A Dynamic Principal Agent Problem with One-sided Commitment

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 HJB equations, parametrized by the agent's type. This dynamic problem is time consistent in 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, due to the presence of different types of agents in our model. The principal may prefer no commitment from the agent, then she can hire a cheaper one from the market at a later time by designing the contract to induce the current agent to quit. Moreover, due to the cost incurring to the agent, the principal will see only finitely many quittings.