The monopolist's free boundary problem in the plane

TL;DR

The paper delivers a comprehensive variational and PDE-based analysis of the multidimensional Monopolist problem on convex planar domains, focusing on the free boundary between bunched and unbunched consumers. By combining variational inequalities, Alexandrov calculus, Legendre duality, and a novel leafwise (ray-based) coordinate system, the authors classify the interior and boundary behavior of the optimizer, establish boundary regularity, and relate the free boundary to an obstacle problem. In the plane, they prove that leafwise regions either extend to the boundary or satisfy a Neumann-type condition, and they provide a complete, bifurcation-driven description of the square-domain solution, revealing transitions from no bunching to targeted and then blunt bunching as the square is shifted away from the origin. These results hinge on the Monge–Ampère framework, obstacle problem regularity, and convex-order localization, offering sharp structural and regularity insights with potential applicability to higher-dimensional monopolist models and related free-boundary problems.

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 Choné's example on the family of squares $Ω= (a,a+1)^2$, 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--Ampére equation, the obstacle problem, and localization for measures in convex-order.

Figures (5)

• Figure 1: Bifurcation to targeted then blunt bunching (Thm \ref{['thm:description-on-square']}) as distance $a\ge0$ of $\Omega=(a,a+1)^2$
• Figure 2: Illustrates the constructions in the proof of Proposition \ref{['prop:neumann-sign']}. Subfigure (A) shows a cross-section of the minimizer $u$, its support $p_{x_0}$, and the perturbation $\hat{u}_t$. Subfigure (B) illustrates that because $\Omega_{t} \subset \{x^1 \geq -t\}$ and $\Omega$ is strictly convex with outer normal $\mathbf{n} = e_1$ at $0$ we have $\overline{\Omega_t} \rightarrow \{0\}$ in the Hausdorff distance.
• Figure 3: An example of the original section and the tilted section. The tilted section is now disjoint from boundary portion $P_{-d}$. The trade off is it may leave the slab $S_{\xi,r}$. Nevertheless $\bar{S}$ is contained in the slightly larger slab $S_{\xi,2r}$.
• Figure 4: Geometry of the constructed set $A$. Note apriori (though not expected) there may be errant leaves such as $\tilde{x_b}$ or those between $\widetilde{\gamma(-\alpha)}$ and $\tilde{x_0}$. However we have constrained the length of such leaves as less than $9R(x_0)/8$ and, when long, their angles are constrained by the outer leaves $\widetilde{\gamma(-\alpha)}$ and $\widetilde{\gamma(\beta)}$ (which other leaves may not intersect). Thus we obtain the (crude) containment estimate indicated by dotted lines.
• Figure 5: (A) Explanation of why $t \mapsto u_1(a+1,t)$ is monotone nondecreasing when leaves make positive angle with the horizontal (i.e. have nonpositive slope). Because $x_1 \in \tilde{x_0}$, $Du(x_0) = Du(x_1)$. Then monotonicity of the gradient of a convex function implies $D_1u(x_2) \geq D_1u(x_1) = D_1u(x_0)$. Thus $t \mapsto D_1u(a+1,t)$ is nondecreasing. (B) Since $Du$ is constant along $\tilde{x_1}$, monotonicity of the gradient implies $D_1u(x) \ge D_1u(x_1)$ and $D_2u(x) \ge D_2u(x_1)$ for all $x \in T$.

Theorems & Definitions (87)

• Theorem 1.1: Partition into foliations by leaves that extend to the boundary
• Remark 1.2: An analytic interface
• Theorem 1.3: Regularity results for the free boundary
• Remark 1.4: Lipschitzianity, convexity, and smoothness
• Theorem 1.5: Blunt bunching is a symptom of a seller's market
• Corollary 1.6: Convexity of solution to, and contact set for, an obstacle problem
• Remark 1.7: Concave nondecreasing profile of stingray's tail
• Remark 1.8: Absence and ordering of blunt vs targeted bunching
• Lemma 2.1: Variational inequalities
• proof