# American Institute of Mathematical Sciences

September  2021, 17(5): 2579-2605. 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  September 2021 Early access  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 and Management Optimization, 2021, 17 (5) : 2579-2605. doi: 10.3934/jimo.2020084
##### 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. [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. [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. [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. [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. [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. [7] D. Fudenberg and L. Rayo, Training and effort dynamics in apprenticeship, American Economic Review, 109 (2019), 3780-3812. [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. [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. [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. [11] B. Holmstrom and P. Milgrom, Aggregation and linearity in the provision of intertemporal incentives, Econometrica, 55 (1987), 303-328.  doi: 10.2307/1913238. [12] H. A. Hopenhayn and A. Jarque, Moral hazard and persistence, Ssrn Electronic Journal, 7 (2007), 1-32.  doi: 10.2139/ssrn.2186649. [13] J. Hörner and L. Samuelson, Incentives for experimenting agents, The RAND Journal of Economics, 44 (2013), 632-663. [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. [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. [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. [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. [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. [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. [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. [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. [22] N. Williams, On dynamic principal-agent problems in continuous time, working paper, University of Wisconsin, Madison, (2009). [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. [24] T.-Y. Wong, Dynamic agency and endogenous risk-taking, Management Science, 65 (2019), 4032-4048. [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.

show all references

##### 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. [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. [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. [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. [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. [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. [7] D. Fudenberg and L. Rayo, Training and effort dynamics in apprenticeship, American Economic Review, 109 (2019), 3780-3812. [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. [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. [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. [11] B. Holmstrom and P. Milgrom, Aggregation and linearity in the provision of intertemporal incentives, Econometrica, 55 (1987), 303-328.  doi: 10.2307/1913238. [12] H. A. Hopenhayn and A. Jarque, Moral hazard and persistence, Ssrn Electronic Journal, 7 (2007), 1-32.  doi: 10.2139/ssrn.2186649. [13] J. Hörner and L. Samuelson, Incentives for experimenting agents, The RAND Journal of Economics, 44 (2013), 632-663. [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. [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. [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. [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. [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. [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. [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. [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. [22] N. Williams, On dynamic principal-agent problems in continuous time, working paper, University of Wisconsin, Madison, (2009). [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. [24] T.-Y. Wong, Dynamic agency and endogenous risk-taking, Management Science, 65 (2019), 4032-4048. [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.
(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 $d^N=ry-\frac{1}{\lambda}\big[K(t)+\ln r -\ln(-q)\big]$ $d^{un}=ry-\frac{1}{\lambda}\big[K_1(t)+\ln r -\ln(-q)\big]$
 [1] Giulia Cavagnari, Antonio Marigonda, Benedetto Piccoli. Optimal synchronization problem for a multi-agent system. Networks and Heterogeneous Media, 2017, 12 (2) : 277-295. doi: 10.3934/nhm.2017012 [2] Richard Carney, Monique Chyba, Chris Gray, George Wilkens, Corey Shanbrom. Multi-agent systems for quadcopters. Journal of Geometric Mechanics, 2022, 14 (1) : 1-28. doi: 10.3934/jgm.2021005 [3] Rui Li, Yingjing Shi. Finite-time optimal consensus control for second-order multi-agent systems. Journal of Industrial and Management Optimization, 2014, 10 (3) : 929-943. doi: 10.3934/jimo.2014.10.929 [4] Brendan Pass. Multi-marginal optimal transport and multi-agent matching problems: Uniqueness and structure of solutions. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1623-1639. doi: 10.3934/dcds.2014.34.1623 [5] Urszula Ledzewicz, Omeiza Olumoye, Heinz Schättler. On optimal chemotherapy with a strongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth. Mathematical Biosciences & Engineering, 2013, 10 (3) : 787-802. doi: 10.3934/mbe.2013.10.787 [6] Hongwei Li, Yuvraj Gajpal, C. R. Bector. A survey of due-date related single-machine with two-agent scheduling problem. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1329-1347. doi: 10.3934/jimo.2019005 [7] Dieter Armbruster, Christian Ringhofer, Andrea Thatcher. A kinetic model for an agent based market simulation. Networks and Heterogeneous Media, 2015, 10 (3) : 527-542. doi: 10.3934/nhm.2015.10.527 [8] Wei-Chieh Chen, Bogdan Kazmierczak. Traveling waves in quadratic autocatalytic systems with complexing agent. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 1827-1842. doi: 10.3934/dcdsb.2020364 [9] Onur Şimşek, O. Erhun Kundakcioglu. Cost of fairness in agent scheduling for contact centers. Journal of Industrial and Management Optimization, 2022, 18 (2) : 873-896. doi: 10.3934/jimo.2021001 [10] Zheng Dou, Shaoyong Lai. Optimal contracts and asset prices in a continuous-time delegated portfolio management problem. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022083 [11] Seung-Yeal Ha, Dohyun Kim, Jaeseung Lee, Se Eun Noh. Emergent dynamics of an orientation flocking model for multi-agent system. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2037-2060. doi: 10.3934/dcds.2020105 [12] Holly Gaff. Preliminary analysis of an agent-based model for a tick-borne disease. Mathematical Biosciences & Engineering, 2011, 8 (2) : 463-473. doi: 10.3934/mbe.2011.8.463 [13] Max-Olivier Hongler, Roger Filliger, Olivier Gallay. Local versus nonlocal barycentric interactions in 1D agent dynamics. Mathematical Biosciences & Engineering, 2014, 11 (2) : 303-315. doi: 10.3934/mbe.2014.11.303 [14] Nadia Loy, Andrea Tosin. Boltzmann-type equations for multi-agent systems with label switching. Kinetic and Related Models, 2021, 14 (5) : 867-894. doi: 10.3934/krm.2021027 [15] Zhongqiang Wu, Zongkui Xie. A multi-objective lion swarm optimization based on multi-agent. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022001 [16] Mei Luo, Jinrong Wang, Yumei Liao. Bounded consensus of double-integrator stochastic multi-agent systems. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022088 [17] Prateek Kunwar, Oleksandr Markovichenko, Monique Chyba, Yuriy Mileyko, Alice Koniges, Thomas Lee. A study of computational and conceptual complexities of compartment and agent based models. Networks and Heterogeneous Media, 2022, 17 (3) : 359-384. doi: 10.3934/nhm.2022011 [18] Ailing Shi, Xingyi Li, Zhongfei Li. Optimal portfolio selection with life insurance under subjective survival belief and habit formation. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022051 [19] Zhiyong Sun, Toshiharu Sugie. Identification of Hessian matrix in distributed gradient-based multi-agent coordination control systems. Numerical Algebra, Control and Optimization, 2019, 9 (3) : 297-318. doi: 10.3934/naco.2019020 [20] Tyrone E. Duncan. Some partially observed multi-agent linear exponential quadratic stochastic differential games. Evolution Equations and Control Theory, 2018, 7 (4) : 587-597. doi: 10.3934/eect.2018028

2020 Impact Factor: 1.801