Directed search with endogenous capacity

We consider a large market with frictions. We are in a directed search type setting, sellers are capacity constrained, post prices, and buyers contact sellers taking into account price and probability of service. We aim to endogenize the sellers' capacities. To this end we formulate a three stage game, where the sellers choose their capacity in the first stage, in the second stage sellers choose their price given the capacities, and in the third stage buyers choose which firm to visit. We analyse price formation when sellers' have different capacities. A non-existence result is introduced for linear costs. We also characterize equilibrium with free entry and increasing and convex costs. In addition we derive the matching function and the social welfare function and show that the capacity chosen under free entry is efficient.