources and be in an advantageous position to produce other commodities not
considered in this simple example. Furthermore, short-run immobihty may lead
to the estabUshment of industry at C which, once established, becomes economic
to operate at C because of relocation costs and other socio-economic factors.
Nonetheless, the validity of the general points we wish to make with this and
the two succeeding figures is not impaired.

27 To be consistent, we assume that: (1) the four firms in region B indicated by
the cross sign ( + ) obtain raw materials from areas not indicated and produce
products wholly consumed within the region; and (2) the three firms indicated
by the double dagger sign (t) use ubiquitous raw materials and produce for the
local market only.

284

LOCATION AND SPACE-ECONOMY

variable is considered. The relative position of the three nations
(regions) A, B, and C is altered. They are assumed to be situated
along a straight line with C in the middle. Trade flows are completely
revamped as indicated in Table IV of Chap. 9. In Fig. 57 we portray
these new flows as they relate to region B. It should be noted that the
industrial structure of B undergoes major change. Steel production

Nation
A

Fig. 56. A commodity flow pattern: international trade.

falls to approximately one-third its former level. Textile output
roughly triples in size. These changes are roughly indicated by the
numbers of firms producing steel (•) and textiles (*) in the indus-
trial district. If we were to probe more deeply, we should unearth
other changes. For example, steel and textiles have different input
requirements. This fact implies change in the structure of subsidiary
industries in region B which feed their outputs into steel and textiles.
Further, steel and textiles generate different levels of income and, in
general, have different internal multiplier effects, whether we consider
industry, employment, or population. As a consequence, the total pat-

PARTIAL GRAPHIC SYNTHESIS AND SUMMARY 285

tern of land use, as well as the pattern of each type of land use — com-
mercial, industrial, residential, etc. — is altered. This is so not only
because steel and textiles have different land requirements and different
competitive potentials in bidding for land but, more important, because
the structures of industry erected upon steel and textiles are signifi-
cantly different and require different quantities and qualities of land
inputs. Thus we see the basic interconnections among urban land-use
patterns (and hence agricultural land-use patterns), commodity flows,
and interregional (international) position.

Nation

Fig. 57. A commodity flow pattern with modified geographic position of
trading nations.

With Fig. 57 we bring to a close the graphic presentation of this
chapter. This presentation, in particular Figs. 52-57, reflects the im-
pact upon land-use patterns and commodity flows of the interaction
of the various location forces. To the extent that a limited number
of diagrams permits, there are embodied in these figures the several
kinds of related forces: (1) those tending to transport-orientation and
to labor and similar forms of orientation; (2) those stemming from
technological change and from economies of scale, localization econo-
mies, and urbanization economies; (3) those leading to the formation
of the standard types as well as the Losch type of market and supply
areas, of modified Thiinen patterns of agricultural land use, and of the
intricate patterns of urban land use; and (4) those, such as are gen-
erated by political boundary lines, which emerge from the broad social
and institutional setting. Together with other similar diagrams which
may be constructed, these figures can embrace different kinds of general
situations — situations which may involve many regions and cities,

286 LOCATION AND SPACE-ECONOMY

where each pair is interconnected by diverse commodity and communi-
cation flows and where the population of each consumes many com-
modities, in the production of each of which many firms may be en-
gaged, where each firm may utilize many raw materials of which each
may be available from many sources.

As already intimated, the above diagrams may be taken to represent
one path of integration, namely, visual integration. In Chap. 10 we
attempt a second contrasting path of integration, one that follows
mathematical lines. There we develop a pervasive and basic location
principle. The marginal rate of substitution between any two transport
inputs or groups of transport inputs, however the transport inputs or
groups of transport inputs may be defined, must equal the reciprocal
of the ratio of their transport rates, social surplus (however defined)
less costs on all other transport inputs being held constant. This prin-
ciple when supported by appropriate postulates implies various location
theories: the transport-orientation dogma of Weber, Fetter-Launhardt
market and supply area analysis, Loschian market area schemata, and
Thiinen agricultural location theory. Thus, this principle demonstrates
a basic unity in much of location theory and permits considerable syn-
thesis of location doctrines. Further, with this principle we are able
to extend and generalize much of this theory to encompass a much
broader range and a more realistic set of situations. Additionally, this
principle and the mathematical formulations of Chap. 10 coupled with
the notion of a spatial transformation function facilitate the fusing of
location theory and production theory.

The above set of summary graphs and discussion bring to a close the
analysis of this book. Needless to say, there is a tremendous amount of
ground yet to be ploughed. The ways in which production theory and
location theory may be interwoven and fused must be spelled out in
considerable detail. The logical relations between trade theory and
location theory need to be more thoroughly explored and a more explicit
synthesis achieved. The transport system and rate structure must be
considered as variables rather than as fixed data ; to do so would facili-
tate the merging of the transport problem as conceived by Koopmans
and the location problem as developed in this book. The concept of
rational behavior must be sharpened via game theory and other con-
ceptual apparatuses; once this concept is defined in concrete and pre-
cise (desirably quantitative) terms, its application to the decision
making process of the firm will permit a more valid statement on the
nature and conditions of locational equilibrium and a deeper under-
standing of the phenomenon of agglomeration, particularly where
pricing policy is a variable and where firms possess considerable
geographic mobility.

PARTIAL GRAPHIC SYNTHESIS AND SUMMARY 287

We need to develop new and superior concepts relating to the spatial
structure of society. Spatial interaction phenomena, as manifested for
example in the various empirical materials on commodity and com-
munication flows and population movement, must be dissected with
tools honed to a much finer sharpness. We especially need to probe
deeply into space preferences, i.e., into man's propensity for intricate
forms and patterns of herd existence and into the socio-psychological
and biological forces which together with economic and other forces
govern the spatial patterns of population settlement. These forces
have a strong bearing upon urban land-use patterns and the mutual
interdependence of the industrial, commercial, and residential sectors.

We must gain further insight into urbanization economies, into the
complex interrelations of the sets of net economy curves, and into the
structure and functioning of metropolitan regions as socio-economic
organisms. How the structure and functioning of any given region are
shaped and limited and how the location decisions of its firms are
restrained by the region's total resources, income, gross product, and
labor productivity, and by the tastes, standards, and expenditure pat-
terns of its populace require thorough investigation. This type of
inquiry is very much in the direction of regional science.

We need to pry into the space-economy with welfare considerations
in mind, to relate spatial structures to social well-being and to intro-
duce political variables and policy decisions as they reflect attempts
to give a concrete basis to values and ideals; and we must study how
they in turn influence location decisions and spatial patterns.

Finally, we need to develop, as will be attempted in a future volume,
operational models to quantify various interrelations and to provide
cutting tools more relevant to policy decisions. Whether we attempt
improvement of regional and interregional input-output models, or
linear programming techniques, or industrial complex analysis, or pro-
jections of gross regional product and its constituents, or gravity
models, or other structural schemata involving ordered arrangements
of groups and sub-groups and of aggregated and disaggregated sectors,
or whether we aim at synthesis of the stronger elements of these models
and analytical techniques, we must be able to present and handle more
effectively the space-economy as a hierarchy of focal points and trans-
port and communication routes. Substitution in the large must be
assigned a much more significant role. Once again such developments
would take us along the channels of regional science. It is our hope
that these channels will be diligently explored.

Author Index

Abbott, Lawrence, 204n

Ackley, Gardner, 165, 165n

Aeroboe, Friedrich, 190n, 243, 243n

Alderson, Wroe, 65n, 67

Allee, Warder C, 84n

Allen, G. R., 57n

Allen, Roy G. D, 118, llSn

Amoroso, Luigi, 161n

Anderson, Theodore R., 65n

Auerbach, F., 55, 55n

Beckerman, W., 209n

Beckmann, Martin, 24n, 167n, 168n, 222n

Benedict, Elizabeth T., 190n, 243n

Benedict, Murray R., 243n

Bergson, Abram, 236n

Bernard, Luther L., 84n

Bogue, Donald J., 68, 68n, 70, 70n, 145n,

251n
Bohm-Bawerk, Eugen, 83n, 85n
Bortkiewicz, Ladislaus, 31, 109, 109n
Boulding, Kenneth E., 81n, 204n, 221n
Breese, Gerald W., 281n
Bright, Margaret L., 65n
Brinkmann, Theodor, 190n, 194n, 199n,

243, 243n, 244, 244n

Capron, William M., 37n
Carrothers, Gerald A. P., 274n
Carver, Thomas N., 28n
Cassel, Gustav, 32, 33, 42

Chamberlin, Edward H., 25, 25n, 27n,
38n, 46, 49n, 50, 50n, 92n, 144, 144n,
162, 162n, 204n

Chipman, John S., 214n

Christaller, Walter, 58n, 60, 60n, 239n

Clark, Colin. 21, 70n

Clark, John B, 82n

Colson, M., 109n

Cournot, Augustin, 161n

Cox, Reavis, 65n, 67

Cumberland, John H., 119n

Dantzig, George B., 222n

Dean, William H., Jr., 15, 30n, 31n, 87n,

93n, 107n, 120, 121, 125, 128n, 132, 225n
Dickinson, Robert E., 273n
Dodd, Stuart C, 65n
Dorfman, Robert, 204n
Dunn, Edgar S., 190, 190n, 192, 193,

193n, 194n, 195n, 198n, 199n, 243, 243n,

244, 244n, 247, 247n

Edgeworth, Francis Y., 161n, 228

Ellis, Howard S., 236n

Englander, Oskar, 15, 24n, 29, 29n, 30n,
31, 32, 32n, 34, 41n, 50, 85n, 93n, 94,
143n, 160, 180, 208, 208n, 231n

Enke, Stephen, 24n, 25n, 167n, 168n

Fetter, Frank A., 25n, 144n, 153, 160,
231n, 239, 286

289

290

AUTHOR INDEX

Fisher, Irving, 83n, 85n, 88n, 89n
Fox, Karl A., 24n, 167n, 169n
Freutel, Guy, 209n, 251n
Friedrich, Carl J, 27n, 128n, 172n, 178n,

222n
Furlan, L. Vladimir, 3 In, 50, 208, 208n

Gibrat, R., 55, 55n

Graham, Frank D., 209, 209n, 210, 212n,

213, 214n, 215n
Greenhut, Melvin L., 144n, 170n, 171n

Haberler, Gottfried, 53, 86n, 215n, 220n

Harrod, Roy F, 215n

Hawley, Amos H., 68, 68n, 145n, 251n,

273n
Hawtrey, Ralph G., 30n
Hayek, Friederich A., 82, 82n, 83n, 85n
Hicks, John R, 25, 25n, 26, 26n, 27, 95n,

102, 118, 118n, 192
Hitchcock, Frank L., 168n
Hoover, Edgar M., Jr., 24n, 30n, 58n,

84n, 86n, 87n, 94n, 105n, 108, 108n,

122, 123. 124n, 125, 127n, 128n, 130n,

139n, 143n, 144n, 148, 148n, 150, 153n.

154n, 155n, 164n, 172, 173n, 174n.

175n, 190n, 194n, 195n, 205n, 210n,

231n, 235, 235n
Hotelling, Harold, 160, 161, 161n, 162,

163n, 164, 164n, 169, 170, 181
Hoyt, Homer, 88n
Hurwicz, Leonid, 165n
Hyson, C. D., 144n, 147n, 157n, 231n,

239
Hyson, W. P., 144n, 147n, 157n, 231n,

239

Isard, Walter, 58n, 119n, 207n, 209n,

211n, 220n, 251n
Isabel!, Eleanor C., 65n

Jurgen, H., 37n, 208n

Kaldor, Nicholas, 82n

Kaysen, Carl, 165n

Knight, Frank H., 81, 81n, 82n, 83n, 86,

89, 89n
Koopmans, Tjalling, 22, 24n, 77n, 168n,

213n, 222n, 286

Lange, Oscar, 25, 25n, 82n

Launhardt, Wilhelm, 24n, 28, 43, 122,
125, 143n, 144n, 153, 160, 161, 210n,
231n, 239, 256, 258, 259, 261, 267, 274,
286

Leontief, Wassily W., 21, 49, 49n, 58n

Lerner, Abba P., 163, 163n, 164n

Losch, August, 15-19, 19n, 42-44, 44n,
45, 48, 48n, 49, 49n, 50, 53n, 54, 58,
58n, 59, 60, 60n, 75n, 78n, 86n, 125,
143, 143n, 150, 150n, 151-153, 153n,
154, 157-159, 159n, 162n, 174, 174n,
190n, 195n, 199n, 208, 231n, 236, 239,
239n, 240-242, 242n, 243n, 244, 247,
247n, 252, 267, 270, 271, 271n, 272,
273, 273n, 274, 274n, 276-278, 285,
286

Lotka, Alfred J., 55, 55n

Luce, R. Duncan, 165n

Machlup, Fritz, 83n
Marschak, Jacob, 165n, 167n
Marshall, Alfred, 24, 24n, 25, 85n, 90,

254
Mayberry, J. P., 165n
McDougall, Wilham, 84n
McKenzie, Roderick D., 68, 68n, 145n
Metzler, Lloyd A., 212n
Moller, H., 164n, 165n
Morgenstern, Oskar, 165, 165n, 166,

166n, 167
Mosak, Jacob L., 25, 25n, 26, 26n

Nash, John F., 165n

Neumann, John von, 165, 165n, 166,

166n, 167
Niederhauser, Elisabeth, 37n, 92n, 109n
Nimkoff, Meyer F., 84n
Nurkse, Ragnar, 82n

Ogburn, Wilham F., 84n

Ohlin, Bertil G., 17, 28n, 50, 51, 52, 52n,

53, 53n, 108n, 127n, 128n, 172, 208,

208n, 215n, 217

Palander, Tord, 24n, 42, 43, 43n, 48, 50,
52n, 86n, 94n, 95n, 108, 108n, 112n,
122, 123, 124n, 125, 130n, 143n. 144n,
161, 162, 163, 163n, 164n, 179n, 180,
210n, 231n, 235, 235n, 239, 256, 256n,
258, 258n, 259, 259n, 261, 262, 263,
263n, 264n, 267, 274

Pareto, Vilfredo, 25, 32, 42, 55n, 56

Peck, Merton J., 71n, 207n

Pick, Georg, 121, 123

Predohl, Andreas, 31, 32, 32n, 33, 33n,
34, 35, 35n, 36 36n, 41n, 42n, 50, 54,
94, 95n, 131, 208, 208n, 254

Proudfoot, Malcolm J., 273n

Raiffa, Howard, 165n
Ravenstein, Ernest G., 64, 64n

AUTHOR INDEX

291

Reilly, William J, 65n

Reiter, Stanley, 16Sn, 213n, 222n

Ricardo, David, 28n, 51n, 89n

Robinson, Austin, 163n

Ritschl, Hans, 15, 30n, 31, 49n, 50, 87n

Roscher, Wilhelm, 15, 28, 28n

Samuelson, Paul A., 24n, 25, 25n, 43n,
103n, 118, 118n, 167n, 168n, 228n, 230n

Schaffle, Albert E. F., 15, 28, 28n

Schmoller, Gustav, 44n, 87n

Schneider, Erich, 160n, 164n, 231n

Schumpeter, Joseph A., 26n, 31

Seedorf, Wilhelm, 37n, 208n

Shubik, Martin, 165n

Singer, H. W., 55, 55n, 163, 163n, 164n

Smithies, Arthur, 82n, 164, 164n, 165n

Stackelberg, Heinrich, 165n

Steiner, Peter 0, 204n

Stewart, John Q., 65, 65n, 66, 66n, 68,
68n, 77, 78

Stippler, H., 243n

Stopler, Wolfgang F., 43n, 49n, 228n

Stouffer, Samuel A, 63, 63n, 64n, 65n

Thomas, Dorothy S., 65n

Thiinen. Johann Heinrich, 3. 15, 16, 17,
18, 19, 19n, 24n, 27, 28, 28n, 29, 33,
33n, 34, 37n, 41n, 51n, 52, 92, 93, 158,
188, 190n, 198, 199, 208n, 210n, 243,
243n, 244n, 249, 249n, 250, 252, 276,
285, 286

Triffin, Robert A.. 50, 50n

Trotter, Wilfred, 84n

Ullman, Edward L., 58n, 60n, 71n, 239n,

281n
Usher, Abbott P, 15, 30n, 31n

Viner, Jacob, 53, 53n, 215n
Vining, D. Rutledge, 22, 57n, 58n, 71n,
77n

Walras, Leon, 25, 32, 33, 42, 43, 48n,
89, 89n

Weber, Alfred, 15, 16, 19, 19n, 23, 27,
27n, 28, 28n, 29, 30n, 31, 36n, 37n, 40,
40n, 43, 50, 52, 53, 54, 91, 92, 92n, 93,
94, 94n, 96, 104n, 108, 108n, 109, 109n,
118, 120, 121, 123, 125, 127, 128n, 130,
130n, 131n, 132, 135, 137n, 141, 143n,
158, 170, 172, 172n, 176, 176n, 177,
178, 179, 179n, 180, 181, 181n, 182,
182n, 183, 188, 189, 196, 208, 210n, 217,
222, 222n, 224, 225n, 228, 252, 252n,
260, 274, 282, 286

Weigmann, Hans, 37, 37n, 38, 39, 40, 41,
41n, 42, 42n, 49, 50, 54, 208, 208n

Wicksell, Knut, 25, 81n

Williams, John H., 208, 208n

Young, Kimball, 84n

Zeuthen, Frederik, 161, 161n, 162, 164n
Zipf, George K., 56, 57, 57n, 60, 61, 61n,
62, 62n, 63, 63n, 64, 65n, 78n, 79n

Subject Index

Abnormal profit, see Profits, surplus
Accessibility, as related to transport outlays
(time-cost) by consumers, 205, 280
effect of competitive land uses on, 280
effect of complementary land uses on, 280
effect on price of urban land, 200-205
effect on urban land use, 200-205, 280
Ackley, cases of discontinuous consumer dis-
tribution, 165
competitive behavior as affected by type of

market discontinuity, 165
determinacy of solution with market dis-
continuity, 165
lack of generalized solutions because of spa-
tially discrete demand, 165
stability of solution with market disconti-
nuity, 165
Activity analysis, as an element of regional
science, 287
as appropriate for short-run trade doctrine,

209
need to synthesize with other techniques,

287
use of, in simple trade-location case, 214n
to determine transport rates and com-
modity flows, 213n
to study structure of space-economy, 287
to treat substitution in the large, 287
Advertising outlays, as affecting urban land
use, 200-201, 281
differences in, and overlapping market

areas, 264
effect on cost curves, 203-204
effect on rent function, 203-204
neglect of, in Launhardt-Palander construc-
tion, 265
Agglomeration, a pattern of, in an urban-
metropolitan region, 278-280
analysis of, as requiring a complex ap-
proach, 205n
as an historical process, 180

Agglomeration — continued

as involving increase in transport outlays,

179, 267
as not affecting industrial distributions by

regions, 172
as similar to complementarity in land use,

205n
caused by iron and steel development, 8,

19n
centers, advantage of existing production

points as, 180
decrease in validity of Weber's assump-
tions with increase in, 179n
effect of differential bargaining abilities

upon, 180-181, 181n
exceptions to Weber's conditions for, 178-

179, 179n
forces of, as basic to location analysis, 139-

140
from economies of scale, 173-176, 265-267
from localization economies, 176-182, 267-

268
from urbanization economies, 182-188, 268-

270
function of economy of, in Weber, 178
in competitive locational equilibrium along

a line, 162, 163n
industrial, differentials in land outlays as

major to, 189
importance of internal spatial dimensions

of, 189
in early stages of settlement, 2
labor locations as centers of, 179
need for critical isodapanes to intersect,

176-178
of steel fabricating activities, 8
of urban activities, as determined by rent

functions, 204-205
point of, and substitution between trans-
port outlays and production outlays,
174-175, 179, 179n, 188, 265, 267, 269

293

294

SUBJECT INDEX

Agglomeration — continued

as affected by replacement deposits, 178
as influenced by side payments, 180-181
as over-all transport cost minimum point,

177-178
as pulled to superior bargainers, 180-181
use of transport-orientation techniques to

identify, 178
Weber's determination of, 177-178

problem of, in multiplant firm, 179, 179n

requisite total output for, 176-178

significance of inherited physical structures
for, 180

significance of relocation costs for, 180-181,
181n

unit, size of and distance from critical iso-
dapane, 178
size of to which producer attracted, 178

use of side payment to induce, 179n

Weber's conditions for, 176-178

see also Localization ; Urbanization
Agglomeration diseconomies, see Deglomera-

tion economies
Agglomeration economies, and decentraliza-
tion policy, 14

and interdependence of sets of net economy
curves, 187-188

and meaningful metropolitan sectors, 14

and need for research on interdependence
of net economy curves, 188, 287

and need for weighting net economy curves,
186-187

and step-by-step migration, 41n

and the definition of industry, 92n

as classified by Hoover, 172

as classified by Ohlin, 172

as excluded in Launhardt-Palander con-
struction, 256

as including economies (diseconomies) of
scale, 139, 172, 265

as including localization economies (dis-
economies), 139, 172, 265, 267

as including urbanization economies (dis-
economies), 139, 172, 265, 268

as ignored by competitive locational equi-
librium models, 169-170

as independent of geographic position, 139-
140

as measured by the critical isodapane, 176n,
178

as misrepresented by a sum of representa-
tive net economy curves, 186-188

as primarily dependent on magnitudes, 139-
140

as they counteract Force of Diversification,
79n

effect of physical environment on, 140

effect on aggregate industrial location pat-
terns, 21, 79n

effect on transport cost surface, 251

failure of Weber to distinguish by types,
176

from interindustry linkage, 21

neglect of, in Losch scheme, 153-154

nonadditive character of, 21, 188

regularity in geographic cost pattern, as
related to transport cost, 139n

spatial pattern of, as derived from inter-
action of other forces, 140

types of, 139, 170

see also Localization economies ; Urbaniza-
tion economies

Agglomeration theory, and need for a sharply
defined concept of rational behavior,
286

as included in an extended trade doctrine,
219

as yielded by the substitution principle,
173-188, 265-269

coalition problem of, as complicated by re-
location costs, 181n

complications, because of different sizes of
agglomeration, 180-181
because of the coalition problem, 181n

difficulties from non-symmetric location
factors, 181n

difficulties of converting to a constant sum
game, 181n

integration with production theory, 173-188

limited application of Weber's, 179, 268-
269

need to develop game theory for, 180,
286

sketches showing fusion of, with other lo-
eation-market-trade doctrines, 256-
285

use of game theory in, 180-181, 181n

Weber's, applicability to entirely new situ-
ations, 179, 268
validity for new area development, 181-

182, 268
validity for regional planning, 181-182,
268

see also Localization theory ; Urbanization
theory
Aggregate demand, see Demand, aggregate
Aggregate demand curve, see Demand curve,

aggregate
Aggregate supply curve, see Supply curve,

aggregate
Aggregation, in location analysis, 21, 92-93,
188-189

industrial, Colin Clark type, 21
Leontief type, 21
useful types of, 21
Aggregative analysis, interrelation with firm
location analysis, 93, 189, 198, 199,
248

use of, to study structure of space-economy,
287
Agricultural enterprise, see Firm, agricul-
tural
Agricultural hinterlands, see Hinterlands,

agricultural
Agricultural land use, and industrial and res-
idential land use, competition be-
tween, 281

and structure of the land market, 40

as affected by raw materials supply, 248-
249, 276

as constrained by regional income and
other total conditions, 199

as interrelated with urban land use, 281,
285

as related to factor mobility, 285

as related to political variable, 285

as related to regional analysis, 199

as related to trade and geographic speciali-
zation, 285

as restricted by cultural values and institu-
tions, 205, 276, 280

as restricted by technology, 205, 280

as yielded by substitution principle, 189-
199, 205-206. 243-253, 275-278, 281

SUBJECT INDEX

295

Agricultural land use — continued

basic factors determining, 2-3, 188-199,

275-276
changes in intensity of, along substitution

paths. 246-247
competition in, as leading to agricultural
location theory, 158
as leading to rent theory, 158
conditions for equilibrium patterns of, 245-

251
definition of type of, by city-region, 249
dependence on industrial location, 8, 19,

19n
diminishing returns from increasing in-
tensity of, 4, 78
effect of differentials in input prices upon,

199n, 275
effect of geographic inequalities of re-
sources upon, 18
effect on industrial location, 7
elements of, in a Thiinen-Losch model, 17-

18
forces causing irregularities in, 276-277
in Ohlin's model, 52

intensity of, as a basic variable in agricul-
tural location theory, 244-247, 245n
as affected by net farm price, 194-195,

244
as affecting market area size, 271
as affecting population density, 271
as falling with distance from core (mar-
ket), 68-70, 155, 155n-157n, 194-
195, 244, 247-248, 271
as related to factor proportions, 194-

195
as related to the distance variable, 194-

195, 244, 285
as related to use of transport inputs, 81
changes in with development, 3, 5, 8, 11
effect of invariance of upon rent func-
tion, 195n
invalidity of constancy assumption, 195n,
244, 247
limits to, as precluding analysis of systems

of supply areas, 158
need to consider changes in intensity of,
to unify firm and industry analysis
in agriculture, 247-248
non-symmetric character of, along sides of

city-region boundary, 249n
optimum pattern, as insured by competi-
tion, 197-199, 281
as involving equation of aggregate sup-
ply and demand, 198-199
as involving problem of firm equilibrium,
198-199, 275-276
patterns of, as affected by multiple mar-
kets, 198-199, 249-251, 276
as determined by rent functions, 195,

197-199, 246, 276
when firm produces a combination of
crops, 199, 276
simple pattern of, 3

sketch of, as fused with sketch of modified

Losch market system, 272, 277-278

for several metropolitan regions, 276-278

zonal intensity of, as rising or falHng with

distance from market, 247n-248n
see also Zones
Agricultural location theory, advantage in use
of inputs of agricultural goods with
distance from city, 245n

Agricultural location theory — continued
and central position of rent differentials in,

189, 196, 275, 276
and conditions for land use equilibrium,

245-251
and crops as a basic variable, 190, 197-

199, 276
and determination of equations of zonal

boundaries, 246-247
and firm indifference to location within

rent yielding hinterland, 197-198
and industrial location theory, common core

of cost differential analysis, 189-190,

275
and production functions invariant with

distance from city, 244
and accepted dualism with Weberian doc-
trine, 92-93, 188-189, 275
and continuity of the location problem, 196
and forcing of firms into efficient substitu-
tion paths, 196-197
and historical approach, 15
and urban land-use theory, competition as

basic to both, 205
complementarity as basic to both, 205
rent functions as basic to both, 205, 280
some dissimilar forces in, 287
and use of rent functions to determine land

use patterns, 195, 197-199, 246, 276
and view of commodity combinations as

single commodities, 244, 276
as a supplement to Losch theory, 16
as a transition from Weberian firm analy-
sis, 189-190, 196-197, 275
as advanced by consideration of firm equi-
librium problem, 198, 275-276
as explicitly considering transport outlays,

205, 280
as involving firm behavior with complete

information, 196-197
as obtainable from analysis along a straight

line from city, 245n, 248n
as reflecting competition in land use, 158,

188-199, 243-251, 277
as related to regional analysis, 199
as substitution between land-use outlays

and other outlays, 33-34, 189-199
as yielded by general location principle,

243-251, 252, 286
as yielded by substitution principle, 189-

199, 205-206, 243-251, 275-276
assumption of given prices in, 210n, 243-

244, 276
assumption of invariance of transport rate

with direction from city, 245
basic elements of, 33n, 188-199, 243-251,

275-278
basic role of space factor in, 188-199
deeper analysis possible than in industrial,

190
disadvantage in use of inputs of industrial

goods with distance from city, 244n-

245n
emphasis of, on cost conditions, 210n
extension of, to cover raw material supply

areas, 248-249
to include multiple markets, 198-199, 249-

251, 276
to incorporate varying unit cost on farm,

244-247
generalized, as assuming prices as given,

249, 276

296

SUBJECT INDEX

Agricultural location theory — continued
as avoiding- definition of social surplus,

249
as avoiding subjective evaluations, 249n
as more desirable than generalized mar-
ket area analysis, 249
to include other location-market-supply
analysis, 250-251

incorporation with Weberian doctrine in
one framework, 92-93, 188-189, 275

intensity of land use as a basic variable in,
244-247, 245n

interconnections with urban land use
theory, 200-206, 280

linkage of individual farm and aggregate
analysis, 189

location of individual producer within, 92-
93, 189-199, 275-276

Losch's approach to, 48

need of general location theory for multi-
commodity framework to encompass,
243

need to consider changing prices in a truly
general system, 243

need to define industry by city-region in,
249

need to study bonds with urban land use
theory, 280

plus elements of Losch and Weber, 16-19

procedure for determining rent, 190-199,
244-251

relative neglect in, of demand, 210n, 243-
244, 275

sketches showing fusion of, with other lo-
cation-market-trade doctrines, 256-
285

statement of problem of, 244

Thiinen and basic location methodology,
27-28

traditional emphasis on aggregative analy-
sis, 92-93, 188-189

transport cost on material inputs in, 244n-
245n, 248

use in, of multicommodity framework, 197-
199, 243-246, 276

value for industrial location analysis, 189

weakness of price assumptions of, 243

see also Zone formation
Agricultural production, see Agricultural land

use
Agricultural stratum, as a foundation for
other strata, 29

effect on early industrialization, 7, 29

formation of, 29

locational pull of, 7, 29

relation to industrial stratum, 29
Agriculture, firm analysis in, see Firm, agri-
cultural

industry analysis in, and firm analysis, case
of meaningless distinction between,
247n
need to consider changes in average unit

cost to unify, 247-248
need to consider changes in land-use in-
tensity to unify, 247-248

in extreme case as firm analysis, 248

interrelation with firm location analysis,
189, 198
Aircraft, and population mobility, 12

and site selection, 12

and topographical barriers, 12

and urban decentralization, 12, 87-88

Aircraft — continued

effect of, on industrial location, 12
on metropolitan structure, 12, 87-88
on space preferences, 13
on trade and trade routes, 12
role in transport network, 12
Airway passenger movements, variation with

population and distance, 62n
Aluminum, as a basic industry, 18-19

location of, and substitution between power
and transport outlays, 189-190
Angle conditions, see Equilibrium point, angle

conditions
Anglo-Saxon bias, against spatial analysis,
24-27, 113
partial correction of, 24n
Atomic energy, and government subsidy,
13
and new industries, 13
and urban decentralization, 13
competitiveness of nuclear power, 12-13
effect of, on existing industries, 13
on interregional trade, 13
on regional development, 13
on space preferences, 13
locational effect of, 13, 79n
Austi'o-Hungary, cities in, rank-size findings

for, 57n
Automobile and bus, dispersion of urban pop-
ulation, 87-88
effect on metropolitan structure, 87-88
Average cost, and marginal cost, same differ-
ence between for all producers in
Losch, 240
as affecting farm output, 190-194
as base for delivered price in Losch scheme,

240
as equated to price in Losch scheme, 240
as varying in the more general case of lo-
cation, 236
changes in, along substitution paths, 246-

247
disadvantages relative to marginal cost in

boundary definition, 236n
equality with net farm price in farm equi-
librium, 197
incorporation of changes in, in agricultural

location theory, 244-247
invalidity of constancy assumption, 195n,

244, 247
need to consider changes in, to unify firm
and industry analysis in agriculture,
247-248
on farm, constancy of, in traditional loca-
tion theory, 244
use of, to determine market boundary in

single firm case, 231-232
see also Cost curves
Average cost curves, see Cost curves
Average cost pricing, see Pricing system

Back hauls, effect on firm's equilibrium loca-
tion, 113n
effect on transport cost surface, 251
Basic form of space-economy, and market
structure, 39-42
and the spatial array of markets, 38-41
Weigmann's concept of, 38-42
Basic industry, and size of urban-metropoli-
tan region, 11, 278
and support of non-basic activities, 11,
128n, 278

SUBJECT INDEX

297

Basic industry — continued

as aflEecting spatial pattern of service activ-
ities, 278n
as generating secondary labor, 128n
by types, different direct land input re-
quirements of, 285
different income effects of, 284
different indirect land input requirements

of, 285
different multiplier effects of, 284
different secondary effects of, 284-285
non-raw material using activities as, 274n-

275n
pull on parasitic industry, 8-9, 128n
service activities as, 274n-275n
using localized raw materials, and asso-
ciated population cluster, 19-20
and change in agricultural land use, 19,

19n
and change in hierarchy of cities, 19, 19n
and change in structure of market-ori-
ented activities, 19, 278
and local multiplier effects, 19
and the Thunen-Losch model, 18-19
as swelling urban population, 278
effect on urban income stream, 278
non-applicability of Losch theory to, 154,

274
plus market-oriented activities as yielding
urban-metropolitan structure, 274-
275, 278-280
pull on ubiquities using activities, 278
Basic-non-basic ratio, 11

variation with type of city, 11
Basic service ratio, see Basic-non-basic

ratio
Beckmann, and continuous geographical in-
tensity distributions of production,
168n
and neglect of certain location forces, 168n-

169n
as generalizing the Enke market problem,

168n
needed extension of, to solve the location
problem, 168n
Behavior patterns, and pattern of settlement,
2, 6, 78
as reflected in space preference, 84-85
as reflected in time preference, 84-85
see also Rationality ; Game theory
Bogue and the impact of distance, on in-
tensity of land use, 68-70
on population density, 68-70
on receipts from services, 68-70
on retail sales, 68-70
on structure of metropolitan region, 68-

70
on value added by manufacture, 68-70
on wholesale sales, 68-70
Bortkiewicz' criticism of Weber's use of ficti-
tious distances, 109
Boundaries, see City-region boundaries ; Mar-
ket boundaries ; Supply area bound-
aries ; Zonal boundaries
Breaks in transport system, see Transport

network, breaks in
Brinkmann, and advantage in use of inputs
of agricultural goods with distance
from city, 245n
and constant unit cost on farm, 244
and the effect of differentials in input
prices on agricultural location, 199n

Brinkmann — continued

and yield per acre as invariant with dis-
tance from city, 244

disadvantage in use of inputs of industrial
goods with distance from city, 244n-
245n

transport costs on inputs in theory of,
244n-245n
Bus passenger movements, and the PvPijD
factor, 61-63

variation with population and distance,
61-63

Canadian cities, rank-size findings for, 57n
Capital, as a location factor, 133

as distinguished from services, 89-90
relation of spatial to other elasticity forms

of, 41
some elements of theory of, 81-90
Capital goods, adaptability of and location, 41
bound, 41

combination-free, 41
market, immobility in, 41
spatial inelasticity in, 41
structure of, 40-41
Capital inputs, and the investment period of
factors, 82-83
and the roundaboutness of production, 82-

83, 255
and the time extent of production, 82
and transport inputs, substitution between,

253
as embodied in transport inputs, 89-90
as ultimately stemming from labor and

land inputs, 81, 255
definition of, 81, 81n
increase in output with increase in use of,

81-82
profit motive and the use of, 81-82
transport inputs contrasted with, 81-85,

255
use of, and the increase in the time period
of investment, 82
Capital outlays, and transport outlays, sub-
stitution between, 33
in Predohl's framework, 33
in terms of use units, 35
Capital structure, as related to spatial extent
of production via substitution prin-
ciple, 253
changes in, with orientation to cheap cap-
ital sites, 133
Capitalism, bound, and immobile labor, 40n-
41n
free, and mobile labor, 40n-41n
Central dependent stratum, as tied to central

organizing stratum, 29
Central organizing stratum, composition of,
29
effect on central dependent stratum, 29
independence of, 29
Central place theories, and frequency distri-
bution of cities, 59-60, 60n
and spatial regularity of cities, 60, 60n
of Christaller and Losch, 60, 60n
Centrifugal effect, and decentralization policy,
13
and space preference, 84-85
from diminishing returns, 4, 78n, 84
Centripetal effect, and lines of force in a
potential field, 78«
from increasing returns, 4

298

SUBJECT INDEX

Chamberlin, solution of, to Hotelling problem
when three or more competitors, 162
tendency for competitors to disperse, 162
see also Monopolistic competition
Cheap labor, see Labor, cheap
Chemicals, as a basic industry, 18-19
Child labor, see Labor, cheap
Chipman, and linear programming in simple

trade-location case, 214n
Christaller, and frequency distribution of
cities, 59-60
and hierarchy of cities, 60n
and spatial regularity of cities, 60n
central place theories of, 60, 60n
resource inequalities and spatial pattern of
cities, 60
Circles, as market boundaries, 145-147, 231,
239
as supply area boundaries, 155
distorted, as superior to regular hexagonal
markets in a general case, 242n-
243n
Cities, advantages and disadvantages of, for
plant location, 183
and size of basic-non-basic ratio, 11
as centers of market-oriented activities, 57
as local peaks of population potential, 66-

67, 78
decline of, 8-9, 17-18, 19
emergence of different sizes of, and econo-
mies of scale, 57-58
frequency distribution of, at shortest dis-
tance intervals, by size classes, 59
growth of, 4-9, 19, 19n, 29

and deglomeration economies, 139
and increase in cost of food supply, 139
and increase of rents, 139
hierarchy of, and regularity of flows over
distance, 58-60
Christaller's, 60n
hierarchy of sites within, in a modified
Losch diagram, 272-273
> network of, as reflecting the joint distribu-
tion of economic activities, 184
desirable changes in, 183
operating economies of, 186-187
optimum hierarchy of, 183
optimum hierarchy of sites within, 183
optimum size of, 12
optimum spatial distribution of, 183
patterns of, as associated with commodity
flows, 281
as associated with population flows, 281
proximity of, as affecting power generation

economies, 185
rank of, 11, 12

rank-size findings on, in Austro-Hungary,
57n
in Canada, 57n
in France, 57n
in Germany, 57n
in India, 57n

in United States, 56-57, 57n
rank-size rule for, mathematical formula-
tion of, 55-56
universality of, 57
validity of, 57

Zipf's interpretation of deviations from,
57n
rise of new, and new basic industry, 19
and new iron and steel location, 19n
and new plant locations, 17, 18

Cities — continued

satellite, and secondary peaks in rent func-
tion, 203
structure of, 12
selection among, 8-9, 19
service activities as basic industry in, 274n-

275n
size of, and deglomeration forces, 78, 139,
186
and dominance in national commodities,

58
and economies of scale in power gener-
ation, 184-185
and geographic inequalities of resources,

78
and hypothetical economies of scale, 186-

187
and size and character of urban trans-
port network, 185-186
and volume and length of population and

commodity flows, 58, 281
as related to basic activities, 278
associated with number of activities in,
57
spatial pattern of, and central place theory,
60, 60n
and hierarchy of cities, 58-60
and resource inequalities, 60
and spatial regularity, 58-60, 60n
and the Forces of Unification and Diver-
sification, 78n-79n
empirical data on, in Iowa, 59
in South Germany, 60n
standardization of, by basic service ratio,
186n
by flow configurations, 186n
by industrial mix, 186n
by land use patterns, 186n
invalidity of, 186-187
variation in power consumption by type
of, 184
City-regions, and need to define industry in
terms of, 249
as focal point for agricultural location

theory, 198-199, 244
boundaries of, as cutting off concentric

zones, 249n
boundaries of, non-symmetric character of

land use along sides of, 249n
commodity trade between, 17-18
effect of multi-, upon agricultural land-use

patterns, 198-199, 249-251
geographic specialization among, 17-18
hinterland boundaries of, as implied by

general location principle, 249n
hinterlands of, 16-17, 249n
see also Cities ; Urban-metropolitan re-
gions ; Urban-metropolitan structure
Classical school, neglect of space, 27, 50, 89,
116
neglect of transport-orientation in trade
theory, 50, 53, 208
Classification of commodities, see Commodi-
ties, classification of
Classification of industry, see Industry, classi-
fication of
Classification of location factors, see Location

factors, classification of
Classification of markets, see Market areas,

classification of
Climate, as a location factor, 2, 3, 133, 138
_ Coal, as a localized raw material. 19

SUBJECT INDEX

299

Coal deposits, relation of, to iron and steel

location, 7-9, 19n, 118n
Commercial activities, importance in urban
structure, 200n-201n
see also Market-oriented activities ; Service
activities
Commercial land use, see Urban land use
Commodities, categories of, in terms of num-
ber of substitution relations, 94
classification of, by dispensability, 93-94
by number of sources, 93-94
by mobility, 93-94
by weight-loss, 93-94
in terms of possible types of substitution,

94
by size market area, 17, 153, 271
conditionally place-bound, 32n
differences in transport rate by type of, 88
Englander's classification, 31n-32n
fixed proportion of, and restraints on
transformation function, 137
and substitution limitations, 131, 136-137
immobility of, and limited competition, 37-

38
markets for, in terms of factor markets, 40
national, 17, 18, 58

and economies of scale, 58
dominance in and size of city, 58
perfect mobility of, 33
place-free, 31n-32n
regional, 17
subnational, 18
subregional, 17
substitution between groups and subgroups

of, 95n, 227-228
supraregional, 17, 18
transport rates by characteristics of, 88,

112, 118n. 227n
unconditionally place-bound, 32n
unique combinations of, and firm analysis

as industry analysis, 248
use of multicommodity framework, in ag-
ricultural location theory, 197-199,
243-246
in Losch analysis, 44-49, 153-154, 270-

271
in urban land use theory, 204-206
in urbanization theory, 185-188, 268-270
view of combinations of, as single com-
modities, 244, 275
Commodity flows, a case of intranational, 282
and composition of trade, 9
as associated with city patterns, 281
as associated with land-use patterns, 281,

282-285
as associated with location patterns, 281
as basic to analysis of space-economy, 22,

281
as determined in a general market-trans-
portation-location model, 168n-169n
as determined in Enke's market problem,

167n-168n
as related to urban-metropolitan structure,

282-2S5
as simultaneously determined with location,

207
distributional stability of, 22
effect of friction of distance, by type, 70,

72n
effect on, of changes in demand, 156n-157n
of changes in distance variable, 215-219,
282-285

Commodity flows — continued

of changes in supply, 156n-157n
of changes in transport rate, 156n-157n
hierarchy of, by volume and length, 58
international, and need to consider trans-
port-orientation, 210
in a simple trade-location example, 210-
219, 283-285
length of, and economies of scale, 58
and market types, 58
and size of city, 58
need for finer analysis of, 287
need to specify demand to determine pat-
tern of, 212
of selected raw materials and finished prod-
uct, 281-285
over distance, 70-75
sketches of, 282, 284-285
variety of in reality, 281
volume of, and size of city, 58

as significant in determining transport
rates, 213n
see also Trade
Comparative advantage, situations of, in a
simple three-country case, 213-219,
282-285
Competition, absence of, and circularity of
market areas, 145
as basic to both agricultural location theory

and land use theory, 205, 280
as eliminating surplus profits, 196, 196n-

197n, 202
as equating price and average cost in

Losch, 240
as forcing farmers into efficient substitu-
tion paths, 196-197
as insuring best farm practices, 197-199
as insuring maximum rent for each site,

196, 197-199
as insuring optimum pattern of agricul-
tural land use, 197-199
as linking individual farm and aggregative

agricultural analysis, 199
degree of, and variation in transport rate,

88
effect on urban land use, 200-205, 280
for sites, effect on price of urban land,

200-205
imperfect, as leading to overlapping mar-
ket boundaries, 264
in agriculture, as involving firm behavior

with complete information, 196-197
land use, as embraced in general equilib-
rium approach, 201n
as problem of firm equilibrium, 198,

280
effect on rent function. 204-205
quality, effect on urban land use, 281
role of, in Losch, 44-45, 240
see also Competitive locational equilibrium ;
Monopolistic competition ; Pure com-
petition
Competitive equilibrium, see Agricultural
land use ; Competitive locational equi-
brium ; Market areas ; Urban land
use
Competitive field, as a spatial array of mar-
kets, 38-39
Competitive locational equihbrium, Ackley's
solutions, 165
and hinterlands of firms, 161-165
and spatial price discrimination, 163n

300

SUBJECT INDEX

Competitive locational equilibrium — continued
and the Cournot problem when firms in-

finitesimally close, 161n
as allied to Koopman's transportation prob-
lem, 167n-169n
as allied to spatial price equilibrium, 167n-

169n
as related to the general transportation

problem, 167n-169n
as yielding less precise results than market

area analysis, 169, 264-265
cases of discontinuous consumer distribu-
tion, 165
Chamberlin's solution when three or more

competitors, 162
competitive behavior as affected by type

of market discontinuity, 165
competitive behavior as unrestricted vari-
able, 166
determinacy of, as related to market dis-
continuity, 165
difficulties in application of present game

theory to, 167, 265
dispersion of competitors in Chamberlin's

analysis, 162
effect on, of economies of scale, 173-176
of size of market, 164
of transport cost, 164
Enke's market problem as a simple case

of, 167n-168n
Hotelling, and simultaneous variation of
price and location, 160-162
agglomeration tendency, 162, 163n
solution (s), assumptions in, 161
borne out by Zeuthen, 161
invalidity of, for autonomous trading

firms, 162-163, 163n
stability of, 161-162
when both firms mobile, 162
when no undercutting, 161
when one producer immobilized, 161
when one producer trades autono-
mously, 161
when producers undercut, 161-162
when production costs are zero, 161
lack of generalized solutions because of

spatially discrete demand, 165
Lerner and Singer, and demand elasticity,
163-164
and more realistic undercutting policy,

163-164
and upper price limit, 163-164
many solutions in, depending on assump-
tions, 160-165, 264-265
models of, as being more sophisticated than
market area analysis, 169, 265
as ignoring various inequalities in re-
source distribution, 169-170
need to allow areal distribution of de-
mand, 169
need to allow areal mobility of the firm,

169
need to allow more realistic pricing

policy, 169
need to consider more realistic cost con-
ditions, 169
need for sharply defined concept of rational

behavior, 286
need to integrate with market and supply

area analysis, 170, 265
need to integrate with Weberian doctrine,
170

Competitive locational equilibrium- — continued
neglect of, in Launhardt-Palander construc-
tion, 264-265
Palander's criticism of Hotelling's agglom-
eration, 162-163, 163n
Palander's solution (s), autonomous and
"superpolitisch" trade in, 163
for autonomously trading firms, 162-163
hintei-land defense, conditions for, 163,

163n
market sharing policy, conditions for,

163, 163n
price fluctuations when firms close, 162-

163
undercutting policy, conditions for, 133,
163n
relevance of game theory for, 165-167, 170,

265
results of, as yielded by substitution prin-
ciple, 170
separation of rivals as insulation from re-
actions, 167
Smithies, and effect of changes in marginal
cost, 165n
and hinterland demand as function of

price and location, 164
and types of competitive behavior, 164n-

165n
solution with linear demand, 164
stability of, as related to market disconti-
nuity, 165
undercutting as related to proximity of

producers, 161
unrealities of game theory postulates for,

167
see also Agricultural land use ; Equilib-
rium points ; Firm agricultural ;
Firm, industrial ; Firm, urban ;
Urban land use
Complementarity of land uses, analysis of,-
as requiring a complex approach,
205n
as basic to both agricultural location and

land use theory, 205n
as embraced in a general equilibrium ap-
proach, 201n
as similar to agglomeration, 205n
effect of, on price of urban land, 200-
205
on production costs, 200-205, 280
on rent functions, 204-205
on sales volume, 200-205
on secondary peaks of sales volume, 201n
on urban land use, 200-205, 280
types of, 200, 205
Complex analysis, as an element of regional
science, 287
need for, in agglomeration theory, 188,
205n
in urban land use theory, 205n
to synthesize with other techniques, 287
use of, to study structure of space-econ-
omy, 287
to treat substitution in the large, 287
Concentration of urban activities as deter-
mined by rent functions, 204-205
Conditions of equilibrium, see Equilibrium

conditions
Congestion, and spatial extent of agricul-
ture, 4
from population growth, 4
increase of, and deglomeration, 139

SUBJECT INDEX

301

Congestion — continued

and urbanization diseconomies, 183, 185-
186
see also Deglomeration economies
Constant cost conditions, market area analy-
sis under, 148, 235-236
supply area analysis under, 155, 235-236
Consumer behavior, and increasing consump-
tion of transport inputs, 87-88
as affected by a fall in time and money cost

of population movement, 87-88
changes in, and the scale effect, 87-88
changes in, and the substitution effect,

87-88
see also Consumption
Consumer distribution, see Population dis-
tribution
Consumer expenditure pattern, see Consump-
tion pattern
Consumer indifference as defining market

boundaries, 231-232, 236, 236n
Consumer surplus, see Surplus, social
Consumer tastes, see Tastes, consumer
Consumption, area of, ability of Launhardt-
Palander construction to treat, 257-
258, 262-264
as a finite number of variables, 237-238
as a set of market areas, 237-242
as an infinity of market points, 237
as related to material sources via indus-
trial producers, 263-264
breakdown by production location, 257-
269
effect of density of, on conditions for max-
imizing social surplus, 233n
level of, as affected by changes in distance
variable, 214-219
in a simple three country trade-location
example, 214-219 ; of new commod-
ity, locational effect of, 4
pattern of, effect of changes in supply on,
156n-157n
effect of changes in transport rate on,

156n-157n
need to study relation with metropolitan
structure, 287
urban, and effect on weighting of net econ-
omy curves, 186-187
variation in density of, and multiple sta-
tionary points in market area analy-
sis, 234
see also Market areas
Corner location, see Equilibrium point, as a
corner ; Equilibrium point, as an end
point
Cost curves, as affected, by distance from
urban core, 202, 202n
by resource content of land, 194, 194n,

202n
by taxes, 194, 194n
change of, with pricing system change,

150n
effect on, of advertising outlays, 203-204
of irregularities in labor cost, 202n
of product quality, 203-204
of service quality, 203-204
of topography, 202n
use of, in agricultural location theory, 190-
194
in determining rent functions, 201-205
in determining urban land use, 201-202
with rent as explicit cost, 191, 193-194

Cost curves — continued

see also Average cost ; Marginal cost
Cost of living, and urban diseconomies, 186

increases in, with city size, 186
Costs, conditions of, as emphasized in Gra-
ham's trade theory, 210
emphasis upon, in traditional location

theory, 21 On
inadequate treatment of, in location theory

for international analysis, 210
inequalities in the geographic pattern of,
as ignored by competitive equilibrium
models, 169-170
by type of, 169
validity of principle of minimization of,

221n
see also Average cost ; Cost curves ; Factor
costs ; Marginal cost ; Opportunity
costs ; Prices
Cournot problem, reduction of Hotelling
problem to, when firms infinitesi-
mally close, 161n
Critical isodapane, see Isodapane, critical
Crops, as a basic variable in agricultural loca-
tion theory, 190, 197-199
changes in, as involving substitution, 197-

199, 275
combinations of, as characteristic of real-
ity, 199
as contained in farm equilibrium analy-
sis, 199, 275
different types of, associated with different

rent functions, 195, 197-199, 276
zones of cultivation of, as determined by
rent functions, 195, 197-199, 275
with multiple markets, 198-199, 275
see also Agricultural land use
Cross hauling, see Market areas, overlapping
Cultivation of crops, see Agricultural land

use
Cultural complexes, patterns and traits, see

Cultural values and institutions
Cultural values and institutions, and cheap
labor, 8-9, 10, 128n
and decentralization policy, 13
and variations in local input costs, 130-131,

138
as a distorting factor in space-economy,

138, 276
as limiting land use, 205-206, 276, 280
as reflected in policy decisions, 287
changes in, from technological advance, 13
effect of, on agricultural zones, 16, 276
on international trade, 75, 283n
on population pattern, 144n-145n, 287
on space-economy, need for deeper study

of, 287
on urban-metropolitan structure, 10, 11,

12, 283n, 287
on urban transit network, 185
locational effect of, 2, 4, 6, 8-9, 10, 12, 13,

21, 138-140, 283n
see also Political variable

Dean, and emphasis on relative gross weights
and relative distances, 121, 225n
and the generalized index test, 121
and the historical approach, 15, 31n
criticism of Weber for overestimating pull

of weight-losing materials, 225n
criticism of Weber for underestimating pull
of pure materials, 225n

302

SUBJECT INDEX

Dean — continued

deviational economies and power orienta-
tion, 132
use of weight triangle in transport-orienta-
tion problem, 120-122
Decentralization (industrial and urban), and
aircraft, 12
and atomic energy, 13

and future metropolitan organization, 13
and technological advance, 10, 12, 79n
of activities, relation to rent functions,

204-205
policy, and agglomeration economies, 14
and demand for transportation and util-
ity services, 14-15
and guiding new industrial growth, 14-15
and historical inertia, 14-15
and input-output and by-product rela-
tions, 14-15
and locational shifts, 14-15
and meaningful metropolitan sectors, 13-

14
and resource conservation, 14-15
and size of consumer market, 14-15
and structure of labor force, 14-15
and structure of labor requirements, 14
as deviational policy, 13
restraints upon, 13-14
Decisions, of urban units, as reflecting prin-
ciples of urban land use theory, 205-
206
policy, see Political variable
rational, see Game theory ; Rationality
Deglomeration, see Agglomeration
Deglomeration diseconomies, see Agglomera-
tion economies
Deglomeration economies, and spread into
space, 78, 255
as associated with the use of transport in-
puts, 255
as independent of geographic position, 139-

140
as primarily dependent on magnitudes, 139-

140
as they counteract Force of Unification,

79n
effect on location patterns, 79n, 84
from congestion, 139
from increase in intensity of land use,

139
from increase in urban population, 139
from rise in cost of food supply, 139
from rise in cost of urban services, 139
from rise in rents, 139
implications for non-herd existence, 84
spatial pattern of, as derived from inter-
action of other forces, 140
see also Agglomeration economies ; Decen-
tralization
Delivered price line, as a transport gradient
line, 148-149
construction of, 148
Delivered prices, equal, loci of points of as
boundaries, 146-147, 237, 240
equality of, from several supply sources,

155, 155n-157n
rise of, with increase in supply area, 155
Demand, aggregate, as related to the labor
market, 40
as related to the land market, 40
equation to aggregate supply for agricul-
tural equilibrium, 198-199

Demand — continued

assumption of fixed pattern of, in long-run

trade theory, 212n
effective, as setting limits to market areas,

145-147
elasticities of, and differentials in revenue

potentials, 126n
need to introduce areal distribution of, into
competitive locational equilibrium
theory, 169
need to specify, to determine exchange
ratios, 212
to determine location pattern, 212
to determine trade pattern, 212
regional, need to consider in general loca-
tion theory, 207
need to consider in trade theory, 207
relative neglect of, in traditional location

theory, 210n
total, as restraining urban land use pat-
tern, 206, 287
see also Demand curve
Demand curve, aggregate, construction of,
156n-157n, 159n
use of with aggregate supply curve, 157n
effect of shift of, on commodity flows,
156n-157n
on output of supply sources, 156n-157n
for the firm, sloping, construction of, 159n

sloping, because of space, 158-159
invalidity of horizontal, 158-159
Demographic energy, concept of, 65
linear relation with state incomes, 68
population and distance as basic variables
of, 65-66
Demographic force, concept of, 65

population and distance as basic variables

of, 65-66
relation to Reilly's law of retail gravitation,
65n
Development processes, Palander's insistence
upon study of, 43
sketch of some basic elements of, 1-15
Deviational economies, see Dean
Differentials, analysis of, as common to both
industrial firm and farm enterprise
location, 189-190, 199, 275
limits to substitution in considering many,

137
outlay and revenue, and substitution, 135-
137, 137n
location effect of, 126-137
need to consider all types of, 135
procedures in considering several, 135-
137, 137n
see also Labor outlays ; Power outlays ; Pro-
duction outlays ; Revenue potentials ;
Transport outlays
Diffusion, see Decentralization
Diminishing returns, and the need for spread
into space, 78, 255
centrifugal effect of, 4, 78n, 84, 255
postponement of, and the use of transport
inputs, 255
Disaggregation, in location analysis, 21, 92-93
industrial, Colin Clark type, 21
Leontief type, 21
useful types of, 21
Disaggregative analysis, see Disaggregation
Discontinuity, see Location, problem of ; Sub-
stitution, in the large ; Transforma-
tion line, discontinuous

SUBJECT INDEX

303

Discounting over space, see Space Discount
Discounting over time, discounting over space

contrasted with, 85-86
Diseconomies of scale, see Deglomeration

economies ; Economies of scale
Dispersion, see Decentralization
Distance, as fostering spatial price discrimi-
nation, 164n
between rivals as insulation from game

theory reaction, 167
effective, as physical distance adjusted in
time-cost dimensions, 200-201, 205n
as related to transport technology, 16
transport network as critical in defini-
tion of, 205
fictitious, invalid use of by Weber, 109,

109n
friction of, and sloping demand curve for
the firm, 158-159
and the need for transport inputs, 79,

80n, 140
effect of by type commodity flows, 70
presence in social system, 75-76
from transport route and increase in mar-
ket area size, 274n
physical, and Weigmann's theory of lim-
ited competition, 38-39
as different from effective distance, 200-
201, 205n
time-cost, 3, 201

unnecessary, and minimization of transport
cost, 96-97
as precluded in the transformation func-
tion, 96-97, lOln, 107
possibility of with discriminatory trans-
port rates, 97n, lOln, 113n, 130n
see also Distance variable ; Space
Distance inputs, replacement of, by the term
transport inputs, 80
use of, instead of transport inputs, pros

and cons, 116
see also Distance variable ; Transport in-
puts
Distance variable (s), and basic regularities
in space-economy, 75-76, 140, 254
and bus passenger movements, 61-63
and Class 1 Railroad shipments, 70-73
and empirical findings by Losch, 60n
and family migration in Cleveland, 63-64
and international commodity flows, 73-75,

208-209, 282-285
and intranational commodity flows, 70-73,

208-209, 282-283
and population density, 68-70, 271-273
and railway express shipments, 60-61
and Ravenstein's finding on migration, 64n
and retail sales, 68-70
and secondary peaks, in rent function, 203

in sales volume curve, 201
and service receipts, 68-70
and shipments, by type I.C.C. commodity

group, 72n
and telephone messages, 61-62
and value added by manufacture, 68-70
and wholesale sales, 68-70
and world ocean-going freight, 73-75
as affecting cost curves, 194, 202
as affecting exchange ratios, 215n, 217
as affecting factor proportions, 275
as affecting farm output, 190-194
as affecting population content of market
areas, 278

D istance variable ( s ) — continued

as affecting price of land input, 275

as affecting structure of industrial districts,
282-285

as affecting substitution points of farm en-
terprise, 197, 275

as basic, in Stewart's social physics, 65-68

as basic to location theory, 35-36

as contained in concept of transport in-
puts, 35, 79, 113-114, 222-223

as related to intensity of land use, 194-195,
244, 247-248, 271-273

as related to the rent function, 194-195,
195n, 197, 201-205, 244

basic role in agricultural location theory,
189-199

effect of, on commodity flows, 215-219,
283-285
on equipotential contours, 66-67
on geographic specialization, 215-219,

282-285
on industrial location, 215-219, 282-285
on levels of consumption, 214-219
on market area size, 271-273, 274n
on price of urban land, 200
on sales volume, 200-201
on trade, 215-219, 282-285
on urban land use, 200, 271, 280, 282-285

emphasis on, in Weberian doctrine by as-
sumption of constant weights, 96

impact on urban-metropolitan structure,
11, 68-70, 270-286

impact on trade and location in simplified
case, 210-219, 282-283

implications of rank-size findings for regu-
larities associated with, 57-60

introduction of, into trade theory, 211-219,
281-282

migration and the intervening opportuni-
ties hypothesis, 64n-65n

need to consider, in calculating opportunity
costs, 211-212, 281-282
to fuse trade and location theory, 209,
281

need to develop gravity models to handle
more adequately, 209

neglect of, in trade theory, 209, 215n

sketches of effect of, 284-285

substitution between, and transport-orien-
tation, 96-112, 222-230

transformation line between, locational
shifts associated with shifts of, 116

transformation relations between, 96-112,
222-223
as embraced by relations between trans-
port inputs, 114-119, 222-223

transport cost restraints on variation in,
in the locational polygon case, 98-
101, lOln

value of, as restrained by spatial trans-
formation function, 223-230

variations in, as derived from transforma-
tion relations between transport in-
puts, 96

variation of, as yielding profit maximiza-
tion in locational polygon case, 226-
230
as yielding profit maximization in loca-
tional triangle case, 222-226

see also Distance ; Space
Dominant weight, and non-existence of Laun-
hardt-Palander construction, 258n

304

SUBJECT INDEX

Dominant weight — continued

and non-existence of weight triangle, 120-

121, 258n
definition of, 120, 225n
Dunn, and graphic analysis under assump-
tions of changing yields and unit

costs, 244n
and non-operational general equilibrium

statement for agriculture, 243n
concept of industry rent function, 195n
constant unit cost on farm in algebraic

statement of, 244, 244n, 247
proper determination of rent, 192-194
proper procedure to determine equilibrium

of farm enterprise, 192-194, 247
yield per acre as invariant with distance

from city in algebraic statement of,

244, 244n, 247
Dynamics, in general location theory, 54
in Weigmann's location theory, 38-41

Economic evolution, see Development proc-
esses
Economic processes, see Production processes
Economies of scale, addition to Launhardt-
Palander construction, 265-267

and boundary changes in Launhardt-Pa-
lander model, 266-267

and emergence of different sizes of cities,
57-58

and extent of market areas, 58, 148-151,
153, 173-174

and graduation of transport rate structure,
105

and hierarchy of cities, 57-58

and integration of power system, 185

and length of commodity flows, 58

and management diseconomies, 185n

and national commodities, 58

and shifts, fi-om transport optimum point,
175n-176n
to labor locations, 175n-176n
to power locations, 175n-176n

and social welfare from production con-
centration, 174

as a basic location variable, 135n, 175-176

as a basis for trade, 17

as affected by the market variable, 175n-
176n

as affecting type of orientation, 175n-176n

as basic to Losch's market area system, 44,
150-151, 153, 174, 267

as eliminating pole line producers, 265-266

as embraced by production theory, 176

as embraced in the substitution principle,
95n, 135n, 173-176, 265-266

as included in agglomeration economies,
139, 172, 265

as involving substitution between transport
and production outlays, 174, 265, 267

as overlapping with localization economies,
182n, 265

as reducing number of market-oriented pro-
ducers in Launhardt-Palander model,
266

as reflected in net economy curves, 186-187

as related to number and spread of plants
via general location principle, 252-
253

as requiring market areas in Launhardt-
Palander model, 266-267

definition of, 172

Economies of scale — continued

effect of, on sources of raw materials, 175n-
176n
on structure of space-economy, 265-267
on substitution between transport out-
lays in different directions, 175n-176n
geographic specialization resulting from, 5
graphic case of impact of, 265-267
in Ohlin's doctrine, 52

in power generation, effect on per capita
consumption, 185n
and size of city, 184-185
and urbanization economies, 184-185
in the location of intermediary establish-
ments, 175n
in the location of marketing establishments,

175n
incorporation in sketches of fused location-
market-trade doctrine, 256-285
introduction into transport-orientation-mar-
ket area framework, 265-267
localization economies as, with multi-plant

firms, 179n
locational shifts from, 173-176, 265-267
use of iso-outlay lines in analysis of, 174,

265
use of outlay-substitution lines in analysis

of, 174, 265
variation of substitution points with, 175-

176, 175n-176n, 265-266
with urban size, hypothetical, 186-187
Educational services as basic urban industry,

274n-275n
Empirical inquiry, value of, for theory, 55,

77
Enclave (s), as an agricultural hinterland,
276n, 277
in agricultural land use patterns, 3
of excluded consumers because of income
and taste differences, 145-146
End point solution, see Equilibrium point, as

an end point
Energy resources, controlling role of, 31n

see also Natural resources
Englander, and fusion of market area analy-
sis and industrial location, 143n
and the evolutionary approach, 15, 29n-30n
and the general theory of "local condition-

ality," 31, 31n-32n, 160
classification of commodities, 31n-32n
conception of immobile commodities, 94
criticism of classical trade theory, 208
criticism of Weber's agglomeration analy-
sis, 180
emphasis on inherited physical structures,

180
emphasis on relocation costs in agglomera-
tion process, 180
failure to consider simultaneous price and
location variations, 160
England, effect of international position of,

on theory, 27
Enke, generalization of his analog solution,
168n
spatial equilibrium problem of, 167n

as a general transportation problem,

167n-168n
as containing the Koopmans transporta-
tion problem, 168n
as encompassing the scale variable, 168n
as excluding the basic location problem,
168n

SUBJECT INDEX

305

Environment, physical, adaptation of human
beings to, 68-70
and economic evolution, 1-2, 6, 12
as a factor in spatial price discrimination,

164n
as restraining urban land use patterns, 206
conditions of, and bearing on labor orienta-
tion, 141
effect of, on agglomeration economies, 140
on urban transit network, 185-187
on weighting of net economy curves,
186-187
in relation to transport technology, 12
locational effect of changes in, 1
non-uniform, and non-circularity of mar-
ket areas, 145-146
see also Natural resources ; Topography
Equalizing differences in labor costs, see
Labor costs, equalizing differences in
Equations, Losch's system of, 47

system of, and sets of spatial co-ordinates,
54
and transport inputs, 54, 222-253
in a general location theory, 54, 222-253
Equilateral triangles, as inferior to regular
hexagons as market forms, 44, 153,
242
as satisfying boundary conditions, 153, 242
in Losch scheme, 44, 153, 241-242
Equilibrium conditions, for agricultural land
use, see Agricultural land use
for farm enterprise, see Firm, agricultural
for industrial firm location, see Equilibrium
point (for firm location) ; Firm, in-
dustrial
for market areas, see Losch (location theory

of) ; Market area analysis
for trade, see Trade

for urban land use, see Urban land use
Equilibrium point (for firm location), alter-
native ways of derivation, 119-124,
224-230
and comparison of relative minimum points,

108, 108n, 124n, 229-230
and diminishing marginal rate of substi-
tution between transport inputs,
116
and the convexity of the transformation

line, 116
and the marginal rate of substitution be-
tween transport inputs, 116-119, 222-
230
angle conditions for corner location, 122,

257
as a minimum transport cost point, 122-

124, 224-230
as a saddle point in game theory, 166
as an equilibrium point of forces (Varig-

non), 121-122, 225-227
as corner of the locational polygon, 107-

112, 122, 224-227, 230
as corner when forces at equilibrium out-
side triangle, 122, 224-227
as end point, because of convex iso-outlay
lines, 107-112, 120
because of graduated rates, 107-112, 120
because of large first zone charge, 107-

112, 230
because of tails on iso-outlay lines, 107-

112
conditions for, 107-108, 122, 224-227,
230

Equilibrium point — continued

as intermediate location, minor importance

of, 108, 113n, 230n
as trough of transport cost surface, 122-

124, 224-230
as yielded by the general location principle,

222-253
at a cheap capital site, as involving further
substitutions, 133
determination of, 133
at a cheap material source, determination

of, 133
at a cheap labor site, as involving further
substitutions, 130, 196
formal conditions for, 129-130
graphic determination of, 129-131
at a cheap power site, as involving further

substitutions, 132
at a higher-price market, determination of,

134-135
change in, with use of continuous trans-
formation curve, 115n
determination of, and substitution between
groups of transport inputs, 227-228
as labor orientation, 129-131, 196
as power orientation, 131-132
by the isodapane technique, 122-124
by use of weight triangle, 121-122, 256-

258
from differentials in revenue potentials,

159
when firm influences prices, 159-171
when no competitive reactions, 159
with competitive reactions, 160-171, 264-

265
with economies of scale, 173-176
with irregular price-ratio lines, llOn,

113n
with iso-outlay lines, 129-137, 159, 170,

174
with iso-revenue-less-outlay lines, 134-

135, 159, 170, 175
with Launhai-dt-Palander construction,

256-269
with outlay-substitution lines, 129-137,

159, 170, 174. 175
with revenue-outlay substitution lines,

134-135, 159, 170. 175
with variations in transport rates. 112
difficulties in determining, under oligopoly,

160-171, 264-265
direct determination by cost comparisons,

137, 140n, 230
effect on, of back hauls, 113n

of breaks in transport network, 110-112,
230
feasibility of labor location as. and critical

isodapane. 130-131
feasibility of power location as, and criti-
cal isodapane, 132
fuU, change in, with change in relative
weights, 104
conditions for, 103-104, 104n, 108, 116,

117. 123-124, 222-239
process of attainment. 103-104. 104n,
123-124. 135-137. 137n. 159, 170,
174, 175
possibility of multiple solutions, 130n,

230
via use of subset isodapanes, 123-124
with realistic rate structures, 105-112,
229-230

306

SUBJECT INDEX

Equilibrium point — continued

minor indeterminacy of, from realistic rate

structures, 107n
need to consider all variables in determina-
tion of, 185-137
partial, change in, with change in relative
weights, 104
conditions for, 102-104, 123-124, 135n,

137n
conditions for, in mathematical terms,
103n
similarity with conditions in produc-
tion theory, 118
first-order condition for, 116
inadequacy of, 135
second-order (stability) condition for,

116
twin solutions, possibility of, 103n
via use of subset isodapanes, 123-124
with realistic rate structures, 105-112
shortcut determination by pole principle,

122
see also Competitive locational equilibrium ;
Firm, agricultural ; General location
principle
Evolutionary approach, as sequences of loca-
tional structures, 30
Englander's contribution, 29n-30n
limitations and virtues of, 30
need for, in urbanization analysis, 183
required in agglomeration analysis, 180
Ritschl's contribution, 30n
to general location theory, 15, 28-30, 31n,

54
Weber's contribution, 15, 27-30
Excess supply functions, potential use of, to

solve the location problem, 168n
Exchange, see Trade

Exchange ratios, as affected by distance, 215n
as affected by transport cost, 215n, 217
in a simple three country case, 212-219
need to specify demand to determine, 212
Export duties, see Import-export duties
Exports, see Commodity flows ; Trade
External economies, see Localization econo-
mies ; Urbanization economies

Factor costs, differentials in, and need to
substitute between outlays, 126-127
as a location factor, 126-137, 189-190
as bearing upon land use patterns, 199n,

275
as contained in the substitution frame-
work, 199n
as excluded in Launhardt-Palander con-
struction, 256
as recognized in agricultural location
theory, 199n
see also Costs ; Labor costs ; Power, costs
of ; Prices ; Transport costs
Factors, classification of, and transport in-
puts, 89-90, 119n
Englander's classification of, 31n-32n
geographic distribution of, and general
equilibrium theory, 32-33
and location, 32-33
in terms of use-units, 34-35
immobility of, in international setting, 283

and limited competition, 37-38
immobility of groups of, and location

theory, 41
local differences in supply of, 52, 52n

Factors — continued

market structure of, 40-41
mobility of, and Ohlin's concepts of regions,
51, 52n
effect of political variable on, 283
effect on geographic specialization, 282-

285
effect on structure of industrial districts,

282-285
effect on trade, 282-285
effect on urban-metropolitan structure,

282-285
effect on urban land use, 282-285
intranationally, 283
perfect mobility of, 33, 53
prices of types of, 86

proportions of, and interrelations with
rent, 192-194, 275
and substitution principle, 95n, 275
as affected by distance variable, 275
as affected by location of farm, 194-195,

275
as affected by net farm price, 194-195
as affected by price of land inputs, 275
as related to farm output, 190-194, 275
as related to intensity of land use, 194-
195
services of, as incorporated in transport in-
puts, 79
Factory prices, differences in, and effect on
market areas of competing firms,
146-147, 239
Farm enterprise, see Firm, agricultural
Fetter, fixed firm locations in market area
analysis, 160
market area theory of, as yielded by gen-
eral location principle, 239
Fictitious distance, see Distance, fictitious
Firm (agricultural), adjustments of, as re-
flected in rent function, 197, 276
when multiple markets, 198-199, 276
analysis of, and industry analysis, case of
meaningless distinction between, 247n
as industry analysis in extreme case,
248
and aggregative analysis, need to consider
changes in land use intensity to
unify, 247-248
need to consider changes in unit cost to
unify, 247-248
and crop changes as involving substitution,

197, 276
and industrial, analysis of differentials as
common to both, 189-190, 196, 199,
275
as forced into efficient operations by com-
petition, 197-199
behavior of, as if having complete knowl-
edge, 196-197
determination of equilibrium output, 190-

194
determination of rent payments for, 190-

194
effect of locational position of, on factor
proportions, 194-195, 275
on farm operations, 194-195, 275
on intensity of land use, 194-195
on net farm price, 194-195
on rent, 194-195, 275
equilibrium of, analysis for in an improved
agricultural location theory, 198-199,
275-276

SUBJECT INDEX

307

Firm (agricultural) — continued

analysis for, when crop combination con-
sidered, 199, 248
and equality of net farm price and mar-
ginal costs, 190-191
and substitution between land inputs and

other inputs, 193, 196, 275
and substitution between rent outlays and

other outlays, 193, 196, 275
as a condition for optimum land use pat-
terns. 198-199, 275-276
many substitution paths to, 193n-194n
importance of internal spatial dimensions

of, 189-190
indifference to location within rent yield-
ing hinterland, 196-197
location of, and substitution between trans-
port and rent outlays, 189-190, 196,
196n, 275
interrelation with diverse factors, 190-
199
operations of, as affected by resource con-
tent of land, 194, 194n, 275
as affected by taxes, 194, 194n
as affected by transport routes, 194
output of, as related to distance from mar-
ket, 190-194, 275
as related to factor proportions, 190-194,

275
as related to market price, 190-194
as related to production costs, 190-194
as related to transport cost, 190-194
if rent were zero, 190-191
profits of, conditions for, 192-194
if rent were zero, 190-191
procedure in determining, 190-194
use of iso-product curves to determine,

191-194
use of price-ratio lines to determine, 192-

194
use of scale lines to determine, 191-194
rent differentials as basic in location of,

189-190, 194-199, 275
shift of, and substitution between rent out-
lays and other outlays, 195-196,
196n
see also Agricultural land use ; Agricultural
location theory
Firm (industrial), analysis of, as interrelated
with regional analysis, 159n-160n,
199
and agricultural, analysis of differentials as
common to both, 189-190, 196, 199,
275
as equal in size in Losch, 240
as restrained by spatial transformation

function, 223-253
assumption of negligible influence on price

as invalid, 158, 265
definition of, 91n

differential bargaining abilities in agglom-
eration, 180-181, 181n
in multifirm market-supply area frame-
work, 235-243
interdependence of, in Triflin's theory, 50n
labor oriented equilibrium, 127-131
localization problem of, in multi-plant case,

179, 179n
location analysis of, as a level of inquiry,
92, 188-189
as interrelated with aggregative analy-
sis. 93. 189, 198, 199, 248

Firm (industrial) — continued

location of, and differentials in net revenue
potentials, 159
and effect upon regional income, 159n-

160n
and substitution between distance vari-
ables, 96-112, 222-230
and substitution between transport in-
puts, 112-124, 222-230
as affected by regional income, 159n-160n
effect of agglomeration economies on,

173-188
effect of economies of scale on, 173-176
effect of localization economies on, 176-

182
effect of raw material supplies on, 175n-

176n
effect of urbanization economies on, 182-

188
in line case, 95-97, 117, 119-120
in polygon case, 98-101, 117, 122-124,

226-230
in triangle case, 97-98, 101-119, 120-122,

222-226
under competitive reactions, 160-171
under different spatial pricing systems,
158-171
location of, use of total cost comparisons
to determine, 137, 137n, 230
when no competitive reaction, 159
with finite number of transport routes,

101-112, 255
with influence on price, 158-171
with realistic transport rate structures,
104-112, 229-230, 255
location situations of, with negligible in-
fluence on price, 158
location theory for, as comparable to that
for agricultural firm, 189-190, 199,
275
multiplant, transport inputs within, 91n
optimum size of, as affected by spatial dis-
tribution of market, 175n-176n
as affected by spatial distribution of ma-
terial sources, 175n-176n
other forms of orientation, 131-137
possibility of location at source of pure

material, 121-122, 225n
power oriented equilibrium, 132
significance of rent differentials for, 189-

190, 199
sloping demand curve of, because of space,

158-159
transport-oriented equilibrium, 91-125, 222-

230
with market areas and supply areas as

variables, 231-235
see also Competitive locational equilibrium ;
Equilibrium point (for firm loca-
tion)
Firm (urban, commercial), analysis of, value
of substitution framework for, 205-
206, 280
determination of equilibrium location, 200-

206
determination of equilibrium operations,

200-206
see also Urban land use
Fixed proportions, see Commodities, fixed

proportions of
Flow phenomena, stability of, and general
location theory, 22

308

SUBJECT INDEX

Flows, see Commodity flows : Population
flows

Food, as a raw material in transport-orienta-
tion, 127n-128n
transport costs on, and equalizing differ-
ences in labor costs, 127n, lS9n

Footloose industry, growth of, and cheap
labor, 8-9
locational tie to basic industry, 8-9

Force of Diversification, and stable interac-
tions over distance, 60
use of, to explain spatial framework, 78n-
79n

Force of Unification, and stable interactions
over distance, 60
effect of technological advance on, 79n
use of, to explain spatial framework, 78n-
79n

Formkoeffizient as a technical concept, 36n

France, cities in, rank-size findings for, 57n

Friction of distance, see Distance ; Distance
variable

Furlan, criticism of classical trade theory,
208
his attempt at general location theory, 31n

Fusion of location-market-production theories
via general location principle, 221-
253, 286

Fusion of location-market-trade doctrines via
sketches. 256-286

Game theory, and a saddle-point solution, 166

and complications of different size agglom-
eration units, 180-181

and difficulties of converting localization
into a constant-sum game, 181n

and rational behavior, 160, 166, 265, 286

and the use of mixed strategies, 166

and types of strategy, 166

as pertaining to situations of interest con-
flict, 165-166

basic elements of, 166-167

coalition problem of, as complicated by re-
location costs, 180-181

conceptual complications of, 167

difficulties in use of, from non-symmetrical
location factors, ISln

need to develop for agglomeration analysis,
180, 286

overemphasis of interdependence of reac-
tions in a spatial setting, 167

postulate of complete information, 166,
166n

postulate of precise value of outcomes,
166

postulate of transferability of utility, 166

postulate of well specified variables, 166

present inadequacy for location analysis,
160, 265

problems of empirical verification, 167

relevance for location equilibrium problem
of firm, 165-166, 170, 265, 286

unrealities of postulates, 167

use of, for a firm and coalition of rivals,
166
in agglomeration analysis, 180-181, 181n

use of relocation costs with, 180-181, ISln

virtues and limitations, 165-167, 265
General equilibrium theory, and a dimen-
sionless economy, 25-26

and the geographic distribution of factors,
32-33

General equilibrium theory — continued

and the siibstitution principle, 32-35

and Triffin's revision of monopolistic com-
petition theory, 50, 50n

as a special case of a general theory of
location and space-economy, 26-27,
33, 36, 53, 254

as including general location theory, 32-33,
35

dependence on premise of pure competition,
43

dynamical stability properties, 43n

emphasis on a one-point economy, 25-27,
33, 42

feasibility of introducing local price dif-
ferences into, 42

feasibility of introducing transport cost
into, 42

Hicksian formulation, 25-26

inapplicability to space-economy, 42-43

inconsistency of pure competition and
transport cost, 43

limitations of static framework, 43

Mosak's trade formulation, 26

neglect of space in, 25-27, 33, 42, 53

Ohlin's use of, in location theory, 51

transport cost as zero, 26, 33, 42, 53

use of, in Gestalt analysis, 38
General location principle, and the assump-
tion of profit maximization, 221n

as a core element of a general location
theory, 222, 252, 286

as applicable to agricultural location with
raw material supply areas, 248-249,
252

as applicable to situation with both supply
and market areas, 235, 238

as fusing various location and market theo-
ries, 222-253

as implying concentric zones, 244-246

as implying various existing location theo-
ries, 222-253, 286

as involving marginal rate of substitution
between transport inputs, 224-253,
286

as involving ratios of transport rates, 224

as involving the substitution principle, 224-
253, 286

as permitting a generalized transport-
orientation, 226-230, 252

as permitting a more generalized agricul-
tural location theory, 252

as permitting a more generalized Losch
scheme, 252

as relating scale economies and number
and spread of plants, 252-253

as relating spatial extent and capital in-
tensity of production, 252-253

as yielding agricultural location theory,
243-251, 252, 286

as yielding equilibrium point of firm, 222-
253

as yielding Losch market area theory, 240-
242, 252, 286

as yielding market boundary conditions,
239. 252

as yielding supply area analysis, 235, 238.
252, 286

as yielding traditional market area theory.
239, 252, 286

as yielding transport-orientation, 222-230,
252, 252n. 286

SUBJECT INDEX

309

General location principle — continued

form of, as unchanging with concept of
social surplus, 234-235

general statement of, 252, 286

need of a multicommodity framework to
apply to agricultural location, 243

pervasiveness of, 221, 286

statement of, for agricultural location prob-
lem. 246
for locational polygon case, 227
for simple Weberian triangle case, 224
for single firm market area case, 233
for two groups of transport inputs,

228
in multifirm, varying unit cost case, 238
which fuses transport-orientation, market

and supply area analysis, 239
with realistic rate structures, 230

use to fuse location-market-production the-
ories, 222-253, 286
General location theory, and aggregative anal-
ysis, 21

and concept of basic form, 38-42

and disaggregation of industries, 21

and geographic distribution of inputs and
outputs, 27, 53

and geographic variations in prices and
costs, 27, 53

and Gestalt analysis, 38-42, 54

and individual and group space prefer-
ences, 22-23

and interindustry linkage, 20-21

and interregional analysis, 21-22

and interrelations of various location theo-
ries, 23

and principle of limited competition, 37-38

and production stage analysis, 19-21

and stability of spatial flows, 22

and statics and dynamics, 39-41

and structure of market, 39-41

and the assumption of profit maximization,
22 In

and the principle of substitution, 32-36,
54, 221

as a dynamic theory, 38-41, 54

as a synthesis of partial location theories,
91, 221-253, 254-287

as a system of market nets, 44-48

as broader than MarshaUian doctrine, 254

as capable of yielding specific location theo-
ries, 91, 252

as containing urban land use theory, 205-
206

as embracing narrowly conceived trade
theory. 53-54, 208

as included in general equilibrium theory,
32-33, 35

as incorporating Weber and Thiinen in
one framework, 92-93, 252-253, 286

as transportation theory, 22, 213n, 221n-
222n, 286

as yielding transport-orientation via spe-
cific assumptions, 252n

conception of, 26-27, 53-54, 254

Englander's contribution to, 29n, 31, 31n-
32n

equivalence with broadly conceived monop-
olistic competition theory, 50, 54, 254

evolutionary approach to, 15, 28-30, 31n, 54

Furlan's attempt at. Sin

general equilibrium theory as a special case
of, 26-27, 38, 86, 58, 254

General location theory — continued

general location principle as a core ele-
ment in, 222, 252, 286
German contributions to, 27
inadequacy of existing, 207
limitations of Losch's system, 48, 48n, 49
Losch approach plus Thunen, 15-16
Losch's contribution to, 43-50
need for, 9-15, 31, 75, 93
need for complete transport-orientation

framework, 113
need for multicommodity framework in, to
encompass agricultural location the-
ory, 243
need for, in urban-metropolitan analysis,

201n
need to consider aggregate regional demand

and income, 207
neglect of, by classical school, 27
Ohlin's contribution to, 51, 53, 208
Palander's contribution to, 42-43
Palander's skepticism of, 42-43
possible approaches to, 15-23
Predohl's contribution to, 32-36
Ritschl's contribution to, 3 On
synonymous with broadly conceived trade

theory, 53, 54, 254
Thiinen and basic methodology, 27-28
Thiinen approach plus Losch and Weber,

16-19
value of a system of equations, 42
value of empirical study for, 36n-37n
value of input-output analysis for, 49
value of monopolistic competition frame-
work, 27n, 37-38, 49-50
value of price-cost analysis for, 49-50
value of Weberian theory for, 36n-37n
Weber's contribution to, 27-30
Weigmann's contribution to, 37-42, 54
see also Location theory
General theory of location, see General loca-
tion theory
General theory of space-economy, see Gen-
eral location theory ; Space-econo-
my
Generalized index, and existence of weight
triangle, 121
definition of, 121
Geographic balance, and decentralization
policy, 14
and vulnerability, 14
Geographic immobility, see Immobilities
Geographic inequalities of resources, see Nat-
ural resources, geographic inequali-
ties of
Geographic shifts, see Locational shifts
Geographic specialization, among metropol-
itan regions, 12, 17-18
and uneven resource endowment, 5
as affected by factor mobility, 282-285
as affecting structure of industrial districts,

282-285
as dictated by rent functions, 197-199
as related to urban land use, 282-285
as related to urban-metropolitan structure,

282-285
changes in, and advance in transport tech-
nology, 22
effect of change in distance variable on,

215-219, 283-285
in a simple three country trade-location ex-
ample, 213-219, 282-285

310

SUBJECT INDEX

Geographic specialization — continued

increase in, from a fall in the transport
rate, 87
German Historical School, contributions to
Raumwirtschaft, 27
study of development stages, 27, 28n
German National Bureau of Statistics, as a
source of international flow data, 73-
75
Germany, cities, rank-size findings for, 57n

South, pattern of settlements in, 60n
Gestalt analysis, and the structure of mar-
kets, 39-42
use of, by Weigmann, 38-42

in a dynamic general location theory, 54
Government subsidy, and aircraft, 12
and atomic energy, 13
and optimum transport network, 9
distribution among areas, 9
Governmental activities as basic urban in-
dustry, 274n-275n
Graduated transport rate structures, see
Transport rate structures, graduated
Graham's trade theory, appeal of, to loca-
tion theorists, 210
as long-run doctrine, 209-218
emphasis of, on supply and cost conditions,

210
fusion of, with location theory, 210-219
multicountry, multicommodity approach of,

210
neglect of transport-orientation in, 210
relative neglect of transport cost in, 210
Gravity models, as an element of regional
science, 287
as appropriate for short-run trade analy-
sis, 209
need to develop, to explain location more
adequately, 209
to explain trade more adequately, 209
to handle distance variable more ade-
quately, 209
to study structure of space-economy, 287
to synthesize with other techniques, 287
to treat substitution in the large, 287
Greenhut and integration of competitive lo-
cational equilibrium and Weberian
doctrine, 171n
Gregarious instinct, and pattern of settle-
ment, 2
and space preference, 84-85
Gross regional product, projection of, as an
element of regional science, 287
as restraining metropolitan structure, 287
need to synthesize with other techniques,

287
use of, to study structure of space-economy,
287
to treat substitution in the large, 287
see also Income, regional
Groups, social and economic, space prefer-
ences of, 23

Haberler and the problem of fusing trade and

location theory, 220n
Hexagons, regular, as a pure theoretical con-
cept, 274n
as derived through substitution principle,

153, 240-242
as derived when Losch's uniformity as-
sumptions used in Launhardt-Palan-
der model, 267

Hexagons — continued

as inferior to distorted circle in a general

case, 242n-243n
as precluded in realistic market area analy-
sis, 274n
as satisfying boundary conditions, 153, 241-

242
as superior to squares and equilateral tri-
angles as market forms, 44, 153,
241-242
as the ideal market form, 44, 153, 242, 242n
distortion of, in modified Losch diagram,

271-273, 274n
in Losch scheme, 44, 153, 241-242
inconsistency of, with resulting population

distribution, 153-154, 271
Losch diagram of system of nets of, 270
modification of, to square with population

distribution, 271-273
size of, as related to economies of scale,
153, 174
as related to transport costs, 153
superimposition of nets of, and hierarchy
of sites, 153-154, 270-271
Hicksian theory, and interrelations of mar-
kets, 25
and the general equilibrium problem, 25-26
as a special case of a general theory of

location and space-economy, 26-27
emphasis on a one-point economy, 25-26,

26n
fii-st and second order equilibrium condi-
tions, 118
rejection of monopolistic competition, 26n
transport cost as zero, 26
Hierarchy, of cities, and economies of scale,
57-58
and regularity of flows over distance, 58-

60
Christaller's, 60n
desirable changes in, 183
optimum, 183

use of transport inputs in the analysis
of, 255
of city-regions, and commodity classifica-
tion, 18
and geographic inequality of resources,

18, 255
as related to new basic industry, 19, 19n
as related to transport cost level, 19n
of flows, and hierarchy of cities, 58-60
and spatial patterning of cities, 58-60
by volume and length, 58
of focal points, as characteristic of realis-
tic space-economy, 230, 251, 273, 287
as related to a multipunctured transport

cost surface, 230, 251
need for improved analysis of, 287
of industrial districts, in an urban-metro-
politan sketch, 278-280
localization economies within, 278-280
of interstitial areas, 11, 183
of regions and spatial substitution in input-
output analysis, 49
of sites, and inconsistency of uniformity
assumptions, 153-154, 271
as related to transport cost level, 19n
as related to transport network, 272-273
desirable changes in, 183
in modified Losch analysis, 272-273
in Losch's scheme, 17, 153-154, 270-271
within city, 12, 183. 272-273

SUBJECT INDEX

311

Hierarchy — continued

within urban-metropolitan reRion, 11, 12,
17, 183, 272-273

of trade relations among rppions, 22

of trade routes, as characteristic of space-
economy, 251, 287
as related to transport cost level, 19n

of transport routes, need for improved
analysis of, 287

of urban areas, sketch of agricultural hin-
terlands of, 276-278

see also Rank-size rule for cities
Hinterland (s), agricultural, as affected by
resource content of land, 276-277
as irregularly bounded, 276-277
boundaries of as yielded by supply area

analysis, 276n
sketch of for urban-metropolitan regions,
276-278

decline of, 17-18

demarcation of, 16-17

effect of geographic inequalities of re-
sources upon, 18

expansion of, 16-18, 78

extent of, 5-6, 17-18

as related to use of transport inputs,
81-82, 255

of city-regions, 3, 16-17, 276-278

of firms in competitive locational equilib-
rium, 161-165

of metropolitan region, and impact of dis-
tance upon, 68-70, 270-286

of towns in competitive locational equilib-
rium, 161-165

rent-yielding and the indifference of the
farm enterprise to location within,
197
Historical approach, see Evolutionary ap-
proach
Historical inertia, and industrial location, 10

and relocation, 14-15
Hoover's location analysis, and categories of
commodities, 94n

and equalizing differences in labor costs
and transport-orientation, 127n-128n

and fusion of market area analysis and in-
dustrial location, 143n

and market areas for two producers, 148-
151

and minor importance of intermediate loca-
tion, 108

and spatial price discrimination in com-
petitive equilibrium, 164n

and the likelihood of concentration at ini-
tial location, 174n

and the location of intermediary establish-
ments, 175n

and the location of marketing establish-
ments, 175n

and the simultaneous existence of point,
line, and areal markets, 235-236

and the use of margin lines, 149-150, 154n

classification of agglomeration economies,
172

definition of isodapane, 122

definition of localization economies, 172

definition of scale economies, 172

definition of urbanization economies, 172

emphasis upon cost conditions, 210n

modifications of, in supply area analysis,
155n-157n

outline of, 3 On

Hoover's location analysis — continued

patterns of markets, with constant cost

producers, 235-236
relative neglect of demand, 210n
rent surface of, as identical with rent func-
tion, 195n
use of isodapanes for transport-orientation

solution, 122-124
use of isotims to construct isodapanes, 122-
123
Hotelling, and agglomeration tendency, 162
and movement along the revenue-outlay

substitution line, 170
and substitution among transport outlays,

170, 170n
and substitution between production and

transport outlays, 170n
and the Cournot problem when firms infini-

tesimally close, 161n
assumption of simplified line case, 160
locational equilibrium when price and loca-
tion vary simultaneously, 160-162
solution (s) of, as borne out by Zeuthen,
160
as yielded by substitution principle, 170,

170n
invalidity of, 162-163, 163n
stability of, 161-162
when both firms mobile, 162
when no undercutting, 160
when one producer immobilized, 160
when one trades autonomously, 160
when producers undercut, 161-162
when production costs are zero, 161
undercutting as related to proximity of
producers, 161
Human ecology, see Agricultural land use ;
Pattern of Settlement; Space-econ-
omy ; Urban land use
Human resources, geographic inequalities in,
and difficulty for Losch theory, 48-49
and the substitution principle, 34-35
and variations in local input costs, 132-

133
as ignored in competitive locational equi-
librium models, 169-170
in OhHn's doctrine, 52
significance of, 19n
uniform distribution of, in Losch theory,
44
Hyperbolas, as market boundaries, 146, 239,
261
as supply area boundaries, 157
Hypercircles, as market area boundaries,
147n, 239
as supply area boundaries, 157
Hyson, definition of market boundaries, 147n,
239
market area theory of, as yielded by general
location principle, 239

I.C.C. data, use of, to measure commodity

fiows, 70-73
Ideal weights, as locational forces in a loca-
tional triangle, 122
conversion of actual weights into, 228
use of, in computing locational weight, 141
to adjust for commodity rates, 109n, 120-

122, 228
to consider substitution between groups
of transport inputs, 228
valid use of, by Weber, 109n

312

SUBJECT INDEX

Immobile commodities, as goods of infinite
weight with infinite weight loss, 32n,
94
Englander's conception, 32n, 94
in a commodity classification, 93-94
Immobilities, and cheap labor, 128n
and Ohlin's concept of region, 51
and secondary labor, 128n
and substitution possibilities in location

analysis, 94
and the locational line case, 96
and the locational triangle case, 97
and Weigmann's principle of Hmited com-
petition, 37-39
as affecting industrial location, 8
as an element in general location theory, 27
as reflecting relocation costs, 283n
basic role in location theory, 37-39
effect on trade and geographic specializa-
tion, 283-285
effect on urban-metropolitan structure and

land use, 283-285
in the capital goods market, 41, 283, 283n
in the classification of commodities, 93-94
in the labor market, 40-41, 283, 283n
in the land market, 40

inadequacy of Classical trade theory prem-
ises on, 208
of individual producers in agricultural loca-
tion theory, 93
spatial, in Ohlin's interdependence system,

51
see also Factors, mobility of
Imperfect competition, see Competition ;

Monopolistic competition
Import-export duties, effect on iso-outlay
lines, 112
possible effect on equilibrium point, 112
Imports, see Trade ; Commodity flows
Income, effects, differences in, from different
basic industry, 284
inequalities of, and enclaves of excluded
consumers, 146n
and noncircularity of market areas, 145-
146
locational effect of geographic patterns of,

21
regional (urban), as it affects firm loca-
tion, 159n-160n
as restraining urban land use, 206
effect of basic activities on, 278
effect of firm's location on, 159n-160n
effect on commercial and service activi-
ties, 278
effect on industries using ubiquities, 278
need to consider in general location

theory, 207
need to consider in trade theory, 207
need to study relations with metropolitan
structure, 287
Increasing cost conditions, market area anal-
ysis under, 149-154
supply area analysis under, 155-158
Increasing returns, and population nuclea-
tion, 2
centripetal effect of, 4
Indeterminacy, in locational line case, 119-
120
minor degree of, from realistic rate struc-
tures, 107n
of competitive equilibrium as related to
market discontinuities, 165

Indeterminacy — continued

of firm location, case of, with scale econo-
mies, 174n
of firm location in oligopolistic situations,

160-171
of locational equilibrium from twin solu-
tions, 103n, 104n
of market boundaries from discrete con-
sumer spread, 146n
Index of labor costs, see Labor, costs of
India, cities in, rank-size findings for, 57n
Indispensability, and substitution possibilities
in location analysis, 94
and the locational line case, 95-96
and the locational polygon case, 98
and the locational triangle case, 97
in the classification of commodities, 93-94
Industrial aggregation, see Aggregation, in-
dustrial
Industrial complex analysis, see Complex

analysis
Industrial complexes within metropolitan re-
gions, 13-14
Industrial concentration, see Production, con-
centration of
Industrial decentralization, see Decentraliza-
tion
Industrial disaggregation, see Disaggregation,

industrial
Industrial districts, a structure of, consistent
with localization economies, 278-280
hierarchy of, in an urban-metropolitan

sketch, 278-280
structure of, as related to distance variable,
282-285
as related to factor mobility, 282-283
as related to geographic specialization,

282-285
as related to political variable, 283-285
as related to trade, 282-285
Industrial land use, see Urban land use
Industrial location, aggregate patterns and
agglomeration economies, 21, 78n-79n
and decentralization policy, 14-15
effect on, of aircraft, 12
of atomic energy, 13

of breaks in transport network, 110-112
of technological advance, 7, 10, 12-13, 22,
79n, 258-259
effect on agricultural location, 7-8
use of activity analysis in determining,

168n-169n
value of agricultural location analysis for,

189
see also Industrial structure ; Location ;
Location theory
Industrial stratum, primary, effect on sec-
ondary industrial stratum, 29
relation to agricultural stratum, 7, 29
secondary, composition of, 29

dependence on primary industrial stra-
tum, 29
Industrial structure, as affecting urban-met-
ropolitan structure, 284-285
as related to agricultural stratum, 7, 29
as related to transport network, 8
impact of atomic energy upon, 13
interrelations within, 29
of nations, as affected by distance variable,

215-219, 282-285
processes in development of, 6-15
sketches of, 262-285

SUBJECT INDEX

313

Industrialization, see Basic industry ; Indus-
trial structure
Industry, agricultural, definition of by city-
region, 249
basic, see Basic industry
classification of, Colin Clark type, 21

Leontief type, 21
definition of, and agglomeration economies,
92n
and monopolistic competition, 92n
in location analysis, 92, 92n
location analysis of, as a level of inquiry,
92
Inelasticities, spatial, as an element in gen-
eral location theory, 27, 37-38
in the capital goods market, 41
in the labor market, 40-41
in the land market, 40
see also Immobilities
Innovation, see Technological advance
Input-output analysis, as appropriate for
short-run trade doctrine, 209
inclusion of, in a broadly conceived monop-
olistic competition theory, 50
incorporation into, of local multiplier ef-
fects, 49
of locational shifts, 49
interregional, as an element of regional
science, 287
need to synthesize -with other techniques,

287
use of to study structure of space-econ-
omy, 287
use of to treat substitution in the large,
287
limits to substitution within, 49
spatial substitution in, via bill of goods
change, 49
via iterative approach and coefficient

change, 49
via matrix change, 40
via regional breakdown, 49
use of, to identify substitution points, 49
value of, in spatial theory, 49
Instability, see Stability

Interest outlays, and substitution with labor
outlays, 36
differentials in, incorporation in an out-
lay-substitution line, 133
Interest rate, as a distorting factor in space-
economy, 138-140
effect of, on supply curve of transport in-
puts, 88n-89n
on the spatial extent of production, 88n-

89n
on the transport rate, 88n-89n
transport rate contrasted with, 88
unsystematic variation with distance,
138
Interlocal trade theory, see Trade theory
Intermediary establishments, location of, see
Location, of intermediary establish-
ments
Intermediate location (s), as a location type
in Launhardt-Palander model, 257-
258, 262-264
economies of scale as reducing number of,
in Launhardt-Palander model, 265-
266
in case of two sources of each of two raw

materials, 262-264
overestimate of, by Weber, 109

Intermediate location (s ) — continued

shifts of, with technological change, 258-

259
small likelihood of, 108, 113n, 230n
International trade, see Trade
Intervening opportunities hypothesis, see

Stouffer
Intraregional trade, see Trade
Iron and steel industry, and economic devel-
opment, 6-7
as a basic industry, 18-19, 128n, 284
as a case of transport-orientation, 21 In,

215-219
as an agglomerating-inducing activity, 8,

19n
as having different income effect than tex-
tiles, 284
as having different land use effect than

textiles, 284-285
as having different multiplier effect than

textiles, 284
as partially labor- and partially transport-
oriented, 217-219
deviating effect of cheap labor upon inter-
national location of, 216-219
effect on hierarchy of cities, 19, 19n, 79n
geographic shift of, 10
international location of, via opportunity

costs, 211-219, 282-285
labor costs as minor in intranational loca-
tion of, 211n
location of, 6-8, 10, 19n, 37n, 211n, and
minimization of transport inputs,
80n, 118n, 215, 216
locational effect on steel fabricating, 8
technological advance in, 7-8, 10
Iron ore deposits and relation to iron and

steel location, 6-8, 10, 19n, 118n
Irrationality, see Rationality
Isodapane technique, applicable to four (or
more) sided polygons, 122n
as embracing realistic transport situations,

122n
cumbersome to handle, 122n
more flexible than weight triangle method,

122n
substitution between transport inputs in-
volving same basic considerations,
123-124
use of, for transport-orientation solution,
122-124
Isodapanes, as contour lines of transport
cost surface, 122-124
construction based on isotims, 122-123
construction based on isovectors, 122-123
critical, definition of in agglomeration anal-
ysis, 176n
definition of in labor orientation, 130n
determinants of, 176n

disadvantage relative to substitution ap-
proach in labor orientation, 130-131,
131n, 259
distance from, and pull on producer,

178
ideal distance from, and best labor site,

130
intersection of, in agglomeration, 176-

178
to determine feasibility of labor loca-
tion. 130-131, 142, 259-260
to determine feasibility of power orienta-
tion, 132, 259

314

Isodapanes — continued

use in analysis of urbanization, 183, 188
use to determine market area of cheap

labor site, 259-260
use to determine market boundary, 259-

260
Weber's use of. 130-131, 132, 176-178
decrease of non-circularity of, with dis-
tance of deviation, 141
definition of, as an incremental concept, by

Weber. 130n
definition of, by Palander. 122
distance between, and likelihood of labor

orientation, 141
effect of locational weight upon distance

between, 141
effect of transport rate upon distance be-
tween, 141
movement from subset to subset of, as in-
volving substitution, 123-124
non-circularity of, 141
subset of. and use by Palander. 123-124
Isolated city-region, see Agricultural location

theory ; City-region
Iso-outlay lines, see Price-ratio (iso-outlay)

lines
Iso-product curve, definition of, 19 In

use of, in agricultural location theory, 191-
194
with scale lines, 191-194
with price-ratio lines, 192-194
Iso-revenue lines, use of. in determination of

firm's location, 159
Iso-revenue-less-outlay lines, as convex (con-
cave) because of firm's influence on
price, 159
construction of. 133-134

for revenue potentials and transport out-
lays, 133-134
use of. to derive Hotelling's solutions. 170,
170n
to determine equilibrium point, 133-135,

159
to determine orientation to higher-price

markets, 133-135
with revenue-outlay substitution line,
134-135, 159. 175
Isotants as market boundary lines, 239
Isotims, as contour lines of transport cost.
122
definition of. 122
use of, by Hoover, 122-123

to construct isodapanes. 122-123
Isovectors. as contour lines of transport cost,
122-123
definition of. 122-123

movement along, as equivalent to substitu-
tion between transport inputs. 123-
124
use of, by Palander, 122-123

to construct isodapanes, 122-123

Journey to work, pattern of, as basic to
analysis of space-economy, 281
effect on urban land use, 281
problem of, increase in with urbanization,
185

Knight, capital theory of, defects of its im-
plications, 83n
distinction between resources and serv-
ices, 89-90

SUBJECT INDEX

Knight — continued

implications of, for theory of space-
economy, 83n

Koopmans transportation problem, see Trans-
portation analysis, Koopmans

Labor, aggregate demand in market for, 40-
41
as a major location factor, 138-140
attraction of, as Alternativattraktion, 130n
as involving discrete geographic jump.
130n
cheap, as a deviating force, 216-219

as a location factor, 8-9. 10. 31n. 128n,

211-219
and minimum subsistence, 128n
causes of, 8-9, 10, 127n-128n
concept of, 127n-128n
effect on international steel location, 216-

219
effect on textile location, 10, 140n, 211-

219
efficiency vs money wages, 127n
from cultural factors, 128n
from low transport cost on food. 127n
in surplus food regions, 127n
possible inclusion in transport-orienta-
tion. 127n
force, structure of, and decentralization

policy, 14-15
geographic distribution of, and economic

activity between men, 53n
immobilities of, see Immobilities
market for, and nature of capitalism. 40n-
41n
and types of migration, 40
productivity, need to study relations with

metropolitan structure, 287
secondary, attraction for parasitic industry,
128n
definition of. 128n
generated by basic industry, 128n
immobility of. 128n
skilled, agglomeration economies in the use
of, 182, 185
as a location factor, 10
increasing cost of from urban congestion,
185
spatial inelasticity of, 40
structure of market for, 40-41
structure of requirements for, and decen-
tralization policy. 14
see also Factors ; Labor costs ; Labor orien-
tation
Labor coefficient, as a measure for different
industries of potential deviation to
labor location, 141
as a technical concept, 36n
as labor cost per locational ton, 141
assumptions in the use of, 141
definition of, 141
general applicability of, 142
power coefficient as a parallel to, 132n,

141-142
pros and cons of ratio of labor cost savings
to additional transport outlays, 142
relation to substitution between transport

and labor outlays, 141-142
to establish priorities for attraction of in-
dustries, 141
use of. to derive ratio of labor cost savings
to additional transport outlays, 142

SUBJECT INDEX

315

Labor cost savings, need to express in terms
of opportunity costs, 217-219
ratio of to additional transport outlays,
advantages over labor coefficient,
142
as derived from labor coefficient, 142
as indicating labor orientation, 142
as measuring savings of labor location,

142
determination of, 142
of less general use than labor coefficient,

142
use to answer basic location questions,
142
Labor costs, as a distorting factor in space-
economy, 138-140, 260
as determining industrial distribution by

regions, 172
as ignored by competitive locational equi-
librium models, 169-170
as major in textile location, 211n
as minor in iron and steel location, 211n
compression of index of, 141
equalizing differences in, as contrasted with
real differences, 127n, 139n
as reflecting transport cost differences on

consumer goods, 127n, 139n
inclusion of in transport-orientation,
127n-128n
for labor intensive firm as comparable to

rent for farm enterprise, 189-190
index of, and labor orientation, 141
real differences in vs. equalizing differences,

139n
regularity in geographic pattern as related

to transport cost, 139n
stable geographic pattern, 138
unsystematic geographic pattern, 138, 139n,

140n, 202n, 259
see also Factor costs ; Labor ; Labor outlays
Labor inputs, as embodied in capital inputs,
81
as embodied in transport inputs, 80n, 81,

89-90
substitution between, at cheap labor site,

131
substitution for other inputs, at cheap

labor site, 131, 131n, 196
substitution with transport inputs, 36, 81-
82
inadequacy of, 126-127
Labor-interest coefficient, difficulties in use

of, 137n
Labor locations, as centers of agglomeration,
179
market area of, as determined by critical

isodapanes, 259-260
shifts to, and the substitution principle,

95n, 135
see also Labor orientation
Labor orientation, agglomeration at points
of, 179
and ideal distance from critical isodapane,

130
and index of labor costs, 141
and locational weight, 141
and percentage of compression of labor

cost index, 141
and relative position of locational figures

and labor locations, 141
and transport rates, 141
as included in extended trade doctrine, 219

Labor orientation — continued

as indicated by the ratio of labor cost sav-
ings to additional transport outlays,
142

as involving additional transport outlays,
127-131, 141-142, 216-219

as involving substitution between transport
and labor outlays, 127-131, 140n. 189-
190, 196, 259, 275

as yielded by substitution principle, 127-
131, 196, 259, 275

cases of internationally, as transport-orien-
tation intranationally, 219

critical isodapane and feasibility of, 130-
131, 259-260

definition of, in international framework,
217

determination of market areas in, 259-260

discontinuities in potential sites of, 130n,
196

effect of economies of scale on, 175n-176n

in international steel location, 217-219

incorporation of, in Launhardt-Palander
construction, 259-260

labor coefficient and priorities for attrac-
tion of industries, 141

labor outlay differentials as central in, 275

likelihood of, and distance between isoda-
panes, 141

need to state in terms of opportunity costs,
210-219

point of, and use of iso-outlay line, 129-
131
and use of outlay-substitution line, 129-
131

possibility of for different industries, as
indicated by labor coefficient, 141

role of replacement deposits in, 131n, 135,
141

sketches showing fusion of, with other lo-
cation-market-trade doctrines, 256-
285

some cases of parasitic industries as ex-
ceptions, 128n

use of market area analysis in, 260-261
Labor outlays, and production outlays, rela-
tions between via substitution prin-
ciple, 259

and transport outlays, substitution between
in boundary formation, 264
substitution between in labor orientation,
127-131, 140n, 189-190, 196, 259, 275

as a variable in transport-orientation anal-
ysis, 113

by types, substitution between, 130, 196,
259

differentials in, as central in labor orienta-
tion, 127-131, 275
incorporation in an outlay-substitution
line, 127-129

in Predohl's framework, 33

in terms of use units, 35

substitution with interest outlays, 36

substitution with transport outlays, 36

see also Labor costs ; Labor orientation ;
Production outlays
Land inputs, and other inputs, substitution
between in operation of farm enter-
prise, 193-194, 275

as embodied in capital inputs, 81

as embodied in transport inputs, 80n, 81,

316

SUBJECT INDEX

Land inputs — continued

different direct requirements of, by type
basic industry, 285

different indirect requirements of, by type
basic industry, 285

price of, as affecting factor proportions,
275
as decreasing with distance from market,

275
see also Rent

see also Agricultural land use
Land market, relation to aggregate demand,
40

spatial inelasticity of, 40

structure of, 40

see also Agricultural land use ; Hinterlands,
agricultural
Land use, see Agricultural land use ; Urban

land use
Land-use outlays, see Land-use units
Land-use theory, see Agricultural location

theory ; Urban land-use theory
Land-use units, and rent outlays, 34

definition of, 34

in Predohl's framework, 33-34

invalid use of, 34-35
Land values, see Rent ; Urban land price
Launhardt, and comprehensive market area
analysis, 143n, 153

and separate analysis for market areas
and production for one-point mar-
ket, 143n

emphasis on cost conditions, 210n

first significant treatment of industrial
location theory, 143n

fixed fii-m locations in market area analysis
of, 160

market area theory of, as yielded by gen-
eral location principle, 239

relative neglect of demand, 210n

use of pole principle for transport-orienta-
tion solution, 122, 256-258

see also Launhardt-HoteUing problem ;
Launhardt-Palander construction
Launhardt-Hotelling problem, 160-163
Launhardt-Palander construction, ability of,
to consider technological change,
258-259

addition to, of economies of scale, 265-267
of localization economies, 267-268
of urbanization economies, 268-270

and discontinuities in market areas, 264

and economies of scale as yielding Losch
theory under uniformity assump-
tions, 267

as applicable to an area of consumers, 256-
258

as emphasizing localized raw material use,
274

as excluding agglomeration economies, 256

as excluding differences in local prices,
256

as implied by substitution principle, 267

as involving angle conditions, 257

as relating consumer districts to material
sources via industrial producers, 263-
264

as yielding an infinite number of produc-
tion points, 257-258

as yielding hexagonal market areas from
use of Losch's uniformity assump-
tions, 267

Launhardt-Palander construction — continued
as yielding market, material, and interme-
diate locations, 257-258, 262-264
changes in boundaries in, from scale econo-
mies, 266-267
gross overstatement in, of number of pro-
ducers, 265
incorporation into, of cheap production
sites, 260
of cheap tax sites, 260
of labor orientation, 259-260
of power orientation, 260
market areas required in, from scale econ-
omies, 266-267
modified, fusion of with modified Losch
scheme to yield urban-metropolitan
structure, 274-275
neglect in, of advertising, 265

of competitive locational equilibrium,

264-265
of firm location policy, 265
of firm pricing policy, 264-265
of oligopolistic behavior, 265
non-existence of, when a weight is domi-
nant, 258n
procedure for developing, 256-257
symmetry of, 257
uniformity assumptions of, 256
use in, of market area analysis, 260-267
of pole lines, 256-258
of weight triangle, 256-258
use of, in analysis of changing weight re-
lations, 259

nalysis of new material source, 259
in dynamic analysis, 258-259

the fusion of location-market-trade
doctrines, 256-285
in transport-orientation, 256-258
to consider several differentials, 259-260
to convert transport-orientation into a
more generalized location problem,
256-269
to delineate consumer districts, 262-264
to delineate market areas, 262-264
to identify locational shifts, 258-259
to portray locational interrelations, 256-

268
with two sources of each of two raw
materials, 262-263
Leontief, see Input-output analysis
Lerner and Singer, demand elasticity in com-
petitive locational equilibrium, 163-
164
effect of market size on solution, 164
effect of transport cost on solution, 164
more realistic undercutting policy, 163-164
price limit in competitive locational equi-

Hbrium, 163-164
solutions in competitive locational equilib-
rium, 163-164
spatial price discrimination in competitive
equilibrium, 164n
Limited competition, see Immobilities ; Mo-
nopolistic competition
Line charges, as causing local minima on
transport cost surface, 229-230
effect of graduation of, on equilibrium con-
ditions, 229-230
on statement of general location princi-
ple, 229-230
graduation of, as affecting industrial loca-
tion, 107, 229-230

SUBJECT INDEX

317

Linear programming, see Activity analysis
Linkage, interindustry, and commodity com-
position of trade, 20-21
and geographic split of production, 20-

21
as generating agglomeration economies,

21
as related to production stage, 21
Loading charges, see Terminal charges
Localization, a pattern of, in an urban-met-
ropolitan region, 278-280
a simple case of, 268

analysis of, as overlapping with urbaniza-
tion analysis, 182
as a historical process, 180, 287-288
as embraced by the substitution principle,

179-182
as involving increase in transport outlays,

179, 267
as involving substitution between produc-
tion and transport outlays, 179, 179n,
267
centers of, advantage of existing produc-
tion points as, 180
as influenced by side payments, 180-181,

181n
as pulled to superior bargainers, 180
clearest analysis of, as yielded by social
welfare approach, 268
for new regional development, 268
complications because of the coalition prob-
lem, 181n
conditions for, 176-178

decrease in validity of Weber's assump-
tions with increase in, 179n
difliculties from non-symmetric location

factors, 181n
difficulties of converting to a constant-sum

game, 181n
effect of differential bargaining abilities on,

180-181, 181n
labor locations as centers of, 179
minimization of transport costs as deter-
mining point of, 177-178
need for critical isodapanes to intersect,

176-178
point of, as affected by replacement de-
posits, 178
difficulty in determining, 267-268
use of transport-orientation techniques
to indentify, 178
problem of, in multiplant firm, 179, 179n
requisite total output for, 176-178
size of, and distance from critical isoda-

pane, 178
significance of inherited physical struc-
tures for, 180
significance of relocation cost for, 180-181,

181n
theory of, and complications of coalition
problem from relocation cost, 181n
sketches showing fusion of with other
location-market-trade doctrines, 256-
285
use of game theory in, 180-181, 181n
use of side payment to induce, 179n
see also Agglomeration ; Agglomeration
theory ; Urbanization
Localization diseconomies, see Deglomeration

economies ; Localization economies
Localization economies, addition of, to Laun-
hardt-Palander model. 267-268

Localization economies — continued

analysis of effect of, 175-182, 267-268

as external economies, 267

as included in agglomeration economies,

139, 172, 265, 267
as scale economies with multiplant firms,

179n
definition of, 172, 267
effect of, on structure of space-economy,

268
in large lot buying and selling, 182
in the use of auxiliary facilities, 182, 267
in the use of skilled labor, 182
incorporation of, in sketches of fused loca-
tion-market-trade doctrines, 256-285
locational shifts from, 268
overlapping with economies of scale, 182n,

265
overlapping with urbanization economies,

182, 265
structure of industrial districts consistent

with, 278-280
see also Agglomeration economies
Localized raw materials, activities using,
plus market-oriented activities as
yielding urban-metropolitan struc-
ture, 274-275, 278-280
and a Thiinen-Losch model, 18-19
and the Zipf theoretical framework, 79n
deposits of, and implications for population

concentrations, 79n
failure of Losch theory to treat, 154, 158,

274
strength of Weberian theory for handling,

158, 274
use of, as emphasized in Launhardt-Palan-
der construction, 274
Location, and adaptability of capital goods,
41
and mobility of groups of factors, 41
and the geographic distribution of factors,

32-33
and the geographic pattern of local supply

prices, 31
and the simultaneous determination of
commodity flows, 207, 215, 219, 282-
285
and trade, a simplified case of, 210-219,
282-285
interrelations of, 6-7, 3 In, 50-54, 167n-

169n, 207-220, 281-285
simultaneous determination of, 6-7, 50-
54, 207, 215, 282-285
as determined by a set of substitution

points, 33-34, 94
case of transport-orientation intranation-
ally as other orientation internation-
ally, 219, 282
continuity of, in agriculture, 196
discontinuity in, 95n, 101-112, 113, 130n,

176n, 196, 230, 251, 264
effect of changes in distance variable upon,

215-219, 283-285
fundamental questions of, 9-15
general principles of, and interrelation of
strata, 30
non-operational character of, 23
use of, 23
international, as affected by the political

variable, 283-285
need to develop gravity models to explain
more fully, 209

318

SUBJECT INDEX

Location — continued

need to specify demand to determine pat-
tern of, 212

non-additive character of, 21

of aluminum industry and substitution be-
tween power and transport outlays,
189-190

of intermediary establishments and substi-
tution among transport outlays, 175n

of intermediary establishments and sub-
stitution between transport and pro-
duction outlays, 175n

of interrelated production stages, 20-21

of iron and steel industry, see Iron and
steel industry

of manufacturing, 52

of marketing establishments and substitu-
tion among transport outlays, 175n

of marketing establishments and substitu-
tion between transport and produc-
tion outlays, 175n

of raw material production, 52

of textile industry, see Textile industry

patterns of, as associated with commodity
flows, 281
as associated with population flows, 281
changes in with technological change,

258-259
under constant cost assumptions, 235-
236

rich and poor sectors of, in Losch, 270-273

spatial coordinates of, 45-46

simultaneous existence of at markets, raw
material sites, and intermediate
points, 235-236

see also Industrial location ; Location the-
ory ; Space-economy, structure of
Location factors, agglomerative and deglom-
erative factors as a major group,
139-140

classification of, into three groups, 138-
140
overlapping of groups, 138, 138n-139n

labor, power, interest, taxes, etc. as a
major group in, 138-139
Location theory, and equilibrium among spa-
tially separated markets, 167n-169n,
213n, 286

and fusion of the several doctrines via sub-
stitution principle, 189-199, 221, 252,
255, 259, 275, 286

and Graham's trade theory, some common
elements of, 210

and need for a sharply defined concept of
rational behavior, 286

and neglect of mobility of groups of
factors, 41

and price theory, 23, 32, 42n

and principles to reduce number of poten-
tial locations, 168n-169n

and production theory, fusion of via gen-
eral location principle, 23, 252-253,
286
fusion of via substitution principle, 113,

221, 252-253, 255, 259
need to spell out fusion in detail, 286
parallel use of substitution, 135-137, 252-

253
similarity of first- and second-order con-
ditions, 118

and trade theory, fusion of, 23, 50-54, 207-
220, 281-282

Location theory — continued

need to explore relations comprehen-
sively, 286
and transport cost as a function of dis-
tance, 35, 138-140, 210
as contained in a total analysis for Ohlin's

district, 52-53, 53n
as contrasted to the Samuelson-Beckmann
market-transportation model, 168n-
169n
as generalized around the Thiinen ap-
proach, 249-251
as generally assuming continuous space-
economy, 251
as generally precluding use of regular hex-
agons, 274n
as related to the Koopmans transportation

problem, 168n-169n
as traditionally exchiding urban land use

theory, 200
as treating separately production for one-
point market and for market area,
143n
existing, as implied by general location

principle, 222-253, 286
extension of, as including developments in
long-run trade theory, 219-220, 282
by fusion of trade doctrine and trans-
port-orientation, 217-219, 281-282
through opportunity cost formulation,

210-219, 281-282
to include production theory through
concept of transport inputs, 118-
119, 252-253
fused location-market-supply statement of,

250-251
generalized Thiinen approach as more de-
sirable than generalized market area
approach, 249
inadequate treatment of costs in, for in-
ternational analysis, 210
incorporation of Weber and Thiinen in
one framework, 92-93, 188-189, 275
most general frameworks of, 249, 252, 285-

286
need to consider transport rate structure

as variable, 213n, 286
need to develop operational models, 287
need to express in terms of opportunity

costs. 210-219
partial, and firm analysis, 91-124

as yielded by general location theory, 91,

252
Dean's contribution, 3 In
fusion of with use of general location

principle, 221-253
Hoover's contribution, 30n
synthesis of in general location theory,

91, 221-253, 254-287
Usher's contribution, 3 In
the Weber-Thiinen dualism in, 92-93, 188-

189, 275
traditional, and assumption of fixed trans-
port rate structures, 213n
and assumption of monopolistic elements

in transportation, 213n
and neglect of transport rates as de-
pendent on flow volumes, 213n
emphasis upon costs in, 210n
relative neglect of demand in, 210n
use with a general market-transportation
model, 168n-169n, 286

SUBJECT INDEX

319

Location theory — continued

see also Agricultural location theory ; Gen-
eral location theory
Locational analysis, as generalized around
the Thunen approach, 249-251

different levels of inquiry, 92-93, 189

firm level of inquiry, 92

generalized Thunen approach as more de-
sirable than generalized market area
approach, 249

industry level of inquiry, 92

most general frameworks of, 249, 252

regional (world) level of inquiry, 92

use of excess supply function in, 168n

use of Launhardt-Palander construction in,
256-268

see also Agricultural location theory ; Gen-
eral location theory ; Location theory
Locational equilibrium, see Equilibrium point
(for firm location) ; Firm, indus-
trial ; Firm, agricultural ; Competi-
tive locational equilibrium ; Firm,
urban ; Agricultural land use ; Urban
land use
Locational forces, as generally precluding
hexagonal markets, 274n

as ideal weights, 121-122

as transport cost per unit distance, 122

equilibrium outside triangle, and angle
conditions, 122

equilibrium outside triangle, and corner lo-
cation, 122

mathematical presentation of equilibrium
of, in simple polygon case, 226-227

mathematical presentation of equilibrium
of, in simple triangle case, 225

use of mechanical model to determine equi-
librium of, 121

use of weight triangle to determine equi-
librium of, 121
Locational line case, as a simplified trans-
port-orientation problem, 96-97, 119-
120

as substitution between distance variables,
96-97

as substitution between transport inputs,
117. 119-120

assumptions of, 95-96

determinacy with use of a weight-losing
material, 120

determinacy with use of an ubiquity, 120

end point solution for, 107, 107n

indeterminacy with graduated transport
rates, 120

most simple situation, indeterminacy of,
119-120

Weberian propositions derived via trans-
formation and price-ratio lines, 119-
120
Locational polygon case, as a simplified trans-
port-orientation problem, 98-101

as handled by the isodapane technique, 122-
124

as substitution between distance variables,
subject to transport cost restraint,
98-101, lOln, 226-227

as substitution between transport inputs,
subject to transport cost restraint,
117, 226-230

assumptions of, 98-100

complications of the substitution problem
in, 98-101. lOln

Locational polygon case — continued

conditions for corner location, 226, 229-230

conditions for minimum point of transport
cost surface in, 226, 229-230

conditions to be satisfied by equilibrium
point, 104, 226-227

end point (corner) solution for, 107-112,
113n, 229-230

first-order conditions for, in terms of modi-
fied transport rates, 229-230

invalid use by Weber of fictitious distances,
109, 109n

possibility of several minimum points, 229-
230

profit maximization in, as reduced to vari-
ation in distance variables, 226

need for direct total transport cost com-
parisons, 230

second-order condition for, in terms of
modified transport rates, 229-230

small likelihood of intermediate location,
108, 113n, 230n

use of realistic rates and weights to de-
rive transport cost restraint, lOOn,
229-230

valid use by Weber of ideal weights, 109n,
228

with finite number of transport routes, 101

with realistic rate structures, 229-230
Locational shifts, and advance in transport
technology, 22

and decentralization policy, 14-15

and guiding new industrial growth, 14-16

and industrialization, 9-10

and national commodity production, 17-18

and substitution between outlays, 33-34, 49,
179, 179n

and substitution between outlays and reve-
nues, 49, 174-175, 175n-176n

and substitution between transport inputs,
49, 251

from focal point to focal point, as substitu-
tion in the large, 251

in terms of substitution and transforma-
tion lines, 175n-176n

incorporation in, in input-output analysis,
49

of agricultural firm and eflfect on farm
operations, 194-195

problems of. 9-10

resulting from localization economies, 268

resulting from scale economies. 265-267

resulting from technological advance, 7,
12-13, 258-259

resulting from urbanization economies, 269-
270, 273-274

role of price changes in, 243

use of Launhardt-Palander construction to
identify, 258-259
Locational triangle case, as a simplified trans-
port-orientation problem. 97-98. 101-
104

as handled by the isodapane technique. 122-
124

as substitution between distance variables,
97-98

as substitution between transport inputs,
114-119, 120-122, 222-226

assumptions of, 97, 102

conditions for corner location, 225-226

conditions for minimum point on transport
cost surface. 223-225

320

SUBJECT INDEX

Locational triangle case — continued

end point (corner) solution for, 107-112,
121-122

mathematical formulation of, 222-226

profit maximization in, as reduced to varia-
tion of three distance variables, 222-
226

situation of single minimum point, 224

smaU likelihood of intermediate location,
108, 113n

solution with the use of price-ratio and
discontinuous transformation lines,
101-112

transport cost surface as convex downward
in, 224

when Dean's index less than unity, and
substitution, 121

when material index less than unity, and
substitution, 120-121

when weight triangle exists, and substitu-
tion, 121-122

with finite number of transport routes,
101-112

with realistic rate structures, 104-112
Locational weight, as a technical concept, 36n

as an element of the labor coefficient, 141

as based on ideal weights, 141

effect on distance between isodapanes, 141

use of, in labor orientation, 141
Long-run trade theory, see Trade theory
Losch (location theory of), and a non-opera-
tional general equilibrium system for
agriculture, 243n

and causes of interregional trade, 17

and constant unit cost on farm, 244, 247

and consumer indifference on market boun-
daries, 46-47, 240

and exhaustion of plain by net of market
areas, 46-47, 241

and frequency distribution of cities, 59-60

and fusion of market area analysis and in-
dustrial location, 143n

and general equilibrium under monopolistic
competition, 43-50

and hierarchy of sites, 17, 153-154, 270-271,
273

and invalidity of pure competition assump-
tions, 158-159

and market area as small as possible, 46-47

and market orientation, 16, 274

and margin lines, 150-151

and maximum coincidence of locations, 271

and maximum number of independent pro-
ducers, 46

and maximum local demand, 271

and minimum shipments, 271

and minimum sum of shortest distances,
271

and natural markets, 150-151

and optimum agricultural pattern, 48

and optimum consumption pattern, 48

and pattern of production centers, 270-271

and price equals average cost, 43, 46-47,
240

and regular hexagons as precluded in real-
istic market analysis, 274n

and sloping demand curve for the firm,
158-159

and substitution between production out-
lays for two producers, 153-154

and substitution between transport inputs,
153-154, 240-242

Losch (location theory of) — continued

and the Chamberlinian tangency solution,
46

and the construction of a firm's aggregate
demand curve, 159n

and the effect of differentials in input
prices on agricultural location, 199n

and the hexagon as a pure theoretical con-
cept, 274n

and the maximizing of social welfare with
hexagonal market areas, 153, 242

and the superiority of hexagonal to equi-
lateral triangular market areas, 242

and the superiority of hexagonal to square
market areas, 241-242

and use of ubiquitous materials, 16, 274

and yield per acre as invariant with dis-
tance from city, 244, 247

anemic character with respect to substitu-
tion points, 49

applicability to urban-metropolitan re-
gions, 154

arrangement of nets about common pro-
duction center, 270-271

as a logical point of departure for regional
analysis, 154

as a supplement to agricultural location
theory, 16-19

as a simplified market area problem, 239-
240

as an outgrowth of Fetter-Launhardt ap-
proach, 153

as applicable to location of service activi-
ties, 154, 274

as derived from Launhardt-Palander con-
struction with economies of scale,
267

as derived from the substitution principle,
153-154, 239-242, 267

as excluding material orientation, 274

as implying regular geometric shapes of
market areas, 241

as implying same size for all producers,
240

as implying straight line boundary between
any two producers, 240-241, 274n

as inapplicable to localized material using
industry, 154, 157, 274

as involving multicommodity framework,
44-49, 153-154, 270-271

as yielded by the general location principle,
240-242, 252, 286

boundary equations of original model, 45-46

central place theories of, 60, 60n

concept of market net, 44, 239, 270-271

concept of system of market nets, 44-45,
143-144, 239, 270-271

concept of the market, 44, 151, 270-271

criticism of Classical trade theory, 208

criticism of Ohlin's trade theory, 53n

criticisms of, 48, 48n, 153-154, 271-273

derivation of hexagonal market area, 44,
153, 174, 239-242

derivation of market net, 44, 153

derivation of system of nets, 44-45, 153,
154

difficulties of, with non-uniformity assump-
tions, 48-49, 153-154

economies of scale as a basic variable, 44,
150-151, 174, 267

emphasis on economic activity between
men, 53n

SUBJECT INDEX

321

Losch (location theory of) — continued

empirical findings and the distance vari-
able, 60n
equality of number of equations and un-
knowns in, 45-47
extension via general location principle, 252
fusion of modified Launhardt-Palander

scheme with modified scheme of, 274-

275
fusion with other schemes in a sketch of

urban-metropolitan structure, 274-

275
inconsistency of market system of, with

resulting population distribution, 271
introduction of space through assumption,

78
modification of market system of, to square

with population distribution, 271-273
modified, sketch of as fused with sketch of

agricultural land use, 272, 277-278
modified system, difficulty in constructing,

274n
neglect of agglomeration economies, 153-

154
regularities of flows and spatial pattern of

cities, 58-60, 60n
resource inequalities and spatial pattern

of cities, 60
rich and poor sectors of production sites

in, 270-273
role of competition and freedom of entry

in, 44-45, 240
simple diagram of system of nets of market

areas, 270
simplified boundary equations in, 240
sketches showing fusion of, with other loca-
tion-market-trade doctrines, 256-285
statement of knowns and unknowns, 45
symbols of spatial arrangement, 45-46
the set of equilibrium conditions, 45-47,

239-243
transport cost as a basic variable, 44, 150-

151, 267
transport costs on raw materials as zero

in, 239-240
transport network in, 270-271
uniformity assumptions, 44, 152-153, 239,

267
urbanization economies in, 270-271
use in general location approaches, 15-20
use of average cost pricing in, 240

Marginal cost, advantages relative to average
cost in boundary definition, 236n
and average cost, same difference between

for all producers in Losch, 240
as affecting farm output, 190-194
effect of changes in, on competitive equi-
librium, 165n
effect on margin line, 150, 149n-150n, 173
equality with net farm price in equilibrium,

190-191, 197
use of, for pricing, and loss possibility for
producer, 236n
to define boundaries in multifirm case,

236-239
to identify optimum space-economy, 236n
see also Cost curves
Marginal cost curves, see Cost curves
Marginal cost pricing, see Pricing system
Margin lines, and definition of competitive
market areas, 150-151

Margin lines — continued

and effect of economies of scale, 173-174
and Losch market area analysis, 150-151
as indicating delivered price variation at
edge of markets of different size,
148-150
changes in, with changes in pricing sys-
tem, 150, 149n-150n
construction of, 148-150, 154n
Market (s), active, 39

and iron and steel location, 10, 118n
allocation of, by use of aggregate supply

curve, 156n-157n
allocation to supply sources, and changes in
demand, 156n-157n
and changes in transport rates, 156n-
157n
as affected by economies of scale, 175n-

176n
classification of, according to structure,

39-41
inactive, 39
local, 58, 58n
locational pull of, 10
major regional, 58, 58n
minor regional, 58, 58n

multiple, as affecting patterns of agricul-
tural land use, 198-199, 249-251, 276
national, 58, 58n
size of, effect on competitive locational

equilibrium, 164
spatial distribution of, effect on economies
of scale, 175n-176n
effect on substitution points, 175n-176n
spatial division of, and price discrimina-
tion, 164n
spatially separated, equilibrium among, and
spatial price equilibrium, 167n-168n
as allied to the location problem, 167n-

169n
as allied to the transportation problem,

167n-169n
as derived by Enke, 167n-168n
as related to activity analysis, 168n-169n
as stated by Samuelson, 168n
structure of, and Weigmann's basic form,

39-42
types of, and city rank, 58, 60

and volume and length of flows, 58
see also Market areas
Market area(s), a Losch diagram of system
of nets of, 270
a modified Losch diagram of systems of,

272
and the competition field, 38-39
and the simultaneous existence of point,

line, and areal markets, 285-236
and the use of space discount, 85-86
as distinct from supply area, 154n
as distorted hexagons in modified Losch

scheme, 271-273
as interrelated with other market areas

and all production sites, 238
as non-hexagonal in an optimum space-
economy if Losch's postulates re-
laxed, 242n-243n
as overlapping with supply areas in the

general case, 235
as required by scale economies in Laun-
hardt-Palander model, 266-267
at the core as points in modified Losch
diagram, 271-273

322

SUBJECT INDEX

Market area(s) — continued

changes in, with technological change, 258-

259
circular, and substitution between transport

inputs in different directions, 147
competitive, and substitution between trans-
port inputs on products of two pro-
ducers, 147-148
as defined by margin lines, 150-151, 173-
176
conditions for non-competitive situation,

151
consumer space preferences as an explana-
tion of transport inputs incurred,
144-145
content of, as changing with conception of

social surplus, 234-235
delineation of, by means of Launhardt-Pa-

lander construction, 262-264
demarcation of, as substitution in the

small, 251
derivation of hexagonal form, 44, 153, 174,

239-242
discontinuity of, in Launhardt-Palander

construction, 264
distortion of, with realistic transport rate

structures, 239n
division of a region into, 236
enclosed, conditions for, 146-147
equilateral triangular, as exhausting do-
main, 241
extent of, and economies of scale, 58, 148-
151, 153, 173-174
and transport cost, 148-151, 153
under different pricing systems, 149n-
150n, 239n
for raw material producers, 154
fusion of modified Losch system of, with
sketch of agricultural land use, 272,
277-278
hexagon as the ideal form, 44, 153, 242,

242n
hexagonal, as exhausting domain, 241

as involving minimization of transport

inputs, 242
as maximizing social welfare, 153, 242
as yielded by the general location prin-
ciple, 242
as yielded by the substitution principle,

153, 240-242
when Losch's uniformity assumptions
used in Launhardt-Palander model,
267
in a market-orientation case with two
sources for each of two raw mate-
rials, 261-262
interrelation of size and number of produc-
tion sites in sector system of, 273
limits to the extent of, 146-147, 150
Losch's, and maximum coincidence of loca-
tions, 271
and maximum local demand, 271
and minimum of shipments, 271
and minimum sum of shortest distances,

271
and rich and poor sectors of production

sites, 270-271
arrangement about common production

center, 270-271
as related to pattern of production

centers, 270-271
as related to transport routes, 271

Market area (s ) — continued

boundary equations of, 45-46, 240, 241
concept of, 44, 150-151, 270-271
concept of net of, 44, 153, 239-242, 270-

271
concept of system of market nets, 44-45,

153, 270-271
inconsistency of with resulting popula-
tion distribution, 153-154, 271
modification of to square with popula-
tion distribution, 271-273
natural, in absence of competition, 150,

231-235
of labor locations, as determined by critical

isodapane, 259-260
overlapping, because of advertising, 264
because of non-price competition, 264
because of oligopolistic behavior, 264
because of price discrimination, 264
pattern of, in multifirm varying unit cost
case, 238
with constant cost producers, 235-236
population content of, as changing with

distance from core, 278
single firm, conditions for circular, 145-146
non-circular, because of directional varia-
tion in transport rates, 23 In
because of geographic irregularities,

145-146
because of income inequalities, 145-146
because of transport scale economies,

145-146
because of uneven consumer spread,
145-146
size of, and commodity classification, 17,
153, 271
as affected by substitution between trans-
port inputs, 233n
as increasing with distance from major

transport route, 274n
as related to intensity of agricultural

land use, 271
as related to intensity of industrial ac-
tivity, 271
as related to population density, 271
increasing with distance from core, 271-
273, 274n
spatial array of, and the basic form, 38-
41
in Ohlin's interdependence system, 51
spatial extent of, and limited competition,

37-38
square, as exhausting domain, 241
structure of, as related to transport net-
work, 272-273
superiority of hexagonal to equilateral tri-
angular, 44, 241-242
superiority of hexagonal to square, 44,

241-242
two competing firms, when factory prices
differ, 146-147
when factory prices the same, 146
when transport rates differ, 146-147
when transport rates the same, 146
when social surplus maximized, multifirm
case, 235-239
single firm case, 231-235
with a straight line boundary, 146, 151-153,

239, 240-241
with an hyperbolic boundary, 146, 239
with hypercircular boundary, 146-147, 147n,
239

SUBJECT INDEX

323

Market area(s) — continued

see also Consumption, areas of ; Market
area analysis ; Market boundaries ;
Markets
Market area analysis, and conditions for max-
imum social surplus, 232-239
and effect of constant consumption density
on conditions for maximum surplus,
233n

and incidence of transport inputs in dif-
ferent producer-consumer situations,
144-145

and marginal rate of substitution between
production outlays of firms, 148

and marginal rate of substitution between
transport inputs, 148, 233-239

and marginal rates under constant cost
production, 148

and relaxation of one-point market assump-
tion, 143, 260

and substitution between production out-
lays of two producers, 148-154, 260-
261

and the defining of market boundary in
terms of consumer indifference, 231-
232, 236, 236n, 237

and the hexagon as a pure theoretical con-
cept, 274n

and transport-orientation framework, com-
plex boundaries in, 262-264
discontinuous markets in, 264
introduction of scale economies into, 265-
267

and use of margin lines, 148-154

and use of Stieltjes integral, 233n

and use of substitution between transport
inputs for groups of consumers.
235n

as involved in transport-orientation with
many market points, 260

as involving explicit use of social surplus,
249

as treated separately from production for
one-point market in traditional the-
ory, 143n

as substitution between transport inputs
on producers' products, 147-154

as yielded by the substitution principle,
147-154, 231-239, 260-267

as yielding more precise results than com-
petitive location analysis, 169

as yielding precise results by abstracting
from complex factors, 169

conditions for reduction to one-point analy-
sis, 145

consumers along a straight line, 148-151

consumers distributed in an area, 151-154

demarcating boundaries as essential core
of, 231

derivation of Losch, from a two firm case,
153

different producer-consumer situations in,
144-145

extended to embrace raw material supply
areas, 235, 238-239

failure of Launhardt to fuse with indus-
trial location theory, 143n

failure of Weberian theory to encompass,
143n

for cases of two sources of each of two
materials, 261-264

for raw material producers, 154

Market area analysis — continued

form of, as invariant with conception of

social surplus, 234-235
fusion of, with transport-orientation, 231-

239
with transport-orientation and supply

area analysis, 235, 238-239
generalized, as less desirable than general-
ized Thiinen approach, 249
in a fused location-supply-production doc-
trine, 252-253, 286
inapplicability to, of transport-orientation

for an infinite number of market

points, 231
inclusion of, in a generalized Thiinen

framework, 250-251
Losch scheme as a simplified case of, 239-

240
mathematical formulation of, in simple

multifirm case, 235-239
in simple single firm case, 231-235
multifirm, under varying unit cost, 236-

239
multiple stationary points in, from varia-
tion in density of consumption, 233
need to consider production sites with

boundaries as variables in, 231
need to integrate with competitive loca-

tional equilibrium, 170
one-point market analysis as a special case

of, 143, 145, 231
realistic, as precluding regular hexagon,

274n
sketches showing fusion of, with other

location-supply-trade doctrines, 256-

285
supply area analysis as reverse of, 155-158
to determine districts served by different

combinations of material sources,

261-264
traditional, as yielded by the general loca-
tion principle, 239, 252, 286
transition from a straight line to an areal

case, 151-153
under different pricing systems, 236n, 239n
under some simple conceptions of social

surplus, 234-235
under uniformity assumptions, 152-153,

239-242
use of boundaries to reduce infinite num-
ber of variables in, 237-238
use of, in labor orientation, 260-261

in Launhardt-Palander construction, 260-

267
where consumer may purchase more than

one unit of product, 234-239
where consumer purchases one unit of

product, 232-233
with realistic transport rate structures,

239n
see also Market areas ; Market boundaries
Market boundai-y (ies), advantage of marginal

cost definition, 236n
and constant cost conditions, 148, 235-236
and increasing cost conditions, 149-154
as a circle, conditions for, 145-146, 239
as a hyperbola, conditions for, 146, 154,

239, 261
as a hypercircle, conditions for, 146, 147n,

239
as a perpendicular bisector, conditions for,

146, 153, 154, 239. 261

324

SUBJECT INDEX

Market boundary (ies) — continued

as a series of boundary stretches, 266-267

as blurred, because of advertising, 264
because of non-price competition, 264
because of oligopolistic behavior, 264
because of price discrimination, 264

as changing with conception of social sur-
plus, 234-235

as defined by maximum price consumer
will pay, 231-232

as determined by margin lines, 150-151

as loci of points of equal delivered prices,
146-147, 237, 240

as non-circular when transport rate varies
with direction, 231n

as straight line perpendicular bisectors in
Losch, 240-241, 274n

changes in, with introduction of economies
of scale, 266-267

complex character of, in transport-orienta-
tion-market area problem, 262-264

conditions of, as yielded by general loca-
tion principle, 239, 252

definition of, 146, 146n, 147n

in terms of consumer indifference, 46-47,
231-232, 236, 236n

delineation of, by means of Launhardt-
Palander construction, 262-264

demarcation of, as essential core of market
area analysis, 231

distortion of, by transport rate structure,
239n

equation of, for isolated monopolist, 231-
232
under different pricing systems, 236n,
239n

for raw material producers, 154, 262

formation of, and substitution between
transport and labor outlays, 264
and substitution between transport in-
puts, 148, 233-239, 264

in a sketch of a Losch system, 270

in a sketch of a modified Losch system, 272

in cases with two sources for each of two
raw materials, 261-264

Losch's equations of, 45-46

minor indeterminacy of, from discrete con-
sumer spread, 146n

need to consider as variables in market
area analysis, 231

shift of, with shift of production sites in
Losch, 241-242

simplified equations of, in Losch scheme,
240

socially efficient, formation of and chang-
ing marginal rates of substitution,
148
formation of and substitution between

production outlays, 148-154, 262
formation of and substitution between
transport inputs, 147-148, 233-239,
262

use of critical isodapane to determine, 259-
260

use of marginal cost to define, 236-239

use to reduce an infinity of variables to a
finite number, 237-238

where social surplus maximized in multi-
firm case, 235-239

where social surplus maximized in single
firm case, 231-235

see also Market area analysis ; Market areas

Market interpenetration, see Market areas,

overlapping
Market-orientation, a case of, involving mar-
ket area analysis, 261-262
and the Losch framework, 16, 274
as a location type in Launhardt-Palander

model, 257-258, 262-264
changes in, with technological change, 258-

259
effect of economies of scale on, 175n-176n
in iron and steel industry, 6-7, 118n
in case of two sources of each of two raw

materials, 261-264
when product weight is dominant, 258n
see also Transport-orientation
Market-oriented activities, as the basis for
cities, 57, 274-275, 274n-275n
effect of basic industry on, 19, 278
number of, and size of city, 57
plus localized material using activities as
yielding urban-metropolitan struc-
ture, 274-275, 278-280
spatial pattern of, as affected by basic ac-
tivity, 278n
Marketing establishment, location of, see
Location of marketing establish-
ments
Marshallian approach, and the incorporation
of transport function within, 90
neglect of space in, 24-25, 254
Material index, as a technical concept, 36n
value of, and existence of weight triangle,
120-121
Material-orientation, as a location type in
Launhardt-Palander construction,
257-258, 262-264
as excluded in Losch scheme, 274
changes in, with technological change, 258-

259
in case of two sources of each of two raw

materials, 262-264
when raw material weight is dominant,

258n
see also Transport-orientation
Medical services as basic urban industry,

274n-275n
Metropolitan region, see Urban-Metropolitan

region
Migration, as varying inversely with distance,
64n
distance of, and the intervening opportuni-
ties hypothesis, 64n, 65n
intercounty, in Sweden and the intervening

opportunities hypothesis, 65n
net interstate and the intervening opportu-
nities hypothesis, 65n
number of families moving varying dis-
tances, 63-64
step-by-step character, 40-41, 40n-41n
time stage of, and the labor market, 40
see also Population flows
Mineral Resources, see Natural resources
Mobilities, see Immobilities
Moller, and spatial price discrimination in
competitive equilibrium, 164n
and stability in competitive locational equi-
librium, 165n
Monetary system, as a cultural institution, 6

locational effect of, 6
Money, see Monetary system
Monopolistic competition (theory of), and
general location theory, 27n

SUBJECT INDEX

325

Monopolistic competition (theory of) — con-
tinued

and price-cost relations for space analysis,
49

and space as a basic variable, 24-25, 25n,
26n, 27n, 50, 54

and the definition of industry, 92n

and the friction of distance, 38-39

and the isolated monopolist, 144-145

and the spatial array of markets, 38-41

Chamberlin's statement of, as a particular
equilibrium theory, 50n

equivalence of broad conception of, and
general location theory, 50, 54, 254

in Losch's general equilibrium framevsrork,
43-50

rejection of, by Hicks, 26n

resemblance of Triffin's approach and spa-
tial substitution, 50

role of, in determining hexagonal market
areas, 44-45

Triffin's general structure of firm interde-
pendence, 50n

Triffin's use of, in general equilibrium
framework, 50, 50n

value for general location theory, 49-50

Weigmann's version of, as limited compe-
tition, 37-39
Morgenstern, see Game theory
Multicommodity framework, see Commodities,

use of multicommodity framework
Multiplier effects, differences in, from dif-
ferent basic industry, 284

local, from new basic industry, 19

incorporation of in input-output analysis,
49
Municipal functions, and urbanization econo-
mies, 185-186

relations with other activities in urban-
metropolitan region, 11-12

National boundaries, see Political variable
National Bureau of Economic Research and

value of empirical inquiry, 77
National commodities, see Commodities, na-
tional
Natural law in locational structures and the

the German Historical School, 28n
Natural resources, accessibility of, and trans-
port innovation, 3 In
complementarity of, and trade among re-
gions, 22
effect on iron and steel location, 7-8
geographic inequalities of, and difficulty
for Losch theory, 48-49, 153
and hierarchy of flows, 58
and need for supply area analysis, 154
and spatial pattern of cities, 58, 60
and the need to use transport inputs, 255
and the resulting spatial spread of so-
ciety, 78, 78n, 84, 255
and the substitution principle, 34-35
and variations in local input costs, 132-

133, 202n
as affecting farm operations, 194n, 202n,

275
as affecting urban land use, 202n
as distorting agricultural zones, 276
as ignored in competitive locational equi-
librium models, 169-170
effect on agricultural land use, 18, 194n,
276-277

Natural resources — continued

effect on agricultural specialization, 18,

194n
effect on hierarchy of sites, 16, 255
effect on hinterlands, 18
effect on market nets, 16
effect on trade among regions, 6, 18, 22,

75
implications for non-herd existence, 84
in Ohlin's doctrine, 52
significance of, 19n, 194n, 202n, 255
locational effect of, 6, 138-140
military and political factors in the use

of, 14
need to study relations with metropolitan

structure, 287
uniform distribution of, in Losch theory,

44, 239
utilization and conservation of, 14
value of, as related to technology, 31n, 258
see also Environment, physical
Net economy curves, as representing econo-
mies of scale, 186-187
difficulty in selecting representative ones,

186-188
in power generation, 184-185

as dependent on transport cost, 187-188
in providing diverse municipal services,
186
as dependent on transport cost, 188
in providing fire and police protection, 186
in providing recreation services, 186
in providing sanitation services, 186
in the use of skilled labor, 185
in urban transit operation, 186

as dependent on power cost, 187-188
interdependence of sets of, 187-188

as a fruitful area for research, 188
invalidity of summing procedure, 186-188
need for deeper analysis of, 287
non-additive character of, 188
summing of, and the problem of weight-
ing, 186-187
summing representative ones to derive ur-
banization economies index, 186-187
weighting of, by consumption patterns of
city, 188
by income of city, 187
by industrial composition of city, 187
by physical environment of city, 188
by social organization of city, 188
Net farm price, as affected by location of
farm, 194-195, 244
as affecting farm output, 190-194
as depressed by transport outlays, 205
definition of, 190

effect on factor proportions, 194-195
equality with average costs in equilibrium,

190-191, 197
equality with marginal costs in equilibrium,
190-191, 197
Net supply price and the aggregate supply

curve, 15on-156n
New town and size of textile factory, 10
New York as the peak of population poten-
tial, 66-67, 78
Nodes, centrality of, and industrial growth, 8
hierarchy of, within metropolitan regions,
11
Nuclear power, see Atomic energy

Occupational Immobility, see Immobilities

326

SUBJECT INDEX

Ocean freight (world), variation of, with dis-
tance, 73-75
Ohlin, and equalizing differences in labor costs
and transport-orientation, 127n-128n

causes of interregional trade, 17

classification of agglomeration economies,
172

concept of district, 52

concept of region, 51

contribution to general location theory,
50-53, 208

criticism of, by Losch, 53n

limitations of his doctrine, 51-53, 53n

limitations of his interlocal trade theory,
52n

outline of his location approach, 51-52

trade theory as part of a general localiza-
tion theory, 50-54, 208

trade theory of, formulated in more con-
crete terms, 217

use of general interdependence framework,
51

use of Weberian dogma, 52
Oligopolistic situations, and indeterminacy of
firm locations, 160-171

as leading to overlapping areas, 264

neglect of, in Launhardt-Palander con-
struction, 265

see also Game theory
Opportunity costs, and the international lo-
cation of iron and steel, 211-219

as relocation costs, 180

extension of, to include transport-oriented
industries, 211-219

in a transport input formulation to yield
superior approach, 215, 219, 281-282

need to consider distance variable in cal-
culating, 211-212

need to consider transport inputs in calcu-
lating, 211-214

need to state labor cost differentials in
terms of, 217-219

need to state labor orientation in terms of,
210-219

need to state location theory in terms of,
210-219, 281-282

need to state transport cost diff'erentials in
terms of, 217-219, 281-282

need to state transport-orientation in terms
of, 210-219, 281-282

table of, in simple three country case, 211
Optimum location patterns, alternative views
of, 221n-222n

and decentralization, 14-15

and individual and group space preferences,
22-23

simultaneous determination of, with opti-
mum transportation system, 22 In-
222n

use of substitution analysis to determine, 36

with fixed transport facilities and rate

structures, 221n-222n
see also Space-economy ; Surplus, social ;
Welfare, social
Optimum spatial patterns, see Optimum loca-
tion patterns ; Space-economy, opti-
mum ; Surplus, social ; Welfare, social
Ore as a localized raw material, 19
Outlay-substitution lines, as incorporating in-
terest cost differentials, 133
as incorporating labor cost differentials,
127-129

Outlay-substitution lines — continued

as incorporating material cost differentials,

133
as incorporating more than two differen-
tials, 136-137, 137n
as incorporating power cost differentials,

131-132
as reflecting for sites two sets of outlays

variables, 127-129
construction of, 127-129, 131-132
for interest and transport outlays as vari-
ables, 133
for labor and transport outlays as variables,

127-129
for power and transport outlays as varia-
bles, 131-132
for raw material and transport outlays as

variables, 133
for transport, labor and interest outlays as

variables, 136-137
for two groups of outlays as variables, 137,

137n
points on, correspondence with transfor-
mation line, 127-128
possibility of positive slope, 130n
use of, in analysis of scale economies, 174,
265
to derive Hotelling's solution, 170, 170n
to determine equilibrium point, 129-137,

259
to determine labor orientation, 129-131
to determine orientation to cheap mate-
rial source, 133
to determine orientation to point of

cheap capital, 133
to determine power orientation, 131-132
with iso-outlay Hne, 129-137, 174, 259,
265
Outlays, see Labor outlays ; Power outlays ;
Production outlays ; Transport out-
lays
Outlays and revenues, substitution between,

see Substitution points
Output, see Production

Palander (location analysis of), and autono-
mous and "superpolitisch" trade, 163

and categories of commodities, 94n

and constant price fluctuation when firms
near each other, 162-163

and fusion of market area analysis and in-
dustrial location, 143n

and hinterland defense, conditions for and
solution, 163, 163n

and inconsistency of Hotelling's agglomera-
tion, 163n

and limitations of general location theory,
42-43

and market sharing policy, conditions for
and solution, 163, 163n

and minor importance of intermediate loca-
tion, 108, 108n

and occurrence of relative minimum points,
108

and relaxing unrealistic general equilibrium
assumptions, 42-43

and substitution possibilities between trans-
port media, 112n

and the Launhardt-Hotelling problem, 160-
163

and undercutting policy, conditions for and
solution, 163, 163n

SUBJECT INDEX

327

Palander (location analysis of) — continued
contribution to general location theory, 42-

43
criticism of Weber's agglomeration analy-
sis, 180
definition of isodapane, 122
emphasis of, on cost conditions, 210n
on inherited physical structures, 180
on relocation cost in agglomeration proc-
ess, 180
inapplicability of general equilibrium the-
ory to space-economy, 43
insistence on studying development proc-
esses, 43
invalid criticisms of Predohl's substitution

principle, 95n
invalidity of Hotelling's solution for auton-
omously trading firms, 162-163
localization economies as scale economies

with multiplant firm, 179n
outline of locational approach of, 43
relative neglect of demand, 210n
use of isodapanes for transport-orientation

solution, 122-124
use of isovector to construct isodapanes,

122-124
use of pole principle for transport-orienta-
tion solution, 122, 256-258
use of subset of isodapanes in isodapane

technique, 123-124
see also Launhardt-Palander construction
Parasitic industry, cases of, as exceptions to
labor orientation, 128n
growth of, and cheap labor, 8-9, 128n
locational tie to basic industry, 8-9, 128n
Pareto's law of income distribution and simi-
larity to rank-size rule for cities, 56
Partial equilibrium location, see Equilibrium

point, partial
Pattern of settlement, and Christaller's cen-
tral place theory, 60, 60n
and deglomerative forces, 78, 84
and ecological processes, 68-70, 144n-145n
and geographic inequalities of resources, 78,

84
as related to topography, 3 In
early stages, 2-7
general processes, 1-15, 78
with industrialization, 8-15, 31n
Zipf's explanation of, 78n-79n
see also Agricultural land use ; Cities ;
Population distribution ; Space-econ-
omy ; Urban land use ; Urban-metro-
politan region ; Urban-metropolitan
structure
P/D factor, see Zipf, the P/D factor
Pi • P2/D factor, see Zipf, the Pi • P2/D factor
Pick's description of Varignon's mechanical

model, 121
Pole principle, as a short cut to determine
equilibrium point, 122
failure to identify transport costs as basic

economic force, 122
use of, in transport-orientation problem,
122, 256-258
Political variable, as reflecting values and
ideals. 287
effect of, on factor mobility, 283

on geographic specialization, 283-285
on international locations, 283-285
on overall structure of space-economy,
19n-20n, 283-285, 287

Political variable — continued

on rank-size distribution of cities, 57n
on structure of industrial districts, 283-

285
on trade, 19n-20n, 283-285
on urban land use, 283-285
on urban-metropolitan structure, 283-285
see also Cultural values and institutions
Population, as basic variable in Stewart's
social physics, 65-68
growth of, and increase in spatial extent

of economy, 78
internal structure of cluster of, 3, 4, 19-

20
nucleus of, and associated hinterland, 3, 78
size of, and bus passenger movements, BI-
OS
and railway express shipments, 60-61
and telephone messages, 61-62
spatial configuration, 3-4
urban, different effects on of different basic
industries, 284
size of, as affecting commercial and serv-
ice activities, 278
size of, as affecting industries using ubiq-
uities, 278
size of, as related to basic activities, 278
Population density, see Population distribu-
tion
Population distribution, and nucleation from
increasing returns, 2-3
and the Forces of Unification and Diversi-
fication, 78n-79n
areal, and market analysis, 151-154
as related to deglomeration forces, 78
as related to industrial distribution, 19-20
as related to resource patterns, 58, 60, 78,

255
density of, as affected by intensity of agri-
cultural land use, 271
as affecting market area size, 271
as decreasing with distance from urban

core, 68-70, 271-273
as related to intensity of industrial ac-
tivity, 271
as related to transport network, 272-273
effect on urban transit system, 185
inconsistency of Losch's uniformity as-
sumption, 15-16, 153-154, 271
straight line, and market analysis, 148-154
uneven, and noncircularity of market areas,

145-146
see also Rank-size rule for cities
Population flows, association of, with city
patterns, 281
with land use patterns, 281
with location patterns, 281
basic to analysis of space-economy, 281
hierarchy of, by volume and length, 58
length of, and size of city, 58
need for finer analysis of, 287
variety of, in reality, 281
volume of, and size of city, 58
see also Migration ; Population mobility
Population mobility, changes in, and rural
population movement, 88n
as reflected in dispersion in metropolitan
region, 87-88
effect on, of aircraft, 12, 87-88
of automobile and bus, 87-88
of street and electric railway, 87-88
see also Immobilities ; Population flows

328

SUBJECT INDEX

Population potential, and deglomeration
forces, 78, 78n
and geographic inequalities of resources,

78, 78n
as an inverted measure of proximity to

people, 66
cities as local peaks, 66-67
concept of, 65-66
falling off with distance from New York,

66-67, 78
linear relation of, with density of rural
non-farm population, 68
with density of rural population, 68
with flow of bank checks, 68
with number of wage earners in manu-
facturing, 68
with railroad mileage per square mile, 68
with rents of rural non-farm dwellings,

68
with rural road mileage per square mile,

68
with value of farmland per acre, 68
population and distance as basic variables

of, 65-66
problems of computation and interpreta-
tion, 66, 66n, 68n
Stewart's map of, for United States, 67
Port development, relation of, to regional

growth, 9, 10
Power, as a distorting factor in space-
economy, 138-140, 260
as a location factor, 12-13, 131-132, 138-140
consumption as varying by type city, 184
costs of, as ignored by competitive loca-
tional equilibrium models, 169-170
as dependent on transport costs, 187-188
for power intensive firms comparable to
rent for farm enterprise, 189-190
economies of scale in generation of, 184-

185, 185n
generation, and urbanization economies,

184-185, 185n
generation economies, as affected by degree
of system integration, 185
as affected by proximity of cities, 185
as affected by size of cities, 184-185, 185n
effect on per capita consumption, 185n
generation, management diseconomies in,

185n
regularity in geographic cost pattern, as
related to transport cost, 138n-139n
stable geographic cost pattern, 138
unsystematic geographic cost pattern, 138,

202n, 259
see also Power orientation
Power coefficient as a parallel concept to

labor coefficient, 132n
Power cost savings, use of ratio of, to addi-
tional transport outlays, 132n
Power inputs and substitution for other

inputs at cheap power site, 132
Power orientation, and substitution between
power outlays and transport outlays,
132, 189-190, 259, 275
as included in extended trade doctrine,

219
as involving additional transport outlays,

132
as yielded by substitution principle, 132,

259, 275
centi-al role of differentials in power out-
lays in, 275

Power orientation — continued

critical isodapane and feasibility of, 132,

259
incorporation of, in Launhardt-Palander

construction, 260
point of, and use of iso-outlay line, 132
and use of outlay-substitution line, 131-
132
role of replacement deposits in, 135
Power outlays, and transport outlays, sub-
stitution between in power orienta-
tion, 132, 189-190, 259, 275
as a variable in transport-orientation anal-
ysis, 113
differentials in, as central to power orienta-
tion, 275
incorporation in an outlay-substitution
hne, 131-132
see also Power, costs of ; Power orienta-
tion ; Production outlays
Predohl, and geographic inequalities of re-
sources, 34
and location theory as price theory, 32-33
and minimum cost location, 33-34
and the distribution of groups of factors,

32-33
and use units, 34-35
contribution to general location theory,

32-36
criticism of classical trade theory, 208
invalidity of Palander's criticism of, 95n
suggested extensions of, 35-36, 54
use of the substitution principle, 32-36, 54,
94, 130, 254
Price discrimination, as leading to overlap-
ping market areas, 264
neglect of, in Launhardt-Palander con-
struction, 265
spatial, and Lerner and Singer's solution
in competitive locational equilibrium,
164n
as fostered by distance, 164n
as fostered by geographic obstacles, 164n
profitability of and number of competi-
tors, 164n
Price equilibrium, spatial, as allied to com-
petitive locational equilibrium, 167n-
169n
as allied to the Koopmans transportation

problem, 167n-169n
as related to the general location prob-
lem, 167n-169n
Beckmann's formulation of, 168n
Samuelson's formulation of, 168n
Price gradients, continuous field of, 48n
Price mark-up, as a relevant variable in
urban rent analysis, 200-202, 200n-
201n
as affecting urban land use, 200-201
as the price of a retail activity, 201n
effect on rent function, 203-204
variation of, and sales volume curves, 200,
201n
Price-ratio (iso-outlay) lines, as a series of
rectangles and squares, 105-112
as convex (concave) because of firm's in-
fluence on price, 159
as reflecting actual weights, 104
based on I.C.C. railroad rate structures,

105-112
change in slope with change in rent, 192-
194

SUBJECT INDEX

329

Price-ratio (iso-outlay) lines — continued
changed content of, with use of transport

inputs as variables, 115-116
convexity of, and end point solutions, 107-
112
because of graduated rates, 106-107, 120
effect on, of breaks in transport network,
110-112
of variations in transport rates, 112, 113n
for labor outlays and transport outlays, 129
for interest outlays and transport outlays,

133
for power outlays and transport outlays,

131-132
for raw material and transport outlays,

133
for transport, labor, and interest outlays,

136-137
for two groups of outlays, 137, 137n
irregularities in, and determination of
equilibrium point, llOn, 113n
when breaks occur, 11 On
problems in the construction of, when

breaks occur, llOn
slope of, dependence upon relative weights,

104, 115
tails of, and end point solutions, 107-112
because of large first zone charge, 106-
108
under realistic rate structures, 105-112,

112n-113n
use of, in analysis of scale economies, 174,
265
to determine equilibrium of farm enter-
prise, 192-194
to determine orientation to cheap capital,

133
to determine orientation to cheap mate-
rial source, 133
to determine point of labor orientation,

129-131
to determine power orientation, 131-132
with iso-product lines, 192-194
with outlay substitution lines to deter-
mine equilibrium point, 129-137, 159,
174, 259, 265
with scale lines, 192-194
with transformation lines to determine
equilibrium location, 101-124
Price theory, relation of, to location theory,

23, 32, 42n, 49-50
Prices, assumed as given in agricultural
location theory, 210n, 243-244, 276
assumed as given in Weber's location the-
ory, 222
changes of, effect on zonal pattern, 243
geographic pattern of, and effect on loca-
tion, 31, 32, 127-128, 133-135, 254
interrelatedness of, and locational equilib-
rium, 160n
local, differences in, and differentials in
revenue potentials, 126n
and general location theory, 27, 53, 254
and international trade, 51
as excluded in Launhardt-Palander con-
struction, 256
as reflecting different markets, 43
as reflecting monopoly, 43
as reflecting product differentiation, 43
in Ohlin's interdependence system, 51
market, changes in and change in rent
functions, 198

Prices — continued

changes in, to equate aggregate supply
and demand, 198
need to consider changes in, in a general

agricultural system, 243
weakness of assumptions on, in agricul-
tural location theory, 243
see also Costs ; Factor costs ; Pricing
system
Pricing policy, and need for sharply defined
concept of rational behavior, 286
differences in, and overlapping market

boundaries, 264
neglect of, in Launhardt-Palander con-
struction, 265
realistic, need for in competitive locational

equilibrium models, 169, 286
see also. Pricing system
Pricing system (s), as a cultural institution, 6
changes in, effect on cost curves, 150n
effect on margin line, 149n-150n, 150
effect on market size, 149n-150n
effect on sales, 149n-150n
locational effect of, 6, 20, 21
spatial, and competitive locational equilib-
rium, 158-171
see also Prices ; Pricing policy
Primary industry, see Basic industry
Principle of Least Effort, and stable inter-
action over distance, 60
use of, to explain spatial framework, 78n-
79n
Product differentiation, and spatial position,
27n
as a reflection of differences in local prices,
43
Production, allocation of, as affected by
changes in transport rates, 156n-
157n
to several market points, 156n-157n
concentration of, as affecting market area
size, 271
as affecting population density, 271
as likely at initial location, 174n, 180
as related to level of transport rate, 87
as related to transport network, 272-

273
as socially desirable, 174
with economies of scale, 173-176
pattern of, in a simple three-country trade-
location example, 214-219
point of concentration, and substitution
between transport outlays and pro-
duction outlays, 174-175
spatial extent of, and relations between
direct and indirect labor, 82n
and the use of transport inputs, 81-82,

255
as affected by the interest rate, 88n-89n
as related to capital intensity, 253
cumulative growth in, 82n
increase in from a fall in transport rate,

87, 255
limitations to increase in, 86
time extent of, and relations between direct
and indirect labor, 82n
and the use of capital inputs, 82
cumulative growth in, 82n
Production coefficients assumed as fixed in

Weberian location theory, 222
Production outlays, and revenues, substitu-
tion between from crop shifts, 197

330

SUBJECT INDEX

Production outlays — continued

and transport outlays, substitution be-
tween, and concentration of produc-
tion, 197
and localization, 179, 179n, 182, 267
and optimum size of firm, 175n-176n
and social welfare, 182, 262
and urbanization, 188, 269
as induced by economies of scale, 174,

265, 267
in determining point of agglomeration,

174-175, 267, 269
in Hotelling's problem, 170n
in location of marketing establishments,
175n
at several sources, substitution among and

social welfare, 155-157, 262
differentials in, and the use of Launhardt-

Palander construction, 260
effect on, of competition in land uses,
286
of complementarity of land uses, 200, 280
for two producers, substitution between,
and Losch location theory, 153-154
substitution between, and market bound-
ary formation, 148-154, 260-261
marginal rate of substitution between, and

market boundary formation, 148
relations by type of, via substitution prin-
ciple, 259
relations with labor outlays via substitu-
tion principle, 259
substitution between, and location equilib-
rium, 159
and optimum size of firm, 175n-176n
from crop shifts, 197
Production processes, as a determinant of a
population cluster, 19-20
conditions for geographic split of, 20
split of, and interindustry linkage, 20-21
and relation to trade, 20
and technological advance, 20
total set of, as a complex substitution
problem in space, 94-95, 95n
Production sites, as potential market sites,
250n
as potential raw material sources, 250n
finite number of, as a special case of zonal

cultivation, 250n
number of intervening, as affected by size

of concentrations, 273
rich and poor sectors of, in Losch scheme,

270-273
size of, as affecting number of intervening

sites, 273
see also Location
Production theory, and location theory, fusion
of via general location principle, 23,
252-253, 286
fusion of via substitution principle, 113,

118-119, 221, 252-253, 255, 259, 286
need to spell out fusion in detail, 286
parallel use of substitution, 135-137, 255
similarity of first and second order con-
ditions, 118
as embracing economies of scale, 176
extension to include location factor via

transport inputs, 91, 118-119, 252
integration of agglomeration theory with,

172-188
substitution processes in, 135-137
transport cost as a variable in, 26n

Profits, maximization of, as basic to general
location theory, 221n
as transport cost minimization in trans-
port-orientation, 222-223
surplus, elimination by competition, 196,

196n-197n, 202
validity of principle of maximization of,

221n
see also Surplus, social
Pure competition, inapplicability to space-
economy, 37-38, 43, 158-159
inconsistency with location theory, 37-38
inconsistency with transport cost, 43
norms of, as meaningless for space-econ-
omy, 50n
Pure material, possibility of location at
source of, 121-122, 225n
underestimate of pull of, by Weber, 225n

Quantity elasticity, definition of, 42n

Radial transport routes, see Transport

routes, radial
Railroad shipments, Class I, variation of with
distance, 70-73
by type I.C.C. commodity group, 72n
Railway express shipments, and the Pi-P2/D
factor, 60-61
variation with population and distance,
60-61
Railway passenger movements, variation with

population and distance, 62n
Rank of cities and sites, see Hierarchy ;

Rank-size rule for cities
Rank-size rule for cities, empirical findings
on, Austro-Hungary, 57n
Canada, 57n
France, 57n
Germany, 57n
India, 57n

United States, 56-57, 57n
implications of findings on, for regularities
associated with distance variable,
57-60
mathematical formulation, 55-56
similarity to Pareto's law of income dis-
tribution, 56
universality of, 57
validity of, 57

Zipf's interpretation of deviations from,
57n
Rationality, and unnecessary distance, 96-97,
97n, lOln, 107, 113n
in behavior and game theory, 160, 166, 265,

286
in behavior, difficulties of identifying, 160,

265, 267, 286
in location decisions, need to study rela-
tions with metropolitan structure,
287
in site selection, 2
need to sharply define and apply to location

theory, 286
similar degrees of, in agricultural location
and urban land use theory, 205
Raumwirtschaft, contributions to, by German

Historical School, 27
Ravenstein and migration as varying in-
versely with distance, 64n
Raw material (s), cheap source of, as a loca-
tion factor, 133
conception of, 274n

SUBJECT INDEX

331

Eaw material (s) — continued

sources of supply, and effect on economies
of scale, 175n-176n
as affected by economies of scale, 175n-

176n
effect on location, 175n-176n
effect on substitution points, 175n-176n
sources, see Supply sources
supply of, as distorting concentric zones,
248
effect on agricultural land use patterns,
248-249
use of new sources in agglomeration, 178
see also Localized raw materials ; Natural
resources ; Supply area analysis ;
Supply sources
Raw material outlays, as ignored in competi-
tive locational equilibrium models,
169-170
attraction of source of cheap, as involving

discrete spatial jump, 133
differentials in, incorporation in an outlay-
substitution line, 133
see also Factor costs ; Production outlays ;
Raw materials
Region (s), changes in geographic specializa-
tion of, 22
complementarity of resources of, 22
concept of, as a competition field, 38-41
effect of advance in transport technology

on, 22
hierarchical order of trade relations

among, 22
hierarchy of, 22

use with input-output analysis, 49
impact of atomic energy on, 13
interrelations of, 21-22
location analysis of, as a level of inquiry,

92
locational shifts within, 22
Ohlin's concept of, 51

planning of, and changes in the hierarchy
of cities, 183
and control of land use intensities, 183
and control of traffic, 183
fundamental questions in, 9-15
optimum hierarchy of cities, 183
optimum spatial distribution of city, 183
validity of Weber's agglomeration analy-
sis for, 181-182
processes in development of, 1-15, 22
trade among, 22
trade within, 22

see also City-regions ; Regional analysis ;
Space-economy ; Urban-metropolitan
region
Regional analysis, and need for research on
interdependence of sets of net econ-
omy curves, 188
as aUied to the general market-transporta-
tion-location problem, 167n-169n
as bearing upon agricultural location

theory, 199
as bearing upon equilibrium land-use pat-
terns, 199
as interrelated with firm and income analy-
sis, 159n-160n, 199
as required to specify total restraints, 199
Losch theory as a point of departure for,

153-154
need to modify Losch's simplified frame-
work, 153-154

Regional analysis — continued

use of activity analysis in, 168n-169n

see also Regional Science ; Space-economy ;

Urban-metropolitan region ; Urban-
metropolitan structure
Regional income, see Income, regional
Regional science, activity analysis as an ele-
ment of, 169n, 287
as involving study of regional income,

resources and consumption as re-
straints, 287
as involving study of restraints upon

urban-metropolitan structure, 287
gravity models as an element of, 287
gross regional product projections as an

element of, 287
industrial complex analysis as an element

of, 287
interregional input-output as an element

of, 287
need to study space-economy as a hierarchy

of focal points, 287
need to study substitution in the large, 287
Reilly's law of retail gravitation and relation

to demographic force, 65n
Relative-maximum, concept of, see Basic

form of space-economy
Relative minimum points, and procedure to

determine equilibrium point, 108,

108n, 124n, 229-230
occurrence of, in transport-orientation, 108,

108n, 124n, 133, 229-230, 251
Relocation, see Locational shifts
Relocation costs, and game theory, 180-181,

181n
as affecting collusive action, 181n
as deviating agglomeration from optimal

transport point, 180-181
as excessive, for urban relocation, 183
as opportunity costs, 180
as reflected in immobilities, 283n
significance of, in agglomeration analysis,

180-181, 181n
Rent(s), and supply area analysis, 155n
and the process of elimination of surplus

profits, 196n-197n, 202
as affected by location of farm, 194-195
as an explicit cost in cost curves, 191, 193-

194
as generated by competition in land use,

158, 202-206
central role of, in agricultural location

theory, 189-199, 275, 280
in urban land use theory, 202-206, 275,

280
determination of, and the entry and exit

of producers, 196n-197n, 202
determination of payments of, for firm,

190-194, 202-206
determination through substitution between

transport inputs, 246n
effect of interest rate on, 88n-89n
effect on farm output when zero, 190-191
emphasis on maximization of, with fixed

prices in agricultural location theory,

210n
for farm enterprise as comparable to labor

cost of labor intensive firms, 189-

190, 275
for farm enterprise as comparable to power

cost of power intensive firms, 189-

190, 275

332

SUBJECT INDEX

Rent (s ) — continued

inadequate treatment of space in tradi-
tional analysis of, 25n
interrelations with factor proportions, 192-

194, 197, 275
interrelations with scale of output, 192-

194, 197, 275

marginal, equation of, in determination of
zonal boundaries, 247

maximum for each site, as insured by com-
petition, 196, 197

rise in, and deglomeration, 139, 183
and urban diseconomies, 183, 186

social, conditions for maximization of, 245-
251
maximization of as affected by raw mate-
rial supply, 248
maximization of in fused location-
market-supply framework, 250-251
substitution between transport inputs in

maximizing, 246
use of rent functions in maximizing,
245-247

see also Rent functions ; Rent outlays ;
Urban land price
Rent functions (marginal), as applying to
crop combinations, 199, 276

as basic to agricultural location and urban
land use theory, 205, 280

as determining patterns of agricultural
land use, 195, 197-199, 246, 276

as determining spread of urban activities,
204-205, 280

as identical with Dunn's industry rent
function, 195n

as identical with Hoover's rent surface,
195n

as involving rent surfaces in the area case,
205n

as reflecting adjustments of farm enter-
prise, 197, 275-276

as related to farm locations, 194-195, 276

as related to the distance variable, 194-

195, 197, 201-205, 244

as restricting an activity to a relatively

few sites, 204-205
as varying, by type crop, 195, 197-199, 276

by type urban land use, 203-205
as yielded by the substitution principle, 276
derivation of, 194-195, 197, 201-204, 244
effect on, of advertising outlays, 203-204
of competition, 204-205
of complementarity, 204-205
of price mark-up, 203-204
of product and service quality, 203-204
intersection of, as determining zonal bound-
aries, 247, 276
secondary peaks in, and secondary peaks of
sales volume, 203
at satellite centers, 203
shift of, from change in market price, 198

to equate supply and demand, 198
straight line, invalidity of, 195n
system of, with multiple markets, 198-199
transition from line to area situation, 205n
use of cost curves to determine, 201-205
use of sales volume curves to determine,

201-205
use of, to determine urban land use, 204-
205, 276, 280
to identify Thiinen rings, 195, 198, 246,
276

Rent functions (marginal) — contintied

when intensity of land use invariant with
distance, 195n
Rent outlays, and other outlays, substitution
between, in operation of farm enter-
prise, 193-196, 196n, 275
substitution between, in urban land use,
205-206, 276
and revenue potentials, substitution be-
tween, in urban land use, 206
and transport outlays, substitution between
in farm enterprise location, 189-190.
196, 196n, 275
as proportional to land-use units, 34-35
differentials in, as a location factor, 133
as basic to agricultural firm location,
189-190, 194-199. 275
see also Rent
Rent surfaces, as a generalized rent func-
tion, 205n
use of, to depict areal land values, 205n
Rent theory, see Agricultural location theory ;
Rent ; Rent functions ; Urban land-
use theory
Replacement deposits, as affecting point of
agglomeration, 178
handling of, as a phase of supply area

analysis, 131n
increasing role of, with increase of devia-

tional distance, 141
role of, in labor orientation, ISln, 135. 141
Residential land use, see Urban land use
Resources, see Natural resources ; Human

resources
Retail sales, decrease of, with distance from
urban core, 68-70
importance in urban structure, 200n-201n
Revenue-outlay substitution line, as incorpo-
rating more than two differentials,
136-137
as incorporating revenue potential differ-
entials, 134
as reflecting for sites the revenue potential

and outlays variables. 134
construction of, 134, 135n
for revenue potential and transport outlays

as variables, 134

movement along in Hotelling's solution. 170

use of, in analysis of scale economies, 175

to derive Hotelling's solution, 170, 170n

to determine equilibrium point, 134-135.

159
to determine orientation to higher-price

market, 134-135
with iso-revenue-less-outlay line, 134-135,
159
Revenue potentials, and rent outlays, substi-
tution between, in urban land use,
206
as affected by transport outlays (time-cost)

by consumer, 205
differentials in, and substitution. 126, 126n
as a location factor, 126-137
incorporation in revenue-outlay substitu-
tion line, 134-135
net, determination of maximum, 159
differences in at different locations, 159
Revenue-substitution lines, as incorporating
more than two differentials, 136-137,
137n
use of, to determine firm equilibrium loca-
tion. 136-137

SUBJECT INDEX

333

Revenues, substitution between, see Substitu-
tion points

Riemann integral, use of, in agricultural
location analysis, 245n

Ritschl and the evolutionary approach, 15,
30n

Roscher and the evolutionary approach, 15,
28n

Sales revenue, changes in, along substitution

paths, 246, 246n
Sales volume, as affected by transport outlays
(time-cost) by consumers, 205
as falling with distance from core, 68-70
curves of, under different assumptions, 200-
201
with n-variables, 201n
desirability of measuring in dollars, 200n-

201n
effect of complementarity of land uses on,

200
secondary peaks of, and need for general
equilibrium approach, 201n
and secondary peaks in rent functions,

203
as affected by complementarity, 201n
as identifiable when urban pattern set,

201n
at different distances from core, 201
difficulty to identify with fluid urban
pattern, 201n
use of curves of, to determine rent func-
tions, 201-205
variations of, at different locations, 200-201
with distance from core, 200-201
Samuelson, and discontinuities in the produc-
tion function, 103n, 118
and dynamic stability properties of a gen-
eral equilibrium system, 43n
and extension of the Enke solution to many

regions, 168n
and the Koopmans transportation problem
as contained in the Enke market
problem, 168n
and the neglect of certain basic loca-
tional forces in market-transporta-
tion model, 168n-169n
needed extension of market-transportation
model for location analysis, 168n
Satellite cities, see Cities, satellite
Scale effect, and consumer behavior, 87-88
and the dispersion of urban population,

87-88
from a fall of transport rate, 87-88, 255
from increase in population mobility, 87-88
Scale lines, definition of, 191n

use of, in agricultural location theory, 191-
194
with iso-product curves, 191-194
with price-ratio lines, 192-194
Scale of output, analysis of, for agricultural
firm, 190-194
as a variable in the Enke market problem,

168n
as fixed in the Koopman's transportation

problem, 168n
interrelations with rent, 192-194, 197, 275
see also Economies of scale
Schaffle and the evolutionary approach, 15,

28n
Schneider, and price policy for spatially fixed
competitors, 160n

Schneider — continued

and spatial price discrimination in competi-
tive equilibrium, lG4n
Scrap and iron and steel location, 10, 118n
Secondary industry, and location analysis,
21
as related to basic industry, 8-9, 19, 128n,

278
relation of, to agricultural stratum, 7
see also Service activities
Sectors, city-rich and city-poor, 270-273
industrial, pattern of, 278-280
interdependence of, and decentralization
policy, 13-15
and Weigmann's basic form, 39-42
in a general location theory, 26-27
meaningful urban-metropolitan, 13-14
urban-metropolitan, as related to transport
network, 271-273
in a modified Losch diagram, 272
in Losch's diagram, 270
Self-sufficiency as related to distance factor, 5
Service activities, as basic urban industry,
274n-275n
as related to urban income, 278
as related to urban population, 278
differential effects on, of different basic in-
dustries, 284
effect of basic activities on, 19, 19n, 278
location of, and applicability of Losch

theory, 154, 274
market oriented plus localized material
using activities as yielding metro-
politan structure, 274-275, 278-280
quality of, as affecting cost functions, 203-
204
as affecting rent functions, 203-204
as affecting urban land use, 200-201
spatial pattern of, as affected by basic in-
dustry, 278n
structure within metropolitan region, 12
Service receipts, decrease of, with distance

from urban core, 68-70
Settlement, see Pattern of settlement
Shopping trip patterns, as basic to analysis
of space-economy, 281
effect on urban land use, 281
Short-run trade theory, see Trade theory
Side payments, use of, in agglomeration
theory, 179n, 180-181, 181n
in game theory, 180-181, 181n
Site selection effect, among cities, 8-9, 19
in national commodity production, 17-18
among port sites, 9
and aircraft, 12
and technological advance, 12
and transport development, 12
from a faU in transport rate, 67
Site selection for initial habitation, 2
Skilled labor, see Labor, skilled
Smithies, competitive locational solutions with
linear demands, 164
effect of changes in marginal cost on com-
petitive equilibrium, 165n
hinterland demand as a function of price

and location, 164
spatial price discrimination in competitive

equilibrium, 164n
types of competitive behavior, 164n-165n
Social physics, see Stewart
Social surplus, see Surplus, social ; Welfare,
social

334

SUBJECT INDEX

Social welfare, see Surplus, social ; Welfare,

social
Soil characteristics, effect of, on agricultural

land use, 276-277
on settlement pattern, 2, 3, 5
Sources, raw material, see Supply sources
Sources, supply, see Supply sources
Space, and overemphasis of interdependence

of game theory reactions, 167
and principle of limited competition, 37-39
and product differentiation, 27n
as a basic factor, 24-27, 76
as a cause of immobilities, 37-38
as a monopoly element, 54
basic role in agricultural location theory,

189-199
importance internally to firm, 189
In monopolistic competition theory, 24-25,

25n, 26n, 27n, 50, 54
limit of, as precluding analysis of systems

of supply areas, 158
neglect of, in economic theory, 24-27, 33,

42
in rent theory, 25n
in trade theory, 25n, 26
see also Distance ; Distance variable
Space axis, need for, in analysis, 77-78
Space discount, as contrasted with time dis-
count, 85-86
associated with concept of transport in-
puts, 255
in market and purchasing area analysis,

85, 85n
in terms of situation advantage, 85n
rate of, as synonymous with transport rate,

85
to compare items spatially separated, 85
Space-economy, a diagram of Losch's struc-
ture of, 270
a modified Losch diagram of, 272
as a hierarchical set of focal points, 230,

251, 287
as a hierarchy of centers, 273
as discontinuous in reality, 251
as distorted by labor, power, interest, taxes,

and other factors, 138-140
as reflecting a multi-punctured transport

cost surface, 230, 251
basic structure of, as implied by spatial

regularities, 75-76, 254
changes in pattern of, with changes in

technology, 7. 10-13, 258-259
changes of structure of, with economies of

scale, 265-267
with localization economies, 268
with urbanization economies, 269-270,

273-274
continuous, as implied by most location-
market analyses, 251
development processes in, 1-15
efiBcient operation of, need to consider

transport inputs in, 35-36, 79-80,

80n, 252
flow phenomena as basic to, 281
focal points of, need for improved analysis

of, 287
general theory of, see General location

theory
hierarchy of flows in, need for improved

analysis of, 287
interaction phenomena in, need for deeper

study of, 287

Space-economy — continued
need for deeper study of social forces

within, 287
need to study welfare aspects of, 287
optimum, as implying hexagonal market
areas with Losch's postulates, 242n-
243n
as involving nonhexagonal market areas

in a general case, 242n-243n
several views of, 221n-222n
use of marginal cost to identify, 236n
order in, and regular variation of trans-
port cost with distance, 138-140, 210
physical structure of, as affecting agglom-
eration, 180
as giving rise to relocation costs, 180
realistic, as involving hierarchy of trade
routes, 251, 287
as involving substitution in the large, 251
sector structure of, and interrelation of
size and number of production sites,
272-273
sketch of agricultural land use in, 276-278
sketch of, as fusion of modified Losch
market system and agricultural land-
use pattern, 272, 277-278
sketch of urban land-use patterns in, 278-

280
sketches of commodity flows in, 282, 284,

285
structure of, as affected by political vari-
able, 282-285, 287
as fusion of modified Launhardt-Palander

and Losch schemes, 274-275
as governed by general location principle,

286
as localized material using activities plus
market-oriented activities, 274-275,
278-280
as precluding regular hexagons, 274n
as related to transport network, 272-273
need of activity analysis to study, 287
need of gravity models to study, 287
need of gross regional product projec-
tions to study, 287
need of industrial complex analysis to

study, 287
need of interregional input-output to

study, 287
need for superior concepts to understand,

287
need to develop operational models of,
287
use of substitution principle to analyze,

254
Weigmann's basic form of, 38-42
see also Agricultural land use ; Urban land
use ; Urban-metropolitan structure
Space preference (s), and centrifugal move-
ment, 84-85, 87-88
and general location theory, 22-23
and hierarchy of sites, 16
and market nets, 16
and optimal spatial patterns, 23
and rural population shifts, 88n
and the gregarious instinct, 84-85
and urban-metropolitan structure, 11, 87-

88, 287
as acquired behavior patterns, 84-85
as contrasted with time preference, 83-85
by income groups, 88n
changes in, from technological advance, 13

SUBJECT INDEX

335

Space preference (s) — continued

consumer, and explanation of spatial pat-
tern of population, 144-145
and explanation of transport inputs in-
curred, 144-145
degree of, and extroverts, 84-85

and introverts, 84
differences in, by type environment, 85n

by type social activity, 85n
effect on location, 21, 22-23, 78
effect on, of aircraft, 13
of atomic energy, 13
of transport cost level, 19n
effect on trade, 22-23

implications of, for herd existence, 84
of groups and location theory, 22-23
need for deeper study of, 287
negative and hermits, 84
relation of, to transport inputs, 255
Spatial arrangement, symbols of, in Losch's

model, 45-46
Spatial coordinates, in a general location sys-
tem, 54
in Losch's model, 45-46
replacement of, by transport inputs, 35-36,

Spatial extent of production, see Production,

spatial extent of
Spatial inelasticities, see Inelasticities, spatial
Spatial lengthening of production, see Pro-
duction, spatial extent of
Spatial pattern of cities, see Cities, spatial

pattern of
Spatial price equilibrium, see Competitive

locational equilibrium ; Prices
Spatial i-egularities, and transport costs as
function of distance, 35, 138-140, 210
as associated ■with, distance variable, 254
as implying basic structure to space-econ-
omy, 75-76, 254
distortion of, by labor, power, interest,
taxes, and other factors, 138-140,
194n, 202n
of flow phenomena, 57-76
within the metropolitan region, 68-70
Squares, as inferior to regular hexagons as
market forms, 44, 153, 241-242
as satisfying boundary conditions, 153, 241-

242
in Losch scheme, 44, 153, 241-242
Stability, case of absence of, with scale econ-
omies, 174n
in competitive locational equilibrium as
related to market discontinuity, 165,
165n
in farm equilibrium, 193-194
in Hotelling's problem, 160-162, 170
Stability conditions, see Equilibrium point

(for firm location)
Stages of production, locational interrela-
tions of, and interindustry linkage,
20-21
Statics in Weigmann's location theory, 38-41
Steam-steel complex, emergence of, 8
Steel fabricating activities, agglomeration of,
8
location with respect to steel, 8
Steel industry, see Iron and steel industry
Stewart, and concept of demographic energy,
65
and concept of demographic force, 65
and concept of population potential, 65-66

Stewart — continued

and social physics, 65-68, 78

map of population potential for United

States, 67
problems in computing population poten-
tials, 66, 66n, 68n
Stieltjes integral, use of, in agricultural lo-
cation analysis, 245n
in market area analysis, 233n
Storage costs, transport costs contrasted

with, 86n
Stouffer, distance, migration, and the inter-
vening opportunities hypothesis, 64n-
65n
number of families moving varying dis-
tances, 63-64
Straight line, as market boundary, 146, 151-
153, 154, 239, 240, 241, 261, 274n
as supply area boundary, 157
Strata, interrelations of, and city growth, 29
and Weigmann's basic form, 39-42
and zone formation, 29
in early development, 7-8
need for general rules, 30
Street and electric railway, and dispersion of
urban population, 87-88
effect on metropolitan structure, 87-88
Subregions, use of, in Ohlin's analysis, 52n,

53
Subsidization, see Government subsidy
Substitution (s), as subject to transport cost
restraints, 98-101, lOln
change in paths of, with introduction of

raw material supply areas, 235

in the large, existence in space-economy,

94, 95n, 130n, 133, 176n, 196, 251, 254

inadequate recognition of, 251-252, 287

need to study in regional science, 287

in the selection among destinations for a

commodity, 94
in the small, existence in space-economy,
94, 130n, 251, 254
in demarcating market areas, 251
in demarcating supply areas, 251
in the use of different commodities in the
production process, 94
the use of several sources of one com-
modity, 94

in the use of weight-losing commodities, 94
limits to, in input-output analysis, 49
paths of, as changing with conception of
social surplus, 234-235
as involving changes in average unit

costs, 246-247
as involving changes in crop revenues,

246, 246n
as involving changes in intensity of cul-
tivation, 246-247
as involving changes in transport outlays

on crops, 246, 246n
as involving zonal shifts, 246-247
in agricultural location theory, 246-251,

276
to farm equilibrium, 193n-194n
possible types of and classification of com-
modities, 94
process of, as identical for both agricultural
and industrial firms, 189-190. 199,
275
spatial, and resemblance to Triffin's substi-
tution analysis, 50
allowance for via bill of goods change, 49

336

SUBJECT INDEX

Substitution (s) — continued

allowance for via matrix change, 49

in input-output framework, 49

the whole production process as a system

of, 94-95, 95n
via division of nation into regions, 49
via iterative approach and coefficient
changes in input-output, 49

see also Substitution effect ; Substitution
points ; Substitution principle ;
Transformation line ; Transforma-
tion function
Substitution effect, and consumer behavior,
87-88

and increase in consumer expenditures on
travel, 88

and the dispersion of urban population,
87-88

component elements of, 87

from a fall of transport rate, 87-88, 255

from increase in population mobility, 87-88
Substitution points, between diverse labor
outlays, 36

between diverse transport outlays, 34

between groups, subgroups, and individual
commodities, 95n, 259

between inputs and outputs, 135-137, 159

between labor inputs at cheap labor site,
131

between labor outlays and interest outlays,
36

between labor outlays and transport out-
lays, 36, 140n, 189-190, 196, 259, 264

between land inputs and other inputs, 193-
194, 275

between outlays, 35, 49, 94, 159, 197, 199,
206, 259, 269

between outlays, need for, 126-127, 135

between outlays and revenues, 35, 49, 54,
94, 140n, 159, 175, 175n-176n, 188,
197, 199, 205-206, 259, 269

between power outlays and transport out-
lays, 189-190

between production outlays and transport
outlays, 170n, 174-175, 175n, 179,
179n, 265, 267

between production outlays at several
sources, 155-157, 262

between production outlays of two pro-
ducers, 148-154, 260-261

between rent outlays and other outlays,
33-34, 193-195, 196n, 205-206, 275

between rent outlays and transport outlays,
189-190, 196, 196n, 275

between revenues, 35, 49, 159, 197, 199,
206, 259, 269

between spatially defined inputs and out-
puts, 95n

between transport and labor inputs, inade-
quacy of, 126-127

between transport and labor outlays, rela-
tion to the labor coefficient, 141-142

between transport inputs, see Transport
inputs

between transport inputs and capital in-
puts, 253

between transport inputs and labor inputs,
36, 81-82

between transport inputs and other inputs,
253

between transport outlays and local capital
and labor outlays, 33-34

Substitution points — continued
change in, from a fall of transport rate,

87-88
with change in crop, 197-199
effect on, of sources of raw materials, 175n-

176n, 262
of spatial distribution of markets, 175n-

176n
inability of Losch theory to identify, 49
interdependence of, 34, 130
of labor inputs for other inputs at cheap

labor site, 131, 131n, 196
of power inputs for other inputs at cheap

power site, 132
technical, and Weberian theory, 36n-37n
use of input-output to identify, 49
use of price-cost relations to identify, 49
variation in, with distance of farm from

market, 197, 276
with economies of scale, 175-176, 175n-

176n, 265-266
see also Substitution ; Substitution principle
Substitution principle, ability of, to consider

many market points, 98-101, 104,

226-230, 255-258, 276
to consider many raw material points,

98-101, 104, 226-230, 255-256
to incorporate realistic rate structures,

108-109, 117, 229-230, 255
advantage relative to critical isodapane in

labor orientation, 130-131, 131n, 259
and discontinuities in the location problem,

95n
and general equilibrium theory, 32-35
and general location theory, 32-36, 54, 221
and locational equilibrium when firm in-
fluences price, 158-171
and optimum resource utilization, 182
as applicable to agricultural location with

raw material supply areas, 248-249,

276
as applicable when crop combination pro-
duced on farm, 199, 275
as basic to a fused location-market-produc-
tion doctrine, 252-253, 286
as embracing scale economies, 95n, 135n,

173-176, 265-267
as embracing localization economies, 179-

182, 268

as embracing urbanization economies, 182-

183, 268-269

as encompassing differentials in input
prices, 199n

as fusing existing location and market
theories, 250-251

as implying Launhardt-Palander construc-
tion, 267

as involved in the general location prin-
ciple, 224-253

as involving movement from a higher value
to a lower value isodapane, 123-124

as involving movement toward trough of
transport cost surface, 123-124, 223-
230

as involving substitution problems within
a substitution problem, 33-34, 130,
275

as relating to competition between land
uses, 281

as relevant to new area development, 182,
268

as relevant to regional planning, 182, 268

SUBJECT INDEX

337

Substitution principle — continued

as yielding agglomeration analysis, 173-
188, 265-269

as yielding agricultural location theory,
33-34, 189-199, 205-206. 243-253. 275-
278, 281

as yielding concentric zones, 244-245, 246n

as yielding Dean's results when his index
is less than unity, 121, 255

as yielding Launhardt's pole principle re-
sults. 122, 255

as yielding location of intermediary estab-
lishments. 175n

as yielding location of marketing establish-
ments. 175n

as yielding Losch location theory, 153-154,
239-242, 267

as yielding market area analysis, 147-154,
231-239, 260-267

as yielding optimum land-use patterns with
multiple markets, 198-199. 276

as yielding orientation to a cheap capital
site. 133

as yielding orientation to a cheap material
source, 133

as yielding orientation to a higher-price
market, 134-135, 259

as yielding patterns of land use by type,
281

as yielding power orientation. 132, 259. 275

as yielding rent functions. 276

as yielding results of competitive locational
equilibrium models, 170

as yielding results of isodapane technique,
122-124. 255

as yielding results of traditional location
doctrines. 119-125, 221-253, 255

as yielding supply area analysis. 154-158.
235. 238-239, 260-262

as yielding transport-orientation, 113-124,
222-230, 255

as yielding urban land use theory. 205-206,
281

as yielding Weber's labor orientation. 127-
131. 196. 259. 275

as yielding Weber's results, for line case.
119-120. 255
when material index less than unity,

120-121
when weight triangle exists, 121-122,
223-226. 255

expanded to include diverse forces, 34

extension to include quantity elasticity,
41n-42n

general value of. for analysis of space-
economy, 35-36, 54, 138-140, 254

invalidity of Palander's criticism of, 95n

limitations in the use of, with fixed pro-
portions, 131, 136-137

limited value for handling labor, power,
and other factors, 140, 140n

limited value for handling spatially irregu-
lar variables, 140, 140n, 202n, 259

need of a multicommodity framework to
apply to agricultural location, 243,
276

need to consider all differentials in costs
and revenues. 135

need to consider historical process. 180. 268-
269

parallel use in production and location
theory, 135-137, 255

Substitution principle — continued

procedure in considering several dilTcren-
tials. 135-137, 137n, 259

relative to total cost comparisons, 137,
137n

special value for handling transport factor,
140

superiority of use of. to Weberian tech-
nique, 108-109

to derive minimum cost location, 33-34

to embrace geographic inequalities of re-
sources, 34

use of, to fuse location and production
theory, 221, 252, 255, 259
to fuse location dogmas, 221, 252, 259
to fuse Thiinen and Weberian doctrines,
189-199, 275

value of, as dependent upon number of
differentials, 137

see also General location principle
Supply, conditions of. as emphasized in Gra-
ham's trade theory. 210
Supply area(s). as a point. 155n

as distinct from market area. 154n

as overlapping with market areas in the
general case. 235. 238-239

circular, conditions for. 155

conditions limiting. 155

definition of. 154n

demarcation of, as substitution in the
small. 251

determination in multifirm, varying unit
cost case, 238-239

difficulties in analysis of systems of, 158

effect on, of differences in supply prices,
157, 157n
of differences in transport rates, 157,
157n

for a given commodity, as reducible to a
finite number of points, 250n

increase in delivered price with increase
of, 155

introduction of, as involving changed paths
of substitution, 235, 238-239

of raw materials, as embraced by an ex-
tended market area analysis, 235,
238-239
as included in agricultural location prob-
lem, 248-249, 276

pattern of, with constant cost producers,
235-236

with constant costs, 155

with increasing costs, 155-157

see also Supply area analysis ; Supply area
boundaries
Supply area analysis, and changes in com-
modity flows, 156n-157n

and demand changes, 156n-157n

and locational rent, 155n

and substitution among production outlays
at several sources, 155-157

and substitution among transport inputs
on materials from several sources,
155-157

and transport rate changes, 156n-157n

as embracing replacement deposits, 13 In

as implied by the general location principle,
235, 238. 252, 286

as market area analysis in reverse, 155-158

as systems of supply areas, and physical
space restrictions, 158
difficulties of, 158

338

SUBJECT INDEX

Supply area analysis — continued

as yielded by the substitution principle,
154-158, 235, 238-239, 260-261

as yielding boundaries of agricultural hin-
terlands of cities, 276n

as yielding more precise results than com-
petitive location analysis, 169

as yielding precise results by abstracting
from complex factors, 169

for relatively few supply sources, 155n-157n

fusion with market area analysis and
transport-orientation, 235, 238

geographic inequalities in raw material de-
posits and need for, 154

in a fused location-market-production doc-
trine, 252-253, 286

in several different situations, 155-158

inclusion in a generalized Thiinen frame-
work, 250-251

inclusion in agricultural location theory,
248-249

need to integrate with competitive loca-
tional equilibrium, 170

sketches showing fusion of, with other loca-
tion-market-trade doctrines, 256-285

use of aggregate supply curve in, 156n-157n

see also Supply area ; Supply area bound-
aries
Supply area boundaries, as circles, under
constant costs, 155
under increasing costs, 155-157

as determined by condition of equal deliv-
ered prices, 155, 155n-157n

as hyperbolas, 157

as hypercircles, 157

as straight lines, 156

equations for, 248-249

for agricultural hinterlands, 276n, 277

socially efficient, formation of, and substi-
tution among production outlays,
155-157
formation of, and substitution among
transport inputs, 155-157

see also Supply area ; Supply area analysis
Supply curve, aggregate, construction of,
155n-156n
equation to aggregate demand for agri-
cultural equilibrium, 198-199
use in analysis of market allocation,

156n-157n
use in supply area analysis, 156n-157n
use of, with aggregate demand curve, 157n

changes in, and output allocation to sev-
eral markets, 156n-157n
effect on commodity flows, 156n-157n
Supply sources, demarcation of consumer
districts served by different combina-
tions of, 261-264

differential prices at, effect on supply area,
157, 157n

effect of shift of demand curve on output
of, 156n-157n

number of, in the classification of com-
modities, 93-94
and substitution possibilities in location
analysis, 94

output of, as related to aggregate supply
and demand curves, 156n-157n
effect on of changes in transport rate,
156n-157n

shift of, and substitution among produc-
tion outlays, 157

Supply sources — continued

and substitution among transport in-
puts, 157
shift of, among producers, and social wel-
fare, 157
Surplus profits, see Profits, surplus
Surplus (social), conception of, as affecting
content of market area, 234-235
as affecting paths of substitution, 234-
235
conditions for maximization of, 232-235
content of conditions for maximization of,
as varying with conception of, 234-
235
determination of maximum, by direct com-
putation, 233, 238
form of condition for maximization of, as
invariant with conception of, 234-
235
in simple multifirm market area case, 235-

239
in simple single firm market area case,

231-235
maximization of, as generally different
from minimization of transport cost,
232n, 233, 233n-234n
in Losch scheme, 240
in multifirm, varying unit cost case, 236-

239
under different pricing systems, 236n,

239n
with both market and supply areas, 235,

238-239
with hexagonal market areas, 242
with raw material supply areas, 235, 238-
239
maximum, effect of constant consumption

density on, 23 3n
measurement of, avoided in generalized

Thiinen approach, 249
non-measurable character of existing con-
ceptions of, 234n
problems in defining, 236n
some simple conceptions of, 234
stationary points in surface of, with vari-
able consumption density, 233
when consumer purchases more than one

unit of product, 234-239
when consumer purchases one unit of

product, 232-233
see also Welfare, social
Synthesis of theories, see Fusion ; General
location principle

Tariff structure, see Transport rate structure
Tastes, consumer, and explanation of spatial
pattern of population, 144n-145n
and explanation of transport inputs in-
curred, 144-145
differences in, and enclaves of excluded
consumers, 146n
and noncircularity of market areas, 145-
146
need to study relations with metropolitan

structure, 287
see also Space preference
Tax outlays, as a distorting factor in space-
economy, 138, 260
differentials in, and the Launhardt-Palan-
der construction, 260
as a location factor, 133, 138-139
effect on farm operations, 194n

SUBJECT INDEX

339

Tax outlays — continued

unsystematic geographic pattern of tax

rates, 138, 140n, 194n, 259
see also Production outlays
Technological advance, and decentralization,
12-13, 79n
and geographic split of production, 20
and geographic unemployment, 8
and revaluation of resources, 7, 22, 79n,

258
and site selection, 12-13

effect of, on cultural values and institu-
tions, 12, 13
on Force of Unification, 79n
on industrial location, 7, 10, 12-13, 22,

79n, 258-259
on market areas, 258-259
on space preference, 13
on spatial structure, 258-259
via Launhardt-Palander construction,
258-259
in iron and steel industry, 7-8, 10
incorporation of, in sketches of fused lo-
cation-market doctrines, 256-285
psychological impact of, 13
see also Transport technology
Technological conditions, and significance of
regional resources, 3 In
as limiting land use, 205, 280
bearing of, on adaptation to environment,

1-2
effect of, on urban transit network, 185
Telephone messages, and the Pi • P2/D factor,
61-62
variation with population and distance, 61-
62
Terminal charges, as affecting industrial loca-
tion, 107-108, 230, 251
as causing local minima on transport cost
surface, 230, 251
Terminal locations, see Equilibrium point

(for firm location), as end point
Tertiary activities, relation to agricultural

stratum, 7
Tertiary industry, and location analysis, 21
relation of, to agricultural stratum, 7
see also Secondary industry ; Service ac-
tivities
Textile Industry, as a basic industry, 284
as a subsidiary industry, 10
as a typical industry in trade doctrine, 211n
as having different income effect than steel,

284
as having different land use impact than

steel, 284-285
as having different multiplier effect than

steel, 284
geographic shift of, 10
international location via opportunity costs,

211-219, 282-285
labor costs as major in location of, 21 In
location of, 10, 140n, 211n

and substitution between transport and
labor outlays, 140n, 189-190
optimum location pattern, 10
transport costs as minor in location of,
211n
Thiinen, location theory of, see Agricultural

location theory
Thiinen rings, see Zone ; Zone formation
Timber resources and relation to iron and
steel location, 6-7

Time axis, need for, in analysis, 77-78
Time-cost distance, see Distance, effective ;

Distance, time-cost
Time preference, as acquiied behavior pat-
tern, 84-85
different forms of, 84-85
space preference contrasted with, 83-85
Time-space continuum, as a proper analytic

framework, 77-78
Topography, as restraining urban land-use
patterns, 206, 276, 280
effect of, on agglomeration economies, 140
on cost curves, 202n
on location, 2, 3, 6, 138
on pattern of settlement, 31n
on transport cost surface, 251
on transport rate, 88, 112
on urban transit system, 185
revaluation by aircraft, 12
see also Environment, physical
Total cost, comparisons of, as superior to
substitution, 137
to determine firm location, 137
Trade, among areas (regions), and geo-
graphic split of production, 20
and interrelations of production stages,

21
effect of advance in transport technology

on, 22
effect of aircraft on, 12
effect of atomic energy on, 13
effect of geographic inequality of re-
sources on, 18, 75
hierarchical order of, 22
and equilibrium among spatially separated

markets, 167n-169n
and location, interrelation of, 6-7, 20, 22,
31n, 50-54, 167n-169n, 207-220, 281
simplified case of, 210-219, 282-285
simultaneous determination of, 6-7, 50-
54, 207, 215, 219, 282-285
and port development, 9
and resources, interrelation of, 6, 75
and the need to analyze spatial relations,

78n
as affected by factor mobility, 282-285
as affecting structure of industrial dis-
tricts, 282-285
as related to urban land use, 282-285
as related to urban-metropolitan structure,

282-285
basis of, and economies of scale, 17

and transport cost, 17
changes in imports and exports with in-
dustrialization, 9
commodity composition of, and geographic
split of production, 20
and interindustry linkage, 20-21
as related to transport cost level, 19n
comparative advantage in, in simple three-
country case, 213-219, 282-285
conditions for, 20, 215n
development of, with industrialization, 7-

8
effect of change in distance variable on,

5-6, 215-219, 282-285
effect of space preferences on, 22-23
external, beginning of, 9
in a Thiinen-Losch framework, 17
in the Enke-Samuelson market problem,

167n-168n
interdependence in intraregional, 22

340

SUBJECT INDEX

Trade — continued

international, and differences in local
prices, 51
and the distance variable, 73-75, 208-
209, 283-285

intranational and the distance variable, 70-
73, 208-209, 282-283

need to develop gravity model to explain
more fully, 209

need to specify demand to determine pat-
tern of, 212

stability of flows and location theory, 22
Trade routes, hierarchy of, as characteristic
of space-economy, 251, 287
as related to transport cost level, 19n
need for improved analysis of, 287

selection among, and fall of transport rate,
67
Trade theory, and location theory, fusion of,
50-54, 207-220, 281-282
need to explore relations comprehen-
sively, 286

and need to consider spatial framev^ork, 209

and spatial considerations, 25n, 27

as derived from total location analysis for
Ohlin's district, 52-53, 53n

as part of a general localization theory, 51,
53-54, 208

broadly conceived, synonymous with gen-
eral location theory, 53-54, 254

Classical, inadequacy of mobility and im-
mobility premises of, 208
neglect of transport-orientation in, 50,
53, 208

complications from the international vari-
able, 208, 283-285

developments in long-run, as included in
an extended location theory, 219-
220, 282

evaluations of difl'erent types of, 209

extended to include commodities sensitive
to transport cost differentials, 217-
219, 281-282

extension of, through formulating location
theory in terms of opportunity cost,
219, 281-282
to include agglomeration analysis, 219
to include labor and power orientation,
219

fusion with transport-orientation, as ex-
tension of location theory, 217-219,
282
in a simplified case, 210-219, 282-285
via transport inputs, 210-219, 281-282

imbalance in development of, 209

in terms of economic activity between
men, 53n

introduction of distance variable into, 211-
219, 281-282

limitations of Ohlin, 51-53, 52n, 53n

long-run, and assumption of fixed con-
sumption pattern, 212n
as exemplified by Graham's doctrine, 209-
218

Mosaic's, for a one point world, 26

need to consider aggregate regional de-
mand and income, 207

need to include transport-orientation in, 53,
210, 281-282

Ohlin doctrine, 17, 50-53, 208

Ohlin's formulation of, in more concrete
terms, 217

Trade theory — continued

past neglect of multicountry, multicom-

modity approach in, 210
short-run and long-run types of, 209
short-run, activity analysis as appropriate
for, 209
gravity models as appropriate for, 209
input-output analysis as appropriate for,
209
sketches showing fusion of, with other loca-
tion-market doctrines, 256-285
transport cost as a basic variable in, 26n
Weber's criticism of, 50, 53, 208
as met, 217
Transfer costs, see Transport costs
Transformation curve, see Transformation

line
Transformation function, as reduced to varia-
tion in three distance variables in
locational triangle case, 222-223
definition of, 96

fixed proportions and restraints of, 137
spatial, as a set of efficient locations, 119
as embracing transport inputs as vari-
ables, 96, 119, 222, 252, 255
as precluding unnecessary distance, 96-

97, 107
as restraining location of firm, 223-230
as restraining transport inputs, 223-253
mathematical representation of, 222
use of, in fusing production and location
theory, 252-253, 255, 286
in location theory, 223-253
see also Ti-ansformation line
Transformation line(s), and definition of
transformation function, 96
as a continuous curve to facilitate analysis

and synthesis of theories, 115
between distance variables, advantage in
use of, 116
and the locational line case, 96-97
and the locational polygon case, 98-101
and the locational triangle case, 97-98
convexity of, 97-100, 107
location shifts associated with shifts of,

116
with finite number of transport routes,
101-112
between pairs of distance variables, the

interrelations of, 97-101
between transport inputs, 96, 113-119

and the locational line case, 117, 119-

120
and the locational polygon case, 117-124
and the locational triangle case, 113-124
convexity of, 116
location shifts not required by shifts of,

116
shift of with change in weights, 115-116
changed content of, with use of transport

inputs as variables, 115-116
continuous, change of equilibrium point

with use of, 115n
convexity of, and diminishing marginal
rate of substitution, 116
second-order (stability) condition, 116
discontinuous, 101-112, 117, 118
points on, correspondence with outlay sub-
stitution line, 127-129
use of, with price-ratio lines, to determine

equilibrium location, 101-119, 159
see also Transformation function

SUBJECT INDEX

341

Transformation relations, between transport
inputs as embracing relations be-
tween distance variables, 114-119
between transport inputs as reduced to
relations between distance variables,
96
see also Substitution ; Substitution points ;
Transformation line
Transportation analysis (Koopmans), as a
problem in activity analysis, 168n,
213n
as allied to competitive locational equilib-
rium, 167n-169n
as allied to spatial price equilibrium, 167n-

169n
as related to general location analysis, 22,

213n, 221n-222n, 286
as taking scale of output as fixed, 168n-
169n
Transport cost, and difficulty of defining a
unit of transportation service, 186
and extent of market, 148-151, 153, 173
and the location of intermediary establish-
ments, 175n
and the location of marketing establish-
ments, 175n
as a major location factor, 138-140
as afl'ecting exchange ratios, 215n, 217
as affecting farm output, 190-194
as an implicit variable in production the-
ory, 26n
as basic to Losch's market area system, 44,

150-151, 153, 267
as contrasted with storage costs, 86n
as determining industrial distribution by

regions, 172
as imparting regularity to space-economy,

35, 138-140, 210
as imparting spatial regularity to agglom-
eration economies, 139n
to labor costs, 127n, 139n
to power costs, 138n-139n
as major in iron and steel location, 211n
as minor in location of textile industry,

211n
as zero, in general equilibrium theory, 26,

33, 42, 53
differentials in, need to express in terms
of opportunity cost, 217-219, 281-
282
effect of breaks in transport network on,

110
effect of, on commodity composition of
trade, 19n
on competitive locational equilibrium, 164
on geographic specialization, 5-6
on hierarchy of sites and city-regions, 19n
on hierarchy of trade routes, 19n
on net economy curves of diverse urban

services, 188
on trade, 17, 51
feasibility of introducing into general equi-
librium theory, 42
handling of irregularities in, 138
inconsistency with pure competition, 43
isodapane contour lines of, 122-123
isotim contour lines of, 123
isovector contour lines of, 122-123
level of, and inter- and intraregional

structure, 22
minimization of, and international trade
theory. 50

Transport cost — continued

and iron and steel location, 7, 19n, 80n
and the hexagonal form, 44-45
and transport-orientation, 95-124
and unnecessary distance, 96-97, 107,

113n
and Zipf's theoretical framework, 68n-

69n
as generally different from maximizing

social surplus, 232n, 233, 233n-234n
as profit maximization in transport-ori-
entation, 222-230
as yielded by the general location prin-
ciple, 222-230
minimum point of, as point of agglomera-
tion, 177-178
minor significance of, in Classical trade

theory, 215n
need to consider as an explicit variable,

26n, 79-80, 80n, 210
on inputs, as basic to Brinkmann's theory,

244n-245n
per unit distance, as locational force in

mechanical model, 122
regular variation with distance and spatial

order, 35, 138-140, 210
relation to zone formation, 3-6
relative neglect of, in Graham's trade doc-
trine, 210, 215n
restraints on total, in the locational poly-
gon case, 98-101, lOln, 117
rise of, and effect on agricultural zones,
19n
and industrial agglomerations, 19n
and significance of resource inequalities,

19n
and significance of space preferences, 19n
scale economies in, and noncircularity of

market areas, 145-146
total, need for direct comparisons of, in

locational polygon, 137, 140n, 230
urban, as dependent on power costs, 187-

188
variation in, predictability over space, 138,

210, 259
see also Transport inputs ; Transport out-
lays ; Transport rate ; Transport rate
structure
Transport cost surface, as convex downward
in simple Weberian triangle case, 224
as having single minimum point in simple

Weberian triangle case, 224
as linearly punctured because of finite num-
ber of routes, 251
as multipunctured, 230, 251
discontinuities in, from rate abnormalities,

230, 251
isodapanes as contour lines of, 122-124
local minima of, from graduated line
charges, 229-230
from loading charges, 230, 251
from special transit privileges, 230, 251
from terminal charges, 230, 251
from transshipment, 230, 251
movement along, as substitution between

transport inputs, 123
realistic, as yielding discontinuous space-
economy, 230, 251
trough of, as equilibrium point in trans-
port-orientation, 122-124, 224-230
Transport inputs, and capital inputs, substi-
tution between, 253

342

SUBJECT INDEX

Transport inputs — continued

and labor inputs, inadequacy of substitu-
tion between, 126-127
substitution between, 36, 81-82

and the classification of factors, 89-90

and the falling of additional product with
increase in the use of, 86

and the Marshallian functional approach,
90, 119n

and the relative importance of the several
types of inputs, 90

and the roundaboutness of production, 82-
83, 255

and the space extent of the economy, 82,
82n, 255

as a basic concept for location theory, 35-
36, 79-90, 113-114, 209, 222, 252,
286

as associated with concept of space dis-
count, 255

as associated with deglomeration econo-
mies, 255

as combinations of land, labor, and capital
services, 87n, 89-90

as contrasted with capital inputs, 81-86,
255

as factor services required to overcome
resistance, 79

as involving distance variation only in
locational triangle case, 222-223

as ultimately stemming from labor and
land inputs, 81, 89, 255

definition of, 79, 79n, 89-90, 113-114, 255

definition of shipping in terms of, 211

demand curve for, as reflected in associated
changes in output, 86, 87n, 255

diminishing marginal rate of substitution
between, and the second-order condi-
tion, 116

effect of interest rate on supply curve of,
88n-89n

explicit incorporation in transformation
function, 222, 252, 255

function of, in the transformation process,
90, 119

general applicability of, 118n

in an opportunity cost formulation as a
superior approach, 215, 219, 281-282

in different directions, susbtitution be-
tween, and circular market areas,
147
and Hotelling's solution, 170
and social welfare, 147-148

incidence of, in different producer-con-
sumer situations, 144-145

increase in hinterland with increase in
use of, 81-82, 255

increase in output with increase in use of,
81-82, 255

incurred by consumers, as explained by
space preference, 144-145

issue of physical productivity of, 86n-87n

marginal rate of substitution between, and
the equilibrium point, 116-119, 224
and the first-order condition, 116
in the general location principle, 224-
253, 286

minimization of, and iron and steel loca-
tion, 80n, 118n, 215
as involved in hexagonal market areas,

242
in weight triangle analysis, 121-122

Transport inputs — continued
momentary character of, 89
need for trade theory to consider, 210-219
need to consider explicitly, 79-80, 80n, 90,

222, 255, 286
need to consider in calculating opportunity

costs, .211-219, 281-282
non-existence of a stock of, 89
non-resource character of, 89-90
on materials from several sources, substitu-
tion among and social welfare, 155-
157
on raw materials and products, relevance of
substitution between, 227-228
substitution between, in agricultural lo-
cation, 248
on raw materials as zero in Losch scheme,

239-240
possible consideration as a factor from a

socio-economic standpoint, 89
profit motive in the use of, 81-82
relation of, to space preference, 255
requirements of, in simple three country

case, 212. 216
service character of, 89-90
shift in supply curve of, and fall in trans-
port rate, 87
substitution between, and friction of dis-
tance, 140
and locational equilibrium, 159
and Losch location theory, 153-154, 240-

242
and market boundary formation, 148,

233-239, 264
and maximization of social surplus, 233-

239
and shifts from focal point to focal

point, 251
and the classification of commodities, 94
and transport-orientation, 36, 113-124,

222-230, 255
as affecting size of market area, 233n
as movement along transport cost sur-
face, 123-124
as synonymous with various formulations

of transport-orientation, 119-125
as yielding market area analysis, 147-154,

233-239, 260-261
as yielding supply area analysis, 154-158,

235, 238-239, 260-261
equivalent to movement along an isovec-

tor, 123-124
equivalent to movement from subset
isodapane to subset isodapane, 123-
124
for different producers in multifirm case,

238-239
in a fused location-mai-ket-supply frame-
work, 250-251
in a generalized Thiinen approach, 250~

251
in maximizing social rent, 246
in the determination of rent, 246n
in zone formation, 246n
involving same basic considerations as

isodapane technique, 123-124
necessity of for understanding spatial

order, 140
similai-ity with substitution in produc-
tion theory, 118
to explain iron and steel location, 118n-
119n

SUBJECT INDEX

343

Transpoi-t inputs — continued

substitution between prroups of, and the
use of ideal weiKhts, 228
in determination of equilibrium point,

227-228
statement of general location principle
for, 228
substitution with other inputs in the trans-
formation process, 90, 119, 253
supply curve of, as reflecting costs of ulti-
mate inputs, 86, 255
transformation line between, location shifts

not required by shifts of, 116
transformation relations between, as em-
bracinpr relations between distance
variables, 114-119, 222-223
transport rate as price of, 86-88, 255
use of, and increase in the spatial extent
of production, 82-83, 255
and postponement of diminishing re-
turns, 255
as associated with exploitation of uneven

resource endowment, 255
as variables and changed content of

price-ratio lines, 115-116
as variables and changed content of
transformation lines, 115-116
a system of equations, 54
analysis of hierarchy of cities, 255
analysis of population distributions,
255
firm location analysis, 91-124
multiplant firm analysis, 91n
instead of distance inputs, pros and cons,

80, 113-114, 116
instead of spatial coordinates, 35-36,

49
to derive Dean's results when his index

less than unity, 121, 255
to derive Launhardt's pole principle re-
sults, 122, 255
to derive results of isodapane technique,

122-124, 255
to derive Weber's results, for line case,
119-120
when material index less than unity,

120-121
when weight triangle exists, 121-122,
255
to extend location theory to include pro-
duction theory, 118-119
to extend production theory, 118-119
to fuse Weberian and production theories,
91, 113, 118-119
values of, as restrained by the spatial

transformation function, 223-253
with respect to groups of consumers, sub-
stitution between, 235n
Transport gradient line(s), as a delivered
price line, 148-149
effect of rate structures on, 148n-149n
use of, to construct aggregate demand
curve, 156n-157n
to construct aggregate supply curve,

155n-156n
to derive net farm price, 194, 194n
Transport junctions, see Transport network,

breaks in
Transport network, and government subsidy,
9, 12
and urban-metropolitan structure, 11, 272-
273

Transport network — continued

as critical in definition of effective distance,

205
as distorting agricultural zones, 276-277
breaks in, effect on industrial location, 110-
112, 230, 251
effect on iso-outlay lines, 110-112
effect on transport cost, 110, 230, 251
types of, 110
changes in, as provoking complex adjust-
ments, 205
geographic inequalities of, 52
in Losch scheme, 270-271
in modified Losch scheme, 272
optimum mix of facilities, 9
optimum, simultaneous determination of,
with optimum location pattern,
221n-222n
optimum spatial pattern, 9, 221n-222n
realistic, as a hierarchy of transport routes,

251
relation of, to hierarchy of sites, 272-273
to industrial traffic, 9
to population density, 272-273
to population movement, 9
to production concentration, 272-273
to structure of market areas, 272-273
to structure of space-economy, 272-273
role of aircraft in, 12

urban, as affected by population distribu-
tion, 185
as affected by physical environment, 185-

187
as affected by social organization, 185
as affected by state of technology, 185
as affected by topography, 185
as dependent on city size, 185-186
economies in operation of subset of, 185
economies in operation of with urbaniza-
tion, 185, 273n
net economy curve in the operation of,
185
variations of transport rates for media of,

112, 227n
see also Transport routes
Transport-orientation, and comparison of
relative minimum points, 108, 108n,
230
and locational equilibrium of the firm, 91-

124, 222-230
and market area framework, complex
boundaries in, 262-264
discontinuous markets in, 264
introduction of scale economies in, 265-
267
and profit maximization as transport cost

minimization, 222-230
and relaxation of certain assumptions, 126
as generalized with use of general location

principle, 226-230, 252
as market location for farm enterprise, 196
as substitution between distance variables,

96-112, 222-230
as substitution between transport inputs,

36, 113-124, 222-230, 255
as the heart of location analysis, 140, 210
as yielded by general location theory via

specific assumptions, 252n
as yielded by general location principle,

222-230, 252, 252n, 286
case of intranationally, as other orientation
internationally, 219, 281-282

344

SUBJECT INDEX

Transport-orientation — continued

conversion into generalized location prob-
lem via Launhardt-Palander con-
struction, 256-269

definition of, in international framework,
217, 281-282

deviations from, because of cheap labor,
216-219
in agglomeration because of side pay-
ments, 180-181. ISln

dominance of variations in transport out-
lays in, 113

effect of economies of scale on, 175n-17Gn

extension of opportunity cost doctrine to
include, 211-215, 281-282

extension of, to include equalizing differ-
ences in labor costs, 127n-128n
to include food as a raw material, 127n-
128n

for infinite number of market points as
market area analysis, 231

fusion of, in location-market-trade doctrine,
252-253, 286
with market area analysis, 231-239
with market and supply area analyses,

235, 238-239
with production theory, 118-119, 255
with trade theory, as a superior ap-
proach, 119, 281-282
in a simphfled case, 210-219, 282-285
via opportunity costs, 210-219, 281-282

improved formulation of, via opportunity
costs, 211-215, 281-282

in iron and steel location, 19n, 80n, 118n,
211n, 215-219

in terms of minimizing ton-kilometers, 36n,
215-216

in terms of technical data, 36n

inclusion in, of many market points, 98-
101, 104, 226-230, 255-258
of many raw material points, 98-101,
104, 226-230, 255-256

international, neglect of in Graham's trade
theory, 210

line case, 95-97, 117, 119-120

different degrees of determinacy, 119-120

mathematical presentation of, 222-230

need for complete framework to fuse theo-
ries, 113

need for concept of transport inputs, 113-
114, 222

need to consider other variables than dis-
tance, 113

need to consider, to understand interna-
tional commodity flows, 210

need to state in terms of opportunity costs,
210-219, 281-282

neglect of, in Classical trade theory, 50, 53,
208

partial deviation from, in international
steel location, 217-219

polygon case, 98-101, 117, 226-230

sketches showing fusion of, with other lo-
cation-market-trade doctrines, 256-
285

solution by Varignon's mechanical model,
121-122

techniques of, to identify point of agglom-
eration, 178

triangle case, 97-98, 101-119, 222-226

under simplified conditions, 95-104, 222-

Transport-orientation — continued

use of isodapanes to derive solution, 122-

124
use of Launhardt-Palander construction in,

256-258
use of pole principle to derive solution, 122,

256-258
use of transformation function in, 222-230,

255
use of weight triangles to derive solution,

120-122
various formulations of, 119-125
Weberian doctrine reformulated, 96-112,

222-230
with finite number of transport routes,

101-112, 255
with fixed markets, as inapplicable to mar-
ket theory, 231
with many market points as involving mar-
ket area analysis, 260
with realistic rate structures, 104-112, 117,
229-230, 255
Transport outlays, additional, as represented
by isodapanes, 130n, 141
incurred in labor orientation, 127-131,

141-142
incurred in power orientation, 132
ratio of labor cost savings to, see Labor
cost savings, ratio of, to additional
transport outlays
and labor outlays, susbtitution between and
boundary formation, 264
substitution between in labor orientation,
127-131, 140n, 189-190, 196, 259, 264,
275
and power outlays, substitution between in
power orientation, 132, 189-190, 259,
275
and production outlays, substitution be-
tween, and concentration of produc-
tion, 174
and Hotelling's solution, 170n
and localization, 179, 179n, 182, 267
and location of marketing establishments,

175n
and optimum size of firm, 175n-176n
and social welfare, 182, 262
and urbanization, 188
as induced by economies of scale, 174,

265, 267
in determining point of agglomeration,
174-175
and rent outlays, substitution between in
farm enterprise location, 189-190,
196, 196n, 275
and substitution with labor outlays, 36
as depressing net farm price, 205
as explicitly considered in agricultural loca-
tion theory, 205, 280
as implicitly considered in urban land use

theory, 205, 280
differentials in, incorporation in outlay-
substitution Hne, 127-129
dominance of variations of, in transport-
orientation, 113
in different directions, substitution between,
and Hotelling's solution, 170, 170n
effect on, of economies of scale, 175n-
176n
in Predohl's framework, 33
localization as involving increase of, 179,
267

SUBJECT INDEX

345

Transport outlays — continued

substitution among, and optimum size of
firm, 175n-176n
in the location of marketing establish-
ments, 175n
time-cost by consumer, as affecting firm
revenues, 205
as basic to urban land use theory, 205
as determining accessibility, 205
Transport rate(s), as contrasted with inter-
e.^t rate, 88
as determined by the supply curve of, and
demand curve for, transport inputs,
86, 255
as the price of a transport input, 35, 86-

88, 255
as the rate of discount in space, 85
assumption by location theory of monopoly

elements in, 213n
assumption of invariance of, with direc-
tion in concentric zone theory, 245
bearing on labor orientation, 141, 142
changes in, and output allocation to several
markets, 156n-157n
effect on commodity flows, 156n-157n
effect on output of supply sources, 156n-
157n
differences in, and effect on market areas
of competing firms, 146-147
and effect on supply areas, 157, 157n
directional variation of, and non-circularity

of market areas, 231n
discriminatory, and the possibility of un-
necessary distance, 97n, lOln, 113n,
130n
effect of interest rate on, 88n-89n
effect on distance between isodapanes,

141
fall of, and change in consumer behavior,
87-88
and dispersion of urban population, 87-88
and increase in geographic specialization,

87
and increase in share of income spent

on travel, 88
and increase in space extent of produc-
tion, 87, 255
and increase in the consumption of trans-
port inputs, 88
and increased concentration of produc-
tion, 87
and population mobility, 87-88, 88n
and scale effects, 87-88, 255
and selection among sites, 87
and selection among trade routes, 87
and substitution between transport in-
puts and other inputs, 87
and substitution effects, 87-88, 255
and urban population shifts by income

groups, 88n
as a decrease in the time and money cost

of population movement, 87-88
from a shift in the supply curve of trans-
port inputs, 87
from advances in transport technology,
87
ratios of, as involved in the general location

principle, 224
use of a single fictitious rate to represent

a multitude of rates, 88
use of ideal weights to adjust for differ-
ences in, 109n, 120, 228

Transport rate(s) — continued

variations in, and adjustment of transport
cost scales, 112
by commodity characteristics, 88, 112,

118n, 227n
by degree of competition, 88
by direction of movement, 112, 227n
by haul characteristics, 88
by nature of topography, 88, 112
by type transport facility, 112, 227n
by type transport route, 112
by volume and speed of movement, 112
effect of on determination of equilibrium

point, 112
incorporation in iso-outlay lines, 112
see also Transport cost ; Transport rate
structure
Transport rate structures, and regular varia-
tion with distance, 138
and the possibility of unnecessary distance,

107, 113n
as dependent on flow volume, neglect of by

traditional location theory, 213n
assumed as fixed by traditional location

theory, 213n
characteristics of, in modern industrialized
societies, 105, 107, 112, 113n, 227n,
229-230
in primitive societies, 105
complications of, and substitution analysis,

36, 104-112
eccentricities in, effect on equilibrium firm

location, 113n
effect of zonal chai-acter of, on price-ratio

lines, 105-112
effect on transport gradient line, 148n-149n
first zone charge, and end point solutions,
107-112, 230
and tails of price-ratio lines, 106-108
graduated, significance of, 105-112, 229-230
graduation of, and convexity of price-ratio
lines, 106-107, 120
and determinacy of locational line case,

120
and end point solutions, 107-112
because of overhead and terminal ex-
penses, 105
in industrialized areas, and end point solu-
tions, 107-112, 230, 230n
need to consider as variable in location

theory, 213n, 286
realistic, and construction of price-ratio
lines, 105-112, 112n-113n
and firm equilibrium locations, 104-112,

113n, 117, 229-230, 255
and modification of first-order conditions,

229-230
and modification of second-order condi-
tions, 229-230, 230n
as distorting market boundaries, 239n
as yielding multipunctured transport cost

surface, 230, 251
embraced by isodapane technique, 122n
inability of Weberian theory to encom-
pass, 108-109
use of ideal weights to adjust for com-
modity rates, 109n, 228
see also Transport rate ; Transport cost
Transport routes, and variations in transport
rates, 112
development of, with industrialization, 7-8
effect of finite number, 3, 16, 113n, 251

346

SUBJECT INDEX

Transport routes — continued
effect on farm operations, 194
effect on intensity of cultivation, 3
finite number, and the transport-orientation

problem, 101-112, 113n
hierarchy of, in realistic space-economy,

251
radial, and urban land-use patterns, 11
as boundaries of sectors, 273n
as centers of sectors, 273, 273n
size of market area as increasing with dis-
tance from, 274n
sketches of, in urban-metropolitan region,

270-279
see also Transport network
Transport system, see Transport network
Transport technology, advance in, and acces-
sibility of resources, 3 In
and effect on agricultural zones, 16, 19n
and effect on internal development of

regions, 22
and effect on interregional trade, 22
and geographic specialization, 22
state of, and inter- and intraregional struc-
ture, 22
and urban-metropolitan structure, 11, 12,
19n
see also Technological advance
Trenton steel location, explanation of, in
terms of substitution between trans-
port inputs, 118n-119n
TrifRn, see Monopolistic competition, theory
of

Ubiquitous materials, and the Losch frame-
work, 16, 274
general locational effects of, 120
implications of, in simplified transport-
orientation, 96n
industries using, as affected by urban in-
come, 278
as affected by urban numbers, 278
as pulled by localized material using in-
dustries, 278
use of, and determinacy in locational line
case, 120
Uniformity assumptions, in Launhardt-Pa-
lander construction, 256
inconsistency with hierarchy of sites, 153-

154, 271
Losch's, and market area analysis, 44, 152-
153, 239
and supply area analysis, 155-157
need to relax in extension of Losch, 153-154
use of Losch's in Launhardt-Palander
model, 267
United States, cities in, rank-size findings
for, 56-57, 57n
map of population potential, 67
metropolitan regions, rank-size findings on,
56
Unnecessary distance, see Distance, unneces-
sary
Urban decentralization, see Decentralization
Urban hierarchy, see Hierarchy
Urban land price (rent), increase in, and
urban diseconomies, 186
with city size, 186
relation of, to accessibility, 200-205

to complementarity of land uses, 200-205
to distance from the core, 200-205
to site competition, 200-205

Urban land price (rent) — continued

use of cost curves to determine, 201-202
Urban land use, along radials, 11

and agricultural land use, competition be-
tween, 281
and the allocation of relatively few sites to

any one activity, 204-205
and the clustering of economic activities,

204-205
as affected by accessibility, 200, 280
as affected by complementarity of land

uses, 200, 280
as affected by distance from core, 11, 200,

280
as affected by price mark-up, 200-201
as affected by service quality, 200-201
as affected by site competition, 200, 280
as associated with population flows, 281
as determined by rent functions, 204-205,

280
as involving substitution among revenues

and outlays, 205-206
as involving substitution in choice of prod-
uct, 205-206
broad classification of types, 280n
by type, as yielded by substitution prin-
ciple, 281
commercial, as related to type of basic in-
dustry, 284-285
determinants of, 11-12, 202-206
effect on, of journey to work pattern, 281
of quality competition, 281
of shopping trip pattern, 281
increase in intensity of, and deglomeration,

139
industrial, as related to type of basic in-
dustry, 284-285
intensities of, and optimum structure of

cities, 183
interdependence of types of, 281, 287
limits to, as precluding analysis of systems

of supply areas, 158
optimal, as yielded by substitution princi-
ple, 205-206, 281
patterns of, as a fusion of localized mate-
rial using industries and modified
Losch market system, 274-275, 278-
280
a simple sketch of, 278-280
desirable changes in, 183
industrial, commercial, residential and

recreational, 278-280
logic for variety of, 206
optimum, need for improved theory to

determine, 280, 287
process of determination of, 202-206
relation of, to advertising outlays, 200-201,
281
to agricultural land use, 281, 285
to distance variable, 282-285
to factor mobility, 282-285
to geographic specialization, 282-285
to the political variable, 283-285
to trade, 282-285
residential, as related to type of basic in-
dustry, 284-285
restraints imposed on, as differing among
areas, 206
by cultural values and institutions, 205,

206, 280
by existing physical structures, 206
by physical environment, 206, 280

SUBJECT INDEX

347

Urban land use — continued
by technology, 205, 280
by total demand, 206
by total Income, 206

structure of, 11-12

transition from line to area case, 205n

use of average cost curves to determine,
201-202

use of marginal cost curves to determine,
201-202

use of sales volume curves to determine,
200-201

see also Urban land-use theory
Urban land-use theory, and agricultural loca-
tion theory, competition as basic to
both, 205
complementarity as basic to both, 205
rent functions as basic to both, 205,

280
some dissimilar forces in, 281

as an integral part of general location
theory, 205-206

as implicitly considering transport outlays,
205, 280

as yielded by the substitution principle,
205-206, 281

interconnections with agricultural location
theory, 200-206, 280

need to study bonds v(fith agricultural loca-
tion theory, 280

sketches shovcing fusion of, with other lo-
cation-market-trade theories, 256-
285

traditionally excluded from location theory,
200

transition from line to area case, 205n

transport outlays (time-cost) by consumers
as basic to, 205, 280

undeveloped state of, 280

use of multicommodity framework in, 204-
206

see also Urban land use
Urban-metropolitan region (s), a modified
Losch diagram of, 272

a simple Losch diagram of, 270

and decentralization policy, 13-15

and distribution of functions, 12

and interactivity relations, 11

and step-by-step migration, 40-41

applicability of Losch's analytic approach,
154

as a hierarchy of sites, 11-12, 183, 273

determination of optimum land-use pat-
terns for, 200-206

differentials in land outlays as major to
analysis of, 189

dispersion within, by income groups, 88n

dynamic processes of growth, 11-12

hierarchy of industrial districts in, 278-280

importance of internal spatial dimensions
of, 189

meaningful sectors of, 13-14, 270-273

need for deeper analysis of, as a socio-
economic organism, 287

population of, as affected by basic indus-
tries, 11, 278

population pattern and consumer tastes,
144n-145n

population pattern and space preferences,
144n-145n, 145

rank-size rule for, 56

relocation of sectors, 13-15

Urban-metropolitan region (s) — continued

service activities as basic industry in, 274n-
275n

sketch of, as fusion of modified Losch mar-
ket system and agricultural land-use
patterns, 272, 277-278

sketch of agricultural hinterlands of, 276-
278

sketch of land-use patterns in, 278-280

spatial configuration of, 11, 68-70

specialization among, 12

structure of core of, 12

tie to basic economic industry, 11

see also Cities ; City-regions ; Regions ;
Space-economy ; Urban-metropolitan
structure ; Urban land use
Urban-metropolitan structure, and aircraft,
12, 87-88

and automobile and bus, 87-88

and clustering of economic activities, 204-
205

and fall in the time and money cost of pop-
ulation movement, 87-88

and street and electric railway, 87-88

applicability of Losch approach, 154

as a hierarchy of centers, 273

as fusion of modified Launhardt-Palander
and Losch schemes, 274-275

as localized material using activities plus
market-oriented activities, 274-275,
278-280

as reflecting the joint geographic distribu-
tion of economic activities, 183-184

as related to distance variable, 282-285

as related to factor mobility, 282-285

as related to geographic specialization, 282-
285

as related to the political variable, 283-285

as related to trade, 282-285

as related to type of basic industry, 284-285

fluidity of, and identification of secondary
peaks of sales volume, 201n

impact of technological change, 12-13

importance of commercial activities in,
200n-201n

importance of retail activities in, 200n-201n

interrelation of size and number of produc-
tion sites in, 273

Losch's simple conception of, 270-271

need for general equilibrium approach in
determining, 201n

need to develop operational models of, 287

need to study relations, with labor produc-
tivity, 287
with location decisions, 287
with regional income, 287
with resources, 287
with tastes and consumption, 287

non-existence of regular hexagons in, 274n

some basic elements of, 11-15

study of restraints on, as involved in re-
gional science, 287

see also Cities ; Space-economy ; Urban land
use ; Urban-metropolitan region
Urban region, see Urban-metropolitan region
Urban structure, see Urban-metropolitan

structure
Urbanization, a simple case of, 268-270

analysis as overlapping with localization
analysis, 182

and comparison of advantages and disad-
vantages by the firm, 183, 188, 269

348

SUBJECT INDEX

Urbanization — continued
and inflexibility of inherited physical struc-
ture, 183
as substitution between outlays and reve-
nues, 188, 269
in a modified Losch approach, 270-275
limited application of Weber's approach

to, 183, 188, 269
need for evolutionary approach in analy-
sis of, 183
the Weber-Launhardt-Palander approach

to, 265-270
theory of, use of multicommodity frame-
work in, 185-188, 268-270
theory, sketches showing fusion of, with
other location-market-trade doctrines,
256-285
use of critical isodapanes in analysis of,
183, 188
Urbanization economies (diseconomies), addi-
tion of, to modified Launhardt-Pa-
lander model, 268-270
and interdependence of net economy curves,

187-188
and need for research on interdependence

of net economy curves, 188
and need for weighting net economy curves,

186-187
and scale economies in power generation,

184-185, 185n
as a location factor, elementary state of

analysis of, 183, 268-269
as included in agglomeration economies,

139, 172, 265, 268
definition of, 172, 268
effect of, on structure of space-economy,

269-270, 273-274
from increase in congestion, 183, 185, 186
from increase in living cost, 183, 185, 186
from increase in rents, 183, 186
from industrial articulation, 182-183
in large lot buying and selling, 182
in providing a subset of transportation

services, 185
in providing diverse urban services, 186
in providing educational services, 186
in providing fire and police protection, 186
in providing recreational services, 186
in providing sanitation services, 186
in providing transportation services, 185,

273n
in the generation of power, 184-185, 185n
in the Losch approach, 270-275
in the use of auxiliary facilities, 182
in the use of skilled labor, 182, 185
in the use of urban apparatus, 182
in transit operations, 185-186, 273n
in transportation, as distorting agricultural

zones, 276-277
incorporation of, in sketches of fused loca-
tion-market-trade doctrines, 256-285
index of, as a sum of representative net

economy curves, 186-187
invalid use of the sum of net economy

curves, 188
locational shifts from, 269-270
need for deeper insights into, 287
non-additive character of, 188
overlapping with localization economies,

182, 265
see also Agglomeration economies ; Deglom-
eration economies ; Urbanization

Usher and the historical approach, 15, 31n

Value added by manufacture, decrease of,
with distance from urban core, 68-70

Varignon mechanical model, and transport
cost as basic location force, 122
use of, to determine equilibrium of location
forces, 121-122

Viner and the effect of transport cost on
trade, 215n

Vining's approach to spatial analysis via dis-
tributional stability of flows, 22

von Neumann and Morgenstern, see Game
theory

Vulnerability, and geographic balance, 14
and urban decentralization, 12

Weber (location theory of), agglomeration
analysis of, as substitution analysis,
179

agglomerative forces as not affecting in-
dustrial distribution by regions, 172

and assumption of fixed production coeffi-
cients, 222

and assumption of given prices, 222

and concept of dominant weight, 120, 225n

and emphasis on firm location, 92-93, 188-
189

and function of economy of agglomeration,
178

and limited aggregative analysis within,
92-93

and minimizing ton-kilometers, 36n

and optimum resource utilization, 182

and size of agglomeration, 178

and the accepted dualism with Thiinen
analysis, 92-93, 188-189, 275

and the classification of commodities, 93-
94, 94n

and the evolutionary approach, 15, 27-30,
40n-41n

and underestimate of significance of corner
locations, 109

as a supplementary empirical theory, 36n-
37n

as element of a general location theory, 23

as a supplement to a Thiinen-Losch model,
16-19, 19n

as based on technical knowledge, 36n-37n

bound capitalism and immobile labor, 40n-
41n

conditions for agglomeration, 176-177

criticism of, by Dean, 225n

for ignoring differential bargaining abil-
ity in agglomeration, 180-181
for ignoring historical process in agglom-
eration, 180
for ignoring relocation costs, 180

criticism of trade theory, 50, 53, 208

criticism of trade theory as met, 217

decrease in validity of assumptions with
size of agglomeration, 179n

definition of critical isodapane in agglom-
eration analysis, 176n

definition of isodapane, 130n

determination of agglomeration center,
177-178, 265-270

emphasis on cost conditions, 210n

emphasis on distance variable by assump-
tion of constant weights, 96, 222-223

emphasis on transport-orientation in inter-
national trade, 50, 53, 208

SUBJECT INDEX

349

Weber (location theory of) — continued

exceptions to conditions for agglomeration,
178-179

failure to distinguish types of agglomera-
tion, 176

failure to treat market area analysis, 143n,
260

firm analysis in, as applicable to farm en-
terprise, 189-190, 196-197, 275

free capitalism and mobile labor, 40n-41n

fusion of, with production theory, 91, 118-
119

greater validity for German economy, 109n

inability to encompass realistic rate struc-
tures, 108, 109

incorporation with Thiinen analysis in one
framework, 92-93, 188-189, 275

invalidity of certain criticisms of, 92-93

invalidity of the use of fictitious distances,
109, 109n

labor cost differentials as determining in-
dustrial distribution by regions, 172

labor locations as centers of agglomera-
tions, 179

limited analysis of urbanization economies,
182-183

limited application of agglomeration analy-
sis of, 179, 268-269

limited application to urbanization analysis,
183, 188, 268-269

mathematical formulation of, 222-230

modified, fusion of in a location-market-
production doctrine, 222-253
fusion of in a location-market-trade doc-
trine, 256-285

need to integrate with competitive loca-
tional equilibrium, 170

overestimate of pull of weight-losing ma-
terials in, 225n

propositions of, for line case as derived via
transformation and price-ratio lines,
119-120

reformulation of labor orientation doctrine,
127-131, 141-142

reformulation of transport-orientation doc-
trine. 96-112, 211-215, 281-282

relative neglect of demand, 210n

role of agglomeration in, 172-188

role of replacement deposits in labor orien-
tation, 131n, 135

sketches showing fusion of, with other loca-
tion-market-trade doctrines, 256-285

sketch of his industrial location problem,
28n, 104n

strength in treating localized raw mate-
rials, 158, 274

transport cost differentials as determining
industrial distribution by region, 172

underestimate of pull of pure materials in,
225n

use by, Ohlin, 52

use of critical isodapane, 130-131

use of labor coefficient, 141-142

use of side payment to induce agglomera-
tion, 179n

valid use of ideal weights to adjust for
commodity rates, 109n

validity of agglomeration analysis of, for
new area development, 179, 181-192,
268
for regional planning, 181-182, 182n, 268

value for general location theory, 36n-37n

Weber (location theory of) — continued

value for iron and steel location analysis,

37n
see also Agglomeration theoi-y ; Labor orien-
tation ; Transport-orientation
Weight (s), actual, and price-ratio lines, 104
and the concept of transport inputs, 35,

79, 113-114, 222
as a variable in transport-orientation analy-
sis, 113, 222-223
ideal, see Ideal weights

relative, as determining slope of price-
ratio lines, 104
effect on firm's equilibrium location, 104
see also Weight triangle
Weight loss, and determinacy in the line
case, 120
and substitution possibilities in location

analysis, 94
general locational effect of, 120
in the classification of commodities, 93-94
infinite, and immobile commodities, 32n, 94
overemphasis of, by Weber, 225n
Weight triangle, as a counterpart to Varig-
non mechanical model, 121
as a shortcut to determine equilibrium

point, 122
existence of, and the generalized index,
121
and the material index, 120-121
failure to identify transport cost as basic

economic force, 122
less flexible than isodapane technique, 122n
need to use ideal weights, 122n
nonexistence of, when a weight is domi-
nant, 120-121, 258n
restricted use of, 122n

use of, in Launhardt-Palander construc-
tion, 256-258
in transport-orientation problem, 120-

122, 256-258
to determine equilibrium of location
forces, 121-122, 256-258
Weigmann, and structure of markets, 39-41
concept of basic form, 38-42
concept of competition field, 38-39
concept of quantity elasticity, 42n
concept of statics and dynamics, 38-41
contribution to general location theory, 37-

42, 54
criticism of classical trade theory, 208
principle of limited competition, 37-39
total localization theory, 41n-42n
use of Gestalt analysis, 38-42
Welfare, analysis of, invalidity of pure com-
petition norms in, 50n
need for monopolistic competition theory
in, 50
individual, conflict with optimum spatial

patterns, 23
social, and production concentration from
scale economies, 174
and shift of consumer allegiance, 147-152
and shift of supply sources among pro-
ducers, 157
and substitution among production out-
lays at several sources, 155-157, 262
and substitution among transport inputs
on materials from several sources,
155-157, 262
and substitution between production and
transport outlays, 182, 262

350

SUBJECT INDEX

Welfare — continued

and substitution between transport in-
puts in different directions, 147
and substitution between transport in-
puts on products of two producers,
147-148
and the use of Weber's agglomeration

analysis, 182
approach of, as yielding clearest localiza-
tion analysis, 268
in simplified market cases, 232-238
maximization of, with hexagonal market

patterns, 153, 252
need to study spatial aspects of, 287
optimum as different from minimization

of transport cost, 232n
problems in defining, 236n
see also Surplus, social
Wholesale sales, decrease of, with distance

from urban core, 68-70
Williams' criticism of mobility and immobility

premises of trade theory, 208
Women labor, see Labor, cheap

Zeuthen as bearing out Hotelling's solutions
on competitive locational equilibrium,
160
Zipf, and concentrative effects of innovation,
79n
and Principle of Least Effort, 60, 78n-79n
empirical findings on rank-size rule, 56-57
explanation of spatial framework, 78n-79n
Forces of Unification and Diversification,

60, 78n-79n
inconsistency of, and deposits of highly

localized raw materials, 79n
interpretation of deviations from rank-
size rule, 57n
needed extensions of, 79n
number of families moving varying dis-
tances, 63-64
on length and number of one-way passen-
ger car trips, 62n
on length and number of one-way truck

trips, 62n
on number of marriage licenses and dis-
tance separating applicants, 62n
the P/D factor and charge accounts of

Jordan Marsh Co., 62n
the Pi • P2/D factor, and airway passenger
movements, 62 n
and average circulation per day of The

New York Times, 62n
and bus passenger movements, 61-63
and number of different news items in

The Chicago Tribune, 62n
and number of obituaries in The New

York Times, 62n
and railway express shipments, 60-61
and railway passenger movements, 62n
and telephone messages, 61-62
Zonal boundaries, as defined by equation of
marginal rents on two crops, 247

Zonal boundaries — continued

as defined by intersection of rent func-
tions, 247, 276
determination of equations of, 246-247
forces generating irregularities in, 276-

277
multiplication of, when each farm produces

unique crop combination, 248
shifts of, along substitution paths, 246-
247
as involving substitution between trans-
port inputs, 246n
simple form of equations of, 246-247
see also Zone formation
Zone(s), concentric, 3, 5, 33n, 52

as cut off by city-region boundaries, 249n,

276-277
as distorted by legal and social institu-
tions, 276
as distorted by physical barriers, 276
as distorted by raw material supply, 248,

276
as distorted by resource content of land,

276
as distorted by scale economies in trans-
portation, 276-277
as distorted by transport network, 276-
277
finite number of production points as a

special case of, 250n
for a given commodity as reducible to a

finite number of points, 250n
intensity of land use among, as rising or
falling with distance from market,
247n-248n
more precise determination of, with firm

equilibrium analysis, 199
multiple, existence of for single crop when

not classified by market, 250n
multiplication of, when each farm pro-
duces unique crop combination, 248
transition of, 3, 4, 5, 8, 16, 18, 19
see also Agricultural land use ; Zonal boun-
daries ; Zone formation
Zone formation, and advance in transport
technology, 16
and interrelations of strata, 29
and Losch's framework, 16
and the development process, 3-6, 8
as obtainable from analysis along a straight

line from a city, 245n, 248n
concentric, as yielded by the general loca-
tion principle, 244-246
based on invariance of transport rate

with direction from city, 245
implied by substitution principle, 244-245
identification of, by use of rent functions,

195, 199
logic of, 244-251
role of competition in, 198-199
role of price change in, 243
substitution between transport inputs in,
246n

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