How Does Green Store Brand Introduction Influence the Effects of Government Subsidy on Supply Chain Performance?

Abstract

With the rising awareness of environmental protection and concern for sustainable development, green products have been highly favored by consumers, enterprises, and the government. As a matter of fact, not only do manufacturers produce green products, but retailers would also like to introduce their green store brands. However, the costly green investment hinders the improvement of the products’ green degree. Therefore, the government may provide financial support to motivate enterprises to increase their products’ green degree. This study investigates how the presence of green store brands and government subsidies affect green supply chain performance. Four models are discussed using the Stackelberg game theoretic approach, and then, the optimal solutions in different cases are compared. The results show that (1) regardless of the government subsidy, the green store brand introduction always reduces the manufacturer’s profit and improves the retailer’s profit and environmental benefit; (2) In most cases, the implementation of a government subsidy can effectively improve the products’ green degree and benefit the supply chain members. However, it is surprising to find that the government subsidy may be detrimental to the manufacturer once the green store brand is introduced; (3) Interestingly, the introduction of green store brand may have an expansion effect, a shrinkage effect or even an inverse effect on the effects of government subsidies on supply chain performance, and these effects become more significant with the increasing green preference of consumers, product substitute, and subsidy rate. The new findings also provide some implications for supply chain members and the government in green supply chain management (GSCM) and green innovation.

1. Introduction

Over the past decades, issues regarding environmental degradation, such as global warming and excessive resource consumption (especially exhaustible resources), have impelled humans to be highly concerned about sustainable development. As a result, the topic of consumption and production of green goods has been a hot one in academic research as well as in government policy. The concept relating to green supply chain was first proposed by the Manufacture Research Council of Michigan State University in 1996. With further research, green supply chain management (GSCM), which refers to the supply chain management practices combined with enterprises’ purpose to minimize negative environmental impact [1], has been established and widely investigated in academic research. Green supply chain management aims to balance economic growth and environmental sustainability, and there is a sense that one of the effective approaches is to produce a green product [2,3]. Green products are recognized to be environmentally cleaner and greatly beneficial to sustainable development by reducing pollution and waste.

Nowadays, a large number of manufacturers and retailers are devoted to the production and sale of green goods. For example, electrical enterprises, such as Media, Haier and Gree, have all pushed their energy-saving products; in the fashion apparel industry, enterprises such as H&M and Levi’s have applied some green technologies into their production processes to minimize carbon emission [4,5]. In addition, the Kodak Company announced that they will reduce both the amount of greenhouse gas emission and water usage by 25% by 2025 through waste degradation, energy saving and emission reduction, and resource reuse, etc. Also, the Dell Company claims that more than 94% of its products are made of reproducible packing materials. Similarly, Coca Cola developed a kind of recyclable PET plastic bottle to replace the disposable ones for environmental protection. In terms of retailers, they have also done a lot to stimulate the sustainable development of the supply chain. The literature [6], based on actual data, summarized 25 major European retailers’ best practice actions in environmental improvement. On the one hand, owing to occupying an intermediate position in the distribution supply chain, retailers directly connect with consumers and manufacturers and have a stronger impact on green consumption than upstream manufacturers. For example, retailers adopt some practices, such as in-store green advertisement, which will effectively push consumers to choose a green product during their decision-making process. On the other hand, retailers also produce and sell their own green store brand products in addition to selling manufacturers’ national brand products, which is highly common in reality. For instance, Walmart declared that some of its store brands are made from recyclable raw materials. Similarly, Sam’s Club also introduced a new private label “Made with Our Member & Planet in Mind”, stressing that this series of store brands has a higher quality and is more consistent with sustainability standards. Moreover, Tesco developed a store brand product of a concentrated foam cleaner, which can reduce the use of plastic packing. Through introducing green store brands, retailers could make a stronger contribution to the green supply chains. This is because they have complete control over the products’ greenness criteria or environmental indices and always deeply participate in the whole production process to ensure the implementation of the requirements. In sum, both manufacturers and retailers contribute to green supply chain development.

However, despite the consumers being willing to pay a price premium for green products [7,8,9,10] and enterprises being proactive in undertaking corporate responsibility for addressing sustainability challenges [11], there are still some negative factors holding back the development of the green product market. Firstly, at the stage of product research and development, enterprises managing to enhance their production technology can improve their green products’ performance [12,13], which helps in the expansion of their market share. For example, automobile enterprises can benefit from better technology to produce high-energy and long-lasting batteries [14]. However, better technology also means higher R&D investment, and this high cost finally hinders the further development of green products. Secondly, the higher unit cost of production will lead to higher wholesale and retail prices. As a result, a green product usually has a price disadvantage on the market while competing with the common one, which negatively affects the demand for the green product. Finally, some empirical studies [15] found that the willingness of consumers to purchase green products varies in different industries and areas. Therefore, for the market segments where the development of green products is at an early stage and the propaganda of green products is not enough, the enterprises are hesitant to improve their products’ greenness due to the relatively low greenness consciousness of consumers.

Faced with the above issues, the government, as a leader of the whole society, makes an enormous effort to stimulate green and sustainable development. Therefore, a series of policies, such as public education and campaigns, taxes, and subsidies, as well as ecological labels and green criteria, are implemented to increase the production and consumption of green products [16,17,18,19,20]. For instance, China started the “Energy-saving Products and Benefit the Citizen Project” in 2009, and since then, the government has distributed an accumulative total of 400 billion CNY to those eligible firms. According to the governmental policy and plan of “Made in China 2025”, the subsidy for green products will remain to ensure the full implementation of green manufacturing. Also, many countries have promulgated and implemented low-carbon policies in carbon tax, cap-and-trade, and low-carbon offset. In this paper, we take subsidy, one of the most popular policies, as our research object to investigate the role of the government in the green supply chain.

A large amount of previous literature has proven the positive effects of government subsidy on the green supply chain without a green store brand. However, with the introduction of a green store brand, the competition between the two brands is not just related to price but also involves the confrontation of greenness. Therefore, it is significant in theory and practice to investigate the pricing and green degree strategies of two competitive products, one national brand and one store brand. Furthermore, there is still the question of how the government subsidy affects the green supply chain with a green store brand. In other words, whether the introduction of green store brands affect the outcome of the subsidy strategy remains uncertain and deserves further exploration.

Based on the above discussions, this paper is motivated to study the impacts of green store brand introduction and government subsidy implementation on the green supply chain and will try to answer the following questions:

What are the optimal decisions under different strategy combinations?

(1)

How does the introduction of green store brands and the implementation of government subsidies affect the supply chain performance?

(2)

How does the introduction of green store brand affect the effects of government subsidies? Do government subsidies always benefit the supply chain members?

(3)

What are the influences of consumers’ green preference and other parameters on the equilibrium results?

Therefore, this paper considers a GSCM system that consists of one manufacturer, one retailer, and the government. The retailer has the option of introducing a green store brand, and the government may provide subsidies for those enterprises that produce green products. This paper builds four game models according to different strategy combinations, and the optimal decisions on products’ prices and green degree(s) are derived in each model. Then, sensitivity analysis and comparative analysis are studied via a mathematical approach or numerical simulation. The main results are summarized as follows. Firstly, regardless of the government subsidy, the green store brand introduction always reduces the manufacturer’s profit and improves the retailer’s profit and environmental benefit. Secondly, the implementation of the government subsidy can usually effectively improve the products’ green degree and benefit the supply chain members. However, it is surprising to find that the government subsidy may be detrimental to the manufacturer once the green store brand is introduced. Consequently, the green store brand introduction may have an expansion, a shrinkage, or even an inverse effect on the impacts of government subsidies on supply chain performance. Finally, with the increasing consumers’ green preference, product substitute, and subsidy rate, the green degree of store brand, the retailer’s profit, and environmental benefit increase while the green degree of national brand and the manufacturer’s profit decline.

This paper contributes to the literature on green supply chain management by investigating the dual competition of price and greenness between one national brand and one store brand and exploring the impacts of green store brand introduction on government subsidy outcomes. In addition, the main conclusions offer some managerial implications for supply chain members and provide a guide for the government to make a series of effective subsidy policies.

The remainder of this paper is organized as follows: Section 2 reviews the relevant literature. Then, we formulate our models in Section 3. We solve the models using a backward induction approach and obtain the optimal solutions in Section 4. In Section 5, we explore the impacts of green store brand introduction and government subsidy implementation on the interaction between the manufacturer and retailer as well as the green supply chain performance. Also, the impacts of green store brand introduction on government subsidy outcomes were studied. Finally, in Section 6, we conclude our findings and propose the managerial implications. We also give the limitations and list the directions for future research.

2. Literature Review

Our paper is closely related to three streams of literature involving green supply chain management, government subsidies, and competition between national and store brands. In this section, we review the related literature from the above three streams and then compare our work with previous papers.

2.1. Green Supply Chain Management

Over the last decades, green supply chain management(GSCM) has rapidly developed in academic research and practical application. There is a large amount of literature focusing on green supply chain management. Styles et al. [6] provided a review of European retailers’ practice actions concerning the green effort, while Tseng et al. [21] systematically reviewed the literature of GSCM and further concluded trends and future challenges for GSCM. These reviews help us briefly understand the condition of GSCM from both aspects of industrial practices and theoretical studies. Many GSCM researchers investigated the relationship between factors with respect to green supply chain management through empirical research methods [22,23,24]. For instance, Roh et al. [22], using the structural equation model technique, validated the direct effect of intellectual property rights and green managerial innovation on sustainable supply chain management. Cousins [24] collected the relevant data from 248 UK manufacturing firms, and the data analysis using moderated hierarchical regression implied that GSCM practices positively affect both environmental and cost performance dimensions. From the perspective of theoretical research, other researchers utilized the MCDM (Multi-Criteria Decision-Making) method to solve the problems of green supply chain performance evaluation [25,26], green supplier selection [27,28,29], or risk analysis of green supply chain [30], and so on. Dou et al. [25] built a grey analytical network process-based model and proved its effectiveness in identifying useful green supplier development programs using a real-world example. Asadabadi [29] took the dynamics of customer needs into the problem of green supplier selection and used the integrated method of analytic network process, quality function deployment, and a Markov chain to solve the problem. As a result, one supplier can be selected as the best one in spite of the changing customer needs.

Additionally, the studies most related to our paper are the studies that apply game theory to investigate how to balance the trade-off between green products’ profitability and environmental protection [31,32,33,34]. Zhang and Liu [31] considered a three-level green supply chain system and constructed one cooperative and three Stackelberg game models. Based on the model analysis, they proposed three coordination mechanisms and confirmed their effectiveness in improving green supply chain performance. Cai et al. [32] modeled three information structures in a green supply chain and studied how the product’s greenness level and wholesale price affect the retailer’s information-sharing choice. Moreover, some researchers looked deeper into the optimal decision on a product’s green degree (e.g., Ghosh and Shah [33,34]). Ghosh and Shah [34] proposed two models of cost-sharing contracts that are proven to improve the green degree of products and realize supply chain coordination.

The above literature assumes that only one product exists in the supply chain; however, competition between products is widespread and cannot be ignored. For a supply chain consisting of one manufacturer and one retailer with a green product and a non-green traditional product, Basiri and Heydari [35] proposed one collaborative contract and confirmed that the proposed collaboration model could enhance the total profit of supply chain as well as achieve Pareto improvement for each member by conducting a numerical study. Yang et al. [14] considered the system framework of two competing firms and the government. Through the game-theoretic approach, they constructed the Cournot models to investigate the conditions under which firms will improve production technology to reduce energy consumption. Differently, some researchers conducted their studies from the perspective of dynamic game theory. Sun et al. [19] developed an evolutionary game theory model for a supply chain with suppliers and manufacturers. The results showed that both the government subsidy and green investment input/output ratio of supply members have impacts on the stable strategy evolution of the system. Similarly, Li et al. [18] explore the dynamic impacts of government policies on EV diffusion by a complex network evolutionary game method. There are also a few studies based on the Stackelberg game theory. Meng et al. [3] used Stackelberg game theory to study the collaborative pricing policies in a dual-channel green supply chain. In addition, they explored the role of government subsidies and found that subsidies reduce green products’ price and promote the sales of green products. Similarly, Lee and Yoon [36] constructed a three-stage Stackelberg game framework to derive equilibrium decisions on pricing and greenness under different distribution channel structures. They also confirmed the positive effect of the government subsidy on the green supply chain. Furthermore, according to different vertical and horizontal cooperation cases between supply chain members, Yang et al. [5] constructed several Stackelberg game models to investigate the dynamic interaction between manufacturers and retailers. The results showed that vertical cooperation can improve the carbon emission reduction rate while the manufacturers’ horizontal cooperation lowers it.

2.2. Government Subsidy

Government subsidies have been widely proven to effectively motivate enterprises to carry out green production and improve products’ green degree, which promotes the development of environmental sustainability [37]. Some researchers compared the effects of the subsidy with other government interventions on supply chain performance. For example, Seyed and Morteza [16] modeled the competitive green and non-green supply chains and studied the interactions between the government and supply chain members. They found that the product’s sustainability and the supply chains’ profits are more sensitive to the subsidy rate than the tax rate. Li et al. [38] simultaneously considered the government tax and subsidy in a supply chain consisting of one supplier and one manufacturer. In order to obtain a more effective strategy, a Stackelberg game between the government and the two-echelon supply chain was built, and the results affirmed that subsidies encourage a greener upstream decision.

Some of the literature also focuses on the comparative analysis of different types of government subsidies [39,40,41,42]. Rong and Xu [39] took two types of government subsidies and altruistic preference into account and built six game-theoretic models for different scenarios. Based on the solutions of models, they explored the impact of government subsidies and altruistic preferences on optimal decisions in the green supply chain. They found that the green R&D subsidy is more effective than the production cost subsidy. Liu et al. [41] conducted a similar framework to Rong and Xu [39]; however, they considered two other types of government subsidies, i.e., subsidy to the manufacturer and subsidy to consumers. By analyzing the optimal decisions of enterprises under different government subsidy modes and altruistic preference situations, they found that both types of subsidies can improve product greenness and corporate profits. Concerning the same two types of subsidies, Bian et al. [42] used the Stackelberg game theory to study the interaction between the government’s decision on environmental subsidy policies and the enterprises’ decisions on developing carbon emission reduction technology. Additionally, Meng et al. [43] added a third type of government subsidy, i.e., the subsidy to the retailer. They found that, although different types of subsidies have different effects on the price of green products, government subsidies always facilitate the sales of green products. Furthermore, from the perspective of consumer heterogeneity, Meng et al. [44] explored the green innovation subsidy strategy and found that the government is inclined to subsidy the manufacturer, and the increase in proportion of green consumers will exacerbate this tendency.

In sum, providing subsidies to producers of green products is one of the most effective approaches for the government to improve green supply chain performance and stimulate the sustainable development of society.

2.3. Store Brand

For decades, researchers have performed a great deal of studies on store brands. Here, we only review the literature focusing on the introduction of store brands. In order to investigate the conditions for retailers to introduce their store brands, Raju et al. [45] built a game model to explore how the basic market demand of the store brand, the amount of on-sale national brands, and the price substitution elasticity between both brands affect the introduction of a store brand. Mills [46] took the first step in applying utility theory to the demand model and found that a retailer will introduce their store brand only if the substitution between both brands is not too small. Morton and Zettelmeyer [47] conducted an empirical investigation and found that a retailer is more likely to introduce a store brand in the categories with large profit margins. Jin et al. [48] considered two channel-structures and two wholesale price schemes; by comparing the results of different cases, they found that both retailers are inclined to introduce store brands when the manufacturer chooses a single-channel (dual-channel) strategy under the uniform (flexible) wholesale price scheme. Moreover, some researchers focused on the impacts of store brand introduction. Most of these studies concluded that the introduction of a store brand could improve the retailer’s profit and the supply chain efficiency (e.g., Xiang Fang et al. [49]), while Chen et al. [50] discovered that introducing a store brand may be detrimental on the overall performance of supply chain. What is interesting is that significantly different from the mainstream conclusion that the introduction of a store band always hurts the manufacturer, Ru et al. [51] showed that one manufacturer may benefit from store brand introduction if the manufacturer is a follower of the supply chain. These papers show the different effects of store brand introduction on supply chain performance under different cases. However, none of them have taken the green supply chain into consideration.

Although some researchers integrated store brands and green supply chains into their studies, to the best of our knowledge, all of these studies only considered the traditional store brand and ignored the burgeoning green store brand. For example, Yang et al. [52] investigated the impacts of a non-green store brand’s introduction on supply chain decisions and found that the presence of the store brand increases the green degree of the national brand; Cheng et al. [53] explored the interaction between store brand and green technology in different sales modes and surprisingly found that one retailer will not introduce the store brand if its quality is higher than a certain threshold. Only a few papers have studied the green store brand. Wu et al. [54] considered a supply chain with a green store brand and found that the green supply chain achieves the best coordination effect when the retailer and manufacturer jointly undertake the initial one-off environmental investment. However, they only considered the green supply chain with a single product and ignored the competition between green products. Zhong and Huo [55] considered a supply chain with one traditional national brand and one green store brand. They studied the conditions in which the retailer can introduce their green store brand under different supply chain power structures. Though the competition between products was put into different models, the competition was limited in pricing strategy instead of both pricing and greenness strategies.

2.4. Research Gaps

We compared our work with the previous studies, and the differences are detailed in

Table 1. Comparison between the previous papers and the current study.

Article	Product Competition and Cross Elasticity of Demand			Store Brand			Green-Level Decision	Government Subsidy
	No Xed	Xed in Prices	Xed in Green-Levels	No SB	Normal SB	Green SB
Seyed and Morteza (2017) [16]		√		√			√	√
Basiri and Heydari (2017) [35]		√	√	√			√
Ranjan and Jha (2019) [2]		√		√			√
Wu et al. (2020) [54]	√					√
Li et al. (2020) [38]	√			√			√	√
Yang et al. (2020) [52]		√			√
Rong and Xu (2021) [39]	√			√			√	√
Yang et al. (2021) [14]		√	√	√				√
Liu et al. (2022) [41]	√			√			√	√
Cheng et al. (2022) [53]		√			√		√
Zhong and Huo (2022) [55]		√				√
Abhijit et al. (2023) [17]		√		√				√
This paper		√	√			√	√	√

. We can see that these papers mainly concentrated on the green supply chain only with national brand(s). Although some papers studied the green supply chain consisting of the national and store brands, they did not consider the greenness level decision of the green store brand. In addition, most of those papers modeling two alternative products ignored the cross elasticity in products’ greenness and only focused on the cross elasticity in prices. However, with the increasing consumers’ greenness preference, the green degree of products also greatly impacts on their demands. Furthermore, according to the previous literature, the government subsidy also has important influences on green supply chain performance, especially on environmental improvement.

Thus, to fill the gaps of previous studies, our paper considers a green supply chain with one manufacturer and one retailer, and the retailer has the option to introduce a green store brand. If the green store brand is introduced, then the dual competition of price and greenness between both brands will be considered. Moreover, we also explore the effects of the government subsidy on supply chain performance and environmental benefits. This work will provide new insights into the supply chain’s green production and governmental subsidy decisions.

3. The Model

This paper models a management system consisting of one manufacture, one retailer, and the government, as shown in Figure 1. In this system, the manufacturer produces a green product and wholesales it to the retailer. Despite distributing the green national brand product, the retailer may also sell her own green store brand to customers. In addition, the government may or may not provide a subsidy to the producers of green products. We also assume that the manufacturer and retailer are rational and have all the information.

The manufacturer wholesales his green national brand product at the price of $w$, and then the retailer distributes the products to customers at the retail price of $p_n$. In addition, the retailer has the ability and option to produce their own green store brand and will sell it at the retail price of $p_s$. While the retailer introduces a green store brand, the demand functions of the national brand and store brand in this paper are linear functions of selling prices and products’ green degrees, which are similar to many previous papers by Basiri and Heydari [35], Yang et al. [14] and Li et al. [56] and presented as follows:

$$\left\{\begin{array}{l}{D}_n=1-p_n+k(p_s-p_n)+\lambda \left[g_n+k(g_n-g_s)\right]\\ D_s=b-p_s+k(p_n-p_s)+\lambda \left[g_s+k(g_s-g_n)\right]\end{array}\right.$$

(1)

In Equation (1), $D_n$ ($D_s$) and $g_n$ ($g_s$) are the demand and green degree of national brand(store brand), respectively. There is an understanding that the basic market demand for the national brand is still larger than the store brand, and this assumption has been used in some of the previous literature (e.g., Raju et al. [45]). In addition, according to the report “Private Label in EMEA: Capturing opportunity and protecting your brand.” published in the Oracle database, more than 70% of European consumers consider the private label when comparing the retail prices of similar products. The ratio of consumers who may purchase a store brand to all consumers also reflects the basic market demand for the green store brand. Therefore, for the sake of simplicity, we normalize the national brand’s basic market demand to be one without loss of generality. Correspondingly, the basic market demand for the store brand is assumed to be $b$, $b\in \left(\frac{1}{2},1\right)$. The parameter $k$ is the cross elasticity coefficient in price and greenness between both brands, and $k\in \left(0,1\right)$ indicates that the demand for one product is more sensitive to its own price and greenness than to those of the rival. Moreover, we denote $\lambda \left(\lambda >0\right)$ as consumers’ preference for the product’s greenness.

Furthermore, referring to the method applied in previous literature (e.g., Abhishek et al. [57] and Shen et al. [58]), we obtain the demand for the national brand when the retailer does not introduce their green store brand by letting $D_s=0$ and solving Equation (1), that is:

$$D_n=1+\frac{bk}{1+k}-\frac{1+2k}{1+k}(p_n-\lambda g_n)$$

(2)

The sunk cost derived from the greenness improvement is assumed to be a quadratic function of the green degree $g_i$, which is given by $\frac{h{g}_i{}^2}{2}$ and $h$ is the coefficient of greenness investment. Such a cost function is used largely in the previous literature (e.g., Ghosh and Shah [33,34]; Basiri and Heydari [35]). Due to the higher cost of a green product than a normal one and the positive attributes of green product in environmental protection, the government may provide financial support to the firm that produces the green product, which is aimed at encouraging firms’ enthusiasm to improve their products’ green degree. We assume that the environmental benefit of product i is positively related to $g_i$ and $D_i$, i.e., $E{B}_i=g_i{D}_i$. For per unit of environmental benefit, the government provides the producer with subsidy $s$. In addition, we assume that $s<\lambda$ to exclude the meaningless case where supply chain members may set a negative price and make profits just from government subsidy. Additionally, this assumption does not affect the main conclusions of this paper. Accordingly, the profit functions of the manufacturer and retailer are presented as follows.

$$\left\{\begin{array}{l}\underset{(w,g_n)}{\max {\pi }_m}=\left(w+xs{g}_n\right)D_n-\frac{h{g}_n{}^2}{2}\\ s.t.\underset{(p_n,p_s,g_s)}{\max {\pi }_r}=(p_n-w)D_n+y\left[\left(p_s+xs{g}_n\right)D_s-\frac{h{g}_s{}^2}{2}\right]\end{array}\right.$$

(3)

where the indicator $x$ equals one if the government provides a subsidy and 0 otherwise. Similarly, the indicator $y$ equals one if the retailer introduces a store brand and 0 otherwise. All the notations in our models are summarized in

Table 2. List of the notations in models.

Notation	Definition
Indices
$i$	Subscript, index of brand, $i=n$ for national brand and $i=s$ for store brand
$j$	Subscript, index of supply chain member, $j=m$ for manufacturer and $j=r$ for retailer
$R$	Superscript, index of retailer’s store brand decision, $R=\left\{N,B\right\}$
$G$	Superscript, index of government subsidy decision, $G=\left\{N,S\right\}$
Parameters
$b$	basic market demand of green store brand
$k$	Coefficient of cross elasticity in price and green degree
$\lambda$	Consumers’ green preference
$h$	Coefficient of cost on greenness improvement
$s$	Subsidy rate for per unit of environment benefit
Decision Variables
$g_i$	Green degree of product $i$
$w$	Wholesale price of national brand product
$p_i$	Retail price of product $i$
Dependent Variables
$D_i$	Demand quantity of product $i$
${\pi }_j$	Profit of supply chain member $j$
EB	Total environmental benefit, $EB=g_n{D}_n+g_s{D}_s$

.

4. The Equilibrium Results

According to the green supply chain observation, green store brands have developed in recent years, while green national brands have a long history. In addition, a report published by Nielsen revealed that 70% of global respondents purchase store brands to save money, and most consumers maintain a preference for national brands (Groznik and Heese [59]; Wang et al. [60]). As a result, retailers set the store brands’ retail prices by referring to the retail prices of national brands [61] to achieve a profitable gap in price. Therefore, to capture the dynamics of interactions between the manufacturer and retailer, we resort to the Stackelberg game model, allowing players to take on a sequential game. Furthermore, considering the manufacturer’s dominant position in the green supply chain, which is consistent with previous research, such as Yang et al. [52] and Cheng et al. [53], we assume that the supply chain members compete in the manufacturer-led Stackelberg game. Additionally, we can solve the aforementioned optimization problems with backward induction. The strategy combination of retailer and manufacturer is denoted by the superscript $RG$, where $R\left(R\in \left\{N,B\right\}\right)$ represents the choice of the retailer on whether or not to introduce a green store brand, and $G\left(G\in \left\{N,S\right\}\right)$ represents the decision of the government on whether or not to provide the subsidy.

4.1. Only National Brand

When the retailer does not introduce a green store brand, only the national brand exits in the supply chain. Consequently, the game sequence is as follows: firstly, the manufacturer decides the wholesale price $w$ and the green degree $g_n$ of the national brand; then, by observing the decisions of the manufacturer on $w$ and $g_n$, the retailer decides the retail price $p_n$ for the national brand to maximize her profit. Next, we resolve the models with and without the government subsidy.

4.1.1. No Government Subsidy (Case “NN”)

In this case, the government does not provide the subsidy, and the profit functions of the retailer and manufacturer are expressed as Equations (4) and (5). We can obtain the optimal solutions through backward induction, which is summarized in Theorem 1. (All proofs are presented in Appendix A).

$${\pi }_m^{NN}=w\left[1+\frac{bk}{1+k}-\frac{1+2k}{1+k}(p_n-\lambda g_n)\right]-\frac{h{g}_n{}^2}{2}$$

(4)

$${\pi }_r^{NN}=(p_n-w)\left[1+\frac{bk}{1+k}-\frac{1+2k}{1+k}(p_n-\lambda g_n)\right]$$

(5)

Theorem 1. 

In the case NN, if $h>\underset{¯}{h_0}=\frac{\left(1+2k\right){\lambda }^2}{4\left(1+k\right)}$ holds, then the unique optimal combination of decision variables $\left(g_n^{NN*}, w^{NN*},p_n^{NN*}\right)$ exists, which are expressed as follows: $g_n{}^{NN*}=\frac{\left(1+k+bk\right)\lambda }{4h\left(1+k\right)-\left(1+2k\right){\lambda }^2}$, $w^{NN*}=\frac{2h\left(1+k\right)\left(1+k+bk\right)}{\left(1+2k\right)\left[4h\left(1+k\right)-\left(1+2k\right){\lambda }^2\right]}$ and $p_n{}^{NN*}=\frac{3h\left(1+k\right)\left(1+b+bk\right)}{\left(1+2k\right)\left[4h\left(1+k\right)-\left(1+2k\right){\lambda }^2\right]}$.

The constraint $h>\underset{¯}{h_0}=\frac{\left(1+2k\right){\lambda }^2}{4\left(1+k\right)}$ may not be satisfied in some cases. However, when the cost of the green improvement is lower than the threshold $\underset{¯}{h_0}$, the manufacturer will increase the green degree of the national brand toward infinity to exploit the demand increase. Referring to the literature [35], we exclude this condition because it is not realistic in a real business environment.

Substituting Theorem 1 into Equations (2), (4), and (5), we can obtain the national brand’s sale quantity and the profits of the manufacturer and retailer. We also calculate the total environmental benefit in this case.

4.1.2. With Government Subsidy (Case “NS”)

In this case, the government provides the subsidy, and the profit functions of the retailer and manufacturer are expressed as Equations (6) and (7), and we can obtain the optimal solutions via backward induction, which is summarized in Theorem 2.

$${\pi }_m^{NN}=\left(w+s{g}_n\right)\left[1+\frac{bk}{1+k}-\frac{1+2k}{1+k}(p_n-\lambda g_n)\right]-\frac{h{g}_n{}^2}{2}$$

(6)

$${\pi }_r^{NN}=(p_n-w)\left[1+\frac{bk}{1+k}-\frac{1+2k}{1+k}(p_n-\lambda g_n)\right]$$

(7)

Theorem 2. 

In the case NS, if $h>\underset{¯}{h_1}=\frac{\left(1+2k\right){\left(\lambda +s\right)}^2}{4\left(1+k\right)}$ holds, then the unique optimal combination of decision variables $\left(g_n^{NS*}, w^{NS*},p_n^{NS*}\right)$ exists, which are expressed as follows: $g_n{}^{NS*}=\frac{\left(1+k+bk\right)\left(\lambda +s\right)}{4h\left(1+k\right)-\left(1+2k\right){\left(\lambda +s\right)}^2}$, $w^{NS*}=\frac{\left(1+k+bk\right)\left[2h\left(1+k\right)-s\left(1+2k\right)\left(\lambda +s\right)\right]}{\left(1+2k\right)\left[4h\left(1+k\right)-\left(1+2k\right){\left(\lambda +s\right)}^2\right]}$ and $p_n{}^{NS*}=\frac{\left(1+k+bk\right)\left[3h\left(1+k\right)-s\left(1+2k\right)\left(\lambda +s\right)\right]}{\left(1+2k\right)\left[4h\left(1+k\right)-\left(1+2k\right){\left(\lambda +s\right)}^2\right]}$.

It is intuitive that $\underset{¯}{h_1}>\underset{¯}{h_0}$, which means if the green cost coefficient satisfies the condition of $\underset{¯}{h_0}<h<\underset{¯}{h_1}$, the manufacturer will increase the green degree of the national brand toward infinity due to the provided government subsidy. It also indicates that the government will provide the subsidy only if the green cost coefficient is relatively high, which agrees with the practical observation.

Substituting Theorem 2 into Equations (2), (6), and (7), we can obtain the national brand’s sale quantity and the profits of the manufacturer and retailer. We also calculate the total environmental benefit of this case.

4.2. Both National Brand and Store Brand

When the retailer introduces a green store brand, then competition between the national brand and the store brand exists. The game sequence is as follows: first, the manufacturer decides the wholesale price $w$ and the green degree $g_n$ of the national brand. Then, by observing the decisions of the manufacturer on $w$ and $g_n$, the retailer decides the retail price $p_n$ for the national brand as well as the retailer price $p_s$ and green degree $g_s$ for the store brand to maximize profits. Next, we resolve the models with and without the government subsidy.

4.2.1. No Government Subsidy (Case “BN”)

In this case, the government does not provide the subsidy, and the profit function of the retailer and manufacturer are expressed as Equations (8) and (9), and we can obtain the optimal solutions via backward induction, which is summarized in Theorem 3.

$${\pi }_m^{NN}=w\left\{1-p_n+k(p_s-p_n)+\lambda \left[g_n+k(g_n-g_s)\right]\right\}-\frac{h{g}_n{}^2}{2}$$

(8)

$$\begin{array}{l}{\pi }_r^{NN}=(p_n-w)\left\{1-p_n+k(p_s-p_n)+\lambda \left[g_n+k(g_n-g_s)\right]\right\}\\ +p_s\left\{b-p_s+k(p_n-p_s)+\lambda \left[g_s+k(g_s-g_n)\right]\right\}-\frac{h{g}_s{}^2}{2}\end{array}$$

(9)

Theorem 3. 

In the case of BN, if $h>\underset{¯}{h_2}=\frac{\left(1+k+bk\right){\lambda }^2}{2}$ holds, then the unique optimal combination of decision variables $\left(g_n^{BN*},g_s^{BN*}, w^{BN*},p_n^{BN*},p_s^{BN*}\right)$ exists, which are expressed as follows:

$$g_n{}^{BN*}=\frac{\left[2h-\left(1+k+bk\right){\lambda }^2\right]\lambda }{8h^2-6h\left(1+k\right){\lambda }^2+\left(1+2k\right){\lambda }^4};$$

$$g_s{}^{BN*}=\frac{\lambda \left\{k\left(2h-{\lambda }^2\right)\left[2h-\left(1+2k\right){\lambda }^2\right]+b\left[8h^2\left(1+k\right)-2h(3+6k+2k^2){\lambda }^2+\left(1+3k+2k^2\right){\lambda }^4\right]\right\}}{\left[2h\left(1+k\right)-\left(1+2k\right){\lambda }^2\right]\left[8h^2-6h\left(1+k\right){\lambda }^2+\left(1+2k\right){\lambda }^4\right]};$$

$$w^{BN*}=\frac{2h\left[2h-\left(1+k\right){\lambda }^2\right]\left[2h-\left(1+k+bk\right){\lambda }^2\right]}{\left[2h\left(1+k\right)-\left(1+2k\right){\lambda }^2\right]\left[8h^2-6h\left(1+k\right){\lambda }^2+\left(1+2k\right){\lambda }^4\right]};$$

$$p_n{}^{BN*}=\frac{h\left[4h^2\left(3+6k+2bk+2\left(1+b\right)k^2\right)-6h\left(1+k+bk\right)\left(2+4k+k^2\right){\lambda }^2+\left(1+2k\right)\left(3+6k+4bk+4\left(1+b\right)k^2\right){\lambda }^4\right]}{\left(1+2k\right)\left[2h\left(1+k\right)-\left(1+2k\right){\lambda }^2\right]\left[8h^2-6h\left(1+k\right){\lambda }^2+\left(1+2k\right){\lambda }^4\right]};$$

$$p_s{}^{BN*}=\frac{h\left\{2k\left[4h^2\left(1+k\right)-h\left(4+8k+3k^2\right){\lambda }^2+\left(1+3k+2k^2\right){\lambda }^4\right]+b\left[8h^2{\left(1+k\right)}^2-6h{\left(1+k\right)}^3{\lambda }^2+\left(1+4k+6k^2+4k^3\right){\lambda }^4\right]\right\}}{\left(1+2k\right)\left[2h\left(1+k\right)-\left(1+2k\right){\lambda }^2\right]\left[8h^2-6h\left(1+k\right){\lambda }^2+\left(1+2k\right){\lambda }^4\right]}.$$

Substituting Theorem 3 into Equations (1), (8), and (9), we can obtain both brands’ sale quantities and the profits of the manufacturer and retailer. We also calculate the total environmental benefit of this case.

4.2.2. With Government Subsidy (Case “BS”)

In this case, the government provides the subsidy, and the profit function of the retailer and manufacturer are expressed as Equations (10) and (11), and we can obtain the optimal solutions via backward induction, which is summarized in Theorem 4.

$${\pi }_m^{BS}=\left(w+s{g}_n\right)\left\{1-p_n+k(p_s-p_n)+\lambda \left[g_n+k(g_n-g_s)\right]\right\}-\frac{h{g}_n{}^2}{2}$$

(10)

$${\pi }_r^{BS}=(p_n-w)\left\{1-p_n+k(p_s-p_n)+\lambda \left[g_n+k(g_n-g_s)\right]\right\}+\left(p_s+s{g}_s\right)\left\{b-p_s+k(p_n-p_s)+\lambda \left[g_s+k(g_s-g_n)\right]\right\}-\frac{h{g}_s{}^2}{2}$$

(11)

Theorem 4. 

In the BS case, if $h>\underset{¯}{h_3}=\frac{\left(1+k+bk\right){\left(\lambda +s\right)}^2}{2}$ holds, then the unique optimal combination of decision variables $\left(g_n^{BS*},g_s^{BS*}, w^{BS*},p_n^{BS*},p_s^{BS*}\right)$ exists, which are expressed as follows: $g_n{}^{BS*}=\frac{\left[2h-\left(1+k+bk\right){\left(\lambda +s\right)}^2\right]\left(\lambda +s\right)}{8h^2-6h\left(1+k\right){\left(\lambda +s\right)}^2+\left(1+2k\right){\left(\lambda +s\right)}^4}$;

$$g_s{}^{BS*}=\frac{\left(\lambda +s\right)\left\{k\left(2h-{\left(\lambda +s\right)}^2\right)\left[2h-\left(1+2k\right){\left(\lambda +s\right)}^2\right]+b\left[8h^2\left(1+k\right)-2h(3+6k+2k^2){\left(\lambda +s\right)}^2+\left(1+3k+2k^2\right){\left(\lambda +s\right)}^4\right]\right\}}{\left[2h\left(1+k\right)-\left(1+2k\right){\left(\lambda +s\right)}^2\right]\left[8h^2-6h\left(1+k\right){\lambda }^2+\left(1+2k\right){\left(\lambda +s\right)}^4\right]};$$

$$w^{BS*}=\frac{2h\left[2h-\left(1+k\right){\left(\lambda +s\right)}^2\right]\left[4h^2+\left(1+2k\right)s{\left(\lambda +s\right)}^3-2h\left(1+k\right){\left(2s^2+3s\lambda +{\lambda }^2\right)}^2\right]}{\left[2h\left(1+k\right)-\left(1+2k\right){\left(\lambda +s\right)}^2\right]\left[8h^2-6h\left(1+k\right){\left(\lambda +s\right)}^2+\left(1+2k\right){\left(\lambda +s\right)}^4\right]};$$

$$p_n{}^{BS*}=\frac{1+k+bk}{2\left(1+2k\right)}+\frac{\left[2h-\left(1+k+bk\right){\left(\lambda +s\right)}^2\right]\left[4h^2-2hs\left(\lambda +s\right)-\left(1+2k\right)\left(\lambda +s\right)\left(2hs-s^3-\lambda s^2+s{\lambda }^2+{\lambda }^3\right)\right]}{2\left[2h\left(1+k\right)-\left(1+2k\right){\left(\lambda +s\right)}^2\right]\left[8h^2-6h\left(1+k\right){\left(\lambda +s\right)}^2+\left(1+2k\right){\left(\lambda +s\right)}^4\right]};$$

$$p_s{}^{BS*}=\frac{\begin{array}{l}b\left[\begin{array}{l}8h^3{\left(1+k\right)}^2-s\left(1+k\right){\left(1+2k\right)}^2{\left(\lambda +s\right)}^5-2\left(\left(7+14k+3k^2\right)s+3\lambda {\left(1+k\right)}^2\right)h^2\left(1+k\right)\left(\lambda +s\right)\\ +h\left(1+2k\right){\left(\lambda +s\right)}^3\left(\left(7+14k+6k^2\right)s+\lambda \left(1+2k+2k^2\right)\right)\end{array}\right]\\ +k\left[\begin{array}{l}8h^3\left(1+k\right)-s{\left(1+2k\right)}^2{\left(\lambda +s\right)}^5-2h\left(1+3k+2k^2\right)\left(\lambda +3s\right){\left(\lambda +s\right)}^3\\ +2h^2\left(\lambda +s\right)\left(3\left(2+4k+k^2\right)s+\lambda \left(4+8k+3k^2\right)\right)\end{array}\right]\end{array}}{\left(1+2k\right)\left[2h\left(1+k\right)-\left(1+2k\right){\left(\lambda +s\right)}^2\right]\left[8h^2-6h\left(1+k\right){\left(\lambda +s\right)}^2+\left(1+2k\right){\left(\lambda +s\right)}^4\right]}.$$

The relationship between two thresholds of $h$ can be expressed as $\underset{¯}{h_3}>\underset{¯}{h_2}$, which implies the aforementioned conclusion still holds under the situation where the green store brand is introduced. Moreover, we find that $\underset{¯}{h_2}>\underset{¯}{h_0}$ and $\underset{¯}{h_3}>\underset{¯}{h_1}$. This is because the competition resulting from the introduction of the store brand stimulates the increase in green degrees of both brands, and they will incline toward infinity if the green cost coefficient is relatively low, i.e., $\underset{¯}{h_0}<h<\underset{¯}{h_2}$ or $\underset{¯}{h_1}<h<\underset{¯}{h_3}$. Therefore, for the effective comparisons between different cases in Section 5, we assume that the green cost coefficient is higher than all of the thresholds corresponding to the selected comparing objects.

Substituting Theorem 4 into Equations (1), (10), and (11), we can obtain both brands’ sale quantities and the profits of the manufacturer and retailer. We also calculate the total environmental benefit of this case.

4.3. Sensitivity Analysis

Now, we examine the sensitivity of greenness and pricing strategies and dependent variables with respect to the key parameters. Before the introduction of the green store brand, we conduct a sensitivity analysis by identifying the sign of partial derivatives. However, the mathematical expressions of variables are too complex to analyze the signal of partial derivatives after the store brand introduction. Therefore, we have to resort to numerical simulation. The results are summarized in

Table 3. Sensitivity analysis in Case NN and Case NS.

	${\mathit{g}}_{\mathit{n}}^{\mathit{N}\mathit{G}}$	${\mathit{w}}^{\mathit{N}\mathit{G}}$	${\mathit{p}}_{\mathit{n}}^{\mathit{N}\mathit{G}}$	${\mathit{D}}_{\mathit{n}}^{\mathit{N}\mathit{G}}$	${\mathit{\pi }}_{\mathit{m}}^{\mathit{N}\mathit{G}}$	${\mathit{\pi }}_{\mathit{r}}^{\mathit{N}\mathit{G}}$	$\mathit{E}{\mathit{C}}^{\mathit{N}\mathit{G}}$
$\lambda$	↑	↑	↑	↑	↑	↑	↑
$h$	↓	↓	↓	↓	↓	↓	↓

,

Table 4. Sensitivity analysis in Case BN.

	${\mathit{g}}_{\mathit{n}}^{\mathit{B}\mathit{N}}$	${\mathit{g}}_{\mathit{s}}^{\mathit{B}\mathit{N}}$	${\mathit{w}}^{\mathit{B}\mathit{N}}$	${\mathit{p}}_{\mathit{n}}^{\mathit{B}\mathit{N}}$	${\mathit{p}}_{\mathit{s}}^{\mathit{B}\mathit{N}}$	${\mathit{D}}_{\mathit{n}}^{\mathit{B}\mathit{N}}$	${\mathit{D}}_{\mathit{s}}^{\mathit{B}\mathit{N}}$	${\mathit{\pi }}_{\mathit{m}}^{\mathit{B}\mathit{N}}$	${\mathit{\pi }}_{\mathit{r}}^{\mathit{B}\mathit{N}}$	$\mathit{E}{\mathit{C}}^{\mathit{B}\mathit{N}}$
$\lambda$	↑↓	↑	↑↓	↑↓	↑	↑↓	↑	↑↓/↓	↑	↑
$h$	↑↓	↓	↑↓	↑↓	↑	↑↓	↓	↑↓	↓	↓
$k$	↓	↑	↓	↓	↑	↓	↑	↓	↑	↑

and

Table 5. Sensitivity analysis in Case BS.

	${\mathit{g}}_{\mathit{n}}^{\mathit{B}\mathit{S}}$	${\mathit{g}}_{\mathit{s}}^{\mathit{B}\mathit{S}}$	${\mathit{w}}^{\mathit{B}\mathit{S}}$	${\mathit{p}}_{\mathit{n}}^{\mathit{B}\mathit{S}}$	${\mathit{p}}_{\mathit{s}}^{\mathit{B}\mathit{S}}$	${\mathit{D}}_{\mathit{n}}^{\mathit{B}\mathit{S}}$	${\mathit{D}}_{\mathit{s}}^{\mathit{B}\mathit{S}}$	${\mathit{\pi }}_{\mathit{m}}^{\mathit{B}\mathit{S}}$	${\mathit{\pi }}_{\mathit{r}}^{\mathit{B}\mathit{S}}$	$\mathit{E}{\mathit{C}}^{\mathit{B}\mathit{S}}$
$\lambda$	↑↓/↓	↑	↑↓/↓	↑↓/↓	↑	↑↓/↓	↑	↑↓/↓	↑	↑
$h$	↑↓	↓	↑↓	↑↓	↑	↑↓	↓	↑↓	↓	↓
$k$	↓	↑	↓	↓	↑	↓	↑	↓	↑	↑

. Among the Tables, $NG=\left\{NN,NS\right\}$; ↑ indicates that the column value will increase as the row parameter increase; ↓ indicates that the column value will decrease as the row parameter increase; ↑↓ indicates that the column value will increase firstly and then decrease as the row parameter increase; / indicates that the sensitivity of the column value to the row parameter depends on the other parameters.

Table 3 indicates that, while there is only a national brand in the supply chain, the green degree and prices of the national brand increase with the increase in the consumers’ green preference and decline with the increase in the green cost coefficient, regardless of the government subsidy. It is straightforward that the higher consumers’ green preference or lower green cost coefficient will encourage the manufacturer to make more effort to improve product’s green degree, and the increasing green degree leads to a higher wholesale price and retail price. Correspondingly, the demand for the national brand increases with the increase in the consumers’ green preference or decrease in the green cost coefficient because the positive effect derived from a higher green degree dominates the negative effect derived from the higher retail price. As a result, the profits of the manufacturer and retailer and the total environmental benefit display the same trend of change.

In the cases of both brands, we find the above conclusions can still be applied to explain the sensitivity of variables related to the store brand, the retailer’s profit, and environmental benefit (see Table 4 and Table 5). However, the effects on variables related to the national brand and the manufacturer’s profit differ. When the consumers’ green preference is higher than a certain threshold or the green cost coefficient is lower than a certain threshold, they will have an opposite impact on the corresponding variable for cases without green store brand. That is, the green degree and prices of the national brand as well as the manufacturer’s profit, unexpectedly decline with the increase in consumers’ green preference and increase with the increase in the green cost coefficient. These counter-intuitive conclusions result from the competition between both brands. The increasing consumers’ green preference or the decreasing green cost coefficient stimulates the retailer to produce and sell a greener store brand, which further cannibalizes national brand sales. As a result, the manufacturer has to lower the green degree and wholesale price of the national brand in hopes of holding back the price-driven sales. This situation will be more likely to occur if the government provides the subsidy. Furthermore, with the government subsidy, the manufacturer’s profit and other variables related to the national brand may even always decline with the increase in consumers’ green preference.

In addition, we also analyze the sensitivity to the parameter of cross-price elasticity between both brands in the case of BN and BS. The results indicate that the manufacturer suffers while the retailer benefits from the increased coefficient of cross-price elasticity. This is intuitive because a more substitute store brand always means a more severe threat to the national brand and lowers the latter’s green degree, prices, and demand.

5. Comparison Analysis and Numerical Simulation

In this section, we compare the equilibrium results of different cases and investigate how the government subsidy implementation and the green store brand introduction affect the optimal decisions in the green supply chain. We denote the $\mathsf{\Delta }$ as the difference value. For example, $\mathsf{\Delta }{g}_n^{NG}$ denotes the difference values of green degree in the case of NS and NN (i.e., $\mathsf{\Delta }{g}_n^{NG}=g_n^{NS}-g_n^{NN}$) and $\mathsf{\Delta }{g}_n^{BG}$ denotes the difference values of green degree in the case of BS and BN (i.e., $\mathsf{\Delta }{g}_n^{BG}=g_n^{BS}-g_n^{BN}$).

5.1. The Effects of the Government Subsidy

5.1.1. Only National Brand

Firstly, we explore how the government subsidy affects the optimal solutions of the supply chain when the green store brand is not introduced. We examine how the equilibrium results in the case of NS change with the increase of the subsidy rate, and the conclusions are summarized in Proposition 1. Furthermore, by comparing the equilibrium results in the case of NS with those in the case of NN, we obtain Proposition 2.

Proposition 1. 

(1) 

$\frac{\partial g_n^{NS}}{\partial s}>0$, $\frac{\partial p_n^{NS}}{\partial s}>0$, $\frac{\partial D_n^{NS}}{\partial s}>0$, $\frac{\partial {\pi }_r^{NS}}{\partial s}>0$, $\frac{\partial {\pi }_m^{NS}}{\partial s}>0$,$\frac{\partial E{B}^{NS}}{\partial s}>0$;

(2) 

If the subsidy rate satisfies $0<s<\min \{\lambda ,\frac{{\left(\sqrt{\left(1+k\right)h}-\sqrt{\left(1+k\right)h-\left(1+2k\right){\lambda }^2}\right)}^2}{\left(1+2k\right)\lambda }\}$, then $\frac{\partial w^{NS}}{\partial s}>0$, otherwise,$\frac{\partial w^{NS}}{\partial s}\le 0$.

Proposition 1 indicates that if there is only a national brand in the supply chain, the product’s green degree and demand and the supply chain members’ profits increase with the increase in government subsidies. In addition, the government also can improve the environmental benefit of the national brand by setting a higher subsidy rate.

Proposition 2. 

(1) 

$\mathsf{\Delta }{g}_n^{NG}>0$; $\mathsf{\Delta }{p}_n^{NG}>0$; $\mathsf{\Delta }{D}_n^{NG}>0$; $\mathsf{\Delta }{\pi }_r^{NG}>0$; $\mathsf{\Delta }{\pi }_m^{NG}>0$; $\mathsf{\Delta }E{B}^{NG}>0$.

(2) 

If the green cost coefficient satisfies $0<s<\min \{\frac{2\sqrt{h\left(1+k\right)\left(1+2k\right)}-\left(1+2k\right)\lambda }{1+2k},\frac{\left(1+2k\right){\lambda }^3}{2h\left(1+k\right)-\left(1+2k\right){\lambda }^2}\}$, then $\mathsf{\Delta }{w}^{NG}>0$, otherwise, $\mathsf{\Delta }{w}^{NG}\le 0$.

Proposition 2 indicates that in the supply chain only with a national brand, the government subsidy can improve the NB’s green degree, retail price, and demand, as well as the supply chain members’ profits. However, the impact on the national brand’s wholesale price depends on the magnitude of the subsidy. If the subsidy rate is low, it is intuitive that the manufacturer will prefer to set a higher wholesale price to compensate for the more expensive cost of the green product. However, if the subsidy rate exceeds a certain threshold, it will be attractive for the manufacturer to promote sales of the national brand by providing the retailer with a preferential wholesale price. This is because the larger demand will provide the manufacturer with more government subsidies.

5.1.2. Both National Brand and Store Brand

Then, we explore how the government subsidy affects the optimal solutions of the supply chain with both national and store brands. Due to the analytical challenge, we have to resort to numerical studies to illustrate the behavior of the models and investigate the effects of subsidy. Referring to the previous literature of Basiri and Heydari [35], the parameters are set as follows: $k=0.25$ or $k=0.75$ which represents the low substitute or high substitute store brand, and $b=2/3$, $h=2$. Since $h>\frac{\left(1+k+bk\right){\left(\lambda +s\right)}^2}{2}$ and $s<\lambda$, we assume that $\lambda \in \left(0,1\right)$ and $s\in \left(0,\min \{\lambda ,\sqrt{\frac{2h}{1+k+bk}}-\lambda \}\right)$. Therefore, all of the values of the parameters can ensure positive solutions. The green degrees, prices, and demands of both brands, as well as the supply chain members’ profits, vary with different sets of values of $\lambda$ and $s$, which are shown in Figure 2, Figure 3, Figure 4 and Figure 5.

Observation 1. 

Figure 2 shows that while the store brand’s green degree always rises after the implementation of a government subsidy, the green degree of the national brand increases only when the consumers’ green preference  $\lambda$  and the subsidy rate $s$  are not too large. Next, we focus on exploring the effects of subsidy rate on both brands’ green degrees in case of BS. As $s$  increasing, the store brand’s green degree always increases while the green degree of the national brand depends on the value of $\lambda$. If $\lambda$  is relatively small, then it increases with $s$; otherwise, it first increases and then declines with $s$. Additionally, as the substitution between two brands becomes more intensive, the green degree of the national brand is more likely to decline due to the provided government subsidy. Similar phenomena can be observed in Figure 3 on the demands of both brands.

Observation 2. 

Figure 4 shows the impact of the government subsidy on the prices of both brands. The wholesale price of the national brand lowers due to the government subsidy implementation and declines with the subsidy rate $s$. However, the impacts of the government subsidy on the retail prices of both brands change with the values of the key parameters. For the national brand’s retail price, the value in the case of BS will incline first and then decline with $s$ if $\lambda$ and $k$ are not too large; otherwise, it will keep declining. As a result, the national brand’s retail price is larger than the value in the case of BN only when the $\lambda$ and $k$ are not too high and the $s$ is low enough. On the other hand, the store brand’s retail price in the case of BS will incline first and then decline with $s$ if the $\lambda$ is medium; otherwise, it will keep increasing (declining) if the $\lambda$ is high (low). As a result, the store brand’s retail price will be larger than the value in the case of BN if $\lambda$ and $s$ are relatively low. Interestingly, the thresholds of $\lambda$ and $s$ will not be highly affected by the value of $k$.

Observation 3. 

Figure 5 shows that the retailer, as well as the environmental benefit, always benefits from the government subsidy and increase with $s$, while the profit of the manufacturer mostly declines with $s$. More specifically, faced with the store brand introduction, the manufacturer can benefit from the government subsidy only when the values of $\lambda$, $k$, and $s$  are low enough.

Corollary 1. 

According to the above observations, in most cases, the governmental subsidy can improve the green degrees and demands of both products and cut down their prices. However, when the consumers’ green preference, substitution between both brands or the subsidy rate is high enough, it is very attractive for the retailer to improve the green degree of their green store brand. In some cases, the store brand has a higher retail price than the national brand and is positioned as a premium product. Faced with a severe threat of store brand, the manufacturer has to lower his product’s green degree to obtain cost and price competition advantages. In addition, the government subsidy is always favorable to the retailer and the same applies to the manufacturer only when the values of $\lambda$, $k$, and $s$  are low.

5.2. The Impacts of the Green Store Brand

5.2.1. Impacts on Equilibrium Results

To explore how the green store brand introduction affects the supply chain performance, we compare the equilibrium results of the cases with and without green store brand and obtain the following proposition.

Proposition 3. 

Compared the equilibrium results in case BG with those in case NG, where $G=\left\{N,S\right\}$, the following holds: $g_n^{BG}<g_n^{NG}$; $w^{BG}<w^{NG}$; $p_n^{BG}<p_n^{NG}$; $D_n^{BG}<D_n^{NG}$; ${\pi }_r^{BG}>{\pi }_r^{NG}$; ${\pi }_m^{BG}<{\pi }_m^{NG}$; $E{B}^{BG}>E{B}^{NG}$.

Proposition 3 implies that, regardless of whether the government subsidy is provided or not, the green store brand introduction lowers the green degree and demand of the national brand, as well as the wholesale and retail prices. This is because the green store brand cannibalizes the market share of the national brand, and the competition between both brands impels the manufacturer to manage to obtain a competitive price for retaining the consumers as much as possible. As a result, the manufacturer has to cut down the green degree and wholesale price of the national brand. Even so, the manufacturer cannot hold back the descending demand of national brand and his profit. On the other hand, the sale of the green store brand brings the extra retailer’s profit and environmental benefit, which recoups the loss from the reduction in the national brand’s demand.

Therefore, no matter whether the government provides the subsidy or not, the retailer will always introduce their green store brand as long as the retailer has the relevant ability and green technology.

5.2.2. Impacts on the Effects of Government Subsidy

From the above analysis, we can obtain that the applied government subsidy will change the equilibrium results of the national brand’s prices and greenness, as well as the profits of supply chain members. Next, we explore the issue of how the store brand introduction affects the effects of government subsidy on supply chain performance through differential analysis. We conduct the numerical studies (see Figure 6, Figure 7, Figure 8 and Figure 9) under the same parameter settings in Section 5.1.2. To simplify, we omit the pictures regarding the national brand’s retailer price and demand and the environmental benefit. However, the impacts of store brand introduction on the effects of government subsidy on all variables are summarized in Proposition 4.

From Figure 6, Figure 7, Figure 8 and Figure 9, we observe that the introduction of store brand shrinks the improvement magnitude of NB’s green degree and expands the reduction magnitude of the NB’s wholesale price and the increase magnitude of the retailer’s profit. In addition, we also find in Figure 6 that the national brand’s green degree always increases due to the government subsidy before the introduction of the store brand but may decrease after the introduction. A similar phenomenon can also be observed in Figure 9. That is, the manufacturer always benefits from the government subsidy before the introduction of the store brand but suffers after the introduction. Therefore, the green store brand introduction also has an inverse effect on the influence of the government subsidy on some variables. Furthermore, the increasing consumers’ green preference strengthens the above shrinkage, expansion, or inverse effect, while the value of the cross-elasticity coefficient has no significant impact on those effects. Based on the above observations, we obtain the following proposition.

Proposition 4. 

The introduction of the green store brand has,

(1) 

a shrinkage effect or even inverse effect on the impacts of government subsidy on the green degree, retail price and demand of the national brand and the profit of the manufacturer;

(2) 

a expansion effect or inverse effect on the impacts of government subsidy on the wholesale price of the national brand;

(3) 

an expansion effect on the impacts of government subsidies on the retailer’s profit and environmental benefit;

Additionally, the above effects are reinforced by the increasing consumers’ green preference, subsidy rate, and the substituent of store brand for national brand.

Proposition 4 implies that the green store brand introduction significantly affects the impacts of the government subsidy on supply chain performance. The introduction further expands the retailer’s extra profit and environmental benefit improvement from government subsidy while it makes the government subsidy less beneficial or even harmful to the manufacturer. More specifically, before the retailer introduces the green store brand, the manufacturer always benefits from the government subsidy, and the benefits are in line with the increasing government subsidy rate. However, it is not the same case once the green store brand is introduced. The government subsidy implementation only makes a slight increase, or even a reduction, in the national brand’s green degree and demand and the manufacturer’s profit. Furthermore, these adverse effects are more and more remarkable with the increase in subsidy rate, product substitute, or consumers’ green preference. These unexpected results result from the mechanism that the government subsidy and consumers’ green preference stimulate the retailer to adopt a greener store brand which means a more severe threat to the national brand. Expecting this tendency, the dominant manufacturer has to lower his national brand’s green degree to obtain a lower production cost and then take advantage of the price to prevent consumers from fleeing to the store brand.

6. Conclusions

Previous studies primarily focused on the green supply chain, only observing the national brand(s). However, with the rising concerns for sustainable development and environmental protection, retailers have also introduced their own green store brands. In addition, the government may provide a subsidy to the producer of the green product to encourage the improvement of the product’s greenness. Motivated by the issues of how the green national brand and green store brand compete with each other in the pricing dimension and greenness dimension and how the store brand introduction influences the effect of government subsidy on supply chain performance, this paper constructs four manufacture-led Stackelberg game models according to the strategy combinations of the retailer whether to introduce a green store brand and the government whether to provide a subsidy to obtain the optimal decision via backward induction.

According to the comparative analysis and numerical simulation, both the green store brand introduction and the government subsidy implementation could improve the retailer’s profit and environmental benefit. On the other hand, the manufacturer may benefit or suffer from the government subsidy, while they always sustain a loss from the green store brand introduction. Moreover, by the comparative analysis and numerical simulation, we also find that the introduction of the green store brand does significantly affect the influences of government subsidy on supply chain performance. More specifically, before the retailer introduces their green store brand, the national brand’s green degree and demand, as well as the manufacturer’s profit, always become larger due to the government subsidy implementation. However, the changes in the value of these variables turn to a minor increase or even a reduction after the introduction of the green store brand. On the contrary, introducing a green store brand will expand the positive effects of the government subsidy on the retailer’s profit and environmental benefits. In addition, with the increasing consumers’ green preference, product substitute, and subsidy rate, the green degree of the store brand, the retailer’s profit, and environmental benefits increase while the green degree of the national brand and manufacturer’s profit decline. These conclusions can be explained by the mechanism that the government subsidy and consumers’ green preference stimulate the retailer to adopt a greener store brand, which leads to the reaction of the dominant manufacturer to lower his national brand’s green degree to obtain a lower production cost and then take advantage of price to prevent the consumers from fleeing to store brands.

Our results offer some meaningful managerial insights to the supply chain members as well as the government.

Firstly, the retailer should actively promote the introduction of the green store brand regardless of the government subsidy. In addition, the retailer should try their best to improve the consumers’ green preference, which will bring them more profit. Furthermore, if the consumers’ green preference and the subsidy rate are relatively high, the retailer could set a higher retail price for their store brand than the national brand, implying that the retailer should position their green store brand as a premium product.

Secondly, the manufacturer can benefit from the government subsidy before the green store brand is introduced. However, this may only sometimes hold after the introduction. In the latter situation, the manufacturer should pay more attention to improving the exclusivity of national brand, which can alleviate the threat from green store brand.

Finally, the subsidy may not be the unique measure for the government to improve the products’ green degree and total environmental benefit. The government could also raise consumers’ green preference by conducting public propaganda or encouraging retailers to promote the store brand development, which can transfer the driving force for producing green products from external financial support to the internal market operation.

There are some limitations of our research that can be extended in the following ways. First, we assume the linear demand functions of products widely used in marketing, and it will be worth investigating these issues using other demand functions. Second, the manufacturer, in reality, may set up a direct channel to sell their products. Therefore, one can extend our work by considering a dual-channel supply chain. Third, concerning the common fact that there are multiple manufacturers and multiple retailers in a particular supply chain system, it will be meaningful to explore the competition between multiple products in a supply chain with multiple members or the competition between supply chains. Fourth, we assume the information is perfect and symmetrical for the manufacturer and retailer; the main issues studied in this paper can be explored under an incomplete information setting. Finally, it is important to validate our results through an empirical study with real data.