On the Maximal Domain Theorem

The maximal domain theorem by Gul and Stacchetti (J. Econ. Theory 87 (1999), 95-124) implies that for markets with indivisible objects and sufficiently many agents, the set of gross substitutable preferences is a largest set for which the existence of a competitive equilibrium is guaranteed, and hence no relaxation of the gross substitutability can ensure the existence of a competitive equilibrium. However, we note that there is a flaw in their proof, and give an example to show that a claim used in the proof may fail to be true. We correct the proof and sharpen the result by showing that even there are only two agents in the market, if the preferences of one agent are not gross substitutable, then gross substitutable preferences can be found for another agent such that no competitive equilibrium exists. Moreover, we introduce the new notion of implicit gross substitutability, which is weaker than the gross substitutability condition and is still sufficient for the existence of a competitive equilibrium when the preferences of some agent are monotone.