As China’s economy continues to develop, environmental protection and ecological construction have become increasingly important in economic and social development planning. Sustainable development is strongly emphasized, particularly a low carbon economy with low energy consumption and low pollution [1J. Chen, "Study on supply chain management in a Low carbon era", J. Syst. Manage., vol. 21, no. 6, pp. 721-728.]. To achieve sustainable development, the study of low carbon supply chain system has attracted considerable attention in academia.

Research on low carbon supply chain system mainly focuses on the pricing and coordination of low carbon supply chain system and decision-making regarding carbon emission reduction levels. Based on the summary of current research on supply chain system management, Chen [1J. Chen, "Study on supply chain management in a Low carbon era", J. Syst. Manage., vol. 21, no. 6, pp. 721-728.] noted several important directions of supply chain system management under a low carbon background. Mao et al. [2Z.F. Mao, S. Zhang, and X.M. Li, "Low carbon supply chain firm integration and firm performance in China", J. Clean. Prod., vol. 153, no. 1, pp. 354-361.
[http://dx.doi.org/10.1016/j.jclepro.2016.07.081] ] analyzed the relationship between internal low carbon integration and external low carbon integration and firm performance and noted that internal low carbon integration improves corporate environmental performance but simultaneously hinders financial performance. Zu et al. [3Y.F. Zu, L.H. Chen, and Y. Fan, "Research on low-carbon strategies in supply chain with environmental regulations based on differential game", J. Clean. Prod., vol. 177, no. 10, pp. 527-546.
[http://dx.doi.org/10.1016/j.jclepro.2017.12.220] ] researched low carbon strategies in a supply chain system with three progressive environmental regulation situations using a Stackelberg game. By considering a two-level supply chain system comprising two manufacturers of low carbon and common products and a single retailer, Xu et al. [4C.Q. Xu, D.Z. Zhao, and B.Y. Yuan, "Study on coordination mechanisms and differential pricing of supply chain in low carbon environment", Oper. Res. Manage. Sci., vol. 24, no. 1, pp. 19-26.] discussed the problem of differential pricing and coordination mechanisms of supply chain systems. Under the single and double sales models, Cao et al. [5B.B. Cao, Z.D. Xiao, and C.Y. Zhu, "Study on supply chain decision under double sales model considering government low carbon policy", Chinese Journal of Management Science, vol. 26, no. 4, pp. 30-40.] studied the mechanism by which government price subsidies, one-time subsidies, and carbon taxes influence decisions in a low carbon supply chain system. Li et al. [6Y Li, and D Z Zhao, "Low-carbonization supply chain coordination with contracts considering fairness preference", Journal of Industrial Engineering / Engineering Management, vol. 29, no. 1, pp. 156-161.] studied the supply chain coordination problem of manufacturers who offer different preferences to retailers with a fair preference under a low carbon environment. Wang et al. [7Q.P. Wang, D.Z. Zhao, and L.F. He, "Contracting emission reduction for supply chains considering market low-carbon preference", J. Clean. Prod., vol. 120, no. 1, pp. 72-84.
[http://dx.doi.org/10.1016/j.jclepro.2015.11.049] ] focused on a dyadic supply chain system with a single manufacturer and retailer to develop a game model for studying issues of carbon emission reduction in the retailer dominant and power balanced cases, respectively. Under the assumption that consumers prefer low carbon products and voluntary emission reduction, Wang et al. [8Q.P. Wang, and D.Z. Zhao, "Revenue-sharing contract of supply chain based on consumer’s preference for low carbon products", Chinese Journal of Management Science, vol. 22, no. 9, pp. 106-113.] studied the decision problem of optimizing order level and reducing carbon emission for a retailer in a two-level supply chain system. Based on preference for low carbon consumption, Sun and Xiao [9J.N. Sun, and Z.D. Xiao, "Research on emission reduction strategy of low carbon supply chain considering consumer’s dual preferences", Chinese Journal of Management Science, vol. 26, no. 4, pp. 49-56.] subdivided consumer channel preferences and provided optimal emission reduction boundary conditions for the low carbon supply chain through the comparison and analysis of the decentralized and the centralized decision-making models. Hang et al. [10L.H. Zhang, G.W. Zhang, and R. Zhang, "Equilibrium contracts strategies of low carbon supply chain and high carbon supply chain", Computer Integrated Manufacturing Systems, vol. 24, no. 3, pp. 763-771.] researched the problem of low carbon technology and coordination strategy in market competition between two supply chain systems, one being high carbon and the other low carbon. The above research all focused more on the low carbon supply chain system decision with neutral manufacturers and retailers, which do not pay attention to risk aversion conditions.

Many manufacturers and retailers in supply chains are risk averse. In research on supply chain system decisions, some scholars have focused on risk aversion and supply chain decision-making. Xiao et al. [11T.J. Xiao, T.M. Choi, D.Q. Yang, and T.C.E. Cheng, "Service commitment strategy and pricing decisions in retail supply chains with risk-averse players", Serv. Sci., vol. 4, no. 3, pp. 236-252.
[http://dx.doi.org/10.1287/serv.1120.0021] ] researched the service commitment strategy and pricing decisions in a single-supplier single-retailer supply chain system where all players (including consumers) are risk averse. Ye et al. [12F. Ye, and Q. Lin, "Revenue sharing contract mechanisms of risk-averse supply chains", Journal of Industrial Engineering/ Engineering Management, vol. 26, no. 1, pp. 113-118.] studied the cooperation incentive problem of a two-level supply chain system, and analyzed the influence degree of the risk aversion coefficient of the supplier and retailer on the decision-making behavior and profit sharing coefficient of the two sides using empirical analysis. Ning et al. [13J. Ning, V. Babich, J. Handley, and J. Keppo, "Risk-aversion and B2B contracting under asymmetric information: Evidence from managed print services", Oper. Res., vol. 66, no. 2, pp. 392-408.
[http://dx.doi.org/10.1287/opre.2017.1673] ] investigated risk-aversion and B2B contracting under asymmetric information using evidence from managed print services. Dai et al. [14J.S. Dai, W.D. Meng, and B. Fan, "Supply chain coordination with risk aversion via buy-back contracts", Journal of Management Science in China, vol. 18, no. 5, pp. 57-65.] used CVaR to study the repurchase contract coordination problem with sales promotion effect and risk aversion supply chain. Xu et al. [15M.L. Xu, X.Z. Nie, and H.Y. Jian, "Pricing decision of dual-channel supply chain with risk preference", Control and Decision, vol. 31, no. 1, pp. 91-98.] explored the impact of risk preference on dual channel supply chain decision and established a pricing decision model of dual channel supply chain system based on the conditional risk value criterion. Liu et al. [16M.Q. Liu, E. Cao, and K. Salifou, "Pricing strategies of a dual-channel supply chain with risk aversion", Transp. Res., Part E Logist. Trans. Rev., vol. 90, no. 6, pp. 108-120.
[http://dx.doi.org/10.1016/j.tre.2015.11.007] ] studied the effect of risk aversion on the optimal policies of a dual-channel supply chain system under complete information and asymmetric information. Zhou et al. [17Y.W. Zhou, J.C. Li, and Y.G. Zhang, "Cooperative advertising and ordering policies in a two-echelon supply chain with risk-averse agents", Omega, vol. 75, no. 3, pp. 97-117.
[http://dx.doi.org/10.1016/j.omega.2017.02.005] ] considered a cooperative advertising and ordering issue in a two-echelon supply chain system in which a risk-averse manufacturer sells a product through a risk-averse retailer. Yan et al. [18B. Yan, Z.J. Jin, Y.P. Liu, and J.B. Yang, "Decision on risk-averse dual-channel supply chain under demand disruption", Commun. Nonlinear Sci. Numer. Simul., vol. 55, no. 2, pp. 206-224.
[http://dx.doi.org/10.1016/j.cnsns.2017.07.003] ] studied a dual-channel supply chain system using centralized and decentralized decision-making models and conducted a comparative analysis of decision-making before and after demand disruption. Yoo [19S.H. Yoo, "Product quality and return policy in a supply chain under risk aversion of a supplier", Int. J. Prod. Econ., vol. 154, no. 8, pp. 146-155.
[http://dx.doi.org/10.1016/j.ijpe.2014.04.012] ] considered an OEM supply chain system, where a buyer decides a return policy for consumers and delegates the product quality decision to a supplier, and identified the relationship between return policy and product quality decision in a decentralized system. In the dual channel supply chain system with random market demand, Zhang and Yao [20L.L. Zhang, and Z. Yao, "Optimal strategy of risk-averse dual-channel supply chain with consumer returns", Computer Integrated Manufacturing Systems, vol. 21, no. 3, pp. 766-775.] considered the optimal product pricing decision problem of the risk aversion dual channel supply chain system when the customer returns. Cao et al. [21X.G. Cao, B.R. Zheng, H.S. Xia, H. Wen, and J.Z. Li, "Analysis on closed-loop supply chain network equilibrium with risk-averse retailers", Control and Decision, vol. 29, no. 4, pp. 659-665.] studied the remanufacturing closed-loop supply chain system network equilibrium given a risk averse retailer under random demand condition. Using CVaR to measure risk, Dai et al. [22J.S. Dai, and W.D. Meng, "Revenue sharing contract for a risk-averse supply chain with promotional effect", Journal of Management Science in China, vol. 17, no. 5, pp. 25-34.] inspected the profit-sharing contract given a risk averse seller and the promotion effect. Gan et al. [23X.H. Gan, and K.F. Ying, "Vendor managed inventory contract model with risk averse", Syst. Eng., vol. 33, no. 1, pp. 116-121.] investigated the optimization and coordination of the supplier management inventory contract under random demand. The above researches pay little attention to the low carbon supply chain system decision problem under different risk aversion models.

Unlike the above literatures, this paper studied the low carbon supply chain decision problem under different risk aversion models. Specifically, the impacts of risk aversion coefficient and consumer preference for low carbon on carbon emission reduction level and the optimal decision of the supply chain system in the two-level low carbon supply chain are investigated. The decision results of the risk aversion low carbon supply chain system under centralized and decentralized decision-making are compared and simulated, and the revenue sharing contract is designed to realize the coordination of the low carbon supply chain system with risk averse participants.

We consider the case of a two-level supply chain comprising a manufacturer and a retailer, where the manufacturer sells only one product through the retailer, and both the manufacturer and the retailer are risk averse. It is assumed that consumers with low carbon preference are willing to pay a premium for low carbon products, even if this means they must increase their consumption investment. As hypothesized in the literature on the cost of carbon emission reduction [24C.T. Zhang, and L.P. Liu, "Research on coordination mechanism in three-level green supply chain under non-cooperative game", Appl. Math. Model., vol. 37, no. 5, pp. 3369-3379.
[http://dx.doi.org/10.1016/j.apm.2012.08.006] ], the cost increases with the size of the reduction, and the quadratic function is used to study the above precondition.

The symbols used in this study are as follows:

K : scale of uncertainty market demand, which has mean µ, and variance δ²;

α : price elasticity coefficient;

p : the retail price per unit of product;

w : the wholesale price per unit of product;

γ : strength of consumer preference for low carbon;

e : carbon emission reduction level;

c_r : retailer selling cost per unit of product;

c_m : manufacturer production cost per unit of product;

C(e) : cost of carbon emission reduction;

η : cost coefficient of carbon emission reduction;

β_m : manufacturer risk aversion coefficient;

β_r : retailer risk aversion coefficient;

π : random profit, the subscripts m, r, respectively, indicate the manufacturer and retailer;

U : utility function.

This paper makes the following basic assumptions: The product’s stochastic demand function is set to D = K - αp + γe; the cost of carbon emission reduction is set to [24C.T. Zhang, and L.P. Liu, "Research on coordination mechanism in three-level green supply chain under non-cooperative game", Appl. Math. Model., vol. 37, no. 5, pp. 3369-3379.
[http://dx.doi.org/10.1016/j.apm.2012.08.006] ]; α, η > 0.

The random profit of a risk averse retailer is,

(1)

The utility function of a risk averse retailer is,

(2)

Similarly, the random profit of a risk averse retailer is,

(3)

The utility function of a risk averse retailer is,

(4)

In the following, the risk aversion supply chain system decision problem under centralized and decentralized decision-making will be analyzed, with the revenue sharing contract being designed to coordinate the risk aversion supply chain system.

Only when the retailer is completely risk averse is the utility of the supply chain equal under the decentralized and centralized decision models. Under other conditions, the supply chain utility is low under the decentralized decision model. To effectively coordinate the operation efficiency of the supply chain system, the incentive mechanism of revenue sharing can be established to maximize the overall utility of the supply chain system. Suppose that a portion of the retailer's income will be transferred to the manufacturer. At this point, the utility functions of the manufacturer and retailer are as follows,

(17)

Since , by setting , the only optimal value p^c* can be obtained as follows

(18)

(19)

(20)

Where M = α + β_rδ², E = K - αc_r- αc_m

Taking the condition of a risk averse manufacturer and risk neutral retailer as an example, when β_r → 0, by substituting Eq.(19) into Eq.(18), we have,

Since w^nc* = c^m, by substituting the above results into Eq.(17), we have,

Where, α > 2β_m δ². That is, when the value interval of the ratio coefficient of income distribution is , the utility of the manufacturer and retailer can be optimized, and the supply chain can achieve perfect coordination.

To more intuitively analyze the impact of parameters on the optimal decision results, a numerical simulation experiment is first performed running on Matlab, with a risk neutral retailer and a risk averse manufacturer. We assume β_r = 0, K = 100, δ = 2, α = 0.25, c_m = 40, c_r = 20.

(1) When η = 0.5, γ = 0.2, under the decentralized and centralized decision models, the change in wholesale price, carbon emission reduction level, and retail price in the manufacturer’s risk aversion coefficient are described below in Fig. (1).

As we can see from Fig. (1), when the manufacturer's risk aversion coefficient is small, the retail price under the decentralized decision model is higher than that under the centralized decision model. When the risk aversion coefficient of the manufacturer is large, the retail price under the decentralized decision model is less than that under the centralized decision model. The wholesale price under the decentralized decision model is higher than that under the centralized decision model. Meanwhile the carbon emission reduction under the decentralized decision model is lower than that under the centralized decision model.

Fig. (1) The influence of βm on w, e, p.

(2) When η = 0.5 and β_m = 0.01 under the decentralized and centralized decision models, the change in wholesale price, carbon emission reduction level, and retail price given the consumer low carbon preference coefficient is described below in Fig. (2).

From Fig. (2), we can see that, with the increase in consumer low carbon preference coefficient, when the manufacturer determines a higher carbon emission reduction level, the wholesale price will increase, and so too will the retail price set by the retailer. When the consumer low carbon preference coefficient is small, the retail price under the decentralized decision model exceeds that under the centralized decision model, and the retail price under the decentralized decision model is smaller than that under the centralized decision model when the consumer low carbon preference coefficient is large. The wholesale price under the decentralized decision model exceeds that under the centralized decision model. Meanwhile, the carbon emission reduction level under the decentralized decision model is lower than that under centralized decision model.

Fig. (2) The influence of γ on w, e, p.

(3) When γ = 0.2, β_m = 0.01, under both the decentralized and centralized decision models, the change in wholesale price, carbon emission reduction level, retail price in the carbon emission reduction cost coefficient can be described in Fig. (3) below.

Fig. (3) The influence of η on w, e, p.

From Fig. (3), we can see that, with the increase of the carbon emission reduction cost coefficient, the wholesale price decreases under the decentralized decision model, and the retail price and carbon emission reduction level decrease under both kinds of decision models. When the cost coefficient of carbon emission reduction is small, the retail price under the centralized decision model exceeds that under the decentralized decision model. When the cost coefficient of carbon emission reduction is large, the carbon emission reduction level under the centralized decision model exceeds that under the decentralized decision model. The wholesale price under the centralized decision model is lower than that under the decentralized decision model.

(4) When η = 0.5, γ = 0.2, before and after coordination, the ratio of income distribution is 0.3, 0.4, 0.5, and the changes in the utility of the retailer and manufacturer with the manufacturer’s risk coefficient are as shown in Figs. (4 and 5).

As can be seen from Figs. (4 and 5), when the retailer is risk neutral and the profit sharing coefficient is fixed, the manufacturer's utility decreases with increasing manufacturer risk aversion coefficient, and the retailer's utility increases with increasing manufacturer risk aversion coefficient. Coordination increases the utilities of the manufacturer and retailer. The utility of the retailer is not related to the risk aversion factor of the manufacturer, or to the increase of the sharing coefficient of income. Given constant manufacturer risk aversion coefficient, the utility of the retailer is increasing, and that of the manufacturer is decreasing.

Fig. (4) The influence of βm on the retailer’s utility.

Fig. (5) The influence of βm on the manufacturer’s utility.

(5) When η = 0.5, γ = 0.2, before and after coordination, the risk aversion coefficient of the manufacturer β_m is set to 0.01, 0.02, 0.03, and the changes in the utility of the retailer and manufacturer with the income sharing coefficient are shown below in Figs. (6 and 7).

As can be seen from Figs. (6 and 7), when the retailer is risk neutral and the manufacturer's risk aversion coefficient is fixed, the utilities of the manufacturer and retailer are greater than without such coordination. The utility of the retailer increases with increasing income share coefficient, and the manufacturer's utility decreases with increasing income sharing coefficient. As the risk aversion coefficient of the manufacturer increases, the utility decreases under the same income sharing coefficient.

Fig. (6) The influence of on the retailer’s utility.

Fig. (7) The influence on the manufacturer’s utility.

In a two-level low carbon supply chain system consisting of a manufacturer and a retailer, the manufacturer first decides the wholesale price and carbon emission reduction level, and then the retailer decides the retail price, with both parties playing a Stackelberg game. Through comparative analysis of the decentralized and centralized decisions, the optimal decision problem of the manufacturer and retailer under different risk aversion models is investigated. Additionally, the influences of consumer low carbon preference coefficient and carbon emission cost coefficient on decision variables of manufacturers and retailers are researched. The results show that: (1) When the manufacturer is risk neutral and the retailer is risk averse, the wholesale price under the decentralized decision model will be less than that under the centralized decision model, given the low carbon preference and carbon emission reduction factor. (2) When the retailer is completely risk averse, the effect equals that under the centralized decision model without any coordination contract. (3) Under the scenario with different risk aversion coefficients, the consumer low carbon preference coefficient and carbon emission cost coefficient equally influence the decision variables of the manufacturer and the retailer, which are positively correlated with the low carbon preference coefficient and negatively correlated with the carbon emission reduction cost coefficient. (4) When the revenue sharing coefficient satisfies a certain range, the revenue sharing contract can perfectly coordinate the supply chain, which optimizes the utilities of the manufacturer and retailer. These results can help the manufacturer and the retailer make decisions under different risk aversion models. This paper considers the problem of low carbon supply chain system decision under information symmetry. The influences of information asymmetry, carbon trading and carbon tax on risk aversion in a low carbon supply chain system require further study.