# A Solvable Time-Inconsistent Principal-Agent Problem

Authors: Chao Li; Zhijian Qiu
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

We consider the dynamic contract model with time inconsistency preference of principal-agent problem to study the influence of the time inconsistency preference on the optimal effort and the optimal reward mechanism. We show that when both the principal and the agent are time-consistent, the optimal effort and the optimal reward are the decreasing functions of the uncertain factor. And when the agent is time-inconsistent, the impatience of the agent has a negative impact on the optimal contract. The higher the discount rate of the agent is, the lower the efforts provided; agents tend to the timely enjoyment. In addition, when both the principal and the agent are time-inconsistent, in a special case, their impatience can offset the impact of uncertainty factor on the optimal contract, but, in turn, their impatience will affect the contract.


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## Body

## 1. Introduction

The principal-agent problem is a classic issue of the optimal contract and is widely used in financial and economical fields. The principal and the agent, as two parties of the contract, interact with each other. Under the constraints of the contract, the agent creates profits for the principal and the principal pays the salary for the agent as incentives. In this paper, we introduce an optimal contract where both the principal and the agent are time-inconsistent to solve the principle-agent problem under moral hazard in a dynamic environment. In general, the agent is regarded as risk-neutral and the principal is risk aversion.In solving the optimal contract with the time-inconsistent principle-agent problem, there are three problems we need to face. The first is the solution to the principal-agent’s problem in the continuous-time. A continuous-time model where the agent controls the Brownian motion drift rate over the time interval is studied by [1]. Later, [2, 3] uses martingale methods to develop the first-order approach of principal-agent problems under moral hazard with exponential utility. Reference [4] shows that the first best sharing rule is also linear in output in the continuous-time principal-agent model with exponential utility. Reference [5, 6] uses the stochastic maximum principle to extended Holmstrom’s model, and discuss the optimal solution of the agent with private information in the continuous-time model. Reference [7, 8] uses the forward-backward stochastic differential equations to consider the optimal contract under moral hazard. Reference [9] systematically expands the problem of continuous-time principal-agent. However, the above methods solved the principal-agent problem in a continuous period of time under the time-consistent preference, which is simplistic for the actual situation. Therefore, it is natural to consider the time inconsistency in the principal-agent model.The second problem is how to find the optimal strategy when the time preference is inconsistent. Reference [10] proposes the optimal contracts for the principal who contracted with the dynamically inconsistent agents in a discrete case. Their study includes exploitative contracts that applied for naive agents to better explain the true contractual arrangements. Based that, [11, 12] takes the neutral agent as a benchmark to study the possibility that the principal manipulates the naive agent. The result shows that the innocence of the agent does not bring benefits to the principal, and the maximum effectiveness of the principal is the same in front of the neutral agent and naive agent. Besides, the definition of naive and neutral agent was firstly mentioned in [13]. Reference [14] takes the discount of the quasi-hyperbolic βδ as the agent’s discount function, then discuss the optimal contract and the profit level of the principal when the agent is neutral or naive. The above mainly deals with the problem of time-inconsistent agents in discrete time. However, less attention is paid in continuous-time because it is complicated to solve the closed-loop solution under nonconstant discounting. Reference [15] proposes the optimal contract models based on the Pontryagin maximum principle for forward-backward stochastic differential equations to study a general continuous-time principal-agent problem in which the utility function is time-inconsistent.The third question is how to find the exact solution of the Hamilton-Jacobi-Bellman (HJB) equation. Using the stochastic control to solve the HJB equation is a complex mathematical process, especially in the case of increasing the control variables, which will be more complex nonlinear partial differential equations. By the Legendre transform, the problem can be transformed into a dual problem that is convenient for analysis, so as to solve some model solving problems. Reference [16] studies the portfolio problem under the general utility function, and prove the effectiveness of using the Legendre dual method to solve the HJB equation. Reference [17] uses Legendre transform-dual theory to solve the optimal investment problem based on hyperbolic absolute risk aversion (HARA) preference under constant elastic variance model. References [18, 19] study the investment-consumption of HARA utility by Legendre method.In this paper, we study the optimal incentive contract under moral hazard in the framework of the principal-agent problem with time inconsistency in continuous-time. Assume both the principal and the agent are time-inconsistent, where the principal is risk-averse with an exponential utility function and the agent is risk-neutral with a linear utility function. To describe the time inconsistency of participants, we assume that the discount rate of participants is a function of time (not a constant) but still takes the form of exponential discounting because the principal’s utility function is the exponential utility function. According to the property of the exponential function, we can divide the discount function into two parts: one part is the traditional discount function (discount rate is constant) and the other part is the uncertainty. Under the moral hazard, the principal can observe the process of output but cannot observe the agent’s efforts and random perturbations. Thus the principal considers a part of the discount function as an unknown factor that affects the output. Reference [20] puts forward that the principal can constantly learn and update his belief from the unknown factor (the uncertain part of the discount function) through the existing information and historical information in the production process. Therefore, we transform the time-inconsistent principal into a principal who has the consistent time and learning process. For time-inconsistent agents, we can employ the Markov subgame perfect Nash equilibrium method [21] to get its time consistency strategy.Through the above assumption, we solve the optimal contract in two cases where the principal is time-consistent and time-inconsistent. When the principal is time consistency, we use the stochastic optimal control method to derive the nonlinear partial differential equation (HJB equation) for the optimal value function of the principal. This partial differential equation is hard to solve for an exact closed-loop solution; however, the original problem can transfer to a dual problem by applying the Legendre transform in some cases. To obtain the exact solution of the optimal contract (closed-loop solution), we use the Legendre transform-dual theory to obtain the explicit solution of the optimal solution and the optimal contract. When the principal is time-inconsistent, we obtain a value function which takes the time, the agent’s personal information, and the utility of agent as variables, so as to derive a three-dimensional nonlinear second-order HJB equation. In this situation, we solve the HJB equation by guess solution.The general structure of this paper is as follows. Section2 presents the model. The incentive compatibility conditions and the given proof are provided in Section 3. Section 4 studies the optimal contract of a time-consistent and time-inconsistent principal. Section 5 provides numerical simulation of optimal strategy. Finally, we made the conclusion in Section 6.


## 2. The Model

### 2.1. The Agent

Suppose a principal made the contract with a time-inconsistent agent to manage a production process (or invest a risk project) and the initial time of contracting period is recorded as 0. Consider an infinite horizon stochastic environment; letWt0 be a standard Brownian motion on the probability space (Ω,F,P0). The risk process that pays a cumulative process Yt evolved on period [0,T] as follows:(1)dYt=et-ctdt+σdWt0assumes that A∈R is a compact set and et∈A is the agent’s effort choice. ct∈R is the salary of the agent (or his consumption). σ is the project’s volatility (constant) where σ>0. The path of Yt is observable both from the principal and the agent, but the path of {Wt0} is observable only from the agent, and the effort choice et is unobservable from the principal.At the initial moment time0, the principal provides the agent with a contract (pay the salary according to the contract). Assume that the salary is composed of two parts, a continuous payment ct and a terminal payment CT. Moreover, we assume that the agent is risk-neutral, u and v ( we will give the explicit functional forms for u and v in the specific question (see (26) and (27) )) are utility functions, and u and v are concave and twice continuously differentiable. And the agent has a discount function h, where h(t)=e-∫0tρ(s)ds (in this paper, for convenience, sometimes h(t) is written as ht) is a general discount function; see [22]; then the agent will be time-inconsistent if ρ(s) is time dependence.The agent’s preferences as of time0 read(2)JA0;e=E∫0Thtuct,etdt+hTvCT


### 2.2. The Principal

In this paper, we assume that the principal is partially naive type ([23] in a discrete time-inconsistent model, disaggregate participants into mature type, naive type, and partially naive type based on the cognitive differences of participants about their own future preference), which means that the principal knows he is time-inconsistent (his discount function is time-variant), but his current perception of the future discount rate is biased against the true value of the future discount rate (at time t, he cannot be sure the value of discount rate δ(τ) (we can further assume that ρ(t),δ(t):[0,T]→[0,1]) when 0≤t<τ≤T). Therefore, the principal will continuously update his belief in the future discount rate based on the past information. The detailed analysis is as follows.The principal’s preferences as of time0 will be(3)JP0;e,c=E∫0TktUet-ctdt+kTL-CTwhere U and L are utility functions by the principle and k(t)=e-∫0tδ(s)ds is the discount function. Assume that the utility function U(et-ct)=-e-λ(et-ct) over salary (consumption) and effort and L(-CT)=-e-λ(-CT), where λ is an absolute risk aversion coefficient. Hence, we rewrite (3) as follows:(4)JP0;e,c=E∫0Te-∫0tδsds·-e-λet-ctdt+e-∫0Tδsds·-e-λ-CT=E∫0Te-δ¯t-e-λet-ct+Ktdt+e-δ¯T-e-λ-CT+KTwhere Kt=∫0tδ(s)ds-δ¯t/λ that t∈[0,T].From (4), we can split the principal’s discount factor δ into two parts under the condition of the exponential utility function: one part is a constant discount rate δ¯ and another part is Kt. The purpose of the above operation is that the principal estimates a suitable constant discount rate δ¯ instead of a time-varying discount rate δ(t), and the principal does not know the exact value of this constant. Therefore, the principal will constantly update the recognition of δ¯ based on past information. ct is the subjective choice of the principal, but Kt indicates an objective reality, reflects the type of principal (time-inconsistent or time-consistent), and does not depend on the subjective choice of the principal. So we can set Kt as a part of the output (investment) process. In this way, we can turn the principal’s time inconsistency problem into an unknown constant discount rate problem. If the principal is time-consistent, namely, the principal’s discount rate is a constant δ0, he can choose δ0 as a constant discount rate δ¯; hence δ0=δ¯ ( or Kt=0). Under the probability measure P of the principal, we can regard that Kt is an intrinsic influence of the risk item and is not subject to the control of the principal but must be considered. Hence the risk process (1) becomes(5)dYt=et-ct+Ktdt+σdWtPAs discussed above, we know that the processYt and the path of Wt0 are observable from the agent; therefore the measure for the agent is P0, which means that the agent does not learn in secretly, so the agent’s beliefs will not be a hidden-state variable ([24]) (this does not mean that the agent cannot mislead the principal by choosing an effort, just that such actions do not cause persistent hidden states according to the agent’s beliefs). From (1) we have(6)dYt=et-ctdt+σdWtP0Equation (5) expresses the principal’s beliefs about the project and (6) expresses the agent’s beliefs. The disagreement between the principal and the agent is caused by principal’s nonindex discount.At timet, the principal knows the exact value of ∫0tδ(s)ds, because δ(s) is his discount rate, but he does not knows the exact value of δ¯; therefore we use a sided Bayesian learning model after signed contract, and we assume that the prior about K at time 0 is normally distributed with mean 0 and variance θ0. The agent does not update his beliefs because he has perfect information. If the agent follows the recommended effort choice e, the principal’s posterior beliefs about Kt depend on Yt and on cumulative effort Et=∫0tesds.According to the Kalman-Bucy filter, see [25]. The conditional expectation K^t=E[Kt∣Yt,Et]=σ-2(Yt-Et)/θt and the precision of filtering θt=E[Kt-K^t-2] satisfy the system of equations (7)dK^t=1σθtdYt-et-ct+K^tdtθt=θ0+σ-2twhere K^0=0 and θ0=0 and Zt is a standard Brownian motion under the measure induced by the effort sequence {es:0≤s≤t} as(8)dZt≜dYt-et-ct+K^tdtσ


## 2.1. The Agent

Suppose a principal made the contract with a time-inconsistent agent to manage a production process (or invest a risk project) and the initial time of contracting period is recorded as 0. Consider an infinite horizon stochastic environment; letWt0 be a standard Brownian motion on the probability space (Ω,F,P0). The risk process that pays a cumulative process Yt evolved on period [0,T] as follows:(1)dYt=et-ctdt+σdWt0assumes that A∈R is a compact set and et∈A is the agent’s effort choice. ct∈R is the salary of the agent (or his consumption). σ is the project’s volatility (constant) where σ>0. The path of Yt is observable both from the principal and the agent, but the path of {Wt0} is observable only from the agent, and the effort choice et is unobservable from the principal.At the initial moment time0, the principal provides the agent with a contract (pay the salary according to the contract). Assume that the salary is composed of two parts, a continuous payment ct and a terminal payment CT. Moreover, we assume that the agent is risk-neutral, u and v ( we will give the explicit functional forms for u and v in the specific question (see (26) and (27) )) are utility functions, and u and v are concave and twice continuously differentiable. And the agent has a discount function h, where h(t)=e-∫0tρ(s)ds (in this paper, for convenience, sometimes h(t) is written as ht) is a general discount function; see [22]; then the agent will be time-inconsistent if ρ(s) is time dependence.The agent’s preferences as of time0 read(2)JA0;e=E∫0Thtuct,etdt+hTvCT


## 2.2. The Principal

In this paper, we assume that the principal is partially naive type ([23] in a discrete time-inconsistent model, disaggregate participants into mature type, naive type, and partially naive type based on the cognitive differences of participants about their own future preference), which means that the principal knows he is time-inconsistent (his discount function is time-variant), but his current perception of the future discount rate is biased against the true value of the future discount rate (at time t, he cannot be sure the value of discount rate δ(τ) (we can further assume that ρ(t),δ(t):[0,T]→[0,1]) when 0≤t<τ≤T). Therefore, the principal will continuously update his belief in the future discount rate based on the past information. The detailed analysis is as follows.The principal’s preferences as of time0 will be(3)JP0;e,c=E∫0TktUet-ctdt+kTL-CTwhere U and L are utility functions by the principle and k(t)=e-∫0tδ(s)ds is the discount function. Assume that the utility function U(et-ct)=-e-λ(et-ct) over salary (consumption) and effort and L(-CT)=-e-λ(-CT), where λ is an absolute risk aversion coefficient. Hence, we rewrite (3) as follows:(4)JP0;e,c=E∫0Te-∫0tδsds·-e-λet-ctdt+e-∫0Tδsds·-e-λ-CT=E∫0Te-δ¯t-e-λet-ct+Ktdt+e-δ¯T-e-λ-CT+KTwhere Kt=∫0tδ(s)ds-δ¯t/λ that t∈[0,T].From (4), we can split the principal’s discount factor δ into two parts under the condition of the exponential utility function: one part is a constant discount rate δ¯ and another part is Kt. The purpose of the above operation is that the principal estimates a suitable constant discount rate δ¯ instead of a time-varying discount rate δ(t), and the principal does not know the exact value of this constant. Therefore, the principal will constantly update the recognition of δ¯ based on past information. ct is the subjective choice of the principal, but Kt indicates an objective reality, reflects the type of principal (time-inconsistent or time-consistent), and does not depend on the subjective choice of the principal. So we can set Kt as a part of the output (investment) process. In this way, we can turn the principal’s time inconsistency problem into an unknown constant discount rate problem. If the principal is time-consistent, namely, the principal’s discount rate is a constant δ0, he can choose δ0 as a constant discount rate δ¯; hence δ0=δ¯ ( or Kt=0). Under the probability measure P of the principal, we can regard that Kt is an intrinsic influence of the risk item and is not subject to the control of the principal but must be considered. Hence the risk process (1) becomes(5)dYt=et-ct+Ktdt+σdWtPAs discussed above, we know that the processYt and the path of Wt0 are observable from the agent; therefore the measure for the agent is P0, which means that the agent does not learn in secretly, so the agent’s beliefs will not be a hidden-state variable ([24]) (this does not mean that the agent cannot mislead the principal by choosing an effort, just that such actions do not cause persistent hidden states according to the agent’s beliefs). From (1) we have(6)dYt=et-ctdt+σdWtP0Equation (5) expresses the principal’s beliefs about the project and (6) expresses the agent’s beliefs. The disagreement between the principal and the agent is caused by principal’s nonindex discount.At timet, the principal knows the exact value of ∫0tδ(s)ds, because δ(s) is his discount rate, but he does not knows the exact value of δ¯; therefore we use a sided Bayesian learning model after signed contract, and we assume that the prior about K at time 0 is normally distributed with mean 0 and variance θ0. The agent does not update his beliefs because he has perfect information. If the agent follows the recommended effort choice e, the principal’s posterior beliefs about Kt depend on Yt and on cumulative effort Et=∫0tesds.According to the Kalman-Bucy filter, see [25]. The conditional expectation K^t=E[Kt∣Yt,Et]=σ-2(Yt-Et)/θt and the precision of filtering θt=E[Kt-K^t-2] satisfy the system of equations (7)dK^t=1σθtdYt-et-ct+K^tdtθt=θ0+σ-2twhere K^0=0 and θ0=0 and Zt is a standard Brownian motion under the measure induced by the effort sequence {es:0≤s≤t} as(8)dZt≜dYt-et-ct+K^tdtσ


## 3. Incentive Compatible Conditions

In this section, we focus on the agent’s problem. Since the agent’s objective function relies on the consumption process{ct}, that is, it relies on the history of the whole output, so it is non-Markov (the specific proof see [6]). Thus, the agent optimization problem cannot be solved by the standard dynamic programming principle. We will employ the stochastic maximum principle of the solution to the weak situation of the agent problem. The main idea is to apply random variational methods; the relative papers [8, 20, 26] used the similar approach.Define the agent’s continuation value (promised utility)qt as the expected discounted utility for remaining in the contract from date t forward(9)qt=maxe∈AE∫tTe-∫tsρτdτucs,Y¯s,esds+e-∫tTρτdτvCT∣Ftewhere Y¯s≜{Yt:0≤t≤s} is the output history. We use Γ to relate the expectation operator Ee· under the measure Qe (because the agent’s objective function depends on the consumption process ct, which is non-Markovian since it depends on the whole output path Y¯t; hence the optimization problem (9) cannot be analyzed with standard methods. So we use a martingale approach. Given a contract (c,e∗), the agent controls the distribution of salary through his choice of effort. For the specific technical treatment see Appendix A). The agent’s objective function can be recast as (10)JA0;e,c¯=Ee∫0Thsucs,Y¯,esds+hTvCT=E∫0TΓt,shsucs,Y¯,esds+Γt,ThTvCTwhere c¯=c(s,Y¯) represents the salary paid by the principal based on output history.After the change of measure, the time-inconsistent agent’s problem only has one control. We apply a stochastic maximum principle to characterize the agent’s optimality condition, and we also use the dynamic programming equation derive a stochastic maximum principle with general time-inconsistent.The agent’s problem is to find an admissible control to maximize the expected rewardJA(0;e,c¯). In other words, the agent needs to solve the problem(11)supe∈A⁡JA0;e,c¯Givenc¯, for all 0≤t≤T, subject to(12)dΓt=Γtσ-1et-ct,Y¯+K^tdZt0next we define the optimal effort for the time-inconsistent agent. Letε>0 and Eε⊆[0,T] be a measurable set whose Lebesgue measure is Eε=ε. Let et∈A be an arbitrary effort choice. We define the following:(13)etε=et∗,t∈0,T∖Eε,et,t∈Eε.with Eε=0,ε, and it is clear that etε∈A. We refer etε as a needle variation of the effort choice et∗. Then, we have following definition.Definition 1.
An effort choiceet∗ is an optimal effort choice for the time-inconsistent agent, for t∈(0,T], if (14)limε→0JA0;e∗,c¯-JA0;eε,c¯ε≥0
The optimal density processΓ∗tt∈[0,T] is a solution of the stochastic differential equation.(15)dΓt∗=Γt∗σ-1et∗-ct,Y¯t+K^t,Yt,Et∗dZt0Through the above technical processing, we convert the time-inconsistent strategy into time consistency optimal strategy of the agent.Next, we analyze the conditions for the implementation of incentive contract. According to the previous analysis, the agent will control the distribution of salary by choosing his effort. The idea of using the distribution of salary as control to solve principal-agent problem goes back to [27] is and expanded by [5, 20]. The learning process of the principal complicates our problem, as the past effort affects not only current salary but also future expectations of the agent and the principal. Therefore, we have to deal with a principal-agent problem with time inconsistency and learning process. In Appendix B, we show how this difficulty can be handled through an extension of the proof by [8, 20]. And the conclusion presents the following.Proposition 2.
The agent’s continuation value can be uniquely represented by the following differential form:(16)dqt=ρtqt-uct,et∗dt+γtσdZtqT=vCTwhere γt is a square integrable predictable process.
The necessary and sufficient conditions foret∗ is the optimal effort choice reads:
(i) Ifet∗ is the optimal effort choice, then for every t∈[0,T] there exists a solution {qt,γt}t∈[0,T] of (16) which satisfies (in this paper, ∂eu and ∂eeu represent function u taking the first-order partial derivative and second-order partial derivative of e, respectively)(17)γt+σ-2θtpt+∂euct,et∗=0where(18)pt≜θt·E-∫tTe-∫tsρτdτγsθs-1ds∣Fte
(ii) For almost allt, if the following inequality holds(19)-2ht∂eeuct,et≥σ2ξtθtwhere ξ is the predictable process defined by (20)E-∫0Te-∫0sρτdτγsσ-2θsds∣Fte∗-E-∫0Te-∫0sρτdτγsσ-2θsds∣F0e∗=∫0tξsσdZsthen et∗ is the optimal effort choice.According to (18), we say that pt is a stochastic process capturing the value of private information and then obtain the solution(21)pt=E∫tThs-t∂eucs,esds∣Ftefor all s∈[t,T].In the following, at any timees∗>0, the process for p reads (22)dpt=ρtpt-∂euct,etdt+ϑtσdZtpT=0.the coefficients ϑ is chosen by the principal to maximize his expected utility (the proof is given by [20])According to (19), the process ξt is the random fluctuation in the discounted sum of marginal utilities evaluated from time 0. Based on the stochastic differential equation of pt, we can obtain that ξt=e-∫0tρ(s)dsσθt. Besides that, ct, et, γt, and ξt are endogenous, which implies that we need to get a contract to satisfy the necessary conditions and then prove that it also meets the sufficient conditions. If the contract has no explicit solution, it will be difficult to prove that the contract also satisfies the sufficient condition. In this paper, the utility function for the principal is exponential function and the utility function for the agent is linear function. In the next section we will employ the exponential utility function to get the closed-loop solution of the contract.


## 4. The Optimal Contract

This section detailedly explains how to solve the principal’s problem and derive the optimal contract in closed form when the principal’s utility is exponential.Eliminating K ^ from the list states. For a given contract (c,e∗), the principal expected utility form data t forward reads(23)Πtc,e∗≜E∫tTe-δ¯s-t-e-λes-cs+K^sds+e-δ¯T-t·-e-λ-CT+K^T∣FtYand defines (24)JPt,c,e∗≜E∫tTe-δ¯s-t·-e-λes-csds+e-δ¯T-t·-e-λ-CT∣FtYand we have the following result.Proposition 3.
(25) m a x c , e  Π t c , e ∗ ⇔ m a x c , e  J P t , c , e ∗Proof in Appendix C.Assume that Propositions2 and 3 hold, so that the necessary condition is also sufficient. The principal’s problem consists of solving for JP(t,c,e) subject to the two promise-keeping constraints (9) and (18) and also to the incentive constraint (17). Given that the posterior mean K^ does not enter directly into any of the constraints, it can be dispensed as a state, only leaving the precision as a belief state. Furthermore, since θt is deterministic, we may index the precision by t. The fact is that the expected value of Kt is immaterial to the principal’s objective and illustrates that incentives are designed to reward effort, not to ability.The Agent’s utility function. To obtain the solution of optimal contract, according to our assumption in Section 2.1, the agent is risk-neutral and the utility functions of the agent are linear, i.e.,(26)uc,e=c-e22; moreover, we make a particular assumption about the terminal utility for the agent, setting(27)vCT=a+ln⁡CTwhere a is a constant. This assumption implies a situation in which an infinitely lived agent retires at the termination date T of the contract and, after retirement, he can consume a permanent annuity derived from CT. We always concentrate on problems where the contracting horizon goes to infinity T→∞, so this particular assumption v is not critical.Incentive providing contracts. Restore the principal’s optimization problem as(28)Vt≜maxc,γ,ϑE∫tTe-δ¯s-t·-e-λes-csds+e-δ¯T-t·-e-λ-CT∣FtYsubject to(29)dqt=ρtqt-uct,etdt+γtσdZt,qT=vCT(30)dpt=ρtpt-∂euct,etdt+ϑσdZt,pT=0(31)γt=-∂euct,et-σ-2θtptSince the state variablesqt and pt are Markovian processes, we can use the HJB equation to analyze the principal’s optimal control problem. Take V(t,q,p) as the principal’s value function; this value function satisfies the HJB equation for 0<t<T: (32)δ¯V-∂tV=maxc,e,γ,ϑ-e-λe-c+∂qV·ρtq-uc,e+∂pV·ρtp-∂euc,e+12σ2∂qqV·γp,c,e2+∂ppV·ϑ2+2∂pqV·ϑγp,c,e


### 4.1. Second Best Contracts for the Time-Consistent Principal

In this section, we mainly consider that the model with the time-consistent principal and setting hidden action means that the principal can observe the processYt but does not know the type of agent and also cannot observe the agent’s effort et. For incentive contracts (e,c):et>0, for any t≥0, the necessary condition for incentive compatibility (17) becomes γt=-∂eu(ct,et). When the principal is time-consistent, it expresses that δt=δ¯ for all t∈[0,T]; hence we say K=0. There is no need to inquire K and there is no influence of belief manipulation, which indicates that the information value p is equal to zero.


#### 4.1.1. The HJB Equation

When the time tends to infinity (T→∞), the agent’s continuous value function q is the only state variable for writing the principal’s HJB equation as follows:(33)δ¯V-∂tV=maxc,e,γ-e-λe-c+∂qV·ρq-c-e22+12σ2∂qqV·γ2c,ewith the terminal condition V(T,qT)=L(-CT), where qT=v(CT). Taking the first-order conditions for (e,c), we have(34)λe-λe-c+∂qV·e+∂qqV·eσ2=0-λe-λe-c-∂qV=0Under full information, the principal can observe the agent’s effort and consumption and there is no private information in this case. Hence, the principal can freely chooseγt as parts of the contract; i.e., γ is independent of c and e, and then we have the following proposition.Proposition 4.
Under full information, the optimal effort for the principal is a constante∗=1. We say e∗=1 is the first best effort level.Under hidden action case, the optimal effort and consumption derived from (34) shows(35)et∗=∂qV∂qV+σ2∂qqV,ct∗=∂qV∂qV+σ2∂qqV+1λln-∂qVλPutting (35) into (33), the HJB equation (33) for the value function V(t,q) is rewritten as(36)δ¯V-∂tV=∂qVλ+∂qV·ρq-∂qV·∂qV∂qV+σ2∂qqV+1λln-∂qVλ+∂qV2·∂qV∂qV+σ2∂qqV2+12σ2∂qqV·∂qV∂qV+σ2∂qqV2Recalling the principal with the exponential utility function, the HJB equation is a complex nonlinear partial differential equation. It is difficult to take the classic separation of the variable method and solve it intuitively. In the next, we will employ the Legendre transform to turn the problem into a dual problem, by solving the dual problem to obtain the optimal solution for the original problem.


#### 4.1.2. Legendre Transform

The dual function ofV is defined by (37)V^t,z=supq>0Vt,q-qzgt,z=infq>0q∣Vt,q≥V^t,z+qzwhere z<0 is the dual variable of q. The function g(t,z) is closely related to V^(t,z) and can be used as a dual function of the function V(t,q). In this paper, g(t,z) is defined as the dual function of V(t,q) and satisfies (38)gt,z=-∂zV^t,zAccording to the definition of the dual function, we havez=∂qV(t,q) and(39)gt,z=qV^t,z=Vt,q-zgt,zBased on the conclusion in [28], the following transformation rules are obtained:(40)∂tV=∂tV^,∂qV=z,∂qqV=-1∂zzV^Define the dual function of the utility function as(41)U^t,z=supx>0Ut,x-xzGt,z=infx>0x∣Ut,x≥U^t,z+xzWith the analysis from [29], the function U(x) and U^(z) can be changed to pass the Legendre conversion (42)U^z=supx>0Ux-xzUx=infz>0U^z+xzThe relationship between the optimal valuesx∗ and z∗ is (43)zt∗=U^′-1-xt∗,xt∗=e∫0tρsdsU′-1zt∗,YT∗-CT∗=L′-1zT∗According to equation (39) and rules (40), the HJB equation for the dual problem is (44)δ¯V^-∂tV^=1λ+ρg-δ¯gz-zλln-zλ-12z2·∂zzV^z·∂zzV^-σ2Taking the derivative ofz and combineing (35), we have(45)-δ¯g+∂tg=-δ¯g-δ¯z·∂zg+ρg+ρz·∂zg-1λln-zλ-12+σ2σ2-z2·∂zzg2z·∂zg+σ22


#### 4.1.3. Solution of the HJB Equation

According to the form of the principle’s utility function, we have(46)GT,z=1λln-zλWe can assume that HJB equation (36) has the following form of solution:(47)gt,z=e∫0tρsdsλϕtln-zλ+φtwith ϕ(T)=h(T),φ(T)=0.(48)∂tg=1λ∂tϕ·h-ϕ·∂thh2ln⁡-zλ+∂tφ∂zg=ϕλhz,∂zzg=-ϕλhz2plug in (45) and separate the variables; then we have (49)0=-1λhlnzλ∂tϕ+h+ϕλhρ-δ¯-12σ2+ϕ/λh+ρφ-∂tφThus, the following two ordinary differential equations are established:(50)∂tϕ+h=0,ϕT=hT(51)∂tφ-ρφ+ϕλhδ¯-ρt+12σ2+ϕ/λh=0,φT=0Proposition 5.
Assume that (i) the principal is time-consistent and the agent is time inconsistent, (ii)u(c,e) and v(C) are as defined in (26) and (27), and (iii) et∗>0 for all t, so that the incentive constraint (17) binds for almost all t∈[0,T]. Then recommended effort and the agent’s consumption is given by(52)ct∗=ϕ/hϕ/h+λσ2+hϕq-hϕφet∗=ϕ/hϕ/h+λσ2where (53)ϕt=hT+∫tThsds,φt=∫tTe-∫tτρsdsϕλhρδ¯-ρ+12σ2+ϕ/λhdτ


### 4.2. Second Best Contracts for the Time-Inconsistent Principal

In this section, we discuss the case when the principal is time-inconsistent, which means the discount rateδ is not a constant. In this case, the principal still cannot observe the agent’s efforts and consumption (moral hazard). Hence, the value of private information is not zero. As described in Section 3, the HJB equation is as follows:(54)δ¯V-∂tV=maxc,e,γ,ϑ-e-λe-c+∂qV·ρq-c+e22+∂pV·ρp+e+12σ2∂qqV·e-σ-2θp2+∂ppV·ϑ2+2∂pqV·ϑe-σ-2θpNow we need to solve this above equation by guessing the solution. Under the first-order conditions for(e,c,ϑ), we have(55)λe-λe-c+∂pV+∂qV+σ2∂ppV·∂qqV-∂pqV2∂ppV·e-∂ppV·∂qqV-∂pqV2∂ppV·pθ=0-λe-λe-c-∂qV=0ϑ=-∂pqV∂ppVγSubstituting (55) into (54), denoting that V~=∂ppV·∂qqV-∂pqV2/∂ppV, we have(56)∂tV-δ¯V+∂qVλ+∂qVρtq-1λln-∂qVλ+∂pV·ρtp-12E~+σ-2p22θ2V~=0where E~=p/θV~+∂qV-∂pV2/σ2V~+∂qV.In particular, we suppose the value function has the following form:(57)Vt,p,q=-eλAtq+gt,pwith A(T)=1 and g(T,p)=0.Hence, for some functionsA(t) and g(t,p), the expressions of optimal effort and consumption are(58)et∗=λA2∂ppgp/θ+A-∂pg∂ppg+λ∂pg2λA2∂ppgσ2+A∂ppg+λA∂pg2ct∗=et∗+lnAλ+Aq+gSubstituting the optimal effort and consumption into (54), we deduce (59)λAt+Aρ-A2q+λ∂tg-δ¯+A-Aln⁡A-λAg+ρλp∂pg+λAσ-42θ2p2-σ-2θp+λσ-2∂pgθp-λ2E~=0where E~=(∂pqg+λ∂pg2)·p/θσ-2A+∂pg-A2/λσ2A2∂pqg+A(∂ppg+λ∂pg2).The following two differential equations can be obtained by eliminating the dependence onq:(60)∂tA+Aρt-A2=0,AT=1(61)∂tg-1λδ¯-A+Aln⁡A-Ag+ρp∂pg+Aσ-42θ2p2-σ-2θp+σ-2∂pgθp-12E~=0,gT,p=0According to (60), we can obtain(62)At=1e-∫tTρsds+∫tTe-∫tτρsdsdτFrom the above analysis, we need to know the specific expressions of g(t,p) to get the explicit expression of effort e(t) and consumption c(t).Let us expandg(t,p) at p=0, (63)gt,p=gt,0+gpt,0p+gppt,02p2+⋯+gnt,0n!pn+⋯≜A~t+B~tp+C~tp2+⋯Here we consider a simple situation; according the principal’s utility function U=-e-λ(e-c) and dp(t)/dt=ρ(t)p(t)+e(t), suppose the structure of the solution of equation (61) is g(t,p)=A~(t)+B~(t)p with the terminal condition A~(T)=0 and B~(T)=ρ(T).Lemma 6.
Suppose the structure of solution of (61) is g(t,p)=A~(t)+B~(t)p, with the terminal condition A~(T)=0 and B~(T)=1, and then A~(t) and B~(t) are, respectively, the solutions of the following differential equations:(64)A~t=-∫tTe-∫tτAsds·DτdτB~t=ρTe∫tTρs-Asdswhere D(τ)=1/λ(δ¯-A(τ)+Aτln⁡A(τ))+A2(τ)+B~2(τ)/2A(τ).It is proved as follows.Substitutingg(t,p)=A~(t)+B~(t)p into (61), we calculate to get (65)∂tA~-AA~-1λδ¯-A+Aln⁡A-A2+B~22A+∂tB~-AB~+ρB~p=0The following two differential equations can be obtained by eliminating the dependence on p:(66)∂tA~-AA~-1λδ¯-A+Aln⁡A-A2+B~22A=0∂tB~-AB~+ρB~=0Lemma 6 conclusion can be provided by solving the above two differential equations.Proposition 7.
When the principal is time-inconsistent, the expressions of optimal effort and consumption for the agent are(67)e∗t=1-B~tAtc∗t=e∗t+ln⁡Atλ+Atqt+A~t+B~tptwhere A(t), A~(t), and B~(t) are given by (62) and (64), respectively.It can be seen from the proposition that the second optimal effort is less than the first optimal effort. The optimal consumption is the linear function of the agent’s promise value and private information.


## 4.1. Second Best Contracts for the Time-Consistent Principal

In this section, we mainly consider that the model with the time-consistent principal and setting hidden action means that the principal can observe the processYt but does not know the type of agent and also cannot observe the agent’s effort et. For incentive contracts (e,c):et>0, for any t≥0, the necessary condition for incentive compatibility (17) becomes γt=-∂eu(ct,et). When the principal is time-consistent, it expresses that δt=δ¯ for all t∈[0,T]; hence we say K=0. There is no need to inquire K and there is no influence of belief manipulation, which indicates that the information value p is equal to zero.


### 4.1.1. The HJB Equation

When the time tends to infinity (T→∞), the agent’s continuous value function q is the only state variable for writing the principal’s HJB equation as follows:(33)δ¯V-∂tV=maxc,e,γ-e-λe-c+∂qV·ρq-c-e22+12σ2∂qqV·γ2c,ewith the terminal condition V(T,qT)=L(-CT), where qT=v(CT). Taking the first-order conditions for (e,c), we have(34)λe-λe-c+∂qV·e+∂qqV·eσ2=0-λe-λe-c-∂qV=0Under full information, the principal can observe the agent’s effort and consumption and there is no private information in this case. Hence, the principal can freely chooseγt as parts of the contract; i.e., γ is independent of c and e, and then we have the following proposition.Proposition 4.
Under full information, the optimal effort for the principal is a constante∗=1. We say e∗=1 is the first best effort level.Under hidden action case, the optimal effort and consumption derived from (34) shows(35)et∗=∂qV∂qV+σ2∂qqV,ct∗=∂qV∂qV+σ2∂qqV+1λln-∂qVλPutting (35) into (33), the HJB equation (33) for the value function V(t,q) is rewritten as(36)δ¯V-∂tV=∂qVλ+∂qV·ρq-∂qV·∂qV∂qV+σ2∂qqV+1λln-∂qVλ+∂qV2·∂qV∂qV+σ2∂qqV2+12σ2∂qqV·∂qV∂qV+σ2∂qqV2Recalling the principal with the exponential utility function, the HJB equation is a complex nonlinear partial differential equation. It is difficult to take the classic separation of the variable method and solve it intuitively. In the next, we will employ the Legendre transform to turn the problem into a dual problem, by solving the dual problem to obtain the optimal solution for the original problem.


### 4.1.2. Legendre Transform

The dual function ofV is defined by (37)V^t,z=supq>0Vt,q-qzgt,z=infq>0q∣Vt,q≥V^t,z+qzwhere z<0 is the dual variable of q. The function g(t,z) is closely related to V^(t,z) and can be used as a dual function of the function V(t,q). In this paper, g(t,z) is defined as the dual function of V(t,q) and satisfies (38)gt,z=-∂zV^t,zAccording to the definition of the dual function, we havez=∂qV(t,q) and(39)gt,z=qV^t,z=Vt,q-zgt,zBased on the conclusion in [28], the following transformation rules are obtained:(40)∂tV=∂tV^,∂qV=z,∂qqV=-1∂zzV^Define the dual function of the utility function as(41)U^t,z=supx>0Ut,x-xzGt,z=infx>0x∣Ut,x≥U^t,z+xzWith the analysis from [29], the function U(x) and U^(z) can be changed to pass the Legendre conversion (42)U^z=supx>0Ux-xzUx=infz>0U^z+xzThe relationship between the optimal valuesx∗ and z∗ is (43)zt∗=U^′-1-xt∗,xt∗=e∫0tρsdsU′-1zt∗,YT∗-CT∗=L′-1zT∗According to equation (39) and rules (40), the HJB equation for the dual problem is (44)δ¯V^-∂tV^=1λ+ρg-δ¯gz-zλln-zλ-12z2·∂zzV^z·∂zzV^-σ2Taking the derivative ofz and combineing (35), we have(45)-δ¯g+∂tg=-δ¯g-δ¯z·∂zg+ρg+ρz·∂zg-1λln-zλ-12+σ2σ2-z2·∂zzg2z·∂zg+σ22


### 4.1.3. Solution of the HJB Equation

According to the form of the principle’s utility function, we have(46)GT,z=1λln-zλWe can assume that HJB equation (36) has the following form of solution:(47)gt,z=e∫0tρsdsλϕtln-zλ+φtwith ϕ(T)=h(T),φ(T)=0.(48)∂tg=1λ∂tϕ·h-ϕ·∂thh2ln⁡-zλ+∂tφ∂zg=ϕλhz,∂zzg=-ϕλhz2plug in (45) and separate the variables; then we have (49)0=-1λhlnzλ∂tϕ+h+ϕλhρ-δ¯-12σ2+ϕ/λh+ρφ-∂tφThus, the following two ordinary differential equations are established:(50)∂tϕ+h=0,ϕT=hT(51)∂tφ-ρφ+ϕλhδ¯-ρt+12σ2+ϕ/λh=0,φT=0Proposition 5.
Assume that (i) the principal is time-consistent and the agent is time inconsistent, (ii)u(c,e) and v(C) are as defined in (26) and (27), and (iii) et∗>0 for all t, so that the incentive constraint (17) binds for almost all t∈[0,T]. Then recommended effort and the agent’s consumption is given by(52)ct∗=ϕ/hϕ/h+λσ2+hϕq-hϕφet∗=ϕ/hϕ/h+λσ2where (53)ϕt=hT+∫tThsds,φt=∫tTe-∫tτρsdsϕλhρδ¯-ρ+12σ2+ϕ/λhdτ


## 4.1.1. The HJB Equation

When the time tends to infinity (T→∞), the agent’s continuous value function q is the only state variable for writing the principal’s HJB equation as follows:(33)δ¯V-∂tV=maxc,e,γ-e-λe-c+∂qV·ρq-c-e22+12σ2∂qqV·γ2c,ewith the terminal condition V(T,qT)=L(-CT), where qT=v(CT). Taking the first-order conditions for (e,c), we have(34)λe-λe-c+∂qV·e+∂qqV·eσ2=0-λe-λe-c-∂qV=0Under full information, the principal can observe the agent’s effort and consumption and there is no private information in this case. Hence, the principal can freely chooseγt as parts of the contract; i.e., γ is independent of c and e, and then we have the following proposition.Proposition 4.
Under full information, the optimal effort for the principal is a constante∗=1. We say e∗=1 is the first best effort level.Under hidden action case, the optimal effort and consumption derived from (34) shows(35)et∗=∂qV∂qV+σ2∂qqV,ct∗=∂qV∂qV+σ2∂qqV+1λln-∂qVλPutting (35) into (33), the HJB equation (33) for the value function V(t,q) is rewritten as(36)δ¯V-∂tV=∂qVλ+∂qV·ρq-∂qV·∂qV∂qV+σ2∂qqV+1λln-∂qVλ+∂qV2·∂qV∂qV+σ2∂qqV2+12σ2∂qqV·∂qV∂qV+σ2∂qqV2Recalling the principal with the exponential utility function, the HJB equation is a complex nonlinear partial differential equation. It is difficult to take the classic separation of the variable method and solve it intuitively. In the next, we will employ the Legendre transform to turn the problem into a dual problem, by solving the dual problem to obtain the optimal solution for the original problem.


## 4.1.2. Legendre Transform

The dual function ofV is defined by (37)V^t,z=supq>0Vt,q-qzgt,z=infq>0q∣Vt,q≥V^t,z+qzwhere z<0 is the dual variable of q. The function g(t,z) is closely related to V^(t,z) and can be used as a dual function of the function V(t,q). In this paper, g(t,z) is defined as the dual function of V(t,q) and satisfies (38)gt,z=-∂zV^t,zAccording to the definition of the dual function, we havez=∂qV(t,q) and(39)gt,z=qV^t,z=Vt,q-zgt,zBased on the conclusion in [28], the following transformation rules are obtained:(40)∂tV=∂tV^,∂qV=z,∂qqV=-1∂zzV^Define the dual function of the utility function as(41)U^t,z=supx>0Ut,x-xzGt,z=infx>0x∣Ut,x≥U^t,z+xzWith the analysis from [29], the function U(x) and U^(z) can be changed to pass the Legendre conversion (42)U^z=supx>0Ux-xzUx=infz>0U^z+xzThe relationship between the optimal valuesx∗ and z∗ is (43)zt∗=U^′-1-xt∗,xt∗=e∫0tρsdsU′-1zt∗,YT∗-CT∗=L′-1zT∗According to equation (39) and rules (40), the HJB equation for the dual problem is (44)δ¯V^-∂tV^=1λ+ρg-δ¯gz-zλln-zλ-12z2·∂zzV^z·∂zzV^-σ2Taking the derivative ofz and combineing (35), we have(45)-δ¯g+∂tg=-δ¯g-δ¯z·∂zg+ρg+ρz·∂zg-1λln-zλ-12+σ2σ2-z2·∂zzg2z·∂zg+σ22


## 4.1.3. Solution of the HJB Equation

According to the form of the principle’s utility function, we have(46)GT,z=1λln-zλWe can assume that HJB equation (36) has the following form of solution:(47)gt,z=e∫0tρsdsλϕtln-zλ+φtwith ϕ(T)=h(T),φ(T)=0.(48)∂tg=1λ∂tϕ·h-ϕ·∂thh2ln⁡-zλ+∂tφ∂zg=ϕλhz,∂zzg=-ϕλhz2plug in (45) and separate the variables; then we have (49)0=-1λhlnzλ∂tϕ+h+ϕλhρ-δ¯-12σ2+ϕ/λh+ρφ-∂tφThus, the following two ordinary differential equations are established:(50)∂tϕ+h=0,ϕT=hT(51)∂tφ-ρφ+ϕλhδ¯-ρt+12σ2+ϕ/λh=0,φT=0Proposition 5.
Assume that (i) the principal is time-consistent and the agent is time inconsistent, (ii)u(c,e) and v(C) are as defined in (26) and (27), and (iii) et∗>0 for all t, so that the incentive constraint (17) binds for almost all t∈[0,T]. Then recommended effort and the agent’s consumption is given by(52)ct∗=ϕ/hϕ/h+λσ2+hϕq-hϕφet∗=ϕ/hϕ/h+λσ2where (53)ϕt=hT+∫tThsds,φt=∫tTe-∫tτρsdsϕλhρδ¯-ρ+12σ2+ϕ/λhdτ


## 4.2. Second Best Contracts for the Time-Inconsistent Principal

In this section, we discuss the case when the principal is time-inconsistent, which means the discount rateδ is not a constant. In this case, the principal still cannot observe the agent’s efforts and consumption (moral hazard). Hence, the value of private information is not zero. As described in Section 3, the HJB equation is as follows:(54)δ¯V-∂tV=maxc,e,γ,ϑ-e-λe-c+∂qV·ρq-c+e22+∂pV·ρp+e+12σ2∂qqV·e-σ-2θp2+∂ppV·ϑ2+2∂pqV·ϑe-σ-2θpNow we need to solve this above equation by guessing the solution. Under the first-order conditions for(e,c,ϑ), we have(55)λe-λe-c+∂pV+∂qV+σ2∂ppV·∂qqV-∂pqV2∂ppV·e-∂ppV·∂qqV-∂pqV2∂ppV·pθ=0-λe-λe-c-∂qV=0ϑ=-∂pqV∂ppVγSubstituting (55) into (54), denoting that V~=∂ppV·∂qqV-∂pqV2/∂ppV, we have(56)∂tV-δ¯V+∂qVλ+∂qVρtq-1λln-∂qVλ+∂pV·ρtp-12E~+σ-2p22θ2V~=0where E~=p/θV~+∂qV-∂pV2/σ2V~+∂qV.In particular, we suppose the value function has the following form:(57)Vt,p,q=-eλAtq+gt,pwith A(T)=1 and g(T,p)=0.Hence, for some functionsA(t) and g(t,p), the expressions of optimal effort and consumption are(58)et∗=λA2∂ppgp/θ+A-∂pg∂ppg+λ∂pg2λA2∂ppgσ2+A∂ppg+λA∂pg2ct∗=et∗+lnAλ+Aq+gSubstituting the optimal effort and consumption into (54), we deduce (59)λAt+Aρ-A2q+λ∂tg-δ¯+A-Aln⁡A-λAg+ρλp∂pg+λAσ-42θ2p2-σ-2θp+λσ-2∂pgθp-λ2E~=0where E~=(∂pqg+λ∂pg2)·p/θσ-2A+∂pg-A2/λσ2A2∂pqg+A(∂ppg+λ∂pg2).The following two differential equations can be obtained by eliminating the dependence onq:(60)∂tA+Aρt-A2=0,AT=1(61)∂tg-1λδ¯-A+Aln⁡A-Ag+ρp∂pg+Aσ-42θ2p2-σ-2θp+σ-2∂pgθp-12E~=0,gT,p=0According to (60), we can obtain(62)At=1e-∫tTρsds+∫tTe-∫tτρsdsdτFrom the above analysis, we need to know the specific expressions of g(t,p) to get the explicit expression of effort e(t) and consumption c(t).Let us expandg(t,p) at p=0, (63)gt,p=gt,0+gpt,0p+gppt,02p2+⋯+gnt,0n!pn+⋯≜A~t+B~tp+C~tp2+⋯Here we consider a simple situation; according the principal’s utility function U=-e-λ(e-c) and dp(t)/dt=ρ(t)p(t)+e(t), suppose the structure of the solution of equation (61) is g(t,p)=A~(t)+B~(t)p with the terminal condition A~(T)=0 and B~(T)=ρ(T).Lemma 6.
Suppose the structure of solution of (61) is g(t,p)=A~(t)+B~(t)p, with the terminal condition A~(T)=0 and B~(T)=1, and then A~(t) and B~(t) are, respectively, the solutions of the following differential equations:(64)A~t=-∫tTe-∫tτAsds·DτdτB~t=ρTe∫tTρs-Asdswhere D(τ)=1/λ(δ¯-A(τ)+Aτln⁡A(τ))+A2(τ)+B~2(τ)/2A(τ).It is proved as follows.Substitutingg(t,p)=A~(t)+B~(t)p into (61), we calculate to get (65)∂tA~-AA~-1λδ¯-A+Aln⁡A-A2+B~22A+∂tB~-AB~+ρB~p=0The following two differential equations can be obtained by eliminating the dependence on p:(66)∂tA~-AA~-1λδ¯-A+Aln⁡A-A2+B~22A=0∂tB~-AB~+ρB~=0Lemma 6 conclusion can be provided by solving the above two differential equations.Proposition 7.
When the principal is time-inconsistent, the expressions of optimal effort and consumption for the agent are(67)e∗t=1-B~tAtc∗t=e∗t+ln⁡Atλ+Atqt+A~t+B~tptwhere A(t), A~(t), and B~(t) are given by (62) and (64), respectively.It can be seen from the proposition that the second optimal effort is less than the first optimal effort. The optimal consumption is the linear function of the agent’s promise value and private information.


## 5. Numerical Simulation

In this section, we provide a numerical simulation to characterize the dynamic behavior of the optimal portfolio strategy derived in the previous section. Firstly, an optimal effort numerical simulation is performed when the principal was time-consistent.As shown in Figure1, the discount rate of the agent is taken as the constant discount rate, namely, ρ(t)=ρ¯. The optimal effort, under different volatility, is reduced with the increase of volatility. It also shows that the greater the uncertainty, the lower the efforts of the agent. In addition, the three curves are almost declining, indicating that effort is a decreasing function of time.Figure 1
The efforts of constant discount rate for different uncertaintyσ.If we take the discount rate of the agent asdρ(t)/dt=ρ¯-ρ(t), i.e., ρ(t)=ρ¯+e-t, where ρ¯=0.1 and σ=0.1; 0.2; 0.5, respectively. The curves of effort variation are drawn in Figure 2.Figure 2
The efforts of variable discount rate for different uncertaintyσ.Analogy with Figure1, although the discount function is different, we still can get the similar conclusion, which means that the greater the uncertainty for any type of agent is (whether he is time-consistent or time-inconsistent), the less effort he provides. The reason is that in the case of moral hazard, the principal cannot distinguish the influence of the agent’s efforts and uncertainty on the risk project’s return.Next, we simulate the optimal consumption (salary) under the specific parameters. Letρ(t)=ρ¯=δ¯, λ=ϕ/hρ, and σ2=0.25. According to the expression of φ, we have (68)φt=1-eρt-T2ρ2+σ2Since q is a stochastic process, the mathematical expectation of q can be expressed as (69)qt=e-∫tTρ-hs/ϕsds·vCT-E∫tTe-∫tτρ-hs/ϕsds·eτ-eτ22-hτϕτφτdτwhere q(T)=v(c(T))≜a+ln⁡c(T) within a constant a. Then we substitute the expressions of e,q and φ into the above expression; the relationship between c(t) and time t can be simulated with the terminal condition c(T)=1.Consumption is initially diminishing because the effort is a function that decreases monotonically over time. After falling to a certain value, the bottoming out of consumption rose and this trend can be explained by the value of terminal condition of consumption setting. The overall consumption trend is shown in Figure3.Figure 3
The consumption of constant discount rate for different uncertaintyσ.As shown in Figure3, in the period of time just after the contract has been performed (for example t∈[0,5]), the greater the volatility, the lower the consumption. The reason is that greater volatility leads to lower efforts. In the latter part of the contract, the situation is just the opposite.Finally, we simulate the optimal effort trend when the principal is time-inconsistent. Assuming that the discount rate for the agent is the constant discount rate, then two different effort curves are both horizontal lines, as Figure4 shows. This indicates that, under the established discount rate, the optimal effort does not change over time. The reason is that we hypothesize the value function of principal as an exponential function under private information. And also, the effort is a decreasing function of the agent discount rate, which means that if the agent pays more attention to the present value (timely enjoyment), the higher discount rate and the less effort provided.Figure 4
The efforts with time-inconsistent principle for different uncertaintyρ.


## 6. Conclusion

In this paper, we were interested in a time-inconsistent principal-agent problem under full information and moral hazard framework. In particular, the optimal contracts we discussed in details assume that the principal is risk aversion and the agent is risk-neutral. There are two main works we have done in the paper. First, we made the technical processing of the time-inconsistent principal and agent, respectively. We transformed the principal by the changing of the time-varying discount rate into the time-consistent. And we used the Markov subgame perfect Nash equilibrium method to get the time consistency strategy for the time-inconsistent agent. Second, we used the Legendre transform duality theory to transform the HJB equation into the dual equation. The solution of the original HJB equation was obtained smoothly, thus obtaining the explicit expression of the optimal effort and optimal consumption. Under moral hazard, we also obtained the exact solution of the original HJB equation by using the guessing solution. We found that the optimal consumption of agent is a linear function of promised valueq and private information p. The optimal effort is the function of the agent discount rate. Eventually, we considered the contractual relationship between the principal and the agent in a special circumstances. The more general situations of time-inconsistent contracts should be considered in future research.



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