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

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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.

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Authors and Affiliations

  1. Caltech, M/C 228-77, 1200 E. California Blvd., Pasadena, CA, 91125, USA

    Jakša Cvitanić

  2. Department of Information and Systems Management, HKUST Business School, Clear Water Bay, Kowloon, Hong Kong

    Xuhu Wan

  3. Department of Mathematics, USC, 3620 S. Vermont Ave., KAP 108, Los Angeles, CA, 90089-2532, USA

    Jianfeng Zhang

Authors
  1. Jakša Cvitanić
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  2. Xuhu Wan
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  3. Jianfeng Zhang
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Correspondence to Jakša Cvitanić.

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Cvitanić, J., Wan, X. & Zhang, J. Optimal Compensation with Hidden Action and Lump-Sum Payment in a Continuous-Time Model. Appl Math Optim 59, 99–146 (2009). https://doi.org/10.1007/s00245-008-9050-0

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  • DOI: https://doi.org/10.1007/s00245-008-9050-0

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