Pricing and matching with frictions

Suppose that n buyers each want one unit and m sellers each have one or more units of a good. Sellers post prices, and then buyers choose sellers. In symmetric equilibrium, similar sellers all post one price, and buyers randomize. Hence, more or fewer buyers may arrive than a seller can accommodate. We call this frictions. We solve for prices and the endogenous matching function for finite n and m and consider the limit as n and m grow. The matching function displays decreasing returns but converges to constant returns. We argue that the standard matching function in the literature is misspecified and discuss implications for the Beveridge curve.