Table Of ContentIndi visibilities
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Hagen Bobzin
Indi visibilities
Microeconomic Theory with Respect to
Indivisible Goods and Factors
With 116 Figures
Physica-Verlag
A Springer-Verlag Company
Series Editors
Werner A. Muller
Martina Bihn
Author
Dr. Hagen Bobzin
FB Wirtschaftswissenschaften
Universitiit - Gesamthochschule Siegen
HOlderlinstr. 3
D-57068 Siegen, Gennany
ISBN-13:978-3-7908-1123-0 e-ISBN-13:978-3-642-47030-1
DOl: 10.1007/978-3-642-47030-1
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Bobzin, Hagen: Indivisibilities: microeconomic theory with respect to indivisible goods and fac
tors I Hagen Bobzin. - Heidelberg: Physica-Verl., 1998
(Contributions to economics)
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Preface
The analysis of this volume represents an attempt to apply modern mathematical
techniques to the problems arising from large and significant indivisibilities. While
the classical microeconomic theory refers to assumptions about the convexity
of production sets and consumer preferences, this book directs the attention to
indivisible commodities. It investigates the influence of the assumed indivisibilities
of factors and goods on the results of the microeconomic theory of the firm, the
theory of the household and market theory.
In order to quantify the relationships between economic variables and among
economic actors the theory is founded on convex analysis. Hence, many results
heavily depend on the approximation of integer sets by their convex hull. As far as
possible numerous figures are provided to develop the reader's geometric intuition.
My intention is to continue with FRANK's (1969) beginnings of a general,
systematic and rigorous analysis of the problems of indivisibilities. The advantage
of the formalized way chosen is that the numerous and detailed properties of the
economic relationships can be deduced on the basis of very few assumptions. It is
not surprising that at least within the market theory the most important assumption
concerning indivisible goods is that at least one commodity is perfectly divisible.
The author's largest debt is to Professor Dr. Walter BUHR, UniversiHit
Gesamthochschule Siegen (University of Siegen). He provided guidance,
suggestions and encouragement at nearly every step along the way. Professor Dr.
Andreas PFINGSTEN, Westfalische Wilhelms-Universitat Munster (University of
Munster), was also very helpful in providing detailed comments and suggestions
on my dissertation.
Further thanks are due to Dr. Thomas CHRISTIAANS, whose comments led to
many improvements of the work. The present manuscript is the translated version
of the German dissertation. Gudrun BARK was especially helpful in reading and
correcting the materials not only in German but also in English.
Siegen Hagen BOBZIN
February 1998
Contents
I. Microeconomic Theory with Respect to Indivisibilities 1
II. Microeconomic Foundations 5
Axiomatic Characterization of Individual Economic Agents . 5
1.1 The Preferences of a Household ... 5
1.2 The Production Technology of a Firm ........ 10
2 Theory of the Firm ..................... 17
2.1 Inverse Representation of the Production Technology 17
2.2 Treatment of Indivisible Goods and Factors . 23
2.2.1 The Concept of Convex Hulls 23
2.2.2 The Assumption of Integer Convexity 32
2.3 Special Production Technologies . . . . . . . 36
2.3.1 Scale Economies . . . . . . . . . . . 36
2.3.2 Additivity of a Production Technology 43
2.3.3 Factor Constraints. . . . . . . . 47
2.4 Optimal Activities . . . . . . . . . . . . . . 48
2.4.1 Technically Efficient Production . . 48
2.4.2 Determination of Optimal Activities 59
2.4.3 Optimal Activities in the Production of One Good . 69
3 Summary ............... 75
4 Appendix . . . . . . . . . . . . . . . 77
4.1 The Concept of Quasi-Concavity 77
4.2 Closedness of the Convex Hull of Input Requirement Sets . 79
III. Microeconomic Theory of Individual Agents 82
The Cost Structure of a Firm . . . . . . . . . . . . . 82
1.1 Dual Statements in the Theory of a Firm . . . . 82
1.2 Determination of the Normalized Cost Function 88
1.3 Reconstruction of the Production Structure . 96
1.4 Properties of Factor Demand . . . . . . . . . . 102
1.4.1 Convex Input Requirement Sets .... 102
1.4.2 Consideration of Indivisible Production Factors 109
1.4.3 The Results under the Assumption of Differentiability. 119
1.5 Summary .......................... 122
1.5.1 Graphical Representation of the Results . . . . . . 122
1.5.2 Results with Respect to the Output Correspondence 127
2 Alternative Representation of the Firm's Cost Structure 129
2.1 The Cost Function . . . . . . . . . . . . . . . . . . 129
2.1.1 Properties of the Cost Function . . . . . . . 129
2.1.2 Reconstruction of the Production Structure. 142
viii Contents
2.1.3 Comparison of the Derived Cost Functions. . . . . . . .. 144
2.2 The Input Distance Function . . . . . . . . . . . . . . . . . . .. 151
2.3 Dual Representation of the Production Structure as Cost Structure 157
2.4 Summary ...................... 169
2.4.1 Schematic Construction of Duality Theory . . . . . 169
2.4.2 Graphical Representation of the Results . . . . . . 173
2.4.3 Results with Respect to the Output Correspondence 182
3 Optimal Activities in the Theory of the Household . . . . 186
3.1 Demand for Commodities. . . . . . . . . . . . . . 186
3.1.1 The Expenditure Structure of a Household . 186
3.1.2 The Individual Demand for Goods 187
3.1.3 The Aggregate Excess Demand. . . . . . . 191
3.2 Special Preference Orderings . . . . . . . . . . . . 194
3.2.1 Implications of the Continuity of Preference Orderings 194
3.2.2 Implications of the Monotonicity of Preference Orderings. 203
3.3 Summary .............................. 207
IV. Theory of Market Equilibria 209
The Problem of General Equilibria . . . . . . . . . . . . . . . . 209
1.1 Approaches to Treating Indivisible Goods . . . . . . . . . 209
1.2 Graphical Representation of Simple Exchange Economies. 216
1.3 The Problem of Unbounded Demand for Goods . 222
1.3.1 Convex Preference Orderings. . . . . . . 222
1.3.2 Consideration of Indivisible Commodities 224
2 Existence of Competitive Equilibria .... 225
2.1 Strictly Convex Preference Orderings. . . 225
2.2 Convex Preference Orderings . . . . . . . 228
2.3 Consideration of Indivisible Commodities 231
2.4 Summary ................. 247
3 Generalization of the Model of an Exchange Economy 249
3.1 Equilibria in Production Economies ..... 249
3.1.1 Description of a Production Economy . 249
3.1.2 Existence of Competitive Equilibria . . 254
3.2 Alternative Criteria for Optimal Market Results 263
3.2.1 The Core of an Exchange Economy . 263
3.2.2 PARETO Optimal Allocations. . . . . . 263
3.2.3 First Theorem of Welfare Economics. . 265
3.2.4 Comments on the Second Theorem of Welfare Economics 267
3.3 Summary .............................. 268
Contents IX
V. Critique 270
Mathematical Appendix
A Basic Concepts of Analysis . . . . . . . . . . . . 280
A.1 Important Properties of the Euclidean Space 280
A.2 Elementary Concepts of Topology 282
A.3 Convergence in Metric Spaces 284
A.4 Compact Sets 287
B Convex Analysis . 289
B.1 Affine Sets . . 289
B.2 Convex Sets . 291
B.3 Separation Theorems 296
C Mappings ........ . 297
C.1 Functions as Single-Valued Mappings 297
C.2 Correspondences as Multi-Valued Mappings . 306
C.3 Fixed-Point Theorems ..... . 311
D Duality Theory ........... . 315
D.1 Duality of Conjugate Functions . 315
D.2 Duality of Polar Gauges. . . . . 322
D.2.1 Properties of the Support Function 322
D.2.2 Properties of the Gauge . 329
D.2.3 Polar Sets and Functions . . . . . 332
Tables
List of Symbols 338
List of Figures . ..... 343
References 346
Index 360
Chapter I.
Microeconomic Theory with Respect
to Indivisibilities
Indivisible goods and factors constitute a subject of economic theory associated
with a series of unsolved problems. Even advanced works on microeconomic theory
like VARIAN (1992) or JEHLE (1991) refrain from the consideration of indivisible
goods and factors to provide a structure for the analysis where relatively simple
mathematical methods can be applied. The Handbook of Mathematical Economics
also does not contain any approach treating explicitly the integer problem. While
GREEN, HELLER (1981) present the instrument of convex analysis with respect
to economic applications in the first chapter, a corresponding work dealing with
the indivisibility of goods and factors is missing. Even in BROWN (1991), who
introduces an equilibrium analysis with nonconvex technologies in Chapter 36, the
problem of indivisibility is merely of minor importance.
However, The New Palgrave: A Dictionary of Economics I contains an explicit
article on indivisibilities written by BAUMOL. Apart from some hints on integer
programming2, the author cites only one work which explicitly deals with analyzing
indivisibilities. In this book, which was already published in 1969, FRANK, as the
first economist, presented a comprehensive analysis of the importance of indivisible
goods in production theory. His approach identifies the problem of indivisibility
with goods which are only available at integer amounts.
While FRANK explicitly picks up the problem of indivisibility, there is a wealth
of approaches including only indirectly the problem of indivisible goods and factors.
For example, ROSEN (1974) describes markets for a class of indivisible goods which
can be distinguished by certain features. At the same time it is assumed that there
is a sufficiently large number of these differentiated goods such that the choice of
characteristics may vary continuously.
Although ROSEN does not pursue the aim to describe a market for a
lSee EATWELL, MILGATE, NEWMAN (1987).
2The mentioned approaches to integer programming of GOMORY (1965) and GOMORY, BAUMOL
(1960) will not be important until Chapter V (oCthe book).
2 Chapter 1. Microeconomic Theory with Respect to Indivisibilities
few indivisible goods, his procedure clarifies various aspects in dealing with
indivisibilities. If the solutions to optimization problems are subject to certain
integer constraints, then the analytical effort ascertaining an optimal solution rises
considerably in relation to "well behaved" convex problems. The alternative
determination of "rounded results" ignoring the required integer values may lead
to considerable errors. Hence, the decision to solve a simple but incorrect problem
or an exact but costly problem depends on the return of the additional effort.3 At
the same time we expect that the return tends to depend on the relative size of the
indivisible goods. The difference between the production of 100000 or 100001
cars is of little significance for an automobile company, whereas a household faces
considerable consequences depending on whether it has got a car or not.4
If, like VIETORISZ (1963), we concentrate on all-or-northing decisions or on
investments where we have to decide on production levels, location, and timing,
then the problem of nonconvexities is not only a mathematical curiosity but it plays
a rather considerable role in daily economic practice. However, microeconomic
theory frequently ignores the problem of indivisible goods and factors in view
of the easier analytical instruments. In particular, such favorable properties as
continuity, convexity, or differentiability of functions justify this procedure as long
as phenomena like indivisibilities are only of minor importance.5 However, when
indivisibilities have great importance, then we usually have to refrain from making
use of the above advantages. The analysis does not only get a new look but also
leads to modified results.
For example, suitable assumptions assure in economic bibliography an exact
duality between the firm's production function and the cost function. Accordingly,
each "well behaved" production function is associated with a unique cost function,
from which we can infer back to the production function in a unique way. If we
now give up the assumption of divisible goods and factors or the corresponding
assumptions of convexity, then the exact one-to-one relation between the above
functions is no longer valid. In view of suitable approximations it can now be
examined to what extent the relationship is abolished. In Chapter III the difference
to the traditional analysis by the modified behavior of the factor demand becomes
most apparent. Chapter IV also indicates that many results of traditional theory
are repudiated under consideration of indivisibilities. Because the existence of
general equilibria bases crucially on corresponding assumptions of convexity -
hence divisible goods - an existence proof considering indivisible goods only
succeeds under very restrictive assumptions. Thus, the analysis concentrates on the
question of how large the fault is when the requirement of integer values is ignored.
If no general equilibrium exists, then these faults yield a measure of the importance
of the indivisibility of goods.
The analysis refers to microeconomic issues, where the consideration of
3Considering the mathematical difficulties, KOOPMANS, BECKMANN (1957) suggest commencing
the analysis with extremely simplified economic problems until more realistic questions are investigated.
4 A similar characterization of indivisible goods is given by BAUMOL (1987).
5DIEWERT (1986) ignores completely the problem of indivisible goods and factors, although the
analysis of welfare effects is based on large and significant investments in infrastructure.
