A duality and free boundary approach to adverse selection

TL;DR

This paper introduces a duality framework for analyzing adverse selection in nonlinear pricing, transforming the problem into a free boundary problem that provides new analytical insights into market segmentation.

Contribution

It develops a duality approach to adverse selection problems, enabling the reduction to a free boundary problem and offering the first analytical description of a specific market segment.

Findings

Duality reduces complex pricing problems to simpler dual problems.

The free boundary approach characterizes market segments analytically.

Application to Rochet-Choné model provides new insights into multidimensional adverse selection.

Abstract

Adverse selection is a version of the principal-agent problem that includes monopolist nonlinear pricing, where a monopolist with known costs seeks a profit-maximizing price menu facing a population of potential consumers whose preferences are known only in the aggregate. For multidimensional spaces of agents and products, Rochet and Chon\'e (1998) reformulated this problem to a concave maximization over the set of convex functions, by assuming agent preferences combine bilinearity in the product and agent parameters with a quasilinear sensitivity to prices. We characterize solutions to this problem by identifying a dual minimization problem. This duality allows us to reduce the solution of the square example of Rochet-Chon\'e to a novel free boundary problem, giving the first analytical description of an overlooked market segment.