Methods of Dealing with Non-Existence of Equilibrium in Economies with Indivisible Goods

Abstract

Kelso and Crawford (1982) showed the existence of competitive equilibrium in economies for which the utility functions of all consumers have the gross substitutes property. However, this condition is fairly restrictive, as it cannot encode complementarities or many other complex relationships between goods. Despite this, it was shown in 1999 by Gul and Stacchetti that the gross substitutes condition is in some sense necessary to guarantee the existence of equilibrium. Given this, and the plethora of economies for which no equilibrium exists, this thesis seeks to investigate possible methods of dealing with such economies. Two methods are proposed. The first method presents and clarifies for the reader results from Azevedo, Weyl, and White (2013) regarding the existence of equilibrium in the finite replication economy. The second method seeks to guarantee equilibrium for a wider class of economies by altering the definition of equilibrium. In particular, we define multiple notions of a "local" equilibrium, which restricts the actions consumers can take after a welfare-maximizing allocation is made. We prove the existence of the various notions of local equilibrium under certain restrictions on utility functions. This investigation provides insights into possible methods of dealing with the problem of non-existence of competitive equilibrium in economies with indivisible goods.