The Max-Convolution Approach to Equilibrium Models with Indivisibilities

This paper studies a competitive market model for trading indivisible commodities. Commodities can be desirable or undesirable. Agents' preferences depend on the bundle of commodities and the quantity of money they hold. We assume that agents have quasi-linear utilities in money. Using the max-convolution approach, we demonstrate that the market has a Walrasian equilibrium if and only if the potential market value function is concave with respect to the total initial endowment of commodities. We then identify sufficient conditions on each individual agent's behavior. In particular, we introduce a class of new utility functions, called the class of max-convolution concavity preservable utility functions. This class of utility functions covers both the class of functions which satisfy the gross substitutes condition of Kelso and Crawford (1982), or the single improvement condition, or the no complementarities condition of Gul and Stacchetti (1999), and the class of discrete concave functions of Murota and Shioura (1999).