The monopolist's free boundary problem in the plane

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 portions 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, nontransversal 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. To do this, we extend the localization for measures in convex-order to accommodate potential discontinuities in the product allocation map at the fixed boundary. We also employ techniques from the study of the Monge--Amp\`ere equation and the obstacle problem