We consider the problem of trade between a price setting party who has private information about the quality of a good and a price taker who may also have private information. We restrict attention to the case in which, under full information, it is efficient to trade only a subset of all qualities. In particular, we assume that trading a low (high) quality is inefficient when the seller (buyer) sets the price. We show that there is a unique equilibrium outcome passing Cho and Kreps (1987) “Never a Weak Best Response”. The refined outcome is always characterized by no trade, although trade would be mutually beneficial in some state of nature. This occurs: 1. Even if the price taker has more precise information than the price setting party, and 2. Even when the information received by both parties is almost perfect. Both results imply that there are inefficiencies due to price setting that are not present in standard markets with adverse selection. We find that the price setting party can always increase her profits through ex-ante delegation of the price choice to an uninformed third party. We discuss applications to professional bodies and the market for unskilled labor.