Dynamic Moral Hazard with Adverse Selection - A Pontryagin Approach

We study the optimal incentive scheme for a long-term project with both moral hazard and adverse selection. The moral hazard issue is due to the fact that the agent's effort, which increases the arrival rate of a Poisson process, is not observable by the principal. In addition, the agent's effort cost, which needs to be reimbursed by the principal, is also the agent's private information. This gives rise to the adverse selection problem. The principal needs to design the optimal menu of contracts, each of which is chosen by the agent with a specific effort cost. We fully characterize the optimal menu in the case of two types of agents. Specifically, the agent with a lower cost is offered a probation contract, which confirms the agent's type if there is an arrival during a probation period; the agent with the higher cost is offered a sign-on-bonus contract with an immediate direct initial payment. We then explore the more general case with continuous types of agents. In particular, we provide an easy-to-compute upper bound on the principal's utility. The upper bound computation also yields a feasible menu of probation and sign-on-bonus contracts, and the corresponding lower bound it generates. We further provide a condition which can be used to verify whether the upper and lower bounds coincide, implying the optimality of our feasible menu of contracts. Numerical studies confirm that the verification condition almost always holds for commonly used probability distributions of the effort cost