Applied Mathematics and Optimization

, Volume 59, Issue 1, pp 99–146

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 action, 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.


Hidden action Moral hazard Second-best optimal contracts and incentives Principal-agent problems Stochastic maximum principle Forward-backward SDEs 


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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Caltech, M/C 228-77PasadenaUSA
  2. 2.Department of Information and Systems ManagementHKUST Business SchoolKowloonHong Kong
  3. 3.Department of MathematicsUSCLos AngelesUSA

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