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

• * Corresponding author: Rui Li

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.

Mathematics Subject Classification: Primary: 91B70, 91B40; Secondary: 91A26.

 Citation: • • Figure 1.  (a) The evolution of the agent's consumption over time $t$ (b) Reduction in the principal's dividend over time $t$

Table 1.  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 $d^N=ry-\frac{1}{\lambda}\big[K(t)+\ln r -\ln(-q)\big]$ $d^{un}=ry-\frac{1}{\lambda}\big[K_1(t)+\ln r -\ln(-q)\big]$
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