Equilibrium Decisions on Pricing and the Greenness Degree in Supply Chains under Single-and Cross-Distribution Channels of Green and Nongreen Products and Government Subsidies

Rapid economic growth and industrialization have brought us material abundance and greater convenience, while also causing socio-environmental problems such as the rapid depletion of resources, unexpected natural disasters, and environmental destruction. For these reasons, environmental issues have attracted public and governmental attention around the world over the past few decades. As a result, environmental sustainability has become a critical indicator for evaluating the success and efciency of supply chain management schemes. For a sustainable supply chain, many governments are employing subsidy policies to encourage consumers to purchase environment-friendly (green) products. Tis article considers a supply chain composed of two competing manufacturers and two retailers. Te frst manufacturer produces a green product, while the second manufacturer produces a nongreen product. Each of the two retailers can sell only a green product, only a nongreen product, or both green and nongreen products. In the market, consumers purchasing the frst manufacturer’s green products can receive certain subsidies from the government. Te amounts of the government subsidies that a consumer receives depend on the greenness degree of the product. Using a three-stage Stackelberg game framework, this article discusses equilibrium decisions on pricing and greenness for four diferent distribution channel structures. Major fndings of this study reveal that (i) the government subsidy has a positive impact on the greenness degree and the demand for green products; (ii) the government subsidy level depends on the man-ufacturers’ distribution channel strategy types; and (iii) the greater the environmentally negative impact of a green product, the lower the level of the government subsidy paid for purchasing green products. Based on these fndings, this article suggests that policymakers focus on establishing the policies that encourage the purchase of green products for the sustainability of a society and a supply chain.


Introduction
Te impact of climate change and global warming on quality of life has recently increased. One of the main causes of this is believed to be rapid economic growth and, as a result, a worsening of the environment. According to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change [1], the Earth's average temperature may increase from a minimum of 1.1∼2.9°C to a maximum of 2.4∼6.4°C during the twenty-frst century, and the rate of global warming is expected to be even faster. With global warming melting sea ice, polar bear populations have plummeted by nearly half in the last decade, and the recent strong storms caused by climate change have had disastrous impacts in South-East Asian countries. Because developing countries are more vulnerable to climate change than advanced countries and given that economic growth and urbanization are expected to progress in the future, extreme weather events threaten the further development of middleincome countries and can even overburden high-income countries. As such, rapid industrialization and economic growth have brought people material abundance and greater convenience while also causing socio-environmental problems such as the rapid depletion of resources, unexpected natural disasters, and environmental destruction. For these reasons, environmental issues have attracted public and governmental attention around the world, and they are also afecting consumer consumption patterns. Consumers have become more aware of environmental degradation and of how their consumption behaviors afect the level of pollution. Tus, consumers' purchasing patterns are becoming more environmentally friendly, and for ecologically minded consumers, protecting the environment has become the frst priority when purchasing products [2].
International environmental regulations, which have emerged as a new nontarif trade barrier and the shift by consumers to environmentally friendly consumption patterns, are demanding environmentally friendly supply chain management strategies for many manufacturers. Proftability improvements and cost leadership have been the main goals of supply chain management over the past few decades. However, as people are becoming more conscious about the protection of the environment from pollutant caused by human beings, businesses are adopting green technology to procure green products to save the environment from pollution [3]. Te increasing rate of environmental deterioration has shifted this focus to socioenvironmental issues; in the context of supply chain research, this has led to more concerns about the sustainability of the supply chain. Gradually, more manufacturers are developing "green" or "sustainable" products made of ecofriendly materials and are increasing their investments in less environmentally harmful manufacturing concepts, such as reuse, refurbishment, recycling, remanufacturing, green product design, product recovery, and waste management. As manufacturers and consumers become more interested in green products, more retailers in the supply chain are also putting green products on their shelves and displaying them. Consumers are now paying attention to sustainable green and organic products and looking for retail outlets that operate in accordance with green methods and green principles. Greening is taking place in many retail sectors, such as clothing and apparel (Ecocentrik Apparel, Natural Clothing Company, and Element Ecowear), furniture (Te Old Wood, Eco Select Furniture, and Vermont Woods Studios), and household cleaning products (Wunder Budder). Due to the unprecedented popularity of the green products, traditional large retailers such as Walmart and Tesco are selling green products to their customers alongside existing nongreen products. Walmart has made signifcant advances in positioning itself to include more green practices in their supply chain operations by enacting a strict policy to cut of suppliers whose manufacturing and distribution methods contribute to environment deterioration. As more retailers deal with green products as well as nongreen products, the distribution channel strategies of manufacturers have a substantial impact on the proftability and sustainability of the supply chain.
Although consumers are aware of the positive impact of green products on the environment, many studies have reported a discrepancy or "gap" between consumers' favorable attitudes and actual purchasing practices [4][5][6][7]. Hughner et al. [8] revealed that while many consumers showed a positive attitude toward buying organic food (67%), only a few consumers (4%) actually bought such products. Similarly, according to Defra [9], 30% of consumers in the UK expressed concerns about the environment, but they rarely translated their concerns into the purchasing of green products. Tis discrepancy is referred to as "green purchasing inconsistency" or the "green attitudebehavior gap," which signifes that consumers' positive attitudes toward green products do not always translate into action. One way to bridge this gap is to encourage consumers to purchase green products by providing subsidies to those who are planning to purchase such products. Recently, car manufacturers have developed environmentally friendly vehicles such as hydrogen-fuel-cell vehicles and plug-in hybrid electric vehicles to mitigate the vast carbon dioxide emissions from cars that use fossil fuels. South Korea will ofer more than 827 million USD in subsidies to people who buy electric and hydrogen-fuel-cell electric vehicles in 2020 to promote the use of eco-friendly vehicles that do not emit the greenhouse gases largely responsible for global warming. Te Chinese government also plans to phase out subsidies for all electrifed vehicles by the end of 2021. Te purpose of these government policies is to protect the environment by subsidizing green products and to enhance the proftability and sustainability of the supply chain at the same time.
As such, establishing greening strategies as part of a supply chain management plan is an important task for the sustainability and proftability of supply chain members. With regard to this article, the main research questions are as follows: (i) Does the government subsidy level afect pricing and greening decisions? (ii) Does the government subsidy level depend on the manufacturers' distribution channel strategies?
(iii) Does the government subsidy level have a positive or negative impact on the proftability and sustainability of the supply chain? (iv) How should the government subsidy level be determined in view of proftability and sustainability in the supply chain?
Te main purpose of this study is to answer to the research questions using a game-theoretical framework. Te detailed objective and contribution of the study are summarized as follows: (i) Tis study investigates the relationship between the government's decision on subsidy and the manufacturer's decision on production. Tis relationship helps supply chain participants establish their own best strategies in supply chains.
(ii) Tis study identifes the relationship between the government's decision on subsidy and the manufacturer's decision on distribution channel design. Tis helps the manufacturers in supply chain distribute their products more efciently.
(iii) Tis study explores the government's socially best decisions on subsidy. Tis helps policymakers formulate the policies that can beneft the sustainability and proftability of supply chains.
In this article, two types of manufacturers are assumed: one producing green (environmentally friendly) products and the other producing nongreen (ordinary) products. Tey compete against each other in a Cournot fashion to maximize their own profts. Cournot competition is an economic model that describes an industry structure in which frms compete on quantity, with decisions made independently of each other and simultaneously [10]. Tis article develops various game models under various distribution channel structures and suggests an equilibrium decision pertaining to each game model. Tis article also investigates the efects of the government subsidy level on pricing and greening decisions. In terms of social welfare, this article determines the equilibrium level of the government subsidy. Various numerical examples are presented to support the major fndings.
Te remainder of this article is organized as follows. Section 2 presents a review of the relevant literature. Section 3 reviews the notations used and the assumptions in the article. Section 4 presents an equilibrium decision of each of the distribution channel structures. Section 5 defnes social welfare and determines the government subsidy level. In Section 6, a brief numerical experiment is presented. Section 7 deals with some managerial insights that managers of the business industry get beneft from the fndings of this study. Te last section provides a summary of the article, presents the conclusion, and provides some directions for future research.

Literature Review
Tis section reviews the relevant literature considering two diferent streams of research in this area: government subsidies and distribution channel strategies.
A government subsidy or government incentive is a form of fnancial aid or support extended to an economic sector (business or individual) generally with the aim of promoting certain economic and social policies. Government subsidies can be cash payments and intermediate goods and services provided freely or at nominal prices from governments to suppliers [11]. In the mainstream of economics, government subsidies can be divided into two types: producer subsidy and consumer subsidy. Producer (production) subsidies ensure producers are better of by supplying market price support or payments to factors of production. Consumer (consumption) subsidies commonly reduce the price of products/services to consumers. Te efect of a subsidy is to boost the supply and demand by the amount of the subsidy. If a producer receives the subsidy, an increase in the price resulting from the marginal subsidy on production leads to the increase in the supply, shifting the supply curve to the right. If a consumer receives the subsidy, a lower price of a product resulting from the marginal subsidy on consumption increases the demand of a product, also shifting the demand curve to the right. Mitra and Webster [12] analyzed the impacts of government subsidies as a means to promote remanufacturing activities and found that the production subsidy increases remanufacturing activities. Tey also found that the manufacturer's proft decreases while the remanufacturer's proft increases when the subsidy is provided only to the remanufacturer. Sheu [13] investigated the negotiation problem between producers and reverse-logistics suppliers under the government fnancial intervention. Using an asymmetrical Nash bargaining game framework, the article showed that the fnancial intervention by the government is not always benefcial for the proftability of the supply chain members as well as the overall social welfare. Ma et al. [14] focused on the infuences of the consumer subsidy on the equilibrium decisions of the members in the dual-channel closed-loop supply chain. Tey revealed that every consumer purchasing a new product is a benefciary of the various levels of the government subsidy and the consumer subsidy is conducive to the expansion of the supply chain. Hu et al. [15] proposed the oligopoly game model to explore the competition in the market with both green and ordinary products and asserted that product characteristics and market structures substantially impact the government tax and subsidy policies. Wang et al. [16] analyzed the impact of four subsidy policies on the development of the recycling and remanufacturing industry and found that the combination of several subsidy policies has much more positive efects on remanufacturing promotion. Hafezalkotob [17] surveyed a price competition model between green and regular supply chains under governmental fnancial intervention, investigating the efects of the environmental protection and revenue-seeking policies by governments on various supply chains. Hafezalkotob [18] extended the work of Hafezalkotob [17], assuming the energy-savings, cooperation, and the sustainable development of the supply chain. Madani and Rasti-Barzoki [19] also studied green supply chains in the context of government intervention and showed that increases in government subsidies lead to increases in the demand for and the degree of greenness of green products. Jena et al. [20] developed four closed-loop supply chain models with which to assess diferent scenarios of government subsidies, suggesting that governments can maximize the proft of the supply chain and incentivize manufacturers who participate in remanufacturing activities by subsidizing only these manufacturers. He et al. [21] investigated a dual-channel closed-loop supply chain in which a government subsidy is provided to consumers purchasing remanufactured products. Teir article showed that when the government subsidy level is relatively low, manufacturers will sell new products directly. Otherwise, manufacturers will sell remanufactured products. Wang et al. [22] investigated the efects of government subsidies on pricing decisions in reverse supply chains of e-waste and found that the remanufacturing utilization rate has a great efect on the equilibrium strategy pertaining to the allocation of the government subsidy. Huang et al. [23] developed game models to address the efects of green loans and government subsidies on the promotion of green innovations by enterprises, showing that government subsidies can efectively incentivize green innovation and improve environmental quality levels. Chen et al. [24] presented game models in which a government determines the amount of its subsidies for a supply chain composed of one manufacturer and one retailer conducting a research joint venture on a sustainable product. Teir article suggested that governments should not use multiple types of subsidies simultaneously for any cost-reduction research and development eforts. Liu et al. [25] examined the decision problems of a retailer-dominated supply chain considering corporate social responsibility (CSR) under a government subsidy and found that a certain level of government subsidy can promote supply chain members to undertake CSR and can enhance the performance of the supply chain and social welfare. Wan and Hong [26] developed several Stackelberg game models to investigate the impacts of subsidy policies and transfer pricing policies on the closed-loop supply chain with dual collection channels and found that either remanufacturing or recycling subsidies stimulate the consumption, increase the recovery, and consequently improve the supply chain members' profts. Recently, Sana [27] dealt with a newsvendor inventory model in view of green product marketing of CSR frms. In Sana [27]'s model, a comparison between green and nongreen marketing is analyzed considering government subsidies and taxes. Shi et al. [28] developed four green supply chain models considering government subsidy policy and retailer's fairness concerns and showed that the government subsidies to both a manufacturer and a retailer lessen the adverse efect of retailers' fairness concerns and enhance the efciency of environmental governance. Evolutionary game models and simulation studies of green supply chains under the subsidy policy can be found in Zhao et al. [29] and the references therein.
Tis work is also related to the literature on distribution channel strategies in supply chains. A variety of distribution channel structures are common in supply chains. Many studies have shown that the proftability of a supply chain depends largely on the structure of the product's distribution channel. McGuire and Staelin [30] discussed the relationship between product substitutability and the equilibrium distribution structure in a duopoly market where each manufacturer distributes its products through a single exclusive retailer. Tis article asserted that a decentralized channel structure is likely to be adopted by a manufacturer when the competition between products intensifes. Choi [31] dealt with a duopoly common retailer channel and studied the efect of price competition among the members of the channel. Te article observed that product diferentiation helps manufacturers, whereas it hurts retailers. Conversely, while store diferentiation helps retailers, it hurts manufacturers. Moner-Colonques et al. [32] examined an asymmetric noncooperative game between two manufacturers who decide upon the number of retailers and suggested that when product diferentiation is strong and brand asymmetry is moderate, the two manufacturers prefer a cross-distribution channel in equilibrium. Wu and Mallik [33] assumed that distribution channel confgurations are determined endogenously by a Nash equilibrium state between manufacturers and retailers. Tey found that manufacturers should strategically use a cross-distribution channel to optimize their profts. Edirisinghe et al. [34] discussed the implications of the distribution channel power on the stability of the supply chain in a setting where multiple manufacturers distribute substitutable products through a common retailer. Teir paper found that an imbalance of the channel power leads to a decrease in supply chain profts, and the more balanced the agents are the greater their profts, regardless of the product competition. Bian et al. [35] studied equilibrium distribution channel strategies in a mixed market, assuming that public frms are concerned with social welfare while private frms maximize their proft. In their paper, they found that the equilibrium channel structure depends on the market competition mode (Bertrand or Cournot competition), the form of vertical contract, and the degree of product substitutability. Bian et al. [36] analyzed manufacturers' distribution channel strategies under environmental taxation and suggested that a monopolistic manufacturer can beneft from a decentralized channel structure when its technology is sufciently polluting and that duopolistic manufacturers are more likely to decentralize their distribution channels when their technologies are more environmentally damaging. Nie et al. [37] investigated the efects of a cross online-and-ofine channel (OOC) on two competing retailers' equilibrium decisions and showed that when the cross-channel efect is insignificantly negative or positive, such retailers prefer the OOC strategy, although they face the prisoners' dilemma. Bian et al. [38] discussed dynamic interactions between manufacturers' distribution channel strategies and collusion incentives. Tey suggested that a single-distribution channel does not always facilitate collusion between manufacturers and held that the selection of the distribution channel mainly depends on the discount factor and on the degree of product Consumer value discount for green products (0 < β < 1) ρ Environmental impact discount of green products (0 < ρ < 1) Functions p i Retail price of manufacturer i's product π mi Manufacturer i's proft π ri Retailer i's proft π s Supply chain proft (π s � π m1 + π m2 + π r1 + π r2 ) SW Social welfare diferentiation. Readers may refer to Tsay and Agrawal [39] to review of this stream of work.
As discussed above, much in-depth research on government subsidies and distribution channel strategies has been conducted over the past few decades. Governmental fnancial intervention is an important factor when evaluating the success and efciency of a supply chain for sustainable growth. In addition, the distribution channel structure plays a very important role in determining the proftability of the supply chain. However, most studies assumed that government subsidies are exogenous. To the best of the author's knowledge, some of them dealt with governmental equilibrium decisions, but they did not consider the impacts of diferent distribution channel structures by manufacturers. Te only paper dealing with all of these issues in a reverse supply chain is that by He et al. [21]. Terefore, this study focuses on how the eco-friendliness of a manufacturer and fnancial intervention by a government afect the profts of supply chain members under various distribution channel structures. Table 1.

Assumptions.
Consider a supply chain composed of two manufacturers and two retailers. In this supply chain, the manufacturers provide consumers with substitutable products. Te frst manufacturer (M1) produces a green (eco-friendly) product, while the second manufacturer (M2) produces a nongreen (ordinary) product. To stimulate the purchase of the green product, the government subsidizes consumers who purchase M1's green product. All products produced by the two manufacturers are distributed via two retailers (R1 and R2). Each manufacturer can distribute their products through either one retailer or both retailers. Under this supply chain confguration, the following assumptions are made: Assumption 1. If a manufacturer chooses to distribute through a single retailer, the channel structure is called a single-distribution channel. A cross-distribution channel represents the channel structure by which a manufacturer distributes its products through both retailers. Hence, the following four diferent distribution channel structures can be defned: the single-single structure (SS), cross-cross structure (CC), cross-single structure (CS), and single-cross structure (SC). Tis study assumes these four diferent distribution channel structures (see Figure 1). A similar assumption can be found in earlier works [33][34][35][36][37][38].

Assumption 2.
Tere is a three-stage game involved in the supply chain. In the frst stage of the game, the government announces the subsidy level s, which is applied to M1's green product. In the second stage, M1 determines its wholesale price w 1 and greenness degree g of its green product. At the same time, M2 sets its wholesale price w 2 . By specifying the values of s, g, w 1 , and w 2 , in the last stage of the game, R1 and R2 independently and simultaneously determine the ordering quantity of the manufacturers' products q 1 and q 2 . All decisions are assumed to be made in a steady-state period.
Assumption 3. According to Assumption 1, each manufacturer can distribute its product through diferent retailers; thus, it follows that q i � q ii + q ij for i � 1, 2 and j � 3 − i, where q ij is manufacturer i's selling quantity distributed through retailer j. We assume that the products ordered from retailers are consumed entirely in the market.

Assumption 4.
Tere are two types of products (green and nongreen products) in the market and the potential market size for each product is assumed to be scaled to one. Consumers have idiosyncratic valuations of nongreen products with respect to their willingness-to-pay v, which follows a uniform distribution in the interval of [0, 1]. Tis assumption is widely accepted when modeling consumers' heterogeneity [40][41][42].
Assumption 5. While green products are suitable for protecting the environment, consumers tend to recognize that the performance of green products is worse than that of ordinary (nongreen) products and that the green products may appear to be more expensive [43]. Due to people's tendency to prefer avoiding losses over making equivalent gains, consumers may hesitate to purchase eco-friendly products [44]. Terefore, it is reasonable to assume that for the green product, consumers value each unit at a discount β ∈ (0, 1). With regard to the sustainability of the supply chain, the government provides a Green product Non-green product Green product Non-green product Green product Non-green product Mathematical Problems in Engineering subsidy to a consumer who buys a green product, which is proportional to the greenness degree of the green product. Terefore, the net utility of a consumer who purchases a green product and a nongreen product equals u 1 � βv − p 1 + gs and u 2 � v − p 2 , respectively, where p i is the retail price of the i th manufacturer's product. From several previous studies [14,21,41,42], the inverse linear demand function of each product is given by (1) In order to ensure non-negative demand for each product, the inequalities gs + βp 2 > p 1 and 1 − gs − β + p 1 > p 2 should be met.
Assumption 6. Te investment in greening the product is assumed to be an increasing and convex function of the greenness degree of M1's product. Hence, the greenness cost can be expressed as c m g 2 /2, where c m is the cost coefcient of the greenness degree. Also assume that manufacturer i incurs fxed cost f i to open another distribution channel. To focus on the efects of the distribution channel and on government intervention in the form of a subsidy, the two manufacturers' production costs are assumed to be constant and normalized to zero. Similarly, the two retailers' operational costs are also normalized to zero. Allowing nonzero production and operational costs will not qualitatively change our results. A similar assumption can be found in Xu et al. [45].
Under Assumptions 1-6, this article develops the gametheoretical models to determine the equilibrium behavior of each participant in the supply chain.

Equilibrium Analysis with a Given Government Subsidy
Tis section presents the equilibrium result of each of the distribution channel structures with a given government subsidy level.

SS Structure: Both M1 and M2 Adopt a Single-Distribution
Channel. First, we discuss the SS structure where M1's green product (M2's nongreen product) is exclusively distributed through R1 (R2) (see Figure 1(a)). Troughout the paper, the SS structure is regarded as a benchmark model. Te proft function of each member in the supply chain is given by With a given government subsidy, the decision sequence in the SS structure is Applying backward induction to equation (3), the equilibrium values of the decision variables for each player are obtained. Let the superscript l ∈ SS, CC, CS, SC { } denote the equilibrium values in the four distribution channel structures. Table 2 presents the equilibrium solutions of equation (3). All proofs in this article are given in Appendix A.

CC Structure: Both M1 and M2 Adopt a Cross-Distribution
Channel. Next, we analyze the CC structure, where each manufacturer distributes its product via both retailers (see Figure 1(b)). Recall that with the CC structure, q i � q ii + q ij , for i � 1, 2 and j � 3 − i because each manufacturer can distribute its product through both retailers. For manufacturer i, additional fxed costs f i are incurred to open another distribution channel. Hence, the proft function of each member in the supply chain is given by With the given government subsidy, the decision sequence in the CC structure is By backward induction, the equilibrium values of the decision variables for each player are obtained, as shown in Table 2. Note that in the CC structure, the profts of the two retailers are identical.

CS Structure: M1 Adopts a Cross-Distribution Channel, While M2 Adopts a Single-Distribution Channel.
In the CS structure, M1 distributes its green products through both R1 and R2, while M2 distributes its nongreen products only through R2 (see Figure 1(c)). In this case, it follows that q 1 � q 11 + q 12 because only green products are distributed via both retailers. For M1, additional fxed costs f 1 are incurred to open another distribution channel. Hence, the proft function of each member in the supply chain is given by 6 Mathematical Problems in Engineering With the given government subsidy, the decision sequence in the CS structure is Via backward induction, the equilibrium values of the decision variables for each player are obtained, as presented in Table 2.

SC Structure: M1 Adopts a Single-Distribution Channel, While M2 Adopts a Cross-Distribution Channel.
In the SC structure, as opposed to the CS structure, M1 distributes its green products only through R1, while M2 distributes its nongreen products through both R1 and R2 (see Figure 1(d)). In this case, it follows that q 2 � q 21 + q 22 because only nongreen products are distributed via both retailers. For M2, additional fxed costs f 2 are incurred to open another distribution channel. Hence, the proft function of each member in the supply chain is given by With the given government subsidy, the decision sequence in the CS structure is With backward induction, the equilibrium values of the decision variables for each player are obtained in Table 2.

Impacts of the Government Subsidy Level.
Tis subsection discusses the impact of the government subsidy level on the equilibrium decisions and profts of the supply chain members and provides some managerial insight. To avoid negative demand for each product, the following inequalities are assumed to hold in each distribution channel structure: Te above equation indicates that the government subsidy should not be too high; otherwise, the green products will squeeze the nongreen products out of the market. At this stage, the following proposition are presented.

Proposition 1. In each of the distribution channel structures,
(1) Te government subsidy level has a positive impact on the greenness degree of a green product (2) Te government subsidy level has a positive (negative) impact on the wholesale price of a green (nongreen) product Table 2: Equilibrium decisions in each distribution channel structure.
Te values of A 0 to A 10 are given in Appendix C.

(3) Te government subsidy level has a positive (negative) impact on the selling quantity of a green (nongreen) product (4) Te government subsidy level has a positive (negative) impact on the retail price of a green (nongreen) product
Proposition 2 presents the important fact that the equilibrium decision of each supply chain member depends on the government subsidy level. Te government subsidy is available only to consumers who purchase M1's green products, but (Cournot) competition between the manufacturers exists in the supply chain, meaning that all decision behaviors of the supply chain members are dependent on the government subsidy. More specially, the greenness degree of M1's product increases with the government subsidy because the higher the government subsidy, the higher the utility (or preference) of the consumer's green product, while the utility of the nongreen product decreases.

Te increased (decreased) utility of M1's green product (M2's green product) directly has a positive (negative) impact on its ordering quantity. In other words, as the government subsidy level increases, consumer preferences for the green product (nongreen product) will increase, which will increase (decrease) the demand for the green product (nongreen product) in the market. If the demand for the nongreen product decreases with an increase in the government subsidy level, it is obvious that M2's proft falls.
To prevent a decline in the demand for the nongreen product, M2 will choose a strategy that lowers the wholesale price of the nongreen product, which in turn lowers its retail price. Once again, the higher the government subsidy level, the higher the greenness degree of M1's product. As indicated in M1's proft functions, the cost of greening the product increases exponentially. To compensate for the increased costs associated with the greening of the product, M1 will utilize a strategy to raise the wholesale price of its green product, which results in an increase in its retail price. Even if the retail price of the green product rises, the high demand for the green product will continue because the government subsidizes the purchase of the green product.
Unlike Proposition 1, the proft of each supply chain member has diferent properties in each distribution channel structure. At this point, Proposition 2 is given.

Proposition 2. Te proft of each of the supply chain members has the following properties:
(1) For all distribution channel structures (i.e., ∀l ∈ SS, CC, CS, SC Te values of t l m1 , t l m2 , and t SS r2 are given in Appendix C. Proposition 2 reveals that the all profts of the supply chain members are functions of the government subsidy. Tis fact is also obvious because the proft consists of the price and demand of a product, which are all afected by the government subsidy. Regardless of the type of distribution channel structure, as the government subsidy level increases, M1's proft initially increases to a certain level; that is, it has a maximum and then decreases drastically. In other words, there exists an optimal value of the government subsidy level t l m1 that maximizes M1's proft in all distribution channel structures. An increasing government subsidy increases M1's sales revenue but exponentially increases its greening cost. For this reason, there exists a threshold of the government subsidy level until which M1's (net) proft increases, and if the government subsidy level exceeds this threshold, the cost of greening the product will be too high, which will adversely afect M1's proft. If M2 distributes its nongreen products through only one retailer, specifcally R2, M2's proft initially decreases to a certain level; that is, it has a minimum and then increases. In the SS and CS structures, there exists a threshold of the government subsidy level, which minimizes M2's proft; therefore, it is not suggested that the government subsidy level approaches t l m2 . On the other hand, if M2 distributes the products through both retailers, its proft decreases consistently with an increase in the government subsidy level. From this fact, the channel choice can be an important decision for M2. In all distribution channel structures, R1's proft always increases with the government subsidy level. In all distribution channel structures, R1 delivers M1's green products to consumers. An increasing government subsidy boosts the demand for green products and R1's proft increases accordingly. If at least one of the manufacturers adopts the cross-distribution channel structure, R2's proft increases consistently with an increase in the government subsidy level. If both manufactures adopt the single-distribution channel structure, R2's proft initially decreases to a certain level; that is, it has a minimum and then increases. If each manufacturer adopts a single channel, the government subsidy level should not approach t SS r2 in order to prevent R2's proft from being minimized. As such, each manufacturer's channel choice has an infuence on R2's proft. Figure 2 is given to illustrate Proposition 2 with the following parameter settings: β � 0.5 and c m � 50.
One interesting question arises at this point: in each of the distribution channel structures, how does the trend of the supply chain proft change when the government subsidy level is increased? Proposition 3 provides an answer to this question.  Note that the supply chain proft is defned as the sum of the profts of all members. Proposition 3 reveals that once at least one of the manufacturers distributes products via both retailers, the supply chain proft is minimal at a certain value of the government subsidy level t l s . Terefore, the government subsidy level should not approach t l s , not only for proftability but also for the sustainability of the supply chain. On the other hand, when each manufacturer distributes its products exclusively via a single retailer, a higher government subsidy level guarantees greater supply chain profts. Figure 3 illustrates Proposition 3 with the following parameter settings: β � 0.9 and c m � 50.

Proposition 4. Consider the situation in which both manufacturers currently adopt the single-distribution channel. For each manufacturer, the decision to open a new distribution channel is as follows:
(1) Both manufacturers are willing to open a new channel at the same time when and

M2 is willing to open a new channel to distribute its nongreen products via another retailer
Te values of A 13 and A 14 are given in Appendix C.

Proposition 5. Consider the situation in which one manufacturer currently adopts the single-distribution channel, while the other uses the cross-distribution channel. For each manufacturer, the decision to open a new distribution channel is as follows:
(

M2 is willing to open a new channel to distribute its nongreen products via another retailer
Propositions 4 and 5 show that the fxed cost of opening a new channel plays a key role in the decision of the manufacturer's channel. If the fxed cost is sufciently high, the manufacturer will be reluctant to distribute its product via a new retailer and will maintain the current distribution channel structure. However, if the conditions in Propositions 4 and 5 are met, opening a new channel guarantees a higher proft and the manufacturer will be motivated to distribute its product via a new retailer. Propositions 4 and 5 also indicate that the manufacturer's best decision with regard to the channel choice depends on the fxed cost. Varying the value of the fxed cost, Figure 4 plots the manufacturers' profts with the following parameter settings: β � 0.15, c m � 250, s � 9.5(a), s � 7.5(b), f 2 � 0(a), and f 1 � 0(b). Take M1's channel choice strategy as an example (see Figure 4(a)). If both manufacturers currently adopt the single-distribution channel and both want to expand their distribution channel, M1's fxed cost of opening a new channel should not exceed θ CC 1 to ensure that π CC m1 > π SS m1 . If only M1 plans to expand, M1's fxed cost of opening a new channel should not exceed θ CS 1 to ensure that π CS m1 > π SS m1 . Finally, if M2 currently maintains the cross-distribution channel and M1 wants to expand its distribution channel, M1's fxed cost of opening a new channel should not exceed ϕ CC 1 to ensure that π CC m1 > π SC m1 . When the fxed cost is greater than θ CS 1 , M1 will then forever abandon any plan to open a new channel, regardless of the channel structure selected by M2. For brevity, we omit M2's channel expansion plan, but similar arguments hold for M2 (see Figure 4(b)). As such, the manufacturers' distribution channel choices difer according to the range of the fxed cost.

Government's Equilibrium Decision of the Subsidy Level
Tus far, this article has regarded the government subsidy level as an exogenous parameter. However, henceforth, the government subsidy level is regarded as an endogenous decision variable. In reality, a government may have a goal when implementing fnancial intervention [18,46]. Assume that the following possible objective is considered by a government: the government wants to maximize social welfare. By determining the government subsidy level s, the government will achieve this objective. All related proofs in Section 5 are given in Appendix B.
Tis section determines the government subsidy level to maximize social welfare. To do this, this section initially formulates the revenue and the cost terms separately and then integrates them to derive the objective function of the government problem, which is social welfare in this case.
(1) Supply chain proft π l s : the supply chain proft is defned as the sum of the profts of all members participating in the supply chain. In each of the distribution channel structures, it has the form of π l s � π l m1 + π l m2 + π l r1 + π l r2 , For ∈ SS, CC, CS, SC { }.
(2) Consumer surplus S l c : in economics, a consumer surplus, also known as an economic surplus and a Marshallian surplus, is the diference between the maximum price consumers are willing to pay and the actual price they pay. Terefore, it can be formulated as where according to equation (1), p l i � p i (q l 1 , q l 2 ) and p max i � p i (0, 0) for i � 1, 2.
(3) Environmental impact cost E l g : the government wants to minimize environmental degradation, while the manufacturers distribute products in the supply chain. Let parameter e and ρ denote the unit environmental impact of nongreen products and the unit environmental impact discount of green products relative to nongreen products, respectively. Tus, the unit environmental impact of green products is ρe. Te lower the value of ρ, the more environmentally benefcial the green products in comparison with nongreen products. A similar assumption can be found in He et l. [21]. Consequently, the environmental impact cost is given by (4) Government expenditure X l g : the government expenditure is defned as the total government subsidies provided to consumers who purchase M1's green products. Terefore, it can be formulated as X l g � g l sq l 1 , For ∈ SS, CC, CS, SC { }.
With equations (11)-(13), social welfare is defned as Before determining the equilibrium government subsidy level, it is important to analyze the properties of the consumer surplus, environmental impact cost, and government expenditure. Propositions 6 and 7 are given.  Proposition 6 says that a higher government subsidy level yields greater consumer surplus. Due to the associated government subsidy, the purchase of green products results in more surplus. As M1's demand for green products increases, the competition between the two manufacturers intensifes, lowering the retail price of nongreen products. Tus, consumers also gain a greater level of surplus from nongreen products. In short, a higher level of the government subsidy increases overall consumer surplus. It is revealed from Proposition 6 that government expenditures increase as the government subsidy level increases. It is obvious that a higher government subsidy level increases not only the greenness degree but also the demand for M1's green product, resulting in increased government expenditures for green products.

Proposition 7.
In each of the distribution channel structures, the environmental impact cost increases (decreases) when ρ > t l g (ρ < t l g ). Te values of t l g are given in Appendix C.
Proposition 7 implies that the increase or decrease in the environmental impact cost depends on the range in which ρ belongs. In other words, the environmental impact cost will decrease (increase) when the unit environmental impact discount of green products is lower (larger) than the threshold t l g . Figure 5 depicts Proposition 7 with the following parameters settings: e � 0.5, β � 0.8, c m � 250, f 1 � f 2 � 0, ρ � 0.1(a), and ρ � 0.8(b). It can be observed from Figure 5 that when ρ is low (high), E l g consistently decreases (increases). As mentioned above, the smaller the value of ρ, the less impact the green product has on the environment. Hence, when the value of ρ is small, an increase in the government subsidy increases the demand for green products, having less of an impact on the environment and resulting in a decrease in the environmental impact cost. On the other hand, when the value of ρ is high, the impact of the green products on the environment becomes similar to that of the nongreen products. If this is the case, an increase in the government subsidy level increases the demand for green products, resulting in an increase in the environmental impact cost.
Let s l g denote the equilibrium level of the government subsidy. For each of the distribution channel structures, s l g is expressed as At this point, the fnal proposition of this article is presented as follows: Or, if ρ > t l g , then s l g � 0, for l ∈ SS, CC, CS, SC { }. Proposition 8 reveals that the government will be motived to provide subsides to consumers when the impact of the green products on environmental degradation is relatively low. Conversely, when the environmental impact of green products is relatively high, the government will not have any incentive to subsidize the green products, even if they can reduce pollution to some extent. Although the distribution of green products is a good way to protect the environment, the government will not create a subsidy policy unless the impact on environmental degradation by green products is low enough. Terefore, when developing and distributing a green product, its manufacturer should consider the environmental impacts of the green product, such as recyclability, remanufacturability, disassemblability, and disposability, among others. Te following corollary supports this argument: Corollary 1. When the government provides subsidies for green products, s l g decreases in ρ.
Corollary 1 implies that the greater the environmental impact of green products, the lesser the government subsidies green products, which not only worsens the consumer surplus but also reduces social welfare. Figure 6 records the equilibrium level of the government subsidy, the consumer surplus, and the social welfare while varying ρ from 0 to 0.2. Figure 6 confrms Corollary 1 for all distribution channel structures.

Numerical Experiment
Te parameter β in this study indicates consumers' value discount for a green product or, equivalently, a fraction of their willingness-to-pay for a green product. Note that if β � 0, consumers are not willing to purchase the green product. If β � 1, consumers recognize that a green product is the perfect substitute for a nongreen product and are willing to pay the same amount for either product. In this section, a numerical experiment is carried out to investigate the efect of β on the equilibrium decisions. Te main dataset used for the analysis is as follows: c m � 50, e � 0.5, f 1 � f 2 � 0, and ρ � 0.1. When varying β from 0.1 to 0.9, Figure 7 records the equilibrium quantities of the decision. As the value of β increases, the followings are observed: increases to a certain level; that is, it has a maximum and then decreases. (iv) In all structures, M2's wholesale price decreases.
(v) In all structures, the selling quantity of M1's green product increases, while that of M's nongreen product decreases.  Figure 7 implies that when each manufacturer distributes its products through a single exclusive retailer, the greater the substitutability of a green product, the more subsidies the government pays. However, if at least one manufacturer decides to distribute its product via both retailers, there exists a maximum government subsidy level. Tis trend can be found in M1's wholesale price. As β increases, more consumers will think that a green product can replace a nongreen product in the market. Tis will increase not only the demand but also the greenness degree of the green product. Conversely, by lowering its wholesale price, M2 attempts to prevent a decline in the demand for its nongreen product. According to the values of β, Figure 8 also plots the obtained profts and social welfare. As the value of β increases, the followings are observed:

Mathematical Problems in Engineering
(i) In the SS structure, M1's proft consistently increases. On the other hand, in the CC, CS, and SC structures, M1's proft initially increases to a certain level; that is, it has a maximum and then decreases.
(iv) In the SS structure, R2's proft consistently decreases. On the other hand, in the CC, CS, and SC structures, M2's proft increases. (v) In all structures, the supply chain proft increases, but its trend is S-shaped. (vi) In all structures, social welfare increases. Figure 8 indicates that the proft trends of the manufacturers are consistent with their wholesale price trends. It should be noted that in the SS structure, R2's proft decreases in β. Tis occurs because R2 cannot deliver M1's green product in the SS structure, which adversely afects R2's proft. Figure 8 also indicates that if a green product is highly substitutable for a nongreen product, the proftability of the supply chain and the level of social welfare will increase. Terefore, it can be suggested that for the sustainability of a society and a supply chain, policymakers should focus on establishing policies that encourage the purchase of green products.

Managerial Insights
Tis section deals with key managerial insights summarized as follows: (i) In all the distribution channels considered in this study, the more environmentally friendly a product is designed, the more government subsidies for the product will beneft the green product producers in the supply chain. Tis insight will lead manufacturers to produce more environmentally friendly products, which has a positive impact on the well-being of the society. (ii) Trough this study, it is understood that the government's subsidy policy has a great infuence on the design of distribution channels in the supply chain. Since the distribution channel structure has a signifcant impact on the proftability and sustainability of the entire supply chain, the government needs to cooperate with the supply chain participants in the process of determining the subsidy level. (iii) Tis study shows that the environmental degradation of the green products afects the government's subsidy level. Even if a product is labeled as an eco-friendly product, the government may be reluctant subsidize the product if it has a negative impact on the environment. Terefore, it is very important for manufacturers to determine the optimal degree of greening when designing ecofriendly products.

Conclusion
Tis paper discussed the equilibrium decisions of pricing and greening in a supply chain consisting of two competing manufacturers and two retailers under a government subsidy policy. Tis paper assumed four diferent distribution channel structures, established a three-stage game model for each of them, and solved the models analytically. Extensive numerical experiments were conducted to support our fndings. Tis paper answers the research questions posed in Section 1: (i) As the government subsidy level increases, a green product becomes greener, leading to increases (decreases) in the wholesale price, the retail price, and the demand for the green product (nongreen product). For more details, see Proposition 1. (ii) For each manufacturer, the fxed cost of opening a new distribution channel has a maximum, which depends on the government subsidy level. For more details, see Propositions 4 and 5. (iii) Te profts of all supply chain members are affected in all cases by the government subsidy; therefore, it is obvious that the supply chain proft depends on the government subsidy level. It should be noted that if at least one of the manufacturers selects a cross-distribution channel, the supply chain proft is minimal at a certain value of the government subsidy level. For more details, see Proposition 3. (iv) To determine the optimal level of the government subsidy, policymakers must consider not only the supply chain proft but also the environmental effects of the products. Te greater the environmental efects of the products are, the lower the government subsidy level should be, thus decreasing consumer surplus and social welfare at the same time. For more details, see Proposition 8 and Corollary 1.
Tis paper provides several recommendations for supply chain participants that undertake sustainability eforts. However, there are also limitations that can be expanded upon and improved in future research. (1) Uncertain demand: while this paper assumes that the retailers sell all products that they order, the assumption of uncertain demand levels for products is more realistic. Tis type of stochastic demand may lead to diferent results. (2) Retailer's sustainability efort: this paper assumes that only manufacturers are involved in sustainability eforts. However, an extended model can assume that retailers can participate in sustainability activities, such as green retailing and/or green marketing. (3) Collusion strategy: this paper does not consider any collusion strategies between manufacturers. A comparative analysis among other supply chain cooperation strategies will provide more meaningful managerial insights. (4) Multiple participants: multiple manufacturers and retailers either collaborate or compete in various supply chains. It would be worthwhile to analyze a supply chain with more complex but realistic conditions.