Equilibrium Stability and Uniqueness with a Large Number of Commodities and Patient Consumers

Abstract

We show that a large effective number of commodities can be a source of equilibrium stability and uniqueness: expanding substitution opportunities strengthens aggregate substitution effects. We study finite dated-commodity exchange economies obtained by truncating a countably infinite-horizon environment with discounted, additively separable utilities. In this setting, the effective number of commodities is the discounted count of dated commodities, so greater patience makes distant commodities more relevant. With an appropriate normalization, equilibrium substitution effects accumulate at the rate of the effective number of commodities. When a preference diversification condition holds, equilibrium income effects grow at a lower rate. The condition is satisfied, for example, when agents have sparse or localized taste differences across commodities, or when their taste profiles become sufficiently heterogeneous as the commodity space expands. Hence, whenever the effective number of commodities is sufficiently large, every equilibrium is locally t\^atonnement stable, which in turn implies equilibrium uniqueness.