Pricing and Signaling with Frictions

We study a large market with directed search and signaling. Each seller chooses an investment that determines the quality of the good which is the seller's private information. A seller also chooses the price of the good and the number of selling sites. After observing sellers' choices of prices and sites, but not quality, buyers choose which price to search. The sites posting the same price and the buyers searching for that price match with each other randomly. In this environment, a seller's choices of prices and sites can direct buyers' search decisions and signal quality ex-ante. After matching, a buyer also receives an imperfectly informative signal about the quality of the good and decides whether to trade at the posted price. When the latter signal received is sufficiently accurate, we prove that there is a unique equilibrium. Moreover, when the quality differential is large, the equilibrium (under private information) implements the socially efficient allocation under public information. When the quality differential is small, the equilibrium is inefficient in the quality of goods produced or/and the number of sites created. This inefficiency is caused by a conflict between the search-directing role and the signaling role of a posted price. We also compare the price-posting equilibrium with the equilibrium under bargaining. The bargaining equilibrium features efficient quality, but inefficient entry. It is superior to the price-posting equilibrium when a seller's bargaining power is intermediate and the quality differential is small