Jin-Chai Lin(林金钗), Ru-Guo Fan(范如国), Yuan-Yuan Wang(王圆缘), and Kang Du(杜康)

1Management School,Chongqing University of Technology,Chongqing 400054,China

2Economics and Management School,Wuhan University,Wuhan 430072,China

Keywords: carbon tax,risk-avers,low-carbon supply chain,complex dynamics

1.Introduction

In recent years, a series of natural disasters caused by global warming has become increasingly prominent,seriously restricting the high-quality development of the global economy.To cope with the global climate change and improve the living environment of mankind,developing low-carbon economy has become the universal consensus of the international community.The carbon emissions of global brands and multinational companies are mainly concentrated in their supply chains, generally accounting for more than 70% of the total carbon emissions of brands, and some even reach 80% or 90%.[1]This poses a great challenge to the living environment,competition rules,cost composition and operation mode of the global supply chain.

Many countries have introduced policies such as carbon tax, carbon quota and carbon trading to regulate the carbon emissions of enterprises.Among them, carbon tax can not only increase government revenue for investment and development of new emission reduction technologies,but also have lower management and operation costs than carbon trading.Scholars have introduced carbon tax into the study of supply chain operation to explore its influence on decision and performance of supply chain.

It is worth noting that the carbon emission reduction decision and pricing decision are management activities mainly conducted by person.In an uncertain environment,the degree of cooperation and related decisions are affected by subjective factors such as people’s psychology, emotions, and their cognitive capabilities.[2-4]With the increase of complexity and uncertainty of the market environment, enterprises often show risk aversion when making decisions.For example, after McKinsey surveyed 1500 executives in 90 countries, it is found that managers show a high level of risk aversion regardless of the size of the investment.[5]Therefore,it is necessary to consider the influence of enterprises’ risk aversion on carbon emission reduction decision-making,pricing,and profit of the supply chain.

It is not difficult to find that in the current study, scholars have abstracted and simplified the social and economic systems such as the supply chain into deterministic and linear systems.In reality, the evolution process of supply chain system often has complex characteristics such as nonlinearity and uncertainty, owing to heterogeneity and autonomy attributes of social subjects, randomness of social environment and other factors.[6,7]Therefore, using static linear model to analyze supply chain system has certain limitations.

Above all, this study introduces the risk aversion behavior of the manufacturer and constructs the game models of low-carbon supply chain.We explore the influence of the manufacturer’s risk aversion on the optimal decision, profit and coordination of the supply chain from the static perspective and dynamic evolution perspective, respectively.Then,the operation of the supply chain system under the decentralized decision-making situation and the centralized decisionmaking situation are compared and analyzed,with the centralized decision-making situation taken as a benchmark.Finally,a carbon emission reduction cost-sharing contract is proposed to improve the carbon emission reduction level and supply chain efficiency,which realizes Pareto improvement.

The rest of this study is organized as follows.In Section 2, the existing reseraches are reviewed.In Section 3, model assumptions and notation are described.In Sections 4 and 5,model construction and analysis are shown in the frame of decentralized decision-making model and centralized decisionmaking,respectively.In Section 6,cost-sharing contract game model is designed.Conclusions and suggestions are presented in Section 7.

（六）实现心理健康教育个性化。针对个别寄宿生出现的厌学、逃学、打架、抽烟等不良现象，我们注重及时发现疏导，由生活指导教师和班主任分别进行个别谈话，细致入微地了解学生的所思所想，找准问题的思想根源，耐心地对学生进行教育引导，避免学生因心理健康问题发现不及时、疏导不到位引发不良后果。

因此一方面对新建、改建110kV及以上的线路参数，要求按照有关基建工程验收规程的规定，在设备投入运行前进行实际参数测试；二方面进一步加强全网主变中性点接地方式的管理；三方面根据双侧电源复杂电网的线路零序电流保护中，零序电流I段作为速动段保护，对于超短线路（2km以下）及短线路（10km以下）宜退出运行的规定，建议将110千伏乃嘎I线、夺城线、柳西双线等长度15km以内共计12条线路的零序电流I段保护功能退出运行。

2.Review on existing researches

Equation (14) is the manufacturer’s decision variable,wheredi,i=1,2 represent the adjustment parameter of wholesale price and carbon emission reduction level, respectively.According to Eq.(6),the dynamic retail price can be expressed as

耕地质量级别监测重点是基于年度内土地资源增多、缩减与建设现状进行。实现地区耕地质量级别与整体生产水平提高的重要技术路线是管理缩减的土地资源，要削减低质土地；要多扩大耕地，扩大的耕地主要保证高；通过耕地修建不断提高耕地质量级别，基本技术路线见图1。

2.1.Carbon emission reduction decision and pricing in supply chain

Considering the influence of consumers’low-carbon preferences on enterprise decision-making is a current research hotspot.[8,9]In addition,the influence of the government’s carbon tax on the decision-making of carbon emission reduction in the supply chain is also the focus of attention of academia and enterprises.Wanget al.[10]found that the carbon tax policy affects the product pricing and emission reduction strategy of enterprises.Huang and Zhang[11]found that the increase of carbon tax rate can increase the emission reduction level of products in low emission industries, but it reduces the emission reduction level of products in high emission industries.Lu and Yi[12]found that with the increase of carbon tax rate,the phenomenon of “less new products and more remanufactured products”appears.

The above studies on carbon emission reduction in supply chain are all based on the risk neutral assumption,ignoring the risk aversion characteristics of decision makers, and the obtained research results can not accurately provide theoretical reference for real supply chain decision makers.

2.2.Risk avoidance in supply chain

Proposition 3 When the cost-sharing contract is implemented,the carbon emission reduction level will beimproved,and it increases with the cost-sharing proportion increasing.

The above-mentioned researches show that risk aversion behavior of enterprises will change the optimal decisions and corporate profits.However, there have been many studies on the influence of the risk aversion behavior of decisionmakers on supply chain financing,[21,22]but few researches involve with the influence of low-carbon supply chain decisionmaking and coordination,which mainly focus on the study of static models.

2.3.Applications of nonlinear dynamical system theory to supply chain

On the construction of low-carbon supply chain game model, the existing literature mainly focuses on static models.However,supply chain management system is a nonlinear system with uncertainty, variability, openness, and dynamics.Xi and Zhang[23]found that the dynamic subsidy of the government to the nitrogen emission reduction behavior of agricultural enterprises will affect the stability of the supply chain system and thus the performance of the supply chain.Tianet al.[24]showed that when consumers’channel preference is too strong,it will damage the profits of supply chain members,no matter how hard enterprises try to improve sales efforts.Maet al.[25]studied dynamic complexity of a multi-channel supply chain considering cap-and-trade regulation.Liet al.[26]incorporated fairness concern into a dynamic model of the closedloop supply chain.This literature only studied the pricing decision and took the carbon emission reduction strategy as an exogenous variable.Unlike this literature, the present study takes carbon emission reduction strategy as a decision variable of the manufacturer.

工程上，风机荷载疲劳一般通过雨流计数法将风机时程疲劳载荷转化为等效疲劳载荷，再利用等效疲劳载荷进行疲劳损伤计算。

3.Model assumptions and notation

Under the carbon tax policy, this study takes a two-level supply chain consisting of a single manufacturer and a single retailer as the research object.We only consider the carbon emission of product production,i.e.,only the manufacturer is affected by the carbon tax.To improve competitiveness and meet consumers’demand for low-carbon products,the manufacturer needs to invest in carbon emission reduction and sells the products to consumers through the retailers.The manufacturer is the leader of the game and first determines optimal carbon emission reduction effort leveleand wholesale pricew.The retailer is the follower and makes optimal retail pricepmaximize its own profit according to the manufacturer’s decision.

Assuming that there are uncertain factors in the market demand.According to Ref.[27], the market demand can be expressed by the following function:

（6）增加与约旦能源合作。约旦页岩油富足，中国资金技术雄厚，且中国已投资阿塔拉特油页岩电站项目。在能源合作方面，中国可以加大投资力度。加快中国“走出去”战略，让约旦对中国友好合作的决心更坚定，从而最大限度地对中国开放。

where ¯a=a+ε,ais the factor determined by the potential demand of the market,εis the random item,which obeys the normal distributionε～N(0,σ2),and the distribution information is common knowledge of both sides;tis the low-carbon preference of consumers.Equation (1)indicates that the increasing of the carbon emission reduction level of the low-carbon product can increase the market demand.

Assume that the initial carbon emission of unit product ise0, and unit carbon tax is set tof, so the manufacturer needs to pay carbon emission tax of unit productf(e0-e).Assume thatcis unit cost when carbon emission reduction investment is not considered, thenw-c-f(e0-e)＞0 will need satisfying in order to ensure that the manufacturer is profitabl.The cost function of carbon emission reduction isC=(1/2)ke2,which has been widely used,[11,28]whereC＇＞0,C＇＇＞0,and the parameterkis the cost coefficient related to emission reduction investment, parametersaandkare much larger than other parameters.

According to the above assumptions, the profit function of the retailer, the manufacturer and the supply chain are as follows:

When the manufacturer shows risk aversion characteristics, owing to market uncertainty, emission reduction investment, and carbon tax, not only should attention be paid to its own profit, but also to how to avoid risks.As for the utility function of risk aversion, this study uses the mean variance method which is widely used in theory and practice to measure the utility function of decision-makers’ risk aversion behavior.[29,30]The manufacturer’s expected utility is expressed asE(Um)=E(πm)-Var(πm)/2Rm, whereRmis the manufacturer’s risk tolerance.According to the mean variance method,the variance of the manufacturer’s expected return is Var(π) = (w-c-f(e0-e))2σ2.Therefore, the manufacturer’s utility function can be expressed as

4.Decentralized decision game model of lowcarbon supply chain

4.1.Static model

The superscript “R∗m” represents the optimal solution of the manufacturer in the case of risk aversion.Reverse induction method is used to solve the Stackelberg game between the manufacturer and the retailer.Solving the second partial derivative of Eq.(5) with respect top, we obtain∂E2(πr)/∂p2=-2＜0.Therefore,πrhas a unique optimal solution with respect top.Letting∂πr/∂p=0, the response function of the retailer is calculated from

Substituting Eq.(6)into Eq.(5)and calculating the firstpartial derivatives of the decision variableswande, from the resulting equation,we obtain

When 4k(Rm+σ2)＞Rm(f+t)2,the Hessian matrix of Eq.(7)is negative-definite,and the utility function of the manufacturer has a unique maximum solution with respect towande.Under this constraint,the optimal decision of the manufacturer can be obtained by solving Eq.(7):

Substituting the above two optimal solutions into Eq.(5),the optimal retail price is as follows:

Substituting the optimal decisions obtained from Eqs.(8)-(10)into Eqs.(2)-(4),respectively,we obtain