Incentive Mechanism Analysis of Environmental Governance Using Multitask Principal–Agent Model

Abstract

In view of the principal–agent relationship between local government and pollution enterprise in environmental governance, this paper established a multitask principal–agent model, gave the optimal contract form of local government incentive to enterprises, examined the influencing factors of the optimal incentive contract, and focused on analyzing the influences of institutional factors, enterprise types and enterprise attributes on the optimal incentive contract. The results show that the optimal incentive contract is affected by the influence coefficient of enterprise economic benefits on local government revenue, the environmental index weight in government performance systems, the types of enterprise, the risk preference of enterprise, the variance in economic benefits, the variance in pollution emission reduction, the ability of economic benefits, the direct cost of emission reduction and the economic cost (income) of emission reduction. In order to realize the coordinated development of economy and environment, the establishment and adjustment of optimal incentive contract should be established or adjusted according to the institutional factors, different types of enterprise, and the enterprise characteristics. Local governments should divide enterprises into traditional enterprises and green innovative enterprises in the process of management. The policy formulation of the local government should be based on the characteristics of the system and cooperate with the national strategy. Local governments should actively collect and master the attribute information of the enterprises, including risk preference, profit model, pollution control technology and ability, management experience and level, production mode and green innovation ability. Local governments should combine the types of enterprises, the characteristics of institutions and the attributes of enterprises, and adjust various policies and measures more flexibly.

1. Introduction

In view of the increasingly serious problem of environmental pollution, the problem of environmental protection is paid more and more attention. The central government of China has repeatedly called for the enhancement of environmental protection, and the public’s concern about environmental quality has also been increasing. In this background, local governments, which have important responsibility for environmental quality in their jurisdictions, are under increasing pressure.

Traditional command-and-control and market-incentive environmental regulation tools emphasize the punishment of environmental illegal enterprises, but punishing, shutting down or relocating the enterprises, etc., often causes local governments to face difficulties in finance, employment, staff resettlement, and it also has a negative impact on local economic development. As a result, when an environmental incident occurs, both the illegal enterprises and the local governments are willing to resolve the problem in a way that is less costly and more acceptable to both parties; for example, the enterprise continues to produce after paying a “symbolic fine”, the result of which is that the environmental pollution problem is not solved.

The coordination between regional economic development and environmental protection should be fundamentally implemented on two aspects, economic benefits and pollution reduction. From the perspective of principal–agent, enterprises, as the agents of the local government, should not only complete the tasks of increasing economic benefits and promoting economic development, but also complete the tasks of pollution reduction and improving environmental quality. Therefore, it will be the effective means to realize the coordinated development of regional economy and environment for local governments to guide and motivate enterprises reasonably through mechanism design, so as to promote enterprises to complete the tasks of economic benefits and pollution reduction.

In recent years research on environmental governance based on principal–agent theory has been increasing, while the core problem in the process of environmental governance is to design and optimize the reasonable incentive models and mechanism. For example, incentive model for third-party remediation of legacy contamination sites left by closed enterprises from the perspective of fully linked accountability [1]; evolutionary game model of the environmental pollution third-party governance considering reward and punishment distribution incentive mechanism [2]; and the information communication model of key pollution source monitoring [3], etc. The design and optimization of reasonable incentive mechanism can effectively solve the problem of information asymmetry in the process of environmental governance, so as to reduce pollution emissions and improve environmental quality. Especially in the context of enterprises undertaking multiple tasks, it is of more theoretical and practical significance to consider introducing multi-task principal–agent theory to analyze environmental governance problems.

2. Literature Review

In order to improve the effect of environmental pollution control, many scholars at home and abroad conducted research on the design of pollution control mechanisms. Under the condition of incomplete information, Dasgupta et al. demonstrated that the use of mechanism design can lead to a Pareto optimal level of pollution control [4]. Xepapadeas proposed a mechanism design to combine fines and subsidies [5]. Huang et al. proposed a monitoring contract that combines supervision, report and fines to supervise and constrain the pollution discharging [6]. Chen and Jiang took the risk attitude characteristics of different enterprises into account, and accordingly established different contractual mechanisms to deal with the enterprises’ lazy behaviour in the process of pollution control [7]. Wang and Meng designed the optimal incentive contract for environmental regulation under adverse selection and limited liability of agents [8].

In recent years, limited research has been conducted under the framework of multitask principal–agent; for example: the design of formal and relational contracts for the outsourcing of applied services; the institutional reform of food safety regulation [9,10]; the financing factor of bank in multi-task principal–agent to study the incentive mechanism between the manufacturer and the supplier [11]; the note on organizational structure and environmental liability [12]; the research on compensation incentive mechanism of doctors in public hospitals [13]; and the chain multitask incentive model for complex weapon equipment delivery based on accountability mechanism [14]. The research covers many aspects of social life.

Moreover, scholars have more often used the relatively complex principal–agent model for their research. For example, Li and Zhang analyzed the optimal environmental regulation of local governments and its fluctuations under the perspective of the optimal incentive contract design, and discussed the role of a third party [15]. By constructing a double-layer principal–agent model, Chen et al. analyzed the adverse selection of enterprises to hide their own pollution information in environmental regulation and the moral hazard by conspiring of local governments [16]. Taking the implementation of the negative list system for industry access in key ecological functional zones as an example, Xu et al. analyzed the theoretical mechanism of policy implementation deviation caused by multi-task conflict, and summarized the local practical experience of correcting policy implementation deviation by adjusting the principal–agent mode [17]. The H&M multi-tasking principal–agent model is adopted to investigate the motivation behind the two tasks of business performance and compliance governance [18]. The multi-task and multi-agent principal–agent model and auction model are used to construct the university budget performance management model under the information asymmetry, and through the theoretical analysis of the model the mathematical expression describing the optimal state relationship of the system is obtained [19]. Using the multi-task principal–agent model, Xi and Han analyzed the effect of Chinese decentralization and industrial policy implementation [20].

By reviewing the existing research we can see that there is not much literature on environmental issues under the principal–agent perspective. Although relevant studies constructed more complex principal–agent models and attempted to introduce more subjects and layers into the models in recent years, they only considered the case of agents engaging in one kind activity, so the studies always limited to design mechanism for environmental regulation and advocated increasing the supervision and punishment of enterprises, ignoring the mutually constraining relationship between environmental governance and economic development. In fact, the production management process of an enterprise has obvious multitask characteristics, so the use of the multitask principal–agent model will be more suitable for analyzing problems and closer to reality. There is still a lack of research applying multitask principal–agent theory to environmental governance.

Accordingly, on the basis of previous studies a multitask principal–agent model between local governments and pollution reduction enterprises is developed, the design of incentive mechanism under the tasks of economic benefit and pollution reduction is examined, the influence of various factors on the optimal incentive contract is focused on in this paper; the more operational incentive mechanism is proposed, theoretical reference for promoting the harmonious development of economy and environment is provided.

3. Multitask Principal–Agent Model Construction and Optimization

3.1. Basic Assumptions

Principal–agent theory aims to design an incentive mechanism to reduce moral hazard so that agents seek to maximize their personal utility while achieving the maximization of the principal’s expected utility. In reality, agents are often engaged in more than one task and their energy allocation between different tasks is conflicting, which led Holmstrom and Milgrom to propose a multitask principal–agent model on the basis of the traditional model [21]. Emission enterprises often undertake two tasks entrusted by local governments in their production process, which is increasing economic benefit and reducing pollution; at the same time, the local governments impose incentives and constraints on the behavioral decision of enterprises, such as economic policy support, tax breaks and so on, as well as subsidies and incentives for reducing pollution or penalties for increased emissions and meeting the requirements of local governments. To facilitate the analysis of the problem, this paper makes the following basic assumptions:

H1: 

Enterprises are entrusted by local governments with two tasks, to improve economic benefit and reduce pollution. Improving economic benefit can promote the growth of the local economy, and reducing pollution can improve local environmental quality. Economic growth and environmental quality improvement are not only an important responsibility of local governments in social governance, but also key tasks in performance assessment. The effort level of the enterprise in economic benefit is $e_1$, $e_1>0$; the effort level of reducing pollution is $e_2$, and when $e_2>0$ it indicates the legal pollution reduction input and pollution control of enterprises; when $e_2\le 0$, it indicates the illegal environmental behaviors of enterprises such as over-standard discharge and concealed emission. Local governments cannot observe the effort level of enterprises, but they can observe the output determined by the effort level. The output function takes linear form and respectively expresses the level of economic benefit ${\pi }_1$, ${\pi }_1=e_1+{\epsilon }_1$; emission reduction ${\pi }_2$, ${\pi }_2=e_2+{\epsilon }_2$. Both ${\epsilon }_1$ and ${\epsilon }_2$ are random variable with normal distribution, whose mean value is 0 and variances are ${\sigma }_1{}^2$ and ${\sigma }_2{}^2$, respectively. The implication is that the level of economic benefit and emission reduction depend not only on the effort level of the enterprises, but are also influenced by external stochastic factors.

H2: 

Local governments pay compensation by using the linear incentive function, which means that the benefit received by the enterprises is expressed as $R({\pi }_1,{\pi }_2)=a+{\delta }_1{\pi }_1+{\delta }_2{\pi }_2$, in which $a$ is the fixed income of enterprises, ${\delta }_1$ and ${\delta }_2$ are, respectively, the incentive intensity (coefficients) of the local government for economic benefits and pollution reduction in the enterprises, which are expressed as policy support, tax exemption, subsidies, sewage fees or fines (when $e_2\le 0$) and so on. In reality, enterprises can neither take no risks (${\delta }_1={\delta }_2=0$), nor bear all risks and incomes (${\delta }_1={\delta }_2=1$), therefore $0<{\delta }_1,{\delta }_2<1$.

H3: 

The enterprise’s effort cost is a strictly increasing convex function with first-order continuous partial derivatives and second-order differentiable. The cost function is set as $C(e_1,e_2)=\frac{r_1}{2}{e}_1{}^2+\frac{r_2}{2}{e}_2{}^2+r_{12}{e}_1{e}_2$. In which, cost factor $r_1=\frac{{\partial }^{2}C(e_1,e_2)}{\partial e_1{}^2}>0$, $r_1$ is the marginal change rate for economic benefit effort cost. The $r_1$ has no effect on the cost of pollution control and can be interpreted as the economic efficiency ability of enterprises, including the knowledge, ideas, experience, methods and other attributes during the production and management of enterprises. The larger the $r_1$, the less the economic benefit ability of enterprises; it means that the economic efficiency of the enterprise becomes worse. Cost factor $r_2=\frac{{\partial }^{2}C(e_1,e_2)}{\partial e_2{}^2}>0$, $r_2$ is the marginal change rate of pollution reduction cost. The $r_2$ has no effect on the cost of economic benefit effort and can be interpreted as the direct cost of pollution control of non-economic impacts. The larger the $r_2$, the larger the direct cost of pollution control, and this means the greater the direct cost of emission reduction for enterprises. Cost factor $r_{12}=\frac{{\partial }^{2}C(e_1,e_2)}{\partial e_1\partial e_2}$, $r_{12}$ indicates the substitute degree between different tasks, which is divided into the following two situations:

Situation I, traditional enterprises. The increase in emission reduction inputs by enterprises may bring about adverse output and economic benefit; under industrial competition, increasing costs lead to higher prices and lower competitiveness. The two aspects above will increase the resistance and difficulty of enterprises to improve economic benefit, showing that the marginal economic benefit effort cost increases with the increase in pollution reduction inputs, i.e.,$\frac{{\partial }^{2}C(e_1,e_2)}{\partial e_1\partial e_2}>0$. The $r_{12}$ portrays the negative impact of pollution reduction inputs on enterprise economic benefit, and can be interpreted as the economic cost of pollution reduction. Situation I reflects a type of “traditional enterprises” whose pollution reduction effort is mainly based on end-of-pipe treatment inputs or passively changing the production factor input combinations to reduce pollution.

Situation II, innovative enterprises. According to the Porter Hypothesis, effective pollution reduction inputs can increase the direct cost of pollution control and can increase enterprises’ benefits through innovation compensation and first-mover advantage at the same time, compensating for, or even exceeding, the direct cost of pollution control which is expressed as the marginal economic benefit effort cost decreases with the increase in pollution reduction inputs, i.e., $\frac{{\partial }^{2}C(e_1,e_2)}{\partial e_1\partial e_2}<0$. The $r_{12}$ portrays the positive impact of pollution reduction inputs on enterprise economic efficiency, and can be interpreted as the economic income of pollution reduction. The smaller $r_{12}$ is, the smaller the marginal economic benefit effort cost brought by the increase in unit emission reduction input is, which means the greater the economic benefit of emission reduction (the stronger the green innovation ability) of the enterprise. Situation II reflects a type of “(green) innovative enterprises”, which have strong awareness of environmental protection and their pollution reduction effort is mainly based on green technology innovation, green product development, or actively adopting green technologies for production to reduce pollution (Intuitively, the efforts to reduce emissions and control pollution have increased economic benefits, made it easier to make profits and brought about new profit growth points. In fact, a lot of green marketing, green innovation is just like this: some energy saving design, some green marketing design, but bring profit increase. One of the purposes of this paper is to design different incentive contracts for these two types of enterprises.).

H4: 

The economic and political benefits of local governments depend on the output of the two tasks of the enterprises, the tax rate and the indicator weights in the performance evaluation system $Y_{\mathrm{L}}={\beta }_1{\pi }_1+{\beta }_2{\pi }_2$, where $L$ denotes local government, ${\beta }_1$ is the influence coefficient of enterprise economic benefit on local government income, the larger ${\beta }_1$ indicates the greater impact of enterprises’ economic performance on local government income, or the greater importance of the enterprises’ economic performance on local government. The ${\beta }_2$ is the environmental index weight in the performance evaluation system. The larger ${\beta }_2$ indicates the greater impact of enterprise’s pollution reduction on local government income, or the greater importance of enterprise’s pollution reduction on local governments. Both ${\beta }_1$ and ${\beta }_2$ are institutional factors, ${\beta }_1>0$, ${\beta }_2>0$.

H5: 

Local governments are neutral risks and their utility function is denoted as $U_{\mathrm{L}}$.

Enterprises are risk-averse and their utility function has a invariant absolute risk-averse characteristic, which is denoted as $u=-e^{-\rho R}$. In which, $\rho$ is the enterprise’s absolute risk-averse degree, $\rho >0$, a larger $\rho$ means that the enterprise is more fearful of risk. Local governments have more adequate information about the jurisdictional enterprises and are therefore fully aware of the absolute risk-averse degree of the enterprises.

3.2. Model Construction and Optimization

The core problem of the principal–agent model is to maximize the utility of the principal through the agent’s profit-maximizing behavior. Therefore, the first problem in building the model is to list the expressions of local government utility and enterprise income, respectively. Based on the above model assumptions, the utility of local government directly depends on the output of two work tasks of enterprises ${\pi }_1$ and ${\pi }_2$, the influence coefficient of enterprise economic benefit on local government income ${\beta }_1$ and the environmental index weight in the performance evaluation system ${\beta }_2$, and the compensation paid to enterprises by local governments deducted $R({\pi }_1,{\pi }_2)=a+{\delta }_1{\pi }_1+{\delta }_2{\pi }_2$. While the income of the enterprise depends directly on what local government pays, the cost of the effort paid by the enterprise $C\left(e_1,e_2\right)=\frac{r_1}{2}{e}_1^2+\frac{r_2}{2}{e}_2^2+r_{12}{e}_1{e}_2$ and the cost of risk enterprises take $\frac{1}{2}\rho \mathrm{var}\left(R\right)=\frac{1}{2}\rho {\sigma }_1^2{\delta }_1^2+\frac{1}{2}\rho {\sigma }_2^2{\delta }_2^2$ [22]. The pursuit of profit maximization by enterprises can be demonstrated by solving the optimal effort level $e_1^{*}$ and $e_2^{*}$. On this basis, the realization of utility maximization by local governments can be demonstrated by solving the optimal incentive intensity ${\delta }_1^{*}$ and ${\delta }_2^{*}$, so as to solve the core problem of the principal–agent model. In addition, the principal must satisfy the personal rational constraint ($IR$) and incentive compatibility constraint ($IC$) when formulating the incentive contract. The following is the specific model construction and solving process:

According to the risk-neutral characteristics of local governments, their expected utility $U_{\mathrm{L}}$ is equal to the expected benefit:

$$\begin{array}{ll}E(U_{\mathrm{L}})& =E(Y_{\mathrm{L}}-R)=E({\beta }_1{\pi }_1+{\beta }_2{\pi }_2-a-{\delta }_1{\pi }_1-{\delta }_2{\pi }_2)\\ & ={\beta }_1{e}_1+{\beta }_2{e}_2-a-{\delta }_1{e}_1-{\delta }_2{e}_2\end{array}$$

(1)

The enterprises’ certainty equivalence benefit ($CE$) is equal to enterprises’ expected income minus the effort and risk cost:

$$\begin{array}{ll}CE& =E(R)-C(e_1,e_2)-\frac{1}{2}\rho \mathrm{var}(R)\\ & =a+{\delta }_1{e}_1+{\delta }_2{e}_2-\frac{r_1}{2}{e}_1{}^2-\frac{r_2}{2}{e}_2{}^2-r_{12}{e}_1{e}_2-\frac{1}{2}\rho {\sigma }_1{}^2{\delta }_1{}^2-\frac{1}{2}\rho {\sigma }_2{}^2{\delta }_2{}^2\end{array}$$

(2)

The main constraints for local governments in the pursuit of expected utility maximization include: individual rationality constraint ($IR$), meaning that enterprises’ $CE$ is no less than its level of retained revenue ${\widehat{Y}}_{\mathrm{E}}$, the subscript $E$ denotes enterprise, otherwise the enterprise will not accept the contract; incentive compatibility constraint ($IC$), meaning that the enterprise will choose optimal effort level $e_1^{*}$ and $e_2^{*}$ to maximize its own $CE$. Therefore, the optimal design model of the incentive contract is:

$$\begin{array}{l}\max E(U_{\mathrm{L}})={\beta }_1{e}_1+{\beta }_2{e}_2-a-{\delta }_1{e}_1-{\delta }_2{e}_2\\ s.t. (\mathrm{IR}) a+{\delta }_1{e}_1+{\delta }_2{e}_2-\frac{r_1}{2}{e}_1{}^2-\frac{r_2}{2}{e}_2{}^2-r_{12}{e}_1{e}_2-\frac{1}{2}\rho {\sigma }_1{}^2{\delta }_1{}^2-\frac{1}{2}\rho {\sigma }_2{}^2{\delta }_2{}^2\ge {\widehat{Y}}_{\mathrm{E}}\\ \left(\mathrm{IC}\right) (e_1,e_2)\in \arg \max a+{\delta }_1{e}_1+{\delta }_2{e}_2-\frac{r_1}{2}{e}_1{}^2-\frac{r_2}{2}{e}_2{}^2-r_{12}{e}_1{e}_2-\frac{1}{2}\rho {\sigma }_1{}^2{\delta }_1{}^2-\frac{1}{2}\rho {\sigma }_2{}^2{\delta }_2{}^2\end{array}$$

According to the first-order condition of the incentive compatibility:

$$\left\{\begin{array}{l}\frac{\partial CE}{\partial e_1}={\delta }_1-r_1{e}_1-r_{12}{e}_2=0\\ \frac{\partial CE}{\partial e_2}={\delta }_2-r_2{e}_2-r_{12}{e}_1=0\end{array}\right.$$

The optimal effort level of enterprises in terms of economic benefit and pollution reduction can be obtained:

$$\left\{\begin{array}{l}{e}_1^{*}=\frac{r_{12}{\delta }_2-r_2{\delta }_1}{r_{12}{}^2-r_1{r}_2}\\ e_2^{*}=\frac{r_{12}{\delta }_1-r_1{\delta }_2}{r_{12}{}^2-r_1{r}_2}\end{array}\right.$$

(3)

The fixed reward of the enterprises will not affect their effort level and incentive intensity, which is determined by the level of retained revenue according to the individual rationality:

$$a={\widehat{Y}}_{\mathrm{E}}-{\delta }_1{e}_1-{\delta }_2{e}_2+\frac{r_1}{2}{e}_1{}^2+\frac{r_2}{2}{e}_2{}^2+r_{12}{e}_1{e}_2+\frac{1}{2}\rho {\sigma }_1{}^2{\delta }_1{}^2+\frac{1}{2}\rho {\sigma }_2{}^2{\delta }_2{}^2$$

(4)

Substituting Formulas (3) and (4) into Formula (1), we get:

$$E(U_{\mathrm{L}})={\beta }_1{e}_1^{*}+{\beta }_2{e}_2^{*}-{\widehat{Y}}_{\mathrm{E}}-\frac{r_1}{2}{(e_1^{*})}^2-\frac{r_2}{2}{(e_2^{*})}^2-r_{12}{e}_1^{*}{e}_2^{*}-\frac{1}{2}\rho {\sigma }_1{}^2{\delta }_1{}^2-\frac{1}{2}\rho {\sigma }_2{}^2{\delta }_2{}^2$$

(5)

Local governments seek to maximize expected utility, according to the first-order condition of Formula (5)

$$\left\{\begin{array}{l}\frac{\partial E(U_{\mathrm{L}})}{\partial {\delta }_1}=0\\ \frac{\partial E(U_{\mathrm{L}})}{\partial {\delta }_2}=0\end{array}\right.$$

The optimal incentive intensity can be obtained by solving for ${\delta }_1$ and ${\delta }_2$:

$$\left\{\begin{array}{l}{\delta }_1^{*}=\frac{(1+\rho {\sigma }_2{}^2{r}_2){\beta }_1-\rho {\sigma }_2{}^2{r}_{12}{\beta }_2}{1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2-{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)}\\ {\delta }_2^{*}=\frac{(1+\rho {\sigma }_1{}^2{r}_1){\beta }_2-\rho {\sigma }_1{}^2{r}_{12}{\beta }_1}{1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2-{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)}\end{array}\right.$$

(6)

4. Influencing Factors Analysis of Optimal Incentive Contracts

According to H3, the cost function $C$ is a strictly increasing convex function and therefore the function satisfies the sufficient conditions of the lower convex function [23]:

$$\frac{{\partial }^{2}C(e_1,e_2)}{\partial e_1{}^2}\ge 0,\frac{{\partial }^{2}C(e_1,e_2)}{\partial e_2{}^2}\ge 0,\frac{{\partial }^{2}C(e_1,e_2)}{\partial e_1{}^2}·\frac{{\partial }^{2}C(e_1,e_2)}{\partial e_2{}^2}-{[\frac{{\partial }^{2}C(e_1,e_2)}{\partial e_1\partial e_2}]}^2\ge 0$$

In which, $\frac{{\partial }^{2}C(e_1,e_2)}{\partial e_1{}^2}=r_1$, $\frac{{\partial }^{2}C(e_1,e_2)}{\partial e_2{}^2}=r_2$, $\frac{{\partial }^{2}C(e_1,e_2)}{\partial e_1\partial e_2}=r_{12}$, and from this it can be seen that $r_{12}{}^2-r_1{r}_2\le 0$.

In Formula (6), $r_{12}{}^2-r_1{r}_2\le 0$, so we get $1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2-{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)>0$, and according to H2 the incentive intensity ${\delta }_1$ and ${\delta }_2$ are positive, so we get $(1+\rho {\sigma }_1{}^2{r}_1){\beta }_2-\rho {\sigma }_1{}^2{r}_{12}{\beta }_1>0$, $(1+\rho {\sigma }_2{}^2{r}_2){\beta }_1-\rho {\sigma }_2{}^2{r}_{12}{\beta }_2>0$.

According to Formula (6), the optimal economic benefit incentive intensity and the optimal pollution reduction incentive intensity are determined by parameters ${\beta }_1$, ${\beta }_2$, $\rho$, ${\sigma }_1{}^2$, ${\sigma }_2{}^2$, $r_1$, $r_2$ and $r_{12}$. We analyze the influence of a single factor change on the optimal incentive contract while keeping other influential factors stable in the following. In the multitasking principal–agent model, the economic and environmental optimal incentive contracts are affected by many factors, but the enterprise-related factors are the main factors affecting the establishment of optimal incentive contracts by local governments. Therefore, based on Stackelberg Model the enterprise in the multitask principal–agent model can be regarded as a “Stackelberg Leader”, so the inverse derivation method is used to analyze the influencing factors of the optimal incentive contract.

4.1. Influencing Factors Analysis of the Optimal Economic Benefit Incentive Intensity

According to Formula (7), it is clear that the optimal economic benefit incentive intensity is an increasing function of ${\beta }_1$. As the influence coefficient of enterprise economic benefit on local government income increases, the optimal economic benefit incentive intensity also increases.

$$\frac{\partial {\delta }_1^{*}}{\partial {\beta }_1}=\frac{1+\rho {\sigma }_2{}^2{r}_2}{1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2-{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)}>0$$

(7)

$$\frac{\partial {\delta }_1^{*}}{\partial {\beta }_2}=\frac{-\rho {\sigma }_2{}^2{r}_{12}}{1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2-{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)}$$

(8)

According to Formula (8), when $r_{12}>0$, $\frac{\partial {\delta }_1^{*}}{\partial {\beta }_2}<0$, the optimal economic benefit incentive intensity is a decreasing function of ${\beta }_2$. As the environmental index weight in the performance evaluation system increases, the optimal economic benefit incentive intensity decreases. When $r_{12}<0$, $\frac{\partial {\delta }_1^{*}}{\partial {\beta }_2}>0$, the optimal economic benefit incentive intensity is an increasing function of ${\beta }_2$. As the environmental index weight in the performance evaluation system increases, the optimal economic benefit incentive intensity increases.

$$\frac{\partial {\delta }_1^{*}}{\partial \rho }=\frac{\begin{array}{l}{\sigma }_1{}^2[2\rho {\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)+{\rho }^2{\sigma }_2{}^4{r}_2(r_{12}{}^2-r_1{r}_2)-r_1]{\beta }_1\\ -{\sigma }_2{}^2{r}_{12}[{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)+1]{\beta }_2\end{array}}{{[1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2-{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)]}^2}$$

(9)

According to Formula (9), when ${\sigma }_1{}^2[2\rho {\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)+{\rho }^2{\sigma }_2{}^4{r}_2(r_{12}{}^2-r_1{r}_2)-r_1]{\beta }_1-{\sigma }_2{}^2{r}_{12}[{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)+1]{\beta }_2>0$, that is when $\frac{{\beta }_1}{{\beta }_2}>\frac{{\sigma }_2{}^2{r}_{12}[{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)+1]}{{\sigma }_1{}^2[2\rho {\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)+{\rho }^2{\sigma }_2{}^4{r}_2(r_{12}{}^2-r_1{r}_2)-r_1]}={\theta }_1^{*}$, $\frac{\partial {\delta }_1^{*}}{\partial \rho }>0$, the optimal economic benefit incentive intensity is a increasing function of $\rho$. As the absolute risk-averse degree of enterprises increases, the optimal economic benefit incentive intensity will gradually increase. When ${\sigma }_1{}^2[2\rho {\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)+{\rho }^2{\sigma }_2{}^4{r}_2(r_{12}{}^2-r_1{r}_2)-r_1]{\beta }_1-{\sigma }_2{}^2{r}_{12}[{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)+1]{\beta }_2<0$, that is when $\frac{{\beta }_1}{{\beta }_2}<\frac{{\sigma }_2{}^2{r}_{12}[{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)+1]}{{\sigma }_1{}^2[2\rho {\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)+{\rho }^2{\sigma }_2{}^4{r}_2(r_{12}{}^2-r_1{r}_2)-r_1]}={\theta }_1^{*}$, $\frac{\partial {\delta }_1^{*}}{\partial \rho }<0$, the optimal economic benefit incentive intensity is a decreasing function of $\rho$. As the absolute risk-averse degree of enterprises increases, the optimal economic benefit incentive intensity will gradually decrease.

$$\frac{\partial {\delta }_1^{*}}{\partial {\sigma }_1{}^2}=\frac{[\rho {\sigma }_2{}^2{r}_{12}{\beta }_2-(1+\rho {\sigma }_2{}^2{r}_2){\beta }_1][\rho r_1-{\rho }^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)]}{{[1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2-{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)]}^2}<0$$

(10)

According to Formula (10), the optimal economic benefit incentive intensity is a decreasing function of ${\sigma }_1{}^2$. As the variance in economic benefit increases, the optimal economic benefit incentive intensity will gradually decrease.

$$\frac{\partial {\delta }_1^{*}}{\partial {\sigma }_2{}^2}=\frac{\rho r_{12}[\rho {\sigma }_1{}^2{r}_{12}{\beta }_1-(1+\rho {\sigma }_1{}^2{r}_1){\beta }_2]}{{[1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2-{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)]}^2}$$

(11)

According to Formula (11), when $r_{12}>0$, $\frac{\partial {\delta }_1^{*}}{\partial {\sigma }_2{}^2}<0$, the optimal economic benefit incentive intensity is a decreasing function of ${\sigma }_2{}^2$. As the variance in emission reduction increases, the optimal economic benefit incentive intensity will gradually decrease. When $r_{12}<0$, $\frac{\partial {\delta }_1^{*}}{\partial {\sigma }_2{}^2}>0$, the optimal economic benefit incentive intensity is an increasing function of ${\sigma }_2{}^2$. As the variance in emission reduction increases, the optimal economic benefit incentive intensity will gradually increase.

$$\frac{\partial {\delta }_1^{*}}{\partial r_1}=\frac{[\rho {\sigma }_2{}^2{r}_{12}{\beta }_2-(1+\rho {\sigma }_2{}^2{r}_2){\beta }_1](\rho {\sigma }_1{}^2+{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2{r}_2)}{{[1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2-{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)]}^2}<0$$

(12)

According to Formula (12), the optimal economic benefit incentive intensity is a decreasing function of $r_1$. As the ability of economic benefit of enterprises decreases ($r_1$↑), the optimal economic benefit incentive intensity will gradually decrease.

$$\frac{\partial {\delta }_1^{*}}{\partial r_2}=\frac{{\rho }^2{\sigma }_2{}^4{r}_{12}[(1+\rho {\sigma }_1{}^2{r}_1){\beta }_2-\rho {\sigma }_1{}^2{r}_{12}{\beta }_1]}{{[1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2-{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)]}^2}$$

(13)

According to Formula (13), when $r_{12}>0$, $\frac{\partial {\delta }_1^{*}}{\partial r_2}>0$, the optimal economic benefit incentive intensity is an increasing function of $r_2$. As the direct cost of pollution control for enterprises increases, the optimal economic benefit incentive intensity will gradually increase. When $r_{12}<0$, $\frac{\partial {\delta }_1^{*}}{\partial r_2}<0$, the optimal economic benefit incentive intensity is a decreasing function of $r_2$. As the direct cost of pollution control for enterprises increases, the optimal economic benefit incentive intensity will gradually decrease.

$$\frac{\partial {\delta }_1^{*}}{\partial r_{12}}=\frac{\begin{array}{l}\rho {\sigma }_2{}^2[2\rho {\sigma }_1{}^2{r}_{12}(1+\rho {\sigma }_2{}^2{r}_2){\beta }_1\\ -(1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2+{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2{r}_1{r}_2+{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2{r}_{12}{}^2){\beta }_2]\end{array}}{{[1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2-{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)]}^2}$$

(14)

According to Formula (14), when $r_{12}>0$, if $2\rho {\sigma }_1{}^2{r}_{12}(1+\rho {\sigma }_2{}^2{r}_2){\beta }_1-(1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2+{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2{r}_1{r}_2+{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2{r}_{12}{}^2){\beta }_2>0$, that is when $\frac{{\beta }_1}{{\beta }_2}>\frac{(1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2+{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2{r}_1{r}_2+{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2{r}_{12}{}^2)}{2\rho {\sigma }_1{}^2{r}_{12}(1+\rho {\sigma }_2{}^2{r}_2)}={\theta }_2^{*}$, $\frac{\partial {\delta }_1^{*}}{\partial r_{12}}>0$, the optimal economic benefit incentive intensity is an increasing function of $r_{12}$. As the economic cost of pollution control for enterprises increases, the optimal economic benefit incentive intensity will gradually increase. If $2\rho {\sigma }_1{}^2{r}_{12}(1+\rho {\sigma }_2{}^2{r}_2){\beta }_1-(1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2+{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2{r}_1{r}_2+{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2{r}_{12}{}^2){\beta }_2<0$, that is when $\frac{{\beta }_1}{{\beta }_2}<\frac{(1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2+{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2{r}_1{r}_2+{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2{r}_{12}{}^2)}{2\rho {\sigma }_1{}^2{r}_{12}(1+\rho {\sigma }_2{}^2{r}_2)}={\theta }_2^{*}$, $\frac{\partial {\delta }_1^{*}}{\partial r_{12}}<0$, the optimal economic benefit incentive intensity is a decreasing function of $r_{12}$. As the economic cost of pollution control for enterprises increases, the optimal economic benefit incentive intensity will gradually decrease.

When $r_{12}<0$, $\frac{\partial {\delta }_1^{*}}{\partial r_{12}}<0$, the optimal economic benefit incentive intensity is a decreasing function of $r_{12}$. As the economic income of pollution control for enterprises increases ($r_{12}$↑), the optimal economic benefit incentive intensity will gradually decrease.

4.2. Influencing Factors Analysis of the Optimal Pollution Reduction Incentive Intensity

According to Formula (15), when $r_{12}>0$, $\frac{\partial {\delta }_2^{*}}{\partial {\beta }_1}<0$, the optimal pollution reduction incentive intensity is a decreasing function of ${\beta }_1$. As the influence coefficient of enterprise economic benefit on local government income increases, the optimal pollution reduction incentive intensity will gradually decrease. When $r_{12}<0$, $\frac{\partial {\delta }_2^{*}}{\partial {\beta }_1}>0$, the optimal pollution reduction incentive intensity is an increasing function of ${\beta }_1$. As the influence coefficient of enterprise economic benefit on local government income increases, the optimal pollution reduction incentive intensity will gradually increase.

$$\frac{\partial {\delta }_2^{*}}{\partial {\beta }_1}=\frac{-\rho {\sigma }_1{}^2{r}_{12}}{1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2-{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)}$$

(15)

$$\frac{\partial {\delta }_2^{*}}{\partial {\beta }_2}=\frac{1+\rho {\sigma }_1{}^2{r}_1}{1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2-{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)}>0$$

(16)

According to Formula (16), the optimal pollution reduction incentive intensity is an increasing function of ${\beta }_2$. As the environmental index weight increases in the performance appraisal system, the optimal pollution reduction incentive intensity will gradually increase.

$$\frac{\partial {\delta }_2^{*}}{\partial \rho }=\frac{\begin{array}{l}{\sigma }_2{}^2[2\rho {\sigma }_1{}^2(r_{12}{}^2-r_1{r}_2)+{\rho }^2{\sigma }_1{}^4{r}_1(r_{12}{}^2-r_1{r}_2)-r_2]{\beta }_2\\ -{\sigma }_1{}^2{r}_{12}[{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)+1]{\beta }_1\end{array}}{{[1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2-{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)]}^2}$$

(17)

According to Formula (17), when ${\sigma }_2{}^2[2\rho {\sigma }_1{}^2(r_{12}{}^2-r_1{r}_2)+{\rho }^2{\sigma }_1{}^4{r}_1(r_{12}{}^2-r_1{r}_2)-r_2]{\beta }_2-{\sigma }_1{}^2{r}_{12}[{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)+1]{\beta }_1>0$, that is when $\frac{{\beta }_2}{{\beta }_1}>\frac{{\sigma }_1{}^2{r}_{12}[{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)+1]}{{\sigma }_2{}^2[2\rho {\sigma }_1{}^2(r_{12}{}^2-r_1{r}_2)+{\rho }^2{\sigma }_1{}^4{r}_1(r_{12}{}^2-r_1{r}_2)-r_2]}={\theta }_3^{*}$, $\frac{\partial {\delta }_2^{*}}{\partial \rho }>0$, the optimal pollution reduction incentive intensity is an increasing function of $\rho$. As the absolute risk-averse degree of enterprises increases, the optimal pollution reduction incentive intensity will gradually increase. When ${\sigma }_2{}^2[2\rho {\sigma }_1{}^2(r_{12}{}^2-r_1{r}_2)+{\rho }^2{\sigma }_1{}^4{r}_1(r_{12}{}^2-r_1{r}_2)-r_2]{\beta }_2-{\sigma }_1{}^2{r}_{12}[{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)+1]{\beta }_1<0$, that is when $\frac{{\beta }_2}{{\beta }_1}<\frac{{\sigma }_1{}^2{r}_{12}[{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)+1]}{{\sigma }_2{}^2[2\rho {\sigma }_1{}^2(r_{12}{}^2-r_1{r}_2)+{\rho }^2{\sigma }_1{}^4{r}_1(r_{12}{}^2-r_1{r}_2)-r_2]}={\theta }_3^{*}$, $\frac{\partial {\delta }_2^{*}}{\partial \rho }<0$, the optimal pollution reduction incentive intensity is a decreasing function of $\rho$. As the absolute risk-averse degree of enterprises increases, the optimal pollution reduction incentive intensity will gradually decrease.

$$\frac{\partial {\delta }_2^{*}}{\partial {\sigma }_1{}^2}=\frac{\rho r_{12}[\rho {\sigma }_2{}^2{r}_{12}{\beta }_2-(1+\rho {\sigma }_2{}^2{r}_2){\beta }_1]}{{[1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2-{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)]}^2}$$

(18)

According to Formula (18), when $r_{12}>0$, $\frac{\partial {\delta }_2^{*}}{\partial {\sigma }_1{}^2}<0$, the optimal pollution reduction incentive intensity is a decreasing function of ${\sigma }_1{}^2$. As the variance in economic benefit increases, the optimal pollution reduction incentive intensity will gradually decrease. When $r_{12}<0$, $\frac{\partial {\delta }_2^{*}}{\partial {\sigma }_1{}^2}>0$, the optimal pollution reduction incentive intensity is an increasing function of ${\sigma }_1{}^2$. As the variance in economic benefit increases, the optimal pollution reduction incentive intensity will gradually increase.

$$\frac{\partial {\delta }_2^{*}}{\partial {\sigma }_2{}^2}=\frac{[\rho {\sigma }_1{}^2{r}_{12}{\beta }_1-(1+\rho {\sigma }_1{}^2{r}_1){\beta }_2][\rho r_2-{\rho }^2{\sigma }_1{}^2(r_{12}{}^2-r_1{r}_2)]}{{[1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2-{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)]}^2}<0$$

(19)

According to Formula (19), the optimal pollution reduction incentive intensity is a decreasing function of ${\sigma }_2{}^2$. As the variance in emission reduction increases, the optimal pollution reduction incentive intensity will gradually decrease.

$$\frac{\partial {\delta }_2^{*}}{\partial r_1}=\frac{{\rho }^2{\sigma }_1{}^4{r}_{12}[(1+\rho {\sigma }_2{}^2{r}_2){\beta }_1-\rho {\sigma }_2{}^2{r}_{12}{\beta }_2]}{{[1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2-{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)]}^2}$$

(20)

According to Formula (20), when $r_{12}>0$, $\frac{\partial {\delta }_2^{*}}{\partial r_1}>0$, the optimal pollution reduction incentive intensity is an increasing function of $r_1$. As the ability of economic benefit of enterprises decreases ($r_1$ increases), the optimal pollution reduction incentive intensity will gradually increase. When $r_{12}<0$, $\frac{\partial {\delta }_2^{*}}{\partial r_1}<0$, the optimal pollution reduction incentive intensity is a decreasing function of $r_1$. As the ability of economic benefit of the enterprise decreases ($r_1$ increases), the optimal pollution reduction incentive intensity will gradually decrease.

$$\frac{\partial {\delta }_2^{*}}{\partial r_2}=\frac{[\rho {\sigma }_1{}^2{r}_{12}{\beta }_1-(1+\rho {\sigma }_1{}^2{r}_1){\beta }_2](\rho {\sigma }_2{}^2+{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2{r}_1)}{{[1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2-{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)]}^2}<0$$

(21)

According to Formula (21), the optimal pollution reduction incentive intensity is a decreasing function of $r_2$. As the direct cost of pollution control for enterprises increases, the optimal pollution reduction incentive intensity will gradually decrease.

$$\frac{\partial {\delta }_2^{*}}{\partial r_{12}}=\frac{\begin{array}{l}\rho {\sigma }_1{}^2[2\rho {\sigma }_2{}^2{r}_{12}(1+\rho {\sigma }_1{}^2{r}_1){\beta }_2\\ -(1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2+{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2{r}_1{r}_2+{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2{r}_{12}{}^2){\beta }_1]\end{array}}{{[1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2-{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2(r_{12}{}^2-r_1{r}_2)]}^2}$$

(22)

According to Formula (22), when $r_{12}>0$, if $2\rho {\sigma }_2{}^2{r}_{12}(1+\rho {\sigma }_1{}^2{r}_1){\beta }_2-(1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2+{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2{r}_1{r}_2+{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2{r}_{12}{}^2){\beta }_1>0$, that is when $\frac{{\beta }_2}{{\beta }_1}>\frac{(1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2+{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2{r}_1{r}_2+{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2{r}_{12}{}^2)}{2\rho {\sigma }_2{}^2{r}_{12}(1+\rho {\sigma }_1{}^2{r}_1)}={\theta }_4^{*}$, $\frac{\partial {\delta }_2^{*}}{\partial r_{12}}>0$, the optimal pollution reduction incentive intensity is an increasing function of $r_{12}$. As the economic cost of pollution control for enterprises increases, the optimal pollution reduction incentive intensity will gradually increase. If $2\rho {\sigma }_2{}^2{r}_{12}(1+\rho {\sigma }_1{}^2{r}_1){\beta }_2-(1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2+{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2{r}_1{r}_2+{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2{r}_{12}{}^2){\beta }_1<0$, that is when $\frac{{\beta }_2}{{\beta }_1}<\frac{(1+\rho {\sigma }_1{}^2{r}_1+\rho {\sigma }_2{}^2{r}_2+{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2{r}_1{r}_2+{\rho }^2{\sigma }_1{}^2{\sigma }_2{}^2{r}_{12}{}^2)}{2\rho {\sigma }_2{}^2{r}_{12}(1+\rho {\sigma }_1{}^2{r}_1)}={\theta }_4^{*}$, $\frac{\partial {\delta }_2^{*}}{\partial r_{12}}<0$, the optimal pollution reduction incentive intensity is a decreasing function of $r_{12}$. As the economic cost of pollution control for enterprises increases, the optimal pollution reduction incentive intensity will gradually decrease.

When $r_{12}<0$, $\frac{\partial {\delta }_2^{*}}{\partial r_{12}}<0$, the optimal pollution reduction incentive intensity is a decreasing function of $r_{12}$. As the economic income of pollution control for enterprises decreases ($r_{12}$↑), the optimal pollution reduction incentive intensity will gradually decrease.

5. Discussion of the Results

Influencing factors of the optimal incentive contracts include the influence coefficient of enterprise economic benefit on local government income ${\beta }_1$, the environmental index weight in the performance appraisal system ${\beta }_2$, the absolute risk-averse degree of enterprises $\rho$, the variance in economic benefit ${\sigma }_1{}^2$, the variance in pollution control ${\sigma }_2{}^2$, the ability of economic benefit of enterprises $r_1$, the direct cost of pollution control for enterprises $r_2$ and the economic cost of pollution control for enterprises $r_{12}$. According to Hypothesis 4, ${\beta }_1$ and ${\beta }_2$, respectively, reflect the importance that local governments attach to the economic ${\beta }_2$ of enterprises and pollution reduction. For easier discussion, the economic meaning of ${\beta }_1/{\beta }_2$ is given as the relative importance that local governments attach to the economic benefit of enterprises; the economic meaning of ${\beta }_2/{\beta }_1$ is the relative importance that local governments attach to pollution reduction by enterprises.

Table 1. Effect of parameter changes on the optimal incentive contract.

Parameter	The Direction of Change	Meaning of Parameter Changes	${\mathit{\delta }}_1^{*}$			${\mathit{\delta }}_2^{*}$
${\beta }_1$	↑	the influence coefficient of enterprise economic benefit on local government income increases	↑			↓	$r_{12}<0$ h,↑
${\beta }_2$	↑	the environmental index weight in the performance appraisal system increases	↓	$r_{12}<0$ h,↑		↑
$\rho$	↑	the absolute risk-averse degree of enterprises increases	${\beta }_1/{\beta }_2>{\theta }_1^{*}$ h		↑	${\beta }_2/{\beta }_1>{\theta }_3^{*}$ h		↑
			${\beta }_1/{\beta }_2<{\theta }_1^{*}$ h		↓	${\beta }_2/{\beta }_1<{\theta }_3^{*}$ h		↓
${\sigma }_1{}^2$	↑	the variance in economic benefit increases	↓			↓	$r_{12}<0$ h,↑
${\sigma }_2{}^2$	↑	the variance in pollution control increases	↓	$r_{12}<0$ h,↑		↓
$r_1$	↑	the ability of economic benefit of enterprises increases	↓			↑	$r_{12}<0$ h,↓
$r_2$	↑	the direct cost of pollution control for enterprises increases	↑	$r_{12}<0$ h,↓		↓
$r_{12}>0$	↑	the economic cost of pollution control for enterprises increases	${\beta }_1/{\beta }_2>{\theta }_2^{*}$ h		↑	${\beta }_2/{\beta }_1>{\theta }_4^{*}$ h		↑
			${\beta }_1/{\beta }_2<{\theta }_2^{*}$ h		↓	${\beta }_2/{\beta }_1<{\theta }_4^{*}$ h		↓
$r_{12}<0$	↑	the economic income of pollution control for enterprises increases	↓			↓

shows the impact of these factors on the optimal incentive contract. It can be seen that local governments should differentiate between types of enterprises, examine their attributes, and design and adjust the optimal incentive contract by combining institutional factors.

When the influence coefficient of the economic benefit of enterprises on local government income increases, the incentive intensity of the economic benefit of enterprises can be increased, prompting them to actively improve their economic benefit. For traditional enterprises, local governments should reduce the incentive intensity of pollution reduction for enterprises, resulting in weakening the incentive for pollution reduction tasks. For innovative enterprises, local governments should increase the incentive intensity of pollution reduction for enterprises, resulting in strengthening the incentive for pollution reduction tasks.

When the environmental index weight in the performance appraisal system increases, the incentive intensity for enterprises to reduce pollution can be increased, prompting enterprises to actively reduce pollution. For traditional enterprises, local governments should reduce the incentive intensity of economic benefit for enterprises, resulting in weakening the incentive for economic benefit tasks. For innovative enterprises, local governments should increase the incentive intensity of economic benefit for enterprises, resulting in strengthening the incentive for economic benefit tasks.

All in all, in order to increase overall benefits local governments should guide traditional enterprises to increase their efforts in two ways: increase the incentive intensity of the task directly; and reduce the opportunity cost of the task. For innovative companies, local governments can increase the incentive intensity of any work task.

If considering the risk preference of enterprise, the local government should combine the institutional factors with the characteristics of enterprise to examine and adjust the incentive contract. For enterprises with high risk aversion, if the relative importance ${\beta }_1/{\beta }_2$ attached by local governments to economic benefit is larger than the threshold value ${\theta }_1^{*}$, the incentive intensity of economic benefit for enterprises should be increased, and if ${\beta }_1/{\beta }_2$ is smaller than ${\theta }_1^{*}$, the incentive intensity of economic benefit for enterprises should be decreased; if ${\beta }_2/{\beta }_1$ attached by local governments to pollution control is larger than the threshold value ${\theta }_3^{*}$, the incentive intensity of pollution control for enterprises should be increased, and if ${\beta }_2/{\beta }_1$ is smaller than ${\theta }_3^{*}$, the incentive intensity of pollution control for enterprises should be decreased.

For enterprises with low risk aversion, the completely opposite incentive strategy can be adopted. The implied economic implication is that for enterprises with high risk aversion at this time, if the relative importance attached by local governments to economic benefit is larger than the threshold value the incentive intensity of economic benefit for enterprises should be increased, and if the relative importance attached by local governments to pollution control is larger than the threshold value the incentive intensity of pollution control for enterprises should be increased; otherwise, more conservative enterprises will be more inclined not to engage in economic benefits or pollution reduction tasks, or to reduce their efforts and investment in the tasks. On the contrary, if the relative importance attached by local governments to economic benefit is smaller than the threshold value or the relative importance attached by local governments to pollution control is smaller than the threshold value, then the incentive to increase economic benefit or reduce pollution reduction can be weakened accordingly.

For the factors of the variance in economic benefit, the variance in pollution reduction, the ability of economic benefit and the direct cost of pollution control, local governments can ignore the institutional factors and the importance it determines for the task, and establish or adjust the incentive contract according to the specifics of these factors and the type of enterprises. For enterprises where the variance in economic benefit is large, whether they are traditional or innovative, local governments should reduce the incentive intensity of their economic benefits. This is because a large variance in economic benefit means that the correlation between the economic benefit efforts of enterprises and the level of economic benefit is not high, and the good or bad of enterprise benefit is more determined by random factors other than enterprise effort (such as business environment, market demand, policy changes, etc.), rather than reflecting the true level of enterprise effort, and thus the level of enterprise benefit tends to show characteristics such as instability and volatility. In this case, intensifying the incentive for enterprises on economic benefit tasks will not achieve the effect of making them work hard. For traditional enterprises, the incentive intensity of pollution reduction should also be reduced to prevent the opportunity cost of economic benefit tasks increasing and further reducing the effort level of the enterprise on economic benefit tasks. For innovative enterprises, the incentive intensity can be increased proportionally to strengthen the incentive for the task of pollution reduction. Conversely, for enterprises with low variance in economic benefit the opposite incentive strategy can be adopted.

For enterprises whose variance in pollution reduction is large, whether traditional or innovative, local governments should reduce the incentive intensity of their pollution reduction. This is because a large variance in pollution reduction implies that the correlation between an enterprise’s efforts to reduce pollution and the amount of pollution reduction is not high, the amount of pollution reduction is more determined by random factors other than enterprise’s efforts (such as geographical factors, climatic factors, other human factors, and so on), and it does not reflect the true level of an enterprises’ effort. In this situation, strengthening the incentive for the task of pollution reduction for enterprises will not achieve the effect of making them work hard. For traditional enterprises, the incentive intensity for economic benefit should also be reduced to prevent the opportunity cost of the pollution reduction task increasing and further reducing the effort level of enterprises on the pollution reduction task. On the other hand, for innovative enterprises the incentive intensity for the economic benefit of enterprises can be increased proportionally to strengthen the incentive for the economic benefit task. Conversely, for enterprises with small variance in pollution reduction, the opposite incentive strategy can be adopted.

For enterprises whose economic benefit capacity is relatively strong, whether traditional and or innovative, local governments should increase the incentive intensity for economic benefit. This is because if enterprises are more capable of economic benefit, such as carrying out production with advanced concepts and methods, abundant knowledge and experience, and a higher level of management, enterprises should further develop their strengths in this area and strengthen incentives for their economic benefit tasks, so as to induce them to devote more time and energy to increasing economic benefit and thereby improving the efficiency of economic output and optimizing resource allocation. For traditional enterprises, the incentive intensity for enterprises to cut pollution can also be reduced to lower the opportunity cost of economic benefit tasks. For innovative enterprises, the incentive intensity for enterprises to reduce pollution can be increased accordingly to strengthen the incentive for the task of pollution reduction. Conversely, for enterprises with relatively poor economic benefit ability the opposite incentive strategy can be adopted.

For enterprises whose direct cost of pollution control is relatively high, whether traditional and or innovative, local governments should reduce the incentive intensity of pollution reduction for enterprises. This is because if the direct cost of pollution control is relatively high, for example, in order to achieve emission reduction targets, enterprises need to directly invest more human, material and financial resources, and the incentive for pollution reduction tasks should be weakened to avoid inefficient use of resources. For traditional enterprises, the incentive intensity for economic benefits can also be increased to raise the opportunity cost of pollution reduction tasks. For innovative enterprises, the incentive intensity for economic benefit can be reduced accordingly to weaken the incentive for economic benefit tasks. Conversely, for enterprises with relatively low direct costs of pollution control, the opposite incentive strategy can be adopted.

For the economic costs of abatement in traditional enterprises, local governments should also combine institutional factors and the importance they determine to the task with the specific economic costs of pollution control, in order to make adjustments to the incentive contract through a comprehensive examination.

For enterprises with a high economic cost of pollution control, if the relative importance ${\beta }_1/{\beta }_2$ attached by local governments to economic benefit is larger than the threshold value ${\theta }_2^{*}$ the incentive intensity of economic benefit for enterprises should be increased, and if ${\beta }_1/{\beta }_2$ is smaller than ${\theta }_2^{*}$ the incentive intensity of economic benefit for enterprises should be decreased; if the relative importance ${\beta }_2/{\beta }_1$ attached by local governments to pollution control is larger than the threshold value ${\theta }_4^{*}$ the incentive intensity of pollution control for enterprises should be increased, and if ${\beta }_2/{\beta }_1$ is smaller than ${\theta }_4^{*}$ the incentive intensity of pollution control for enterprises should be decreased. The implied economic meaning is that for enterprises with relatively high economic costs of pollution control, such as those that are mainly industrial in their production and development methods and more dependent on environmental resource exploitation in their use of resources, or those that are more sensitive to changes in the intensity of pollution reduction and those where the negative impact of investment in pollution reduction on economic benefit is greater, at this time, if the relative importance that local governments attach to the economic benefits of enterprises is higher than the threshold value, the incentive intensity for the economic benefits of enterprises should be strengthened, and if the relative importance that local governments attach to the pollution reduction in enterprises is higher than the threshold value, the incentive intensity for the pollution reduction in enterprises should be strengthened.

Once local governments start to pay attention to environmental issues, if they increase the incentive intensity for enterprises to reduce pollution and induce them to treat and reduce emissions, they can also obtain significant improvements in environmental quality at the cost of abandoning economic benefits. Therefore, adjusting the incentive contract based on the relative importance local governments attach to the task can significantly increase the total utility of local governments. Conversely, if the relative importance that local governments attach to the economic benefits of enterprises is below the threshold value, or the relative importance that they attach to pollution reduction in enterprises is below the threshold value, the incentives for the economic benefits task or the pollution reduction task can be weakened accordingly. As for enterprises with relatively low economic costs of pollution control, such as those whose sources of economic benefits are less dependent on environmental resources, whose production and development methods are dominated by agriculture or service industries with smaller pollution, and who are less affected by the pressure to reduce pollution, at this time the incentive for the task of increasing economic benefits should be weakened if the relative importance that local governments attach to the economic benefits of the enterprises is higher than the threshold value, and the incentive for the task of reducing pollution should be weakened if the relative importance that local governments attach to the pollution reduction in the enterprises is higher than the threshold value. For such enterprises whose constraints between economic efficiency and pollution abatement are weak, the output effect of either strengthening incentives for the economic benefit task or for the pollution reduction task is not particularly significant. Therefore, the more importance local governments attach to either economic efficiency or pollution reduction, the weaker the incentives for both tasks should be so that the total utility of local governments can be increased instead. Conversely, if the relative importance that local governments attach to the economic benefit of enterprises is lower than the threshold value, or the relative importance that they attach to the pollution reduction in enterprises is lower than the threshold value, then the incentives for the economic benefit task or the pollution reduction task can be strengthened correspondingly.

For the economic benefit of pollution control for innovative enterprises, local governments can ignore the institutional factors and the importance they determine for the task. The higher the economic benefit of pollution control, the stronger the incentive for economic benefit or pollution control; the lower the economic benefit of pollution control, the lower the incentive for economic benefit or pollution control. This is because for such enterprises, there are different levels of synergy between the two tasks, with higher economic benefit of pollution control indicating this synergy is stronger and therefore local government incentives for either task will increase the output level of the work task.

Previous studies often emphasized incentives to enterprises on control pollution and punishments for illegal emissions, and most of the policy suggestions that have been put forward were single-dimensional, ignoring the restrictive relationship between economic development and environmental protection. Compared with earlier studies the differences in this study are: based on the multitask principal–agent model, this paper comprehensively examines economic development and environmental protection, and introduces the three influencing factors of institutional characteristics, enterprise types and enterprise attributes. It not only examines the influence of a single factor on the optimal incentive contract, but also comprehensively analyzes the interaction of multiple factors, so as to put forward more operational policy suggestions: the establishment of optimal incentive contract cannot simply rely on rewarding or punishing enterprises, but should consider various factors, adjustable policies and measures flexibly.

6. Conclusions

This paper constructed a multitask principal–agent model of local governments and emission enterprises, studied the incentive problem of enterprises under two task dimensions of economic benefit and pollution reduction, analyzed the influence of relevant factors on the optimal incentive contract, and the results of the study showed that in designing incentive mechanisms for enterprises local governments should comprehensively consider three factors: institutional characteristics, type of enterprises and enterprises’ characteristics. In order to promote the harmonious development between regional economy and the environment, the design of the mechanism based on the results of this study included eight specific aspects:

• When the economic index weight, tax rate or tax retention ratio in the performance appraisal system increases, it can increase the incentive intensity of economic benefit and reduce (increase) the incentive intensity of pollution reduction for traditional (innovative) enterprises; when the environmental index weight in the performance appraisal system increases, it can increase the incentive intensity of pollution reduction and reduce (increase) the incentive intensity of economic benefit for traditional (innovative) enterprises;
• For enterprises with a high risk-averse degree, if local governments attach more importance to economic benefit, they should strengthen the incentive intensity of economic benefit; if local governments attach more importance to pollution reduction, they should strengthen the incentive intensity of pollution reduction. For enterprises with a low risk-averse degree, the opposite incentive strategy can be adopted;
• For enterprises with large variance in economic benefit, whether traditional or innovative, local governments should reduce the incentive intensity for economic benefit. For traditional enterprises incentives for pollution reduction should also be reduced, while for innovative enterprises incentives for pollution reduction can be increased correspondingly. The opposite incentive strategy can be adopted for enterprises with low variance in economic benefit;
• For enterprises with large variance in emission reductions, whether traditional or innovative, local governments should reduce the incentive intensity for pollution reduction. For traditional enterprises the incentive intensity for economic benefit should also be reduced, while for innovative enterprises the incentive intensity for economic benefit can be increased correspondingly. The opposite incentive strategy can be adopted for enterprises with small variance in emission reductions;
• For enterprises with strong ability of economic benefit, whether traditional or innovative, local governments should increase the incentive intensity for economic benefit. For traditional enterprises the incentive intensity for pollution reduction can also be reduced, while for innovative enterprises the incentive intensity for pollution reduction can be increased correspondingly. The opposite incentive strategy could be adopted for enterprises with less ability of economic benefit;
• For enterprises with a relatively high direct cost of pollution control, whether traditional or innovative, local governments should reduce the incentive intensity for pollution reduction. For traditional enterprises the incentive intensity for economic benefit can be increased, while for innovative enterprises the incentive intensity for economic benefit can be reduced correspondingly. The opposite incentive strategy could be adopted for enterprises with a relatively low direct cost of pollution control;
• For traditional enterprises with relatively high economic cost of pollution control, if the local governments attach more importance to economic benefit the incentive intensity of economic benefit should be strengthened; if the local governments attach more importance to pollution reduction, the incentive intensity of pollution reduction should be strengthened. For traditional enterprises with relatively low economic cost of pollution control, the opposite incentive strategy can be adopted;
• For innovative enterprises, the stronger the green innovation ability, the stronger the incentive intensity of economic benefit or pollution reduction; the weaker the green innovation ability, the lower the incentive intensity of economic benefit or pollution reduction.

According to the research conclusions above, the following policy suggestions can be put forward: Firstly, local governments should classify enterprises into basic types in the management process, including traditional enterprises and green innovative enterprises, so as to take different incentives or punishment measures for different types of enterprises. Secondly, the local government’s policy making should be based on the institutional characteristics of the system, learning and understanding the spirit of the central policy so as to cooperate with the national strategy to govern the area. Thirdly, local governments should actively collect and master enterprise attribute information, including risk preference, profit model, pollution control technology and ability, management experience and level, production mode and green innovation ability, etc., which will be very conducive to improving the policy effect. Finally, local governments should combine enterprise types, institutional characteristics and enterprise attributes, flexibly adjusting various policies and measures.

In the future research we can continue to expand the following directions: firstly, introducing the factors of public participation and media attention into the model, so as to further analyze the direct and interactive influences of the two on the optimal incentive contract; and secondly, collecting data through investigation and research, carrying out empirical research, and designing more accurate policy measures based on quantitative analysis.