The monopolist's free boundary problem in the plane
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910917593137152 |
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| author | McCann, Robert J. Rankin, Cale Zhang, Kelvin Shuangjian |
| author_facet | McCann, Robert J. Rankin, Cale Zhang, Kelvin Shuangjian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15505 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The monopolist's free boundary problem in the plane McCann, Robert J. Rankin, Cale Zhang, Kelvin Shuangjian Analysis of PDEs Optimization and Control 35R35, 49N10, 91B41 (Primary) 35Q91, 49Q22, 90B50, 91A65, 91B43 (Secondary) 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. |
| title | The monopolist's free boundary problem in the plane |
| topic | Analysis of PDEs Optimization and Control 35R35, 49N10, 91B41 (Primary) 35Q91, 49Q22, 90B50, 91A65, 91B43 (Secondary) |
| url | https://arxiv.org/abs/2412.15505 |