Convexity and multi-dimensional screening for spaces with different dimensions

TL;DR

This paper investigates the principal-agent problem in multi-dimensional spaces, revealing how convexity conditions influence optimal product offerings and the encoding of economic information across different dimensional settings.

Contribution

It establishes the necessity of $b$-convexity for convex reformulation and characterizes optimal product spaces when dimensions differ.

Findings

When $m > n$, extra dimensions lack independent economic information.

For $m < n$, optimal offerings are confined to a specific subset of goods.

$b$-convexity ensures the problem can be formulated as a convex maximization.

Abstract

We study the principal-agent problem. We show that $b$-convexity of the space of products, a condition which appears in a recent paper by Figalli, Kim and McCann \cite{fkm}, is necessary to formulate the problem as a maximization over a convex set. We then show that when the dimension $m$ of the space of types is larger than the dimension $n$ of the space of products, this condition implies that the extra dimensions do not encode independent economic information. When $m$ is smaller than $n$, we show that under $b$-convexity of the space of products, it is always optimal for the principal to offer goods only from a certain prescribed subset. We show that this is equivalent to offering an $m$-dimensional space of goods.