Intertemporal Price Discrimination: Structure and Computation of Optimal Policies

We study a firm's optimal pricing policy under commitment. The firm's objective is to maximize its long-term average revenue given a steady arrival of strategic customers. In particular, customers arrive over time, are strategic in timing their purchases, and are heterogeneous along two dimensions: their valuation for the firm's product and their willingness to wait before purchasing or leaving. The customers' patience and valuation may be correlated in an arbitrary fashion. For this general formulation, we prove that the firm may restrict attention to cyclic pricing policies, which have length, at most, twice the maximum willingness to wait of the customer population. To efficiently compute optimal policies, we develop a dynamic programming approach that uses a novel state space that is general, capable of handling arbitrary problem primitives, and that generalizes to finite horizon problems with nonstationary parameters. We analyze the class of monotone pricing policies and establish their suboptimality in general. Optimal policies are, in a typical scenario, characterized by nested sales, where the firm offers partial discounts throughout each cycle, offers a significant discount halfway through the cycle, and holds its largest discount at the end of the cycle. We further establish a form of equivalence between the problem of pricing for a stream of heterogeneous strategic customers and pricing for a pool of heterogeneous customers who may stockpile units of the product.