Optimal multi-unit mechanisms with private demands

A seller can produce multiple units of a single good. The buyer has constant marginal value for each unit she receives up to a demand, and zero marginal value for units beyond the demand. The marginal value and the demand are drawn from a distribution and are privately known to the buyer. We show that under natural regularity conditions on the distribution, the optimal (revenue-maximizing) selling mechanism is deterministic. It is a price schedule that specifies the payment based on the number of units purchased. Further, under the same conditions, the revenue as a function of the price schedule is concave, which in turn implies that the optimal price schedule can be found in polynomial time. We give a more detailed characterization of the optimal prices when there are only two possible demands.