Core-Walras Equivalence in Economies with a Continuum of Agents and Commodities

Abstract

This paper contains the following results for economies with infinite dimensional commodity spaces. (i) He establish a core-Walras equivalence theorem for economies with an atomless measure space of agents and with an ordered separable Banach commodity space whose positive cone has a non-empty norm interior. This result includes as a special case the Aumann (1964) and Hildenbrand (1974) finite dimensional theorems. (ii) We provide a counterexample which shows that the above result fails in ordered Banach spaces whose positive cone has an empty interior even if preferences are strictly convex, monotone and weak* continuous and initial endowments are strictly positive. (iii) After introducing a new assumption on preferences called "commodity pair desirability," (which is automatically satisfied whenever preferences are monotone and the positive cone of the commodity space has a non-empty interior), we establish core-Walras equivalence in any arbitrary separable Banach lattice whose positive cone may have an empty (norm) interior. (iv) We provide a proof that in some concrete spaces whose positive cone may have an empty interior, the assumption of an extremely desirable commodity or uniform properness suffices for core-Walras equivalence. Finally, (v) we indicate how our methods can be used to obtain core-Walras equivalence results for the space M(~) of measures on a compact metric space.