Can every allocation in the core of an economy be decentralized by a suitably chosen price system? Werner Hildenbrand shows that the answer is yes if the economy has "many" participating agents and if the influence of every individual agent on collective actions is "negligible." To give a general and precise definition of economics with this property he considers both economies with a continuum of agents, and a sequence of economies with an increasing number of participants. In both cases this leads to a measure theoretic formulation of economic equilibrium analysis.
In the first part of the book the relevant mathematics is developed. In the second part the continuity and convexity properties of the total demand of a consumption sector are investigated. An important result is the equivalence between the core and the set of Walras equilibria for an exchange economy with a continuum of agents. The author then deals with limit theorems on the core for purely competitive sequences of exchange economies. In the last chapter the core and the set of Walras equilibria for a coalition production economy and the relation between these two equilibrium concepts are studied.
Originally published in 1974.
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Core and Equilibria of a Large Economy
By Werner HildenbrandPRINCETON UNIVERSITY PRESS
Copyright © 1974 Princeton University Press
All rights reserved.
ISBN: 978-0-691-04189-6Contents
PREFACE, vii,
PART I MATHEMATICS,
A. Notation in Set Theory, 3,
B. Elementary Facts in Metric Spaces, 8,
I. Generalities, 8,
II. The Space of Subsets of a Metric Space, 15,
III. Continuous Correspondences, 21,
C. Miscellany in Euclidean Spaces, 35,
I. Notation, 35,
II. Convex Sets, 36,
III. Fixed Point Theorems for Correspondences, 39,
D. Some Measure Theory, 40,
I. Definitions and Standard Results, 40,
II. The Integral of a Correspondence, 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, 148,
2.3. APPENDIX, 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,
CHAPTER 4 ECONOMIES WITH PRODUCTION, 209,
4.1. Introduction, 209,
4.2. Coalition Production Economies, 210,
4.3. Pareto Efficient Allocations, 229,
SUMMARY OF NOTATION, 234,
BIBLIOGRAPHY, 236,
NAME INDEX, 247,
SUBJECT INDEX, 249,
CHAPTER 1
DEMAND
1.1. Introduction
The following standard economic concepts are introduced: commodity; commodity space; consumption plan; consumption set; preference relation; prices; wealth; budget set; demand set.
COMMODITY SPACE Rl
A commodity is a good or a service. It is characterized by its physical characteristics (properties) and the date and the location at which it will be available. Thus, given physical characteristics (e.g. wheat of a specified type) made available at different dates and/or different locations will be treated as different commodities. Typical examples of commodities are consumption goods related to food, housing, and clothing. A typical example of a service is labor. The physical characteristic of labor is the task performed.
The quantity of a commodity can be expressed by a number. The physical characteristics, which define a commodity, are assumed to be homogeneous (i.e., equal quantities of the same commodity are interchangeable in all their uses). A commodity is assumed to be infinitely divisible (i.e., the quantity can be any nonnegative real number).
It is assumed that there is a finite number of distinguishable commodities: these are indicated by an index h running from 1 to l. If every unit vector 1h = (0, ..., 1, ..., 0) of the linear space Rl is identified with one unit of the commodity h (h = 1, ..., l), then the linear space Rl is called a commodity space. Every bundle of commodities can be represented by a vector in the commodity space Rl.
We emphasize that the introduction of the date and location of availability in defining a commodity will have its drawbacks: the meaningfulness and acceptability of some assumptions made later — in particular on preferences — should always be discussed from the viewpoint of different dates and locations.
CONSUMPTION PLAN x [member of] Rl
CONSUMPTION SET X [subset] Rl
A consumption plan for an economic agent specifies the quantity
