Impacts of CSR Undertaking Modes on Technological Innovation and Carbon-Emission-Reduction Decisions of Supply Chain

Abstract

Taking a low-carbon supply chain composed of one producer and one retailer as the object, this paper explores the impacts of different corporate social responsibility (CSR)-undertaking-mode choices on the carbon-emission reduction and technological innovation decisions of the low-carbon supply chain. Under three CSR-undertaking modes, the optimal decisions of carbon-emission-reduction intensity and technological-innovation level of low-carbon supply chain are explored by using the Stackelberg game. Through a comparative analysis of the optimal equilibrium solutions, we found that, with the enhancement of CSR awareness of member firms, the carbon-emission-reduction intensity, technological innovation level, and product sales volume of the low-carbon supply chain under different CSR-undertaking modes are all improved, and the total profits and total welfare of the low-carbon supply chain are all increased. The carbon-emission-reduction intensity, technological innovation level, and product sales volume are all the highest under the mode of the retailer undertaking CSR, the second highest under the mode of the producer and retailer jointly undertaking CSR, and the lowest under the mode of the producer undertaking CSR mode. Therefore, under the background of technological innovation carried out by the producer, it is most conducive to promoting the development of low-carbon supply chain for the retailer to undertake social responsibility.

1. Introduction

With the continuous growth of industrial production and automobile consumption, the carbon dioxide emissions have soared. According to a report released by the Dutch Environmental Assessment Agency (PBL) in 2020, total global greenhouse gas emissions have grown by an average annual rate of 1.4% since 2010, reaching a record high of 59.1 billion tons of CO₂ equivalent in 2019 [1]. Huge carbon dioxide emissions have led to frequent climate anomalies and extreme climate disasters around the world. Tackling the global climate crisis has become a common challenge faced by human society, and the low-carbon transition of economy has become the common consensus of all countries in the world. In 2008, the European Commission on Climate Change issued the “Climate Change and Energy Package Act”, calling for a reduction in carbon dioxide gas emissions by 20% compared with 1990 by 2020 [2]. In 2009, the United States enacted the American Clean Energy Security Act, which requires the disclosure of carbon-emission information during the production and sale of products, and the implementation of the “carbon tariff” policy from 2020. In 2020, the Chinese government clearly put forward the goal of “carbon peak” by 2030 and “carbon neutrality” by 2060 [3]. With the government’s improvement of environmental protection standards and the strengthening of public awareness of environmental protection, consumers also prefer to buy the environmental-protection version of products [4]. Vanday et al. used the case-study method to find that, in Australia, people prefer to buy environmental products rather than ordinary products [5]. Firms are the micro-subject of the market economy and actual practitioner and promoter of the development of a low-carbon economy; hence, how to reasonably guide firms to develop low-carbon production is a hot topic of common concern in the academic and business circles.

Meanwhile, with the development and progress of the society, corporate social responsibility (CSR) has attracted extensive attention from all sectors of society. Corporate social responsibility means that a company should not take profit creation as its sole business objective; it should also take responsibility for consumers, the environment, and society. In real life, social responsibility can be assumed by producers, such as IBM and Sany Heavy Industry; it can also be undertaken by retailers, such as Walmart and Suning Tesco. Furthermore, it can be jointly undertaken by the producer and the retailer. For example, JD.com sells Hewlett-Packard’s notebook computers, and both the producer and the seller bear social responsibilities. According to a survey by KPMG, three-quarters of the leading one hundred firms in Europe and the United States published social-responsibility reports in 2015. According to the China CSR report released by Golden Bee, a total of 1926 companies in China issued CSR reports in 2021, nearly double the number in 2011.

In addition, when producing low-carbon products, most firms need to pay a high initial cost for low-carbon transformation of production technology or process, resulting in a high production cost and market price of low-carbon products, thus hampering the market promotion of low-carbon products and making the goal of developing a low-carbon economy less supported by the market. Reducing the production cost of low-carbon products through technological innovation is the basis for the promotion and popularization of low-carbon products. However, there are no works in the literature exploring the impact of undertaking social responsibility on technological-innovation decisions of the low-carbon supply chain hitherto.

Therefore, this study focused on the following questions: (1) Under the three social-responsibility-undertaking modes, what are the effects of member firms undertaking social responsibility on the carbon-emission-reduction intensity and technological-innovation level of the low-carbon supply chain, respectively? (2) Among the three modes of social-responsibility undertaking, which mode is most conducive to promoting the technological-innovation level and carbon-emission-reduction intensity of the low-carbon supply chain? (3) How does the social-responsibility-undertaking mode affect the member firms’ net profits and social welfare and that of the whole low-carbon supply chain? The study of the above problems can provide a theoretical reference for the selection of the appropriate social-responsibility-undertaking mode for the low-carbon supply chain and promote the scientific development of the low-carbon supply chain.

The main contribution of this paper is embodied in two aspects: Firstly, it respectively investigates the impacts of undertaking social responsibility on the producer’s optimal decisions of carbon-emission intensity and technical-innovation level in the low-carbon supply chain under three modes of social-responsibility undertaking, further discussing the impacts of undertaking social responsibility on the low-carbon supply chain’s product sales volume, total profit, and total welfare. Secondly, by comparing and analyzing the optimal equilibrium decisions of the low-carbon supply chain under the three social-responsibility-undertaking modes, it is found that the retailer-undertaking-social-responsibility mode is most conducive to encouraging the producer to carry out technological innovation, improve carbon-emission-reduction level, and expand product market sales.

The rest of this paper is arranged as follows: Section 2 reviews the research related to this study. Section 3 states the research questions and model assumptions. In Section 4, the decision-making models under the three modes of social-responsibility undertaking are constructed and solved. Section 5 compares and analyzes the equilibrium solution of decision models. Section 6 provides the numerical simulation analysis of the relevant conclusions. Section 7 describes the research conclusion and future research direction of this paper.

2. Review of Literature and Motivations

The previous literature closely related to this study mainly involved three areas, namely low-carbon supply chain management, supply chain decisions considering CSR, and supply chain management considering technological innovation.

2.1. Low-Carbon Supply Chain Management

Numerous works from the literature have explored the management issues of the low-carbon supply chain from different perspectives. Taking Hyundai Motor Corporation of South Korea as an example, Lee [2] discussed the distribution and management strategy of product carbon footprint among member firms of automotive supply chain. Dong et al. [6] investigated the financing problem of member firms in low-carbon supply chain and found that supply chain member firms can solve the financing problem by using the composed strategy. Du et al. [7] investigated the impact of consumers’ low-carbon consciousness on member firms’ decision-making and found that enhancing consumers’ low-carbon consciousness can effectively improve the whole profit of the supply chain. Wu and Yang [8] found that, in the market environment, where consumers have strong low-carbon awareness, an appropriate cost-undertaking ratio is beneficial to reduce carbon emissions. Xia et al. [9] investigated the impact of carbon trading policy on low-carbon supply chain decision-making and found that carbon trading can increase the products’ market price and expand the products’ sales volume. Du et al. [10] found that the enhancement of consumers’ low-carbon concern can encourage the producer and the retailer to reduce carbon emission at the same time. Zhou et al. [11] studied the improvement strategy of the supply chain performance through contract design and found that the design of a reasonable contract can enhance the supply chain’s performance. Moreover, the retailer’s fairness-concern attribute may, in some cases, alter the coordination ability of the cooperative advertising contract and the carbon-reduction cost-undertaking contract. Zu et al. [12] utilized the differential games to explore the influence of the consumers’ environmental concern, government-intervention effect, and the coordination contract on the operation of low-carbon supply chain and revealed that the coordination contract led to greater emission-reduction efforts from the supplier. Wang et al. [13] developed decision models to study the emission-reduction decisions of the low-carbon supply chain under different conditions, namely the non-cooperation scenario, the producer supporting the supplier’s emission reduction, and the cooperation contract scenario, the results indicate that the member firms’ profits and products’ emission reductions are all the largest under the two-way cooperation contract scenario. Jiang and Chen [14] analyzed the decisions and coordination of member firms in the low-carbon supply chain and revealed that the proper contract can improve the profit of the whole supply chain. Chen and Hao [15] considered the incentive of a carbon tax on firm’s product pricing and production strategies and found that a higher carbon tax can result in a higher reduction of carbon emissions. Yu et al. [16] investigated the impact of a carbon tax on the producers’ carbon-emission-reduction efforts and product-pricing decisions in two competing chains and found that imposing appropriate carbon taxes can encourage producers to optimize the products’ abatement rate, and the carbon tax made the carbon emission and retail price of the products drop under supply chain’s member firms’ vertical collaboration. Yuan et al. [17] investigated the influence of member firm’s risk aversion on the optimal strategies and supply chain coordination when the customers prefer low-carbon products and found that the carbon-reduction efforts by the producer can make the retailer increase advertising effort and the order quantity. Xu et al. [18] studied the decision problems of production and pricing and found that find that the optimal number of kinds of products is jointly determined by the emission prices and the cap; the optimal tax rate is either increasing or constant. Du et al. [19] explored the influence of emission caps on the optimal decisions of member firms and found that the whole and the member firm’s net profits increase with the emission cap. Li et al. [20] explored the impact of government participation on the operation decisions in the low-carbon supply chain and found that construct a kind of mechanism can improve the performance of carbon-emission reduction. Liu and Chen [21] investigated the optimal decision-making problem of the three-tier low-carbon supply chain in the context of shared manufacturing and found that the implementation of a two-way cost-sharing contract can promote the effort level of member firms and the products’ carbon-emission-reduction level. The above studies mainly focused on the impact of consumers’ concern over low-carbon products and decision-makers’ behavior and government regulatory measures on low-carbon supply chain operation management, and they seldom paid attention to the impact of producers’ innovation behavior. Furthermore, existing studies did not involve the influence of the CSR awareness of the member firms of the low-carbon supply chain.

2.2. Supply-Chain Decisions Considering CSR

In recent years, the research of supply chain management considering the social-responsibility consciousness of member firms has become a hot topic. Panda [22] explored the decision problems when the producer or retailer undertakes social responsibility alone and realized supply chain coordination by means of revenue undertaking contract. Furthermore, Panda et al. [23] discussed the three-layer supply chain decision, taking social responsibility into consideration; coordinated the supply chain with Nash bargaining contract; and realized the reasonable distribution of consumer surplus. On this basis, Panda and Modak [24] explored the impact of producer or retailer undertaking social responsibility on member firms’ decision-making and found that the retailer’s pursuit of social-welfare maximization can alleviate channel conflicts in the supply chain. Through an empirical analysis, Liu et al. [25] found that firms willing to assume social responsibility are more likely to obtain competitive advantages in market competition. Ni and Li [26] explored the CSR-undertaking strategies of supply chain firms under different decision-making structures and found that, in some cases, the incentive mechanism of social responsibility could realize a win-win situation for supply chain nodal firms. Servaes and Tamayo [27] pointed out that firms with a sense of social responsibility will gain greater corporate value. Pino et al. [28] analyzed the positive significance of firms undertaking social responsibility from the perspective of consumers. Wu et al. [29] studied the influence of undertaking social responsibility on the decision-making of node firms in the supply chain and the coordination strategies and realized perfect coordination of the supply chain by using an improved wholesale price contract. Panda et al. [30] confirmed that the node firms in a closed-loop supply chain take the initiative to undertake social responsibility, which is conducive to improving their own profits. Vosooghidizaji et al. [31] investigated the problem of coordinating CSR under information asymmetry and found that, when the information is asymmetrical, the whole profits decreases, and the coordination mechanism leader–leader Stackelberg models can improve the whole profit of the supply chain. Liu et al. [32] explored the optimal strategies of member firms under two kinds of power structure and found that the optimal strategy is more likely to ensure a lower violation probability. Wang et al. [33] explored the influence of fairness preference and information asymmetry on the optimal decisions of the closed loop supply chain and found that the producer’s CSR concern is influenced by its fairness preference; the value of information is conducive for the retailer and not conducive for the producer. Zhao et al. [34] investigated the impact of CSR on the operation decisions of a closed-loop supply chain led by retailers and found that the manufacturer’s CSR input can increase the product recovery rate. The above studies seldom paid attention to the impact of CSR on low-carbon supply chain management and did not consider the relationship between CSR and producers’ innovation decisions.

2.3. Supply Chain Management Considering Technological Innovation

Nowadays, the importance of innovation has been recognized by all sectors of society. As a member of the supply chain, the influencing factors and mechanism of a firm’s innovation behavior have attracted the attention of academic circles. Gilbert and Cvsa [35] investigated the impact of the price decision of firms in the supply chain on the technological innovation of upstream firms. Wang and Shin [36] studied the impact of three different strategies on member firms’ innovation and found that the revenue-undertaking contract can coordinate supply chain decisions when the producer carries out innovation. Yang [37] investigated the dual-channel supply chain pricing decisions under the simultaneous action of price and technological innovation in the dual-channel supply chain structure of suppliers and found that, with the improvement of the technological-innovation level, the price, demand, and profit of suppliers and producers’ respective channels will increase accordingly. Li and Wan [38] considered the influence of observability of innovation behavior on producers’ innovation decisions and found that, when the innovation behavior is not observable, competition among producers will reduce their motivation for technological innovation. Zhou et al. [39] investigated the impact of multiple factors on the choice of remanufacturing strategies and found that the remanufacturing-strategy choice is mainly related to the price of carbon trading. The existing research studies mainly focused on the related factors and mechanism of technological innovation in conventional supply chain, dual-channel supply chain, or closed-loop supply chain and seldom paid attention to technological innovation in the low-carbon supply chain. In addition, there is a lack of research on the impact of undertaking social responsibility on technological innovation.

3. Model Description and Assumptions

The research background of this paper is as follows: A low-carbon supply chain consists of one low-carbon producer and one retailer. Before the low-carbon products launch on the market, the producer invests in upgrading production equipment and technology to produce low-carbon products, meanwhile carrying out technological innovation to reduce the production cost per unit product. Members of the low-carbon supply chain have a sense of social responsibility, and social responsibility can be undertaken by the producer, the retailer, or both of them. The producer plays a dominant role in the low-carbon supply chain, while the retailer plays a subordinate role in the low-carbon supply chain. The decision-making procedure of the low-carbon supply chain is as follows: the producer first makes the decision of wholesale price, carbon-emission-reduction intensity, and technological innovation level, and then the retailer makes the decision of retail price after observing the producer’s decision.

In order to make this study more realistic and feasible, this paper makes the following assumptions. In addition, the main symbols, parameters, and variables used in this article are summarized in

Table 1. Notation definition.

Symbols	The Definitions of Parameters
$m,r,s$	The producer, the retailer, the low-carbon supply chain, respectively
$MU$	The producer undertaking social responsibility
$RU$	The retailer undertaking social responsibility
$JU$	The producer and retailer jointly undertaking social responsibility
Parameters	The Definitions of Parameters
$a$	The potential market demand
$c$	The production cost per unit product
$\beta$	The carbon preference coefficient of consumers
$\mu$	The cost coefficient of carbon-emission reduction
$k$	The cost coefficient of technological innovation
${\pi }_i^j$	The expected profit of supply chain member firm i in case j, $i=\left\{m,r,s\right\}$, $j=\left\{MU,RU,JU\right\}$
$V_i^j$	The social welfare of supply chain member firm i in scenario j, $i=\left\{m,r,s\right\}$, $j=\left\{MU,RU,JU\right\}$
Decision Variables	The Definitions of Decision Variables
$w$	The wholesale price of the product
$p$	The retail price of the product
$\theta$	The carbon-emission-reduction intensity of the product
$x$	The technological-innovation level of the product

.

Assumption 1. 

The market-demand volume of low-carbon products. The consumers have low carbon preferences, and the products’ market demand volume is influenced by both the price of the product and the intensity of carbon-emission reduction, and the impacts of product price and the intensity of carbon-emission reduction on market demand are considered to be linear [6,10,16]. Thus, the demand for low-carbon products can be obtained as follows:

$$q\left(p,\theta \right)=a-p+\beta \theta$$

(1)

To make the study realistic, the products’ market demand should be positive, which means$q\left(p,\theta \right)>0$. Because the products’ sales price is always greater than the production cost of the product, $p>c$, so we can obtain the following:

$$a-p+\beta \theta >a-p>a-c>0$$

Assumption 2. 

The cost of carbon-emission reduction. The higherof the products’ carbon-reduction intensity, thelargerthe cost of the emission-reduction inputby the producer. With the increase of carbon-emission-reduction intensity, the cost of carbon-emission-reduction input also increases.Therefore, we define the carbon-emission-reduction investment-cost functionas$C\left(\theta \right)=\frac{1}{2}\mu {\theta }^2$, where $\mu$represents the investment cost coefficient of carbon-emission reduction [7,9].

Assumption 3. 

The social-welfare function of afirm. Weassume that social welfare is the degree of concern of the supply chain’s memberfirms compared to consumer surplus; if the concern degree is$\tau (1>\tau >0)$, according to theconsumer surplus functionestablishedby Pandaand Modak, the consumer-surplus function can be expressedas$CS={\int }_{p_{min}}^{p_{max}\int dp=\frac{{(a-p+\beta \theta )}^2}{2}}\left(a-p+\beta \theta \right)$, and then the social-welfare function is$\tau CS$ [30].

Assumption 4. 

The cost of technology innovation. The higher the level of technological innovation, the higher the cost of technological innovation. According to the research by Li and Wan, this study defines the difficulty coefficient of the producer’s technological innovation as$k$if the technological-innovation level of the product is$x$, meaning that the production cost per unit product is reduced by$x$units through technological innovation; then the producer needs to invest a fixed cost for technological innovation,$k{x}^2$ [38].

Assumption 5. 

Assume that$\frac{{\beta }^2}{\mu }$is the carbon coefficient [40].

4. Decision Models of Low-Carbon Supply Chain under Three Modes of Social-Responsibility Undertaking

The decision-making models of low-carbon supply chain under three corporate social-responsibility-undertaking modes are as below. In these three models, the producer and the retailer play a Stackelberg game, in which the producer is the dominant player, and the retailer is the subordinate.

4.1. The Producer Undertaking Social Responsibilities: Model MU

When the producer undertakes social responsibilities, the producer’s decision goal is to maximize its own social welfare, and the decision problem is to decide the wholesale price, carbon-emission-reduction intensity, and technological-innovation level simultaneously. The retailer’s decision goal is to maximize its own net profit, and the decision problem is to decide the retail price. Based on the assumptions in Part 3, the decision objective functions of the producer and the retailer can be expressed as follows, respectively:

$$V_m^{MU}\left(w,\theta ,x\right)={\pi }_m^{MU}+\tau CS=\left(w-c+x\right)\left(a-p+\beta \theta \right)-\frac{1}{2}\mu {\theta }^2-k{x}^2+\frac{\tau {(a-p+\beta \theta )}^2}{2}$$

(2)

$${\pi }_r^{MU}\left(p\right)=\left(p-w\right)\left(a-p+\beta \theta \right)$$

(3)

The above decision model is solved by backward induction method. Firstly, let the first-order condition of Equation (3) to be zero, and the optimal reaction function of the retailer is obtained as follows:

$$p^{MU}\left(w,\theta \right)=\frac{1}{2}\left(a+\beta \theta +w\right)$$

(4)

Substituting Equation (4) into Equation (2), the necessary condition for obtaining the maximum value of Equation (2) is that the negative definite of the Hessian matrix is satisfied; that is, $8k\mu -2k{\beta }^2-2k\mu \tau -\mu >0$. Under this condition, let the three first partial derivatives of Equation (2) be zero, solve the equation system that comprises the above three equations, and we can obtain the optimal $w^{MU*}$, ${\theta }^{MU*}$, and $x^{MU*}$. Then we substitute $w^{MU*}$, ${\theta }^{MU*}$, and $x^{MU*}$ into Equations (1)–(4), and the following proposition is obtained.

Proposition 1. 

Under the scenario of producer undertaking social responsibilities, the optimal equilibrium resultsof the wholesale price, the retail price, the carbon-emission-reduction intensity, the technological innovation level, and the market demand can be obtained as follows:

$$w^{MU*}=\frac{4k\mu \left(a+c\right)-a\mu \left(1+2k\tau \right)-2k{\beta }^{2}c}{8k\mu -2k{\beta }^2-2k\mu \tau -\mu }$$

(5)

$$p^{MU*}=\frac{2k\mu \left(3a+c\right)-a\mu \left(1+2k\tau \right)-2k{\beta }^{2}c}{8k\mu -2k{\beta }^2-2k\mu \tau -\mu }$$

(6)

$${\theta }^{MU*}=\frac{2k\beta \left(a-c\right)}{8k\mu -2k{\beta }^2-2k\mu \tau -\mu }$$

(7)

$$x^{MU*}=\frac{\mu \left(a-c\right)}{8k\mu -2k{\beta }^2-2k\mu \tau -\mu }$$

(8)

$$q^{MU*}=\frac{2k\mu \left(a-c\right)}{8k\mu -2k{\beta }^2-2k\mu \tau -\mu }$$

(9)

$${\pi }_r^{MU}=\frac{4k^2{\mu }^2{(a-c)}^2}{{(8k\mu -2k{\beta }^2-2k\mu \tau -\mu )}^2}$$

(10)

$${\pi }_m^{MU}=\frac{k\mu \left(8k\mu -2k{\beta }^2-4k\mu \tau -\mu \right){(a-c)}^2}{{(8k\mu -2k{\beta }^2-2k\mu \tau -\mu )}^2}$$

(11)

$${\pi }_s^{MU}=\frac{k\mu \left(12k\mu -2k{\beta }^2-4k\mu \tau -\mu \right){(a-c)}^2}{{(8k\mu -2k{\beta }^2-2k\mu \tau -\mu )}^2}$$

(12)

$$V_m^{MU}=\frac{k\mu {(a-c)}^2}{8k\mu -2k{\beta }^2-2k\mu \tau -\mu }$$

(13)

$$V_s^{MU}=V_m^{MU}+{\pi }_r^{MU}=\frac{k\mu \left(12k\mu -2k{\beta }^2-2k\mu \tau -\mu \right){(a-c)}^2}{{(8k\mu -2k{\beta }^2-2k\mu \tau -\mu )}^2}$$

(14)

4.2. The Retailer Undertaking Social Responsibilities: Model RU

When the retailer undertakes social responsibilities, the producer’s decision goal is to maximize its own net profit, and the decision problem is to decide the wholesale price, carbon-emission-reduction intensity, and technological-innovation level simultaneously. The retailer’s decision goal is to maximize its own social welfare, and the decision problem is to decide the retail price. Based on the assumptions in Part 3, the decision objective functions of the producer and the retailer can be expressed as follows, respectively:

$${\pi }_m^{RU}\left(w,\theta ,x\right)=\left(w-c+x\right)\left(a-p+\beta \theta \right)-\frac{1}{2}\mu {\theta }^2-k{x}^2$$

(15)

$$V_r^{RU}\left(p\right)={\pi }_r^{RU}+\tau CS=\left(p-w\right)\left(a-p+\beta \theta \right)+\frac{\tau {(a-p+\beta \theta )}^2}{2}$$

(16)

The above decision model is solved by using the backward induction method. Firstly, let the first partial derivative of Equation (16) to be zero, and the optimal reaction function of the retailer can be obtained as follows:

$$p^{RU}\left(w,\theta \right)=\frac{\left(a+\beta \theta \right)\left(1-\tau \right)+w}{2-\tau }$$

(17)

When substituting Equation (17) into Equation (15), the necessary condition for Equation (15) to obtain the maximum value is that its Hessian matrix is negative definite; that is, the follow condition is satisfied: $8k\mu -2k{\beta }^2-4k\mu \tau -\mu >0$. Under this condition, let the three first-order partial derivatives of Equation (15) be zero, solve the equation system that comprises the above three equations, and we can obtain the optimal $w^{RU*}$, ${\theta }^{RU*}$, and $x^{RU*}$. Then we substitute $w^{RU*}$, ${\theta }^{RU*}$, and $x^{RU*}$ into Equations (1) and (15)–(17), and the following proposition is obtained.

Proposition 2. 

Under the scenario of retailer undertaking social responsibilities, themember firms’optimalstrategiesof the wholesale price, the retail price, the carbon-emission-reduction intensity, the technological-innovation level, and the market demand can be obtained as follows:

$$w^{RU*}=\frac{2k\mu \left(2-\tau \right)\left(a+c\right)-2k{\beta }^{2}c-a\mu }{8k\mu -2k{\beta }^2-4k\mu \tau -\mu }$$

(18)

$${\theta }^{RU*}=\frac{2k\beta \left(a-c\right)}{8k\mu -2k{\beta }^2-4k\mu \tau -\mu }$$

(19)

$$x^{RU*}=\frac{\mu \left(a-c\right)}{8k\mu -2k{\beta }^2-4k\mu \tau -\mu }$$

(20)

$$p^{RU*}=\frac{2k\mu \left(3a+c\right)-a\mu \left(1+4k\tau \right)-2k{\beta }^{2}c}{8k\mu -2k{\beta }^2-4k\mu \tau -\mu }$$

(21)

$$q^{RU*}=\frac{2k\mu \left(a-c\right)}{8k\mu -2k{\beta }^2-4k\mu \tau -\mu }$$

(22)

$${\pi }_m^{RU}=\frac{k\mu {(a-c)}^2}{8k\mu -2k{\beta }^2-4k\mu \tau -\mu }$$

(23)

$${\pi }_r^{RU}=\frac{4k^2{\mu }^2\left(1-\tau \right){(a-c)}^2}{{(8k\mu -2k{\beta }^2-4k\mu \tau -\mu )}^2}$$

(24)

$${\pi }_s^{RU}=\frac{k\mu \left(12k\mu -2k{\beta }^2-8k\mu \tau -\mu \right){(a-c)}^2}{{(8k\mu -2k{\beta }^2-4k\mu \tau -\mu )}^2}$$

(25)

$$V_r^{RU}=\frac{2k^2{\mu }^2\left(2-\tau \right){(a-c)}^2}{{(8k\mu -2k{\beta }^2-4k\mu \tau -\mu )}^2}$$

(26)

$$V_s^{RU}=V_r^{RU}+{\pi }_m^{RU}=\frac{k\mu \left(12k\mu -2k{\beta }^2-6k\mu \tau -\mu \right){(a-c)}^2}{{(8k\mu -2k{\beta }^2-4k\mu \tau -\mu )}^2}$$

(27)

4.3. The Producer and the Retailer Jointly Undertaking Social Responsibilities: Model JU

When the producer and the retailer jointly undertake social responsibilities, both of their decision goals are to maximize their own social welfares; the producer’s decision problem is to decide the wholesale price, carbon-emission-reduction intensity, and technological-innovation level simultaneously; and the retailer’s decision problem is to decide the retail price. Assume that the producer’s share of social responsibility is $\lambda (1>\lambda >0)$, then the retailer’s share is $1-\lambda$. Based on the assumptions in Part 3, the decision objective functions of the member firms can be obtained as shown below:

$$V_m^{JU}\left(w,\theta ,x\right)={\pi }_m^{JU}+\lambda \tau CS=\left(w-c+x\right)\left(a-p+\beta \theta \right)-\frac{1}{2}\mu {\theta }^2-k{x}^2+\frac{\lambda \tau {(a-p+\beta \theta )}^2}{2}$$

(28)

$$V_r^{JU}\left(p\right)={\pi }_r^{JU}+\left(1-\lambda \right)\tau CS=\left(p-w\right)\left(a-p+\beta \theta \right)+\frac{\left(1-\lambda \right)\tau {(a-p+\beta \theta )}^2}{2}$$

(29)

The above decision model is solved by using the backward induction method. Firstly, let the first partial derivative of Equation (29) be zero, and the optimal reaction function of the retailer is obtained as follows:

$$p^{JU}\left(w,\theta \right)=\frac{\left(a+\beta \theta \right)\left(1-\tau +\lambda \tau \right)+w}{2-\tau +\lambda \tau }$$

(30)

Substitute Equation (30) into Equation (28), and the necessary condition for maximizing Equation (28) is that its Hessian matrix is negative definite; that is, it satisfies the following condition: $8k\mu -2k{\beta }^2-2k\mu \tau \left(2-\lambda \right)-\mu >0$. Let the three first-order partial derivatives of Equation (28) be zero, solve the equation system that comprises the above three equations, and we can then obtain the optimal $w^{JU*}$, ${\theta }^{JU*}$, and $x^{JU*}$. Then we substitute $w^{JU*}$, ${\theta }^{JU*}$, and $x^{JU*}$ into Equations (1) and (28)–(30); thus, the following proposition can be obtained.

Proposition 3. 

Under the scenario in which the producer and the retailer jointly undertake responsibilities, the member firms’ optimal decisionsof the wholesale price, the retail price, the carbon-emission-reduction intensity, the technological-innovation level, and the market demand can be calculated as follows:

$$w^{JU*}=\frac{2k\mu \left(2-\tau \right)\left(a+c\right)+2kc\left(\lambda \mu \tau -{\beta }^2\right)-a\mu }{8k\mu -2k{\beta }^2-2k\mu \tau \left(2-\lambda \right)-\mu }$$

(31)

$${\theta }^{JU*}=\frac{2k\beta \left(a-c\right)}{8k\mu -2k{\beta }^2-2k\mu \tau \left(2-\lambda \right)-\mu }$$

(32)

$$x^{JU*}=\frac{\mu \left(a-c\right)}{8k\mu -2k{\beta }^2-2k\mu \tau \left(2-\lambda \right)-\mu }$$

(33)

$$p^{JU*}=\frac{2k\mu \left(3a+c\right)-2ak\mu \tau \left(2-\lambda \right)-2k{\beta }^{2}c-a\mu }{8k\mu -2k{\beta }^2-2k\mu \tau \left(2-\lambda \right)-\mu }$$

(34)

$$q^{JU*}=\frac{2k\mu \left(a-c\right)}{8k\mu -2k{\beta }^2-2k\mu \tau \left(2-\lambda \right)-\mu }$$

(35)

$${\pi }_r^{JU}=\frac{4k^2{\mu }^2\left(1-\tau +\lambda \tau \right){(a-c)}^2}{{[8k\mu -2k{\beta }^2-2k\mu \tau \left(2-\lambda \right)-\mu ]}^2}$$

(36)

$${\pi }_m^{JU}=\frac{k\mu \left(8k\mu -2k{\beta }^2-4k\mu \tau -\mu \right){(a-c)}^2}{{[8k\mu -2k{\beta }^2-2k\mu \tau \left(2-\lambda \right)-\mu ]}^2}$$

(37)

$${\pi }_s^{JU}=\frac{k\mu \left[12k\mu -2k{\beta }^2-4k\mu \tau \left(2-\lambda \right)-\mu \right]{(a-c)}^2}{{[8k\mu -2k{\beta }^2-2k\mu \tau \left(2-\lambda \right)-\mu ]}^2}$$

(38)

$$V_r^{JU}=\frac{2k^2{\mu }^2\left(2-\tau +\lambda \tau \right){(a-c)}^2}{{[8k\mu -2k{\beta }^2-2k\mu \tau \left(2-\lambda \right)-\mu ]}^2}$$

(39)

$$V_m^{JU}=\frac{k\mu {(a-c)}^2}{8k\mu -2k{\beta }^2-2k\mu \tau \left(2-\lambda \right)-\mu }$$

(40)

$$V_s^{JU}=V_r^{JU}+V_m^{JU}=\frac{k\mu \left[12k\mu -2k{\beta }^2-2k\mu \tau \left(3-2\lambda \right)-\mu \right]{(a-c)}^2}{{[8k\mu -2k{\beta }^2-2k\mu \tau \left(2-\lambda \right)-\mu ]}^2}$$

(41)

5. Analysis of Equilibrium Results

In this section, we comparatively analyze the optimal decisions under three social-responsibility-undertaking scenarios, that is, those in which the producer, the retailer, or both of them agree to undertake social responsibility. Comparing the optimal decisions under three modes, the impact of social-responsibility-undertaking modes on the optimal decisions of low-carbon supply chain firms can be revealed.

Theorem 1. 

The influence ofsupply-chain member firms’ social-responsibility consciousness on optimal equilibrium decisions of the low-carbon supply chain. According to the optimal equilibrium-decision results of the low-carbon supply chain under three social-responsibility-undertakingmodes, the following results can be observed:$\frac{\partial {\theta }^L}{\partial \tau }>0$,$\frac{\partial x^L}{\partial \tau }>0$, and$\frac{\partial q^L}{\partial \tau }>0$. When $\frac{{\beta }^2}{\mu }>1$,then$\frac{\partial p^L}{\partial \tau }>0$; otherwise, $\frac{\partial p^L}{\partial \tau }<0$,$L=\left\{MU,RU,JU\right\}$.

The proof is given in Appendix B.

Theorem 1 shows that, under three modes of social responsibility undertaking, with the increasing concern of supply chain members firms on social responsibility, the carbon-emission reduction per unit product increased, the technological-innovation level of products improved, and the market sales of low-carbon products increased; while the changing trend of retail price and wholesale price of products depends on the size of carbon coefficient of products, when the carbon coefficient of products is greater than 1, the retail price of products rise; and when the carbon coefficient of products is less than 1, the retail price of products fall.

Theorem 1 reveals that, under three social-responsibility-undertaking modes, the CSR-concern behavior of member firms of the low-carbon supply chain all have a strengthening effect on carbon-emission-reduction intensity and technological-innovation level of producers. When the carbon coefficient of the product is large, the retail price is increasing with the increasing concern of social responsibility, at this time because the positive impact of the increasing of carbon-emission-reduction intensity and technological-innovation level exceeds the negative impact of the increasing of retail price on market demand, so the market demand still increasing. Du et al. [10] and Chen and Hao [15] respectively confirmed that enhancing the consumers’ low-carbon preference and charge carbon emission tax can increase carbon-emission reduction, the research of this paper further found that enhancing the social-responsibility concern of member firms in the low-carbon supply chain can promote the carbon-emission reduction.

Theorem 2. 

The influence of social-responsibility consciousness on optimal profit or social welfare of member firms of the low-carbon supply chain. According to the optimal profit or social welfare of the low-carbon supply chain’s member firms under three social-responsibility-undertaking modes, the following results can be observed:$\frac{\partial {\pi }_s^L}{\partial \tau }>0$,$\frac{\partial V_s^L}{\partial \tau }>0$, and$L=\left\{MU,RU,JU\right\}$. Moreover,$\frac{\partial {\pi }_r^{MU}}{\partial \tau }>0$,$\frac{\partial {\pi }_m^{MU}}{\partial \tau }<0$, and$\frac{\partial V_m^{MU}}{\partial \tau }>0$. Furthermore,$\frac{\partial {\pi }_m^{RU}}{\partial \tau }>0$,$\frac{\partial V_r^{RU}}{\partial \tau }>0$, andif$\frac{{\beta }^2}{\mu }>\frac{4k\tau -1}{2k}$, then$\frac{\partial {\pi }_r^{RU}}{\partial \tau }>0$. If$\frac{{\beta }^2}{\mu }>\frac{4k\tau +2k{\lambda }^2\tau +\lambda -6k\lambda \tau -4k\lambda -1}{2k\left(1-\lambda \right)}$, then$\frac{\partial {\pi }_r^{JU}}{\partial \tau }>0$; if$\lambda <\frac{8k\mu -2k{\beta }^2-4k\mu \tau -\mu }{8k\mu -2k{\beta }^2-2k\mu \tau -\mu }$, then$\frac{\partial {\pi }_m^{JU}}{\partial \tau }>0$;$\frac{\partial V_r^{JU}}{\partial \tau }>0$, and$\frac{\partial V_m^{JU}}{\partial \tau }>0$.

The proof is given in Appendix C.

Theorem 2 shows that, under three modes of social-responsibility undertaking, with the increasing concern of the supply chain members in regard to social responsibility, the whole profit of the low-carbon supply chain increases, and the total welfare of the low-carbon supply chain increases. Under the scenario of the producer undertaking social responsibility, with the increasing of the social-responsibility-concern degree of the producer, the profit of the retailer increases, the profit of the producer decreases, and the social welfare of the producer increases. Under the retailer undertaking social responsibility scenario, with the increasing of social-responsibility-concern degree of the retailer, the net profit of the producer and the social welfare of the retailer all increase, when the carbon coefficient of the products is larger than a certain critical value, the profit of the retailer increase. Under the scenario in which the producer and retailer are jointly undertaking social responsibility, the social welfare of the producer and the retailer increases; when the carbon coefficient of the products is larger than a certain critical value, the net profit of the retailer increases; and when the producer’s share of social responsibility is less than a certain critical value, the net profit of the producer increases.

Theorem 2 reveals that, when the producer carries out technological innovation aimed at reducing costs, under the three modes of social-responsibility undertaking, member firms’ social-responsibility behaviors all improve the whole profit of the low-carbon supply chain and the whole welfare level of the low-carbon supply chain. Under the producer undertaking social responsibility mode, the producer improves its social welfare by giving benefits to retailers, leading to the increase of the retailer’s net profit and the decrease of the producer’s net profit. Under the retailer-undertaking-social-responsibility mode, the retailer improve its social welfare by giving benefits to the producer, thus leading to the increase of the producer’s net profit; when the carbon coefficient is large, according to Theorem 1, the retailer’s social-responsibility consciousness makes the market demand increase, which further leads to the increase of the retailer’s net profit, because this positive effect is upstream the negative effect of undertaking social responsibility on the retailer’s net profit, so the net profit of the retailer still increases. Under the mode of jointly undertaking social responsibility, when the carbon coefficient and the producer’s share of social responsibility meet certain conditions, the positive effect of social-responsibility consciousness on the net profits of the producer and the retailer upstream the negative effect of undertaking social responsibility, so the net profits of the producer and the retailer still increase, and the social welfare of the producer and retailer all increase. Du et al. [7] found that enhancing consumers’ low-carbon consciousness can improve the whole profit of the supply chain; this research further revealed that enhancing member firms’ concern of social responsibility can also improve the whole profit of the low-carbon supply chain.

Theorem 3. 

The influence of social-responsibility-undertaking modes on optimal equilibrium decisions of the low-carbon supply chain. Based on the a comparison among the three social-responsibility-undertaking modes, the following relationship can be obtained:${\theta }^{RU}>{\theta }^{JU}>{\theta }^{MU}$,$x^{RU}>x^{JU}>x^{MU}$, and$q^{RU}>q^{JU}>q^{MU}$ If$\frac{{\beta }^2}{\mu }>1$, then$p^{RU}>p^{JU}>p^{MU}$.

The proof is given in Appendix D.

Theorem 3 shows that the products’ carbon-emission reduction, technological-innovation level, and market demand of the low-carbon supply chain are all the highest under the mode of retailer undertaking social responsibility, followed by the mode of jointly undertaking social responsibility mode; and they are all the lowest in the mode of the producer undertaking social responsibility. Under the three social-responsibility-undertaking modes, the order of product retail prices depends on the carbon coefficient of products, when the carbon coefficient of products is greater than 1, the retail price of low-carbon products is the highest under the mode of the retailer undertaking social responsibility, followed by the mode of jointly undertaking social responsibility, and the lowest under the mode of the producer undertaking social responsibility mode.

This conclusion can be easily understood as follows. The producer needs input costs to engage in technological innovation. Under the mode of the producer undertaking social responsibility, the producer improves its social welfare by giving benefits to the retailer, and in order not to reduce its own net profit too much, it will choose a lower level of carbon intensity and technological innovation, which, in turn, leads to lower market demand; meanwhile, under the retailer-undertaking-social-responsibility mode, the opposite occurs. When the carbon coefficient of products is large, the products’ market demand will increase greatly due to the concern of the social responsibility by supply chain member firms, under the retailer undertaking social responsibility mode, it gives benefit to the producer and choose a higher retail price to compensate, while under the producer undertaking social responsibility mode, the opposite occurs. Previous studies rarely compared the effects of the three modes of social-responsibility undertaking and did not explore the impact of undertaking social responsibility on producers’ technological-innovation decision. Therefore, this conclusion reveals that the retailer-undertaking-social-responsibility mode is the most conducive to motivate the producer to carry out technological innovation and carbon-emission reduction, which is an extension of the existing research.

Theorem 4. 

The influence of social-responsibility-undertaking modes on the optimal net profits of the member firms of the low-carbon supply chain. Based on a comparison of net profits under three social-responsibility-undertaking modes, the following relationship can be obtained:${\pi }_m^{RU}>{\pi }_m^{JU}>{\pi }_m^{MU}$. When the following condition$\tau \in \left({\tau }_1,{\tau }_2\right)$is true,${\pi }_r^{RU}>{\pi }_r^{MU}$, where ${\tau }_1=\frac{5k\mu -2k{\beta }^2-\mu -\sqrt{4{\beta }^2{k}^2\mu -7k^2{\mu }^2+2k{\mu }^2}}{2k\mu }$and ${\tau }_2=\frac{5k\mu -2k{\beta }^2-\mu +\sqrt{4{\beta }^2{k}^2\mu -7k^2{\mu }^2+2k{\mu }^2}}{2k\mu }$; when the condition$\lambda <{\lambda }_1$is true,${\pi }_r^{MU}>{\pi }_r^{JU}$, where${\lambda }_1=\frac{\left(4k\mu -2k{\beta }^2-\mu \right)\left(8k\mu -4k\mu \tau -2k{\beta }^2-\mu \right)-\left(1-\tau \right)4k^2{\mu }^2\tau }{4k^2{\mu }^2\tau }$; and when the condition$\lambda <{\lambda }_2$is true,${\pi }_r^{JU}>{\pi }_r^{RU}$, where${\lambda }_2=\frac{\left(4k\mu -2k{\beta }^2-\mu \right)\left(8k\mu -4k\mu \tau -2k{\beta }^2-\mu \right)}{4k^2{\mu }^2\tau \left(1-\tau \right)}$.

The proof is given in Appendix E.

Theorem 4 shows that, among the three social-responsibility-undertaking modes, the producer’s net profit is the highest under the mode wherein the retailer undertakes social responsibility, followed by the mode of jointly undertaking social responsibility; and it is the lowest under the mode of the producer undertaking social responsibility. Among the three social-responsibility-undertaking modes, the relative size of the retailer’s net profit is not fixed, and it is related to the social-responsibility-concern degree of the member firms and the share of the producer’s social responsibility. When the social-responsibility concern of member firms lies in a certain range, the retailer’s net profit is higher under the retailer-undertaking-social-responsibility mode than under the producer-undertaking-social-responsibility mode. When the producer’s share of social responsibility is lower than a certain threshold ${\lambda }_1$, the retailer’s net profit is higher under the mode of the producer undertaking social responsibility than under the mode of jointly undertaking social responsibility; when the producer’s share of social responsibility is lower than a certain threshold ${\lambda }_2$, the retailer’s net profit is higher under the mode of jointly undertaking social responsibility than that under the mode of the retailer undertaking social responsibility.

Theorem 4 reveals that, from the perspective of improving its own net profits, the producer prefers the mode of the retailer undertaking social responsibility, while the retailer’s preferred mode of social-responsibility undertaking depends on the level of social-responsibility concern of member firms and the share of social responsibility undertaken by the producer. Combined with Proposition 3, it can be found that the retailer-undertaking-social-responsibility mode not only conforms to the preference of the dominant producer in the supply chain, but can also better improve the carbon-emission-reduction intensity and technological-innovation level of products and expand product market sales, so as to better realize the goal of developing low-carbon economy. For the retailer, economic profits should not be the only goal to pursue; initiatively undertaking social responsibilities is more conducive to the stable and healthy development of the low-carbon supply chain. Meanwhile, the retailer can optimize its own net profits by adjusting its social-responsibility concern.

Theorem 5. 

Based on the comparison of the producer’s and the retailer’s net profits under the modes of the producer or the retailer undertaking social responsibility, the following results can be obtained:${\pi }_r^{RU}<{\pi }_m^{RU}$; when$\tau \to 0$, ${\pi }_r^{MU}<{\pi }_m^{MU}$;when $\lambda \tau \to 0$, ${\pi }_r^{JU}<{\pi }_m^{JU}$.

The proof is given in Appendix F.

Theorem 5 shows that, under the retailer-undertaking-social-responsibility mode, the net profits of the retailer are lower than those of the producer. Under the producer-undertaking-social-responsibility mode, the relative size of net profit between the retailer and the producer depends on the producer’s degree of concern for social responsibility. When the producer’s degree of concern for social responsibility approaches zero, the net profit of the retailer is lower than that of the producer; otherwise, the net profit of the retailer is higher than that of the producer. Under the mode of jointly undertaking social responsibility, the relative size of net profits between the retailer and the producer is not only related to the social-responsibility concern of member firms, but it is also related to the ratio of social-responsibility undertaking by the producer. When the member firms’ degree of concern for social responsibility approaches zero, or the producer’s ratio of social responsibility approaches zero, the net profit of the retailer is lower than that of the producer; otherwise, the net profit of the retailer is higher than that of the producer.

Theorem 5 reveals that, under the retailer-undertaking-social-responsibility mode, the retailer benefits the producer, and since the producer is the dominant player in the low-carbon supply chain, the net profit of the retailer is lower than that of the producer. Under the producer-undertaking-social-responsibility mode, when the producer’s degree of concern for social responsibility is lower, it benefits the retailer to a lesser extent, and since the producer is the dominant player in the supply chain, the net profit of the retailer is still lower than that of the producer. However, when the producer’s concern of social responsibility is higher, the retailer’s net profit is higher than that of the producer because the producer gives more benefits to the retailer due to its undertaking social responsibility. Under the mode of jointly undertaking social responsibility, when the member firms’ sense of social responsibility is high and the producer is undertaking a larger ratio of social responsibility, the producer gives more profits to the retailer, resulting in a higher net profit for the retailer than for the producer. On the contrary, if the producer gives less profits to the retailer, it can use the dominant power to obtain higher profit undertaking than the retailer.

Theorem 6. 

Based on the comparison between the producer’s and the retailer’s social welfare under three social-responsibility-undertaking modes, the following results can be obtained:$V_r^{JU}<V_r^{RU},$and$V_m^{JU}>V_m^{MU}$. When$\tau >\frac{4k\mu -2k{\beta }^2-\mu }{2k\mu }$,$V_r^{JU}>V_m^{JU}$;otherwise,$V_r^{JU}<V_m^{JU}$.

The proof is given in Appendix G.

Theorem 6 shows that from the perspective of maximizing the retailer’s social welfare, the retailer undertaking social responsibility mode is better than the jointly undertaking social responsibility mode. From the perspective of maximizing the producer’s social welfare, the mode of jointly undertaking social responsibility is better than the mode of the producer undertaking social responsibility. Under the mode of jointly undertaking social responsibility mode, when the social-responsibility concern of member firms is greater than a certain threshold, the social welfare of the retailer is higher than that of the producer; otherwise, the social welfare of the retailer is lower than that of the producer.

Theorem 6 reveals that, in terms of maximizing their own social welfare, the producer and the retailer have different preferences for the social-responsibility-undertaking mode of the low-carbon supply chain. The retailer prefers the mode wherein the retailer undertakes social responsibility, while the producer prefers the mode of jointly undertaking social responsibility. Under the mode of jointly undertaking social responsibility, when member firms pay more attention to social responsibility, according to Proposition 1, the carbon-emission-reduction intensity and technological-innovation level of products are also higher; although the producer is in a dominant position in the low-carbon supply chain, it needs to invest high fixed costs in carbon-emission reduction and technological innovation, thus leading to excessive losses of the producer’s profits caused by undertaking social responsibility and ultimately resulting in its having lower social welfare than the retailer.

6. Numerical Simulation and Analysis

In this section, we verify the correctness of the above propositions and theorems through some numerical analyses. According to the data in References [16,18], the parameter values are set as follows: $a=100,c=5,\mu =30,\beta =7,k=3$, $\tau \in \left[0, 1\right],\lambda \in \left[0, 1\right]$. It was proven that, under the given parameters, the conditions for achieving the optimal value of decision objective function under different social-responsibility-undertaking modes are all met. The above conclusions are verified with numerical examples shown in

Table 2. The correlation between the technological-innovation level, $x$, and the $\tau$ and $\lambda$ under three modes of social-responsibility undertaking.

		$\mathit{\tau } = 0.1$	$\mathit{\tau } = 0.3$	$\mathit{\tau } = 0.5$	$\mathit{\tau } = 0.7$	$\mathit{\tau } = 0.9$
$x^{MU}$		7.54	8.33	9.31	10.56	12.18
$x^{RU}$		7.92	9.90	13.19	19.79	39.58
$x^{JU}$	$\lambda =0.1$	7.88	9.71	12.67	18.20	32.31
	$\lambda =0.3$	7.80	9.37	11.73	15.68	23.63
	$\lambda =0.5$	7.72	9.05	10.92	13.77	18.63
	$\lambda =0.7$	7.65	8.75	10.22	12.27	15.37
	$\lambda =0.9$	7.58	8.47	9.60	11.07	13.09

and

Table 3. The correlation between the carbon-emission-reduction intensity, $\theta$, and the $\tau$ and $\lambda$ under three modes of social-responsibility undertaking.

		$\mathit{\tau } = 0.1$	$\mathit{\tau } = 0.3$	$\mathit{\tau } = 0.5$	$\mathit{\tau } = 0.7$	$\mathit{\tau } = 0.9$
${\theta }^{MU}$		10.56	11.67	13.04	14.78	17.05
${\theta }^{RU}$		11.08	13.85	18.47	27.71	55.42
${\theta }^{JU}$	$\lambda =0.1$	11.03	13.60	17.73	25.48	45.24
	$\lambda =0.3$	10.92	13.12	16.42	21.95	33.08
	$\lambda =0.5$	10.81	12.67	15.29	19.28	26.08
	$\lambda =0.7$	10.71	12.25	14.30	17.18	21.52
	$\lambda =0.9$	10.61	11.85	13.43	15.50	18.32

and Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5.

It can be seen from Table 2 and Table 3 that, under the three modes of social responsibility undertaking, the producer’s technological-innovation level and carbon-emission-reduction intensity all increase as the concern of social responsibility increases for the member forms of the low-carbon supply chain. Under the mode of jointly undertaking social responsibility, the technological-innovation level and carbon-emission-reduction intensity of the producer decrease as the ratio of the producer’s social responsibility allocation increase. The technological-innovation level and carbon-emission-reduction intensity of the producer are both the highest under the mode of the retailer undertaking social responsibility, followed by the mode of jointly undertaking social responsibility; and they are the lowest under the mode of the producer undertaking social responsibility. Under the mode of jointly undertaking social responsibility, the smaller the ratio of the producer’s social responsibility allocation, the closer the producer’s technological-innovation level and carbon-emission-reduction intensity are to the mode of the retailer undertaking social responsibility; and the larger the ratio of the producer’s social-responsibility allocation, the closer the producer’s technological-innovation level and carbon-emission-reduction intensity are to the producer-undertaking-social-responsibility mode. These conclusions partly verified Theory 1 and Theory 3. Therefore, the government should take relevant measures to promote the awareness of social responsibility of member firms in the low-carbon supply chain. For the leader in the low-carbon supply chain, the producer should take effective measures to guide the retailer to undertake social responsibility. In this way, the producer can be effectively promoted to improve the level of technological innovation, reduce the production cost of unit product, create conditions for the promotion and popularization of low-carbon products, and improve the intensity of carbon-emission reduction, so as to better achieve the goal of developing a low-carbon economy.

As can be seen from Figure 1, when the producer undertakes social responsibility alone, the relative size of net profits of the producer and the retailer depends on the producer’s attention to social responsibility. When the producer’s degree of concern for social responsibility is lower than a certain threshold value, the net profit of the producer is higher than that of the retailer; and when the producer’s degree of concern for social responsibility is higher than the threshold, the opposite occurs. Thus, Theorem 5 is partly verified. This is because the firm’s undertaking of social responsibility is benefiting the stakeholders of the firm; in essence, since the producer is the dominant player in the low-carbon supply chain, it can use the right of prior pricing to obtain a higher profit share than the retailer. When the producer has a weak sense of social responsibility, it gives less benefit to the retailer, so its net profit is still higher than that of the retailer; meanwhile, when the producer has a strong sense of social responsibility, it gives larger benefit to the retailer, so its net profit is lower than that of the retailer.

Figure 2 shows that, under the mode of the retailer undertaking social responsibility, the producer’s net profit is always higher than the retailer’s net profit, regardless of the retailer’s social-responsibility-concern degree, and the higher the retailer’s social-responsibility-concern degree, the higher the producer’s net profit is than the retailer. This conclusion partly verifies Theorem 5. This is because, on the one hand, the producer is in a leading position in the supply chain and can use the prior pricing power to make favorable decisions to obtain a higher profit share. On the other hand, the retailer enhances its social welfare by benefiting the producer; the stronger the retailer’s sense of social responsibility, the greater the effort to benefit the producer, thus making the net-profit gap between the producer and the retailer wider.

Figure 3 shows that, under the mode of jointly undertaking social responsibility, the relative size of the net profits of the producer and the retailer is not only related to the member firms’ degree of concern for social responsibility, but it also related to the ratio of social-responsibility allocation of the producer. When the concern degree of member firms’ social responsibility tends toward zero or the producer’s ratio of social-responsibility allocation tends toward zero, the producer gives less benefit to the retailer, and because the producer is the dominant player in the low-carbon supply chain, the producer can take advantage of its dominance to obtain a higher percentage of net profit than the producer; otherwise, the opposite occurs. Thus, Theorem 5 is partly verified.

Figure 4 exhibits the whole profit of the low-carbon supply chain is the highest under the mode of the retailer undertaking social responsibility, followed by the mode of jointly undertaking social responsibility; and it is the lowest under the mode of the producer undertaking social responsibility. With the increasing concern of social responsibility of the low-carbon supply chain’s member firms, the whole profit of the low-carbon supply chain increases gradually under both the mode of the retailer undertaking social responsibility mode and the mode of the producer undertaking social responsibility, but the total profit increases faster under the mode of the retailer undertaking social responsibility. The whole profit of the low-carbon supply chain under the mode of jointly undertaking social responsibility shows a trend of gradually increasing, but the increasing rate is affected by both the degree of social responsibility concern and the ratio of social-responsibility distribution; the lower the ratio of the producer undertaking social responsibility, the higher the growth rate of the whole profit of the low-carbon supply chain. Then Theorem 2 is partly verified. Therefore, among the three modes of social-responsibility undertaking, the retailer-undertaking-social-responsibility mode is the mode that is most conducive to optimizing the whole profit of the low-carbon supply chain.

Figure 5 exhibits that the welfare of the whole low-carbon supply chain is the highest under the mode of the retailer undertaking social responsibility, followed by the mode of jointly undertaking social responsibility; and it is the lowest under the mode of the producer undertaking social responsibility. With the increasing concern of social responsibility of the supply chain’s member firms, the welfare of the whole low-carbon supply chain under the retailer-undertaking-social-responsibility mode and the producer-undertaking-social-responsibility mode gradually increases, but the increase rate under the retailer-undertaking-social-responsibility mode is faster. Under the mode of jointly undertaking social responsibility, the welfare of the whole low-carbon supply chain gradually increases, but the increase rate is affected by both of the concern of social responsibility and the ratio of social-responsibility allocation. The lower the ratio of the producer’s social-responsibility allocation is, the faster the welfare of the whole low-carbon supply chain increases; conversely, the welfare of the whole low-carbon supply chain increases slower. So, Theorem 2 is partly verified. Therefore, the retailer-undertaking-social-responsibility mode is the most conducive to optimizing the whole welfare of the low-carbon supply chain.

7. Conclusions

This study investigates the impact of social-responsibility-undertaking mode on technological innovation and carbon-reduction decision of the low-carbon supply chain’s member firms. Under three social-responsibility-undertaking modes, that is, the producer undertaking social responsibility, the retailer undertaking social responsibility, and the producer and the retailer jointly undertaking social responsibility, the decision models of member firms are established considering the technological innovation behavior of the producer, and the equilibrium solutions of the three decision models are comparatively analyzed.

Following the analytical results of the equilibrium solutions of the three decision models, this study obtained the following conclusions: (1) Under the three social-responsibility-undertaking modes, the concern of social responsibility of the supply-chain member firms can help to improve the innovation level of the producer, improve the carbon-emission-reduction intensity of the products, expand the sales volume of products, and improve the whole profits and welfare level of the low-carbon supply chain. (2) The producer’s technological-innovation level, carbon-emission-reduction intensity, and product sales volume are all the highest under the mode of the retailer undertaking social responsibility, followed by the mode of jointly undertaking social responsibility; and it is the lowest under the mode of the producer undertaking social responsibility. (3) The producer’s net profit is the highest under the mode of the retailer undertaking social responsibility, followed by the mode of jointly undertaking social responsibility mode; and it is the lowest under the mode of the producer undertaking social responsibility. Meanwhile, the rank of the retailer’s net profit under three social-responsibility-undertaking modes depends on the member firms’ degree of concern for social responsibility and the producer’s ratio of social-responsibility undertaking.

From the above conclusions, we can obtain the following management implications. Firstly, the government management departments should strengthen publicity and supervision and guide the member firms of the low-carbon supply chain to enhance their awareness of social responsibility, which is conducive to the enhancement of technological-innovation level and carbon-emission-reduction intensity, thus promoting the development of a low-carbon economy. Secondly, under the background of technological innovation carried out by the producer, adopting the retailer-undertaking-social-responsibility mode is more favorable for the sustainable development of the low-carbon supply chain. Finally, under the condition of technological innovation carried out by the producer, it is necessary to guide the retailer to undertake social responsibility; the retailer also need to change its mindset, for it is more conducive to the stable development of the low-carbon supply chain to initiatively undertaking social responsibility rather than just maximizing its own profit.

Future research can be carried out from the following aspects: Firstly, this paper assumes that the market demand is a constant, but in reality, the situation of random market demand is relatively common. Therefore, it is of great practical significance to study the impact of undertaking social responsibility on technological innovation and emission-reduction decision of the low-carbon supply chain in the case of random market demand. Secondly, this paper assumes that technological innovation and carbon-emission reduction are the independent actions of the producer, ignoring the intervention measures of government departments. Therefore, the impact of undertaking social responsibility under the government regulation on technological innovation and carbon-emission-reduction decision of the low-carbon supply chain is worth further discussion. Finally, the low-carbon supply chain in this paper comprised a single producer and a single retailer, with a relatively simple structure. In reality, the structure of the low-carbon supply chain is often complex. Therefore, the influence of social-responsibility-undertaking mode on the decision-making of the multilayer low-carbon supply chain is also worth further discussion.