Collusive Equilibria in Directed Search Models

We consider a model of directed search where the sellers are allowed to post general mechanisms. Regardless of the number of buyers and sellers, the sellers are able extract all the surplus of the buyers by introducing entry fees and making their price schedule positively sloped in the number of buyers arriving to their shops. Therefore, shopping card membership fees - similar to our entry fee component - may act as a collusion device between sellers. If there is a lower bound on the prices or there is an upper bound on the entry fees that can be charged, then the equilibrium with full rent extraction may not exist any more. However, the seller optimal equilibrium still features positive entry fees and increasing price schedules. When the constraint is on the entry fees in the seller optimal equilibrium all sellers offer auctions. This implies that if the firms can coordinate on an equilibrium they would not choose a fixed price one (studied by Burdett, Shi and Wright (2001)) or one without entry fees (as in Coles and Eeckhout (2003)). When the market becomes infinitely large, and the sellers face constraints on the prices or entry fees they can choose, then any equilibrium yields the same profits and the sellers cannot sustain collusion in equilibrium