# 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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##### References:
 [1] T. Adrian and M. M. Westerfield, Disagreement and learning in a dynamic contracting model, The Review of Financial Studies, 22 (2009), 3873-3906.   Google Scholar [2] D. Bergemann and U. Hege, Venture capital financing, moral hazard, and learning, Journal of Banking and Finance, 22 (1998), 703-735.  doi: 10.1016/S0378-4266(98)00017-X.  Google Scholar [3] J.-M. Bismut, Conjugate convex functions in optimal stochastic control, Journal of Mathematical Analysis and Applications, 44 (1973), 384-404.  doi: 10.1016/0022-247X(73)90066-8.  Google Scholar [4] J.-M. Bismut, Duality methods in the control of densities, SIAM Journal on Control and Optimization, 16 (1978), 771-777.  doi: 10.1137/0316052.  Google Scholar [5] K. Chen, X. Wang, M. Huang and W.-K. Ching, Salesforce contract design, joint pricing and production planning with asymmetric overconfidence sales agent, Journal of Industrial and Management Optimization, 13 (2017), 873-899.  doi: 10.3934/jimo.2016051.  Google Scholar [6] J. Cvitanić, X. Wan and J. Zhang, Optimal compensation with hidden action and lump-sum payment in a continuous-time model, Applied Mathematics and Optimization, 59 (2009), 99-146.  doi: 10.1007/s00245-008-9050-0.  Google Scholar [7] D. Fudenberg and L. Rayo, Training and effort dynamics in apprenticeship, American Economic Review, 109 (2019), 3780-3812.   Google Scholar [8] M. Fujisaki, G. Kallianpur and H. Kunita, Stochastic differential equations for the non linear filtering problem, Osaka Journal of Mathematics, 9 (1972), 19-40.   Google Scholar [9] Y. Giat, S. T. Hackman and A. Subramanian, Investment under uncertainty, heterogeneous beliefs, and agency conflicts, The Review of Financial Studies, 23 (2009), 1360-1404.   Google Scholar [10] Z. He, B. Wei, J. Yu and F. Gao, Optimal long-term contracting with learning, The Review of Financial Studies, 30 (2017), 2006-2065.   Google Scholar [11] B. Holmstrom and P. Milgrom, Aggregation and linearity in the provision of intertemporal incentives, Econometrica, 55 (1987), 303-328.  doi: 10.2307/1913238.  Google Scholar [12] H. A. Hopenhayn and A. Jarque, Moral hazard and persistence, Ssrn Electronic Journal, 7 (2007), 1-32.  doi: 10.2139/ssrn.2186649.  Google Scholar [13] J. Hörner and L. Samuelson, Incentives for experimenting agents, The RAND Journal of Economics, 44 (2013), 632-663.   Google Scholar [14] J. Mirlees, The optimal structure of incentives and authority within an organization, Bell Journal of Economics, 7 (1976), 105-131.  doi: 10.2307/3003192.  Google Scholar [15] M. Mitchell and Y. Zhang, Unemployment insurance with hidden savings, Journal of Economic Theory, 145 (2010), 2078-2107.  doi: 10.1016/j.jet.2010.03.016.  Google Scholar [16] J. Prat and B. Jovanovic, Dynamic contracts when the agent's quality is unknown, Theoretical Economics, 9 (2014), 865-914.  doi: 10.3982/TE1439.  Google Scholar [17] Y. Sannikov, A continuous-time version of the principal-agent problem, The Review of Economic Studies, 75 (2008), 957-984.  doi: 10.1111/j.1467-937X.2008.00486.x.  Google Scholar [18] H. Schättler and J. Sung, The first-order approach to the continuous-time principal–agent problem with exponential utility, Journal of Economic Theory, 61 (1993), 331-371.  doi: 10.1006/jeth.1993.1072.  Google Scholar [19] K. Uğurlu, Dynamic optimal contract under parameter uncertainty with risk-averse agent and principal, Turkish Journal of Mathematics, 42 (2018), 977-992.  doi: 10.3906/mat-1703-102.  Google Scholar [20] C. Wang and Y. Yang, Optimal self-enforcement and termination, Journal of Economic Dynamics and Control, 101 (2019), 161-186.  doi: 10.1016/j.jedc.2018.12.010.  Google Scholar [21] X. Wang, Y. Lan and W. Tang, An uncertain wage contract model for risk-averse worker under bilateral moral hazard, Journal of Industrial and Management Optimization, 13 (2017), 1815-1840.  doi: 10.3934/jimo.2017020.  Google Scholar [22] N. Williams, On dynamic principal-agent problems in continuous time, working paper, University of Wisconsin, Madison, (2009). Google Scholar [23] N. Williams, A solvable continuous time dynamic principal–agent model, Journal of Economic Theory, 159 (2015), 989-1015.  doi: 10.1016/j.jet.2015.07.006.  Google Scholar [24] T.-Y. Wong, Dynamic agency and endogenous risk-taking, Management Science, 65 (2019), 4032-4048.   Google Scholar [25] J. Yong and X. Y. Zhou, Stochastic controls: Hamiltonian systems and HJB equations, vol. 43, Springer Science and Business Media, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar
(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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