The Limits of Interval-Regulated Price Discrimination

In this paper, we study third-degree price discrimination in a model first presented by Bergemann, Brooks, and Morris [2015]. Since such price discrimination might create market segments with vastly different posted prices, we consider regulating these prices, specifically, by restricting them to lie within an interval. Given a price interval, we consider segmentations of the market where a seller, who is oblivious to the existence of such regulation, still posts prices within the price interval. We show the following surprising result: For any market and price interval where such segmentation is feasible, there is always a different segmentation that optimally transfers all excess surplus to the consumers. In addition, we characterize the entire space of buyer and seller surplus that is achievable by such segmentation, including maximizing seller surplus, and simultaneously minimizing buyer and seller surplus. A key technical challenge is that the classical segmentation method of Bergemann, Brooks, and Morris [2015] fails under price constraints. To address this, we develop three intuitive but fundamentally distinct segmentation constructions, each tailored to a different surplus objective. These constructions maintain different invariants, reflect different economic intuitions, and collectively form the core of our regulated surplus characterization.