COST-SHARING STRATEGY FOR CARBON EMISSION REDUCTION AND SALES EFFORT: A NASH GAME WITH GOVERNMENT SUBSIDY

. We investigate the cost-sharing strategies of a retailer and a manufacturer in a Nash game considering government subsidy, consumers’ green preference and retailer’s sales eﬀort. We provide a function to describe the demand for green products considering the eﬀect of green preference of con- sumers and the sales eﬀort of the retailer. Next, we construct proﬁt functions of the manufacturer and the retailer considering government subsidy for four scenarios: no sharing of cost (NSC), sharing of carbon emission reduction cost (SCERC), sharing of sales eﬀort cost (SSEC), and sharing both carbon emis- sion reduction cost and sales eﬀort cost (SBC). Furthermore, we determine the optimal policies of price, sales eﬀort level, wholesale price and carbon emission reduction eﬀort level for the four scenarios by maximizing the proﬁts of the manufacturer and the retailer in the Nash game. We ﬁnd that the sales eﬀort cost-sharing ratio and the carbon emission reduction cost-sharing ratio can af-fect the optimal policies of the manufacturer and the retailer, and the trends and extent of eﬀects may be diﬀerent. Our results show that it is advanta-geous for the manufacturer and the retailer to consider the cost-sharing eﬀects of sales eﬀort and carbon emission reduction eﬀort, and the optimal policies of the retailer and the manufacturer are diﬀerent for diﬀerent scenarios.


1.
Introduction. Green products generally refer to a kind of energy-saving and environment-friendly products whose demand is sensitive to the consumers' green preference, see [11,20,40,45]. Green products are widespread and include such products as all-electric vehicles, energy-efficient refrigerators, solar heaters and recycled paper. To protect and improve the environment, the government usually provides the subsidy to encourage manufacturers to produce or retailers to sell green products, see [45,24]. For the manufacturer that produces green products with carbon emission reduction effort, the higher the carbon emission reduction effort level is, the higher the green degree of the products is, and further the higher the demand is, see [40,32,35]. For the retailer that undertakes the sales effort, the higher the sales effort level is, the higher the demand is, see [3,4]. Usually, the manufacturer needs to pay the carbon emission reduction cost and the retailer needs to pay the sales effort cost. In reality, to maximize the profit, the manufacturer and the retailer may share each others cost, see [9,27]. Thus, how to determine the optimal cost-sharing strategies of the retailer and the manufacturer is an important research question.
The key to this problem is: how to describe and depict the demand for green products with the retailer's sales effort and the manufacturer's carbon emission reduction effort, how to construct the profit functions with government subsidy, how to obtain the optimal policies of the retailer and the manufacturer, what are the impacts of cost-sharing ratios on the decisions of the retailer and the manufacturer, and how to determine the cost-sharing strategy.
The motivation of this study is four-fold. First, the retailer that sells green products usually puts in sales effort to increase demand. Thus, in the demand analysis, it is necessary to consider the impacts of consumers' green preference and sales effort on the demand. Second, the government usually provides a subsidy to manufacturers or retailers in the green supply chain. Third, the existing research pays little attention to the optimal policies of price, sales effort level, wholesale price and carbon emission reduction effort considering sales effort cost-sharing and carbon emission reduction effort cost-sharing, whereas in reality, the manufacturer and the retailer usually need to determine the optimal policies. Fourth, the manufacturer and retailer need to determine the optimal cost-sharing strategies for different scenarios.
This study contributes to the literature in the following three aspects. First, we analyze and describe the demand for green products considering consumers' green preference and retailer's sales effort, and we construct a green sensitivity function and a demand function. Second, we construct the profit functions of the retailer and the manufacturer for the following four scenarios: no sharing of cost (NSC), sharing of carbon emission reduction cost (SCERC), sharing of sales effort cost (SSEC), and sharing both carbon emission reduction cost and sales effort cost (SBC). Then, we determine the optimal policies of price, sales effort level, wholesale price and carbon emission reduction effort level. Third, we analyze the impacts of cost-sharing ratios on the optimal policies, and find that cost-sharing ratios can affect the optimal policies of the retailer and the manufacturer to varying degrees. Fourth, we determine and analyze the optimal cost-sharing strategies of the manufacturer and the retailer from different perspectives for the scenarios.
The rest of this paper is organized as follows. Section 2 reviews the existing related literature. Section 3 provides a problem description, demand function and profit functions for the four scenarios: NSC, SCERC, SSEC and SBC, and then obtains the optimal policies of the manufacturer and the retailer by simultaneously maximizing their profits according to a Nash game. In Section 4, we conduct a numerical study to show the impacts of cost-sharing ratios on the optimal policies of the manufacturer and the retailer, and then we analyze the optimal cost-sharing strategies of the manufacturer and the retailer for the different scenarios. In Section 5, we discuss the managerial contributions of our study. In Section 6, we conclude and suggest possible extensions for further research. All the proofs are provided in the Appendix.
2. Literature review. There is a substantial amount of research on cost-sharing [7,13,30,34], consumers' green preference [20,16,25,31,43], sales effort in the supply chain [3,6,15,17,33,38,39,41], and government subsidy [1,5,8,21,26,36], but research on cost-sharing strategy in a Nash game considering simultaneously government subsidy, consumers' green preference and retailer's sales effort is still lacking. As the key to our study is to determine the carbon emission reduction and sales effort cost-sharing strategy in the green supply chain, we mainly focus on the most relevant literature on the cost-sharing strategy in some fields. The related literature primarily focuses on the cost-sharing contract, cost-sharing game and other cost-sharing problems.
1) Cost-sharing contract: Lee and Cho [18] design a contract for vendor-managed inventory (VMI) with consignment stock considering stockout-cost sharing. The authors find that VMI may result in significant cost-sharing saving for both the retailer and the manufacturer, and that the value of information sharing is also important for VMI contracting. Lee et al. [19] focus on supply chain coordination with stockout-cost sharing in a VMI system when the storage capacity is limited, and provide a condition under which VMI can coordinate the supply chain. Ghosh and Shah [12] design a cost-sharing contract to coordinate the green supply chain, and explore the impact of the cost-sharing contract on the decisions of supply chain players. Zhou et al. [46] design a cost-sharing contract with respect to co-op advertising and emission reduction cost in a low carbon supply chain considering fairness concerns. The authors find that in some cases, the retailer's fairness concerns can change the coordination of co-op advertising and emission reduction cost sharing contracts. Bai et al. [2] propose revenue and promotional cost-sharing contract and a two-part tariff contract to coordinate a two-tier sustainable supply chain system. The authors find that the two contracts can lead to perfect coordination, and that the revenue and promotional cost-sharing contract is less robust than the two-tier tariff contract. Dai et al. [7] analyze cartelization and cost-sharing contract, and find that the cost-sharing contract yields more profit than the non-cooperative mode does, and that cartelization is Pareto improvement in some certain conditions.
2) Cost-sharing game: Gopalakrishnan et al. [14] study the problem of designing distribution rules to share welfare (cost or revenue), and show that the potential games are necessary to ensure pure Nash equilibrium in cost-sharing games. Panda et al. [30] investigate the coordination and profit division of the supply chain with cost sharing and bargaining, and design mechanisms to resolve channel conflict. The authors show that the channel members' profits are a win-win in the mechanism. Zhang et al. [44] focus on the reciprocity between buyer's cost sharing and supplier's technology sharing, and find that buyer's cost sharing is reciprocated with supplier's sharing of new technology, while supplier's sharing was found to be reciprocated by the buyers. Gkatzelis et al. [13] investigate a policymaker that can set a priori rules to minimize the inefficiency induced by selfish players. The authors prescribe desirable properties of the cost-sharing method and prove that the costsharing method induced by the Shapley value minimizes the worst-case inefficiency of equilibria. Zeng et al. [42] study the problem in which independent operators of queuing systems cooperate to generate a win-win solution through capacity transfer among each other. The authors propose a greedy heuristic to find approximate solutions and design cost-allocation rules for the corresponding game.
3) Other cost-sharing problems: Tsao and Sheen [34] incorporate the idea of the sales learning curve into promotion cost, and analyze the effects of a promotion cost-sharing policy under retailer competition and promotional effort with a sales learning curve. The authors also discuss how retailer competition and the sales learning curve affect channel decisions and profits. Elmachtoub and Levi [10] study a general class of online selection problems, and provide a general framework to develop online algorithms for the problems based on the cost-sharing mechanisms. Deleire et al. [8] study the cost-sharing subsidies of consumers with respect to insurance selection plans, and show that consumers are highly sensitive to the value of cost-sharing reductions when selecting insurance plans.
The existing literature has made great contributions to research on cost-sharing in some fields, but does not involve the research on cost-sharing strategy in a Nash game simultaneously considering government subsidy, consumers' green preference and retailers sales effort. Thus, the existing research results are not suitable to solve the problem in our study. Hence, it is necessary to conduct the research to solve the problem in this study.
3. Profit functions and solutions. In this section, we first present a problem description for the cost-sharing strategies of a retailer and a manufacturer in a Nash game in the case of considering government subsidy, consumers' green preference and retailer's sales effort. Then, we give the notation about the decision variables, parameters, functions and optimal values involved in this study. Next, we build the demand function by analyzing the impacts of the retail price, sales effort level and carbon emission reduction effort level on the demand. On this basis, we construct the corresponding profit functions for the four scenarios: NSC, SCERC, SSEC and SBC, and determine the optimal policies of the manufacturer and the retailer by maximizing their profits based on the Nash game.
3.1. Problem description and notation. We consider a cost-sharing decision problem in a dyadic supply chain with government subsidy, consumers' green preference and retailer's sales effort. In the problem, the manufacturer and the retailer are profit-seeking decision makers. Before the selling season, the manufacturer decides the wholesale price and the carbon emission reduction effort level, while the retailer decides the price and sales effort level. Usually, the retailer's price is higher than the manufacturer's wholesale price, and the wholesale price is higher than the cost of unit product, that is, p > w > c ≥ 0 . The retailer has only one opportunity to order from the manufacturer, and the manufacturer has enough capacity to supply products according to the retailer's order, and can deliver them to the retailer on time. The government usually provides subsidies to the manufacturer and the retailer in the environment-friendly supply chain, and here the subsidies are considered to be constant, see [24,41,22]. The structure and interaction process of green supply chain is shown in Figure 1.
The demand is related to not only the price but also the retailer's sales effort level. Moreover, for green products, demand is also related to the manufacturer's carbon emission reduction effort level. For the sales effort level, the retailer has to pay a certain sales effort cost. The higher the sales effort level is, the higher the sales effort cost is. The manufacturer can share the sales effort cost of the retailer at a set ratio. Usually, the sales effort cost-sharing ratio is given by the manufacturer. The motivation of sales effort cost sharing is to maximize the manufacturer's profit. On the other hand, for the carbon emission reduction effort level, the manufacturer has to pay a certain carbon emission reduction cost. The higher the carbon emission reduction effort level is, the higher the carbon emission reduction cost is. The retailer can share the carbon emission reduction cost of the manufacturer at a set ratio. Usually, the carbon emission reduction cost-sharing ratio is given by the retailer. The motivation of carbon emission reduction cost sharing is to maximize the retailer's profit. The problem analyzed in this study is how to determine the cost-sharing strategy regarding sales effort cost and carbon emission reduction cost considering government subsidy, consumers' green preference and retailer's sales effort in the four scenarios of NSC, SCERC, SSEC and SBC. The problem focuses on the following five aspects: (1) How to describe the demand for green products with sales effort level and carbon emission reduction effort level?
(2) How to construct the profit functions of the manufacturer and retailer for the four scenarios of NSC, SCERC, SSEC and SBC?
(3) How to determine the optimal policy of the wholesale price and carbon emission reduction effort level of the manufacturer and the optimal policy of the price and sales effort level of the retailer?
(4) How do cost-sharing parameters (i.e., sales effort cost sharing ratio and carbon emission reduction cost sharing ratio) affect the optimal decisions of the manufacturer and retailer for NSC, SCERC, SSEC and SBC? (5) How to determine the cost-sharing strategies of the manufacturer and the retailer regarding sales effort cost and carbon emission reduction cost? The notations used in this study are as follows: Decision variables: p i : price of unit product, which is determined by the retailer, where the price consists of wholesale price w i of the manufacturer and unit margin x i of the retailer [22], that is, p i = w i + x i , x i is used to describe the retailer's profit for unit sold product, p i > 0, i = 1, 2, 3, 4, where i corresponds to the four scenarios: NSC, SCERC, SSEC and SBC.
A i : sales effort level of the retailer, A i ≥ 0, i = 1, 2, 3, 4. w i : wholesale price of unit product, which is determined by the manufacturer w i > 0, i = 1, 2, 3, 4. e i : carbon emission reduction effort level of the manufacturer, which is related to the green degree of the product; the higher the carbon emission reduction effort level is, the higher the product's green degree is, e i ≥ 0, i = 1, 2, 3, 4.

Parameters:
c: production cost of unit product without carbon emission reduction related cost, c > 0, c < w i < p i . β: production cost for carbon emission reduction of unit product with respect to carbon emission reduction effort level e i , β ≥ 0, c + βe i < w i < p i . a: market size of green products, which is related to the function and quality of green products, a ≥ 0. b: price elasticity, which is related to consumers' sensitivity to price, b > 0. m: effectiveness of sales effort, m > 0. n: consumers' low-carbon preferences, n > 0. T r : subsidy of the government for the retailer, which is considered a constant value, T r > 0. T m : subsidy of the government for the manufacturer, which is considered a constant value, T m > 0. γ A : sales effort cost-sharing ratio of the manufacturer, which is generally decided by the manufacturer, 0 < γ A < 1. γ e : carbon emission reduction cost-sharing ratio of the retailer, which is generally decided by the retailer, 0 < γ e < 1.

Functions:
F Ai : sales effort cost of the retailer, F Ai = αA i 2 2, where α denotes the sales effort cost factor, α > 0, i = 1, 2, 3, 4. F ei : fixed carbon emission reduction cost of the manufacturer, such as facility cost, F ei = δe i 2 2, where δ denotes the carbon emission reduction cost factor, δ > 0, i = 1, 2, 3, 4. E (e i ): green sensitivity demand, which is related to the carbon emission reduction effort, E (e i ) = ei−ē e−eD e , e i ∈ [e,ē], where e andē denote the lower and upper bounds of the carbon emission reduction effort level, respectively. In particular, when e i = e, the green sensitivity demand is not related to consumers green preference; when e < e i <ē, the green sensitivity demand is directly proportional to consumers green preference; when e i =ē, the green sensitivity demand has the maximum valueD e , i = 1, 2, 3, 4. D i : market demand for green products, i = 1, 2, 3, 4. π i R : profit of the retailer, i = 1, 2, 3, 4. π i M : profit of the manufacturer, i = 1, 2, 3, 4. π i sc : profit of the green supply chain, π i sc = π i R + π i M , i = 1, 2, 3, 4.
Optimal values: p * i : optimal price of the retailer for four scenarios, i = 1, 2, 3, 4. A * i : optimal sales effort level of the retailer for four scenarios, i = 1, 2, 3, 4. w * i : optimal wholesale price of the manufacturer for four scenarios, i = 1, 2, 3, 4. e * i : optimal carbon emission reduction effort level of the manufacturer for four scenarios, i = 1, 2, 3, 4.
3.2. Demand function. In reality, the demand for green products can be affected by the retail price and the carbon emission reduction effort level, see [39,23,29]. In addition, the retailer's sales effort level can affect demand. On this basis, we consider that the demand function is where E (e i ) is the green sensitivity demand, as shown in Figure 2, which is defined to describe the trend and extent of the impact of the carbon emission reduction effort level on demand [28,37]. The green sensitivity demand E (e i ) can be described as  According to Eqs. (1) and (2), the demand function for green products can be converted into

) Sharing Both Carbon Emission Reduction Cost and
Sales Effort Cost. Then, the Nash game theory is used as the applied optimization tool, and the optimization computation is made to obtain the optimal policies by using the mathematical software, i.e., MATLAB 2010a. In the following, the specific analysis is provided.

Scenario 1: No Sharing Cost (NSC)
In this scenario, the retailer is responsible for the sales effort cost, and the manufacturer bears the carbon emission reduction cost. According to the demand function, that is, Eq. (3), the profit π 1 R of the retailer can be determined, that is, Similarly, the profit π 1 M of the manufacturer can be determined, that is, According to Eqs. (4) and (5), we can obtain the following Lemma 3.1 and Theorem 3.2.
R is concave with respect to price p 1 and sales effort level M is concave with respect to wholesale price w 1 and carbon emission reduction effort level e 1 .
The proof can be seen in the Appendix. Based on Lemma 3.1, we obtain Theorem 3.2.
Theorem 3.2. The optimal price and optimal sales effort level of the retailer are The optimal wholesale price and optimal carbon emission reduction effort level of the manufacturer are where, The proof can be seen in the Appendix. Furthermore, the optimal profits of the retailer, manufacturer and green supply chain, that is, π 1 * R , π 1 * M and π 1 * SC , respectively, can be determined, i.e.,

Scenario 2: Sharing Carbon Emission Reduction Cost (SCERC)
In this scenario, the retailer shares the emission reduction cost of the manufacturer, but the manufacturer does not share the sales effort cost of the retailer. According to the demand function, that is, Eq. (3), the profit π 2 R of the retailer can be determined, i.e., Similarly, the profit π 2 M of the manufacturer can be determined, i.e., According to Eqs. (13) and (14), we can obtain the following Lemma 3.3, Theorem 3.4, and Corollary 1.
R is concave with respect to price p 2 and sales effort level A 2 ; profit π 2 M is concave with respect to wholesale price w 2 and carbon emission reduction effort level e 2 .
The proof can be seen in the Appendix. Based on Lemma 3.3, we obtain Theorem 3.4.
Theorem 3.4. The optimal price and optimal sales effort level of the retailer are The optimal wholesale price and optimal carbon emission reduction effort level of the manufacturer are where, The proof can be seen in the Appendix.
Corollary 1. The optimal policy for scenario SCERC degenerates into the one for scenario NSC when γ e = 0.
Proof. From Theorems 3.2 and 3.4, it is easy to see that the corollary is true, and thus, we do not give the redundant illustration here.
Furthermore, the optimal profits of the retailer, manufacturer and green supply chain, that is, π 2 * R , π 2 * M and π 2 * SC , respectively, can be determined, i.e.,

Scenario 3: Sharing Sales Effort Cost (SSEC)
In this scenario, the manufacturer shares the sales effort cost of the retailer, but the retailer does not share the carbon emission reduction cost of the manufacturer. According to Eq. (3), the profit π 3 R of the retailer can be determined, i.e., Similarly, the profit π 3 M of the manufacturer can be determined, i.e., According to Eqs. (22) and (23), we can obtain the following Lemma 3.5, Theorem 3.6, and Corollary 2.
R is concave with respect to price p 3 and sales effort level M is concave with respect to wholesale price w 3 and carbon emission reduction effort level e 3 .
The proof can be seen in the Appendix. Based on Lemma 3.5, we obtain Theorem 3.6.
Theorem 3.6. The optimal price and optimal sales effort level of the retailer are The optimal wholesale price and optimal carbon emission reduction effort level of the manufacturer are where, The proof can be seen in the Appendix.
Corollary 2. The optimal policy for scenario SSEC degenerates into the one for scenario NSC when γ A = 0.
Proof. From Theorems 3.2 and 3.6, it is easy to see that the corollary is true, and thus, we do not give the redundant illustration here. Furthermore, the optimal profits of the retailer, manufacturer and green supply chain, that is, π 3 * R , π 3 * M and π 3 * SC , respectively, can be determined, i.e.,

Scenario 4: Sharing Both Carbon Emission Reduction Cost and Sales Effort Cost (SBC)
In this scenario, the manufacturer shares the sales effort cost of the retailer, and the retailer shares the carbon emission reduction cost of the manufacturer. According to Eq. (3), the profit π 4 R of retailer can be determined, i.e., Similarly, the profit π 4 M of the manufacturer can be determined, i.e., According to Eqs. (31) and (32), we can obtain the following Lemma 3.7, Theorem 3.8, and Corollary 3. R is concave with respect to price p 4 and sales effort level A 4 ; profit π 4 M is concave with respect to wholesale price w 4 and carbon emission reduction effort level e 4 .
The proof can be seen in the Appendix. Based on Lemma 3.7, we obtain Theorem 3.8.
Theorem 3.8. The optimal price and optimal sales effort level of the retailer are The optimal wholesale price and optimal carbon emission reduction effort level of the manufacturer are where, The proof can be seen in the Appendix.
Corollary 3. The optimal policy for scenario SBC degenerates into the one for scenario SCERC when γ A = 0; the optimal policy for scenario SBC degenerates into the one for scenario SSEC when γ e = 0.
Proof. From Theorems 3.4-3.8, it is easy to see that the corollary is true, and thus, we do not give the redundant illustration here.
Furthermore, the optimal profits of the retailer, manufacturer and green supply chain, that is, π 4 * R , π 4 * M and π 4 * SC , respectively, can be determined, i.e., 4. Cost-sharing strategy analysis. Based on the analysis in Section 3, we conduct the analysis on the optimal cost-sharing strategies of the retailer and manufacturer. Since the complexity of the profit functions of the retailer and the manufacturer is high, it is difficult to analyze the cost-sharing strategy directly. Thus, in this section, we provide a numerical study. In specific numerical study, given that the complexity of the solution process and the convenience of the mathematical software, we use the mathematical software, i.e., MATLAB 2010a, as the executive system, and make the optimization computation.
Since the optimal policies for the NSC scenario are not related to the sales effort cost-sharing ratio and the carbon emission reduction effort cost-sharing ratio, we do not involve scenario NSC in this section. Since the optimal policies for scenarios SCERC and SSEC are special cases of scenario SBC, the analysis results for scenario SBC are more general than those for scenarios SCERC and SSEC. Hence, we conduct the numerical study for scenario SBC. In addition, given that the subsidy does not affect the impacts of the cost-sharing ratios on the profits of the retailer and the manufacturer, we pay little attention to the parameter analysis of subsidy. In the specific analysis, we first analyze the impacts of cost-sharing ratios γ A and γ e on the optimal policies of the retailer and manufacturer, and then, we analyze the optimal cost-sharing strategy.

4.1.
The impacts of cost-sharing ratios. According to the obtained theorems, we analyze the impacts of cost-sharing ratios γ A and γ e on price, sales effort level, wholesale price, carbon reduction emission effort level, and profits of the retailer, manufacturer and supply chain. The values of the involved parameters are taken as a = 300, b = 20, c = 20, m = 15, n = 12, e = 0,ē = 100,D e = 300, α = 1, β = 1, δ = 1, T r = 5000 and T m = 10000. It is necessary to point out that, values of the parameters are not taken randomly, they are determined by reference to the operation data of specific products (such as new energy vehicles) and the existing related literatures [3,25]. Given γ A ∈ [0, 1] and γ e ∈ [0, 1], the impacts can be determined as shown in Figures 3-9.
It can be seen from Figure 3 that the optimal price p * 4 can be affected by ratios γ A and γ e . Specifically, p * 4 generally decreases with ratio γ e when γ A = 1, but has no significant change with ratio γ e when γ A = 1. Price p * 4 may increase or decrease with ratio γ A , this depends on the value of ratio γ e . For a lower value of ratio γ e , p * 4 decreases slowly with ratio γ A ; for a greater value of ratio γ e , p * 4 increases sharply with ratio γ A . Price p * 4 has the maximum value when γ e = 0 and γ A = 0. The impact of ratio γ e (γ A ) on p * 4 is related to the value of ratio γ A (γ e ). For different values of ratio γ e (γ A ), the trends and extent of the impact of ratio γ A (γ e ) on p * 4 may be different.
Specifically, for scenario SCERC (i.e.,γ A = 0), price p * 4 sharply decreases with ratio γ e ; for scenario SSEC (i.e., γ e = 0), p * 4 slowly decreases with ratio γ A . It can be seen from Figure 4 that the optimal sales effort level A * 4 can be affected by ratios γ A and γ e . Specifically, for a lower value of ratio γ A , A * 4 increases slowly with ratio γ e ; for a higher value of ratio γ A , A * 4 increases quickly with ratio γ e ; for γ A = 1, A * 4 has no significant change with ratio γ e . Similarly, for γ e = 1, A * 4 decreases with ratio γ A ; the higher ratio γ e is, the faster the decline is.  It can be seen from Figure 5 that the optimal wholesale price w * 4 can be affected by ratios γ e and γ A . Specifically, w * 4 decreases with ratio γ e for γ A = 1 . Similarly, w * 4 decreases with ratio γ A for γ e = 1.Moreover, w * 4 has no significant change with ratio γ A for γ e = 1, and has no significant change with ratio γ e for γ A = 1. The wholesale price w * 4 has the maximum value when γ e = 0 and γ A = 0. The impact of ratio γ e (γ A ) on w * 4 is related to the value of ratio γ A (γ e ). For different values of ratio γ e (γ A ), the impact trends of ratio γ A (γ e ) on w * 4 may differ. Specifically, for scenario SCERC (i.e.,γ A = 0), the wholesale price w * 4 decreases with ratio γ e ; for scenario SSEC (i.e., γ e = 0), w * 4 decreases with ratio γ A .  It can be seen from Figure 6 that the optimal carbon emission reduction effort level e * 4 can be affected by ratios γ A and γ e .Specifically, e * 4 increases with ratio γ A for γ A = 1, but decreases with ratio γ A for γ e = 1. Moreover, e * 4 has no significant change with ratio γ A for γ e = 1. Similarly, e * 4 has no significant change with ratio 2016

XUE-YAN WU, ZHI-PING FAN AND BING-BING CAO
γ e for γ A = 1. The carbon emission reduction effort level e * 4 has the maximum value when γ e = 1 and γ A = 0.The impact of ratio γ e (γ A ) on e * 4 is related to the value of ratio γ A (γ e ). For different values of ratio γ e (γ A ), the impact trends of ratio γ A (γ e ) on e * 4 may be different. Specifically, for scenario SCERC (i.e., γ A = 0), the carbon emission reduction effort level e * 4 sharply increases with ratio γ e ; for scenario SSEC (i.e., γ e = 0), e * 4 slowly decreases with ratio γ A . It can be seen from Figure 7 that the retailer's optimal profit π 4 * R can be affected by ratios γ e and γ A . Specifically, π 4 * R first increases then decreases with ratio γ e for γ A = 1, but increases with ratio γ e for γ A = 1). Similarly, π 4 * R decreases with ratio γ A for lower values of ratio γ e ; first increases then decreases with ratio γ A for greater values of ratio γ e ; but has no significant change with ratio γ A for γ e = 1. Specifically, the retailer's profit π 4 * R has the maximum value when γ e = 0.6 and γ A = 0. It is necessary to point out that the retailer's profit π 4 * R mainly comes from the government subsidy when γ e = 0 and γ A = 0. The impact of ratio γ e (γ A ) on π 4 * R is related to the value of ratio γ A (γ e ). For different values of ratio γ e (γ A ), the impact trends of ratio γ A (γ e ) on π 4 * R may be different.  Specifically, for scenario SCERC (i.e., γ A = 0), profit π 4 * R first sharply increases then sharply decreases with ratio γ e ; for scenario SSEC (i.e., γ e = 0), π 4 * R sharply decreases with ratio γ A .
It can be seen from Figure 8 that the manufacturer's optimal profit π 4 * M can be affected by ratios γ e and γ A . Specifically, π 4 * M first increases then decreases with ratio γ e for γ A = 1, but has no significant change with ratio γ e for γ A = 1. Similarly, π 4 * M may increase or decrease with ratio γ A , this depends on the value of γ e . For a lower value of γ e , π 4 * M increases with γ A , the lower the ratio γ e is, and the faster profit π 4 * M increases; for a greater value of γ e , π 4 * M decreases slowly with γ A . Specifically, the manufacturer's profit π 4 * M has a maximum value when γ e = 0.67 and γ A = 0. It is necessary to point out that the manufacturer's profit π 4 * M mainly comes from the government subsidy when γ e = 0 and γ A = 0. The impact of ratio γ e (γ A ) on π 4 * M is related to the value of ratio γ A (γ e ). For different values of ratio γ e (γ A ), the impact trends of ratio γ A (γ e ) on π 4 * M may be different. Specifically, for scenario SCERC (i.e., γ A = 0), profit π 4 * M first sharply increases and then slowly decreases with ratio γ e ; for scenario SSEC (i.e., γ e = 0), π 4 * M sharply increases with ratio γ A .
It can be seen from Figure 9 that the supply chain's optimal profit π 4 * SC can be affected by ratios γ A and γ e . Specifically, π 4 * SC first increases and then decreases with ratio γ e for γ A = 1, but has no significant change with ratio γ e for γ A = 1. Similarly, π 4 * SC may increase or decrease with ratio γ A this depends on the value of γ e . For a lower value of γ e , π 4 * SC decreases with γ A ; for a greater value of γ e , π 4 * SC increases with γ A ; specifically, π 4 * SC has no significant change with the increases of ratio γ A when γ e = 1. The supply chain's profit π 4 * SC has the maximum value when γ e = 0.55 and γ A = 0. The impact of ratio γ e (γ A ) on π 4 * SC is related to the value of ratio γ A (γ e ). For different values of ratio γ e (γ A ), the impact trends of ratio γ A (γ e ) on π 4 * SC may differ. Specifically, for scenario SCERC (i.e., γ A = 0), profit π 4 * SC first sharply increases then sharply decreases with ratio γ e ; for scenario SSEC (i.e., γ e = 0), π 4 * SC slowly decreases with ratio γ A .

4.2.
The cost-sharing strategies. According to the impact analysis in Subsection 4.1, we conduct the analysis on cost-sharing strategies as follows.
(1) Figure 7 shows that the retailer's cost-sharing strategy is directly related to the sales effort ratio γ A of the manufacturer. When ratio γ A is lower, the retailer's cost-sharing strategy is to share more than half of the carbon emission reduction effort cost of the manufacturer, and the retailer's cost-sharing ratio γ e increases with ratio γ A . It is necessary to point out that the profit of the retailer takes the maximum value when γ e = 0.6 and γ A = 0. Obviously, from the perspective of the retailer, the optimal cost-sharing strategy is for the retailer to share 60% of the carbon emission reduction cost and the manufacturer not to share the sales effort  Figure 9. The impacts of sharing ratios γ A and γ e on profit π SC of the supply chain.
cost. Specifically, for scenario SCERC, the retailer obtains the maximum profit when the retailer shares 60% of the carbon emission reduction cost, and thus, the optimal cost-sharing strategy of retailer is for the retailer to share 60% of the carbon emission reduction cost; for scenario SSEC, the retailer obtains the maximum profit when the manufacturer does not share the sales effort cost, and the retailer does not get involved in the cost-sharing strategy.
(2) According to Figure 8, we show that the manufacturer's cost-sharing strategy is directly related to the carbon emission reduction effort ratio γ e of the retailer. When ratio γ e is lower, the manufacturer's cost-sharing strategy is to share as much of the sales effort cost of the retailer as possible; when ratio γ e is greater, the manufacturer's cost-sharing strategy is to share the sales effort cost of the retailer as little as possible. It is necessary to point out that the profit of the manufacturer takes the maximum value when γ e = 0.67 and γ A = 0. Obviously, from the perspective of the manufacturer, the optimal cost-sharing strategy is that the retailer shares 67% of the carbon emission reduction cost and the manufacturer does not share the sales effort cost. Specifically, for scenario SCERC, the manufacturer obtains the maximum profit when the retailer shares 67% of the carbon emission reduction cost, but the manufacturer does not get involved in the cost-sharing strategy; for scenario SSEC, the manufacturer obtains the maximum profit when the manufacturer shares as much of the sales effort cost as possible, and thus, the optimal cost-sharing strategy of the manufacturer is to share the sales effort cost as much as possible.
(3) According to Figure 9, the cost-sharing strategy of the supply chain is related to those of the retailer and the manufacturer. When ratio γ A is lower, the retailer's cost-sharing strategy is to share approximately half of the carbon emission reduction effort cost of the manufacturer, and the retailer's cost-sharing ratio γ e increases with ratio γ A . It is necessary to point out that the profit of the supply chain takes the maximum value when γ e = 0.55 and γ A = 0. Obviously, from the perspective of the supply chain, the optimal cost-sharing strategy is for the retailer to share 55% of the carbon emission reduction cost and for the manufacturer not to share the sales effort cost. Specifically, for scenario SCERC, the supply chain has the maximum profit when the retailer shares 55% of the carbon emission reduction cost, and thus, the optimal cost-sharing strategy of the supply chain is for the retailer to share 55% of the carbon emission reduction cost; for scenario SSEC, the supply chain has the maximum profit when the manufacturer does not share the sales effort cost, and thus, the optimal cost-sharing strategy of the supply chain is for the manufacturer not to share the sales effort cost.
5. Managerial insights. The main managerial contributions of our study are summarized as follows.
(1) The manufacturer's carbon emission reduction effort and the retailer's sales effort can affect the market demand of green products. In the analysis of the demand of green products, the manufacturer's carbon emission reduction effort, the retailer's sales effort and consumers' green preference should be considered simultaneously.
(2) The optimal policies of the manufacturer and the retailer are related to the cost-sharing strategies of the manufacturer and the retailer. To obtain the optimal profit, the manufacturer and the retailer should first determine their optimal costsharing strategy for the scenarios SCERC, SSEC and SBC, respectively.
(3) When the government provides a fixed subsidy to the manufacturer and the retailer, the subsidy cannot affect the optimal policies of the manufacturer and the retailer. In this way, the subsidy does not affect the optimal cost-sharing strategy of the manufacturer and the retailer.
(4) From the perspective of the retailer for the scenarios SCERC, SSEC and SBC, when the cost-sharing strategy of the manufacturer is fixed, the optimal costsharing strategy of the retailer is uniquely determined, and tends to share more than half of the carbon emission reduction cost of the manufacturer. Specifically, for scenario SCERC, the retailer's optimal cost-sharing strategy is to share more than half of the carbon emission reduction cost; for scenario SSEC, the retailer obtains the maximum profit when the manufacturer does not share the sales effort cost.
(5) From the perspective of the manufacturer for scenarios SCERC, SSEC and SBC, when the cost-sharing strategy of the retailer is fixed, the optimal cost-sharing strategy of the manufacturer is uniquely determined, and tends not to share the sales effort cost of the retailer. Specifically, for scenario SCERC, the manufacturer obtains the maximum profit when the retailer shares more than half of the carbon emission reduction cost; for scenario SSEC, the manufacturers optimal cost-sharing strategy is to share as much of the sales effort cost as possible.
(6) From the perspective of the supply chain for scenarios SCERC, SSEC and SBC, the optimal cost-sharing strategy is that the retailer shares more than half of the manufacturer's carbon emission reduction cost, and the manufacturer does not share the sales effort cost of the retailer. Specifically, for scenario SCERC, the optimal cost-sharing strategy of the supply chain is that the retailer shares more than half of the carbon emission reduction cost; for scenario SSEC, the optimal cost-sharing strategy of the supply chain is that the manufacturer does not share the sales effort cost.
6. Conclusions and further research. In this paper, we studied a cost-sharing strategy in a Nash game considering government subsidy, consumers' green preference and retailer's sales effort. Specifically, we analyzed the green-sensitive demand considering consumers' green preference, and further constructed a demand function for four cost-sharing scenarios. On the basis, we constructed the profit functions of the retailer and the manufacturer considering government subsidy, consumers' green preference and retailer's sales effort for four cost-sharing scenarios. Then, we determined the optimal policies of the retailer and the manufacturer according to a Nash game. By the analysis, we showed the cost-sharing strategies from the perspectives of the retailer, manufacturer and supply chain, respectively. Furthermore, we investigated the impacts of the sales effort cost-sharing ratio and the carbon emission reduction cost-sharing ratio on the optimal policies of the retailer and the manufacturer. We found that the sales effort cost-sharing ratio and the carbon emission reduction cost-sharing ratio could affect the retailer's optimal price and sales effort level and the manufacturer's wholesale price and carbon emission reduction effort level to varying degrees. From different perspectives, the optimal cost-sharing strategies are different. We also found that the fixed subsidy would not affect the optimal cost-sharing strategies of the retailer and the manufacturer.
Compared with existing studies that do not consider the sales effort cost-sharing ratio and the carbon emission reduction cost-sharing ratio or studies that consider only one of sales effort cost-sharing ratio and carbon emission reduction cost-sharing ratio, our study reflects not only the impacts of both ratios but also the interactive effects of both ratios. In addition, our study provides the optimal cost-sharing strategy for the retailer and the manufacturer from different perspectives.
In future research, we seek to analyze the cost-sharing strategy based on bounded rationality and to examine the effects of bounded rationality on the optimal policies and cost-sharing strategies in a dyadic supply chain.

Appendix.
Proof of Lemma 3.1. For scenario NSC, according to Eq. (4), the first-order derivatives of profit π 1 R with respect to unit margin x 1 and sales effort level A 1 can be determined, i.e., Similarly, according to Eq. (5), the first-order derivatives of profit π 1 M with respect to wholesale price w 1 and carbon emission reduction effort level e 1 can be determined, i.e.,  .4), the corresponding second order derivatives can be determined, i.e., (A.8) Since b > 0, α > 0, δ > 0, β > 0, n > 0,D e > 0 andē > e, we have ∂ 2 π 1 R ∂x 1 2 < 0, ∂ 2 π 1 R ∂A 1 2 < 0, ∂ 2 π 1 M ∂w 1 2 < 0 and ∂ 2 π 1 M ∂e 1 2 < 0. Then, we know that profit π 1 R is a concave function with respect to price p 1 and sales effort level A 1 , and that profit π 1 M is a concave function with respect to wholesale price w 1 and carbon emission reduction effort level e 1 .
Proof of Lemma 3.3. For scenario SCERC, according to Eq. (13), the first-order derivatives of profit π 2 R with respect to unit margin x 2 and sales effort level A 2 can be determined, i.e., Similarly, according to Eq. (14), the first-order derivatives of profit π 2 M with respect to wholesale price w 2 and carbon emission reduction effort level e 2 can be determined, i.e., Since b > 0, α > 0, β > 0, n > 0,D e > 0, 0 < γ e < 1 andē > e, we have ∂ 2 π 2 R ∂x 2 2 < 0, ∂ 2 π 2 R ∂A 2 2 < 0, ∂ 2 π 2 M ∂w 2 2 < 0 and ∂ 2 π 2 M ∂e 2 2 < 0 . Then, we know that profit π 2 R is a concave function with respect to price p 2 and sales effort level A 2 , and that profit π 2 M is a concave function with respect to wholesale price w 2 and carbon emission reduction effort level e 2 .
Proof of Theorem 3.4. For scenario SCERC, according to the Nash game, we can determine the optimal policies of the retailer and the manufacturer by ∂π 2 R ∂x 2 = 0, ∂π 2 R ∂A 2 = 0 , ∂π 2 M ∂w 2 = 0 and ∂π 2 M ∂e 2 = 0. By simplification, the optimal policies of the retailer and the manufacturer are On that basis, since p 2 = w 2 + x 2 , we can determine the optimal price of retailer p * 2 , i.e., where, 2 −D e n δ − bβ 2 γ e e 2 −ēe . Obviously, Theorem 3.4 holds.
Proof of Theorem 3.8. For scenario SBC, according to the Nash game, we can determine the optimal policies of the retailer and the manufacturer by ∂π 4 R ∂x 4 = 0,∂π 4 R ∂A 4 = 0, ∂π 4 M ∂w 4 = 0 and ∂π 4 M ∂e 4 = 0. By simplification, the optimal policies of the retailer and the manufacturer are On that basis, since , we can determine the optimal price of retailer p 4 = w 4 +x 4 , i.e.,