Effects of Carbon Policy on Carbon Emission Reduction in Supply Chain under Uncertain Demand

Abstract

Although there are many articles on carbon emission reduction of sustainable supply chain, most of them study the carbon emission reduction efficiency of supply chain in the case of single carbon policy or demand determination. Based on previous studies, this paper considered a supply chain consisting of a single manufacturer and a single retailer in an uncertain demand market. The effects of demand randomness and different carbon policies on carbon emission reduction level and optimal decision in supply chain were studied by constructing mean-variance utility function and Stackelberg game. Due to the difficulty of data acquisition, this paper verified the equalization results by numerical simulation. The results show that: (1) cap-and-trade policy, government subsidy policy and carbon tax policy can promote the carbon emission reduction investment of supply chain, while carbon tax policy will lead to the decline of the overall profit of supply chain; (2) For the manufacturer and the retailer, adopting a strategy with a low degree of risk avoidance will increase its own profits; (3) For the supply chain as a whole, it is more advantageous for manufacturers to adopt higher risk avoidance strategies, while retailers to adopt lower risk avoidance strategies. In addition, in the conclusion, this paper puts forward management implications related to stakeholders, thus providing help for the development of sustainable supply chain.

1. Introduction

As global warming increases, climate issues are becoming a major concern for all countries. Under such circumstances, the task of carbon emissions abatement in the world is even more urgent. As early as 1997, the Kyoto Protocol was formulated with the goal of “stabilizing greenhouse gas levels in the atmosphere at an appropriate level”. Nowadays, the European Union, Japan, South Korea and more than 110 other countries have committed to becoming carbon neutral by 2050 [1]. However, although carbon reduction efforts are in order around the world, the implementation of companies and supply chains still puts a lot of pressure on them. From the economic side, companies are not willing to undertake carbon reduction efforts, because it means they have to add a larger amount of spending to invest in carbon reduction. As a result, some policies have been issued to encourage companies to actively invest in carbon reduction.

The first is the active use of government subsidies. Government subsidy means that the government subsidizes enterprises that reduce emissions or consumers who buy low-carbon products to encourage enterprises to actively carry out the activities about carbon emission reduction. In the “Climate Protection Plan 2030” issued by the German government, 40% of the replacement cost subsidy is provided for residents to switch to more environmentally friendly equipment or renewable energy heating [2]. The Chinese government’s subsidy budget for new energy vehicles in 2021 is 37.58529 billion yuan, up 234% year-on-year [3]. Subsidizing the cost of carbon emission reduction can offset some effects of environmental tax: refusing to use carbon emission reduction technology and the loss of social welfare [4].

In addition to government subsidies as an incentive policy for carbon emission reduction, carbon cap-and-trade mechanisms and carbon taxes are also widely used in various countries. The cap-and-trade mechanism is generally considered as one of the most effective market mechanisms to curb carbon emissions [5]. The EU ETS, established in 2005, allocates emission reduction targets from the Kyoto Protocol to member states and allows trading in its carbon trading system. Under this mechanism, the government usually sets an emissions cap for carbon emitting companies and allows them to buy excess carbon permits or sell excess carbon permits on the carbon trading market [6]. The data shows that since online trading was officially launched in China’s carbon market on 16 July 2021, the performance rate of the first performance cycle was 99.5%, and the total carbon quota was 4.5 billion tons, but the turnover rate was only 3%, compared with 417% in the EU carbon trading market; the trading in China’s carbon market was generally inactive [7].

As an environmental tax, the starting point of carbon tax is to solve the negative externalities of the environment and reduce the use of fossil fuels and greenhouse gas emissions. Finland was the first country in the world to levy carbon tax, which began in 1990 [8]. On 1 July 2008, British Columbia and Canada, began to collect carbon tax for end consumers, which is also the first such tax category in North America [9]. Government subsidies, cap-and-trade and carbon tax are all formulated to promote carbon emission reduction, but the policy implementation effects of these three carbon policies are not necessarily the same. In addition, the price of low-carbon products produced by carbon emission reduction in supply chain is generally higher than that of ordinary products of the same type, and the market demand for low-carbon products is unstable and random due to various reasons such as consumer preference and market competition. If there are sources of uncertainty such as demand uncertainty in the supply chain environment, the performance of the supply chain will also become uncertain [10].

Through the introduction of the above background, based on the concept of sustainable supply chain, this paper raises the research perspective from the micro level of enterprises’ response to external risks to the macro level of supply chain. In short, this paper had two main purposes. On the one hand, through the study of the supply chain’s response to different carbon policies and carbon emission reduction levels, the profit and carbon emission reduction level of the supply chain can be balanced to achieve optimal operation efficiency, so as to improve the overall sustainable competitiveness. On the other hand, it coordinates the competitive relationship within the supply chain to deal with the uncertain demand, so that the supply chain can operate under the long-term stable structure and obtain the optimal performance. In the long run, for the whole social development, both the economy and low carbon have been optimized to some extent, which is conducive to long-term sustainability. In this paper, we consider the carbon emission reduction decisions under three policies of random market demand, and mainly solve the following problems:

1.

How does the risk disturbance coefficient affect the equilibrium results and profits of supply chain members?

2.

How does the risk aversion preferences of the manufacturer and the retailer affect the equilibrium results and profits?

3.

When the government faces different types of supply chain members, what kind of policies can be adopted to encourage manufacturers to actively reduce carbon emissions?

In order to solve the above problems, this paper constructs a two-echelon supply chain Stackelberg game model composed of the manufacturer and the retailer, in which the manufacturer is the leader of the supply chain, the manufacturer undertakes the task of carbon emission reduction in the supply chain and invests in carbon emission reduction technology, and retailers wholesale products from the manufacturer and sell them to the final product market. Because of the risks in the market, the market demand is random. Faced with uncertain market demand, both manufacturers and retailers have a certain risk aversion tendency. According to different carbon emission reduction policies, this paper considers four policy scenarios: anarchic intervention, cap-and-trade policy, government subsidy policy and carbon tax policy.

The structure of the rest of the paper is as follows: The second section reviews the relevant literature. The third section is the basic assumption and model construction of the paper. The fourth section is numerical analysis, the fifth section is sensitivity analysis and the sixth section is discussion and managerial implications. The last section is conclusions and future research directions.

2. Literature Review

The theory of sustainable development originated from the concept of triple bottom line (TBL) proposed by Elkington [11], which mainly includes three dimensions: environment, economy and society. There is no doubt that TBL is the cornerstone of the sustainable supply chain studied in this paper. However, with the change of environment, the perspective of sustainable supply chain research is also changing. In recent years, in addition to the above three dimensions, many scholars have extended the research content of sustainability theory to low-carbon operation and uncertain risk management. Therefore, based on the above considerations, the literature review in this section briefly reviews relevant literature on low-carbon supply chain, carbon emission reduction policy and supply chain operation under uncertain demand.

2.1. Low Carbon Supply Chain

In order to implement sustainable development, green and low-carbon has gradually formed the situation of government guidance, enterprise implementation, social supervision and consumer trend. The research on green transportation, green economy, clean energy consumption and carbon dioxide emission has become a hot topic [12,13,14,15].

Cong et al. [16] investigated the optimal strategies for a capital constrained low-carbon supply chain. The results indicate that the green finance subsidy and the low-carbon subsidy have positive impacts on the carbon emission reduction, but the yield uncertainty has negative impacts. Zou et al. [17] discussed the impact of fair concerns of manufacturers or retailers on emission reduction rates and profit of low-carbon supply chain. Wang et al. [18] thought that retailers’ altruistic preference can improve the profits and system efficiency of small and medium-sized manufacturers in the low-carbon supply chain. Liu et al. [19] considered the emission reduction in both single-channel and dual-channel situations, and considered that the implementation of the low-carbon strategy in the supply chain depends on the substitutability of low-carbon products and the basic demand of supply chain channels. Du et al. [20] studied the influence of consumers’ low-carbon preference on low-carbon decision-making in a low-carbon supply chain. Wang et al. [21] discussed the emission reduction of manufacturers and retailers in dual supply chain with low carbon preference through contract cooperation, and explained the emission reduction and profit of supply chain under different power structures and contracts. He et al. [22] discussed the influence of consumers’ free-riding behavior on carbon emissions in the product life cycle in a two-channel closed-loop supply chain. Li et al. [23] studied the green development behavior and performance mechanism of industrial enterprises based on a partial least squares structural equation model.

The above-mentioned research on low-carbon supply chain mostly focused on the influence of internal factors of supply chains on low-carbon supply chain decision-making. This paper considers the influence of external policies on carbon emission reduction of supply chains.

2.2. Related Policies of Carbon Emission Reduction

In order to promote the steady development of carbon emission reduction, the government has successively issued a variety of policies, among which, the implementation of the following three policies has achieved considerable results: government subsidies, cap-and-trade and carbon tax policy.

Government subsidy refers to a carbon emission reduction policy in which the government subsidizes enterprises according to their carbon emission reduction investment costs or emissions reductions in order to encourage supply chain members to actively carry out carbon emission reduction tasks. Xue et al. [24] studied the influence of government subsidies based on product energy saving level on the decision-making of supply chain in social welfare and other aspects in the case of manufacturers’ R&D of energy saving and emission reduction technologies. Lou et al. [25] studied the optimal investment and pricing decision of a two-stage supply chain in the context of allowing emission trading and emission reduction technology investment, and pointed out that the government can promote emission reduction technology investment and achieve the supply chain emission reduction target by providing technology investment subsidies. Ma et al. [26] considered a supply chain system consisting of a manufacturer investing in emission reduction technology and a retailer investing in promoting green products. The study shows that government subsidies to manufacturers’ emission reduction costs can increase supply chain emission reduction and performance. Mondal and Giri [27] analyzed the closed-loop green supply chain composed of a manufacturer and two competing retailers under government subsidies, and the study showed that government subsidies can effectively mobilize the enthusiasm of supply chain emission reduction.

Cap-and-trade policies and carbon taxes constrain companies to meet their carbon reduction targets through carbon caps and tax costs. Wang et al. [28] studied the impact of cost-sharing contract and joint emission reduction mode under cap-and-trade policy on production and carbon emission reduction decisions of two-echelon low-carbon supply chain. Ji et al. [29] studied the emission reduction cooperation strategies of manufacturers and retailers in the dual-channel supply chain under the cap-and-trade policy, and found that the carbon cap-and-trade policy is effective for the supply chain only when the consumer’s low-carbon preference coefficient reaches a certain value. Yang et al. [30] studied the cooperation and carbon emission reduction decisions of two supply chains composed of manufacturers and retailers under the carbon cap-and-trade policy. Ding et al. [31] analyzed the impact of carbon tax on remanufacturing enterprises’ production and carbon emission reduction decisions, and believed that increasing carbon tax would reduce consumer surplus and even cause environmental damage. Martin et al. [32] found that the collection of carbon tax helps the manufacturing industry to save electricity and reduce the energy intensity of the manufacturing industry. Wang et al. [33] studied the operation of enterprises under centralized and decentralized supply chains and the government’s decision on carbon tax. They believed that excessive carbon tax would increase the price of products in the supply chain, thus negatively affecting social welfare.

In addition to the above-mentioned impacts of individual carbon emission reduction policies, some scholars have also made some comparisons between different carbon emission reduction policies. Yuyin and Jinxi [34] believe that when the initial emission reduction level of manufacturers is low, both carbon tax and government subsidy can improve the carbon emission reduction level, but when the initial emission reduction level is high, only government subsidy policy can promote the emission reduction of manufacturers. Zakeri et al. [35] found that carbon cap-and-trade policy can improve the performance of supply chain better than carbon tax policy, but in general, the effect of carbon trading policy is more susceptible to uncertain factors. Hu et al. [36] studied carbon tax and carbon cap-and-trade policies in the closed-loop supply chain and found that carbon cap-and-trade policies are more suitable for remanufacturing enterprises. This paper compares the effects of these three policies on carbon emission reduction.

2.3. Supply Chain Operation under Stochastic Demand

Since the market demand is fluctuating, the linear market demand function cannot well fit the real market, and the random market demand will also have a certain impact on the production, transportation and carbon emission reduction of products in the supply chain. Therefore, many researchers use stochastic demand functions to study the decisions and activities of members in the supply chain.

Peng et al. [37] studied carbon reduction decisions and order quantity decisions in a dyadic supply chain under stochastic demand. Assuming stochastic demand. L. Wang et al. [38] discussed the level of supply chain emission reduction under stochastic market demand, low carbon preference and carbon tax. Jauhari et al. [39] studied the production and carbon reduction decision problem in a green closed-loop supply chain with stochastic demand. X. Chen and Wang [40] studied the optimal ordering and transportation mode decisions of retailers under different carbon reduction policies under the stochastic demand of customers. Babagolzadeh et al. [41] studied the impact of demand uncertainty and carbon tax policies on carbon emissions in the refrigerated supply chain. In the study of supply chains with stochastic demand, many scholars set the stochastic factor in market demand as a parameter that obeys a probability distribution function [42,43]. In addition, some scholars also use separable multiplication to represent the stochastic demand function [44,45]. The mean-variance model used in this paper is also frequently used in the field of supply chain. Bai et al. [46] studied the impact of risk aversion and sustainability investment on two-stage supply chain coordination under a carbon tax policy in a stochastic demand setting with a mean-variance model. Q. Wang and He [47] developed a supply chain game model considering risk aversion using a mean-variance model to investigate the impact of risk aversion of suppliers and manufacturers on the performance of low-carbon supply chains.

Table 1. The differences between this study and previous studies.

Author(s)	Carbon Emission Reduction	Stochastic Demand		Carbon Emission Reduction Policy
		Mean-Variance	Others	Cap and Trade	Carbon Tax	Government Subsidy
Ji et al. (2016) [29]	✓			✓
Ding et al. (2020) [31]	✓				✓
Ma et al. (2021) [26]	✓					✓
Zakeri et al. (2014) [35]	✓			✓	✓
Yi et al. (2018) [34]	✓				✓	✓
Mondal and Giri (2020) [27]	✓			✓		✓
Hu et al. (2020) [36]	✓			✓	✓	✓
Chen et al. (2015) [40]	✓		✓	✓	✓
Wang et al. (2016) [21]	✓		✓	✓
Wang et al. (2017) [33]	✓		✓		✓
Peng et al. (2020) [37]	✓		✓
Kumar et al. (2020) [43]	✓		✓	✓
Yu et al. (2021) [44]	✓		✓	✓		✓
Chiu et al. (2016) [10]		✓
Wang et al. (2018) [47]	✓	✓
Bai et al. (2020) [46]	✓	✓			✓
Our model	✓	✓		✓	✓	✓

shows the differences between this study and previous studies.

From the above literature, it can be seen that research on the aspects of low-carbon supply chains, supply chain carbon reduction policies and supply chain decision making under stochastic demand has begun to bear fruit. However, the existing literature does not provide a comparative analysis of these three carbon reduction policies in a stochastic setting. To this end, this paper incorporated stochastic demand into the low-carbon supply chain model, used a mean-variance model to fit the fluctuation of demand, and considered the risk-averse behavior of supply chain members due to demand fluctuation. Based on this, the carbon reduction strategies of the supply chain under the three carbon reduction policies (government subsidy, cap-and-trade and carbon tax) were investigated.

3. Mathematical Model and Analysis

This paper studies the strategies of the manufacturer and the retailer in the supply chain to cope with different carbon policies and uncertain demand. The Stackelberg game model (leader-follower) is an extensive research method based on a large number of literature studies. In our model, the manufacturer should respond to different carbon policies of the government on the one hand, and set the wholesale price of products in response to uncertain demand on the other hand. Therefore, it is reasonable to evaluate the leading role of manufacturers in decision-making. Figure 1 illustrates the interaction between stakeholders in the supply chain. Therefore, we developed the models, and analyzed the optimal results.

3.1. Notations and Assumptions

We considered a distribution channel with one manufacturer and one retailer. The manufacturer produces a low carbon product and sells it to consumers through the retailer. In making pricing and investment decisions, the manufacturer acts as the Stackelberg leader, and the retailer is the follower. The following symbols are derived from

Table 2. The following notations are used to develop the models.

Variable	Definition
$\tilde{a}$	The potential market demand to the retailer
$p$	Retail price per unit of retailer
$e$	Carbon emission for unit product
$w$	Wholesale price per unit of manufacturer
$\lambda$	Carbon emission reduction cost investment coefficient
$c_e$	The trading price per unit of carbon in a carbon market
$E$	Carbon cap given by the government to the manufacturer
$\mu$	The government subsidizes manufacturers’ carbon reduction cost coefficient
$k$	Carbon tax adjustment factor to the manufacturer
$e_0$	Initial carbon emission per unit of product
$A_m$	The risk-averse level of manufacturer
$A_r$	The risk-averse level of retailer
${\pi }_m{}^I$	Profit of the manufacturer in model I, (I = A, B, C, D)
${\pi }_r{}^I$	Profit of the retailer in model I, (I = A, B, C, D)

. We made the following assumptions to the proposed models:

A1. It is assumed that the product demand function of the market is a linear function of price and carbon emission reduction, and the potential market demand is uncertain, i.e., $D=\tilde{a}-\beta p+\gamma e$, where a is intrinsic demand, $\xi$ the random variable and $\tilde{a}=a+\xi$. The parameters $\beta$ and $\gamma$ represent the selling price sensitivity and the sensitivity of carbon emission reduction level, respectively [48]. We assume that $\xi$ follows normal distribution $N\left(0,{\sigma }^2\right)$.

A2. It is assumed that in the operation of supply chain, carbon trading price, government subsidy coefficient and carbon tax coefficient are exogenous variables determined by carbon trading market and government, respectively.

A3. Similar to [27] and [49], here we consider that the manufacturer pays the cost $c_e\left(\left(e_0-e\right)D-E\right)$ to buy carbon in the carbon market, where $c_e$ is the cost of emission permit per unit and $e_0$ is the initial carbon emissions per unit product. If $\left(e_0-e\right)D-E>0$, the manufacturer has to pay $c_e\left(\left(e_0-e\right)D-E\right)$ to produce; otherwise, the extra revenue of carbon trading market is $c_e\left(E-\left(e_0-e\right)D\right)$.

A4. The function for the carbon emission reduction cost of the manufacturer is $C\left(e\right)=\frac{1}{2}\lambda e^2$, which is an extensively accepted assumption [50] because the increased carbon emission reduction efforts carry increased costs. In the government subsidy policy, the government will subsidize the manufacturer’s carbon emission reduction cost $\frac{1}{2}\mu \lambda e^2$, where $\mu$ is proportion of government subsidy.

A5. In the carbon tax policy model, the government charges a carbon tax $k\left(e_0-e\right)D$ for companies that do not actively reduce carbon emissions, where $k$ is the adjustment factor [31].

We are interested in the impact of different carbon policies implemented by governments on supply chain performance; we mainly studied the different behaviors of supply chain members when the manufacturer is a Stackelberg leader and the retailer is a follower in the case of uncertain demand. The supply chain can implement the following four scenarios:

(1)

Anarchic intervention scenario (A)—Manufacturer conducts carbon-reduction operations without government control (Figure 1A);

(2)

Cap-and-trade policy scenario (B)—Manufacturer has to buy or sell carbon through the carbon market to keep their businesses running (Figure 1B);

(3)

Government subsidy policy scenario(C)—The government subsidizes the cost of retailer that actively reduce carbon emissions (Figure 1C);

(4)

Carbon tax policy scenario (D)—The government taxes the manufacturer based on their carbon emissions (Figure 1D).

3.2. Anarchic Intervention Scenario (Model A)

When the supply chain system is not intervened by the government, the supply chain is responsible for the production of low-carbon products, and the retailer is responsible for the sales of low-carbon products. At this time, the profit functions of the manufacturer and the retailer in the supply chain system are respectively:

$${{\pi }_m}^A\left(w,e\right)=w\left(a+\xi -\beta p+\gamma e\right)-\frac{1}{2}\lambda e^2$$

(1)

$${{\pi }_r}^A\left(p\right)=\left(p-w\right)\left(a+\xi -\beta p+\gamma e\right)$$

(2)

In supply chain operations with uncertain demand, if supply chain members want to obtain optimal profits, they must consider the mean and variance of profits at the same time, so as to maximize the mean value and minimize the variance of profits. Ray and Jenamani established a mean-variance model to study the disruption risk of the supply chain, introduced the supply chain risk into the model and ensured the scientific nature of the research question [51]. We assumed a utility function $U=E\left[\pi \right]-A\sqrt{\mathrm{var}\pi }$ to represent the decision function of supply chain members, where $A\left(0<A\le 1\right)$ represents the risk aversion level of supply chain members.

The utility functions for the manufacturer and the retailer are shown below:

$${U_m}^A=w\left(a-\beta p+\gamma e\right)-\frac{1}{2}\lambda e^2-A_{m}w\sigma$$

(3)

$${U_r}^A=\left(p-w\right)\left(a-\beta p+\gamma e\right)-A_r\left(p-w\right)\sigma$$

(4)

Proposition 1 can be obtained from the utility functions of manufacturer and retailer.

Proposition 1.

If $4\beta \lambda -{\gamma }^2>0$, the Model A has the following unique solution

$$e^A*=\frac{\gamma \left(a-2 A_m \sigma +A_r \sigma \right)}{4 \beta \lambda -{\gamma }^2}$$

$$w^A*=\frac{2 \lambda \left(a-2 A_m \sigma +A_r \sigma \right)}{4 \beta \lambda -{\gamma }^2}$$

$$p^A*=\frac{\left(A_r {\gamma }^2-A_m {\gamma }^2-2 A_m \beta \lambda -A_r \beta \lambda \right) \sigma +3 a \beta \lambda }{\beta \left(4 \beta \lambda -{\gamma }^2\right)}$$

Proof. 

Solving the equation $\frac{\partial U_r^A}{\partial p}=0$, we get the optimal solution as ${\overline{p}}^A=\frac{a-A_r \sigma +e \gamma +\beta w}{2 \beta }$. Substituting the value in the manufacture’s utility function (3), we get the utility function of the manufacturer as follows:

$${U_m}^A\left(w,e\right)= \frac{w}{2}\left(a+A_r \sigma +e \gamma -\beta w\right)-\frac{e^2 \lambda }{2}-A_m \sigma w$$

□

The Hessian matrix for the manufacturer is

$$H^A=\left(\begin{array}{cc}\frac{{\partial }^2{U_m}^A}{\partial w^2}& \frac{{\partial }^2{U_m}^A}{\partial w\partial e}\\ \frac{{\partial }^2{U_m}^A}{\partial e\partial w}& \frac{{\partial }^2{U_m}^A}{\partial e^2}\end{array}\right)=\left(\begin{array}{cc}-\beta & \frac{\gamma }{2}\\ \frac{\gamma }{2}& -\lambda \end{array}\right)$$

Now, if $4\beta \lambda -{\gamma }^2>0$ and $\beta >0$, the Hessian matrix $H^A$ is strictly negative definite. Using the first order conditions for optimality i.e., $\frac{{U_m}^A}{\partial w}=0$ and $\frac{{U_m}^A}{\partial e}=0$, the optimal decisions of the manufacturer can be obtained, and by putting these decisions in the retailers’ utility function, the optimal decisions of the retailer can also be obtained, which are given in Proposition 1.

Theorem 1 (Model A).

When the supply chain system is free from government intervention, the equilibrium profits of supply chain members are given by:

$$\left\{\begin{array}{l}{{\pi }_m}^{A*}=\frac{\lambda \left(-4 {A_m}^2 {\sigma }^2+{A_r}^2 {\sigma }^2+2 A_r a \sigma +a^2\right)}{2 \left(4 \beta \lambda -{\gamma }^2\right)}\\ {{\pi }_m}^{A*}=\frac{\left(a \beta \lambda +\left(2 A_m+A_r\right) \beta \lambda \sigma -A_m \sigma {\gamma }^2\right){\mathsf{\Psi }}_1}{\beta {\left(4 \beta \lambda -{\gamma }^2\right)}^2}\end{array}\right.$$

$${\mathsf{\Psi }}_1=a \beta \lambda -A_m {\gamma }^2 \sigma +A_r {\gamma }^2 \sigma +2 A_m \beta \lambda \sigma -3 A_r \beta \lambda \sigma$$

3.3. Cap-and-Trade Policy Scenario (Model B)

When the supply chain system is under the background of cap-and-trade policy, the manufacturer purchases or sells carbon through the carbon trading market to ensure the normal operation of the enterprise. Therefore, based on Model A, the profit function of the manufacturer and retailer is improved as follows:

$${\mathsf{\Psi }}_1=a \beta \lambda -A_m {\gamma }^2 \sigma +A_r {\gamma }^2 \sigma +2 A_m \beta \lambda \sigma -3 A_r \beta \lambda \sigma$$

(5)

$${{\pi }_r}^B\left(p\right)=\left(p-w\right)\left(a+\xi -\beta p+\gamma e\right)$$

(6)

The utility functions for the manufacturer and the retailer are shown below:

$${U_m}^B=w\left(a-\beta p+\gamma e\right)-c_e\left[\left(e_0-e\right)\left(a-\beta p+\gamma e\right)-E\right]-\frac{1}{2}\lambda e^2-A_m\left[w-c_e\left(e_0-e\right)\right]\sigma$$

(7)

$${U_r}^B=\left(p-w\right)\left(a-\beta p+\gamma e\right)-A_r\left(p-w\right)\sigma$$

(8)

As with model A, proposition 2 can be obtained by backward induction:

Proposition 2.

If $4 \lambda \beta -{\beta }^2 c_e{}^2-2 \beta c_e \gamma -{\gamma }^2>0$, in cap-and-trade policy, the optimal decisions are given by

$$e^B*=\frac{\left(\gamma +\beta c_e\right) \left(a-2 A_m \sigma +A_r \sigma -\beta c_e e_0\right)}{4 \lambda \beta -{\beta }^2 {c_e}^2-2 \beta c_e \gamma -{\gamma }^2}$$

$$w^B*=\frac{[\left(2 A_m-A_r\right)\sigma -a] \left(\beta {c_e}^2+\gamma c_e-2 \lambda \right)-\beta e_0 {c_e}^2 \gamma -e_0 c_e {\gamma }^2+2 \beta e_0 \lambda c_e}{4 \lambda \beta -{\beta }^2 {c_e}^2-2 \beta c_e \gamma -{\gamma }^2}$$

$$p^B*=\frac{\left(3 \lambda \beta -{\beta }^2 {c_e}^2-\gamma \beta c_e\right) a-\left({\mathsf{\Psi }}_2 \sigma +e_0 {\beta }^2 {c_e}^2 \gamma -e_0 \lambda {\beta }^2 c_e+e_0 \beta c_e {\gamma }^2\right)}{\beta \left(4 \lambda \beta -{\beta }^2 {c_e}^2-2 \beta c_e \gamma -{\gamma }^2\right)}$$

where ${\mathsf{\Psi }}_2=A_m {\gamma }^2-A_r {\gamma }^2-A_m {\beta }^2 {c_e}^2+2 A_m \beta \lambda +A_r \beta \lambda -A_r \beta c_e \gamma$

The proof process of Proposition 2 is similar to the Proposition 1, so it will not be repeated.

Theorem 2 (Model B).

When the supply chain system is under the background of cap-and-trade policy, the equilibrium profits of supply chain members are given by:

$$\left\{\begin{array}{l}{{\pi }_m}^{B*}=\frac{\lambda [{\left(a+A_r \sigma -\beta c_e e_0\right)}^2-4{A_m}^2{\sigma }^2]}{2 \left(4 \lambda \beta -{\beta }^2 {c_e}^2-2 \beta c_e \gamma -{\gamma }^2\right)}+c_{e}E\\ {{\pi }_r}^{B*}=\frac{{\mathsf{\Psi }}_3\left[A_m\sigma {\left(\gamma +\beta c_e\right)}^2+\beta \lambda \left(\beta c_e{e}_0-a\right)-\sigma \beta \lambda \left(2A_m+A_r\right)\right]}{\beta {\left({\beta }^2 {c_e}^2+2 \beta c_e \gamma -4 \lambda \beta +{\gamma }^2\right)}^2}\end{array}\right.$$

$${\mathsf{\Psi }}_3={\left(\gamma +\beta c_e\right)}^2\left(A_r-A_m\right)\sigma +\beta \lambda \sigma \left(2A_m-3A_r\right)+a \beta \lambda -{\beta }^2 c_e e_0 \lambda$$

3.4. Government Subsidy Policy Scenario (Model C)

In the case of government-subsidized supply chain operation, the government will bear part of the cost of carbon emission reduction in order to promote manufacturers to actively reduce carbon emission. Therefore, in Model C, the profit functions of the manufacturer and the retailer are shown as follows:

$${{\pi }_m}^C\left(w,e\right)=w\left(a+\xi -\beta p+\gamma e\right)-\frac{1}{2}\left(1-\mu \right)\lambda e^2$$

(9)

$${{\pi }_r}^C\left(p\right)=\left(p-w\right)\left(a+\xi -\beta p+\gamma e\right)$$

(10)

The utility functions for the manufacturer and the retailer are shown below:

$${U_m}^C=w\left(a-\beta p+\gamma e\right)-\frac{1}{2}\left(1-\mu \right)\lambda e^2-A_{m}w\sigma$$

(11)

$${U_r}^C=\left(p-w\right)\left(a-\beta p+\gamma e\right)-A_r\left(p-w\right)\sigma$$

(12)

As with model A, Proposition 3 can be obtained by backward induction:

Preposition 3.

If $4 \beta \lambda \left(1-\mu \right)-{\gamma }^2>0$, in Government subsidy policy, the optimal decisions are given by

$$e^{C*}=\frac{\gamma \left(a-2 A_m \sigma +A_r \sigma \right)}{4 \beta \lambda -{\gamma }^2-4 \beta \lambda \mu }$$

$$w^{C*}=\frac{2 \lambda \left(\mu -1\right) \left(a-2 A_m \sigma +A_r \sigma \right)}{{\gamma }^2-4 \beta \lambda +4 \beta \lambda \mu }$$

$$p^{C*}=\frac{\left(A_m {\gamma }^2-A_r {\gamma }^2+2 A_m \beta \lambda +A_r \beta \lambda -2 A_m \beta \lambda \mu -A_r \beta \lambda \mu \right) \sigma +3 a \beta \lambda \mu -3 a \beta \lambda }{\beta \left({\gamma }^2-4 \beta \lambda +4 \beta \lambda \mu \right)}$$

The proof process of Proposition 3 is similar to the Proposition 1, so it will not be repeated.

Theorem 3 (Model C).

When the supply chain system is under the background of government subsidy policy, the equilibrium profits of supply chain members are given by:

$$\left\{\begin{array}{l}{\pi }_m{}^{C*}=\frac{\lambda \left(\mu -1\right) \left(-4 {A_m}^2 {\sigma }^2+{A_r}^2 {\sigma }^2+2 A_r a \sigma +a^2\right)}{2 \left({\gamma }^2-4 \beta \lambda +4 \beta \lambda \mu \right)}\\ {\pi }_r{}^{C*}=\frac{\left[\left(2 A_m \beta \lambda +A_r \beta \lambda \right)\left(1-\mu \right)\sigma -A_m {\gamma }^2\sigma +a \beta \lambda \left(1-\mu \right)\right]{\mathsf{\Psi }}_4}{\beta {\left({\gamma }^2-4 \beta \lambda +4 \beta \lambda \mu \right)}^2}\end{array}\right.$$

$${\mathsf{\Psi }}_4=\left(3 A_r \beta \lambda -2 A_m \beta \lambda \right) \left(\mu -1\right)\sigma +\left(A_r-A_m \right){\gamma }^2\sigma +a \beta \lambda \left(1-\mu \right)$$

3.5. Carbon Tax Policy Scenario (Model D)

When the supply chain system is under the carbon tax policy, the government will charge carbon tax $k\left(e_0-e\right)\left(\tilde{a}-\beta p+\gamma e\right)$ according to the carbon emissions of the manufacturer. Therefore, based on Model A, we improved the profit function of the manufacturer as shown below.

$${{\pi }_m}^D\left(w,e\right)=\left(w-k\left(e_0-e\right)\right)\left(a+\xi -\beta p+\gamma e\right)-\frac{1}{2}\lambda e^2$$

(13)

$${{\pi }_r}^D\left(p\right)=\left(p-w\right)\left(a+\xi -\beta p+\gamma e\right)$$

(14)

The utility functions for the manufacturer and the retailer are shown below:

$${U_m}^D=\left(w-k\left(e_0-e\right)\right)\left(a-\beta p+\gamma e\right)-\frac{1}{2}\lambda e^2-A_m\left[w-k\left(e_0-e\right)\right]\sigma$$

(15)

$${U_r}^D=\left(p-w\right)\left(a-\beta p+\gamma e\right)-A_r\left(p-w\right)\sigma$$

(16)

As with Model A, Proposition 4 can be obtained by backward induction:

Proposition 4.

If $4 \lambda \beta -{\beta }^2 k^2-2 \beta \gamma k-{\gamma }^2>0$, in the carbon tax policy, the optimal decisions are given by

$$e^{D*}=\frac{\left(\gamma +\beta k\right) \left(a-2 A_m \sigma +A_r \sigma -\beta e_0 k\right)}{4 \lambda \beta -{\beta }^2 k^2-2 \beta \gamma k-{\gamma }^2}$$

$$w^{D*}=\frac{\left(2 A_m-A_r\right) \left(\beta k^2+\gamma k-2 \lambda \right)\sigma +\left(2 a+2 \beta e_0 k\right) \lambda -k \left(a+e_0 \gamma \right) \left(\gamma +\beta k\right)}{4 \lambda \beta -{\beta }^2 k^2-2 \beta \gamma k-{\gamma }^2}$$

$$p^{D*}=\frac{\left(A_r {\gamma }^2-A_m {\gamma }^2+A_m {\beta }^2 k^2-2 A_m \beta \lambda -A_r \beta \lambda +A_r \beta \gamma k\right)\sigma +\left(e_0 k {\beta }^2+3 a \beta \right) \lambda -\beta k \left(a+e_0 \gamma \right) \left(\gamma +\beta k\right)}{\beta \left(4 \lambda \beta -{\beta }^2 k^2-2 \beta \gamma k-{\gamma }^2\right)}$$

The proof process of Proposition 4 is similar to Proposition 1, so it will not be repeated.

Theorem 4 (Model D).

When the supply chain system is under the background of carbon tax policy, the equilibrium profits of supply chain members are given by:

$$\left\{\begin{array}{l}{\pi }_m{}^{D*}=\frac{\lambda \left[\left({A_r}^2-4 {A_m}^2\right) {\sigma }^2+\left(2 A_r a-2 A_r \beta e_0 k\right) \sigma +{\left(a-\beta e_0 k\right)}^2\right]}{2 \left(4 \lambda \beta -{\beta }^2 k^2-2 \beta \gamma k-{\gamma }^2\right)}\\ {\pi }_r{}^{D*}=\frac{\left[A_m {\left(\gamma +\beta k\right)}^2\sigma -\beta \lambda \left(2 A_m+A_r\right)\right]{\mathsf{\Psi }}_5}{\beta {\left({\beta }^2 k^2+2 \beta \gamma k-4 \lambda \beta +{\gamma }^2\right)}^2}\end{array}\right.$$

$${\mathsf{\Psi }}_5=\left(A_m-A_r\right) {\left(\gamma +\beta k\right)}^2\sigma -\beta \lambda \left(2 A_m-3 A_r\right)\sigma -a \beta \lambda +{\beta }^2 e_0 k \lambda$$

4. Numerical Analysis

Through the solving process in the previous section, we obtained the optimal equilibrium decision and profit as shown in Theorem 1–4 and Proposition 1–4. Due to the complexity of the optimal solution, the performance of the four models in different values of parameters is analyzed numerically in this section. We made some adjustments to data sets similar to [27] and [52].

Here, Set 1–5 studies the influence of uncertain demand on supply chain members’ decision-making and supply chain performance. Sets 1–3 respectively indicate that uncertainty factor σ is at a low, medium and high level. Set 4 indicates that the manufacturer’s risk aversion degree is lower than that of the retailer, i.e., the manufacturer is a risk-preference enterprise; Set 5 indicates that the manufacturer’s risk aversion degree is higher than that of the retailer, i.e., the retailer is a risk-preference enterprise. Based on this, we test the necessary conditions for the existence of optimal solutions of four models, that is, the Hessian matrices are strictly negative definite. We use MATLAB R2020a software to derive the optimal decision and supply chain profit as shown in Theorem 1–4.

$$\mathbf{Set} \mathbf{1}: a=500;\beta =0.8;\gamma =0.2;\lambda =300;A_m=0.5;A_r=0.5;k=0.5;e_0=1;c_e=2;E=500;\mu =0.5;\sigma =30$$

$$\mathbf{Set} \mathbf{2}: a=500;\beta =0.8;\gamma =0.2;\lambda =300;A_m=0.5;A_r=0.5;k=0.5;e_0=1;c_e=2;E=500;\mu =0.5;\sigma =70$$

$$\mathbf{Set} \mathbf{3}: a=500;\beta =0.8;\gamma =0.2;\lambda =300;A_m=0.5;A_r=0.5;k=0.5;e_0=1;c_e=2;E=500;\mu =0.5;\sigma =120$$

$$\mathbf{Set} \mathbf{4}: a=500;\beta =0.8;\gamma =0.2;\lambda =300;A_m=0.1;A_r=0.8;k=0.5;e_0=1;c_e=2;E=500;\mu =0.5;\sigma =70$$

$$\mathbf{Set} \mathbf{5}: a=500;\beta =0.8;\gamma =0.2;\lambda =300;A_m=0.8;A_r=0.1;k=0.5;e_0=1;c_e=2;E=500;\mu =0.5;\sigma =70$$

From

Table 3. Optimal results under uncertainty.

Policy	Set	${\mathit{w}}^{*}$	${\mathit{e}}^{*}$	${\mathit{p}}^{*}$	${\mathit{\pi }}_{\mathit{m}}{}^{*}$	${\mathit{\pi }}_{\mathit{r}}{}^{*}$	${\mathit{\pi }}_{\mathit{s}\mathit{c}}{}^{*}$
Anarchic intervention	Set 1	303.138	0.102	454.707	41,303	20,652	61,955
	Set 2	290.637	0.097	435.956	43,959	21,980	65,939
	Set 3	275.011	0.092	412.517	46,752	23,377	70,129
	Set 4	338.764	0.113	446.896	48,274	15,409	63,683
	Set 5	246.885	0.082	431.578	38,205	28,582	66,787
Cap-and-trade policy	Set 1	303.329	0.909	454.903	42,183	20,653	62,836
	Set 2	290.862	0.872	436.165	44,838	21,976	66,814
	Set 3	275.278	0.825	412.742	47,628	23,365	70,993
	Set 4	338.860	1.017	447.057	49,157	15,424	64,581
	Set 5	247.227	0.740	431.831	39,079	28,555	67,634
Government subsidy policy	Set 1	303.150	0.202	454.725	41,304	20,654	61,958
	Set 2	290.649	0.194	435.974	43,961	21,982	65,942
	Set 3	275.023	0.183	412.534	46,754	23,378	70,132
	Set 4	338.778	0.226	446.917	48,276	15,411	63,687
	Set 5	246.896	0.165	431.593	38,207	28,584	66,791
Carbon tax policy	Set 1	303.337	0.303	454.831	41,252	20,633	61,885
	Set 2	290.839	0.290	436.081	43,907	21,960	65,866
	Set 3	275.216	0.275	412.642	46,698	23,354	70,052
	Set 4	338.958	0.339	447.021	48,220	15,394	63,614
	Set 5	247.094	0.247	431.703	38,155	28,557	66,711

, we noted that a high level of demand uncertainty performs better than a low level when both manufacturers and retailers are risk-neutral decision makers. In this scenario, consumers can obtain low-carbon products at a relatively low price, and the low selling price increases the market demand. As a result, total profits across the supply chain increase. When considering manufacturers’ and retailers’ risk aversion behaviors, we set a medium level of demand uncertainty, and note that a high level of manufacturers’ risk aversion is profitable for retailers and the whole supply chain. The reason behind this result is that when the retailer has a low degree of risk aversion, it will lower the selling price of the product and increase the demand, thus increasing the profit of the retailer and the whole supply chain. However, when the retailer’s risk aversion is much higher than that of the manufacturer, the retailer will raise the selling price of the product to reduce the demand, resulting in a decrease in the total profit of the supply chain. Therefore, it is better for manufacturers and retailers to adopt a risk preference strategy rather than a risk aversion strategy in the face of uncertain demand. As for the supply chain as a whole, manufacturers, as the leader of the supply chain, should adopt the risk avoidance strategy, while retailers should adopt the risk preference strategy, which is beneficial to the interests of the supply chain. Thus, from Table 3, we have the following understanding of manufacturers’ and retailers’ wholesale prices, carbon emission reduction levels, selling prices and supply chain profits: (1) Optimal decision and optimal profit have the same trend in the four models we studied. With the increase of demand uncertainty factor σ, wholesale price, selling price and carbon emission reduction level are constantly decreasing, while manufacturer’s profit, retailer’s profit and total profit of supply chain are constantly increasing. (2) When the demand uncertainty is set at a medium level and the manufacturer’s risk aversion coefficient is much lower than that of the retailer, the manufacturer’s profit is better than that gained when the manufacturer and retailer are risk aversion neutral, but the supply chain’s profit will decrease. (3) When the demand uncertainty is set at a medium level and the retailer’s risk aversion coefficient is much lower than that of the manufacturer, the retailer’s profit and the total profit of the supply chain will increase relative to other conditions. (4) In the carbon emission reduction level of the four models studied, cap-and-trade policy, government subsidy policy and carbon tax policy are superior to the supply chain without government intervention.

Table 4. Optimal results under the three policies.

Policy	Set	${\mathit{w}}^{*}$	${\mathit{e}}^{*}$	${\mathit{p}}^{*}$	${\mathit{\pi }}_{\mathit{m}}{}^{*}$	${\mathit{\pi }}_{\mathit{r}}{}^{*}$	${\mathit{\pi }}_{\mathit{s}\mathit{c}}{}^{*}$
Cap-and-trade policy	$c_e=1.5$	290.951	0.678	436.185	44,596	21,958	66,554
	$c_e=2$	290.862	0.872	436.165	44,838	21,976	66,814
Government subsidy policy	$\mu =0.2$	290.64	0.121	435.96	43,959	21,981	65,940
	$\mu =0.8$	290.686	0.484	436.028	43,966	21,987	65,953
Carbon tax policy	$k=0.5$	290.839	0.290	436.081	43907	21,960	65,866
	$k=1$	290.943	0.484	436.157	43,869	21,952	65,821

represents the optimal results of data Set 6 in carbon trading price c_e, carbon tax coefficient k and government subsidy coefficient $\mu$. In the case of cap-and-trade policy, the total profit and carbon emission reduction level of supply chain increase with the increase of carbon price. It is a wise choice for manufacturers to increase carbon emission reduction levels in response to the increase of carbon price. However, faced with the increase of carbon emission reduction cost, manufacturers will try to improve market demand to ensure that profits do not decrease, so the wholesale price and selling price will decrease. In the case of government subsidy policy, with the increase of the proportion of government subsidy, the profits of the manufacturer and the retailer will increase. Part of the cost of carbon reduction for manufacturers is borne by the government. Therefore, in pursuit of more profits, the manufacturer will raise wholesale prices and the retailer will raise selling prices for their own profits. In the case of carbon tax policy, the carbon emission reduction level increases with the increase of carbon tax coefficient, and the profits of the manufacturer and the retailer decrease with the increase of carbon tax coefficient. The manufacturers bear the cost of carbon emission reduction on the one hand and the pressure of carbon tax on the other hand. Therefore, with the increase of carbon tax coefficient, manufacturers will actively undertake carbon emission reduction in order to reduce the pressure of carbon tax.

$$\mathbf{Set} \mathbf{6}: a=500;\beta =0.8;\gamma =0.2;\lambda =300;A_m=0.5;A_r=0.5;e_0=1;E=500;\sigma =70$$

It can be seen from Table 3 and Table 4 that the performance of the three policies is better than that of anarchic intervention. From the perspective of carbon emission reduction, cap-and-trade policy, government subsidy policy and carbon tax policy can promote the manufacturer to actively reduce carbon emission. From the perspective of supply chain performance, both cap-and-trade policy and government subsidy can increase the total profit of supply chain. From the perspective of consumers, cap-and-trade policy can pay a lower price to buy more high-quality and low-carbon products.

5. Sensitivity Analysis

In this section, we discuss the sensitivity of some key model parameters and study the influence of these parameters on optimal results. We leave the other parameters the same, changing their values one at a time, and here we consider the values given in data Set 6. Figure 2, Figure 3 and Figure 4 represent the sensitivity parameters of $c_e$, $\mu$ and k, and their impacts on the carbon emission reduction level of supply chain members and the profitability of the supply chain respectively. Figure 5 represents the influence of risk aversion coefficient of the manufacturer and the retailer on carbon emission reduction level and profit of supply chain.

In the case that the manufacturer and the retailer have different degrees of risk avoidance, i.e., $A_m=0.2,A_r=0.8;A_m=0.5,A_r=0.5;A_m=0.8,A_r=0.2$, we first analyzed the impact of carbon price c_e on manufacturer’s carbon emission reduction level, manufacturer’s profit, retailer’s profit and the total profit of supply chain in the scenario of cap-and-trade policy. In the second step, we analyzed the influence of government subsidy coefficient $\mu$ on manufacturer’s carbon emission reduction level, manufacturer’s profit, retailer’s profit and the total profit of supply chain in the scenario of government subsidy policy. Finally, we analyzed the influence of carbon tax coefficient k on manufacturer’s carbon emission reduction level, manufacturer’s profit, retailer’s profit and the total profit of supply chain in the scenario of carbon tax policy.

Obviously, a higher carbon trading price c_e has a positive effect on manufacturers’ active carbon emission reduction. Meanwhile, we observed that when retailers’ risk aversion level is higher than manufacturers’, the manufacturers will actively undertake carbon emission reduction. The reasoning behind this is that higher levels of carbon reduction can both increase demand and reduce the cost of buying carbon. On the other hand, the higher carbon price c_e has a positive effect on the total profit of the supply chain, and the total profit of the supply chain is the largest when the risk aversion degree of the manufacturer is much higher than that of the retailer. However, for the manufacturer, when the manufacturer’s risk aversion degree is at a low level, its own profit is the largest.

It can be observed from Figure 3 that the higher the proportion of government subsidies is, the more likely it is to promote manufacturers to actively reduce carbon emissions, but it has little influence on improving the profits of supply chain. This is because the government bears part of the cost of carbon emission reduction for the manufacturer, and the manufacturer can increase market demand without bearing more cost of carbon emission reduction by increasing the level of carbon emission reduction. Through the comparison of different risk avoidance degrees, we found that when the manufacturer’s risk avoidance degree is higher, the supply chain can get more profits. Low risk aversion is a better option for manufacturers themselves.

It can be observed from Figure 4 that the increase of carbon tax ratio can promote the manufacturer to actively reduce carbon emission, but also lead to the decline of the total profit of the supply chain. The reason behind this is that the manufacturer bears the burden of the carbon tax. For the manufacturer in the process of supply chain operations, as a leader in the supply chain, the pressure of a carbon tax will be passed on to the retailer and consumers, the manufacturer to wholesale price increases, to increase their profits, then the retailer, in order to maintain their own profits per unit product, can raise the selling prices of products; final consumers need to pay a higher price to buy low carbon products. Similar to cap-and-trade policy and government subsidy policy, in the scenario of carbon tax policy, the supply chain can get more profits when the manufacturer’s risk avoidance degree is higher, while for the manufacturer, the supply chain can get more profits when the risk avoidance degree is lower.

In addition, another thing we are concerned about is how the degree of risk aversion of the manufacturer and the retailer affects the total profit of the supply chain and the carbon emission reduction level under the four models. From Table 3 and Table 4, we observed that there is almost no gap in the supply chain profits of the four models. Therefore, we studied the three-dimensional graph of the profit of manufacturer, retailer and the whole supply chain in Model A as the profit changes with the degree of risk aversion. It can be observed from Figure 5 that in the scenarios of the four models, the manufacturer’s carbon emission reduction level is the best in the cap-and-trade policy, followed by the carbon tax policy and government subsidy policy, and the carbon emission reduction level is the lowest in the scenario of anarchic intervention. i.e., $e^{B*}>e^{D*}>e^{C*}>e^{A*}$. When the risk avoidance degree of manufacturer is minimum and the risk avoidance degree of retailer is maximum, i.e., $A_m=0,A_r=1$, the carbon emission reduction level of the manufacturer is the highest and the profit of manufacturer is the largest. When the manufacturer has the highest degree of risk aversion and the retailer has the lowest degree of risk aversion, i.e., $A_m=1,A_r=0$, the retailer and the supply chain have the highest profit. The reason behind this result is that when the manufacturer has a low degree of risk aversion, they will increase their investment in carbon emission reduction, and market demand and manufacturers’ profits will increase accordingly. When the manufacturer has a high degree of risk aversion and the retailer has a low degree of risk aversion, the retailer will reduce the price of the product to increase the market demand, and the higher market demand will improve the profit of the manufacturer and the whole supply chain. Based on the above analysis, for supply chain members themselves, adopting a low risk avoidance strategy can increase their own profits, while for the whole supply chain, manufacturers adopting a high risk avoidance strategy and retailers adopting a low risk avoidance strategy are more beneficial to the whole supply chain.

6. Discussion and Managerial Implications

In this section, we compare our results with those of previous authors by discussing the above numerical analysis.

(1) This paper finds that uncertain demand and risk appetite of supply chain stakeholders have significant impact on supply chain performance and carbon emission reduction level. Peng, Q et al. [37] found that both one-way option contract and two-way option contract can effectively improve corporate profits and curb carbon emissions when dealing with uncertain demand risks. Wang, L et al. [38] studied that carbon tax and consumers’ low-carbon preference can encourage supply chain members to reach contracts in the case of uncertain demand. Jauhari, W.A et al. [39] found that in the case of uncertain demand, recycling incentives and manufacturers’ investment in collection efforts can effectively improve the recovery rate of used products. Babagolzadeh et al. [41] studied that demand uncertainty and carbon tax policy have a positive impact on carbon emissions of the refrigerated supply chain. However, they did not clearly point out the role of uncertain demand disturbance factor and supply chain members’ preference level to deal with risk on supply chain performance. This paper makes up for this deficiency objectively.

(2) This paper finds that cap-and-trade policy, government subsidy policy and carbon tax policy all have a significant impact on improving the carbon reduction level of supply chains. Wang et al. [28,29,30] found that the cap-and-trade policy has a positive impact on supply chain members’ cooperation in emission reduction and profit improvement. Ding et al. [31,32,33] found that carbon tax has a positive impact on improving carbon emission reduction levels, but has a negative impact on social welfare and product price. Xue et al. [24,25,26,27] found that government subsidy can effectively mobilize the enthusiasm of supply chain emission reduction. On this basis, some scholars also compared the impact of different carbon policies on the supply chain. Yuyin and Jinxi et al. [34,35,36] compared the impact of two policies. However, few scholars have compared the impact of the three policies on the supply chain in the face of uncertain risks. This study makes up for this deficiency.

Based on the above discussion, we have the following important observations:

1. Cap-and-trade policy, government subsidy policy and carbon tax policy have a positive impact on promoting the carbon emission reduction level of the whole supply chain. It is worth noting that cap-and-trade policy and government subsidy policy have positive effects on improving the overall profit of the supply chain, while carbon tax policy has negative effects on the profit of the supply chain. Therefore, governments that formulate carbon policies should make policies based on different goals to encourage enterprises.

2. In the four models dealing with uncertain demand, when supply chain members have normal risk aversion preference, the higher the value of uncertain demand factor σ, the lower the wholesale price and sales price of low-carbon products, and the higher the profit of supply chain. This indicates that stakeholders in the supply chain can formulate strategic decisions that do not change when responding to uncertain risks.

3. In the actual operation of the supply chain, in order to improve their respective profits, manufacturers and retailers should adopt low risk aversion strategies. In order to improve the profits of the whole supply chain, manufacturers should adopt higher risk avoidance strategies, while retailers should adopt lower risk avoidance strategies.

7. Conclusions and Future Research Directions

The purpose of this paper is to study the impact of different carbon policies on the decision-making of stakeholders in the supply chain under uncertain demand. Based on the triple bottom line (TBL) theory of sustainable development, the low-carbon preference of consumers and the risk aversion of supply chain members were considered. Through the construction of mean-variance utility function and Stackelberg game models among supply chain members, this paper specifically studied: (1) anarchic intervention policy, (2) cap-and-trade policy, (3) government subsidy policy and (4) carbon tax policy. Through horizontal comparison and numerical analysis, the following conclusions can be drawn:

(1) The carbon emission reduction level under the three carbon policies is higher than that under the anarchic intervention. In addition, under the cap-and-trade policy, supply chain members can not only increase their own profits, but also improve the carbon reduction level of the supply chain. Government subsidies and carbon taxes can be used as short-term incentives for companies to actively reduce carbon emissions.

(2) From the perspective of supply chain performance, both cap-and-trade policy and government subsidy can play a positive role. Cap-and-trade policy can not only improve the profit of supply chain, but also improve the carbon emission reduction level of enterprises.

(3) From the perspective of dealing with the risk of uncertain demand, for duopoly of the manufacturer and the retailer, the change of uncertain demand factors will not pose a great threat to the enterprise’s income, but in order to pursue profits, the risk avoidance strategy adopted by enterprises will reduce the profits of the whole supply chain system.

Although this work is well supported by the literature and combines uncertain demand, government subsidies, carbon taxes and cap-and-trade, there are some limitations due to some assumptions. Some behaviors of the carbon trading market and the government appear as exogenous variables, which can hardly reflect the carbon emission reduction behaviors taken by manufacturers in the face of the government and the carbon trading market. This can be used as a research direction in the future. Further analysis of multi-party games between government, the carbon trading market and carbon emission reduction manufacturers can be established in the future. In addition, in this paper, we only studied the low-carbon supply chain decision-making of manufacturers as leaders, which can be extended to the influence of different power structures on stakeholders in low-carbon supply chains in the future.