A Dynamic Principal Agent Problem with One-sided Commitment

The principal agent problem in the standard literature is typically time inconsistent, in the sense that an optimal contract may not remain optimal if the principal reconsiders the problem at a later time. Such time inconsistency issue becomes highly relevant when one or both parties do not commit to the contract. In this paper we consider a model where the current agent can quit before the expiration date, and then the principal will hire a new agent from the market, possibly with a different type. Both parties are aware of the non-commitment of the agent when considering the contract in the beginning. We shall show that the new dynamic problem is time consistent in certain sense and we characterize the principal's optimal utility through an infinite dimensional system of HJB equations, parametrized with the agent's type. However, the dynamic value function of the principal could be discontinuous on the boundary, and thus it is characterized as the minimum solution of the HJB system. Two interesting observations are worth mentioning. First, we assume there are a family of agents with different types in the market (as often in reality). It is possible that the principal could hire a cheaper agent when the current one quits, and ends up with a larger optimal utility. So, the principal may have the incentive to design a contract to encourage the agent to quit before the expiration date. Next, we assume the agent (and the principal) will bear some cost when he quits. As a consequence, the principal will see only finitely many quittings within the finite contract period.