Speaker
Description
The principal-agent problem is an important paradigm in economic theory for studying the value of private information: the nonlinear pricing problem faced by a monopolist is one example; others include optimal taxation and auction design.
For multidimensional spaces of consumers (=agents) and products, Rochet and Chone (1998) reformulated this problem as a concave maximization over the set of convex functions, by assuming agent preferences are bilinear in the product and agent parameters. This optimization corresponds mathematically to a convexity-constrained obstacle problem whose endogenous obstacle has limited regularity. The solution is divided into multiple regions, according to the rank of the Hessian of the optimizer.
If the monopolists costs grow quadratically with the product type, the map from agents to products is locally Lipschitz. If it is globally Lipschitz, we show a partially smooth free boundary delineates the region where it becomes efficient to customize products for individual buyers, and give the first complete solution of the problem on square domains. We discover new transitions from unbunched to targeted and from targeted to blunt bunching as market conditions become more and more favorable to the seller.
Based on Math. Models Methods Appl. Sci. 34 (2024) 2351-2394, J. Convex Anal. 32 (2025) 579-584, Adv. Math. 478 (2025) 110396, 22 pp, arxiv 2412.15505, and arXiv 2603.14100.