# Optimal Decisions for Carbon Emission Reduction through Technological Innovation in a Hybrid-Channel Supply Chain with Consumers’ Channel Preferences

Authors: Chong Xin; Yunzhu Zhou; Xiaochen Zhu; Lin Li; Xin Chen
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

This paper integrates carbon emission reduction via technological innovation with consumer channel preferences in both single- and dual-channel supply chains selling low-carbon products. Linear demand functions which simultaneously reflect the consumers’ channel preferences and low-carbon sensitivity are developed by considering the consumers’ segmentation. On this basis, we present two Stackelberg game models: one for each of the single- and dual-channel supply chains. In the first, the manufacturer sells low-carbon products through a traditional retailer who has a physical store, while in the second the manufacturer opens an online direct channel to compete with the traditional retailer. For the two models developed, the optimal pricing decisions, carbon emission reduction level, and profits are derived and discussed. Numerical examples are given to verify the effectiveness and practicality of the proposed models and solutions. The results show that supply chain members’ profits are affected by system parameters such as the carbon price, consumers’ low-carbon sensitivity, channel preference, etc. Furthermore, although the aforementioned parameters stimulate the manufacturer to reduce carbon emission, this does not always benefit the retailer. Comparison of the two models indicates that dual-channel selling is only the better choice for both the manufacturer and the retailer under certain conditions.


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## Body

## 1. Introduction

In recent years, increasing greenhouse gas emissions have had a disastrous impact on both society and the environment. From the Kyoto Protocol adopted in 1997 to the Copenhagen climate conference held in 2010, global climate change—which is mainly caused by carbon emissions from human society and economic activities—has been paid increasing attention by the international community [1, 2]. Meanwhile, with the dramatic growth of energy consumption and the ongoing development in the concept of sustainable development, an increasing number of firms in many countries have taken measures to reduce carbon emissions with the aim of achieving low-carbon economies. For example, Wal-Mart focuses on the product category with the highest carbon footprint and works with suppliers to reduce the greenhouse gas emissions in the product life cycle, including raw material procurement, manufacturing, transportation, and product use or disposal [3]. Similarly, Huawei is committed to promoting the ICT (Information and Communication Technology) industry, enabling other industries to save energy and reduce emissions, and providing society with smart energy and clean energy solutions. Huawei also focuses on helping customers to save energy and reduce emissions, promoting the construction of a low-carbon society in order to contribute to building a green world [4]. In order to encourage the building of a low-carbon supply chain management standard system and the creation of pioneer enterprises, governments have also introduced a series of subsidy policies to guide enterprises regarding environmental protection and resource conservation throughout the entire production process.As effective choices to curb carbon emissions and mitigate greenhouse effects, carbon cap-and-trade [5, 6], carbon tax [7, 8], and carbon labelling [9] have all been adopted by both governments and firms. In particular, the cap-and-trade regulation has proven to be an effective way to achieve emission reduction goals [10]. In the cap-and-trade system,cap means that the government limits the total amount of carbon emissions quotas, andtrade means that firms can sell/buy carbon quotas in carbon trading marketing if they have a surplus/lack of quotas [2]. Therefore, it is necessary for firms to make a tradeoff between spending in carbon trade marketing and reducing carbon emissions, since the cap-and-trade regulation also provides an important financial opportunity for those firms who commit to reduce their carbon emissions [1]. Zakeri et al. stated that carbon cap-and-trade provides a flexible market mechanism for carbon emissions control that seems to lead to better environmental and economic performance of the supply chain [11]. Specifically, firms can obtain additional benefits by selling their remaining carbon permits if actual emissions are less than a given carbon cap. Otherwise, firms need to purchase carbon emission permits from the carbon trading market, which will increase their operating costs [12].From the consumer perspective, the global green ecological revolution is increasing low-carbon awareness [13], which means that the market for low-carbon products is also growing. It has been shown that consumers have obvious preferences for low-carbon products and are willing to pay more for those products [14]. The stronger the preference is, the more consumers are willing to pay more for environmentally friendly products, which stimulates manufacturers to proactively engage in low-carbon production. Moreover, the rapid rise of the Internet economy has changed peoples’ traditional shopping methods. E-commerce provides consumers with another way to get products and helps manufacturers to attract different types of consumers and increase their stickiness [15]. Recognizing the inevitable trend of the online retail channel, manufacturers in the apparel, electronics, and general merchandise sectors have chosen dual-channel models for product sales [16]. However, consumers’ channel preference may influence the options available to them. Both consumers’ channel preference and low-carbon preference prompted us to explore integrated models for low-carbon production and operation in hybrid-channel supply chains. It would be interesting to find a balance wherein the manufacturer can achieve the goal of reducing carbon emissions without damaging their own or the retailer’s profits. This could also encourage the manufacturer to increase the carbon emission reduction rate and then increase the overall profitability of the supply chain, which is also likely to benefit consumers [17].This paper aims to analyze the impact of carbon emissions reduction through technological innovation on supply chain members’ optimal decisions and profits, by considering two different low-carbon supply chain systems. One is the traditional single-channel supply chain, which sells low-carbon products through a retailer with a physical store; the other is the dual-channel supply chain where low-carbon products are sold through both physical and online channels. Specifically, we aimed to answer the following research questions:(1) After considering consumers’ channel preferences and low-carbon sensitivity, how should the manufacturer and the retailer make their optimal decisions? (2) What is the impact of carbon cap-and-trade regulations on the manufacturer’s carbon emission reduction level and supply chain members’ optimal pricing decisions and profits? (3) Under a low-carbon environment, does opening an online direct channel benefit the manufacturer and/or the retailer despite leading to competition between them?To address these questions, linear demand functions which simultaneously reflect the consumers’ channel preferences and low-carbon sensitivity are developed, and then two Stackelberg game modes are presented for the single- and dual-channel supply chains consisting of one manufacturer and one retailer, respectively. Without loss of generality, we assume that the manufacturer is the leader and the retailer is the follower for each model. In the traditional supply chain, the manufacturer decides the wholesale price of the low-carbon products sold to the retailer and the carbon emission reduction level, and then the retailer decides on the retail price. In the dual-channel supply chain, the manufacturer decides the online direct price of the low-carbon products sold to consumers through their online channel, in addition to setting the wholesale price and the carbon emission reduction level, while the retailer decides the retail price of the products. All of the optimal decisions are derived by backward deduction.The rest of the paper is organized as follows: we briefly review the related literature in Section2. In Section 3, we introduce notations and assumptions and develop models. Furthermore, the equilibrium solutions for each model are presented. In Section 4, numerical examples are given to illustrate the proposed models and solutions. In the last section, we summarize and conclude the paper and present the future research directions.


## 2. Literature Review

In recent years, the global climate change problem caused by carbon emissions has become an important environmental issue and has severely affected sustainable development. Managers are facing pressure from both the government and the public. The public requires managers to take more social responsibility, especially for environmental protection [1]. Therefore, low-carbon policies have received a great deal of attention from academia, industry, and government sectors. The attraction of a sustainable economy in business practices encourages both academia and industry to focus on creating a sustainable supply chain. The intense pressure of global warming and the shortage of natural resources highlight the importance of low-carbon supply chain management, which has driven many studies to focus on firms’ operational decisions under low-carbon policies, including pricing, production, inventory, and carbon emissions reduction. This paper focuses on the reduction of supply chain carbon emissions and pricing decisions affected by consumers’ channel preferences and low-carbon sensitivity. To highlight the contributions of this paper, we review only literature regarding carbon emissions reduction in traditional single- and dual-channel supply chains.In the field of carbon emissions reduction, many scholars are concerned with joint emissions reduction strategies and contract coordination. Swami and Shah studied the coordination problem of a manufacturer and a retailer in a green supply chain and found that the ratio of the optimal greening efforts put in by the manufacturer and retailer was equal to the ratio of their green sensitivity ratios and greening cost ratios, while profits and efforts were higher in an integrated channel as compared to a decentralized channel [18]. They proposed a two-part tariff contract to produce channel coordination. Chen et al. [19] established cooperation decision models for a mixed carbon policy of carbon trading-carbon tax in a two-stage supply chain and investigated the influence of mixed carbon policy with the constraint of reduction targets on supply chain decisions. They modelled a cost-sharing contract and analyzed its impact on a green supply chain. Zhou et al. [13] analyzed how the co-op advertising contract and the co-op advertising and emissions reduction cost sharing contracts impact the low-carbon supply chain’s optimal decision-making, taking into account the low-carbon supply chain channels of a manufacturer and a retailer. Wang et al. [17] developed a game model and found that, with cost-sharing and wholesale price premium contracts, retailers can achieve the goal of reducing carbon emissions together with manufacturers, which can encourage manufacturers to increase carbon emission reduction rates and improve the profits of the supply chain. Xu et al. [20] also analyzed the decision behavior and coordination mechanisms for a two-echelon sustainable supply chain under a cap-and-trade regulation. By analyzing the conditions for a win–win outcome, they proved that only a two-part tariff contract can lead to perfect coordination. Recently, Bai et al. [21] integrated carbon emission reduction into the MTO supply chain and tested the performance of the supply chain by comparing its profit with carbon emissions under centralized and decentralized scenarios. The results showed that the profit of the supply chain in the decentralized scenario could be improved and carbon emissions could be reduced. Furthermore, they proposed a revenue and investment sharing contract to coordinate the supply chain.With the development of the low-carbon economy, the environmental awareness of consumers plays an important role in the management of low-carbon supply chains—the stronger the consumers’ environmental preference is, the more willing they are to pay higher prices for a low-carbon product [22]. It has been proven that consumers have a preference for low carbon dioxide emissions in their choice of a daily shopping method [23]. When investigating the impact of consumers’ environmental consciousness on supply chain operation, Simpson et al. [24] discussed the moderating impact of relationship conditions existing between a customer and their suppliers on the uptake and effectiveness of the customer’s environmental performance requirements. Yalabik and Fairchild [25] incorporated emission-sensitive demand, or consumer environmental awareness, in their own initiative, and found that competition over environmentally sensitive customers can improve the effectiveness of environmental pressures. Liu et al. [26] used two-stage Stackelberg game models to investigate the dynamics among the supply chain players and found that as consumers’ environmental awareness increases, retailers and manufacturers with superior eco-friendly operations will benefit. Nouira et al. [27] found that manufacturers must focus on the environmental impacts of manufacturing activities and integrate the environmental performance of finished products. Zhao et al. [28] used system dynamics to examine consumer responses to carbon label products and found that public awareness, education levels, key premiums, and perceived effectiveness were key factors that affect customer purchasing behavior.Zhang et al. [29] studied the impact of consumer environmental awareness on order quantities and channel coordination within a one-manufacturer and one-retailer supply chain, based on the multiproduct newsvendor model. Du et al. [30] studied the impact of consumers’ low-carbon preferences in the emission-concerned supply chain and found that the decision-maker of the supply chain will choose different emission reduction strategies for different cases. Du et al. [31] studied the impacts of carbon footprint and low-carbon preferences on the production decisions of emission-dependent firms in the cap-and-trade system. Their research indicated that consumers’ low-carbon preferences influence their purchase decisions, which in turn affects business decision-making. Du et al. [32] further extended their previous works by proposing a carbon-related price-discount sharing-like scheme to coordinate the decentralized supply chain and discussed the possibility of Pareto improvement. Incorporating the competition between the two rival manufacturers in the demand function, Chen et al. [33] studied optimal pricing and carbon reduction policies. By considering consumer environmental awareness, Cheng et al. [9] built a game theoretic model of a supply chain with one manufacturer and one retailer to investigate the manufacturer’s and retailer’s pricing and investment decisions for products with different initial carbon footprints.Note that all of the above studies are based on the traditional single-channel supply chain. With the development of e-commerce, more and more researchers are paying attention to dual-channel issues. Chiang et al. [34] found that it is beneficial for a supplier to set up a direct channel to compete with its retailer. Khouja et al. [35] analyzed the channel selection and price setting of a manufacturer who has several distribution options. Khouja and Wang [36] explored the impact of a digital channel for goods on the profitability and behavior of players in the supply chain. They found that the dual distribution channel is most profitable. Jay and Brown [37] performed a comprehensive comparison of carbon emissions resulting from conventional shopping versus e-commerce-based online retailing. Interestingly, Loon et al. [38] found that online retailing can lower the environmental impact of shopping under specific circumstances.However, research on carbon emission reduction problems in a dual-channel supply chain under a low-carbon environment is limited. Carrillo et al. [39] formulated a dual-channel model for a retailer that can access online and traditional market outlets in order to analyze the impact of customer environmental sensitivity. Zhou et al. [13] introduced a coefficient for a consumer’s low-carbon preference in the demand function and investigated how the consumer’s low-carbon preference affects channel decisions. Li et al. [40] discussed the pricing and greening strategies in a dual-channel supply chain. Considering consumers’ low-carbon preferences, Ji et al. [1] introduced cap-and-trade regulation into dual-channel supply chain management and developed models to investigate supply chain members’ pricing and emission reduction decisions under a low-carbon environment. Similarly, Zhou et al. [41] analyzed the optimal equilibrium strategies in centralized and decentralized dual-channel supply chains in a low-carbon environment. Wang et al. [2] discussed the impacts of cap-and-trade regulation and consumers’ low-carbon preferences on carbon emission reduction in the framework of a dual-channel supply chain.Similar to the studies of Ji et al. [1] and Wang et al. [2] that discuss the carbon emission reduction and pricing decisions in a dual-channel supply chain, this paper investigates the impact of carbon emission reduction through technological innovation on supply chain members’ decisions and corresponding profits. The main contributions of this paper are as follows: first, we modeled consumers’ channel preferences through the demand discount coefficient due to channel mismatch, which is different from the works of Ji et al. [1] and Wang et al. [2]. Specifically, we divided consumers into two segments according to their channel preference: (1) the segment that prefers to buy in a physical store and (2) the segment that prefers to buy online. Secondly, linear demand functions, which simultaneously reflect consumers’ channel preferences and low-carbon sensitivity, were developed for both the traditional single- and dual-channel supply chains. Thirdly, we constructed two Stackelberg game models, one for each of the single- and dual-channel supply chains, and the optimal carbon emission reduction and pricing decisions were derived. Furthermore, we also discuss whether opening an online direct channel benefits the manufacturer and/or the retailer.


## 3. Model Development

In this paper, we consider a supply chain consisting of a traditional retailer who has a physical store and a manufacturer who produces low-carbon products. We focus on carbon emission reduction through technological innovation and its impact on supply chain decisions, taking both consumers’ channel preferences and low-carbon sensitivity into account. Two different supply chain systems are surveyed: (I) the manufacturer sells low-carbon products through the traditional retail channel operated by the manufacturer and retailer, which is referred to as the traditional single-channel supply chain (see Figure1(a)); and (II) the manufacturer opens an online direct channel to compete with the retailer through a direct pricing strategy, which is referred to as the dual-channel supply chain (see Figure 1(b)). We model the problems as Stackelberg games in which the manufacturer is the leader of the supply chain in pricing, while the retailer is the follower.Figure 1
Traditional single- and dual-channel supply chains.
(a) (b)We developed linear demand functions in the spirit of Cai et al. [42], Khouja et al. [35], Wang et al. [17], and Wang et al. [2]. To reflect channel preferences, the consumers were divided into two segments: those who prefer shopping in a physical store (denoted as PS consumers) and those who prefer shopping online (denoted as PO consumers). Consumers’ purchase decisions are simultaneously influenced by channel preference, product price, and the low-carbon preference, which can be observed by the demand functions proposed in Sections 3.1 and 3.2, respectively. Before presenting the models, the main symbols and notation used in this paper are summarized in Table 1.Table 1
Symbols and notations.
Symbols and notation Explanation αs Potential number of consumers who prefer shopping in a physical store α o Potential number of consumers who prefer shopping online θ s Discount coefficient of the retailer’s offline demand due to the consumers’ channel mismatch θ o Discount coefficient of the online demand due to the consumers’ channel mismatch β Consumer price sensitivity coefficient η Consumer price gap sensitivity coefficient ρ r Low-carbon sensitivity coefficient in the retail channel ρ m Low-carbon sensitivity coefficient in the online direct channel p r Retail price p m Direct selling price of the manufacturer c r Unit operating cost of the retailer c s Unit production cost of the manufacturer c Carbon permit price in the carbon market K Initial carbon permits allocation e 0 Unit carbon emission level before carbon emission reduction e Carbon emission reduction level or unit amount of emission reduction k Cost coefficient of carbon emission reduction


### 3.1. Model of the Traditional Single-Channel Supply Chain

Before the manufacturer opens the online direct channel, the demand of the traditional supply chain can be described as(1)DrI=αs1-βpr+αoθo1-βpr+ρre,where the first part on the right of (1) measures the demand from consumers who prefer shopping in a physical store, while the second part reflects the demand from those who prefer shopping online. Note that the parameter θo∈(0,1) is introduced to reflect the degree of consumers’ acceptance of their less-preferred channels and to model the extent to which consumers’ purchase decisions are influenced by their channel preferences. The last term measures the demand increment due to the carbon emission reduction (i.e., ρr measures consumers’ low-carbon preference). The higher the value of ρr is, the more sensitive the consumers are. Without loss of generality, we assume ρr<(αs+α0θ0)β, which means that the supply chain’s price effect is larger than the low-carbon degree effect.Based on the demand function (1), the retailer’s profit is given below:(2)πrIpr=pr-w-crDrI=pr-w-crαs+αoθo-αs+αoθoβρr+ρre.A manufacturer who needs to make low-carbon products will employ technological innovation to achieve carbon emission reduction, which incurs an investment cost. According to Ji et al. [1] and Wang et al. [2], we define the investment cost as a quadratic function of the carbon emission reduction level (i.e., C(e)=ke2/2). Unsurprisingly, a higher degree of emission reduction raises the difficulty of reducing carbon emissions, such that a higher cost is incurred. Moreover, under the carbon cap-and-trade regulation, the manufacturer is initially allocated free carbon permits by the government. However, the manufacturer can buy permits from the carbon trading market with the permit price c when their total carbon emissions exceed the initial permitted allocation K (K≥0). On the contrary, the manufacturer can also earn money by selling their excessive permits in the carbon market. Let e0 represent the carbon emission level per unit product before implementing carbon emission reduction, then the function E(e;DrI)=c[(e0-e)DrI-K] measures the purchase cost or earnings from the carbon trade market. Thus, the manufacturer’s profit can be described as follows:(3)πmIw,e=w-csDrI-Ce-Ee;DrI=w-cs-ce0-eαs1-βρr+αoθo1-βρr+ρre-ke22+cK.


### 3.2. Model of the Dual-Channel Supply Chain

In the dual-channel supply chain, the manufacturer opens an online direct channel to compete with the traditional retailer. In this case, the consumers can buy the product through the traditional channel or online, depending on their channel preferences. After the manufacturer enters the online direct channel, the retailer’s demand is(4)DrII=αs1-βpr-ηpr-pm+αoθo1-βpr-ηpr-pm+ρre,where the gap (pr-pm) in (4) reflects the competition between the manufacturer and the retailer. The larger the value of the retailer’s price pr, the lower the retailer’s demand, and vice versa. The term 1-βpr-η(pr-pm) is the percentage of PS consumers who buy the product from the retail channel, and θo(1-βpr)-η(pr-pm) is the percentage of PO consumers who buy the product from the retail channel, respectively. We assume ρr<(αs+αoθo)β+(αs+αo)η to make sure that the retailer’s price effect is larger than the low-carbon degree effect.The manufacturer’s online demand is(5)DmII=αsθs1-βpm-ηpm-pr+αo1-βpm-ηpm-pr+ρme,where the term θs(1-βpm)-η(pm-pr) is the percentage of PS consumers who buy the product from the online direct channel, and (1-βpm)-η(pm-pr) is the percentage of PO consumers who buy the product from the online direct channel. Similarly, we assume ρm<(αo+αsθs)β+(αs+αo)η to ensure that the manufacturer’s price effect is larger than the low-carbon degree effect. Without loss of generality, we also suppose ρm<ρr, which implies that the level of improvement in carbon emissions has a greater influence on the retail channel’s demand than on the demand of the online direct channel because the low-carbon products purchased from the online channel cannot be examined physically [2].Based on demand functions (4) and (5), and following the same analysis as in Section 3.1, the manufacturer’s profit can be described as follows:(6)πmIIpm,w,e=w-csDrII+pm-csDmII-Ce-Ee;DrII,DmII=w-cs-ce0-eDrII+pm-cs-ce0-eDmII-ke22+cK.As the leader, the manufacturer first chooses the optimal online pricepm, wholesale price w, and carbon emission level e to maximize their profit function (6), based on which the retailer (as the follower) seeks to maximize its profit function defined below by choosing the optimal retail price:(7)πrIIpr=pr-w-crDrII=pr-w-crαs1-βρr-ηpr-pm+αoθo1-βρr-ηpr-pm+ρre.


## 3.1. Model of the Traditional Single-Channel Supply Chain

Before the manufacturer opens the online direct channel, the demand of the traditional supply chain can be described as(1)DrI=αs1-βpr+αoθo1-βpr+ρre,where the first part on the right of (1) measures the demand from consumers who prefer shopping in a physical store, while the second part reflects the demand from those who prefer shopping online. Note that the parameter θo∈(0,1) is introduced to reflect the degree of consumers’ acceptance of their less-preferred channels and to model the extent to which consumers’ purchase decisions are influenced by their channel preferences. The last term measures the demand increment due to the carbon emission reduction (i.e., ρr measures consumers’ low-carbon preference). The higher the value of ρr is, the more sensitive the consumers are. Without loss of generality, we assume ρr<(αs+α0θ0)β, which means that the supply chain’s price effect is larger than the low-carbon degree effect.Based on the demand function (1), the retailer’s profit is given below:(2)πrIpr=pr-w-crDrI=pr-w-crαs+αoθo-αs+αoθoβρr+ρre.A manufacturer who needs to make low-carbon products will employ technological innovation to achieve carbon emission reduction, which incurs an investment cost. According to Ji et al. [1] and Wang et al. [2], we define the investment cost as a quadratic function of the carbon emission reduction level (i.e., C(e)=ke2/2). Unsurprisingly, a higher degree of emission reduction raises the difficulty of reducing carbon emissions, such that a higher cost is incurred. Moreover, under the carbon cap-and-trade regulation, the manufacturer is initially allocated free carbon permits by the government. However, the manufacturer can buy permits from the carbon trading market with the permit price c when their total carbon emissions exceed the initial permitted allocation K (K≥0). On the contrary, the manufacturer can also earn money by selling their excessive permits in the carbon market. Let e0 represent the carbon emission level per unit product before implementing carbon emission reduction, then the function E(e;DrI)=c[(e0-e)DrI-K] measures the purchase cost or earnings from the carbon trade market. Thus, the manufacturer’s profit can be described as follows:(3)πmIw,e=w-csDrI-Ce-Ee;DrI=w-cs-ce0-eαs1-βρr+αoθo1-βρr+ρre-ke22+cK.


## 3.2. Model of the Dual-Channel Supply Chain

In the dual-channel supply chain, the manufacturer opens an online direct channel to compete with the traditional retailer. In this case, the consumers can buy the product through the traditional channel or online, depending on their channel preferences. After the manufacturer enters the online direct channel, the retailer’s demand is(4)DrII=αs1-βpr-ηpr-pm+αoθo1-βpr-ηpr-pm+ρre,where the gap (pr-pm) in (4) reflects the competition between the manufacturer and the retailer. The larger the value of the retailer’s price pr, the lower the retailer’s demand, and vice versa. The term 1-βpr-η(pr-pm) is the percentage of PS consumers who buy the product from the retail channel, and θo(1-βpr)-η(pr-pm) is the percentage of PO consumers who buy the product from the retail channel, respectively. We assume ρr<(αs+αoθo)β+(αs+αo)η to make sure that the retailer’s price effect is larger than the low-carbon degree effect.The manufacturer’s online demand is(5)DmII=αsθs1-βpm-ηpm-pr+αo1-βpm-ηpm-pr+ρme,where the term θs(1-βpm)-η(pm-pr) is the percentage of PS consumers who buy the product from the online direct channel, and (1-βpm)-η(pm-pr) is the percentage of PO consumers who buy the product from the online direct channel. Similarly, we assume ρm<(αo+αsθs)β+(αs+αo)η to ensure that the manufacturer’s price effect is larger than the low-carbon degree effect. Without loss of generality, we also suppose ρm<ρr, which implies that the level of improvement in carbon emissions has a greater influence on the retail channel’s demand than on the demand of the online direct channel because the low-carbon products purchased from the online channel cannot be examined physically [2].Based on demand functions (4) and (5), and following the same analysis as in Section 3.1, the manufacturer’s profit can be described as follows:(6)πmIIpm,w,e=w-csDrII+pm-csDmII-Ce-Ee;DrII,DmII=w-cs-ce0-eDrII+pm-cs-ce0-eDmII-ke22+cK.As the leader, the manufacturer first chooses the optimal online pricepm, wholesale price w, and carbon emission level e to maximize their profit function (6), based on which the retailer (as the follower) seeks to maximize its profit function defined below by choosing the optimal retail price:(7)πrIIpr=pr-w-crDrII=pr-w-crαs1-βρr-ηpr-pm+αoθo1-βρr-ηpr-pm+ρre.


## 4. Solutions and Equilibrium Analysis

In this section, we derive the optimal decisions for both the manufacturer and the retailer and then discuss the equilibrium solutions. For the convenience of description, the models developed in Sections3.1 and 3.2 are referred to as Model I and Model II, respectively.


### 4.1. Solutions for Model I (Traditional Single-Channel Supply Chain)

In a traditional supply chain, the manufacturer and the retailer seek to maximize their profit by optimizing their own decisions. The problem is modeled as a Stackelberg game in which the manufacturer is the leader and the retailer is the follower. Backward induction is used to find the equilibrium of the game. We then have the following proposition to show the analytical results. All the proof can be found in The Appendix.Proposition 1.
For a traditional supply chain, given the manufacturer’s decisions(w,e), the retailer’s optimal retail price response function can be determined as(8)pr∗w,e=αs+αoθo1+w+crβ+ρre2αs+αoθoβ.
It can be found from (8) that the retailer’s price increases with the manufacturer’s wholesale price w and the carbon emission level e, since ∂pr∗/∂w=1/2>0 and ∂pr∗/∂e=ρr/2(αs+αoθo)β>0. Although increasing the wholesale price will inevitably push up the retailer’s price and reduce the demand in return, it can be compensated for by increasing the carbon emission reduction level.Corollary 2.
For a traditional supply chain, given the manufacturer’s decisions (w,e), the retailer’s optimal retail price has the following elasticity with respect to low-carbon sensitivity coefficient in the retail channel, numbers of PS and PO consumers, and discount coefficient of the online demand due to the consumers’ channel mismatch, respectively:(9)∂pr∗∂ρr<0,∂pr∗∂αs<0,∂pr∗∂αo<0,∂pr∗∂θo<0.
Corollary2 can be easily proved by taking the first derivative of pr∗(w,e) with respect to ρr, αs, αo, and θo, respectively. Note that although ρr, αs, αo, and θo have a positive influence on the demand, they affect the retail price in a different way. Interestingly, the retail price increases with ρr, while it decreases with αs, αo, and θo. Based on the retailer’s response function (8), we obtain the manufacturer’s optimal decisions in Proposition 3.Proposition 3.
For a traditional supply chain, if the retailer makes their price decisions according to (8), then the manufacturer’s optimal decisions can be determined as follows:
(i) if[ρr-β(αs+αoθo)c]2≤4β(αs+αoθo)(k-cρr), then the manufacturer’s optimal wholesale price and carbon emission reduction level are(10)w∗=2αs+αoθo1+βcs-cr+ce0k-cρr-αs+αoθo1-βcrc-ρrcs+ce0αs+αoθoβc-ρr4αs+αoθoβk-cρr-αs+αoθoβc-ρr2,(11)e∗=2αs+αoθo1+βcrc-ρrcs+ce0αs+αoθoβ-αs+αoθo1+βcs-cr+ce0αs+αoθoβc-ρr4αs+αoθoβk-cρr-αs+αoθoβc-ρr2;
(ii) if[ρr-β(αs+αoθo)c]2>4β(αs+αoθo)(k-cρr), then the manufacturer’s optimal carbon emission reduction level is(12)e∗=αs+αoθoρr+αs+αoθoβc1-cr+cs+ce0β4kβαs+αoθo-ρr+αs+αoθoβc2,k>ρr+αs+αoθoβc24βαs+αoθoe0,k≤ρr+αs+αoθoβc24βαs+αoθoand the optimal wholesale price is(13)w∗=2αs+αoθo1+βcs-cr+ce0-αs+αoθoβc-ρre∗2αs+αoθoβ.Proposition3 displays two different methods for the manufacturer to determine the wholesale price and carbon emission reduction level, which depends on whether the Hessian matrix of profit function (3) is negative definite or not. Specifically, the condition [ρr-β(αs+αoθo)c]2≤4β(αs+αoθo)(k-cρr) ensures that the manufacturer’s profit function (3) is jointly concave with (w,e). Then, we have the optimal wholesale price and carbon emission reduction level as shown in (10) and (11), respectively. However, when focusing on the case that [ρr-β(αs+αoθo)c]2>4β(αs+αoθo)(k-cρr), we have different decision results. It can be found from (12) that when the carbon emission reduction cost is low, i.e., k≤[ρr+(αs+αoθo)βc]2/4β(αs+αoθo), the manufacturer will choose to reduce its carbon emission completely. When the carbon emission reduction cost is high, i.e., k>[ρr+(αs+αoθo)βc]2/4β(αs+αoθo), the manufacturer’s optimal carbon emission reduction level depends on many factors such as carbon price, low-carbon sensitivity, and so on. In the following, we focus on the case when [ρr-β(αs+αoθo)c]2>4β(αs+αoθo)(k-cρr) and k>[ρr+(αs+αoθo)βc]2/4β(αs+αoθo) since it is more general and complicated.Corollary 4.
If conditions[ρr-β(αs+αoθo)c]2>4β(αs+αoθo)(k-cρr) and k>[ρr+(αs+α0θ0)βc]2/4β(αs+α0θ0) hold, we have ∂e∗/∂ρr>0 and (14)∂e∗∂ρr>0,iffc>0=0,iffc=0<0,otherwisewhere fc=4kαβ(αβ¯+ρre0)+(αβ¯-ρre0)(ρr+αβc)2+8kα2β2e0c,α=αs+αoθo,β¯=1-β(cr+cs).Corollary4 indicates that the higher carbon price can stimulate the manufacturer to emit less carbon only when a certain condition, that is, fc>0 is satisfied. In addition, the higher the low-carbon sensitivity is, the more the manufacturer is willing to implement the carbon emission reduction to gain competition advantage by enhancing its environmental concerns.


### 4.2. Solutions for Model II (Dual-Channel Supply Chain)

In a dual-channel supply chain, the manufacturer who operates the online direct channel competes with the retailer in pricing. As with Model I, backward induction is also used here to derive the equilibrium of the Stackelberg game. Proposition5 illustrates the retailer’s optimal price response function.Proposition 5.
For a dual-channel supply chain, given the manufacturer’s decisions(pm,w,e), the retailer’s optimal price response function is(15)pr∗pm,w,e=αs+αoθo+ηpmαs+αo+ρre+w+crαs+αoθoβ+αs+αoη2αs+αoθoβ+αs+αoθoη.
The proof of Proposition5 is similar to that of Proposition 1, and therefore we omit it here. To simplify the description, we rewrite the demand function (4) as follows:(16)DrII=a-bpr+vpm+ρre,where a=αs+αoθo, b=(αs+αoθo)β+(αs+αo)η, v=αsη+αoη. Similarly, the demand function (5) can be rewritten as(17)DmII=a-+b-pr-v-pm+ρme,where a-=αo+αsθs, b¯=η(αs+α0), v-=(αo+αsθs)β+(αo+αs)η. Thus, (15) can be reformulated as(18)pr∗w,e=a+bcr2b+v2bpm+12w+ρr2be.It can also be found from (18) that the retailer’s optimal price pr∗ increases with the manufacturer’s online price pm, wholesale price w, and carbon emission level e, respectively, since ∂pr∗/∂pm>0, ∂pr∗/∂w>0, and ∂pr∗/∂e>0.Now, consider the following manufacturer’s profit maximization problem by substituting (18) into (6):(19)max⁡πmIIpm,w,e=w-csDrII+pm-csDmII-Ce-Ee;DrII,DmII=w-cs-ce0-eDrII+pm-cs-ce0-eDmII-ke22+cKThe Hessian matrix of the functionπmII(pm,w,e) is (20)H=∂2πmII∂pm2∂2πmII∂pm∂w∂2πmII∂pm∂e∂2πmII∂w∂pm∂2πmII∂w2∂2πmII∂w∂e∂2πmII∂e∂pm∂2πmII∂e∂w∂2πmII∂e2=b-vb-2v-v+b-2vcb-+b+b-ρr+2bpm2b-cv-v+b¯2-bρr-bc+b-c2vcb-+b+b-ρr+2bpm2b-cv-ρr-bc+b-c2cb¯+bρr+2bpmb-k.Clearly, H1=b-v/b-2v-=(-v2-2[aa-β2+a+a-βv)/b<0, H2=-(b-v-2bv-)-(v+b-)2/4=2[aa-β2+(a+a-)βv]>0. However, H3=H may be positive or negative, which indicates that H is not always a negative definite. Therefore, the manufacturer’s optimal decisions cannot be directly derived according to the first-order conditions. Moreover, we assume that the wholesale price cannot be higher than the manufacturer’s online direct price to avoid the retailer buying the product through the online channel. We discuss the manufacturer’s decisions in two cases.Case 1 (H3=H≤0).
In this case, the manufacturer’s profit function is jointly concave with respect to(pm,w,e), then we have Proposition 6.Proposition 6.
For a dual-channel supply chain, if the retailer makes their price decisions according to (15), then the manufacturer’s optimal decisions can be determined as follows:(21)pm∗=γδ^ε--δ-ε^+εδ-γ^-δ^γ-+δγ-ε^-ε-γ^γλ-ε^-λ^ε-+λε-γ^-γ-ε^+ελ^γ--λ-γ^,(22)w∗=λδ-ε^-ε-δ^+ελ-δ^-λ^δ-+δλ^ε--λ-ε^γλ-ε^-λ^ε-+λε-γ^-γ-ε^+ελ^γ--λ-γ^,λ+γδ^ε--δ-ε^+λ^+γ^εδ--δε-+λ-+γ-δε^-εδ^>0,pm∗,otherwise,(23)e∗=γλ^δ--λ-δ^+λγ-δ^-δ-γ^+δλ-γ^-λ^γ-γλ-ε^-λ^ε-+λε-γ^-γ-ε^+ελ^γ--λ-γ^,where(24)γ=b-+v2,λ=b-v-2bv-b,ε=bvc+b-v-2bv-c+b-ρr+2bpm2b,δ=2a-b+b-a+bcr-vb+b--2bv-cs+ce02b,γ-=-b,λ-=v+b-2,ε-=ρr-b-b-c2,δ-=a-bcr+cs+ce0b-b-2,γ^=cb--b+ρr2,λ^=vcb--b+2bcv-v-+b-ρr+2ρmb2b,ε^=cρrb+b-+2bcρmb-k,δ^=2bca+a-+cb--ba+bcr-cs+ce0b+b-ρr+2bρm2b.
Note that the condition(λ+γ)(δ^ε--δ-ε^)+(λ^+γ^)(εδ--δε-)+(λ-+γ-)(δε^-εδ^)>0 in Proposition 6 ensures that the wholesale price cannot be higher than the manufacturer’s online direct price. The condition (λ+γ)(δ^ε--δ-ε^)+(λ^+γ^)(εδ--δε-)+(λ-+γ-)(δε^-εδ^)≤0 implies that the optimal wholesale price is higher than the direct price. In this case, the manufacturer will set the wholesale price equal to the direct price.Case 2 (H3=|H|>0).
In this case, the manufacturer’s profit function is not jointly concave with respect to (pm,w,e). However, we find that, for a given carbon emission reduction level e, the function πmII(pm,w,e) is jointly concave with respect to (pm,w). We then have the following Proposition 7.Proposition 7.
For a dual-channel supply chain, if the retailer makes their price decisions according to (15), then the manufacturer’s optimal carbon emission reduction level is (25)e∗=Ξ2T0+cx1+2T-0+cx2-k,k>2T0+cx1+2T-0+cx2e0,k≤2T0+cx1+2T-0+cx2,where(26)Ξ=-T0+ca+vT¯1-T¯0+ca--v-T¯1-T0+cb-T¯0+cb¯a+bcr2b+vT-12b+T12-T1x1-T¯1x2+x1+x2cs+ce0,x1=ρr+vT-0-bvT-0+ρr2b+T02,x2=ρm-vT-0+bvT-0+ρr2b+T02,T0=ε-λ-ελ-γλ--γ-λ,T1=δ-λ-δλ-γλ--γ-λ,T-0=εγ--γε-γλ--γ-λ,T-1=δγ--γδ¯γλ--γ-λ.The manufacturer’s optimal online direct price and wholesale price can be determined as follows:(27)pm∗=εγ--γε-γλ--γ-λe∗+δγ--γδ-γλ--γ-λ,(28)w∗=ε-λ-ελ-γλ--γ-λe∗+δ-λ-δλ-γλ--γ-λ,e∗>δ-λ+γ-δλ-+γ-ελ-+γ--ε-λ+γpm∗,otherwise.Note that the manufacturer may sell their products either through the traditional physical channel or the dual-channel model, which depends on the profits that can be obtained. In the following, we are interested in when the manufacturer should open an online channel to compete with the traditional retailer, that is, whether the manufacturer will benefit from introducing competition mechanisms. We focus on the comparison between the profits that the supply chain members obtain in a traditional supply chain and those obtained in a dual-channel supply chain, which correspond to Model I and Model II, respectively. To simplify the description, let (prI∗,wI∗,eI∗) represent the optimal decisions in Model I, which yield the optimal profits πrI∗ and πmI∗, respectively. Let (prII∗,pmII∗,wII∗,eII∗) represent the optimal decisions in Model II, which yield the optimal profits πrII∗ and πmII∗, respectively.In general, the larger the difference between the profits that the manufacturer obtains in Model II and in Model I, the more likely the manufacturer is to sell the product through dual channels. The conditionΔπm=πmII∗-πmI∗>0 implies the rationality of opening an online channel. However, the retailer’s profit may be encroached upon, as part of the demand will be shifted to the manufacturer’s online channel. Therefore, the gap between πrII∗ and πrI∗ may be less than zero: Δπr=πrII∗-πrI∗<0. Both Δπm and Δπr depend on the values of the system parameters. Due to the complexities of the developed models and the corresponding decisions, in the next section, we conduct numerical experiments to explore whether opening an online channel will benefit the manufacturer or the retailer.


## 4.1. Solutions for Model I (Traditional Single-Channel Supply Chain)

In a traditional supply chain, the manufacturer and the retailer seek to maximize their profit by optimizing their own decisions. The problem is modeled as a Stackelberg game in which the manufacturer is the leader and the retailer is the follower. Backward induction is used to find the equilibrium of the game. We then have the following proposition to show the analytical results. All the proof can be found in The Appendix.Proposition 1.
For a traditional supply chain, given the manufacturer’s decisions(w,e), the retailer’s optimal retail price response function can be determined as(8)pr∗w,e=αs+αoθo1+w+crβ+ρre2αs+αoθoβ.
It can be found from (8) that the retailer’s price increases with the manufacturer’s wholesale price w and the carbon emission level e, since ∂pr∗/∂w=1/2>0 and ∂pr∗/∂e=ρr/2(αs+αoθo)β>0. Although increasing the wholesale price will inevitably push up the retailer’s price and reduce the demand in return, it can be compensated for by increasing the carbon emission reduction level.Corollary 2.
For a traditional supply chain, given the manufacturer’s decisions (w,e), the retailer’s optimal retail price has the following elasticity with respect to low-carbon sensitivity coefficient in the retail channel, numbers of PS and PO consumers, and discount coefficient of the online demand due to the consumers’ channel mismatch, respectively:(9)∂pr∗∂ρr<0,∂pr∗∂αs<0,∂pr∗∂αo<0,∂pr∗∂θo<0.
Corollary2 can be easily proved by taking the first derivative of pr∗(w,e) with respect to ρr, αs, αo, and θo, respectively. Note that although ρr, αs, αo, and θo have a positive influence on the demand, they affect the retail price in a different way. Interestingly, the retail price increases with ρr, while it decreases with αs, αo, and θo. Based on the retailer’s response function (8), we obtain the manufacturer’s optimal decisions in Proposition 3.Proposition 3.
For a traditional supply chain, if the retailer makes their price decisions according to (8), then the manufacturer’s optimal decisions can be determined as follows:
(i) if[ρr-β(αs+αoθo)c]2≤4β(αs+αoθo)(k-cρr), then the manufacturer’s optimal wholesale price and carbon emission reduction level are(10)w∗=2αs+αoθo1+βcs-cr+ce0k-cρr-αs+αoθo1-βcrc-ρrcs+ce0αs+αoθoβc-ρr4αs+αoθoβk-cρr-αs+αoθoβc-ρr2,(11)e∗=2αs+αoθo1+βcrc-ρrcs+ce0αs+αoθoβ-αs+αoθo1+βcs-cr+ce0αs+αoθoβc-ρr4αs+αoθoβk-cρr-αs+αoθoβc-ρr2;
(ii) if[ρr-β(αs+αoθo)c]2>4β(αs+αoθo)(k-cρr), then the manufacturer’s optimal carbon emission reduction level is(12)e∗=αs+αoθoρr+αs+αoθoβc1-cr+cs+ce0β4kβαs+αoθo-ρr+αs+αoθoβc2,k>ρr+αs+αoθoβc24βαs+αoθoe0,k≤ρr+αs+αoθoβc24βαs+αoθoand the optimal wholesale price is(13)w∗=2αs+αoθo1+βcs-cr+ce0-αs+αoθoβc-ρre∗2αs+αoθoβ.Proposition3 displays two different methods for the manufacturer to determine the wholesale price and carbon emission reduction level, which depends on whether the Hessian matrix of profit function (3) is negative definite or not. Specifically, the condition [ρr-β(αs+αoθo)c]2≤4β(αs+αoθo)(k-cρr) ensures that the manufacturer’s profit function (3) is jointly concave with (w,e). Then, we have the optimal wholesale price and carbon emission reduction level as shown in (10) and (11), respectively. However, when focusing on the case that [ρr-β(αs+αoθo)c]2>4β(αs+αoθo)(k-cρr), we have different decision results. It can be found from (12) that when the carbon emission reduction cost is low, i.e., k≤[ρr+(αs+αoθo)βc]2/4β(αs+αoθo), the manufacturer will choose to reduce its carbon emission completely. When the carbon emission reduction cost is high, i.e., k>[ρr+(αs+αoθo)βc]2/4β(αs+αoθo), the manufacturer’s optimal carbon emission reduction level depends on many factors such as carbon price, low-carbon sensitivity, and so on. In the following, we focus on the case when [ρr-β(αs+αoθo)c]2>4β(αs+αoθo)(k-cρr) and k>[ρr+(αs+αoθo)βc]2/4β(αs+αoθo) since it is more general and complicated.Corollary 4.
If conditions[ρr-β(αs+αoθo)c]2>4β(αs+αoθo)(k-cρr) and k>[ρr+(αs+α0θ0)βc]2/4β(αs+α0θ0) hold, we have ∂e∗/∂ρr>0 and (14)∂e∗∂ρr>0,iffc>0=0,iffc=0<0,otherwisewhere fc=4kαβ(αβ¯+ρre0)+(αβ¯-ρre0)(ρr+αβc)2+8kα2β2e0c,α=αs+αoθo,β¯=1-β(cr+cs).Corollary4 indicates that the higher carbon price can stimulate the manufacturer to emit less carbon only when a certain condition, that is, fc>0 is satisfied. In addition, the higher the low-carbon sensitivity is, the more the manufacturer is willing to implement the carbon emission reduction to gain competition advantage by enhancing its environmental concerns.


## 4.2. Solutions for Model II (Dual-Channel Supply Chain)

In a dual-channel supply chain, the manufacturer who operates the online direct channel competes with the retailer in pricing. As with Model I, backward induction is also used here to derive the equilibrium of the Stackelberg game. Proposition5 illustrates the retailer’s optimal price response function.Proposition 5.
For a dual-channel supply chain, given the manufacturer’s decisions(pm,w,e), the retailer’s optimal price response function is(15)pr∗pm,w,e=αs+αoθo+ηpmαs+αo+ρre+w+crαs+αoθoβ+αs+αoη2αs+αoθoβ+αs+αoθoη.
The proof of Proposition5 is similar to that of Proposition 1, and therefore we omit it here. To simplify the description, we rewrite the demand function (4) as follows:(16)DrII=a-bpr+vpm+ρre,where a=αs+αoθo, b=(αs+αoθo)β+(αs+αo)η, v=αsη+αoη. Similarly, the demand function (5) can be rewritten as(17)DmII=a-+b-pr-v-pm+ρme,where a-=αo+αsθs, b¯=η(αs+α0), v-=(αo+αsθs)β+(αo+αs)η. Thus, (15) can be reformulated as(18)pr∗w,e=a+bcr2b+v2bpm+12w+ρr2be.It can also be found from (18) that the retailer’s optimal price pr∗ increases with the manufacturer’s online price pm, wholesale price w, and carbon emission level e, respectively, since ∂pr∗/∂pm>0, ∂pr∗/∂w>0, and ∂pr∗/∂e>0.Now, consider the following manufacturer’s profit maximization problem by substituting (18) into (6):(19)max⁡πmIIpm,w,e=w-csDrII+pm-csDmII-Ce-Ee;DrII,DmII=w-cs-ce0-eDrII+pm-cs-ce0-eDmII-ke22+cKThe Hessian matrix of the functionπmII(pm,w,e) is (20)H=∂2πmII∂pm2∂2πmII∂pm∂w∂2πmII∂pm∂e∂2πmII∂w∂pm∂2πmII∂w2∂2πmII∂w∂e∂2πmII∂e∂pm∂2πmII∂e∂w∂2πmII∂e2=b-vb-2v-v+b-2vcb-+b+b-ρr+2bpm2b-cv-v+b¯2-bρr-bc+b-c2vcb-+b+b-ρr+2bpm2b-cv-ρr-bc+b-c2cb¯+bρr+2bpmb-k.Clearly, H1=b-v/b-2v-=(-v2-2[aa-β2+a+a-βv)/b<0, H2=-(b-v-2bv-)-(v+b-)2/4=2[aa-β2+(a+a-)βv]>0. However, H3=H may be positive or negative, which indicates that H is not always a negative definite. Therefore, the manufacturer’s optimal decisions cannot be directly derived according to the first-order conditions. Moreover, we assume that the wholesale price cannot be higher than the manufacturer’s online direct price to avoid the retailer buying the product through the online channel. We discuss the manufacturer’s decisions in two cases.Case 1 (H3=H≤0).
In this case, the manufacturer’s profit function is jointly concave with respect to(pm,w,e), then we have Proposition 6.Proposition 6.
For a dual-channel supply chain, if the retailer makes their price decisions according to (15), then the manufacturer’s optimal decisions can be determined as follows:(21)pm∗=γδ^ε--δ-ε^+εδ-γ^-δ^γ-+δγ-ε^-ε-γ^γλ-ε^-λ^ε-+λε-γ^-γ-ε^+ελ^γ--λ-γ^,(22)w∗=λδ-ε^-ε-δ^+ελ-δ^-λ^δ-+δλ^ε--λ-ε^γλ-ε^-λ^ε-+λε-γ^-γ-ε^+ελ^γ--λ-γ^,λ+γδ^ε--δ-ε^+λ^+γ^εδ--δε-+λ-+γ-δε^-εδ^>0,pm∗,otherwise,(23)e∗=γλ^δ--λ-δ^+λγ-δ^-δ-γ^+δλ-γ^-λ^γ-γλ-ε^-λ^ε-+λε-γ^-γ-ε^+ελ^γ--λ-γ^,where(24)γ=b-+v2,λ=b-v-2bv-b,ε=bvc+b-v-2bv-c+b-ρr+2bpm2b,δ=2a-b+b-a+bcr-vb+b--2bv-cs+ce02b,γ-=-b,λ-=v+b-2,ε-=ρr-b-b-c2,δ-=a-bcr+cs+ce0b-b-2,γ^=cb--b+ρr2,λ^=vcb--b+2bcv-v-+b-ρr+2ρmb2b,ε^=cρrb+b-+2bcρmb-k,δ^=2bca+a-+cb--ba+bcr-cs+ce0b+b-ρr+2bρm2b.
Note that the condition(λ+γ)(δ^ε--δ-ε^)+(λ^+γ^)(εδ--δε-)+(λ-+γ-)(δε^-εδ^)>0 in Proposition 6 ensures that the wholesale price cannot be higher than the manufacturer’s online direct price. The condition (λ+γ)(δ^ε--δ-ε^)+(λ^+γ^)(εδ--δε-)+(λ-+γ-)(δε^-εδ^)≤0 implies that the optimal wholesale price is higher than the direct price. In this case, the manufacturer will set the wholesale price equal to the direct price.Case 2 (H3=|H|>0).
In this case, the manufacturer’s profit function is not jointly concave with respect to (pm,w,e). However, we find that, for a given carbon emission reduction level e, the function πmII(pm,w,e) is jointly concave with respect to (pm,w). We then have the following Proposition 7.Proposition 7.
For a dual-channel supply chain, if the retailer makes their price decisions according to (15), then the manufacturer’s optimal carbon emission reduction level is (25)e∗=Ξ2T0+cx1+2T-0+cx2-k,k>2T0+cx1+2T-0+cx2e0,k≤2T0+cx1+2T-0+cx2,where(26)Ξ=-T0+ca+vT¯1-T¯0+ca--v-T¯1-T0+cb-T¯0+cb¯a+bcr2b+vT-12b+T12-T1x1-T¯1x2+x1+x2cs+ce0,x1=ρr+vT-0-bvT-0+ρr2b+T02,x2=ρm-vT-0+bvT-0+ρr2b+T02,T0=ε-λ-ελ-γλ--γ-λ,T1=δ-λ-δλ-γλ--γ-λ,T-0=εγ--γε-γλ--γ-λ,T-1=δγ--γδ¯γλ--γ-λ.The manufacturer’s optimal online direct price and wholesale price can be determined as follows:(27)pm∗=εγ--γε-γλ--γ-λe∗+δγ--γδ-γλ--γ-λ,(28)w∗=ε-λ-ελ-γλ--γ-λe∗+δ-λ-δλ-γλ--γ-λ,e∗>δ-λ+γ-δλ-+γ-ελ-+γ--ε-λ+γpm∗,otherwise.Note that the manufacturer may sell their products either through the traditional physical channel or the dual-channel model, which depends on the profits that can be obtained. In the following, we are interested in when the manufacturer should open an online channel to compete with the traditional retailer, that is, whether the manufacturer will benefit from introducing competition mechanisms. We focus on the comparison between the profits that the supply chain members obtain in a traditional supply chain and those obtained in a dual-channel supply chain, which correspond to Model I and Model II, respectively. To simplify the description, let (prI∗,wI∗,eI∗) represent the optimal decisions in Model I, which yield the optimal profits πrI∗ and πmI∗, respectively. Let (prII∗,pmII∗,wII∗,eII∗) represent the optimal decisions in Model II, which yield the optimal profits πrII∗ and πmII∗, respectively.In general, the larger the difference between the profits that the manufacturer obtains in Model II and in Model I, the more likely the manufacturer is to sell the product through dual channels. The conditionΔπm=πmII∗-πmI∗>0 implies the rationality of opening an online channel. However, the retailer’s profit may be encroached upon, as part of the demand will be shifted to the manufacturer’s online channel. Therefore, the gap between πrII∗ and πrI∗ may be less than zero: Δπr=πrII∗-πrI∗<0. Both Δπm and Δπr depend on the values of the system parameters. Due to the complexities of the developed models and the corresponding decisions, in the next section, we conduct numerical experiments to explore whether opening an online channel will benefit the manufacturer or the retailer.


## 5. Further Extension

In Section3, two models are developed for the traditional single- and dual-channel supply chain by assuming consumers in both channels are sensitive to low-carbon products. However, it is not always the case that consumers are innately environmentalists, which indicates that the manufacturer and/or government needs to make efforts to make consumers more sensitive to environmental problems. In this section, we further discuss the situation that the manufacturer is empowered to allocate some of their resources to cultivate low-carbon sensitive consumers and analyze how this behavior affects the supply chain decisions.Note that whenρr=ρm=0, the developed models in Section 3 can be regarded as the case in which consumers have no low-carbon preference. To make consumers more sensitive to low-carbon products, the manufacturer needs to take some measures such as advertising and marketing activities, etc., which inevitably incur extra cost. Similarly, we measure such cost by using a quadratic function, i.e., C^(ρi)=κρi2/2, i=r,m. Note that C^(0)=0; that is, the cost of manipulating consumers’ low-carbon preference is eliminated if consumers are not sensitive to low-carbon products. However, the manufacturer may be unwilling to spend money on advertising and marketing since this could reduce profits. To avoid this situation, we assume that the manufacturer can be granted a government subsidy t for per unit low-carbon products sold [43, 44]. Then, the manufacturer’s profits in (3) and (6) can be modified as(29)πmIw,e=w-cs+tDrI-Ce-Ee,DrI=w-cs-ce0-e+tαs1-βρr+αoθo1-βρr+ρre-ke22-κρr22+cK,(30)πmIIpm,w,e=w-cs+tDrII+pm-cs+tDmII-Ce-Ee;DrII,DmII-C^ρr-C^ρm=w-cs-ce0-e+tDrII+pm-cs-ce0-e+tDmII-ke22-κρr2+ρm22+cKClearly, the more low-carbon sensitive consumers are, the higher the cost the manufacturer would have to pay. To simplify the resolution process, both ρr and ρm are assumed to be parameters rather than variables, which implies that all decisions derived in Section 4 still hold with cs=cs-t.Corollary 8.
For a traditional supply chain, the manufacturer’s optimal wholesale price and carbon emission reduction level have the following elasticity with respect to the government subsidy:(31)∂w∗∂t<0,∂e∗∂t>0.
Substitutecs with cs-t in Proposition 3. Corollary 8 can be proved easily by taking the first-order derivative of w∗ and e∗ with respect to t. It can be seen from Corollary 8 that the higher the government subsidy, the more the manufacturer is willing to implement the carbon emission reduction. Moreover, the manufacturer will decrease the wholesale price as government subsidies are granted for low-carbon products, which will reduce the retail price in turn, thus increasing the demand.Note that when the manufacturer opens an online channel to compete with the traditional retailer, decisions can be made according to Propositions6 or 7. However, we cannot obtain the straightforward impact tendencies of government subsidies on the online price, wholesale price, and carbon emission reduction level, due to the complexity of analytical solutions. In the next section, some numerical examples will be given to illustrate how the government subsidy influences the retailer’s and the manufacturer’s optimal decisions as well as their profits.


## 6. Numerical Examples

In this section, we outline some numerical experiments that were performed to investigate the developed models and further intuitively illustrate the results we obtained. For the traditional single- and dual-channel supply chain models (Model I and Model II), the optimal decisions and the corresponding performances were analyzed under different carbon prices (c), low-carbon sensitivity (ρr,ρm), and channel preferences reflected by demand discount coefficients due to consumers’ channel mismatches (θs,θo) and the potential numbers of PS and PO consumers (αs,αo). For the problem instances in the numerical study, the price sensitivity and price-gap sensitivity were set to β=0.15 and η=0.02, respectively. Furthermore, cr=0, cs=0.4, e0=3, k=3.5, and K=10 were used.


### 6.1. Analysis of the Traditional Single-Channel Supply Chain

In order to observe the changing tendencies of the optimal supply chain decisions and the corresponding profits of the retailer and manufacturer, the unit carbon price was set to vary from 0 to 1 in increments of 0.1. The potential numbers of the consumers who prefer shopping in-store and online were set asαs=3 and αo=5, respectively. The demand discount coefficient and the low-carbon sensitivity were set as θo=0.30 and ρr=0.7. According to Propositions 1 and 3, the results are shown in Figures 2 and 3. From Figure 2(a), it can be seen that the manufacturer’s optimal carbon emission reduction level first increased and then decreased as the carbon price increased, which means there was a carbon price threshold (c=0.8) that the manufacturer could use to determine whether they should increase or decrease their carbon emission reduction level. From Figure 2(b), it can be seen that both the retailer’s price and the manufacturer’s wholesale price increased as the carbon price increased. This may be caused by the fact that implementing technological innovations to reduce carbon emission inevitably induces extra costs, which prompts the manufacturer to raise the wholesale price. In response, the retailer will then raise their retail price, which is consistent with the analysis of Proposition 1. As expected, the manufacturer would benefit from a carbon emissions reduction. It can be observed from Figure 3 that the manufacturer’s profit increased with the carbon price. However, the retailer’s profit decreased with the carbon price, because the wholesale price grew faster than the retail price.Figure 2
Impact of the carbon price on supply chain decisions.
(a) (b)Figure 3
Impact of the carbon price on supply chain members’ profits.To further analyze the impact of low-carbon sensitivity and the demand discount coefficient on the supply chain decisions and profits of the retailer and manufacturer, more experiments were conducted according to Propositions1 and 3. The results are presented in Figures 4–6, where the figures (a) illustrate the effect of low-carbon sensitivity when θo=0.30, and the figures (b) illustrate the effect of the online demand discount coefficient when ρr=0.7. Figure 4 shows the increasing tendencies of the optimal carbon emission reduction level with respect to low-carbon sensitivity ρr and the online demand discount coefficient θo. However, Figures 5(a) and 5(b) display the opposite tendencies of the retail and wholesale price, which implies that low-carbon sensitivity and the demand discount coefficient had different effects on pricing decisions. Interestingly, it can be seen from Figure 6 that the profits of the manufacturer and retailer increased with low-carbon sensitivity and the demand discount coefficient, which indicates that higher low-carbon preference and online demand discount coefficient not only motivate the manufacturer to reduce carbon emissions, but also benefit both the manufacturer and the retailer.Figure 4
Impact of low-carbon sensitivity and channel preference on the carbon emission reduction level whenc=0.5.
(a) (b)Figure 5
Impact of low-carbon sensitivity and channel preference on the retail and wholesale price whenc=0.5.
(a) (b)Figure 6
Impact of low-carbon sensitivity and channel preference on supply chain members’ profits whenc=0.5.
(a) (b)In order to analyze the impact of the consumers’ channel preference on the supply chain decisions and the supply chain members’ profits, we assumedθo=0.30, c=0.50, and ρr=0.50. When investigating the effect of the potential number of PS (PO) consumers, αo (αs) was set as αo=5 (αs=3). The results are shown in Figures 7–9. It can be seen that both the numbers of PS and PO consumers had positive effects on the optimal carbon emission reduction level and the profits of the supply chain members. We can then conclude that greater numbers of PS or PO consumers would stimulate the manufacturer to reduce carbon emissions and benefit the supply chain members’ profits as well. However, it can be seen from Figure 8 that the retail price and wholesale price both decreased with the increase of αs or αo. Furthermore, the effect of αs was more significant than that of αo, which can be seen from the rates of decline in the retail and wholesale price, as well as the increment rates of the retailer’s and manufacturer’s profits. This was caused by the fact that there was no online channel in the traditional supply chain.Figure 7
Impact of potential numbers of PS (prefer shopping in a physical store) and PO (prefer shopping online) consumers on the manufacturer’s carbon emission reduction level.
(a) (b)Figure 8
Impact of potential numbers of PS and PO consumers on supply chain members’ profits.
(a) (b)Figure 9
Impact of potential numbers of PS and PO consumers on supply chain members’ profits.
(a) (b)


### 6.2. Analysis of the Dual-Channel Supply Chain

In this section, we focus on the analysis of the carbon price, low-carbon sensitivity, and channel preferences as they pertain to dual-channel supply chain decisions and the profits of supply chain members. To analyze the effect of the carbon price, we setαs=3, αo=5, θs=θo=0.30, ρr=0.7, and ρm=0.5. The results are shown in Figures 10 and 11. It can be seen from Figure 10 that both the carbon emission reduction level and the supply chain pricing decisions increased with the carbon price. As the carbon price increased, the retailer’s profit first decreased and then increased. When c<0.2, it implies that e∗<1.13, πr∗ decreases with c. When c≥0.2, it implies that e∗≥1.13, πr∗ increases with c. However, the manufacturer’s profit monotonously increased with the carbon price. Recall that, in the traditional supply chain scenario, the carbon price had a positive (negative) effect on the manufacturer’s (retailer’s) profit, indicating that carbon emissions reduction would always benefit the manufacturer.Figure 10
Impact of the carbon price on supply chain decisions.
(a) (b)Figure 11
Impact of the carbon price on supply chain members’ profits.
(a) (b)To analyze the impacts of low-carbon preference in different channels on the supply chain decisions and the profits of the supply chain members, more experiments were conducted by assumingc=0.5, ρm=0.5, and ρr=0.7, respectively. The results are shown in Figures 12–14. As expected, the manufacturer’s optimal carbon emission reduction levels increased with both ρr and ρm (see Figure 12). Similar increasing trends of pr∗, pm∗, and w∗ with ρr and ρm can also be seen in Figure 13. However, the retailer’s price pr∗ (pm∗) was more sensitive to ρr(ρm) because ρr(ρm) only reflects the consumers’ low-carbon preference in the traditional channel (manufacturer’s online channel). In particular, we found that pm∗=w∗, which was caused by the fact that although the Hessian matrix of the manufacturer’s profit function was jointly concave with respect to (pr,pm,w), the parameter values could not ensure that the manufacturer’s online direct price was higher than the wholesale price. To avoid the retailer buying the products through the manufacturer’s online channel, the manufacturer would set the wholesale price the same as the online direct price. Unsurprisingly, the retailer’s and the manufacturer’s profits would both increase with ρr and ρm due to the increasing e∗, and ρr or ρm would increase with the offline and online demands and then benefit the supply chain members.Figure 12
Impact of low-carbon sensitivity on the manufacturer’s carbon emission reduction level.
(a) (b)Figure 13
Impact of low-carbon sensitivity on dual-channel supply chain decisions.
(a) (b)Figure 14
Impact of low-carbon sensitivity on supply chain members’ profits.
(a) (b)Figures15–17 show the effects of both the offline and online demand discount coefficients due to the channel mismatch on the supply chain optimal decisions as well as the profits of the retailer and the manufacturer. The results were obtained by assuming c=0.5, αs=3, αo=5, ρm=0.5, and ρr=0.7. For the figures (a), θo=0.3, while for the figures (b), θs=0.3. Similar to Figure 12, Figure 15 shows that the optimal carbon emission reduction level e∗ increased with θs and θo, respectively, and so did the profits of the retailer and manufacturer. Note that increasing e∗ and θs or θo will increase both the offline and online demands, which in turn benefits the supply chain members. Interestingly, compared with the tendency towards increases in Figure 13, Figure 16 shows that supply chain pricing decisions (pr∗, pm∗, and w∗) slightly decreased with θs and θo, which implies that the low-carbon preference and the demand discount coefficient influenced supply chain decisions differently, although all of them had positive effects on the supply chain members’ profits.Figure 15
Impact of channel preference on the manufacturer’s carbon emission reduction level.
(a) (b)Figure 16
Impact of channel preference on supply chain pricing decisions.
(a) (b)Figure 17
Impact of channel preference on supply chain members’ profits.
(a) (b)Similar change tendencies to those in Figures15–17 can also be found in Figures 18–20 when investigating the impacts of the potential numbers of PS and PO consumers. From Figure 18, we find that the carbon emission reduction level significantly increased with αs and αo, which implies that the manufacturer’s willingness to cut carbon emissions increased as the number of PS/PO consumers increased. Unsurprisingly, an increasing market size would in turn enhance the profits of both the manufacturer and the retailer, which can be seen in Figure 19. Note that a lower wholesale price would simultaneously reduce the retail price, which can further increase the retailer’s demand, and a lower online price can likewise increase the online demand. Therefore, with the increase of the market size, the manufacturer and the retailer would both be more willing to reduce their prices to share the positive effects of the larger demand on their profits. Comparing Figures 16 and 17 and Figures 19 and 20, we find that either the supply chain pricing decisions or the supply chain members’ profits were more influenced by the potential number of PS or PO consumers, which indicates that the decision-maker should pay more attention to increasing the potential market size.Figure 18
Impact of the potential numbers of PS and PO consumers on the manufacturers’ carbon emission reduction levels.
(a) (b)Figure 19
Impact of the potential numbers of PS and PO consumers on supply chain pricing decisions.
(a) (b)Figure 20
Impact of the potential numbers of PS and PO consumers on supply chain members’ profits.
(a) (b)


### 6.3. Comparison between the Traditional Single- and Dual-Channel Supply Chains

In this section, some more numerical experiments are outlined that were conducted to analyze when the manufacturer should open an online channel. We also analyze whether there is benefit to the retailer (i.e., profit) when it is beneficial for the manufacturer to operate with a dual-channel model. To calculate the values ofΔπm and Δπr, the system parameters were set as αs=3, αo=5, θs=θo=0.3, ρr=0.7, ρm=0.5, c=0.5, cr=0, cs=0.4, e0=3, k=3.5, and K=10. We obtained Δπm=10.44>0 and Δπr=0.14>0, which indicates that opening an online direct channel would benefit both the manufacturer and the retailer.Next, in order to further investigate the conditions under which the traditional single-channel is superior to the dual-channel, some of the parameter values were allowed to change within certain ranges. Specifically, the carbon pricec was set from 0 to 2 in increments of 0.1. αs and αo (θs and θo) were set to start from 1 (0.1) to 5 (0.5) in increments of 1 (0.1), respectively. By calculating the gap between the profits obtained in the dual-channel and traditional channel models, we found that opening an online channel did not benefit the manufacturer in two cases. The results are shown in Table 2 and show that the dual-channel model would be more attractive to the retailer since they could gain much higher profits. However, it did have a negative influence on the manufacturer’s profit. Moreover, it was only beneficial for the manufacturer to open an online channel when (c,αs,αo,θs,θo) = (0.1, 5, 5, 0.1, 0.1), which yielded Δπm=9.45 and Δπr=-0.02. The above results indicate that dual-channel selling would only be the best choice for both the manufacturer and the retailer under certain conditions.Table 2
The profit gaps when only the retailer benefits from a dual-channel model.
Profits gap Case 1 Case 2 (c,αs,αo,θs,θo)=(1.6, 1, 5, 0.5, 0.2) (c,αs,αo,θs,θo)=(1.6, 2, 5, 0.4, 0.1) Δπm -1.51 -1.00 Δπr 110.19 201.65


### 6.4. The Impact of Government Subsidy on Dual-Channel Supply Chain Operations

To further analyze the impacts of government subsidy on both the retailer’s and the manufacturer’s optimal decisions and their profits in a dual-channel supply chain, more experiments were executed by setting the system parameters asαs=3, αo=5, θs=θo=0.3, β=0.15, η=0.02, ρr=0.7, ρm=0.5, c=0.5, cr=0, cs=0.4, e0=3, k=3.5, and K=10. The results are shown in Figures 20 and 21.Figure 21
Impact of the government subsidy on supply chain members’ decisions.
(a) (b)It can be seen from Figure21(a) that the manufacturer’s optimal carbon emission reduction level increased with the government subsidy, which indicates that such a subsidy mechanism will effectively stimulate the manufacturer to implement carbon emission reduction. Furthermore, the manufacturer will reduce its online price and wholesale price since it can obtain subsidies from the government, which incurs a lower retail price in the traditional channel (see Figure 21(b)). Interestingly, Figure 22 displays positive effects of the government subsidy on both the retailer’s and the manufacturer’s profit, which means that introducing the government subsidy mechanism can not only encourage the manufacturer to implement green production and invest some resources to cultivate consumers’ low-carbon awareness, but also improve supply chain members’ profits. The results in Figures 21 and 22 also provide evidence that the government should give some subsidies to manufacturers engaged in green production.Figure 22
Impact of the government subsidy on supply chain members’ profits.
(a) (b)


## 6.1. Analysis of the Traditional Single-Channel Supply Chain

In order to observe the changing tendencies of the optimal supply chain decisions and the corresponding profits of the retailer and manufacturer, the unit carbon price was set to vary from 0 to 1 in increments of 0.1. The potential numbers of the consumers who prefer shopping in-store and online were set asαs=3 and αo=5, respectively. The demand discount coefficient and the low-carbon sensitivity were set as θo=0.30 and ρr=0.7. According to Propositions 1 and 3, the results are shown in Figures 2 and 3. From Figure 2(a), it can be seen that the manufacturer’s optimal carbon emission reduction level first increased and then decreased as the carbon price increased, which means there was a carbon price threshold (c=0.8) that the manufacturer could use to determine whether they should increase or decrease their carbon emission reduction level. From Figure 2(b), it can be seen that both the retailer’s price and the manufacturer’s wholesale price increased as the carbon price increased. This may be caused by the fact that implementing technological innovations to reduce carbon emission inevitably induces extra costs, which prompts the manufacturer to raise the wholesale price. In response, the retailer will then raise their retail price, which is consistent with the analysis of Proposition 1. As expected, the manufacturer would benefit from a carbon emissions reduction. It can be observed from Figure 3 that the manufacturer’s profit increased with the carbon price. However, the retailer’s profit decreased with the carbon price, because the wholesale price grew faster than the retail price.Figure 2
Impact of the carbon price on supply chain decisions.
(a) (b)Figure 3
Impact of the carbon price on supply chain members’ profits.To further analyze the impact of low-carbon sensitivity and the demand discount coefficient on the supply chain decisions and profits of the retailer and manufacturer, more experiments were conducted according to Propositions1 and 3. The results are presented in Figures 4–6, where the figures (a) illustrate the effect of low-carbon sensitivity when θo=0.30, and the figures (b) illustrate the effect of the online demand discount coefficient when ρr=0.7. Figure 4 shows the increasing tendencies of the optimal carbon emission reduction level with respect to low-carbon sensitivity ρr and the online demand discount coefficient θo. However, Figures 5(a) and 5(b) display the opposite tendencies of the retail and wholesale price, which implies that low-carbon sensitivity and the demand discount coefficient had different effects on pricing decisions. Interestingly, it can be seen from Figure 6 that the profits of the manufacturer and retailer increased with low-carbon sensitivity and the demand discount coefficient, which indicates that higher low-carbon preference and online demand discount coefficient not only motivate the manufacturer to reduce carbon emissions, but also benefit both the manufacturer and the retailer.Figure 4
Impact of low-carbon sensitivity and channel preference on the carbon emission reduction level whenc=0.5.
(a) (b)Figure 5
Impact of low-carbon sensitivity and channel preference on the retail and wholesale price whenc=0.5.
(a) (b)Figure 6
Impact of low-carbon sensitivity and channel preference on supply chain members’ profits whenc=0.5.
(a) (b)In order to analyze the impact of the consumers’ channel preference on the supply chain decisions and the supply chain members’ profits, we assumedθo=0.30, c=0.50, and ρr=0.50. When investigating the effect of the potential number of PS (PO) consumers, αo (αs) was set as αo=5 (αs=3). The results are shown in Figures 7–9. It can be seen that both the numbers of PS and PO consumers had positive effects on the optimal carbon emission reduction level and the profits of the supply chain members. We can then conclude that greater numbers of PS or PO consumers would stimulate the manufacturer to reduce carbon emissions and benefit the supply chain members’ profits as well. However, it can be seen from Figure 8 that the retail price and wholesale price both decreased with the increase of αs or αo. Furthermore, the effect of αs was more significant than that of αo, which can be seen from the rates of decline in the retail and wholesale price, as well as the increment rates of the retailer’s and manufacturer’s profits. This was caused by the fact that there was no online channel in the traditional supply chain.Figure 7
Impact of potential numbers of PS (prefer shopping in a physical store) and PO (prefer shopping online) consumers on the manufacturer’s carbon emission reduction level.
(a) (b)Figure 8
Impact of potential numbers of PS and PO consumers on supply chain members’ profits.
(a) (b)Figure 9
Impact of potential numbers of PS and PO consumers on supply chain members’ profits.
(a) (b)


## 6.2. Analysis of the Dual-Channel Supply Chain

In this section, we focus on the analysis of the carbon price, low-carbon sensitivity, and channel preferences as they pertain to dual-channel supply chain decisions and the profits of supply chain members. To analyze the effect of the carbon price, we setαs=3, αo=5, θs=θo=0.30, ρr=0.7, and ρm=0.5. The results are shown in Figures 10 and 11. It can be seen from Figure 10 that both the carbon emission reduction level and the supply chain pricing decisions increased with the carbon price. As the carbon price increased, the retailer’s profit first decreased and then increased. When c<0.2, it implies that e∗<1.13, πr∗ decreases with c. When c≥0.2, it implies that e∗≥1.13, πr∗ increases with c. However, the manufacturer’s profit monotonously increased with the carbon price. Recall that, in the traditional supply chain scenario, the carbon price had a positive (negative) effect on the manufacturer’s (retailer’s) profit, indicating that carbon emissions reduction would always benefit the manufacturer.Figure 10
Impact of the carbon price on supply chain decisions.
(a) (b)Figure 11
Impact of the carbon price on supply chain members’ profits.
(a) (b)To analyze the impacts of low-carbon preference in different channels on the supply chain decisions and the profits of the supply chain members, more experiments were conducted by assumingc=0.5, ρm=0.5, and ρr=0.7, respectively. The results are shown in Figures 12–14. As expected, the manufacturer’s optimal carbon emission reduction levels increased with both ρr and ρm (see Figure 12). Similar increasing trends of pr∗, pm∗, and w∗ with ρr and ρm can also be seen in Figure 13. However, the retailer’s price pr∗ (pm∗) was more sensitive to ρr(ρm) because ρr(ρm) only reflects the consumers’ low-carbon preference in the traditional channel (manufacturer’s online channel). In particular, we found that pm∗=w∗, which was caused by the fact that although the Hessian matrix of the manufacturer’s profit function was jointly concave with respect to (pr,pm,w), the parameter values could not ensure that the manufacturer’s online direct price was higher than the wholesale price. To avoid the retailer buying the products through the manufacturer’s online channel, the manufacturer would set the wholesale price the same as the online direct price. Unsurprisingly, the retailer’s and the manufacturer’s profits would both increase with ρr and ρm due to the increasing e∗, and ρr or ρm would increase with the offline and online demands and then benefit the supply chain members.Figure 12
Impact of low-carbon sensitivity on the manufacturer’s carbon emission reduction level.
(a) (b)Figure 13
Impact of low-carbon sensitivity on dual-channel supply chain decisions.
(a) (b)Figure 14
Impact of low-carbon sensitivity on supply chain members’ profits.
(a) (b)Figures15–17 show the effects of both the offline and online demand discount coefficients due to the channel mismatch on the supply chain optimal decisions as well as the profits of the retailer and the manufacturer. The results were obtained by assuming c=0.5, αs=3, αo=5, ρm=0.5, and ρr=0.7. For the figures (a), θo=0.3, while for the figures (b), θs=0.3. Similar to Figure 12, Figure 15 shows that the optimal carbon emission reduction level e∗ increased with θs and θo, respectively, and so did the profits of the retailer and manufacturer. Note that increasing e∗ and θs or θo will increase both the offline and online demands, which in turn benefits the supply chain members. Interestingly, compared with the tendency towards increases in Figure 13, Figure 16 shows that supply chain pricing decisions (pr∗, pm∗, and w∗) slightly decreased with θs and θo, which implies that the low-carbon preference and the demand discount coefficient influenced supply chain decisions differently, although all of them had positive effects on the supply chain members’ profits.Figure 15
Impact of channel preference on the manufacturer’s carbon emission reduction level.
(a) (b)Figure 16
Impact of channel preference on supply chain pricing decisions.
(a) (b)Figure 17
Impact of channel preference on supply chain members’ profits.
(a) (b)Similar change tendencies to those in Figures15–17 can also be found in Figures 18–20 when investigating the impacts of the potential numbers of PS and PO consumers. From Figure 18, we find that the carbon emission reduction level significantly increased with αs and αo, which implies that the manufacturer’s willingness to cut carbon emissions increased as the number of PS/PO consumers increased. Unsurprisingly, an increasing market size would in turn enhance the profits of both the manufacturer and the retailer, which can be seen in Figure 19. Note that a lower wholesale price would simultaneously reduce the retail price, which can further increase the retailer’s demand, and a lower online price can likewise increase the online demand. Therefore, with the increase of the market size, the manufacturer and the retailer would both be more willing to reduce their prices to share the positive effects of the larger demand on their profits. Comparing Figures 16 and 17 and Figures 19 and 20, we find that either the supply chain pricing decisions or the supply chain members’ profits were more influenced by the potential number of PS or PO consumers, which indicates that the decision-maker should pay more attention to increasing the potential market size.Figure 18
Impact of the potential numbers of PS and PO consumers on the manufacturers’ carbon emission reduction levels.
(a) (b)Figure 19
Impact of the potential numbers of PS and PO consumers on supply chain pricing decisions.
(a) (b)Figure 20
Impact of the potential numbers of PS and PO consumers on supply chain members’ profits.
(a) (b)


## 6.3. Comparison between the Traditional Single- and Dual-Channel Supply Chains

In this section, some more numerical experiments are outlined that were conducted to analyze when the manufacturer should open an online channel. We also analyze whether there is benefit to the retailer (i.e., profit) when it is beneficial for the manufacturer to operate with a dual-channel model. To calculate the values ofΔπm and Δπr, the system parameters were set as αs=3, αo=5, θs=θo=0.3, ρr=0.7, ρm=0.5, c=0.5, cr=0, cs=0.4, e0=3, k=3.5, and K=10. We obtained Δπm=10.44>0 and Δπr=0.14>0, which indicates that opening an online direct channel would benefit both the manufacturer and the retailer.Next, in order to further investigate the conditions under which the traditional single-channel is superior to the dual-channel, some of the parameter values were allowed to change within certain ranges. Specifically, the carbon pricec was set from 0 to 2 in increments of 0.1. αs and αo (θs and θo) were set to start from 1 (0.1) to 5 (0.5) in increments of 1 (0.1), respectively. By calculating the gap between the profits obtained in the dual-channel and traditional channel models, we found that opening an online channel did not benefit the manufacturer in two cases. The results are shown in Table 2 and show that the dual-channel model would be more attractive to the retailer since they could gain much higher profits. However, it did have a negative influence on the manufacturer’s profit. Moreover, it was only beneficial for the manufacturer to open an online channel when (c,αs,αo,θs,θo) = (0.1, 5, 5, 0.1, 0.1), which yielded Δπm=9.45 and Δπr=-0.02. The above results indicate that dual-channel selling would only be the best choice for both the manufacturer and the retailer under certain conditions.Table 2
The profit gaps when only the retailer benefits from a dual-channel model.
Profits gap Case 1 Case 2 (c,αs,αo,θs,θo)=(1.6, 1, 5, 0.5, 0.2) (c,αs,αo,θs,θo)=(1.6, 2, 5, 0.4, 0.1) Δπm -1.51 -1.00 Δπr 110.19 201.65


## 6.4. The Impact of Government Subsidy on Dual-Channel Supply Chain Operations

To further analyze the impacts of government subsidy on both the retailer’s and the manufacturer’s optimal decisions and their profits in a dual-channel supply chain, more experiments were executed by setting the system parameters asαs=3, αo=5, θs=θo=0.3, β=0.15, η=0.02, ρr=0.7, ρm=0.5, c=0.5, cr=0, cs=0.4, e0=3, k=3.5, and K=10. The results are shown in Figures 20 and 21.Figure 21
Impact of the government subsidy on supply chain members’ decisions.
(a) (b)It can be seen from Figure21(a) that the manufacturer’s optimal carbon emission reduction level increased with the government subsidy, which indicates that such a subsidy mechanism will effectively stimulate the manufacturer to implement carbon emission reduction. Furthermore, the manufacturer will reduce its online price and wholesale price since it can obtain subsidies from the government, which incurs a lower retail price in the traditional channel (see Figure 21(b)). Interestingly, Figure 22 displays positive effects of the government subsidy on both the retailer’s and the manufacturer’s profit, which means that introducing the government subsidy mechanism can not only encourage the manufacturer to implement green production and invest some resources to cultivate consumers’ low-carbon awareness, but also improve supply chain members’ profits. The results in Figures 21 and 22 also provide evidence that the government should give some subsidies to manufacturers engaged in green production.Figure 22
Impact of the government subsidy on supply chain members’ profits.
(a) (b)


## 7. Conclusions

In this paper, we integrate carbon emission reduction through technological innovation and consumers’ channel preference in traditional single- and dual-channel supply chains. By developing linear demand functions which simultaneously reflect the consumer channel and low-carbon preferences, two Stackelberg game models are presented for the single- and dual-channel supply chains. For each model, backward induction is used to find the equilibrium of the Stackelberg game. Some numerical examples were conducted to verify the effectiveness and practicality of the proposed models and solutions. The results showed that the supply chain members’ profits are influenced by the cap-and-trade regulations, consumers’ low-carbon sensitivity, and channel preferences. In particular, although it is beneficial for the manufacturer to implement carbon emission reductions through technological innovation, it is not always the case for the retailer. This runs in contrast to the results in Wang et al. showing that higher carbon prices may benefit the manufacturer only at a high clean production level [2]. This may be caused by consumer segmentation and demand discounts due to channel mismatch. Furthermore, when comparing the single- and dual-channel selling models, the results indicate that opening an online channel is only the best choice for the supply chain members if certain conditions are satisfied.The findings in this paper also provide useful practical suggestions. On one hand, they may aid manufacturers in deciding whether they should open an online channel to compete with traditional retailers. On the other hand, since consumers’ low-carbon preferences in both the traditional single- and dual-channel supply chains have positive influences on the supply chain members’ profits, the manufacturer and the retailer should take measures to promote consumers’ low-carbon sensitivity. This can also be regarded as a policy instrument for governments and could be used to facilitate the development of a sustainable economy. For example, government-funded publicity campaigns advocating for green consumption could promote consumers’ low-carbon preferences in both selling channels. Recall that although a higher carbon price can stimulate the manufacturer to cut carbon emissions, it reduces the retailer’s profit when the carbon price is too high. Fortunately, this can be compensated for by increasing consumers’ low-carbon sensitivity.It should be noted that this paper has some limitations which can be considered for future research. First, this paper considers a supply chain as consisting of only one manufacturer and one retailer; this could be extended by considering multiple competitive retailers. Second, responding to the manufacturer’s online direct channel, the retailer can also open their own online selling channel. In this case, the physical store and the online channel may be operated by different departments of the retailer, which could be modeled as a Bertrand or Stackelberg game. Moreover, it would be more interesting to discuss when the retailer’s online and offline prices should be the same, which reflects the characteristics of an O2O (Online to Offline) operation. Third, this paper only considers profit-maximizing problems in the framework of Stackelberg game. Designing a contract that can completely coordinate the supply chain would further benefit the supply chain members.



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