Core and Equilibria of a Large Economy. (PSME-5) 9781400869473

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CORE AND EQUILIBRIA OF A LARGE ECONOMY

PRINCETON STUDIES IN MATHEMATICAL ECONOMICS Edited by David Gale, Harold W. Kuhn, and H. Nikaido I. Spectral Analysis of Economic Time Series, by C. W. J. Granger and M. Hatanaka 2. The Economics of Uncertainty, by Karl Henrik Borch 3. Production Theory and Indivisible Commodities, by Charles Frank, Jr. 4. Theory of Cost and Production Functions, by Ronald W. Shephard 5. Core and Equilibria of a Large Economy, by Werner Hildenbrand

CORE AND EQUILIBRIA OF A LARGE ECONOMY WERNER H I L D E N B R A N D

PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY

Copyright © 1974 by Princeton University Press Published by Princeton University Press, Princeton and London All Rights Reserved LCC: 72-12112 ISBN: 0-Q691-04189-X This book has been composed in Times New Roman Library of Congress Cataloging in Publication data may be found on the last printed page of this book. Printed in the United States of America by Princeton University Press, Princeton, New Jersey

CONTENTS

PREFACE

vii PART I MATHEMATICS

A. NOTATION IN SET THEORY

3

B. ELEMENTARY FACTS IN METRIC SPACES

8

I. GENERALITIES II. THE SPACE OF SUBSETS OF A METRIC SPACE III. CONTINUOUS CORRESPONDENCES C. MISCELLANY IN EUCLIDEAN SPACES

I. NOTATION II. CONVEX SETS III. FIXED POINT THEOREMS FOR CORRESPONDENCES D. SOME MEASURE THEORY

I. II.

8 15 21 35

35 36 39 40

DEFINITIONS AND STANDARD RESULTS THE INTEGRAL OF A CORRESPONDENCE

40 53

PART II ECONOMICS CHAPTER 1 DEMAND

83

1.1. INTRODUCTION

83

1.2. INDIVIDUAL DEMAND

95

1.3. MEAN DEMAND

109

CHAPTER 2 EXCHANGE

123

2.1. CORE AND WALRAS EQUILIBRIA

123

2.2. DETERMINATENESS OF EQUILIBRIA 2.3. APPENDIX

148 168

CHAPTER 3 LIMIT THEOREMS ON THE CORE

177

3.1. INTRODUCTION

177

3.2. LIMIT THEOREMS WITH STRONGLY CONVEX PREFERENCES

178

3.3. LIMIT THEOREMS WITHOUT CONVEXITY OF PREFERENCES

199

ν

CONTENTS

CHAPTER 4 ECONOMIES WITH PRODUCTION 4.1. INTRODUCTION 4.2. COALITION PRODUCTION ECONOMIES 4.3. PARETO EFFICIENT ALLOCATIONS

209 209 210 229

SUMMARY OF NOTATION

234

BIBLIOGRAPHY

236

NAME INDEX

247

SUBJECT INDEX

249

PREFACE

The work reported in this book has been developed mainly in the last ten years and is a direct outgrowth of two basic contributions: a paper by Debreu and Scarf (1963) "A Limit Theorem on the Core of an Economy," and a paper by Aumann (1964) "Markets with a Continuum of Traders." The central problem of the theory that this book presents is the relationship between two fundamental concepts of equilibrium for an economy: the core, which is a cooperative equilibrium concept, and Walras equilibrium, which is a noncooperative concept. Both concepts of equilibrium for an economy have a long tra­ dition in economic theory. In 1874, L. Walras gave the first formu­ lation of a general economic equilibrium. The core and its relation to the Walras equilibria were analyzed by F. Edgeworth as early as 1881. Both concepts of equilibrium have been revived recently with the development of game theory. I believe that the essential ideas in the book which are relevant to economic theory can be understood on an intuitive level without familiarity with all the mathematics given in Part I. As a general rule, 1 recommend that the reader should, at least in a first reading, leaf through Part I, Mathematics, and go without further ado to Chapters 2 and 3, which constitute the core of the book. Most of the results given in the text under the heading of "Theorems" and "Proposi­ tions" are stated and proved in a simple situation. In the text I have not tried to be as general as possible. Generalizations of most of the results, as well as additional results, are treated as "Problems" at the end of each section. Indeed, to solve these problems one often needs much more mathematics than is required to understand the text. At this point, a careful reading of Part I may be helpful. I am afraid that it is psychologically awkward to begin a book on mathe­ matical economics with a lengthy chapter on mathematics: the economist may be discouraged and the mathematician may give

PREFACE

too much importance to the mathematical aspects of the matter. However, it seems to me convenient for the average reader to have collected, and partly proved, mathematical concepts and results which come from quite different fields in mathematics. When it comes to the acknowledgments I want first of all to express my deepest gratitude to Gerard Debreu, who introduced me to the subject I treat in this book. Without his constant encouragement and constructive criticism I would, most probably, not have written this book. I have used many of his ideas and suggestions without ex­ plicit reference. In the last four years I have had particularly close collaboration with Robert Aumann, Truman Bewley, Birgit Grodal, Yakar Kannai, Jean-Francois Mertens, David Schmeidler, and Shmuel Zamir. Their contributions shaped the book essentially, and I would like to thank all of them for their friendly collaboration. During recent years I have taught on the subject of the book at the University of California, Berkeley; Center for Operations Re­ search and Econometrics, Louvain; Stanford University; and Uni­ versity of Bonn. It is a pleasure to acknowledge my indebtedness to many colleagues and students. I am grateful to Kurt Hildenbrand, who contributed the Appen­ dix to Chapter 2. Egbert Dierker, Jerry Green, Kurt Hildenbrand, Alan Kirman, Wilhelm Neuefeind, Dieter Sondermann, and Walter Trockel read the different versions of the manuscript or extensive parts of it. Their help and comments are gratefully acknowledged. I am grateful also to Conny Reschke, who typed with great patience, accuracy, and skill the various versions of the manucript. Finally I gratefully acknowledge the financial support of the Ford Foundation, Fond de la Recherche Fondamentale Collective, and Deutsche Forschungsgemeinschaft. W. H. Bonn, February 1973

PART I

MATHEMATICS

A. Notation in Set Theory The notion of a set is taken here as a primitive concept. The objects constituting a set are called elements of the set. means x is an element of the set S (x belongs to S). ' means x is not an element of the set S. means every element of the set T is also an element of the set S (T is a subset of S, or T is contained in S). T — S means

and ,

(S and T are equal).

t denotes the set without any element (empty set). {x,y,z,...} denotes the set whose elements are those listed inside the brackets, that is x, y, z,... . Order and repetitions in listing the elements are immaterial. Thus, denotes the set of positive integers, denotes the set of all x for which the proposition P(x) is true. denotes the set of all x which belong to S and for which the proposition P(x) is true. U: The union of the sets S and T is the set it is denoted by S U T.

UNION

INTERSECTION

The intersection of the sets S and T is the set it is denoted by ,

COMPLEMENT

\ : The complement of T with respect to S is the set it is denoted by S\T.

(•, •) : The formal mathematical method of associating the object y to the object x is to form the ordered pair (x,y). The notion of an ordered pair is taken here as a primitive concept. Two ordered pairs

ORDERED PAIRS

CARTESIAN PRODUCT

X: The cartesian product of the sets S and T is 3

MATHEMATICS

the set whose elements are odered pairs (x; y) where y £ T; it is denoted by

and

The mathematical description of a rule which associates certain objects with other objects is the set of ordered pairs (x,y) such that y is associated to x by the rule. This leads to the following definition: A relation

χ or Iim x„ = χ The point χ (which by def" η inition of a metric is unique) is called the limit of the sequence (χ*). CONTINUITY OF A MAPPING: A mapping / of a metric space (M,d) into a metric space (M',d!) is said to be continuous at χ if / (xn) —> f(x) whenever x„ —> x. The mapping / is said to be continη η uous if it is continuous at every point χ in M. HOMEOMORPHISM: Two metric spaces (M,d) and are said to be homeomorphic if there exists a one-to-one mapping / of M onto M' such that / and /-1 are continuous. The mapping / is then called a homeomorphism. OPEN SETS: A subset G of a metric space (M,d) is said to be open if it is empty or if, for every χ £ G, there exists an open ball with center χ and nonzero radius contained in G. The class of open subsets in a metric space (M,d) has the follow­ ing properties:

(i) (ii) (iii) (iv)

Every union (finite or infinite) of open sets is open. Every finite intersection of open sets is open. The whole space M and the empty set are open. For every X 1 , x 2 £ Μ , χ ι ^ x 2 , there are two disjoint open sets Gi and G2 such that Xi 6 GI and x2 ζ G2.

TOPOLOGY: Every collection 3 of subsets of a set M which has prop­ erties (i), (ii), (iii) is called a topology on the set M, and the pair (Λ/,3) is called a topological space. The elements in 3 are called open sets. If the topology also has property (iv), it is called sepa­ rated (Hausdorff). The topology derived from the metric d on M is called the metric topology of (M,d). Not every separated topology on a set M can be derived from a metric d on M. A topological space (Μ,3) is called metrizable if there exists a metric d on M such that the class of open sets derived from d coincides with 3. The same topology (i.e., the same collection of open sets) on a set M may be associated with many different metrics (these metrics

Β. ELEMENTARY FACTS IN METRIC SPACES

are then called equivalent). A metric does not therefore constitute an intrinsic tool for the study of convergence and continuity. CLOSED SETS: A subset F in a topological space (Af,3) is called closed

if its complement M\F is open. (1) In a metric space (M,d) a subset F is closed if and only if every converging sequence in F has its limit in F. CLOSURE OF A SET: The intersection of all closed sets containing the

set A is called the closure of Ai and is denoted by A or clA. NEIGHBORHOODS: A neighborhood U of a point χ in a topological space

is a subset of M containing an open set which contains the point x. A neighborhood of a set A in (M,3) is a subset of M containing an open set which contains the set A.

(M,3)

We now reformulate the definition of convergence of a sequence and of continuity of a mapping in the language of neighborhoods. (2) A sequence (Xn) in the topological space (M,3) is said to con­ verge to χ ζ M if for every neighborhood U of χ there exists an integer η such that x„ £ U for every η > η. (3) A mapping f of the topological space (M,3) into the topological space (M'ff) is said to be continuous at the point χ ξ. M if for every neighborhood U of f (x) there exists a neighborhood V of χ such that f(z) € U for every ζ ζ V. (4) The mapping f of (Mi3) into (M',3') is continuous if and only if the set f~'(G) is open for every open set G in M'. INTERIOR POINT OF A SET: The point Λ: is called an interior point of the

subset A of (Mi3) if A contains a neighborhood of x. The set of all interior points of A is called the interior of A and is denoted by A or int(/l). BOUNDARY OF A SET: The boundary dA of a subset A of (M,3) is the

set of points, each of whose neighborhoods contains at least one point of A and of M\A. For every subset A of (MirS) one has dA = A\A.

MATHEMATICS DENSE SUBSETS:

The subset D of (M,s) is said to be dense in M if

topological space is said to be separable if there is a countable dense subset in M. (5) A metric space (M,d) is separable if and only if there exists a countable base of open sets, i.e., there exists a countable family ® of open sets such that every open set in M is union of sets in (&.

SEPARABLE SPACES: A

A topological space (M, every subset is open.

DISCRETE SPACES:

3)

is called to be discrete if

A subset A of a topological space (M,3) is called a subspace if A is endowed with the topology 3A whose open sets are the intersections with A of the open sets in M. Let A be a subset of the metric space The restriction dA of d on A X A is clearly a metric on A.

SUBSPACES:

(6) The topology on A derived from the metric dA coincides with the subspace topology 3A(7) In order that every open (closed) set in the subspace A of (M,3) be an open {closed) set in M, it is necessary and sufficient that A be open (closed) in M. (8) Every subspace of a separable metric space is a separable metric space. Let be topological spaces. An open rectangle in set of the form where G, is an open set in M,. Let 3 denote the collection of subsets in M which is obtained by taking arbitrary unions of open rectangles. It follows that 3 satisfies axioms (i), (ii), and (iii). The topological space 1 defined, is called the topological product of the space and 3 is called the product topology.

PRODUCT SPACES:

(9) Let be metric spaces and Define en d is a metric on M and the topology of the metric space (M,d) is the product topology. 12

Β. ELEMENTARY FACTS I N METRIC S P A C E S

(10) The finite topological product of separable metric spaces is a separable metrizable space. (11) T h e m a p p i n g f o f a topological space T into the topological product space M = JJf=, M, is continuous at χ if and only if each of the coordinate mappings f of T into Mi is continuous at x. (12) The sequence (x\,x™)?=1,... of points in the topological space

M = JJfLlM, converges to (xl,..., xm) € M if and only if for every i the sequence (Xii)i-It... converges to xl. COMPACT METRIC SPACES

A topological space ( M , a ) is said to be compact if from every open covering of M one can select a finite subcovering of M. A subset K of a topological space is said to be compact if the subspace (K,3K) is compact. A subset A of a topological space is said to be relative compact if the closure A is compact. Examples Every finite set and every closed and bounded set in Rm is compact. The extended real line R (which is obtained by adding to R two ad­ ditional points + oo and — 00 with the property: —