Optimal contract design via relaxation: application to the problem of brokerage fee for a client with private signal

TL;DR

This paper introduces a relaxation-based method to prove the existence of optimal contracts under information asymmetry, specifically applied to brokerage fee design with private signals, applicable to broader contract problems with certain structure.

Contribution

It develops a relaxation technique for establishing optimal contracts in models with information asymmetry, extending to cases with linear drift control and compact contract spaces.

Findings

Proves existence of optimal brokerage fees in models with private signals.

Demonstrates the method's applicability to broader contract problems.

Provides a framework for handling information asymmetry in contract design.

Abstract

In this paper we show how the relaxation techniques can be used to establish the existence of an optimal contract in presence of information asymmetry. The method we illustrate was initially motivated by the problem of designing optimal brokerage fees, but it does apply to other optimal contract problems, in which (i) the agent controls linearly the drift of a diffusion process, (ii) the direct dependence of the principal's and the agent's objectives on the strategy of the agent is of a special form, and (iii) the space of admissible contracts is compact. This method is then applied to establish existence of an optimal brokerage fee in a market model with a private trading signal observed by the broker's client but not by the broker.