# Principal-Agent Problem with Common Agency without Communication

## TL;DR

This paper analyzes a continuous-time principal-agent model with multiple Principals and a common Agent, deriving equilibrium conditions, solving coupled HJB equations, and extending classical results to a linear-quadratic setting.

## Contribution

It introduces a framework for optimal contracts in common agency without communication, including existence conditions and an extension of Bernheim and Whinston's results.

## Key findings

- Optimal contracts satisfy equilibrium conditions.
- Coupled HJB equations can be solved under certain risk-neutral assumptions.
- In the linear-quadratic model, the first-best effort coincides with the aggregate effort only in the first-best case.

## Abstract

In this paper, we consider a problem of contract theory in which several Principals hire a common Agent and we study the model in the continuous time setting. We show that optimal contracts should satisfy some equilibrium conditions and we reduce the optimisation problem of the Principals to a system of coupled Hamilton-Jacobi-Bellman (HJB) equations. We provide conditions ensuring that for risk-neutral Principals, the system of coupled HJB equations admits a solution. Further, we apply our study in a more specific linear-quadratic model where two interacting Principals hire one common Agent. In this continuous time model, we extend the result of Bernheim and Whinston (1986) in which the authors compare the optimal effort of the Agent in a non-cooperative Principals model and that in the aggregate model, by showing that these two optimisations coincide only in the first best case. We also study the sensibility of the optimal effort and the optimal remunerations with respect to appetence parameters and the correlation between the projects.