Research on cooperation strategy between government and green supply chain based on differential game

Abstract Based on the “three bottom line” and stakeholder theory, the paper considers the relationship and cooperation strategy between the government and the supplier and manufacturer of the green supply chain. By constructing the dynamic differential game model, the paper discusses the differences in the optimal effort level, green degree of product, reputation and the optimal benefit under the three situations of noncooperation, government promotion and collaborative cooperation. The results show that the optimal effort level, green degree of product, reputation and the optimal benefit in collaborative cooperation are obviously higher than the situations of non-cooperation and government promotion, and the cooperation of the three parties can promote the development of green supply chain. Government promotion is better than noncooperation. The government plays an active role in improving the optimal benefit and reputation of green supply chain. Finally, the reliability of the proposed proposition is verified by an example analysis, which provides an important reference for improving the efficiency of green supply chain.


Introduction
China's economic transformation is taking place. It has been recognized that the development of traditional rough type has caused serious environmental pollution and ecological damage [1]. The three bottom line (TBL) theory holds that enterprise development must take into account economic, environmental and social bene ts [2]. It provides a basis for the transformation and development of Chinese enterprises. Therefore, in the "13th Five-Year plan" , China put forward the new development concept of innovation, coordination, green, openness and sharing. The formation of green supply chain ts the new development concept. "Actively promoting the green supply chain and strengthening the green supervision of the supply chain" is also clearly required in the guidance of the Chinese government to actively promote the innovation and application of supply chain. However, the green supply chain of China is still in the development stage, and the relevant governance mechanism is not perfect. Therefore, based on stakeholder theory, this paper discusses the cooperation strategy between the government and the green supply chain, and considers the di erences between the government and the green supply chain in di erent game situations, which will help to improve the government's governance of the green supply chain.
Green Supply Chain Management (GSCM) has become a hot topic in academic circles, and many scholars have done related research [3,4]. Among them, the green supply chain coordination, consumer willingness, government subsidies have attracted the attention of scholars. With regard to the coordination of green supply chain, Huang et al. [5] use game theory to study the relationship between green supply chain coordination and greenhouse gas emission reduction, and nd that product line design, supplier selection, transport mode selection and pricing strategy will a ect greenhouse gas emissions. Price and green level sensitivity will a ect the coordination of green supply chain, and the cost sharing contract in green supply chain has a positive impact on improving product green level [6]. The development strategy selection of green supply chain is analyzed by di erential game model, and it is found that centralized decision can be achieved through contract design to achieve the coordination of green supply chain [7]. On the study of the consumer behavior of the green supply chain, consumers' green sensitivity plays an important role in the green supply chain of smart phones, and consumers are highly sensitive to the green level of smart phones [8]. Wang et al. [9] nd that the cost driven elements and consumer driven elements play an important role in the management practice of the green supply chain, and the role of consumer driven for small companies is more obvious. At the government level, Sheu [10] believes the government's scal intervention will have a signi cant impact on the strategy choices of members of the green supply chain, and government intervention may have a negative e ect on corporate pro ts and social welfare. Yang and Xiao [11] discuss the relationship between product pricing and green level in green supply chain under government intervention through three game models, it is found that when government intervention increases, the level of green will rise, but government intervention is not always conducive to manufacturers of green supply chains. Sinayi and Rasti-Barzoki [12] construct a two-level model of government and supply chain, analyze the role of government in the green development of supply chain, and point out that di erent government policies have a signi cant e ect on the pro t and environment of supply chain members. Guo et al. [13] compare the impact of di erent government subsidy policies (subsidies for manufacturers' green e orts and price subsidies to green products) on the green manufacturing of supply chain, and nd that the choice of government subsidy policy depends on the consumer sensitivity to the price of green products. Government subsidies can increase the pro ts of green supply chain entities and promote the development of green supply chain [14].
To sum up, the studies of green supply chain by scholars have laid a solid foundation for subsequent research, but there are still have some knowledge gaps. First of all, the previous researches usually show the role of government as an impact parameter in consideration of the role of the government, and do not bring the government as a stakeholder into the research framework, which lead to a certain degree of deviation in the analysis of the role of the government. Secondly, the green supply chain should re ect the combination of economic, social and environmental bene ts. Most of the previous studies consider the problem of the coordination of green supply chain from the perspective of maximizing the economic pro t. Finally, the cooperation of the green supply chain is a dynamic process. The green degree of product and reputation of the supplier and manufacturer will change according to the change of the state variables, and it is a long-term process. The previous researches are mostly based on the static perspective, and the dynamic changes of the characteristics of the green supply chain are not fully considered. Therefore, based on the di erential game theory, this paper constructs the dynamic game model between the government and the supplier and manufacturer of the green supply chain, and analyzes the three situations of the noncooperation, the government promotion and the collaborative cooperation. We hope to reveal the speci c role of the government in the green supply chain, and then provide suggestions for the development of green supply chain. This paper is arranged as follows. Section 2 presents the problem description and research assumptions. In Section 3, three di erential game models are formulated according to the three situations. Section 4 presents a comparison of the di erent situations. Simulations and example analysis are conducted in Section 5, and conclusions are presented in Section 6.

Problem description and research assumptions
A simple system is used to describe the relationship between the government and the supplier and manufacturer of the green supply chain. In the green supply chain, the supplier and the manufacturer constitute a two-level supply chain. The main body of the supply chain needs to improve and update the raw materials and equipment, so as to better satisfy the relevant conditions of green innovation. Supplier provide raw materials to manufacturer, and manufacturer provide green products to consumers through production, assembly and sales [7]. Due to the growing awareness of consumers' environmental protection, a preference for green products is formed [15]. The consumers play an important role in the game because their environmental awareness has a signi cant in uence on market demand. The role of government in green supply chain is to provide a series of policies and subsidies to supplier and manufacturer. The government's green supervision will also promote the green production of the supply chain, further reduce energy consumption and environmental pollution. We assume that government behavior will encourage green supply chain to carry out green innovation and enhance the green degree of product. It is conducive to the uni cation of economic, environmental and social bene ts [15].
In green supply chain, supplier and manufacturer will make e orts to enhance the green degree of product and reputation. Let N S (t) denotes the e ort level of supplier at time t, and let N M (t) denotes the e ort level of manufacturer at time t. According to the assumption of cost in previous studies [17,18], we assume that the cost of upgrading the green degree of product and reputation is a convex function of e ort.
Therefore, the cost of supplier and manufacturer of green supply chain can be written as C S (t) = λ S N S (t) and C M (t) = λ M N M (t), where λ S , λ M are the cost coe cients of the supplier and manufacturer at time t, respectively. As a stakeholder, the government will make corresponding e orts to improve the green degree of product and reputation of the supplier and manufacturer, because the government wants the green supply chain to develop rapidly. Let N G (t) and N G (t) denote the e ort level of government for supplier and manufacturer. The cost of government can be written as C G (t) = λ G N G (t) + N G (t) , where λ G is the cost coe cient of the government at time t. It is proposed that the promotion of product green degree is a time-varied dynamic process. It depends on the e orts of the supplier, manufacturer and government toward the promotion of product green degree. The e orts of green supply chain and government during the production process contribute to the promotion of product green degree, which evolves according to the Nerlove and Arrow model (1962). When there is no e ort, the green degree of product will decay because the related equipment will depreciate with time. In addition, the green degree of the supplier's product can a ect the green degree of the manufacturer's product. Let D S (t) and D M (t) respectively denote the product green degree of supplier and manufacturer at time t. The di erential equation of the product green degree of supplier can be expressed as For the manufacturer, the green degree of the product supplied by the supplier will a ect the green degree of the manufacturer's product to a certain extent. According to formula (1), the di erential equation of green degree of manufacturer's product can be expressed as where α and α are respectively the in uence coe cient of supplier and manufacturer's e ort level on product green degree, β and β are respectively the in uence coe cient of government e ort level on product green degree of green supply chain. η and η are respectively the attenuation coe cient of product green degree for supplier and manufacturer. µ is the promotion coe cient of supplier product to the green degree of manufacturer product. In the initial state, Considering that green products are environmentally friendly and can be recognized by consumers, the promotion of reputation will also bene t the green supply chain and the whole system. Based on this, the consideration of reputation e ect will further improve the research of green supply chain. Reputation also depends on the e orts of green supply chains and government [19]. Similar to the product green degree, reputation also decays over time without e ort. Let Q S (t) and Q M (t) respectively denote the reputation of supplier and manufacturer at time t. The di erential equation for the reputation of supplier and manufacturer can be written as Q where δ and δ are respectively the in uence coe cient of supplier and manufacturer e ort level on reputation, φ and φ are respectively the in uence coe cient of government e ort level on reputation of green supply chain. γ and γ are respectively the attenuation coe cient of reputation for supplier and manufacturer. In the initial state, Let π S (t) and π M (t) denote the bene ts (including economic bene t, environmental bene t and social bene t) of supplier and manufacturer at time t. As we all know, the bene ts come from the e orts of manufacturer, supplier and government, the green dgree of products and reputation. According to literature [15], the bene t function of supplier and manufacturer can be expressed as where p and p are the in uence coe cient of supplier and manufacturer e ort level on bene t, k and k are the in uence coe cient of government e ort level on bene t of green supply chain. l and l are the in uence coe cient of product green degree on bene t, b and b are the in uence coe cient of reputation on bene t.

Di erential game equilibrium analysis
Cooperation between government and green supply chain is a long-term dynamic process, and the e ects of the supplier, manufacturer and government can be inter-temporal. Supplier, manufacturer and government are far sighted, they seek bene t maximization in the long run. It is useful to introduce the dynamic framework into the research on cooperation between government and green supply chain. Therefore, we use the di erential game to analyze this problem and discuss the di erences under the three situations of non-cooperation, government promotion and collaborative cooperation.

. Situation of non-cooperation
In the situation of non-cooperation, both supplier and manufacturer will independently decide their level of e ort to maximize their own bene t. The only regulatory mechanism in this situation is based on the market. Namely, the government will supervise green supply chain and provide corresponding e orts. However, the government does not subsidize the cost of product development of supplier and manufacturer of green supply chain. The bene t of green supply chain will be allocated between the government and the green supply chain. Let θ i ( ≤ θ i ≤ , i = , ) respectively denote the bene t coe cient of the government from supplier and manufacturer, let − θ denotes the bene t coe cient of the supplier, let − θ denotes the bene t coe cient of the manufacturer. λ S , λ M and λ G are the cost coe cients of the supplier, manufacturer and government at time t, respectively. We assume that the discount rate r is the same. In this situation, the objective functions (net bene t) of the supplier, manufacturer and government be expressed as For convenience in writing and understanding, time t is omitted in the next analysis [20].

Proposition 1
In the situation of non-cooperation, the feedback non-cooperative game Nash equilibria are Proof According to the optimal control theory and the su cient conditions of the Nash equilibrium strategy, the three parties of the supplier, manufacturer of the green supply chain and government have bene t optimal value functions.
respectively denote the bene t optimal value functions of the supplier, manufacturer and government. For all D i ≥ and Q S ≥ , i ∈ {S, M} , the bene t optimal value functions satisfy the following Hamilton-Jacobi-Bellman equation For solving formula (14), (15) and (17), using extreme conditions and searching for the optimal value of N S , N M , N G and N G by setting the all partial derivative equal to zero, we can get Substituting the results of (17), (18), (19) and (20) into (14), (15) and (17), we can obtain Through the above formulas and the optimal control theory, we can see that the linear optimal function of the D S , D M , Q S , Q M is the solution of the HJB equation. Based on the predecessor's literature [16,21], we can get where σ i , q i , g i (i = , , , ) and ω i (i = , , ) are the constants to be solved. We can have Substituting the result of (24) and (25) into (21), (22) and (23), we can get Using the D S , D M , Q S , Q M ≥ to (26), (27) and (28), parameter values of the optimal value function can be expressed as follows Substituting the results of (29), (30), (31) and (32) into (24), we can obtain In this case, the bene t of the whole system is Solving the partial derivative of (33), (34) and (35), and substituting the results into (17), (18), (19) and (20), we can get the optimal e ort level of supplier and manufacturer and government as the proposition 1.
In the situation of non-cooperation, according to formula (1) and (3), we use the general solution method of di erential equations, and can get the supplier's product green degree and reputation as Similar to supplier, according to formulas (2) and (4), the manufacturer's product green degree and reputation are From proposition 1, we can nd that when supplier, manufacturer and government only consider maximizing their own bene ts, the optimal e ort level is related to some parameters. For the supplier and manufacturer of the green supply chain, p i , α i , l i , δ i , b i (i = , ) are positively related to the optimal level of e ort, η i , γ i (i = , ) , λ S , λ M , r are negatively related to the optimal level of e ort, k i (i = , ) are not related to the optimal level of e ort, it shows that government e orts have no impact on the optimal level of e ort of supplier and manufacturer.
For the government, and µ are positively related to the optimal level of e ort, η i , γ i (i = , ) , λ G and r are negatively related to the optimal level of e ort. Government needs to integrate various factors to formulate corresponding incentive policies and regulatory mechanisms.

. Situation of government promotion
The government plays an important role in promoting green supply chain cooperation. Cost subsidy is a common measure for the government to promote the development of green supply chain. Therefore, in the situation of government promotion, we focus on the cost sharing of the government to the green supply chain. We assume that the cost sharing coe cient for the supplier and manufacturer are ε and ε . The government is a leader, the rst to make the cost sharing coe cient, the supplier and manufacturer in the green supply chain as the followers, and then decide how much e ort should be made. So, the situation of the Stackelberg master-slave game is formed. In this situation, the objective functions (net bene t) of the supplier, manufacturer and government be expressed as Proposition 2 In the situation of government promotion, the feedback Stackelberg master-slave equilibria are (for the proof, see the Appendix) Similar to proposition 1, in the situation of government promotion, we can get the supplier's product green degree and reputation as The manufacturer's product green degree and reputation are From proposition 2, in the case of government promotion, we nd that the e orts of supplier and manufacturer in green supply chain are positively related to the government's bene t distribution coe cient. The more bene t the government gains, the higher the e orts of the supplier and manufacturer, which re ects the characteristics of the three parties driven by the government. The government is a leader, the rst to make the cost sharing coe cient, the supplier and manufacturer in the green supply chain as the followers, if the government expects to get more bene t, it will enhance the subsidy level and promote the e ciency of green supply chain. When the bene t distribution coe cient is too small, the government will not provide a cost subsidy strategy to the green supply chain. The other in uence coe cients are the same as the situation of non-cooperation.

. Situation of collaborative cooperation
In the situation of collaborative cooperation, the government and the supplier and manufacturer of the green supply chain will choose their optimal e ort levels based on maximization of their total bene t. Therefore, product green degree and reputation can be further improved through cooperation between the government and the supplier and manufacturer of the green supply chain.

Proposition 3
In the situation of collaborative cooperation, the feedback cooperative game equilibria are (for the proof, see the Appendix) Similar to proposition 1, in the situation of collaborative cooperation, we can get the supplier's product green degree and reputation as The manufacturer's product green degree and reputation are From proposition 3, in the case of collaborative cooperation, we nd that the bene t allocation coe cient no longer a ects the decisions of supplier, manufacturer and government on the optimal e ort level. The reason is that supplier, manufacturer and government work together to maximize the bene t of the whole system. It is no longer binding on the bene t obtained by each other. Therefore, bene t allocation coe cient does not a ect the optimal e ort level. The other in uence coe cients are the same as the situation of noncooperation and government promotion.

Comparison between situations
According to the analysis of the three situations, we get di erent optimal e ort level, optimal bene t, product green degree and reputation. So, we develop four propositions for the di erent game equilibriums in the aspects of di erent optimal e ort level, optimal bene t, product green degree and reputation.
According to the < θ ≤ , we can getN ** S − N * S > ,N *** S − N ** S ≥ . From formula (11), (45) and (55), we can get From formula (13), (47) and (57), we can get N * G = N ** G , So, the cost sharing coe cient of the government are ε ** = In proposition 4, we can nd that the optimal e ort of supplier and manufacturer has been improved in the situation of government promotion, and this promotion has a signi cant correlation with the government's cost subsidy coe cient, but the optimal e ort level of government has not changed. The reason is that the government will work out the level of e ort based on the related factors of its own bene t, which is the same as that in the non-cooperative situation, so the optimal e ort level of the government has not changed. In the situation of collaborative cooperation, government, supplier and manufacturer will both make higher levels of e ort to improve the product green degree and reputation.

Proposition 5
If the bene t distribution coe cient from the government satis es the condition < θ ≤ , then the steady-state product green degree by the supplier hold for D * S ≤ D ** S ≤ D *** S , the steady-state product green degree by the manufacturer hold for D * M ≤ D ** M ≤ D *** M (for the proof, see the Appendix). In proposition 5, we can see that government cost subsidies play an important role in promoting the product green degree of green supply chain. Both supplier and manufacturer's product green degree has been increased. Collaborative cooperation is optimal in three situations.

Proposition 6
If the bene t distribution coe cient from the government satis es the condition < θ ≤ , then the steady-state reputation by the supplier hold for Q * S ≤ Q ** S ≤ Q *** S , the steady-state reputation by the manufacturer hold for Q * M ≤ Q ** M ≤ Q *** M (for the proof, see the Appendix). In proposition 6, we can see that government cost subsidies play an important role in promoting the reputation of green supply chain. Both supplier and manufacturer's reputation has been increased. Collaborative cooperation is optimal in three situations.

Proposition 7
If the bene t distribution coe cient from the government satis es the condition < θ ≤ , then the steady-state bene t by the supplier hold for V * S ≤ V ** S , the steady-state bene t by the manufacturer hold for V * M ≤ V ** M , the steady-state bene t by the government hold for V * G < V ** G , the steady-state bene t by the whole system hold for V * T ≤ V ** T ≤ V *** T (for the proof, see the Appendix). In proposition 7, we can see that the bene ts of government, supplier, manufacturer and the whole system have been increased in the situations of government promotion and collaborative cooperation. Government cost subsidies play an important role in promoting the reputation of green supply chain.

Simulation and example analysis
To better understand the long-term relationship between the equilibriums of three situations, simulation and example analysis of the models are conducted in this section. According to the assignment method of relevant literature [22,23], through the assignment of relevant parameters, three situations can be compared. The speci c parameter assignment is shown in Table 1.
We calculate the optimal e ort level of the government and the supplier and manufacturer of the green supply chain in three situations, as shown in Table 2.
In the three situations, we can nd that the level of e ort of supplier, manufacturer and government is the lowest in the non-cooperative di erential game. The government's cost subsidy strategy enhances the level of e ort of supplier and manufacturer in green supply chain. The level of e ort supplier, manufacturer and government have greatly improved in the situation of collaborative cooperation. This veri es the truth of proposition 4. Based on the results of Table 2, we can further calculate the product green degree, reputation and bene t, as shown in Table 3.
In order to make more intuitive comparison, we use Origin to draw out the trend of product green degree and reputation changing with time, as shown in Figures 1 and 2. As time goes on, the green degree of product and reputation of the supplier and manufacturer are growing rapidly in the initial period, then the speed decreases, and nally tends to be smooth. In the case of collaborative cooperation, supplier and manufacturer in green supply chain have the highest level of e ort, and government promotion is higher than non-cooperation. This veri es propositions 5 and 6.

Situation
Product green degree, reputation and bene t non cooperation government promotion We compare the bene ts of supplier, manufacturer, government and the whole system in di erent situations, as shown in Figure 3. When the government adopts the cost subsidy strategy, the bene t of the supplier and manufacturer of the green supply chain will be greatly improved than the non-cooperation situation. It shows that the government plays an important role in promoting the development of green supply chain. A relatively suitable strategy can e ectively improve the bene ts of the green supply chain and the government. Figure 3 (d) shows the whole system bene ts of government, supplier and manufacturer in di erent situations. It can be found that the whole system bene t reaches the maximum in the situation of collaborative cooperation, followed by the situation of government promotion, and the lowest one is the non-cooperation situation.

Conclusion
In this paper, we have shown a di erential game model under the three situations of non-cooperation, government promotion and cooperative cooperation. We consider the relationship and cooperation strategy between the government and the supplier and manufacturer of the green supply chain, and discuss the di erences in the optimal e ort level, green degree of product, reputation and the optimal bene t of each subject. By comparing and analyzing of equilibrium results, we have known that the optimal e ort level, product green degree, reputation and optimal bene t are obviously higher in the situation of collaborative cooperation than that under non-cooperation and government promotion. The government's cost subsidy strategy enhances the level of e ort of supplier and manufacturer in green supply chain. The cost coe cient, the natural attenuation coe cient of product green degree, the discount rate and the negative in uence factors of reputation are negatively correlated with the optimal level of e ort. The in uence coe cient of the level of e ort, the in uence coe cient of product green degree on the bene t, the in uence coe cient of reputation on the bene t, the in uence coe cient of the degree of e ort on the green degree of the product, and the in uence coe cient of the e ort on the reputation are positively related to the optimal level of e ort.

Proof of Proposition 3
In order to obtain the cooperative equilibrium, there exists a bene t optimal value function of