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$C^{1,1}$ regularity for principal-agent problems

Robert J. McCann, Cale Rankin, Kelvin Shuangjian Zhang

Abstract

We prove the interior $C^{1,1}$ regularity of the indirect utilities which solve a subclass of principal-agent problems originally considered by Figalli, Kim, and McCann. Our approach is based on construction of a suitable comparison function which, essentially, allows one to pinch the solution between parabolas. The original ideas for this proof arise from an earlier, unpublished, result of Caffarelli and Lions for bilinear preferences which we extend here to general quasilinear benefit functions. We give a simple example which shows the $C^{1,1}$ regularity is optimal.

$C^{1,1}$ regularity for principal-agent problems

Abstract

We prove the interior regularity of the indirect utilities which solve a subclass of principal-agent problems originally considered by Figalli, Kim, and McCann. Our approach is based on construction of a suitable comparison function which, essentially, allows one to pinch the solution between parabolas. The original ideas for this proof arise from an earlier, unpublished, result of Caffarelli and Lions for bilinear preferences which we extend here to general quasilinear benefit functions. We give a simple example which shows the regularity is optimal.
Paper Structure (8 sections, 5 theorems, 77 equations, 1 figure)

This paper contains 8 sections, 5 theorems, 77 equations, 1 figure.

Table of Contents

  1. Setting and statement of results
  2. Economic model
  3. Mathematical formulation
  4. The simplest case; an exposition of Caffarelli and Lions's result
  5. Background material: convexification and localization
  6. Proof of Theorem \ref{['thm:main']} assuming a key lemma
  7. Proof of Lemma \ref{['lem:main']}: Section containment and energy estimate
  8. Proof of Lemma \ref{['lem:transformations']}

Key Result

Theorem 3

Assume $b$ satisfies B1,B2,B3. Assume that $c$ is uniformly $b^*$-convex and $\mu = f \ dx$ where $f \in C^{0,1}(\mathcal{X})$ satisfies $0 < \lambda \leq f(x)$. Then the solution of the principal-agent problem, $u$, satisfies $u \in C^{1,1}_{\text{loc}}(\mathcal{X})$ .

Figures (1)

  • Figure 1: Two geometric arguments. In (a) we illustrate how the estimate $\left\lvert \frac{h}{2r}e_1 - y_{*} \right\rvert \leq \varepsilon |y_{*}|^2$ yields the esimate $\sin \theta \leq \varepsilon |y_{*}|$ (recall trigonometry). In (b) we illustrate that if the normal, $\hat{n}$, to a support plane to the convex body $S$ makes angle $\sin \theta \leq Cr$ with the $e_1$ axis $S \subset \{x ; x^1 \leq r + (\sin\theta)\text{diam}(\mathcal{X}) \leq Kr\}$.

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Remark 4
  • Remark 5
  • Theorem 6
  • proof
  • Lemma 7
  • Theorem 8
  • Lemma 9: Geometry of a carefully chosen trial function
  • ...and 3 more