Relaxed Large Economies with Infinite-Dimensional Commodity Spaces: The Existence of Walrasian Equilibria

TL;DR

This paper extends Walrasian equilibrium theory to large economies with infinite-dimensional commodity spaces by introducing relaxed economies, removing convexity assumptions under saturation, and unifying existing results.

Contribution

It develops a comprehensive framework for relaxed economies and equilibria in infinite-dimensional spaces, generalizing classical results without convexity assumptions.

Findings

Convexity can be removed under the saturation hypothesis.

Relaxed economies admit Walrasian equilibria in infinite-dimensional spaces.

Existing results are recovered as special cases.

Abstract

Whereas "convexification by aggregation" is a well-understood procedure in mathematical economics, "convexification by randomization" has largely been limited to theories of statistical decision-making, optimal control and non-cooperative games. In this paper, in the context of classical Walrasian general equilibrium theory, we offer a comprehensive treatment of {\it relaxed economies} and their {\it relaxed Walrasian equilibria}: our results pertain to a setting with a finite or a continuum of agents, and a continuum of commodities modeled either as an ordered separable Banach space or as an $L^\infty$-space. As a substantive consequence, we demonstrate that the convexity hypothesis can be removed from the original large economy under the saturation hypothesis, and that existing results in the antecedent literature can be effortlessly recovered.