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Optimal Contract for the Principal-Agent Under Knightian Uncertainty

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Abstract

Under the Knightian uncertainty, this paper constructs the optimal principal (he)-agent (she) contract model based on the principal’s expected profit and the agent’s expected utility function by using the sublinear expectation theory. The output process in the model is provided by the agent’s continuous efforts and the principal cannot directly observe the agent’s efforts. In the process of work, risk-averse agent will have the opportunity to make external choices. In order to promote the agent’s continuous efforts, the principal will continuously provide the agents with consumption according to the observable output process after the probation period. In this paper, the Hamilton–Jacobi–Bellman equation is deduced by using the optimality principle under sublinear expectation while the smoothness viscosity condition of the principal-agent optimal contract is given. Moreover, the continuation value of the agent is taken as the state variable to characterize the optimal expected profit of the principal, the agent’s effort and the consumption level under different degrees of Knightian uncertainty. Finally, the behavioral economics is used to analyze the simulation results. The research findings are that the increasing Knightian uncertainty incurs the decline of the principal’s maximum profit; within the probation period, the increasing Knightian uncertainty leads to the shortening of probation period and makes the agent give higher effort when she faces the outside option; what’s more, after the smooth completion of the probation period for the agent, the agent’s consumption level will rise and her effort level will drop as Knightian uncertainty increasing.

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References

  1. Spear, S., Srivastava, S.: On repeated moral hazard with discounting. Rev. Econ. Stud. 54(4), 599–617 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  2. Sung, J.: Optimal contracts under adverse selection and moral hazard: a continuous-time approach. Rev. Financ. Stud. 18(3), 1021–1073 (2005)

    Article  Google Scholar 

  3. DeMarzo, P.M., Sannikov, Y.: Optimal security design and dynamic capital structure in a continuous-time agency model. J. Finance 61(6), 2681–2724 (2006)

    Article  Google Scholar 

  4. Sannikov, Y.: Games with imperfectly observable actions in continuous time. Econometrica 75(5), 1285–1329 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Sannikov, Y.: A continuous-time version of the principal-agent problem. Rev. Econ. Stud. 75(3), 957–984 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. DeMarzo, P.M., Fishman, M.J., He, Z.: Dynamic agency and the \(q\) theory of investment. J. Finance 67(6), 2295–2340 (2012)

    Google Scholar 

  7. Edmans, A., Gabaix, X., Sadzik, T., Sannikov, Y.: Dynamic CEO compensation. J. Finance 67(5), 1603–1647 (2012)

    Article  Google Scholar 

  8. Zhang, Y.: Dynamic contracting with persistent shocks. J. Econ. Theory 144(2), 635–675 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Capponi, A., Cvitanić, J., Yolcu, T.: Optimal contracting with effort and misvaluation. Math. Financ. Econ. 7(1), 93–128 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Koufopoulos, K., Kozhan, R.: Optimal insurance under adverse selection and ambiguity aversion. Econ. Theory 62(4), 659–687 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Mastrolia, T., Ren, Z.: Principal-agent problem with common agency without communication. J. Finan. Math. 9(2), 775–799 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bolton, P., Wang, N., Yang, J.: Optimal contracting, corporate finance, and valuation with inalienable human capital. J. Finance 74(3), 1363–1429 (2019)

    Article  Google Scholar 

  13. Choi, S.M.: On Sannikov’s Continuous-time Principal-agent Problem, A dissertation. University of California, Berkeley (2014)

  14. Wang, X., Lin, X., Guo, S.: The compensation incentive model for constructors based on principle-agent theory. J. Eng. Manag. 27(5), 12–17 (2013). (in Chinese)

    Google Scholar 

  15. Li, H., Li, E., Sun, Q., Wang, H.: Research on the incentive contract based on bargaining games. Syst. Eng. Theory Pract. 35(9), 2280–2287 (2015). (in Chinese)

    Google Scholar 

  16. Yang, J., Song, L., Cui, X., Li, X.: Moral hazard governance of design-build mode based on principal-agent theory. J. Eng. Manag. 32(1), 35–40 (2018). (in Chinese)

    Google Scholar 

  17. Li, E., Li, Q., Bi, R., Li, H.: Game analysis of government and enterprise innovation subsidy based on contract theory. J. Ind. Technol. Econ. 37(11), 12–18 (2018). (in Chinese)

    Google Scholar 

  18. Huang, Z., Li, G., Wang, T.: Relational contract between small and medium-sized energy service companies and banks under double moral hazard. Chin. J. Manag. Sci. 24(8), 10–17 (2016). (in Chinese)

    Google Scholar 

  19. Wang, W., Zhao, X., Zang, P., Lu, Z., Chen, M.: Incentive mechanism of closed-loop supply chain under double information asymmetry. Chin. J. Manag. Sci. 24(10), 69–77 (2016). (in Chinese)

    Google Scholar 

  20. Ma, G., Meng, W., Dai, J.: The principal-agent contract design based on fairness preference and learning effect. Syst. Eng. Theory Pract. 37(6), 1548–1556 (2017). (in Chinese)

    Google Scholar 

  21. Liu, C., Huang, W., Deng, B., Yang, C.: The decision-making of manufacturer stimulating supplier innovation considering marketing effort and innovation capability. Syst. Eng. Theory Pract. 37(12), 3040–3051 (2017). (in Chinese)

    Google Scholar 

  22. Wang, Q., Xie, C., Cui, Y.: Quality incentives based on newsvendor model under asymmetric information. Manag. Rev. 30(6), 227–237 (2018). (in Chinese)

    Google Scholar 

  23. Fei, W., Zhang, F., Yang, X.: The impact of inflation on executives equity incentive and work effort. Chin. J. Manag. Sci. https://doi.org/10.16381/j.cnki.issn1003-207x.2018.0741. (in Chinese)

  24. Jin, L.: Optimal online referral strategy and incentive mechanism in offline to online (O2O) supply chain. Manag. Rev. 31(5), 242–253 (2019). (in Chinese)

    Google Scholar 

  25. Peng, S.: Theory, methods and meaning of nonlinear expectation theory. Sci. Sin. (Math.) 47(10), 1223–1254 (2017). (in Chinese)

    Article  Google Scholar 

  26. Chen, Z., Epstein, L.: Ambiguity risk and asset returns in continuous time. Econometrica 70(4), 1403–1443 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Fei, W.: Optimal portfolio choice based on \(\alpha \)-MEU under ambiguity. Stoch. Models 25(3), 455–482 (2009)

    MathSciNet  MATH  Google Scholar 

  28. Fei, W., Chen, C., Liang, Y.: Optimal consumption-portfolio and retirement problem with disutility under Knightian uncertainty. Chin. J. Appl. Probab. Stat. 29(1), 53–63 (2013). (in Chinese)

    MathSciNet  MATH  Google Scholar 

  29. Meng, W., Zang, J., Zang, H.: Asymmetric relationship between monetary policy and credit quality in Knight uncertainty environment numerical analysis based on the models of default probability. Syst. Eng. 33(5), 17–24 (2015). (in Chinese)

    Google Scholar 

  30. Gong, X., Yang, S., Hu, J., Zang, N.: Non-linear expectation theory and risk measurement based on model ambiguity. Econ. Res. J. 50(11), 133–147 (2015). (in Chinese)

    Google Scholar 

  31. Gao, Y.: The impacts of the supplier’s optimism levels on pricing decisions under Knightian uncertainty. Syst. Eng. Theory Pract. 37(9), 2297–2305 (2017). (in Chinese)

    Google Scholar 

  32. Fei, W., Yang, S., Liang, Y.: Continuous-time contracting problems with one-sided limited commitment under Knightian uncertainty. J. Univ. Sci. Technol. China 50(2), 110–119 (2020). (in Chinese)

    Google Scholar 

  33. Fei, C., Yu, P., Fei, W., Yan, L.: Dynamics of contract design with moral hazards under Knightian uncertainty. J. Manag. Sci. China 22(6), 86–96 (2019). (in Chinese)

    Google Scholar 

  34. Fei, C., Yu, P., Fei, W., Yang, X., Yan, L.: Contract design with Knightian uncertainty and adverse selection. Syst. Eng. Theory Pract. 40(9), 2302–2313 (2020). (in Chinese)

    Google Scholar 

  35. Fei, W., Fei, C.: Optimal stochastic control and optimal consumption and portfolio with \(G\)-Brownian motion (2013). arxiv:1309.0209v1

  36. Peng, S., Song, Y., Zhang, J.: A complete representation theorem for G-martingales. Stoch. Int. J. Probab. Stoch. Process. 86(4), 609–631 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Peng, S.: Nonlinear Expectations and Stochastic Calculus Under Uncertainty. Springer, Berlin (2019)

    Book  MATH  Google Scholar 

  38. Wang, C.: Termination of dynamic contracts in an equilibrium labor market model. J. Econ. Theory 146(1), 74–110 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Financial Engineering, Anhui Polytechnic University, Wuhu, 241000, Anhui, China

    Kun-Lun Wang & Wei-Yin Fei

  2. Glorious Sun School of Business and Management, Donghua University, Shanghai, 200051, China

    Chen Fei

Authors
  1. Kun-Lun Wang
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  2. Chen Fei
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  3. Wei-Yin Fei
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Correspondence to Wei-Yin Fei.

Additional information

This research was supported by the National Natural Science Foundation of China (No. 71571001).

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Wang, KL., Fei, C. & Fei, WY. Optimal Contract for the Principal-Agent Under Knightian Uncertainty. J. Oper. Res. Soc. China 8, 637–654 (2020). https://doi.org/10.1007/s40305-020-00316-7

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  • DOI: https://doi.org/10.1007/s40305-020-00316-7

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