A Note on 'The Limits of Price Discrimination' by Bergemann, Brooks, and Morris

TL;DR

This paper critically examines the analysis of third-degree price discrimination by Bergemann et al. (2015), identifying a flaw in their claim about market segmentation equivalence and proposing an alternative definition.

Contribution

It corrects a key assumption in previous work on price discrimination and offers a revised framework for understanding market segmentation with deterministic pricing.

Findings

Counterexamples show the original claim is incorrect under the given definition.

An alternative definition of market segmentation is proposed to address the issue.

Implications of the new definition affect the main results of prior analysis.

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.