The monopolist's free boundary problem in the plane: an excursion into the economic value of private information

Abstract: The principal-agent problem is an important paradigm in economic theory for studying the value of private information: the nonlinear pricing problem faced by a monopolist is one example; others include optimal taxation and auction design. For multidimensional spaces of consumers (i.e. agents) and products, Rochet and Chone (1998) reformulated this problem as a concave maximization over the set of convex functions, by assuming agent preferences are bilinear in the product and agent parameters. This optimization corresponds mathematically to a convexity-constrained obstacle problem. The solution is divided into multiple regions, according to the rank of the Hessian of the optimizer.

Apart from four possible pathologies, if the monopolist’s costs grow quadratically with the product type we show that a smooth free boundary delineates the region where it becomes efficient to customize products for individual buyers. We give the first complete solution of the problem on square domains, and discover new transitions from unbunched to targeted and from targeted to blunt bunching as market conditions become more and more favorable to the seller.

Based on work with Cale Rankin (Monash University) and Kelvin Shuangjian Zhang (Fudan University) arxiv.org/abs/2412.15505.

analysis of PDEs

Audience: researchers in the discipline

( paper )

Comments: Chair: Marcelo Disconzi, marcelo.disconzi@vanderbilt.edu

---

Seminar In the Analysis and Methods of PDE (SIAM PDE)

Series comments: The Editorial Board of SIMA and the officers of the SIAM Activity Group on Analysis of Partial Differential Equations (SIAG/APDE) are launching the Seminar In the Analysis and Methods of PDE (SIAM PDE) to be held on the first Thursday of the month at 11:30am ET, except in August and January. The COVID-19 pandemic and consequent social distancing call for online venues of research dissemination. This webinar will serve as a way to recognize achievements in our area, and it will be expected to help promoting the standing of both SIMA and APDE. Further, these webinars should be beneficial to mathematicians who are in relatively isolated places.

This webinar is intended to continue well beyond this social distancing period. There are two type of SIAM PDE webinars: (i) Colloquium-type webinars by outstanding speakers presenting a survey of results on a topic where critical progress was recently achieved (and not necessarily published in SIMA); (ii) Webinars by outstanding authors based on some of the best articles accepted by SIMA. To launch the webinar series the initial six seminars will be of the Colloquium-type.

The talks are open to the public and, due to security reasons, all attendees have to register by following the link siam.zoom.us/webinar/register/WN_wShvDipBQw6WafNN6QO5wg

The link will contain detailed information about the talks. The registration is quick (asks only for your name and email) and, once registered, you will receive an email with the link to the webinars, which is unique to you, so please do not share that email. The registration is valid for all this webinar series, and you will receive a reminder about future talks.

Further, you will receive a follow-up email up to 7 days after each webinar, with a link to the on-demand SIAM PDE YouTube playlist recording.

Updated information and schedule will be available at sinews.siam.org/Details-Page/seminar-in-the-analysis-and-methods-of-pde

The SIAM PDE Webinar was initiated on June 2020:

By the SIMA Editors:

Robert Lipton, SIMA Editor-in-Chief (Louisiana State University, USA)

Athanasios Tzavaras (KAUST, Saudi Arabia)

By the SIAG/APDE Officers:

---