A Dynamic Principal-Agent Problem with One-Sided Commitment

Abstract

In this paper, we consider a principal-agent problem where the agent is allowed to quit by incurring a cost. When the current agent quits the job, the principal will hire a new one, possibly with a different type. We characterize the principal’s dynamic value function, which could be discontinuous at the boundary, as the (unique) minimal solution of an infinite dimensional system of Hamilton-Jacobi-Bellman equations, parameterized by the agent’s type. This dynamic problem is time consistent in a certain sense. Some interesting findings are worth mentioning. First, self-enforcing contracts are typically suboptimal. The principal would rather let the agent quit and hire a new one. Next, the standard contract for a committed agent may also be suboptimal because of the presence of different types of agents in our model. The principal may prefer no commitment from the agent; then, the principal can hire a cheaper one from the market at a later time by designing the contract to induce the current agent to quit. Moreover, because of the cost incurred to the agent, the principal will see only finitely many quittings. Funding: J. Zhang is supported in part by the NSF [Grants DMS-1908665 and DMS-2205972]. Z. Zhu is supported by The Hong Kong University of Science and Technology (Guangzhou) Start-Up Fund [Grant G0101000240] and the Guangzhou-HKUST(GZ) Joint Funding Program [Grant 2024A03J0630].