A Note on 'The Limits of Price Discrimination' by Bergemann, Brooks, and Morris
Keita Kuwahara
TL;DR
This note critically revisits Bergemann, Brooks, and Morris's analysis of third-degree price discrimination, showing that their Proposition 2 is not generally valid under the original segmentation definition. It provides concrete counterexamples with binary and ternary valuation sets to demonstrate the failure, and then proposes revised segmentation definitions to restore a direct-segmentation correspondence. Despite the counterexamples, the authors argue that the main welfare result—the Surplus Triangle—remains practically applicable, and they delineate precise conditions under which direct segmentation can replicate the full set of feasible surplus pairs. The work clarifies the role of segmentation definitions in the analysis and offers a path to consistent interpretation of the model’s welfare implications in applications.
Abstract
This note revisits the analysis of third-degree price discrimination developed by Bergemann et al. (2015), which characterizes the set of consumer-producer surplus pairs that can be achieved through market segmentation. This was proved by means of market segmentation with random prices, but it was claimed that any segmentation with possibly random pricing has a corresponding direct segmentation, where a deterministic price is charged in each market segment. However, the latter claim is not correct under the definition of market segmentation given in the paper, and we provide counterexamples. We then propose an alternative definition to resolve this issue and examine the implications of the difference between the two definitions in terms of the main result of their paper.