# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020084

## The optimal solution to a principal-agent problem with unknown agent ability

 1 School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, Kent Street, Bentley, Perth, Western Australia 6102 2 School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China 3 School of Management and Economics, University of Electronic Science and Technology of China, No.2006, Xiyuan Avenue, West Hi-Tech Zone, Chengdu 611731, China

* Corresponding author: Rui Li

Received  September 2019 Revised  February 2020 Published  April 2020

Fund Project: This work is supported by the National Natural Science Foundation of China (No.11871302) and the Australian Research Council for the research

We investigate a principal-agent model featured with unknown agent ability. Under the exponential utilities, the necessary and sufficient conditions of the incentive contract are derived by utilizing the martingale and variational methods, and the solutions of the optimal contracts are obtained by using the stochastic maximum principle. The ability uncertainty reduces the principal's ability of incentive provision. It is shown that as time goes by, the information about the ability accumulates, giving the agent less space for belief manipulation, and incentive provision will become easier. Namely, as the contractual time tends to infinity (long-term), the agent ability is revealed completely, the ability uncertainty disappears, and the optimal contracts under known and unknown ability become identical.

Citation: Chong Lai, Lishan Liu, Rui Li. The optimal solution to a principal-agent problem with unknown agent ability. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020084
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(a) The evolution of the agent's consumption over time $t$ (b) Reduction in the principal's dividend over time $t$
Comparison of the optimal consumption and dividend under known and unknown ability
 Known ability Unknown ability Consumption $c^N=\mu M-\frac{1}{\lambda}\left[\ln k+\ln(-q)\right]$ $c^{un}=\mu M-\frac{1}{\lambda}\big[\ln {k^T(t)}+\ln(-q)\big]$ Dividend $d^N=ry-\frac{1}{\lambda}\big[K(t)+\ln r -\ln(-q)\big]$ $d^{un}=ry-\frac{1}{\lambda}\big[K_1(t)+\ln r -\ln(-q)\big]$
 Known ability Unknown ability Consumption $c^N=\mu M-\frac{1}{\lambda}\left[\ln k+\ln(-q)\right]$ $c^{un}=\mu M-\frac{1}{\lambda}\big[\ln {k^T(t)}+\ln(-q)\big]$ Dividend $d^N=ry-\frac{1}{\lambda}\big[K(t)+\ln r -\ln(-q)\big]$ $d^{un}=ry-\frac{1}{\lambda}\big[K_1(t)+\ln r -\ln(-q)\big]$
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