Competitive equilibrium for indivisible objects

We study an exchange economy with indivisible objects that may not be substitutes for each other, and we introduce the p-substitutability condition, a relaxation of the gross substitutes condition of Kelso and Crawford (1982), in which a parameter vector p is adopted to permit complicated types of complementarity. We prove that for any economy E, there exists a corresponding vector p^{E} such that the p^{E}-substitutability condition is sufficient to guarantee the existence of a competitive equilibrium, and that the largest competitive price of each object is equal to its contribution to the social welfare. Our approach relies on a classification result that partitions economies into disjoint similarity classes such that whenever a similarity class contains an economy with an equilibrium, each economy in this class also has an equilibrium.