Optimal Contract to Induce Continued Effort

We consider a basic model of a risk-neutral principal incentivizing a risk neutral agent to exert effort in order to raise the arrival rate of a Poisson process. The effort is costly to the agent, unobservable to the principal, and affects the instantaneous arrival rate. Each arrival yields a constant revenue to the principal. The principal, therefore, devises a mechanism involving payments and a potential stopping time in order to motivate the agent to always exert effort. We formulate this problem as a stochastic optimal control model with an incentive constraint in continuous time over an infinite horizon. Although we allow payments to take general forms contingent upon past arrival times, the optimal contract has a simple and intuitive structure, which depends on whether the agent is as patient as or less patient than the principal towards future income.