Gerard Debreu

EXISTENCE OF AN EQUILIBRIUM FOR A COMPETITIVE ECONOMY

A. Wald has presented a model of production and a model of exchange and proofs of the existence of an equilibrium for each of them. Here proofs of the existence of an equilibrium are given for an integrated model of production, exchange and consumption. In addition the assumptions made on the technologies of producers and the tastes of consumers are significantly weaker than Walds. Finally a simplification of the structure of the proofs has been made possible through use of the concept of an abstract economy, a generalization of that of a game. Introduction L. Walras [ 24 ] first formulated the state of the economic system at any point of time as the solution of a system of simultaneous equations representing the demand for goods by consumers, the supply of goods by producers, and the equilibrium condition that supply equal demand on every market. It was assumed that each consumer acts so as to maximize his utility, each producer acts so as to maximize his profit, and perfect competition prevails, in the sense that each producer and consumer regards the prices paid and received as independent of his own choices. Walras did not, however, give any conclusive arguments to show that the equations, as given, have a solution.

NEW CONCEPTS AND TECHNIQUES FOR EQUILIBRIUM ANALYSIS

Abstract : In the study of the existence of an equilibrium for a private ownership economy, one meets with the basic mathematical difficulty that the demand correspondence of a consumer may not be upper semi-continuous when his wealth equals the minimum compatible with his consumption set. One can prevent this minimum-wealth situation from ever arising by suitable assumptions on the economy; for example, it is postulated that free disposal prevails and that every consumer can dispose of a positive quantity of every commodity from his resources and still have a possible consumption. However, assumptions of this type have not been readily accepted on account of their strength, and this in spite of the simplicity that they give to the analysis. The first purpose of this study is to attempt to unify these various approaches.

Existence of competitive equilibrium

Publisher Summary This chapter discusses that the mathematical model of a competitive economy conceived as an attempt to explain the state of equilibrium reached by a large number of small agents interacting through markets. Four distinct, but closely related, approaches to the existence problem are recognized. At first, proofs of existence of an economic equilibrium were uniformly obtained by application of a fixed-point theorem of the Brouwer type or Kakutani type or by analogous arguments. This approach, which has remained of central importance to the present, is the subject of the chapter. Second, in the past decade, efficient algorithms of a combinatorial nature for the computation of an approximate economic equilibrium were developed. Third, recently, the theory of the fixed-point index of a map and the degree theory of maps were used to establish the existence of an economic equilibrium, and finally a differential process was proposed whose generic convergence to an economic equilibrium provides an alternative constructive solution of the existence problem.

Additively decomposed quasiconvex functions

Letf be a real-valued function defined on the product ofm finite-dimensional open convex setsX1, ⋯,Xm.Assume thatf is quasiconvex and is the sum of nonconstant functionsf1, ⋯,fm defined on the respective factor sets. Then everyfi is continuous; with at most one exception every functionfi is convex; if the exception arises, all the other functions have a strict convexity property and the nonconvex function has several of the differentiability properties of a convex function.We define the convexity index of a functionfi appearing as a term in an additive decomposition of a quasiconvex function, and we study the properties of that index. In particular, in the case of two one-dimensional factor sets, we characterize the quasiconvexity of an additively decomposed functionf either in terms of the nonnegativity of the sum of the convexity indices off1 andf2, or, equivalently, in terms of the separation of the graphs off1 andf2 by means of a logarithmic function. We investigate the extension of these results to the case ofm factor sets of arbitrary finite dimensions. The introduction discusses applications to economic theory.

Theoretic Models: Mathematical Form and Economic Content

The steady course on which mathematical economics has held for the past four decades sharply contrasts with its progress during the preceding century, which was marked by several major scientific accidents. One of them occurred in 1838, at the beginning of that period, with the publication of Augustin Cournot’s Recherches sur les Principes Mathematiques de la Theorie des Richesses. By its mathematical form and by its economic content, his book stands in splendid isolation in time; and in explaining its date historians of economic analysis in the first half of the nineteenth century must use a wide confidence interval.

Mathematical Economics: A classical tax-subsidy problem

Of Section 1: In the economic system 1 commodities (given quantities of which are initially available) are transformed into each other by n productionunits (the technological knowledge of which is limited) and consumed by m consumption-units (the preferences of which are represented by m satisfaction functions). Of Section 2: Given a set of values (si, * , s, * , Sm) of the individual satisfactions, the economic efficiency of this situation is defined as follows: the quantities of all the available resources are multiplied by a number such that using these new resources and the same technological knowledge as before it is still possible to achieve for the ith individual (i = 1, ***, m) a satisfaction at least equal to si . The smallest of the numbers satisfying this condition is p, the coefficient of resource utilization of the economy in this situation2 [3]; p equals one for a Pareto optimal situation, is smaller than one for a nonoptimal situation. In the last case the loss of efficiency is 1 p; a more interesting quantity (immediately derived from 1 p) is the money value of the resources which might be thrown away while still permitting the achievement of the same individual satisfactions. This latter quantity will be referred to as the economic loss associated with the situation (s, *, Si, ... , sm). This definition of the loss has been critically compared in [3] with other definitions. We will only recall here that these took as a reference an arbitrarily selected optimal state (which was, in the problems of Sections 4-5, the initial state); moreover they could associate with two states of the system, where all satisfactions were the same, two different values of the economic loss.3 The one exception to which these criticisms do not apply is the definition given by M. Allais [1] p. 638, but the latter rests on the arbitrary choice of a particular commodity. The tastes, the technology, and the available resources are fixed in this paper and p is thus a function p(s,, * ... Sm) of the individual levels of satisfaction.

Preference Functions on Measure Spaces of Economic Agents

Given a set of economic agents and a set of coalitions, a non-empty family of subsets of the first set closed under the formation of countable unions and complements, an allocation is a countably additive function from the set of coalitions to the closed positive orthant of the commodity space. To describe preferences in this context, one can either introduce a positive, finite real measure defined on the set of coalitions and specify, for each agent, a relation of preference or difference on the closed positive orthant of the commodity space, or specify, for each coalition, a relation of preference or indifference on the set of allocations. This article studies the extent to which these two approaches are equivalent.

Existence of General Equilibrium

Leon Walras provided in his Elements d’economie politique pure (1874–7) an answer to an outstanding scientific question raised by several of his predecessors. Notably, Adam Smith had asked in An Inquiry into the Nature and Causes of the Wealth of Nations (1776) why a large number of agents motivated by self-interest and making independent decisions do not create social chaos in a private ownership economy. Smith himself had gained a deep insight into the impersonal coordination of those decisions by markets for commodities. Only a mathematical model, however, could take into full account the interdependence of the variables involved. In constructing such a model Walras founded the theory of general economic equilibrium.

Economic Theory in the Mathematical Mode

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On the Preferences Characterization of Additively Separable Utility

The keynote address surveys the hystorical background for constructing scalar-valued objective functions in the late 1950’s with special attention to the characterization of additively separable utility in terms of the preferences that it represents.