# Research on the Dynamics Game Model in a Green Supply Chain: Government Subsidy Strategies under the Retailer’s Selling Effort Level

Authors: Xigang Yuan; Xiaoqing Zhang; Dalin Zhang
Journal: Complexity
 (2020)

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## Abstract

Based on dynamic game theory and the principal-agent theory, this paper examined different government subsidy strategies in green supply chain management. Assuming that the retailer’s level of selling effort involved asymmetric information, this study analyzed the impact of different government subsidy strategies on the wholesale price, the product greenness level, retail price, the level of selling effort, the manufacturer’s profit, and the retailer’s profit. The results showed that (1) the government’s subsidy strategy can effectively not only improve the product greenness level but also increase the profits of an enterprise in a green supply chain, which helps the retailer to enhance their selling effort; (2) regardless of whether the retailer’s level of selling effort was high or low, as the government’s subsidy coefficient increased, the wholesale price continued to decrease, and the product greenness level and retailer’s selling effort level also increased.


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## Body

## 1. Introduction

With the rapid development of the economy and society, living standards are improving and there is a trend of rapid growth with respect to energy consumption, all of which have led to intense pressure to control the emissions of harmful gases and reduce the impact of the greenhouse effect and cub environmental harm [1]. Thus, the development of the green economy and the control of greenhouse gas emissions have gradually become a hot topic. Green development and the environmental protection have become indispensable factors to measure healthy economic and societal development. Green supply chain management (GSCM) has also become a key point in academic research [2]. GSCM aims to enhance a firm’s profit and maintain ecological efficiency at each stage of the product life cycle [3]. GSCM can enable firms to realize their coordination and development efforts. The integration of a green supply chain encourages enterprises to share information and cooperate with each other. The relationship between the supply chain and the environment means that GSCM represents an inevitable trend in the area of sustainable economic development.More and more enterprises have focused on GSCM and have implemented it in practice. As a result, enterprises pay greater attention to environmental protection and seek to reduce environmental pollution. For example, in terms of sustainable development, Lenovo explored the construction and implementation of GSCM, and the company strives to develop leading environmental products so as to minimize the impact on environment. Huawei has begun to work on GSCM and integrate the green concept into their entire procurement and production process. Apple Inc. encourages suppliers to participate in GSCM, implements energy transformation, and uses recycled or renewable materials as much as possible. Wal-Mart announced that, in the near future, the company will utilize only renewable energy sources, employ a zero waste strategy, and sell products that satisfy environmental standards.However, as more and more enterprises participate in GSCM activities, they must also incur higher costs. Therefore, it is far from enough to rely on enterprises to promote GSCM practice, and the government should offer substantially more subsidies to firms that implement the GSCM model. The environmental laws and regulations that have been introduced by the government have encouraged enterprises to attach greater importance to GSCM activities. For example, “The directive restricting the use of hazardous substances in electrical and electronic equipment” was implemented in the European Union in 2006, while “The energy-saving products for the benefit of people” was carried out in China in 2009. It provided financial subsidies for ten categories of high-efficiency energy-saving products which have an energy efficiency grade one or above, and it is advantageous as it promotes the development of energy-saving products. It is evident that the government plays a very important role in GSCM activity. On the one hand, the government can either reward or penalize the manufacturer with respect to their carbon emissions from their production processes, and this reward-penalty mechanism can be used to reduce carbon emissions. On the other hand, in order to improve the sale of green products, the government can subsidize retailers based on their selling effort. Thus, the impact of different types of government subsidy strategies on the green supply chain management should be worthy of considering.The mainly purpose of this study is to discuss the impact of different types of government subsidy strategies on the green supply chain management. Thus, we consider a two-level green supply chain system with one manufacturer and one retailer. Assuming that the retailer’s level of selling effort involves asymmetric information, we examined the impact of different government subsidy strategies on their decision-making behavior by using dynamic game theory. The objectives of this paper include the following: (1) to explore how the different government subsidy strategies influence enterprise decision-making behavior and (2) to identify which types of government subsidy strategies are more effective for implementing GSCM.The main contributions of this paper are as follows: (1) we build three dynamic game models, i.e., a game model without a government subsidy strategy (NG model), a game model with a government subsidized manufacturer (MG model), and a game model with a government subsidized retailer (RG model) in GSCM. Moreover, the retailer’s level of selling effort is represented by asymmetric information. (2) We discuss the impact of different government subsidy strategies on decision-making behavior by using dynamic game theory. Moreover, we aimed to explore how different government subsidy strategies influence enterprise decision-making behavior. (3) We identify which types of government subsidy strategies are more effective for implementing GSCM.The remainder of this paper was organized as follows: Section2 reviews the relevant literature; the problem description and model assumptions are provided in Section 3; Section 4 presents the dynamics game model which was developed to analyze the impact of different government subsidy strategies on decision-making behavior; the simulation analysis and management implications are outlined in Section 5; and Section 6 presents the conclusions.


## 2. Literature Review

The government subsidy strategy has been thoroughly studied in the context of GSCM. The literature related to government subsidy strategies and the application of dynamic game theory in SCM and GSCM is reviewed and analyzed in this section. GSCM refers to the process of considering environmental problems at each link in the supply chain, and an emphasis is placed on environmental protection while promoting the coordinated development of the economy and the environment.


### 2.1. Government Subsidy Strategy in SCM and GSCM

The government plays a very important role in SCM and in GSCM. The government can implement a reward-penalty mechanism or subsidy strategy in order to reduce the cost of producing green products and improve the product greenness level. Runkel [4] analyzed the impact of government subsidy policies on GSCM. Subsequently, Aksen et al. [5] and Sheu [6] expanded the scope of research in this area by examining the influence of government policy on GSCM. Aksen et al. [5] studied the different impacts of the government’s supportive policy and legislative policy on GSCM. Sheu [6] pointed out that the government’s green legislation and the financial intervention policy could achieve the enterprise’s interest coordination mechanism of GSCM. Ma et al. [7] studied how the consumption-subsidy strategy impacts dual channel GSCM. Benjaafar et al. [8] used low-carbon factors to solve operational management problems in GSCM. Toptal et al. [9] discussed the carbon emission reduction investment strategy problem in GSCM. Chen et al. [10] analyzed the retailer’s optimal decision with green technology investment in GSCM. Chen et al. [11] developed an evolutionary game theory model of the interaction between governments and manufacturers based on static carbon taxes and subsidies. The results showed that the static carbon tax and subsidy mechanism implemented by the governments could not provide the needed positive impact on manufacturers decision-making. Kang et al. [12] examined supply chain operations when the supply chain participants acted individually and cooperatively in a “green poverty alleviation” supply chain in which the manufacturer initiated product “greening” and provided microfinance to alleviate the poverty of poor raw material suppliers by using dynamics game model. He et al. [13] considered a dual-channel closed-loop supply chain (CLSC) where a manufacturer could distribute new products through an independent retailer and sell remanufactured products via a third-party firm or platform (3P) in the presence of possible government subsidy. They found that government could encourage the manufacturer to adopt desired channel structures by setting appropriate subsidy levels.Some scholars concentrated on the coordination problem under different government strategies. Huang et al. [14] discussed the government’s subsidy strategy and highlighted how customers’ bargaining power can influence customer’ welfare. Zakeri et al. [15] analyzed the effect of carbon emission policies on operational management in the case of GSCM. Li et al. [16] studied the relationship between the manufacturer and retailer within the context of government carbon regulations, and the results showed that the investment cost contract can reduce carbon emissions in GSCM. Li et al. [17] analyzed the impact of government subsidy strategies on welfare by employing Nash game theory. Li and Hu [18] developed the Stackelberg game model, which considered the government reward-penalty mechanism, and analyzed the optimal solution under different situations. Zhang et al. [19] established four game models under fairness preferences in GSCM and compared the impact of parameters on the decision-making process. Gharaei et al. [20] proposed a green supply chain game model under a reward-penalty mechanism and green investment policy.The above literature discussed the impact of different government strategies on GSCM. However, fewer scholars discuss this problem within the context of asymmetric information. In contrast to previous studies, this paper examined the effect of government subsidy strategies on the decision-making process in GSCM under the situation of asymmetric information.


### 2.2. Application of Dynamic Game Theory in SCM and GSCM

Dynamic game theory proposes that the actions of the players are in sequence, and that the latter can observe the choice of the former and make corresponding choices. Dynamic game theory has become a very important method that can be employed in theoretical and applied research. Many scholars discuss human economic decision-making behavior and solve operational management problems [21].Although an extensive amount of research has studied many problems in different fields, few papers have examined the impact of government subsidy strategies on GSCM by using dynamic game theory. Wang et al. [22] developed a GSCM model with a dual collection channel without the government’s reward-penalty mechanism and compared it with the government’s reward-penalty mechanism. Su et al. [23] considered the impact of the environmental protection input on the whole green supply chain and analyzed the effect of centralized decision-making and decentralized decision-making on the returns and pricing strategies of each participant by using dynamic game theory. Sun et al. [24] studied green investment strategies under a government subsidy policy in a two-echelon green supply chain by employing dynamic game theory. He et al. [25] considered an online-to-offline (O2O) tourism supply chain (TSC) consisting of an offline tourism service provider (TSP) and an online travel agency (OTA) who has a corporate social responsibility (CSR). They built three game-theoretic models to explore the optimal online selling model and pricing decisions for the TSC and its members. The result showed that the TSC should pay more attention to CSR and improve consumers’ channel acceptance for the reselling channel to create more total utility. Chen et al. [26] introduced the concept of equity as the incentive mechanism to coordinate the GSCM and established a new network equilibrium model using dynamic game theory. Zhang et al. [27] built three closed-loop supply chain dynamic game models using dynamics game theory and principal-agent theory. Moreover, they discussed the impact of government’s reward-penalty mechanism on the decision variables. He et al. [28] investigated a service supply chain (SC) consisting of a service provider (SP) who was in charge of carbon emission reduction and service, and a service integrator (SI) who was responsible for low-carbon advertising, considering corporate social responsibility (CSR). They found that consumers’ low-carbon preference and chain members’ marginal profits and CSR behaviors significantly influence the optimal solutions.Based on the above analysis, large numbers of studies have considered governmental strategies in their research on GSCM. However, the aforementioned studies simply assumed that the information between all enterprises is symmetric. On the contrary, this paper considered a two-level green supply chain system with one manufacturer and one retailer, and the retailer’s level of selling effort is represented by asymmetric information. We build three dynamic game models, i.e., a game model without a government subsidy strategy (NG model), a game model with a government subsidized manufacturer (MG model), and a game model with a government subsidized retailer (RG model) in GSCM. We discuss the impact of different government subsidy strategies on decision-making behavior by using dynamic game theory. Moreover, we aimed to explore how different government subsidy strategies influence enterprise decision-making behavior. The results showed that government’s subsidy strategy effectively improves not only the product greenness level but also the profits of an enterprise in GSCM, which helps the retailer to improve their own selling effort level.


## 2.1. Government Subsidy Strategy in SCM and GSCM

The government plays a very important role in SCM and in GSCM. The government can implement a reward-penalty mechanism or subsidy strategy in order to reduce the cost of producing green products and improve the product greenness level. Runkel [4] analyzed the impact of government subsidy policies on GSCM. Subsequently, Aksen et al. [5] and Sheu [6] expanded the scope of research in this area by examining the influence of government policy on GSCM. Aksen et al. [5] studied the different impacts of the government’s supportive policy and legislative policy on GSCM. Sheu [6] pointed out that the government’s green legislation and the financial intervention policy could achieve the enterprise’s interest coordination mechanism of GSCM. Ma et al. [7] studied how the consumption-subsidy strategy impacts dual channel GSCM. Benjaafar et al. [8] used low-carbon factors to solve operational management problems in GSCM. Toptal et al. [9] discussed the carbon emission reduction investment strategy problem in GSCM. Chen et al. [10] analyzed the retailer’s optimal decision with green technology investment in GSCM. Chen et al. [11] developed an evolutionary game theory model of the interaction between governments and manufacturers based on static carbon taxes and subsidies. The results showed that the static carbon tax and subsidy mechanism implemented by the governments could not provide the needed positive impact on manufacturers decision-making. Kang et al. [12] examined supply chain operations when the supply chain participants acted individually and cooperatively in a “green poverty alleviation” supply chain in which the manufacturer initiated product “greening” and provided microfinance to alleviate the poverty of poor raw material suppliers by using dynamics game model. He et al. [13] considered a dual-channel closed-loop supply chain (CLSC) where a manufacturer could distribute new products through an independent retailer and sell remanufactured products via a third-party firm or platform (3P) in the presence of possible government subsidy. They found that government could encourage the manufacturer to adopt desired channel structures by setting appropriate subsidy levels.Some scholars concentrated on the coordination problem under different government strategies. Huang et al. [14] discussed the government’s subsidy strategy and highlighted how customers’ bargaining power can influence customer’ welfare. Zakeri et al. [15] analyzed the effect of carbon emission policies on operational management in the case of GSCM. Li et al. [16] studied the relationship between the manufacturer and retailer within the context of government carbon regulations, and the results showed that the investment cost contract can reduce carbon emissions in GSCM. Li et al. [17] analyzed the impact of government subsidy strategies on welfare by employing Nash game theory. Li and Hu [18] developed the Stackelberg game model, which considered the government reward-penalty mechanism, and analyzed the optimal solution under different situations. Zhang et al. [19] established four game models under fairness preferences in GSCM and compared the impact of parameters on the decision-making process. Gharaei et al. [20] proposed a green supply chain game model under a reward-penalty mechanism and green investment policy.The above literature discussed the impact of different government strategies on GSCM. However, fewer scholars discuss this problem within the context of asymmetric information. In contrast to previous studies, this paper examined the effect of government subsidy strategies on the decision-making process in GSCM under the situation of asymmetric information.


## 2.2. Application of Dynamic Game Theory in SCM and GSCM

Dynamic game theory proposes that the actions of the players are in sequence, and that the latter can observe the choice of the former and make corresponding choices. Dynamic game theory has become a very important method that can be employed in theoretical and applied research. Many scholars discuss human economic decision-making behavior and solve operational management problems [21].Although an extensive amount of research has studied many problems in different fields, few papers have examined the impact of government subsidy strategies on GSCM by using dynamic game theory. Wang et al. [22] developed a GSCM model with a dual collection channel without the government’s reward-penalty mechanism and compared it with the government’s reward-penalty mechanism. Su et al. [23] considered the impact of the environmental protection input on the whole green supply chain and analyzed the effect of centralized decision-making and decentralized decision-making on the returns and pricing strategies of each participant by using dynamic game theory. Sun et al. [24] studied green investment strategies under a government subsidy policy in a two-echelon green supply chain by employing dynamic game theory. He et al. [25] considered an online-to-offline (O2O) tourism supply chain (TSC) consisting of an offline tourism service provider (TSP) and an online travel agency (OTA) who has a corporate social responsibility (CSR). They built three game-theoretic models to explore the optimal online selling model and pricing decisions for the TSC and its members. The result showed that the TSC should pay more attention to CSR and improve consumers’ channel acceptance for the reselling channel to create more total utility. Chen et al. [26] introduced the concept of equity as the incentive mechanism to coordinate the GSCM and established a new network equilibrium model using dynamic game theory. Zhang et al. [27] built three closed-loop supply chain dynamic game models using dynamics game theory and principal-agent theory. Moreover, they discussed the impact of government’s reward-penalty mechanism on the decision variables. He et al. [28] investigated a service supply chain (SC) consisting of a service provider (SP) who was in charge of carbon emission reduction and service, and a service integrator (SI) who was responsible for low-carbon advertising, considering corporate social responsibility (CSR). They found that consumers’ low-carbon preference and chain members’ marginal profits and CSR behaviors significantly influence the optimal solutions.Based on the above analysis, large numbers of studies have considered governmental strategies in their research on GSCM. However, the aforementioned studies simply assumed that the information between all enterprises is symmetric. On the contrary, this paper considered a two-level green supply chain system with one manufacturer and one retailer, and the retailer’s level of selling effort is represented by asymmetric information. We build three dynamic game models, i.e., a game model without a government subsidy strategy (NG model), a game model with a government subsidized manufacturer (MG model), and a game model with a government subsidized retailer (RG model) in GSCM. We discuss the impact of different government subsidy strategies on decision-making behavior by using dynamic game theory. Moreover, we aimed to explore how different government subsidy strategies influence enterprise decision-making behavior. The results showed that government’s subsidy strategy effectively improves not only the product greenness level but also the profits of an enterprise in GSCM, which helps the retailer to improve their own selling effort level.


## 3. Problem Description and Model Assumptions

### 3.1. Problem Description

In this section, we consider a two-level green supply chain system with one manufacturer and one retailer (as shown in Figure1). The manufacturer (as leader) sells green products to the retailer at the wholesale price w, and the retailer (as follower) sells green products to the consumer at pricep. Assume that the level of selling effort represents asymmetric information. Furthermore, the retailer’s selling cost not only corresponds to their own level of selling effort but also relates to the greenness level, and it can be expressed as cr=e2g. To simplify the analysis, the selling cost is divided into two types: H-type and L-type. While the manufacturer does not know the real information about the retailer’s selling cost, it knows the probability distribution as peH=vandpeL=1−v. In order to identify the retailer’s true information about the selling cost, the manufacturer can design a screening contract, thus permitting the retailer to make a rational choice by using the information screening method. The government can formulate different subsidy strategies for the manufacturer and the retailer. Therefore, the government can incentivize the manufacturer to produce green products while also encouraging the retailer to improve their selling effort.Figure 1
The theoretical model of the green supply chain considering government’s subsidy strategy under asymmetric information.The following symbols are used in the paper:cm: the manufacturer’s production costwi: the manufacturer’s wholesale price which is the decision variable, i∈H,L, where wH expresses the high wholesale price and wL expresses the low wholesale priced: consumer’s demand for green products from retailera: potential demand in consumer market, where a>0b: the sensitivity of market demand to price, where b>0pi: the retailer’s selling price which is the decision variable, i∈H,L, where pH expresses the high selling price and pL expresses the low selling priceλ: consumer preference for green productsgi: product greenness level which is the decision variable, i∈H,L, where gH expresses a high greenness level and gL expresses a low greenness level (Ghosh and Shah [29, 30], Yang and Xiao [31], and Zhu and Dou [32])cri: the retailer’s selling cost, i∈H,L, where crH expresses a high selling cost and crL expresses a low selling costei: the retailer’s selling effort, i∈H,L, where eH expresses a high selling effort and eL expresses a selling effortθ: the coefficient of the unit product subsidy received by the manufacturerμ: the coefficient of selling cost subsidy received by the retailerz: manufacturer’s research and develop impact factor


### 3.2. Model Assumption

To formulate the problem, several assumptions were made:(1)
The manufacturer and the retailer are participants in the Stackelberg game, whereby the manufacturer is the leader and the retailer is the follower, and they can achieve optimal equilibrium by using dynamic game theory.(2)
Both the manufacturer and the retailer are risk neutral, and they pursue the maximization of their own interests.(3)
The retailer’s level of selling effort is expressed asei, i∈H,L, eL<eH, which indicates that the retailer’s high level of selling effort is higher than the retailer’s low level of selling effort.(4)
To improve the greenness level, the government subsidizes the manufacturer. According to Yang and Xiao [31], Li et al. [33]; Yan et al. [34], and Zhang et al. [35], θg expresses the unit product subsidy received by the manufacturer.(5)
The government subsidizes the retailer on the basis of the selling cost, andμe2g expresses the total amount of subsidies received by the retailer.(6)
The manufacturer’s production capacity is infinite, and it can meet the market demand.


## 3.1. Problem Description

In this section, we consider a two-level green supply chain system with one manufacturer and one retailer (as shown in Figure1). The manufacturer (as leader) sells green products to the retailer at the wholesale price w, and the retailer (as follower) sells green products to the consumer at pricep. Assume that the level of selling effort represents asymmetric information. Furthermore, the retailer’s selling cost not only corresponds to their own level of selling effort but also relates to the greenness level, and it can be expressed as cr=e2g. To simplify the analysis, the selling cost is divided into two types: H-type and L-type. While the manufacturer does not know the real information about the retailer’s selling cost, it knows the probability distribution as peH=vandpeL=1−v. In order to identify the retailer’s true information about the selling cost, the manufacturer can design a screening contract, thus permitting the retailer to make a rational choice by using the information screening method. The government can formulate different subsidy strategies for the manufacturer and the retailer. Therefore, the government can incentivize the manufacturer to produce green products while also encouraging the retailer to improve their selling effort.Figure 1
The theoretical model of the green supply chain considering government’s subsidy strategy under asymmetric information.The following symbols are used in the paper:cm: the manufacturer’s production costwi: the manufacturer’s wholesale price which is the decision variable, i∈H,L, where wH expresses the high wholesale price and wL expresses the low wholesale priced: consumer’s demand for green products from retailera: potential demand in consumer market, where a>0b: the sensitivity of market demand to price, where b>0pi: the retailer’s selling price which is the decision variable, i∈H,L, where pH expresses the high selling price and pL expresses the low selling priceλ: consumer preference for green productsgi: product greenness level which is the decision variable, i∈H,L, where gH expresses a high greenness level and gL expresses a low greenness level (Ghosh and Shah [29, 30], Yang and Xiao [31], and Zhu and Dou [32])cri: the retailer’s selling cost, i∈H,L, where crH expresses a high selling cost and crL expresses a low selling costei: the retailer’s selling effort, i∈H,L, where eH expresses a high selling effort and eL expresses a selling effortθ: the coefficient of the unit product subsidy received by the manufacturerμ: the coefficient of selling cost subsidy received by the retailerz: manufacturer’s research and develop impact factor


## 3.2. Model Assumption

To formulate the problem, several assumptions were made:(1)
The manufacturer and the retailer are participants in the Stackelberg game, whereby the manufacturer is the leader and the retailer is the follower, and they can achieve optimal equilibrium by using dynamic game theory.(2)
Both the manufacturer and the retailer are risk neutral, and they pursue the maximization of their own interests.(3)
The retailer’s level of selling effort is expressed asei, i∈H,L, eL<eH, which indicates that the retailer’s high level of selling effort is higher than the retailer’s low level of selling effort.(4)
To improve the greenness level, the government subsidizes the manufacturer. According to Yang and Xiao [31], Li et al. [33]; Yan et al. [34], and Zhang et al. [35], θg expresses the unit product subsidy received by the manufacturer.(5)
The government subsidizes the retailer on the basis of the selling cost, andμe2g expresses the total amount of subsidies received by the retailer.(6)
The manufacturer’s production capacity is infinite, and it can meet the market demand.


## 4. Dynamics Game Model in GSCM

In the paper, the retailer’s level of selling effort, which represents asymmetric information, can form a principle-agent relationship between the manufacturer and the retailer. Information discrimination theory can be used to identify the type of the retailer’s true selling cost. In addition, the contract design, which is expressed aseH,wL,eL,wH,eH,wL, indicates that the high level of the retailer’s selling effort prompts the manufacturer to offer a low wholesale price to the retailer. eL,wH indicates that the low level of the retailer’s selling effort prompts the manufacturer to offer a high wholesale price to the retailer. The government subsidy strategy was introduced into the game model. The government can implement a subsidy strategy for the manufacturer or retailer. We also carried out a comparative analysis with the nongovernment subsidy strategy.


### 4.1. GSCM Game Model without Government Subsidy Strategy (NG Model)

This game model does not consider the government subsidy strategy, and the manufacturer and the retailer pursue the maximization of their own interests. As the entrusting party, the manufacturer identifies the type of the retailer’s true selling cost and the contract design aseH,wL,eL,wH. The game model is constructed as follows:When the retailer’s selling cost is high, its profit function can be expressed as(1)maxpH,eHπRHNG=pH−ωLa−bpH+λgH+eH−eH2gH.When the retailer’s selling cost is low, its profit function can be expressed as(2)maxpL,eLπRLNG=pL−ωHa−bpL+λgL+eL−eL2gL.The manufacturer’s profit function can be expressed as(3)maxgH,gL,ωH,ωLπMNG=vωLa−bpH+λgH+eH−zgH2+1−vωHa−bpL+λgL+eL−zgL2.The constraints can be expressed as(4)pH−ωLa−bpH+λgH+eH−eH2gH≥π0,(5)pL−ωHa−bpL+λgL+eL−eL2gL≥π0,(6)pH−ωLa−bpH+λgH+eH−eH2gH≥pL−ωHa−bpL+λgL+eL−eL2gL,(7)pL−ωHa−bpL+λgL+eL−eL2gL≥pH−ωLa−bpH+λgH+eH−eH2gH,where π0 is the retailer’s retained profit, and constraints (4) and (5) are the participation constraints which indicate that the retailer can accept the manufacturer’s commission when the retailer’s profit is greater than the retained profit. Constraints (6) and (7) are incentive constraints and indicate that the retailer can maximize profit by choosing a contract that corresponds to their own type of selling cost.Proposition 1.
(a)
The manufacturer’s optimal wholesale price and the optimal product greenness level are expressed as follows:(8)ωHNG∗=aλ+b2v+12v2+aa2+8bz,ωLNG∗=aλ+av−12b+v2a2+8bz,gHNG∗=4azλ+1−va+4bv2a2+8bz,gLNG∗=4az1−v4zv+bv2a2+8bz.(b)
The retailer’s optimal level of selling effort and the optimal selling price are expressed as follows:(9)eHNG∗=3λ+2bvv+1a2+8bz,eLNG∗=λ+bvv+1a2+8bz,pLNG∗=12ba+4λaz1−v4zv+bv2a2+8bz+abλ+abv−12b+v2a2+8bz,pHNG∗=12ba+4λaz+1−va+4bv2a2+8bz+abλ+b22v+12v2+aa2+8bz.Proof.
First, we can obtain the first and second derivatives ofpHNG,eHNG,pLNG,eLNG from functions (1) and (2):(10)∂πRHNG∂pHNG=−2bpHNG+a+λgHNG+eHNG+bwLNG;∂2πRHNG∂pHNG2=−2b<0,∂πRHNG∂eHNG=pHNG−ωLNG−2eHNGgHNG;∂2πRHNG∂eHNG2=−2gHNG<0,∂πRLNG∂pLNG=−2bpLNG+a+λgLNG+eLNG+bwHNG;∂2πRLNG∂pLNG2=−2b<0,∂πRLNG∂eLNG=pLNG−ωHNG−2eLNGgLNG;∂2πRLNG∂eLNG2=−2gLNG<0.
Assuming that the first derivative of thepHNG,eHNG,pLNG,eLNG is equal to zero, we can derive the following:(11)pHNG=a+cg−bwL4bg−1,eHNG=a−bwH+λg4bg−1,pLNG=2ag−wL+2λg2+2bgwL4bg−1,eLNG=2ag−wH+2λg2+2bwHg4bg−1.
Taking equation (11) into the manufacturer’s profit, we can determine that(12)maxg,ωH,ωLπMNG=vωLa−ba+cg−bwL4bg−1+λg+a−bwH+λg4bg−1−zg2+1−vωHa−b2ag−wL+2λg2+2bgwL4bg−1+λg+2ag−wH+2λg2+2bwHg4bg−1−zg2.
From equation (12), we can obtain the first and second derivatives of ωHNG and gHNG, and the Hessian matrix H1 can be expressed as follows:(13)H1=bv−1−λv−12−λv−122zv−1.
Ifλ2−2bz<0, the Hessian matrix H1 is negative, such that πMNG is the concave function, while equation (12) has the maximum, so we can obtain the optimal value ωHNG∗ and gHNG∗. On the contrary, from equation (12), we can obtain the first and the second derivatives of ωLNG and gLNG, and the Hessian matrix H2 can be expressed as follows:(14)H2=bv−1−λv−12−λv−122zv−1
Ifλ2−2bz<0, the Hessian matrix H2 is negative, such that πMNG is the concave function, while equation (12) has the maximum, so we can obtain the optimal value ωLNG∗ and gLNG∗. Thus, we can determine Proposition 1.Theorem 1.
Assume that the retailer’s level of selling effort is represented by asymmetric information. Without the government subsidy strategy and with an increasing green preference among consumers, the wholesale price, manufacturer’s product greenness level, retailer’s selling effort level, and retailer’s selling price continue to increase.According to Proposition1 and Theorem 1, we obtain the following managerial insight. Without the government subsidy strategy, the level of selling effort is the retailer’s private information, in order to obtain more profit, the manufacturer should invest more in the production of green products and strive to improve the greenness level of the product. The manufacturer should communicate with the retailer so as to motivate them to sell more green products and make more profits.


### 4.2. GSCM Game Model with Government Subsidization of the Manufacturer (MG Model)

In order to encourage the manufacturer to produce more green products, the government should subsidize the manufacturers. According to Dai et al. (2017) and Yang and Xiao [31], θg expresses the unit product subsidy received by the manufacturer. The game model can be constructed as follows.When the retailer’s selling cost is high, its profit function can be expressed as(15)maxpH,eHπRHNG=pH−ωLa−bpH+λgH+eH−eH2gH.When the retailer’s selling cost is low, its profit function can be expressed as(16)maxpL,eLπRLNG=pL−ωHa−bpL+λgL+eL−eL2gL.The manufacturer’s profit function can be expressed as(17)maxgH,gL,ωL,ωHπMMG=vωL+θgHa−bpH+λgH+eH−zgH2+1−vωH+θgLa−bpL+λgL+eL−zgL2.The constraints can be expressed as follows:(18)pH−ωLa−bpH+λgH+eH−eH2gH≥π0,(19)pL−ωHa−bpL+λgL+eL−eL2gL≥π0,(20)pH−ωLa−bpH+λgH+eH−eH2gH≥pL−ωHa−bpL+λgL+eL−eL2gL,(21)pL−ωHa−bpL+λgL+eL−eL2gL≥pH−ωLa−bpH+λgH+eH−eH2gH,where π0 is the retailer’s retained profit; constraints (18) and (19) are the participation constraints, which indicate that the retailer can accept the manufacturer’s commission when the retailer’s profit is greater than the retained profit. Constraints (20) and (21) are incentive constraints, which indicate that the retailer can maximize profit by choosing a contract that corresponds to their own type of selling cost.Proposition 2.
(a)
The manufacturer’s optimal wholesale price and the optimal product greenness level are expressed as follows:(22)ωHMG∗=aλ−bθ+b2v+12v2+aa2+8bz,ωLMG∗=aλ−bθ+av−12b+v2a2+8bz,gHMG∗=4zbθλ+4az+1−va+4bv2a2+8bz,gLMG∗=4zbλθ+4az4zv+bv2a2+8bz.(b)
The retailer’s optimal selling effort level and the optimal selling price can be expressed as follows:(23)eHMG∗=3λ+2bvv+1+4vθ2a2+8bz,eLMG∗=λ+bvv+1+2vθa2+8bz,pLMG∗=12ba+4zbλ2θ+4azλ4zv+bv2a2+8bz+abλ+4b2θ+abv−12b+v2a2+8bz,pHMG∗=12ba+4zbθλ2+4azλ+λ1−va+4bv2a2+8bz+abλ+4b2θ+b22v+12v2+aa2+8bz.
The proof process is similar to that of Proposition1.Theorem 2.
Assume that the retailer’s level of selling effort is represented by asymmetric information. Considering that the government subsidizes the manufacturer, and given increasing green preferences among consumers, there is an increase in the wholesale price, the manufacturer’s product greenness level, the retailer’s level of selling effort, and retailer’s selling price.Theorem 3.
Assume that the retailer’s level of selling effort is represented by asymmetric information. Considering that the government subsidizes the manufacturer, and given a higher unit product subsidy coefficient, there is an increase in the manufacturer’s product greenness level, the retailer’s selling effort level, and the retailer’s selling price. However, the manufacturer’s wholesale price decreases.According to Proposition2 and Theorems 2 and 3, we obtain the following managerial insight. When the government subsidizes the manufacturer, the manufacturer obtains the subsidy so that the manufacturer can reduce the wholesale price of green products. In this way, the purchase cost of the retailer can be reduced so that the retailer can be more effectively encouraged to sell more green products. As a result, the manufacturer can obtain much more profit.


### 4.3. GSCM Game Model with Government Subsidization of the Retailer (MG Model)

The government encourages the retailer to sell more green products. Based on the retailer’s selling cost, the government subsidizes the retailer. The total amount of subsidies received by the retailer can be expressed asμe2g. The game model can be constructed as follows.When the retailer’s selling cost is high, its profit function can be expressed as(24)maxpH,eHπRHNG=pH−ωLa−bpH+λgH+eH+μ−1eH2gH.When the retailer’s selling cost is low, its profit function can be expressed as(25)maxpL,eLπRLNG=pL−ωHa−bpL+λgL+eL+μ−1eL2gL.The manufacturer’s profit function can be expressed as(26)maxgH,gL,ωL,ωHπMMG=vωLa−bpH+λgH+eH−zgH2+1−vωHa−bpL+λgL+eL−zgL2.The constraints can be expressed as follows:(27)pH−ωLa−bpH+λgH+eH+μ−1eH2gH≥π0,(28)pL−ωHa−bpL+λgL+eL+μ−1eL2gL≥π0,(29)pH−ωLa−bpH+λgH+eH+μ−1eH2gH≥pL−ωHa−bpL+λgL+eL+μ−1eL2gL,(30)pL−ωHa−bpL+λgL+eL+μ−1eL2gL≥pH−ωLa−bpH+λgH+eH+μ−1eH2gH,where π0 is the retailer’s retained profit; constraints (27) and (28) are the participation constraints which indicate that the retailer can accept the manufacturer’s commission when the retailer’s profit is greater than the retained profit. Constraints (29) and (30) are incentive constraints which indicate that the retailer can maximize profit by choosing a contract that corresponds to their own type of selling cost.Proposition 3.
(a)
The manufacturer’s optimal wholesale price and the optimal product greenness level can be expressed as follows:(31)ωHRG∗=aλ−4bμ+b2v+12v2+aa2+8bz,ωLRG∗=aλ−4bμ+av−12b+v2a2+8bz,gHRG∗=2λμ+4az+1−va+4bv2a2+8bz,gLRG∗=2λμ+4az4zv+bv2a2+8bz.(b)
The retailer’s optimal level of selling effort and the optimal selling price are expressed as follows:(32)eHRG∗=3λ+2bvv+1+2vμa2+8bz,eLRG∗=λ+bvv+1+vμa2+8bz,pLRG∗=12ba+2λ2μ+4azλ4zv+bv2a2+8bz+abλ+b2λμ+abv−12b+v2a2+8bz,pHRG∗=12ba+2λ2μ+4azλ+λ1−va+4bv2a2+8bz+abλ+b2λμ+b22v+12v2+aa2+8bz.
The proof process is similar that of Proposition1.Theorem 4.
Assume that the retailer’s selling effort is represented by asymmetric information. Considering that the government subsidizes the retailer, and given increasing green preferences among consumers, there is an increase in the wholesale price, the manufacturer’s product greenness level, the retailer’s selling effort level, and the retailer’s selling price.Theorem 5.
Assume that the retailer’s level of selling effort is represented by asymmetric information. Considering that the government subsidizes the retailer, and given the increasing coefficient of the selling cost subsidy, there is an increase in the manufacturer’s product greenness level, the retailer’s selling effort level, and the retailer’s selling price. However, the manufacturer’s wholesale price decreases.According to Proposition3 and Theorems 4 and 5, we obtain the following managerial insight. When the green preferences among consumers is higher, on the one hand, the manufacturer should improve the investment and increase the greenness level of the product, so that it can meet the consumer’s demand. On the other hand, the retailer gets the government’s subsidy, and it can improve the selling effort level and enlarge the selling quantity of green products. As a result, the manufacturer and the retailer can obtain much more profit.


## 4.1. GSCM Game Model without Government Subsidy Strategy (NG Model)

This game model does not consider the government subsidy strategy, and the manufacturer and the retailer pursue the maximization of their own interests. As the entrusting party, the manufacturer identifies the type of the retailer’s true selling cost and the contract design aseH,wL,eL,wH. The game model is constructed as follows:When the retailer’s selling cost is high, its profit function can be expressed as(1)maxpH,eHπRHNG=pH−ωLa−bpH+λgH+eH−eH2gH.When the retailer’s selling cost is low, its profit function can be expressed as(2)maxpL,eLπRLNG=pL−ωHa−bpL+λgL+eL−eL2gL.The manufacturer’s profit function can be expressed as(3)maxgH,gL,ωH,ωLπMNG=vωLa−bpH+λgH+eH−zgH2+1−vωHa−bpL+λgL+eL−zgL2.The constraints can be expressed as(4)pH−ωLa−bpH+λgH+eH−eH2gH≥π0,(5)pL−ωHa−bpL+λgL+eL−eL2gL≥π0,(6)pH−ωLa−bpH+λgH+eH−eH2gH≥pL−ωHa−bpL+λgL+eL−eL2gL,(7)pL−ωHa−bpL+λgL+eL−eL2gL≥pH−ωLa−bpH+λgH+eH−eH2gH,where π0 is the retailer’s retained profit, and constraints (4) and (5) are the participation constraints which indicate that the retailer can accept the manufacturer’s commission when the retailer’s profit is greater than the retained profit. Constraints (6) and (7) are incentive constraints and indicate that the retailer can maximize profit by choosing a contract that corresponds to their own type of selling cost.Proposition 1.
(a)
The manufacturer’s optimal wholesale price and the optimal product greenness level are expressed as follows:(8)ωHNG∗=aλ+b2v+12v2+aa2+8bz,ωLNG∗=aλ+av−12b+v2a2+8bz,gHNG∗=4azλ+1−va+4bv2a2+8bz,gLNG∗=4az1−v4zv+bv2a2+8bz.(b)
The retailer’s optimal level of selling effort and the optimal selling price are expressed as follows:(9)eHNG∗=3λ+2bvv+1a2+8bz,eLNG∗=λ+bvv+1a2+8bz,pLNG∗=12ba+4λaz1−v4zv+bv2a2+8bz+abλ+abv−12b+v2a2+8bz,pHNG∗=12ba+4λaz+1−va+4bv2a2+8bz+abλ+b22v+12v2+aa2+8bz.Proof.
First, we can obtain the first and second derivatives ofpHNG,eHNG,pLNG,eLNG from functions (1) and (2):(10)∂πRHNG∂pHNG=−2bpHNG+a+λgHNG+eHNG+bwLNG;∂2πRHNG∂pHNG2=−2b<0,∂πRHNG∂eHNG=pHNG−ωLNG−2eHNGgHNG;∂2πRHNG∂eHNG2=−2gHNG<0,∂πRLNG∂pLNG=−2bpLNG+a+λgLNG+eLNG+bwHNG;∂2πRLNG∂pLNG2=−2b<0,∂πRLNG∂eLNG=pLNG−ωHNG−2eLNGgLNG;∂2πRLNG∂eLNG2=−2gLNG<0.
Assuming that the first derivative of thepHNG,eHNG,pLNG,eLNG is equal to zero, we can derive the following:(11)pHNG=a+cg−bwL4bg−1,eHNG=a−bwH+λg4bg−1,pLNG=2ag−wL+2λg2+2bgwL4bg−1,eLNG=2ag−wH+2λg2+2bwHg4bg−1.
Taking equation (11) into the manufacturer’s profit, we can determine that(12)maxg,ωH,ωLπMNG=vωLa−ba+cg−bwL4bg−1+λg+a−bwH+λg4bg−1−zg2+1−vωHa−b2ag−wL+2λg2+2bgwL4bg−1+λg+2ag−wH+2λg2+2bwHg4bg−1−zg2.
From equation (12), we can obtain the first and second derivatives of ωHNG and gHNG, and the Hessian matrix H1 can be expressed as follows:(13)H1=bv−1−λv−12−λv−122zv−1.
Ifλ2−2bz<0, the Hessian matrix H1 is negative, such that πMNG is the concave function, while equation (12) has the maximum, so we can obtain the optimal value ωHNG∗ and gHNG∗. On the contrary, from equation (12), we can obtain the first and the second derivatives of ωLNG and gLNG, and the Hessian matrix H2 can be expressed as follows:(14)H2=bv−1−λv−12−λv−122zv−1
Ifλ2−2bz<0, the Hessian matrix H2 is negative, such that πMNG is the concave function, while equation (12) has the maximum, so we can obtain the optimal value ωLNG∗ and gLNG∗. Thus, we can determine Proposition 1.Theorem 1.
Assume that the retailer’s level of selling effort is represented by asymmetric information. Without the government subsidy strategy and with an increasing green preference among consumers, the wholesale price, manufacturer’s product greenness level, retailer’s selling effort level, and retailer’s selling price continue to increase.According to Proposition1 and Theorem 1, we obtain the following managerial insight. Without the government subsidy strategy, the level of selling effort is the retailer’s private information, in order to obtain more profit, the manufacturer should invest more in the production of green products and strive to improve the greenness level of the product. The manufacturer should communicate with the retailer so as to motivate them to sell more green products and make more profits.


## 4.2. GSCM Game Model with Government Subsidization of the Manufacturer (MG Model)

In order to encourage the manufacturer to produce more green products, the government should subsidize the manufacturers. According to Dai et al. (2017) and Yang and Xiao [31], θg expresses the unit product subsidy received by the manufacturer. The game model can be constructed as follows.When the retailer’s selling cost is high, its profit function can be expressed as(15)maxpH,eHπRHNG=pH−ωLa−bpH+λgH+eH−eH2gH.When the retailer’s selling cost is low, its profit function can be expressed as(16)maxpL,eLπRLNG=pL−ωHa−bpL+λgL+eL−eL2gL.The manufacturer’s profit function can be expressed as(17)maxgH,gL,ωL,ωHπMMG=vωL+θgHa−bpH+λgH+eH−zgH2+1−vωH+θgLa−bpL+λgL+eL−zgL2.The constraints can be expressed as follows:(18)pH−ωLa−bpH+λgH+eH−eH2gH≥π0,(19)pL−ωHa−bpL+λgL+eL−eL2gL≥π0,(20)pH−ωLa−bpH+λgH+eH−eH2gH≥pL−ωHa−bpL+λgL+eL−eL2gL,(21)pL−ωHa−bpL+λgL+eL−eL2gL≥pH−ωLa−bpH+λgH+eH−eH2gH,where π0 is the retailer’s retained profit; constraints (18) and (19) are the participation constraints, which indicate that the retailer can accept the manufacturer’s commission when the retailer’s profit is greater than the retained profit. Constraints (20) and (21) are incentive constraints, which indicate that the retailer can maximize profit by choosing a contract that corresponds to their own type of selling cost.Proposition 2.
(a)
The manufacturer’s optimal wholesale price and the optimal product greenness level are expressed as follows:(22)ωHMG∗=aλ−bθ+b2v+12v2+aa2+8bz,ωLMG∗=aλ−bθ+av−12b+v2a2+8bz,gHMG∗=4zbθλ+4az+1−va+4bv2a2+8bz,gLMG∗=4zbλθ+4az4zv+bv2a2+8bz.(b)
The retailer’s optimal selling effort level and the optimal selling price can be expressed as follows:(23)eHMG∗=3λ+2bvv+1+4vθ2a2+8bz,eLMG∗=λ+bvv+1+2vθa2+8bz,pLMG∗=12ba+4zbλ2θ+4azλ4zv+bv2a2+8bz+abλ+4b2θ+abv−12b+v2a2+8bz,pHMG∗=12ba+4zbθλ2+4azλ+λ1−va+4bv2a2+8bz+abλ+4b2θ+b22v+12v2+aa2+8bz.
The proof process is similar to that of Proposition1.Theorem 2.
Assume that the retailer’s level of selling effort is represented by asymmetric information. Considering that the government subsidizes the manufacturer, and given increasing green preferences among consumers, there is an increase in the wholesale price, the manufacturer’s product greenness level, the retailer’s level of selling effort, and retailer’s selling price.Theorem 3.
Assume that the retailer’s level of selling effort is represented by asymmetric information. Considering that the government subsidizes the manufacturer, and given a higher unit product subsidy coefficient, there is an increase in the manufacturer’s product greenness level, the retailer’s selling effort level, and the retailer’s selling price. However, the manufacturer’s wholesale price decreases.According to Proposition2 and Theorems 2 and 3, we obtain the following managerial insight. When the government subsidizes the manufacturer, the manufacturer obtains the subsidy so that the manufacturer can reduce the wholesale price of green products. In this way, the purchase cost of the retailer can be reduced so that the retailer can be more effectively encouraged to sell more green products. As a result, the manufacturer can obtain much more profit.


## 4.3. GSCM Game Model with Government Subsidization of the Retailer (MG Model)

The government encourages the retailer to sell more green products. Based on the retailer’s selling cost, the government subsidizes the retailer. The total amount of subsidies received by the retailer can be expressed asμe2g. The game model can be constructed as follows.When the retailer’s selling cost is high, its profit function can be expressed as(24)maxpH,eHπRHNG=pH−ωLa−bpH+λgH+eH+μ−1eH2gH.When the retailer’s selling cost is low, its profit function can be expressed as(25)maxpL,eLπRLNG=pL−ωHa−bpL+λgL+eL+μ−1eL2gL.The manufacturer’s profit function can be expressed as(26)maxgH,gL,ωL,ωHπMMG=vωLa−bpH+λgH+eH−zgH2+1−vωHa−bpL+λgL+eL−zgL2.The constraints can be expressed as follows:(27)pH−ωLa−bpH+λgH+eH+μ−1eH2gH≥π0,(28)pL−ωHa−bpL+λgL+eL+μ−1eL2gL≥π0,(29)pH−ωLa−bpH+λgH+eH+μ−1eH2gH≥pL−ωHa−bpL+λgL+eL+μ−1eL2gL,(30)pL−ωHa−bpL+λgL+eL+μ−1eL2gL≥pH−ωLa−bpH+λgH+eH+μ−1eH2gH,where π0 is the retailer’s retained profit; constraints (27) and (28) are the participation constraints which indicate that the retailer can accept the manufacturer’s commission when the retailer’s profit is greater than the retained profit. Constraints (29) and (30) are incentive constraints which indicate that the retailer can maximize profit by choosing a contract that corresponds to their own type of selling cost.Proposition 3.
(a)
The manufacturer’s optimal wholesale price and the optimal product greenness level can be expressed as follows:(31)ωHRG∗=aλ−4bμ+b2v+12v2+aa2+8bz,ωLRG∗=aλ−4bμ+av−12b+v2a2+8bz,gHRG∗=2λμ+4az+1−va+4bv2a2+8bz,gLRG∗=2λμ+4az4zv+bv2a2+8bz.(b)
The retailer’s optimal level of selling effort and the optimal selling price are expressed as follows:(32)eHRG∗=3λ+2bvv+1+2vμa2+8bz,eLRG∗=λ+bvv+1+vμa2+8bz,pLRG∗=12ba+2λ2μ+4azλ4zv+bv2a2+8bz+abλ+b2λμ+abv−12b+v2a2+8bz,pHRG∗=12ba+2λ2μ+4azλ+λ1−va+4bv2a2+8bz+abλ+b2λμ+b22v+12v2+aa2+8bz.
The proof process is similar that of Proposition1.Theorem 4.
Assume that the retailer’s selling effort is represented by asymmetric information. Considering that the government subsidizes the retailer, and given increasing green preferences among consumers, there is an increase in the wholesale price, the manufacturer’s product greenness level, the retailer’s selling effort level, and the retailer’s selling price.Theorem 5.
Assume that the retailer’s level of selling effort is represented by asymmetric information. Considering that the government subsidizes the retailer, and given the increasing coefficient of the selling cost subsidy, there is an increase in the manufacturer’s product greenness level, the retailer’s selling effort level, and the retailer’s selling price. However, the manufacturer’s wholesale price decreases.According to Proposition3 and Theorems 4 and 5, we obtain the following managerial insight. When the green preferences among consumers is higher, on the one hand, the manufacturer should improve the investment and increase the greenness level of the product, so that it can meet the consumer’s demand. On the other hand, the retailer gets the government’s subsidy, and it can improve the selling effort level and enlarge the selling quantity of green products. As a result, the manufacturer and the retailer can obtain much more profit.


## 5. Result Analysis and Management Implications

### 5.1. Comparative Basis

Based on Theorems3 and 5, the government subsidy not only improves the degree of greenness but also increases the retailer’s level of selling effort and the retailer’s selling price. Based on the same government subsidy expenditure, we can compare and analyze the different effects of the two types of subsidy strategies. The total government subsidy expenditure under the two types of subsidy strategies can be expressed as follows:(33)EθMG∗=θa+3bλ+b2+2v+1v2+2a+8bz3,EμRG∗=μ3λ+2bvv+1+2v4az+1−va+4bv2a2+8bz3.In order to get the same amount of government subsidy expenditure, it makes the comparison more meaningful. The parametersθandμ should satisfy a quantitative relationship. Assuming that EθMG∗=EμRG∗, we can obtain the value μθRG∗ which can be expressed as follows:(34)μθRG∗=θa+3bλ+b2+2v+1v2+23λ+2bvv+1+2v4az+1−va+4bv2.


### 5.2. Comparative Analysis

The reasons as to why the government would subsidize the enterprise in GSCM can be explained as follows: on the one hand, the government can improve the greenness of the products produced by the manufacturer; on the other hand, it can encourage the retailer to sell more green products. Based on the above analysis, the optimal solutions of GSCM in each of the three situations can be compared and analyzed, and it is beneficial to examine the different effects of government subsidy strategies. We can derive the following corollaries.Corollary 1.
Under the same amount of government subsidy expenditure, when the retailer’s level of selling effort is high, there is a negative relationship between the coefficient of the unit product subsidy received by manufacturer and the wholesale price. There is a positive relationship between the coefficient of the unit product subsidy received by the manufacturer and the product greenness level, the retailer’s level of selling effort, and the retailer’s selling price. When the retailer’s selling effort is low, we arrive at the same conclusion. In addition, when the retailer’s level of selling effort is high, there is a negative relationship between the coefficient of the selling cost subsidy received by the retailer and the wholesale price. There is a positive relationship between the coefficient of the selling cost subsidy received by the retailer and the product greenness level, the retailer’s level of selling effort, and the retailer’s selling price. When the retailer’s selling effort is low, we arrive at the same conclusion.Proof.
We can derive the first derivatives ofwHMG∗,wLMG∗,gHMG∗,gLMG∗,eHMG∗,eLMG∗,pHMG∗,andpLMG∗ from Proposition 2, and we can obtain the following:(35)∂ωHMG∗∂θ=−ba2+8bz<0,∂ωLMG∗∂θ=−ba2+8bz<0,∂gHMG∗∂θ=4zbλa2+8bz>0,∂gLMG∗∂θ=4zbλa2+8bz>0,∂eHMG∗∂θ=8vθa2+8bz>0,∂eLMG∗∂θ=2va2+8bz>0,∂pLMG∗∂θ=2zλ2+2ba2+8bz>0,∂pHMG∗∂θ=2zλ2+2ba2+8bz>0.
We can derive the first derivatives ofwHRG∗,wLRG∗,gHRG∗,gLRG∗,eHRG∗,eLRG∗,pHRG∗,andpLRG∗ from Proposition 3, and we can determine the following:(36)∂ωHRG∗∂μ=−4ba2+8bz<0,∂ωLRG∗∂μ=−4ba2+8bz<0,∂gHRG∗∂μ=2λa2+8bz>0,∂gLRG∗∂θ=2λa2+8bz>0,∂eHRG∗∂μ=2va2+8bz>0,∂eLRG∗∂μ=va2+8bz>0,∂pLRG∗∂μ=12b2λ2+b2λa2+8bz>0,∂pHRG∗∂θ=12b2λ2+b2λa2+8bz>0.
Thus, we can obtain Corollary1.Corollary 2.
Under the same amount of government subsidy expenditure, and when the retailer’s level of selling effort is high, under different situations, the manufacturer’s optimal wholesale price has the following inequalityωHNG∗>ωHRG∗>ωHMG∗. When the retailer’s level of selling effort is low, under different situations, the manufacturer’s optimal wholesale price has the following inequality ωLNG∗>ωLRG∗>ωLMG∗.Corollary2 shows that the government’s subsidy strategy can lower the manufacturer’s wholesale price. On the one hand, without the government subsidy strategy, the manufacturer must improve the wholesale price in order to obtain higher profits. Under the government subsidy strategy, the manufacturer can receive certain economic compensation from the government so that the manufacturer can lower the wholesale price. However, when the government subsidizes the manufacturer, the wholesale price is lower than that which would be observed in the case of government subsidization of the retailer.Corollary 3.
Under the same amount of government subsidy expenditure, when the retailer’s level of selling effort is high, under different situations, the manufacturer’s optimal level of greenness has the following inequalitygHNG∗<gHRG∗<gHMG∗. When the retailer’s level of selling effort is low, under different situations, the manufacturer’s optimal greenness level has the following inequality gLNG∗<gLRG∗<gLMG∗.Corollary3 shows that the government’s subsidy strategy can improve the manufacturer’s greenness level. On the one hand, without the government subsidy strategy, the manufacturer can reduce the cost to earn more money, which can result in a lower greenness level. Under the government subsidy strategy, the manufacturer can receive certain economic compensation from the government so that the manufacturer can improve the greenness level. On the other hand, when the government subsidizes the manufacturer, the product greenness level is higher than that which would be observed when the government subsidizes the retailer. From the perspective of the level of product greenness, the government’s subsidization of the manufacturer is more effective than subsidization of the retailer.Corollary 4.
Under the same amount of government subsidy expenditure, when the retailer’s level of selling effort is high, the retailer’s optimal selling effort has the following inequalityeHNG∗<eHMG∗<eHRG∗. When the retailer’s level of selling effort is low, the retailer’s optimal selling effort has the following inequality eLNG∗<eLMG∗<eLRG∗.Corollary4 shows that the government’s subsidy strategy can improve the retailer’s selling effort. In order to increase the quantity of sales, the retailer needs to improve their level of selling effort, which can lead to higher selling costs for the retailer. Under the government subsidy strategy, the retailer is able to improve their selling effort. However, under the same amount of government subsidy expenditure, the retailer can benefit from the government subsidy strategy. Thus, it can strongly motivate the retailer to sell more products. From the perspective of the level of selling effort, government subsidization of the retailer is more effective than subsidization of the manufacturer.


### 5.3. Numerical Analysis

The impact of the government subsidy strategy on the manufacturer’s and retailer’s profit can be analyzed using numerical analysis. Assuminga=200, b=10, ρ=0.4, z=2, and v=4. The corresponding value of subsidy parameters is shown in Table 1.Table 1
The corresponding value of subsidy parameters under the same government subsidy expenditure.
θ0.10.20.30.40.50.60.70.80.91μθRG∗0.150.270.310.440.5210.6840.7810.880.9141.21Under two different types of government subsidy strategy, the impact of different subsidy parameters on the wholesale price, greenness level, selling price, level of selling effort, and manufacturer’s and retailer’s profit is shown in Figures2‒7.Figure 2
The impact of different government subsidy coefficients on the manufacturer’s wholesale price under two models.Figure 3
The impact of different government subsidy coefficients on the manufacturer’s greenness level under two models.Figure 4
The impact of different government subsidy coefficients on retailer’s selling price under two models.Figure 5
The impact of different government subsidy coefficients on retailer’s level of selling effort under two models.Figure 6
The impact of different government subsidy coefficients on the enterprise’s profit when the selling effort level is low.Figure 7
The impact of different government subsidy coefficients on the enterprise’s profit when the level of selling effort is high.As shown in Figure2, whether the retailer’s level of selling effort is either high or low, the manufacturer’s wholesale price shows a downward trend in the MG and RG models. Moreover, when the retailer’s level of selling effort is high, the manufacturer’s wholesale price is higher than when the retailer’s level of selling effort is low. The manufacturer can receive some economic compensation. Thus, the manufacturer’s wholesale price can decrease under the government subsidy strategy. Furthermore, the manufacturer’s wholesale price is at the highest level in the RG model, while it is at its lowest level in the MG model. This indicates that government subsidization of the manufacturer is more effective than subsidization of the retailer.As shown in Figure3, regardless of whether the retailer’s level of selling effort is high or low, as the government subsidy coefficient increases, the greenness level increases in the MG and RG models. Moreover, when the retailer’s level of selling effort is high, the greenness level is higher than when the retailer’s level of selling effort is low. The manufacturer can benefit from two types of government subsidy strategy. The manufacturer is in a position to increase its investment in the research and development of green products, and it can improve the product greenness level. However, the greenness level in the MG model is higher than that in the RG model. From the perspective of improving the greenness level, this indicates that the government’s subsidization of the manufacturer is more effective than subsidization of the retailer.As shown in Figure4, regardless of whether the retailer’s level of selling effort is high or low, as the government subsidy coefficient increases, the retailer’s selling price shows a downward trend in the MG and RG models. The retailer’s selling price is at the lowest level under the RG model and at the highest level under the MG model. In addition, when the retailer’s level of selling effort is high, the retailer’s selling price is higher than when the retailer’s level of selling effort is low. From the perspective of improving the retailer’s selling price, this indicates that the government’s subsidization of the retailer is more effective than subsidization of the manufacturer.As shown in Figure5, as the government subsidy coefficient increases, the retailer’s level of selling effort increases under two types of government subsidy strategy. The retailer’s level of selling effort under the RG model is higher than that under the MG model. It is more obvious that the government subsidizes the retailer so that it can improve the retailer’s level of selling effort.As shown in Figures6 and 7, when the retailer’s level of selling effort is low, and as the government subsidy coefficient increases, both the manufacturer and retailer’s profit increases. For the manufacturer, its profit in the RG model is higher than that in the MG model. On the contrary, for the retailer, when it satisfies the condition θ∈0,0.8, the retailer’s profit in the MG model is higher than that in the RG model. When it satisfies the condition θ∈0.8,1, the retailer’s profit in the RG model is higher than that in the MG model. Moreover, when the retailer’s level of selling effort is high, the manufacturer’s profit in the MG model is higher than that in the RG model; the retailer’s profit in the MG model is higher than that in the RG model.


## 5.1. Comparative Basis

Based on Theorems3 and 5, the government subsidy not only improves the degree of greenness but also increases the retailer’s level of selling effort and the retailer’s selling price. Based on the same government subsidy expenditure, we can compare and analyze the different effects of the two types of subsidy strategies. The total government subsidy expenditure under the two types of subsidy strategies can be expressed as follows:(33)EθMG∗=θa+3bλ+b2+2v+1v2+2a+8bz3,EμRG∗=μ3λ+2bvv+1+2v4az+1−va+4bv2a2+8bz3.In order to get the same amount of government subsidy expenditure, it makes the comparison more meaningful. The parametersθandμ should satisfy a quantitative relationship. Assuming that EθMG∗=EμRG∗, we can obtain the value μθRG∗ which can be expressed as follows:(34)μθRG∗=θa+3bλ+b2+2v+1v2+23λ+2bvv+1+2v4az+1−va+4bv2.


## 5.2. Comparative Analysis

The reasons as to why the government would subsidize the enterprise in GSCM can be explained as follows: on the one hand, the government can improve the greenness of the products produced by the manufacturer; on the other hand, it can encourage the retailer to sell more green products. Based on the above analysis, the optimal solutions of GSCM in each of the three situations can be compared and analyzed, and it is beneficial to examine the different effects of government subsidy strategies. We can derive the following corollaries.Corollary 1.
Under the same amount of government subsidy expenditure, when the retailer’s level of selling effort is high, there is a negative relationship between the coefficient of the unit product subsidy received by manufacturer and the wholesale price. There is a positive relationship between the coefficient of the unit product subsidy received by the manufacturer and the product greenness level, the retailer’s level of selling effort, and the retailer’s selling price. When the retailer’s selling effort is low, we arrive at the same conclusion. In addition, when the retailer’s level of selling effort is high, there is a negative relationship between the coefficient of the selling cost subsidy received by the retailer and the wholesale price. There is a positive relationship between the coefficient of the selling cost subsidy received by the retailer and the product greenness level, the retailer’s level of selling effort, and the retailer’s selling price. When the retailer’s selling effort is low, we arrive at the same conclusion.Proof.
We can derive the first derivatives ofwHMG∗,wLMG∗,gHMG∗,gLMG∗,eHMG∗,eLMG∗,pHMG∗,andpLMG∗ from Proposition 2, and we can obtain the following:(35)∂ωHMG∗∂θ=−ba2+8bz<0,∂ωLMG∗∂θ=−ba2+8bz<0,∂gHMG∗∂θ=4zbλa2+8bz>0,∂gLMG∗∂θ=4zbλa2+8bz>0,∂eHMG∗∂θ=8vθa2+8bz>0,∂eLMG∗∂θ=2va2+8bz>0,∂pLMG∗∂θ=2zλ2+2ba2+8bz>0,∂pHMG∗∂θ=2zλ2+2ba2+8bz>0.
We can derive the first derivatives ofwHRG∗,wLRG∗,gHRG∗,gLRG∗,eHRG∗,eLRG∗,pHRG∗,andpLRG∗ from Proposition 3, and we can determine the following:(36)∂ωHRG∗∂μ=−4ba2+8bz<0,∂ωLRG∗∂μ=−4ba2+8bz<0,∂gHRG∗∂μ=2λa2+8bz>0,∂gLRG∗∂θ=2λa2+8bz>0,∂eHRG∗∂μ=2va2+8bz>0,∂eLRG∗∂μ=va2+8bz>0,∂pLRG∗∂μ=12b2λ2+b2λa2+8bz>0,∂pHRG∗∂θ=12b2λ2+b2λa2+8bz>0.
Thus, we can obtain Corollary1.Corollary 2.
Under the same amount of government subsidy expenditure, and when the retailer’s level of selling effort is high, under different situations, the manufacturer’s optimal wholesale price has the following inequalityωHNG∗>ωHRG∗>ωHMG∗. When the retailer’s level of selling effort is low, under different situations, the manufacturer’s optimal wholesale price has the following inequality ωLNG∗>ωLRG∗>ωLMG∗.Corollary2 shows that the government’s subsidy strategy can lower the manufacturer’s wholesale price. On the one hand, without the government subsidy strategy, the manufacturer must improve the wholesale price in order to obtain higher profits. Under the government subsidy strategy, the manufacturer can receive certain economic compensation from the government so that the manufacturer can lower the wholesale price. However, when the government subsidizes the manufacturer, the wholesale price is lower than that which would be observed in the case of government subsidization of the retailer.Corollary 3.
Under the same amount of government subsidy expenditure, when the retailer’s level of selling effort is high, under different situations, the manufacturer’s optimal level of greenness has the following inequalitygHNG∗<gHRG∗<gHMG∗. When the retailer’s level of selling effort is low, under different situations, the manufacturer’s optimal greenness level has the following inequality gLNG∗<gLRG∗<gLMG∗.Corollary3 shows that the government’s subsidy strategy can improve the manufacturer’s greenness level. On the one hand, without the government subsidy strategy, the manufacturer can reduce the cost to earn more money, which can result in a lower greenness level. Under the government subsidy strategy, the manufacturer can receive certain economic compensation from the government so that the manufacturer can improve the greenness level. On the other hand, when the government subsidizes the manufacturer, the product greenness level is higher than that which would be observed when the government subsidizes the retailer. From the perspective of the level of product greenness, the government’s subsidization of the manufacturer is more effective than subsidization of the retailer.Corollary 4.
Under the same amount of government subsidy expenditure, when the retailer’s level of selling effort is high, the retailer’s optimal selling effort has the following inequalityeHNG∗<eHMG∗<eHRG∗. When the retailer’s level of selling effort is low, the retailer’s optimal selling effort has the following inequality eLNG∗<eLMG∗<eLRG∗.Corollary4 shows that the government’s subsidy strategy can improve the retailer’s selling effort. In order to increase the quantity of sales, the retailer needs to improve their level of selling effort, which can lead to higher selling costs for the retailer. Under the government subsidy strategy, the retailer is able to improve their selling effort. However, under the same amount of government subsidy expenditure, the retailer can benefit from the government subsidy strategy. Thus, it can strongly motivate the retailer to sell more products. From the perspective of the level of selling effort, government subsidization of the retailer is more effective than subsidization of the manufacturer.


## 5.3. Numerical Analysis

The impact of the government subsidy strategy on the manufacturer’s and retailer’s profit can be analyzed using numerical analysis. Assuminga=200, b=10, ρ=0.4, z=2, and v=4. The corresponding value of subsidy parameters is shown in Table 1.Table 1
The corresponding value of subsidy parameters under the same government subsidy expenditure.
θ0.10.20.30.40.50.60.70.80.91μθRG∗0.150.270.310.440.5210.6840.7810.880.9141.21Under two different types of government subsidy strategy, the impact of different subsidy parameters on the wholesale price, greenness level, selling price, level of selling effort, and manufacturer’s and retailer’s profit is shown in Figures2‒7.Figure 2
The impact of different government subsidy coefficients on the manufacturer’s wholesale price under two models.Figure 3
The impact of different government subsidy coefficients on the manufacturer’s greenness level under two models.Figure 4
The impact of different government subsidy coefficients on retailer’s selling price under two models.Figure 5
The impact of different government subsidy coefficients on retailer’s level of selling effort under two models.Figure 6
The impact of different government subsidy coefficients on the enterprise’s profit when the selling effort level is low.Figure 7
The impact of different government subsidy coefficients on the enterprise’s profit when the level of selling effort is high.As shown in Figure2, whether the retailer’s level of selling effort is either high or low, the manufacturer’s wholesale price shows a downward trend in the MG and RG models. Moreover, when the retailer’s level of selling effort is high, the manufacturer’s wholesale price is higher than when the retailer’s level of selling effort is low. The manufacturer can receive some economic compensation. Thus, the manufacturer’s wholesale price can decrease under the government subsidy strategy. Furthermore, the manufacturer’s wholesale price is at the highest level in the RG model, while it is at its lowest level in the MG model. This indicates that government subsidization of the manufacturer is more effective than subsidization of the retailer.As shown in Figure3, regardless of whether the retailer’s level of selling effort is high or low, as the government subsidy coefficient increases, the greenness level increases in the MG and RG models. Moreover, when the retailer’s level of selling effort is high, the greenness level is higher than when the retailer’s level of selling effort is low. The manufacturer can benefit from two types of government subsidy strategy. The manufacturer is in a position to increase its investment in the research and development of green products, and it can improve the product greenness level. However, the greenness level in the MG model is higher than that in the RG model. From the perspective of improving the greenness level, this indicates that the government’s subsidization of the manufacturer is more effective than subsidization of the retailer.As shown in Figure4, regardless of whether the retailer’s level of selling effort is high or low, as the government subsidy coefficient increases, the retailer’s selling price shows a downward trend in the MG and RG models. The retailer’s selling price is at the lowest level under the RG model and at the highest level under the MG model. In addition, when the retailer’s level of selling effort is high, the retailer’s selling price is higher than when the retailer’s level of selling effort is low. From the perspective of improving the retailer’s selling price, this indicates that the government’s subsidization of the retailer is more effective than subsidization of the manufacturer.As shown in Figure5, as the government subsidy coefficient increases, the retailer’s level of selling effort increases under two types of government subsidy strategy. The retailer’s level of selling effort under the RG model is higher than that under the MG model. It is more obvious that the government subsidizes the retailer so that it can improve the retailer’s level of selling effort.As shown in Figures6 and 7, when the retailer’s level of selling effort is low, and as the government subsidy coefficient increases, both the manufacturer and retailer’s profit increases. For the manufacturer, its profit in the RG model is higher than that in the MG model. On the contrary, for the retailer, when it satisfies the condition θ∈0,0.8, the retailer’s profit in the MG model is higher than that in the RG model. When it satisfies the condition θ∈0.8,1, the retailer’s profit in the RG model is higher than that in the MG model. Moreover, when the retailer’s level of selling effort is high, the manufacturer’s profit in the MG model is higher than that in the RG model; the retailer’s profit in the MG model is higher than that in the RG model.


## 6. Conclusions and Future Research

This study examined a two level green supply chain system with one manufacturer and one retailer, in which the market demand was mainly affected by the product greenness level, the level of selling effort, and the selling price. First, it was assumed that the retailer’s level of selling effort was represented by asymmetric information, and that the manufacturer designed incentive contracts to induce the retailer to disclose private information by using the principal-agent theory. Second, the game models, both without a government subsidy strategy and with two types of government subsidy strategies, were constructed. The optimal solution of the wholesale price, the greenness level, the selling price, the level of selling effort, and the manufacturer and the retailer’s profit was obtained. Finally, we arrived at the following conclusions: (1) the government subsidy strategy improves not only the greenness level but also the manufacturer and retailer’s profit; (2) regardless of whether the retailer’s level of selling effort is high or low, as the government subsidy coefficient increases, the wholesale price, greenness level, and the level of selling effort increases; (3) when the retailer’s level of selling effort is low, and as the government subsidy coefficient increases, the manufacturer and retailer’s profit increase. For the manufacturer, its profit in theRG model is higher than that in the MG model. Moreover, when the retailer’s level of selling effort is high, the manufacturer’s profit in the MG model is higher than that in the RG model; the retailer’s profit in the MG model is higher than that in the RG model.In the paper, we only considered the green supply chain game dominated by the manufacturer. We did not consider the green supply chain game dominated by the retailer. This study explored the impact of government subsidy strategies on the green supply chain. This paper did not consider the impact of a government carbon tax strategy or a reward-penalty mechanism on the green supply chain. We aim to evaluate these issues in future research.



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*Source: 3083761-2020-06-16.xml*