The monopolist's free boundary problem in the plane

TL;DR

This paper analyzes the Monopolist's free boundary problem in the plane, providing a full solution for square domains and classifying boundary regularity and bifurcations in consumer-product allocation.

Contribution

It offers the first complete solution to a classical economic problem on square domains, detailing boundary structure, regularity, and bifurcation phenomena.

Findings

The product allocation map is Lipschitz up to most of the boundary.

Interior free boundary can only fail to be smooth in four specific ways.

Bifurcations from targeted to blunt bunching occur as the domain shifts away from the origin.

Abstract

We study the Monopolist's problem with a focus on the free boundary separating bunched from unbunched consumers, especially in the plane, and give a full description of its solution for the family of square domains $\{(a,a+1)^2\}_{a \ge 0}$. The Monopolist's problem is fundamental in economics, yet widely considered analytically intractable when both consumers and products have more than one degree of heterogeneity. Mathematically, the problem is to minimize a smooth, uniformly convex Lagrangian over the space of nonnegative convex functions. What results is a free boundary problem between the regions of strict and nonstrict convexity. Our work is divided into three parts: a study of the structure of the free boundary problem on convex domains in $\mathbf{R}^n$ showing that the product allocation map remains Lipschitz up to most of the fixed boundary and that each bunch extends to this boundary; a proof in $\mathbf{R}^2$ that the interior free boundary can only fail to be smooth in one of four specific ways (cusp, high frequency oscillations, stray bunch, nontranversal bunch); and, finally, the first complete solution to Rochet and Chon\'e's example on the family of squares $\Omega = (a,a+1)^2$, where we discover bifurcations first to targeted and then to blunt bunching as the distance $a \ge 0$ to the origin is increased. We use techniques from the study of the Monge--Amp\'ere equation, the obstacle problem, and localization for measures in convex-order.