Research ArticleStudy on Green Supply Chain Cooperation and Carbon TaxPolicy considering Consumerrsquos Behavior

Jian Liu and Chao Hu

School of Information Technology Jiangxi University of Finance and Economics Nanchang 330032 China

Correspondence should be addressed to Chao Hu hc173067300163com

Received 23 June 2020 Revised 11 August 2020 Accepted 4 December 2020 Published 29 December 2020

Academic Editor Rong-Chang Chen

Copyright copy 2020 Jian Liu and Chao Hu )is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Carbon tax policy has been shown to be an effective incentive for the reduction of carbon emissions and it also profoundlyinfluences supply chain cooperation )is paper explores the interaction between carbon taxes and green supply chain coop-eration Specifically we analyze the impact of a carbon tax on green supply chain coordination and further optimize the carbon taxto achieve a win-win situation for both the supply chain and the environment Because consumerrsquos behavior has a significantimpact on green product demand we consider the problems above under two types of consumerrsquos behavior characteristicsconsumerrsquos environmental awareness and consumerrsquos reference behavior A game-theoretic model is employed to describe a greensupply chain consisting of a manufacturer and a retailer combining important factors such as the carbon tax rate green in-vestment coefficient and degree of reference effect )en we obtain the optimal carbon tax rate by balancing the total tax revenueand product greenness A revenue-sharing contract is introduced to achieve green supply chain coordination and the impact ofthe carbon tax on coordination is analyzed )e results show the following (1) )e carbon tax rate and the difference between thepower of the manufacturer and retailer are the main factors determining green supply chain coordination (2) Maximumgreenness can be achieved when development costs are higher while the maximum tax revenue is obtained when the developmentcost is lower but with the loss of greenness (3) If the power of the manufacturer is low coordination can be achieved under theoptimal carbon tax If the power of the manufacturer is at a medium level coordination can be achieved by increasing the carbontax as a result increased greenness will be realized but with the loss of tax revenue However when the power of the manufactureris strong coordination cannot be achieved (4) Price reference behavior can promote supply chain coordination but consumerrsquosenvironmental awareness cannot

1 Introduction

With the degradation of the environment increasing at-tention has been directed toward global warming Forsustainable development many countries have been com-mitted to reducing carbon emissions For example at the2009 United Nations Climate Conference in Copenhagenthe Chinese government declared that carbon dioxideemissions per unit of GDP would be decreased by 40ndash50in 2020 compared with the levels in 2005 [1] Carbon dioxideis widely emitted by the transportation and manufacturingsectors [2] Carbon tax policy has been proven to be effectivefor emissions reduction [3ndash5] but it also exerts some sideeffects on enterprises [6] From the perspective of the green

supply chain carbon tax policies imposed on an enterprisecould decrease the profit within the supply chain thus af-fecting supply chain cooperation However previous re-search [7ndash9] has mostly focused on decision-making andcooperation problems within the supply chain under a givencarbon tax level rather than considering the interactionbetween carbon taxes and supply chain cooperation)erefore it is important to examine the interaction betweencarbon taxes and supply chain cooperation

Currently to meet the requirements of carbon taxregulations a growing number of enterprises have beenstriving for sustainability by committing to designingproducing and promoting green products to reduce carbonemissions [10] In this context green products have been

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 4131936 17 pageshttpsdoiorg10115520204131936

regarded as one of the important factors in achieving eco-nomic growth energy conservation and environmentalsustainability [11] Benjaafar et al inferred that introducingcarbon emissions into a supply chain optimization modelscan promote emissions reduction in the supply chain [12]Meanwhile carbon taxes increase production costs thusimposing burdens on enterprises Consequently it has beenincreasingly challenging for governments to enact appro-priate policies to reduce carbon emissions and improvesupply chain performance at the same time [13] In additionconsumerrsquos behavior is a crucial factor affecting productdemand and sustainable decisions In this study we considertwo types of consumerrsquos behavior characteristics consumerenvironmental awareness (CEA) and consumerrsquos referencebehavior )e former is an important factor that motivatesfirms to develop green products [14 15] )e latter is acrucial factor that affects green product demand [16 17] Inreality many enterprises are concerned about carbonemissions for example HP IBM Ge etc )ey are not onlybeginning to design green products but also enhancingsupply chain management and cooperation to achieve thegoal of emission reduction However it is hard to achievesupply chain cooperation )erefore it is meaningful tostudy supply chain green decisions and cooperation con-sidering consumer environmental awareness under a carbontax policy )erefore we consider the interaction betweenthe carbon tax rate and supply chain cooperation and op-timize the tax rate in the context of CEA and consumerrsquosreference behavior Specifically we aim to answer the fol-lowing questions

(i) Is carbon tax policy favorable for improving thegreenness of products

(ii) Can carbon tax policy promote supply chain co-ordination to incentivize supply chain members tocooperate

(iii) Can carbon tax policy be beneficial for both productgreenness and supply chain coordinationsimultaneously

To investigate the above problems a two-echelon supplychain is introduced )is is used to explore the interactionbetween green supply chain cooperation and carbon taxpolicy where a manufacturer acts as the leader and deter-mines the productrsquos greenness and wholesale price and aretailer acts as the follower and determines the retail price ofgreen products Moreover traditional products withoutgreen attributes (ie greenness) compete with greenproducts in the market)rough consumer utility we obtainthe demand functions for both green and traditionalproducts In addition the government decides the carbontax rate to limit carbon emissions Unit carbon emissions areassociated with the greenness of products )e higher thegreenness the higher the product RampD cost and the lowerthe unit carbon emission Finally the impact of the carbontax on supply chain cooperation is analyzed by introducing arevenue-sharing contract

)e results show the following (1) When the carbon taxand development cost are at high levels the greenness

increases with the carbon tax otherwise the greennessdecreases with increasing carbon tax )erefore there existsan optimal carbon tax to maximize the greenness when thedevelopment cost is high (2) Cooperation can be promotedwhen the government increases the carbon tax because theretailer plays a crucial role in coordination and the region ofcooperation is expanded when the carbon tax increases (3)When the manufacturerrsquos power is relatively low optimalcarbon tax and supply chain cooperation can be achievedsimultaneously whereas when the manufacturerrsquos power isrelatively high cooperation cannot be achieved In additionwhen the manufacturerrsquos power is at a moderate level co-operation can be achieved but with a loss of tax revenue insuch a case

)e main contributions of this paper are as follows Weinvestigated a green supply chain cooperation problem andthe interaction between carbon taxes and supply chaincooperation taking consumerrsquos reference behavior intoconsideration which fills a research gap in supply chaincooperation Our result provides a reference for pricing andgreen product design and a meaningful reference forpolicymakers

)e remainder of this paper is organized as follows Aliterature review is presented in Section 2 Section 3 in-troduces the basic assumptions and notations and thesupply chain model is formulated then the results of themodel are analyzed and we summarize the major conclu-sions of the numerical analysis Finally Section 4 summa-rizes the main research content and results

2 Literature Review

)ree streams of research are closely related to our workFirst we review the research on carbon taxes in green supplychains Second our work is related to research on referencebehaviors in operational management )ird relevant re-search on supply chain coordination is reviewed Finally wedistinguish our study from the three streams of researchmentioned above

21 Carbon Tax Reduction of greenhouse gas emissions isbecoming a vital issue and almost all developed and de-veloping countries are now implementing policies for car-bon emission reduction [18] Numerous studies havefocused on carbon taxes Carbon taxes restrict the demandfor fuels and thereby reduce the emissions of harmfulgreenhouse gases Generally carbon taxes are not always ashigh as possible For example a falling tax rate encouragesmanufacturers to produce and reduce emissions [19] )eoptimization problem of carbon taxes has previously beenstudied [20] Modak and Kelle integrated corporate socialresponsibility (CSR) investments into the supply chainstrategy and operations and concluded that the optimalrecycling rate and appropriate investment in recycling ac-tivities increase with an increase in the carbon tax rate [21])ere has also been other research regarding carbon taxesFor example Ulph and Ulph analyzed the optimal time pathfor a carbon tax and the numerical results suggested that a

2 Mathematical Problems in Engineering

carbon tax should initially rise and then fall [22] Kverndokkconsidered the optimal extraction of exhaustible resourcesand came to the same conclusion the optimal carbon taxshould initially rise and eventually fall [23] Some scholarshave suggested that the optimal tax should increasemonotonically or follow a U-shaped pattern [24 25]However the above studies did not consider the carbon taxin the context of a supply chain

)ere are some research gaps in the studies on optimalcarbon taxes for green supply chains Most of the studieshave focused on the decisions and cooperation problems ofthe supply chain under a carbon tax Hariga et al presentedthree operational models to determine the optimal lot-sizing and shipping quantities to reduce carbon emissionsIn those experiments a minor increase in operational costwith carbon tax regulation is outweighed by the costsavings resulting from carbon-related costs [26] Turkenet al investigated the effect of environmental regulations inthe form of a carbon tax on the plant capacity and locationdecisions of a firm )ey proposed two novel policy op-tions (1) a per unit per mile transportation penalty and (2)a collective transportation emissions policy with a limit ontotal transportation emissions Turken et al also revealedthat stricter regulations without high penalties would notensure compliance as the benefits from the increasing scaleassociated with a centralized plant frequently outweigh theregulatory penalties and a per unit carbon tax had no effecton regional production of emissions [27] Xu et al in-vestigated the joint production and pricing of amanufacturing firm with multiple products under cap-and-trade and carbon tax regulations )eir results showed thatthe optimal quantity of products produced under a carbontax regulation is determined by the emissionsrsquo tradingprices and the tax rate [9] Yu and Han studied the impactof a carbon tax on carbon emissions and retail prices in atwo-echelon supply chain consisting of a manufacturer anda retailer )e results indicated that with an increase in thecarbon tax both the optimal emission reduction level andthe optimal retail price initially increase and then remainstable [28] Sinha and Modak developed an economicproduction quantity model that elucidates a new side ofCO2 emissions reduction [29] Zhang investigated theimpact of the carbon tax on enterprise operation andobtained the coopetition supply chain and carbon taxmechanism [30] Chen et al investigated how a carbonemissions taxation scheme can be designed to reducecarbon emissions [31]

In general there are two streams in the previous liter-ature the optimization of the carbon tax and the relationshipbetween the carbon tax and operation decisions in thesupply chain However there exists a research gap in theexisting literature few papers focus on the relationshipbetween the optimal carbon tax and supply chain coordi-nation Our results show that by adjusting the carbon tax thegovernment can promote the coordination of supply chainsand reduce carbon emissions )e optimal carbon tax policyto promote coordination between the supplier and retailer isobtained by balancing the tax revenue and the productgreenness

22 Consumerrsquos Reference Behavior Consumers are alwaysconcerned about product value when choosing products onshelves )e final decision of consumption is a function ofgains and losses with respect to a reference outcome [32]Consumerrsquos reference behavior plays an important role inthis process of comparison thereby influencing firmsrsquo op-erational decisions such as product pricing and decisionsregarding product greenness [33 34]

Kopalle et al studied a novel household heterogeneitytranslation model considering consumersrsquo price referencebehavior and developed a normative pricing policy for re-tailers that maximizes category profit using individual-levelestimates [35] Hsieh and Dye investigated an inventorymodel based on price reference effects and established anoptimal dynamic pricing model to determine a pricingstrategy that maximizes the discounted total profit [36] )eresults suggested that the strength of the memory factor isimportant for a retailer to measure because a high memoryfactor value represents consumers with a longer memory ofperceived gains or losses )e optimal discounted total profitinitially increases as the memory factor increases but de-creases when the memory factor is relatively high Somedynamic price studies have also considered price referencebehavior [37 38] In these two previous studies the pricereference effect dominated the optimal pricing and inven-tory policy of the firm )e expected steady-state referenceprice was compared to the steady-state reference price in amodel with a deterministic reference price effect and theresults showed that the former was always higher Con-sumerrsquos environmental awareness is a common behavior inreal life At present consumers are frequently concernedabout environmental protection Green preference andgreen product design have attracted extensive attention fromscholars Zhang et al investigate the impacts of consumerenvironmental awareness and retailerrsquos fairness concerns onenvironmental quality [39] Chen studied product designand marketing decisions based on consumer preferences forenvironmental attributes [40] )e development of greenproducts depends heavily on the joint efforts of both thesupply chain and the government )erefore the govern-ment should create a regulatory environment that is benignto green product innovation Chitra inferred that greenconsumers affect marketing issues and a preference forgreenness will promote the purchase of green products [41])e higher the consumer environmental preference is thehigher the price will be that the consumer is willing to pay forlow-carbon products Consumerrsquos green preference is infavor of supply chain performance on the environment [42]Moreover as competition intensifies the profits of manu-facturers with inferior eco-friendly operations will alwaysdecrease Li et al infer that consumers should avoid ex-cessive pursuit of green product design otherwise they hurtthe environment by investigating the impact of consumerpreference for green product design [43]

Consumerrsquos reference behavior plays an important rolein firmsrsquo operational decisions and consumer environ-mental awareness is a common behavior in real life )eliterature above is about the influence of consumersrsquo greenpreference on enterprisesrsquo decision-making such as pricing

Mathematical Problems in Engineering 3

and green manufacturing )ere exists a gap in the existingliterature which is that the consumerrsquos behavior has notbeen concerned In our research the green preference andprice reference behavior are considered

23 Cooperation Supply chain cooperation is defined asldquolong-term relationships where participants generally co-operate share information and work together to plan andeven modify their business practices to improve joint per-formancerdquo [44] In general supply chain cooperation meansachieving better performance In the supply chain there aremany coordination strategies to choose from such as rev-enue sharing buybacks quantity discounts and two-parttariff contracts Among these contracts revenue sharing hasattracted the attention of many scholars and is widely used inactual supply chains [45] For example Xu et al proposed atwo-way revenue sharing contract to coordinate multipledistributors in a dual-channel supply chain and the resultsshowed that the manufacturer could prompt the retailer tocooperate by providing this contract [46] Shi et al studiedreverse revenue-sharing contracts in a closed-loop systemand proposed a function to calculate the optimal ratio of thetransfer collection price )e results also suggested thatreverse revenue-sharing contracts are more attractive formanufacturers than a two-part tariff [47] Panda et al ex-plored channel coordination in a socially responsiblemanufacturerndashretailer closed-loop supply chain and foundthat a revenue-sharing contract resolved channel conflict[48] Modak et al used the subgame perfect equilibrium andalternative offer bargaining strategy to resolve channelconflict and distribute surplus profit [49] Wang and Zhaodesigned a revenue-sharing contract to reduce carbonemissions and both the supplier and the retailer achievedPareto improvement In addition they developed a functionto determine the revenue sharing ratio using the Rubinsteinbargaining model [50] Yu et al considered a cooperationproblem in the low-carbon supply chain and found that theenvironmental awareness of consumers and tax rates con-siderably affect the emission reduction [51] )ere have alsobeen some supply chain coordination studies conductedunder carbon policies Revenue-sharing contracts have beendesigned to improve the performance of supply chainmembers based on different carbon policies [52] Xu et alstudied the coordination problem in a two-echelon supplychain and the effect of government policy-making ondistributing the optimal emission quota was investigated)e results showed that a reasonable revenue-sharingcontract is essential to increase supply chain membersrsquoprofits even under low-carbon conditions [53] Modak et alconcluded that the optimal recycling rate increases with theCSR activity of the manufacturer and a profit-sharingcontract provides the best channel performance in a closed-loop distribution channel consisting of a socially responsiblemanufacturer multiple retailers and a third-party collector[54] Feng infers that win-win results can be achieved byestablishing profit-sharing contracts considering the pref-erence of green consumers [55] Previous papers often focuson two aspects including the choice of contract and the

conditions of the contract However the above literature stillhas a gap between cooperation and consumerrsquos behavior)erefore we investigate the supply chain cooperationproblem under the context of consumerrsquos reference behaviorand the carbon tax

In summary this study examines the interaction be-tween supply chain cooperation and carbon taxes in a two-echelon supply chain considering consumerrsquos behaviorSome important factors should be considered simulta-neously to study this problem such as the optimal carbon taxand consumerrsquos behavior however previous research hasonly considered these factors separately We also investigatethe interaction between coordination and the carbon tax)e difference between our study and others in the literatureis presented in Table 1

3 Supply Chain Model

In this section a two-echelon supply chain model is intro-duced to study green supply chain cooperation and carbon taxpolicy )e optimal decisions of the supply chain and thegovernment tax are addressed We then introduce the rev-enue-sharing contract used to coordinate the supply chain

31 Model Assumptions To answer the first question (iscarbon tax policy favorable for improving the greenness ofproducts) a two-echelon supply chain model consisting ofa single supplier and a single retailer without contracts isestablished )e manufacturer determines the productrsquosgreenness and wholesale price and the retailer determinesthe retail price of the green product )e green productcompetes for market share with traditional productsCompared with traditional products green products havethe characteristics of low pollution and being environ-mentally friendly but they may be less functional such aselectric vehicles which has poor endurance and slow speedsIn our study we focus on green products that are lessfunctional than traditional products For example Belloset al noted that manufacturers offer vehicles with poorperformance for customers who focus on fuel efficiency [56]In real life greenness reflects the environmental attributes ofthe product and it is commonly used to measure howenvironmentally friendly a product is In this study we use g

to denote the greenness degree of green products as ameasure of their environmental attributes which is acommon practice [38 41]

Consumers are environmentally conscious and haveenvironmental awareness )e utility gained from envi-ronmental attributes is assumed to be kg where k is thesensitivity of the consumer to the greenness of a greenproduct [38] We use V to denote the utility obtained by aconsumer from a traditional product It is commonly as-sumed that V is uniformly distributed on [0 1] to simplifythe problem without affecting the conclusion [57] )en αV

is the utility from the green product where α isin (0 1) is thefunctional attribute coefficient of the green product thatreflects its weak functional performance compared with thetraditional product

4 Mathematical Problems in Engineering

In real life consumers usually compare prices betweentwo similar products Let β be the consumersrsquo recognitionlevel of the reference price [35] )erefore the utility ofgreen products is obtained from four parts a positive partfrom the basic utility (αV) a negative part from the price (p)a positive part from the environmental consciousness (kg)and a negative part from the price reference (β(p minus pn)))eutility to a consumer of green products and traditionalproducts can thus be expressed as follows

un V minus pn

ug αV minus p + kg minus β p minus pn( 1113857

⎧⎨

⎩ (1)

Table 2 summarizes the notation used in this study

32 Model and Solution Consumers choose between greenand traditional products by comparing utility when ug gt un

and ug gt 0 consumers will purchase the green productwhereas when un gt ug and un gt 0 consumers will purchasethe traditional product As a result demand is obtained asshown in equation (2) )e proof of the demand function isgiven in Appendix A

q (α + β)pn minus (1 + β)p + kg

α(1 minus α)

qn 1 minus(1 + β) pn minus p( 1113857 + kg

1 minus α

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(2)

)e manufacturer determines the wholesale price andgreenness of the green product Let c denote the cost rate oftechnology development then the total development cost is(12)cg2 which is convexly increasing with the greenness[40] In addition (e minus g) is a linear function of the unitcarbon emissions for green products [58] )erefore theprofit function for the manufacturer is obtained from threeparts a positive part from the wholesale ((w minus c)q) anegative part from green technology development ((cg22))and a negative part from the carbon tax (t(e minus g)q) )emanufacturerrsquos decision model is as follows

Maxwg

πM (w minus c minus t(e minus g))q minuscg

2

2 (3)

)e retailerrsquos decision problem is then formulated asfollows

Maxp

πR (p minus w)q (4)

)e optimal solutions for the retailer and manufacturerare derived as the following theorem by substituting thedemand function into the equation above )e proof ispresented in Appendix A

Theorem 1 -e optimal solutions for both parties are

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

g (1 + β)(te + c) minus (α + β)pn minus ke( 1113857t

t2(1 + β) minus 4cα(1 minus α)

p 3cpnα

3+ 2A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)

)e corresponding profits are

πm minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2c

2 t2β + t

2+ 4(minus 1 + α)cα1113872 1113873(β + 1)

πr minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2(minus 1 + α)αc

2

4α2c minus 4αc +(β + 1)t2

1113872 11138732(β + 1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

According to this theorem price and greenness decisionsare directly affected by the carbon tax rate Facing a highercarbon tax and development cost the manufacturer andretailer will reduce the greenness of products and the saleprice with increasing carbon tax However when facing alower carbon tax the greenness and price will increase withincreasing carbon tax (the details are shown in Appendix Bequations (1) and (5)) Next we investigate the effects ofthese parameters on the product greenness )e proofs areprovided in Appendix B

Proposition 1 -e effects of the carbon tax rate pricerecognition level development cost and functional attributecoefficient on product greenness are as follows

(1) -ere exist some thresholds 1113954t1 and 1113954c1 for which g

decreases with increasing t when cgt 1113954c1 and tgt1113954t1 elseg increases with increasing t

Table 1 Differences between our study and the available literature

Literature Decisions under the carbon tax Optimal carbon tax Consumerrsquos behavior CoordinationYu and Han [28] radic radicUlph and Ulph [22] radicHsieh and Dye [34] radicCao et al [47] radic radicXu et al [48] radic radicOur paper radic radic radic radic

Mathematical Problems in Engineering 5

(2) -ere exist some thresholds 1113954t2 and 1113954c2 for which g

decreases with increasing β when cgt 1113954c2 and tgt1113954t2 elseg increases with increasing β

(3) g increases with increasing k decreases with in-creasing c and initially increases and then decreaseswith increasing α

We can conclude that the carbon tax development costprice recognition level greenness sensitivity and functionalattribute coefficient all influence the greenness in Proposi-tion 1 When the carbon tax is low the manufacturer incursless cost to improve the greenness of the products As aresult the greenness increases with the carbon tax In ad-dition if the carbon tax is high and the development cost islow the greenness of the products will increase with in-creasing carbon tax because the tax is the crucial factor indecisions In contrast if the carbon tax is high and devel-opment costs are also high the greenness of products willdecrease because the manufacturer will choose to reducedevelopment costs )erefore pollution will not be reducedas the carbon tax increases in some cases Facing high de-velopment costs the manufacturer will not improvegreenness unless it is promoted by a lower carbon tax Anumerical simulation of this scenario is shown in Figure 1

Similarly the results show that greenness is affected bythe consumersrsquo price reference behavior When the carbontax is low the negative effect of price increases when con-sumers have higher concerns about price )erefore themanufacturer will offset this negative effect by improving thegreenness of products When the carbon tax is high and thedevelopment cost is low the greenness of products willincrease under a high consumer concern about price Incontrast if the carbon tax is high and the development cost ishigh the greenness will decrease because the manufacturerwill choose to reduce development costs Numerical sim-ulations of these scenarios are shown in Figure 2

In addition greenness sensitivity and the functional attri-bute coefficient also affect greenness )e greenness increaseswith increasing greenness sensitivity of consumers Highgreenness sensitivity can promote more green products )ismeans that the government can promote green production by

increasing the green consciousness of consumersMoreover thegreenness decreases with increasing development cost A higherdegree of greenness is obtained when manufacturers face lowerdevelopment costs In addition greenness first increases andthen decreases with the increase in the functional attributecoefficient When the functional attribute of the green productis low themanufacturer will produce a lower greenness productwith increases in the functional attribute coefficient In contrastthe manufacturer will produce a product with higher greennesswith an increase in the functional attribute after a functionalthreshold has been reached Numerical simulations of thesescenarios are shown in Figure 3

Overall consumersrsquo reference behavior plays a positive rolein promoting green production However the carbon tax is noteffective for improving green products in some cases Forexample if the carbon tax is high and the development cost isalso high the greenness of products will decrease)erefore wenext investigate the optimal carbon tax

First an optimal carbon tax is defined as that whichprovides the maximum greenness without losing total taxrevenue )e result is provided in Proposition 2 and theproofs can be found in Appendix C

Table 2 Notations

Parameters Definitionp pn Prices of the green product and traditional product respectivelyq qn Demand for the green product and traditional product respectivelyw Wholesale priceg Product greennesst Carbon tax ratec Cost rate of technology developmentc Unit product cost of the green producte Initial unit carbon emissionsα Functional attribute coefficient of the green product functionβ Consumersrsquo recognition level of the reference pricek Sensitivity of consumers to product greennessV Utility obtained by a consumer from the traditional productη Proportion of the total revenue obtained by the retailerλ Member strength in the supply chain

10

08

06g

04

02

0000 01 02 03

t

t = t1

04 05

010

008

006

004

002

000

γ lt γ1γ gt γ1

Figure 1 Effects of carbon taxes on greenness

6 Mathematical Problems in Engineering

Proposition 2 -e optimal carbon tax is defined as follows

tlowast

M minusM

2minus N

1113968

e(1 + β) 0lt clt 1113954c3

P minus

P2

minus Q

1113969

2(1 + β)M 1113954c3 lt clt 1113954c1

1113954t1 1113954c1 lt c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

)e optimal carbon tax is obtained in Proposition 2based on the manufacturerrsquos development cost to maximizegreenness without losing the total carbon tax revenue Wecan observe that the crucial factor is the development costAn appropriate carbon tax policy should be chosen whenfacing different production technologies to ensure greennessand total tax revenue In addition it is easily observed thatthe maximum greenness is obtained when the developmentcost is high whereas the maximum tax is obtained when thedevelopment cost is lower but some of the greenness is lostA numerical simulation of this scenario is shown in Figure 4

33 Supply Chain Coordination It is also important forsupply chains to establish green supply chain models byintegrating internal and external resources to make deci-sions which enables supply chains to achieve better per-formance by improving cooperation [59 60] However moststudies on coordination have mainly focused reducingemissions and improving profit Few studies have investi-gated the relationship between carbon taxes and supplychain cooperation

In this section we study the impact of carbon taxes andconsumerrsquos behavior on supply chain coordination wherethe manufacturer is the leader and the retailer is the fol-lower A revenue-sharing contract is introduced to achieve

coordination Let η denote the proportion of total revenueobtained by the retailer then the manufacturerrsquos decisionmodel is as follows

Maxwg

πCM ((1 minus η)p + w minus c minus t(e minus g))q minus

cg2

2 (8)

)e retailerrsquos decision problem can be described asfollows

Maxp

πCR (ηp minus w)q (9)

)e optimal solutions for the retailer and manufacturerare derived with the following theorem by substituting thedemand function into the equation above )e proof isfound in Appendix D

Theorem 2 -e greenness is defined as follows

g t minus αpn + et(β + 1) + c minus pn( 1113857β minus ke + c( 1113857

2c(1 + η)α2 minus 2c(1 + η)α + t2(β + 1)

(10)

-en the profits of both parties are the following

πm minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2c

2 t2β + t

2+ 4(minus 1 + α)cα1113872 1113873(β + 1)

πr minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2(minus 1 + α)αc

2

4α2c minus 4αc +(β + 1)t2

1113872 11138732(β + 1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

We investigate the impact of the carbon tax on pro-moting the coordination of the supply chain by comparingthe profit changes of the two partners after the introductionof the revenue-sharing contract in Proposition 3 )e proofsare provided in Appendix E

Proposition 3 Coordination can be promoted by increasingthe carbon tax

01204

01202

01200g

01198

01196

00 01 02 03β

04 05

11900 times 10ndash5

11895 times 10ndash5

11898 times 10ndash5

11892 times 10ndash5

11890 times 10ndash5

11888 times 10ndash5

11885 times 10ndash5

11882 times 10ndash5

11880 times 10ndash5

γ lt γ2 t lt t2γ gt γ2 t lt t2

(a)

096

095

094

093g

092

091

090

08900 01 02 03

β04 05

0035

0040

0045

0050

0055

0060

γ lt γ2 t gt t2γ gt γ2 t gt t2

(b)

Figure 2 Effects of consumersrsquo price reference behavior on greenness (a) Low-carbon tax rate (b) High carbon tax rate

Mathematical Problems in Engineering 7

When the profits of both the retailer and the manu-facturer improve simultaneously coordination is achievedBy comparing equations (11) and (6) we find that therevenue-sharing contract improves the retailerrsquos profit onlywhen η isin (η1 1) whereas the manufacturer always benefitsfrom the contract )e range of η depends on the carbon taxrate When η is in the given range the retailer and man-ufacturer achieve Pareto improvement Furthermore therange expands as the carbon tax rate increases)erefore thegovernment can promote supply chain collaboration byincreasing the carbon tax Based on the previous results forthe optimal carbon tax we next investigate how to achievecoordination when the supply chain faces the optimalcarbon tax Figure 5 shows the Pareto improvement region

According to the assumptions described above themanufacturer acts as the leader and the retailer acts as afollower It is reasonable that the proportion of profitreceived by the retailer is decided by the manufacturer)erefore it is necessary to investigate whether theproportion decided by the leader is in the region in whichcoordination is achieved A parameter λ is introduced todenote the strength of a partner in the supply chain Let

λm isin (05 1) and λr 1 minus λm represent the power of themanufacturer and the retailer respectively )en weconsider a simple function (λrλm) to determine η whichis the proportion of revenue received by the retailer )eretailer retains the maximum revenue when the power ofthe two parties is equal (ie λm λr 05) whereas all ofthe benefits go to the leader when the power of themanufacturer is overwhelming compared with that of theretailer (ie λm 1) Similar to Proposition 2 the optimalcarbon tax is obtained under the revenue-sharing contractas follows

tlowast

M minus

M2

minus (1 + η2)N

1113969

e(1 + β) 0lt clt 1113954c

C2

(1 + η2)P minus

(1 + η2)2P2

minus (1 + η2)Q

1113969

2(1 + β)M 1113954c

C2 lt clt 1113954c1

1113954t1 1113954c1 lt c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

05

04

03g

02

01

0010 15 20 25

k3530 40

g

(a)

10

08

06g

04

02

0010 09 0708 06

γ030405

g

(b)

029

027

028

025

026

g

024

022

023

021035030 040 045 050

α060 065055 070

g

(c)

Figure 3 Effects of parameters on greenness (a) Greenness sensitivity (b) Development cost (c) Functional attribute coefficient

8 Mathematical Problems in Engineering

Proposition 4 is designed to answer the question ofwhether cooperation will be achieved the proofs are foundin Appendix F

Proposition 4 Coordination will be achieved if the power ofthe manufacturer is low (ie 05lt λm lt 1113954λ1) Moreover whenthe power of the manufacturer is at a medium level (ie1113954λ1 lt λm lt 1113954λ2) coordination can be achieved but part of thetotal tax revenue will be lost if the development cost is low(ie clt 1113954c1) In other cases (ie cgt 1113954c1 or 1113954λlt λm) coordi-nation cannot be achieved because the power of the manu-facturer is overwhelming

Coordination situations for different manufacturerpower levels are present in Table 3 )e achievement ofcoordination is decided by the power of the manufacturerand the optimal carbon tax faced by the supply chain Basedon Proposition 3 coordination can be promoted if thegovernment adjusts the carbon tax On the other handcoordination can easily be achieved under the optimalcarbon tax if the power of the manufacturer is low Inaddition there are two possibilities when the power of themanufacturer is at a medium level If the development cost istoo high adjusting the tax will lead to an unknown result thegreenness and total tax revenue may both decline at the same

10 014

012

010

008

006

004

002

000

08

06

Gre

enne

ss

Tota

l tax

04

02

00005000 010 015 020

t

γ lt γ3

t = ta

025 030

GreennessTotal tax

(a)

10 030

025

020

015

010

005

000

08

06

Gre

enne

ss

Tota

l tax

04

02

000100 02 03 04

t

γ3 lt γ lt γ1

t = tc

05 06

GreennessTotal tax

(b)

005 040

035

025

020

015

010

005

000

004

003

Gre

enne

ss

Tota

l tax

002

001

0000100 02 03 04

t

γ1 lt γ

t = t1

06 08

030

05 07

GreennessTotal tax

(c)

Figure 4 Determination of the optimal carbon tax (a) 0lt clt 1113954c3 (b) 1113954c3 lt clt 1113954c1 (c) 1113954c1 lt c

005t

η

010 015 020 025 030

Pareto improvement

035 040

Figure 5 Pareto improvement region with increasing carbon tax

Mathematical Problems in Engineering 9

time If the development cost is lower the greenness may beimproved with a loss of revenue by increasing the carbon taxHowever when the power of the manufacturer is strong theregion δ (η1 1) cannot be achieved

According to our theory the government can promotethe coordination between the manufacturer and the retailerby adjusting the carbon tax When the power of the man-ufacturer increases the government still achieves coordi-nation by raising the carbon tax In general cooperation isachieved and the government plays a key role in promotingcooperation

)e effects of consumersrsquo behavior on supply chaincoordination are also investigated in the next proposition)e proofs are provided in Appendix G

Proposition 5 Price reference behavior can promote supplychain coordination but green preference cannot

From the previous results price reference has a compleximpact on the supply chain First the increasing sensitivityof customers to price will promote supply chain coordi-nation because this negative impact can be mitigated bycooperation In addition price reference behavior also af-fects the green decisions of the supply chain in a complexmanner In contrast the mechanism by which consumerrsquogreen preferences influence the supply chain is relativelysimple Greenness increases with increasing greennesspreference and green preference has no effect on the co-ordination of the supply chain

4 Conclusion

)is study examines the interaction between supply chaincooperation and the carbon tax problem in a two-echelonsupply chain under consumerrsquos reference behavior )e opti-mal carbon tax policy is obtained based on analysis of thecarbon tax green investment coefficient and degree of con-sumerrsquos price reference )e optimal carbon tax is defined asthe simultaneous optimization of total tax revenue and productgreenness In addition coordination is achieved by introducingrevenue-sharing contracts)e impact of consumersrsquo referencebehavior and the carbon tax on supply chain coordination isalso investigated )e results are as follows

(1) )e greenness increases with increasing carbon taxwhen the carbon tax is low In addition if the carbontax is higher and the development cost is low thegreenness of products will increase with increasingcarbon tax Conversely if the carbon tax is higherand the development cost is also high the greennessof products will decrease

(2) When the carbon tax is low the negative effect ofprice increases as consumer concerns about priceincrease )erefore the manufacturer offsets thenegative effect by improving the greenness Whenthe carbon tax is higher and development cost is lowthe greenness of products will increase with highconsumerrsquos concern about price In contrast if thecarbon tax is higher and the development cost ishigh the greenness will decrease

(3) )e greenness increases with an increasing greennesspreference of consumers )eir preference behaviorcan promote more green products )is means thatthe government can promote green production bypromoting the green consciousness of consumersMoreover the greenness decreases with increasingdevelopment cost A higher greenness is obtainedwhen manufacturers face lower development costsIn addition greenness first increases and then de-creases with increases in the functional attributecoefficient When the functional attribute of thegreen product is low the manufacturer will producea lower greenness product with an increase in thefunctional attribute In contrast the manufacturerwill produce a higher greenness product with anincrease in the functional attribute after a func-tionality threshold

(4) We investigated the impact of the carbon tax onpromoting the coordination of the supply chain bycomparing the profit changes of two partners afterthe introduction of a revenue-sharing contract inProposition 3 We found that coordination could bepromoted by increasing the carbon tax

(5) )e achievement of coordination depends on thetype of manufacturer If the power of the manu-facturer is low coordination can be achievedunder the optimal carbon tax If the power of themanufacturer is at a medium level coordinationcan be achieved by increasing the carbon tax andimproved greenness will be realized with a loss ofrevenue However when the power of the man-ufacturer is strong coordination cannot beachieved

In this study we focused on the interaction between acarbon tax and supply chain cooperation However othercarbon policies (eg a cap-and-trade policy) and supplychain structures are worth exploring For example it wouldbe interesting to investigate the effects on a supply chainstructure consisting of the retailer as the leader under acomplex carbon policy

Table 3 Coordination situations for different manufacturer power levels

Power of the manufacturer Coordination Greenness Total carbon tax revenue05lt λm lt 1113954λ1 Achieve Improve Improve1113954λ1 lt λm lt 1113954λ2 clt 1113954c1 Achieve Improve Decline1113954λ1 lt λm lt 1113954λ2 clt 1113954c1 Cannot achievecgt 1113954c1 Cannot achieve

10 Mathematical Problems in Engineering

Appendix

A Proof of Theorem 1

)e demand function can be written as

q P ug ge un ug ge 01113966 1113967

qn P un gt ug un ge 01113966 1113967

⎧⎪⎨

⎪⎩(A1)

After replacing un and ug we have

q P αv minus p + kg minus β p minus pn( 1113857 minus v minus pn( 1113857ge 0 αv minus p + kg minus β p minus pn( 1113857ge 01113864 1113865

qn P v minus pn minus αv minus p + kg minus β p minus pn( 1113857( 1113857gt 0 v minus pn ge 01113864 11138651113896 (A2)

Given that V is uniformly distributed on [0 1] thedemand function is

q (α + β)pn minus (1 + β)p + kg

α(1 minus α)

qn 1 minus(1 + β) pn minus p( 1113857 + kg

1 minus α

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A3)

)e condition kgt ((1 + β)p minus (α + β)png) is necessaryto ensure nonnegativity )e retailer problem is addressedfirst and we have

p (α + β)pn +(1 + β)w + kg

2(1 + β) (A4)

)en substitute p into the manufacturerrsquos profit func-tion Solving the optimal w and g simultaneously yields

w (α + β)pn +(1 + β)(c + t(e minus g)) + kg

2(1 + β)

g t (α + β)pn minus (1 + β)w + ke( 1113857

2cα(1 minus α)

(A5)

)en we have

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

g (1 + β)(te + c) minus (α + β)pn minus ke( 1113857t

t2(1 + β) minus 4cα(1 minus α)

p 3cpnα

3+ 2A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1 (1 + β)(c + te) + ke minus (1 minus β)pn

A2 (1 + β)(c + te) + 3ke minus (3 minus β)pn

⎧⎪⎨

⎪⎩

B1 minus 2c (1 + β)(c + te) + ke + βpn( 1113857 + t2(1 + β)pn

B2 minus c (1 + β)(c + te) + 3ke + 3βpn( 1113857 + t2(1 + β)pn

⎧⎪⎨

⎪⎩

(A6)

Notes that we have (α + β)pn + ke minus (1 + β)(te + c)gt 0)e concavity condition is 4αc(1 minus α) minus (1 + β)t2 gt 0

B Proof of Proposition 1

(1) )e first derivative of g in t is

zg

zt

minus 4cα3pn + 4cA3α2

+ B3α minus t2(1 + β) (1 + β)c minus ke minus βpn( 1113857

t2(1 minus β) minus 4cα(1 minus α)1113872 1113873

2

A3 (1 + β)(c + 2te) minus ke +(1 minus β)pn

B3 minus 4c (1 + β)(c + 2te) minus ke minus βpn( 1113857 + t2(1 + β)pn

⎧⎨

⎩

(B1)

We have a threshold

1113954c1 minust2(1 + β) (α + β)pn + ke minus (1 + β)c( 1113857

4α(1 minus α) (α + β)pn + ke minus (1 + β)(c + 2te)( 1113857

1113954t1 (α + β)pn + ke minus (1 + β)c

2(1 + β)e

(B2)

When cgt 1113954c1 and tgt1113954t1 g is decreasing in t Bycontrast g is increasing in t in other cases

(2) )e first derivative of g in β is

zg

zβ

minus 4α(1 minus α) te + c minus pn( 1113857c + t2

ke + αpn minus pn( 1113857

t2(1 + β) minus 4α(1 minus α)c1113872 1113873

2

(B3)

We have a threshold

1113954c2 minust2

ke minus (1 minus α)pn( 1113857

4α(1 minus α) te + c minus pn( 1113857

1113954t2 pn minus c

e

(B4)

Review we have (α + β)pn + ke minus (1 + β)(te + c)gt 0therefore we can find that

tltpn minus c

e+

ke minus (1 minus α)pn

(1 + β)e (B5)

(i) When ke minus (1 minus α)pn lt 0 then tlt1113954t2 is inevitable)erefore (zgzβ)gt 0 is correct

(ii) When ke minus (1 minus α)pn gt 0 we can find that(zgzβ)lt 0 if cgt 1113954c2 and tgt1113954t2 (zgzβ)lt 0 in othercases

(3) )e first derivative of g in k and c is

Mathematical Problems in Engineering 11

zg

zc minus

4α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

4cα(1 minus α) minus t2(1 + β)1113872 1113873

2 lt 0

zg

zk

te

4cα(1 minus α) minus t2(1 + β)gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(B6)

(4) )e first derivative of g in α is

zg

zα

minus Xt

4α(1 minus α)c minus t2(1 + β)1113872 1113873

2 (B7)

And X minus 4α2cpn + (4(1 + β)(te + c) minus 4ke minus 4pn)

(2α + 1)c + pnt2(1 + β) is a quadratic function )erange of α in 4αc(1 minus α) minus (1 + β)t2 gt 0 is

α1 c +

c2

minus ct2(1 + β)

1113969

2c

α2 c minus

c2

minus ct2(1 + β)

1113969

2c

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(B8)

By substituting those into X we have

X|αα1 4

c2

minus ct2(1 + β)

1113969

T minus 2 c minus t2(1 + β)1113872 1113873pn

X|αα2 minus 4

c2

minus ct2(1 + β)

1113969

T minus 2 c minus t2(1 + β)1113872 1113873pn lt 0

⎧⎪⎪⎨

⎪⎪⎩

(B9)

And T (1 + β)(te + c) minus ke minus ((12) + β)pn lt 0We can find

X|αα1 times X|αα2 minus 4 c minus t2(1 + β)1113872 1113873

middot 4 βpn + ke minus (1 + β)(te + c)( 11138572

1113872

+ p2nt

2(1 + β)1113873lt 0

(B10)

)erefore X|αα1 gt 0 )ere only exist a 1113954α isin (α2 α1)that makes (zgzα) 0 And (zgzα) gt 0 ifα isin (α2 1113954α) (zgzα)lt 0 if α isin (1113954α α1)

(5) )e first derivative of p in t is

zp

zt minus

2α(α minus 1) e(1 + β)t221113872 1113873 + minus pnα minus ke + c minus pn( 1113857β + c( 1113857t minus 2eαc(α minus 1)1113872 1113873c

(1 + β)t2

+ 4αc(α minus 1)1113872 11138732

A3 (1 + β)(c + 2te) minus ke +(1 minus β)pn

B3 minus 4c (1 + β)(c + 2te) minus ke minus βpn( 1113857 + t2(1 + β)pn

⎧⎪⎨

⎪⎩

(B11)

)e discriminant of quadratic function

e(1 + β)t2

2+ minus pnα minus ke + c minus pn( 1113857β + c( 1113857t minus 2eαc(α minus 1)

(B12)

is

Δ (minus β minus 1)c + ke + βpn + pnα( 11138572

+ 4e2(1 + β)t

2αc(α minus 1)

(B13)

We have a threshold

1113954c minus(minus β minus 1)c + ke + βpn + pnα( 1113857

2

4e2(1 + β)t

2α(α minus 1) (B14)

Review we have(α + β)pn + ke minus (1 + β)(te + c)gt 0therefore we can find that

tm 2

minus (1 + β)αc(α minus 1)

1113968

1 + β (B15)

Substituting it into above quadratic function

2 c

radicminus pnα minus ke + c minus pn( 1113857β + c( 1113857

minus (1 + β)α(α minus 1)

1113968minus 4eαc(α minus 1)(1 + β)

1 + β (B16)

And we have a same threshold

1113954c minus(minus β minus 1)c + ke + βpn + pnα( 1113857

2

4e2(1 + β)t

2α(α minus 1) (B17)

It could be found that(zpzt) gt 0 because Δlt 0 whenclt 1113954c And when cgt 1113954c we have another threshold 1113954t andzpzt|

t1113954t 0 because Δgt 0 and zpzt|tmlt 0)erefore when

cgt 1113954c and tgt1113954t p is decreasing in t And when cgt 1113954c and tlt1113954tp is increasing in t

12 Mathematical Problems in Engineering

C Proof of Proposition 2

(1) Let R t(e minus g)q denote total carbon tax and thefirst derivative of R in t is

zR

zt2c A2t

2+ B2t + c1113872 1113873 A3t

2+ B3t + c1113872 1113873

4cα(1 minus α) minus t2(1 + β)1113872 1113873

3 (C1)

Among them

A2 (1 + β)e

2gt 0

B2 minus (α + β)pn + ke minus (1 + β)c( 1113857lt 0

C2 2ceα(1 minus α)gt 0

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

A3 (α + β)pn + ke minus (1 + β)c( 1113857(1 + β)gt 0

B3 minus 8ceα(1 minus α)(1 + β)lt 0

C3 4cα(1 minus α) (α + β)pn + ke minus (1 + β)c( 1113857gt 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(C2)

Let Φ A2t2 + B2t + C2 and Ψ A3t

2 + B3t + C3 andwe can find that zRzt and Φ times Ψ have a same sign Bysolving Δ1 and Δ2 a threshold is

1113954c3 (α + β)pn + ke minus (1 + β)c( 1113857

2

4e2α(1 minus α)(1 + β)

(C3)

)ere are two positive real numbers of solutions forΦ ifclt 1113954c3 and let 0lt ta lt tb denote it Similarly let0lt tc lt td denote the solutions for Ψ if cgt 1113954c3 Reviewwe have 4αc(1 minus α) minus (1 + β)t2 gt 0 therefore lettm

4cα(1 minus a)1 + β

1113968 Substituting tm into Φ and Ψ

Φ|tm minus 2

cα(1 minus α)

1 + β

1113971

(α + β)pn + ke minus (1 + β)c( 1113857 + 4ecα(1 minus α)

Ψ|tm minus 8cα(1 minus α) 2(1 + β)e

α(1 minus α)

1 + βc

1113971

minus (α + β)pn + ke minus (1 + β)c( 1113857⎛⎝ ⎞⎠

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(C4)

If clt 1113954c3 Φ|tmlt 0 and Ψ|tm

gt 0 then zRzt|tmlt 0 If

cgt 1113954c3 Φ|tmgt 0 and Ψ|tm

lt 0 then zRzt|tmlt 0 )ere-

fore we have tm isin (ta tb) or tm isin (tc td) FurthermoreR t(e minus g)q is maximized when t ta if clt 1113954c3 R

t(e minus g)q is maximized when t tc if cgt 1113954c3 Totally wehave

tlowast

ta clt 1113954c3

tc cgt 1113954c31113896 (C5)

(2) Review Proposition 1 If cgt 1113954c1 g is maximized whentlowast 1113954t1 If clt 1113954c1 g is maximized when tlowast tmWhen tgt1113954t1 1113954c1 gt 1113954c5 And 1113954c1 lt 0lt 1113954c5 if tlt1113954t1Substituting 1113954t1 into Φ

Φ|1113954t1 minus

3 (α + β)pn + ke minus (1 + β)c( 11138572

8e(1 + β)+ 2ecα(1 minus α)

(C6)

A threshold is 1113954c4 (34)1113954c3 )erefore we have (i)when 0lt clt 1113954c4 tlowast ta because of Φ|1113954t1

lt 0 and 1113954t1 gt ta

(ii) when 1113954c4 lt clt 1113954c3 tlowast ta because of 1113954t1 lt ta (iii) tlowast

tc when 1113954c3 lt clt 1113954c1 and (iv) tlowast tc when 1113954c1 lt c )eresult is obtained by solving Φ and Ψ

D Proof of Theorem 2

Similar to Appendix A )e retailer problem is addressedfirst and we have

p αηpn + βηpn + ηkg + βw + w

2η(β + 1) (D1)

)en substituting p into the manufacturerrsquos profitfunction Solving the optimal w and g simultaneously yields

w αηpn + βet + βηpn minus βgt + eηk + cβ + et minus gt + c( 1113857η

(1 + η)(β + 1)

g minusαηpn + βηpn + eηk minus βw minus w( 1113857t

2(α minus 1)cαη

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(D2)

Mathematical Problems in Engineering 13

By solving the equation above we have

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

g t minus αpn + et(β + 1) + c minus pn( 1113857β minus ke + c( 1113857

t2(1 + β) minus (1 + η)cα(1 minus α)

p (2η + 1)cpnα

3+ A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1 et + ηpn + c( 1113857β + et + ke minus pn( 1113857η + c

A2 (2η + 1)pn + et + c( 1113857β +(minus 2η minus 1)pn + et +(2ηk + k)e + c

⎧⎪⎪⎨

⎪⎪⎩

B1 minus 2et minus 2ηpn minus 2c( 1113857β minus 2eηk minus 2et minus 2c( 1113857c + t2pn(β + 1)

B2 (minus 2η minus 1)pn minus et minus c( 1113857β minus et +(minus 2ηk minus k)e minus c( 1113857c + t2pn(β + 1)

⎧⎪⎪⎨

⎪⎪⎩

(D3)

Note that we have minus αpn + et(β + 1) + (c minus pn)β minus ke +

clt 0 )e concavity condition is (1 + β)(t2(1 + β) minus (1+

η)cα(1 minus α))lt 0

Substituting solutions into the profit function we have

πm minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2c

2 t2β + t

2+ 4(minus 1 + α)cα1113872 1113873(β + 1)

πr minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2(minus 1 + α)αc

2

4α2c minus 4αc +(β + 1)t2

1113872 11138732(β + 1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(D4)

E Proof of Proposition 3

πM (α + β)pn + ke minus (1 + β)c( 1113857

2c

2(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCM

(α + β)pn + ke minus (1 + β)c( 11138572c

2(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(E1)

By comparing the profits of two partners change after therevenue-sharing contract introduced we found that themanufacturer always benefits from cooperation

πR c2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCR

ηc2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(E2)

A threshold is

η1 2cα(1 minus α) minus (1 + β)t2

2cα(1 minus α)1113888 1113889

2

(E3)

When η is in the region δ (η1 1) the retailer willaccept cooperation And the first derivative of η1 in t is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t(β + 1)

α2c2(minus 1 + α)

2 lt 0 (E4)

)erefore η1 is decreasing in t )at means the regionδ (η1 1) is expanding when tax rises And the revenue-sharing contract is accepted easily than before

F Proof of Proposition 4

We consider a simple function (λrλm) to decide thepractical η where λm isin (05 1) and λr 1 minus λm Given the

14 Mathematical Problems in Engineering

carbon tax tlowast we have a region δ (η(tlowast) 1) of cooperationIt is clear that the gap of strength between two partitions inthe supply chain is a crucial factor in cooperation Bycomparing practice η decided by strength and η(tlowast) de-cided by the carbon tax we have

1λm

minus 1 2α2c minus 2αc + t

2(β + 1)1113872 1113873

2

4α2c2(minus 1 + α)

2 (F1)

A threshold is

1113954λ1 4α2c2

(minus 1 + α)2

8α4c2minus 16α3c2

+ 4 2c + t2(β + 1)1113872 1113873cα2 minus 4t

2c(β + 1)α + t

4(β + 1)

2 (F2)

When 05lt λm lt 1113954λ1 the practical η is in the regionδ (η(tlowast) 1) therefore the cooperation is achieved facingthe optimal carbon tax

However the practical η is out of the range δ (η(tlowast) 1)

when 1113954λ1 lt λm Review previous result the region δ (η1 1)

is expanding when tax rises )erefore there exists a pos-sibility that the government increasing carbon tax to co-ordinate the supply chain We assume a realistic thing thegovernment is willing to boost the greenness of the producteven reducing part of the revenue from the carbon tax

Based on it the carbon up to 1113954t1 which means thegreenness is maximized )erefore )e cooperation isachieved when clt 1113954c1 because the greenness is decreasing inthe carbon tax when 1113954c1 lt c )en we have the secondthreshold

1113954λ2 1

η1 1113954t1( 1113857 + 1 (F3)

)e cooperation could not be achieved when λm gt 1113954λ2)e conclusion is as follows (1) Coordination would be

achieved when the manufacturer is democratic (ie05lt λm lt 1113954λ1) (2) Coordination could be achieved but partof the total tax will lose if development cost is lower when themanufacturer is moderate (ie 1113954λ1 lt λm lt 1113954λ2) (3) )e co-operation could not be achieved if the manufacturer is adictator (ie λm gt 1113954λ2)

G Proof of Proposition 5

)e first derivative of η1 in β is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t

2

2α2c2(minus 1 + α)

2 lt 0 (G1)

Similarly η1 is decreasing in β )at means the regionδ (η1 1) is expanding when β rises

And it is apparent that η1 (2cα(1 minus α) minus (1 + β)t22cα(1 minus α))2 is independent with k

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is research was supported by the National Natural ScienceFoundation of China (no 71761015) and the Science andTechnology Research Project of Jiangxi Education Depart-ment of China (no GJJ150475)

References

[1] Y Li M K Lim J Hu and M-L Tseng ldquoInvestigating theeffect of carbon tax and carbon quota policy to achieve lowcarbon logistics operationsrdquo Resources Conservation andRecycling vol 154 p 104535 2020

[2] M Abedi-Varaki ldquoStudy of carbon dioxide gas treatmentbased on equations of kinetics in plasma discharge reactorrdquoModern Physics Letters B vol 31 no 22 p 1750210 2017

[3] S Meng M Siriwardana and J McNeill ldquo)e environmentaland economic impact of the carbon tax in Australiardquo Envi-ronmental and Resource Economics vol 54 no 3 pp 313ndash332 2013

[4] Y Chen and C L Tseng ldquoInducing clean technology in theelectricity sector tradable permits or carbon tax policiesrdquo-eEnergy Journal vol 32 no 3 2011

[5] H-B Duan L Zhu and Y Fan ldquoOptimal carbon taxes incarbon-constrained China a logistic-induced energy eco-nomic hybrid modelrdquo Energy vol 69 pp 345ndash356 2014

[6] C Lu Q Tong and X Liu ldquo)e impacts of carbon tax andcomplementary policies on Chinese economyrdquo Energy Policyvol 38 no 11 pp 7278ndash7285 2010

[7] K Cao B Xu and J Wang ldquoOptimal trade-in and warrantyperiod strategies for new and remanufactured products undercarbon tax policyrdquo International Journal of Production Re-search vol 58 no 1 pp 180ndash199 2020

[8] P He W Zhang X Xu and Y Bian ldquoProduction lot-sizingand carbon emissions under cap-and-trade and carbon taxregulationsrdquo Journal of Cleaner Production vol 103pp 241ndash248 2015

[9] X Xu X Xu and P He ldquoJoint production and pricing de-cisions for multiple products with cap-and-trade and carbontax regulationsrdquo Journal of Cleaner Production vol 112pp 4093ndash4106 2016

Mathematical Problems in Engineering 15

[10] Q Li X Guan T Shi and W Jiao ldquoGreen product designwith competition and fairness concerns in the circulareconomy erardquo International Journal of Production Researchvol 58 no 1 pp 165ndash179 2020

[11] A Ranjan and J K Jha ldquoPricing and coordination strategiesof a dual-channel supply chain considering green quality andsales effortrdquo Journal of Cleaner Production vol 218pp 409ndash424 2019

[12] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineeringvol 10 no 1 pp 99ndash116 2012

[13] X Yin X Chen X Xu and L Zhang ldquoTax or subsidyOptimal carbon emission policy a supply chain perspectiverdquoSustainability vol 12 no 4 p 1548 2020

[14] B B Schlegelmilch G M Bohlen and A Diamantopoulosldquo)e link between green purchasing decisions and measuresof environmental consciousnessrdquo European Journal of Mar-keting vol 30 no 5 pp 35ndash55 1996

[15] M S Hopkins ldquoWhat the rsquogreenrsquo consumer wantsrdquo MITSloan Management Review vol 50 no 4 pp 87ndash89 2009

[16] G Kalyanaram and R S Winer ldquoEmpirical generalizationsfrom reference price researchrdquo Marketing Science vol 14no 3 pp G161ndashG169 1995

[17] E A Greenleaf ldquo)e impact of reference price effects on theprofitability of price promotionsrdquo Marketing Science vol 14no 1 pp 82ndash104 1995

[18] N M Modak D K Ghosh S Panda and S S SanaldquoManaging green house gas emission cost and pricing policiesin a two-echelon supply chainrdquo CIRP Journal ofManufacturing Science and Technology vol 20 pp 1ndash11 2018

[19] P Sinclair ldquoHigh does nothing and rising is worse carbontaxes should keep declining to cut harmful emissionsrdquo -eManchester School vol 60 no 1 pp 41ndash52 1992

[20] G M Grossman and A B Krueger ldquoEconomic growth andthe environmentrdquo -e Quarterly Journal of Economicsvol 110 no 2 pp 353ndash377 1995

[21] N M Modak and P Kelle ldquoUsing social work donation as atool of corporate social responsibility in a closed-loop supplychain considering carbon emissions tax and demand uncer-taintyrdquo Journal of the Operational Research Society pp 1ndash172019

[22] A Ulph and D Ulph ldquo)e optimal time path of a carbon taxrdquoOxford Economic Papers vol 46 no S1 pp 857ndash868 1994

[23] M Hoel and S Kverndokk ldquoDepletion of fossil fuels and theimpact of global warmingrdquo Resource and Energy Economicsvol 18 no 2 pp 115ndash136 1996

[24] Y H Farzin and O Tahvonen ldquoGlobal carbon cycle and theoptimal time path of a carbon taxrdquo Oxford Economic Papersvol 48 no 4 pp 515ndash536 1996

[25] V Bosetti C Carraro R Duval andM Tavoni ldquoWhat shouldwe expect from innovation A model-based assessment of theenvironmental and mitigation cost implications of climate-related RampDrdquo Energy Economics vol 33 no 6 pp 1313ndash13202011

[26] M Hariga R Asrsquoad and A Shamayleh ldquoIntegrated economicand environmental models for a multi stage cold supply chainunder carbon tax regulationrdquo Journal of Cleaner Productionvol 166 pp 1357ndash1371 2017

[27] N Turken J Carrillo and V Verter ldquoFacility location andcapacity acquisition under carbon tax and emissions limits tocentralize or to decentralizerdquo International Journal of Pro-duction Economics vol 187 pp 126ndash141 2017

[28] W Yu and R Han ldquoCoordinating a two-echelon supply chainunder carbon taxrdquo Sustainability vol 9 no 12 p 2360 2017

[29] S Sinha and N M Modak ldquoAn EPQmodel in the perspectiveof carbon emission reductionrdquo International Journal ofMathematics in Operational Research vol 14 no 3pp 338ndash358 2019

[30] H Zhang P Li H Zheng and Y Zhang ldquoImpact of carbontax on enterprise operation and production strategy for low-carbon products in a co-opetition supply chainrdquo Journal ofCleaner Production Article ID 125058 2020

[31] X Chen H Yang X Wang et al ldquoOptimal carbon tax designfor achieving low carbon supply chainsrdquo Annals of OperationsResearch pp 1ndash28 2020

[32] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Handbook of the Fundamentals of Fi-nancial Decision Making Part I pp 99ndash127 )e EconometricSociety Cleveland OH USA 2013

[33] W Kim and M Kim ldquoReference quality-based competitivemarket structure for innovation driven marketsrdquo Interna-tional Journal of Research in Marketing vol 32 no 3pp 284ndash296 2015

[34] I Popescu and Y Wu ldquoDynamic pricing strategies withreference effectsrdquo Operations Research vol 55 no 3pp 413ndash429 2007

[35] P K Kopalle P K Kannan L B Boldt and N Arora ldquo)eimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Re-tailing vol 88 no 1 pp 102ndash114 2012

[36] T-P Hsieh and C-Y Dye ldquoOptimal dynamic pricing fordeteriorating items with reference price effects when inven-tories stimulate demandrdquo European Journal of OperationalResearch vol 262 no 1 pp 136ndash150 2017

[37] X Chen Z-Y Hu and Y-H Zhang ldquoDynamic pricing withstochastic reference price effectrdquo Journal of the OperationsResearch Society of China vol 7 no 1 pp 107ndash125 2019

[38] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquo Opera-tions Research vol 64 no 6 pp 1529ndash1536 2016

[39] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[40] C Chen ldquoDesign for the environment a quality-based modelfor green product developmentrdquo Management Sciencevol 47 no 2 pp 250ndash263 2001

[41] K Chitra ldquoIn search of the green consumers a perceptualstudyrdquo Journal of Services Research vol 7 no 1 2007

[42] Z Liu T D Anderson and J M Cruz ldquoConsumer envi-ronmental awareness and competition in two-stage supplychainsrdquo European Journal of Operational Research vol 218no 3 pp 602ndash613 2012

[43] B Li Y Wang and Z Wang ldquoManaging a closed-loop supplychain with take-back legislation and consumer preference forgreen designrdquo Journal of Cleaner Production Article ID124481 2020

[44] J Rantanen D B Grant and W Piotrowicz ldquoInvestigatingsupply chain cooperation in Finnish grocery retailrdquo ResearchJournal of the University of Gdansk Transport Economics andLogistics vol 71 pp 19ndash34 2017

[45] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[46] G Xu B Dan X Zhang and C Liu ldquoCoordinating a dual-channel supply chain with risk-averse under a two-way

16 Mathematical Problems in Engineering

revenue sharing contractrdquo International Journal of ProductionEconomics vol 147 pp 171ndash179 2014

[47] Z Shi N Wang T Jia et al ldquoReverse revenue sharingcontract versus two-part tariff contract under a closed-loopsupply chain systemrdquoMathematical Problems in Engineeringvol 2016 Article ID 5464570 15 pages 2016

[48] S Panda N M Modak and L E Cardenas-Barron ldquoCo-ordinating a socially responsible closed-loop supply chainwith product recyclingrdquo International Journal of ProductionEconomics vol 188 pp 11ndash21 2017

[49] N M Modak N Modak S Panda and S S Sana ldquoAnalyzingstructure of two-echelon closed-loop supply chain for pricingquality and recycling managementrdquo Journal of Cleaner Pro-duction vol 171 pp 512ndash528 2018

[50] Q P Wang and D Z Zhao ldquoRevenue-sharing contract ofsupply chain based on consumerrsquos preference for low carbonproductsrdquo Chinese Journal of Management Science vol 22no 9 pp 106ndash113 2014

[51] B Yu J Wang X Lu and H Yang ldquoCollaboration in a low-carbon supply chain with reference emission and cost learningeffects cost sharing versus revenue sharing strategiesrdquo Journalof Cleaner Production vol 250 Article ID 119460 2020

[52] J Cao X Zhang and G Zhou ldquoSupply chain coordinationwith revenue-sharing contracts considering carbon emissionsand governmental policy makingrdquo Environmental Progress ampSustainable Energy vol 35 no 2 pp 479ndash488 2016

[53] X Xu P He H Xu and Q Zhang ldquoSupply chain coordi-nation with green technology under cap-and-trade regula-tionrdquo International Journal of Production Economics vol 183pp 433ndash442 2017

[54] N M Modak S Sinha S Panda et al ldquoAnalyzing a sociallyresponsible closed-loop distribution channel with recyclingfacilityrdquo SN Applied Sciences vol 1 no 10 p 1189 2019

[55] D Feng L Ma Y Ding G Wu and Y Zhang ldquoDecisions ofthe dual-channel supply chain under double policy consid-ering remanufacturingrdquo International Journal of Environ-mental Research and Public Health vol 16 no 3 p 465 2019

[56] I Bellos M Ferguson and L B Toktay ldquo)e car sharingeconomy interaction of business model choice and productline designrdquo Manufacturing amp Service Operations Manage-ment vol 19 no 2 pp 185ndash201 2017

[57] G Ferrer and J M Swaminathan ldquoManaging new andremanufactured productsrdquo Management Science vol 52no 1 pp 15ndash26 2006

[58] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2pp 519ndash527 2011

[59] L Cui S Guo and H Zhang ldquoCoordinating a green agri-foodsupply chain with revenue-sharing contracts consideringretailersrsquo green marketing effortsrdquo Sustainability vol 12no 4 p 1289 2020

[60] J Heydari K Govindan and A Aslani ldquoPricing and greeningdecisions in a three-tier dual channel supply chainrdquo Inter-national Journal of Production Economics vol 217 pp 185ndash196 2019

Mathematical Problems in Engineering 17

carbon tax should initially rise and then fall [22] Kverndokkconsidered the optimal extraction of exhaustible resourcesand came to the same conclusion the optimal carbon taxshould initially rise and eventually fall [23] Some scholarshave suggested that the optimal tax should increasemonotonically or follow a U-shaped pattern [24 25]However the above studies did not consider the carbon taxin the context of a supply chain

)ere are some research gaps in the studies on optimalcarbon taxes for green supply chains Most of the studieshave focused on the decisions and cooperation problems ofthe supply chain under a carbon tax Hariga et al presentedthree operational models to determine the optimal lot-sizing and shipping quantities to reduce carbon emissionsIn those experiments a minor increase in operational costwith carbon tax regulation is outweighed by the costsavings resulting from carbon-related costs [26] Turkenet al investigated the effect of environmental regulations inthe form of a carbon tax on the plant capacity and locationdecisions of a firm )ey proposed two novel policy op-tions (1) a per unit per mile transportation penalty and (2)a collective transportation emissions policy with a limit ontotal transportation emissions Turken et al also revealedthat stricter regulations without high penalties would notensure compliance as the benefits from the increasing scaleassociated with a centralized plant frequently outweigh theregulatory penalties and a per unit carbon tax had no effecton regional production of emissions [27] Xu et al in-vestigated the joint production and pricing of amanufacturing firm with multiple products under cap-and-trade and carbon tax regulations )eir results showed thatthe optimal quantity of products produced under a carbontax regulation is determined by the emissionsrsquo tradingprices and the tax rate [9] Yu and Han studied the impactof a carbon tax on carbon emissions and retail prices in atwo-echelon supply chain consisting of a manufacturer anda retailer )e results indicated that with an increase in thecarbon tax both the optimal emission reduction level andthe optimal retail price initially increase and then remainstable [28] Sinha and Modak developed an economicproduction quantity model that elucidates a new side ofCO2 emissions reduction [29] Zhang investigated theimpact of the carbon tax on enterprise operation andobtained the coopetition supply chain and carbon taxmechanism [30] Chen et al investigated how a carbonemissions taxation scheme can be designed to reducecarbon emissions [31]

In general there are two streams in the previous liter-ature the optimization of the carbon tax and the relationshipbetween the carbon tax and operation decisions in thesupply chain However there exists a research gap in theexisting literature few papers focus on the relationshipbetween the optimal carbon tax and supply chain coordi-nation Our results show that by adjusting the carbon tax thegovernment can promote the coordination of supply chainsand reduce carbon emissions )e optimal carbon tax policyto promote coordination between the supplier and retailer isobtained by balancing the tax revenue and the productgreenness

22 Consumerrsquos Reference Behavior Consumers are alwaysconcerned about product value when choosing products onshelves )e final decision of consumption is a function ofgains and losses with respect to a reference outcome [32]Consumerrsquos reference behavior plays an important role inthis process of comparison thereby influencing firmsrsquo op-erational decisions such as product pricing and decisionsregarding product greenness [33 34]

Kopalle et al studied a novel household heterogeneitytranslation model considering consumersrsquo price referencebehavior and developed a normative pricing policy for re-tailers that maximizes category profit using individual-levelestimates [35] Hsieh and Dye investigated an inventorymodel based on price reference effects and established anoptimal dynamic pricing model to determine a pricingstrategy that maximizes the discounted total profit [36] )eresults suggested that the strength of the memory factor isimportant for a retailer to measure because a high memoryfactor value represents consumers with a longer memory ofperceived gains or losses )e optimal discounted total profitinitially increases as the memory factor increases but de-creases when the memory factor is relatively high Somedynamic price studies have also considered price referencebehavior [37 38] In these two previous studies the pricereference effect dominated the optimal pricing and inven-tory policy of the firm )e expected steady-state referenceprice was compared to the steady-state reference price in amodel with a deterministic reference price effect and theresults showed that the former was always higher Con-sumerrsquos environmental awareness is a common behavior inreal life At present consumers are frequently concernedabout environmental protection Green preference andgreen product design have attracted extensive attention fromscholars Zhang et al investigate the impacts of consumerenvironmental awareness and retailerrsquos fairness concerns onenvironmental quality [39] Chen studied product designand marketing decisions based on consumer preferences forenvironmental attributes [40] )e development of greenproducts depends heavily on the joint efforts of both thesupply chain and the government )erefore the govern-ment should create a regulatory environment that is benignto green product innovation Chitra inferred that greenconsumers affect marketing issues and a preference forgreenness will promote the purchase of green products [41])e higher the consumer environmental preference is thehigher the price will be that the consumer is willing to pay forlow-carbon products Consumerrsquos green preference is infavor of supply chain performance on the environment [42]Moreover as competition intensifies the profits of manu-facturers with inferior eco-friendly operations will alwaysdecrease Li et al infer that consumers should avoid ex-cessive pursuit of green product design otherwise they hurtthe environment by investigating the impact of consumerpreference for green product design [43]

Consumerrsquos reference behavior plays an important rolein firmsrsquo operational decisions and consumer environ-mental awareness is a common behavior in real life )eliterature above is about the influence of consumersrsquo greenpreference on enterprisesrsquo decision-making such as pricing

Mathematical Problems in Engineering 3

and green manufacturing )ere exists a gap in the existingliterature which is that the consumerrsquos behavior has notbeen concerned In our research the green preference andprice reference behavior are considered

23 Cooperation Supply chain cooperation is defined asldquolong-term relationships where participants generally co-operate share information and work together to plan andeven modify their business practices to improve joint per-formancerdquo [44] In general supply chain cooperation meansachieving better performance In the supply chain there aremany coordination strategies to choose from such as rev-enue sharing buybacks quantity discounts and two-parttariff contracts Among these contracts revenue sharing hasattracted the attention of many scholars and is widely used inactual supply chains [45] For example Xu et al proposed atwo-way revenue sharing contract to coordinate multipledistributors in a dual-channel supply chain and the resultsshowed that the manufacturer could prompt the retailer tocooperate by providing this contract [46] Shi et al studiedreverse revenue-sharing contracts in a closed-loop systemand proposed a function to calculate the optimal ratio of thetransfer collection price )e results also suggested thatreverse revenue-sharing contracts are more attractive formanufacturers than a two-part tariff [47] Panda et al ex-plored channel coordination in a socially responsiblemanufacturerndashretailer closed-loop supply chain and foundthat a revenue-sharing contract resolved channel conflict[48] Modak et al used the subgame perfect equilibrium andalternative offer bargaining strategy to resolve channelconflict and distribute surplus profit [49] Wang and Zhaodesigned a revenue-sharing contract to reduce carbonemissions and both the supplier and the retailer achievedPareto improvement In addition they developed a functionto determine the revenue sharing ratio using the Rubinsteinbargaining model [50] Yu et al considered a cooperationproblem in the low-carbon supply chain and found that theenvironmental awareness of consumers and tax rates con-siderably affect the emission reduction [51] )ere have alsobeen some supply chain coordination studies conductedunder carbon policies Revenue-sharing contracts have beendesigned to improve the performance of supply chainmembers based on different carbon policies [52] Xu et alstudied the coordination problem in a two-echelon supplychain and the effect of government policy-making ondistributing the optimal emission quota was investigated)e results showed that a reasonable revenue-sharingcontract is essential to increase supply chain membersrsquoprofits even under low-carbon conditions [53] Modak et alconcluded that the optimal recycling rate increases with theCSR activity of the manufacturer and a profit-sharingcontract provides the best channel performance in a closed-loop distribution channel consisting of a socially responsiblemanufacturer multiple retailers and a third-party collector[54] Feng infers that win-win results can be achieved byestablishing profit-sharing contracts considering the pref-erence of green consumers [55] Previous papers often focuson two aspects including the choice of contract and the

conditions of the contract However the above literature stillhas a gap between cooperation and consumerrsquos behavior)erefore we investigate the supply chain cooperationproblem under the context of consumerrsquos reference behaviorand the carbon tax

In summary this study examines the interaction be-tween supply chain cooperation and carbon taxes in a two-echelon supply chain considering consumerrsquos behaviorSome important factors should be considered simulta-neously to study this problem such as the optimal carbon taxand consumerrsquos behavior however previous research hasonly considered these factors separately We also investigatethe interaction between coordination and the carbon tax)e difference between our study and others in the literatureis presented in Table 1

3 Supply Chain Model

In this section a two-echelon supply chain model is intro-duced to study green supply chain cooperation and carbon taxpolicy )e optimal decisions of the supply chain and thegovernment tax are addressed We then introduce the rev-enue-sharing contract used to coordinate the supply chain

31 Model Assumptions To answer the first question (iscarbon tax policy favorable for improving the greenness ofproducts) a two-echelon supply chain model consisting ofa single supplier and a single retailer without contracts isestablished )e manufacturer determines the productrsquosgreenness and wholesale price and the retailer determinesthe retail price of the green product )e green productcompetes for market share with traditional productsCompared with traditional products green products havethe characteristics of low pollution and being environ-mentally friendly but they may be less functional such aselectric vehicles which has poor endurance and slow speedsIn our study we focus on green products that are lessfunctional than traditional products For example Belloset al noted that manufacturers offer vehicles with poorperformance for customers who focus on fuel efficiency [56]In real life greenness reflects the environmental attributes ofthe product and it is commonly used to measure howenvironmentally friendly a product is In this study we use g

to denote the greenness degree of green products as ameasure of their environmental attributes which is acommon practice [38 41]

Consumers are environmentally conscious and haveenvironmental awareness )e utility gained from envi-ronmental attributes is assumed to be kg where k is thesensitivity of the consumer to the greenness of a greenproduct [38] We use V to denote the utility obtained by aconsumer from a traditional product It is commonly as-sumed that V is uniformly distributed on [0 1] to simplifythe problem without affecting the conclusion [57] )en αV

is the utility from the green product where α isin (0 1) is thefunctional attribute coefficient of the green product thatreflects its weak functional performance compared with thetraditional product

4 Mathematical Problems in Engineering

In real life consumers usually compare prices betweentwo similar products Let β be the consumersrsquo recognitionlevel of the reference price [35] )erefore the utility ofgreen products is obtained from four parts a positive partfrom the basic utility (αV) a negative part from the price (p)a positive part from the environmental consciousness (kg)and a negative part from the price reference (β(p minus pn)))eutility to a consumer of green products and traditionalproducts can thus be expressed as follows

un V minus pn

ug αV minus p + kg minus β p minus pn( 1113857

⎧⎨

⎩ (1)

Table 2 summarizes the notation used in this study

32 Model and Solution Consumers choose between greenand traditional products by comparing utility when ug gt un

and ug gt 0 consumers will purchase the green productwhereas when un gt ug and un gt 0 consumers will purchasethe traditional product As a result demand is obtained asshown in equation (2) )e proof of the demand function isgiven in Appendix A

q (α + β)pn minus (1 + β)p + kg

α(1 minus α)

qn 1 minus(1 + β) pn minus p( 1113857 + kg

1 minus α

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(2)

)e manufacturer determines the wholesale price andgreenness of the green product Let c denote the cost rate oftechnology development then the total development cost is(12)cg2 which is convexly increasing with the greenness[40] In addition (e minus g) is a linear function of the unitcarbon emissions for green products [58] )erefore theprofit function for the manufacturer is obtained from threeparts a positive part from the wholesale ((w minus c)q) anegative part from green technology development ((cg22))and a negative part from the carbon tax (t(e minus g)q) )emanufacturerrsquos decision model is as follows

Maxwg

πM (w minus c minus t(e minus g))q minuscg

2

2 (3)

)e retailerrsquos decision problem is then formulated asfollows

Maxp

πR (p minus w)q (4)

)e optimal solutions for the retailer and manufacturerare derived as the following theorem by substituting thedemand function into the equation above )e proof ispresented in Appendix A

Theorem 1 -e optimal solutions for both parties are

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

g (1 + β)(te + c) minus (α + β)pn minus ke( 1113857t

t2(1 + β) minus 4cα(1 minus α)

p 3cpnα

3+ 2A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)

)e corresponding profits are

πm minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2c

2 t2β + t

2+ 4(minus 1 + α)cα1113872 1113873(β + 1)

πr minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2(minus 1 + α)αc

2

4α2c minus 4αc +(β + 1)t2

1113872 11138732(β + 1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

According to this theorem price and greenness decisionsare directly affected by the carbon tax rate Facing a highercarbon tax and development cost the manufacturer andretailer will reduce the greenness of products and the saleprice with increasing carbon tax However when facing alower carbon tax the greenness and price will increase withincreasing carbon tax (the details are shown in Appendix Bequations (1) and (5)) Next we investigate the effects ofthese parameters on the product greenness )e proofs areprovided in Appendix B

Proposition 1 -e effects of the carbon tax rate pricerecognition level development cost and functional attributecoefficient on product greenness are as follows

(1) -ere exist some thresholds 1113954t1 and 1113954c1 for which g

decreases with increasing t when cgt 1113954c1 and tgt1113954t1 elseg increases with increasing t

Table 1 Differences between our study and the available literature

Literature Decisions under the carbon tax Optimal carbon tax Consumerrsquos behavior CoordinationYu and Han [28] radic radicUlph and Ulph [22] radicHsieh and Dye [34] radicCao et al [47] radic radicXu et al [48] radic radicOur paper radic radic radic radic

Mathematical Problems in Engineering 5

(2) -ere exist some thresholds 1113954t2 and 1113954c2 for which g

decreases with increasing β when cgt 1113954c2 and tgt1113954t2 elseg increases with increasing β

(3) g increases with increasing k decreases with in-creasing c and initially increases and then decreaseswith increasing α

We can conclude that the carbon tax development costprice recognition level greenness sensitivity and functionalattribute coefficient all influence the greenness in Proposi-tion 1 When the carbon tax is low the manufacturer incursless cost to improve the greenness of the products As aresult the greenness increases with the carbon tax In ad-dition if the carbon tax is high and the development cost islow the greenness of the products will increase with in-creasing carbon tax because the tax is the crucial factor indecisions In contrast if the carbon tax is high and devel-opment costs are also high the greenness of products willdecrease because the manufacturer will choose to reducedevelopment costs )erefore pollution will not be reducedas the carbon tax increases in some cases Facing high de-velopment costs the manufacturer will not improvegreenness unless it is promoted by a lower carbon tax Anumerical simulation of this scenario is shown in Figure 1

Similarly the results show that greenness is affected bythe consumersrsquo price reference behavior When the carbontax is low the negative effect of price increases when con-sumers have higher concerns about price )erefore themanufacturer will offset this negative effect by improving thegreenness of products When the carbon tax is high and thedevelopment cost is low the greenness of products willincrease under a high consumer concern about price Incontrast if the carbon tax is high and the development cost ishigh the greenness will decrease because the manufacturerwill choose to reduce development costs Numerical sim-ulations of these scenarios are shown in Figure 2

In addition greenness sensitivity and the functional attri-bute coefficient also affect greenness )e greenness increaseswith increasing greenness sensitivity of consumers Highgreenness sensitivity can promote more green products )ismeans that the government can promote green production by

increasing the green consciousness of consumersMoreover thegreenness decreases with increasing development cost A higherdegree of greenness is obtained when manufacturers face lowerdevelopment costs In addition greenness first increases andthen decreases with the increase in the functional attributecoefficient When the functional attribute of the green productis low themanufacturer will produce a lower greenness productwith increases in the functional attribute coefficient In contrastthe manufacturer will produce a product with higher greennesswith an increase in the functional attribute after a functionalthreshold has been reached Numerical simulations of thesescenarios are shown in Figure 3

Overall consumersrsquo reference behavior plays a positive rolein promoting green production However the carbon tax is noteffective for improving green products in some cases Forexample if the carbon tax is high and the development cost isalso high the greenness of products will decrease)erefore wenext investigate the optimal carbon tax

First an optimal carbon tax is defined as that whichprovides the maximum greenness without losing total taxrevenue )e result is provided in Proposition 2 and theproofs can be found in Appendix C

Table 2 Notations

Parameters Definitionp pn Prices of the green product and traditional product respectivelyq qn Demand for the green product and traditional product respectivelyw Wholesale priceg Product greennesst Carbon tax ratec Cost rate of technology developmentc Unit product cost of the green producte Initial unit carbon emissionsα Functional attribute coefficient of the green product functionβ Consumersrsquo recognition level of the reference pricek Sensitivity of consumers to product greennessV Utility obtained by a consumer from the traditional productη Proportion of the total revenue obtained by the retailerλ Member strength in the supply chain

10

08

06g

04

02

0000 01 02 03

t

t = t1

04 05

010

008

006

004

002

000

γ lt γ1γ gt γ1

Figure 1 Effects of carbon taxes on greenness

6 Mathematical Problems in Engineering

Proposition 2 -e optimal carbon tax is defined as follows

tlowast

M minusM

2minus N

1113968

e(1 + β) 0lt clt 1113954c3

P minus

P2

minus Q

1113969

2(1 + β)M 1113954c3 lt clt 1113954c1

1113954t1 1113954c1 lt c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

)e optimal carbon tax is obtained in Proposition 2based on the manufacturerrsquos development cost to maximizegreenness without losing the total carbon tax revenue Wecan observe that the crucial factor is the development costAn appropriate carbon tax policy should be chosen whenfacing different production technologies to ensure greennessand total tax revenue In addition it is easily observed thatthe maximum greenness is obtained when the developmentcost is high whereas the maximum tax is obtained when thedevelopment cost is lower but some of the greenness is lostA numerical simulation of this scenario is shown in Figure 4

33 Supply Chain Coordination It is also important forsupply chains to establish green supply chain models byintegrating internal and external resources to make deci-sions which enables supply chains to achieve better per-formance by improving cooperation [59 60] However moststudies on coordination have mainly focused reducingemissions and improving profit Few studies have investi-gated the relationship between carbon taxes and supplychain cooperation

In this section we study the impact of carbon taxes andconsumerrsquos behavior on supply chain coordination wherethe manufacturer is the leader and the retailer is the fol-lower A revenue-sharing contract is introduced to achieve

coordination Let η denote the proportion of total revenueobtained by the retailer then the manufacturerrsquos decisionmodel is as follows

Maxwg

πCM ((1 minus η)p + w minus c minus t(e minus g))q minus

cg2

2 (8)

)e retailerrsquos decision problem can be described asfollows

Maxp

πCR (ηp minus w)q (9)

)e optimal solutions for the retailer and manufacturerare derived with the following theorem by substituting thedemand function into the equation above )e proof isfound in Appendix D

Theorem 2 -e greenness is defined as follows

g t minus αpn + et(β + 1) + c minus pn( 1113857β minus ke + c( 1113857

2c(1 + η)α2 minus 2c(1 + η)α + t2(β + 1)

(10)

-en the profits of both parties are the following

πm minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2c

2 t2β + t

2+ 4(minus 1 + α)cα1113872 1113873(β + 1)

πr minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2(minus 1 + α)αc

2

4α2c minus 4αc +(β + 1)t2

1113872 11138732(β + 1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

We investigate the impact of the carbon tax on pro-moting the coordination of the supply chain by comparingthe profit changes of the two partners after the introductionof the revenue-sharing contract in Proposition 3 )e proofsare provided in Appendix E

Proposition 3 Coordination can be promoted by increasingthe carbon tax

01204

01202

01200g

01198

01196

00 01 02 03β

04 05

11900 times 10ndash5

11895 times 10ndash5

11898 times 10ndash5

11892 times 10ndash5

11890 times 10ndash5

11888 times 10ndash5

11885 times 10ndash5

11882 times 10ndash5

11880 times 10ndash5

γ lt γ2 t lt t2γ gt γ2 t lt t2

(a)

096

095

094

093g

092

091

090

08900 01 02 03

β04 05

0035

0040

0045

0050

0055

0060

γ lt γ2 t gt t2γ gt γ2 t gt t2

(b)

Figure 2 Effects of consumersrsquo price reference behavior on greenness (a) Low-carbon tax rate (b) High carbon tax rate

Mathematical Problems in Engineering 7

When the profits of both the retailer and the manu-facturer improve simultaneously coordination is achievedBy comparing equations (11) and (6) we find that therevenue-sharing contract improves the retailerrsquos profit onlywhen η isin (η1 1) whereas the manufacturer always benefitsfrom the contract )e range of η depends on the carbon taxrate When η is in the given range the retailer and man-ufacturer achieve Pareto improvement Furthermore therange expands as the carbon tax rate increases)erefore thegovernment can promote supply chain collaboration byincreasing the carbon tax Based on the previous results forthe optimal carbon tax we next investigate how to achievecoordination when the supply chain faces the optimalcarbon tax Figure 5 shows the Pareto improvement region

According to the assumptions described above themanufacturer acts as the leader and the retailer acts as afollower It is reasonable that the proportion of profitreceived by the retailer is decided by the manufacturer)erefore it is necessary to investigate whether theproportion decided by the leader is in the region in whichcoordination is achieved A parameter λ is introduced todenote the strength of a partner in the supply chain Let

λm isin (05 1) and λr 1 minus λm represent the power of themanufacturer and the retailer respectively )en weconsider a simple function (λrλm) to determine η whichis the proportion of revenue received by the retailer )eretailer retains the maximum revenue when the power ofthe two parties is equal (ie λm λr 05) whereas all ofthe benefits go to the leader when the power of themanufacturer is overwhelming compared with that of theretailer (ie λm 1) Similar to Proposition 2 the optimalcarbon tax is obtained under the revenue-sharing contractas follows

tlowast

M minus

M2

minus (1 + η2)N

1113969

e(1 + β) 0lt clt 1113954c

C2

(1 + η2)P minus

(1 + η2)2P2

minus (1 + η2)Q

1113969

2(1 + β)M 1113954c

C2 lt clt 1113954c1

1113954t1 1113954c1 lt c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

05

04

03g

02

01

0010 15 20 25

k3530 40

g

(a)

10

08

06g

04

02

0010 09 0708 06

γ030405

g

(b)

029

027

028

025

026

g

024

022

023

021035030 040 045 050

α060 065055 070

g

(c)

Figure 3 Effects of parameters on greenness (a) Greenness sensitivity (b) Development cost (c) Functional attribute coefficient

8 Mathematical Problems in Engineering

Proposition 4 is designed to answer the question ofwhether cooperation will be achieved the proofs are foundin Appendix F

Proposition 4 Coordination will be achieved if the power ofthe manufacturer is low (ie 05lt λm lt 1113954λ1) Moreover whenthe power of the manufacturer is at a medium level (ie1113954λ1 lt λm lt 1113954λ2) coordination can be achieved but part of thetotal tax revenue will be lost if the development cost is low(ie clt 1113954c1) In other cases (ie cgt 1113954c1 or 1113954λlt λm) coordi-nation cannot be achieved because the power of the manu-facturer is overwhelming

Coordination situations for different manufacturerpower levels are present in Table 3 )e achievement ofcoordination is decided by the power of the manufacturerand the optimal carbon tax faced by the supply chain Basedon Proposition 3 coordination can be promoted if thegovernment adjusts the carbon tax On the other handcoordination can easily be achieved under the optimalcarbon tax if the power of the manufacturer is low Inaddition there are two possibilities when the power of themanufacturer is at a medium level If the development cost istoo high adjusting the tax will lead to an unknown result thegreenness and total tax revenue may both decline at the same

10 014

012

010

008

006

004

002

000

08

06

Gre

enne

ss

Tota

l tax

04

02

00005000 010 015 020

t

γ lt γ3

t = ta

025 030

GreennessTotal tax

(a)

10 030

025

020

015

010

005

000

08

06

Gre

enne

ss

Tota

l tax

04

02

000100 02 03 04

t

γ3 lt γ lt γ1

t = tc

05 06

GreennessTotal tax

(b)

005 040

035

025

020

015

010

005

000

004

003

Gre

enne

ss

Tota

l tax

002

001

0000100 02 03 04

t

γ1 lt γ

t = t1

06 08

030

05 07

GreennessTotal tax

(c)

Figure 4 Determination of the optimal carbon tax (a) 0lt clt 1113954c3 (b) 1113954c3 lt clt 1113954c1 (c) 1113954c1 lt c

005t

η

010 015 020 025 030

Pareto improvement

035 040

Figure 5 Pareto improvement region with increasing carbon tax

Mathematical Problems in Engineering 9

time If the development cost is lower the greenness may beimproved with a loss of revenue by increasing the carbon taxHowever when the power of the manufacturer is strong theregion δ (η1 1) cannot be achieved

According to our theory the government can promotethe coordination between the manufacturer and the retailerby adjusting the carbon tax When the power of the man-ufacturer increases the government still achieves coordi-nation by raising the carbon tax In general cooperation isachieved and the government plays a key role in promotingcooperation

)e effects of consumersrsquo behavior on supply chaincoordination are also investigated in the next proposition)e proofs are provided in Appendix G

Proposition 5 Price reference behavior can promote supplychain coordination but green preference cannot

From the previous results price reference has a compleximpact on the supply chain First the increasing sensitivityof customers to price will promote supply chain coordi-nation because this negative impact can be mitigated bycooperation In addition price reference behavior also af-fects the green decisions of the supply chain in a complexmanner In contrast the mechanism by which consumerrsquogreen preferences influence the supply chain is relativelysimple Greenness increases with increasing greennesspreference and green preference has no effect on the co-ordination of the supply chain

4 Conclusion

)is study examines the interaction between supply chaincooperation and the carbon tax problem in a two-echelonsupply chain under consumerrsquos reference behavior )e opti-mal carbon tax policy is obtained based on analysis of thecarbon tax green investment coefficient and degree of con-sumerrsquos price reference )e optimal carbon tax is defined asthe simultaneous optimization of total tax revenue and productgreenness In addition coordination is achieved by introducingrevenue-sharing contracts)e impact of consumersrsquo referencebehavior and the carbon tax on supply chain coordination isalso investigated )e results are as follows

(1) )e greenness increases with increasing carbon taxwhen the carbon tax is low In addition if the carbontax is higher and the development cost is low thegreenness of products will increase with increasingcarbon tax Conversely if the carbon tax is higherand the development cost is also high the greennessof products will decrease

(2) When the carbon tax is low the negative effect ofprice increases as consumer concerns about priceincrease )erefore the manufacturer offsets thenegative effect by improving the greenness Whenthe carbon tax is higher and development cost is lowthe greenness of products will increase with highconsumerrsquos concern about price In contrast if thecarbon tax is higher and the development cost ishigh the greenness will decrease

(3) )e greenness increases with an increasing greennesspreference of consumers )eir preference behaviorcan promote more green products )is means thatthe government can promote green production bypromoting the green consciousness of consumersMoreover the greenness decreases with increasingdevelopment cost A higher greenness is obtainedwhen manufacturers face lower development costsIn addition greenness first increases and then de-creases with increases in the functional attributecoefficient When the functional attribute of thegreen product is low the manufacturer will producea lower greenness product with an increase in thefunctional attribute In contrast the manufacturerwill produce a higher greenness product with anincrease in the functional attribute after a func-tionality threshold

(4) We investigated the impact of the carbon tax onpromoting the coordination of the supply chain bycomparing the profit changes of two partners afterthe introduction of a revenue-sharing contract inProposition 3 We found that coordination could bepromoted by increasing the carbon tax

(5) )e achievement of coordination depends on thetype of manufacturer If the power of the manu-facturer is low coordination can be achievedunder the optimal carbon tax If the power of themanufacturer is at a medium level coordinationcan be achieved by increasing the carbon tax andimproved greenness will be realized with a loss ofrevenue However when the power of the man-ufacturer is strong coordination cannot beachieved

In this study we focused on the interaction between acarbon tax and supply chain cooperation However othercarbon policies (eg a cap-and-trade policy) and supplychain structures are worth exploring For example it wouldbe interesting to investigate the effects on a supply chainstructure consisting of the retailer as the leader under acomplex carbon policy

Table 3 Coordination situations for different manufacturer power levels

Power of the manufacturer Coordination Greenness Total carbon tax revenue05lt λm lt 1113954λ1 Achieve Improve Improve1113954λ1 lt λm lt 1113954λ2 clt 1113954c1 Achieve Improve Decline1113954λ1 lt λm lt 1113954λ2 clt 1113954c1 Cannot achievecgt 1113954c1 Cannot achieve

10 Mathematical Problems in Engineering

Appendix

A Proof of Theorem 1

)e demand function can be written as

q P ug ge un ug ge 01113966 1113967

qn P un gt ug un ge 01113966 1113967

⎧⎪⎨

⎪⎩(A1)

After replacing un and ug we have

q P αv minus p + kg minus β p minus pn( 1113857 minus v minus pn( 1113857ge 0 αv minus p + kg minus β p minus pn( 1113857ge 01113864 1113865

qn P v minus pn minus αv minus p + kg minus β p minus pn( 1113857( 1113857gt 0 v minus pn ge 01113864 11138651113896 (A2)

Given that V is uniformly distributed on [0 1] thedemand function is

q (α + β)pn minus (1 + β)p + kg

α(1 minus α)

qn 1 minus(1 + β) pn minus p( 1113857 + kg

1 minus α

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A3)

)e condition kgt ((1 + β)p minus (α + β)png) is necessaryto ensure nonnegativity )e retailer problem is addressedfirst and we have

p (α + β)pn +(1 + β)w + kg

2(1 + β) (A4)

)en substitute p into the manufacturerrsquos profit func-tion Solving the optimal w and g simultaneously yields

w (α + β)pn +(1 + β)(c + t(e minus g)) + kg

2(1 + β)

g t (α + β)pn minus (1 + β)w + ke( 1113857

2cα(1 minus α)

(A5)

)en we have

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

g (1 + β)(te + c) minus (α + β)pn minus ke( 1113857t

t2(1 + β) minus 4cα(1 minus α)

p 3cpnα

3+ 2A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1 (1 + β)(c + te) + ke minus (1 minus β)pn

A2 (1 + β)(c + te) + 3ke minus (3 minus β)pn

⎧⎪⎨

⎪⎩

B1 minus 2c (1 + β)(c + te) + ke + βpn( 1113857 + t2(1 + β)pn

B2 minus c (1 + β)(c + te) + 3ke + 3βpn( 1113857 + t2(1 + β)pn

⎧⎪⎨

⎪⎩

(A6)

Notes that we have (α + β)pn + ke minus (1 + β)(te + c)gt 0)e concavity condition is 4αc(1 minus α) minus (1 + β)t2 gt 0

B Proof of Proposition 1

(1) )e first derivative of g in t is

zg

zt

minus 4cα3pn + 4cA3α2

+ B3α minus t2(1 + β) (1 + β)c minus ke minus βpn( 1113857

t2(1 minus β) minus 4cα(1 minus α)1113872 1113873

2

A3 (1 + β)(c + 2te) minus ke +(1 minus β)pn

B3 minus 4c (1 + β)(c + 2te) minus ke minus βpn( 1113857 + t2(1 + β)pn

⎧⎨

⎩

(B1)

We have a threshold

1113954c1 minust2(1 + β) (α + β)pn + ke minus (1 + β)c( 1113857

4α(1 minus α) (α + β)pn + ke minus (1 + β)(c + 2te)( 1113857

1113954t1 (α + β)pn + ke minus (1 + β)c

2(1 + β)e

(B2)

When cgt 1113954c1 and tgt1113954t1 g is decreasing in t Bycontrast g is increasing in t in other cases

(2) )e first derivative of g in β is

zg

zβ

minus 4α(1 minus α) te + c minus pn( 1113857c + t2

ke + αpn minus pn( 1113857

t2(1 + β) minus 4α(1 minus α)c1113872 1113873

2

(B3)

We have a threshold

1113954c2 minust2

ke minus (1 minus α)pn( 1113857

4α(1 minus α) te + c minus pn( 1113857

1113954t2 pn minus c

e

(B4)

Review we have (α + β)pn + ke minus (1 + β)(te + c)gt 0therefore we can find that

tltpn minus c

e+

ke minus (1 minus α)pn

(1 + β)e (B5)

(i) When ke minus (1 minus α)pn lt 0 then tlt1113954t2 is inevitable)erefore (zgzβ)gt 0 is correct

(ii) When ke minus (1 minus α)pn gt 0 we can find that(zgzβ)lt 0 if cgt 1113954c2 and tgt1113954t2 (zgzβ)lt 0 in othercases

(3) )e first derivative of g in k and c is

Mathematical Problems in Engineering 11

zg

zc minus

4α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

4cα(1 minus α) minus t2(1 + β)1113872 1113873

2 lt 0

zg

zk

te

4cα(1 minus α) minus t2(1 + β)gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(B6)

(4) )e first derivative of g in α is

zg

zα

minus Xt

4α(1 minus α)c minus t2(1 + β)1113872 1113873

2 (B7)

And X minus 4α2cpn + (4(1 + β)(te + c) minus 4ke minus 4pn)

(2α + 1)c + pnt2(1 + β) is a quadratic function )erange of α in 4αc(1 minus α) minus (1 + β)t2 gt 0 is

α1 c +

c2

minus ct2(1 + β)

1113969

2c

α2 c minus

c2

minus ct2(1 + β)

1113969

2c

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(B8)

By substituting those into X we have

X|αα1 4

c2

minus ct2(1 + β)

1113969

T minus 2 c minus t2(1 + β)1113872 1113873pn

X|αα2 minus 4

c2

minus ct2(1 + β)

1113969

T minus 2 c minus t2(1 + β)1113872 1113873pn lt 0

⎧⎪⎪⎨

⎪⎪⎩

(B9)

And T (1 + β)(te + c) minus ke minus ((12) + β)pn lt 0We can find

X|αα1 times X|αα2 minus 4 c minus t2(1 + β)1113872 1113873

middot 4 βpn + ke minus (1 + β)(te + c)( 11138572

1113872

+ p2nt

2(1 + β)1113873lt 0

(B10)

)erefore X|αα1 gt 0 )ere only exist a 1113954α isin (α2 α1)that makes (zgzα) 0 And (zgzα) gt 0 ifα isin (α2 1113954α) (zgzα)lt 0 if α isin (1113954α α1)

(5) )e first derivative of p in t is

zp

zt minus

2α(α minus 1) e(1 + β)t221113872 1113873 + minus pnα minus ke + c minus pn( 1113857β + c( 1113857t minus 2eαc(α minus 1)1113872 1113873c

(1 + β)t2

+ 4αc(α minus 1)1113872 11138732

A3 (1 + β)(c + 2te) minus ke +(1 minus β)pn

B3 minus 4c (1 + β)(c + 2te) minus ke minus βpn( 1113857 + t2(1 + β)pn

⎧⎪⎨

⎪⎩

(B11)

)e discriminant of quadratic function

e(1 + β)t2

2+ minus pnα minus ke + c minus pn( 1113857β + c( 1113857t minus 2eαc(α minus 1)

(B12)

is

Δ (minus β minus 1)c + ke + βpn + pnα( 11138572

+ 4e2(1 + β)t

2αc(α minus 1)

(B13)

We have a threshold

1113954c minus(minus β minus 1)c + ke + βpn + pnα( 1113857

2

4e2(1 + β)t

2α(α minus 1) (B14)

Review we have(α + β)pn + ke minus (1 + β)(te + c)gt 0therefore we can find that

tm 2

minus (1 + β)αc(α minus 1)

1113968

1 + β (B15)

Substituting it into above quadratic function

2 c

radicminus pnα minus ke + c minus pn( 1113857β + c( 1113857

minus (1 + β)α(α minus 1)

1113968minus 4eαc(α minus 1)(1 + β)

1 + β (B16)

And we have a same threshold

1113954c minus(minus β minus 1)c + ke + βpn + pnα( 1113857

2

4e2(1 + β)t

2α(α minus 1) (B17)

It could be found that(zpzt) gt 0 because Δlt 0 whenclt 1113954c And when cgt 1113954c we have another threshold 1113954t andzpzt|

t1113954t 0 because Δgt 0 and zpzt|tmlt 0)erefore when

cgt 1113954c and tgt1113954t p is decreasing in t And when cgt 1113954c and tlt1113954tp is increasing in t

12 Mathematical Problems in Engineering

C Proof of Proposition 2

(1) Let R t(e minus g)q denote total carbon tax and thefirst derivative of R in t is

zR

zt2c A2t

2+ B2t + c1113872 1113873 A3t

2+ B3t + c1113872 1113873

4cα(1 minus α) minus t2(1 + β)1113872 1113873

3 (C1)

Among them

A2 (1 + β)e

2gt 0

B2 minus (α + β)pn + ke minus (1 + β)c( 1113857lt 0

C2 2ceα(1 minus α)gt 0

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

A3 (α + β)pn + ke minus (1 + β)c( 1113857(1 + β)gt 0

B3 minus 8ceα(1 minus α)(1 + β)lt 0

C3 4cα(1 minus α) (α + β)pn + ke minus (1 + β)c( 1113857gt 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(C2)

Let Φ A2t2 + B2t + C2 and Ψ A3t

2 + B3t + C3 andwe can find that zRzt and Φ times Ψ have a same sign Bysolving Δ1 and Δ2 a threshold is

1113954c3 (α + β)pn + ke minus (1 + β)c( 1113857

2

4e2α(1 minus α)(1 + β)

(C3)

)ere are two positive real numbers of solutions forΦ ifclt 1113954c3 and let 0lt ta lt tb denote it Similarly let0lt tc lt td denote the solutions for Ψ if cgt 1113954c3 Reviewwe have 4αc(1 minus α) minus (1 + β)t2 gt 0 therefore lettm

4cα(1 minus a)1 + β

1113968 Substituting tm into Φ and Ψ

Φ|tm minus 2

cα(1 minus α)

1 + β

1113971

(α + β)pn + ke minus (1 + β)c( 1113857 + 4ecα(1 minus α)

Ψ|tm minus 8cα(1 minus α) 2(1 + β)e

α(1 minus α)

1 + βc

1113971

minus (α + β)pn + ke minus (1 + β)c( 1113857⎛⎝ ⎞⎠

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(C4)

If clt 1113954c3 Φ|tmlt 0 and Ψ|tm

gt 0 then zRzt|tmlt 0 If

cgt 1113954c3 Φ|tmgt 0 and Ψ|tm

lt 0 then zRzt|tmlt 0 )ere-

fore we have tm isin (ta tb) or tm isin (tc td) FurthermoreR t(e minus g)q is maximized when t ta if clt 1113954c3 R

t(e minus g)q is maximized when t tc if cgt 1113954c3 Totally wehave

tlowast

ta clt 1113954c3

tc cgt 1113954c31113896 (C5)

(2) Review Proposition 1 If cgt 1113954c1 g is maximized whentlowast 1113954t1 If clt 1113954c1 g is maximized when tlowast tmWhen tgt1113954t1 1113954c1 gt 1113954c5 And 1113954c1 lt 0lt 1113954c5 if tlt1113954t1Substituting 1113954t1 into Φ

Φ|1113954t1 minus

3 (α + β)pn + ke minus (1 + β)c( 11138572

8e(1 + β)+ 2ecα(1 minus α)

(C6)

A threshold is 1113954c4 (34)1113954c3 )erefore we have (i)when 0lt clt 1113954c4 tlowast ta because of Φ|1113954t1

lt 0 and 1113954t1 gt ta

(ii) when 1113954c4 lt clt 1113954c3 tlowast ta because of 1113954t1 lt ta (iii) tlowast

tc when 1113954c3 lt clt 1113954c1 and (iv) tlowast tc when 1113954c1 lt c )eresult is obtained by solving Φ and Ψ

D Proof of Theorem 2

Similar to Appendix A )e retailer problem is addressedfirst and we have

p αηpn + βηpn + ηkg + βw + w

2η(β + 1) (D1)

)en substituting p into the manufacturerrsquos profitfunction Solving the optimal w and g simultaneously yields

w αηpn + βet + βηpn minus βgt + eηk + cβ + et minus gt + c( 1113857η

(1 + η)(β + 1)

g minusαηpn + βηpn + eηk minus βw minus w( 1113857t

2(α minus 1)cαη

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(D2)

Mathematical Problems in Engineering 13

By solving the equation above we have

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

g t minus αpn + et(β + 1) + c minus pn( 1113857β minus ke + c( 1113857

t2(1 + β) minus (1 + η)cα(1 minus α)

p (2η + 1)cpnα

3+ A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1 et + ηpn + c( 1113857β + et + ke minus pn( 1113857η + c

A2 (2η + 1)pn + et + c( 1113857β +(minus 2η minus 1)pn + et +(2ηk + k)e + c

⎧⎪⎪⎨

⎪⎪⎩

B1 minus 2et minus 2ηpn minus 2c( 1113857β minus 2eηk minus 2et minus 2c( 1113857c + t2pn(β + 1)

B2 (minus 2η minus 1)pn minus et minus c( 1113857β minus et +(minus 2ηk minus k)e minus c( 1113857c + t2pn(β + 1)

⎧⎪⎪⎨

⎪⎪⎩

(D3)

Note that we have minus αpn + et(β + 1) + (c minus pn)β minus ke +

clt 0 )e concavity condition is (1 + β)(t2(1 + β) minus (1+

η)cα(1 minus α))lt 0

Substituting solutions into the profit function we have

πm minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2c

2 t2β + t

2+ 4(minus 1 + α)cα1113872 1113873(β + 1)

πr minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2(minus 1 + α)αc

2

4α2c minus 4αc +(β + 1)t2

1113872 11138732(β + 1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(D4)

E Proof of Proposition 3

πM (α + β)pn + ke minus (1 + β)c( 1113857

2c

2(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCM

(α + β)pn + ke minus (1 + β)c( 11138572c

2(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(E1)

By comparing the profits of two partners change after therevenue-sharing contract introduced we found that themanufacturer always benefits from cooperation

πR c2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCR

ηc2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(E2)

A threshold is

η1 2cα(1 minus α) minus (1 + β)t2

2cα(1 minus α)1113888 1113889

2

(E3)

When η is in the region δ (η1 1) the retailer willaccept cooperation And the first derivative of η1 in t is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t(β + 1)

α2c2(minus 1 + α)

2 lt 0 (E4)

)erefore η1 is decreasing in t )at means the regionδ (η1 1) is expanding when tax rises And the revenue-sharing contract is accepted easily than before

F Proof of Proposition 4

We consider a simple function (λrλm) to decide thepractical η where λm isin (05 1) and λr 1 minus λm Given the

14 Mathematical Problems in Engineering

carbon tax tlowast we have a region δ (η(tlowast) 1) of cooperationIt is clear that the gap of strength between two partitions inthe supply chain is a crucial factor in cooperation Bycomparing practice η decided by strength and η(tlowast) de-cided by the carbon tax we have

1λm

minus 1 2α2c minus 2αc + t

2(β + 1)1113872 1113873

2

4α2c2(minus 1 + α)

2 (F1)

A threshold is

1113954λ1 4α2c2

(minus 1 + α)2

8α4c2minus 16α3c2

+ 4 2c + t2(β + 1)1113872 1113873cα2 minus 4t

2c(β + 1)α + t

4(β + 1)

2 (F2)

When 05lt λm lt 1113954λ1 the practical η is in the regionδ (η(tlowast) 1) therefore the cooperation is achieved facingthe optimal carbon tax

However the practical η is out of the range δ (η(tlowast) 1)

when 1113954λ1 lt λm Review previous result the region δ (η1 1)

is expanding when tax rises )erefore there exists a pos-sibility that the government increasing carbon tax to co-ordinate the supply chain We assume a realistic thing thegovernment is willing to boost the greenness of the producteven reducing part of the revenue from the carbon tax

Based on it the carbon up to 1113954t1 which means thegreenness is maximized )erefore )e cooperation isachieved when clt 1113954c1 because the greenness is decreasing inthe carbon tax when 1113954c1 lt c )en we have the secondthreshold

1113954λ2 1

η1 1113954t1( 1113857 + 1 (F3)

)e cooperation could not be achieved when λm gt 1113954λ2)e conclusion is as follows (1) Coordination would be

achieved when the manufacturer is democratic (ie05lt λm lt 1113954λ1) (2) Coordination could be achieved but partof the total tax will lose if development cost is lower when themanufacturer is moderate (ie 1113954λ1 lt λm lt 1113954λ2) (3) )e co-operation could not be achieved if the manufacturer is adictator (ie λm gt 1113954λ2)

G Proof of Proposition 5

)e first derivative of η1 in β is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t

2

2α2c2(minus 1 + α)

2 lt 0 (G1)

Similarly η1 is decreasing in β )at means the regionδ (η1 1) is expanding when β rises

And it is apparent that η1 (2cα(1 minus α) minus (1 + β)t22cα(1 minus α))2 is independent with k

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is research was supported by the National Natural ScienceFoundation of China (no 71761015) and the Science andTechnology Research Project of Jiangxi Education Depart-ment of China (no GJJ150475)

References

[1] Y Li M K Lim J Hu and M-L Tseng ldquoInvestigating theeffect of carbon tax and carbon quota policy to achieve lowcarbon logistics operationsrdquo Resources Conservation andRecycling vol 154 p 104535 2020

[2] M Abedi-Varaki ldquoStudy of carbon dioxide gas treatmentbased on equations of kinetics in plasma discharge reactorrdquoModern Physics Letters B vol 31 no 22 p 1750210 2017

[3] S Meng M Siriwardana and J McNeill ldquo)e environmentaland economic impact of the carbon tax in Australiardquo Envi-ronmental and Resource Economics vol 54 no 3 pp 313ndash332 2013

[4] Y Chen and C L Tseng ldquoInducing clean technology in theelectricity sector tradable permits or carbon tax policiesrdquo-eEnergy Journal vol 32 no 3 2011

[5] H-B Duan L Zhu and Y Fan ldquoOptimal carbon taxes incarbon-constrained China a logistic-induced energy eco-nomic hybrid modelrdquo Energy vol 69 pp 345ndash356 2014

[6] C Lu Q Tong and X Liu ldquo)e impacts of carbon tax andcomplementary policies on Chinese economyrdquo Energy Policyvol 38 no 11 pp 7278ndash7285 2010

[7] K Cao B Xu and J Wang ldquoOptimal trade-in and warrantyperiod strategies for new and remanufactured products undercarbon tax policyrdquo International Journal of Production Re-search vol 58 no 1 pp 180ndash199 2020

[8] P He W Zhang X Xu and Y Bian ldquoProduction lot-sizingand carbon emissions under cap-and-trade and carbon taxregulationsrdquo Journal of Cleaner Production vol 103pp 241ndash248 2015

[9] X Xu X Xu and P He ldquoJoint production and pricing de-cisions for multiple products with cap-and-trade and carbontax regulationsrdquo Journal of Cleaner Production vol 112pp 4093ndash4106 2016

Mathematical Problems in Engineering 15

[10] Q Li X Guan T Shi and W Jiao ldquoGreen product designwith competition and fairness concerns in the circulareconomy erardquo International Journal of Production Researchvol 58 no 1 pp 165ndash179 2020

[11] A Ranjan and J K Jha ldquoPricing and coordination strategiesof a dual-channel supply chain considering green quality andsales effortrdquo Journal of Cleaner Production vol 218pp 409ndash424 2019

[12] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineeringvol 10 no 1 pp 99ndash116 2012

[13] X Yin X Chen X Xu and L Zhang ldquoTax or subsidyOptimal carbon emission policy a supply chain perspectiverdquoSustainability vol 12 no 4 p 1548 2020

[14] B B Schlegelmilch G M Bohlen and A Diamantopoulosldquo)e link between green purchasing decisions and measuresof environmental consciousnessrdquo European Journal of Mar-keting vol 30 no 5 pp 35ndash55 1996

[15] M S Hopkins ldquoWhat the rsquogreenrsquo consumer wantsrdquo MITSloan Management Review vol 50 no 4 pp 87ndash89 2009

[16] G Kalyanaram and R S Winer ldquoEmpirical generalizationsfrom reference price researchrdquo Marketing Science vol 14no 3 pp G161ndashG169 1995

[17] E A Greenleaf ldquo)e impact of reference price effects on theprofitability of price promotionsrdquo Marketing Science vol 14no 1 pp 82ndash104 1995

[18] N M Modak D K Ghosh S Panda and S S SanaldquoManaging green house gas emission cost and pricing policiesin a two-echelon supply chainrdquo CIRP Journal ofManufacturing Science and Technology vol 20 pp 1ndash11 2018

[19] P Sinclair ldquoHigh does nothing and rising is worse carbontaxes should keep declining to cut harmful emissionsrdquo -eManchester School vol 60 no 1 pp 41ndash52 1992

[20] G M Grossman and A B Krueger ldquoEconomic growth andthe environmentrdquo -e Quarterly Journal of Economicsvol 110 no 2 pp 353ndash377 1995

[21] N M Modak and P Kelle ldquoUsing social work donation as atool of corporate social responsibility in a closed-loop supplychain considering carbon emissions tax and demand uncer-taintyrdquo Journal of the Operational Research Society pp 1ndash172019

[22] A Ulph and D Ulph ldquo)e optimal time path of a carbon taxrdquoOxford Economic Papers vol 46 no S1 pp 857ndash868 1994

[23] M Hoel and S Kverndokk ldquoDepletion of fossil fuels and theimpact of global warmingrdquo Resource and Energy Economicsvol 18 no 2 pp 115ndash136 1996

[24] Y H Farzin and O Tahvonen ldquoGlobal carbon cycle and theoptimal time path of a carbon taxrdquo Oxford Economic Papersvol 48 no 4 pp 515ndash536 1996

[25] V Bosetti C Carraro R Duval andM Tavoni ldquoWhat shouldwe expect from innovation A model-based assessment of theenvironmental and mitigation cost implications of climate-related RampDrdquo Energy Economics vol 33 no 6 pp 1313ndash13202011

[26] M Hariga R Asrsquoad and A Shamayleh ldquoIntegrated economicand environmental models for a multi stage cold supply chainunder carbon tax regulationrdquo Journal of Cleaner Productionvol 166 pp 1357ndash1371 2017

[27] N Turken J Carrillo and V Verter ldquoFacility location andcapacity acquisition under carbon tax and emissions limits tocentralize or to decentralizerdquo International Journal of Pro-duction Economics vol 187 pp 126ndash141 2017

[28] W Yu and R Han ldquoCoordinating a two-echelon supply chainunder carbon taxrdquo Sustainability vol 9 no 12 p 2360 2017

[29] S Sinha and N M Modak ldquoAn EPQmodel in the perspectiveof carbon emission reductionrdquo International Journal ofMathematics in Operational Research vol 14 no 3pp 338ndash358 2019

[30] H Zhang P Li H Zheng and Y Zhang ldquoImpact of carbontax on enterprise operation and production strategy for low-carbon products in a co-opetition supply chainrdquo Journal ofCleaner Production Article ID 125058 2020

[31] X Chen H Yang X Wang et al ldquoOptimal carbon tax designfor achieving low carbon supply chainsrdquo Annals of OperationsResearch pp 1ndash28 2020

[32] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Handbook of the Fundamentals of Fi-nancial Decision Making Part I pp 99ndash127 )e EconometricSociety Cleveland OH USA 2013

[33] W Kim and M Kim ldquoReference quality-based competitivemarket structure for innovation driven marketsrdquo Interna-tional Journal of Research in Marketing vol 32 no 3pp 284ndash296 2015

[34] I Popescu and Y Wu ldquoDynamic pricing strategies withreference effectsrdquo Operations Research vol 55 no 3pp 413ndash429 2007

[35] P K Kopalle P K Kannan L B Boldt and N Arora ldquo)eimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Re-tailing vol 88 no 1 pp 102ndash114 2012

[36] T-P Hsieh and C-Y Dye ldquoOptimal dynamic pricing fordeteriorating items with reference price effects when inven-tories stimulate demandrdquo European Journal of OperationalResearch vol 262 no 1 pp 136ndash150 2017

[37] X Chen Z-Y Hu and Y-H Zhang ldquoDynamic pricing withstochastic reference price effectrdquo Journal of the OperationsResearch Society of China vol 7 no 1 pp 107ndash125 2019

[38] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquo Opera-tions Research vol 64 no 6 pp 1529ndash1536 2016

[39] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[40] C Chen ldquoDesign for the environment a quality-based modelfor green product developmentrdquo Management Sciencevol 47 no 2 pp 250ndash263 2001

[41] K Chitra ldquoIn search of the green consumers a perceptualstudyrdquo Journal of Services Research vol 7 no 1 2007

[42] Z Liu T D Anderson and J M Cruz ldquoConsumer envi-ronmental awareness and competition in two-stage supplychainsrdquo European Journal of Operational Research vol 218no 3 pp 602ndash613 2012

[43] B Li Y Wang and Z Wang ldquoManaging a closed-loop supplychain with take-back legislation and consumer preference forgreen designrdquo Journal of Cleaner Production Article ID124481 2020

[44] J Rantanen D B Grant and W Piotrowicz ldquoInvestigatingsupply chain cooperation in Finnish grocery retailrdquo ResearchJournal of the University of Gdansk Transport Economics andLogistics vol 71 pp 19ndash34 2017

[45] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[46] G Xu B Dan X Zhang and C Liu ldquoCoordinating a dual-channel supply chain with risk-averse under a two-way

16 Mathematical Problems in Engineering

revenue sharing contractrdquo International Journal of ProductionEconomics vol 147 pp 171ndash179 2014

[47] Z Shi N Wang T Jia et al ldquoReverse revenue sharingcontract versus two-part tariff contract under a closed-loopsupply chain systemrdquoMathematical Problems in Engineeringvol 2016 Article ID 5464570 15 pages 2016

[48] S Panda N M Modak and L E Cardenas-Barron ldquoCo-ordinating a socially responsible closed-loop supply chainwith product recyclingrdquo International Journal of ProductionEconomics vol 188 pp 11ndash21 2017

[49] N M Modak N Modak S Panda and S S Sana ldquoAnalyzingstructure of two-echelon closed-loop supply chain for pricingquality and recycling managementrdquo Journal of Cleaner Pro-duction vol 171 pp 512ndash528 2018

[50] Q P Wang and D Z Zhao ldquoRevenue-sharing contract ofsupply chain based on consumerrsquos preference for low carbonproductsrdquo Chinese Journal of Management Science vol 22no 9 pp 106ndash113 2014

[51] B Yu J Wang X Lu and H Yang ldquoCollaboration in a low-carbon supply chain with reference emission and cost learningeffects cost sharing versus revenue sharing strategiesrdquo Journalof Cleaner Production vol 250 Article ID 119460 2020

[52] J Cao X Zhang and G Zhou ldquoSupply chain coordinationwith revenue-sharing contracts considering carbon emissionsand governmental policy makingrdquo Environmental Progress ampSustainable Energy vol 35 no 2 pp 479ndash488 2016

[53] X Xu P He H Xu and Q Zhang ldquoSupply chain coordi-nation with green technology under cap-and-trade regula-tionrdquo International Journal of Production Economics vol 183pp 433ndash442 2017

[54] N M Modak S Sinha S Panda et al ldquoAnalyzing a sociallyresponsible closed-loop distribution channel with recyclingfacilityrdquo SN Applied Sciences vol 1 no 10 p 1189 2019

[55] D Feng L Ma Y Ding G Wu and Y Zhang ldquoDecisions ofthe dual-channel supply chain under double policy consid-ering remanufacturingrdquo International Journal of Environ-mental Research and Public Health vol 16 no 3 p 465 2019

[56] I Bellos M Ferguson and L B Toktay ldquo)e car sharingeconomy interaction of business model choice and productline designrdquo Manufacturing amp Service Operations Manage-ment vol 19 no 2 pp 185ndash201 2017

[57] G Ferrer and J M Swaminathan ldquoManaging new andremanufactured productsrdquo Management Science vol 52no 1 pp 15ndash26 2006

[58] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2pp 519ndash527 2011

[59] L Cui S Guo and H Zhang ldquoCoordinating a green agri-foodsupply chain with revenue-sharing contracts consideringretailersrsquo green marketing effortsrdquo Sustainability vol 12no 4 p 1289 2020

[60] J Heydari K Govindan and A Aslani ldquoPricing and greeningdecisions in a three-tier dual channel supply chainrdquo Inter-national Journal of Production Economics vol 217 pp 185ndash196 2019

Mathematical Problems in Engineering 17

and green manufacturing )ere exists a gap in the existingliterature which is that the consumerrsquos behavior has notbeen concerned In our research the green preference andprice reference behavior are considered

23 Cooperation Supply chain cooperation is defined asldquolong-term relationships where participants generally co-operate share information and work together to plan andeven modify their business practices to improve joint per-formancerdquo [44] In general supply chain cooperation meansachieving better performance In the supply chain there aremany coordination strategies to choose from such as rev-enue sharing buybacks quantity discounts and two-parttariff contracts Among these contracts revenue sharing hasattracted the attention of many scholars and is widely used inactual supply chains [45] For example Xu et al proposed atwo-way revenue sharing contract to coordinate multipledistributors in a dual-channel supply chain and the resultsshowed that the manufacturer could prompt the retailer tocooperate by providing this contract [46] Shi et al studiedreverse revenue-sharing contracts in a closed-loop systemand proposed a function to calculate the optimal ratio of thetransfer collection price )e results also suggested thatreverse revenue-sharing contracts are more attractive formanufacturers than a two-part tariff [47] Panda et al ex-plored channel coordination in a socially responsiblemanufacturerndashretailer closed-loop supply chain and foundthat a revenue-sharing contract resolved channel conflict[48] Modak et al used the subgame perfect equilibrium andalternative offer bargaining strategy to resolve channelconflict and distribute surplus profit [49] Wang and Zhaodesigned a revenue-sharing contract to reduce carbonemissions and both the supplier and the retailer achievedPareto improvement In addition they developed a functionto determine the revenue sharing ratio using the Rubinsteinbargaining model [50] Yu et al considered a cooperationproblem in the low-carbon supply chain and found that theenvironmental awareness of consumers and tax rates con-siderably affect the emission reduction [51] )ere have alsobeen some supply chain coordination studies conductedunder carbon policies Revenue-sharing contracts have beendesigned to improve the performance of supply chainmembers based on different carbon policies [52] Xu et alstudied the coordination problem in a two-echelon supplychain and the effect of government policy-making ondistributing the optimal emission quota was investigated)e results showed that a reasonable revenue-sharingcontract is essential to increase supply chain membersrsquoprofits even under low-carbon conditions [53] Modak et alconcluded that the optimal recycling rate increases with theCSR activity of the manufacturer and a profit-sharingcontract provides the best channel performance in a closed-loop distribution channel consisting of a socially responsiblemanufacturer multiple retailers and a third-party collector[54] Feng infers that win-win results can be achieved byestablishing profit-sharing contracts considering the pref-erence of green consumers [55] Previous papers often focuson two aspects including the choice of contract and the

conditions of the contract However the above literature stillhas a gap between cooperation and consumerrsquos behavior)erefore we investigate the supply chain cooperationproblem under the context of consumerrsquos reference behaviorand the carbon tax

In summary this study examines the interaction be-tween supply chain cooperation and carbon taxes in a two-echelon supply chain considering consumerrsquos behaviorSome important factors should be considered simulta-neously to study this problem such as the optimal carbon taxand consumerrsquos behavior however previous research hasonly considered these factors separately We also investigatethe interaction between coordination and the carbon tax)e difference between our study and others in the literatureis presented in Table 1

3 Supply Chain Model

In this section a two-echelon supply chain model is intro-duced to study green supply chain cooperation and carbon taxpolicy )e optimal decisions of the supply chain and thegovernment tax are addressed We then introduce the rev-enue-sharing contract used to coordinate the supply chain

31 Model Assumptions To answer the first question (iscarbon tax policy favorable for improving the greenness ofproducts) a two-echelon supply chain model consisting ofa single supplier and a single retailer without contracts isestablished )e manufacturer determines the productrsquosgreenness and wholesale price and the retailer determinesthe retail price of the green product )e green productcompetes for market share with traditional productsCompared with traditional products green products havethe characteristics of low pollution and being environ-mentally friendly but they may be less functional such aselectric vehicles which has poor endurance and slow speedsIn our study we focus on green products that are lessfunctional than traditional products For example Belloset al noted that manufacturers offer vehicles with poorperformance for customers who focus on fuel efficiency [56]In real life greenness reflects the environmental attributes ofthe product and it is commonly used to measure howenvironmentally friendly a product is In this study we use g

to denote the greenness degree of green products as ameasure of their environmental attributes which is acommon practice [38 41]

Consumers are environmentally conscious and haveenvironmental awareness )e utility gained from envi-ronmental attributes is assumed to be kg where k is thesensitivity of the consumer to the greenness of a greenproduct [38] We use V to denote the utility obtained by aconsumer from a traditional product It is commonly as-sumed that V is uniformly distributed on [0 1] to simplifythe problem without affecting the conclusion [57] )en αV

is the utility from the green product where α isin (0 1) is thefunctional attribute coefficient of the green product thatreflects its weak functional performance compared with thetraditional product

4 Mathematical Problems in Engineering

In real life consumers usually compare prices betweentwo similar products Let β be the consumersrsquo recognitionlevel of the reference price [35] )erefore the utility ofgreen products is obtained from four parts a positive partfrom the basic utility (αV) a negative part from the price (p)a positive part from the environmental consciousness (kg)and a negative part from the price reference (β(p minus pn)))eutility to a consumer of green products and traditionalproducts can thus be expressed as follows

un V minus pn

ug αV minus p + kg minus β p minus pn( 1113857

⎧⎨

⎩ (1)

Table 2 summarizes the notation used in this study

32 Model and Solution Consumers choose between greenand traditional products by comparing utility when ug gt un

and ug gt 0 consumers will purchase the green productwhereas when un gt ug and un gt 0 consumers will purchasethe traditional product As a result demand is obtained asshown in equation (2) )e proof of the demand function isgiven in Appendix A

q (α + β)pn minus (1 + β)p + kg

α(1 minus α)

qn 1 minus(1 + β) pn minus p( 1113857 + kg

1 minus α

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(2)

)e manufacturer determines the wholesale price andgreenness of the green product Let c denote the cost rate oftechnology development then the total development cost is(12)cg2 which is convexly increasing with the greenness[40] In addition (e minus g) is a linear function of the unitcarbon emissions for green products [58] )erefore theprofit function for the manufacturer is obtained from threeparts a positive part from the wholesale ((w minus c)q) anegative part from green technology development ((cg22))and a negative part from the carbon tax (t(e minus g)q) )emanufacturerrsquos decision model is as follows

Maxwg

πM (w minus c minus t(e minus g))q minuscg

2

2 (3)

)e retailerrsquos decision problem is then formulated asfollows

Maxp

πR (p minus w)q (4)

)e optimal solutions for the retailer and manufacturerare derived as the following theorem by substituting thedemand function into the equation above )e proof ispresented in Appendix A

Theorem 1 -e optimal solutions for both parties are

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

g (1 + β)(te + c) minus (α + β)pn minus ke( 1113857t

t2(1 + β) minus 4cα(1 minus α)

p 3cpnα

3+ 2A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)

)e corresponding profits are

πm minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2c

2 t2β + t

2+ 4(minus 1 + α)cα1113872 1113873(β + 1)

πr minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2(minus 1 + α)αc

2

4α2c minus 4αc +(β + 1)t2

1113872 11138732(β + 1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

According to this theorem price and greenness decisionsare directly affected by the carbon tax rate Facing a highercarbon tax and development cost the manufacturer andretailer will reduce the greenness of products and the saleprice with increasing carbon tax However when facing alower carbon tax the greenness and price will increase withincreasing carbon tax (the details are shown in Appendix Bequations (1) and (5)) Next we investigate the effects ofthese parameters on the product greenness )e proofs areprovided in Appendix B

Proposition 1 -e effects of the carbon tax rate pricerecognition level development cost and functional attributecoefficient on product greenness are as follows

(1) -ere exist some thresholds 1113954t1 and 1113954c1 for which g

decreases with increasing t when cgt 1113954c1 and tgt1113954t1 elseg increases with increasing t

Table 1 Differences between our study and the available literature

Literature Decisions under the carbon tax Optimal carbon tax Consumerrsquos behavior CoordinationYu and Han [28] radic radicUlph and Ulph [22] radicHsieh and Dye [34] radicCao et al [47] radic radicXu et al [48] radic radicOur paper radic radic radic radic

Mathematical Problems in Engineering 5

(2) -ere exist some thresholds 1113954t2 and 1113954c2 for which g

decreases with increasing β when cgt 1113954c2 and tgt1113954t2 elseg increases with increasing β

(3) g increases with increasing k decreases with in-creasing c and initially increases and then decreaseswith increasing α

We can conclude that the carbon tax development costprice recognition level greenness sensitivity and functionalattribute coefficient all influence the greenness in Proposi-tion 1 When the carbon tax is low the manufacturer incursless cost to improve the greenness of the products As aresult the greenness increases with the carbon tax In ad-dition if the carbon tax is high and the development cost islow the greenness of the products will increase with in-creasing carbon tax because the tax is the crucial factor indecisions In contrast if the carbon tax is high and devel-opment costs are also high the greenness of products willdecrease because the manufacturer will choose to reducedevelopment costs )erefore pollution will not be reducedas the carbon tax increases in some cases Facing high de-velopment costs the manufacturer will not improvegreenness unless it is promoted by a lower carbon tax Anumerical simulation of this scenario is shown in Figure 1

Similarly the results show that greenness is affected bythe consumersrsquo price reference behavior When the carbontax is low the negative effect of price increases when con-sumers have higher concerns about price )erefore themanufacturer will offset this negative effect by improving thegreenness of products When the carbon tax is high and thedevelopment cost is low the greenness of products willincrease under a high consumer concern about price Incontrast if the carbon tax is high and the development cost ishigh the greenness will decrease because the manufacturerwill choose to reduce development costs Numerical sim-ulations of these scenarios are shown in Figure 2

In addition greenness sensitivity and the functional attri-bute coefficient also affect greenness )e greenness increaseswith increasing greenness sensitivity of consumers Highgreenness sensitivity can promote more green products )ismeans that the government can promote green production by

increasing the green consciousness of consumersMoreover thegreenness decreases with increasing development cost A higherdegree of greenness is obtained when manufacturers face lowerdevelopment costs In addition greenness first increases andthen decreases with the increase in the functional attributecoefficient When the functional attribute of the green productis low themanufacturer will produce a lower greenness productwith increases in the functional attribute coefficient In contrastthe manufacturer will produce a product with higher greennesswith an increase in the functional attribute after a functionalthreshold has been reached Numerical simulations of thesescenarios are shown in Figure 3

Overall consumersrsquo reference behavior plays a positive rolein promoting green production However the carbon tax is noteffective for improving green products in some cases Forexample if the carbon tax is high and the development cost isalso high the greenness of products will decrease)erefore wenext investigate the optimal carbon tax

First an optimal carbon tax is defined as that whichprovides the maximum greenness without losing total taxrevenue )e result is provided in Proposition 2 and theproofs can be found in Appendix C

Table 2 Notations

Parameters Definitionp pn Prices of the green product and traditional product respectivelyq qn Demand for the green product and traditional product respectivelyw Wholesale priceg Product greennesst Carbon tax ratec Cost rate of technology developmentc Unit product cost of the green producte Initial unit carbon emissionsα Functional attribute coefficient of the green product functionβ Consumersrsquo recognition level of the reference pricek Sensitivity of consumers to product greennessV Utility obtained by a consumer from the traditional productη Proportion of the total revenue obtained by the retailerλ Member strength in the supply chain

10

08

06g

04

02

0000 01 02 03

t

t = t1

04 05

010

008

006

004

002

000

γ lt γ1γ gt γ1

Figure 1 Effects of carbon taxes on greenness

6 Mathematical Problems in Engineering

Proposition 2 -e optimal carbon tax is defined as follows

tlowast

M minusM

2minus N

1113968

e(1 + β) 0lt clt 1113954c3

P minus

P2

minus Q

1113969

2(1 + β)M 1113954c3 lt clt 1113954c1

1113954t1 1113954c1 lt c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

)e optimal carbon tax is obtained in Proposition 2based on the manufacturerrsquos development cost to maximizegreenness without losing the total carbon tax revenue Wecan observe that the crucial factor is the development costAn appropriate carbon tax policy should be chosen whenfacing different production technologies to ensure greennessand total tax revenue In addition it is easily observed thatthe maximum greenness is obtained when the developmentcost is high whereas the maximum tax is obtained when thedevelopment cost is lower but some of the greenness is lostA numerical simulation of this scenario is shown in Figure 4

33 Supply Chain Coordination It is also important forsupply chains to establish green supply chain models byintegrating internal and external resources to make deci-sions which enables supply chains to achieve better per-formance by improving cooperation [59 60] However moststudies on coordination have mainly focused reducingemissions and improving profit Few studies have investi-gated the relationship between carbon taxes and supplychain cooperation

In this section we study the impact of carbon taxes andconsumerrsquos behavior on supply chain coordination wherethe manufacturer is the leader and the retailer is the fol-lower A revenue-sharing contract is introduced to achieve

coordination Let η denote the proportion of total revenueobtained by the retailer then the manufacturerrsquos decisionmodel is as follows

Maxwg

πCM ((1 minus η)p + w minus c minus t(e minus g))q minus

cg2

2 (8)

)e retailerrsquos decision problem can be described asfollows

Maxp

πCR (ηp minus w)q (9)

)e optimal solutions for the retailer and manufacturerare derived with the following theorem by substituting thedemand function into the equation above )e proof isfound in Appendix D

Theorem 2 -e greenness is defined as follows

g t minus αpn + et(β + 1) + c minus pn( 1113857β minus ke + c( 1113857

2c(1 + η)α2 minus 2c(1 + η)α + t2(β + 1)

(10)

-en the profits of both parties are the following

πm minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2c

2 t2β + t

2+ 4(minus 1 + α)cα1113872 1113873(β + 1)

πr minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2(minus 1 + α)αc

2

4α2c minus 4αc +(β + 1)t2

1113872 11138732(β + 1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

We investigate the impact of the carbon tax on pro-moting the coordination of the supply chain by comparingthe profit changes of the two partners after the introductionof the revenue-sharing contract in Proposition 3 )e proofsare provided in Appendix E

Proposition 3 Coordination can be promoted by increasingthe carbon tax

01204

01202

01200g

01198

01196

00 01 02 03β

04 05

11900 times 10ndash5

11895 times 10ndash5

11898 times 10ndash5

11892 times 10ndash5

11890 times 10ndash5

11888 times 10ndash5

11885 times 10ndash5

11882 times 10ndash5

11880 times 10ndash5

γ lt γ2 t lt t2γ gt γ2 t lt t2

(a)

096

095

094

093g

092

091

090

08900 01 02 03

β04 05

0035

0040

0045

0050

0055

0060

γ lt γ2 t gt t2γ gt γ2 t gt t2

(b)

Figure 2 Effects of consumersrsquo price reference behavior on greenness (a) Low-carbon tax rate (b) High carbon tax rate

Mathematical Problems in Engineering 7

When the profits of both the retailer and the manu-facturer improve simultaneously coordination is achievedBy comparing equations (11) and (6) we find that therevenue-sharing contract improves the retailerrsquos profit onlywhen η isin (η1 1) whereas the manufacturer always benefitsfrom the contract )e range of η depends on the carbon taxrate When η is in the given range the retailer and man-ufacturer achieve Pareto improvement Furthermore therange expands as the carbon tax rate increases)erefore thegovernment can promote supply chain collaboration byincreasing the carbon tax Based on the previous results forthe optimal carbon tax we next investigate how to achievecoordination when the supply chain faces the optimalcarbon tax Figure 5 shows the Pareto improvement region

According to the assumptions described above themanufacturer acts as the leader and the retailer acts as afollower It is reasonable that the proportion of profitreceived by the retailer is decided by the manufacturer)erefore it is necessary to investigate whether theproportion decided by the leader is in the region in whichcoordination is achieved A parameter λ is introduced todenote the strength of a partner in the supply chain Let

λm isin (05 1) and λr 1 minus λm represent the power of themanufacturer and the retailer respectively )en weconsider a simple function (λrλm) to determine η whichis the proportion of revenue received by the retailer )eretailer retains the maximum revenue when the power ofthe two parties is equal (ie λm λr 05) whereas all ofthe benefits go to the leader when the power of themanufacturer is overwhelming compared with that of theretailer (ie λm 1) Similar to Proposition 2 the optimalcarbon tax is obtained under the revenue-sharing contractas follows

tlowast

M minus

M2

minus (1 + η2)N

1113969

e(1 + β) 0lt clt 1113954c

C2

(1 + η2)P minus

(1 + η2)2P2

minus (1 + η2)Q

1113969

2(1 + β)M 1113954c

C2 lt clt 1113954c1

1113954t1 1113954c1 lt c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

05

04

03g

02

01

0010 15 20 25

k3530 40

g

(a)

10

08

06g

04

02

0010 09 0708 06

γ030405

g

(b)

029

027

028

025

026

g

024

022

023

021035030 040 045 050

α060 065055 070

g

(c)

Figure 3 Effects of parameters on greenness (a) Greenness sensitivity (b) Development cost (c) Functional attribute coefficient

8 Mathematical Problems in Engineering

Proposition 4 is designed to answer the question ofwhether cooperation will be achieved the proofs are foundin Appendix F

Proposition 4 Coordination will be achieved if the power ofthe manufacturer is low (ie 05lt λm lt 1113954λ1) Moreover whenthe power of the manufacturer is at a medium level (ie1113954λ1 lt λm lt 1113954λ2) coordination can be achieved but part of thetotal tax revenue will be lost if the development cost is low(ie clt 1113954c1) In other cases (ie cgt 1113954c1 or 1113954λlt λm) coordi-nation cannot be achieved because the power of the manu-facturer is overwhelming

Coordination situations for different manufacturerpower levels are present in Table 3 )e achievement ofcoordination is decided by the power of the manufacturerand the optimal carbon tax faced by the supply chain Basedon Proposition 3 coordination can be promoted if thegovernment adjusts the carbon tax On the other handcoordination can easily be achieved under the optimalcarbon tax if the power of the manufacturer is low Inaddition there are two possibilities when the power of themanufacturer is at a medium level If the development cost istoo high adjusting the tax will lead to an unknown result thegreenness and total tax revenue may both decline at the same

10 014

012

010

008

006

004

002

000

08

06

Gre

enne

ss

Tota

l tax

04

02

00005000 010 015 020

t

γ lt γ3

t = ta

025 030

GreennessTotal tax

(a)

10 030

025

020

015

010

005

000

08

06

Gre

enne

ss

Tota

l tax

04

02

000100 02 03 04

t

γ3 lt γ lt γ1

t = tc

05 06

GreennessTotal tax

(b)

005 040

035

025

020

015

010

005

000

004

003

Gre

enne

ss

Tota

l tax

002

001

0000100 02 03 04

t

γ1 lt γ

t = t1

06 08

030

05 07

GreennessTotal tax

(c)

Figure 4 Determination of the optimal carbon tax (a) 0lt clt 1113954c3 (b) 1113954c3 lt clt 1113954c1 (c) 1113954c1 lt c

005t

η

010 015 020 025 030

Pareto improvement

035 040

Figure 5 Pareto improvement region with increasing carbon tax

Mathematical Problems in Engineering 9

time If the development cost is lower the greenness may beimproved with a loss of revenue by increasing the carbon taxHowever when the power of the manufacturer is strong theregion δ (η1 1) cannot be achieved

According to our theory the government can promotethe coordination between the manufacturer and the retailerby adjusting the carbon tax When the power of the man-ufacturer increases the government still achieves coordi-nation by raising the carbon tax In general cooperation isachieved and the government plays a key role in promotingcooperation

)e effects of consumersrsquo behavior on supply chaincoordination are also investigated in the next proposition)e proofs are provided in Appendix G

Proposition 5 Price reference behavior can promote supplychain coordination but green preference cannot

From the previous results price reference has a compleximpact on the supply chain First the increasing sensitivityof customers to price will promote supply chain coordi-nation because this negative impact can be mitigated bycooperation In addition price reference behavior also af-fects the green decisions of the supply chain in a complexmanner In contrast the mechanism by which consumerrsquogreen preferences influence the supply chain is relativelysimple Greenness increases with increasing greennesspreference and green preference has no effect on the co-ordination of the supply chain

4 Conclusion

)is study examines the interaction between supply chaincooperation and the carbon tax problem in a two-echelonsupply chain under consumerrsquos reference behavior )e opti-mal carbon tax policy is obtained based on analysis of thecarbon tax green investment coefficient and degree of con-sumerrsquos price reference )e optimal carbon tax is defined asthe simultaneous optimization of total tax revenue and productgreenness In addition coordination is achieved by introducingrevenue-sharing contracts)e impact of consumersrsquo referencebehavior and the carbon tax on supply chain coordination isalso investigated )e results are as follows

(1) )e greenness increases with increasing carbon taxwhen the carbon tax is low In addition if the carbontax is higher and the development cost is low thegreenness of products will increase with increasingcarbon tax Conversely if the carbon tax is higherand the development cost is also high the greennessof products will decrease

(2) When the carbon tax is low the negative effect ofprice increases as consumer concerns about priceincrease )erefore the manufacturer offsets thenegative effect by improving the greenness Whenthe carbon tax is higher and development cost is lowthe greenness of products will increase with highconsumerrsquos concern about price In contrast if thecarbon tax is higher and the development cost ishigh the greenness will decrease

(3) )e greenness increases with an increasing greennesspreference of consumers )eir preference behaviorcan promote more green products )is means thatthe government can promote green production bypromoting the green consciousness of consumersMoreover the greenness decreases with increasingdevelopment cost A higher greenness is obtainedwhen manufacturers face lower development costsIn addition greenness first increases and then de-creases with increases in the functional attributecoefficient When the functional attribute of thegreen product is low the manufacturer will producea lower greenness product with an increase in thefunctional attribute In contrast the manufacturerwill produce a higher greenness product with anincrease in the functional attribute after a func-tionality threshold

(4) We investigated the impact of the carbon tax onpromoting the coordination of the supply chain bycomparing the profit changes of two partners afterthe introduction of a revenue-sharing contract inProposition 3 We found that coordination could bepromoted by increasing the carbon tax

(5) )e achievement of coordination depends on thetype of manufacturer If the power of the manu-facturer is low coordination can be achievedunder the optimal carbon tax If the power of themanufacturer is at a medium level coordinationcan be achieved by increasing the carbon tax andimproved greenness will be realized with a loss ofrevenue However when the power of the man-ufacturer is strong coordination cannot beachieved

In this study we focused on the interaction between acarbon tax and supply chain cooperation However othercarbon policies (eg a cap-and-trade policy) and supplychain structures are worth exploring For example it wouldbe interesting to investigate the effects on a supply chainstructure consisting of the retailer as the leader under acomplex carbon policy

Table 3 Coordination situations for different manufacturer power levels

Power of the manufacturer Coordination Greenness Total carbon tax revenue05lt λm lt 1113954λ1 Achieve Improve Improve1113954λ1 lt λm lt 1113954λ2 clt 1113954c1 Achieve Improve Decline1113954λ1 lt λm lt 1113954λ2 clt 1113954c1 Cannot achievecgt 1113954c1 Cannot achieve

10 Mathematical Problems in Engineering

Appendix

A Proof of Theorem 1

)e demand function can be written as

q P ug ge un ug ge 01113966 1113967

qn P un gt ug un ge 01113966 1113967

⎧⎪⎨

⎪⎩(A1)

After replacing un and ug we have

q P αv minus p + kg minus β p minus pn( 1113857 minus v minus pn( 1113857ge 0 αv minus p + kg minus β p minus pn( 1113857ge 01113864 1113865

qn P v minus pn minus αv minus p + kg minus β p minus pn( 1113857( 1113857gt 0 v minus pn ge 01113864 11138651113896 (A2)

Given that V is uniformly distributed on [0 1] thedemand function is

q (α + β)pn minus (1 + β)p + kg

α(1 minus α)

qn 1 minus(1 + β) pn minus p( 1113857 + kg

1 minus α

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A3)

)e condition kgt ((1 + β)p minus (α + β)png) is necessaryto ensure nonnegativity )e retailer problem is addressedfirst and we have

p (α + β)pn +(1 + β)w + kg

2(1 + β) (A4)

)en substitute p into the manufacturerrsquos profit func-tion Solving the optimal w and g simultaneously yields

w (α + β)pn +(1 + β)(c + t(e minus g)) + kg

2(1 + β)

g t (α + β)pn minus (1 + β)w + ke( 1113857

2cα(1 minus α)

(A5)

)en we have

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

g (1 + β)(te + c) minus (α + β)pn minus ke( 1113857t

t2(1 + β) minus 4cα(1 minus α)

p 3cpnα

3+ 2A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1 (1 + β)(c + te) + ke minus (1 minus β)pn

A2 (1 + β)(c + te) + 3ke minus (3 minus β)pn

⎧⎪⎨

⎪⎩

B1 minus 2c (1 + β)(c + te) + ke + βpn( 1113857 + t2(1 + β)pn

B2 minus c (1 + β)(c + te) + 3ke + 3βpn( 1113857 + t2(1 + β)pn

⎧⎪⎨

⎪⎩

(A6)

Notes that we have (α + β)pn + ke minus (1 + β)(te + c)gt 0)e concavity condition is 4αc(1 minus α) minus (1 + β)t2 gt 0

B Proof of Proposition 1

(1) )e first derivative of g in t is

zg

zt

minus 4cα3pn + 4cA3α2

+ B3α minus t2(1 + β) (1 + β)c minus ke minus βpn( 1113857

t2(1 minus β) minus 4cα(1 minus α)1113872 1113873

2

A3 (1 + β)(c + 2te) minus ke +(1 minus β)pn

B3 minus 4c (1 + β)(c + 2te) minus ke minus βpn( 1113857 + t2(1 + β)pn

⎧⎨

⎩

(B1)

We have a threshold

1113954c1 minust2(1 + β) (α + β)pn + ke minus (1 + β)c( 1113857

4α(1 minus α) (α + β)pn + ke minus (1 + β)(c + 2te)( 1113857

1113954t1 (α + β)pn + ke minus (1 + β)c

2(1 + β)e

(B2)

When cgt 1113954c1 and tgt1113954t1 g is decreasing in t Bycontrast g is increasing in t in other cases

(2) )e first derivative of g in β is

zg

zβ

minus 4α(1 minus α) te + c minus pn( 1113857c + t2

ke + αpn minus pn( 1113857

t2(1 + β) minus 4α(1 minus α)c1113872 1113873

2

(B3)

We have a threshold

1113954c2 minust2

ke minus (1 minus α)pn( 1113857

4α(1 minus α) te + c minus pn( 1113857

1113954t2 pn minus c

e

(B4)

Review we have (α + β)pn + ke minus (1 + β)(te + c)gt 0therefore we can find that

tltpn minus c

e+

ke minus (1 minus α)pn

(1 + β)e (B5)

(i) When ke minus (1 minus α)pn lt 0 then tlt1113954t2 is inevitable)erefore (zgzβ)gt 0 is correct

(ii) When ke minus (1 minus α)pn gt 0 we can find that(zgzβ)lt 0 if cgt 1113954c2 and tgt1113954t2 (zgzβ)lt 0 in othercases

(3) )e first derivative of g in k and c is

Mathematical Problems in Engineering 11

zg

zc minus

4α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

4cα(1 minus α) minus t2(1 + β)1113872 1113873

2 lt 0

zg

zk

te

4cα(1 minus α) minus t2(1 + β)gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(B6)

(4) )e first derivative of g in α is

zg

zα

minus Xt

4α(1 minus α)c minus t2(1 + β)1113872 1113873

2 (B7)

And X minus 4α2cpn + (4(1 + β)(te + c) minus 4ke minus 4pn)

(2α + 1)c + pnt2(1 + β) is a quadratic function )erange of α in 4αc(1 minus α) minus (1 + β)t2 gt 0 is

α1 c +

c2

minus ct2(1 + β)

1113969

2c

α2 c minus

c2

minus ct2(1 + β)

1113969

2c

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(B8)

By substituting those into X we have

X|αα1 4

c2

minus ct2(1 + β)

1113969

T minus 2 c minus t2(1 + β)1113872 1113873pn

X|αα2 minus 4

c2

minus ct2(1 + β)

1113969

T minus 2 c minus t2(1 + β)1113872 1113873pn lt 0

⎧⎪⎪⎨

⎪⎪⎩

(B9)

And T (1 + β)(te + c) minus ke minus ((12) + β)pn lt 0We can find

X|αα1 times X|αα2 minus 4 c minus t2(1 + β)1113872 1113873

middot 4 βpn + ke minus (1 + β)(te + c)( 11138572

1113872

+ p2nt

2(1 + β)1113873lt 0

(B10)

)erefore X|αα1 gt 0 )ere only exist a 1113954α isin (α2 α1)that makes (zgzα) 0 And (zgzα) gt 0 ifα isin (α2 1113954α) (zgzα)lt 0 if α isin (1113954α α1)

(5) )e first derivative of p in t is

zp

zt minus

2α(α minus 1) e(1 + β)t221113872 1113873 + minus pnα minus ke + c minus pn( 1113857β + c( 1113857t minus 2eαc(α minus 1)1113872 1113873c

(1 + β)t2

+ 4αc(α minus 1)1113872 11138732

A3 (1 + β)(c + 2te) minus ke +(1 minus β)pn

B3 minus 4c (1 + β)(c + 2te) minus ke minus βpn( 1113857 + t2(1 + β)pn

⎧⎪⎨

⎪⎩

(B11)

)e discriminant of quadratic function

e(1 + β)t2

2+ minus pnα minus ke + c minus pn( 1113857β + c( 1113857t minus 2eαc(α minus 1)

(B12)

is

Δ (minus β minus 1)c + ke + βpn + pnα( 11138572

+ 4e2(1 + β)t

2αc(α minus 1)

(B13)

We have a threshold

1113954c minus(minus β minus 1)c + ke + βpn + pnα( 1113857

2

4e2(1 + β)t

2α(α minus 1) (B14)

Review we have(α + β)pn + ke minus (1 + β)(te + c)gt 0therefore we can find that

tm 2

minus (1 + β)αc(α minus 1)

1113968

1 + β (B15)

Substituting it into above quadratic function

2 c

radicminus pnα minus ke + c minus pn( 1113857β + c( 1113857

minus (1 + β)α(α minus 1)

1113968minus 4eαc(α minus 1)(1 + β)

1 + β (B16)

And we have a same threshold

1113954c minus(minus β minus 1)c + ke + βpn + pnα( 1113857

2

4e2(1 + β)t

2α(α minus 1) (B17)

It could be found that(zpzt) gt 0 because Δlt 0 whenclt 1113954c And when cgt 1113954c we have another threshold 1113954t andzpzt|

t1113954t 0 because Δgt 0 and zpzt|tmlt 0)erefore when

cgt 1113954c and tgt1113954t p is decreasing in t And when cgt 1113954c and tlt1113954tp is increasing in t

12 Mathematical Problems in Engineering

C Proof of Proposition 2

(1) Let R t(e minus g)q denote total carbon tax and thefirst derivative of R in t is

zR

zt2c A2t

2+ B2t + c1113872 1113873 A3t

2+ B3t + c1113872 1113873

4cα(1 minus α) minus t2(1 + β)1113872 1113873

3 (C1)

Among them

A2 (1 + β)e

2gt 0

B2 minus (α + β)pn + ke minus (1 + β)c( 1113857lt 0

C2 2ceα(1 minus α)gt 0

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

A3 (α + β)pn + ke minus (1 + β)c( 1113857(1 + β)gt 0

B3 minus 8ceα(1 minus α)(1 + β)lt 0

C3 4cα(1 minus α) (α + β)pn + ke minus (1 + β)c( 1113857gt 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(C2)

Let Φ A2t2 + B2t + C2 and Ψ A3t

2 + B3t + C3 andwe can find that zRzt and Φ times Ψ have a same sign Bysolving Δ1 and Δ2 a threshold is

1113954c3 (α + β)pn + ke minus (1 + β)c( 1113857

2

4e2α(1 minus α)(1 + β)

(C3)

)ere are two positive real numbers of solutions forΦ ifclt 1113954c3 and let 0lt ta lt tb denote it Similarly let0lt tc lt td denote the solutions for Ψ if cgt 1113954c3 Reviewwe have 4αc(1 minus α) minus (1 + β)t2 gt 0 therefore lettm

4cα(1 minus a)1 + β

1113968 Substituting tm into Φ and Ψ

Φ|tm minus 2

cα(1 minus α)

1 + β

1113971

(α + β)pn + ke minus (1 + β)c( 1113857 + 4ecα(1 minus α)

Ψ|tm minus 8cα(1 minus α) 2(1 + β)e

α(1 minus α)

1 + βc

1113971

minus (α + β)pn + ke minus (1 + β)c( 1113857⎛⎝ ⎞⎠

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(C4)

If clt 1113954c3 Φ|tmlt 0 and Ψ|tm

gt 0 then zRzt|tmlt 0 If

cgt 1113954c3 Φ|tmgt 0 and Ψ|tm

lt 0 then zRzt|tmlt 0 )ere-

fore we have tm isin (ta tb) or tm isin (tc td) FurthermoreR t(e minus g)q is maximized when t ta if clt 1113954c3 R

t(e minus g)q is maximized when t tc if cgt 1113954c3 Totally wehave

tlowast

ta clt 1113954c3

tc cgt 1113954c31113896 (C5)

(2) Review Proposition 1 If cgt 1113954c1 g is maximized whentlowast 1113954t1 If clt 1113954c1 g is maximized when tlowast tmWhen tgt1113954t1 1113954c1 gt 1113954c5 And 1113954c1 lt 0lt 1113954c5 if tlt1113954t1Substituting 1113954t1 into Φ

Φ|1113954t1 minus

3 (α + β)pn + ke minus (1 + β)c( 11138572

8e(1 + β)+ 2ecα(1 minus α)

(C6)

A threshold is 1113954c4 (34)1113954c3 )erefore we have (i)when 0lt clt 1113954c4 tlowast ta because of Φ|1113954t1

lt 0 and 1113954t1 gt ta

(ii) when 1113954c4 lt clt 1113954c3 tlowast ta because of 1113954t1 lt ta (iii) tlowast

tc when 1113954c3 lt clt 1113954c1 and (iv) tlowast tc when 1113954c1 lt c )eresult is obtained by solving Φ and Ψ

D Proof of Theorem 2

Similar to Appendix A )e retailer problem is addressedfirst and we have

p αηpn + βηpn + ηkg + βw + w

2η(β + 1) (D1)

)en substituting p into the manufacturerrsquos profitfunction Solving the optimal w and g simultaneously yields

w αηpn + βet + βηpn minus βgt + eηk + cβ + et minus gt + c( 1113857η

(1 + η)(β + 1)

g minusαηpn + βηpn + eηk minus βw minus w( 1113857t

2(α minus 1)cαη

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(D2)

Mathematical Problems in Engineering 13

By solving the equation above we have

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

g t minus αpn + et(β + 1) + c minus pn( 1113857β minus ke + c( 1113857

t2(1 + β) minus (1 + η)cα(1 minus α)

p (2η + 1)cpnα

3+ A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1 et + ηpn + c( 1113857β + et + ke minus pn( 1113857η + c

A2 (2η + 1)pn + et + c( 1113857β +(minus 2η minus 1)pn + et +(2ηk + k)e + c

⎧⎪⎪⎨

⎪⎪⎩

B1 minus 2et minus 2ηpn minus 2c( 1113857β minus 2eηk minus 2et minus 2c( 1113857c + t2pn(β + 1)

B2 (minus 2η minus 1)pn minus et minus c( 1113857β minus et +(minus 2ηk minus k)e minus c( 1113857c + t2pn(β + 1)

⎧⎪⎪⎨

⎪⎪⎩

(D3)

Note that we have minus αpn + et(β + 1) + (c minus pn)β minus ke +

clt 0 )e concavity condition is (1 + β)(t2(1 + β) minus (1+

η)cα(1 minus α))lt 0

Substituting solutions into the profit function we have

πm minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2c

2 t2β + t

2+ 4(minus 1 + α)cα1113872 1113873(β + 1)

πr minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2(minus 1 + α)αc

2

4α2c minus 4αc +(β + 1)t2

1113872 11138732(β + 1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(D4)

E Proof of Proposition 3

πM (α + β)pn + ke minus (1 + β)c( 1113857

2c

2(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCM

(α + β)pn + ke minus (1 + β)c( 11138572c

2(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(E1)

By comparing the profits of two partners change after therevenue-sharing contract introduced we found that themanufacturer always benefits from cooperation

πR c2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCR

ηc2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(E2)

A threshold is

η1 2cα(1 minus α) minus (1 + β)t2

2cα(1 minus α)1113888 1113889

2

(E3)

When η is in the region δ (η1 1) the retailer willaccept cooperation And the first derivative of η1 in t is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t(β + 1)

α2c2(minus 1 + α)

2 lt 0 (E4)

)erefore η1 is decreasing in t )at means the regionδ (η1 1) is expanding when tax rises And the revenue-sharing contract is accepted easily than before

F Proof of Proposition 4

We consider a simple function (λrλm) to decide thepractical η where λm isin (05 1) and λr 1 minus λm Given the

14 Mathematical Problems in Engineering

carbon tax tlowast we have a region δ (η(tlowast) 1) of cooperationIt is clear that the gap of strength between two partitions inthe supply chain is a crucial factor in cooperation Bycomparing practice η decided by strength and η(tlowast) de-cided by the carbon tax we have

1λm

minus 1 2α2c minus 2αc + t

2(β + 1)1113872 1113873

2

4α2c2(minus 1 + α)

2 (F1)

A threshold is

1113954λ1 4α2c2

(minus 1 + α)2

8α4c2minus 16α3c2

+ 4 2c + t2(β + 1)1113872 1113873cα2 minus 4t

2c(β + 1)α + t

4(β + 1)

2 (F2)

When 05lt λm lt 1113954λ1 the practical η is in the regionδ (η(tlowast) 1) therefore the cooperation is achieved facingthe optimal carbon tax

However the practical η is out of the range δ (η(tlowast) 1)

when 1113954λ1 lt λm Review previous result the region δ (η1 1)

is expanding when tax rises )erefore there exists a pos-sibility that the government increasing carbon tax to co-ordinate the supply chain We assume a realistic thing thegovernment is willing to boost the greenness of the producteven reducing part of the revenue from the carbon tax

Based on it the carbon up to 1113954t1 which means thegreenness is maximized )erefore )e cooperation isachieved when clt 1113954c1 because the greenness is decreasing inthe carbon tax when 1113954c1 lt c )en we have the secondthreshold

1113954λ2 1

η1 1113954t1( 1113857 + 1 (F3)

)e cooperation could not be achieved when λm gt 1113954λ2)e conclusion is as follows (1) Coordination would be

achieved when the manufacturer is democratic (ie05lt λm lt 1113954λ1) (2) Coordination could be achieved but partof the total tax will lose if development cost is lower when themanufacturer is moderate (ie 1113954λ1 lt λm lt 1113954λ2) (3) )e co-operation could not be achieved if the manufacturer is adictator (ie λm gt 1113954λ2)

G Proof of Proposition 5

)e first derivative of η1 in β is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t

2

2α2c2(minus 1 + α)

2 lt 0 (G1)

Similarly η1 is decreasing in β )at means the regionδ (η1 1) is expanding when β rises

And it is apparent that η1 (2cα(1 minus α) minus (1 + β)t22cα(1 minus α))2 is independent with k

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is research was supported by the National Natural ScienceFoundation of China (no 71761015) and the Science andTechnology Research Project of Jiangxi Education Depart-ment of China (no GJJ150475)

References

[1] Y Li M K Lim J Hu and M-L Tseng ldquoInvestigating theeffect of carbon tax and carbon quota policy to achieve lowcarbon logistics operationsrdquo Resources Conservation andRecycling vol 154 p 104535 2020

[2] M Abedi-Varaki ldquoStudy of carbon dioxide gas treatmentbased on equations of kinetics in plasma discharge reactorrdquoModern Physics Letters B vol 31 no 22 p 1750210 2017

[3] S Meng M Siriwardana and J McNeill ldquo)e environmentaland economic impact of the carbon tax in Australiardquo Envi-ronmental and Resource Economics vol 54 no 3 pp 313ndash332 2013

[4] Y Chen and C L Tseng ldquoInducing clean technology in theelectricity sector tradable permits or carbon tax policiesrdquo-eEnergy Journal vol 32 no 3 2011

[5] H-B Duan L Zhu and Y Fan ldquoOptimal carbon taxes incarbon-constrained China a logistic-induced energy eco-nomic hybrid modelrdquo Energy vol 69 pp 345ndash356 2014

[6] C Lu Q Tong and X Liu ldquo)e impacts of carbon tax andcomplementary policies on Chinese economyrdquo Energy Policyvol 38 no 11 pp 7278ndash7285 2010

[7] K Cao B Xu and J Wang ldquoOptimal trade-in and warrantyperiod strategies for new and remanufactured products undercarbon tax policyrdquo International Journal of Production Re-search vol 58 no 1 pp 180ndash199 2020

[8] P He W Zhang X Xu and Y Bian ldquoProduction lot-sizingand carbon emissions under cap-and-trade and carbon taxregulationsrdquo Journal of Cleaner Production vol 103pp 241ndash248 2015

[9] X Xu X Xu and P He ldquoJoint production and pricing de-cisions for multiple products with cap-and-trade and carbontax regulationsrdquo Journal of Cleaner Production vol 112pp 4093ndash4106 2016

Mathematical Problems in Engineering 15

[10] Q Li X Guan T Shi and W Jiao ldquoGreen product designwith competition and fairness concerns in the circulareconomy erardquo International Journal of Production Researchvol 58 no 1 pp 165ndash179 2020

[11] A Ranjan and J K Jha ldquoPricing and coordination strategiesof a dual-channel supply chain considering green quality andsales effortrdquo Journal of Cleaner Production vol 218pp 409ndash424 2019

[12] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineeringvol 10 no 1 pp 99ndash116 2012

[13] X Yin X Chen X Xu and L Zhang ldquoTax or subsidyOptimal carbon emission policy a supply chain perspectiverdquoSustainability vol 12 no 4 p 1548 2020

[14] B B Schlegelmilch G M Bohlen and A Diamantopoulosldquo)e link between green purchasing decisions and measuresof environmental consciousnessrdquo European Journal of Mar-keting vol 30 no 5 pp 35ndash55 1996

[15] M S Hopkins ldquoWhat the rsquogreenrsquo consumer wantsrdquo MITSloan Management Review vol 50 no 4 pp 87ndash89 2009

[16] G Kalyanaram and R S Winer ldquoEmpirical generalizationsfrom reference price researchrdquo Marketing Science vol 14no 3 pp G161ndashG169 1995

[17] E A Greenleaf ldquo)e impact of reference price effects on theprofitability of price promotionsrdquo Marketing Science vol 14no 1 pp 82ndash104 1995

[18] N M Modak D K Ghosh S Panda and S S SanaldquoManaging green house gas emission cost and pricing policiesin a two-echelon supply chainrdquo CIRP Journal ofManufacturing Science and Technology vol 20 pp 1ndash11 2018

[19] P Sinclair ldquoHigh does nothing and rising is worse carbontaxes should keep declining to cut harmful emissionsrdquo -eManchester School vol 60 no 1 pp 41ndash52 1992

[20] G M Grossman and A B Krueger ldquoEconomic growth andthe environmentrdquo -e Quarterly Journal of Economicsvol 110 no 2 pp 353ndash377 1995

[21] N M Modak and P Kelle ldquoUsing social work donation as atool of corporate social responsibility in a closed-loop supplychain considering carbon emissions tax and demand uncer-taintyrdquo Journal of the Operational Research Society pp 1ndash172019

[22] A Ulph and D Ulph ldquo)e optimal time path of a carbon taxrdquoOxford Economic Papers vol 46 no S1 pp 857ndash868 1994

[23] M Hoel and S Kverndokk ldquoDepletion of fossil fuels and theimpact of global warmingrdquo Resource and Energy Economicsvol 18 no 2 pp 115ndash136 1996

[24] Y H Farzin and O Tahvonen ldquoGlobal carbon cycle and theoptimal time path of a carbon taxrdquo Oxford Economic Papersvol 48 no 4 pp 515ndash536 1996

[25] V Bosetti C Carraro R Duval andM Tavoni ldquoWhat shouldwe expect from innovation A model-based assessment of theenvironmental and mitigation cost implications of climate-related RampDrdquo Energy Economics vol 33 no 6 pp 1313ndash13202011

[26] M Hariga R Asrsquoad and A Shamayleh ldquoIntegrated economicand environmental models for a multi stage cold supply chainunder carbon tax regulationrdquo Journal of Cleaner Productionvol 166 pp 1357ndash1371 2017

[27] N Turken J Carrillo and V Verter ldquoFacility location andcapacity acquisition under carbon tax and emissions limits tocentralize or to decentralizerdquo International Journal of Pro-duction Economics vol 187 pp 126ndash141 2017

[28] W Yu and R Han ldquoCoordinating a two-echelon supply chainunder carbon taxrdquo Sustainability vol 9 no 12 p 2360 2017

[29] S Sinha and N M Modak ldquoAn EPQmodel in the perspectiveof carbon emission reductionrdquo International Journal ofMathematics in Operational Research vol 14 no 3pp 338ndash358 2019

[30] H Zhang P Li H Zheng and Y Zhang ldquoImpact of carbontax on enterprise operation and production strategy for low-carbon products in a co-opetition supply chainrdquo Journal ofCleaner Production Article ID 125058 2020

[31] X Chen H Yang X Wang et al ldquoOptimal carbon tax designfor achieving low carbon supply chainsrdquo Annals of OperationsResearch pp 1ndash28 2020

[32] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Handbook of the Fundamentals of Fi-nancial Decision Making Part I pp 99ndash127 )e EconometricSociety Cleveland OH USA 2013

[33] W Kim and M Kim ldquoReference quality-based competitivemarket structure for innovation driven marketsrdquo Interna-tional Journal of Research in Marketing vol 32 no 3pp 284ndash296 2015

[34] I Popescu and Y Wu ldquoDynamic pricing strategies withreference effectsrdquo Operations Research vol 55 no 3pp 413ndash429 2007

[35] P K Kopalle P K Kannan L B Boldt and N Arora ldquo)eimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Re-tailing vol 88 no 1 pp 102ndash114 2012

[36] T-P Hsieh and C-Y Dye ldquoOptimal dynamic pricing fordeteriorating items with reference price effects when inven-tories stimulate demandrdquo European Journal of OperationalResearch vol 262 no 1 pp 136ndash150 2017

[37] X Chen Z-Y Hu and Y-H Zhang ldquoDynamic pricing withstochastic reference price effectrdquo Journal of the OperationsResearch Society of China vol 7 no 1 pp 107ndash125 2019

[38] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquo Opera-tions Research vol 64 no 6 pp 1529ndash1536 2016

[39] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[40] C Chen ldquoDesign for the environment a quality-based modelfor green product developmentrdquo Management Sciencevol 47 no 2 pp 250ndash263 2001

[41] K Chitra ldquoIn search of the green consumers a perceptualstudyrdquo Journal of Services Research vol 7 no 1 2007

[42] Z Liu T D Anderson and J M Cruz ldquoConsumer envi-ronmental awareness and competition in two-stage supplychainsrdquo European Journal of Operational Research vol 218no 3 pp 602ndash613 2012

[43] B Li Y Wang and Z Wang ldquoManaging a closed-loop supplychain with take-back legislation and consumer preference forgreen designrdquo Journal of Cleaner Production Article ID124481 2020

[44] J Rantanen D B Grant and W Piotrowicz ldquoInvestigatingsupply chain cooperation in Finnish grocery retailrdquo ResearchJournal of the University of Gdansk Transport Economics andLogistics vol 71 pp 19ndash34 2017

[45] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[46] G Xu B Dan X Zhang and C Liu ldquoCoordinating a dual-channel supply chain with risk-averse under a two-way

16 Mathematical Problems in Engineering

revenue sharing contractrdquo International Journal of ProductionEconomics vol 147 pp 171ndash179 2014

[47] Z Shi N Wang T Jia et al ldquoReverse revenue sharingcontract versus two-part tariff contract under a closed-loopsupply chain systemrdquoMathematical Problems in Engineeringvol 2016 Article ID 5464570 15 pages 2016

[48] S Panda N M Modak and L E Cardenas-Barron ldquoCo-ordinating a socially responsible closed-loop supply chainwith product recyclingrdquo International Journal of ProductionEconomics vol 188 pp 11ndash21 2017

[49] N M Modak N Modak S Panda and S S Sana ldquoAnalyzingstructure of two-echelon closed-loop supply chain for pricingquality and recycling managementrdquo Journal of Cleaner Pro-duction vol 171 pp 512ndash528 2018

[50] Q P Wang and D Z Zhao ldquoRevenue-sharing contract ofsupply chain based on consumerrsquos preference for low carbonproductsrdquo Chinese Journal of Management Science vol 22no 9 pp 106ndash113 2014

[51] B Yu J Wang X Lu and H Yang ldquoCollaboration in a low-carbon supply chain with reference emission and cost learningeffects cost sharing versus revenue sharing strategiesrdquo Journalof Cleaner Production vol 250 Article ID 119460 2020

[52] J Cao X Zhang and G Zhou ldquoSupply chain coordinationwith revenue-sharing contracts considering carbon emissionsand governmental policy makingrdquo Environmental Progress ampSustainable Energy vol 35 no 2 pp 479ndash488 2016

[53] X Xu P He H Xu and Q Zhang ldquoSupply chain coordi-nation with green technology under cap-and-trade regula-tionrdquo International Journal of Production Economics vol 183pp 433ndash442 2017

[54] N M Modak S Sinha S Panda et al ldquoAnalyzing a sociallyresponsible closed-loop distribution channel with recyclingfacilityrdquo SN Applied Sciences vol 1 no 10 p 1189 2019

[55] D Feng L Ma Y Ding G Wu and Y Zhang ldquoDecisions ofthe dual-channel supply chain under double policy consid-ering remanufacturingrdquo International Journal of Environ-mental Research and Public Health vol 16 no 3 p 465 2019

[56] I Bellos M Ferguson and L B Toktay ldquo)e car sharingeconomy interaction of business model choice and productline designrdquo Manufacturing amp Service Operations Manage-ment vol 19 no 2 pp 185ndash201 2017

[57] G Ferrer and J M Swaminathan ldquoManaging new andremanufactured productsrdquo Management Science vol 52no 1 pp 15ndash26 2006

[58] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2pp 519ndash527 2011

[59] L Cui S Guo and H Zhang ldquoCoordinating a green agri-foodsupply chain with revenue-sharing contracts consideringretailersrsquo green marketing effortsrdquo Sustainability vol 12no 4 p 1289 2020

[60] J Heydari K Govindan and A Aslani ldquoPricing and greeningdecisions in a three-tier dual channel supply chainrdquo Inter-national Journal of Production Economics vol 217 pp 185ndash196 2019

Mathematical Problems in Engineering 17

In real life consumers usually compare prices betweentwo similar products Let β be the consumersrsquo recognitionlevel of the reference price [35] )erefore the utility ofgreen products is obtained from four parts a positive partfrom the basic utility (αV) a negative part from the price (p)a positive part from the environmental consciousness (kg)and a negative part from the price reference (β(p minus pn)))eutility to a consumer of green products and traditionalproducts can thus be expressed as follows

un V minus pn

ug αV minus p + kg minus β p minus pn( 1113857

⎧⎨

⎩ (1)

Table 2 summarizes the notation used in this study

32 Model and Solution Consumers choose between greenand traditional products by comparing utility when ug gt un

and ug gt 0 consumers will purchase the green productwhereas when un gt ug and un gt 0 consumers will purchasethe traditional product As a result demand is obtained asshown in equation (2) )e proof of the demand function isgiven in Appendix A

q (α + β)pn minus (1 + β)p + kg

α(1 minus α)

qn 1 minus(1 + β) pn minus p( 1113857 + kg

1 minus α

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(2)

)e manufacturer determines the wholesale price andgreenness of the green product Let c denote the cost rate oftechnology development then the total development cost is(12)cg2 which is convexly increasing with the greenness[40] In addition (e minus g) is a linear function of the unitcarbon emissions for green products [58] )erefore theprofit function for the manufacturer is obtained from threeparts a positive part from the wholesale ((w minus c)q) anegative part from green technology development ((cg22))and a negative part from the carbon tax (t(e minus g)q) )emanufacturerrsquos decision model is as follows

Maxwg

πM (w minus c minus t(e minus g))q minuscg

2

2 (3)

)e retailerrsquos decision problem is then formulated asfollows

Maxp

πR (p minus w)q (4)

)e optimal solutions for the retailer and manufacturerare derived as the following theorem by substituting thedemand function into the equation above )e proof ispresented in Appendix A

Theorem 1 -e optimal solutions for both parties are

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

g (1 + β)(te + c) minus (α + β)pn minus ke( 1113857t

t2(1 + β) minus 4cα(1 minus α)

p 3cpnα

3+ 2A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)

)e corresponding profits are

πm minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2c

2 t2β + t

2+ 4(minus 1 + α)cα1113872 1113873(β + 1)

πr minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2(minus 1 + α)αc

2

4α2c minus 4αc +(β + 1)t2

1113872 11138732(β + 1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

According to this theorem price and greenness decisionsare directly affected by the carbon tax rate Facing a highercarbon tax and development cost the manufacturer andretailer will reduce the greenness of products and the saleprice with increasing carbon tax However when facing alower carbon tax the greenness and price will increase withincreasing carbon tax (the details are shown in Appendix Bequations (1) and (5)) Next we investigate the effects ofthese parameters on the product greenness )e proofs areprovided in Appendix B

Proposition 1 -e effects of the carbon tax rate pricerecognition level development cost and functional attributecoefficient on product greenness are as follows

(1) -ere exist some thresholds 1113954t1 and 1113954c1 for which g

decreases with increasing t when cgt 1113954c1 and tgt1113954t1 elseg increases with increasing t

Table 1 Differences between our study and the available literature

Literature Decisions under the carbon tax Optimal carbon tax Consumerrsquos behavior CoordinationYu and Han [28] radic radicUlph and Ulph [22] radicHsieh and Dye [34] radicCao et al [47] radic radicXu et al [48] radic radicOur paper radic radic radic radic

Mathematical Problems in Engineering 5

(2) -ere exist some thresholds 1113954t2 and 1113954c2 for which g

decreases with increasing β when cgt 1113954c2 and tgt1113954t2 elseg increases with increasing β

(3) g increases with increasing k decreases with in-creasing c and initially increases and then decreaseswith increasing α

We can conclude that the carbon tax development costprice recognition level greenness sensitivity and functionalattribute coefficient all influence the greenness in Proposi-tion 1 When the carbon tax is low the manufacturer incursless cost to improve the greenness of the products As aresult the greenness increases with the carbon tax In ad-dition if the carbon tax is high and the development cost islow the greenness of the products will increase with in-creasing carbon tax because the tax is the crucial factor indecisions In contrast if the carbon tax is high and devel-opment costs are also high the greenness of products willdecrease because the manufacturer will choose to reducedevelopment costs )erefore pollution will not be reducedas the carbon tax increases in some cases Facing high de-velopment costs the manufacturer will not improvegreenness unless it is promoted by a lower carbon tax Anumerical simulation of this scenario is shown in Figure 1

Similarly the results show that greenness is affected bythe consumersrsquo price reference behavior When the carbontax is low the negative effect of price increases when con-sumers have higher concerns about price )erefore themanufacturer will offset this negative effect by improving thegreenness of products When the carbon tax is high and thedevelopment cost is low the greenness of products willincrease under a high consumer concern about price Incontrast if the carbon tax is high and the development cost ishigh the greenness will decrease because the manufacturerwill choose to reduce development costs Numerical sim-ulations of these scenarios are shown in Figure 2

In addition greenness sensitivity and the functional attri-bute coefficient also affect greenness )e greenness increaseswith increasing greenness sensitivity of consumers Highgreenness sensitivity can promote more green products )ismeans that the government can promote green production by

increasing the green consciousness of consumersMoreover thegreenness decreases with increasing development cost A higherdegree of greenness is obtained when manufacturers face lowerdevelopment costs In addition greenness first increases andthen decreases with the increase in the functional attributecoefficient When the functional attribute of the green productis low themanufacturer will produce a lower greenness productwith increases in the functional attribute coefficient In contrastthe manufacturer will produce a product with higher greennesswith an increase in the functional attribute after a functionalthreshold has been reached Numerical simulations of thesescenarios are shown in Figure 3

Overall consumersrsquo reference behavior plays a positive rolein promoting green production However the carbon tax is noteffective for improving green products in some cases Forexample if the carbon tax is high and the development cost isalso high the greenness of products will decrease)erefore wenext investigate the optimal carbon tax

First an optimal carbon tax is defined as that whichprovides the maximum greenness without losing total taxrevenue )e result is provided in Proposition 2 and theproofs can be found in Appendix C

Table 2 Notations

Parameters Definitionp pn Prices of the green product and traditional product respectivelyq qn Demand for the green product and traditional product respectivelyw Wholesale priceg Product greennesst Carbon tax ratec Cost rate of technology developmentc Unit product cost of the green producte Initial unit carbon emissionsα Functional attribute coefficient of the green product functionβ Consumersrsquo recognition level of the reference pricek Sensitivity of consumers to product greennessV Utility obtained by a consumer from the traditional productη Proportion of the total revenue obtained by the retailerλ Member strength in the supply chain

10

08

06g

04

02

0000 01 02 03

t

t = t1

04 05

010

008

006

004

002

000

γ lt γ1γ gt γ1

Figure 1 Effects of carbon taxes on greenness

6 Mathematical Problems in Engineering

Proposition 2 -e optimal carbon tax is defined as follows

tlowast

M minusM

2minus N

1113968

e(1 + β) 0lt clt 1113954c3

P minus

P2

minus Q

1113969

2(1 + β)M 1113954c3 lt clt 1113954c1

1113954t1 1113954c1 lt c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

)e optimal carbon tax is obtained in Proposition 2based on the manufacturerrsquos development cost to maximizegreenness without losing the total carbon tax revenue Wecan observe that the crucial factor is the development costAn appropriate carbon tax policy should be chosen whenfacing different production technologies to ensure greennessand total tax revenue In addition it is easily observed thatthe maximum greenness is obtained when the developmentcost is high whereas the maximum tax is obtained when thedevelopment cost is lower but some of the greenness is lostA numerical simulation of this scenario is shown in Figure 4

33 Supply Chain Coordination It is also important forsupply chains to establish green supply chain models byintegrating internal and external resources to make deci-sions which enables supply chains to achieve better per-formance by improving cooperation [59 60] However moststudies on coordination have mainly focused reducingemissions and improving profit Few studies have investi-gated the relationship between carbon taxes and supplychain cooperation

In this section we study the impact of carbon taxes andconsumerrsquos behavior on supply chain coordination wherethe manufacturer is the leader and the retailer is the fol-lower A revenue-sharing contract is introduced to achieve

coordination Let η denote the proportion of total revenueobtained by the retailer then the manufacturerrsquos decisionmodel is as follows

Maxwg

πCM ((1 minus η)p + w minus c minus t(e minus g))q minus

cg2

2 (8)

)e retailerrsquos decision problem can be described asfollows

Maxp

πCR (ηp minus w)q (9)

)e optimal solutions for the retailer and manufacturerare derived with the following theorem by substituting thedemand function into the equation above )e proof isfound in Appendix D

Theorem 2 -e greenness is defined as follows

g t minus αpn + et(β + 1) + c minus pn( 1113857β minus ke + c( 1113857

2c(1 + η)α2 minus 2c(1 + η)α + t2(β + 1)

(10)

-en the profits of both parties are the following

πm minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2c

2 t2β + t

2+ 4(minus 1 + α)cα1113872 1113873(β + 1)

πr minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2(minus 1 + α)αc

2

4α2c minus 4αc +(β + 1)t2

1113872 11138732(β + 1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

We investigate the impact of the carbon tax on pro-moting the coordination of the supply chain by comparingthe profit changes of the two partners after the introductionof the revenue-sharing contract in Proposition 3 )e proofsare provided in Appendix E

Proposition 3 Coordination can be promoted by increasingthe carbon tax

01204

01202

01200g

01198

01196

00 01 02 03β

04 05

11900 times 10ndash5

11895 times 10ndash5

11898 times 10ndash5

11892 times 10ndash5

11890 times 10ndash5

11888 times 10ndash5

11885 times 10ndash5

11882 times 10ndash5

11880 times 10ndash5

γ lt γ2 t lt t2γ gt γ2 t lt t2

(a)

096

095

094

093g

092

091

090

08900 01 02 03

β04 05

0035

0040

0045

0050

0055

0060

γ lt γ2 t gt t2γ gt γ2 t gt t2

(b)

Figure 2 Effects of consumersrsquo price reference behavior on greenness (a) Low-carbon tax rate (b) High carbon tax rate

Mathematical Problems in Engineering 7

When the profits of both the retailer and the manu-facturer improve simultaneously coordination is achievedBy comparing equations (11) and (6) we find that therevenue-sharing contract improves the retailerrsquos profit onlywhen η isin (η1 1) whereas the manufacturer always benefitsfrom the contract )e range of η depends on the carbon taxrate When η is in the given range the retailer and man-ufacturer achieve Pareto improvement Furthermore therange expands as the carbon tax rate increases)erefore thegovernment can promote supply chain collaboration byincreasing the carbon tax Based on the previous results forthe optimal carbon tax we next investigate how to achievecoordination when the supply chain faces the optimalcarbon tax Figure 5 shows the Pareto improvement region

According to the assumptions described above themanufacturer acts as the leader and the retailer acts as afollower It is reasonable that the proportion of profitreceived by the retailer is decided by the manufacturer)erefore it is necessary to investigate whether theproportion decided by the leader is in the region in whichcoordination is achieved A parameter λ is introduced todenote the strength of a partner in the supply chain Let

λm isin (05 1) and λr 1 minus λm represent the power of themanufacturer and the retailer respectively )en weconsider a simple function (λrλm) to determine η whichis the proportion of revenue received by the retailer )eretailer retains the maximum revenue when the power ofthe two parties is equal (ie λm λr 05) whereas all ofthe benefits go to the leader when the power of themanufacturer is overwhelming compared with that of theretailer (ie λm 1) Similar to Proposition 2 the optimalcarbon tax is obtained under the revenue-sharing contractas follows

tlowast

M minus

M2

minus (1 + η2)N

1113969

e(1 + β) 0lt clt 1113954c

C2

(1 + η2)P minus

(1 + η2)2P2

minus (1 + η2)Q

1113969

2(1 + β)M 1113954c

C2 lt clt 1113954c1

1113954t1 1113954c1 lt c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

05

04

03g

02

01

0010 15 20 25

k3530 40

g

(a)

10

08

06g

04

02

0010 09 0708 06

γ030405

g

(b)

029

027

028

025

026

g

024

022

023

021035030 040 045 050

α060 065055 070

g

(c)

Figure 3 Effects of parameters on greenness (a) Greenness sensitivity (b) Development cost (c) Functional attribute coefficient

8 Mathematical Problems in Engineering

Proposition 4 is designed to answer the question ofwhether cooperation will be achieved the proofs are foundin Appendix F

Proposition 4 Coordination will be achieved if the power ofthe manufacturer is low (ie 05lt λm lt 1113954λ1) Moreover whenthe power of the manufacturer is at a medium level (ie1113954λ1 lt λm lt 1113954λ2) coordination can be achieved but part of thetotal tax revenue will be lost if the development cost is low(ie clt 1113954c1) In other cases (ie cgt 1113954c1 or 1113954λlt λm) coordi-nation cannot be achieved because the power of the manu-facturer is overwhelming

Coordination situations for different manufacturerpower levels are present in Table 3 )e achievement ofcoordination is decided by the power of the manufacturerand the optimal carbon tax faced by the supply chain Basedon Proposition 3 coordination can be promoted if thegovernment adjusts the carbon tax On the other handcoordination can easily be achieved under the optimalcarbon tax if the power of the manufacturer is low Inaddition there are two possibilities when the power of themanufacturer is at a medium level If the development cost istoo high adjusting the tax will lead to an unknown result thegreenness and total tax revenue may both decline at the same

10 014

012

010

008

006

004

002

000

08

06

Gre

enne

ss

Tota

l tax

04

02

00005000 010 015 020

t

γ lt γ3

t = ta

025 030

GreennessTotal tax

(a)

10 030

025

020

015

010

005

000

08

06

Gre

enne

ss

Tota

l tax

04

02

000100 02 03 04

t

γ3 lt γ lt γ1

t = tc

05 06

GreennessTotal tax

(b)

005 040

035

025

020

015

010

005

000

004

003

Gre

enne

ss

Tota

l tax

002

001

0000100 02 03 04

t

γ1 lt γ

t = t1

06 08

030

05 07

GreennessTotal tax

(c)

Figure 4 Determination of the optimal carbon tax (a) 0lt clt 1113954c3 (b) 1113954c3 lt clt 1113954c1 (c) 1113954c1 lt c

005t

η

010 015 020 025 030

Pareto improvement

035 040

Figure 5 Pareto improvement region with increasing carbon tax

Mathematical Problems in Engineering 9

time If the development cost is lower the greenness may beimproved with a loss of revenue by increasing the carbon taxHowever when the power of the manufacturer is strong theregion δ (η1 1) cannot be achieved

According to our theory the government can promotethe coordination between the manufacturer and the retailerby adjusting the carbon tax When the power of the man-ufacturer increases the government still achieves coordi-nation by raising the carbon tax In general cooperation isachieved and the government plays a key role in promotingcooperation

)e effects of consumersrsquo behavior on supply chaincoordination are also investigated in the next proposition)e proofs are provided in Appendix G

Proposition 5 Price reference behavior can promote supplychain coordination but green preference cannot

From the previous results price reference has a compleximpact on the supply chain First the increasing sensitivityof customers to price will promote supply chain coordi-nation because this negative impact can be mitigated bycooperation In addition price reference behavior also af-fects the green decisions of the supply chain in a complexmanner In contrast the mechanism by which consumerrsquogreen preferences influence the supply chain is relativelysimple Greenness increases with increasing greennesspreference and green preference has no effect on the co-ordination of the supply chain

4 Conclusion

)is study examines the interaction between supply chaincooperation and the carbon tax problem in a two-echelonsupply chain under consumerrsquos reference behavior )e opti-mal carbon tax policy is obtained based on analysis of thecarbon tax green investment coefficient and degree of con-sumerrsquos price reference )e optimal carbon tax is defined asthe simultaneous optimization of total tax revenue and productgreenness In addition coordination is achieved by introducingrevenue-sharing contracts)e impact of consumersrsquo referencebehavior and the carbon tax on supply chain coordination isalso investigated )e results are as follows

(1) )e greenness increases with increasing carbon taxwhen the carbon tax is low In addition if the carbontax is higher and the development cost is low thegreenness of products will increase with increasingcarbon tax Conversely if the carbon tax is higherand the development cost is also high the greennessof products will decrease

(2) When the carbon tax is low the negative effect ofprice increases as consumer concerns about priceincrease )erefore the manufacturer offsets thenegative effect by improving the greenness Whenthe carbon tax is higher and development cost is lowthe greenness of products will increase with highconsumerrsquos concern about price In contrast if thecarbon tax is higher and the development cost ishigh the greenness will decrease

(3) )e greenness increases with an increasing greennesspreference of consumers )eir preference behaviorcan promote more green products )is means thatthe government can promote green production bypromoting the green consciousness of consumersMoreover the greenness decreases with increasingdevelopment cost A higher greenness is obtainedwhen manufacturers face lower development costsIn addition greenness first increases and then de-creases with increases in the functional attributecoefficient When the functional attribute of thegreen product is low the manufacturer will producea lower greenness product with an increase in thefunctional attribute In contrast the manufacturerwill produce a higher greenness product with anincrease in the functional attribute after a func-tionality threshold

(4) We investigated the impact of the carbon tax onpromoting the coordination of the supply chain bycomparing the profit changes of two partners afterthe introduction of a revenue-sharing contract inProposition 3 We found that coordination could bepromoted by increasing the carbon tax

(5) )e achievement of coordination depends on thetype of manufacturer If the power of the manu-facturer is low coordination can be achievedunder the optimal carbon tax If the power of themanufacturer is at a medium level coordinationcan be achieved by increasing the carbon tax andimproved greenness will be realized with a loss ofrevenue However when the power of the man-ufacturer is strong coordination cannot beachieved

In this study we focused on the interaction between acarbon tax and supply chain cooperation However othercarbon policies (eg a cap-and-trade policy) and supplychain structures are worth exploring For example it wouldbe interesting to investigate the effects on a supply chainstructure consisting of the retailer as the leader under acomplex carbon policy

Table 3 Coordination situations for different manufacturer power levels

Power of the manufacturer Coordination Greenness Total carbon tax revenue05lt λm lt 1113954λ1 Achieve Improve Improve1113954λ1 lt λm lt 1113954λ2 clt 1113954c1 Achieve Improve Decline1113954λ1 lt λm lt 1113954λ2 clt 1113954c1 Cannot achievecgt 1113954c1 Cannot achieve

10 Mathematical Problems in Engineering

Appendix

A Proof of Theorem 1

)e demand function can be written as

q P ug ge un ug ge 01113966 1113967

qn P un gt ug un ge 01113966 1113967

⎧⎪⎨

⎪⎩(A1)

After replacing un and ug we have

q P αv minus p + kg minus β p minus pn( 1113857 minus v minus pn( 1113857ge 0 αv minus p + kg minus β p minus pn( 1113857ge 01113864 1113865

qn P v minus pn minus αv minus p + kg minus β p minus pn( 1113857( 1113857gt 0 v minus pn ge 01113864 11138651113896 (A2)

Given that V is uniformly distributed on [0 1] thedemand function is

q (α + β)pn minus (1 + β)p + kg

α(1 minus α)

qn 1 minus(1 + β) pn minus p( 1113857 + kg

1 minus α

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A3)

)e condition kgt ((1 + β)p minus (α + β)png) is necessaryto ensure nonnegativity )e retailer problem is addressedfirst and we have

p (α + β)pn +(1 + β)w + kg

2(1 + β) (A4)

)en substitute p into the manufacturerrsquos profit func-tion Solving the optimal w and g simultaneously yields

w (α + β)pn +(1 + β)(c + t(e minus g)) + kg

2(1 + β)

g t (α + β)pn minus (1 + β)w + ke( 1113857

2cα(1 minus α)

(A5)

)en we have

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

g (1 + β)(te + c) minus (α + β)pn minus ke( 1113857t

t2(1 + β) minus 4cα(1 minus α)

p 3cpnα

3+ 2A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1 (1 + β)(c + te) + ke minus (1 minus β)pn

A2 (1 + β)(c + te) + 3ke minus (3 minus β)pn

⎧⎪⎨

⎪⎩

B1 minus 2c (1 + β)(c + te) + ke + βpn( 1113857 + t2(1 + β)pn

B2 minus c (1 + β)(c + te) + 3ke + 3βpn( 1113857 + t2(1 + β)pn

⎧⎪⎨

⎪⎩

(A6)

Notes that we have (α + β)pn + ke minus (1 + β)(te + c)gt 0)e concavity condition is 4αc(1 minus α) minus (1 + β)t2 gt 0

B Proof of Proposition 1

(1) )e first derivative of g in t is

zg

zt

minus 4cα3pn + 4cA3α2

+ B3α minus t2(1 + β) (1 + β)c minus ke minus βpn( 1113857

t2(1 minus β) minus 4cα(1 minus α)1113872 1113873

2

A3 (1 + β)(c + 2te) minus ke +(1 minus β)pn

B3 minus 4c (1 + β)(c + 2te) minus ke minus βpn( 1113857 + t2(1 + β)pn

⎧⎨

⎩

(B1)

We have a threshold

1113954c1 minust2(1 + β) (α + β)pn + ke minus (1 + β)c( 1113857

4α(1 minus α) (α + β)pn + ke minus (1 + β)(c + 2te)( 1113857

1113954t1 (α + β)pn + ke minus (1 + β)c

2(1 + β)e

(B2)

When cgt 1113954c1 and tgt1113954t1 g is decreasing in t Bycontrast g is increasing in t in other cases

(2) )e first derivative of g in β is

zg

zβ

minus 4α(1 minus α) te + c minus pn( 1113857c + t2

ke + αpn minus pn( 1113857

t2(1 + β) minus 4α(1 minus α)c1113872 1113873

2

(B3)

We have a threshold

1113954c2 minust2

ke minus (1 minus α)pn( 1113857

4α(1 minus α) te + c minus pn( 1113857

1113954t2 pn minus c

e

(B4)

Review we have (α + β)pn + ke minus (1 + β)(te + c)gt 0therefore we can find that

tltpn minus c

e+

ke minus (1 minus α)pn

(1 + β)e (B5)

(i) When ke minus (1 minus α)pn lt 0 then tlt1113954t2 is inevitable)erefore (zgzβ)gt 0 is correct

(ii) When ke minus (1 minus α)pn gt 0 we can find that(zgzβ)lt 0 if cgt 1113954c2 and tgt1113954t2 (zgzβ)lt 0 in othercases

(3) )e first derivative of g in k and c is

Mathematical Problems in Engineering 11

zg

zc minus

4α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

4cα(1 minus α) minus t2(1 + β)1113872 1113873

2 lt 0

zg

zk

te

4cα(1 minus α) minus t2(1 + β)gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(B6)

(4) )e first derivative of g in α is

zg

zα

minus Xt

4α(1 minus α)c minus t2(1 + β)1113872 1113873

2 (B7)

And X minus 4α2cpn + (4(1 + β)(te + c) minus 4ke minus 4pn)

(2α + 1)c + pnt2(1 + β) is a quadratic function )erange of α in 4αc(1 minus α) minus (1 + β)t2 gt 0 is

α1 c +

c2

minus ct2(1 + β)

1113969

2c

α2 c minus

c2

minus ct2(1 + β)

1113969

2c

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(B8)

By substituting those into X we have

X|αα1 4

c2

minus ct2(1 + β)

1113969

T minus 2 c minus t2(1 + β)1113872 1113873pn

X|αα2 minus 4

c2

minus ct2(1 + β)

1113969

T minus 2 c minus t2(1 + β)1113872 1113873pn lt 0

⎧⎪⎪⎨

⎪⎪⎩

(B9)

And T (1 + β)(te + c) minus ke minus ((12) + β)pn lt 0We can find

X|αα1 times X|αα2 minus 4 c minus t2(1 + β)1113872 1113873

middot 4 βpn + ke minus (1 + β)(te + c)( 11138572

1113872

+ p2nt

2(1 + β)1113873lt 0

(B10)

)erefore X|αα1 gt 0 )ere only exist a 1113954α isin (α2 α1)that makes (zgzα) 0 And (zgzα) gt 0 ifα isin (α2 1113954α) (zgzα)lt 0 if α isin (1113954α α1)

(5) )e first derivative of p in t is

zp

zt minus

2α(α minus 1) e(1 + β)t221113872 1113873 + minus pnα minus ke + c minus pn( 1113857β + c( 1113857t minus 2eαc(α minus 1)1113872 1113873c

(1 + β)t2

+ 4αc(α minus 1)1113872 11138732

A3 (1 + β)(c + 2te) minus ke +(1 minus β)pn

B3 minus 4c (1 + β)(c + 2te) minus ke minus βpn( 1113857 + t2(1 + β)pn

⎧⎪⎨

⎪⎩

(B11)

)e discriminant of quadratic function

e(1 + β)t2

2+ minus pnα minus ke + c minus pn( 1113857β + c( 1113857t minus 2eαc(α minus 1)

(B12)

is

Δ (minus β minus 1)c + ke + βpn + pnα( 11138572

+ 4e2(1 + β)t

2αc(α minus 1)

(B13)

We have a threshold

1113954c minus(minus β minus 1)c + ke + βpn + pnα( 1113857

2

4e2(1 + β)t

2α(α minus 1) (B14)

Review we have(α + β)pn + ke minus (1 + β)(te + c)gt 0therefore we can find that

tm 2

minus (1 + β)αc(α minus 1)

1113968

1 + β (B15)

Substituting it into above quadratic function

2 c

radicminus pnα minus ke + c minus pn( 1113857β + c( 1113857

minus (1 + β)α(α minus 1)

1113968minus 4eαc(α minus 1)(1 + β)

1 + β (B16)

And we have a same threshold

1113954c minus(minus β minus 1)c + ke + βpn + pnα( 1113857

2

4e2(1 + β)t

2α(α minus 1) (B17)

It could be found that(zpzt) gt 0 because Δlt 0 whenclt 1113954c And when cgt 1113954c we have another threshold 1113954t andzpzt|

t1113954t 0 because Δgt 0 and zpzt|tmlt 0)erefore when

cgt 1113954c and tgt1113954t p is decreasing in t And when cgt 1113954c and tlt1113954tp is increasing in t

12 Mathematical Problems in Engineering

C Proof of Proposition 2

(1) Let R t(e minus g)q denote total carbon tax and thefirst derivative of R in t is

zR

zt2c A2t

2+ B2t + c1113872 1113873 A3t

2+ B3t + c1113872 1113873

4cα(1 minus α) minus t2(1 + β)1113872 1113873

3 (C1)

Among them

A2 (1 + β)e

2gt 0

B2 minus (α + β)pn + ke minus (1 + β)c( 1113857lt 0

C2 2ceα(1 minus α)gt 0

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

A3 (α + β)pn + ke minus (1 + β)c( 1113857(1 + β)gt 0

B3 minus 8ceα(1 minus α)(1 + β)lt 0

C3 4cα(1 minus α) (α + β)pn + ke minus (1 + β)c( 1113857gt 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(C2)

Let Φ A2t2 + B2t + C2 and Ψ A3t

2 + B3t + C3 andwe can find that zRzt and Φ times Ψ have a same sign Bysolving Δ1 and Δ2 a threshold is

1113954c3 (α + β)pn + ke minus (1 + β)c( 1113857

2

4e2α(1 minus α)(1 + β)

(C3)

)ere are two positive real numbers of solutions forΦ ifclt 1113954c3 and let 0lt ta lt tb denote it Similarly let0lt tc lt td denote the solutions for Ψ if cgt 1113954c3 Reviewwe have 4αc(1 minus α) minus (1 + β)t2 gt 0 therefore lettm

4cα(1 minus a)1 + β

1113968 Substituting tm into Φ and Ψ

Φ|tm minus 2

cα(1 minus α)

1 + β

1113971

(α + β)pn + ke minus (1 + β)c( 1113857 + 4ecα(1 minus α)

Ψ|tm minus 8cα(1 minus α) 2(1 + β)e

α(1 minus α)

1 + βc

1113971

minus (α + β)pn + ke minus (1 + β)c( 1113857⎛⎝ ⎞⎠

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(C4)

If clt 1113954c3 Φ|tmlt 0 and Ψ|tm

gt 0 then zRzt|tmlt 0 If

cgt 1113954c3 Φ|tmgt 0 and Ψ|tm

lt 0 then zRzt|tmlt 0 )ere-

fore we have tm isin (ta tb) or tm isin (tc td) FurthermoreR t(e minus g)q is maximized when t ta if clt 1113954c3 R

t(e minus g)q is maximized when t tc if cgt 1113954c3 Totally wehave

tlowast

ta clt 1113954c3

tc cgt 1113954c31113896 (C5)

(2) Review Proposition 1 If cgt 1113954c1 g is maximized whentlowast 1113954t1 If clt 1113954c1 g is maximized when tlowast tmWhen tgt1113954t1 1113954c1 gt 1113954c5 And 1113954c1 lt 0lt 1113954c5 if tlt1113954t1Substituting 1113954t1 into Φ

Φ|1113954t1 minus

3 (α + β)pn + ke minus (1 + β)c( 11138572

8e(1 + β)+ 2ecα(1 minus α)

(C6)

A threshold is 1113954c4 (34)1113954c3 )erefore we have (i)when 0lt clt 1113954c4 tlowast ta because of Φ|1113954t1

lt 0 and 1113954t1 gt ta

(ii) when 1113954c4 lt clt 1113954c3 tlowast ta because of 1113954t1 lt ta (iii) tlowast

tc when 1113954c3 lt clt 1113954c1 and (iv) tlowast tc when 1113954c1 lt c )eresult is obtained by solving Φ and Ψ

D Proof of Theorem 2

Similar to Appendix A )e retailer problem is addressedfirst and we have

p αηpn + βηpn + ηkg + βw + w

2η(β + 1) (D1)

)en substituting p into the manufacturerrsquos profitfunction Solving the optimal w and g simultaneously yields

w αηpn + βet + βηpn minus βgt + eηk + cβ + et minus gt + c( 1113857η

(1 + η)(β + 1)

g minusαηpn + βηpn + eηk minus βw minus w( 1113857t

2(α minus 1)cαη

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(D2)

Mathematical Problems in Engineering 13

By solving the equation above we have

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

g t minus αpn + et(β + 1) + c minus pn( 1113857β minus ke + c( 1113857

t2(1 + β) minus (1 + η)cα(1 minus α)

p (2η + 1)cpnα

3+ A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1 et + ηpn + c( 1113857β + et + ke minus pn( 1113857η + c

A2 (2η + 1)pn + et + c( 1113857β +(minus 2η minus 1)pn + et +(2ηk + k)e + c

⎧⎪⎪⎨

⎪⎪⎩

B1 minus 2et minus 2ηpn minus 2c( 1113857β minus 2eηk minus 2et minus 2c( 1113857c + t2pn(β + 1)

B2 (minus 2η minus 1)pn minus et minus c( 1113857β minus et +(minus 2ηk minus k)e minus c( 1113857c + t2pn(β + 1)

⎧⎪⎪⎨

⎪⎪⎩

(D3)

Note that we have minus αpn + et(β + 1) + (c minus pn)β minus ke +

clt 0 )e concavity condition is (1 + β)(t2(1 + β) minus (1+

η)cα(1 minus α))lt 0

Substituting solutions into the profit function we have

πm minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2c

2 t2β + t

2+ 4(minus 1 + α)cα1113872 1113873(β + 1)

πr minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2(minus 1 + α)αc

2

4α2c minus 4αc +(β + 1)t2

1113872 11138732(β + 1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(D4)

E Proof of Proposition 3

πM (α + β)pn + ke minus (1 + β)c( 1113857

2c

2(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCM

(α + β)pn + ke minus (1 + β)c( 11138572c

2(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(E1)

By comparing the profits of two partners change after therevenue-sharing contract introduced we found that themanufacturer always benefits from cooperation

πR c2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCR

ηc2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(E2)

A threshold is

η1 2cα(1 minus α) minus (1 + β)t2

2cα(1 minus α)1113888 1113889

2

(E3)

When η is in the region δ (η1 1) the retailer willaccept cooperation And the first derivative of η1 in t is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t(β + 1)

α2c2(minus 1 + α)

2 lt 0 (E4)

)erefore η1 is decreasing in t )at means the regionδ (η1 1) is expanding when tax rises And the revenue-sharing contract is accepted easily than before

F Proof of Proposition 4

We consider a simple function (λrλm) to decide thepractical η where λm isin (05 1) and λr 1 minus λm Given the

14 Mathematical Problems in Engineering

carbon tax tlowast we have a region δ (η(tlowast) 1) of cooperationIt is clear that the gap of strength between two partitions inthe supply chain is a crucial factor in cooperation Bycomparing practice η decided by strength and η(tlowast) de-cided by the carbon tax we have

1λm

minus 1 2α2c minus 2αc + t

2(β + 1)1113872 1113873

2

4α2c2(minus 1 + α)

2 (F1)

A threshold is

1113954λ1 4α2c2

(minus 1 + α)2

8α4c2minus 16α3c2

+ 4 2c + t2(β + 1)1113872 1113873cα2 minus 4t

2c(β + 1)α + t

4(β + 1)

2 (F2)

When 05lt λm lt 1113954λ1 the practical η is in the regionδ (η(tlowast) 1) therefore the cooperation is achieved facingthe optimal carbon tax

However the practical η is out of the range δ (η(tlowast) 1)

when 1113954λ1 lt λm Review previous result the region δ (η1 1)

is expanding when tax rises )erefore there exists a pos-sibility that the government increasing carbon tax to co-ordinate the supply chain We assume a realistic thing thegovernment is willing to boost the greenness of the producteven reducing part of the revenue from the carbon tax

Based on it the carbon up to 1113954t1 which means thegreenness is maximized )erefore )e cooperation isachieved when clt 1113954c1 because the greenness is decreasing inthe carbon tax when 1113954c1 lt c )en we have the secondthreshold

1113954λ2 1

η1 1113954t1( 1113857 + 1 (F3)

)e cooperation could not be achieved when λm gt 1113954λ2)e conclusion is as follows (1) Coordination would be

achieved when the manufacturer is democratic (ie05lt λm lt 1113954λ1) (2) Coordination could be achieved but partof the total tax will lose if development cost is lower when themanufacturer is moderate (ie 1113954λ1 lt λm lt 1113954λ2) (3) )e co-operation could not be achieved if the manufacturer is adictator (ie λm gt 1113954λ2)

G Proof of Proposition 5

)e first derivative of η1 in β is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t

2

2α2c2(minus 1 + α)

2 lt 0 (G1)

Similarly η1 is decreasing in β )at means the regionδ (η1 1) is expanding when β rises

And it is apparent that η1 (2cα(1 minus α) minus (1 + β)t22cα(1 minus α))2 is independent with k

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is research was supported by the National Natural ScienceFoundation of China (no 71761015) and the Science andTechnology Research Project of Jiangxi Education Depart-ment of China (no GJJ150475)

References

[1] Y Li M K Lim J Hu and M-L Tseng ldquoInvestigating theeffect of carbon tax and carbon quota policy to achieve lowcarbon logistics operationsrdquo Resources Conservation andRecycling vol 154 p 104535 2020

[2] M Abedi-Varaki ldquoStudy of carbon dioxide gas treatmentbased on equations of kinetics in plasma discharge reactorrdquoModern Physics Letters B vol 31 no 22 p 1750210 2017

[3] S Meng M Siriwardana and J McNeill ldquo)e environmentaland economic impact of the carbon tax in Australiardquo Envi-ronmental and Resource Economics vol 54 no 3 pp 313ndash332 2013

[4] Y Chen and C L Tseng ldquoInducing clean technology in theelectricity sector tradable permits or carbon tax policiesrdquo-eEnergy Journal vol 32 no 3 2011

[5] H-B Duan L Zhu and Y Fan ldquoOptimal carbon taxes incarbon-constrained China a logistic-induced energy eco-nomic hybrid modelrdquo Energy vol 69 pp 345ndash356 2014

[6] C Lu Q Tong and X Liu ldquo)e impacts of carbon tax andcomplementary policies on Chinese economyrdquo Energy Policyvol 38 no 11 pp 7278ndash7285 2010

[7] K Cao B Xu and J Wang ldquoOptimal trade-in and warrantyperiod strategies for new and remanufactured products undercarbon tax policyrdquo International Journal of Production Re-search vol 58 no 1 pp 180ndash199 2020

[8] P He W Zhang X Xu and Y Bian ldquoProduction lot-sizingand carbon emissions under cap-and-trade and carbon taxregulationsrdquo Journal of Cleaner Production vol 103pp 241ndash248 2015

[9] X Xu X Xu and P He ldquoJoint production and pricing de-cisions for multiple products with cap-and-trade and carbontax regulationsrdquo Journal of Cleaner Production vol 112pp 4093ndash4106 2016

Mathematical Problems in Engineering 15

[10] Q Li X Guan T Shi and W Jiao ldquoGreen product designwith competition and fairness concerns in the circulareconomy erardquo International Journal of Production Researchvol 58 no 1 pp 165ndash179 2020

[11] A Ranjan and J K Jha ldquoPricing and coordination strategiesof a dual-channel supply chain considering green quality andsales effortrdquo Journal of Cleaner Production vol 218pp 409ndash424 2019

[12] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineeringvol 10 no 1 pp 99ndash116 2012

[13] X Yin X Chen X Xu and L Zhang ldquoTax or subsidyOptimal carbon emission policy a supply chain perspectiverdquoSustainability vol 12 no 4 p 1548 2020

[14] B B Schlegelmilch G M Bohlen and A Diamantopoulosldquo)e link between green purchasing decisions and measuresof environmental consciousnessrdquo European Journal of Mar-keting vol 30 no 5 pp 35ndash55 1996

[15] M S Hopkins ldquoWhat the rsquogreenrsquo consumer wantsrdquo MITSloan Management Review vol 50 no 4 pp 87ndash89 2009

[16] G Kalyanaram and R S Winer ldquoEmpirical generalizationsfrom reference price researchrdquo Marketing Science vol 14no 3 pp G161ndashG169 1995

[17] E A Greenleaf ldquo)e impact of reference price effects on theprofitability of price promotionsrdquo Marketing Science vol 14no 1 pp 82ndash104 1995

[18] N M Modak D K Ghosh S Panda and S S SanaldquoManaging green house gas emission cost and pricing policiesin a two-echelon supply chainrdquo CIRP Journal ofManufacturing Science and Technology vol 20 pp 1ndash11 2018

[19] P Sinclair ldquoHigh does nothing and rising is worse carbontaxes should keep declining to cut harmful emissionsrdquo -eManchester School vol 60 no 1 pp 41ndash52 1992

[20] G M Grossman and A B Krueger ldquoEconomic growth andthe environmentrdquo -e Quarterly Journal of Economicsvol 110 no 2 pp 353ndash377 1995

[21] N M Modak and P Kelle ldquoUsing social work donation as atool of corporate social responsibility in a closed-loop supplychain considering carbon emissions tax and demand uncer-taintyrdquo Journal of the Operational Research Society pp 1ndash172019

[22] A Ulph and D Ulph ldquo)e optimal time path of a carbon taxrdquoOxford Economic Papers vol 46 no S1 pp 857ndash868 1994

[23] M Hoel and S Kverndokk ldquoDepletion of fossil fuels and theimpact of global warmingrdquo Resource and Energy Economicsvol 18 no 2 pp 115ndash136 1996

[24] Y H Farzin and O Tahvonen ldquoGlobal carbon cycle and theoptimal time path of a carbon taxrdquo Oxford Economic Papersvol 48 no 4 pp 515ndash536 1996

[25] V Bosetti C Carraro R Duval andM Tavoni ldquoWhat shouldwe expect from innovation A model-based assessment of theenvironmental and mitigation cost implications of climate-related RampDrdquo Energy Economics vol 33 no 6 pp 1313ndash13202011

[26] M Hariga R Asrsquoad and A Shamayleh ldquoIntegrated economicand environmental models for a multi stage cold supply chainunder carbon tax regulationrdquo Journal of Cleaner Productionvol 166 pp 1357ndash1371 2017

[27] N Turken J Carrillo and V Verter ldquoFacility location andcapacity acquisition under carbon tax and emissions limits tocentralize or to decentralizerdquo International Journal of Pro-duction Economics vol 187 pp 126ndash141 2017

[28] W Yu and R Han ldquoCoordinating a two-echelon supply chainunder carbon taxrdquo Sustainability vol 9 no 12 p 2360 2017

[29] S Sinha and N M Modak ldquoAn EPQmodel in the perspectiveof carbon emission reductionrdquo International Journal ofMathematics in Operational Research vol 14 no 3pp 338ndash358 2019

[30] H Zhang P Li H Zheng and Y Zhang ldquoImpact of carbontax on enterprise operation and production strategy for low-carbon products in a co-opetition supply chainrdquo Journal ofCleaner Production Article ID 125058 2020

[31] X Chen H Yang X Wang et al ldquoOptimal carbon tax designfor achieving low carbon supply chainsrdquo Annals of OperationsResearch pp 1ndash28 2020

[32] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Handbook of the Fundamentals of Fi-nancial Decision Making Part I pp 99ndash127 )e EconometricSociety Cleveland OH USA 2013

[33] W Kim and M Kim ldquoReference quality-based competitivemarket structure for innovation driven marketsrdquo Interna-tional Journal of Research in Marketing vol 32 no 3pp 284ndash296 2015

[34] I Popescu and Y Wu ldquoDynamic pricing strategies withreference effectsrdquo Operations Research vol 55 no 3pp 413ndash429 2007

[35] P K Kopalle P K Kannan L B Boldt and N Arora ldquo)eimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Re-tailing vol 88 no 1 pp 102ndash114 2012

[36] T-P Hsieh and C-Y Dye ldquoOptimal dynamic pricing fordeteriorating items with reference price effects when inven-tories stimulate demandrdquo European Journal of OperationalResearch vol 262 no 1 pp 136ndash150 2017

[37] X Chen Z-Y Hu and Y-H Zhang ldquoDynamic pricing withstochastic reference price effectrdquo Journal of the OperationsResearch Society of China vol 7 no 1 pp 107ndash125 2019

[38] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquo Opera-tions Research vol 64 no 6 pp 1529ndash1536 2016

[39] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[40] C Chen ldquoDesign for the environment a quality-based modelfor green product developmentrdquo Management Sciencevol 47 no 2 pp 250ndash263 2001

[41] K Chitra ldquoIn search of the green consumers a perceptualstudyrdquo Journal of Services Research vol 7 no 1 2007

[42] Z Liu T D Anderson and J M Cruz ldquoConsumer envi-ronmental awareness and competition in two-stage supplychainsrdquo European Journal of Operational Research vol 218no 3 pp 602ndash613 2012

[43] B Li Y Wang and Z Wang ldquoManaging a closed-loop supplychain with take-back legislation and consumer preference forgreen designrdquo Journal of Cleaner Production Article ID124481 2020

[44] J Rantanen D B Grant and W Piotrowicz ldquoInvestigatingsupply chain cooperation in Finnish grocery retailrdquo ResearchJournal of the University of Gdansk Transport Economics andLogistics vol 71 pp 19ndash34 2017

[45] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[46] G Xu B Dan X Zhang and C Liu ldquoCoordinating a dual-channel supply chain with risk-averse under a two-way

16 Mathematical Problems in Engineering

revenue sharing contractrdquo International Journal of ProductionEconomics vol 147 pp 171ndash179 2014

[47] Z Shi N Wang T Jia et al ldquoReverse revenue sharingcontract versus two-part tariff contract under a closed-loopsupply chain systemrdquoMathematical Problems in Engineeringvol 2016 Article ID 5464570 15 pages 2016

[48] S Panda N M Modak and L E Cardenas-Barron ldquoCo-ordinating a socially responsible closed-loop supply chainwith product recyclingrdquo International Journal of ProductionEconomics vol 188 pp 11ndash21 2017

[49] N M Modak N Modak S Panda and S S Sana ldquoAnalyzingstructure of two-echelon closed-loop supply chain for pricingquality and recycling managementrdquo Journal of Cleaner Pro-duction vol 171 pp 512ndash528 2018

[50] Q P Wang and D Z Zhao ldquoRevenue-sharing contract ofsupply chain based on consumerrsquos preference for low carbonproductsrdquo Chinese Journal of Management Science vol 22no 9 pp 106ndash113 2014

[51] B Yu J Wang X Lu and H Yang ldquoCollaboration in a low-carbon supply chain with reference emission and cost learningeffects cost sharing versus revenue sharing strategiesrdquo Journalof Cleaner Production vol 250 Article ID 119460 2020

[52] J Cao X Zhang and G Zhou ldquoSupply chain coordinationwith revenue-sharing contracts considering carbon emissionsand governmental policy makingrdquo Environmental Progress ampSustainable Energy vol 35 no 2 pp 479ndash488 2016

[53] X Xu P He H Xu and Q Zhang ldquoSupply chain coordi-nation with green technology under cap-and-trade regula-tionrdquo International Journal of Production Economics vol 183pp 433ndash442 2017

[54] N M Modak S Sinha S Panda et al ldquoAnalyzing a sociallyresponsible closed-loop distribution channel with recyclingfacilityrdquo SN Applied Sciences vol 1 no 10 p 1189 2019

[55] D Feng L Ma Y Ding G Wu and Y Zhang ldquoDecisions ofthe dual-channel supply chain under double policy consid-ering remanufacturingrdquo International Journal of Environ-mental Research and Public Health vol 16 no 3 p 465 2019

[56] I Bellos M Ferguson and L B Toktay ldquo)e car sharingeconomy interaction of business model choice and productline designrdquo Manufacturing amp Service Operations Manage-ment vol 19 no 2 pp 185ndash201 2017

[57] G Ferrer and J M Swaminathan ldquoManaging new andremanufactured productsrdquo Management Science vol 52no 1 pp 15ndash26 2006

[58] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2pp 519ndash527 2011

[59] L Cui S Guo and H Zhang ldquoCoordinating a green agri-foodsupply chain with revenue-sharing contracts consideringretailersrsquo green marketing effortsrdquo Sustainability vol 12no 4 p 1289 2020

[60] J Heydari K Govindan and A Aslani ldquoPricing and greeningdecisions in a three-tier dual channel supply chainrdquo Inter-national Journal of Production Economics vol 217 pp 185ndash196 2019

Mathematical Problems in Engineering 17

(2) -ere exist some thresholds 1113954t2 and 1113954c2 for which g

decreases with increasing β when cgt 1113954c2 and tgt1113954t2 elseg increases with increasing β

(3) g increases with increasing k decreases with in-creasing c and initially increases and then decreaseswith increasing α

We can conclude that the carbon tax development costprice recognition level greenness sensitivity and functionalattribute coefficient all influence the greenness in Proposi-tion 1 When the carbon tax is low the manufacturer incursless cost to improve the greenness of the products As aresult the greenness increases with the carbon tax In ad-dition if the carbon tax is high and the development cost islow the greenness of the products will increase with in-creasing carbon tax because the tax is the crucial factor indecisions In contrast if the carbon tax is high and devel-opment costs are also high the greenness of products willdecrease because the manufacturer will choose to reducedevelopment costs )erefore pollution will not be reducedas the carbon tax increases in some cases Facing high de-velopment costs the manufacturer will not improvegreenness unless it is promoted by a lower carbon tax Anumerical simulation of this scenario is shown in Figure 1

Similarly the results show that greenness is affected bythe consumersrsquo price reference behavior When the carbontax is low the negative effect of price increases when con-sumers have higher concerns about price )erefore themanufacturer will offset this negative effect by improving thegreenness of products When the carbon tax is high and thedevelopment cost is low the greenness of products willincrease under a high consumer concern about price Incontrast if the carbon tax is high and the development cost ishigh the greenness will decrease because the manufacturerwill choose to reduce development costs Numerical sim-ulations of these scenarios are shown in Figure 2

In addition greenness sensitivity and the functional attri-bute coefficient also affect greenness )e greenness increaseswith increasing greenness sensitivity of consumers Highgreenness sensitivity can promote more green products )ismeans that the government can promote green production by

increasing the green consciousness of consumersMoreover thegreenness decreases with increasing development cost A higherdegree of greenness is obtained when manufacturers face lowerdevelopment costs In addition greenness first increases andthen decreases with the increase in the functional attributecoefficient When the functional attribute of the green productis low themanufacturer will produce a lower greenness productwith increases in the functional attribute coefficient In contrastthe manufacturer will produce a product with higher greennesswith an increase in the functional attribute after a functionalthreshold has been reached Numerical simulations of thesescenarios are shown in Figure 3

Overall consumersrsquo reference behavior plays a positive rolein promoting green production However the carbon tax is noteffective for improving green products in some cases Forexample if the carbon tax is high and the development cost isalso high the greenness of products will decrease)erefore wenext investigate the optimal carbon tax

First an optimal carbon tax is defined as that whichprovides the maximum greenness without losing total taxrevenue )e result is provided in Proposition 2 and theproofs can be found in Appendix C

Table 2 Notations

Parameters Definitionp pn Prices of the green product and traditional product respectivelyq qn Demand for the green product and traditional product respectivelyw Wholesale priceg Product greennesst Carbon tax ratec Cost rate of technology developmentc Unit product cost of the green producte Initial unit carbon emissionsα Functional attribute coefficient of the green product functionβ Consumersrsquo recognition level of the reference pricek Sensitivity of consumers to product greennessV Utility obtained by a consumer from the traditional productη Proportion of the total revenue obtained by the retailerλ Member strength in the supply chain

10

08

06g

04

02

0000 01 02 03

t

t = t1

04 05

010

008

006

004

002

000

γ lt γ1γ gt γ1

Figure 1 Effects of carbon taxes on greenness

6 Mathematical Problems in Engineering

Proposition 2 -e optimal carbon tax is defined as follows

tlowast

M minusM

2minus N

1113968

e(1 + β) 0lt clt 1113954c3

P minus

P2

minus Q

1113969

2(1 + β)M 1113954c3 lt clt 1113954c1

1113954t1 1113954c1 lt c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

)e optimal carbon tax is obtained in Proposition 2based on the manufacturerrsquos development cost to maximizegreenness without losing the total carbon tax revenue Wecan observe that the crucial factor is the development costAn appropriate carbon tax policy should be chosen whenfacing different production technologies to ensure greennessand total tax revenue In addition it is easily observed thatthe maximum greenness is obtained when the developmentcost is high whereas the maximum tax is obtained when thedevelopment cost is lower but some of the greenness is lostA numerical simulation of this scenario is shown in Figure 4

33 Supply Chain Coordination It is also important forsupply chains to establish green supply chain models byintegrating internal and external resources to make deci-sions which enables supply chains to achieve better per-formance by improving cooperation [59 60] However moststudies on coordination have mainly focused reducingemissions and improving profit Few studies have investi-gated the relationship between carbon taxes and supplychain cooperation

In this section we study the impact of carbon taxes andconsumerrsquos behavior on supply chain coordination wherethe manufacturer is the leader and the retailer is the fol-lower A revenue-sharing contract is introduced to achieve

coordination Let η denote the proportion of total revenueobtained by the retailer then the manufacturerrsquos decisionmodel is as follows

Maxwg

πCM ((1 minus η)p + w minus c minus t(e minus g))q minus

cg2

2 (8)

)e retailerrsquos decision problem can be described asfollows

Maxp

πCR (ηp minus w)q (9)

)e optimal solutions for the retailer and manufacturerare derived with the following theorem by substituting thedemand function into the equation above )e proof isfound in Appendix D

Theorem 2 -e greenness is defined as follows

g t minus αpn + et(β + 1) + c minus pn( 1113857β minus ke + c( 1113857

2c(1 + η)α2 minus 2c(1 + η)α + t2(β + 1)

(10)

-en the profits of both parties are the following

πm minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2c

2 t2β + t

2+ 4(minus 1 + α)cα1113872 1113873(β + 1)

πr minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2(minus 1 + α)αc

2

4α2c minus 4αc +(β + 1)t2

1113872 11138732(β + 1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

We investigate the impact of the carbon tax on pro-moting the coordination of the supply chain by comparingthe profit changes of the two partners after the introductionof the revenue-sharing contract in Proposition 3 )e proofsare provided in Appendix E

Proposition 3 Coordination can be promoted by increasingthe carbon tax

01204

01202

01200g

01198

01196

00 01 02 03β

04 05

11900 times 10ndash5

11895 times 10ndash5

11898 times 10ndash5

11892 times 10ndash5

11890 times 10ndash5

11888 times 10ndash5

11885 times 10ndash5

11882 times 10ndash5

11880 times 10ndash5

γ lt γ2 t lt t2γ gt γ2 t lt t2

(a)

096

095

094

093g

092

091

090

08900 01 02 03

β04 05

0035

0040

0045

0050

0055

0060

γ lt γ2 t gt t2γ gt γ2 t gt t2

(b)

Figure 2 Effects of consumersrsquo price reference behavior on greenness (a) Low-carbon tax rate (b) High carbon tax rate

Mathematical Problems in Engineering 7

When the profits of both the retailer and the manu-facturer improve simultaneously coordination is achievedBy comparing equations (11) and (6) we find that therevenue-sharing contract improves the retailerrsquos profit onlywhen η isin (η1 1) whereas the manufacturer always benefitsfrom the contract )e range of η depends on the carbon taxrate When η is in the given range the retailer and man-ufacturer achieve Pareto improvement Furthermore therange expands as the carbon tax rate increases)erefore thegovernment can promote supply chain collaboration byincreasing the carbon tax Based on the previous results forthe optimal carbon tax we next investigate how to achievecoordination when the supply chain faces the optimalcarbon tax Figure 5 shows the Pareto improvement region

According to the assumptions described above themanufacturer acts as the leader and the retailer acts as afollower It is reasonable that the proportion of profitreceived by the retailer is decided by the manufacturer)erefore it is necessary to investigate whether theproportion decided by the leader is in the region in whichcoordination is achieved A parameter λ is introduced todenote the strength of a partner in the supply chain Let

λm isin (05 1) and λr 1 minus λm represent the power of themanufacturer and the retailer respectively )en weconsider a simple function (λrλm) to determine η whichis the proportion of revenue received by the retailer )eretailer retains the maximum revenue when the power ofthe two parties is equal (ie λm λr 05) whereas all ofthe benefits go to the leader when the power of themanufacturer is overwhelming compared with that of theretailer (ie λm 1) Similar to Proposition 2 the optimalcarbon tax is obtained under the revenue-sharing contractas follows

tlowast

M minus

M2

minus (1 + η2)N

1113969

e(1 + β) 0lt clt 1113954c

C2

(1 + η2)P minus

(1 + η2)2P2

minus (1 + η2)Q

1113969

2(1 + β)M 1113954c

C2 lt clt 1113954c1

1113954t1 1113954c1 lt c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

05

04

03g

02

01

0010 15 20 25

k3530 40

g

(a)

10

08

06g

04

02

0010 09 0708 06

γ030405

g

(b)

029

027

028

025

026

g

024

022

023

021035030 040 045 050

α060 065055 070

g

(c)

Figure 3 Effects of parameters on greenness (a) Greenness sensitivity (b) Development cost (c) Functional attribute coefficient

8 Mathematical Problems in Engineering

Proposition 4 is designed to answer the question ofwhether cooperation will be achieved the proofs are foundin Appendix F

Proposition 4 Coordination will be achieved if the power ofthe manufacturer is low (ie 05lt λm lt 1113954λ1) Moreover whenthe power of the manufacturer is at a medium level (ie1113954λ1 lt λm lt 1113954λ2) coordination can be achieved but part of thetotal tax revenue will be lost if the development cost is low(ie clt 1113954c1) In other cases (ie cgt 1113954c1 or 1113954λlt λm) coordi-nation cannot be achieved because the power of the manu-facturer is overwhelming

Coordination situations for different manufacturerpower levels are present in Table 3 )e achievement ofcoordination is decided by the power of the manufacturerand the optimal carbon tax faced by the supply chain Basedon Proposition 3 coordination can be promoted if thegovernment adjusts the carbon tax On the other handcoordination can easily be achieved under the optimalcarbon tax if the power of the manufacturer is low Inaddition there are two possibilities when the power of themanufacturer is at a medium level If the development cost istoo high adjusting the tax will lead to an unknown result thegreenness and total tax revenue may both decline at the same

10 014

012

010

008

006

004

002

000

08

06

Gre

enne

ss

Tota

l tax

04

02

00005000 010 015 020

t

γ lt γ3

t = ta

025 030

GreennessTotal tax

(a)

10 030

025

020

015

010

005

000

08

06

Gre

enne

ss

Tota

l tax

04

02

000100 02 03 04

t

γ3 lt γ lt γ1

t = tc

05 06

GreennessTotal tax

(b)

005 040

035

025

020

015

010

005

000

004

003

Gre

enne

ss

Tota

l tax

002

001

0000100 02 03 04

t

γ1 lt γ

t = t1

06 08

030

05 07

GreennessTotal tax

(c)

Figure 4 Determination of the optimal carbon tax (a) 0lt clt 1113954c3 (b) 1113954c3 lt clt 1113954c1 (c) 1113954c1 lt c

005t

η

010 015 020 025 030

Pareto improvement

035 040

Figure 5 Pareto improvement region with increasing carbon tax

Mathematical Problems in Engineering 9

time If the development cost is lower the greenness may beimproved with a loss of revenue by increasing the carbon taxHowever when the power of the manufacturer is strong theregion δ (η1 1) cannot be achieved

According to our theory the government can promotethe coordination between the manufacturer and the retailerby adjusting the carbon tax When the power of the man-ufacturer increases the government still achieves coordi-nation by raising the carbon tax In general cooperation isachieved and the government plays a key role in promotingcooperation

)e effects of consumersrsquo behavior on supply chaincoordination are also investigated in the next proposition)e proofs are provided in Appendix G

Proposition 5 Price reference behavior can promote supplychain coordination but green preference cannot

From the previous results price reference has a compleximpact on the supply chain First the increasing sensitivityof customers to price will promote supply chain coordi-nation because this negative impact can be mitigated bycooperation In addition price reference behavior also af-fects the green decisions of the supply chain in a complexmanner In contrast the mechanism by which consumerrsquogreen preferences influence the supply chain is relativelysimple Greenness increases with increasing greennesspreference and green preference has no effect on the co-ordination of the supply chain

4 Conclusion

)is study examines the interaction between supply chaincooperation and the carbon tax problem in a two-echelonsupply chain under consumerrsquos reference behavior )e opti-mal carbon tax policy is obtained based on analysis of thecarbon tax green investment coefficient and degree of con-sumerrsquos price reference )e optimal carbon tax is defined asthe simultaneous optimization of total tax revenue and productgreenness In addition coordination is achieved by introducingrevenue-sharing contracts)e impact of consumersrsquo referencebehavior and the carbon tax on supply chain coordination isalso investigated )e results are as follows

(1) )e greenness increases with increasing carbon taxwhen the carbon tax is low In addition if the carbontax is higher and the development cost is low thegreenness of products will increase with increasingcarbon tax Conversely if the carbon tax is higherand the development cost is also high the greennessof products will decrease

(2) When the carbon tax is low the negative effect ofprice increases as consumer concerns about priceincrease )erefore the manufacturer offsets thenegative effect by improving the greenness Whenthe carbon tax is higher and development cost is lowthe greenness of products will increase with highconsumerrsquos concern about price In contrast if thecarbon tax is higher and the development cost ishigh the greenness will decrease

(3) )e greenness increases with an increasing greennesspreference of consumers )eir preference behaviorcan promote more green products )is means thatthe government can promote green production bypromoting the green consciousness of consumersMoreover the greenness decreases with increasingdevelopment cost A higher greenness is obtainedwhen manufacturers face lower development costsIn addition greenness first increases and then de-creases with increases in the functional attributecoefficient When the functional attribute of thegreen product is low the manufacturer will producea lower greenness product with an increase in thefunctional attribute In contrast the manufacturerwill produce a higher greenness product with anincrease in the functional attribute after a func-tionality threshold

(4) We investigated the impact of the carbon tax onpromoting the coordination of the supply chain bycomparing the profit changes of two partners afterthe introduction of a revenue-sharing contract inProposition 3 We found that coordination could bepromoted by increasing the carbon tax

(5) )e achievement of coordination depends on thetype of manufacturer If the power of the manu-facturer is low coordination can be achievedunder the optimal carbon tax If the power of themanufacturer is at a medium level coordinationcan be achieved by increasing the carbon tax andimproved greenness will be realized with a loss ofrevenue However when the power of the man-ufacturer is strong coordination cannot beachieved

In this study we focused on the interaction between acarbon tax and supply chain cooperation However othercarbon policies (eg a cap-and-trade policy) and supplychain structures are worth exploring For example it wouldbe interesting to investigate the effects on a supply chainstructure consisting of the retailer as the leader under acomplex carbon policy

Table 3 Coordination situations for different manufacturer power levels

Power of the manufacturer Coordination Greenness Total carbon tax revenue05lt λm lt 1113954λ1 Achieve Improve Improve1113954λ1 lt λm lt 1113954λ2 clt 1113954c1 Achieve Improve Decline1113954λ1 lt λm lt 1113954λ2 clt 1113954c1 Cannot achievecgt 1113954c1 Cannot achieve

10 Mathematical Problems in Engineering

Appendix

A Proof of Theorem 1

)e demand function can be written as

q P ug ge un ug ge 01113966 1113967

qn P un gt ug un ge 01113966 1113967

⎧⎪⎨

⎪⎩(A1)

After replacing un and ug we have

q P αv minus p + kg minus β p minus pn( 1113857 minus v minus pn( 1113857ge 0 αv minus p + kg minus β p minus pn( 1113857ge 01113864 1113865

qn P v minus pn minus αv minus p + kg minus β p minus pn( 1113857( 1113857gt 0 v minus pn ge 01113864 11138651113896 (A2)

Given that V is uniformly distributed on [0 1] thedemand function is

q (α + β)pn minus (1 + β)p + kg

α(1 minus α)

qn 1 minus(1 + β) pn minus p( 1113857 + kg

1 minus α

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A3)

)e condition kgt ((1 + β)p minus (α + β)png) is necessaryto ensure nonnegativity )e retailer problem is addressedfirst and we have

p (α + β)pn +(1 + β)w + kg

2(1 + β) (A4)

)en substitute p into the manufacturerrsquos profit func-tion Solving the optimal w and g simultaneously yields

w (α + β)pn +(1 + β)(c + t(e minus g)) + kg

2(1 + β)

g t (α + β)pn minus (1 + β)w + ke( 1113857

2cα(1 minus α)

(A5)

)en we have

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

g (1 + β)(te + c) minus (α + β)pn minus ke( 1113857t

t2(1 + β) minus 4cα(1 minus α)

p 3cpnα

3+ 2A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1 (1 + β)(c + te) + ke minus (1 minus β)pn

A2 (1 + β)(c + te) + 3ke minus (3 minus β)pn

⎧⎪⎨

⎪⎩

B1 minus 2c (1 + β)(c + te) + ke + βpn( 1113857 + t2(1 + β)pn

B2 minus c (1 + β)(c + te) + 3ke + 3βpn( 1113857 + t2(1 + β)pn

⎧⎪⎨

⎪⎩

(A6)

Notes that we have (α + β)pn + ke minus (1 + β)(te + c)gt 0)e concavity condition is 4αc(1 minus α) minus (1 + β)t2 gt 0

B Proof of Proposition 1

(1) )e first derivative of g in t is

zg

zt

minus 4cα3pn + 4cA3α2

+ B3α minus t2(1 + β) (1 + β)c minus ke minus βpn( 1113857

t2(1 minus β) minus 4cα(1 minus α)1113872 1113873

2

A3 (1 + β)(c + 2te) minus ke +(1 minus β)pn

B3 minus 4c (1 + β)(c + 2te) minus ke minus βpn( 1113857 + t2(1 + β)pn

⎧⎨

⎩

(B1)

We have a threshold

1113954c1 minust2(1 + β) (α + β)pn + ke minus (1 + β)c( 1113857

4α(1 minus α) (α + β)pn + ke minus (1 + β)(c + 2te)( 1113857

1113954t1 (α + β)pn + ke minus (1 + β)c

2(1 + β)e

(B2)

When cgt 1113954c1 and tgt1113954t1 g is decreasing in t Bycontrast g is increasing in t in other cases

(2) )e first derivative of g in β is

zg

zβ

minus 4α(1 minus α) te + c minus pn( 1113857c + t2

ke + αpn minus pn( 1113857

t2(1 + β) minus 4α(1 minus α)c1113872 1113873

2

(B3)

We have a threshold

1113954c2 minust2

ke minus (1 minus α)pn( 1113857

4α(1 minus α) te + c minus pn( 1113857

1113954t2 pn minus c

e

(B4)

Review we have (α + β)pn + ke minus (1 + β)(te + c)gt 0therefore we can find that

tltpn minus c

e+

ke minus (1 minus α)pn

(1 + β)e (B5)

(i) When ke minus (1 minus α)pn lt 0 then tlt1113954t2 is inevitable)erefore (zgzβ)gt 0 is correct

(ii) When ke minus (1 minus α)pn gt 0 we can find that(zgzβ)lt 0 if cgt 1113954c2 and tgt1113954t2 (zgzβ)lt 0 in othercases

(3) )e first derivative of g in k and c is

Mathematical Problems in Engineering 11

zg

zc minus

4α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

4cα(1 minus α) minus t2(1 + β)1113872 1113873

2 lt 0

zg

zk

te

4cα(1 minus α) minus t2(1 + β)gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(B6)

(4) )e first derivative of g in α is

zg

zα

minus Xt

4α(1 minus α)c minus t2(1 + β)1113872 1113873

2 (B7)

And X minus 4α2cpn + (4(1 + β)(te + c) minus 4ke minus 4pn)

(2α + 1)c + pnt2(1 + β) is a quadratic function )erange of α in 4αc(1 minus α) minus (1 + β)t2 gt 0 is

α1 c +

c2

minus ct2(1 + β)

1113969

2c

α2 c minus

c2

minus ct2(1 + β)

1113969

2c

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(B8)

By substituting those into X we have

X|αα1 4

c2

minus ct2(1 + β)

1113969

T minus 2 c minus t2(1 + β)1113872 1113873pn

X|αα2 minus 4

c2

minus ct2(1 + β)

1113969

T minus 2 c minus t2(1 + β)1113872 1113873pn lt 0

⎧⎪⎪⎨

⎪⎪⎩

(B9)

And T (1 + β)(te + c) minus ke minus ((12) + β)pn lt 0We can find

X|αα1 times X|αα2 minus 4 c minus t2(1 + β)1113872 1113873

middot 4 βpn + ke minus (1 + β)(te + c)( 11138572

1113872

+ p2nt

2(1 + β)1113873lt 0

(B10)

)erefore X|αα1 gt 0 )ere only exist a 1113954α isin (α2 α1)that makes (zgzα) 0 And (zgzα) gt 0 ifα isin (α2 1113954α) (zgzα)lt 0 if α isin (1113954α α1)

(5) )e first derivative of p in t is

zp

zt minus

2α(α minus 1) e(1 + β)t221113872 1113873 + minus pnα minus ke + c minus pn( 1113857β + c( 1113857t minus 2eαc(α minus 1)1113872 1113873c

(1 + β)t2

+ 4αc(α minus 1)1113872 11138732

A3 (1 + β)(c + 2te) minus ke +(1 minus β)pn

B3 minus 4c (1 + β)(c + 2te) minus ke minus βpn( 1113857 + t2(1 + β)pn

⎧⎪⎨

⎪⎩

(B11)

)e discriminant of quadratic function

e(1 + β)t2

2+ minus pnα minus ke + c minus pn( 1113857β + c( 1113857t minus 2eαc(α minus 1)

(B12)

is

Δ (minus β minus 1)c + ke + βpn + pnα( 11138572

+ 4e2(1 + β)t

2αc(α minus 1)

(B13)

We have a threshold

1113954c minus(minus β minus 1)c + ke + βpn + pnα( 1113857

2

4e2(1 + β)t

2α(α minus 1) (B14)

Review we have(α + β)pn + ke minus (1 + β)(te + c)gt 0therefore we can find that

tm 2

minus (1 + β)αc(α minus 1)

1113968

1 + β (B15)

Substituting it into above quadratic function

2 c

radicminus pnα minus ke + c minus pn( 1113857β + c( 1113857

minus (1 + β)α(α minus 1)

1113968minus 4eαc(α minus 1)(1 + β)

1 + β (B16)

And we have a same threshold

1113954c minus(minus β minus 1)c + ke + βpn + pnα( 1113857

2

4e2(1 + β)t

2α(α minus 1) (B17)

It could be found that(zpzt) gt 0 because Δlt 0 whenclt 1113954c And when cgt 1113954c we have another threshold 1113954t andzpzt|

t1113954t 0 because Δgt 0 and zpzt|tmlt 0)erefore when

cgt 1113954c and tgt1113954t p is decreasing in t And when cgt 1113954c and tlt1113954tp is increasing in t

12 Mathematical Problems in Engineering

C Proof of Proposition 2

(1) Let R t(e minus g)q denote total carbon tax and thefirst derivative of R in t is

zR

zt2c A2t

2+ B2t + c1113872 1113873 A3t

2+ B3t + c1113872 1113873

4cα(1 minus α) minus t2(1 + β)1113872 1113873

3 (C1)

Among them

A2 (1 + β)e

2gt 0

B2 minus (α + β)pn + ke minus (1 + β)c( 1113857lt 0

C2 2ceα(1 minus α)gt 0

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

A3 (α + β)pn + ke minus (1 + β)c( 1113857(1 + β)gt 0

B3 minus 8ceα(1 minus α)(1 + β)lt 0

C3 4cα(1 minus α) (α + β)pn + ke minus (1 + β)c( 1113857gt 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(C2)

Let Φ A2t2 + B2t + C2 and Ψ A3t

2 + B3t + C3 andwe can find that zRzt and Φ times Ψ have a same sign Bysolving Δ1 and Δ2 a threshold is

1113954c3 (α + β)pn + ke minus (1 + β)c( 1113857

2

4e2α(1 minus α)(1 + β)

(C3)

)ere are two positive real numbers of solutions forΦ ifclt 1113954c3 and let 0lt ta lt tb denote it Similarly let0lt tc lt td denote the solutions for Ψ if cgt 1113954c3 Reviewwe have 4αc(1 minus α) minus (1 + β)t2 gt 0 therefore lettm

4cα(1 minus a)1 + β

1113968 Substituting tm into Φ and Ψ

Φ|tm minus 2

cα(1 minus α)

1 + β

1113971

(α + β)pn + ke minus (1 + β)c( 1113857 + 4ecα(1 minus α)

Ψ|tm minus 8cα(1 minus α) 2(1 + β)e

α(1 minus α)

1 + βc

1113971

minus (α + β)pn + ke minus (1 + β)c( 1113857⎛⎝ ⎞⎠

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(C4)

If clt 1113954c3 Φ|tmlt 0 and Ψ|tm

gt 0 then zRzt|tmlt 0 If

cgt 1113954c3 Φ|tmgt 0 and Ψ|tm

lt 0 then zRzt|tmlt 0 )ere-

fore we have tm isin (ta tb) or tm isin (tc td) FurthermoreR t(e minus g)q is maximized when t ta if clt 1113954c3 R

t(e minus g)q is maximized when t tc if cgt 1113954c3 Totally wehave

tlowast

ta clt 1113954c3

tc cgt 1113954c31113896 (C5)

(2) Review Proposition 1 If cgt 1113954c1 g is maximized whentlowast 1113954t1 If clt 1113954c1 g is maximized when tlowast tmWhen tgt1113954t1 1113954c1 gt 1113954c5 And 1113954c1 lt 0lt 1113954c5 if tlt1113954t1Substituting 1113954t1 into Φ

Φ|1113954t1 minus

3 (α + β)pn + ke minus (1 + β)c( 11138572

8e(1 + β)+ 2ecα(1 minus α)

(C6)

A threshold is 1113954c4 (34)1113954c3 )erefore we have (i)when 0lt clt 1113954c4 tlowast ta because of Φ|1113954t1

lt 0 and 1113954t1 gt ta

(ii) when 1113954c4 lt clt 1113954c3 tlowast ta because of 1113954t1 lt ta (iii) tlowast

tc when 1113954c3 lt clt 1113954c1 and (iv) tlowast tc when 1113954c1 lt c )eresult is obtained by solving Φ and Ψ

D Proof of Theorem 2

Similar to Appendix A )e retailer problem is addressedfirst and we have

p αηpn + βηpn + ηkg + βw + w

2η(β + 1) (D1)

)en substituting p into the manufacturerrsquos profitfunction Solving the optimal w and g simultaneously yields

w αηpn + βet + βηpn minus βgt + eηk + cβ + et minus gt + c( 1113857η

(1 + η)(β + 1)

g minusαηpn + βηpn + eηk minus βw minus w( 1113857t

2(α minus 1)cαη

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(D2)

Mathematical Problems in Engineering 13

By solving the equation above we have

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

g t minus αpn + et(β + 1) + c minus pn( 1113857β minus ke + c( 1113857

t2(1 + β) minus (1 + η)cα(1 minus α)

p (2η + 1)cpnα

3+ A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1 et + ηpn + c( 1113857β + et + ke minus pn( 1113857η + c

A2 (2η + 1)pn + et + c( 1113857β +(minus 2η minus 1)pn + et +(2ηk + k)e + c

⎧⎪⎪⎨

⎪⎪⎩

B1 minus 2et minus 2ηpn minus 2c( 1113857β minus 2eηk minus 2et minus 2c( 1113857c + t2pn(β + 1)

B2 (minus 2η minus 1)pn minus et minus c( 1113857β minus et +(minus 2ηk minus k)e minus c( 1113857c + t2pn(β + 1)

⎧⎪⎪⎨

⎪⎪⎩

(D3)

Note that we have minus αpn + et(β + 1) + (c minus pn)β minus ke +

clt 0 )e concavity condition is (1 + β)(t2(1 + β) minus (1+

η)cα(1 minus α))lt 0

Substituting solutions into the profit function we have

πm minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2c

2 t2β + t

2+ 4(minus 1 + α)cα1113872 1113873(β + 1)

πr minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2(minus 1 + α)αc

2

4α2c minus 4αc +(β + 1)t2

1113872 11138732(β + 1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(D4)

E Proof of Proposition 3

πM (α + β)pn + ke minus (1 + β)c( 1113857

2c

2(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCM

(α + β)pn + ke minus (1 + β)c( 11138572c

2(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(E1)

By comparing the profits of two partners change after therevenue-sharing contract introduced we found that themanufacturer always benefits from cooperation

πR c2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCR

ηc2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(E2)

A threshold is

η1 2cα(1 minus α) minus (1 + β)t2

2cα(1 minus α)1113888 1113889

2

(E3)

When η is in the region δ (η1 1) the retailer willaccept cooperation And the first derivative of η1 in t is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t(β + 1)

α2c2(minus 1 + α)

2 lt 0 (E4)

)erefore η1 is decreasing in t )at means the regionδ (η1 1) is expanding when tax rises And the revenue-sharing contract is accepted easily than before

F Proof of Proposition 4

We consider a simple function (λrλm) to decide thepractical η where λm isin (05 1) and λr 1 minus λm Given the

14 Mathematical Problems in Engineering

carbon tax tlowast we have a region δ (η(tlowast) 1) of cooperationIt is clear that the gap of strength between two partitions inthe supply chain is a crucial factor in cooperation Bycomparing practice η decided by strength and η(tlowast) de-cided by the carbon tax we have

1λm

minus 1 2α2c minus 2αc + t

2(β + 1)1113872 1113873

2

4α2c2(minus 1 + α)

2 (F1)

A threshold is

1113954λ1 4α2c2

(minus 1 + α)2

8α4c2minus 16α3c2

+ 4 2c + t2(β + 1)1113872 1113873cα2 minus 4t

2c(β + 1)α + t

4(β + 1)

2 (F2)

When 05lt λm lt 1113954λ1 the practical η is in the regionδ (η(tlowast) 1) therefore the cooperation is achieved facingthe optimal carbon tax

However the practical η is out of the range δ (η(tlowast) 1)

when 1113954λ1 lt λm Review previous result the region δ (η1 1)

is expanding when tax rises )erefore there exists a pos-sibility that the government increasing carbon tax to co-ordinate the supply chain We assume a realistic thing thegovernment is willing to boost the greenness of the producteven reducing part of the revenue from the carbon tax

Based on it the carbon up to 1113954t1 which means thegreenness is maximized )erefore )e cooperation isachieved when clt 1113954c1 because the greenness is decreasing inthe carbon tax when 1113954c1 lt c )en we have the secondthreshold

1113954λ2 1

η1 1113954t1( 1113857 + 1 (F3)

)e cooperation could not be achieved when λm gt 1113954λ2)e conclusion is as follows (1) Coordination would be

achieved when the manufacturer is democratic (ie05lt λm lt 1113954λ1) (2) Coordination could be achieved but partof the total tax will lose if development cost is lower when themanufacturer is moderate (ie 1113954λ1 lt λm lt 1113954λ2) (3) )e co-operation could not be achieved if the manufacturer is adictator (ie λm gt 1113954λ2)

G Proof of Proposition 5

)e first derivative of η1 in β is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t

2

2α2c2(minus 1 + α)

2 lt 0 (G1)

Similarly η1 is decreasing in β )at means the regionδ (η1 1) is expanding when β rises

And it is apparent that η1 (2cα(1 minus α) minus (1 + β)t22cα(1 minus α))2 is independent with k

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is research was supported by the National Natural ScienceFoundation of China (no 71761015) and the Science andTechnology Research Project of Jiangxi Education Depart-ment of China (no GJJ150475)

References

[1] Y Li M K Lim J Hu and M-L Tseng ldquoInvestigating theeffect of carbon tax and carbon quota policy to achieve lowcarbon logistics operationsrdquo Resources Conservation andRecycling vol 154 p 104535 2020

[2] M Abedi-Varaki ldquoStudy of carbon dioxide gas treatmentbased on equations of kinetics in plasma discharge reactorrdquoModern Physics Letters B vol 31 no 22 p 1750210 2017

[3] S Meng M Siriwardana and J McNeill ldquo)e environmentaland economic impact of the carbon tax in Australiardquo Envi-ronmental and Resource Economics vol 54 no 3 pp 313ndash332 2013

[4] Y Chen and C L Tseng ldquoInducing clean technology in theelectricity sector tradable permits or carbon tax policiesrdquo-eEnergy Journal vol 32 no 3 2011

[5] H-B Duan L Zhu and Y Fan ldquoOptimal carbon taxes incarbon-constrained China a logistic-induced energy eco-nomic hybrid modelrdquo Energy vol 69 pp 345ndash356 2014

[6] C Lu Q Tong and X Liu ldquo)e impacts of carbon tax andcomplementary policies on Chinese economyrdquo Energy Policyvol 38 no 11 pp 7278ndash7285 2010

[7] K Cao B Xu and J Wang ldquoOptimal trade-in and warrantyperiod strategies for new and remanufactured products undercarbon tax policyrdquo International Journal of Production Re-search vol 58 no 1 pp 180ndash199 2020

[8] P He W Zhang X Xu and Y Bian ldquoProduction lot-sizingand carbon emissions under cap-and-trade and carbon taxregulationsrdquo Journal of Cleaner Production vol 103pp 241ndash248 2015

[9] X Xu X Xu and P He ldquoJoint production and pricing de-cisions for multiple products with cap-and-trade and carbontax regulationsrdquo Journal of Cleaner Production vol 112pp 4093ndash4106 2016

Mathematical Problems in Engineering 15

[10] Q Li X Guan T Shi and W Jiao ldquoGreen product designwith competition and fairness concerns in the circulareconomy erardquo International Journal of Production Researchvol 58 no 1 pp 165ndash179 2020

[11] A Ranjan and J K Jha ldquoPricing and coordination strategiesof a dual-channel supply chain considering green quality andsales effortrdquo Journal of Cleaner Production vol 218pp 409ndash424 2019

[12] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineeringvol 10 no 1 pp 99ndash116 2012

[13] X Yin X Chen X Xu and L Zhang ldquoTax or subsidyOptimal carbon emission policy a supply chain perspectiverdquoSustainability vol 12 no 4 p 1548 2020

[14] B B Schlegelmilch G M Bohlen and A Diamantopoulosldquo)e link between green purchasing decisions and measuresof environmental consciousnessrdquo European Journal of Mar-keting vol 30 no 5 pp 35ndash55 1996

[15] M S Hopkins ldquoWhat the rsquogreenrsquo consumer wantsrdquo MITSloan Management Review vol 50 no 4 pp 87ndash89 2009

[16] G Kalyanaram and R S Winer ldquoEmpirical generalizationsfrom reference price researchrdquo Marketing Science vol 14no 3 pp G161ndashG169 1995

[17] E A Greenleaf ldquo)e impact of reference price effects on theprofitability of price promotionsrdquo Marketing Science vol 14no 1 pp 82ndash104 1995

[18] N M Modak D K Ghosh S Panda and S S SanaldquoManaging green house gas emission cost and pricing policiesin a two-echelon supply chainrdquo CIRP Journal ofManufacturing Science and Technology vol 20 pp 1ndash11 2018

[19] P Sinclair ldquoHigh does nothing and rising is worse carbontaxes should keep declining to cut harmful emissionsrdquo -eManchester School vol 60 no 1 pp 41ndash52 1992

[20] G M Grossman and A B Krueger ldquoEconomic growth andthe environmentrdquo -e Quarterly Journal of Economicsvol 110 no 2 pp 353ndash377 1995

[21] N M Modak and P Kelle ldquoUsing social work donation as atool of corporate social responsibility in a closed-loop supplychain considering carbon emissions tax and demand uncer-taintyrdquo Journal of the Operational Research Society pp 1ndash172019

[22] A Ulph and D Ulph ldquo)e optimal time path of a carbon taxrdquoOxford Economic Papers vol 46 no S1 pp 857ndash868 1994

[23] M Hoel and S Kverndokk ldquoDepletion of fossil fuels and theimpact of global warmingrdquo Resource and Energy Economicsvol 18 no 2 pp 115ndash136 1996

[24] Y H Farzin and O Tahvonen ldquoGlobal carbon cycle and theoptimal time path of a carbon taxrdquo Oxford Economic Papersvol 48 no 4 pp 515ndash536 1996

[25] V Bosetti C Carraro R Duval andM Tavoni ldquoWhat shouldwe expect from innovation A model-based assessment of theenvironmental and mitigation cost implications of climate-related RampDrdquo Energy Economics vol 33 no 6 pp 1313ndash13202011

[26] M Hariga R Asrsquoad and A Shamayleh ldquoIntegrated economicand environmental models for a multi stage cold supply chainunder carbon tax regulationrdquo Journal of Cleaner Productionvol 166 pp 1357ndash1371 2017

[27] N Turken J Carrillo and V Verter ldquoFacility location andcapacity acquisition under carbon tax and emissions limits tocentralize or to decentralizerdquo International Journal of Pro-duction Economics vol 187 pp 126ndash141 2017

[28] W Yu and R Han ldquoCoordinating a two-echelon supply chainunder carbon taxrdquo Sustainability vol 9 no 12 p 2360 2017

[29] S Sinha and N M Modak ldquoAn EPQmodel in the perspectiveof carbon emission reductionrdquo International Journal ofMathematics in Operational Research vol 14 no 3pp 338ndash358 2019

[30] H Zhang P Li H Zheng and Y Zhang ldquoImpact of carbontax on enterprise operation and production strategy for low-carbon products in a co-opetition supply chainrdquo Journal ofCleaner Production Article ID 125058 2020

[31] X Chen H Yang X Wang et al ldquoOptimal carbon tax designfor achieving low carbon supply chainsrdquo Annals of OperationsResearch pp 1ndash28 2020

[32] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Handbook of the Fundamentals of Fi-nancial Decision Making Part I pp 99ndash127 )e EconometricSociety Cleveland OH USA 2013

[33] W Kim and M Kim ldquoReference quality-based competitivemarket structure for innovation driven marketsrdquo Interna-tional Journal of Research in Marketing vol 32 no 3pp 284ndash296 2015

[34] I Popescu and Y Wu ldquoDynamic pricing strategies withreference effectsrdquo Operations Research vol 55 no 3pp 413ndash429 2007

[35] P K Kopalle P K Kannan L B Boldt and N Arora ldquo)eimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Re-tailing vol 88 no 1 pp 102ndash114 2012

[36] T-P Hsieh and C-Y Dye ldquoOptimal dynamic pricing fordeteriorating items with reference price effects when inven-tories stimulate demandrdquo European Journal of OperationalResearch vol 262 no 1 pp 136ndash150 2017

[37] X Chen Z-Y Hu and Y-H Zhang ldquoDynamic pricing withstochastic reference price effectrdquo Journal of the OperationsResearch Society of China vol 7 no 1 pp 107ndash125 2019

[38] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquo Opera-tions Research vol 64 no 6 pp 1529ndash1536 2016

[39] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[40] C Chen ldquoDesign for the environment a quality-based modelfor green product developmentrdquo Management Sciencevol 47 no 2 pp 250ndash263 2001

[41] K Chitra ldquoIn search of the green consumers a perceptualstudyrdquo Journal of Services Research vol 7 no 1 2007

[42] Z Liu T D Anderson and J M Cruz ldquoConsumer envi-ronmental awareness and competition in two-stage supplychainsrdquo European Journal of Operational Research vol 218no 3 pp 602ndash613 2012

[43] B Li Y Wang and Z Wang ldquoManaging a closed-loop supplychain with take-back legislation and consumer preference forgreen designrdquo Journal of Cleaner Production Article ID124481 2020

[44] J Rantanen D B Grant and W Piotrowicz ldquoInvestigatingsupply chain cooperation in Finnish grocery retailrdquo ResearchJournal of the University of Gdansk Transport Economics andLogistics vol 71 pp 19ndash34 2017

[45] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[46] G Xu B Dan X Zhang and C Liu ldquoCoordinating a dual-channel supply chain with risk-averse under a two-way

16 Mathematical Problems in Engineering

revenue sharing contractrdquo International Journal of ProductionEconomics vol 147 pp 171ndash179 2014

[47] Z Shi N Wang T Jia et al ldquoReverse revenue sharingcontract versus two-part tariff contract under a closed-loopsupply chain systemrdquoMathematical Problems in Engineeringvol 2016 Article ID 5464570 15 pages 2016

[48] S Panda N M Modak and L E Cardenas-Barron ldquoCo-ordinating a socially responsible closed-loop supply chainwith product recyclingrdquo International Journal of ProductionEconomics vol 188 pp 11ndash21 2017

[49] N M Modak N Modak S Panda and S S Sana ldquoAnalyzingstructure of two-echelon closed-loop supply chain for pricingquality and recycling managementrdquo Journal of Cleaner Pro-duction vol 171 pp 512ndash528 2018

[50] Q P Wang and D Z Zhao ldquoRevenue-sharing contract ofsupply chain based on consumerrsquos preference for low carbonproductsrdquo Chinese Journal of Management Science vol 22no 9 pp 106ndash113 2014

[51] B Yu J Wang X Lu and H Yang ldquoCollaboration in a low-carbon supply chain with reference emission and cost learningeffects cost sharing versus revenue sharing strategiesrdquo Journalof Cleaner Production vol 250 Article ID 119460 2020

[52] J Cao X Zhang and G Zhou ldquoSupply chain coordinationwith revenue-sharing contracts considering carbon emissionsand governmental policy makingrdquo Environmental Progress ampSustainable Energy vol 35 no 2 pp 479ndash488 2016

[53] X Xu P He H Xu and Q Zhang ldquoSupply chain coordi-nation with green technology under cap-and-trade regula-tionrdquo International Journal of Production Economics vol 183pp 433ndash442 2017

[54] N M Modak S Sinha S Panda et al ldquoAnalyzing a sociallyresponsible closed-loop distribution channel with recyclingfacilityrdquo SN Applied Sciences vol 1 no 10 p 1189 2019

[55] D Feng L Ma Y Ding G Wu and Y Zhang ldquoDecisions ofthe dual-channel supply chain under double policy consid-ering remanufacturingrdquo International Journal of Environ-mental Research and Public Health vol 16 no 3 p 465 2019

[56] I Bellos M Ferguson and L B Toktay ldquo)e car sharingeconomy interaction of business model choice and productline designrdquo Manufacturing amp Service Operations Manage-ment vol 19 no 2 pp 185ndash201 2017

[57] G Ferrer and J M Swaminathan ldquoManaging new andremanufactured productsrdquo Management Science vol 52no 1 pp 15ndash26 2006

[58] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2pp 519ndash527 2011

[59] L Cui S Guo and H Zhang ldquoCoordinating a green agri-foodsupply chain with revenue-sharing contracts consideringretailersrsquo green marketing effortsrdquo Sustainability vol 12no 4 p 1289 2020

[60] J Heydari K Govindan and A Aslani ldquoPricing and greeningdecisions in a three-tier dual channel supply chainrdquo Inter-national Journal of Production Economics vol 217 pp 185ndash196 2019

Mathematical Problems in Engineering 17

Proposition 2 -e optimal carbon tax is defined as follows

tlowast

M minusM

2minus N

1113968

e(1 + β) 0lt clt 1113954c3

P minus

P2

minus Q

1113969

2(1 + β)M 1113954c3 lt clt 1113954c1

1113954t1 1113954c1 lt c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

)e optimal carbon tax is obtained in Proposition 2based on the manufacturerrsquos development cost to maximizegreenness without losing the total carbon tax revenue Wecan observe that the crucial factor is the development costAn appropriate carbon tax policy should be chosen whenfacing different production technologies to ensure greennessand total tax revenue In addition it is easily observed thatthe maximum greenness is obtained when the developmentcost is high whereas the maximum tax is obtained when thedevelopment cost is lower but some of the greenness is lostA numerical simulation of this scenario is shown in Figure 4

33 Supply Chain Coordination It is also important forsupply chains to establish green supply chain models byintegrating internal and external resources to make deci-sions which enables supply chains to achieve better per-formance by improving cooperation [59 60] However moststudies on coordination have mainly focused reducingemissions and improving profit Few studies have investi-gated the relationship between carbon taxes and supplychain cooperation

In this section we study the impact of carbon taxes andconsumerrsquos behavior on supply chain coordination wherethe manufacturer is the leader and the retailer is the fol-lower A revenue-sharing contract is introduced to achieve

coordination Let η denote the proportion of total revenueobtained by the retailer then the manufacturerrsquos decisionmodel is as follows

Maxwg

πCM ((1 minus η)p + w minus c minus t(e minus g))q minus

cg2

2 (8)

)e retailerrsquos decision problem can be described asfollows

Maxp

πCR (ηp minus w)q (9)

)e optimal solutions for the retailer and manufacturerare derived with the following theorem by substituting thedemand function into the equation above )e proof isfound in Appendix D

Theorem 2 -e greenness is defined as follows

g t minus αpn + et(β + 1) + c minus pn( 1113857β minus ke + c( 1113857

2c(1 + η)α2 minus 2c(1 + η)α + t2(β + 1)

(10)

-en the profits of both parties are the following

πm minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2c

2 t2β + t

2+ 4(minus 1 + α)cα1113872 1113873(β + 1)

πr minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2(minus 1 + α)αc

2

4α2c minus 4αc +(β + 1)t2

1113872 11138732(β + 1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

We investigate the impact of the carbon tax on pro-moting the coordination of the supply chain by comparingthe profit changes of the two partners after the introductionof the revenue-sharing contract in Proposition 3 )e proofsare provided in Appendix E

Proposition 3 Coordination can be promoted by increasingthe carbon tax

01204

01202

01200g

01198

01196

00 01 02 03β

04 05

11900 times 10ndash5

11895 times 10ndash5

11898 times 10ndash5

11892 times 10ndash5

11890 times 10ndash5

11888 times 10ndash5

11885 times 10ndash5

11882 times 10ndash5

11880 times 10ndash5

γ lt γ2 t lt t2γ gt γ2 t lt t2

(a)

096

095

094

093g

092

091

090

08900 01 02 03

β04 05

0035

0040

0045

0050

0055

0060

γ lt γ2 t gt t2γ gt γ2 t gt t2

(b)

Figure 2 Effects of consumersrsquo price reference behavior on greenness (a) Low-carbon tax rate (b) High carbon tax rate

Mathematical Problems in Engineering 7

When the profits of both the retailer and the manu-facturer improve simultaneously coordination is achievedBy comparing equations (11) and (6) we find that therevenue-sharing contract improves the retailerrsquos profit onlywhen η isin (η1 1) whereas the manufacturer always benefitsfrom the contract )e range of η depends on the carbon taxrate When η is in the given range the retailer and man-ufacturer achieve Pareto improvement Furthermore therange expands as the carbon tax rate increases)erefore thegovernment can promote supply chain collaboration byincreasing the carbon tax Based on the previous results forthe optimal carbon tax we next investigate how to achievecoordination when the supply chain faces the optimalcarbon tax Figure 5 shows the Pareto improvement region

According to the assumptions described above themanufacturer acts as the leader and the retailer acts as afollower It is reasonable that the proportion of profitreceived by the retailer is decided by the manufacturer)erefore it is necessary to investigate whether theproportion decided by the leader is in the region in whichcoordination is achieved A parameter λ is introduced todenote the strength of a partner in the supply chain Let

λm isin (05 1) and λr 1 minus λm represent the power of themanufacturer and the retailer respectively )en weconsider a simple function (λrλm) to determine η whichis the proportion of revenue received by the retailer )eretailer retains the maximum revenue when the power ofthe two parties is equal (ie λm λr 05) whereas all ofthe benefits go to the leader when the power of themanufacturer is overwhelming compared with that of theretailer (ie λm 1) Similar to Proposition 2 the optimalcarbon tax is obtained under the revenue-sharing contractas follows

tlowast

M minus

M2

minus (1 + η2)N

1113969

e(1 + β) 0lt clt 1113954c

C2

(1 + η2)P minus

(1 + η2)2P2

minus (1 + η2)Q

1113969

2(1 + β)M 1113954c

C2 lt clt 1113954c1

1113954t1 1113954c1 lt c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

05

04

03g

02

01

0010 15 20 25

k3530 40

g

(a)

10

08

06g

04

02

0010 09 0708 06

γ030405

g

(b)

029

027

028

025

026

g

024

022

023

021035030 040 045 050

α060 065055 070

g

(c)

Figure 3 Effects of parameters on greenness (a) Greenness sensitivity (b) Development cost (c) Functional attribute coefficient

8 Mathematical Problems in Engineering

Proposition 4 is designed to answer the question ofwhether cooperation will be achieved the proofs are foundin Appendix F

Proposition 4 Coordination will be achieved if the power ofthe manufacturer is low (ie 05lt λm lt 1113954λ1) Moreover whenthe power of the manufacturer is at a medium level (ie1113954λ1 lt λm lt 1113954λ2) coordination can be achieved but part of thetotal tax revenue will be lost if the development cost is low(ie clt 1113954c1) In other cases (ie cgt 1113954c1 or 1113954λlt λm) coordi-nation cannot be achieved because the power of the manu-facturer is overwhelming

Coordination situations for different manufacturerpower levels are present in Table 3 )e achievement ofcoordination is decided by the power of the manufacturerand the optimal carbon tax faced by the supply chain Basedon Proposition 3 coordination can be promoted if thegovernment adjusts the carbon tax On the other handcoordination can easily be achieved under the optimalcarbon tax if the power of the manufacturer is low Inaddition there are two possibilities when the power of themanufacturer is at a medium level If the development cost istoo high adjusting the tax will lead to an unknown result thegreenness and total tax revenue may both decline at the same

10 014

012

010

008

006

004

002

000

08

06

Gre

enne

ss

Tota

l tax

04

02

00005000 010 015 020

t

γ lt γ3

t = ta

025 030

GreennessTotal tax

(a)

10 030

025

020

015

010

005

000

08

06

Gre

enne

ss

Tota

l tax

04

02

000100 02 03 04

t

γ3 lt γ lt γ1

t = tc

05 06

GreennessTotal tax

(b)

005 040

035

025

020

015

010

005

000

004

003

Gre

enne

ss

Tota

l tax

002

001

0000100 02 03 04

t

γ1 lt γ

t = t1

06 08

030

05 07

GreennessTotal tax

(c)

Figure 4 Determination of the optimal carbon tax (a) 0lt clt 1113954c3 (b) 1113954c3 lt clt 1113954c1 (c) 1113954c1 lt c

005t

η

010 015 020 025 030

Pareto improvement

035 040

Figure 5 Pareto improvement region with increasing carbon tax

Mathematical Problems in Engineering 9

time If the development cost is lower the greenness may beimproved with a loss of revenue by increasing the carbon taxHowever when the power of the manufacturer is strong theregion δ (η1 1) cannot be achieved

According to our theory the government can promotethe coordination between the manufacturer and the retailerby adjusting the carbon tax When the power of the man-ufacturer increases the government still achieves coordi-nation by raising the carbon tax In general cooperation isachieved and the government plays a key role in promotingcooperation

)e effects of consumersrsquo behavior on supply chaincoordination are also investigated in the next proposition)e proofs are provided in Appendix G

Proposition 5 Price reference behavior can promote supplychain coordination but green preference cannot

From the previous results price reference has a compleximpact on the supply chain First the increasing sensitivityof customers to price will promote supply chain coordi-nation because this negative impact can be mitigated bycooperation In addition price reference behavior also af-fects the green decisions of the supply chain in a complexmanner In contrast the mechanism by which consumerrsquogreen preferences influence the supply chain is relativelysimple Greenness increases with increasing greennesspreference and green preference has no effect on the co-ordination of the supply chain

4 Conclusion

)is study examines the interaction between supply chaincooperation and the carbon tax problem in a two-echelonsupply chain under consumerrsquos reference behavior )e opti-mal carbon tax policy is obtained based on analysis of thecarbon tax green investment coefficient and degree of con-sumerrsquos price reference )e optimal carbon tax is defined asthe simultaneous optimization of total tax revenue and productgreenness In addition coordination is achieved by introducingrevenue-sharing contracts)e impact of consumersrsquo referencebehavior and the carbon tax on supply chain coordination isalso investigated )e results are as follows

(1) )e greenness increases with increasing carbon taxwhen the carbon tax is low In addition if the carbontax is higher and the development cost is low thegreenness of products will increase with increasingcarbon tax Conversely if the carbon tax is higherand the development cost is also high the greennessof products will decrease

(2) When the carbon tax is low the negative effect ofprice increases as consumer concerns about priceincrease )erefore the manufacturer offsets thenegative effect by improving the greenness Whenthe carbon tax is higher and development cost is lowthe greenness of products will increase with highconsumerrsquos concern about price In contrast if thecarbon tax is higher and the development cost ishigh the greenness will decrease

(3) )e greenness increases with an increasing greennesspreference of consumers )eir preference behaviorcan promote more green products )is means thatthe government can promote green production bypromoting the green consciousness of consumersMoreover the greenness decreases with increasingdevelopment cost A higher greenness is obtainedwhen manufacturers face lower development costsIn addition greenness first increases and then de-creases with increases in the functional attributecoefficient When the functional attribute of thegreen product is low the manufacturer will producea lower greenness product with an increase in thefunctional attribute In contrast the manufacturerwill produce a higher greenness product with anincrease in the functional attribute after a func-tionality threshold

(4) We investigated the impact of the carbon tax onpromoting the coordination of the supply chain bycomparing the profit changes of two partners afterthe introduction of a revenue-sharing contract inProposition 3 We found that coordination could bepromoted by increasing the carbon tax

(5) )e achievement of coordination depends on thetype of manufacturer If the power of the manu-facturer is low coordination can be achievedunder the optimal carbon tax If the power of themanufacturer is at a medium level coordinationcan be achieved by increasing the carbon tax andimproved greenness will be realized with a loss ofrevenue However when the power of the man-ufacturer is strong coordination cannot beachieved

In this study we focused on the interaction between acarbon tax and supply chain cooperation However othercarbon policies (eg a cap-and-trade policy) and supplychain structures are worth exploring For example it wouldbe interesting to investigate the effects on a supply chainstructure consisting of the retailer as the leader under acomplex carbon policy

Table 3 Coordination situations for different manufacturer power levels

Power of the manufacturer Coordination Greenness Total carbon tax revenue05lt λm lt 1113954λ1 Achieve Improve Improve1113954λ1 lt λm lt 1113954λ2 clt 1113954c1 Achieve Improve Decline1113954λ1 lt λm lt 1113954λ2 clt 1113954c1 Cannot achievecgt 1113954c1 Cannot achieve

10 Mathematical Problems in Engineering

Appendix

A Proof of Theorem 1

)e demand function can be written as

q P ug ge un ug ge 01113966 1113967

qn P un gt ug un ge 01113966 1113967

⎧⎪⎨

⎪⎩(A1)

After replacing un and ug we have

q P αv minus p + kg minus β p minus pn( 1113857 minus v minus pn( 1113857ge 0 αv minus p + kg minus β p minus pn( 1113857ge 01113864 1113865

qn P v minus pn minus αv minus p + kg minus β p minus pn( 1113857( 1113857gt 0 v minus pn ge 01113864 11138651113896 (A2)

Given that V is uniformly distributed on [0 1] thedemand function is

q (α + β)pn minus (1 + β)p + kg

α(1 minus α)

qn 1 minus(1 + β) pn minus p( 1113857 + kg

1 minus α

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A3)

)e condition kgt ((1 + β)p minus (α + β)png) is necessaryto ensure nonnegativity )e retailer problem is addressedfirst and we have

p (α + β)pn +(1 + β)w + kg

2(1 + β) (A4)

)en substitute p into the manufacturerrsquos profit func-tion Solving the optimal w and g simultaneously yields

w (α + β)pn +(1 + β)(c + t(e minus g)) + kg

2(1 + β)

g t (α + β)pn minus (1 + β)w + ke( 1113857

2cα(1 minus α)

(A5)

)en we have

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

g (1 + β)(te + c) minus (α + β)pn minus ke( 1113857t

t2(1 + β) minus 4cα(1 minus α)

p 3cpnα

3+ 2A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1 (1 + β)(c + te) + ke minus (1 minus β)pn

A2 (1 + β)(c + te) + 3ke minus (3 minus β)pn

⎧⎪⎨

⎪⎩

B1 minus 2c (1 + β)(c + te) + ke + βpn( 1113857 + t2(1 + β)pn

B2 minus c (1 + β)(c + te) + 3ke + 3βpn( 1113857 + t2(1 + β)pn

⎧⎪⎨

⎪⎩

(A6)

Notes that we have (α + β)pn + ke minus (1 + β)(te + c)gt 0)e concavity condition is 4αc(1 minus α) minus (1 + β)t2 gt 0

B Proof of Proposition 1

(1) )e first derivative of g in t is

zg

zt

minus 4cα3pn + 4cA3α2

+ B3α minus t2(1 + β) (1 + β)c minus ke minus βpn( 1113857

t2(1 minus β) minus 4cα(1 minus α)1113872 1113873

2

A3 (1 + β)(c + 2te) minus ke +(1 minus β)pn

B3 minus 4c (1 + β)(c + 2te) minus ke minus βpn( 1113857 + t2(1 + β)pn

⎧⎨

⎩

(B1)

We have a threshold

1113954c1 minust2(1 + β) (α + β)pn + ke minus (1 + β)c( 1113857

4α(1 minus α) (α + β)pn + ke minus (1 + β)(c + 2te)( 1113857

1113954t1 (α + β)pn + ke minus (1 + β)c

2(1 + β)e

(B2)

When cgt 1113954c1 and tgt1113954t1 g is decreasing in t Bycontrast g is increasing in t in other cases

(2) )e first derivative of g in β is

zg

zβ

minus 4α(1 minus α) te + c minus pn( 1113857c + t2

ke + αpn minus pn( 1113857

t2(1 + β) minus 4α(1 minus α)c1113872 1113873

2

(B3)

We have a threshold

1113954c2 minust2

ke minus (1 minus α)pn( 1113857

4α(1 minus α) te + c minus pn( 1113857

1113954t2 pn minus c

e

(B4)

Review we have (α + β)pn + ke minus (1 + β)(te + c)gt 0therefore we can find that

tltpn minus c

e+

ke minus (1 minus α)pn

(1 + β)e (B5)

(i) When ke minus (1 minus α)pn lt 0 then tlt1113954t2 is inevitable)erefore (zgzβ)gt 0 is correct

(ii) When ke minus (1 minus α)pn gt 0 we can find that(zgzβ)lt 0 if cgt 1113954c2 and tgt1113954t2 (zgzβ)lt 0 in othercases

(3) )e first derivative of g in k and c is

Mathematical Problems in Engineering 11

zg

zc minus

4α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

4cα(1 minus α) minus t2(1 + β)1113872 1113873

2 lt 0

zg

zk

te

4cα(1 minus α) minus t2(1 + β)gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(B6)

(4) )e first derivative of g in α is

zg

zα

minus Xt

4α(1 minus α)c minus t2(1 + β)1113872 1113873

2 (B7)

And X minus 4α2cpn + (4(1 + β)(te + c) minus 4ke minus 4pn)

(2α + 1)c + pnt2(1 + β) is a quadratic function )erange of α in 4αc(1 minus α) minus (1 + β)t2 gt 0 is

α1 c +

c2

minus ct2(1 + β)

1113969

2c

α2 c minus

c2

minus ct2(1 + β)

1113969

2c

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(B8)

By substituting those into X we have

X|αα1 4

c2

minus ct2(1 + β)

1113969

T minus 2 c minus t2(1 + β)1113872 1113873pn

X|αα2 minus 4

c2

minus ct2(1 + β)

1113969

T minus 2 c minus t2(1 + β)1113872 1113873pn lt 0

⎧⎪⎪⎨

⎪⎪⎩

(B9)

And T (1 + β)(te + c) minus ke minus ((12) + β)pn lt 0We can find

X|αα1 times X|αα2 minus 4 c minus t2(1 + β)1113872 1113873

middot 4 βpn + ke minus (1 + β)(te + c)( 11138572

1113872

+ p2nt

2(1 + β)1113873lt 0

(B10)

)erefore X|αα1 gt 0 )ere only exist a 1113954α isin (α2 α1)that makes (zgzα) 0 And (zgzα) gt 0 ifα isin (α2 1113954α) (zgzα)lt 0 if α isin (1113954α α1)

(5) )e first derivative of p in t is

zp

zt minus

2α(α minus 1) e(1 + β)t221113872 1113873 + minus pnα minus ke + c minus pn( 1113857β + c( 1113857t minus 2eαc(α minus 1)1113872 1113873c

(1 + β)t2

+ 4αc(α minus 1)1113872 11138732

A3 (1 + β)(c + 2te) minus ke +(1 minus β)pn

B3 minus 4c (1 + β)(c + 2te) minus ke minus βpn( 1113857 + t2(1 + β)pn

⎧⎪⎨

⎪⎩

(B11)

)e discriminant of quadratic function

e(1 + β)t2

2+ minus pnα minus ke + c minus pn( 1113857β + c( 1113857t minus 2eαc(α minus 1)

(B12)

is

Δ (minus β minus 1)c + ke + βpn + pnα( 11138572

+ 4e2(1 + β)t

2αc(α minus 1)

(B13)

We have a threshold

1113954c minus(minus β minus 1)c + ke + βpn + pnα( 1113857

2

4e2(1 + β)t

2α(α minus 1) (B14)

Review we have(α + β)pn + ke minus (1 + β)(te + c)gt 0therefore we can find that

tm 2

minus (1 + β)αc(α minus 1)

1113968

1 + β (B15)

Substituting it into above quadratic function

2 c

radicminus pnα minus ke + c minus pn( 1113857β + c( 1113857

minus (1 + β)α(α minus 1)

1113968minus 4eαc(α minus 1)(1 + β)

1 + β (B16)

And we have a same threshold

1113954c minus(minus β minus 1)c + ke + βpn + pnα( 1113857

2

4e2(1 + β)t

2α(α minus 1) (B17)

It could be found that(zpzt) gt 0 because Δlt 0 whenclt 1113954c And when cgt 1113954c we have another threshold 1113954t andzpzt|

t1113954t 0 because Δgt 0 and zpzt|tmlt 0)erefore when

cgt 1113954c and tgt1113954t p is decreasing in t And when cgt 1113954c and tlt1113954tp is increasing in t

12 Mathematical Problems in Engineering

C Proof of Proposition 2

(1) Let R t(e minus g)q denote total carbon tax and thefirst derivative of R in t is

zR

zt2c A2t

2+ B2t + c1113872 1113873 A3t

2+ B3t + c1113872 1113873

4cα(1 minus α) minus t2(1 + β)1113872 1113873

3 (C1)

Among them

A2 (1 + β)e

2gt 0

B2 minus (α + β)pn + ke minus (1 + β)c( 1113857lt 0

C2 2ceα(1 minus α)gt 0

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

A3 (α + β)pn + ke minus (1 + β)c( 1113857(1 + β)gt 0

B3 minus 8ceα(1 minus α)(1 + β)lt 0

C3 4cα(1 minus α) (α + β)pn + ke minus (1 + β)c( 1113857gt 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(C2)

Let Φ A2t2 + B2t + C2 and Ψ A3t

2 + B3t + C3 andwe can find that zRzt and Φ times Ψ have a same sign Bysolving Δ1 and Δ2 a threshold is

1113954c3 (α + β)pn + ke minus (1 + β)c( 1113857

2

4e2α(1 minus α)(1 + β)

(C3)

)ere are two positive real numbers of solutions forΦ ifclt 1113954c3 and let 0lt ta lt tb denote it Similarly let0lt tc lt td denote the solutions for Ψ if cgt 1113954c3 Reviewwe have 4αc(1 minus α) minus (1 + β)t2 gt 0 therefore lettm

4cα(1 minus a)1 + β

1113968 Substituting tm into Φ and Ψ

Φ|tm minus 2

cα(1 minus α)

1 + β

1113971

(α + β)pn + ke minus (1 + β)c( 1113857 + 4ecα(1 minus α)

Ψ|tm minus 8cα(1 minus α) 2(1 + β)e

α(1 minus α)

1 + βc

1113971

minus (α + β)pn + ke minus (1 + β)c( 1113857⎛⎝ ⎞⎠

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(C4)

If clt 1113954c3 Φ|tmlt 0 and Ψ|tm

gt 0 then zRzt|tmlt 0 If

cgt 1113954c3 Φ|tmgt 0 and Ψ|tm

lt 0 then zRzt|tmlt 0 )ere-

fore we have tm isin (ta tb) or tm isin (tc td) FurthermoreR t(e minus g)q is maximized when t ta if clt 1113954c3 R

t(e minus g)q is maximized when t tc if cgt 1113954c3 Totally wehave

tlowast

ta clt 1113954c3

tc cgt 1113954c31113896 (C5)

(2) Review Proposition 1 If cgt 1113954c1 g is maximized whentlowast 1113954t1 If clt 1113954c1 g is maximized when tlowast tmWhen tgt1113954t1 1113954c1 gt 1113954c5 And 1113954c1 lt 0lt 1113954c5 if tlt1113954t1Substituting 1113954t1 into Φ

Φ|1113954t1 minus

3 (α + β)pn + ke minus (1 + β)c( 11138572

8e(1 + β)+ 2ecα(1 minus α)

(C6)

A threshold is 1113954c4 (34)1113954c3 )erefore we have (i)when 0lt clt 1113954c4 tlowast ta because of Φ|1113954t1

lt 0 and 1113954t1 gt ta

(ii) when 1113954c4 lt clt 1113954c3 tlowast ta because of 1113954t1 lt ta (iii) tlowast

tc when 1113954c3 lt clt 1113954c1 and (iv) tlowast tc when 1113954c1 lt c )eresult is obtained by solving Φ and Ψ

D Proof of Theorem 2

Similar to Appendix A )e retailer problem is addressedfirst and we have

p αηpn + βηpn + ηkg + βw + w

2η(β + 1) (D1)

)en substituting p into the manufacturerrsquos profitfunction Solving the optimal w and g simultaneously yields

w αηpn + βet + βηpn minus βgt + eηk + cβ + et minus gt + c( 1113857η

(1 + η)(β + 1)

g minusαηpn + βηpn + eηk minus βw minus w( 1113857t

2(α minus 1)cαη

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(D2)

Mathematical Problems in Engineering 13

By solving the equation above we have

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

g t minus αpn + et(β + 1) + c minus pn( 1113857β minus ke + c( 1113857

t2(1 + β) minus (1 + η)cα(1 minus α)

p (2η + 1)cpnα

3+ A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1 et + ηpn + c( 1113857β + et + ke minus pn( 1113857η + c

A2 (2η + 1)pn + et + c( 1113857β +(minus 2η minus 1)pn + et +(2ηk + k)e + c

⎧⎪⎪⎨

⎪⎪⎩

B1 minus 2et minus 2ηpn minus 2c( 1113857β minus 2eηk minus 2et minus 2c( 1113857c + t2pn(β + 1)

B2 (minus 2η minus 1)pn minus et minus c( 1113857β minus et +(minus 2ηk minus k)e minus c( 1113857c + t2pn(β + 1)

⎧⎪⎪⎨

⎪⎪⎩

(D3)

Note that we have minus αpn + et(β + 1) + (c minus pn)β minus ke +

clt 0 )e concavity condition is (1 + β)(t2(1 + β) minus (1+

η)cα(1 minus α))lt 0

Substituting solutions into the profit function we have

πm minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2c

2 t2β + t

2+ 4(minus 1 + α)cα1113872 1113873(β + 1)

πr minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2(minus 1 + α)αc

2

4α2c minus 4αc +(β + 1)t2

1113872 11138732(β + 1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(D4)

E Proof of Proposition 3

πM (α + β)pn + ke minus (1 + β)c( 1113857

2c

2(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCM

(α + β)pn + ke minus (1 + β)c( 11138572c

2(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(E1)

By comparing the profits of two partners change after therevenue-sharing contract introduced we found that themanufacturer always benefits from cooperation

πR c2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCR

ηc2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(E2)

A threshold is

η1 2cα(1 minus α) minus (1 + β)t2

2cα(1 minus α)1113888 1113889

2

(E3)

When η is in the region δ (η1 1) the retailer willaccept cooperation And the first derivative of η1 in t is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t(β + 1)

α2c2(minus 1 + α)

2 lt 0 (E4)

)erefore η1 is decreasing in t )at means the regionδ (η1 1) is expanding when tax rises And the revenue-sharing contract is accepted easily than before

F Proof of Proposition 4

We consider a simple function (λrλm) to decide thepractical η where λm isin (05 1) and λr 1 minus λm Given the

14 Mathematical Problems in Engineering

carbon tax tlowast we have a region δ (η(tlowast) 1) of cooperationIt is clear that the gap of strength between two partitions inthe supply chain is a crucial factor in cooperation Bycomparing practice η decided by strength and η(tlowast) de-cided by the carbon tax we have

1λm

minus 1 2α2c minus 2αc + t

2(β + 1)1113872 1113873

2

4α2c2(minus 1 + α)

2 (F1)

A threshold is

1113954λ1 4α2c2

(minus 1 + α)2

8α4c2minus 16α3c2

+ 4 2c + t2(β + 1)1113872 1113873cα2 minus 4t

2c(β + 1)α + t

4(β + 1)

2 (F2)

When 05lt λm lt 1113954λ1 the practical η is in the regionδ (η(tlowast) 1) therefore the cooperation is achieved facingthe optimal carbon tax

However the practical η is out of the range δ (η(tlowast) 1)

when 1113954λ1 lt λm Review previous result the region δ (η1 1)

is expanding when tax rises )erefore there exists a pos-sibility that the government increasing carbon tax to co-ordinate the supply chain We assume a realistic thing thegovernment is willing to boost the greenness of the producteven reducing part of the revenue from the carbon tax

Based on it the carbon up to 1113954t1 which means thegreenness is maximized )erefore )e cooperation isachieved when clt 1113954c1 because the greenness is decreasing inthe carbon tax when 1113954c1 lt c )en we have the secondthreshold

1113954λ2 1

η1 1113954t1( 1113857 + 1 (F3)

)e cooperation could not be achieved when λm gt 1113954λ2)e conclusion is as follows (1) Coordination would be

achieved when the manufacturer is democratic (ie05lt λm lt 1113954λ1) (2) Coordination could be achieved but partof the total tax will lose if development cost is lower when themanufacturer is moderate (ie 1113954λ1 lt λm lt 1113954λ2) (3) )e co-operation could not be achieved if the manufacturer is adictator (ie λm gt 1113954λ2)

G Proof of Proposition 5

)e first derivative of η1 in β is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t

2

2α2c2(minus 1 + α)

2 lt 0 (G1)

Similarly η1 is decreasing in β )at means the regionδ (η1 1) is expanding when β rises

And it is apparent that η1 (2cα(1 minus α) minus (1 + β)t22cα(1 minus α))2 is independent with k

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is research was supported by the National Natural ScienceFoundation of China (no 71761015) and the Science andTechnology Research Project of Jiangxi Education Depart-ment of China (no GJJ150475)

References

[1] Y Li M K Lim J Hu and M-L Tseng ldquoInvestigating theeffect of carbon tax and carbon quota policy to achieve lowcarbon logistics operationsrdquo Resources Conservation andRecycling vol 154 p 104535 2020

[2] M Abedi-Varaki ldquoStudy of carbon dioxide gas treatmentbased on equations of kinetics in plasma discharge reactorrdquoModern Physics Letters B vol 31 no 22 p 1750210 2017

[3] S Meng M Siriwardana and J McNeill ldquo)e environmentaland economic impact of the carbon tax in Australiardquo Envi-ronmental and Resource Economics vol 54 no 3 pp 313ndash332 2013

[4] Y Chen and C L Tseng ldquoInducing clean technology in theelectricity sector tradable permits or carbon tax policiesrdquo-eEnergy Journal vol 32 no 3 2011

[5] H-B Duan L Zhu and Y Fan ldquoOptimal carbon taxes incarbon-constrained China a logistic-induced energy eco-nomic hybrid modelrdquo Energy vol 69 pp 345ndash356 2014

[6] C Lu Q Tong and X Liu ldquo)e impacts of carbon tax andcomplementary policies on Chinese economyrdquo Energy Policyvol 38 no 11 pp 7278ndash7285 2010

[7] K Cao B Xu and J Wang ldquoOptimal trade-in and warrantyperiod strategies for new and remanufactured products undercarbon tax policyrdquo International Journal of Production Re-search vol 58 no 1 pp 180ndash199 2020

[8] P He W Zhang X Xu and Y Bian ldquoProduction lot-sizingand carbon emissions under cap-and-trade and carbon taxregulationsrdquo Journal of Cleaner Production vol 103pp 241ndash248 2015

[9] X Xu X Xu and P He ldquoJoint production and pricing de-cisions for multiple products with cap-and-trade and carbontax regulationsrdquo Journal of Cleaner Production vol 112pp 4093ndash4106 2016

Mathematical Problems in Engineering 15

[10] Q Li X Guan T Shi and W Jiao ldquoGreen product designwith competition and fairness concerns in the circulareconomy erardquo International Journal of Production Researchvol 58 no 1 pp 165ndash179 2020

[11] A Ranjan and J K Jha ldquoPricing and coordination strategiesof a dual-channel supply chain considering green quality andsales effortrdquo Journal of Cleaner Production vol 218pp 409ndash424 2019

[12] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineeringvol 10 no 1 pp 99ndash116 2012

[13] X Yin X Chen X Xu and L Zhang ldquoTax or subsidyOptimal carbon emission policy a supply chain perspectiverdquoSustainability vol 12 no 4 p 1548 2020

[14] B B Schlegelmilch G M Bohlen and A Diamantopoulosldquo)e link between green purchasing decisions and measuresof environmental consciousnessrdquo European Journal of Mar-keting vol 30 no 5 pp 35ndash55 1996

[15] M S Hopkins ldquoWhat the rsquogreenrsquo consumer wantsrdquo MITSloan Management Review vol 50 no 4 pp 87ndash89 2009

[16] G Kalyanaram and R S Winer ldquoEmpirical generalizationsfrom reference price researchrdquo Marketing Science vol 14no 3 pp G161ndashG169 1995

[17] E A Greenleaf ldquo)e impact of reference price effects on theprofitability of price promotionsrdquo Marketing Science vol 14no 1 pp 82ndash104 1995

[18] N M Modak D K Ghosh S Panda and S S SanaldquoManaging green house gas emission cost and pricing policiesin a two-echelon supply chainrdquo CIRP Journal ofManufacturing Science and Technology vol 20 pp 1ndash11 2018

[19] P Sinclair ldquoHigh does nothing and rising is worse carbontaxes should keep declining to cut harmful emissionsrdquo -eManchester School vol 60 no 1 pp 41ndash52 1992

[20] G M Grossman and A B Krueger ldquoEconomic growth andthe environmentrdquo -e Quarterly Journal of Economicsvol 110 no 2 pp 353ndash377 1995

[21] N M Modak and P Kelle ldquoUsing social work donation as atool of corporate social responsibility in a closed-loop supplychain considering carbon emissions tax and demand uncer-taintyrdquo Journal of the Operational Research Society pp 1ndash172019

[22] A Ulph and D Ulph ldquo)e optimal time path of a carbon taxrdquoOxford Economic Papers vol 46 no S1 pp 857ndash868 1994

[23] M Hoel and S Kverndokk ldquoDepletion of fossil fuels and theimpact of global warmingrdquo Resource and Energy Economicsvol 18 no 2 pp 115ndash136 1996

[24] Y H Farzin and O Tahvonen ldquoGlobal carbon cycle and theoptimal time path of a carbon taxrdquo Oxford Economic Papersvol 48 no 4 pp 515ndash536 1996

[25] V Bosetti C Carraro R Duval andM Tavoni ldquoWhat shouldwe expect from innovation A model-based assessment of theenvironmental and mitigation cost implications of climate-related RampDrdquo Energy Economics vol 33 no 6 pp 1313ndash13202011

[26] M Hariga R Asrsquoad and A Shamayleh ldquoIntegrated economicand environmental models for a multi stage cold supply chainunder carbon tax regulationrdquo Journal of Cleaner Productionvol 166 pp 1357ndash1371 2017

[27] N Turken J Carrillo and V Verter ldquoFacility location andcapacity acquisition under carbon tax and emissions limits tocentralize or to decentralizerdquo International Journal of Pro-duction Economics vol 187 pp 126ndash141 2017

[28] W Yu and R Han ldquoCoordinating a two-echelon supply chainunder carbon taxrdquo Sustainability vol 9 no 12 p 2360 2017

[29] S Sinha and N M Modak ldquoAn EPQmodel in the perspectiveof carbon emission reductionrdquo International Journal ofMathematics in Operational Research vol 14 no 3pp 338ndash358 2019

[30] H Zhang P Li H Zheng and Y Zhang ldquoImpact of carbontax on enterprise operation and production strategy for low-carbon products in a co-opetition supply chainrdquo Journal ofCleaner Production Article ID 125058 2020

[31] X Chen H Yang X Wang et al ldquoOptimal carbon tax designfor achieving low carbon supply chainsrdquo Annals of OperationsResearch pp 1ndash28 2020

[32] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Handbook of the Fundamentals of Fi-nancial Decision Making Part I pp 99ndash127 )e EconometricSociety Cleveland OH USA 2013

[33] W Kim and M Kim ldquoReference quality-based competitivemarket structure for innovation driven marketsrdquo Interna-tional Journal of Research in Marketing vol 32 no 3pp 284ndash296 2015

[34] I Popescu and Y Wu ldquoDynamic pricing strategies withreference effectsrdquo Operations Research vol 55 no 3pp 413ndash429 2007

[35] P K Kopalle P K Kannan L B Boldt and N Arora ldquo)eimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Re-tailing vol 88 no 1 pp 102ndash114 2012

[36] T-P Hsieh and C-Y Dye ldquoOptimal dynamic pricing fordeteriorating items with reference price effects when inven-tories stimulate demandrdquo European Journal of OperationalResearch vol 262 no 1 pp 136ndash150 2017

[37] X Chen Z-Y Hu and Y-H Zhang ldquoDynamic pricing withstochastic reference price effectrdquo Journal of the OperationsResearch Society of China vol 7 no 1 pp 107ndash125 2019

[38] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquo Opera-tions Research vol 64 no 6 pp 1529ndash1536 2016

[39] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[40] C Chen ldquoDesign for the environment a quality-based modelfor green product developmentrdquo Management Sciencevol 47 no 2 pp 250ndash263 2001

[41] K Chitra ldquoIn search of the green consumers a perceptualstudyrdquo Journal of Services Research vol 7 no 1 2007

[42] Z Liu T D Anderson and J M Cruz ldquoConsumer envi-ronmental awareness and competition in two-stage supplychainsrdquo European Journal of Operational Research vol 218no 3 pp 602ndash613 2012

[43] B Li Y Wang and Z Wang ldquoManaging a closed-loop supplychain with take-back legislation and consumer preference forgreen designrdquo Journal of Cleaner Production Article ID124481 2020

[44] J Rantanen D B Grant and W Piotrowicz ldquoInvestigatingsupply chain cooperation in Finnish grocery retailrdquo ResearchJournal of the University of Gdansk Transport Economics andLogistics vol 71 pp 19ndash34 2017

[45] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[46] G Xu B Dan X Zhang and C Liu ldquoCoordinating a dual-channel supply chain with risk-averse under a two-way

16 Mathematical Problems in Engineering

revenue sharing contractrdquo International Journal of ProductionEconomics vol 147 pp 171ndash179 2014

[47] Z Shi N Wang T Jia et al ldquoReverse revenue sharingcontract versus two-part tariff contract under a closed-loopsupply chain systemrdquoMathematical Problems in Engineeringvol 2016 Article ID 5464570 15 pages 2016

[48] S Panda N M Modak and L E Cardenas-Barron ldquoCo-ordinating a socially responsible closed-loop supply chainwith product recyclingrdquo International Journal of ProductionEconomics vol 188 pp 11ndash21 2017

[49] N M Modak N Modak S Panda and S S Sana ldquoAnalyzingstructure of two-echelon closed-loop supply chain for pricingquality and recycling managementrdquo Journal of Cleaner Pro-duction vol 171 pp 512ndash528 2018

[50] Q P Wang and D Z Zhao ldquoRevenue-sharing contract ofsupply chain based on consumerrsquos preference for low carbonproductsrdquo Chinese Journal of Management Science vol 22no 9 pp 106ndash113 2014

[51] B Yu J Wang X Lu and H Yang ldquoCollaboration in a low-carbon supply chain with reference emission and cost learningeffects cost sharing versus revenue sharing strategiesrdquo Journalof Cleaner Production vol 250 Article ID 119460 2020

[52] J Cao X Zhang and G Zhou ldquoSupply chain coordinationwith revenue-sharing contracts considering carbon emissionsand governmental policy makingrdquo Environmental Progress ampSustainable Energy vol 35 no 2 pp 479ndash488 2016

[53] X Xu P He H Xu and Q Zhang ldquoSupply chain coordi-nation with green technology under cap-and-trade regula-tionrdquo International Journal of Production Economics vol 183pp 433ndash442 2017

[54] N M Modak S Sinha S Panda et al ldquoAnalyzing a sociallyresponsible closed-loop distribution channel with recyclingfacilityrdquo SN Applied Sciences vol 1 no 10 p 1189 2019

[55] D Feng L Ma Y Ding G Wu and Y Zhang ldquoDecisions ofthe dual-channel supply chain under double policy consid-ering remanufacturingrdquo International Journal of Environ-mental Research and Public Health vol 16 no 3 p 465 2019

[56] I Bellos M Ferguson and L B Toktay ldquo)e car sharingeconomy interaction of business model choice and productline designrdquo Manufacturing amp Service Operations Manage-ment vol 19 no 2 pp 185ndash201 2017

[57] G Ferrer and J M Swaminathan ldquoManaging new andremanufactured productsrdquo Management Science vol 52no 1 pp 15ndash26 2006

[58] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2pp 519ndash527 2011

[59] L Cui S Guo and H Zhang ldquoCoordinating a green agri-foodsupply chain with revenue-sharing contracts consideringretailersrsquo green marketing effortsrdquo Sustainability vol 12no 4 p 1289 2020

[60] J Heydari K Govindan and A Aslani ldquoPricing and greeningdecisions in a three-tier dual channel supply chainrdquo Inter-national Journal of Production Economics vol 217 pp 185ndash196 2019

Mathematical Problems in Engineering 17

When the profits of both the retailer and the manu-facturer improve simultaneously coordination is achievedBy comparing equations (11) and (6) we find that therevenue-sharing contract improves the retailerrsquos profit onlywhen η isin (η1 1) whereas the manufacturer always benefitsfrom the contract )e range of η depends on the carbon taxrate When η is in the given range the retailer and man-ufacturer achieve Pareto improvement Furthermore therange expands as the carbon tax rate increases)erefore thegovernment can promote supply chain collaboration byincreasing the carbon tax Based on the previous results forthe optimal carbon tax we next investigate how to achievecoordination when the supply chain faces the optimalcarbon tax Figure 5 shows the Pareto improvement region

According to the assumptions described above themanufacturer acts as the leader and the retailer acts as afollower It is reasonable that the proportion of profitreceived by the retailer is decided by the manufacturer)erefore it is necessary to investigate whether theproportion decided by the leader is in the region in whichcoordination is achieved A parameter λ is introduced todenote the strength of a partner in the supply chain Let

λm isin (05 1) and λr 1 minus λm represent the power of themanufacturer and the retailer respectively )en weconsider a simple function (λrλm) to determine η whichis the proportion of revenue received by the retailer )eretailer retains the maximum revenue when the power ofthe two parties is equal (ie λm λr 05) whereas all ofthe benefits go to the leader when the power of themanufacturer is overwhelming compared with that of theretailer (ie λm 1) Similar to Proposition 2 the optimalcarbon tax is obtained under the revenue-sharing contractas follows

tlowast

M minus

M2

minus (1 + η2)N

1113969

e(1 + β) 0lt clt 1113954c

C2

(1 + η2)P minus

(1 + η2)2P2

minus (1 + η2)Q

1113969

2(1 + β)M 1113954c

C2 lt clt 1113954c1

1113954t1 1113954c1 lt c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

05

04

03g

02

01

0010 15 20 25

k3530 40

g

(a)

10

08

06g

04

02

0010 09 0708 06

γ030405

g

(b)

029

027

028

025

026

g

024

022

023

021035030 040 045 050

α060 065055 070

g

(c)

Figure 3 Effects of parameters on greenness (a) Greenness sensitivity (b) Development cost (c) Functional attribute coefficient

8 Mathematical Problems in Engineering

Proposition 4 is designed to answer the question ofwhether cooperation will be achieved the proofs are foundin Appendix F

Proposition 4 Coordination will be achieved if the power ofthe manufacturer is low (ie 05lt λm lt 1113954λ1) Moreover whenthe power of the manufacturer is at a medium level (ie1113954λ1 lt λm lt 1113954λ2) coordination can be achieved but part of thetotal tax revenue will be lost if the development cost is low(ie clt 1113954c1) In other cases (ie cgt 1113954c1 or 1113954λlt λm) coordi-nation cannot be achieved because the power of the manu-facturer is overwhelming

Coordination situations for different manufacturerpower levels are present in Table 3 )e achievement ofcoordination is decided by the power of the manufacturerand the optimal carbon tax faced by the supply chain Basedon Proposition 3 coordination can be promoted if thegovernment adjusts the carbon tax On the other handcoordination can easily be achieved under the optimalcarbon tax if the power of the manufacturer is low Inaddition there are two possibilities when the power of themanufacturer is at a medium level If the development cost istoo high adjusting the tax will lead to an unknown result thegreenness and total tax revenue may both decline at the same

10 014

012

010

008

006

004

002

000

08

06

Gre

enne

ss

Tota

l tax

04

02

00005000 010 015 020

t

γ lt γ3

t = ta

025 030

GreennessTotal tax

(a)

10 030

025

020

015

010

005

000

08

06

Gre

enne

ss

Tota

l tax

04

02

000100 02 03 04

t

γ3 lt γ lt γ1

t = tc

05 06

GreennessTotal tax

(b)

005 040

035

025

020

015

010

005

000

004

003

Gre

enne

ss

Tota

l tax

002

001

0000100 02 03 04

t

γ1 lt γ

t = t1

06 08

030

05 07

GreennessTotal tax

(c)

Figure 4 Determination of the optimal carbon tax (a) 0lt clt 1113954c3 (b) 1113954c3 lt clt 1113954c1 (c) 1113954c1 lt c

005t

η

010 015 020 025 030

Pareto improvement

035 040

Figure 5 Pareto improvement region with increasing carbon tax

Mathematical Problems in Engineering 9

time If the development cost is lower the greenness may beimproved with a loss of revenue by increasing the carbon taxHowever when the power of the manufacturer is strong theregion δ (η1 1) cannot be achieved

According to our theory the government can promotethe coordination between the manufacturer and the retailerby adjusting the carbon tax When the power of the man-ufacturer increases the government still achieves coordi-nation by raising the carbon tax In general cooperation isachieved and the government plays a key role in promotingcooperation

)e effects of consumersrsquo behavior on supply chaincoordination are also investigated in the next proposition)e proofs are provided in Appendix G

Proposition 5 Price reference behavior can promote supplychain coordination but green preference cannot

From the previous results price reference has a compleximpact on the supply chain First the increasing sensitivityof customers to price will promote supply chain coordi-nation because this negative impact can be mitigated bycooperation In addition price reference behavior also af-fects the green decisions of the supply chain in a complexmanner In contrast the mechanism by which consumerrsquogreen preferences influence the supply chain is relativelysimple Greenness increases with increasing greennesspreference and green preference has no effect on the co-ordination of the supply chain

4 Conclusion

)is study examines the interaction between supply chaincooperation and the carbon tax problem in a two-echelonsupply chain under consumerrsquos reference behavior )e opti-mal carbon tax policy is obtained based on analysis of thecarbon tax green investment coefficient and degree of con-sumerrsquos price reference )e optimal carbon tax is defined asthe simultaneous optimization of total tax revenue and productgreenness In addition coordination is achieved by introducingrevenue-sharing contracts)e impact of consumersrsquo referencebehavior and the carbon tax on supply chain coordination isalso investigated )e results are as follows

(1) )e greenness increases with increasing carbon taxwhen the carbon tax is low In addition if the carbontax is higher and the development cost is low thegreenness of products will increase with increasingcarbon tax Conversely if the carbon tax is higherand the development cost is also high the greennessof products will decrease

(2) When the carbon tax is low the negative effect ofprice increases as consumer concerns about priceincrease )erefore the manufacturer offsets thenegative effect by improving the greenness Whenthe carbon tax is higher and development cost is lowthe greenness of products will increase with highconsumerrsquos concern about price In contrast if thecarbon tax is higher and the development cost ishigh the greenness will decrease

(3) )e greenness increases with an increasing greennesspreference of consumers )eir preference behaviorcan promote more green products )is means thatthe government can promote green production bypromoting the green consciousness of consumersMoreover the greenness decreases with increasingdevelopment cost A higher greenness is obtainedwhen manufacturers face lower development costsIn addition greenness first increases and then de-creases with increases in the functional attributecoefficient When the functional attribute of thegreen product is low the manufacturer will producea lower greenness product with an increase in thefunctional attribute In contrast the manufacturerwill produce a higher greenness product with anincrease in the functional attribute after a func-tionality threshold

(4) We investigated the impact of the carbon tax onpromoting the coordination of the supply chain bycomparing the profit changes of two partners afterthe introduction of a revenue-sharing contract inProposition 3 We found that coordination could bepromoted by increasing the carbon tax

(5) )e achievement of coordination depends on thetype of manufacturer If the power of the manu-facturer is low coordination can be achievedunder the optimal carbon tax If the power of themanufacturer is at a medium level coordinationcan be achieved by increasing the carbon tax andimproved greenness will be realized with a loss ofrevenue However when the power of the man-ufacturer is strong coordination cannot beachieved

In this study we focused on the interaction between acarbon tax and supply chain cooperation However othercarbon policies (eg a cap-and-trade policy) and supplychain structures are worth exploring For example it wouldbe interesting to investigate the effects on a supply chainstructure consisting of the retailer as the leader under acomplex carbon policy

Table 3 Coordination situations for different manufacturer power levels

Power of the manufacturer Coordination Greenness Total carbon tax revenue05lt λm lt 1113954λ1 Achieve Improve Improve1113954λ1 lt λm lt 1113954λ2 clt 1113954c1 Achieve Improve Decline1113954λ1 lt λm lt 1113954λ2 clt 1113954c1 Cannot achievecgt 1113954c1 Cannot achieve

10 Mathematical Problems in Engineering

Appendix

A Proof of Theorem 1

)e demand function can be written as

q P ug ge un ug ge 01113966 1113967

qn P un gt ug un ge 01113966 1113967

⎧⎪⎨

⎪⎩(A1)

After replacing un and ug we have

q P αv minus p + kg minus β p minus pn( 1113857 minus v minus pn( 1113857ge 0 αv minus p + kg minus β p minus pn( 1113857ge 01113864 1113865

qn P v minus pn minus αv minus p + kg minus β p minus pn( 1113857( 1113857gt 0 v minus pn ge 01113864 11138651113896 (A2)

Given that V is uniformly distributed on [0 1] thedemand function is

q (α + β)pn minus (1 + β)p + kg

α(1 minus α)

qn 1 minus(1 + β) pn minus p( 1113857 + kg

1 minus α

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A3)

)e condition kgt ((1 + β)p minus (α + β)png) is necessaryto ensure nonnegativity )e retailer problem is addressedfirst and we have

p (α + β)pn +(1 + β)w + kg

2(1 + β) (A4)

)en substitute p into the manufacturerrsquos profit func-tion Solving the optimal w and g simultaneously yields

w (α + β)pn +(1 + β)(c + t(e minus g)) + kg

2(1 + β)

g t (α + β)pn minus (1 + β)w + ke( 1113857

2cα(1 minus α)

(A5)

)en we have

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

g (1 + β)(te + c) minus (α + β)pn minus ke( 1113857t

t2(1 + β) minus 4cα(1 minus α)

p 3cpnα

3+ 2A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1 (1 + β)(c + te) + ke minus (1 minus β)pn

A2 (1 + β)(c + te) + 3ke minus (3 minus β)pn

⎧⎪⎨

⎪⎩

B1 minus 2c (1 + β)(c + te) + ke + βpn( 1113857 + t2(1 + β)pn

B2 minus c (1 + β)(c + te) + 3ke + 3βpn( 1113857 + t2(1 + β)pn

⎧⎪⎨

⎪⎩

(A6)

Notes that we have (α + β)pn + ke minus (1 + β)(te + c)gt 0)e concavity condition is 4αc(1 minus α) minus (1 + β)t2 gt 0

B Proof of Proposition 1

(1) )e first derivative of g in t is

zg

zt

minus 4cα3pn + 4cA3α2

+ B3α minus t2(1 + β) (1 + β)c minus ke minus βpn( 1113857

t2(1 minus β) minus 4cα(1 minus α)1113872 1113873

2

A3 (1 + β)(c + 2te) minus ke +(1 minus β)pn

B3 minus 4c (1 + β)(c + 2te) minus ke minus βpn( 1113857 + t2(1 + β)pn

⎧⎨

⎩

(B1)

We have a threshold

1113954c1 minust2(1 + β) (α + β)pn + ke minus (1 + β)c( 1113857

4α(1 minus α) (α + β)pn + ke minus (1 + β)(c + 2te)( 1113857

1113954t1 (α + β)pn + ke minus (1 + β)c

2(1 + β)e

(B2)

When cgt 1113954c1 and tgt1113954t1 g is decreasing in t Bycontrast g is increasing in t in other cases

(2) )e first derivative of g in β is

zg

zβ

minus 4α(1 minus α) te + c minus pn( 1113857c + t2

ke + αpn minus pn( 1113857

t2(1 + β) minus 4α(1 minus α)c1113872 1113873

2

(B3)

We have a threshold

1113954c2 minust2

ke minus (1 minus α)pn( 1113857

4α(1 minus α) te + c minus pn( 1113857

1113954t2 pn minus c

e

(B4)

Review we have (α + β)pn + ke minus (1 + β)(te + c)gt 0therefore we can find that

tltpn minus c

e+

ke minus (1 minus α)pn

(1 + β)e (B5)

(i) When ke minus (1 minus α)pn lt 0 then tlt1113954t2 is inevitable)erefore (zgzβ)gt 0 is correct

(ii) When ke minus (1 minus α)pn gt 0 we can find that(zgzβ)lt 0 if cgt 1113954c2 and tgt1113954t2 (zgzβ)lt 0 in othercases

(3) )e first derivative of g in k and c is

Mathematical Problems in Engineering 11

zg

zc minus

4α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

4cα(1 minus α) minus t2(1 + β)1113872 1113873

2 lt 0

zg

zk

te

4cα(1 minus α) minus t2(1 + β)gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(B6)

(4) )e first derivative of g in α is

zg

zα

minus Xt

4α(1 minus α)c minus t2(1 + β)1113872 1113873

2 (B7)

And X minus 4α2cpn + (4(1 + β)(te + c) minus 4ke minus 4pn)

(2α + 1)c + pnt2(1 + β) is a quadratic function )erange of α in 4αc(1 minus α) minus (1 + β)t2 gt 0 is

α1 c +

c2

minus ct2(1 + β)

1113969

2c

α2 c minus

c2

minus ct2(1 + β)

1113969

2c

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(B8)

By substituting those into X we have

X|αα1 4

c2

minus ct2(1 + β)

1113969

T minus 2 c minus t2(1 + β)1113872 1113873pn

X|αα2 minus 4

c2

minus ct2(1 + β)

1113969

T minus 2 c minus t2(1 + β)1113872 1113873pn lt 0

⎧⎪⎪⎨

⎪⎪⎩

(B9)

And T (1 + β)(te + c) minus ke minus ((12) + β)pn lt 0We can find

X|αα1 times X|αα2 minus 4 c minus t2(1 + β)1113872 1113873

middot 4 βpn + ke minus (1 + β)(te + c)( 11138572

1113872

+ p2nt

2(1 + β)1113873lt 0

(B10)

)erefore X|αα1 gt 0 )ere only exist a 1113954α isin (α2 α1)that makes (zgzα) 0 And (zgzα) gt 0 ifα isin (α2 1113954α) (zgzα)lt 0 if α isin (1113954α α1)

(5) )e first derivative of p in t is

zp

zt minus

2α(α minus 1) e(1 + β)t221113872 1113873 + minus pnα minus ke + c minus pn( 1113857β + c( 1113857t minus 2eαc(α minus 1)1113872 1113873c

(1 + β)t2

+ 4αc(α minus 1)1113872 11138732

A3 (1 + β)(c + 2te) minus ke +(1 minus β)pn

B3 minus 4c (1 + β)(c + 2te) minus ke minus βpn( 1113857 + t2(1 + β)pn

⎧⎪⎨

⎪⎩

(B11)

)e discriminant of quadratic function

e(1 + β)t2

2+ minus pnα minus ke + c minus pn( 1113857β + c( 1113857t minus 2eαc(α minus 1)

(B12)

is

Δ (minus β minus 1)c + ke + βpn + pnα( 11138572

+ 4e2(1 + β)t

2αc(α minus 1)

(B13)

We have a threshold

1113954c minus(minus β minus 1)c + ke + βpn + pnα( 1113857

2

4e2(1 + β)t

2α(α minus 1) (B14)

Review we have(α + β)pn + ke minus (1 + β)(te + c)gt 0therefore we can find that

tm 2

minus (1 + β)αc(α minus 1)

1113968

1 + β (B15)

Substituting it into above quadratic function

2 c

radicminus pnα minus ke + c minus pn( 1113857β + c( 1113857

minus (1 + β)α(α minus 1)

1113968minus 4eαc(α minus 1)(1 + β)

1 + β (B16)

And we have a same threshold

1113954c minus(minus β minus 1)c + ke + βpn + pnα( 1113857

2

4e2(1 + β)t

2α(α minus 1) (B17)

It could be found that(zpzt) gt 0 because Δlt 0 whenclt 1113954c And when cgt 1113954c we have another threshold 1113954t andzpzt|

t1113954t 0 because Δgt 0 and zpzt|tmlt 0)erefore when

cgt 1113954c and tgt1113954t p is decreasing in t And when cgt 1113954c and tlt1113954tp is increasing in t

12 Mathematical Problems in Engineering

C Proof of Proposition 2

(1) Let R t(e minus g)q denote total carbon tax and thefirst derivative of R in t is

zR

zt2c A2t

2+ B2t + c1113872 1113873 A3t

2+ B3t + c1113872 1113873

4cα(1 minus α) minus t2(1 + β)1113872 1113873

3 (C1)

Among them

A2 (1 + β)e

2gt 0

B2 minus (α + β)pn + ke minus (1 + β)c( 1113857lt 0

C2 2ceα(1 minus α)gt 0

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

A3 (α + β)pn + ke minus (1 + β)c( 1113857(1 + β)gt 0

B3 minus 8ceα(1 minus α)(1 + β)lt 0

C3 4cα(1 minus α) (α + β)pn + ke minus (1 + β)c( 1113857gt 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(C2)

Let Φ A2t2 + B2t + C2 and Ψ A3t

2 + B3t + C3 andwe can find that zRzt and Φ times Ψ have a same sign Bysolving Δ1 and Δ2 a threshold is

1113954c3 (α + β)pn + ke minus (1 + β)c( 1113857

2

4e2α(1 minus α)(1 + β)

(C3)

)ere are two positive real numbers of solutions forΦ ifclt 1113954c3 and let 0lt ta lt tb denote it Similarly let0lt tc lt td denote the solutions for Ψ if cgt 1113954c3 Reviewwe have 4αc(1 minus α) minus (1 + β)t2 gt 0 therefore lettm

4cα(1 minus a)1 + β

1113968 Substituting tm into Φ and Ψ

Φ|tm minus 2

cα(1 minus α)

1 + β

1113971

(α + β)pn + ke minus (1 + β)c( 1113857 + 4ecα(1 minus α)

Ψ|tm minus 8cα(1 minus α) 2(1 + β)e

α(1 minus α)

1 + βc

1113971

minus (α + β)pn + ke minus (1 + β)c( 1113857⎛⎝ ⎞⎠

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(C4)

If clt 1113954c3 Φ|tmlt 0 and Ψ|tm

gt 0 then zRzt|tmlt 0 If

cgt 1113954c3 Φ|tmgt 0 and Ψ|tm

lt 0 then zRzt|tmlt 0 )ere-

fore we have tm isin (ta tb) or tm isin (tc td) FurthermoreR t(e minus g)q is maximized when t ta if clt 1113954c3 R

t(e minus g)q is maximized when t tc if cgt 1113954c3 Totally wehave

tlowast

ta clt 1113954c3

tc cgt 1113954c31113896 (C5)

(2) Review Proposition 1 If cgt 1113954c1 g is maximized whentlowast 1113954t1 If clt 1113954c1 g is maximized when tlowast tmWhen tgt1113954t1 1113954c1 gt 1113954c5 And 1113954c1 lt 0lt 1113954c5 if tlt1113954t1Substituting 1113954t1 into Φ

Φ|1113954t1 minus

3 (α + β)pn + ke minus (1 + β)c( 11138572

8e(1 + β)+ 2ecα(1 minus α)

(C6)

A threshold is 1113954c4 (34)1113954c3 )erefore we have (i)when 0lt clt 1113954c4 tlowast ta because of Φ|1113954t1

lt 0 and 1113954t1 gt ta

(ii) when 1113954c4 lt clt 1113954c3 tlowast ta because of 1113954t1 lt ta (iii) tlowast

tc when 1113954c3 lt clt 1113954c1 and (iv) tlowast tc when 1113954c1 lt c )eresult is obtained by solving Φ and Ψ

D Proof of Theorem 2

Similar to Appendix A )e retailer problem is addressedfirst and we have

p αηpn + βηpn + ηkg + βw + w

2η(β + 1) (D1)

)en substituting p into the manufacturerrsquos profitfunction Solving the optimal w and g simultaneously yields

w αηpn + βet + βηpn minus βgt + eηk + cβ + et minus gt + c( 1113857η

(1 + η)(β + 1)

g minusαηpn + βηpn + eηk minus βw minus w( 1113857t

2(α minus 1)cαη

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(D2)

Mathematical Problems in Engineering 13

By solving the equation above we have

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

g t minus αpn + et(β + 1) + c minus pn( 1113857β minus ke + c( 1113857

t2(1 + β) minus (1 + η)cα(1 minus α)

p (2η + 1)cpnα

3+ A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1 et + ηpn + c( 1113857β + et + ke minus pn( 1113857η + c

A2 (2η + 1)pn + et + c( 1113857β +(minus 2η minus 1)pn + et +(2ηk + k)e + c

⎧⎪⎪⎨

⎪⎪⎩

B1 minus 2et minus 2ηpn minus 2c( 1113857β minus 2eηk minus 2et minus 2c( 1113857c + t2pn(β + 1)

B2 (minus 2η minus 1)pn minus et minus c( 1113857β minus et +(minus 2ηk minus k)e minus c( 1113857c + t2pn(β + 1)

⎧⎪⎪⎨

⎪⎪⎩

(D3)

Note that we have minus αpn + et(β + 1) + (c minus pn)β minus ke +

clt 0 )e concavity condition is (1 + β)(t2(1 + β) minus (1+

η)cα(1 minus α))lt 0

Substituting solutions into the profit function we have

πm minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2c

2 t2β + t

2+ 4(minus 1 + α)cα1113872 1113873(β + 1)

πr minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2(minus 1 + α)αc

2

4α2c minus 4αc +(β + 1)t2

1113872 11138732(β + 1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(D4)

E Proof of Proposition 3

πM (α + β)pn + ke minus (1 + β)c( 1113857

2c

2(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCM

(α + β)pn + ke minus (1 + β)c( 11138572c

2(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(E1)

By comparing the profits of two partners change after therevenue-sharing contract introduced we found that themanufacturer always benefits from cooperation

πR c2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCR

ηc2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(E2)

A threshold is

η1 2cα(1 minus α) minus (1 + β)t2

2cα(1 minus α)1113888 1113889

2

(E3)

When η is in the region δ (η1 1) the retailer willaccept cooperation And the first derivative of η1 in t is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t(β + 1)

α2c2(minus 1 + α)

2 lt 0 (E4)

)erefore η1 is decreasing in t )at means the regionδ (η1 1) is expanding when tax rises And the revenue-sharing contract is accepted easily than before

F Proof of Proposition 4

We consider a simple function (λrλm) to decide thepractical η where λm isin (05 1) and λr 1 minus λm Given the

14 Mathematical Problems in Engineering

carbon tax tlowast we have a region δ (η(tlowast) 1) of cooperationIt is clear that the gap of strength between two partitions inthe supply chain is a crucial factor in cooperation Bycomparing practice η decided by strength and η(tlowast) de-cided by the carbon tax we have

1λm

minus 1 2α2c minus 2αc + t

2(β + 1)1113872 1113873

2

4α2c2(minus 1 + α)

2 (F1)

A threshold is

1113954λ1 4α2c2

(minus 1 + α)2

8α4c2minus 16α3c2

+ 4 2c + t2(β + 1)1113872 1113873cα2 minus 4t

2c(β + 1)α + t

4(β + 1)

2 (F2)

When 05lt λm lt 1113954λ1 the practical η is in the regionδ (η(tlowast) 1) therefore the cooperation is achieved facingthe optimal carbon tax

However the practical η is out of the range δ (η(tlowast) 1)

when 1113954λ1 lt λm Review previous result the region δ (η1 1)

is expanding when tax rises )erefore there exists a pos-sibility that the government increasing carbon tax to co-ordinate the supply chain We assume a realistic thing thegovernment is willing to boost the greenness of the producteven reducing part of the revenue from the carbon tax

Based on it the carbon up to 1113954t1 which means thegreenness is maximized )erefore )e cooperation isachieved when clt 1113954c1 because the greenness is decreasing inthe carbon tax when 1113954c1 lt c )en we have the secondthreshold

1113954λ2 1

η1 1113954t1( 1113857 + 1 (F3)

)e cooperation could not be achieved when λm gt 1113954λ2)e conclusion is as follows (1) Coordination would be

achieved when the manufacturer is democratic (ie05lt λm lt 1113954λ1) (2) Coordination could be achieved but partof the total tax will lose if development cost is lower when themanufacturer is moderate (ie 1113954λ1 lt λm lt 1113954λ2) (3) )e co-operation could not be achieved if the manufacturer is adictator (ie λm gt 1113954λ2)

G Proof of Proposition 5

)e first derivative of η1 in β is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t

2

2α2c2(minus 1 + α)

2 lt 0 (G1)

Similarly η1 is decreasing in β )at means the regionδ (η1 1) is expanding when β rises

And it is apparent that η1 (2cα(1 minus α) minus (1 + β)t22cα(1 minus α))2 is independent with k

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is research was supported by the National Natural ScienceFoundation of China (no 71761015) and the Science andTechnology Research Project of Jiangxi Education Depart-ment of China (no GJJ150475)

References

[1] Y Li M K Lim J Hu and M-L Tseng ldquoInvestigating theeffect of carbon tax and carbon quota policy to achieve lowcarbon logistics operationsrdquo Resources Conservation andRecycling vol 154 p 104535 2020

[2] M Abedi-Varaki ldquoStudy of carbon dioxide gas treatmentbased on equations of kinetics in plasma discharge reactorrdquoModern Physics Letters B vol 31 no 22 p 1750210 2017

[3] S Meng M Siriwardana and J McNeill ldquo)e environmentaland economic impact of the carbon tax in Australiardquo Envi-ronmental and Resource Economics vol 54 no 3 pp 313ndash332 2013

[4] Y Chen and C L Tseng ldquoInducing clean technology in theelectricity sector tradable permits or carbon tax policiesrdquo-eEnergy Journal vol 32 no 3 2011

[5] H-B Duan L Zhu and Y Fan ldquoOptimal carbon taxes incarbon-constrained China a logistic-induced energy eco-nomic hybrid modelrdquo Energy vol 69 pp 345ndash356 2014

[6] C Lu Q Tong and X Liu ldquo)e impacts of carbon tax andcomplementary policies on Chinese economyrdquo Energy Policyvol 38 no 11 pp 7278ndash7285 2010

[7] K Cao B Xu and J Wang ldquoOptimal trade-in and warrantyperiod strategies for new and remanufactured products undercarbon tax policyrdquo International Journal of Production Re-search vol 58 no 1 pp 180ndash199 2020

[8] P He W Zhang X Xu and Y Bian ldquoProduction lot-sizingand carbon emissions under cap-and-trade and carbon taxregulationsrdquo Journal of Cleaner Production vol 103pp 241ndash248 2015

[9] X Xu X Xu and P He ldquoJoint production and pricing de-cisions for multiple products with cap-and-trade and carbontax regulationsrdquo Journal of Cleaner Production vol 112pp 4093ndash4106 2016

Mathematical Problems in Engineering 15

[10] Q Li X Guan T Shi and W Jiao ldquoGreen product designwith competition and fairness concerns in the circulareconomy erardquo International Journal of Production Researchvol 58 no 1 pp 165ndash179 2020

[11] A Ranjan and J K Jha ldquoPricing and coordination strategiesof a dual-channel supply chain considering green quality andsales effortrdquo Journal of Cleaner Production vol 218pp 409ndash424 2019

[12] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineeringvol 10 no 1 pp 99ndash116 2012

[13] X Yin X Chen X Xu and L Zhang ldquoTax or subsidyOptimal carbon emission policy a supply chain perspectiverdquoSustainability vol 12 no 4 p 1548 2020

[14] B B Schlegelmilch G M Bohlen and A Diamantopoulosldquo)e link between green purchasing decisions and measuresof environmental consciousnessrdquo European Journal of Mar-keting vol 30 no 5 pp 35ndash55 1996

[15] M S Hopkins ldquoWhat the rsquogreenrsquo consumer wantsrdquo MITSloan Management Review vol 50 no 4 pp 87ndash89 2009

[16] G Kalyanaram and R S Winer ldquoEmpirical generalizationsfrom reference price researchrdquo Marketing Science vol 14no 3 pp G161ndashG169 1995

[17] E A Greenleaf ldquo)e impact of reference price effects on theprofitability of price promotionsrdquo Marketing Science vol 14no 1 pp 82ndash104 1995

[18] N M Modak D K Ghosh S Panda and S S SanaldquoManaging green house gas emission cost and pricing policiesin a two-echelon supply chainrdquo CIRP Journal ofManufacturing Science and Technology vol 20 pp 1ndash11 2018

[19] P Sinclair ldquoHigh does nothing and rising is worse carbontaxes should keep declining to cut harmful emissionsrdquo -eManchester School vol 60 no 1 pp 41ndash52 1992

[20] G M Grossman and A B Krueger ldquoEconomic growth andthe environmentrdquo -e Quarterly Journal of Economicsvol 110 no 2 pp 353ndash377 1995

[21] N M Modak and P Kelle ldquoUsing social work donation as atool of corporate social responsibility in a closed-loop supplychain considering carbon emissions tax and demand uncer-taintyrdquo Journal of the Operational Research Society pp 1ndash172019

[22] A Ulph and D Ulph ldquo)e optimal time path of a carbon taxrdquoOxford Economic Papers vol 46 no S1 pp 857ndash868 1994

[23] M Hoel and S Kverndokk ldquoDepletion of fossil fuels and theimpact of global warmingrdquo Resource and Energy Economicsvol 18 no 2 pp 115ndash136 1996

[24] Y H Farzin and O Tahvonen ldquoGlobal carbon cycle and theoptimal time path of a carbon taxrdquo Oxford Economic Papersvol 48 no 4 pp 515ndash536 1996

[25] V Bosetti C Carraro R Duval andM Tavoni ldquoWhat shouldwe expect from innovation A model-based assessment of theenvironmental and mitigation cost implications of climate-related RampDrdquo Energy Economics vol 33 no 6 pp 1313ndash13202011

[26] M Hariga R Asrsquoad and A Shamayleh ldquoIntegrated economicand environmental models for a multi stage cold supply chainunder carbon tax regulationrdquo Journal of Cleaner Productionvol 166 pp 1357ndash1371 2017

[27] N Turken J Carrillo and V Verter ldquoFacility location andcapacity acquisition under carbon tax and emissions limits tocentralize or to decentralizerdquo International Journal of Pro-duction Economics vol 187 pp 126ndash141 2017

[28] W Yu and R Han ldquoCoordinating a two-echelon supply chainunder carbon taxrdquo Sustainability vol 9 no 12 p 2360 2017

[29] S Sinha and N M Modak ldquoAn EPQmodel in the perspectiveof carbon emission reductionrdquo International Journal ofMathematics in Operational Research vol 14 no 3pp 338ndash358 2019

[30] H Zhang P Li H Zheng and Y Zhang ldquoImpact of carbontax on enterprise operation and production strategy for low-carbon products in a co-opetition supply chainrdquo Journal ofCleaner Production Article ID 125058 2020

[31] X Chen H Yang X Wang et al ldquoOptimal carbon tax designfor achieving low carbon supply chainsrdquo Annals of OperationsResearch pp 1ndash28 2020

[32] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Handbook of the Fundamentals of Fi-nancial Decision Making Part I pp 99ndash127 )e EconometricSociety Cleveland OH USA 2013

[33] W Kim and M Kim ldquoReference quality-based competitivemarket structure for innovation driven marketsrdquo Interna-tional Journal of Research in Marketing vol 32 no 3pp 284ndash296 2015

[34] I Popescu and Y Wu ldquoDynamic pricing strategies withreference effectsrdquo Operations Research vol 55 no 3pp 413ndash429 2007

[35] P K Kopalle P K Kannan L B Boldt and N Arora ldquo)eimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Re-tailing vol 88 no 1 pp 102ndash114 2012

[36] T-P Hsieh and C-Y Dye ldquoOptimal dynamic pricing fordeteriorating items with reference price effects when inven-tories stimulate demandrdquo European Journal of OperationalResearch vol 262 no 1 pp 136ndash150 2017

[37] X Chen Z-Y Hu and Y-H Zhang ldquoDynamic pricing withstochastic reference price effectrdquo Journal of the OperationsResearch Society of China vol 7 no 1 pp 107ndash125 2019

[38] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquo Opera-tions Research vol 64 no 6 pp 1529ndash1536 2016

[39] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[40] C Chen ldquoDesign for the environment a quality-based modelfor green product developmentrdquo Management Sciencevol 47 no 2 pp 250ndash263 2001

[41] K Chitra ldquoIn search of the green consumers a perceptualstudyrdquo Journal of Services Research vol 7 no 1 2007

[42] Z Liu T D Anderson and J M Cruz ldquoConsumer envi-ronmental awareness and competition in two-stage supplychainsrdquo European Journal of Operational Research vol 218no 3 pp 602ndash613 2012

[43] B Li Y Wang and Z Wang ldquoManaging a closed-loop supplychain with take-back legislation and consumer preference forgreen designrdquo Journal of Cleaner Production Article ID124481 2020

[44] J Rantanen D B Grant and W Piotrowicz ldquoInvestigatingsupply chain cooperation in Finnish grocery retailrdquo ResearchJournal of the University of Gdansk Transport Economics andLogistics vol 71 pp 19ndash34 2017

[45] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[46] G Xu B Dan X Zhang and C Liu ldquoCoordinating a dual-channel supply chain with risk-averse under a two-way

16 Mathematical Problems in Engineering

revenue sharing contractrdquo International Journal of ProductionEconomics vol 147 pp 171ndash179 2014

[47] Z Shi N Wang T Jia et al ldquoReverse revenue sharingcontract versus two-part tariff contract under a closed-loopsupply chain systemrdquoMathematical Problems in Engineeringvol 2016 Article ID 5464570 15 pages 2016

[48] S Panda N M Modak and L E Cardenas-Barron ldquoCo-ordinating a socially responsible closed-loop supply chainwith product recyclingrdquo International Journal of ProductionEconomics vol 188 pp 11ndash21 2017

[49] N M Modak N Modak S Panda and S S Sana ldquoAnalyzingstructure of two-echelon closed-loop supply chain for pricingquality and recycling managementrdquo Journal of Cleaner Pro-duction vol 171 pp 512ndash528 2018

[50] Q P Wang and D Z Zhao ldquoRevenue-sharing contract ofsupply chain based on consumerrsquos preference for low carbonproductsrdquo Chinese Journal of Management Science vol 22no 9 pp 106ndash113 2014

[51] B Yu J Wang X Lu and H Yang ldquoCollaboration in a low-carbon supply chain with reference emission and cost learningeffects cost sharing versus revenue sharing strategiesrdquo Journalof Cleaner Production vol 250 Article ID 119460 2020

[52] J Cao X Zhang and G Zhou ldquoSupply chain coordinationwith revenue-sharing contracts considering carbon emissionsand governmental policy makingrdquo Environmental Progress ampSustainable Energy vol 35 no 2 pp 479ndash488 2016

[53] X Xu P He H Xu and Q Zhang ldquoSupply chain coordi-nation with green technology under cap-and-trade regula-tionrdquo International Journal of Production Economics vol 183pp 433ndash442 2017

[54] N M Modak S Sinha S Panda et al ldquoAnalyzing a sociallyresponsible closed-loop distribution channel with recyclingfacilityrdquo SN Applied Sciences vol 1 no 10 p 1189 2019

[55] D Feng L Ma Y Ding G Wu and Y Zhang ldquoDecisions ofthe dual-channel supply chain under double policy consid-ering remanufacturingrdquo International Journal of Environ-mental Research and Public Health vol 16 no 3 p 465 2019

[56] I Bellos M Ferguson and L B Toktay ldquo)e car sharingeconomy interaction of business model choice and productline designrdquo Manufacturing amp Service Operations Manage-ment vol 19 no 2 pp 185ndash201 2017

[57] G Ferrer and J M Swaminathan ldquoManaging new andremanufactured productsrdquo Management Science vol 52no 1 pp 15ndash26 2006

[58] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2pp 519ndash527 2011

[59] L Cui S Guo and H Zhang ldquoCoordinating a green agri-foodsupply chain with revenue-sharing contracts consideringretailersrsquo green marketing effortsrdquo Sustainability vol 12no 4 p 1289 2020

[60] J Heydari K Govindan and A Aslani ldquoPricing and greeningdecisions in a three-tier dual channel supply chainrdquo Inter-national Journal of Production Economics vol 217 pp 185ndash196 2019

Mathematical Problems in Engineering 17

Proposition 4 is designed to answer the question ofwhether cooperation will be achieved the proofs are foundin Appendix F

Proposition 4 Coordination will be achieved if the power ofthe manufacturer is low (ie 05lt λm lt 1113954λ1) Moreover whenthe power of the manufacturer is at a medium level (ie1113954λ1 lt λm lt 1113954λ2) coordination can be achieved but part of thetotal tax revenue will be lost if the development cost is low(ie clt 1113954c1) In other cases (ie cgt 1113954c1 or 1113954λlt λm) coordi-nation cannot be achieved because the power of the manu-facturer is overwhelming

Coordination situations for different manufacturerpower levels are present in Table 3 )e achievement ofcoordination is decided by the power of the manufacturerand the optimal carbon tax faced by the supply chain Basedon Proposition 3 coordination can be promoted if thegovernment adjusts the carbon tax On the other handcoordination can easily be achieved under the optimalcarbon tax if the power of the manufacturer is low Inaddition there are two possibilities when the power of themanufacturer is at a medium level If the development cost istoo high adjusting the tax will lead to an unknown result thegreenness and total tax revenue may both decline at the same

10 014

012

010

008

006

004

002

000

08

06

Gre

enne

ss

Tota

l tax

04

02

00005000 010 015 020

t

γ lt γ3

t = ta

025 030

GreennessTotal tax

(a)

10 030

025

020

015

010

005

000

08

06

Gre

enne

ss

Tota

l tax

04

02

000100 02 03 04

t

γ3 lt γ lt γ1

t = tc

05 06

GreennessTotal tax

(b)

005 040

035

025

020

015

010

005

000

004

003

Gre

enne

ss

Tota

l tax

002

001

0000100 02 03 04

t

γ1 lt γ

t = t1

06 08

030

05 07

GreennessTotal tax

(c)

Figure 4 Determination of the optimal carbon tax (a) 0lt clt 1113954c3 (b) 1113954c3 lt clt 1113954c1 (c) 1113954c1 lt c

005t

η

010 015 020 025 030

Pareto improvement

035 040

Figure 5 Pareto improvement region with increasing carbon tax

Mathematical Problems in Engineering 9

time If the development cost is lower the greenness may beimproved with a loss of revenue by increasing the carbon taxHowever when the power of the manufacturer is strong theregion δ (η1 1) cannot be achieved

According to our theory the government can promotethe coordination between the manufacturer and the retailerby adjusting the carbon tax When the power of the man-ufacturer increases the government still achieves coordi-nation by raising the carbon tax In general cooperation isachieved and the government plays a key role in promotingcooperation

)e effects of consumersrsquo behavior on supply chaincoordination are also investigated in the next proposition)e proofs are provided in Appendix G

Proposition 5 Price reference behavior can promote supplychain coordination but green preference cannot

From the previous results price reference has a compleximpact on the supply chain First the increasing sensitivityof customers to price will promote supply chain coordi-nation because this negative impact can be mitigated bycooperation In addition price reference behavior also af-fects the green decisions of the supply chain in a complexmanner In contrast the mechanism by which consumerrsquogreen preferences influence the supply chain is relativelysimple Greenness increases with increasing greennesspreference and green preference has no effect on the co-ordination of the supply chain

4 Conclusion

)is study examines the interaction between supply chaincooperation and the carbon tax problem in a two-echelonsupply chain under consumerrsquos reference behavior )e opti-mal carbon tax policy is obtained based on analysis of thecarbon tax green investment coefficient and degree of con-sumerrsquos price reference )e optimal carbon tax is defined asthe simultaneous optimization of total tax revenue and productgreenness In addition coordination is achieved by introducingrevenue-sharing contracts)e impact of consumersrsquo referencebehavior and the carbon tax on supply chain coordination isalso investigated )e results are as follows

(1) )e greenness increases with increasing carbon taxwhen the carbon tax is low In addition if the carbontax is higher and the development cost is low thegreenness of products will increase with increasingcarbon tax Conversely if the carbon tax is higherand the development cost is also high the greennessof products will decrease

(2) When the carbon tax is low the negative effect ofprice increases as consumer concerns about priceincrease )erefore the manufacturer offsets thenegative effect by improving the greenness Whenthe carbon tax is higher and development cost is lowthe greenness of products will increase with highconsumerrsquos concern about price In contrast if thecarbon tax is higher and the development cost ishigh the greenness will decrease

(3) )e greenness increases with an increasing greennesspreference of consumers )eir preference behaviorcan promote more green products )is means thatthe government can promote green production bypromoting the green consciousness of consumersMoreover the greenness decreases with increasingdevelopment cost A higher greenness is obtainedwhen manufacturers face lower development costsIn addition greenness first increases and then de-creases with increases in the functional attributecoefficient When the functional attribute of thegreen product is low the manufacturer will producea lower greenness product with an increase in thefunctional attribute In contrast the manufacturerwill produce a higher greenness product with anincrease in the functional attribute after a func-tionality threshold

(4) We investigated the impact of the carbon tax onpromoting the coordination of the supply chain bycomparing the profit changes of two partners afterthe introduction of a revenue-sharing contract inProposition 3 We found that coordination could bepromoted by increasing the carbon tax

(5) )e achievement of coordination depends on thetype of manufacturer If the power of the manu-facturer is low coordination can be achievedunder the optimal carbon tax If the power of themanufacturer is at a medium level coordinationcan be achieved by increasing the carbon tax andimproved greenness will be realized with a loss ofrevenue However when the power of the man-ufacturer is strong coordination cannot beachieved

In this study we focused on the interaction between acarbon tax and supply chain cooperation However othercarbon policies (eg a cap-and-trade policy) and supplychain structures are worth exploring For example it wouldbe interesting to investigate the effects on a supply chainstructure consisting of the retailer as the leader under acomplex carbon policy

Table 3 Coordination situations for different manufacturer power levels

Power of the manufacturer Coordination Greenness Total carbon tax revenue05lt λm lt 1113954λ1 Achieve Improve Improve1113954λ1 lt λm lt 1113954λ2 clt 1113954c1 Achieve Improve Decline1113954λ1 lt λm lt 1113954λ2 clt 1113954c1 Cannot achievecgt 1113954c1 Cannot achieve

10 Mathematical Problems in Engineering

Appendix

A Proof of Theorem 1

)e demand function can be written as

q P ug ge un ug ge 01113966 1113967

qn P un gt ug un ge 01113966 1113967

⎧⎪⎨

⎪⎩(A1)

After replacing un and ug we have

q P αv minus p + kg minus β p minus pn( 1113857 minus v minus pn( 1113857ge 0 αv minus p + kg minus β p minus pn( 1113857ge 01113864 1113865

qn P v minus pn minus αv minus p + kg minus β p minus pn( 1113857( 1113857gt 0 v minus pn ge 01113864 11138651113896 (A2)

Given that V is uniformly distributed on [0 1] thedemand function is

q (α + β)pn minus (1 + β)p + kg

α(1 minus α)

qn 1 minus(1 + β) pn minus p( 1113857 + kg

1 minus α

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A3)

)e condition kgt ((1 + β)p minus (α + β)png) is necessaryto ensure nonnegativity )e retailer problem is addressedfirst and we have

p (α + β)pn +(1 + β)w + kg

2(1 + β) (A4)

)en substitute p into the manufacturerrsquos profit func-tion Solving the optimal w and g simultaneously yields

w (α + β)pn +(1 + β)(c + t(e minus g)) + kg

2(1 + β)

g t (α + β)pn minus (1 + β)w + ke( 1113857

2cα(1 minus α)

(A5)

)en we have

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

g (1 + β)(te + c) minus (α + β)pn minus ke( 1113857t

t2(1 + β) minus 4cα(1 minus α)

p 3cpnα

3+ 2A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1 (1 + β)(c + te) + ke minus (1 minus β)pn

A2 (1 + β)(c + te) + 3ke minus (3 minus β)pn

⎧⎪⎨

⎪⎩

B1 minus 2c (1 + β)(c + te) + ke + βpn( 1113857 + t2(1 + β)pn

B2 minus c (1 + β)(c + te) + 3ke + 3βpn( 1113857 + t2(1 + β)pn

⎧⎪⎨

⎪⎩

(A6)

Notes that we have (α + β)pn + ke minus (1 + β)(te + c)gt 0)e concavity condition is 4αc(1 minus α) minus (1 + β)t2 gt 0

B Proof of Proposition 1

(1) )e first derivative of g in t is

zg

zt

minus 4cα3pn + 4cA3α2

+ B3α minus t2(1 + β) (1 + β)c minus ke minus βpn( 1113857

t2(1 minus β) minus 4cα(1 minus α)1113872 1113873

2

A3 (1 + β)(c + 2te) minus ke +(1 minus β)pn

B3 minus 4c (1 + β)(c + 2te) minus ke minus βpn( 1113857 + t2(1 + β)pn

⎧⎨

⎩

(B1)

We have a threshold

1113954c1 minust2(1 + β) (α + β)pn + ke minus (1 + β)c( 1113857

4α(1 minus α) (α + β)pn + ke minus (1 + β)(c + 2te)( 1113857

1113954t1 (α + β)pn + ke minus (1 + β)c

2(1 + β)e

(B2)

When cgt 1113954c1 and tgt1113954t1 g is decreasing in t Bycontrast g is increasing in t in other cases

(2) )e first derivative of g in β is

zg

zβ

minus 4α(1 minus α) te + c minus pn( 1113857c + t2

ke + αpn minus pn( 1113857

t2(1 + β) minus 4α(1 minus α)c1113872 1113873

2

(B3)

We have a threshold

1113954c2 minust2

ke minus (1 minus α)pn( 1113857

4α(1 minus α) te + c minus pn( 1113857

1113954t2 pn minus c

e

(B4)

Review we have (α + β)pn + ke minus (1 + β)(te + c)gt 0therefore we can find that

tltpn minus c

e+

ke minus (1 minus α)pn

(1 + β)e (B5)

(i) When ke minus (1 minus α)pn lt 0 then tlt1113954t2 is inevitable)erefore (zgzβ)gt 0 is correct

(ii) When ke minus (1 minus α)pn gt 0 we can find that(zgzβ)lt 0 if cgt 1113954c2 and tgt1113954t2 (zgzβ)lt 0 in othercases

(3) )e first derivative of g in k and c is

Mathematical Problems in Engineering 11

zg

zc minus

4α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

4cα(1 minus α) minus t2(1 + β)1113872 1113873

2 lt 0

zg

zk

te

4cα(1 minus α) minus t2(1 + β)gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(B6)

(4) )e first derivative of g in α is

zg

zα

minus Xt

4α(1 minus α)c minus t2(1 + β)1113872 1113873

2 (B7)

And X minus 4α2cpn + (4(1 + β)(te + c) minus 4ke minus 4pn)

(2α + 1)c + pnt2(1 + β) is a quadratic function )erange of α in 4αc(1 minus α) minus (1 + β)t2 gt 0 is

α1 c +

c2

minus ct2(1 + β)

1113969

2c

α2 c minus

c2

minus ct2(1 + β)

1113969

2c

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(B8)

By substituting those into X we have

X|αα1 4

c2

minus ct2(1 + β)

1113969

T minus 2 c minus t2(1 + β)1113872 1113873pn

X|αα2 minus 4

c2

minus ct2(1 + β)

1113969

T minus 2 c minus t2(1 + β)1113872 1113873pn lt 0

⎧⎪⎪⎨

⎪⎪⎩

(B9)

And T (1 + β)(te + c) minus ke minus ((12) + β)pn lt 0We can find

X|αα1 times X|αα2 minus 4 c minus t2(1 + β)1113872 1113873

middot 4 βpn + ke minus (1 + β)(te + c)( 11138572

1113872

+ p2nt

2(1 + β)1113873lt 0

(B10)

)erefore X|αα1 gt 0 )ere only exist a 1113954α isin (α2 α1)that makes (zgzα) 0 And (zgzα) gt 0 ifα isin (α2 1113954α) (zgzα)lt 0 if α isin (1113954α α1)

(5) )e first derivative of p in t is

zp

zt minus

2α(α minus 1) e(1 + β)t221113872 1113873 + minus pnα minus ke + c minus pn( 1113857β + c( 1113857t minus 2eαc(α minus 1)1113872 1113873c

(1 + β)t2

+ 4αc(α minus 1)1113872 11138732

A3 (1 + β)(c + 2te) minus ke +(1 minus β)pn

B3 minus 4c (1 + β)(c + 2te) minus ke minus βpn( 1113857 + t2(1 + β)pn

⎧⎪⎨

⎪⎩

(B11)

)e discriminant of quadratic function

e(1 + β)t2

2+ minus pnα minus ke + c minus pn( 1113857β + c( 1113857t minus 2eαc(α minus 1)

(B12)

is

Δ (minus β minus 1)c + ke + βpn + pnα( 11138572

+ 4e2(1 + β)t

2αc(α minus 1)

(B13)

We have a threshold

1113954c minus(minus β minus 1)c + ke + βpn + pnα( 1113857

2

4e2(1 + β)t

2α(α minus 1) (B14)

Review we have(α + β)pn + ke minus (1 + β)(te + c)gt 0therefore we can find that

tm 2

minus (1 + β)αc(α minus 1)

1113968

1 + β (B15)

Substituting it into above quadratic function

2 c

radicminus pnα minus ke + c minus pn( 1113857β + c( 1113857

minus (1 + β)α(α minus 1)

1113968minus 4eαc(α minus 1)(1 + β)

1 + β (B16)

And we have a same threshold

1113954c minus(minus β minus 1)c + ke + βpn + pnα( 1113857

2

4e2(1 + β)t

2α(α minus 1) (B17)

It could be found that(zpzt) gt 0 because Δlt 0 whenclt 1113954c And when cgt 1113954c we have another threshold 1113954t andzpzt|

t1113954t 0 because Δgt 0 and zpzt|tmlt 0)erefore when

cgt 1113954c and tgt1113954t p is decreasing in t And when cgt 1113954c and tlt1113954tp is increasing in t

12 Mathematical Problems in Engineering

C Proof of Proposition 2

(1) Let R t(e minus g)q denote total carbon tax and thefirst derivative of R in t is

zR

zt2c A2t

2+ B2t + c1113872 1113873 A3t

2+ B3t + c1113872 1113873

4cα(1 minus α) minus t2(1 + β)1113872 1113873

3 (C1)

Among them

A2 (1 + β)e

2gt 0

B2 minus (α + β)pn + ke minus (1 + β)c( 1113857lt 0

C2 2ceα(1 minus α)gt 0

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

A3 (α + β)pn + ke minus (1 + β)c( 1113857(1 + β)gt 0

B3 minus 8ceα(1 minus α)(1 + β)lt 0

C3 4cα(1 minus α) (α + β)pn + ke minus (1 + β)c( 1113857gt 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(C2)

Let Φ A2t2 + B2t + C2 and Ψ A3t

2 + B3t + C3 andwe can find that zRzt and Φ times Ψ have a same sign Bysolving Δ1 and Δ2 a threshold is

1113954c3 (α + β)pn + ke minus (1 + β)c( 1113857

2

4e2α(1 minus α)(1 + β)

(C3)

)ere are two positive real numbers of solutions forΦ ifclt 1113954c3 and let 0lt ta lt tb denote it Similarly let0lt tc lt td denote the solutions for Ψ if cgt 1113954c3 Reviewwe have 4αc(1 minus α) minus (1 + β)t2 gt 0 therefore lettm

4cα(1 minus a)1 + β

1113968 Substituting tm into Φ and Ψ

Φ|tm minus 2

cα(1 minus α)

1 + β

1113971

(α + β)pn + ke minus (1 + β)c( 1113857 + 4ecα(1 minus α)

Ψ|tm minus 8cα(1 minus α) 2(1 + β)e

α(1 minus α)

1 + βc

1113971

minus (α + β)pn + ke minus (1 + β)c( 1113857⎛⎝ ⎞⎠

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(C4)

If clt 1113954c3 Φ|tmlt 0 and Ψ|tm

gt 0 then zRzt|tmlt 0 If

cgt 1113954c3 Φ|tmgt 0 and Ψ|tm

lt 0 then zRzt|tmlt 0 )ere-

fore we have tm isin (ta tb) or tm isin (tc td) FurthermoreR t(e minus g)q is maximized when t ta if clt 1113954c3 R

t(e minus g)q is maximized when t tc if cgt 1113954c3 Totally wehave

tlowast

ta clt 1113954c3

tc cgt 1113954c31113896 (C5)

(2) Review Proposition 1 If cgt 1113954c1 g is maximized whentlowast 1113954t1 If clt 1113954c1 g is maximized when tlowast tmWhen tgt1113954t1 1113954c1 gt 1113954c5 And 1113954c1 lt 0lt 1113954c5 if tlt1113954t1Substituting 1113954t1 into Φ

Φ|1113954t1 minus

3 (α + β)pn + ke minus (1 + β)c( 11138572

8e(1 + β)+ 2ecα(1 minus α)

(C6)

A threshold is 1113954c4 (34)1113954c3 )erefore we have (i)when 0lt clt 1113954c4 tlowast ta because of Φ|1113954t1

lt 0 and 1113954t1 gt ta

(ii) when 1113954c4 lt clt 1113954c3 tlowast ta because of 1113954t1 lt ta (iii) tlowast

tc when 1113954c3 lt clt 1113954c1 and (iv) tlowast tc when 1113954c1 lt c )eresult is obtained by solving Φ and Ψ

D Proof of Theorem 2

Similar to Appendix A )e retailer problem is addressedfirst and we have

p αηpn + βηpn + ηkg + βw + w

2η(β + 1) (D1)

)en substituting p into the manufacturerrsquos profitfunction Solving the optimal w and g simultaneously yields

w αηpn + βet + βηpn minus βgt + eηk + cβ + et minus gt + c( 1113857η

(1 + η)(β + 1)

g minusαηpn + βηpn + eηk minus βw minus w( 1113857t

2(α minus 1)cαη

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(D2)

Mathematical Problems in Engineering 13

By solving the equation above we have

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

g t minus αpn + et(β + 1) + c minus pn( 1113857β minus ke + c( 1113857

t2(1 + β) minus (1 + η)cα(1 minus α)

p (2η + 1)cpnα

3+ A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1 et + ηpn + c( 1113857β + et + ke minus pn( 1113857η + c

A2 (2η + 1)pn + et + c( 1113857β +(minus 2η minus 1)pn + et +(2ηk + k)e + c

⎧⎪⎪⎨

⎪⎪⎩

B1 minus 2et minus 2ηpn minus 2c( 1113857β minus 2eηk minus 2et minus 2c( 1113857c + t2pn(β + 1)

B2 (minus 2η minus 1)pn minus et minus c( 1113857β minus et +(minus 2ηk minus k)e minus c( 1113857c + t2pn(β + 1)

⎧⎪⎪⎨

⎪⎪⎩

(D3)

Note that we have minus αpn + et(β + 1) + (c minus pn)β minus ke +

clt 0 )e concavity condition is (1 + β)(t2(1 + β) minus (1+

η)cα(1 minus α))lt 0

Substituting solutions into the profit function we have

πm minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2c

2 t2β + t

2+ 4(minus 1 + α)cα1113872 1113873(β + 1)

πr minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2(minus 1 + α)αc

2

4α2c minus 4αc +(β + 1)t2

1113872 11138732(β + 1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(D4)

E Proof of Proposition 3

πM (α + β)pn + ke minus (1 + β)c( 1113857

2c

2(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCM

(α + β)pn + ke minus (1 + β)c( 11138572c

2(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(E1)

By comparing the profits of two partners change after therevenue-sharing contract introduced we found that themanufacturer always benefits from cooperation

πR c2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCR

ηc2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(E2)

A threshold is

η1 2cα(1 minus α) minus (1 + β)t2

2cα(1 minus α)1113888 1113889

2

(E3)

When η is in the region δ (η1 1) the retailer willaccept cooperation And the first derivative of η1 in t is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t(β + 1)

α2c2(minus 1 + α)

2 lt 0 (E4)

)erefore η1 is decreasing in t )at means the regionδ (η1 1) is expanding when tax rises And the revenue-sharing contract is accepted easily than before

F Proof of Proposition 4

We consider a simple function (λrλm) to decide thepractical η where λm isin (05 1) and λr 1 minus λm Given the

14 Mathematical Problems in Engineering

carbon tax tlowast we have a region δ (η(tlowast) 1) of cooperationIt is clear that the gap of strength between two partitions inthe supply chain is a crucial factor in cooperation Bycomparing practice η decided by strength and η(tlowast) de-cided by the carbon tax we have

1λm

minus 1 2α2c minus 2αc + t

2(β + 1)1113872 1113873

2

4α2c2(minus 1 + α)

2 (F1)

A threshold is

1113954λ1 4α2c2

(minus 1 + α)2

8α4c2minus 16α3c2

+ 4 2c + t2(β + 1)1113872 1113873cα2 minus 4t

2c(β + 1)α + t

4(β + 1)

2 (F2)

When 05lt λm lt 1113954λ1 the practical η is in the regionδ (η(tlowast) 1) therefore the cooperation is achieved facingthe optimal carbon tax

However the practical η is out of the range δ (η(tlowast) 1)

when 1113954λ1 lt λm Review previous result the region δ (η1 1)

is expanding when tax rises )erefore there exists a pos-sibility that the government increasing carbon tax to co-ordinate the supply chain We assume a realistic thing thegovernment is willing to boost the greenness of the producteven reducing part of the revenue from the carbon tax

Based on it the carbon up to 1113954t1 which means thegreenness is maximized )erefore )e cooperation isachieved when clt 1113954c1 because the greenness is decreasing inthe carbon tax when 1113954c1 lt c )en we have the secondthreshold

1113954λ2 1

η1 1113954t1( 1113857 + 1 (F3)

)e cooperation could not be achieved when λm gt 1113954λ2)e conclusion is as follows (1) Coordination would be

achieved when the manufacturer is democratic (ie05lt λm lt 1113954λ1) (2) Coordination could be achieved but partof the total tax will lose if development cost is lower when themanufacturer is moderate (ie 1113954λ1 lt λm lt 1113954λ2) (3) )e co-operation could not be achieved if the manufacturer is adictator (ie λm gt 1113954λ2)

G Proof of Proposition 5

)e first derivative of η1 in β is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t

2

2α2c2(minus 1 + α)

2 lt 0 (G1)

Similarly η1 is decreasing in β )at means the regionδ (η1 1) is expanding when β rises

And it is apparent that η1 (2cα(1 minus α) minus (1 + β)t22cα(1 minus α))2 is independent with k

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is research was supported by the National Natural ScienceFoundation of China (no 71761015) and the Science andTechnology Research Project of Jiangxi Education Depart-ment of China (no GJJ150475)

References

[1] Y Li M K Lim J Hu and M-L Tseng ldquoInvestigating theeffect of carbon tax and carbon quota policy to achieve lowcarbon logistics operationsrdquo Resources Conservation andRecycling vol 154 p 104535 2020

[2] M Abedi-Varaki ldquoStudy of carbon dioxide gas treatmentbased on equations of kinetics in plasma discharge reactorrdquoModern Physics Letters B vol 31 no 22 p 1750210 2017

[3] S Meng M Siriwardana and J McNeill ldquo)e environmentaland economic impact of the carbon tax in Australiardquo Envi-ronmental and Resource Economics vol 54 no 3 pp 313ndash332 2013

[4] Y Chen and C L Tseng ldquoInducing clean technology in theelectricity sector tradable permits or carbon tax policiesrdquo-eEnergy Journal vol 32 no 3 2011

[5] H-B Duan L Zhu and Y Fan ldquoOptimal carbon taxes incarbon-constrained China a logistic-induced energy eco-nomic hybrid modelrdquo Energy vol 69 pp 345ndash356 2014

[6] C Lu Q Tong and X Liu ldquo)e impacts of carbon tax andcomplementary policies on Chinese economyrdquo Energy Policyvol 38 no 11 pp 7278ndash7285 2010

[7] K Cao B Xu and J Wang ldquoOptimal trade-in and warrantyperiod strategies for new and remanufactured products undercarbon tax policyrdquo International Journal of Production Re-search vol 58 no 1 pp 180ndash199 2020

[8] P He W Zhang X Xu and Y Bian ldquoProduction lot-sizingand carbon emissions under cap-and-trade and carbon taxregulationsrdquo Journal of Cleaner Production vol 103pp 241ndash248 2015

[9] X Xu X Xu and P He ldquoJoint production and pricing de-cisions for multiple products with cap-and-trade and carbontax regulationsrdquo Journal of Cleaner Production vol 112pp 4093ndash4106 2016

Mathematical Problems in Engineering 15

[10] Q Li X Guan T Shi and W Jiao ldquoGreen product designwith competition and fairness concerns in the circulareconomy erardquo International Journal of Production Researchvol 58 no 1 pp 165ndash179 2020

[11] A Ranjan and J K Jha ldquoPricing and coordination strategiesof a dual-channel supply chain considering green quality andsales effortrdquo Journal of Cleaner Production vol 218pp 409ndash424 2019

[12] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineeringvol 10 no 1 pp 99ndash116 2012

[13] X Yin X Chen X Xu and L Zhang ldquoTax or subsidyOptimal carbon emission policy a supply chain perspectiverdquoSustainability vol 12 no 4 p 1548 2020

[14] B B Schlegelmilch G M Bohlen and A Diamantopoulosldquo)e link between green purchasing decisions and measuresof environmental consciousnessrdquo European Journal of Mar-keting vol 30 no 5 pp 35ndash55 1996

[15] M S Hopkins ldquoWhat the rsquogreenrsquo consumer wantsrdquo MITSloan Management Review vol 50 no 4 pp 87ndash89 2009

[16] G Kalyanaram and R S Winer ldquoEmpirical generalizationsfrom reference price researchrdquo Marketing Science vol 14no 3 pp G161ndashG169 1995

[17] E A Greenleaf ldquo)e impact of reference price effects on theprofitability of price promotionsrdquo Marketing Science vol 14no 1 pp 82ndash104 1995

[18] N M Modak D K Ghosh S Panda and S S SanaldquoManaging green house gas emission cost and pricing policiesin a two-echelon supply chainrdquo CIRP Journal ofManufacturing Science and Technology vol 20 pp 1ndash11 2018

[19] P Sinclair ldquoHigh does nothing and rising is worse carbontaxes should keep declining to cut harmful emissionsrdquo -eManchester School vol 60 no 1 pp 41ndash52 1992

[20] G M Grossman and A B Krueger ldquoEconomic growth andthe environmentrdquo -e Quarterly Journal of Economicsvol 110 no 2 pp 353ndash377 1995

[21] N M Modak and P Kelle ldquoUsing social work donation as atool of corporate social responsibility in a closed-loop supplychain considering carbon emissions tax and demand uncer-taintyrdquo Journal of the Operational Research Society pp 1ndash172019

[22] A Ulph and D Ulph ldquo)e optimal time path of a carbon taxrdquoOxford Economic Papers vol 46 no S1 pp 857ndash868 1994

[23] M Hoel and S Kverndokk ldquoDepletion of fossil fuels and theimpact of global warmingrdquo Resource and Energy Economicsvol 18 no 2 pp 115ndash136 1996

[24] Y H Farzin and O Tahvonen ldquoGlobal carbon cycle and theoptimal time path of a carbon taxrdquo Oxford Economic Papersvol 48 no 4 pp 515ndash536 1996

[25] V Bosetti C Carraro R Duval andM Tavoni ldquoWhat shouldwe expect from innovation A model-based assessment of theenvironmental and mitigation cost implications of climate-related RampDrdquo Energy Economics vol 33 no 6 pp 1313ndash13202011

[26] M Hariga R Asrsquoad and A Shamayleh ldquoIntegrated economicand environmental models for a multi stage cold supply chainunder carbon tax regulationrdquo Journal of Cleaner Productionvol 166 pp 1357ndash1371 2017

[27] N Turken J Carrillo and V Verter ldquoFacility location andcapacity acquisition under carbon tax and emissions limits tocentralize or to decentralizerdquo International Journal of Pro-duction Economics vol 187 pp 126ndash141 2017

[28] W Yu and R Han ldquoCoordinating a two-echelon supply chainunder carbon taxrdquo Sustainability vol 9 no 12 p 2360 2017

[29] S Sinha and N M Modak ldquoAn EPQmodel in the perspectiveof carbon emission reductionrdquo International Journal ofMathematics in Operational Research vol 14 no 3pp 338ndash358 2019

[30] H Zhang P Li H Zheng and Y Zhang ldquoImpact of carbontax on enterprise operation and production strategy for low-carbon products in a co-opetition supply chainrdquo Journal ofCleaner Production Article ID 125058 2020

[31] X Chen H Yang X Wang et al ldquoOptimal carbon tax designfor achieving low carbon supply chainsrdquo Annals of OperationsResearch pp 1ndash28 2020

[32] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Handbook of the Fundamentals of Fi-nancial Decision Making Part I pp 99ndash127 )e EconometricSociety Cleveland OH USA 2013

[33] W Kim and M Kim ldquoReference quality-based competitivemarket structure for innovation driven marketsrdquo Interna-tional Journal of Research in Marketing vol 32 no 3pp 284ndash296 2015

[34] I Popescu and Y Wu ldquoDynamic pricing strategies withreference effectsrdquo Operations Research vol 55 no 3pp 413ndash429 2007

[35] P K Kopalle P K Kannan L B Boldt and N Arora ldquo)eimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Re-tailing vol 88 no 1 pp 102ndash114 2012

[36] T-P Hsieh and C-Y Dye ldquoOptimal dynamic pricing fordeteriorating items with reference price effects when inven-tories stimulate demandrdquo European Journal of OperationalResearch vol 262 no 1 pp 136ndash150 2017

[37] X Chen Z-Y Hu and Y-H Zhang ldquoDynamic pricing withstochastic reference price effectrdquo Journal of the OperationsResearch Society of China vol 7 no 1 pp 107ndash125 2019

[38] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquo Opera-tions Research vol 64 no 6 pp 1529ndash1536 2016

[39] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[40] C Chen ldquoDesign for the environment a quality-based modelfor green product developmentrdquo Management Sciencevol 47 no 2 pp 250ndash263 2001

[41] K Chitra ldquoIn search of the green consumers a perceptualstudyrdquo Journal of Services Research vol 7 no 1 2007

[42] Z Liu T D Anderson and J M Cruz ldquoConsumer envi-ronmental awareness and competition in two-stage supplychainsrdquo European Journal of Operational Research vol 218no 3 pp 602ndash613 2012

[43] B Li Y Wang and Z Wang ldquoManaging a closed-loop supplychain with take-back legislation and consumer preference forgreen designrdquo Journal of Cleaner Production Article ID124481 2020

[44] J Rantanen D B Grant and W Piotrowicz ldquoInvestigatingsupply chain cooperation in Finnish grocery retailrdquo ResearchJournal of the University of Gdansk Transport Economics andLogistics vol 71 pp 19ndash34 2017

[45] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[46] G Xu B Dan X Zhang and C Liu ldquoCoordinating a dual-channel supply chain with risk-averse under a two-way

16 Mathematical Problems in Engineering

revenue sharing contractrdquo International Journal of ProductionEconomics vol 147 pp 171ndash179 2014

[47] Z Shi N Wang T Jia et al ldquoReverse revenue sharingcontract versus two-part tariff contract under a closed-loopsupply chain systemrdquoMathematical Problems in Engineeringvol 2016 Article ID 5464570 15 pages 2016

[48] S Panda N M Modak and L E Cardenas-Barron ldquoCo-ordinating a socially responsible closed-loop supply chainwith product recyclingrdquo International Journal of ProductionEconomics vol 188 pp 11ndash21 2017

[49] N M Modak N Modak S Panda and S S Sana ldquoAnalyzingstructure of two-echelon closed-loop supply chain for pricingquality and recycling managementrdquo Journal of Cleaner Pro-duction vol 171 pp 512ndash528 2018

[50] Q P Wang and D Z Zhao ldquoRevenue-sharing contract ofsupply chain based on consumerrsquos preference for low carbonproductsrdquo Chinese Journal of Management Science vol 22no 9 pp 106ndash113 2014

[51] B Yu J Wang X Lu and H Yang ldquoCollaboration in a low-carbon supply chain with reference emission and cost learningeffects cost sharing versus revenue sharing strategiesrdquo Journalof Cleaner Production vol 250 Article ID 119460 2020

[52] J Cao X Zhang and G Zhou ldquoSupply chain coordinationwith revenue-sharing contracts considering carbon emissionsand governmental policy makingrdquo Environmental Progress ampSustainable Energy vol 35 no 2 pp 479ndash488 2016

[53] X Xu P He H Xu and Q Zhang ldquoSupply chain coordi-nation with green technology under cap-and-trade regula-tionrdquo International Journal of Production Economics vol 183pp 433ndash442 2017

[54] N M Modak S Sinha S Panda et al ldquoAnalyzing a sociallyresponsible closed-loop distribution channel with recyclingfacilityrdquo SN Applied Sciences vol 1 no 10 p 1189 2019

[55] D Feng L Ma Y Ding G Wu and Y Zhang ldquoDecisions ofthe dual-channel supply chain under double policy consid-ering remanufacturingrdquo International Journal of Environ-mental Research and Public Health vol 16 no 3 p 465 2019

[56] I Bellos M Ferguson and L B Toktay ldquo)e car sharingeconomy interaction of business model choice and productline designrdquo Manufacturing amp Service Operations Manage-ment vol 19 no 2 pp 185ndash201 2017

[57] G Ferrer and J M Swaminathan ldquoManaging new andremanufactured productsrdquo Management Science vol 52no 1 pp 15ndash26 2006

[58] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2pp 519ndash527 2011

[59] L Cui S Guo and H Zhang ldquoCoordinating a green agri-foodsupply chain with revenue-sharing contracts consideringretailersrsquo green marketing effortsrdquo Sustainability vol 12no 4 p 1289 2020

[60] J Heydari K Govindan and A Aslani ldquoPricing and greeningdecisions in a three-tier dual channel supply chainrdquo Inter-national Journal of Production Economics vol 217 pp 185ndash196 2019

Mathematical Problems in Engineering 17

time If the development cost is lower the greenness may beimproved with a loss of revenue by increasing the carbon taxHowever when the power of the manufacturer is strong theregion δ (η1 1) cannot be achieved

According to our theory the government can promotethe coordination between the manufacturer and the retailerby adjusting the carbon tax When the power of the man-ufacturer increases the government still achieves coordi-nation by raising the carbon tax In general cooperation isachieved and the government plays a key role in promotingcooperation

)e effects of consumersrsquo behavior on supply chaincoordination are also investigated in the next proposition)e proofs are provided in Appendix G

Proposition 5 Price reference behavior can promote supplychain coordination but green preference cannot

From the previous results price reference has a compleximpact on the supply chain First the increasing sensitivityof customers to price will promote supply chain coordi-nation because this negative impact can be mitigated bycooperation In addition price reference behavior also af-fects the green decisions of the supply chain in a complexmanner In contrast the mechanism by which consumerrsquogreen preferences influence the supply chain is relativelysimple Greenness increases with increasing greennesspreference and green preference has no effect on the co-ordination of the supply chain

4 Conclusion

)is study examines the interaction between supply chaincooperation and the carbon tax problem in a two-echelonsupply chain under consumerrsquos reference behavior )e opti-mal carbon tax policy is obtained based on analysis of thecarbon tax green investment coefficient and degree of con-sumerrsquos price reference )e optimal carbon tax is defined asthe simultaneous optimization of total tax revenue and productgreenness In addition coordination is achieved by introducingrevenue-sharing contracts)e impact of consumersrsquo referencebehavior and the carbon tax on supply chain coordination isalso investigated )e results are as follows

(1) )e greenness increases with increasing carbon taxwhen the carbon tax is low In addition if the carbontax is higher and the development cost is low thegreenness of products will increase with increasingcarbon tax Conversely if the carbon tax is higherand the development cost is also high the greennessof products will decrease

(2) When the carbon tax is low the negative effect ofprice increases as consumer concerns about priceincrease )erefore the manufacturer offsets thenegative effect by improving the greenness Whenthe carbon tax is higher and development cost is lowthe greenness of products will increase with highconsumerrsquos concern about price In contrast if thecarbon tax is higher and the development cost ishigh the greenness will decrease

(3) )e greenness increases with an increasing greennesspreference of consumers )eir preference behaviorcan promote more green products )is means thatthe government can promote green production bypromoting the green consciousness of consumersMoreover the greenness decreases with increasingdevelopment cost A higher greenness is obtainedwhen manufacturers face lower development costsIn addition greenness first increases and then de-creases with increases in the functional attributecoefficient When the functional attribute of thegreen product is low the manufacturer will producea lower greenness product with an increase in thefunctional attribute In contrast the manufacturerwill produce a higher greenness product with anincrease in the functional attribute after a func-tionality threshold

(4) We investigated the impact of the carbon tax onpromoting the coordination of the supply chain bycomparing the profit changes of two partners afterthe introduction of a revenue-sharing contract inProposition 3 We found that coordination could bepromoted by increasing the carbon tax

(5) )e achievement of coordination depends on thetype of manufacturer If the power of the manu-facturer is low coordination can be achievedunder the optimal carbon tax If the power of themanufacturer is at a medium level coordinationcan be achieved by increasing the carbon tax andimproved greenness will be realized with a loss ofrevenue However when the power of the man-ufacturer is strong coordination cannot beachieved

In this study we focused on the interaction between acarbon tax and supply chain cooperation However othercarbon policies (eg a cap-and-trade policy) and supplychain structures are worth exploring For example it wouldbe interesting to investigate the effects on a supply chainstructure consisting of the retailer as the leader under acomplex carbon policy

Table 3 Coordination situations for different manufacturer power levels

Power of the manufacturer Coordination Greenness Total carbon tax revenue05lt λm lt 1113954λ1 Achieve Improve Improve1113954λ1 lt λm lt 1113954λ2 clt 1113954c1 Achieve Improve Decline1113954λ1 lt λm lt 1113954λ2 clt 1113954c1 Cannot achievecgt 1113954c1 Cannot achieve

10 Mathematical Problems in Engineering

Appendix

A Proof of Theorem 1

)e demand function can be written as

q P ug ge un ug ge 01113966 1113967

qn P un gt ug un ge 01113966 1113967

⎧⎪⎨

⎪⎩(A1)

After replacing un and ug we have

q P αv minus p + kg minus β p minus pn( 1113857 minus v minus pn( 1113857ge 0 αv minus p + kg minus β p minus pn( 1113857ge 01113864 1113865

qn P v minus pn minus αv minus p + kg minus β p minus pn( 1113857( 1113857gt 0 v minus pn ge 01113864 11138651113896 (A2)

Given that V is uniformly distributed on [0 1] thedemand function is

q (α + β)pn minus (1 + β)p + kg

α(1 minus α)

qn 1 minus(1 + β) pn minus p( 1113857 + kg

1 minus α

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A3)

)e condition kgt ((1 + β)p minus (α + β)png) is necessaryto ensure nonnegativity )e retailer problem is addressedfirst and we have

p (α + β)pn +(1 + β)w + kg

2(1 + β) (A4)

)en substitute p into the manufacturerrsquos profit func-tion Solving the optimal w and g simultaneously yields

w (α + β)pn +(1 + β)(c + t(e minus g)) + kg

2(1 + β)

g t (α + β)pn minus (1 + β)w + ke( 1113857

2cα(1 minus α)

(A5)

)en we have

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

g (1 + β)(te + c) minus (α + β)pn minus ke( 1113857t

t2(1 + β) minus 4cα(1 minus α)

p 3cpnα

3+ 2A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1 (1 + β)(c + te) + ke minus (1 minus β)pn

A2 (1 + β)(c + te) + 3ke minus (3 minus β)pn

⎧⎪⎨

⎪⎩

B1 minus 2c (1 + β)(c + te) + ke + βpn( 1113857 + t2(1 + β)pn

B2 minus c (1 + β)(c + te) + 3ke + 3βpn( 1113857 + t2(1 + β)pn

⎧⎪⎨

⎪⎩

(A6)

Notes that we have (α + β)pn + ke minus (1 + β)(te + c)gt 0)e concavity condition is 4αc(1 minus α) minus (1 + β)t2 gt 0

B Proof of Proposition 1

(1) )e first derivative of g in t is

zg

zt

minus 4cα3pn + 4cA3α2

+ B3α minus t2(1 + β) (1 + β)c minus ke minus βpn( 1113857

t2(1 minus β) minus 4cα(1 minus α)1113872 1113873

2

A3 (1 + β)(c + 2te) minus ke +(1 minus β)pn

B3 minus 4c (1 + β)(c + 2te) minus ke minus βpn( 1113857 + t2(1 + β)pn

⎧⎨

⎩

(B1)

We have a threshold

1113954c1 minust2(1 + β) (α + β)pn + ke minus (1 + β)c( 1113857

4α(1 minus α) (α + β)pn + ke minus (1 + β)(c + 2te)( 1113857

1113954t1 (α + β)pn + ke minus (1 + β)c

2(1 + β)e

(B2)

When cgt 1113954c1 and tgt1113954t1 g is decreasing in t Bycontrast g is increasing in t in other cases

(2) )e first derivative of g in β is

zg

zβ

minus 4α(1 minus α) te + c minus pn( 1113857c + t2

ke + αpn minus pn( 1113857

t2(1 + β) minus 4α(1 minus α)c1113872 1113873

2

(B3)

We have a threshold

1113954c2 minust2

ke minus (1 minus α)pn( 1113857

4α(1 minus α) te + c minus pn( 1113857

1113954t2 pn minus c

e

(B4)

Review we have (α + β)pn + ke minus (1 + β)(te + c)gt 0therefore we can find that

tltpn minus c

e+

ke minus (1 minus α)pn

(1 + β)e (B5)

(i) When ke minus (1 minus α)pn lt 0 then tlt1113954t2 is inevitable)erefore (zgzβ)gt 0 is correct

(ii) When ke minus (1 minus α)pn gt 0 we can find that(zgzβ)lt 0 if cgt 1113954c2 and tgt1113954t2 (zgzβ)lt 0 in othercases

(3) )e first derivative of g in k and c is

Mathematical Problems in Engineering 11

zg

zc minus

4α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

4cα(1 minus α) minus t2(1 + β)1113872 1113873

2 lt 0

zg

zk

te

4cα(1 minus α) minus t2(1 + β)gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(B6)

(4) )e first derivative of g in α is

zg

zα

minus Xt

4α(1 minus α)c minus t2(1 + β)1113872 1113873

2 (B7)

And X minus 4α2cpn + (4(1 + β)(te + c) minus 4ke minus 4pn)

(2α + 1)c + pnt2(1 + β) is a quadratic function )erange of α in 4αc(1 minus α) minus (1 + β)t2 gt 0 is

α1 c +

c2

minus ct2(1 + β)

1113969

2c

α2 c minus

c2

minus ct2(1 + β)

1113969

2c

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(B8)

By substituting those into X we have

X|αα1 4

c2

minus ct2(1 + β)

1113969

T minus 2 c minus t2(1 + β)1113872 1113873pn

X|αα2 minus 4

c2

minus ct2(1 + β)

1113969

T minus 2 c minus t2(1 + β)1113872 1113873pn lt 0

⎧⎪⎪⎨

⎪⎪⎩

(B9)

And T (1 + β)(te + c) minus ke minus ((12) + β)pn lt 0We can find

X|αα1 times X|αα2 minus 4 c minus t2(1 + β)1113872 1113873

middot 4 βpn + ke minus (1 + β)(te + c)( 11138572

1113872

+ p2nt

2(1 + β)1113873lt 0

(B10)

)erefore X|αα1 gt 0 )ere only exist a 1113954α isin (α2 α1)that makes (zgzα) 0 And (zgzα) gt 0 ifα isin (α2 1113954α) (zgzα)lt 0 if α isin (1113954α α1)

(5) )e first derivative of p in t is

zp

zt minus

2α(α minus 1) e(1 + β)t221113872 1113873 + minus pnα minus ke + c minus pn( 1113857β + c( 1113857t minus 2eαc(α minus 1)1113872 1113873c

(1 + β)t2

+ 4αc(α minus 1)1113872 11138732

A3 (1 + β)(c + 2te) minus ke +(1 minus β)pn

B3 minus 4c (1 + β)(c + 2te) minus ke minus βpn( 1113857 + t2(1 + β)pn

⎧⎪⎨

⎪⎩

(B11)

)e discriminant of quadratic function

e(1 + β)t2

2+ minus pnα minus ke + c minus pn( 1113857β + c( 1113857t minus 2eαc(α minus 1)

(B12)

is

Δ (minus β minus 1)c + ke + βpn + pnα( 11138572

+ 4e2(1 + β)t

2αc(α minus 1)

(B13)

We have a threshold

1113954c minus(minus β minus 1)c + ke + βpn + pnα( 1113857

2

4e2(1 + β)t

2α(α minus 1) (B14)

Review we have(α + β)pn + ke minus (1 + β)(te + c)gt 0therefore we can find that

tm 2

minus (1 + β)αc(α minus 1)

1113968

1 + β (B15)

Substituting it into above quadratic function

2 c

radicminus pnα minus ke + c minus pn( 1113857β + c( 1113857

minus (1 + β)α(α minus 1)

1113968minus 4eαc(α minus 1)(1 + β)

1 + β (B16)

And we have a same threshold

1113954c minus(minus β minus 1)c + ke + βpn + pnα( 1113857

2

4e2(1 + β)t

2α(α minus 1) (B17)

It could be found that(zpzt) gt 0 because Δlt 0 whenclt 1113954c And when cgt 1113954c we have another threshold 1113954t andzpzt|

t1113954t 0 because Δgt 0 and zpzt|tmlt 0)erefore when

cgt 1113954c and tgt1113954t p is decreasing in t And when cgt 1113954c and tlt1113954tp is increasing in t

12 Mathematical Problems in Engineering

C Proof of Proposition 2

(1) Let R t(e minus g)q denote total carbon tax and thefirst derivative of R in t is

zR

zt2c A2t

2+ B2t + c1113872 1113873 A3t

2+ B3t + c1113872 1113873

4cα(1 minus α) minus t2(1 + β)1113872 1113873

3 (C1)

Among them

A2 (1 + β)e

2gt 0

B2 minus (α + β)pn + ke minus (1 + β)c( 1113857lt 0

C2 2ceα(1 minus α)gt 0

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

A3 (α + β)pn + ke minus (1 + β)c( 1113857(1 + β)gt 0

B3 minus 8ceα(1 minus α)(1 + β)lt 0

C3 4cα(1 minus α) (α + β)pn + ke minus (1 + β)c( 1113857gt 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(C2)

Let Φ A2t2 + B2t + C2 and Ψ A3t

2 + B3t + C3 andwe can find that zRzt and Φ times Ψ have a same sign Bysolving Δ1 and Δ2 a threshold is

1113954c3 (α + β)pn + ke minus (1 + β)c( 1113857

2

4e2α(1 minus α)(1 + β)

(C3)

)ere are two positive real numbers of solutions forΦ ifclt 1113954c3 and let 0lt ta lt tb denote it Similarly let0lt tc lt td denote the solutions for Ψ if cgt 1113954c3 Reviewwe have 4αc(1 minus α) minus (1 + β)t2 gt 0 therefore lettm

4cα(1 minus a)1 + β

1113968 Substituting tm into Φ and Ψ

Φ|tm minus 2

cα(1 minus α)

1 + β

1113971

(α + β)pn + ke minus (1 + β)c( 1113857 + 4ecα(1 minus α)

Ψ|tm minus 8cα(1 minus α) 2(1 + β)e

α(1 minus α)

1 + βc

1113971

minus (α + β)pn + ke minus (1 + β)c( 1113857⎛⎝ ⎞⎠

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(C4)

If clt 1113954c3 Φ|tmlt 0 and Ψ|tm

gt 0 then zRzt|tmlt 0 If

cgt 1113954c3 Φ|tmgt 0 and Ψ|tm

lt 0 then zRzt|tmlt 0 )ere-

fore we have tm isin (ta tb) or tm isin (tc td) FurthermoreR t(e minus g)q is maximized when t ta if clt 1113954c3 R

t(e minus g)q is maximized when t tc if cgt 1113954c3 Totally wehave

tlowast

ta clt 1113954c3

tc cgt 1113954c31113896 (C5)

(2) Review Proposition 1 If cgt 1113954c1 g is maximized whentlowast 1113954t1 If clt 1113954c1 g is maximized when tlowast tmWhen tgt1113954t1 1113954c1 gt 1113954c5 And 1113954c1 lt 0lt 1113954c5 if tlt1113954t1Substituting 1113954t1 into Φ

Φ|1113954t1 minus

3 (α + β)pn + ke minus (1 + β)c( 11138572

8e(1 + β)+ 2ecα(1 minus α)

(C6)

A threshold is 1113954c4 (34)1113954c3 )erefore we have (i)when 0lt clt 1113954c4 tlowast ta because of Φ|1113954t1

lt 0 and 1113954t1 gt ta

(ii) when 1113954c4 lt clt 1113954c3 tlowast ta because of 1113954t1 lt ta (iii) tlowast

tc when 1113954c3 lt clt 1113954c1 and (iv) tlowast tc when 1113954c1 lt c )eresult is obtained by solving Φ and Ψ

D Proof of Theorem 2

Similar to Appendix A )e retailer problem is addressedfirst and we have

p αηpn + βηpn + ηkg + βw + w

2η(β + 1) (D1)

)en substituting p into the manufacturerrsquos profitfunction Solving the optimal w and g simultaneously yields

w αηpn + βet + βηpn minus βgt + eηk + cβ + et minus gt + c( 1113857η

(1 + η)(β + 1)

g minusαηpn + βηpn + eηk minus βw minus w( 1113857t

2(α minus 1)cαη

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(D2)

Mathematical Problems in Engineering 13

By solving the equation above we have

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

g t minus αpn + et(β + 1) + c minus pn( 1113857β minus ke + c( 1113857

t2(1 + β) minus (1 + η)cα(1 minus α)

p (2η + 1)cpnα

3+ A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1 et + ηpn + c( 1113857β + et + ke minus pn( 1113857η + c

A2 (2η + 1)pn + et + c( 1113857β +(minus 2η minus 1)pn + et +(2ηk + k)e + c

⎧⎪⎪⎨

⎪⎪⎩

B1 minus 2et minus 2ηpn minus 2c( 1113857β minus 2eηk minus 2et minus 2c( 1113857c + t2pn(β + 1)

B2 (minus 2η minus 1)pn minus et minus c( 1113857β minus et +(minus 2ηk minus k)e minus c( 1113857c + t2pn(β + 1)

⎧⎪⎪⎨

⎪⎪⎩

(D3)

Note that we have minus αpn + et(β + 1) + (c minus pn)β minus ke +

clt 0 )e concavity condition is (1 + β)(t2(1 + β) minus (1+

η)cα(1 minus α))lt 0

Substituting solutions into the profit function we have

πm minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2c

2 t2β + t

2+ 4(minus 1 + α)cα1113872 1113873(β + 1)

πr minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2(minus 1 + α)αc

2

4α2c minus 4αc +(β + 1)t2

1113872 11138732(β + 1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(D4)

E Proof of Proposition 3

πM (α + β)pn + ke minus (1 + β)c( 1113857

2c

2(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCM

(α + β)pn + ke minus (1 + β)c( 11138572c

2(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(E1)

By comparing the profits of two partners change after therevenue-sharing contract introduced we found that themanufacturer always benefits from cooperation

πR c2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCR

ηc2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(E2)

A threshold is

η1 2cα(1 minus α) minus (1 + β)t2

2cα(1 minus α)1113888 1113889

2

(E3)

When η is in the region δ (η1 1) the retailer willaccept cooperation And the first derivative of η1 in t is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t(β + 1)

α2c2(minus 1 + α)

2 lt 0 (E4)

)erefore η1 is decreasing in t )at means the regionδ (η1 1) is expanding when tax rises And the revenue-sharing contract is accepted easily than before

F Proof of Proposition 4

We consider a simple function (λrλm) to decide thepractical η where λm isin (05 1) and λr 1 minus λm Given the

14 Mathematical Problems in Engineering

carbon tax tlowast we have a region δ (η(tlowast) 1) of cooperationIt is clear that the gap of strength between two partitions inthe supply chain is a crucial factor in cooperation Bycomparing practice η decided by strength and η(tlowast) de-cided by the carbon tax we have

1λm

minus 1 2α2c minus 2αc + t

2(β + 1)1113872 1113873

2

4α2c2(minus 1 + α)

2 (F1)

A threshold is

1113954λ1 4α2c2

(minus 1 + α)2

8α4c2minus 16α3c2

+ 4 2c + t2(β + 1)1113872 1113873cα2 minus 4t

2c(β + 1)α + t

4(β + 1)

2 (F2)

When 05lt λm lt 1113954λ1 the practical η is in the regionδ (η(tlowast) 1) therefore the cooperation is achieved facingthe optimal carbon tax

However the practical η is out of the range δ (η(tlowast) 1)

when 1113954λ1 lt λm Review previous result the region δ (η1 1)

is expanding when tax rises )erefore there exists a pos-sibility that the government increasing carbon tax to co-ordinate the supply chain We assume a realistic thing thegovernment is willing to boost the greenness of the producteven reducing part of the revenue from the carbon tax

Based on it the carbon up to 1113954t1 which means thegreenness is maximized )erefore )e cooperation isachieved when clt 1113954c1 because the greenness is decreasing inthe carbon tax when 1113954c1 lt c )en we have the secondthreshold

1113954λ2 1

η1 1113954t1( 1113857 + 1 (F3)

)e cooperation could not be achieved when λm gt 1113954λ2)e conclusion is as follows (1) Coordination would be

achieved when the manufacturer is democratic (ie05lt λm lt 1113954λ1) (2) Coordination could be achieved but partof the total tax will lose if development cost is lower when themanufacturer is moderate (ie 1113954λ1 lt λm lt 1113954λ2) (3) )e co-operation could not be achieved if the manufacturer is adictator (ie λm gt 1113954λ2)

G Proof of Proposition 5

)e first derivative of η1 in β is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t

2

2α2c2(minus 1 + α)

2 lt 0 (G1)

Similarly η1 is decreasing in β )at means the regionδ (η1 1) is expanding when β rises

And it is apparent that η1 (2cα(1 minus α) minus (1 + β)t22cα(1 minus α))2 is independent with k

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is research was supported by the National Natural ScienceFoundation of China (no 71761015) and the Science andTechnology Research Project of Jiangxi Education Depart-ment of China (no GJJ150475)

References

[1] Y Li M K Lim J Hu and M-L Tseng ldquoInvestigating theeffect of carbon tax and carbon quota policy to achieve lowcarbon logistics operationsrdquo Resources Conservation andRecycling vol 154 p 104535 2020

[2] M Abedi-Varaki ldquoStudy of carbon dioxide gas treatmentbased on equations of kinetics in plasma discharge reactorrdquoModern Physics Letters B vol 31 no 22 p 1750210 2017

[3] S Meng M Siriwardana and J McNeill ldquo)e environmentaland economic impact of the carbon tax in Australiardquo Envi-ronmental and Resource Economics vol 54 no 3 pp 313ndash332 2013

[4] Y Chen and C L Tseng ldquoInducing clean technology in theelectricity sector tradable permits or carbon tax policiesrdquo-eEnergy Journal vol 32 no 3 2011

[5] H-B Duan L Zhu and Y Fan ldquoOptimal carbon taxes incarbon-constrained China a logistic-induced energy eco-nomic hybrid modelrdquo Energy vol 69 pp 345ndash356 2014

[6] C Lu Q Tong and X Liu ldquo)e impacts of carbon tax andcomplementary policies on Chinese economyrdquo Energy Policyvol 38 no 11 pp 7278ndash7285 2010

[7] K Cao B Xu and J Wang ldquoOptimal trade-in and warrantyperiod strategies for new and remanufactured products undercarbon tax policyrdquo International Journal of Production Re-search vol 58 no 1 pp 180ndash199 2020

[8] P He W Zhang X Xu and Y Bian ldquoProduction lot-sizingand carbon emissions under cap-and-trade and carbon taxregulationsrdquo Journal of Cleaner Production vol 103pp 241ndash248 2015

[9] X Xu X Xu and P He ldquoJoint production and pricing de-cisions for multiple products with cap-and-trade and carbontax regulationsrdquo Journal of Cleaner Production vol 112pp 4093ndash4106 2016

Mathematical Problems in Engineering 15

[10] Q Li X Guan T Shi and W Jiao ldquoGreen product designwith competition and fairness concerns in the circulareconomy erardquo International Journal of Production Researchvol 58 no 1 pp 165ndash179 2020

[11] A Ranjan and J K Jha ldquoPricing and coordination strategiesof a dual-channel supply chain considering green quality andsales effortrdquo Journal of Cleaner Production vol 218pp 409ndash424 2019

[12] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineeringvol 10 no 1 pp 99ndash116 2012

[13] X Yin X Chen X Xu and L Zhang ldquoTax or subsidyOptimal carbon emission policy a supply chain perspectiverdquoSustainability vol 12 no 4 p 1548 2020

[14] B B Schlegelmilch G M Bohlen and A Diamantopoulosldquo)e link between green purchasing decisions and measuresof environmental consciousnessrdquo European Journal of Mar-keting vol 30 no 5 pp 35ndash55 1996

[15] M S Hopkins ldquoWhat the rsquogreenrsquo consumer wantsrdquo MITSloan Management Review vol 50 no 4 pp 87ndash89 2009

[16] G Kalyanaram and R S Winer ldquoEmpirical generalizationsfrom reference price researchrdquo Marketing Science vol 14no 3 pp G161ndashG169 1995

[17] E A Greenleaf ldquo)e impact of reference price effects on theprofitability of price promotionsrdquo Marketing Science vol 14no 1 pp 82ndash104 1995

[18] N M Modak D K Ghosh S Panda and S S SanaldquoManaging green house gas emission cost and pricing policiesin a two-echelon supply chainrdquo CIRP Journal ofManufacturing Science and Technology vol 20 pp 1ndash11 2018

[19] P Sinclair ldquoHigh does nothing and rising is worse carbontaxes should keep declining to cut harmful emissionsrdquo -eManchester School vol 60 no 1 pp 41ndash52 1992

[20] G M Grossman and A B Krueger ldquoEconomic growth andthe environmentrdquo -e Quarterly Journal of Economicsvol 110 no 2 pp 353ndash377 1995

[21] N M Modak and P Kelle ldquoUsing social work donation as atool of corporate social responsibility in a closed-loop supplychain considering carbon emissions tax and demand uncer-taintyrdquo Journal of the Operational Research Society pp 1ndash172019

[22] A Ulph and D Ulph ldquo)e optimal time path of a carbon taxrdquoOxford Economic Papers vol 46 no S1 pp 857ndash868 1994

[23] M Hoel and S Kverndokk ldquoDepletion of fossil fuels and theimpact of global warmingrdquo Resource and Energy Economicsvol 18 no 2 pp 115ndash136 1996

[24] Y H Farzin and O Tahvonen ldquoGlobal carbon cycle and theoptimal time path of a carbon taxrdquo Oxford Economic Papersvol 48 no 4 pp 515ndash536 1996

[25] V Bosetti C Carraro R Duval andM Tavoni ldquoWhat shouldwe expect from innovation A model-based assessment of theenvironmental and mitigation cost implications of climate-related RampDrdquo Energy Economics vol 33 no 6 pp 1313ndash13202011

[26] M Hariga R Asrsquoad and A Shamayleh ldquoIntegrated economicand environmental models for a multi stage cold supply chainunder carbon tax regulationrdquo Journal of Cleaner Productionvol 166 pp 1357ndash1371 2017

[27] N Turken J Carrillo and V Verter ldquoFacility location andcapacity acquisition under carbon tax and emissions limits tocentralize or to decentralizerdquo International Journal of Pro-duction Economics vol 187 pp 126ndash141 2017

[28] W Yu and R Han ldquoCoordinating a two-echelon supply chainunder carbon taxrdquo Sustainability vol 9 no 12 p 2360 2017

[29] S Sinha and N M Modak ldquoAn EPQmodel in the perspectiveof carbon emission reductionrdquo International Journal ofMathematics in Operational Research vol 14 no 3pp 338ndash358 2019

[30] H Zhang P Li H Zheng and Y Zhang ldquoImpact of carbontax on enterprise operation and production strategy for low-carbon products in a co-opetition supply chainrdquo Journal ofCleaner Production Article ID 125058 2020

[31] X Chen H Yang X Wang et al ldquoOptimal carbon tax designfor achieving low carbon supply chainsrdquo Annals of OperationsResearch pp 1ndash28 2020

[32] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Handbook of the Fundamentals of Fi-nancial Decision Making Part I pp 99ndash127 )e EconometricSociety Cleveland OH USA 2013

[33] W Kim and M Kim ldquoReference quality-based competitivemarket structure for innovation driven marketsrdquo Interna-tional Journal of Research in Marketing vol 32 no 3pp 284ndash296 2015

[34] I Popescu and Y Wu ldquoDynamic pricing strategies withreference effectsrdquo Operations Research vol 55 no 3pp 413ndash429 2007

[35] P K Kopalle P K Kannan L B Boldt and N Arora ldquo)eimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Re-tailing vol 88 no 1 pp 102ndash114 2012

[36] T-P Hsieh and C-Y Dye ldquoOptimal dynamic pricing fordeteriorating items with reference price effects when inven-tories stimulate demandrdquo European Journal of OperationalResearch vol 262 no 1 pp 136ndash150 2017

[37] X Chen Z-Y Hu and Y-H Zhang ldquoDynamic pricing withstochastic reference price effectrdquo Journal of the OperationsResearch Society of China vol 7 no 1 pp 107ndash125 2019

[38] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquo Opera-tions Research vol 64 no 6 pp 1529ndash1536 2016

[39] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[40] C Chen ldquoDesign for the environment a quality-based modelfor green product developmentrdquo Management Sciencevol 47 no 2 pp 250ndash263 2001

[41] K Chitra ldquoIn search of the green consumers a perceptualstudyrdquo Journal of Services Research vol 7 no 1 2007

[42] Z Liu T D Anderson and J M Cruz ldquoConsumer envi-ronmental awareness and competition in two-stage supplychainsrdquo European Journal of Operational Research vol 218no 3 pp 602ndash613 2012

[43] B Li Y Wang and Z Wang ldquoManaging a closed-loop supplychain with take-back legislation and consumer preference forgreen designrdquo Journal of Cleaner Production Article ID124481 2020

[44] J Rantanen D B Grant and W Piotrowicz ldquoInvestigatingsupply chain cooperation in Finnish grocery retailrdquo ResearchJournal of the University of Gdansk Transport Economics andLogistics vol 71 pp 19ndash34 2017

[45] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[46] G Xu B Dan X Zhang and C Liu ldquoCoordinating a dual-channel supply chain with risk-averse under a two-way

16 Mathematical Problems in Engineering

revenue sharing contractrdquo International Journal of ProductionEconomics vol 147 pp 171ndash179 2014

[47] Z Shi N Wang T Jia et al ldquoReverse revenue sharingcontract versus two-part tariff contract under a closed-loopsupply chain systemrdquoMathematical Problems in Engineeringvol 2016 Article ID 5464570 15 pages 2016

[48] S Panda N M Modak and L E Cardenas-Barron ldquoCo-ordinating a socially responsible closed-loop supply chainwith product recyclingrdquo International Journal of ProductionEconomics vol 188 pp 11ndash21 2017

[49] N M Modak N Modak S Panda and S S Sana ldquoAnalyzingstructure of two-echelon closed-loop supply chain for pricingquality and recycling managementrdquo Journal of Cleaner Pro-duction vol 171 pp 512ndash528 2018

[50] Q P Wang and D Z Zhao ldquoRevenue-sharing contract ofsupply chain based on consumerrsquos preference for low carbonproductsrdquo Chinese Journal of Management Science vol 22no 9 pp 106ndash113 2014

[51] B Yu J Wang X Lu and H Yang ldquoCollaboration in a low-carbon supply chain with reference emission and cost learningeffects cost sharing versus revenue sharing strategiesrdquo Journalof Cleaner Production vol 250 Article ID 119460 2020

[52] J Cao X Zhang and G Zhou ldquoSupply chain coordinationwith revenue-sharing contracts considering carbon emissionsand governmental policy makingrdquo Environmental Progress ampSustainable Energy vol 35 no 2 pp 479ndash488 2016

[53] X Xu P He H Xu and Q Zhang ldquoSupply chain coordi-nation with green technology under cap-and-trade regula-tionrdquo International Journal of Production Economics vol 183pp 433ndash442 2017

[54] N M Modak S Sinha S Panda et al ldquoAnalyzing a sociallyresponsible closed-loop distribution channel with recyclingfacilityrdquo SN Applied Sciences vol 1 no 10 p 1189 2019

[55] D Feng L Ma Y Ding G Wu and Y Zhang ldquoDecisions ofthe dual-channel supply chain under double policy consid-ering remanufacturingrdquo International Journal of Environ-mental Research and Public Health vol 16 no 3 p 465 2019

[56] I Bellos M Ferguson and L B Toktay ldquo)e car sharingeconomy interaction of business model choice and productline designrdquo Manufacturing amp Service Operations Manage-ment vol 19 no 2 pp 185ndash201 2017

[57] G Ferrer and J M Swaminathan ldquoManaging new andremanufactured productsrdquo Management Science vol 52no 1 pp 15ndash26 2006

[58] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2pp 519ndash527 2011

[59] L Cui S Guo and H Zhang ldquoCoordinating a green agri-foodsupply chain with revenue-sharing contracts consideringretailersrsquo green marketing effortsrdquo Sustainability vol 12no 4 p 1289 2020

[60] J Heydari K Govindan and A Aslani ldquoPricing and greeningdecisions in a three-tier dual channel supply chainrdquo Inter-national Journal of Production Economics vol 217 pp 185ndash196 2019

Mathematical Problems in Engineering 17

Appendix

A Proof of Theorem 1

)e demand function can be written as

q P ug ge un ug ge 01113966 1113967

qn P un gt ug un ge 01113966 1113967

⎧⎪⎨

⎪⎩(A1)

After replacing un and ug we have

q P αv minus p + kg minus β p minus pn( 1113857 minus v minus pn( 1113857ge 0 αv minus p + kg minus β p minus pn( 1113857ge 01113864 1113865

qn P v minus pn minus αv minus p + kg minus β p minus pn( 1113857( 1113857gt 0 v minus pn ge 01113864 11138651113896 (A2)

Given that V is uniformly distributed on [0 1] thedemand function is

q (α + β)pn minus (1 + β)p + kg

α(1 minus α)

qn 1 minus(1 + β) pn minus p( 1113857 + kg

1 minus α

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A3)

)e condition kgt ((1 + β)p minus (α + β)png) is necessaryto ensure nonnegativity )e retailer problem is addressedfirst and we have

p (α + β)pn +(1 + β)w + kg

2(1 + β) (A4)

)en substitute p into the manufacturerrsquos profit func-tion Solving the optimal w and g simultaneously yields

w (α + β)pn +(1 + β)(c + t(e minus g)) + kg

2(1 + β)

g t (α + β)pn minus (1 + β)w + ke( 1113857

2cα(1 minus α)

(A5)

)en we have

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

g (1 + β)(te + c) minus (α + β)pn minus ke( 1113857t

t2(1 + β) minus 4cα(1 minus α)

p 3cpnα

3+ 2A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus 4cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1 (1 + β)(c + te) + ke minus (1 minus β)pn

A2 (1 + β)(c + te) + 3ke minus (3 minus β)pn

⎧⎪⎨

⎪⎩

B1 minus 2c (1 + β)(c + te) + ke + βpn( 1113857 + t2(1 + β)pn

B2 minus c (1 + β)(c + te) + 3ke + 3βpn( 1113857 + t2(1 + β)pn

⎧⎪⎨

⎪⎩

(A6)

Notes that we have (α + β)pn + ke minus (1 + β)(te + c)gt 0)e concavity condition is 4αc(1 minus α) minus (1 + β)t2 gt 0

B Proof of Proposition 1

(1) )e first derivative of g in t is

zg

zt

minus 4cα3pn + 4cA3α2

+ B3α minus t2(1 + β) (1 + β)c minus ke minus βpn( 1113857

t2(1 minus β) minus 4cα(1 minus α)1113872 1113873

2

A3 (1 + β)(c + 2te) minus ke +(1 minus β)pn

B3 minus 4c (1 + β)(c + 2te) minus ke minus βpn( 1113857 + t2(1 + β)pn

⎧⎨

⎩

(B1)

We have a threshold

1113954c1 minust2(1 + β) (α + β)pn + ke minus (1 + β)c( 1113857

4α(1 minus α) (α + β)pn + ke minus (1 + β)(c + 2te)( 1113857

1113954t1 (α + β)pn + ke minus (1 + β)c

2(1 + β)e

(B2)

When cgt 1113954c1 and tgt1113954t1 g is decreasing in t Bycontrast g is increasing in t in other cases

(2) )e first derivative of g in β is

zg

zβ

minus 4α(1 minus α) te + c minus pn( 1113857c + t2

ke + αpn minus pn( 1113857

t2(1 + β) minus 4α(1 minus α)c1113872 1113873

2

(B3)

We have a threshold

1113954c2 minust2

ke minus (1 minus α)pn( 1113857

4α(1 minus α) te + c minus pn( 1113857

1113954t2 pn minus c

e

(B4)

Review we have (α + β)pn + ke minus (1 + β)(te + c)gt 0therefore we can find that

tltpn minus c

e+

ke minus (1 minus α)pn

(1 + β)e (B5)

(i) When ke minus (1 minus α)pn lt 0 then tlt1113954t2 is inevitable)erefore (zgzβ)gt 0 is correct

(ii) When ke minus (1 minus α)pn gt 0 we can find that(zgzβ)lt 0 if cgt 1113954c2 and tgt1113954t2 (zgzβ)lt 0 in othercases

(3) )e first derivative of g in k and c is

Mathematical Problems in Engineering 11

zg

zc minus

4α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

4cα(1 minus α) minus t2(1 + β)1113872 1113873

2 lt 0

zg

zk

te

4cα(1 minus α) minus t2(1 + β)gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(B6)

(4) )e first derivative of g in α is

zg

zα

minus Xt

4α(1 minus α)c minus t2(1 + β)1113872 1113873

2 (B7)

And X minus 4α2cpn + (4(1 + β)(te + c) minus 4ke minus 4pn)

(2α + 1)c + pnt2(1 + β) is a quadratic function )erange of α in 4αc(1 minus α) minus (1 + β)t2 gt 0 is

α1 c +

c2

minus ct2(1 + β)

1113969

2c

α2 c minus

c2

minus ct2(1 + β)

1113969

2c

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(B8)

By substituting those into X we have

X|αα1 4

c2

minus ct2(1 + β)

1113969

T minus 2 c minus t2(1 + β)1113872 1113873pn

X|αα2 minus 4

c2

minus ct2(1 + β)

1113969

T minus 2 c minus t2(1 + β)1113872 1113873pn lt 0

⎧⎪⎪⎨

⎪⎪⎩

(B9)

And T (1 + β)(te + c) minus ke minus ((12) + β)pn lt 0We can find

X|αα1 times X|αα2 minus 4 c minus t2(1 + β)1113872 1113873

middot 4 βpn + ke minus (1 + β)(te + c)( 11138572

1113872

+ p2nt

2(1 + β)1113873lt 0

(B10)

)erefore X|αα1 gt 0 )ere only exist a 1113954α isin (α2 α1)that makes (zgzα) 0 And (zgzα) gt 0 ifα isin (α2 1113954α) (zgzα)lt 0 if α isin (1113954α α1)

(5) )e first derivative of p in t is

zp

zt minus

2α(α minus 1) e(1 + β)t221113872 1113873 + minus pnα minus ke + c minus pn( 1113857β + c( 1113857t minus 2eαc(α minus 1)1113872 1113873c

(1 + β)t2

+ 4αc(α minus 1)1113872 11138732

A3 (1 + β)(c + 2te) minus ke +(1 minus β)pn

B3 minus 4c (1 + β)(c + 2te) minus ke minus βpn( 1113857 + t2(1 + β)pn

⎧⎪⎨

⎪⎩

(B11)

)e discriminant of quadratic function

e(1 + β)t2

2+ minus pnα minus ke + c minus pn( 1113857β + c( 1113857t minus 2eαc(α minus 1)

(B12)

is

Δ (minus β minus 1)c + ke + βpn + pnα( 11138572

+ 4e2(1 + β)t

2αc(α minus 1)

(B13)

We have a threshold

1113954c minus(minus β minus 1)c + ke + βpn + pnα( 1113857

2

4e2(1 + β)t

2α(α minus 1) (B14)

Review we have(α + β)pn + ke minus (1 + β)(te + c)gt 0therefore we can find that

tm 2

minus (1 + β)αc(α minus 1)

1113968

1 + β (B15)

Substituting it into above quadratic function

2 c

radicminus pnα minus ke + c minus pn( 1113857β + c( 1113857

minus (1 + β)α(α minus 1)

1113968minus 4eαc(α minus 1)(1 + β)

1 + β (B16)

And we have a same threshold

1113954c minus(minus β minus 1)c + ke + βpn + pnα( 1113857

2

4e2(1 + β)t

2α(α minus 1) (B17)

It could be found that(zpzt) gt 0 because Δlt 0 whenclt 1113954c And when cgt 1113954c we have another threshold 1113954t andzpzt|

t1113954t 0 because Δgt 0 and zpzt|tmlt 0)erefore when

cgt 1113954c and tgt1113954t p is decreasing in t And when cgt 1113954c and tlt1113954tp is increasing in t

12 Mathematical Problems in Engineering

C Proof of Proposition 2

(1) Let R t(e minus g)q denote total carbon tax and thefirst derivative of R in t is

zR

zt2c A2t

2+ B2t + c1113872 1113873 A3t

2+ B3t + c1113872 1113873

4cα(1 minus α) minus t2(1 + β)1113872 1113873

3 (C1)

Among them

A2 (1 + β)e

2gt 0

B2 minus (α + β)pn + ke minus (1 + β)c( 1113857lt 0

C2 2ceα(1 minus α)gt 0

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

A3 (α + β)pn + ke minus (1 + β)c( 1113857(1 + β)gt 0

B3 minus 8ceα(1 minus α)(1 + β)lt 0

C3 4cα(1 minus α) (α + β)pn + ke minus (1 + β)c( 1113857gt 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(C2)

Let Φ A2t2 + B2t + C2 and Ψ A3t

2 + B3t + C3 andwe can find that zRzt and Φ times Ψ have a same sign Bysolving Δ1 and Δ2 a threshold is

1113954c3 (α + β)pn + ke minus (1 + β)c( 1113857

2

4e2α(1 minus α)(1 + β)

(C3)

)ere are two positive real numbers of solutions forΦ ifclt 1113954c3 and let 0lt ta lt tb denote it Similarly let0lt tc lt td denote the solutions for Ψ if cgt 1113954c3 Reviewwe have 4αc(1 minus α) minus (1 + β)t2 gt 0 therefore lettm

4cα(1 minus a)1 + β

1113968 Substituting tm into Φ and Ψ

Φ|tm minus 2

cα(1 minus α)

1 + β

1113971

(α + β)pn + ke minus (1 + β)c( 1113857 + 4ecα(1 minus α)

Ψ|tm minus 8cα(1 minus α) 2(1 + β)e

α(1 minus α)

1 + βc

1113971

minus (α + β)pn + ke minus (1 + β)c( 1113857⎛⎝ ⎞⎠

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(C4)

If clt 1113954c3 Φ|tmlt 0 and Ψ|tm

gt 0 then zRzt|tmlt 0 If

cgt 1113954c3 Φ|tmgt 0 and Ψ|tm

lt 0 then zRzt|tmlt 0 )ere-

fore we have tm isin (ta tb) or tm isin (tc td) FurthermoreR t(e minus g)q is maximized when t ta if clt 1113954c3 R

t(e minus g)q is maximized when t tc if cgt 1113954c3 Totally wehave

tlowast

ta clt 1113954c3

tc cgt 1113954c31113896 (C5)

(2) Review Proposition 1 If cgt 1113954c1 g is maximized whentlowast 1113954t1 If clt 1113954c1 g is maximized when tlowast tmWhen tgt1113954t1 1113954c1 gt 1113954c5 And 1113954c1 lt 0lt 1113954c5 if tlt1113954t1Substituting 1113954t1 into Φ

Φ|1113954t1 minus

3 (α + β)pn + ke minus (1 + β)c( 11138572

8e(1 + β)+ 2ecα(1 minus α)

(C6)

A threshold is 1113954c4 (34)1113954c3 )erefore we have (i)when 0lt clt 1113954c4 tlowast ta because of Φ|1113954t1

lt 0 and 1113954t1 gt ta

(ii) when 1113954c4 lt clt 1113954c3 tlowast ta because of 1113954t1 lt ta (iii) tlowast

tc when 1113954c3 lt clt 1113954c1 and (iv) tlowast tc when 1113954c1 lt c )eresult is obtained by solving Φ and Ψ

D Proof of Theorem 2

Similar to Appendix A )e retailer problem is addressedfirst and we have

p αηpn + βηpn + ηkg + βw + w

2η(β + 1) (D1)

)en substituting p into the manufacturerrsquos profitfunction Solving the optimal w and g simultaneously yields

w αηpn + βet + βηpn minus βgt + eηk + cβ + et minus gt + c( 1113857η

(1 + η)(β + 1)

g minusαηpn + βηpn + eηk minus βw minus w( 1113857t

2(α minus 1)cαη

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(D2)

Mathematical Problems in Engineering 13

By solving the equation above we have

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

g t minus αpn + et(β + 1) + c minus pn( 1113857β minus ke + c( 1113857

t2(1 + β) minus (1 + η)cα(1 minus α)

p (2η + 1)cpnα

3+ A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1 et + ηpn + c( 1113857β + et + ke minus pn( 1113857η + c

A2 (2η + 1)pn + et + c( 1113857β +(minus 2η minus 1)pn + et +(2ηk + k)e + c

⎧⎪⎪⎨

⎪⎪⎩

B1 minus 2et minus 2ηpn minus 2c( 1113857β minus 2eηk minus 2et minus 2c( 1113857c + t2pn(β + 1)

B2 (minus 2η minus 1)pn minus et minus c( 1113857β minus et +(minus 2ηk minus k)e minus c( 1113857c + t2pn(β + 1)

⎧⎪⎪⎨

⎪⎪⎩

(D3)

Note that we have minus αpn + et(β + 1) + (c minus pn)β minus ke +

clt 0 )e concavity condition is (1 + β)(t2(1 + β) minus (1+

η)cα(1 minus α))lt 0

Substituting solutions into the profit function we have

πm minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2c

2 t2β + t

2+ 4(minus 1 + α)cα1113872 1113873(β + 1)

πr minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2(minus 1 + α)αc

2

4α2c minus 4αc +(β + 1)t2

1113872 11138732(β + 1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(D4)

E Proof of Proposition 3

πM (α + β)pn + ke minus (1 + β)c( 1113857

2c

2(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCM

(α + β)pn + ke minus (1 + β)c( 11138572c

2(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(E1)

By comparing the profits of two partners change after therevenue-sharing contract introduced we found that themanufacturer always benefits from cooperation

πR c2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCR

ηc2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(E2)

A threshold is

η1 2cα(1 minus α) minus (1 + β)t2

2cα(1 minus α)1113888 1113889

2

(E3)

When η is in the region δ (η1 1) the retailer willaccept cooperation And the first derivative of η1 in t is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t(β + 1)

α2c2(minus 1 + α)

2 lt 0 (E4)

)erefore η1 is decreasing in t )at means the regionδ (η1 1) is expanding when tax rises And the revenue-sharing contract is accepted easily than before

F Proof of Proposition 4

We consider a simple function (λrλm) to decide thepractical η where λm isin (05 1) and λr 1 minus λm Given the

14 Mathematical Problems in Engineering

carbon tax tlowast we have a region δ (η(tlowast) 1) of cooperationIt is clear that the gap of strength between two partitions inthe supply chain is a crucial factor in cooperation Bycomparing practice η decided by strength and η(tlowast) de-cided by the carbon tax we have

1λm

minus 1 2α2c minus 2αc + t

2(β + 1)1113872 1113873

2

4α2c2(minus 1 + α)

2 (F1)

A threshold is

1113954λ1 4α2c2

(minus 1 + α)2

8α4c2minus 16α3c2

+ 4 2c + t2(β + 1)1113872 1113873cα2 minus 4t

2c(β + 1)α + t

4(β + 1)

2 (F2)

When 05lt λm lt 1113954λ1 the practical η is in the regionδ (η(tlowast) 1) therefore the cooperation is achieved facingthe optimal carbon tax

However the practical η is out of the range δ (η(tlowast) 1)

when 1113954λ1 lt λm Review previous result the region δ (η1 1)

is expanding when tax rises )erefore there exists a pos-sibility that the government increasing carbon tax to co-ordinate the supply chain We assume a realistic thing thegovernment is willing to boost the greenness of the producteven reducing part of the revenue from the carbon tax

Based on it the carbon up to 1113954t1 which means thegreenness is maximized )erefore )e cooperation isachieved when clt 1113954c1 because the greenness is decreasing inthe carbon tax when 1113954c1 lt c )en we have the secondthreshold

1113954λ2 1

η1 1113954t1( 1113857 + 1 (F3)

)e cooperation could not be achieved when λm gt 1113954λ2)e conclusion is as follows (1) Coordination would be

achieved when the manufacturer is democratic (ie05lt λm lt 1113954λ1) (2) Coordination could be achieved but partof the total tax will lose if development cost is lower when themanufacturer is moderate (ie 1113954λ1 lt λm lt 1113954λ2) (3) )e co-operation could not be achieved if the manufacturer is adictator (ie λm gt 1113954λ2)

G Proof of Proposition 5

)e first derivative of η1 in β is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t

2

2α2c2(minus 1 + α)

2 lt 0 (G1)

Similarly η1 is decreasing in β )at means the regionδ (η1 1) is expanding when β rises

And it is apparent that η1 (2cα(1 minus α) minus (1 + β)t22cα(1 minus α))2 is independent with k

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is research was supported by the National Natural ScienceFoundation of China (no 71761015) and the Science andTechnology Research Project of Jiangxi Education Depart-ment of China (no GJJ150475)

References

[1] Y Li M K Lim J Hu and M-L Tseng ldquoInvestigating theeffect of carbon tax and carbon quota policy to achieve lowcarbon logistics operationsrdquo Resources Conservation andRecycling vol 154 p 104535 2020

[2] M Abedi-Varaki ldquoStudy of carbon dioxide gas treatmentbased on equations of kinetics in plasma discharge reactorrdquoModern Physics Letters B vol 31 no 22 p 1750210 2017

[3] S Meng M Siriwardana and J McNeill ldquo)e environmentaland economic impact of the carbon tax in Australiardquo Envi-ronmental and Resource Economics vol 54 no 3 pp 313ndash332 2013

[4] Y Chen and C L Tseng ldquoInducing clean technology in theelectricity sector tradable permits or carbon tax policiesrdquo-eEnergy Journal vol 32 no 3 2011

[5] H-B Duan L Zhu and Y Fan ldquoOptimal carbon taxes incarbon-constrained China a logistic-induced energy eco-nomic hybrid modelrdquo Energy vol 69 pp 345ndash356 2014

[6] C Lu Q Tong and X Liu ldquo)e impacts of carbon tax andcomplementary policies on Chinese economyrdquo Energy Policyvol 38 no 11 pp 7278ndash7285 2010

[7] K Cao B Xu and J Wang ldquoOptimal trade-in and warrantyperiod strategies for new and remanufactured products undercarbon tax policyrdquo International Journal of Production Re-search vol 58 no 1 pp 180ndash199 2020

[8] P He W Zhang X Xu and Y Bian ldquoProduction lot-sizingand carbon emissions under cap-and-trade and carbon taxregulationsrdquo Journal of Cleaner Production vol 103pp 241ndash248 2015

[9] X Xu X Xu and P He ldquoJoint production and pricing de-cisions for multiple products with cap-and-trade and carbontax regulationsrdquo Journal of Cleaner Production vol 112pp 4093ndash4106 2016

Mathematical Problems in Engineering 15

[10] Q Li X Guan T Shi and W Jiao ldquoGreen product designwith competition and fairness concerns in the circulareconomy erardquo International Journal of Production Researchvol 58 no 1 pp 165ndash179 2020

[11] A Ranjan and J K Jha ldquoPricing and coordination strategiesof a dual-channel supply chain considering green quality andsales effortrdquo Journal of Cleaner Production vol 218pp 409ndash424 2019

[12] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineeringvol 10 no 1 pp 99ndash116 2012

[13] X Yin X Chen X Xu and L Zhang ldquoTax or subsidyOptimal carbon emission policy a supply chain perspectiverdquoSustainability vol 12 no 4 p 1548 2020

[14] B B Schlegelmilch G M Bohlen and A Diamantopoulosldquo)e link between green purchasing decisions and measuresof environmental consciousnessrdquo European Journal of Mar-keting vol 30 no 5 pp 35ndash55 1996

[15] M S Hopkins ldquoWhat the rsquogreenrsquo consumer wantsrdquo MITSloan Management Review vol 50 no 4 pp 87ndash89 2009

[16] G Kalyanaram and R S Winer ldquoEmpirical generalizationsfrom reference price researchrdquo Marketing Science vol 14no 3 pp G161ndashG169 1995

[17] E A Greenleaf ldquo)e impact of reference price effects on theprofitability of price promotionsrdquo Marketing Science vol 14no 1 pp 82ndash104 1995

[18] N M Modak D K Ghosh S Panda and S S SanaldquoManaging green house gas emission cost and pricing policiesin a two-echelon supply chainrdquo CIRP Journal ofManufacturing Science and Technology vol 20 pp 1ndash11 2018

[19] P Sinclair ldquoHigh does nothing and rising is worse carbontaxes should keep declining to cut harmful emissionsrdquo -eManchester School vol 60 no 1 pp 41ndash52 1992

[20] G M Grossman and A B Krueger ldquoEconomic growth andthe environmentrdquo -e Quarterly Journal of Economicsvol 110 no 2 pp 353ndash377 1995

[21] N M Modak and P Kelle ldquoUsing social work donation as atool of corporate social responsibility in a closed-loop supplychain considering carbon emissions tax and demand uncer-taintyrdquo Journal of the Operational Research Society pp 1ndash172019

[22] A Ulph and D Ulph ldquo)e optimal time path of a carbon taxrdquoOxford Economic Papers vol 46 no S1 pp 857ndash868 1994

[23] M Hoel and S Kverndokk ldquoDepletion of fossil fuels and theimpact of global warmingrdquo Resource and Energy Economicsvol 18 no 2 pp 115ndash136 1996

[24] Y H Farzin and O Tahvonen ldquoGlobal carbon cycle and theoptimal time path of a carbon taxrdquo Oxford Economic Papersvol 48 no 4 pp 515ndash536 1996

[25] V Bosetti C Carraro R Duval andM Tavoni ldquoWhat shouldwe expect from innovation A model-based assessment of theenvironmental and mitigation cost implications of climate-related RampDrdquo Energy Economics vol 33 no 6 pp 1313ndash13202011

[26] M Hariga R Asrsquoad and A Shamayleh ldquoIntegrated economicand environmental models for a multi stage cold supply chainunder carbon tax regulationrdquo Journal of Cleaner Productionvol 166 pp 1357ndash1371 2017

[27] N Turken J Carrillo and V Verter ldquoFacility location andcapacity acquisition under carbon tax and emissions limits tocentralize or to decentralizerdquo International Journal of Pro-duction Economics vol 187 pp 126ndash141 2017

[28] W Yu and R Han ldquoCoordinating a two-echelon supply chainunder carbon taxrdquo Sustainability vol 9 no 12 p 2360 2017

[29] S Sinha and N M Modak ldquoAn EPQmodel in the perspectiveof carbon emission reductionrdquo International Journal ofMathematics in Operational Research vol 14 no 3pp 338ndash358 2019

[30] H Zhang P Li H Zheng and Y Zhang ldquoImpact of carbontax on enterprise operation and production strategy for low-carbon products in a co-opetition supply chainrdquo Journal ofCleaner Production Article ID 125058 2020

[31] X Chen H Yang X Wang et al ldquoOptimal carbon tax designfor achieving low carbon supply chainsrdquo Annals of OperationsResearch pp 1ndash28 2020

[32] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Handbook of the Fundamentals of Fi-nancial Decision Making Part I pp 99ndash127 )e EconometricSociety Cleveland OH USA 2013

[33] W Kim and M Kim ldquoReference quality-based competitivemarket structure for innovation driven marketsrdquo Interna-tional Journal of Research in Marketing vol 32 no 3pp 284ndash296 2015

[34] I Popescu and Y Wu ldquoDynamic pricing strategies withreference effectsrdquo Operations Research vol 55 no 3pp 413ndash429 2007

[35] P K Kopalle P K Kannan L B Boldt and N Arora ldquo)eimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Re-tailing vol 88 no 1 pp 102ndash114 2012

[36] T-P Hsieh and C-Y Dye ldquoOptimal dynamic pricing fordeteriorating items with reference price effects when inven-tories stimulate demandrdquo European Journal of OperationalResearch vol 262 no 1 pp 136ndash150 2017

[37] X Chen Z-Y Hu and Y-H Zhang ldquoDynamic pricing withstochastic reference price effectrdquo Journal of the OperationsResearch Society of China vol 7 no 1 pp 107ndash125 2019

[38] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquo Opera-tions Research vol 64 no 6 pp 1529ndash1536 2016

[39] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[40] C Chen ldquoDesign for the environment a quality-based modelfor green product developmentrdquo Management Sciencevol 47 no 2 pp 250ndash263 2001

[41] K Chitra ldquoIn search of the green consumers a perceptualstudyrdquo Journal of Services Research vol 7 no 1 2007

[42] Z Liu T D Anderson and J M Cruz ldquoConsumer envi-ronmental awareness and competition in two-stage supplychainsrdquo European Journal of Operational Research vol 218no 3 pp 602ndash613 2012

[43] B Li Y Wang and Z Wang ldquoManaging a closed-loop supplychain with take-back legislation and consumer preference forgreen designrdquo Journal of Cleaner Production Article ID124481 2020

[44] J Rantanen D B Grant and W Piotrowicz ldquoInvestigatingsupply chain cooperation in Finnish grocery retailrdquo ResearchJournal of the University of Gdansk Transport Economics andLogistics vol 71 pp 19ndash34 2017

[45] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[46] G Xu B Dan X Zhang and C Liu ldquoCoordinating a dual-channel supply chain with risk-averse under a two-way

16 Mathematical Problems in Engineering

revenue sharing contractrdquo International Journal of ProductionEconomics vol 147 pp 171ndash179 2014

[47] Z Shi N Wang T Jia et al ldquoReverse revenue sharingcontract versus two-part tariff contract under a closed-loopsupply chain systemrdquoMathematical Problems in Engineeringvol 2016 Article ID 5464570 15 pages 2016

[48] S Panda N M Modak and L E Cardenas-Barron ldquoCo-ordinating a socially responsible closed-loop supply chainwith product recyclingrdquo International Journal of ProductionEconomics vol 188 pp 11ndash21 2017

[49] N M Modak N Modak S Panda and S S Sana ldquoAnalyzingstructure of two-echelon closed-loop supply chain for pricingquality and recycling managementrdquo Journal of Cleaner Pro-duction vol 171 pp 512ndash528 2018

[50] Q P Wang and D Z Zhao ldquoRevenue-sharing contract ofsupply chain based on consumerrsquos preference for low carbonproductsrdquo Chinese Journal of Management Science vol 22no 9 pp 106ndash113 2014

[51] B Yu J Wang X Lu and H Yang ldquoCollaboration in a low-carbon supply chain with reference emission and cost learningeffects cost sharing versus revenue sharing strategiesrdquo Journalof Cleaner Production vol 250 Article ID 119460 2020

[52] J Cao X Zhang and G Zhou ldquoSupply chain coordinationwith revenue-sharing contracts considering carbon emissionsand governmental policy makingrdquo Environmental Progress ampSustainable Energy vol 35 no 2 pp 479ndash488 2016

[53] X Xu P He H Xu and Q Zhang ldquoSupply chain coordi-nation with green technology under cap-and-trade regula-tionrdquo International Journal of Production Economics vol 183pp 433ndash442 2017

[54] N M Modak S Sinha S Panda et al ldquoAnalyzing a sociallyresponsible closed-loop distribution channel with recyclingfacilityrdquo SN Applied Sciences vol 1 no 10 p 1189 2019

[55] D Feng L Ma Y Ding G Wu and Y Zhang ldquoDecisions ofthe dual-channel supply chain under double policy consid-ering remanufacturingrdquo International Journal of Environ-mental Research and Public Health vol 16 no 3 p 465 2019

[56] I Bellos M Ferguson and L B Toktay ldquo)e car sharingeconomy interaction of business model choice and productline designrdquo Manufacturing amp Service Operations Manage-ment vol 19 no 2 pp 185ndash201 2017

[57] G Ferrer and J M Swaminathan ldquoManaging new andremanufactured productsrdquo Management Science vol 52no 1 pp 15ndash26 2006

[58] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2pp 519ndash527 2011

[59] L Cui S Guo and H Zhang ldquoCoordinating a green agri-foodsupply chain with revenue-sharing contracts consideringretailersrsquo green marketing effortsrdquo Sustainability vol 12no 4 p 1289 2020

[60] J Heydari K Govindan and A Aslani ldquoPricing and greeningdecisions in a three-tier dual channel supply chainrdquo Inter-national Journal of Production Economics vol 217 pp 185ndash196 2019

Mathematical Problems in Engineering 17

zg

zc minus

4α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

4cα(1 minus α) minus t2(1 + β)1113872 1113873

2 lt 0

zg

zk

te

4cα(1 minus α) minus t2(1 + β)gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(B6)

(4) )e first derivative of g in α is

zg

zα

minus Xt

4α(1 minus α)c minus t2(1 + β)1113872 1113873

2 (B7)

And X minus 4α2cpn + (4(1 + β)(te + c) minus 4ke minus 4pn)

(2α + 1)c + pnt2(1 + β) is a quadratic function )erange of α in 4αc(1 minus α) minus (1 + β)t2 gt 0 is

α1 c +

c2

minus ct2(1 + β)

1113969

2c

α2 c minus

c2

minus ct2(1 + β)

1113969

2c

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(B8)

By substituting those into X we have

X|αα1 4

c2

minus ct2(1 + β)

1113969

T minus 2 c minus t2(1 + β)1113872 1113873pn

X|αα2 minus 4

c2

minus ct2(1 + β)

1113969

T minus 2 c minus t2(1 + β)1113872 1113873pn lt 0

⎧⎪⎪⎨

⎪⎪⎩

(B9)

And T (1 + β)(te + c) minus ke minus ((12) + β)pn lt 0We can find

X|αα1 times X|αα2 minus 4 c minus t2(1 + β)1113872 1113873

middot 4 βpn + ke minus (1 + β)(te + c)( 11138572

1113872

+ p2nt

2(1 + β)1113873lt 0

(B10)

)erefore X|αα1 gt 0 )ere only exist a 1113954α isin (α2 α1)that makes (zgzα) 0 And (zgzα) gt 0 ifα isin (α2 1113954α) (zgzα)lt 0 if α isin (1113954α α1)

(5) )e first derivative of p in t is

zp

zt minus

2α(α minus 1) e(1 + β)t221113872 1113873 + minus pnα minus ke + c minus pn( 1113857β + c( 1113857t minus 2eαc(α minus 1)1113872 1113873c

(1 + β)t2

+ 4αc(α minus 1)1113872 11138732

A3 (1 + β)(c + 2te) minus ke +(1 minus β)pn

B3 minus 4c (1 + β)(c + 2te) minus ke minus βpn( 1113857 + t2(1 + β)pn

⎧⎪⎨

⎪⎩

(B11)

)e discriminant of quadratic function

e(1 + β)t2

2+ minus pnα minus ke + c minus pn( 1113857β + c( 1113857t minus 2eαc(α minus 1)

(B12)

is

Δ (minus β minus 1)c + ke + βpn + pnα( 11138572

+ 4e2(1 + β)t

2αc(α minus 1)

(B13)

We have a threshold

1113954c minus(minus β minus 1)c + ke + βpn + pnα( 1113857

2

4e2(1 + β)t

2α(α minus 1) (B14)

Review we have(α + β)pn + ke minus (1 + β)(te + c)gt 0therefore we can find that

tm 2

minus (1 + β)αc(α minus 1)

1113968

1 + β (B15)

Substituting it into above quadratic function

2 c

radicminus pnα minus ke + c minus pn( 1113857β + c( 1113857

minus (1 + β)α(α minus 1)

1113968minus 4eαc(α minus 1)(1 + β)

1 + β (B16)

And we have a same threshold

1113954c minus(minus β minus 1)c + ke + βpn + pnα( 1113857

2

4e2(1 + β)t

2α(α minus 1) (B17)

It could be found that(zpzt) gt 0 because Δlt 0 whenclt 1113954c And when cgt 1113954c we have another threshold 1113954t andzpzt|

t1113954t 0 because Δgt 0 and zpzt|tmlt 0)erefore when

cgt 1113954c and tgt1113954t p is decreasing in t And when cgt 1113954c and tlt1113954tp is increasing in t

12 Mathematical Problems in Engineering

C Proof of Proposition 2

(1) Let R t(e minus g)q denote total carbon tax and thefirst derivative of R in t is

zR

zt2c A2t

2+ B2t + c1113872 1113873 A3t

2+ B3t + c1113872 1113873

4cα(1 minus α) minus t2(1 + β)1113872 1113873

3 (C1)

Among them

A2 (1 + β)e

2gt 0

B2 minus (α + β)pn + ke minus (1 + β)c( 1113857lt 0

C2 2ceα(1 minus α)gt 0

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

A3 (α + β)pn + ke minus (1 + β)c( 1113857(1 + β)gt 0

B3 minus 8ceα(1 minus α)(1 + β)lt 0

C3 4cα(1 minus α) (α + β)pn + ke minus (1 + β)c( 1113857gt 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(C2)

Let Φ A2t2 + B2t + C2 and Ψ A3t

2 + B3t + C3 andwe can find that zRzt and Φ times Ψ have a same sign Bysolving Δ1 and Δ2 a threshold is

1113954c3 (α + β)pn + ke minus (1 + β)c( 1113857

2

4e2α(1 minus α)(1 + β)

(C3)

)ere are two positive real numbers of solutions forΦ ifclt 1113954c3 and let 0lt ta lt tb denote it Similarly let0lt tc lt td denote the solutions for Ψ if cgt 1113954c3 Reviewwe have 4αc(1 minus α) minus (1 + β)t2 gt 0 therefore lettm

4cα(1 minus a)1 + β

1113968 Substituting tm into Φ and Ψ

Φ|tm minus 2

cα(1 minus α)

1 + β

1113971

(α + β)pn + ke minus (1 + β)c( 1113857 + 4ecα(1 minus α)

Ψ|tm minus 8cα(1 minus α) 2(1 + β)e

α(1 minus α)

1 + βc

1113971

minus (α + β)pn + ke minus (1 + β)c( 1113857⎛⎝ ⎞⎠

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(C4)

If clt 1113954c3 Φ|tmlt 0 and Ψ|tm

gt 0 then zRzt|tmlt 0 If

cgt 1113954c3 Φ|tmgt 0 and Ψ|tm

lt 0 then zRzt|tmlt 0 )ere-

fore we have tm isin (ta tb) or tm isin (tc td) FurthermoreR t(e minus g)q is maximized when t ta if clt 1113954c3 R

t(e minus g)q is maximized when t tc if cgt 1113954c3 Totally wehave

tlowast

ta clt 1113954c3

tc cgt 1113954c31113896 (C5)

(2) Review Proposition 1 If cgt 1113954c1 g is maximized whentlowast 1113954t1 If clt 1113954c1 g is maximized when tlowast tmWhen tgt1113954t1 1113954c1 gt 1113954c5 And 1113954c1 lt 0lt 1113954c5 if tlt1113954t1Substituting 1113954t1 into Φ

Φ|1113954t1 minus

3 (α + β)pn + ke minus (1 + β)c( 11138572

8e(1 + β)+ 2ecα(1 minus α)

(C6)

A threshold is 1113954c4 (34)1113954c3 )erefore we have (i)when 0lt clt 1113954c4 tlowast ta because of Φ|1113954t1

lt 0 and 1113954t1 gt ta

(ii) when 1113954c4 lt clt 1113954c3 tlowast ta because of 1113954t1 lt ta (iii) tlowast

tc when 1113954c3 lt clt 1113954c1 and (iv) tlowast tc when 1113954c1 lt c )eresult is obtained by solving Φ and Ψ

D Proof of Theorem 2

Similar to Appendix A )e retailer problem is addressedfirst and we have

p αηpn + βηpn + ηkg + βw + w

2η(β + 1) (D1)

)en substituting p into the manufacturerrsquos profitfunction Solving the optimal w and g simultaneously yields

w αηpn + βet + βηpn minus βgt + eηk + cβ + et minus gt + c( 1113857η

(1 + η)(β + 1)

g minusαηpn + βηpn + eηk minus βw minus w( 1113857t

2(α minus 1)cαη

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(D2)

Mathematical Problems in Engineering 13

By solving the equation above we have

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

g t minus αpn + et(β + 1) + c minus pn( 1113857β minus ke + c( 1113857

t2(1 + β) minus (1 + η)cα(1 minus α)

p (2η + 1)cpnα

3+ A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1 et + ηpn + c( 1113857β + et + ke minus pn( 1113857η + c

A2 (2η + 1)pn + et + c( 1113857β +(minus 2η minus 1)pn + et +(2ηk + k)e + c

⎧⎪⎪⎨

⎪⎪⎩

B1 minus 2et minus 2ηpn minus 2c( 1113857β minus 2eηk minus 2et minus 2c( 1113857c + t2pn(β + 1)

B2 (minus 2η minus 1)pn minus et minus c( 1113857β minus et +(minus 2ηk minus k)e minus c( 1113857c + t2pn(β + 1)

⎧⎪⎪⎨

⎪⎪⎩

(D3)

Note that we have minus αpn + et(β + 1) + (c minus pn)β minus ke +

clt 0 )e concavity condition is (1 + β)(t2(1 + β) minus (1+

η)cα(1 minus α))lt 0

Substituting solutions into the profit function we have

πm minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2c

2 t2β + t

2+ 4(minus 1 + α)cα1113872 1113873(β + 1)

πr minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2(minus 1 + α)αc

2

4α2c minus 4αc +(β + 1)t2

1113872 11138732(β + 1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(D4)

E Proof of Proposition 3

πM (α + β)pn + ke minus (1 + β)c( 1113857

2c

2(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCM

(α + β)pn + ke minus (1 + β)c( 11138572c

2(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(E1)

By comparing the profits of two partners change after therevenue-sharing contract introduced we found that themanufacturer always benefits from cooperation

πR c2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCR

ηc2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(E2)

A threshold is

η1 2cα(1 minus α) minus (1 + β)t2

2cα(1 minus α)1113888 1113889

2

(E3)

When η is in the region δ (η1 1) the retailer willaccept cooperation And the first derivative of η1 in t is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t(β + 1)

α2c2(minus 1 + α)

2 lt 0 (E4)

)erefore η1 is decreasing in t )at means the regionδ (η1 1) is expanding when tax rises And the revenue-sharing contract is accepted easily than before

F Proof of Proposition 4

We consider a simple function (λrλm) to decide thepractical η where λm isin (05 1) and λr 1 minus λm Given the

14 Mathematical Problems in Engineering

carbon tax tlowast we have a region δ (η(tlowast) 1) of cooperationIt is clear that the gap of strength between two partitions inthe supply chain is a crucial factor in cooperation Bycomparing practice η decided by strength and η(tlowast) de-cided by the carbon tax we have

1λm

minus 1 2α2c minus 2αc + t

2(β + 1)1113872 1113873

2

4α2c2(minus 1 + α)

2 (F1)

A threshold is

1113954λ1 4α2c2

(minus 1 + α)2

8α4c2minus 16α3c2

+ 4 2c + t2(β + 1)1113872 1113873cα2 minus 4t

2c(β + 1)α + t

4(β + 1)

2 (F2)

When 05lt λm lt 1113954λ1 the practical η is in the regionδ (η(tlowast) 1) therefore the cooperation is achieved facingthe optimal carbon tax

However the practical η is out of the range δ (η(tlowast) 1)

when 1113954λ1 lt λm Review previous result the region δ (η1 1)

is expanding when tax rises )erefore there exists a pos-sibility that the government increasing carbon tax to co-ordinate the supply chain We assume a realistic thing thegovernment is willing to boost the greenness of the producteven reducing part of the revenue from the carbon tax

Based on it the carbon up to 1113954t1 which means thegreenness is maximized )erefore )e cooperation isachieved when clt 1113954c1 because the greenness is decreasing inthe carbon tax when 1113954c1 lt c )en we have the secondthreshold

1113954λ2 1

η1 1113954t1( 1113857 + 1 (F3)

)e cooperation could not be achieved when λm gt 1113954λ2)e conclusion is as follows (1) Coordination would be

achieved when the manufacturer is democratic (ie05lt λm lt 1113954λ1) (2) Coordination could be achieved but partof the total tax will lose if development cost is lower when themanufacturer is moderate (ie 1113954λ1 lt λm lt 1113954λ2) (3) )e co-operation could not be achieved if the manufacturer is adictator (ie λm gt 1113954λ2)

G Proof of Proposition 5

)e first derivative of η1 in β is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t

2

2α2c2(minus 1 + α)

2 lt 0 (G1)

Similarly η1 is decreasing in β )at means the regionδ (η1 1) is expanding when β rises

And it is apparent that η1 (2cα(1 minus α) minus (1 + β)t22cα(1 minus α))2 is independent with k

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is research was supported by the National Natural ScienceFoundation of China (no 71761015) and the Science andTechnology Research Project of Jiangxi Education Depart-ment of China (no GJJ150475)

References

[1] Y Li M K Lim J Hu and M-L Tseng ldquoInvestigating theeffect of carbon tax and carbon quota policy to achieve lowcarbon logistics operationsrdquo Resources Conservation andRecycling vol 154 p 104535 2020

[2] M Abedi-Varaki ldquoStudy of carbon dioxide gas treatmentbased on equations of kinetics in plasma discharge reactorrdquoModern Physics Letters B vol 31 no 22 p 1750210 2017

[3] S Meng M Siriwardana and J McNeill ldquo)e environmentaland economic impact of the carbon tax in Australiardquo Envi-ronmental and Resource Economics vol 54 no 3 pp 313ndash332 2013

[4] Y Chen and C L Tseng ldquoInducing clean technology in theelectricity sector tradable permits or carbon tax policiesrdquo-eEnergy Journal vol 32 no 3 2011

[5] H-B Duan L Zhu and Y Fan ldquoOptimal carbon taxes incarbon-constrained China a logistic-induced energy eco-nomic hybrid modelrdquo Energy vol 69 pp 345ndash356 2014

[6] C Lu Q Tong and X Liu ldquo)e impacts of carbon tax andcomplementary policies on Chinese economyrdquo Energy Policyvol 38 no 11 pp 7278ndash7285 2010

[7] K Cao B Xu and J Wang ldquoOptimal trade-in and warrantyperiod strategies for new and remanufactured products undercarbon tax policyrdquo International Journal of Production Re-search vol 58 no 1 pp 180ndash199 2020

[8] P He W Zhang X Xu and Y Bian ldquoProduction lot-sizingand carbon emissions under cap-and-trade and carbon taxregulationsrdquo Journal of Cleaner Production vol 103pp 241ndash248 2015

[9] X Xu X Xu and P He ldquoJoint production and pricing de-cisions for multiple products with cap-and-trade and carbontax regulationsrdquo Journal of Cleaner Production vol 112pp 4093ndash4106 2016

Mathematical Problems in Engineering 15

[10] Q Li X Guan T Shi and W Jiao ldquoGreen product designwith competition and fairness concerns in the circulareconomy erardquo International Journal of Production Researchvol 58 no 1 pp 165ndash179 2020

[11] A Ranjan and J K Jha ldquoPricing and coordination strategiesof a dual-channel supply chain considering green quality andsales effortrdquo Journal of Cleaner Production vol 218pp 409ndash424 2019

[12] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineeringvol 10 no 1 pp 99ndash116 2012

[13] X Yin X Chen X Xu and L Zhang ldquoTax or subsidyOptimal carbon emission policy a supply chain perspectiverdquoSustainability vol 12 no 4 p 1548 2020

[14] B B Schlegelmilch G M Bohlen and A Diamantopoulosldquo)e link between green purchasing decisions and measuresof environmental consciousnessrdquo European Journal of Mar-keting vol 30 no 5 pp 35ndash55 1996

[15] M S Hopkins ldquoWhat the rsquogreenrsquo consumer wantsrdquo MITSloan Management Review vol 50 no 4 pp 87ndash89 2009

[16] G Kalyanaram and R S Winer ldquoEmpirical generalizationsfrom reference price researchrdquo Marketing Science vol 14no 3 pp G161ndashG169 1995

[17] E A Greenleaf ldquo)e impact of reference price effects on theprofitability of price promotionsrdquo Marketing Science vol 14no 1 pp 82ndash104 1995

[18] N M Modak D K Ghosh S Panda and S S SanaldquoManaging green house gas emission cost and pricing policiesin a two-echelon supply chainrdquo CIRP Journal ofManufacturing Science and Technology vol 20 pp 1ndash11 2018

[19] P Sinclair ldquoHigh does nothing and rising is worse carbontaxes should keep declining to cut harmful emissionsrdquo -eManchester School vol 60 no 1 pp 41ndash52 1992

[20] G M Grossman and A B Krueger ldquoEconomic growth andthe environmentrdquo -e Quarterly Journal of Economicsvol 110 no 2 pp 353ndash377 1995

[21] N M Modak and P Kelle ldquoUsing social work donation as atool of corporate social responsibility in a closed-loop supplychain considering carbon emissions tax and demand uncer-taintyrdquo Journal of the Operational Research Society pp 1ndash172019

[22] A Ulph and D Ulph ldquo)e optimal time path of a carbon taxrdquoOxford Economic Papers vol 46 no S1 pp 857ndash868 1994

[23] M Hoel and S Kverndokk ldquoDepletion of fossil fuels and theimpact of global warmingrdquo Resource and Energy Economicsvol 18 no 2 pp 115ndash136 1996

[24] Y H Farzin and O Tahvonen ldquoGlobal carbon cycle and theoptimal time path of a carbon taxrdquo Oxford Economic Papersvol 48 no 4 pp 515ndash536 1996

[25] V Bosetti C Carraro R Duval andM Tavoni ldquoWhat shouldwe expect from innovation A model-based assessment of theenvironmental and mitigation cost implications of climate-related RampDrdquo Energy Economics vol 33 no 6 pp 1313ndash13202011

[26] M Hariga R Asrsquoad and A Shamayleh ldquoIntegrated economicand environmental models for a multi stage cold supply chainunder carbon tax regulationrdquo Journal of Cleaner Productionvol 166 pp 1357ndash1371 2017

[27] N Turken J Carrillo and V Verter ldquoFacility location andcapacity acquisition under carbon tax and emissions limits tocentralize or to decentralizerdquo International Journal of Pro-duction Economics vol 187 pp 126ndash141 2017

[28] W Yu and R Han ldquoCoordinating a two-echelon supply chainunder carbon taxrdquo Sustainability vol 9 no 12 p 2360 2017

[29] S Sinha and N M Modak ldquoAn EPQmodel in the perspectiveof carbon emission reductionrdquo International Journal ofMathematics in Operational Research vol 14 no 3pp 338ndash358 2019

[30] H Zhang P Li H Zheng and Y Zhang ldquoImpact of carbontax on enterprise operation and production strategy for low-carbon products in a co-opetition supply chainrdquo Journal ofCleaner Production Article ID 125058 2020

[31] X Chen H Yang X Wang et al ldquoOptimal carbon tax designfor achieving low carbon supply chainsrdquo Annals of OperationsResearch pp 1ndash28 2020

[32] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Handbook of the Fundamentals of Fi-nancial Decision Making Part I pp 99ndash127 )e EconometricSociety Cleveland OH USA 2013

[33] W Kim and M Kim ldquoReference quality-based competitivemarket structure for innovation driven marketsrdquo Interna-tional Journal of Research in Marketing vol 32 no 3pp 284ndash296 2015

[34] I Popescu and Y Wu ldquoDynamic pricing strategies withreference effectsrdquo Operations Research vol 55 no 3pp 413ndash429 2007

[35] P K Kopalle P K Kannan L B Boldt and N Arora ldquo)eimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Re-tailing vol 88 no 1 pp 102ndash114 2012

[36] T-P Hsieh and C-Y Dye ldquoOptimal dynamic pricing fordeteriorating items with reference price effects when inven-tories stimulate demandrdquo European Journal of OperationalResearch vol 262 no 1 pp 136ndash150 2017

[37] X Chen Z-Y Hu and Y-H Zhang ldquoDynamic pricing withstochastic reference price effectrdquo Journal of the OperationsResearch Society of China vol 7 no 1 pp 107ndash125 2019

[38] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquo Opera-tions Research vol 64 no 6 pp 1529ndash1536 2016

[39] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[40] C Chen ldquoDesign for the environment a quality-based modelfor green product developmentrdquo Management Sciencevol 47 no 2 pp 250ndash263 2001

[41] K Chitra ldquoIn search of the green consumers a perceptualstudyrdquo Journal of Services Research vol 7 no 1 2007

[42] Z Liu T D Anderson and J M Cruz ldquoConsumer envi-ronmental awareness and competition in two-stage supplychainsrdquo European Journal of Operational Research vol 218no 3 pp 602ndash613 2012

[43] B Li Y Wang and Z Wang ldquoManaging a closed-loop supplychain with take-back legislation and consumer preference forgreen designrdquo Journal of Cleaner Production Article ID124481 2020

[44] J Rantanen D B Grant and W Piotrowicz ldquoInvestigatingsupply chain cooperation in Finnish grocery retailrdquo ResearchJournal of the University of Gdansk Transport Economics andLogistics vol 71 pp 19ndash34 2017

[45] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[46] G Xu B Dan X Zhang and C Liu ldquoCoordinating a dual-channel supply chain with risk-averse under a two-way

16 Mathematical Problems in Engineering

revenue sharing contractrdquo International Journal of ProductionEconomics vol 147 pp 171ndash179 2014

[47] Z Shi N Wang T Jia et al ldquoReverse revenue sharingcontract versus two-part tariff contract under a closed-loopsupply chain systemrdquoMathematical Problems in Engineeringvol 2016 Article ID 5464570 15 pages 2016

[48] S Panda N M Modak and L E Cardenas-Barron ldquoCo-ordinating a socially responsible closed-loop supply chainwith product recyclingrdquo International Journal of ProductionEconomics vol 188 pp 11ndash21 2017

[49] N M Modak N Modak S Panda and S S Sana ldquoAnalyzingstructure of two-echelon closed-loop supply chain for pricingquality and recycling managementrdquo Journal of Cleaner Pro-duction vol 171 pp 512ndash528 2018

[50] Q P Wang and D Z Zhao ldquoRevenue-sharing contract ofsupply chain based on consumerrsquos preference for low carbonproductsrdquo Chinese Journal of Management Science vol 22no 9 pp 106ndash113 2014

[51] B Yu J Wang X Lu and H Yang ldquoCollaboration in a low-carbon supply chain with reference emission and cost learningeffects cost sharing versus revenue sharing strategiesrdquo Journalof Cleaner Production vol 250 Article ID 119460 2020

[52] J Cao X Zhang and G Zhou ldquoSupply chain coordinationwith revenue-sharing contracts considering carbon emissionsand governmental policy makingrdquo Environmental Progress ampSustainable Energy vol 35 no 2 pp 479ndash488 2016

[53] X Xu P He H Xu and Q Zhang ldquoSupply chain coordi-nation with green technology under cap-and-trade regula-tionrdquo International Journal of Production Economics vol 183pp 433ndash442 2017

[54] N M Modak S Sinha S Panda et al ldquoAnalyzing a sociallyresponsible closed-loop distribution channel with recyclingfacilityrdquo SN Applied Sciences vol 1 no 10 p 1189 2019

[55] D Feng L Ma Y Ding G Wu and Y Zhang ldquoDecisions ofthe dual-channel supply chain under double policy consid-ering remanufacturingrdquo International Journal of Environ-mental Research and Public Health vol 16 no 3 p 465 2019

[56] I Bellos M Ferguson and L B Toktay ldquo)e car sharingeconomy interaction of business model choice and productline designrdquo Manufacturing amp Service Operations Manage-ment vol 19 no 2 pp 185ndash201 2017

[57] G Ferrer and J M Swaminathan ldquoManaging new andremanufactured productsrdquo Management Science vol 52no 1 pp 15ndash26 2006

[58] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2pp 519ndash527 2011

[59] L Cui S Guo and H Zhang ldquoCoordinating a green agri-foodsupply chain with revenue-sharing contracts consideringretailersrsquo green marketing effortsrdquo Sustainability vol 12no 4 p 1289 2020

[60] J Heydari K Govindan and A Aslani ldquoPricing and greeningdecisions in a three-tier dual channel supply chainrdquo Inter-national Journal of Production Economics vol 217 pp 185ndash196 2019

Mathematical Problems in Engineering 17

C Proof of Proposition 2

(1) Let R t(e minus g)q denote total carbon tax and thefirst derivative of R in t is

zR

zt2c A2t

2+ B2t + c1113872 1113873 A3t

2+ B3t + c1113872 1113873

4cα(1 minus α) minus t2(1 + β)1113872 1113873

3 (C1)

Among them

A2 (1 + β)e

2gt 0

B2 minus (α + β)pn + ke minus (1 + β)c( 1113857lt 0

C2 2ceα(1 minus α)gt 0

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

A3 (α + β)pn + ke minus (1 + β)c( 1113857(1 + β)gt 0

B3 minus 8ceα(1 minus α)(1 + β)lt 0

C3 4cα(1 minus α) (α + β)pn + ke minus (1 + β)c( 1113857gt 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(C2)

Let Φ A2t2 + B2t + C2 and Ψ A3t

2 + B3t + C3 andwe can find that zRzt and Φ times Ψ have a same sign Bysolving Δ1 and Δ2 a threshold is

1113954c3 (α + β)pn + ke minus (1 + β)c( 1113857

2

4e2α(1 minus α)(1 + β)

(C3)

)ere are two positive real numbers of solutions forΦ ifclt 1113954c3 and let 0lt ta lt tb denote it Similarly let0lt tc lt td denote the solutions for Ψ if cgt 1113954c3 Reviewwe have 4αc(1 minus α) minus (1 + β)t2 gt 0 therefore lettm

4cα(1 minus a)1 + β

1113968 Substituting tm into Φ and Ψ

Φ|tm minus 2

cα(1 minus α)

1 + β

1113971

(α + β)pn + ke minus (1 + β)c( 1113857 + 4ecα(1 minus α)

Ψ|tm minus 8cα(1 minus α) 2(1 + β)e

α(1 minus α)

1 + βc

1113971

minus (α + β)pn + ke minus (1 + β)c( 1113857⎛⎝ ⎞⎠

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(C4)

If clt 1113954c3 Φ|tmlt 0 and Ψ|tm

gt 0 then zRzt|tmlt 0 If

cgt 1113954c3 Φ|tmgt 0 and Ψ|tm

lt 0 then zRzt|tmlt 0 )ere-

fore we have tm isin (ta tb) or tm isin (tc td) FurthermoreR t(e minus g)q is maximized when t ta if clt 1113954c3 R

t(e minus g)q is maximized when t tc if cgt 1113954c3 Totally wehave

tlowast

ta clt 1113954c3

tc cgt 1113954c31113896 (C5)

(2) Review Proposition 1 If cgt 1113954c1 g is maximized whentlowast 1113954t1 If clt 1113954c1 g is maximized when tlowast tmWhen tgt1113954t1 1113954c1 gt 1113954c5 And 1113954c1 lt 0lt 1113954c5 if tlt1113954t1Substituting 1113954t1 into Φ

Φ|1113954t1 minus

3 (α + β)pn + ke minus (1 + β)c( 11138572

8e(1 + β)+ 2ecα(1 minus α)

(C6)

A threshold is 1113954c4 (34)1113954c3 )erefore we have (i)when 0lt clt 1113954c4 tlowast ta because of Φ|1113954t1

lt 0 and 1113954t1 gt ta

(ii) when 1113954c4 lt clt 1113954c3 tlowast ta because of 1113954t1 lt ta (iii) tlowast

tc when 1113954c3 lt clt 1113954c1 and (iv) tlowast tc when 1113954c1 lt c )eresult is obtained by solving Φ and Ψ

D Proof of Theorem 2

Similar to Appendix A )e retailer problem is addressedfirst and we have

p αηpn + βηpn + ηkg + βw + w

2η(β + 1) (D1)

)en substituting p into the manufacturerrsquos profitfunction Solving the optimal w and g simultaneously yields

w αηpn + βet + βηpn minus βgt + eηk + cβ + et minus gt + c( 1113857η

(1 + η)(β + 1)

g minusαηpn + βηpn + eηk minus βw minus w( 1113857t

2(α minus 1)cαη

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(D2)

Mathematical Problems in Engineering 13

By solving the equation above we have

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

g t minus αpn + et(β + 1) + c minus pn( 1113857β minus ke + c( 1113857

t2(1 + β) minus (1 + η)cα(1 minus α)

p (2η + 1)cpnα

3+ A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1 et + ηpn + c( 1113857β + et + ke minus pn( 1113857η + c

A2 (2η + 1)pn + et + c( 1113857β +(minus 2η minus 1)pn + et +(2ηk + k)e + c

⎧⎪⎪⎨

⎪⎪⎩

B1 minus 2et minus 2ηpn minus 2c( 1113857β minus 2eηk minus 2et minus 2c( 1113857c + t2pn(β + 1)

B2 (minus 2η minus 1)pn minus et minus c( 1113857β minus et +(minus 2ηk minus k)e minus c( 1113857c + t2pn(β + 1)

⎧⎪⎪⎨

⎪⎪⎩

(D3)

Note that we have minus αpn + et(β + 1) + (c minus pn)β minus ke +

clt 0 )e concavity condition is (1 + β)(t2(1 + β) minus (1+

η)cα(1 minus α))lt 0

Substituting solutions into the profit function we have

πm minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2c

2 t2β + t

2+ 4(minus 1 + α)cα1113872 1113873(β + 1)

πr minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2(minus 1 + α)αc

2

4α2c minus 4αc +(β + 1)t2

1113872 11138732(β + 1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(D4)

E Proof of Proposition 3

πM (α + β)pn + ke minus (1 + β)c( 1113857

2c

2(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCM

(α + β)pn + ke minus (1 + β)c( 11138572c

2(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(E1)

By comparing the profits of two partners change after therevenue-sharing contract introduced we found that themanufacturer always benefits from cooperation

πR c2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCR

ηc2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(E2)

A threshold is

η1 2cα(1 minus α) minus (1 + β)t2

2cα(1 minus α)1113888 1113889

2

(E3)

When η is in the region δ (η1 1) the retailer willaccept cooperation And the first derivative of η1 in t is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t(β + 1)

α2c2(minus 1 + α)

2 lt 0 (E4)

)erefore η1 is decreasing in t )at means the regionδ (η1 1) is expanding when tax rises And the revenue-sharing contract is accepted easily than before

F Proof of Proposition 4

We consider a simple function (λrλm) to decide thepractical η where λm isin (05 1) and λr 1 minus λm Given the

14 Mathematical Problems in Engineering

carbon tax tlowast we have a region δ (η(tlowast) 1) of cooperationIt is clear that the gap of strength between two partitions inthe supply chain is a crucial factor in cooperation Bycomparing practice η decided by strength and η(tlowast) de-cided by the carbon tax we have

1λm

minus 1 2α2c minus 2αc + t

2(β + 1)1113872 1113873

2

4α2c2(minus 1 + α)

2 (F1)

A threshold is

1113954λ1 4α2c2

(minus 1 + α)2

8α4c2minus 16α3c2

+ 4 2c + t2(β + 1)1113872 1113873cα2 minus 4t

2c(β + 1)α + t

4(β + 1)

2 (F2)

When 05lt λm lt 1113954λ1 the practical η is in the regionδ (η(tlowast) 1) therefore the cooperation is achieved facingthe optimal carbon tax

However the practical η is out of the range δ (η(tlowast) 1)

when 1113954λ1 lt λm Review previous result the region δ (η1 1)

is expanding when tax rises )erefore there exists a pos-sibility that the government increasing carbon tax to co-ordinate the supply chain We assume a realistic thing thegovernment is willing to boost the greenness of the producteven reducing part of the revenue from the carbon tax

Based on it the carbon up to 1113954t1 which means thegreenness is maximized )erefore )e cooperation isachieved when clt 1113954c1 because the greenness is decreasing inthe carbon tax when 1113954c1 lt c )en we have the secondthreshold

1113954λ2 1

η1 1113954t1( 1113857 + 1 (F3)

)e cooperation could not be achieved when λm gt 1113954λ2)e conclusion is as follows (1) Coordination would be

achieved when the manufacturer is democratic (ie05lt λm lt 1113954λ1) (2) Coordination could be achieved but partof the total tax will lose if development cost is lower when themanufacturer is moderate (ie 1113954λ1 lt λm lt 1113954λ2) (3) )e co-operation could not be achieved if the manufacturer is adictator (ie λm gt 1113954λ2)

G Proof of Proposition 5

)e first derivative of η1 in β is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t

2

2α2c2(minus 1 + α)

2 lt 0 (G1)

Similarly η1 is decreasing in β )at means the regionδ (η1 1) is expanding when β rises

And it is apparent that η1 (2cα(1 minus α) minus (1 + β)t22cα(1 minus α))2 is independent with k

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is research was supported by the National Natural ScienceFoundation of China (no 71761015) and the Science andTechnology Research Project of Jiangxi Education Depart-ment of China (no GJJ150475)

References

[1] Y Li M K Lim J Hu and M-L Tseng ldquoInvestigating theeffect of carbon tax and carbon quota policy to achieve lowcarbon logistics operationsrdquo Resources Conservation andRecycling vol 154 p 104535 2020

[2] M Abedi-Varaki ldquoStudy of carbon dioxide gas treatmentbased on equations of kinetics in plasma discharge reactorrdquoModern Physics Letters B vol 31 no 22 p 1750210 2017

[3] S Meng M Siriwardana and J McNeill ldquo)e environmentaland economic impact of the carbon tax in Australiardquo Envi-ronmental and Resource Economics vol 54 no 3 pp 313ndash332 2013

[4] Y Chen and C L Tseng ldquoInducing clean technology in theelectricity sector tradable permits or carbon tax policiesrdquo-eEnergy Journal vol 32 no 3 2011

[5] H-B Duan L Zhu and Y Fan ldquoOptimal carbon taxes incarbon-constrained China a logistic-induced energy eco-nomic hybrid modelrdquo Energy vol 69 pp 345ndash356 2014

[6] C Lu Q Tong and X Liu ldquo)e impacts of carbon tax andcomplementary policies on Chinese economyrdquo Energy Policyvol 38 no 11 pp 7278ndash7285 2010

[7] K Cao B Xu and J Wang ldquoOptimal trade-in and warrantyperiod strategies for new and remanufactured products undercarbon tax policyrdquo International Journal of Production Re-search vol 58 no 1 pp 180ndash199 2020

[8] P He W Zhang X Xu and Y Bian ldquoProduction lot-sizingand carbon emissions under cap-and-trade and carbon taxregulationsrdquo Journal of Cleaner Production vol 103pp 241ndash248 2015

[9] X Xu X Xu and P He ldquoJoint production and pricing de-cisions for multiple products with cap-and-trade and carbontax regulationsrdquo Journal of Cleaner Production vol 112pp 4093ndash4106 2016

Mathematical Problems in Engineering 15

[10] Q Li X Guan T Shi and W Jiao ldquoGreen product designwith competition and fairness concerns in the circulareconomy erardquo International Journal of Production Researchvol 58 no 1 pp 165ndash179 2020

[11] A Ranjan and J K Jha ldquoPricing and coordination strategiesof a dual-channel supply chain considering green quality andsales effortrdquo Journal of Cleaner Production vol 218pp 409ndash424 2019

[12] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineeringvol 10 no 1 pp 99ndash116 2012

[13] X Yin X Chen X Xu and L Zhang ldquoTax or subsidyOptimal carbon emission policy a supply chain perspectiverdquoSustainability vol 12 no 4 p 1548 2020

[14] B B Schlegelmilch G M Bohlen and A Diamantopoulosldquo)e link between green purchasing decisions and measuresof environmental consciousnessrdquo European Journal of Mar-keting vol 30 no 5 pp 35ndash55 1996

[15] M S Hopkins ldquoWhat the rsquogreenrsquo consumer wantsrdquo MITSloan Management Review vol 50 no 4 pp 87ndash89 2009

[16] G Kalyanaram and R S Winer ldquoEmpirical generalizationsfrom reference price researchrdquo Marketing Science vol 14no 3 pp G161ndashG169 1995

[17] E A Greenleaf ldquo)e impact of reference price effects on theprofitability of price promotionsrdquo Marketing Science vol 14no 1 pp 82ndash104 1995

[18] N M Modak D K Ghosh S Panda and S S SanaldquoManaging green house gas emission cost and pricing policiesin a two-echelon supply chainrdquo CIRP Journal ofManufacturing Science and Technology vol 20 pp 1ndash11 2018

[19] P Sinclair ldquoHigh does nothing and rising is worse carbontaxes should keep declining to cut harmful emissionsrdquo -eManchester School vol 60 no 1 pp 41ndash52 1992

[20] G M Grossman and A B Krueger ldquoEconomic growth andthe environmentrdquo -e Quarterly Journal of Economicsvol 110 no 2 pp 353ndash377 1995

[21] N M Modak and P Kelle ldquoUsing social work donation as atool of corporate social responsibility in a closed-loop supplychain considering carbon emissions tax and demand uncer-taintyrdquo Journal of the Operational Research Society pp 1ndash172019

[22] A Ulph and D Ulph ldquo)e optimal time path of a carbon taxrdquoOxford Economic Papers vol 46 no S1 pp 857ndash868 1994

[23] M Hoel and S Kverndokk ldquoDepletion of fossil fuels and theimpact of global warmingrdquo Resource and Energy Economicsvol 18 no 2 pp 115ndash136 1996

[24] Y H Farzin and O Tahvonen ldquoGlobal carbon cycle and theoptimal time path of a carbon taxrdquo Oxford Economic Papersvol 48 no 4 pp 515ndash536 1996

[25] V Bosetti C Carraro R Duval andM Tavoni ldquoWhat shouldwe expect from innovation A model-based assessment of theenvironmental and mitigation cost implications of climate-related RampDrdquo Energy Economics vol 33 no 6 pp 1313ndash13202011

[26] M Hariga R Asrsquoad and A Shamayleh ldquoIntegrated economicand environmental models for a multi stage cold supply chainunder carbon tax regulationrdquo Journal of Cleaner Productionvol 166 pp 1357ndash1371 2017

[27] N Turken J Carrillo and V Verter ldquoFacility location andcapacity acquisition under carbon tax and emissions limits tocentralize or to decentralizerdquo International Journal of Pro-duction Economics vol 187 pp 126ndash141 2017

[28] W Yu and R Han ldquoCoordinating a two-echelon supply chainunder carbon taxrdquo Sustainability vol 9 no 12 p 2360 2017

[29] S Sinha and N M Modak ldquoAn EPQmodel in the perspectiveof carbon emission reductionrdquo International Journal ofMathematics in Operational Research vol 14 no 3pp 338ndash358 2019

[30] H Zhang P Li H Zheng and Y Zhang ldquoImpact of carbontax on enterprise operation and production strategy for low-carbon products in a co-opetition supply chainrdquo Journal ofCleaner Production Article ID 125058 2020

[31] X Chen H Yang X Wang et al ldquoOptimal carbon tax designfor achieving low carbon supply chainsrdquo Annals of OperationsResearch pp 1ndash28 2020

[32] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Handbook of the Fundamentals of Fi-nancial Decision Making Part I pp 99ndash127 )e EconometricSociety Cleveland OH USA 2013

[33] W Kim and M Kim ldquoReference quality-based competitivemarket structure for innovation driven marketsrdquo Interna-tional Journal of Research in Marketing vol 32 no 3pp 284ndash296 2015

[34] I Popescu and Y Wu ldquoDynamic pricing strategies withreference effectsrdquo Operations Research vol 55 no 3pp 413ndash429 2007

[35] P K Kopalle P K Kannan L B Boldt and N Arora ldquo)eimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Re-tailing vol 88 no 1 pp 102ndash114 2012

[36] T-P Hsieh and C-Y Dye ldquoOptimal dynamic pricing fordeteriorating items with reference price effects when inven-tories stimulate demandrdquo European Journal of OperationalResearch vol 262 no 1 pp 136ndash150 2017

[37] X Chen Z-Y Hu and Y-H Zhang ldquoDynamic pricing withstochastic reference price effectrdquo Journal of the OperationsResearch Society of China vol 7 no 1 pp 107ndash125 2019

[38] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquo Opera-tions Research vol 64 no 6 pp 1529ndash1536 2016

[39] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[40] C Chen ldquoDesign for the environment a quality-based modelfor green product developmentrdquo Management Sciencevol 47 no 2 pp 250ndash263 2001

[41] K Chitra ldquoIn search of the green consumers a perceptualstudyrdquo Journal of Services Research vol 7 no 1 2007

[42] Z Liu T D Anderson and J M Cruz ldquoConsumer envi-ronmental awareness and competition in two-stage supplychainsrdquo European Journal of Operational Research vol 218no 3 pp 602ndash613 2012

[43] B Li Y Wang and Z Wang ldquoManaging a closed-loop supplychain with take-back legislation and consumer preference forgreen designrdquo Journal of Cleaner Production Article ID124481 2020

[44] J Rantanen D B Grant and W Piotrowicz ldquoInvestigatingsupply chain cooperation in Finnish grocery retailrdquo ResearchJournal of the University of Gdansk Transport Economics andLogistics vol 71 pp 19ndash34 2017

[45] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[46] G Xu B Dan X Zhang and C Liu ldquoCoordinating a dual-channel supply chain with risk-averse under a two-way

16 Mathematical Problems in Engineering

revenue sharing contractrdquo International Journal of ProductionEconomics vol 147 pp 171ndash179 2014

[47] Z Shi N Wang T Jia et al ldquoReverse revenue sharingcontract versus two-part tariff contract under a closed-loopsupply chain systemrdquoMathematical Problems in Engineeringvol 2016 Article ID 5464570 15 pages 2016

[48] S Panda N M Modak and L E Cardenas-Barron ldquoCo-ordinating a socially responsible closed-loop supply chainwith product recyclingrdquo International Journal of ProductionEconomics vol 188 pp 11ndash21 2017

[49] N M Modak N Modak S Panda and S S Sana ldquoAnalyzingstructure of two-echelon closed-loop supply chain for pricingquality and recycling managementrdquo Journal of Cleaner Pro-duction vol 171 pp 512ndash528 2018

[50] Q P Wang and D Z Zhao ldquoRevenue-sharing contract ofsupply chain based on consumerrsquos preference for low carbonproductsrdquo Chinese Journal of Management Science vol 22no 9 pp 106ndash113 2014

[51] B Yu J Wang X Lu and H Yang ldquoCollaboration in a low-carbon supply chain with reference emission and cost learningeffects cost sharing versus revenue sharing strategiesrdquo Journalof Cleaner Production vol 250 Article ID 119460 2020

[52] J Cao X Zhang and G Zhou ldquoSupply chain coordinationwith revenue-sharing contracts considering carbon emissionsand governmental policy makingrdquo Environmental Progress ampSustainable Energy vol 35 no 2 pp 479ndash488 2016

[53] X Xu P He H Xu and Q Zhang ldquoSupply chain coordi-nation with green technology under cap-and-trade regula-tionrdquo International Journal of Production Economics vol 183pp 433ndash442 2017

[54] N M Modak S Sinha S Panda et al ldquoAnalyzing a sociallyresponsible closed-loop distribution channel with recyclingfacilityrdquo SN Applied Sciences vol 1 no 10 p 1189 2019

[55] D Feng L Ma Y Ding G Wu and Y Zhang ldquoDecisions ofthe dual-channel supply chain under double policy consid-ering remanufacturingrdquo International Journal of Environ-mental Research and Public Health vol 16 no 3 p 465 2019

[56] I Bellos M Ferguson and L B Toktay ldquo)e car sharingeconomy interaction of business model choice and productline designrdquo Manufacturing amp Service Operations Manage-ment vol 19 no 2 pp 185ndash201 2017

[57] G Ferrer and J M Swaminathan ldquoManaging new andremanufactured productsrdquo Management Science vol 52no 1 pp 15ndash26 2006

[58] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2pp 519ndash527 2011

[59] L Cui S Guo and H Zhang ldquoCoordinating a green agri-foodsupply chain with revenue-sharing contracts consideringretailersrsquo green marketing effortsrdquo Sustainability vol 12no 4 p 1289 2020

[60] J Heydari K Govindan and A Aslani ldquoPricing and greeningdecisions in a three-tier dual channel supply chainrdquo Inter-national Journal of Production Economics vol 217 pp 185ndash196 2019

Mathematical Problems in Engineering 17

By solving the equation above we have

w 2cpnα

3+ 2A1cα

2+ B1α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

g t minus αpn + et(β + 1) + c minus pn( 1113857β minus ke + c( 1113857

t2(1 + β) minus (1 + η)cα(1 minus α)

p (2η + 1)cpnα

3+ A2cα

2+ B2α + t

2(1 + β) βpn + ke( 1113857

(1 + β) t2(1 + β) minus (1 + η)cα(1 minus α)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1 et + ηpn + c( 1113857β + et + ke minus pn( 1113857η + c

A2 (2η + 1)pn + et + c( 1113857β +(minus 2η minus 1)pn + et +(2ηk + k)e + c

⎧⎪⎪⎨

⎪⎪⎩

B1 minus 2et minus 2ηpn minus 2c( 1113857β minus 2eηk minus 2et minus 2c( 1113857c + t2pn(β + 1)

B2 (minus 2η minus 1)pn minus et minus c( 1113857β minus et +(minus 2ηk minus k)e minus c( 1113857c + t2pn(β + 1)

⎧⎪⎪⎨

⎪⎪⎩

(D3)

Note that we have minus αpn + et(β + 1) + (c minus pn)β minus ke +

clt 0 )e concavity condition is (1 + β)(t2(1 + β) minus (1+

η)cα(1 minus α))lt 0

Substituting solutions into the profit function we have

πm minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2c

2 t2β + t

2+ 4(minus 1 + α)cα1113872 1113873(β + 1)

πr minuste + c minus pn( 1113857β +(minus k + t)e minus αpn + c( 1113857

2(minus 1 + α)αc

2

4α2c minus 4αc +(β + 1)t2

1113872 11138732(β + 1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(D4)

E Proof of Proposition 3

πM (α + β)pn + ke minus (1 + β)c( 1113857

2c

2(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCM

(α + β)pn + ke minus (1 + β)c( 11138572c

2(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(E1)

By comparing the profits of two partners change after therevenue-sharing contract introduced we found that themanufacturer always benefits from cooperation

πR c2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 4cα(1 minus α) minus (1 + β)t2

1113872 11138732

πCR

ηc2α(1 minus α) (α + β)pn + ke minus (1 + β)(te + c)( 1113857

2

(1 + β) 2(1 + η)cα(1 minus α) minus (1 + β)t2

1113872 11138732

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(E2)

A threshold is

η1 2cα(1 minus α) minus (1 + β)t2

2cα(1 minus α)1113888 1113889

2

(E3)

When η is in the region δ (η1 1) the retailer willaccept cooperation And the first derivative of η1 in t is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t(β + 1)

α2c2(minus 1 + α)

2 lt 0 (E4)

)erefore η1 is decreasing in t )at means the regionδ (η1 1) is expanding when tax rises And the revenue-sharing contract is accepted easily than before

F Proof of Proposition 4

We consider a simple function (λrλm) to decide thepractical η where λm isin (05 1) and λr 1 minus λm Given the

14 Mathematical Problems in Engineering

carbon tax tlowast we have a region δ (η(tlowast) 1) of cooperationIt is clear that the gap of strength between two partitions inthe supply chain is a crucial factor in cooperation Bycomparing practice η decided by strength and η(tlowast) de-cided by the carbon tax we have

1λm

minus 1 2α2c minus 2αc + t

2(β + 1)1113872 1113873

2

4α2c2(minus 1 + α)

2 (F1)

A threshold is

1113954λ1 4α2c2

(minus 1 + α)2

8α4c2minus 16α3c2

+ 4 2c + t2(β + 1)1113872 1113873cα2 minus 4t

2c(β + 1)α + t

4(β + 1)

2 (F2)

When 05lt λm lt 1113954λ1 the practical η is in the regionδ (η(tlowast) 1) therefore the cooperation is achieved facingthe optimal carbon tax

However the practical η is out of the range δ (η(tlowast) 1)

when 1113954λ1 lt λm Review previous result the region δ (η1 1)

is expanding when tax rises )erefore there exists a pos-sibility that the government increasing carbon tax to co-ordinate the supply chain We assume a realistic thing thegovernment is willing to boost the greenness of the producteven reducing part of the revenue from the carbon tax

Based on it the carbon up to 1113954t1 which means thegreenness is maximized )erefore )e cooperation isachieved when clt 1113954c1 because the greenness is decreasing inthe carbon tax when 1113954c1 lt c )en we have the secondthreshold

1113954λ2 1

η1 1113954t1( 1113857 + 1 (F3)

)e cooperation could not be achieved when λm gt 1113954λ2)e conclusion is as follows (1) Coordination would be

achieved when the manufacturer is democratic (ie05lt λm lt 1113954λ1) (2) Coordination could be achieved but partof the total tax will lose if development cost is lower when themanufacturer is moderate (ie 1113954λ1 lt λm lt 1113954λ2) (3) )e co-operation could not be achieved if the manufacturer is adictator (ie λm gt 1113954λ2)

G Proof of Proposition 5

)e first derivative of η1 in β is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t

2

2α2c2(minus 1 + α)

2 lt 0 (G1)

Similarly η1 is decreasing in β )at means the regionδ (η1 1) is expanding when β rises

And it is apparent that η1 (2cα(1 minus α) minus (1 + β)t22cα(1 minus α))2 is independent with k

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is research was supported by the National Natural ScienceFoundation of China (no 71761015) and the Science andTechnology Research Project of Jiangxi Education Depart-ment of China (no GJJ150475)

References

[1] Y Li M K Lim J Hu and M-L Tseng ldquoInvestigating theeffect of carbon tax and carbon quota policy to achieve lowcarbon logistics operationsrdquo Resources Conservation andRecycling vol 154 p 104535 2020

[2] M Abedi-Varaki ldquoStudy of carbon dioxide gas treatmentbased on equations of kinetics in plasma discharge reactorrdquoModern Physics Letters B vol 31 no 22 p 1750210 2017

[3] S Meng M Siriwardana and J McNeill ldquo)e environmentaland economic impact of the carbon tax in Australiardquo Envi-ronmental and Resource Economics vol 54 no 3 pp 313ndash332 2013

[4] Y Chen and C L Tseng ldquoInducing clean technology in theelectricity sector tradable permits or carbon tax policiesrdquo-eEnergy Journal vol 32 no 3 2011

[5] H-B Duan L Zhu and Y Fan ldquoOptimal carbon taxes incarbon-constrained China a logistic-induced energy eco-nomic hybrid modelrdquo Energy vol 69 pp 345ndash356 2014

[6] C Lu Q Tong and X Liu ldquo)e impacts of carbon tax andcomplementary policies on Chinese economyrdquo Energy Policyvol 38 no 11 pp 7278ndash7285 2010

[7] K Cao B Xu and J Wang ldquoOptimal trade-in and warrantyperiod strategies for new and remanufactured products undercarbon tax policyrdquo International Journal of Production Re-search vol 58 no 1 pp 180ndash199 2020

[8] P He W Zhang X Xu and Y Bian ldquoProduction lot-sizingand carbon emissions under cap-and-trade and carbon taxregulationsrdquo Journal of Cleaner Production vol 103pp 241ndash248 2015

[9] X Xu X Xu and P He ldquoJoint production and pricing de-cisions for multiple products with cap-and-trade and carbontax regulationsrdquo Journal of Cleaner Production vol 112pp 4093ndash4106 2016

Mathematical Problems in Engineering 15

[10] Q Li X Guan T Shi and W Jiao ldquoGreen product designwith competition and fairness concerns in the circulareconomy erardquo International Journal of Production Researchvol 58 no 1 pp 165ndash179 2020

[11] A Ranjan and J K Jha ldquoPricing and coordination strategiesof a dual-channel supply chain considering green quality andsales effortrdquo Journal of Cleaner Production vol 218pp 409ndash424 2019

[12] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineeringvol 10 no 1 pp 99ndash116 2012

[13] X Yin X Chen X Xu and L Zhang ldquoTax or subsidyOptimal carbon emission policy a supply chain perspectiverdquoSustainability vol 12 no 4 p 1548 2020

[14] B B Schlegelmilch G M Bohlen and A Diamantopoulosldquo)e link between green purchasing decisions and measuresof environmental consciousnessrdquo European Journal of Mar-keting vol 30 no 5 pp 35ndash55 1996

[15] M S Hopkins ldquoWhat the rsquogreenrsquo consumer wantsrdquo MITSloan Management Review vol 50 no 4 pp 87ndash89 2009

[16] G Kalyanaram and R S Winer ldquoEmpirical generalizationsfrom reference price researchrdquo Marketing Science vol 14no 3 pp G161ndashG169 1995

[17] E A Greenleaf ldquo)e impact of reference price effects on theprofitability of price promotionsrdquo Marketing Science vol 14no 1 pp 82ndash104 1995

[18] N M Modak D K Ghosh S Panda and S S SanaldquoManaging green house gas emission cost and pricing policiesin a two-echelon supply chainrdquo CIRP Journal ofManufacturing Science and Technology vol 20 pp 1ndash11 2018

[19] P Sinclair ldquoHigh does nothing and rising is worse carbontaxes should keep declining to cut harmful emissionsrdquo -eManchester School vol 60 no 1 pp 41ndash52 1992

[20] G M Grossman and A B Krueger ldquoEconomic growth andthe environmentrdquo -e Quarterly Journal of Economicsvol 110 no 2 pp 353ndash377 1995

[21] N M Modak and P Kelle ldquoUsing social work donation as atool of corporate social responsibility in a closed-loop supplychain considering carbon emissions tax and demand uncer-taintyrdquo Journal of the Operational Research Society pp 1ndash172019

[22] A Ulph and D Ulph ldquo)e optimal time path of a carbon taxrdquoOxford Economic Papers vol 46 no S1 pp 857ndash868 1994

[23] M Hoel and S Kverndokk ldquoDepletion of fossil fuels and theimpact of global warmingrdquo Resource and Energy Economicsvol 18 no 2 pp 115ndash136 1996

[24] Y H Farzin and O Tahvonen ldquoGlobal carbon cycle and theoptimal time path of a carbon taxrdquo Oxford Economic Papersvol 48 no 4 pp 515ndash536 1996

[25] V Bosetti C Carraro R Duval andM Tavoni ldquoWhat shouldwe expect from innovation A model-based assessment of theenvironmental and mitigation cost implications of climate-related RampDrdquo Energy Economics vol 33 no 6 pp 1313ndash13202011

[26] M Hariga R Asrsquoad and A Shamayleh ldquoIntegrated economicand environmental models for a multi stage cold supply chainunder carbon tax regulationrdquo Journal of Cleaner Productionvol 166 pp 1357ndash1371 2017

[27] N Turken J Carrillo and V Verter ldquoFacility location andcapacity acquisition under carbon tax and emissions limits tocentralize or to decentralizerdquo International Journal of Pro-duction Economics vol 187 pp 126ndash141 2017

[28] W Yu and R Han ldquoCoordinating a two-echelon supply chainunder carbon taxrdquo Sustainability vol 9 no 12 p 2360 2017

[29] S Sinha and N M Modak ldquoAn EPQmodel in the perspectiveof carbon emission reductionrdquo International Journal ofMathematics in Operational Research vol 14 no 3pp 338ndash358 2019

[30] H Zhang P Li H Zheng and Y Zhang ldquoImpact of carbontax on enterprise operation and production strategy for low-carbon products in a co-opetition supply chainrdquo Journal ofCleaner Production Article ID 125058 2020

[31] X Chen H Yang X Wang et al ldquoOptimal carbon tax designfor achieving low carbon supply chainsrdquo Annals of OperationsResearch pp 1ndash28 2020

[32] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Handbook of the Fundamentals of Fi-nancial Decision Making Part I pp 99ndash127 )e EconometricSociety Cleveland OH USA 2013

[33] W Kim and M Kim ldquoReference quality-based competitivemarket structure for innovation driven marketsrdquo Interna-tional Journal of Research in Marketing vol 32 no 3pp 284ndash296 2015

[34] I Popescu and Y Wu ldquoDynamic pricing strategies withreference effectsrdquo Operations Research vol 55 no 3pp 413ndash429 2007

[35] P K Kopalle P K Kannan L B Boldt and N Arora ldquo)eimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Re-tailing vol 88 no 1 pp 102ndash114 2012

[36] T-P Hsieh and C-Y Dye ldquoOptimal dynamic pricing fordeteriorating items with reference price effects when inven-tories stimulate demandrdquo European Journal of OperationalResearch vol 262 no 1 pp 136ndash150 2017

[37] X Chen Z-Y Hu and Y-H Zhang ldquoDynamic pricing withstochastic reference price effectrdquo Journal of the OperationsResearch Society of China vol 7 no 1 pp 107ndash125 2019

[38] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquo Opera-tions Research vol 64 no 6 pp 1529ndash1536 2016

[39] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[40] C Chen ldquoDesign for the environment a quality-based modelfor green product developmentrdquo Management Sciencevol 47 no 2 pp 250ndash263 2001

[41] K Chitra ldquoIn search of the green consumers a perceptualstudyrdquo Journal of Services Research vol 7 no 1 2007

[42] Z Liu T D Anderson and J M Cruz ldquoConsumer envi-ronmental awareness and competition in two-stage supplychainsrdquo European Journal of Operational Research vol 218no 3 pp 602ndash613 2012

[43] B Li Y Wang and Z Wang ldquoManaging a closed-loop supplychain with take-back legislation and consumer preference forgreen designrdquo Journal of Cleaner Production Article ID124481 2020

[44] J Rantanen D B Grant and W Piotrowicz ldquoInvestigatingsupply chain cooperation in Finnish grocery retailrdquo ResearchJournal of the University of Gdansk Transport Economics andLogistics vol 71 pp 19ndash34 2017

[45] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[46] G Xu B Dan X Zhang and C Liu ldquoCoordinating a dual-channel supply chain with risk-averse under a two-way

16 Mathematical Problems in Engineering

revenue sharing contractrdquo International Journal of ProductionEconomics vol 147 pp 171ndash179 2014

[47] Z Shi N Wang T Jia et al ldquoReverse revenue sharingcontract versus two-part tariff contract under a closed-loopsupply chain systemrdquoMathematical Problems in Engineeringvol 2016 Article ID 5464570 15 pages 2016

[48] S Panda N M Modak and L E Cardenas-Barron ldquoCo-ordinating a socially responsible closed-loop supply chainwith product recyclingrdquo International Journal of ProductionEconomics vol 188 pp 11ndash21 2017

[49] N M Modak N Modak S Panda and S S Sana ldquoAnalyzingstructure of two-echelon closed-loop supply chain for pricingquality and recycling managementrdquo Journal of Cleaner Pro-duction vol 171 pp 512ndash528 2018

[50] Q P Wang and D Z Zhao ldquoRevenue-sharing contract ofsupply chain based on consumerrsquos preference for low carbonproductsrdquo Chinese Journal of Management Science vol 22no 9 pp 106ndash113 2014

[51] B Yu J Wang X Lu and H Yang ldquoCollaboration in a low-carbon supply chain with reference emission and cost learningeffects cost sharing versus revenue sharing strategiesrdquo Journalof Cleaner Production vol 250 Article ID 119460 2020

[52] J Cao X Zhang and G Zhou ldquoSupply chain coordinationwith revenue-sharing contracts considering carbon emissionsand governmental policy makingrdquo Environmental Progress ampSustainable Energy vol 35 no 2 pp 479ndash488 2016

[53] X Xu P He H Xu and Q Zhang ldquoSupply chain coordi-nation with green technology under cap-and-trade regula-tionrdquo International Journal of Production Economics vol 183pp 433ndash442 2017

[54] N M Modak S Sinha S Panda et al ldquoAnalyzing a sociallyresponsible closed-loop distribution channel with recyclingfacilityrdquo SN Applied Sciences vol 1 no 10 p 1189 2019

[55] D Feng L Ma Y Ding G Wu and Y Zhang ldquoDecisions ofthe dual-channel supply chain under double policy consid-ering remanufacturingrdquo International Journal of Environ-mental Research and Public Health vol 16 no 3 p 465 2019

[56] I Bellos M Ferguson and L B Toktay ldquo)e car sharingeconomy interaction of business model choice and productline designrdquo Manufacturing amp Service Operations Manage-ment vol 19 no 2 pp 185ndash201 2017

[57] G Ferrer and J M Swaminathan ldquoManaging new andremanufactured productsrdquo Management Science vol 52no 1 pp 15ndash26 2006

[58] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2pp 519ndash527 2011

[59] L Cui S Guo and H Zhang ldquoCoordinating a green agri-foodsupply chain with revenue-sharing contracts consideringretailersrsquo green marketing effortsrdquo Sustainability vol 12no 4 p 1289 2020

[60] J Heydari K Govindan and A Aslani ldquoPricing and greeningdecisions in a three-tier dual channel supply chainrdquo Inter-national Journal of Production Economics vol 217 pp 185ndash196 2019

Mathematical Problems in Engineering 17

carbon tax tlowast we have a region δ (η(tlowast) 1) of cooperationIt is clear that the gap of strength between two partitions inthe supply chain is a crucial factor in cooperation Bycomparing practice η decided by strength and η(tlowast) de-cided by the carbon tax we have

1λm

minus 1 2α2c minus 2αc + t

2(β + 1)1113872 1113873

2

4α2c2(minus 1 + α)

2 (F1)

A threshold is

1113954λ1 4α2c2

(minus 1 + α)2

8α4c2minus 16α3c2

+ 4 2c + t2(β + 1)1113872 1113873cα2 minus 4t

2c(β + 1)α + t

4(β + 1)

2 (F2)

When 05lt λm lt 1113954λ1 the practical η is in the regionδ (η(tlowast) 1) therefore the cooperation is achieved facingthe optimal carbon tax

However the practical η is out of the range δ (η(tlowast) 1)

when 1113954λ1 lt λm Review previous result the region δ (η1 1)

is expanding when tax rises )erefore there exists a pos-sibility that the government increasing carbon tax to co-ordinate the supply chain We assume a realistic thing thegovernment is willing to boost the greenness of the producteven reducing part of the revenue from the carbon tax

Based on it the carbon up to 1113954t1 which means thegreenness is maximized )erefore )e cooperation isachieved when clt 1113954c1 because the greenness is decreasing inthe carbon tax when 1113954c1 lt c )en we have the secondthreshold

1113954λ2 1

η1 1113954t1( 1113857 + 1 (F3)

)e cooperation could not be achieved when λm gt 1113954λ2)e conclusion is as follows (1) Coordination would be

achieved when the manufacturer is democratic (ie05lt λm lt 1113954λ1) (2) Coordination could be achieved but partof the total tax will lose if development cost is lower when themanufacturer is moderate (ie 1113954λ1 lt λm lt 1113954λ2) (3) )e co-operation could not be achieved if the manufacturer is adictator (ie λm gt 1113954λ2)

G Proof of Proposition 5

)e first derivative of η1 in β is

zη1zt

2α2c minus 2αc + t

2(β + 1)1113872 1113873t

2

2α2c2(minus 1 + α)

2 lt 0 (G1)

Similarly η1 is decreasing in β )at means the regionδ (η1 1) is expanding when β rises

And it is apparent that η1 (2cα(1 minus α) minus (1 + β)t22cα(1 minus α))2 is independent with k

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is research was supported by the National Natural ScienceFoundation of China (no 71761015) and the Science andTechnology Research Project of Jiangxi Education Depart-ment of China (no GJJ150475)

References

[1] Y Li M K Lim J Hu and M-L Tseng ldquoInvestigating theeffect of carbon tax and carbon quota policy to achieve lowcarbon logistics operationsrdquo Resources Conservation andRecycling vol 154 p 104535 2020

[2] M Abedi-Varaki ldquoStudy of carbon dioxide gas treatmentbased on equations of kinetics in plasma discharge reactorrdquoModern Physics Letters B vol 31 no 22 p 1750210 2017

[3] S Meng M Siriwardana and J McNeill ldquo)e environmentaland economic impact of the carbon tax in Australiardquo Envi-ronmental and Resource Economics vol 54 no 3 pp 313ndash332 2013

[4] Y Chen and C L Tseng ldquoInducing clean technology in theelectricity sector tradable permits or carbon tax policiesrdquo-eEnergy Journal vol 32 no 3 2011

[5] H-B Duan L Zhu and Y Fan ldquoOptimal carbon taxes incarbon-constrained China a logistic-induced energy eco-nomic hybrid modelrdquo Energy vol 69 pp 345ndash356 2014

[6] C Lu Q Tong and X Liu ldquo)e impacts of carbon tax andcomplementary policies on Chinese economyrdquo Energy Policyvol 38 no 11 pp 7278ndash7285 2010

[7] K Cao B Xu and J Wang ldquoOptimal trade-in and warrantyperiod strategies for new and remanufactured products undercarbon tax policyrdquo International Journal of Production Re-search vol 58 no 1 pp 180ndash199 2020

[8] P He W Zhang X Xu and Y Bian ldquoProduction lot-sizingand carbon emissions under cap-and-trade and carbon taxregulationsrdquo Journal of Cleaner Production vol 103pp 241ndash248 2015

[9] X Xu X Xu and P He ldquoJoint production and pricing de-cisions for multiple products with cap-and-trade and carbontax regulationsrdquo Journal of Cleaner Production vol 112pp 4093ndash4106 2016

Mathematical Problems in Engineering 15

[10] Q Li X Guan T Shi and W Jiao ldquoGreen product designwith competition and fairness concerns in the circulareconomy erardquo International Journal of Production Researchvol 58 no 1 pp 165ndash179 2020

[11] A Ranjan and J K Jha ldquoPricing and coordination strategiesof a dual-channel supply chain considering green quality andsales effortrdquo Journal of Cleaner Production vol 218pp 409ndash424 2019

[12] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineeringvol 10 no 1 pp 99ndash116 2012

[13] X Yin X Chen X Xu and L Zhang ldquoTax or subsidyOptimal carbon emission policy a supply chain perspectiverdquoSustainability vol 12 no 4 p 1548 2020

[14] B B Schlegelmilch G M Bohlen and A Diamantopoulosldquo)e link between green purchasing decisions and measuresof environmental consciousnessrdquo European Journal of Mar-keting vol 30 no 5 pp 35ndash55 1996

[15] M S Hopkins ldquoWhat the rsquogreenrsquo consumer wantsrdquo MITSloan Management Review vol 50 no 4 pp 87ndash89 2009

[16] G Kalyanaram and R S Winer ldquoEmpirical generalizationsfrom reference price researchrdquo Marketing Science vol 14no 3 pp G161ndashG169 1995

[17] E A Greenleaf ldquo)e impact of reference price effects on theprofitability of price promotionsrdquo Marketing Science vol 14no 1 pp 82ndash104 1995

[18] N M Modak D K Ghosh S Panda and S S SanaldquoManaging green house gas emission cost and pricing policiesin a two-echelon supply chainrdquo CIRP Journal ofManufacturing Science and Technology vol 20 pp 1ndash11 2018

[19] P Sinclair ldquoHigh does nothing and rising is worse carbontaxes should keep declining to cut harmful emissionsrdquo -eManchester School vol 60 no 1 pp 41ndash52 1992

[20] G M Grossman and A B Krueger ldquoEconomic growth andthe environmentrdquo -e Quarterly Journal of Economicsvol 110 no 2 pp 353ndash377 1995

[21] N M Modak and P Kelle ldquoUsing social work donation as atool of corporate social responsibility in a closed-loop supplychain considering carbon emissions tax and demand uncer-taintyrdquo Journal of the Operational Research Society pp 1ndash172019

[22] A Ulph and D Ulph ldquo)e optimal time path of a carbon taxrdquoOxford Economic Papers vol 46 no S1 pp 857ndash868 1994

[23] M Hoel and S Kverndokk ldquoDepletion of fossil fuels and theimpact of global warmingrdquo Resource and Energy Economicsvol 18 no 2 pp 115ndash136 1996

[24] Y H Farzin and O Tahvonen ldquoGlobal carbon cycle and theoptimal time path of a carbon taxrdquo Oxford Economic Papersvol 48 no 4 pp 515ndash536 1996

[25] V Bosetti C Carraro R Duval andM Tavoni ldquoWhat shouldwe expect from innovation A model-based assessment of theenvironmental and mitigation cost implications of climate-related RampDrdquo Energy Economics vol 33 no 6 pp 1313ndash13202011

[26] M Hariga R Asrsquoad and A Shamayleh ldquoIntegrated economicand environmental models for a multi stage cold supply chainunder carbon tax regulationrdquo Journal of Cleaner Productionvol 166 pp 1357ndash1371 2017

[27] N Turken J Carrillo and V Verter ldquoFacility location andcapacity acquisition under carbon tax and emissions limits tocentralize or to decentralizerdquo International Journal of Pro-duction Economics vol 187 pp 126ndash141 2017

[28] W Yu and R Han ldquoCoordinating a two-echelon supply chainunder carbon taxrdquo Sustainability vol 9 no 12 p 2360 2017

[29] S Sinha and N M Modak ldquoAn EPQmodel in the perspectiveof carbon emission reductionrdquo International Journal ofMathematics in Operational Research vol 14 no 3pp 338ndash358 2019

[30] H Zhang P Li H Zheng and Y Zhang ldquoImpact of carbontax on enterprise operation and production strategy for low-carbon products in a co-opetition supply chainrdquo Journal ofCleaner Production Article ID 125058 2020

[31] X Chen H Yang X Wang et al ldquoOptimal carbon tax designfor achieving low carbon supply chainsrdquo Annals of OperationsResearch pp 1ndash28 2020

[32] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Handbook of the Fundamentals of Fi-nancial Decision Making Part I pp 99ndash127 )e EconometricSociety Cleveland OH USA 2013

[33] W Kim and M Kim ldquoReference quality-based competitivemarket structure for innovation driven marketsrdquo Interna-tional Journal of Research in Marketing vol 32 no 3pp 284ndash296 2015

[34] I Popescu and Y Wu ldquoDynamic pricing strategies withreference effectsrdquo Operations Research vol 55 no 3pp 413ndash429 2007

[35] P K Kopalle P K Kannan L B Boldt and N Arora ldquo)eimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Re-tailing vol 88 no 1 pp 102ndash114 2012

[36] T-P Hsieh and C-Y Dye ldquoOptimal dynamic pricing fordeteriorating items with reference price effects when inven-tories stimulate demandrdquo European Journal of OperationalResearch vol 262 no 1 pp 136ndash150 2017

[37] X Chen Z-Y Hu and Y-H Zhang ldquoDynamic pricing withstochastic reference price effectrdquo Journal of the OperationsResearch Society of China vol 7 no 1 pp 107ndash125 2019

[38] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquo Opera-tions Research vol 64 no 6 pp 1529ndash1536 2016

[39] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[40] C Chen ldquoDesign for the environment a quality-based modelfor green product developmentrdquo Management Sciencevol 47 no 2 pp 250ndash263 2001

[41] K Chitra ldquoIn search of the green consumers a perceptualstudyrdquo Journal of Services Research vol 7 no 1 2007

[42] Z Liu T D Anderson and J M Cruz ldquoConsumer envi-ronmental awareness and competition in two-stage supplychainsrdquo European Journal of Operational Research vol 218no 3 pp 602ndash613 2012

[43] B Li Y Wang and Z Wang ldquoManaging a closed-loop supplychain with take-back legislation and consumer preference forgreen designrdquo Journal of Cleaner Production Article ID124481 2020

[44] J Rantanen D B Grant and W Piotrowicz ldquoInvestigatingsupply chain cooperation in Finnish grocery retailrdquo ResearchJournal of the University of Gdansk Transport Economics andLogistics vol 71 pp 19ndash34 2017

[45] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[46] G Xu B Dan X Zhang and C Liu ldquoCoordinating a dual-channel supply chain with risk-averse under a two-way

16 Mathematical Problems in Engineering

revenue sharing contractrdquo International Journal of ProductionEconomics vol 147 pp 171ndash179 2014

[47] Z Shi N Wang T Jia et al ldquoReverse revenue sharingcontract versus two-part tariff contract under a closed-loopsupply chain systemrdquoMathematical Problems in Engineeringvol 2016 Article ID 5464570 15 pages 2016

[48] S Panda N M Modak and L E Cardenas-Barron ldquoCo-ordinating a socially responsible closed-loop supply chainwith product recyclingrdquo International Journal of ProductionEconomics vol 188 pp 11ndash21 2017

[49] N M Modak N Modak S Panda and S S Sana ldquoAnalyzingstructure of two-echelon closed-loop supply chain for pricingquality and recycling managementrdquo Journal of Cleaner Pro-duction vol 171 pp 512ndash528 2018

[50] Q P Wang and D Z Zhao ldquoRevenue-sharing contract ofsupply chain based on consumerrsquos preference for low carbonproductsrdquo Chinese Journal of Management Science vol 22no 9 pp 106ndash113 2014

[51] B Yu J Wang X Lu and H Yang ldquoCollaboration in a low-carbon supply chain with reference emission and cost learningeffects cost sharing versus revenue sharing strategiesrdquo Journalof Cleaner Production vol 250 Article ID 119460 2020

[52] J Cao X Zhang and G Zhou ldquoSupply chain coordinationwith revenue-sharing contracts considering carbon emissionsand governmental policy makingrdquo Environmental Progress ampSustainable Energy vol 35 no 2 pp 479ndash488 2016

[53] X Xu P He H Xu and Q Zhang ldquoSupply chain coordi-nation with green technology under cap-and-trade regula-tionrdquo International Journal of Production Economics vol 183pp 433ndash442 2017

[54] N M Modak S Sinha S Panda et al ldquoAnalyzing a sociallyresponsible closed-loop distribution channel with recyclingfacilityrdquo SN Applied Sciences vol 1 no 10 p 1189 2019

[55] D Feng L Ma Y Ding G Wu and Y Zhang ldquoDecisions ofthe dual-channel supply chain under double policy consid-ering remanufacturingrdquo International Journal of Environ-mental Research and Public Health vol 16 no 3 p 465 2019

[56] I Bellos M Ferguson and L B Toktay ldquo)e car sharingeconomy interaction of business model choice and productline designrdquo Manufacturing amp Service Operations Manage-ment vol 19 no 2 pp 185ndash201 2017

[57] G Ferrer and J M Swaminathan ldquoManaging new andremanufactured productsrdquo Management Science vol 52no 1 pp 15ndash26 2006

[58] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2pp 519ndash527 2011

[59] L Cui S Guo and H Zhang ldquoCoordinating a green agri-foodsupply chain with revenue-sharing contracts consideringretailersrsquo green marketing effortsrdquo Sustainability vol 12no 4 p 1289 2020

[60] J Heydari K Govindan and A Aslani ldquoPricing and greeningdecisions in a three-tier dual channel supply chainrdquo Inter-national Journal of Production Economics vol 217 pp 185ndash196 2019

Mathematical Problems in Engineering 17

[10] Q Li X Guan T Shi and W Jiao ldquoGreen product designwith competition and fairness concerns in the circulareconomy erardquo International Journal of Production Researchvol 58 no 1 pp 165ndash179 2020

[11] A Ranjan and J K Jha ldquoPricing and coordination strategiesof a dual-channel supply chain considering green quality andsales effortrdquo Journal of Cleaner Production vol 218pp 409ndash424 2019

[12] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineeringvol 10 no 1 pp 99ndash116 2012

[13] X Yin X Chen X Xu and L Zhang ldquoTax or subsidyOptimal carbon emission policy a supply chain perspectiverdquoSustainability vol 12 no 4 p 1548 2020

[14] B B Schlegelmilch G M Bohlen and A Diamantopoulosldquo)e link between green purchasing decisions and measuresof environmental consciousnessrdquo European Journal of Mar-keting vol 30 no 5 pp 35ndash55 1996

[15] M S Hopkins ldquoWhat the rsquogreenrsquo consumer wantsrdquo MITSloan Management Review vol 50 no 4 pp 87ndash89 2009

[16] G Kalyanaram and R S Winer ldquoEmpirical generalizationsfrom reference price researchrdquo Marketing Science vol 14no 3 pp G161ndashG169 1995

[17] E A Greenleaf ldquo)e impact of reference price effects on theprofitability of price promotionsrdquo Marketing Science vol 14no 1 pp 82ndash104 1995

[18] N M Modak D K Ghosh S Panda and S S SanaldquoManaging green house gas emission cost and pricing policiesin a two-echelon supply chainrdquo CIRP Journal ofManufacturing Science and Technology vol 20 pp 1ndash11 2018

[19] P Sinclair ldquoHigh does nothing and rising is worse carbontaxes should keep declining to cut harmful emissionsrdquo -eManchester School vol 60 no 1 pp 41ndash52 1992

[20] G M Grossman and A B Krueger ldquoEconomic growth andthe environmentrdquo -e Quarterly Journal of Economicsvol 110 no 2 pp 353ndash377 1995

[21] N M Modak and P Kelle ldquoUsing social work donation as atool of corporate social responsibility in a closed-loop supplychain considering carbon emissions tax and demand uncer-taintyrdquo Journal of the Operational Research Society pp 1ndash172019

[22] A Ulph and D Ulph ldquo)e optimal time path of a carbon taxrdquoOxford Economic Papers vol 46 no S1 pp 857ndash868 1994

[23] M Hoel and S Kverndokk ldquoDepletion of fossil fuels and theimpact of global warmingrdquo Resource and Energy Economicsvol 18 no 2 pp 115ndash136 1996

[24] Y H Farzin and O Tahvonen ldquoGlobal carbon cycle and theoptimal time path of a carbon taxrdquo Oxford Economic Papersvol 48 no 4 pp 515ndash536 1996

[25] V Bosetti C Carraro R Duval andM Tavoni ldquoWhat shouldwe expect from innovation A model-based assessment of theenvironmental and mitigation cost implications of climate-related RampDrdquo Energy Economics vol 33 no 6 pp 1313ndash13202011

[26] M Hariga R Asrsquoad and A Shamayleh ldquoIntegrated economicand environmental models for a multi stage cold supply chainunder carbon tax regulationrdquo Journal of Cleaner Productionvol 166 pp 1357ndash1371 2017

[27] N Turken J Carrillo and V Verter ldquoFacility location andcapacity acquisition under carbon tax and emissions limits tocentralize or to decentralizerdquo International Journal of Pro-duction Economics vol 187 pp 126ndash141 2017

[28] W Yu and R Han ldquoCoordinating a two-echelon supply chainunder carbon taxrdquo Sustainability vol 9 no 12 p 2360 2017

[29] S Sinha and N M Modak ldquoAn EPQmodel in the perspectiveof carbon emission reductionrdquo International Journal ofMathematics in Operational Research vol 14 no 3pp 338ndash358 2019

[30] H Zhang P Li H Zheng and Y Zhang ldquoImpact of carbontax on enterprise operation and production strategy for low-carbon products in a co-opetition supply chainrdquo Journal ofCleaner Production Article ID 125058 2020

[31] X Chen H Yang X Wang et al ldquoOptimal carbon tax designfor achieving low carbon supply chainsrdquo Annals of OperationsResearch pp 1ndash28 2020

[32] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Handbook of the Fundamentals of Fi-nancial Decision Making Part I pp 99ndash127 )e EconometricSociety Cleveland OH USA 2013

[33] W Kim and M Kim ldquoReference quality-based competitivemarket structure for innovation driven marketsrdquo Interna-tional Journal of Research in Marketing vol 32 no 3pp 284ndash296 2015

[34] I Popescu and Y Wu ldquoDynamic pricing strategies withreference effectsrdquo Operations Research vol 55 no 3pp 413ndash429 2007

[35] P K Kopalle P K Kannan L B Boldt and N Arora ldquo)eimpact of household level heterogeneity in reference priceeffects on optimal retailer pricing policiesrdquo Journal of Re-tailing vol 88 no 1 pp 102ndash114 2012

[36] T-P Hsieh and C-Y Dye ldquoOptimal dynamic pricing fordeteriorating items with reference price effects when inven-tories stimulate demandrdquo European Journal of OperationalResearch vol 262 no 1 pp 136ndash150 2017

[37] X Chen Z-Y Hu and Y-H Zhang ldquoDynamic pricing withstochastic reference price effectrdquo Journal of the OperationsResearch Society of China vol 7 no 1 pp 107ndash125 2019

[38] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquo Opera-tions Research vol 64 no 6 pp 1529ndash1536 2016

[39] L Zhang H Zhou Y Liu and R Lu ldquoOptimal environmentalquality and price with consumer environmental awarenessand retailerrsquos fairness concerns in supply chainrdquo Journal ofCleaner Production vol 213 pp 1063ndash1079 2019

[40] C Chen ldquoDesign for the environment a quality-based modelfor green product developmentrdquo Management Sciencevol 47 no 2 pp 250ndash263 2001

[41] K Chitra ldquoIn search of the green consumers a perceptualstudyrdquo Journal of Services Research vol 7 no 1 2007

[42] Z Liu T D Anderson and J M Cruz ldquoConsumer envi-ronmental awareness and competition in two-stage supplychainsrdquo European Journal of Operational Research vol 218no 3 pp 602ndash613 2012

[43] B Li Y Wang and Z Wang ldquoManaging a closed-loop supplychain with take-back legislation and consumer preference forgreen designrdquo Journal of Cleaner Production Article ID124481 2020

[44] J Rantanen D B Grant and W Piotrowicz ldquoInvestigatingsupply chain cooperation in Finnish grocery retailrdquo ResearchJournal of the University of Gdansk Transport Economics andLogistics vol 71 pp 19ndash34 2017

[45] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[46] G Xu B Dan X Zhang and C Liu ldquoCoordinating a dual-channel supply chain with risk-averse under a two-way

16 Mathematical Problems in Engineering

revenue sharing contractrdquo International Journal of ProductionEconomics vol 147 pp 171ndash179 2014

[47] Z Shi N Wang T Jia et al ldquoReverse revenue sharingcontract versus two-part tariff contract under a closed-loopsupply chain systemrdquoMathematical Problems in Engineeringvol 2016 Article ID 5464570 15 pages 2016

[48] S Panda N M Modak and L E Cardenas-Barron ldquoCo-ordinating a socially responsible closed-loop supply chainwith product recyclingrdquo International Journal of ProductionEconomics vol 188 pp 11ndash21 2017

[49] N M Modak N Modak S Panda and S S Sana ldquoAnalyzingstructure of two-echelon closed-loop supply chain for pricingquality and recycling managementrdquo Journal of Cleaner Pro-duction vol 171 pp 512ndash528 2018

[50] Q P Wang and D Z Zhao ldquoRevenue-sharing contract ofsupply chain based on consumerrsquos preference for low carbonproductsrdquo Chinese Journal of Management Science vol 22no 9 pp 106ndash113 2014

[51] B Yu J Wang X Lu and H Yang ldquoCollaboration in a low-carbon supply chain with reference emission and cost learningeffects cost sharing versus revenue sharing strategiesrdquo Journalof Cleaner Production vol 250 Article ID 119460 2020

[52] J Cao X Zhang and G Zhou ldquoSupply chain coordinationwith revenue-sharing contracts considering carbon emissionsand governmental policy makingrdquo Environmental Progress ampSustainable Energy vol 35 no 2 pp 479ndash488 2016

[53] X Xu P He H Xu and Q Zhang ldquoSupply chain coordi-nation with green technology under cap-and-trade regula-tionrdquo International Journal of Production Economics vol 183pp 433ndash442 2017

[54] N M Modak S Sinha S Panda et al ldquoAnalyzing a sociallyresponsible closed-loop distribution channel with recyclingfacilityrdquo SN Applied Sciences vol 1 no 10 p 1189 2019

[55] D Feng L Ma Y Ding G Wu and Y Zhang ldquoDecisions ofthe dual-channel supply chain under double policy consid-ering remanufacturingrdquo International Journal of Environ-mental Research and Public Health vol 16 no 3 p 465 2019

[56] I Bellos M Ferguson and L B Toktay ldquo)e car sharingeconomy interaction of business model choice and productline designrdquo Manufacturing amp Service Operations Manage-ment vol 19 no 2 pp 185ndash201 2017

[57] G Ferrer and J M Swaminathan ldquoManaging new andremanufactured productsrdquo Management Science vol 52no 1 pp 15ndash26 2006

[58] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2pp 519ndash527 2011

[59] L Cui S Guo and H Zhang ldquoCoordinating a green agri-foodsupply chain with revenue-sharing contracts consideringretailersrsquo green marketing effortsrdquo Sustainability vol 12no 4 p 1289 2020

[60] J Heydari K Govindan and A Aslani ldquoPricing and greeningdecisions in a three-tier dual channel supply chainrdquo Inter-national Journal of Production Economics vol 217 pp 185ndash196 2019

Mathematical Problems in Engineering 17

revenue sharing contractrdquo International Journal of ProductionEconomics vol 147 pp 171ndash179 2014

[47] Z Shi N Wang T Jia et al ldquoReverse revenue sharingcontract versus two-part tariff contract under a closed-loopsupply chain systemrdquoMathematical Problems in Engineeringvol 2016 Article ID 5464570 15 pages 2016

[48] S Panda N M Modak and L E Cardenas-Barron ldquoCo-ordinating a socially responsible closed-loop supply chainwith product recyclingrdquo International Journal of ProductionEconomics vol 188 pp 11ndash21 2017

[49] N M Modak N Modak S Panda and S S Sana ldquoAnalyzingstructure of two-echelon closed-loop supply chain for pricingquality and recycling managementrdquo Journal of Cleaner Pro-duction vol 171 pp 512ndash528 2018

[50] Q P Wang and D Z Zhao ldquoRevenue-sharing contract ofsupply chain based on consumerrsquos preference for low carbonproductsrdquo Chinese Journal of Management Science vol 22no 9 pp 106ndash113 2014

[51] B Yu J Wang X Lu and H Yang ldquoCollaboration in a low-carbon supply chain with reference emission and cost learningeffects cost sharing versus revenue sharing strategiesrdquo Journalof Cleaner Production vol 250 Article ID 119460 2020

[52] J Cao X Zhang and G Zhou ldquoSupply chain coordinationwith revenue-sharing contracts considering carbon emissionsand governmental policy makingrdquo Environmental Progress ampSustainable Energy vol 35 no 2 pp 479ndash488 2016

[53] X Xu P He H Xu and Q Zhang ldquoSupply chain coordi-nation with green technology under cap-and-trade regula-tionrdquo International Journal of Production Economics vol 183pp 433ndash442 2017

[54] N M Modak S Sinha S Panda et al ldquoAnalyzing a sociallyresponsible closed-loop distribution channel with recyclingfacilityrdquo SN Applied Sciences vol 1 no 10 p 1189 2019

[55] D Feng L Ma Y Ding G Wu and Y Zhang ldquoDecisions ofthe dual-channel supply chain under double policy consid-ering remanufacturingrdquo International Journal of Environ-mental Research and Public Health vol 16 no 3 p 465 2019

[56] I Bellos M Ferguson and L B Toktay ldquo)e car sharingeconomy interaction of business model choice and productline designrdquo Manufacturing amp Service Operations Manage-ment vol 19 no 2 pp 185ndash201 2017

[57] G Ferrer and J M Swaminathan ldquoManaging new andremanufactured productsrdquo Management Science vol 52no 1 pp 15ndash26 2006

[58] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2pp 519ndash527 2011

[59] L Cui S Guo and H Zhang ldquoCoordinating a green agri-foodsupply chain with revenue-sharing contracts consideringretailersrsquo green marketing effortsrdquo Sustainability vol 12no 4 p 1289 2020

[60] J Heydari K Govindan and A Aslani ldquoPricing and greeningdecisions in a three-tier dual channel supply chainrdquo Inter-national Journal of Production Economics vol 217 pp 185ndash196 2019

Mathematical Problems in Engineering 17