Optimal pricing, private information and search for an outside offer

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