Monopoly Pricing, Optimal Randomization, and Resale

Would a seller set non-market clearing prices and worry about resale that results from randomization and reduces inefficiency? We show that the optimal mechanism for selling a given quantity of a homogeneous good involves rationing if and only if the revenue function is convex at this quantity. With vertically differentiated goods, like seats in an arena or hotels on a booking platform, the optimal selling mechanism involves conflating goods of differing quality, rationing, and opaque pricing. Although resale harms the seller, and possibly consumers, the optimal selling mechanism involves non-market clearing prices if resale is unavoidable but not too effective