Extension of Stiemke's lemma and equilibrium in economies with infinite-dimensional commodity space and incomplete financial markets

Abstract

This paper analyses the general equilibrium existence problem in a two-period economy with infinite-dimensional space R L × Π L l = 1 l ∞ and incomplete financial markets. It is assumed that trading takes place in the sequence of spot markets and futures markets for securities payable in units of account. Unlimited short-selling in securities is allowed. The Extension of Stiemke;s Lemma is the Arbitrage Pricing Theory in the case ( l 1, l ∞), the present value of the securities prices at date 0 is the value of their returns over all countably infinite possible states of nature at date 1. A general equilibrium is the set of current and future prices (contingent upon uncertain events) and the set of individual plans such that all markets are cleared. The existence of such an equilibrium is proved under the following conditions: Mackey continuity, weak convexity, strict monotonicity, complete preferences, and strictly positive endowments.