Optimal Contract for the Principal-Agent Under Knightian Uncertainty
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Optimal Contract for the Principal-Agent Under Knightian Uncertainty Kun-Lun Wang1 · Chen Fei 2 · Wei-Yin Fei1 Received: 4 October 2019 / Revised: 5 June 2020 / Accepted: 28 June 2020 © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag GmbH Germany, part of Springer Nature 2020
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
This research was supported by the National Natural Science Foundation of China (No. 71571001).
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Wei-Yin Fei [email protected] Kun-Lun Wang [email protected] Chen Fei [email protected]
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Department of Financial Engineering, Anhui Polytechnic University, Wuhu 241000, Anhui, China
2
Glorious Sun School of Business and Management, Donghua University, Shanghai 200051, China
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K.-L. Wang et al.
probation period for the agent, the agent’s consumption level will rise and her effort level will drop as Knightian uncertainty increasing. Keywords Knightian uncertainty · Principal-agent · Sublinear expectation · HJB equation · Behavioral economics Mathematics Subject Classification 91B70 · 93C39
1 Introduction The results in line with incomplete contract theory and principal-agent contract incentive mechanism that is achieved by Oliver Hart from Harvard University and Bengt Holmstrom from MIT have been widely used to solve the principal-agent contract problems, such as between state-owned enterprise managers and grassroots, company executives and employees, etc. In real life, since the information between principal and agent is often asymmetric, the
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