Competitive equilibria with infinitely many commodities (II)

Abstract

In this paper we state and prove five theorems concerning the existence of competitive equilibria in production economies with finitely many “agents” and infinitely many commodities. The first theorem is an extension of a theorem of K. Arrow and G. Debreu ([l], Theorem I, p. 272). The second and third are extensions of a theorem of T. Bewley (cf. [2], Theorem 1 and 3, pp. 13 and 19). The last two are extensions of theorems of G. Debreu (cf. [3], Theorems 6.3, 6.4, pp. 94-95). Similar theorems for exchange economies were established in [7]. The need to prove theorems such as Theorems l-5 below arise in the study of resource allocation in economies which operate over infinitely many periods and in economies which operate over finitely many periods in an uncertain world in which one of a denumerable infinity of states of nature might occur. To simlify our notation we will argue as if the economies we consider are faced with the problem of allocating resources over infinitely many periods in a world of perfect certainty with respect to the state of nature. To state our results we must first introduce a lot of symbols. Let R” be the set of all n-dimensional real vectors: let