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

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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

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

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Publisher
Springer-Verlag
Copyright
Copyright © 2009 by Springer Science+Business Media, LLC
Subject
Mathematics; Numerical and Computational Methods ; Mathematical Methods in Physics; Mathematical and Computational Physics; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-008-9050-0
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Feb 1, 2009

References

  • Dynamic security design: convergence to continuous time and asset pricing implications
    Biais, B.; Mariotti, T.; Plantin, G.; Rochet, J.C.

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