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

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.