The existence of Walrasian equilibrium: infinitely many commodities, measure space of agents, and discontinuous preferences

Abstract

This study establishes two equilibrium existence results for large economies with infinitely many commodities. The novel results allow for nontransitive, incomplete, discontinuous, and price-dependent preferences and do not require an interiority condition on initial endowments. The first result is an existence result when the positive cone of the commodity space has a nonempty interior. The second result is an existence result under a nonsatiation condition, including the case of the empty interior of the positive cone. The second result covers infinite-dimensional commodity spaces which could not be covered before due to the interiority condition, such as the space of square integrable functions. Specifically, we employ a saturated measure space of agents to appeal to the convexifying effect of aggregation. The notion of the continuous inclusion property introduced for finite-agent economies is applied to large economies, enabling us to dispense with the continuity assumption regarding preferences. In addition, we provide examples of Walrasian equilibrium and infinite-dimensional commodity spaces newly covered by our results.