# Optimal R&D Investment Strategy of Pollution Abatement and Incentive Mechanism Design under Asymmetric Information

Authors: Chao Li; Na Zuo
Journal: Discrete Dynamics in Nature and Society
 (2021)

---

## Abstract

The study examines optimal pollution control R&D investment strategy of firms under asymmetric information and further analyzes the impact of government incentive mechanism on it. We use stochastic optimal control theory to get the exact solution of R&D investment strategy and incentive mechanism. Our analysis reveals that if there is no supervisor, firms choose not to cooperate, but the government can take appropriate incentive compensation to make firms reach Nash equilibrium. If there are supervisors, the optimal strategy of the enterprise is to choose cooperation, and there will be Pareto optimum among the firms. Furthermore, the R&D investment level decreases with increasing environmental uncertainty.


---

## Body

## 1. Introduction

With the increasingly serious problem of environmental pollution, many countries attach great importance to the treatment of environmental pollution. On the one hand, the government is increasing investment in pollution control; on the other hand, a taxation and compensation mechanism is established for environmental pollution, firms are guided to carry out technological innovation activities, energy efficiency is improved, and pollution emissions are increased ( see [1–5]). In fact, at present, China’s high-pollution energy-consuming firms are in a transition period, the government incentive mechanism is in the development stage, and there is a certain imbalance in the regulatory system; in addition, companies are usually far away from the daily management scope of the government, making the problem of information asymmetry prominent. Therefore, it is more likely to cause “stealing behavior” and “free rider behavior” of firms, which will affect the efficiency of pollution control (see [6–10]).Many scholars have reached different conclusions on the impact of government environmental policies on firms’ investment decisions on emission reduction. For example, the authors in [11] analyzed the impact of environmental policies on pollution control under an infinite time and stable random framework. The research proves that under the parameterization of the real model, the optimal environmental policy subsidies increase with the increase in uncertainty rise. The authors in [12] studied the impact of pollution emission trading policies and low-carbon subsidy policies on manufacturers’ production and carbon emission reduction levels and discussed which policies are better for society. The results show that the amount of carbon emission reduction increases with the increase in carbon trading volume, and the price increases, regardless of the unit low-carbon subsidy. Interestingly, the carbon trading price does not always have a negative impact on the manufacturer’s profits. The authors in [13] studied the impact of R&D taxation and subsidy environmental policies on multinational firms, and they proved that more generous tax incentives in one country increase MNEs’ R&D investments and emission reduction investments in affiliates located there. Further research shows that when the environmental damage coefficient is less than the threshold, the low-carbon subsidy policy is more beneficial to society, while in other cases, pollution emissions policies are more beneficial. However, the study of incentive mechanism for emission reduction investment has failed to consider firm’s unverifiable private information.To fill the aforementioned gap in the literature, this paper proposes a government firm principal-agent model under the asymmetric information framework, which aims to study the optimal pollution control R&D investment of firms and the optimal incentive compensation mechanism of the government in order to maximize the social welfare function. Assuming that there is a government and multiple firms in the society and the government and firms are risk aversion, the increase in pollution control investment by firms can increase the amount of pollution reduction. First, we study how the government designs an incentive mechanism when each enterprise invests separately so that there is a Nash equilibrium investment strategy between firms. Then, we add a supervisor to the model. The supervisor regards all firms as a whole, and we will prove that there is a Pareto optimal investment strategy between firms. This paper emphasizes the R&D investment of two firms (which can be extended to multiple firms) under the government’s incentive mechanism. Furthermore, the conclusions of our model can help to study the cooperative emission reduction mechanism of upstream and downstream firms in the green supply chain.The structure of our paper is as follows. Section2 presents the continuous-time multifirm model. We discuss the design of the optimal compensation under the hidden-R&D investment case in Section 3, and we analyze the optimal compensation for some special cases through the numerical simulation. Finally, we conclude this paper in Section 4.


## 2. The Model

In this section, we first define the dynamics of the pollution reduction process of firms. Second, we define the objective function of each enterprise, as well as one government.In our model, we assume that there areN firms and one government, in which government acts as principal and firms act as agents. To better describe the principal-agent relationship, let us first give some notations.


### 2.1. Pollution Reduction Process

We consider a finite time horizon0,T, and all pollution reduction processes are uncertain; therefore, we give a probability space Ω,ℱ,ℙ and on which define an N-dimensional Brownian motion Bt=Bt1,…,BtN⊤t∈0,T, and ℱt is a suitably augmented filtration generated by the Brownian motion Bt.Next, let us define the pollution emission reduction per unit of productYt (pollution reduction processes) which evolves over time as follows:(1)dYt=ΣtdBt,where Yt=Yt1,…,YtN⊤ is an N-dimensional outcome vector at time t, and each process Ytjj=1,…,N is influenced by only one uncertainty σj. Namely,(2)Σt=σ10⋯00σ2⋯0⋮⋮⋱⋮00⋯σN,Σt is a N×N volatility matrix, and we assume that Σt is bounded and invertible.If every firm does not participate in pollution abatement R&D investment, then the pollution emission reduction will be zero, that is,EYt=0. ui,j is firm j’s emissions abatement R&D investment on product i.And the parametersai,j measure the emissions abatement R&D investment efficiency of firm i on process j and denote matrix A=a11⋯a1N⋮⋱⋮aN1⋯aNN.Ifai,j=0 or ui,j=0,i≠j, it means that firm i’s emissions abatement R&D investment affects only its own emission reduction rate.According to the above analysis, by pollution abatement R&D investment, the firms joint changes the measure fromℙ to ℙu, with(3)dℙudℙ=exp∫0TΣ−1bt,A,ut⊤dBt−12∫0TΣ−1bt,A,ut2dt≜Λ∫0TΣ−1bt,A,ut⊤dBt=ΛTu,where b:0,T×RN×RN⟶R is a bounded and continuous function of u and Lipschitz uniformly continuous with respect to Yt.By Girsanov’s theorem, underℙu,(4)Btu=Bt−∫0tΣ−1bs,A,usds,0≤t≤T,which is a vector of Brownian motion.Therefore, we can rewrite (1) as follows:(5)dYt1⋮dYtN=a11ut1,1+⋯+a1Nut1,N⋮aN1utN,1+⋯+aNNutN,N+Σ⋅dBtu,1⋮dBtu,N.Then, the total emission reduction of all products isYt1+⋯+YtN at time t.We consider two information structures: full information and hidden-R&D investment. Full information means that all the information in the model are symmetric and exact, while under hidden-R&D investment, the government can monitor the emission reduction processYt but cannot observe the R&D investments of all firms or the uncertainty. So, the government cannot distinguish the level of influence of R&D investment and uncertainty on the emission reduction process.


### 2.2. The Objective Function of Each Firm

Suppose that the government gives firms one-time compensation for R&D investment, an amountC=C1,…,CN at time T. At time 0, the government issued an incentive policy to encourage firms to participate in emission abatement R&D investment. For each firm, choose whether to accept the incentive mechanism; if the firm accepts the incentive mechanism, then he will choose a ut to maximize his utility. The firms’ utility is composed of government compensation and the cost of R&D investment.Therefore, we assume that the utilities of the firms are exponential; fori=1,…,N, the utility function of firm i is(6)UiCi,ui=Eℙu−e−riCi−GTi,where GTi represents the cumulative cost of firm i’s R&D investment and defines as follows:(7)GTi=∫0Tgiutidt=∫0T∑j=1Ngj,iutj,idt=∫0T∑j=1Nki2utj,i2dt,ri>0 is a risk aversion constant.


### 2.3. The Objective Function of Government

The government wants to maximize his expected utility function by choosing a suitable compensation and a recommended R&D investment. We defined the expected utility function for government by(8)UPYT,C=Eℙu−e−rP∑i=1NK⋅YTi−Ci,where rP>0 is the risk aversion of the government and K is the social benefit per unit of emission reduction; without loss of generality, we can assume K=1. The suitable compensation C¯=C¯1,…,C¯N and a recommended R&D investment u¯ should meet the following three conditions:(1)
The suitable compensationC¯, to maximize his expected utility, is(9)C¯∈argmaxCUPYT,C.(2)
The government wants each firm to participate in R&D investment; this means that the compensation provided by the government should ensure that the utility of each firm is not less than its reservation utilityU¯i (Participation Constraint), i.e., for i=1,…,N,(10)UiCi,ui≥U¯i.(3)
Under hidden-R&D investment, the government sets a recommend R&D investmentu¯ but cannot directly verify if the firms were to deviate and thus should structure the compensation C to provide incentives for firms to meet their R&D investment target (Incentive Constraint), i.e., for i=1,…,N,(11)u¯i∈argmaxuiUiCi,ui.


## 2.1. Pollution Reduction Process

We consider a finite time horizon0,T, and all pollution reduction processes are uncertain; therefore, we give a probability space Ω,ℱ,ℙ and on which define an N-dimensional Brownian motion Bt=Bt1,…,BtN⊤t∈0,T, and ℱt is a suitably augmented filtration generated by the Brownian motion Bt.Next, let us define the pollution emission reduction per unit of productYt (pollution reduction processes) which evolves over time as follows:(1)dYt=ΣtdBt,where Yt=Yt1,…,YtN⊤ is an N-dimensional outcome vector at time t, and each process Ytjj=1,…,N is influenced by only one uncertainty σj. Namely,(2)Σt=σ10⋯00σ2⋯0⋮⋮⋱⋮00⋯σN,Σt is a N×N volatility matrix, and we assume that Σt is bounded and invertible.If every firm does not participate in pollution abatement R&D investment, then the pollution emission reduction will be zero, that is,EYt=0. ui,j is firm j’s emissions abatement R&D investment on product i.And the parametersai,j measure the emissions abatement R&D investment efficiency of firm i on process j and denote matrix A=a11⋯a1N⋮⋱⋮aN1⋯aNN.Ifai,j=0 or ui,j=0,i≠j, it means that firm i’s emissions abatement R&D investment affects only its own emission reduction rate.According to the above analysis, by pollution abatement R&D investment, the firms joint changes the measure fromℙ to ℙu, with(3)dℙudℙ=exp∫0TΣ−1bt,A,ut⊤dBt−12∫0TΣ−1bt,A,ut2dt≜Λ∫0TΣ−1bt,A,ut⊤dBt=ΛTu,where b:0,T×RN×RN⟶R is a bounded and continuous function of u and Lipschitz uniformly continuous with respect to Yt.By Girsanov’s theorem, underℙu,(4)Btu=Bt−∫0tΣ−1bs,A,usds,0≤t≤T,which is a vector of Brownian motion.Therefore, we can rewrite (1) as follows:(5)dYt1⋮dYtN=a11ut1,1+⋯+a1Nut1,N⋮aN1utN,1+⋯+aNNutN,N+Σ⋅dBtu,1⋮dBtu,N.Then, the total emission reduction of all products isYt1+⋯+YtN at time t.We consider two information structures: full information and hidden-R&D investment. Full information means that all the information in the model are symmetric and exact, while under hidden-R&D investment, the government can monitor the emission reduction processYt but cannot observe the R&D investments of all firms or the uncertainty. So, the government cannot distinguish the level of influence of R&D investment and uncertainty on the emission reduction process.


## 2.2. The Objective Function of Each Firm

Suppose that the government gives firms one-time compensation for R&D investment, an amountC=C1,…,CN at time T. At time 0, the government issued an incentive policy to encourage firms to participate in emission abatement R&D investment. For each firm, choose whether to accept the incentive mechanism; if the firm accepts the incentive mechanism, then he will choose a ut to maximize his utility. The firms’ utility is composed of government compensation and the cost of R&D investment.Therefore, we assume that the utilities of the firms are exponential; fori=1,…,N, the utility function of firm i is(6)UiCi,ui=Eℙu−e−riCi−GTi,where GTi represents the cumulative cost of firm i’s R&D investment and defines as follows:(7)GTi=∫0Tgiutidt=∫0T∑j=1Ngj,iutj,idt=∫0T∑j=1Nki2utj,i2dt,ri>0 is a risk aversion constant.


## 2.3. The Objective Function of Government

The government wants to maximize his expected utility function by choosing a suitable compensation and a recommended R&D investment. We defined the expected utility function for government by(8)UPYT,C=Eℙu−e−rP∑i=1NK⋅YTi−Ci,where rP>0 is the risk aversion of the government and K is the social benefit per unit of emission reduction; without loss of generality, we can assume K=1. The suitable compensation C¯=C¯1,…,C¯N and a recommended R&D investment u¯ should meet the following three conditions:(1)
The suitable compensationC¯, to maximize his expected utility, is(9)C¯∈argmaxCUPYT,C.(2)
The government wants each firm to participate in R&D investment; this means that the compensation provided by the government should ensure that the utility of each firm is not less than its reservation utilityU¯i (Participation Constraint), i.e., for i=1,…,N,(10)UiCi,ui≥U¯i.(3)
Under hidden-R&D investment, the government sets a recommend R&D investmentu¯ but cannot directly verify if the firms were to deviate and thus should structure the compensation C to provide incentives for firms to meet their R&D investment target (Incentive Constraint), i.e., for i=1,…,N,(11)u¯i∈argmaxuiUiCi,ui.


## 3. Implementable Compensations

In this paper, we only consider the case which the firms’ R&D investments cannot be observed by the government (hidden-R&D investment). Compared with the one-agent model of [14], in our paper, the firms’ R&D investments will affect each other’s utility. Therefore, given a compensation, we should have to find an equilibrium resulting of the interactions between the firms. In this section, we assume that the firms are acting simultaneously and we will find a Nash equilibrium or a Pareto equilibrium.


### 3.1. Incentive Compatibility without Supervisor

To solve the optimization problem for the government in Section3.2, we define the Nash equilibrium R&D investments and Pareto optimal R&D investments for the firms first, and next we find the sufficient and necessary conditions for optimality and finally maximize government’s utility with the implementable compensation.Definition 1 (refer [15, 16] for the same definition).
Given a compensationC=C1,…,CN, we define u¯1,…,u¯N as a Nash equilibrium R&D investment for firms; for i=1,…,N, we have(12)UiCi,u¯i,u¯−i=supui∈AiUiCi,ui,u¯−i.
According to the game theory, Nash equilibrium is not optimal in Prisoners’ Dilemma. And in the multiagent model, the prisoner’s dilemma may be present. Therefore, in this paper, we will focus on Pareto optimality. We give the definition of Pareto optimal as follows.Definition 2 (see [17]).
Given a compensationC, we define u¯ is a Pareto optimal R&D investment for firm if it does not exist u such that UiC,u¯≥UiC,u, for i=1,…,N, and UiC,u¯>UiC,u for at least one i.
We will discuss the existence of Pareto’s optimal R&D investment in Section3.3.
In order to find the Nash equilibrium R&D investment, we need to do some mathematical analysis. Given a compensationC=Ci,C−i and the other firm’s R&D investment strategy u−i=u1,…,ui−1,ui+1,…,uN∈A−i, the firm i aims solving(13)U0iCi,ui,u−i≜supu∈AEℙu−e−riCi−∫0Tgiusids,and for any u∈A and any t∈0,T, define the value function of the firm i, i=1,2, by(14)UtiCi,ui,u−i≜Eℙu−e−riCi−∫tTgiusids|ℱt.
Then, we know thateri∫0tgiusids⋅UtiCi,ui,u−i is an F,ℙu-martingale. By the martingale representation property, there is a F-predictable process Z˜i=Z˜1,i,Z˜2,i,…,Z˜N,i such that(15)eri∫0tgiusids⋅UtiCi,ui,u−i=eri∫0Tgiusids⋅UTiCi,ui,u−i−∫tTZ˜iΣsdBsui,u−i,and using Ito’s formula, we have(16)UtiCi,ui,u−i=−e−riCi+∫tTrigiusiUsiCi,ui,u−ids−∫tTe−ri∫0sgiuτidτZ˜siΣsdBsui,u−i,0≤t≤T.
By the definition ofBtu,(17)UtiCi,ui,u−i=−e−riCi+∫tTriUsiCi,ui,u−iZsiΣsdBs−∫tTriUsiCi,ui,u−iZsibs,A,us−giusids,0≤t≤T,where Zi=−e−ri∫0tgiusidsZ˜i/riUtiCi,ui,u−i.
We define that(18)Xti,ui≜−ln−UtiCi,ui,u−iri,0≤t≤T.
By Ito’s formula, for anyt∈0,T,(19)Xti,ui=Ci−∫tTZsiΣsdBs+∫tT−ri2ZsiΣs2+Zsibs,A,us−giusids,and next we define a map fi:0,T×RN⟶R by (in this paper, the symbol Mn represents the sum of the n-th power of the elements of the matrix M, which means that Mn=∑i,jmi,jn, n=1,2,…)(20)fis,Zsi=−ri2ZsiΣs2+supui∈AiZsibs,A,us−giusi.
Then, a BSDE with terminal conditionXTi=Ci is defined as follows:(21)Xti=Ci+∫tTfis,Zsids−∫tTZsiΣsdBs.
The link between existence of Nash equilibrium R&D investments between the firms and existence of solutions to BSDE (21) is given in the following lemma.Lemma 1 (see Theorem 4.1 in [15]).
There exists a Nash equilibrium R&D investmentu¯t1,u¯t2 if and only if it is a unique solution of BSDE (21).Lemma1 indicates the existence of a Nash equilibrium R&D investment, which is connected to the existence of a unique solution of BSDE (21). As a consequence of Lemma 1, we know that for Ci, i=1,…,N, there exist a pair X0i,Zi with(22)Ci=X0i−∫0Tfis,Zsids+∫0TZsiΣsdBs.Becausefi is quadratic linearity about ui, taking the first order derivative will obtain the optimal R&D investment:(23)u¯s=u¯s1⋯u¯sN=u¯s1,1⋯u¯s1,N⋮⋱⋮u¯sN,1⋯u¯sN,N=a11k1Zs1,1⋯a1NkNZs1,N⋮⋱⋮aN1k1ZsN,1⋯aNNkNZsN,N.


### 3.2. Problem of the Government

The goal of the government is to maximize its utility function and recall the government's utility function:(24)UPYT,C=Eℙu¯Z−e−rP⋅∑i=1NYTi−Ci.According equations (5) and (22), we get(25)UPYT,C=Eℙu¯Z−e−rP∫0Tbs,A,u¯sds+∫0TΣsdBsu¯⋅e−rP−X0−12∫0TrZsΣs2+ku¯s2ds−∫0TZsΣsdBsu¯=Eℙu¯Z−e−rP∫0TΣs−ZsΣsdBsu¯+rP2∫0TΣs−ZsΣs2ds⋅e−rP∫0Tbs,A,u¯s−X0−12rZsΣs2+ku¯s2−rP2Σs−ZsΣs2ds=Eℙu¯Z−e−rP∫0TΣs−ZsΣsdBs−rP2∫0TΣs−ZsΣs2ds⋅e−rP∫0Tbs,A,u¯s−X0−12rZsΣs2+ku¯s2−rP2Σs−ZsΣs2ds=Eℙu¯Z−Λ−rP⋅∫0TΣs−ZsΣsdBs⋅e−rP⋅∫0Tgs,Zsds,where(26)gs,Zs=∑j=1N∑i=1Naji2kiZj,i−∑i=1NX0i−∑j=1N∑i=1Naji22kiZj,i2−∑j=1Nσj22⋅∑i=1NriZj,i2−rP2⋅∑j=1Nσj2⋅1−∑i=1NZj,i2.The above process shows that the government maximizing its utility function is equivalent to the maximizing functiong, and it is easy to computations that g is convex about Zj,i. Taking the first derivative with respect to Zj,i,(27)rPσj2⋅∑l=1NZj,l−1+σj2riZj,i+aji2kiZj,i=aji2ki.That is, for eachj=1,…,N, Zj,i satisfies the following linear equations:(28)rPσj2+σj2r1+aj12k1Zj,1+rPσj2Zj,2+⋯+rPσj2Zj,N=aj12k1+rPσj2rPσj2Zj,1+rPσj2+σj2r2+aj22k2Zj,2+⋯+rPσj2Zj,N=aj22k2+rPσj2⋮rPσj2Zj,1+rPσj2Zj,2+⋯+rPσj2+σj2rN+ajN2kNZj,N=ajN2kN+rPσj2.Therefore, we can solve the optimal control variable denoted byZNAj,i as follows:(29)ZNAj,1ZNAj,2⋮ZNAj,N=rPσj2+σj2r1+aj12k1rPσj2⋯rPσj2rPσj2rPσj2+σj2r2+aj22k2⋯rPσj2⋮⋮⋱⋮rPσj2rPσj2⋯rPσj2+σj2rN+ajN2kN−1⋅aj12k1+rPσj2aj22k2+rPσj2⋮ajN2kN+rPσj2.According to the above derivation and equation (22), we have the following result.Proposition 1.
The Nash equilibriumR & D investment for the firms is as follows:(30)uNA=uNA1uNA2⋯uNAN=a11k1ZNA1,1a12k2ZNA1,2⋯a1NkNZNA1,Na21k1ZNA2,1a22k2ZNA2,2⋯a2NkNZNA2,N⋮⋮⋱⋮aN1k1ZNAN,1aN2k2ZNAN,2⋯aNNkNZNAN,N,and the optimal utility of the government is(31)UPYT,C=−e−rPTgNAZNA,and the optimal compensations given to firm i are(32)CNAi=Li+riT2⋅∑j=1NσjZNAj,i2+T2ki⋅∑j=1NajiZNAj,i2+∑j=1NZNAj,iYTj−T⋅∑j=1NZNAj,i⋅∑l=1Najl2klZNAj,l.Through the above analysis and deduction, we have obtained the exact expressions of the Nash optimal R&D investments and the optimal incentive compensations when the firms are rational and noncooperative and obtained the government’s expected utility in this case.


### 3.3. Incentive Compatibility in the Presence of Supervisor

In this section, we will consider the case where the firm is a unit. That is, we assume that the government hires a supervisor to manage the firms’s investments and compensations. The supervisory considers the firms’ expected utility as an overall expectation function, then finds the optimal R&D investments for the firms, and imposes the optimal R&D investments to the firms.Obviously, when the supervisor considers the expected utility of firms as a whole (the sum of nonlinear functions), we will face a multiobjective optimization problem. Before analyzing the multiobjective optimization problem, we will give the following lemma.Lemma 2.
There areN real numbers λi≥0, and they satisfied ∑i=1N=1; if u^=u^1,…,u^N is a solution to the following problem:(33)supu∈AEℙuλ1U1+λ2U2+⋯+λNUN,then u^ is Pareto optimal.We denote that the Pareto optimal solutionu^=uPA=uPA1,…,uPAN. And we select a special weight λ=1/N (it is reasonable that we choose λ=1/N because, in the team, both firms are equally “significant”). Therefore, the multiobjective optimization problem is as follows:(34)supu∈AEℙu1NU1+1NU2+⋯+1NUN=1Nsupu∈AEℙuU1+U2+⋯+UN⇔supu∈AEℙuU1+U2+⋯+UN.The optimization problem that the supervisor considers is as follows.Problem 1.
(multiobjective optimization problem). The utility function of the firmi is Ui. When all firms cooperate as a whole, the supervisor’s problem is to choose an appropriate R&D investment after a given compensation to maximize the expected utility of the team:(35)supu∈AEℙuU1+U2+⋯+UN=supu∈AEℙu−e−r1C1−GT1−er2C2−GT2−⋯−erNCN−GTN,such that(36)1dYt=bt,A,utdt+ΣdBtu,2supu∈AEℙuU1+U2+⋯+UN≥−e−r1L1+e−r2L2+⋯+e−rNLN.
According to equation (18), for i=1,…,N, X0i,ui=−ln−U0iC,ui,u−i/ri; therefore,(37)supu∈AEℙuU1+U2+⋯+UN=supu∈AEℙu−e−r1X01,u1−er2X02,u2−⋯−erNX0N,uN.
To solve the optimal solution of problem (37), we will apply Boach's theorem [18], as follows:Lemma 3 (Borch theorem).
A set of deterministic equivalentX^01,u1,X^02,u2,…,X^0N,uN maximizes problem (37) if and only if r1e−r1X^01,u1=r2e−r2X^02,u2=⋯=rNe−rNX^0N,uN. And X^01,u1,X^02,u2,…,X^0N,uN is Pareto optimal.A brief proof of Borch’s theorem can be found in [19].We denote thatC^=C^1,C^2,…,C^N is Pareto optimal compensation and u^=u^1,u^2,…,u^N is Pareto optimal R&D investment. According Lemma 3, we have(38)−r1e−r1C^1−G^T1u^1=−r2e−r2C^2−G^T2u^2=⋯=−rNe−rNC^N−G^TNu^N.Similar to the analysis in Section3.1, we know that there is a set of F-predictable process ZPA1,ZPA2,…,ZPA2 and a set of deterministic equivalent X^01,X^02,…,X^0N that satisfy the following conditions:(39)C^1=X^01−∫0T−r12ZPA1Σ2+ZPA1bs,A,u^−g1u^1ds+∫0TZPA1ΣdBsC^2=X^02−∫0T−r22ZPA2Σ2+ZPA2bs,A,u^−g2u^2ds+∫0TZPA2ΣdBs⋮C^N=X^0N−∫0T−rN2ZPANΣ2+ZPANbs,A,u^−gNu^Nds+∫0TZPANΣdBs−r1e−r1C^1−G^T1u^1=−r2e−r2C^2−G^T2u^2=⋯=−rNe−rNC^N−G^TNu^N.From the above conditions, we can deduce that(40)r1ZPA1=r2ZPA2=⋯=rNZPAN,r1X^01−lnr1=r2X^02−lnr2=⋯=rNX^0N−lnrN.According to equation (30), ui is a linear function of ZNAi; further, according to equation (29), ZNAi is linear with respect to ZNA−i. Therefore, we can assume that for every i=1,2,…,N, we have ui=ϕiZPA1,ZPA2,…,ZPAN. While on the basis of equation (40), we can assume that there are N×N linear functions φj,ij,i=1,2,…,N, and ui=φj,iZPAj. Moreover, assume that there are N−1×N−1 linear functions ψj,i (j,i=1,2,…,N, and j≠i), such that ui=ψj,iuj (or uj=ψj,1−1u1=⋯=ψj,N−1uN).Next, we will discuss the specific functional relationship betweenu and ZPA and find an explicit expression of ZPA.We first use lexicographic ordering method (the lexicographic ordering method is a method to find Pareto optimal solution; for more details on the lexicographic ordering method, see [20]) to translate a multiobjective optimization problem into a multiobjective optimization problem. Thus, Problem 1 translates into the following.Problem 2 (lexicographic order problem).
(41)supu∈AEℙuU1C1,u1such that:supu∈AEℙuU2C2,u2⋮supu∈AEℙuUNCN,uNdYt=a11u1,1+a12ψ1,2u1,1+⋯+a1,Nψ1,Nu1,1⋮aN1uN,1+aN2ψ1,2uN,1+⋯+aNNψ1,NuN,1dt+ΣdBtuEℙuU1C1,u1≥−e−r1L1⋮EℙuUNCN,uN≥−e−rNLN.
We have the following lemma to describe the relationship between the solutions of Problems1 and 2.Lemma 4.
The solution of Problem2 (lexicographic order problem) is Pareto solution of Problem 1 (multiobjective optimization problem).For the proof details of the lemma, see [21].Following the method of solving the optimal R&D investment in Section3.1, we can obtain the optimal R&D investment for the first single-objective problem in lexicographic ordering problem.(42)ZPA1a1+a2ψ1,2′+⋯+aNψ1,N′=k1u^1.Here,aj=a1ja2j⋯aNJ⊤ and u^1=u^1,1u^2,1⋯u^N,1⊤.Similarly, the optimal R&D investment of the second single-objective problem is as follows:(43)ZPA2a1ψ1,2−1′+a2+⋯+aNψ2,N′=k2u^2.By doing so, we can get the optimal R&D investment up to theN single-objective problem:(44)ZPANa1ψ1,N−1′+a2ψ2,N−1′+⋯+aN=kNu^N.According to equations (42)–(44), we can get the conditions r1ZPA1=r2ZPA2=⋯=rNZPAN. According to the linear property of ψj,I, we know that ψj,i−1′=ψj,i′−1. After calculation, we can get the following system of differential equations: 1≤j<i≤N.(45)ψj,iu^jψj,i′u^j=rjkjrikiu^j.So, if we solve those differential equations, we getu^i=ψj,iu^j=rjkj/rikiu^j.Substituting the above formula into equations (42)–(44), we can get Pareto optimal R&D investment u^ as follows:(46)u^1u^2⋯u^N=∑l=1Na1lr1k1/rlklk1ZPA1,1∑l=1Na1lr2k2/rlklk2ZPA1,2⋯∑l=1Na1lrNkN/rlklkNZPA1,N∑l=1Na2lr1k1/rlklk1ZPA2,1∑l=1Na2lr2k2/rlklk2ZPA2,2⋯∑l=1Na2lrNkN/rlklkNZPA2,N⋮⋮⋱⋮∑l=1NaNlr1k1/rlklk1ZPAN,1∑l=1NaNlr2k2/rlklk2ZPAN,2⋯∑l=1NaNlrNkN/rlklkNZPAN,N.Next, we will solve for the exact expression of Pareto’s optimal R&D investment. We recall the government’s problem as follows:(47)UPAPYT,C=supC1,C2,…,CNEℙu−e−rP∑j=1NYTj−Cj.Similar to the derivation in Section3.2, we have(48)UPAPYT,C=supZEℙu−Λ−rP⋅∫0TΣs−ZsΣsdBs⋅e−rP⋅∫0TgPAs,Zsds,with(49)gPAs,ZPA1=−∑j=1NX^0j+∑j=1N∑i=1Naj,i⋅∑l=1Najlriki/rlklki⋅ZPAj,i−∑j=1N∑i=1N∑l=1Najlriki/rlkl22ki⋅ZPAj,i2−∑j=1Nσj22⋅∑i=1NriZj,i2−rP2⋅∑j=1Nσj2⋅1−∑i=1NZj,i2.Obviously, the functiongPA is convex with respect to ZPAj,i. Therefore, taking the first derivative of gPA about ZPAj,i, we can obtain the exact solution of Pareto’s optimal R&D investment. The first-order condition of ZPAj,i is as follows:(50)rPσj2⋅∑l=1NZj,l−1+σj2riZj,i+∑l=1Najlriki/rlkl2kiZj,i=aji⋅∑l=1Najlriki/rlklki.That is, for eachj=1,2…,N, Zj,i satisfies the following linear equations:(51)ZPAj,1⋮ZPAj,N=rPσj2+σj2r1+∑l=1Najlr1k1/rlkl2k1⋯rPσj2⋮⋱⋮rPσj2⋯rPσj2+σj2rN+∑l=1NajlrNkN/rlkl2kN−1⋅aj1⋅∑l=1Najlr1k1/rlklk1+rPσj2⋮ajN⋅∑l=1NajlrNkN/rlklkN+rPσj2.We substitute the above optimal control variableZPAj,i into the expected utility function of the government and get the optimal utility of the government as follows:(52)UPAPYT,C=−e−rP⋅T⋅gPAZPA.In the case of supervision unit, the optimal incentive compensation obtained by firmi is as follows:(53)CPAi=X^0i+riT2⋅∑j=1NσjZPAj,i2+T2ki⋅∑j=1N∑l=1Najlrikirlkl⋅ZPAj,i2+∑j=1NZPAj,iYTj−T⋅∑j=1N∑κ=1Nrjajκrκkκ⋅∑l=1Najlrikirlkl⋅Zj,i2.Note: if−rje−rjLj=maxi=1,2,…,N−rie−riLi, then X^0j=Lj; when i≠j, X^0i=rj/riLj+1/rilnri/rj.Based on the above analysis, we have discussed the optimal incentive compensation for firms in the presence of a supervisor. Next, we will describe our results with special examples and numerical simulation methods.


### 3.4. Special Case

The results of theoretical analysis are shown more clearly and more intuitively, especially to show the different optimal compensations under the different conditions and the comparison between the different expected utilities of the government. In this subsection, we discuss the explicit expression of the optimal compensations under specific assumptions.Next, we perform numerical simulation analysis; assume thataij=a=1, ri=rP=r=0.5, ki=k=1, and Li=L=0 (for a description of the parameter assumptions, see [22]). Substituting these parameters into equations (28) and (51), then(54)ZNA1,1=ZNA1,2=σ14+4σ12+43σ14+8σ12+4,ZNA2,1=ZNA2,2=σ24+4σ22+43σ24+8σ22+4,ZPA1,1=ZPA1,2=σ12+43σ12+8,ZPA2,1=ZPA2,2=σ22+43σ22+8.According to the conclusions in Proposition1 and Problem 1, we can get the tendency of Nash equilibrium R&D investments uNA and Pareto optimal R&D investments uPA to change with uncertainty σ as shown in Figures 1 and 2.Figure 1
The Nash equilibrium R&D investments for firm 1.Figure 2
The Nash equilibrium R&D investments for firm 2.As can be seen from Figures1–4, the R&D investments provided by the firms are a decreasing function of uncertainty. When the uncertainty is 0 (full information), all R&D investments reach the maximum value of 1. Here, because we assume that there are no differences between the two firms, the corresponding R&D investments they provide are also the same.Figure 3
The Pareto optimal R&D investments for firm 1.Figure 4
The Pareto optimal R&D investments for firm 2.Figures5–10, respectively, depict that the expected utility of the government and the optimal compensations of the firms under two kinds of R&D investments and the comparison of the expected utility and optimal compensations under these two kinds of R&D investments.Figure 5
The functiong for the government with firms that choose Nash equilibrium strategy.Figure 6
The functiong for the government with firms that choose Pareto equilibrium strategy.Figure 7
The optimal compensation with firms that choose Nash equilibrium strategy.Figure 8
The optimal compensation with firms that choose Pareto equilibrium strategy.Figure 9
The comparison of optimal compensations.Figure 10
The comparison of the functiong.According to the analysis and numerical simulation of Section3, we can get the following conclusions. For the firms, the R&D investments provided by the firms are a decreasing function of the uncertainty factor σ; of course, the compensation received by the firm is also a decreasing function of the uncertainty factor. The firms that choose Pareto optimal R&D investment can improve their compensations compared with the case they choose the Nash equilibrium R&D investments. For the government, the firms that choose Pareto optimal R&D investments can improve the value of government, but if the government only hires one firm (in [17], the author refers to this condition as weak Pareto optimality), then the value of government cannot be improved compared with the case with the firms that choose Nash equilibrium R&D investments.From the above analysis, we can see that the firms that choose the Pareto optimal R&D investments are more beneficial to both the government and the firms. However, in general, firms will only consider their own interests and will not consider their coworkers’ interests together with their own interests. According to the analysis in [17], the government should hire a supervisor who manages the firm’s R&D investments and acts for the welfare of the firms. Therefore, there will be two problems: first, if the supervisor is exogenous, how does the government provide the supervisor with the optimal compensation; second, if the supervisor is endogenous (supervisor is the firm himself), how does the government design the optimal compensation for firms. There are more special cases; for example, in [16], if the government is risk-neutral and provided the linear compensations for the firms, then there will be a prisoner’s dilemma among firms. In other words, one firm chooses Pareto optimal R&D investment (cooperation) and another firm chooses Nash equilibrium R&D investment (independent work).Therefore, we should design a reasonable incentive mechanism to ensure that firms will choose Pareto optimal R&D investments (i.e., firms choose cooperation).


## 3.1. Incentive Compatibility without Supervisor

To solve the optimization problem for the government in Section3.2, we define the Nash equilibrium R&D investments and Pareto optimal R&D investments for the firms first, and next we find the sufficient and necessary conditions for optimality and finally maximize government’s utility with the implementable compensation.Definition 1 (refer [15, 16] for the same definition).
Given a compensationC=C1,…,CN, we define u¯1,…,u¯N as a Nash equilibrium R&D investment for firms; for i=1,…,N, we have(12)UiCi,u¯i,u¯−i=supui∈AiUiCi,ui,u¯−i.
According to the game theory, Nash equilibrium is not optimal in Prisoners’ Dilemma. And in the multiagent model, the prisoner’s dilemma may be present. Therefore, in this paper, we will focus on Pareto optimality. We give the definition of Pareto optimal as follows.Definition 2 (see [17]).
Given a compensationC, we define u¯ is a Pareto optimal R&D investment for firm if it does not exist u such that UiC,u¯≥UiC,u, for i=1,…,N, and UiC,u¯>UiC,u for at least one i.
We will discuss the existence of Pareto’s optimal R&D investment in Section3.3.
In order to find the Nash equilibrium R&D investment, we need to do some mathematical analysis. Given a compensationC=Ci,C−i and the other firm’s R&D investment strategy u−i=u1,…,ui−1,ui+1,…,uN∈A−i, the firm i aims solving(13)U0iCi,ui,u−i≜supu∈AEℙu−e−riCi−∫0Tgiusids,and for any u∈A and any t∈0,T, define the value function of the firm i, i=1,2, by(14)UtiCi,ui,u−i≜Eℙu−e−riCi−∫tTgiusids|ℱt.
Then, we know thateri∫0tgiusids⋅UtiCi,ui,u−i is an F,ℙu-martingale. By the martingale representation property, there is a F-predictable process Z˜i=Z˜1,i,Z˜2,i,…,Z˜N,i such that(15)eri∫0tgiusids⋅UtiCi,ui,u−i=eri∫0Tgiusids⋅UTiCi,ui,u−i−∫tTZ˜iΣsdBsui,u−i,and using Ito’s formula, we have(16)UtiCi,ui,u−i=−e−riCi+∫tTrigiusiUsiCi,ui,u−ids−∫tTe−ri∫0sgiuτidτZ˜siΣsdBsui,u−i,0≤t≤T.
By the definition ofBtu,(17)UtiCi,ui,u−i=−e−riCi+∫tTriUsiCi,ui,u−iZsiΣsdBs−∫tTriUsiCi,ui,u−iZsibs,A,us−giusids,0≤t≤T,where Zi=−e−ri∫0tgiusidsZ˜i/riUtiCi,ui,u−i.
We define that(18)Xti,ui≜−ln−UtiCi,ui,u−iri,0≤t≤T.
By Ito’s formula, for anyt∈0,T,(19)Xti,ui=Ci−∫tTZsiΣsdBs+∫tT−ri2ZsiΣs2+Zsibs,A,us−giusids,and next we define a map fi:0,T×RN⟶R by (in this paper, the symbol Mn represents the sum of the n-th power of the elements of the matrix M, which means that Mn=∑i,jmi,jn, n=1,2,…)(20)fis,Zsi=−ri2ZsiΣs2+supui∈AiZsibs,A,us−giusi.
Then, a BSDE with terminal conditionXTi=Ci is defined as follows:(21)Xti=Ci+∫tTfis,Zsids−∫tTZsiΣsdBs.
The link between existence of Nash equilibrium R&D investments between the firms and existence of solutions to BSDE (21) is given in the following lemma.Lemma 1 (see Theorem 4.1 in [15]).
There exists a Nash equilibrium R&D investmentu¯t1,u¯t2 if and only if it is a unique solution of BSDE (21).Lemma1 indicates the existence of a Nash equilibrium R&D investment, which is connected to the existence of a unique solution of BSDE (21). As a consequence of Lemma 1, we know that for Ci, i=1,…,N, there exist a pair X0i,Zi with(22)Ci=X0i−∫0Tfis,Zsids+∫0TZsiΣsdBs.Becausefi is quadratic linearity about ui, taking the first order derivative will obtain the optimal R&D investment:(23)u¯s=u¯s1⋯u¯sN=u¯s1,1⋯u¯s1,N⋮⋱⋮u¯sN,1⋯u¯sN,N=a11k1Zs1,1⋯a1NkNZs1,N⋮⋱⋮aN1k1ZsN,1⋯aNNkNZsN,N.


## 3.2. Problem of the Government

The goal of the government is to maximize its utility function and recall the government's utility function:(24)UPYT,C=Eℙu¯Z−e−rP⋅∑i=1NYTi−Ci.According equations (5) and (22), we get(25)UPYT,C=Eℙu¯Z−e−rP∫0Tbs,A,u¯sds+∫0TΣsdBsu¯⋅e−rP−X0−12∫0TrZsΣs2+ku¯s2ds−∫0TZsΣsdBsu¯=Eℙu¯Z−e−rP∫0TΣs−ZsΣsdBsu¯+rP2∫0TΣs−ZsΣs2ds⋅e−rP∫0Tbs,A,u¯s−X0−12rZsΣs2+ku¯s2−rP2Σs−ZsΣs2ds=Eℙu¯Z−e−rP∫0TΣs−ZsΣsdBs−rP2∫0TΣs−ZsΣs2ds⋅e−rP∫0Tbs,A,u¯s−X0−12rZsΣs2+ku¯s2−rP2Σs−ZsΣs2ds=Eℙu¯Z−Λ−rP⋅∫0TΣs−ZsΣsdBs⋅e−rP⋅∫0Tgs,Zsds,where(26)gs,Zs=∑j=1N∑i=1Naji2kiZj,i−∑i=1NX0i−∑j=1N∑i=1Naji22kiZj,i2−∑j=1Nσj22⋅∑i=1NriZj,i2−rP2⋅∑j=1Nσj2⋅1−∑i=1NZj,i2.The above process shows that the government maximizing its utility function is equivalent to the maximizing functiong, and it is easy to computations that g is convex about Zj,i. Taking the first derivative with respect to Zj,i,(27)rPσj2⋅∑l=1NZj,l−1+σj2riZj,i+aji2kiZj,i=aji2ki.That is, for eachj=1,…,N, Zj,i satisfies the following linear equations:(28)rPσj2+σj2r1+aj12k1Zj,1+rPσj2Zj,2+⋯+rPσj2Zj,N=aj12k1+rPσj2rPσj2Zj,1+rPσj2+σj2r2+aj22k2Zj,2+⋯+rPσj2Zj,N=aj22k2+rPσj2⋮rPσj2Zj,1+rPσj2Zj,2+⋯+rPσj2+σj2rN+ajN2kNZj,N=ajN2kN+rPσj2.Therefore, we can solve the optimal control variable denoted byZNAj,i as follows:(29)ZNAj,1ZNAj,2⋮ZNAj,N=rPσj2+σj2r1+aj12k1rPσj2⋯rPσj2rPσj2rPσj2+σj2r2+aj22k2⋯rPσj2⋮⋮⋱⋮rPσj2rPσj2⋯rPσj2+σj2rN+ajN2kN−1⋅aj12k1+rPσj2aj22k2+rPσj2⋮ajN2kN+rPσj2.According to the above derivation and equation (22), we have the following result.Proposition 1.
The Nash equilibriumR & D investment for the firms is as follows:(30)uNA=uNA1uNA2⋯uNAN=a11k1ZNA1,1a12k2ZNA1,2⋯a1NkNZNA1,Na21k1ZNA2,1a22k2ZNA2,2⋯a2NkNZNA2,N⋮⋮⋱⋮aN1k1ZNAN,1aN2k2ZNAN,2⋯aNNkNZNAN,N,and the optimal utility of the government is(31)UPYT,C=−e−rPTgNAZNA,and the optimal compensations given to firm i are(32)CNAi=Li+riT2⋅∑j=1NσjZNAj,i2+T2ki⋅∑j=1NajiZNAj,i2+∑j=1NZNAj,iYTj−T⋅∑j=1NZNAj,i⋅∑l=1Najl2klZNAj,l.Through the above analysis and deduction, we have obtained the exact expressions of the Nash optimal R&D investments and the optimal incentive compensations when the firms are rational and noncooperative and obtained the government’s expected utility in this case.


## 3.3. Incentive Compatibility in the Presence of Supervisor

In this section, we will consider the case where the firm is a unit. That is, we assume that the government hires a supervisor to manage the firms’s investments and compensations. The supervisory considers the firms’ expected utility as an overall expectation function, then finds the optimal R&D investments for the firms, and imposes the optimal R&D investments to the firms.Obviously, when the supervisor considers the expected utility of firms as a whole (the sum of nonlinear functions), we will face a multiobjective optimization problem. Before analyzing the multiobjective optimization problem, we will give the following lemma.Lemma 2.
There areN real numbers λi≥0, and they satisfied ∑i=1N=1; if u^=u^1,…,u^N is a solution to the following problem:(33)supu∈AEℙuλ1U1+λ2U2+⋯+λNUN,then u^ is Pareto optimal.We denote that the Pareto optimal solutionu^=uPA=uPA1,…,uPAN. And we select a special weight λ=1/N (it is reasonable that we choose λ=1/N because, in the team, both firms are equally “significant”). Therefore, the multiobjective optimization problem is as follows:(34)supu∈AEℙu1NU1+1NU2+⋯+1NUN=1Nsupu∈AEℙuU1+U2+⋯+UN⇔supu∈AEℙuU1+U2+⋯+UN.The optimization problem that the supervisor considers is as follows.Problem 1.
(multiobjective optimization problem). The utility function of the firmi is Ui. When all firms cooperate as a whole, the supervisor’s problem is to choose an appropriate R&D investment after a given compensation to maximize the expected utility of the team:(35)supu∈AEℙuU1+U2+⋯+UN=supu∈AEℙu−e−r1C1−GT1−er2C2−GT2−⋯−erNCN−GTN,such that(36)1dYt=bt,A,utdt+ΣdBtu,2supu∈AEℙuU1+U2+⋯+UN≥−e−r1L1+e−r2L2+⋯+e−rNLN.
According to equation (18), for i=1,…,N, X0i,ui=−ln−U0iC,ui,u−i/ri; therefore,(37)supu∈AEℙuU1+U2+⋯+UN=supu∈AEℙu−e−r1X01,u1−er2X02,u2−⋯−erNX0N,uN.
To solve the optimal solution of problem (37), we will apply Boach's theorem [18], as follows:Lemma 3 (Borch theorem).
A set of deterministic equivalentX^01,u1,X^02,u2,…,X^0N,uN maximizes problem (37) if and only if r1e−r1X^01,u1=r2e−r2X^02,u2=⋯=rNe−rNX^0N,uN. And X^01,u1,X^02,u2,…,X^0N,uN is Pareto optimal.A brief proof of Borch’s theorem can be found in [19].We denote thatC^=C^1,C^2,…,C^N is Pareto optimal compensation and u^=u^1,u^2,…,u^N is Pareto optimal R&D investment. According Lemma 3, we have(38)−r1e−r1C^1−G^T1u^1=−r2e−r2C^2−G^T2u^2=⋯=−rNe−rNC^N−G^TNu^N.Similar to the analysis in Section3.1, we know that there is a set of F-predictable process ZPA1,ZPA2,…,ZPA2 and a set of deterministic equivalent X^01,X^02,…,X^0N that satisfy the following conditions:(39)C^1=X^01−∫0T−r12ZPA1Σ2+ZPA1bs,A,u^−g1u^1ds+∫0TZPA1ΣdBsC^2=X^02−∫0T−r22ZPA2Σ2+ZPA2bs,A,u^−g2u^2ds+∫0TZPA2ΣdBs⋮C^N=X^0N−∫0T−rN2ZPANΣ2+ZPANbs,A,u^−gNu^Nds+∫0TZPANΣdBs−r1e−r1C^1−G^T1u^1=−r2e−r2C^2−G^T2u^2=⋯=−rNe−rNC^N−G^TNu^N.From the above conditions, we can deduce that(40)r1ZPA1=r2ZPA2=⋯=rNZPAN,r1X^01−lnr1=r2X^02−lnr2=⋯=rNX^0N−lnrN.According to equation (30), ui is a linear function of ZNAi; further, according to equation (29), ZNAi is linear with respect to ZNA−i. Therefore, we can assume that for every i=1,2,…,N, we have ui=ϕiZPA1,ZPA2,…,ZPAN. While on the basis of equation (40), we can assume that there are N×N linear functions φj,ij,i=1,2,…,N, and ui=φj,iZPAj. Moreover, assume that there are N−1×N−1 linear functions ψj,i (j,i=1,2,…,N, and j≠i), such that ui=ψj,iuj (or uj=ψj,1−1u1=⋯=ψj,N−1uN).Next, we will discuss the specific functional relationship betweenu and ZPA and find an explicit expression of ZPA.We first use lexicographic ordering method (the lexicographic ordering method is a method to find Pareto optimal solution; for more details on the lexicographic ordering method, see [20]) to translate a multiobjective optimization problem into a multiobjective optimization problem. Thus, Problem 1 translates into the following.Problem 2 (lexicographic order problem).
(41)supu∈AEℙuU1C1,u1such that:supu∈AEℙuU2C2,u2⋮supu∈AEℙuUNCN,uNdYt=a11u1,1+a12ψ1,2u1,1+⋯+a1,Nψ1,Nu1,1⋮aN1uN,1+aN2ψ1,2uN,1+⋯+aNNψ1,NuN,1dt+ΣdBtuEℙuU1C1,u1≥−e−r1L1⋮EℙuUNCN,uN≥−e−rNLN.
We have the following lemma to describe the relationship between the solutions of Problems1 and 2.Lemma 4.
The solution of Problem2 (lexicographic order problem) is Pareto solution of Problem 1 (multiobjective optimization problem).For the proof details of the lemma, see [21].Following the method of solving the optimal R&D investment in Section3.1, we can obtain the optimal R&D investment for the first single-objective problem in lexicographic ordering problem.(42)ZPA1a1+a2ψ1,2′+⋯+aNψ1,N′=k1u^1.Here,aj=a1ja2j⋯aNJ⊤ and u^1=u^1,1u^2,1⋯u^N,1⊤.Similarly, the optimal R&D investment of the second single-objective problem is as follows:(43)ZPA2a1ψ1,2−1′+a2+⋯+aNψ2,N′=k2u^2.By doing so, we can get the optimal R&D investment up to theN single-objective problem:(44)ZPANa1ψ1,N−1′+a2ψ2,N−1′+⋯+aN=kNu^N.According to equations (42)–(44), we can get the conditions r1ZPA1=r2ZPA2=⋯=rNZPAN. According to the linear property of ψj,I, we know that ψj,i−1′=ψj,i′−1. After calculation, we can get the following system of differential equations: 1≤j<i≤N.(45)ψj,iu^jψj,i′u^j=rjkjrikiu^j.So, if we solve those differential equations, we getu^i=ψj,iu^j=rjkj/rikiu^j.Substituting the above formula into equations (42)–(44), we can get Pareto optimal R&D investment u^ as follows:(46)u^1u^2⋯u^N=∑l=1Na1lr1k1/rlklk1ZPA1,1∑l=1Na1lr2k2/rlklk2ZPA1,2⋯∑l=1Na1lrNkN/rlklkNZPA1,N∑l=1Na2lr1k1/rlklk1ZPA2,1∑l=1Na2lr2k2/rlklk2ZPA2,2⋯∑l=1Na2lrNkN/rlklkNZPA2,N⋮⋮⋱⋮∑l=1NaNlr1k1/rlklk1ZPAN,1∑l=1NaNlr2k2/rlklk2ZPAN,2⋯∑l=1NaNlrNkN/rlklkNZPAN,N.Next, we will solve for the exact expression of Pareto’s optimal R&D investment. We recall the government’s problem as follows:(47)UPAPYT,C=supC1,C2,…,CNEℙu−e−rP∑j=1NYTj−Cj.Similar to the derivation in Section3.2, we have(48)UPAPYT,C=supZEℙu−Λ−rP⋅∫0TΣs−ZsΣsdBs⋅e−rP⋅∫0TgPAs,Zsds,with(49)gPAs,ZPA1=−∑j=1NX^0j+∑j=1N∑i=1Naj,i⋅∑l=1Najlriki/rlklki⋅ZPAj,i−∑j=1N∑i=1N∑l=1Najlriki/rlkl22ki⋅ZPAj,i2−∑j=1Nσj22⋅∑i=1NriZj,i2−rP2⋅∑j=1Nσj2⋅1−∑i=1NZj,i2.Obviously, the functiongPA is convex with respect to ZPAj,i. Therefore, taking the first derivative of gPA about ZPAj,i, we can obtain the exact solution of Pareto’s optimal R&D investment. The first-order condition of ZPAj,i is as follows:(50)rPσj2⋅∑l=1NZj,l−1+σj2riZj,i+∑l=1Najlriki/rlkl2kiZj,i=aji⋅∑l=1Najlriki/rlklki.That is, for eachj=1,2…,N, Zj,i satisfies the following linear equations:(51)ZPAj,1⋮ZPAj,N=rPσj2+σj2r1+∑l=1Najlr1k1/rlkl2k1⋯rPσj2⋮⋱⋮rPσj2⋯rPσj2+σj2rN+∑l=1NajlrNkN/rlkl2kN−1⋅aj1⋅∑l=1Najlr1k1/rlklk1+rPσj2⋮ajN⋅∑l=1NajlrNkN/rlklkN+rPσj2.We substitute the above optimal control variableZPAj,i into the expected utility function of the government and get the optimal utility of the government as follows:(52)UPAPYT,C=−e−rP⋅T⋅gPAZPA.In the case of supervision unit, the optimal incentive compensation obtained by firmi is as follows:(53)CPAi=X^0i+riT2⋅∑j=1NσjZPAj,i2+T2ki⋅∑j=1N∑l=1Najlrikirlkl⋅ZPAj,i2+∑j=1NZPAj,iYTj−T⋅∑j=1N∑κ=1Nrjajκrκkκ⋅∑l=1Najlrikirlkl⋅Zj,i2.Note: if−rje−rjLj=maxi=1,2,…,N−rie−riLi, then X^0j=Lj; when i≠j, X^0i=rj/riLj+1/rilnri/rj.Based on the above analysis, we have discussed the optimal incentive compensation for firms in the presence of a supervisor. Next, we will describe our results with special examples and numerical simulation methods.


## 3.4. Special Case

The results of theoretical analysis are shown more clearly and more intuitively, especially to show the different optimal compensations under the different conditions and the comparison between the different expected utilities of the government. In this subsection, we discuss the explicit expression of the optimal compensations under specific assumptions.Next, we perform numerical simulation analysis; assume thataij=a=1, ri=rP=r=0.5, ki=k=1, and Li=L=0 (for a description of the parameter assumptions, see [22]). Substituting these parameters into equations (28) and (51), then(54)ZNA1,1=ZNA1,2=σ14+4σ12+43σ14+8σ12+4,ZNA2,1=ZNA2,2=σ24+4σ22+43σ24+8σ22+4,ZPA1,1=ZPA1,2=σ12+43σ12+8,ZPA2,1=ZPA2,2=σ22+43σ22+8.According to the conclusions in Proposition1 and Problem 1, we can get the tendency of Nash equilibrium R&D investments uNA and Pareto optimal R&D investments uPA to change with uncertainty σ as shown in Figures 1 and 2.Figure 1
The Nash equilibrium R&D investments for firm 1.Figure 2
The Nash equilibrium R&D investments for firm 2.As can be seen from Figures1–4, the R&D investments provided by the firms are a decreasing function of uncertainty. When the uncertainty is 0 (full information), all R&D investments reach the maximum value of 1. Here, because we assume that there are no differences between the two firms, the corresponding R&D investments they provide are also the same.Figure 3
The Pareto optimal R&D investments for firm 1.Figure 4
The Pareto optimal R&D investments for firm 2.Figures5–10, respectively, depict that the expected utility of the government and the optimal compensations of the firms under two kinds of R&D investments and the comparison of the expected utility and optimal compensations under these two kinds of R&D investments.Figure 5
The functiong for the government with firms that choose Nash equilibrium strategy.Figure 6
The functiong for the government with firms that choose Pareto equilibrium strategy.Figure 7
The optimal compensation with firms that choose Nash equilibrium strategy.Figure 8
The optimal compensation with firms that choose Pareto equilibrium strategy.Figure 9
The comparison of optimal compensations.Figure 10
The comparison of the functiong.According to the analysis and numerical simulation of Section3, we can get the following conclusions. For the firms, the R&D investments provided by the firms are a decreasing function of the uncertainty factor σ; of course, the compensation received by the firm is also a decreasing function of the uncertainty factor. The firms that choose Pareto optimal R&D investment can improve their compensations compared with the case they choose the Nash equilibrium R&D investments. For the government, the firms that choose Pareto optimal R&D investments can improve the value of government, but if the government only hires one firm (in [17], the author refers to this condition as weak Pareto optimality), then the value of government cannot be improved compared with the case with the firms that choose Nash equilibrium R&D investments.From the above analysis, we can see that the firms that choose the Pareto optimal R&D investments are more beneficial to both the government and the firms. However, in general, firms will only consider their own interests and will not consider their coworkers’ interests together with their own interests. According to the analysis in [17], the government should hire a supervisor who manages the firm’s R&D investments and acts for the welfare of the firms. Therefore, there will be two problems: first, if the supervisor is exogenous, how does the government provide the supervisor with the optimal compensation; second, if the supervisor is endogenous (supervisor is the firm himself), how does the government design the optimal compensation for firms. There are more special cases; for example, in [16], if the government is risk-neutral and provided the linear compensations for the firms, then there will be a prisoner’s dilemma among firms. In other words, one firm chooses Pareto optimal R&D investment (cooperation) and another firm chooses Nash equilibrium R&D investment (independent work).Therefore, we should design a reasonable incentive mechanism to ensure that firms will choose Pareto optimal R&D investments (i.e., firms choose cooperation).


## 4. Conclusion

In this paper, we research the optimal incentive compensations in a standard continuous-time multiagent problem. Under the assumption of risk aversion, both the government and the firms have classical exponential utility function. First, we discussed optimal compensation in the case in which the firms are rational, noncooperative, and compute themselves their optimal R&D investment. We call this incentive compensation as Nash equilibrium compensation. Then, we consider the multiobjective optimization problem with supervisor. In this case, a supervisor considers the expected utility of the all firms as a whole, and the optimal compensation we get is called the Pareto optimal compensation, and then we compare the government’s expected utility in these two situations.In order to make the theoretical analysis more clear and intuitive, this paper adopts the numerical simulation method and obtains the following conclusions:(a)
For the firms, the optimal R&D investments they provided are a decreasing function of uncertainty. The Nash equilibrium R&D investments they provided are less than Pareto optimal R&D investments. And the Nash equilibrium compensations received by the firms are also less than the Pareto optimal compensation. Therefore, it is better for firms to provide Pareto optimal R&D investments.(b)
We have proved that it is advantageous from the government point of view, to a supervisor in the model who imposes Pareto optimal R&D investments to the firms, compared with the case in which the firms are provided the Nash equilibrium R&D investments. Therefore, the government can hire a supervisor to improve the government’s expected utility. However, if the government hires a supervisor, then the cost of employment should be considered. If the cost of hiring a supervisor is higher than the benefit brought from the supervisor, the government will not consider the hiring supervisor.



---

*Source: 1042791-2021-09-26.xml*