Equilibria in Incomplete Assets Economies with Infinite Dimensional Spot Markets

The paper studies the two period incomplete markets model where assets are claims on state contingent commodity bundles and there are no bounds on portfolio trading. The important results on the existence of equilibrium in this model assume that there is a finite number of commodities traded in each spot market and that preferences are given by smooth utility functions. With these assumptions an equilibrium exists outside an exceptional set of assets structures and initial endowments. The present paper extends these results by allowing for general infinite dimensional commodity spaces in each spot market. These include all the important commodity spaces studied in the literature on the existence of Walrasian equilibrium in each spot market the consumption sets are the positive cone of an arbitrary locally solid Riesz space or of an ordered topological vector space with order unit or of a locally solid Riesz space with quasi-interior point. The paper establishes that even with our very general commodity spaces there exists an equilibrium for a very dense set of assets structures. Our approach is in the main convex analytic and the results do not require that preferences be smooth or complete or transitive. The typical situation in infinite dimensional commodity spaces does not readily allow for the kind of differential analysis and smoothness assumptions used in the finite dimensional setting. In the general settings that we study it seems that one is restricted to convex analytic techniques and assumptions. Therefore, the concepts and techniques studied in this paper also have important finite dimensional applications.