Optimal Compensation with Hidden Action and Lump-Sum Payment in a Continuous-Time Model

We consider a problem of finding optimal contracts in continuous time, when the agent’s actions are unobservable by the principal, who pays the agent with a one-time payoff at the end of the contract. We fully solve the case of quadratic cost and separable utility, for general utility functions. The optimal contract is, in general, a nonlinear function of the final outcome only, while in the previously solved cases, for exponential and linear utility functions, the optimal contract is linear in the final output value. In a specific example we compute, the first-best principal’s utility is infinite, while it becomes finite with hidden actions, which is increasing in value of the output. In the second part of the paper we formulate a general mathematical theory for the problem. We apply the stochastic maximum principle to give necessary conditions for optimal contracts. Sufficient conditions are hard to establish, but we suggest a way to check sufficiency using non-convex optimization. ∗Research supported in part by NSF grants DMS 04-03575, DMS 06-31298 and DMS 06-31366, and through the Programme ”GUEST” of the National Foundation For Science, Higher Education and Technological Development of the Republic of Croatia. We are solely responsible for any remaining errors, and the opinions, findings and conclusions or suggestions in this article do not necessarily reflect anyone’s opinions but the authors’. We are grateful to the editor and the anonymous referees for helpful suggestions that improved the exposition of the paper. †Corresponding Author; Caltech, M/C 228-77, 1200 E. California Blvd. Pasadena, CA 91125. Ph: (626) 395-1784. E-mail: [email protected] ‡Department of Information and Systems Management, HKUST Business School , Clear Water Bay, Kowloon , Hong Kong. Ph: +852 2358-7731. Fax: +852 2359-1908. E-mail: [email protected]. §Department of Mathematics, USC, 3620 S Vermont Ave, MC 2532, Los Angeles, CA 90089-1113. Ph: (213) 740-9805. E-mail: [email protected].