The structure of competitive equilibria in an assignment market

We study the structure of the set of competitive equilibria in a generalized assignment market. When all indivisible goods are homogeneous, it holds, called non-simultaneous multiplicity, that if there are multiple competitive prices, the equilibrium quantity supplied is unique; equivalently, if there are multiple equilibrium quantities, the competitive price is unique. We show that even if we allow commodity differentiation, the non-simultaneous multiplicity holds separately for each type of an indivisible good. Based on this result, we can evaluate the sizes of the sets of competitive prices and quantities for each good. As an application, we give a sufficient condition for the set of competitive prices to shrink to a unique price when markets are large and dense. We also argue that it would be difficult to extend the non-simultaneous multiplicity result to a market model where each buyer may demand more than one unit of an indivisible good.