Existence of Equilibrium in Infinite Economies with Production

Abstract

THE MAJOR INTEREST in the recent literature on economies has lain in the results about but finite economies that have been derived from the results proved for economies. It is thus important to find simple yet general proofs for economies. In this article we wish to provide a simple yet general proof of the existence of a competitive equilibrium in an infinite, nonstandard economy with production. The simplicity of our proof comes from the fact that nonstandard analysis can deal with large and small quantities very much as ordinary analysis deals with finite quantities.2 As a result, it is possible to follow very closely the proof of existence for an economy with a finite number of traders, such as that in G. Debreu's classic Theory of Value [6]. In fact, if one is willing to believe that nonstandard analysis permits us to manipulate quantities as claimed above, then no further knowledge of nonstandard analysis is required in order to follow the proof. As examples of the simplicity of nonstandard analysis, it may be pointed out that no analogue of the Fatou-Schmeidler lemma [7, p. 69], a fairly difficult mathematical theorem, is required; nor is it necessary to prove separately that preserves upper-semicontinuity, a proof that Aumann [1] has recently simplified, because integration in the nonstandard model consists of an infinite summation, hence an appeal to 1.9.4 of Debreu [6] suffices to establish this point. As our main objective is to obtain results about but finite economies, it is a welcome bonus to find out that no further effort is needed to obtain these desired theorems. This arises because of the following property of nonstandard analysis. Consider a sequence of real numbers {an} which tends to zero. If we could extend this sequence to the integers, it would surely be a necessary property of the values of {an} at the integers that they are all infinitely close to zero. What makes nonstandard analysis powerful is that the above line of reasoning can be reversed, so to speak. Suppose we have a sequence which