The Study Of Optimal Contracts In Continuous Time Principal-Agent Models

In order to deal with the problem of the conflict of interest and information asymmetry between principal and agent,it is important to design an effective incentive contract.Under the continuous time principal-agent model with exponential-linear structure,Holmstrom and Milgrom(1987)obtained the linear optimal contract.Based on the model of Holmstrom and Milgrom(1987),many scholars have studied the optimal contract problem for complex models.In the case where both marginal productivity and volatility are constants,Williams(2015)studied the optimal contract problems under three information structures: full information,hidden actions and hidden savings,and analyzed the influence of information friction on optimal contracts.In this thesis,we will study the optimal contract problem with linear marginal productivity,endogenous technology,unknown quality and size dependence.Specifically,we focus on the study of four aspects to the above mentioned problems.(1)In a continuous time principal-agent model,the optimal contract problem is studied when the agent’s marginal productivity is a linear decreasing function of effort.Using exponential utility and linear production technology,three different information structures: full information,hidden actions and hidden savings are considered.Applying dynamic programming principle and stochastic maximum principle,the explicit solutions of optimal contracts are found.Through the explicit solutions and numerical simulations,we analyze the influence of information friction on optimal contracts.In hidden action case,the main effect of information friction is a reduction of agent’s effort,but a smaller effect on the agent’s consumption.In hidden saving case,the intertemporal distortion of agent’s consumption almost vanishes due to the agent’s effective rate of return equals to the risk-free rate of return.Since the agent’s effort distortion is expanded,the agent’s consumption decreases substantially.Compared with the case of full information,the principal’s consumption decreases in both hidden action and hidden saving cases,and the principal’s consumption decreases more greatly in hidden savings.In our setting,the agent’s optimal effort is reduced with the decline of marginal productivity.(2)Under three different information structures,the optimal contracts are studied and the explicit solutions of the contracts are obtained when the agent’s cost is affected by endogenous technology.Under the assumption of exponential preferences,optimal contract is linear in a dynamic setting.Under full information,the agent’s effort and consumption will increase due to the choice of advanced technology.In hidden action case,information friction reduces the agent’s effort,and the small reduction of the effective rate of return offsets the distortion of agent’s consumption partly which is caused by information friction.In hidden savings,information friction leads to a larger reduction of the agent’s effort,and the principal’s ability of providing intertemporal incentives for the agent is limited because the effective rate of return equals to the risk-free rate of return,and then the agent’s consumption decreases greatly.In both hidden action and hidden saving cases,the advanced technology improves the production efficiency and reduces the cost of agent’s effort,so that the agent’s effort increases,and then an increase of agent’s consumption occurs.Compared with the case of full information,the principal’s consumption decreases in both hidden action and hidden saving cases,and the principal’s consumption decreases more greatly in hidden savings.In our settings,the optimal technology in three different information structures always exists.(3)We study the optimal contracts with agent’s quality known and unknown when the model contains the agent’s quality.In our settings,both the necessary and the sufficient conditions of incentive contract are derived and the explicit solutions of the optimal contracts are obtained.When the agent’s quality is known and unknown,the agent always puts forth his best effort.The unknown agent’s quality leads to a reduction of the principal’s ability of providing incentives for the agent.As the execution time of the contract goes on,the agent’s quality is progressively revealed and then the principal’s power of providing incentives for the agent becomes stronger.Thus the payment received by agent is higher in the earlier stage than that in the later stage.When the contract expires or the execution time of the contract is long enough,the agent’s quality is almost revealed completely,and the case where the agent’s quality is unknown degenerates into the case where the agent’s quality is known.(4)The optimal contracts under full information and hidden actions are studied when volatility is affected by firm size.In our model,the explicit solutions of the optimal contract are found under full information and the existence and uniqueness of the optimal contract solution are stated under hidden action case.The amount of the agent’s effort with full information equals to his marginal productivity.The information friction of hidden actions results in a reduction of the agent’s effort.As the firm size is small enough,the agent’s effort level of hidden actions is approximately equal to the first-best level.