Optimal pricing, private information and search for an outside offer

A buyer can either buy a good at a local monopolist or search for it in the market. The more intensely the buyer searches, the more likely he will find the good in the market; if his search fails, he can still buy it from the local monopolist. We show that a buyer with a higher willingness to pay searches (weakly) more intensely. This skews the distribution of types buying at the local monopolist towards lower valuations and exerts pressure on the local monopolist to reduce his price. Despite this effect, offering the monopoly price remains weakly optimal in equilibrium.