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Overview
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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Product Details
| ISBN-13: | 9780691618784 |
|---|---|
| Publisher: | Princeton University Press |
| Publication date: | 03/08/2015 |
| Series: | Princeton Studies in Mathematical Economics , #1492 |
| Pages: | 262 |
| Product dimensions: | 6.00(w) x 9.10(h) x 0.70(d) |
Read an Excerpt
Core and Equilibria of a Large Economy
By Werner Hildenbrand
PRINCETON UNIVERSITY PRESS
Copyright © 1974 Princeton University PressAll rights reserved.
ISBN: 978-0-691-04189-6
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 of each commodity which he consumes (i.e., which has to be made available to him), and the quantity of (possibly several types of) labor which he will make available. We shall use the convention that the quantity of a commodity which is made available by an agent is represented by a negative number, and the quantity of a commodity which has to be made available to an agent is represented by a positive number. Then every consumption plan can be represented by a vector x in the commodity space Rl.
For obvious physical, physiological, or institutional reasons, not every vector in Rl can be interpreted meaningfully as a consumption plan. For example, impossible combinations of commodities, such as the supplying of several types of labor to an amount totaling more than 24 hours a day or the consumption of a bundle of commodities insufficient to maintain life. It is assumed that for every consumer a there is a nonempty closed subset Xa in Rl, the consumption set of agent a, which describes the set of a priori possible consumption plans. Here a priori possible means that, ignoring budget considerations, the agent can carry out the consumption plan. That is to say, he can make available the labor and is able to consume the consumption goods as specified in x [member of] Xa.
Although a consumption set X will typically belong to a subspace of Rl, the dimension of which is essentially less than l, it still will be, in general, of quite high dimension. An intersection of a typical consumption set X with a plane parallel to a labor- and food-coordinate or two consumption good-coordinates is visualized in Figure 1-1:
As explained before, only those coordinates of a consumption plan can be negative which refer to the various types of labor. Since a day has only 24 hours it follows therefore — whatever the length of the periods used to define the commodity space — that there is a universal lower bound b for all possible consumption plans of every economic agent.
Likewise it is clear from the interpretation of a consumption set that there will be a (possibly large) number k such that every economic agent can survive if his consumption plans were restricted to be in every commodity less than k. That is to say, we assume the existence of a universal constant k such that for every consumption set X the set {x [member of] X | x ≤ (k)} is nonempty.
Finally, for technical reasons, which will become clear in later chapters, we shall assume that the consumption sets X are convex (for a typical reasoning where this assumption is used see the proof of Proposition 1).
We summarize the above assumptions in the following:
Definition 1: A consumption set X is a nonempty subset of the commodity space which is closed, convex and bounded from below. Given a vector b [member of] Rland a compact subset K [subset] Rl, we denote by X the set of all consumption sets X which have the properties: b ≤ X and X [intersection] K ≠ φ.
We shall fix, once and for all, the vector b and the compact set K; their particular numerical value is immaterial. To restrict in this way the "universe" of consumption sets is no restriction for the economic analysis; however, it simplifies the mathematical presentation (the set X will turn out to be compact [1.2, Theorem 1]).
PREFERENCES >
We say that an economic agent prefers the consumption plan x to the consumption plan x' if he wants to choose x whenever he is offered the alternatives x and x'. The binary relation "preferred" becomes a powerful tool for economic analysis if the behavior of the economic agents reveals a certain "consistency" of choices. Absolute consistency of choices, to take the ideal case, is described as a transitive preference relation. This is not the place to discuss the empirical content of this hypothesis.
In defining the commodity space, we assumed that all commodities are infinitely divisible. Consequently, to give the concept of a preference relation its full analytical power, we assume that a preference relation is continuous, that is to say, if x is preferred to y, then for a sufficiently small change of x to x' and y to y', x' is still preferred to y'. This is a technical assumption; it cannot be falsified by experiments.
The foregoing motivates the following:
Definition 2: A preference relation is defined by a pair (X, >) where X is a consumption set and > [subset] X X X is a transitive and irreflexive binary relation on X such that > is open in X X X. The set of all preference relations with X [member of] X is denoted by P.
Instead of (x,y) [member of] > we shall use the more suggestive notation x > y. Thus x [??] y means (x,y) [not member of] >.
To every preference relation (X, >) in P we associate the set
F: = {(x,y) [member of] Rl X Rl | x, y [member of] X and x [??] y}.
The set F is characterized by the following properties:
(i) F is a closed subset in Rl X Rl
(ii) the set {x [member of] Rl | there is y with (x,y) [member of] F} belongs to X
(iii) (reflexivity) (x,y) [member of] F implies (x,x) [member of] F and (y,y) [member of] F
(iv) (negative transitivity) (x,y) [not member of] F and (y,z) [not member of] F imply (x,z) [not member of] F.
Given such a set F, we obtain the corresponding preference relation by setting X:= (x [member of] Rl | (x,x) [member of] F} and > := (X X X)\F.
In order to prove the results of the following chapters one often requires additional properties of preference relations. We shall now define subsets of the set P of preference relations by one or several of the following properties:
LOCAL NONSATIATION
A preference relation (X, >) is said to be locally nonsatiated if for every x [member of] X and every neighborhood U of x there exists x' [member of] X [intersection] U such that x' >x.
The set of all locally nonsatiated preference relations in P is denoted by Plns.
MONOTONICITY
A preference relation (X, >) is said to be monotonic if the consumption set X is equal to the positive orthant Rl+ and 0 ≤ x ≤ y, x ≠ y imply y > x. The set of all monotonic preference relations in P is denoted by Pmo.
Clearly one can define monotonicity of preferences also if the consumption set X is not the positive orthant. However, in models where one has to consider general consumption sets, for example, for economies with production [Chapter 4], the assumption of monotonic preferences is in any case too strong. For monotonic preferences in Pmo we shall write > [member of] Pmo instead of (Rl+, >) [member of] Pmo.
TRANSITIVE INDIFFERENCE
A preference relation (X, >) is said to be negatively transitive if for every x, y, z [member of] X with and x [??] y and y [??] z it follows that x [??] z.
The set of all negatively transitive preference relations in P is denoted by P*.
For a preference relation in P* one defines an indifference relation by x ~ y if and only if x [??] y and x [??] y. The indifference relation ~ on X is reflexive, transitive and symmetric. The relation [??] is then written as [??]. The preference-indifference relation [??] is reflexive, transitive and complete.
Preference relations in P* are easily visualized by drawing their indifference surfaces as in Figure 1-2:
[FIGURE 1-2 OMITTED]
Surely one can also define the relation ~ for a preference relation in P. However, one can make no use of this relation since it is not transitive.
CONVEXITY
A preference relation (X, [??]) in P* is called
(a) convex if for every z [member of] X the set {x [member of] X | z [??] x} is convex
(b) strongly convex if for every x ~ x', x ≠ x', and every λ, 0 < λ < 1 it follows that λx + (1 - λ) x' > x.
The set of all convex (strongly convex) preference relations in P* is denoted by (P*co (P*sco).
We defined convexity of preferences only for relations in P* since for relations in <P, not belonging to P*, one can make no use of the convexity property.
Finally we use the self-explaining notation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
PARETIAN UTILITY
Sometimes it is convenient to represent a preference relation by a real-valued function. Let (X, >) be a preference relation. A Paretian utility for (X, >) is a continuous function u : X ->R such that x > y if and only if u(x) > u(y). Clearly, if the preference relation (X, >) admits a continuous utility representation, then the relation > is negatively transitive and the corresponding preference-indifference relation [??] is a continuous complete preordering. Conversely, it has been shown (for References, see Notes 1.1), that every continuous complete preordering on a connected, separable topological space X (or a topological space with a countable base of open sets) possesses a Paretian utility. Since we shall use a continuous representation of a preference relation only incidentally in this book (proof of Theorem 3 in Chapter 4.3) we give no details on the construction of Paretian utilities.
PRICES p [member of] Rl
A price system p associates to every commodity h a real number ph, its price. Thus p can be considered as an element in Rl.To say that in an economy the price system p prevails means that an economic agent has to make available (at the specified date and location) the amount ph'/ph of commodity h in order to obtain one unit of commodity h' (at the specified date and location in h'). It is clear from this interpretation of the price system that the economy is assumed to work without the help of a good (money) serving as a medium of exchange. If the price system p prevails, the commodity bundle x [member of] Rl, can be exchanged into the commodity bundle x' [member of] Rl, if their values of exchange are the same, i.e., p x x = p x x'. We emphasize that one obtains interest and discount rates by comparing prices at the same location at different dates, and that one obtains exchange rates by comparing prices at the same date at different locations.
The price of a commodity may be positive (scarce commodity), zero (free commodity), or negative (noxious commodity).
BUDGET-SET β(X, w, p) [subset] Rl
If the price system p prevails, then the choice of a consumption plan in the consumption set Xa of agent a is further restricted: the exchange value p x x of the consumption plan x cannot exceed a certain amount wa, the wealth of agent a. The real number wa represents the exchange value of all possessions of agent a (real estate, cars, furniture, ..., stocks, bonds, ...). Thus, the wealth w is typically a function of the prevailing price system. It will be convenient to treat the wealth (in the theory of demand) as an independent argument.
We emphasize that the value of the labor that is made available by agent a, i.e., his labor-income, is not included in wa. The quantity of labor offered by agent a is part of his consumption plan. Consequently, given the price system, the labor-income of an agent is determined by the consumption plan he chooses.
(Continues...)
Excerpted from Core and Equilibria of a Large Economy by Werner Hildenbrand. Copyright © 1974 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Table of Contents
- Frontmatter, pg. i
- CONTENTS, pg. v
- PREFACE, pg. viii
- PART I. MATHEMATICS, pg. 1
- PART II. ECONOMICS, pg. 81
- SUMMARY OF NOTATION, pg. 234
- BIBLIOGRAPHY, pg. 236
- NAME INDEX, pg. 247
- SUBJECT INDEX, pg. 249