Intertemporal Price Discrimination: Structure and Computation of Optimal Policies

TLDR

We study a firm’s optimal pricing policy under commitment to maximize long‑term average revenue from a steady stream of strategic customers. Customers arrive over time, are strategic in timing purchases, and differ in valuation and patience—possibly correlated—and the authors develop a dynamic‑programming method with a novel state space that handles arbitrary primitives and generalizes to finite‑horizon, nonstationary settings. The authors prove that optimal policies can be restricted to cyclic schedules of bounded length, show that monotone pricing is generally suboptimal, and find that typical optimal policies feature nested sales with increasing discounts culminating at the cycle’s end, while also establishing an equivalence to pricing for a stockpiling customer pool. The paper was accepted by Yossi Aviv, Operations Management.

Abstract

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. This paper was accepted by Yossi Aviv, operations management.