Continuous-Time Principal-Agent Problems with Hidden Action: The Weak Formulation

We consider the problem of optimal contracts in continuous time, when the agent’s actions are unobservable by the principal. We apply the stochastic maximum principle to give necessary conditions for optimal contracts, both for the case of the utility from the payoff being separable and not separable from the cost of the agent’s effort. The necessary conditions are shown also to be sufficient for the case of quadratic cost and separable utility, for general utility functions. The solution for the latter case is almost explicit, and the optimal contract is a function of the final outcome only, the fact which was known before for exponential and linear utility functions. For general non-separable utility functions, sufficient conditions are hard to establish, but we suggest a way to check sufficiency using non-convex optimization. Unlike previous work on the subject, we use the weak formulation both for the agent’s problem and the principal’s problem, thus avoiding tricky measurability issues.