An Empirical Analysis of Optimal Nonlinear Pricing

Abstract

In "continuous choice" settings, consumers decide not only on whether to purchase a product, but also on how much to purchase. As a result, firms should optimize a full price schedule rather than a single price point. This paper provides a methodology to empirically estimate the optimal schedule under multi-dimensional consumer heterogeneity. We apply our method to novel data from an educational-services firm that contains purchase-size information not only for deals that materialized, but also for potential deals that eventually failed. We show that the optimal seconddegree price discrimination (i.e., optimal nonlinear tariff) improves the firm's profit upon linear pricing by about 7.9%. That said, this second-degree price discrimination scheme only recovers 7.4% of the gap between the profitability of linear pricing (i.e., no price discrimination) and that of infeasible first degree price discrimination. We also conduct several further counterfactual analyses (i) comparing the role of demand-v.s. cost-side factors in shaping the optimal price schedule, (ii) examining third-degree price discrimination, and (iii) empirically quantifying the magnitude by which incentive-compatibility constraints impact the optimal pricing and profits. * We thank Saman Ghili for his advice on optimization methods and literature. We also thank Yufeng Huang, Yewon Kim, Tesary Lin, Wenting Yu, and various conference and seminar participants for their helpful comments. Ghili acknowledges financial support from the Yale Center for Customer Insights. We thank Wanxi Zhou for outstanding research assistance. All errors are our own.