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 at a market price. The more intensely the buyer searches, the more likely he will find the good in the market, whereas 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: depending on the parameters, the local monopolist either chooses the monopoly price with probability one or he randomizes over a set of prices with the monopoly price as the upper bound of the support. Interestingly, a higher market price can make it more likely that the local monopolist prices below the monopoly level