Robert McCann, University of Toronto

The monopolist’s free boundary problem in the plane: an excursion into the economic value of private information

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 (i.e. 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. The solution is divided into multiple regions, according to the rank of the Hessian of the optimizer.
Apart from four possible pathologies,  if the monopolists costs grow quadratically with the product type we show that a smooth free boundary delineates the region where it becomes efficient to customize products for individual buyers.  We give the first complete solution of the problem on square domains, and discover new transitions from unbunched to targeted and from targeted to blunt bunching as market conditions become more and more favorable to the seller.

Leonardo Abbrescia, Georgia Tech

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Ronnie Pavlov, University of Denver

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Mark de Cataldo, Stony Brook University

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