Intertemporal Pricing via Nonparametric Estimation: Integrating Reference Effects and Consumer Heterogeneity

Abstract

We consider intertemporal pricing in the presence of reference effects and consumer heterogeneity. Our research question encompasses how to estimate heterogeneous consumer reference effects from data and how to efficiently compute the optimal pricing policy. Understanding reference effects is essential for designing pricing policies in modern retailing. Our work contributes to this area by further incorporating consumer heterogeneity under arbitrary distributions. We propose a demand model that allows arbitrary joint distributions of valuations, responsiveness to prices, and responsiveness to reference prices among consumers. To learn consumer heterogeneity from transaction data, we use a nonparametric estimation method. We formulate the pricing optimization as an infinite horizon dynamic programming problem and solve it by applying a modified policy iteration algorithm. We investigate the structure of optimal pricing policies and prove the sub-optimality of constant pricing policies even when all consumers are loss-averse according to the classical definition. Our numerical studies further show that our estimation and optimization framework improves the expected revenue of retailers via accounting for heterogeneity. We validate our model using real data from JD.com, a large E-commerce retailer, and find empirical evidence of consumer heterogeneity. In practice, ignoring consumer heterogeneity may lead to a significant loss of revenue. Furthermore, the existence of heterogeneous reference effects offers a strong motive for promotions and price fluctuations.