As one of the main themes of global sustainable development, reducing greenhouse gas emissions has been extensively studied. Some scholars consider carbon emission reduction strategies under a low carbon regulation, like capandtrade regulation (e.g., [1]), low carbon subsidies (e.g., [2]), and carbon tax (e.g., [3]). Some one demonstrate the coordination in low carbon supply chain (e.g., [4]). We focus on the Stackelberg games in the low carbon decisions. For example, Zhang et al. [5] find out that social welfare is not affected by the power structure of supply chain when manufacturer taking green technology strategy. Wang et al. [6] show that manufacturer Stackelberg (MS) is not conducive to improve manufacturer's profit, while the other Stackelberg games help the improvement of manufacturer's profit. More research references for lowcarbon supply chain under power structure see [7, 8].
In reality, competition is everywhere. The existence of multiple manufacturers can prevent monopoly and effectively stimulate product upgrading. Multiretailer project is conducive to multipoint sales of products. Zhou et al. [9] point out that compared with other competing retailers, the demand for retailers selling more environmentally friendly products will increase as the carbon tax rate increases. Zhang et al. [10] find that different green technologies will ease competition among manufacturers. Liu et al. [11] show that fierce competition for emissions reduction will harm the members of the supply chain and the environment under the capandtrade regulation.
In fact, competition will inevitably increase the pressure on companies to resist risks. At the same time, the uncertainty of demand will also increase the company's operational risks. Hence, companies often adopt a riskaverse attitude and adopt conservative decisions to reduce risks. Wang and He [12] constructe a lowcarbon supply chain game model considering risk aversion through the mean variance method, and pointe out that the manufacturer's risk aversion has a positive impact on the wholesale price, while the supplier's impact is negative. Bai et al. [13] show that if the underlying manufacturer's degree of risk aversion is not deep, low carbon emissions can be achieved.
In order to promote the longterm development of the transaction and ensure the continuous supply of goods, retailers may bring benefits to their partners. At this point, they often express their altruistic preference attitudes. Fan et al. [14] point out that retailer's altruistic behavior is always beneficial to manufacturer under both Stackelberg game lowcarbon model and vertical Nash game lowcarbon model. Wang et al. [15] show that the retailer's altruistic preference helps to improve the profit and system efficiency of small and mediumsized manufacturers, but it will reduce the retailer's profit.
In view of the fact of lowcarbon supply chain transactions, companies may be competitive and have risk aversion behaviors, moreover retailers are more likely to be altruistic. However, the previous literature has not adequately investigated the lowcarbon decisions when these issues are considered altogether. Motivated by the research gap, considering supply chain with altruistic retailer, we investigate four Stackelberg models, including manufacturerled with competitive/noncompetitive, retailerled with competitive/noncompetitive, to analysis the optimal decisions. We contribute to the literature in the following three aspects.
First, we show that the emission reduction level decrease with manufacturers' risk aversion. Second, we indicate that with the transfer of powers from the retailers to the manufacturers, the price will increase. At last, we point out that competition can bring a higher price, a deeper degree of low carbon, and more profits to manufacturers and retailers via a series of numerical experiments.
The remainder of this paper is structured as follows. Section 2 presents the assumptions and notations. Section 3 discusses the models. Section 4 analysis the results. Section 5 investigates the impact of competition. Section 6 concludes the paper.
In the distribution of lowcarbon products, manufacturers adopt carbon emission reduction technology to improve the green degree of products. Altruistic retailers sell products purchased from manufacturers to consumers. At this time, the consumer market is affected by retail price, carbon emission reduction level and random demand shock. Following the existing literature, e.g., [16], we denote, the cost of carbon emission reduction is
The notions and parameters
Retail price  
Wholesale price  
Retail margin 

Emission reduction level  
Unit production cost  
Demand shock, with expected function 

Risk aversion coefficient  
Firms' profit  
Superscripts, represent manufacturer Stackelberg model  
Superscripts, represent retailer Stackelberg model  
*  Superscript, represents the optimality 
Subscript, represents the firm, 

Subscript, represents the manufacturer  
Subscript, represents the retailer 
This section discusses four models, the first model and second model are two stage supply chain composed of a single manufacturer and a single retailer. The third model and fourth model are competitive supply chains composed of two competitive manufacturers and two retailers. The first model and third model are manufacturer Stackelberg game, the manufacturer is a leader and decisions wholesale price and emission reduction level, the retailer acts a follower and sets the retail price. The second model and fourth model are retailer Stackelberg game, the retailer is the leader and sets the retail margin, the manufacturer acts the follower and decides wholesale price and emission reduction level.
Noting that both the manufacturer and the retailer are risk averse, in line with [17], we consider using the mean variance (MV) method to characterize their respective utility profits.
It also note that the retailer holds the attitude of altruistic preference, in line with [18], the retailer's utility function satisfies that
In the non competitive supply chain, the supply chain' operation is shown in Figure 1. We consider two Stackelberg games: MS model and RS model. The demand follows that 1 −
In this subsection, the expected profits of the manufacturer and the retailer are as follow:
Then the MV's functions of the manufacturer and the retailer are as follow:
Thus, the altruistic retailer's utility function satisfies that
First, we consider the retailer's decision. Since
Second, we consider the manufacturer's decisions. Substituting Eq. (8) into Eq. (6), we have
By assuming that
Therefore the Hessian Matrix of
Substituting Eqs. (10) and (11) into Eq. (8), we can obtain that
In this subsection, noting that the retailer's margin
Thus, the altruistic retailer's utility function satisfies that
First, we consider the manufacturer's decisions. Since
Thus the Hessian Matrix of
Second, we consider the retailer's decision. Substituting Eq. (16) and Eq. (17) into Eq. (15), we have
Considering the first derivative order condition, i.e.,
Substituting Eqs. (19) into Eqs. (16) and (17), respectively. We can obtain that
Thus, we have
In the competitive supply chain, as shown in Figure 2, the demand of supply chain
In this subsection, the expected profits of the manufacturers and the retailers are as follow:
Then the MV's functions of the manufacturers and the retailers are as follow:
Thus, the altruistic retailers' utility function satisfy that
Similarly, we also consider the retailers' decisions at first. Since
Then by solving
Second, we consider the manufacturers' decisions. Substituting Eq. (28) into Eq. (26), we have
By assuming that
Therefore the Hessian Matrix is negative definite, if
Substituting Eqs. (29) and (30) into Eq. (28), we can obtain that
In this subsection, the MV's functions of the manufacturers and the retailers are as follow:
Therefore, the altruistic retailers' utility function are
Similar to subsection 3.1.2, we consider the manufacturers' decisions at first, then consider the retailers' decisions. Since second derivative order of
Then we can obtain the wholesale prices and emission reduction levels satisfy that
Substituting Eq. (35) and Eq. (36) into Eq. (34), we have
Substituting Eqs. (37) into Eqs. (35) and (36), respectively. We can obtain that
Thus, we have
In this section, we discuss the impact of risk aversion and power structure on the decisions.
By assuming that
For RS game, the equilibrium retail price, the equilibrium wholesale price and the equilibrium emission reduction level are
Theorem 1 indicates that the higher the risk aversion of the retailer, the higher the wholesale price and emission reduction level of the noncompetitive supply chain, and the lower the retail price of the competitive supply chain.
Theorem 2 reflects that the more risk aversion manufacturers are, the more they will restrain price increases and reduce emissions reduction inputs at the same time.
Theorem 3 shows that with the transfer of powers from the retailers to the manufacturers, the price will increase.
Theorem 4 reflects that compared with the retailerled model, the manufacturerled model is conducive to improving the level of carbon emission reduction. However, in a noncompetitive supply chain, when the degree of altruistic behavior of the retailer is low, the emission reduction effect led by the manufacturer is not as good as the one led by the retailer.
In this section, a series of numerical studies will be conducted. We assume that the variables follow
Figure 3 shows that no matter in RS modes or in MS modes, the retail price in competitive supply chain is higher than it in the noncompetitive model, i.e.
Figure 6 and Figure 7 respectively points out that the MV function of the manufacturer and the altruistic utility of the retailer in a competitive supply chain are both higher than that in the case of noncompetition supply chain. Figure 6 and Figure 7 also demonstrate that competition can bring more profits to manufacturers and retailers.
Considering a low carbon supply chain with risk aversion, this paper establishes four models under retailer's altruistic preference. The first is that manufacturer Stackelberg of noncompetitive supply chain, the second is retailer Stackelberg of noncompetitive supply chain, the third is that manufacturerled competitive supply chain, and the last is that retailerled competitive supply chain. The results are as follows:
First, we analyze the role of risk aversion. Results show that the more risk aversion manufacturers are, the lower the price and emission reduction level are; The more risk averse retailers are, the higher the wholesale price and emission reduction level in the non competitive model, and the lower the retail price in the competitive model.
Second, we compare the difference in power structure, and find that with the transfer of powers from the retailers to the manufacturers, the price will increase.
At last, we make a series of numerical experiments. It points out that competition can bring a higher price, a deeper degree of low carbon, and more profits to manufacturers and retailers.
The notions and parameters
Retail price  
Wholesale price  
Retail margin 

Emission reduction level  
Unit production cost  
Demand shock, with expected function 

Risk aversion coefficient  
Firms' profit  
Superscripts, represent manufacturer Stackelberg model  
Superscripts, represent retailer Stackelberg model  
*  Superscript, represents the optimality 
Subscript, represents the firm, 

Subscript, represents the manufacturer  
Subscript, represents the retailer 
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