Buying several indivisible goods

This paper studies economies in which agents exchange indivisible goods and money. The indivisible goods are differentiated and agents have potential use for all of them. We assume that agents have quasi-linear utilities in money, have sufficient money endowments to afford any group of objects priced below their reservation values, have reservation values which are submodular and satisfy the cardinality condition. This cardinality condition requires that for each agent the marginal utility of an object depends only on the number of objects to which it is added, not on their characteristics. Under these assumptions, we show that the set of competitive equilibrium prices is a non-empty lattice and that, in any equilibrium, the price of an object is between the social value of the object and its value in its second best use.