Optimal Mechanism for Selling Substitutes

We study a problem of a multiproduct monopolist selling substitutable goods to a buyer with unknown valuations. Under the standard distributional assumptions we find that in the optimal menu every nontrivial contract delivers some good with certainty. Using this result we apply control-theoretic tools to the case of two goods and solve a number of examples. The optimal menus generally have a simple structure and sometimes are substantially more profitable than the deterministic menus. We also extend the approach to the case when the buyer desires more than a single unit of the good.