Research ArticleOptimal Strategies for Low Carbon Supply Chain with StrategicCustomer Behavior and Green Technology Investment

Wen Jiang1 and Xu Chen2

1College of Architecture and Urban-Rural Planning Sichuan Agricultural University Chengdu 611830 China2School of Management and Economics University of Electronic Science and Technology of China Chengdu 611731 China

Correspondence should be addressed to Xu Chen xchenxchen263net

Received 5 November 2015 Accepted 13 January 2016

Academic Editor Jorge J Julvez

Copyright copy 2016 W Jiang and X Chen This 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 is properlycited

Climate change is mainly caused by excessive emissions of carbon dioxide and other greenhouse gases In order to reduce carbonemissions cap and trade policy is implemented by governments inmany countries which has significant impacts on the decisions ofcompanies at all levels of the low carbon supply chainThis paper investigates the decision-making and coordination of a low carbonsupply chain consisting of a low carbon manufacturer who produces one product and is allowed to invest in green technology toreduce carbon emissions in production and a retailer who faces stochastic demands formed by homogeneous strategic customersWe investigate the optimal production pricing carbon trading and green technology investment strategies of the low carbon supplychain in centralized (including Rational Expected Equilibrium scenario and quantity commitment scenario) and decentralizedsettings It is demonstrated that quantity commitment strategy can improve the profit of the low carbon supply chain with strategiccustomer behavior We also show that the performance of decentralized supply chain is lower than that of quantity commitmentscenarioWe prove that the low carbon supply chain cannot be coordinated by revenue sharing contract but by revenue sharing-costsharing contract

1 Introduction

In recent years global warming resulting from excess emis-sions of carbon dioxide and other greenhouse gases hasbeen challenging the survival and development of humanbeings leading to serious consequences like droughts heatwaves sea level rise intense rainfall and so forth Researchshows that global warming is mainly caused by humanactivities (at least 90 probability) [1] Therefore achievinga sustainable low carbon economy by changing the modeof human production and life has become the focus ofglobal attention [2] Developing low carbon economy alsochallenges the supply chain management In addition tothe development of carbon reduction technology and newenergy technologies increasing number of researchers andentrepreneurs pay attention to the optimization of supplychain operation strategy to reduce carbon emissions

In order to meet the goal of carbon emissions reductionthe governments all over the world tend to implement carbon

emission regulation policies Comparedwith other regulationpolicies cap and trade policy is effective in reducing corpo-rate carbon emissions without increasing costs significantly[3] and has obvious advantages in feasibility fairness andbusiness participation [4] Cap and trade policy has becomethe preference of governments as it achieves the goal ofeffective carbon reduction through dual means of regulationand market [5]

The implementation of cap and trade policy makes thecarbon emission permits the essential factors of productionFor supply chain businesses in addition to carbon emissionstrading investing in production process improvement car-bon capture and storage technologies of production processis another way to get carbon emission permits [6 7] Greentechnology investment will increase production cost but alsocan save carbon emissions for businesses and get additionalrevenue Furthermore customers tend to buy low carbonproducts with the gradual increase of the environmentalawareness among them [8] By investing in green technology

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016 Article ID 9645087 13 pageshttpdxdoiorg10115520169645087

2 Discrete Dynamics in Nature and Society

customers canmeet the low carbon demand of customers andobtain the corresponding competitive advantage Businessesin supply chain needs to trade off the costs and benefits ofgreen technology investment to decide whether to investTherefore under the cap and trade policy study on lowcarbon supply chain operation strategies optimization withgreen technology investment is of great theoretical value

In recent years scholars and practitioners have takengreater interests in strategic customer behavior which isshown in the increasing marketing and operations researchliterature on the problem Strategic customer behavior refersto the behavior of customers who do not buy products atfull price but wait until the products markdown Researchshows that the ignorance of the strategic customer behavior inoperation strategies would be harmful to the performance ofsupply chain [9 10] Strategic customer behavior has becomea common phenomenon in perishable product sales processwhich made it a regular assumption in many literatures [11ndash13] Therefore considering strategic customer behavior tostudy low carbon supply chain strategies and coordination ismore realistic

The production pricing carbon trading and greentechnology investment problems are examined in a supplychain setting made up of a retailer and a manufacturer Themanufacturer can invest in green technology to reduce unitcarbon emissions of the product and distribute the product tostrategic customers through the retailer Three key questionsare addressed in the paper

(1) In centralized supply chain setting what is theoptimal production quantity pricing carbon tradingstrategy and green technology investment

(2) In centralized supply chain setting can quantitycommitment improve the supply chain performanceand what effect does the quantity commitment haveon the supply chain strategies

(3) In decentralized supply chain setting what is theoptimal strategies for the retailer and manufacturerrespectively How to coordinate the decentralizedsupply chain to achieve the performance in quantitycommitment scenario

After a review of the literature in Section 2 we presentthe model descriptions and assumptions in Section 3 InSection 4 the centralized supply chain setting models arediscussed We obtain the optimal strategies for the central-ized decision-maker in scenarios of Rational ExpectationsEquilibrium (RE Equilibrium) and quantity commitment(QC) respectively In Section 5 the optimal strategies for themanufacturer and the retailer in decentralized supply chainsetting are discussed We design supply chain coordinationmechanism in Section 6 In Section 7 we present conclusionsof our findings and highlight possible future work

2 Literature Review

There are two streams of literature closely related to ourwork literature on low carbon supply chainmanagement and

literature on supply chain business decisions with strategiccustomer behavior

The literature on low carbon supply chain managementcan be further divided into two classes One is the researchon centralized supply chain that is single business decisionsand the other is the research on decentralized supply chaindecisions and coordination As to the literature on singlebusiness decisions a lot of researchers examined the impactof cap and trade policy without green technology investmentsuch as Dobos 2005 [14] Chen et al 2015 [15] Chen andWang 2015 [16] and Chang et al 2015 [17] This streamof literature explores the research framework that examinesthe effect of cap and trade policy on low carbon supplychain decisions but the green technology investment is notincluded Meanwhile a lot of researchers have introducedgreen technology investment into low carbon supply chainmanagement Zhao et al 2010 [18] developed a carbonemissions allowance allocation systemwith green technologyinvestment in electric power market Yalabik and Fairchild2011 [19] investigated the manufacturerrsquos investment deci-sion on environmentally friendly product with customerbehavior and government regulation They showed that themanufacturer had an incentive to carry out green technologyinvestments to reduce carbon emissions when the customerdemand was emissions-sensitive Toptal et al 2014 [20]investigated the joint decisions of procurement and greentechnology investment under carbon tax and cap and tradepolicy respectively They also discussed the effect of differentcarbon policies on the decision of green technology invest-ment In Manikas and Kroes 2015 [21] a forward buyingheuristic is designed for those firmswho get carbon emissionsby auctions As to the literature on decentralized supplychain decisions and coordination without green technologyinvestment Benjaafar et al 2013 [22] introduced carbonemissions into the simple supply chain system and inves-tigated the procurement production inventory and greentechnology investment decisions with cap and trade policyXu et al 2015 [23] studied the production and pricingproblem of a MTO supply chain consisting of a two-productmanufacturer and a retailerThemanufacturer is constrainedby cap and trade policy and determines the wholesale priceof the two products Zhen et al 2015 [24] examined theretailerrsquos optimal pricing ordering and transportation modestrategies with cap and trade policy They showed thatthe retailer preferred low carbon transportation mode withcap and trade policy under certain conditions There are afew literatures on decentralized supply chain decisions andcoordination with green technology investment Swami andShah 2013 [25] considered the customerrsquos environmentalawareness that is the green technology investments of firmsin the supply chain would affect the customer demand anddesign a coordination mechanism for the supply chain basedon costs sharing contract

As to the literature on supply chain business deci-sions with strategic customer behavior Coase 1972 [26]first paid attention to strategic customer behavior in eco-nomics He found that when the monopoly company facedstrategic customer the monopoly company would set themargin cost as the price and earn zero profit We review

Discrete Dynamics in Nature and Society 3

this stream literature from two aspects of centralized anddecentralized supply chain decisions with strategic customerbehavior Su 2007 [27] investigated the dynamic pricingof a monopolist selling a finite inventory over a finite timeperiodThe demand was endogenous and intertemporalThecustomers were heterogeneous and strategic The researchshows that the heterogeneous valuation and patience ofcustomers determine the optimal pricing policies structureZhang and Cooper 2008 [28] considered a firm that sellsa single product over two periods and examined the effectsof strategic customer behavior on the firmrsquos rationing andpricing decisionsTheir research shows that when prices werefixed in advance rationing could improve revenue Whang2015 [29] considered heterogeneous strategic customers withdifferent reservation value and studied the effect of demanduncertainly on the retailerrsquos markdown policy Du et al 2015[30] taking strategic customers into consideration with riskpreference and decreasing value studied the joint stock andpricing decision problem Compared with classical newsboymodels the ordering quantity and total profit got lower whenstrategic customer behavior was taken into considerationAs to the effects of strategic customer behavior on thedecentralized supply chain performance and coordinationstrategy Su and Zhang 2008 [31] examined the effect ofstrategic customer behavior on the performance of supplychain and the valve of commitment They characterizedRational Expected Equilibrium between strategic customersand the retailer They showed that quantity commitmentcould improve the sellerrsquos profit and decentralization couldimprove the supply chain performance Yang 2012 [32]studied the effect of competition and discounting on theperformance of decentralized supply chain with strategiccustomer behavior It showed that the performance of a cen-tralized supply chain was lower than that of a decentralizedsupply chain when strategic customers existed It showed thatretailer competition and the firm and customer discountingwere also driving factors of higher decentralized supply chainperformance except for double marginalization effect Yanget al 2015 [33] addressed the effect of quick response onthe performance of supply chain when strategic customerbehavior was taken into consideration They compared thequick response value in different supply chain structuresThey showed that a decentralized supply chain with revenuesharing contracts could achieve the performance of a central-ized supply chain but allocating the profit arbitrary in supplychain members could not be realized here

Nevertheless despite the increased attention on lowcarbon supply chainmanagement in operationsmanagementliterature very few studies have been carried by consideringgreen technology investments and cap and trade policysimultaneously To the best of our knowledge there are nopublished works that introduce strategic customer behaviorinto low carbon supply chain framework to investigate theoptimal production pricing carbon trading and green tech-nology investment decisions One distinction of our modelis examining the optimal strategies of low carbon supplychain by considering cap and trade policy green technologyinvestment and strategic customer behavior simultaneouslyAnother distinction is systematic consideration of optimizing

low carbon strategies in centralized and decentralized sce-narios The contributions of our work are as follows firstwe obtain the optimal production pricing carbon tradingand green technology investment strategies for companies atall levels of low carbon supply chain in different scenariosSecond we analyze the impact of quantity commitment onthe operation strategies of a centralized supply chain andits performance Third coordination mechanisms for lowcarbon supply chain with strategic customer behavior aredesigned based on revenue sharing-cost sharing contract

3 Model Descriptions and Assumptions

We examine a two-echelon low carbon supply chain madeup of a low carbon manufacturer and a retailer The manu-facturer produces a product and distributes it to customersthrough the retailer We divide the wholesales period of theretailer into two phases The retailer sells the product at fullprice in phase one and at salvage price in phase two If theproduct is sold out in phase one phase two does not existThe customers are homogeneous strategic customers whowill take into account the possibility of buying the productat salvage price to choose buying the product at full price orwait to purchase the product at salvage price to maximize theexpected surplus Each customer purchases one product atmost

Variables and parameters for model developmentadopted in this study are denoted as in Notations section

Because the parameters must meet certain conditions tomake sense we assume the following

(1) 119901 le 119903 Only when the retail price is no more thanthe customer reservation price the customer maypurchase the product at full price

(2) V gt 119901 gt 119908 gt 119888 gt 119904 gt 0 This conditionensures that there is a positive profit margin for themanufacturer retailer and customers when a productis sold to customers In addition the production costis greater than the salvage price which indicates thatthe retailer will lose money when the product fails tosell at full price This prompts the retailer to orderthe products according to the customersrsquo demandbecause the excess inventories generate losses

(3) 119908 gt 119888+119896 This condition states that the manufactureris willing to produce products with cap and tradepolicy Otherwise the manufacturer will not produceproducts but sell carbon emissions to earn profits

(4) The shortage cost of the manufacturer and operatingcost of the retailer are not taken into consideration

(5) We assume that the manufacturer and the retailer areall rational and self-interested that is both of themaim tomaximize their profitWe also assume that theyare risk neutral

(6) The carbon emissions in production is the mainsource of the total carbon emissions So we onlyconsider the carbon emissions of production

4 Discrete Dynamics in Nature and Society

(7) The green technology investment is the function of120591 denoted by 119868(120578) We assume that 119868(120578) ge 01198681015840(120578) gt 0 and 119868

10158401015840(120578) gt 0 This is consistent with

the practice The conditions show that the margingreen technology investment is increasing in carbonemissions reduction rate Referring to drsquoAspremontand Jacquemin 1988 [34] we set 119868(120578) = (12)119905120578

2where 119905 represents the efficiency of green technologyinvestments

Variables and parameters related to low carbon include119890 (a decision variable which represents the carbon emissionstrading policy) 120578 (a decision variable which represents thegreen technology investment policy) and 119864 and 119896 (representthe initial carbon emissions and unit price of carbon emis-sions trading resp)

4 Centralized Supply Chain Model

In this section we assume that themanufacturer is authorizedto determine the production pricing carbon trading andgreen technology investment strategies under cap and tradepolicy to maximize the profit of the centralized supply chainAt the beginning the government allocates a certain amountof free carbon emissions to the manufacturer During theproducing process if the carbon emission is limited themanufacturer should buy extra carbon emissions from theexternalmarket Otherwise if the carbon emission is enoughthe manufacturer could sell extra carbon emissions to gainrevenue At the end of the period carbon emissions of themanufacturer must not exceed the carbon emission rights itholds

The sequence of events in this part is as follows first themanufacturer forms the belief of customersrsquo reservation price120585119903and then decides the retail selling price product quantity

carbon trading volume and green technology investmentsecond the customers form the beliefs 120585prob of probabilityof the product sold at salvage price 119904 according to theinformation of market price and then form the reservationprice 119903 third the customersrsquo demand is satisfied and theproducts are sold at full price119901 finally all remaining productsare sold to the external market at salvage price 119904

41 Rational Expectations Equilibrium Scenario We charac-terize the RE Equilibrium between the manufacturer and thestrategic customers Muth 1961 [35] first proposed rationalexpectations hypothesis which refers to the situation thatthere is no systematic bias between the actual economic resultand peoplersquos expectations Then Su and Zhang 2008 [31]introduced it into operations management to analyze thedecision problems of the enterprises when strategic customerbehavior is taken into account Since then the rationalexpectations hypothesis has been adopted by scholars all overthe world [13 27 30]

First we examine the decision problem of strategiccustomers The customers choose to buy the product imme-diately at price 119901 or wait for markdown to maximize theirexpected surplus The customer surplus is V minus 119901 when thecustomer buys the product at full price and (V minus 119904)120585prob at

salvage price Therefore the maximum expected surplus ofthe customer is max(V minus 119904)120585prob V minus 119901 If and only if (V minus119904)120585prob le Vminus119901 the customer will buy the product at full priceSo given 120585prob we can obtain the customerrsquos reservation price119903(120585prob) = V minus (V minus 119904)120585prob

Then we examine the decision problem of the manufac-turer The profit function of the manufacturer with cap andtrade policy and green technology investment denoted by120587(119902 119901 119890 120578) is

120587 (119902 119901 119890 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909) minus (119888 minus 119904) 119902

minus 119896119890 minus1

21199051205782

(1)

First two items represent the manufacturerrsquos expectedprofit without cap and trade policy and green technologyinvestmentThis is the same with the newsvendormodelThethird item represents the carbon emissions trading costprofitof the manufacturer The forth item represents the cost ofmanufacturer investing in green technology

The carbon trading volume after green technology invest-ment under cap and trade policy is

119890 = (1 minus 120578) 119902 minus 119864 (2)

First term represents the manufacturerrsquos carbon emis-sions in production after green technology investment thesecond term represents initial free carbon emission owned bythe manufacturer

Then 120587(119902 119901 119890 120578) can be transformed into

120587 (119902 119901 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + 119896 (1 minus 120578)) 119902 + 119896119864 minus1

21199051205782

(3)

The beliefs of the manufacturer over the customerrsquosreservation price are 120585

119903 Obviously the manufacturer will

set 119901 = 120585119903 119902 and 120578 are the maximizer max

119902120578120587(119902 119901 120578) when

given 119901 According to the definition of RE Equilibrium theRE Equilibrium solution (119901 119902 120578 119903 120585

119903 120585prob)must meet

119903 = V minus (V minus 119904) 120585prob (4)

119901 = 120585119903 (5)

(119902 120578) = argmax119902120578

120587 (119902 119901 120578) (6)

120585prob = 119865 (119902) (7)

120585119903= 119903 (8)

Conditions (4) (5) and (6) indicate that the manufac-turer and customers will choose the action to maximizetheir own utility Conditions (7) and (8) can ensure that thesolution meets rational expectations hypothesis that is theactual situation of economic operation in line with peoplersquosexpectations

Discrete Dynamics in Nature and Society 5

The Rational Expectations Equilibrium makes

119901 = V minus (V minus 119904) 119865 (119902) (9)

The production pricing and green technology invest-ment decision model of the manufacturer with cap and tradepolicy and green technology investment is

max119902120578

120587 (119902 119901 120578)

st (1 minus 120578) 119902 le 119864

(10)

Defining 1205791(119902) = (1(1 minus 120578))[(119901 minus 119904)119865(119902) minus (119888 minus 119904)]

1205792(120578) = 119905120578119902 120579

1(119902) represents themargin profit of unit carbon

emission that is the profit gained by the manufacturerwith one unit carbon emission input in production 120579

2(120578)

represents the margin cost of unit carbon emission that isthe cost that the manufacturer invests in green technology toget one unit carbon emission reduction

Lemma 1 Given 119901 and (V minus 119904)119891(119902)119865(119902)119905 minus 1198962 gt 0 120587(119902 119901 120578)is a joint concave function of 120578 and 119902 in RE Equilibrium

Proof Given 119901 according to formula (3) 120597120587(119902 119901 120578)120597119902 =

(119901minus119904)119865(119902)minus(119888minus119904)minus(1minus120578)119896 1205972120587(119902 119901 120578)1205971199022 = minus(119901minus119904)119891(119902) lt0 120597120587(119902 119901 120578)120597120578 = 119902119896 minus 119905120578 1205972120587(119902 119901 120578)1205971205782 = minus119905 lt 0 and1205972120587(119902 119901 120578)120597119902120597120578 = 120597

2120587(119902 119901 120578)120597120578120597119902 = 119896

When 119905(V minus 119904)119891(119902)119865(119902) minus 1198962 gt 0 under RE Equilibriumwe get 119905(119901minus119904)119891(119902)minus1198962 gt 0 then we have 1205972120587(119902 119901 120578)120597119902120597120578 =1205972120587(119902 119901 120578)120597120578120597119902 = 119896 and

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205972120587 (119902 119901 120578)

1205971199022

1205972120587 (119902 119901 120578)

120597119902120597120578

1205972120587 (119902 119901 120578)

120597120578120597119902

1205972120587 (119902 119901 120578)

1205971205782

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 119905 (119901 minus 119904) 119891 (119902) minus 1198962

gt 0

(11)

The Hessian Matrix of the problem is negative definite andthen we can prove that 120587(119902 119901 120578) is joint concave function of119902 and 120578 when given 119901 and (V minus 119904)119891(119902)119865(119902)119905 minus 119896

2gt 0 This

completes the proof

As to the manufacturerrsquos optimal production quantity(denoted by 119902

lowast) pricing (denoted by 119901lowast) carbon trading

(denoted by 119890lowast) and green technology investment (denotedby 120578lowast) the following proposition is obtained

Proposition 2 If (Vminus119904)119891(119902)119865(119902)119905minus1198962 gt 0 themanufacturerrsquosoptimal production quantity (119902lowast) pricing (119901lowast) carbon trading(119890lowast) and green technology investment (120578lowast) strategies satisfy

1205791(119902lowast) = 1205792(120578lowast) = 119896

119901lowast= 119904 + (V minus 119904) 119865 (119902lowast)

119890lowast= (1 minus 120578

lowast) 119902lowastminus 119864

0 lt 120578lowastlt 1

(12)

Proof According to Lemma 1 if (Vminus 119904)119891(119902)119865(119902)119905 minus 1198962 gt 0 let120597120587(119902 119901 120578)120597119902 = 0 120597120587(119902 119901 120578)120597120578 = 0 and combine (9) and(2) we get

(119901 minus 119904) 119865 (119902) minus (119888 minus 119904) minus (1 minus 120578) 119896 = 0

119902119896 minus 119905120578 = 0

119901 = V minus (V minus 119904) 119865 (119902)

119890 = (1 minus 120578) 119902 minus 119864

0 lt 120578 lt 1

(13)

Solving the equations above can derive the optimalproduction pricing carbon trading and green technologyinvestment strategies of themanufacturerThis completes theproof

Proposition 2 shows that considering strategic customerbehavior the manufacturerrsquos optimal production quantitypricing carbon trading volume and green technology invest-ment strategywith cap and trade policy and green technologyinvestment exist and are unique The inherent implicationof the proposition is very intuitive 120579

1(119902) represents the

margin profit of unit carbon emission 1205792(120578) represents the

margin cost of unit carbon emission and 119896 is the unit carbonemission trading price which can be seen as marginal costor profit by carbon emission trading The optimal strategiesof the manufacturer are that the margin profit of unitcarbon emission is equal to the margin cost of unit carbonemission The marginal cost of getting unit carbon emissionfrom different way (green technology investment or buying)is equal The marginal profit of unit carbon emission fordifferent uses (production or sale) is equal

Substitute 119902lowast 119901lowast and 120578lowast into (3) we can obtain the max-imum expected profit of themanufacturer with cap and tradepolicy and green technology investment 120587(119902lowast 119901lowast 120578lowast) =

(119901lowastminus119904)(119902lowastminusint119902lowast

0119865(119909)119889119909)minus(119888minus119904+119896(1minus120578

lowast))119902lowast+119896119864minus(12)119905120578

lowast2

42 Quantity Commitment Scenario In traditional literatureon strategic customer behavior quantity commitment canimprove the manufacturerrsquos expected profit Quantity com-mitment refers to the action in which the manufacturerpromises customers that only a certain number of productsis produced and sold In low carbon supply chain settingcan the maximum expected profit be improved by quantitycommitment We attempt to answer the question in thispaper

We assume that the manufacturer can convince cus-tomers by appropriate means that they only can obtain 119902

units of the product in the wholesales period At this timethe strategic customers no longer need to anticipate theprobability of getting the product at salvage price When 119902

is given the probability of being able to get the product atprice 119904 is 119865(119902) The reservation price is 119901(119902) = Vminus (Vminus 119904)119865(119902)which also is the optimal pricing of the manufacturer Wehave 119890 = (1 minus 120578)119902 minus 119864 then we can obtain the expected profit

6 Discrete Dynamics in Nature and Society

function of themanufacturer with respect to 119902 and 120578 denotedby 120587119902(119902 120578)

120587119902(119902 120578) = (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + 119896 (1 minus 120578)) 119902 + 119896119864 minus1

21199051205782

(14)

So the manufacturerrsquos optimal production quantity andcarbon reduction rate in QC scenario are (119902

119902lowast 120578119902lowast) =

argmax119902ge00lt120578lt1

120587119902(119902 120578) the optimal price is 119901119902lowast = V minus (V minus

119904)119865(119902119902lowast) and the optimal carbon trading volume is 119890119902lowast =

(1 minus 120578119902lowast)119902119902lowastminus 119864

Lemma 3 Given 119902 the optimal green technology investmentstrategy of the manufacturer denoted by 120578119902lowast satisfies

120578119902lowastequiv 120578 (119902) =

119896119902

119905 0 lt 120578

119902lowastlt 1 (15)

Proof Wehave 120597120587119902(119902 120578)120597120578 = 119902119896minus119905120578 1205972120587119902(119902 120578)1205971205782 = minus119905 lt0 Let 120597120587119902(119902 120578)120597120578 = 0 we get 120578119902lowast equiv 120578(119902) = 119896119902119905 In additionaccording to the assumption in Section 3 we know 0 lt 120578

119902lowastlt

1 This completes the proof

Substituting 120578119902lowast = 120578(119902) into (14)

120587119902(119902) equiv 120587

119902(119902 120578 (119902))

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + 119896) 119902 + 119896119864 +11989621199022

2119905

(16)

Then the two-variable optimization problem is simplifiedas a single variable optimization problem

max119902ge0

120587119902(119902) (17)

Lemma4 If 1198962119905minus(Vminus119904)[3119891(119902)119865(119902)+1198911015840(119902)(119902minusint1199020119865(119909)119889119909)] lt

0 120587119902(119902) is a concave function of 119902

Proof The first-order and second-order derivatives of 120587119902(119902)are

119889120587119902(119902)

119889119902= (V minus 119904) [119865

2

(119902) minus 119891 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) +1198962

119905119902

1198892120587119902(119902)

1198891199022=1198962

119905minus (V minus 119904)

sdot [3119891 (119902) 119865 (119902) + 1198911015840(119902) (119902 minus int

119902

0

119865 (119909) 119889119909)] lt 0

(18)

Then we can obtain that 120587119902(119902) is a concave function of 119902This completes the proof

As to the manufacturerrsquos optimal production quantity inQC scenario (denoted by 119902119902lowast) the following proposition isobtained

Proposition 5 If 1198962119905 minus (V minus 119904)[3119891(119902)119865(119902) + 1198911015840(119902)(119902 minus

int119902

0119865(119909)119889119909)] lt 0 the optimal production quantity in QC

scenario (119902119902lowast) satisfies

(V minus 119904) [1198652

(119902) minus 119891 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) +1198962

119905119902 = 0

(19)

Proof It can be directly derived according to Lemma 4 Thiscompletes the proof

Proposition 5 shows that under certain condition theoptimal production quantity of the manufacturer with capand trade and green technology investment exist and areunique

43 The Effect of Quantity Commitment The effect of QCon the optimal strategies and the maximum expected profitof the manufacturer is analyzed by comparing the optimalstrategies and the maximum expected profit of the manufac-turer in RE Equilibrium scenario and QC scenario

Proposition 6 Consider 119902119902lowast lt 119902lowast 119901119902lowast gt 119901lowast 120578119902lowast lt 120578lowast

Proof In order to ensure that the manufacturerrsquos optimalstrategies in the two scenarios exist the condition 119896

2119905 lt

min(Vminus119904)119891(119902)119865(119902) (Vminus119904)[3119891(119902)119865(119902)+1198911015840(119902)(119902minusint1199020119865(119909)119889119909)]

must be held Then 120587119902(119902) is a concave function of 119902According to Proposition 2 we have 120579

1(119902lowast) = 1205792(120578lowast) = 119896

1205792(120578lowast) = 119896 can be written as 120578lowast = 119896119902

lowast119905 Then we also can

rearrange 1205791(119902lowast) = 119896 to (Vminus119904)1198652(119902lowast)minus(119888minus119904+119896)+(1198962119905)119902lowast = 0

We can obtain (119889120587119902(119902)119889119902)|

119902=119902lowast = (V minus 119904)[119865

2

(119902lowast) minus

119891(119902lowast)(119902lowastminus int119902lowast

0119865(119909)119889119909)] minus (119888 minus 119904 + 119896) + (119896

2119905)119902lowast= minus(V minus

119904)119891(119902lowast)(119902lowastminusint119902lowast

0119865(119909)119889119909) lt 0 Then we get 119902119902lowast lt 119902lowast Because

120578 = 119896119902119905 and 119901 = Vminus(Vminus119904)119865(119902) are held in the two scenarioswe can obtain 119901119902lowast gt 119901

lowast and 120578119902lowast lt 120578lowast according to 119902119902lowast lt 119902

lowastThis completes the proof

Proposition 6 shows that compared with RE Equilibriumscenario the manufacturerrsquos optimal production quantity islower the optimal pricing is higher and the optimal carbonreduction rate is lower in QC scenario

Proposition 7 Consider 120587119902(119902119902lowast 120578119902lowast) gt 120587(119902lowast 119901lowast 120578lowast)

Proof In order to ensure that the manufacturerrsquos optimalstrategies in the two scenarios exist the condition 119896

2119905 lt

min(Vminus119904)119891(119902)119865(119902) (Vminus119904)[3119891(119902)119865(119902)+1198911015840(119902)(119902minusint1199020119865(119909)119889119909)]

must be held Then 120587119902(119902) is a concave function of 119902Substitute 119902lowast 119901lowast and 120578

lowast= 119896119902

lowast119905 into (3) we have

120587(119902lowast 119901lowast 120578lowast) = 120587

119902(119902lowast) We know that 120587119902(119902) is a concave

function of 119902 and (119889120587119902(119902)119889119902)|119902=119902119902lowast = 0 Because of 119902119902lowast lt 119902lowast

Discrete Dynamics in Nature and Society 7

120587119902(119902lowast) lt 120587

119902(119902119902lowast) that is 120587(119902lowast 119901lowast 120578lowast) lt 120587

119902(119902119902lowast 120578119902lowast) This

completes the proof

Proposition 7 shows that QC strategy can improve themanufacturerrsquos maximum expected profit when green tech-nology investment and cap and trade policy are taken intoaccount

5 Decentralized Supply Chain Model

In this section the retailer and the manufacturer make deci-sions independently to maximize their own expected profitThe manufacturer is the Stackelberg leader and determinesthe wholesale price and the green technology investmentunder cap and trade policy The retailer is the follower anddetermines the retail price and the order quantity of theproduct

The sequence of events in this section is as follows firstthe manufacturer determines the optimal wholesale pricecarbon trading and green technology investment strategiessecond the retailer forms the belief of customersrsquo reservationprice and then decides the optimal retail price and orderquantity third the manufacture produces products anddelivers to the retailer fourth the customers form the beliefs120585prob of probability of the product sold at salvage price119904 according to the information of market price and thenform the reservation price 119903 fifth the customersrsquo demand issatisfied and the products are sold at full price 119901 finally allremaining products are sold to the external market at salvageprice 119904

51 The Optimal Strategies First we examine the decisionproblem of the retailer Recall that 119908 is the manufacturerrsquoswholesale price The retailerrsquos expected profit denoted by120587119903(119902 119901) is

120587119903(119902 119901) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909) minus (119908 minus 119904) 119902 (20)

Lemma 8 When 119901 is given 120587119903(119902 119901) is a concave function of119902

Proof When 119901 is given according to (20) we have120597120587119903(119902 119901)120597119902 = (119901 minus 119904)119865(119902) minus (119908 minus 119904) and 1205972120587119903(119902 119901)1205971199022 =

minus(119901 minus 119904)119891(119902) lt 0 Then we know that 120587119903(119902 119901) is a concavefunction of 119902 when 119901 is given This completes the proof

As to the retailerrsquos optimal strategies in RE Equilibriumwe can obtain the following proposition

Proposition 9 When the wholesale price is given the retailerrsquosoptimal order strategies (119902119903lowast) and optimal pricing strategies(119901119903lowast) with strategic customer behavior are

119902119903lowast= 119865minus1

(radic119908 minus 119904

V minus 119904) (21)

119901119903lowast= 119904 + radic(119908 minus 119904) (V minus 119904) (22)

Proof In RE Equilibrium (9) still holds Let 120597120587119903(119902 119901)120597119902 = 0and combining with (9) we have

119901 = V minus (V minus 119904) 119865 (119902)

(119901 minus 119904) 119865 (119902) minus (119908 minus 119904) = 0

(23)

According to Lemma 8 we can obtain the optimal orderand pricing strategies of the retailer by solving the equationsabove This completes the proof

Proposition 9 shows that the retailerrsquos optimal order andpricing strategies in REEquilibriumwith cap and trade policyexist and are unique

Second we investigate the manufacturerrsquos decision prob-lem The expected profit function of the manufacturer withgreen technology investment and cap and trade policy is120587119898(119908 119890 120578) = (119908 minus 119888)119902 minus 119896119890 minus (12)119905120578

2According to (21) with green technology investment and

cap and trade policy in wholesale price contract the optimalorder quantity of the retailer 119902119903lowast and the optimal wholesaleprice of the manufacturer 119908119898lowast is a one-to-one relationship

119908 = 119904 + (V minus 119904) 1198652

(119902) (24)

Combined with 119890 = (1 minus 120578)119902 minus 119864 the manufacturerrsquosexpected profit function can be converted into

120587119898(119902 120578) = [(V minus 119904) 119865

2

(119902) minus (119888 minus 119904 + (1 minus 120578) 119896)] 119902

+ 119896119864 minus1

21199051205782

(25)

When using wholesale price contract the total profit ofthe supply chain denoted by 120587sc

(119902 119901 119890 120578) is

120587sc(119902 119901 119890 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909) minus (119888 minus 119904) 119902

minus 119896119890 minus1

21199051205782

(26)

According to (9) and 119890 = (1 minus 120578)119902 minus 119864 the total supplychain profit function in RE Equilibrium can be simplified as

120587sc(119902 120578) = (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578) 119896) 119902 + 119896119864 minus1

21199051205782

(27)

Lemma 10 Given 119902 the optimal green technology investmentstrategy of the manufacturer denoted by 120578119898lowast satisfies

120578119898lowast

equiv 120578 (119902) =119896119902

1199050 lt 120578119898lowast

lt 1 (28)

Proof The first-order and second-order partial derivatives of120587119898(119902 120578) with respect to 120578 are 120597120587119898(119902 120578)120597120578 = 119902119896 minus 119905120578 and

1205972120587119898(119902 120578)120597120578

2= minus119905 lt 0 Let 120597120587119898(119902 120578)120597120578 = 0 we get

120578119898lowast

equiv 120578(119902) = 119896119902119905 In addition according to the assumptionin Section 3 we know 0 lt 120578

119898lowastlt 1 This completes the

proof

8 Discrete Dynamics in Nature and Society

Substituting 120578119898lowast = 120578(119902) into (25)

120587119898(119902) equiv 120587

119898(119902 120578 (119902))

= [(V minus 119904) 1198652

(119902) minus (119888 minus 119904 + 119896)] 119902 + 119896119864

+11989621199022

2119905

(29)

Then the two-variable optimization problem is simplifiedas a single variable optimization problem

max119902ge0

120587119898(119902) (30)

Lemma 11 If 1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus

1199021198912(119902)] lt 0 120587119898(119902) is a concave function of 119902

Proof The first-order and second-order derivatives of 120587119898(119902)with respect to 119902 are 119889120587

119898(119902)119889119902 = (V minus 119904)[119865

2

(119902) minus

2119902119891(119902)119865(119902)] minus (119888 minus 119904 + 119896) + (1198962119905)119902 and 119889

2120587119898(119902)119889119902

2=

1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus 1199021198912(119902)] lt 0 Then

we have that 120587119898(119902) is a concave function of 119902 This completesthe proof

In decentralized supply chain as to the manufacturerrsquosoptimal wholesale price (denoted by 119908

119898lowast) the followingproposition is obtained

Proposition 12 If 1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus1199021198912(119902)] lt 0 the manufacturerrsquos optimal wholesale price in

decentralized supply chain (119908119898lowast) satisfies 119908119898lowast = 119904 + (V minus119904)1198652

(119902119903lowast) where 119902119903lowast represents the retailerrsquos optimal order

quantity and satisfies

(V minus 119904) [1198652

(119902) minus 2119902119891 (119902) 119865 (119902)] minus (119888 minus 119904 + 119896) +1198962

119905119902

= 0

(31)

Proof According to Lemma 11 we can obtain that 119902119903lowastwhichmaximizes themanufacturerrsquos expected profit satisfies(31) Combining with (24) we know that the manufacturerrsquosoptimal wholesale price satisfies 119908119898lowast = 119904 + (V minus 119904)119865

2

(119902119903lowast) at

this time This completes the proof

Proposition 12 shows that the optimal commitment quan-tity of the manufacturer in decentralized supply chain withgreen technology investment and cap and trade policy existand are unique The optimal wholesale price does not relateto the cap allocated by the government but relates to thecarbon trading price This is because the production of themanufacturer is not affected by the cap but is affected bythe margin cost of the product However the margin cost ofthe product varies with the change of carbon trading pricewhen carbon trading is allowed The change results in thatthe optimal commitment quantity is not related to the cap butrelate to the carbon trading price

Substitute 119908119898lowast into (22) we derive the retailerrsquos optimalpricing in decentralized situation119901119903lowast = 119904+radic(119908119898lowast minus 119904)(V minus 119904)

According to Lemma 10 we get the optimal green technologyinvestment of the manufacturer 120578119898lowast = (119896119905)119902119903lowast

Substitute 119902119903lowast 119901119903lowast 119908119898lowast and 120578119898lowast into 120587

119903(119902 119901) 120587119898(119902)

and 120587sc(119902 120578) we obtain the maximum expected profit of

the retailer the manufacturer and the supply chain indecentralized channel

120587119903(119902119903lowast 119901119903lowast) = (119901

119903lowastminus 119904) (119902

119903lowastminus int

119902119903lowast

0

119865 (119909) 119889119909)

minus (119908119898lowast

minus 119904) 119902119903lowast

120587119898(119902119903lowast) = [(V minus 119904) 119865

2

(119902119903lowast) minus (119888 minus 119904 + 119896)] 119902

119903lowast

+ 119896119864 +1198962119902119903lowast2

2119905

120587sc(119902119903lowast 120578119898lowast) = (V minus 119904) 119865 (119902119903lowast) (119902119903lowast minus int

119902119903lowast

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578119898lowast) 119896) 119902119903lowast+ 119896119864

minus1

2119905120578119898lowast2

(32)

52 The Effect of Decentralization We examine the impact ofdecentralization on the supply chain optimal strategies andmaximumexpected profitwhen green technology investmentis taken into consideration by comparing the optimal strate-gies and the maximum expected profit in centralized anddecentralized channel with green technology investment andcap and trade policy

As the effect of decentralization of supply chain we canobtain the following proposition

Proposition 13 Consider 119902119903lowast lt 119902119902lowast

lt 119902lowast 119901119903lowast gt 119901

119902lowastgt 119901lowast

120578119898lowast

lt 120578119902lowastlt 120578lowast

Proof In Proposition 6 119902119902lowast lt 119902lowast 119901119902lowast gt 119901

lowast and 120578119902lowast lt 120578lowast

have been provedTherefore we only need to prove 119902119903lowast lt 119902119902lowast119901119903lowastgt 119901119902lowast and 120578119898lowast lt 120578119902lowast

Define 119867(119902) = 120587119902(119902) minus 120587119898(119902) then

119867(119902)

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

minus (V minus 119904) 1199021198652

(119902)

1198671015840(119902)

= (V minus 119904) 119891 (119902) [2119902119865 (119902) minus (119902 minus int119902

0

119865 (119909) 119889119909)]

(33)

Define 119866(119902) = 2119902119865(119902) minus (119902 minus int119902

0119865(119909)119889119909) then 1198661015840(119902) =

119865(119902) minus 2119902119891(119902) Because 119866(0) = 0 1198661015840(0) = 1 and 119865 satisfiesIFR we know that the value of 119866(119902) starts from 0 increasingfirst and then decreasing in 119902 That is 119866(119902) = 0 has uniquesolution and we denote it as So when 119902 lt 119866(119902) gt 0 that

Discrete Dynamics in Nature and Society 9

is119867(119902) is increasing in 119902 when 119902 gt 119866(119902) lt 0 that is119867(119902)is decreasing in 119902 Then we have that119867(119902) is a quasi-concavefunction of 119902 and 1198661015840 () = 119865() minus 2119891() lt 0 According tothe analysis above we know that1198671015840() = 0 that is 2119865() =( minus int

0119865(119909)119889119909) Substitute into 119889120587119902(119902)119889119902 we have

119889120587119902(119902)

119889119902

100381610038161003816100381610038161003816100381610038161003816119902=

= (V minus 119904) [1198652

() minus 119891 () ( minus int

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) +1198962

119905

= (V minus 119904) 119865 () [119865 () minus 2119891 ()] minus (119888 minus 119904 + 119896)

+1198962

119905lt (V minus 119904) 119865 () [119865 () minus 2119891 ()] lt 0

(34)

According to Lemma 4 we have that 120587119902(119902) is a concavefunction of 119902 so we get 119902119902lowast lt Because 2119865() = ( minus

int

0119865(119909)119889119909) and119867(119902) is a quasi-concave function of 119902 thenwe

get that 119889120587119898(119902)119889119902 lt 119889120587119902(119902)119889119902 if 119902 lt and 119889120587119898(119902)119889119902 gt

119889120587119902(119902)119889119902 if 119902 gt Therefore the only possible orderings for

119902119902lowast 119902119903lowast and are lt 119902

119902lowastlt 119902119903lowast and 119902119903lowast lt 119902

119902lowastlt We

have proved 119902119902lowast lt so we have 119902119903lowast lt 119902119902lowast We get 119901119903lowast gt 119901119902lowastbecause 119901 and 119902 satisfy 119901 = V minus (V minus 119904)119865(119902) in two situationsWe get 120578119898lowast lt 120578

119902lowast because 120578 and 119902 satisfy 120578 = 119896119902119905 in twosituations This completes the proof

Proposition 13 shows that for the decision of a decen-tralized supply chain with green technology investment theoptimal production quantity is increasing the optimal priceis decreasing and the green technology investment (iecarbon reduction rate) is increasing respectively in the threesituations of decentralized supply chain QC scenario and REEquilibrium scenario

The reason is that themanufacturer prefers higher whole-sale price (ie lower production and less green technologyinvestment) to guarantee its ownprofitmaximization thoughthis may harm the retailer even the whole supply chain

In Proposition 7 we have found 120587119902(119902119902lowast 120578119902lowast) gt

120587(119902lowast 119901lowast 120578lowast) that is the manufacturerrsquos maximum profit in

QC scenario is always greater than that in RE EquilibriumTherefore the section analyzes the effect of decentralizationon supply chain performance based on the QC scenario toprovide benchmark for designing of subsequent coordinationcontract

As to the effect of decentralization on themaximumprofitof supply chain we have the following proposition

Proposition 14 Consider 120587sc(119902119903lowast 120578119898lowast) lt 120587119902(119902119902lowast 120578119902lowast)

Proof The optimal green technology investment in decen-tralized supply chain and QC scenario are 120578(119902) = 119896119902119905

according to Lemmas 3 and 10 So the profit function in thetwo situations above can be converted into a same singlevariable function with respect to 119902 that is (16)

According to Lemma 4 we know that 120587119902(119902) is a concavefunction and reaches the maximum value at 119902119902lowast Accordingto Proposition 13 we have 119902119903lowast lt 119902

119902lowast then we get 120587119902(119902119903lowast) lt120587119902(119902119902lowast) that is 120587sc

(119902119903lowast 120578119898lowast) lt 120587

119902(119902119902lowast 120578119902lowast) This completes

the proof

Proposition 14 shows that the maximum profit of decen-tralized supply chain is always lower than that in QC scenariowhich indicates that the profit of the decentralized supplychain can get higher This is the benchmark of supply chaincoordination

6 Supply Chain Coordination

This section coordinates the supply chain with green tech-nology investment and cap and trade policy based on theQC scenario to improve the profit of the decentralized supplychain

61 Coordination Contract Based on Revenue and CostsSharing Contract With revenue sharing contract we assumethat 120601 (0 le 120601 le 1) represents the ratio of revenue kept by theretailer and (1 minus 120601) represents the ratio of revenue deliveredto the manufacturer

The retailerrsquos expected profit function with revenue shar-ing contract denoted by 120587119903

119904(119902 119901 120601) is

120587119903

119904(119902 119901 120601) = 120601 (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119908 minus 120601119904) 119902

(35)

Themanufacturerrsquos expected profit functionwith revenuesharing contract denoted by 120587119898

119904(119908 119890 120578 120601) is

120587119898

119904(119908 119890 120578 120601) = (119908 minus 119888) 119902 + (1 minus 120601)

sdot [int

119902

0

[119901119909 + 119904 (119902 minus 119909)] 119891 (119909) 119889119909

+ int

infin

119902

119901119902119891 (119909) 119889119909] minus 119896119890 minus1

21199051205782

(36)

Substitute 119890 = (1 minus 120578)119902 minus 119864 into the equation above andwe have

120587119898

119904(119908 120578 120601) = (1 minus 120601) (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

+ (119908 minus 119888 minus (1 minus 120578) 119896 + (1 minus 120601) 119904) 119902

+ 119896119864 minus1

21199051205782

(37)

The total expected profit of the whole supply chaindenoted by 120587sc

119904(119902 119901 120578) is

120587sc119904(119902 119901 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578) 119896) 119902 + 119896119864 minus1

21199051205782

(38)

10 Discrete Dynamics in Nature and Society

The subscript 119904 represents the situation of supply chaincoordination using revenue sharing contracts

As to the retailerrsquos optimal order strategy (denoted by119902lowast

119904) and pricing strategy (denoted by 119901lowast

119904) of the supply chain

with revenue sharing contract the following proposition isobtained

Proposition 15 Under revenue sharing contracts the retailerrsquosoptimal order strategy (119902lowast

119904) and optimal pricing strategy (119901lowast

119904)

are

119902lowast

119904= 119865minus1

(radic119908 minus 120601119904

120601 (V minus 119904))

119901lowast

119904= 119904 + radic

(119908 minus 120601119904) ((V minus 119904))120601

(39)

Proof According to the definition of RE Equilibrium wealso have 119901 = V minus (V minus 119904)119865(119902) under revenue sharingcontracts

The first-order and second-order derivative of 120587119903119904(119902 119901 120601)

with respect to 119902 are 120597120587119903

119904(119902 119901 120601)120597119902 = 120601(119901 minus 119904)119865(119902) minus

(119908 minus 120601119904) and 1205972120587119903

119904(119902 119901 120601)120597119902

2= minus120601(119901 minus 119904)119891(119902) lt

0 So given 119901 the retailerrsquos expected profit function is aconcave function of 119902 Let 120597120587119903

119904(119902 119901 120601)120597119902 = 0 and combine

with 119901 = V minus (V minus 119904)119865(119902) we can solve the retailerrsquos optimalstrategies under revenue sharing contractThis completes theproof

Proposition 15 shows that the optimal order and pricingstrategies of the retailer based on revenue sharing contract inRE Equilibrium exist and are uniqueThis is also the retailerrsquosoptimal response function with respect to 119908

Substituting (39) into (43)

120587119898

119904(119902 120578 120601)

= (1 minus 120601) (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ [120601 (V minus 119904) 1198652

(119902) minus (119888 minus 119904 + (1 minus 120578) 119896)] 119902 + 119896119864

minus1

21199051205782

120597120587119898

119904(119902 120578 120601)

120597120578= 119902119896 minus 119905120578

1205972120587119898

119904(119902 120578 120601)

1205971205782= minus119905 lt 0

(40)

Given 119902 and 120601 the manufacturerrsquos green technologyinvestment strategies with revenue sharing contracts exist

and are unique Let 120597120587119898119904(119902 120578 120601)120597120578 = 0 we get 120578119898lowast

119904equiv 120578(119902) =

119902119896119905 and substitute it into 120587119898119904(119902 120578 120601)

120587119898

119904(119902 120601)

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ 120601 (V minus 119904) 119865 (119902) [119902119865 (119902) minus (119902 minus int119902

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) 119902 + 119896119864 +11989621199022

2119905

(41)

Proposition 16 The supply chain with green technologyinvestment and cap and trade policy can not be coordinated byrevenue sharing contract

Proof The derivative of 120587119898119904(119902 120601) with respect to 119902 can be

arranged as follows

120597120587119898

119904(119902 120601)

120597119902=120597120587119902(119902)

120597119902+ 120601(

120597120587119898(119902)

120597119902minus120597120587119902(119902)

120597119902) (42)

Only when 120601 = 0 are the optimal strategies of supplychain under revenue sharing contracts equal to that in QCscenario Let 119908lowast

119904represent the optimal wholesale price of

the manufacture and 119902lowast119904represent the retailerrsquos optimal order

quantity Then 120587119903

119904(119902 119901 120601) = minus119908

lowast

119904119902lowast

119904lt 0 So the revenue

sharing contract cannot realize the supply chain coordinationat this time This completes the proof

Proposition 16 shows that revenue sharing contractscan not realize the supply chain coordination with greentechnology investment and cap and trade policy The rea-son is that the manufacturer undertakes all the costs ofgreen technology investment when using revenue sharingcontracts thus making the manufacturer tend to invest lessin green technology Only when the manufacturer owns allthe revenue will themanufacturerrsquos optimal green technologyinvestment under revenue sharing contracts be equal to thatin QC scenario However the maximum profit of the retaileris negative at this time and it will not participate

62 Coordination Contract Based on Revenue Sharing-CostSharing Contract Proposition 16 states that the supply chainwith green technology investment and cap and trade policycan not be coordinated by revenue sharing contracts becausethe costs of green technology investment are undertaken bythe manufacturer and the game of retailer and customersis characterized by RE Equilibrium So we consider costssharing of green technology investment and carbon tradingand quantity commitment made by the manufacturer basedon revenue sharing contracts Let 120601 (0 lt 120601 le 1) represent theratio of revenue kept by the retailer and let (1minus120601) represent theratio of revenue delivered to the manufacturer 120572 (0 le 120572 le 1)represents the ratio of costs of green technology investmentand carbon trading undertaken by the manufacturer and(1 minus 120572) represents the ratio of costs of green technologyinvestment and carbon trading delivered to the retailer

Discrete Dynamics in Nature and Society 11

The retailerrsquos expected profit function with revenuesharing-cost sharing contract is

120587119903

120572(119902 120601 120572) = 120601 (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119908 minus 120601119904) 119902

minus (1 minus 120572) 119896 [(1 minus 120578) 119902 minus 119864]

minus1

2(1 minus 120572) 119905120578

2

(43)

We have 119890 = (1minus120578)119902minus119864 then themanufacturerrsquos expectedprofit function denoted by 120587119898

120572(119908 120578 120601 120572) is

120587119898

120572(119908 120578 120601 120572)

= (1 minus 120601) (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ (119908 minus 119888 minus 120572 (1 minus 120578) 119896 + (1 minus 120601) 119904) 119902 + 120572119896119864

minus1

21205721199051205782

(44)

The subscript 120572 represents the situation that using rev-enue sharing-cost sharing contract coordinates the supplychain

Proposition 17 If 0 le 120582 le 1 the parameters of revenuesharing-cost sharing (119908 120601 120572) satisfy

119908 = 120582119888 + (1 minus 120578) 119896

120601 = 120582

120572 = 1 minus 120582

(45)

The supply chainwith green technology investment and cap andtrade policy is coordinated

Proof When the parameters of revenue sharing-cost sharingsatisfy (45) the expected profit function of the retailer andthe manufacturer can be written as follows

120587119903

120572(119902 119901 120601 120572) = 120582120587

119902(119902 120578)

120587119898

120572(119908 120578 120601 120572) = (1 minus 120582) 120587

119902(119902 120578)

(46)

The optimal order of decentralized supply chain is equalto that of QC scenario Substitute 119901 = V minus (V minus 119904)119865(119902) into120587sc119888(119902 119901 120578) we get that the profit function of decentralized

supply chain at this time is the same as that of QC scenarioTherefore when the parameters of revenue sharing-costsharing (119908 120601 120572) satisfy (45) the decentralized supply chaincoordination is realizedThemanufacturer earns all the profitif 120582 = 0 and the retailer earns all the profit if 120582 = 1Arbitrary allocation of the supply chain profit between themanufacturer and the retailer is allowed This completes theproof

Proposition 17 shows that revenue sharing-cost sharingcontract can realize the supply chain coordination on condi-tion that the manufacturer makes a quantity commitment to

the customers The optimal production quantity and greentechnology investment in decentralized situation are lowerthan that in QC scenario The quantity commitment of themanufacturer can increase the green technology investmentand production quantity and also reduce the retailer priceThat is the quantity commitment is beneficial for the man-ufacturer the retailer and customers Thus the quantitycommitment is credible to customers now

7 Conclusions and Suggestions for FurtherResearch

This paper investigates the production pricing carbon trad-ing and green technology investment strategies and thecoordination of low carbon supply chain made up of a lowcarbon manufacturer and a retailer The low carbon manu-facturer produces one product under cap and trade policyand performs green technology investment to reduce carbonemissions in production process The retailer orders fromthemanufacturer and distributes the product to homogenouscustomers To the best of our knowledge this is the firststudy on the management of low carbon supply chainswith strategic customer behaviorThis paper provides severalinteresting observations

Observation 1 There are unique optimal production pricingcarbon trading and green technology investment strategiesin centralized (include RE Equilibrium scenario and QCscenario) and decentralized supply chain Therefore byderiving the optimal production pricing carbon trading andgreen technology investment strategies in different situationsthe low carbon manufacturer and the retailer can behaveappropriately based on our findings to maximize their profit

Observation 2 Our findings also show that the centralizedsupply chain can improve its performance by quantity com-mitment However the manufacturerrsquos production and greentechnology investment are reduced while the optimal priceis increased at this time Therefore quantity commitment isbeneficial for the supply chain but is harmful to customers

Observation 3 We also show that the performance of decen-tralized supply chain is lower than that in QC scenario Wefind that the decentralization reduces the manufacturersquos opti-mal production quantity and green technology investmentand increases the retailerrsquos optimal price Besides we find thatthe optimal production quantity is increasing the optimalprice is decreasing and the green technology investment isincreasing respectively in the three situations of decentral-ized supply chain QC scenario andREEquilibrium scenario

Observation 4 We demonstrate that the traditional revenuesharing contracts can not realize the low carbon supply chaincoordination when green technology investment is takenin consideration We also find that the revenue sharing-cost sharing contract realizes the decentralized supply chain

12 Discrete Dynamics in Nature and Society

coordination when the manufacturer makes a quantity com-mitment It is beneficial to the manufacturer the retailerand customers if themanufacturermakes a quantity commit-ment So the quantity commitment is credible to the customerat this time

We assume the customers are homogeneous strategiccustomers But themyopic customers and strategic customersoften coexist and strategic customers also have differenttolerance in real-life Extending this assumption will havea knock-on effect on supply chain decisions So one keyresearch direction is to consider that the customers aremixed customers myopic customers and strategic customersSecond we assume that the supply chain only producesone product While this simplified setting provides someinteresting insights we have to acknowledge that a manufac-turer usually produces two or more productsThese productswill substitute or complement each other So one importantextension of our work is to assume that the manufacturerproduces two or more products which have different unitcarbon emissions in production

Notations

119863 The random demand and119863 ge 0

119891(sdot) Probability density function of119863119865(sdot) Distribution function of119863 We assume

that 119865 satisfy IFR and define 119865 = 1 minus 119865

119888 Unit production cost119904 Unit salvage price of the product which

is an exogenous variableV The customersrsquo utility from consuming

the product119908 Unit wholesale price119901 Unit retail selling price We assume that

it can be observed by the customers119902 The manufacturerrsquos production

quantitythe retailerrsquos order quantityWe assume that it cannot be observedby the customers

119903 The customersrsquo reservation price whichis customersrsquo private information andcannot be observed by the retailer

120585119903 The beliefs of the manufacturer over the

customerrsquo reservation price120585prob The beliefs of customers over their

chances of obtaining the product atphase two

119864 Initial free carbon emission allocated bygovernment

119896 Unit carbon emission trading price119890 Carbon trading volume120578 Carbon emissions reduction rate

Conflict of Interests

The authors declare no conflict of interests

Acknowledgments

This research is partially supported by National Natural Sci-ence Foundation of China (nos 71432003 and 71272128) theProgram forNewCentury Excellent Talents inUniversity (noNCET-12-0087) and Specialized Research Fund for the Doc-toral Program of Higher Education (no 20130185110006)

References

[1] R Alley T Berntsen N L Bindoff et al ldquoSummary forpolicymakersrdquo in Climate Change 2007 The Physical ScienceBasis vol 6 pp 1ndash18 IPCC 2007

[2] X Chen and G Hao ldquoSustainable pricing and productionpolicies for two competing firms with carbon emissions taxrdquoInternational Journal of Production Research vol 53 pp 6408ndash6420 2015

[3] X Chen S Benjaafar and A Elomri ldquoThe carbon-constrainedEOQrdquo Operations Research Letters vol 41 no 2 pp 172ndash1792013

[4] N O Keohane ldquoCap and trade rehabilitated using tradablepermits to control US greenhouse gasesrdquo Review of Environ-mental Economics and Policy vol 3 no 1 pp 42ndash62 2009

[5] B Liu M Holmbom A Segerstedt and W Chen ldquoEffectsof carbon emission regulations on remanufacturing decisionswith limited information of demand distributionrdquo InternationalJournal of Production Research vol 53 no 2 pp 532ndash548 2015

[6] L Zhu and Y Fan ldquoA real options-based CCS investmentevaluation model case study of Chinarsquos power generationsectorrdquo Applied Energy vol 88 no 12 pp 4320ndash4333 2011

[7] XWang and LDu ldquoStudy on carbon capture and storage (CCS)investment decision-making based on real options for Chinarsquoscoal-fired power plantsrdquo Journal of Cleaner Production vol 112part 5 pp 4123ndash4131 2016

[8] W Jiang and X Chen ldquoOptimal strategies for manufacturerwith strategic customer behavior under carbon emissions-sensitive randomdemandrdquo Industrial Management amp Data Sys-tems 2015

[9] Y Levin J McGill and M Nediak ldquoDynamic pricing in thepresence of strategic consumers and oligopolistic competitionrdquoManagement Science vol 55 no 1 pp 32ndash46 2009

[10] A K Parlakturk ldquoThe value of product variety when sellingto strategic consumersrdquo Manufacturing amp Service OperationsManagement vol 14 no 3 pp 371ndash385 2012

[11] X Su and F Zhang ldquoOn the value of commitment andavailability guarantees when selling to strategic consumersrdquoManagement Science vol 55 no 5 pp 713ndash726 2009

[12] W Elmaghraby S A Lippman C S Tang and R Yin ldquoWillmore purchasing options benefit customersrdquo Production andOperations Management vol 18 no 4 pp 381ndash401 2009

[13] W Jiang and X Chen ldquoManufacturersquos production and pricingstrategies with carbon tax policy and strategic customer behav-iorrdquoManagement Science and Engineering vol 9 no 1 pp 30ndash35 2015

[14] I Dobos ldquoThe effects of emission trading on production andinventories in the Arrow-Karlin modelrdquo International Journalof Production Economics vol 93-94 pp 301ndash308 2005

[15] X Chen C K Chan and Y C E Lee ldquoResponsible productionpolicies with substitution and carbon emissions tradingrdquo Jour-nal of Cleaner Production 2015

Discrete Dynamics in Nature and Society 13

[16] X Chen and X Wang ldquoEffects of carbon emission reductionpolicies on transportation mode selections with stochasticdemandrdquo Transportation Research Part E Logistics and Trans-portation Review 2015

[17] X Chang H Xia H Zhu T Fan and H Zhao ldquoProductiondecisions in a hybrid manufacturingndashremanufacturing systemwith carbon cap and trade mechanismrdquo International Journal ofProduction Economics vol 162 pp 160ndash173 2015

[18] J Zhao B F Hobbs and J-S Pang ldquoLong-run equilibriummodeling of emissions allowance allocation systems in electricpowermarketsrdquoOperations Research vol 58 no 3 pp 529ndash5482010

[19] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2 pp519ndash527 2011

[20] A Toptal H Ozlu and D Konur ldquoJoint decisions on inventoryreplenishment and emission reduction investment under dif-ferent emission regulationsrdquo International Journal of ProductionResearch vol 52 no 1 pp 243ndash269 2014

[21] A S Manikas and J R Kroes ldquoA newsvendor approach tocompliance and production under cap and trade emissionsregulationrdquo International Journal of Production Economics vol159 pp 274ndash284 2015

[22] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineering vol10 no 1 pp 99ndash116 2013

[23] X Xu W Zhang P He and X Xu ldquoProduction and pricingproblems in make-to-order supply chain with cap-and-traderegulationrdquo Omega 2015

[24] Y Zhen X Yang and W Jiang ldquoThe impact of transport modeand carbon policy on low-carbon retailerrdquoDiscrete Dynamics inNature and Society vol 2015 Article ID 964305 12 pages 2015

[25] S Swami and J Shah ldquoChannel coordination in green supplychainmanagementrdquo Journal of the Operational Research Societyvol 64 no 3 pp 336ndash351 2013

[26] R H Coase ldquoDurability andmonopolyrdquoThe Journal of Law andEconomics vol 15 no 1 pp 143ndash149 1972

[27] X Su ldquoIntertemporal pricing with strategic customer behaviorrdquoManagement Science vol 53 no 5 pp 726ndash741 2007

[28] D Zhang and W L Cooper ldquoManaging clearance sales in thepresence of strategic customersrdquo Production and OperationsManagement vol 17 no 4 pp 416ndash431 2008

[29] S Whang ldquoDemand uncertainty and the Bayesian effect inmarkdown pricing with strategic customersrdquoManufacturing ampService Operations Management vol 17 no 1 pp 66ndash77 2015

[30] J Du J Zhang and G Hua ldquoPricing and inventory manage-ment in the presence of strategic customers with risk preferenceand decreasing valuerdquo International Journal of Production Eco-nomics vol 164 pp 160ndash166 2015

[31] X Su and F Zhang ldquoStrategic customer behavior commitmentand supply chain performancerdquo Management Science vol 54no 10 pp 1759ndash1773 2008

[32] H Yang ldquoImpact of discounting and competition on benefit ofdecentralization with strategic customersrdquo Operations ResearchLetters vol 40 no 2 pp 123ndash127 2012

[33] D Yang E Qi and Y Li ldquoQuick response and supply chainstructure with strategic consumersrdquo Omega vol 52 pp 1ndash142015

[34] C drsquoAspremont and A Jacquemin ldquoCooperative and nonco-operative R amp D in duopoly with spilloversrdquo The AmericanEconomic Review vol 78 no 5 pp 1133ndash1137 1988

[35] J F Muth ldquoRational expectations and the theory of pricemovementsrdquo Econometrica vol 29 no 3 pp 315ndash335 1961

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

customers canmeet the low carbon demand of customers andobtain the corresponding competitive advantage Businessesin supply chain needs to trade off the costs and benefits ofgreen technology investment to decide whether to investTherefore under the cap and trade policy study on lowcarbon supply chain operation strategies optimization withgreen technology investment is of great theoretical value

In recent years scholars and practitioners have takengreater interests in strategic customer behavior which isshown in the increasing marketing and operations researchliterature on the problem Strategic customer behavior refersto the behavior of customers who do not buy products atfull price but wait until the products markdown Researchshows that the ignorance of the strategic customer behavior inoperation strategies would be harmful to the performance ofsupply chain [9 10] Strategic customer behavior has becomea common phenomenon in perishable product sales processwhich made it a regular assumption in many literatures [11ndash13] Therefore considering strategic customer behavior tostudy low carbon supply chain strategies and coordination ismore realistic

The production pricing carbon trading and greentechnology investment problems are examined in a supplychain setting made up of a retailer and a manufacturer Themanufacturer can invest in green technology to reduce unitcarbon emissions of the product and distribute the product tostrategic customers through the retailer Three key questionsare addressed in the paper

(1) In centralized supply chain setting what is theoptimal production quantity pricing carbon tradingstrategy and green technology investment

(2) In centralized supply chain setting can quantitycommitment improve the supply chain performanceand what effect does the quantity commitment haveon the supply chain strategies

(3) In decentralized supply chain setting what is theoptimal strategies for the retailer and manufacturerrespectively How to coordinate the decentralizedsupply chain to achieve the performance in quantitycommitment scenario

After a review of the literature in Section 2 we presentthe model descriptions and assumptions in Section 3 InSection 4 the centralized supply chain setting models arediscussed We obtain the optimal strategies for the central-ized decision-maker in scenarios of Rational ExpectationsEquilibrium (RE Equilibrium) and quantity commitment(QC) respectively In Section 5 the optimal strategies for themanufacturer and the retailer in decentralized supply chainsetting are discussed We design supply chain coordinationmechanism in Section 6 In Section 7 we present conclusionsof our findings and highlight possible future work

2 Literature Review

There are two streams of literature closely related to ourwork literature on low carbon supply chainmanagement and

literature on supply chain business decisions with strategiccustomer behavior

The literature on low carbon supply chain managementcan be further divided into two classes One is the researchon centralized supply chain that is single business decisionsand the other is the research on decentralized supply chaindecisions and coordination As to the literature on singlebusiness decisions a lot of researchers examined the impactof cap and trade policy without green technology investmentsuch as Dobos 2005 [14] Chen et al 2015 [15] Chen andWang 2015 [16] and Chang et al 2015 [17] This streamof literature explores the research framework that examinesthe effect of cap and trade policy on low carbon supplychain decisions but the green technology investment is notincluded Meanwhile a lot of researchers have introducedgreen technology investment into low carbon supply chainmanagement Zhao et al 2010 [18] developed a carbonemissions allowance allocation systemwith green technologyinvestment in electric power market Yalabik and Fairchild2011 [19] investigated the manufacturerrsquos investment deci-sion on environmentally friendly product with customerbehavior and government regulation They showed that themanufacturer had an incentive to carry out green technologyinvestments to reduce carbon emissions when the customerdemand was emissions-sensitive Toptal et al 2014 [20]investigated the joint decisions of procurement and greentechnology investment under carbon tax and cap and tradepolicy respectively They also discussed the effect of differentcarbon policies on the decision of green technology invest-ment In Manikas and Kroes 2015 [21] a forward buyingheuristic is designed for those firmswho get carbon emissionsby auctions As to the literature on decentralized supplychain decisions and coordination without green technologyinvestment Benjaafar et al 2013 [22] introduced carbonemissions into the simple supply chain system and inves-tigated the procurement production inventory and greentechnology investment decisions with cap and trade policyXu et al 2015 [23] studied the production and pricingproblem of a MTO supply chain consisting of a two-productmanufacturer and a retailerThemanufacturer is constrainedby cap and trade policy and determines the wholesale priceof the two products Zhen et al 2015 [24] examined theretailerrsquos optimal pricing ordering and transportation modestrategies with cap and trade policy They showed thatthe retailer preferred low carbon transportation mode withcap and trade policy under certain conditions There are afew literatures on decentralized supply chain decisions andcoordination with green technology investment Swami andShah 2013 [25] considered the customerrsquos environmentalawareness that is the green technology investments of firmsin the supply chain would affect the customer demand anddesign a coordination mechanism for the supply chain basedon costs sharing contract

As to the literature on supply chain business deci-sions with strategic customer behavior Coase 1972 [26]first paid attention to strategic customer behavior in eco-nomics He found that when the monopoly company facedstrategic customer the monopoly company would set themargin cost as the price and earn zero profit We review

Discrete Dynamics in Nature and Society 3

this stream literature from two aspects of centralized anddecentralized supply chain decisions with strategic customerbehavior Su 2007 [27] investigated the dynamic pricingof a monopolist selling a finite inventory over a finite timeperiodThe demand was endogenous and intertemporalThecustomers were heterogeneous and strategic The researchshows that the heterogeneous valuation and patience ofcustomers determine the optimal pricing policies structureZhang and Cooper 2008 [28] considered a firm that sellsa single product over two periods and examined the effectsof strategic customer behavior on the firmrsquos rationing andpricing decisionsTheir research shows that when prices werefixed in advance rationing could improve revenue Whang2015 [29] considered heterogeneous strategic customers withdifferent reservation value and studied the effect of demanduncertainly on the retailerrsquos markdown policy Du et al 2015[30] taking strategic customers into consideration with riskpreference and decreasing value studied the joint stock andpricing decision problem Compared with classical newsboymodels the ordering quantity and total profit got lower whenstrategic customer behavior was taken into considerationAs to the effects of strategic customer behavior on thedecentralized supply chain performance and coordinationstrategy Su and Zhang 2008 [31] examined the effect ofstrategic customer behavior on the performance of supplychain and the valve of commitment They characterizedRational Expected Equilibrium between strategic customersand the retailer They showed that quantity commitmentcould improve the sellerrsquos profit and decentralization couldimprove the supply chain performance Yang 2012 [32]studied the effect of competition and discounting on theperformance of decentralized supply chain with strategiccustomer behavior It showed that the performance of a cen-tralized supply chain was lower than that of a decentralizedsupply chain when strategic customers existed It showed thatretailer competition and the firm and customer discountingwere also driving factors of higher decentralized supply chainperformance except for double marginalization effect Yanget al 2015 [33] addressed the effect of quick response onthe performance of supply chain when strategic customerbehavior was taken into consideration They compared thequick response value in different supply chain structuresThey showed that a decentralized supply chain with revenuesharing contracts could achieve the performance of a central-ized supply chain but allocating the profit arbitrary in supplychain members could not be realized here

Nevertheless despite the increased attention on lowcarbon supply chainmanagement in operationsmanagementliterature very few studies have been carried by consideringgreen technology investments and cap and trade policysimultaneously To the best of our knowledge there are nopublished works that introduce strategic customer behaviorinto low carbon supply chain framework to investigate theoptimal production pricing carbon trading and green tech-nology investment decisions One distinction of our modelis examining the optimal strategies of low carbon supplychain by considering cap and trade policy green technologyinvestment and strategic customer behavior simultaneouslyAnother distinction is systematic consideration of optimizing

low carbon strategies in centralized and decentralized sce-narios The contributions of our work are as follows firstwe obtain the optimal production pricing carbon tradingand green technology investment strategies for companies atall levels of low carbon supply chain in different scenariosSecond we analyze the impact of quantity commitment onthe operation strategies of a centralized supply chain andits performance Third coordination mechanisms for lowcarbon supply chain with strategic customer behavior aredesigned based on revenue sharing-cost sharing contract

3 Model Descriptions and Assumptions

We examine a two-echelon low carbon supply chain madeup of a low carbon manufacturer and a retailer The manu-facturer produces a product and distributes it to customersthrough the retailer We divide the wholesales period of theretailer into two phases The retailer sells the product at fullprice in phase one and at salvage price in phase two If theproduct is sold out in phase one phase two does not existThe customers are homogeneous strategic customers whowill take into account the possibility of buying the productat salvage price to choose buying the product at full price orwait to purchase the product at salvage price to maximize theexpected surplus Each customer purchases one product atmost

Variables and parameters for model developmentadopted in this study are denoted as in Notations section

Because the parameters must meet certain conditions tomake sense we assume the following

(1) 119901 le 119903 Only when the retail price is no more thanthe customer reservation price the customer maypurchase the product at full price

(2) V gt 119901 gt 119908 gt 119888 gt 119904 gt 0 This conditionensures that there is a positive profit margin for themanufacturer retailer and customers when a productis sold to customers In addition the production costis greater than the salvage price which indicates thatthe retailer will lose money when the product fails tosell at full price This prompts the retailer to orderthe products according to the customersrsquo demandbecause the excess inventories generate losses

(3) 119908 gt 119888+119896 This condition states that the manufactureris willing to produce products with cap and tradepolicy Otherwise the manufacturer will not produceproducts but sell carbon emissions to earn profits

(4) The shortage cost of the manufacturer and operatingcost of the retailer are not taken into consideration

(5) We assume that the manufacturer and the retailer areall rational and self-interested that is both of themaim tomaximize their profitWe also assume that theyare risk neutral

(6) The carbon emissions in production is the mainsource of the total carbon emissions So we onlyconsider the carbon emissions of production

4 Discrete Dynamics in Nature and Society

(7) The green technology investment is the function of120591 denoted by 119868(120578) We assume that 119868(120578) ge 01198681015840(120578) gt 0 and 119868

10158401015840(120578) gt 0 This is consistent with

the practice The conditions show that the margingreen technology investment is increasing in carbonemissions reduction rate Referring to drsquoAspremontand Jacquemin 1988 [34] we set 119868(120578) = (12)119905120578

2where 119905 represents the efficiency of green technologyinvestments

Variables and parameters related to low carbon include119890 (a decision variable which represents the carbon emissionstrading policy) 120578 (a decision variable which represents thegreen technology investment policy) and 119864 and 119896 (representthe initial carbon emissions and unit price of carbon emis-sions trading resp)

4 Centralized Supply Chain Model

In this section we assume that themanufacturer is authorizedto determine the production pricing carbon trading andgreen technology investment strategies under cap and tradepolicy to maximize the profit of the centralized supply chainAt the beginning the government allocates a certain amountof free carbon emissions to the manufacturer During theproducing process if the carbon emission is limited themanufacturer should buy extra carbon emissions from theexternalmarket Otherwise if the carbon emission is enoughthe manufacturer could sell extra carbon emissions to gainrevenue At the end of the period carbon emissions of themanufacturer must not exceed the carbon emission rights itholds

The sequence of events in this part is as follows first themanufacturer forms the belief of customersrsquo reservation price120585119903and then decides the retail selling price product quantity

carbon trading volume and green technology investmentsecond the customers form the beliefs 120585prob of probabilityof the product sold at salvage price 119904 according to theinformation of market price and then form the reservationprice 119903 third the customersrsquo demand is satisfied and theproducts are sold at full price119901 finally all remaining productsare sold to the external market at salvage price 119904

41 Rational Expectations Equilibrium Scenario We charac-terize the RE Equilibrium between the manufacturer and thestrategic customers Muth 1961 [35] first proposed rationalexpectations hypothesis which refers to the situation thatthere is no systematic bias between the actual economic resultand peoplersquos expectations Then Su and Zhang 2008 [31]introduced it into operations management to analyze thedecision problems of the enterprises when strategic customerbehavior is taken into account Since then the rationalexpectations hypothesis has been adopted by scholars all overthe world [13 27 30]

First we examine the decision problem of strategiccustomers The customers choose to buy the product imme-diately at price 119901 or wait for markdown to maximize theirexpected surplus The customer surplus is V minus 119901 when thecustomer buys the product at full price and (V minus 119904)120585prob at

salvage price Therefore the maximum expected surplus ofthe customer is max(V minus 119904)120585prob V minus 119901 If and only if (V minus119904)120585prob le Vminus119901 the customer will buy the product at full priceSo given 120585prob we can obtain the customerrsquos reservation price119903(120585prob) = V minus (V minus 119904)120585prob

Then we examine the decision problem of the manufac-turer The profit function of the manufacturer with cap andtrade policy and green technology investment denoted by120587(119902 119901 119890 120578) is

120587 (119902 119901 119890 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909) minus (119888 minus 119904) 119902

minus 119896119890 minus1

21199051205782

(1)

First two items represent the manufacturerrsquos expectedprofit without cap and trade policy and green technologyinvestmentThis is the same with the newsvendormodelThethird item represents the carbon emissions trading costprofitof the manufacturer The forth item represents the cost ofmanufacturer investing in green technology

The carbon trading volume after green technology invest-ment under cap and trade policy is

119890 = (1 minus 120578) 119902 minus 119864 (2)

First term represents the manufacturerrsquos carbon emis-sions in production after green technology investment thesecond term represents initial free carbon emission owned bythe manufacturer

Then 120587(119902 119901 119890 120578) can be transformed into

120587 (119902 119901 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + 119896 (1 minus 120578)) 119902 + 119896119864 minus1

21199051205782

(3)

The beliefs of the manufacturer over the customerrsquosreservation price are 120585

119903 Obviously the manufacturer will

set 119901 = 120585119903 119902 and 120578 are the maximizer max

119902120578120587(119902 119901 120578) when

given 119901 According to the definition of RE Equilibrium theRE Equilibrium solution (119901 119902 120578 119903 120585

119903 120585prob)must meet

119903 = V minus (V minus 119904) 120585prob (4)

119901 = 120585119903 (5)

(119902 120578) = argmax119902120578

120587 (119902 119901 120578) (6)

120585prob = 119865 (119902) (7)

120585119903= 119903 (8)

Conditions (4) (5) and (6) indicate that the manufac-turer and customers will choose the action to maximizetheir own utility Conditions (7) and (8) can ensure that thesolution meets rational expectations hypothesis that is theactual situation of economic operation in line with peoplersquosexpectations

Discrete Dynamics in Nature and Society 5

The Rational Expectations Equilibrium makes

119901 = V minus (V minus 119904) 119865 (119902) (9)

The production pricing and green technology invest-ment decision model of the manufacturer with cap and tradepolicy and green technology investment is

max119902120578

120587 (119902 119901 120578)

st (1 minus 120578) 119902 le 119864

(10)

Defining 1205791(119902) = (1(1 minus 120578))[(119901 minus 119904)119865(119902) minus (119888 minus 119904)]

1205792(120578) = 119905120578119902 120579

1(119902) represents themargin profit of unit carbon

emission that is the profit gained by the manufacturerwith one unit carbon emission input in production 120579

2(120578)

represents the margin cost of unit carbon emission that isthe cost that the manufacturer invests in green technology toget one unit carbon emission reduction

Lemma 1 Given 119901 and (V minus 119904)119891(119902)119865(119902)119905 minus 1198962 gt 0 120587(119902 119901 120578)is a joint concave function of 120578 and 119902 in RE Equilibrium

Proof Given 119901 according to formula (3) 120597120587(119902 119901 120578)120597119902 =

(119901minus119904)119865(119902)minus(119888minus119904)minus(1minus120578)119896 1205972120587(119902 119901 120578)1205971199022 = minus(119901minus119904)119891(119902) lt0 120597120587(119902 119901 120578)120597120578 = 119902119896 minus 119905120578 1205972120587(119902 119901 120578)1205971205782 = minus119905 lt 0 and1205972120587(119902 119901 120578)120597119902120597120578 = 120597

2120587(119902 119901 120578)120597120578120597119902 = 119896

When 119905(V minus 119904)119891(119902)119865(119902) minus 1198962 gt 0 under RE Equilibriumwe get 119905(119901minus119904)119891(119902)minus1198962 gt 0 then we have 1205972120587(119902 119901 120578)120597119902120597120578 =1205972120587(119902 119901 120578)120597120578120597119902 = 119896 and

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205972120587 (119902 119901 120578)

1205971199022

1205972120587 (119902 119901 120578)

120597119902120597120578

1205972120587 (119902 119901 120578)

120597120578120597119902

1205972120587 (119902 119901 120578)

1205971205782

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 119905 (119901 minus 119904) 119891 (119902) minus 1198962

gt 0

(11)

The Hessian Matrix of the problem is negative definite andthen we can prove that 120587(119902 119901 120578) is joint concave function of119902 and 120578 when given 119901 and (V minus 119904)119891(119902)119865(119902)119905 minus 119896

2gt 0 This

completes the proof

As to the manufacturerrsquos optimal production quantity(denoted by 119902

lowast) pricing (denoted by 119901lowast) carbon trading

(denoted by 119890lowast) and green technology investment (denotedby 120578lowast) the following proposition is obtained

Proposition 2 If (Vminus119904)119891(119902)119865(119902)119905minus1198962 gt 0 themanufacturerrsquosoptimal production quantity (119902lowast) pricing (119901lowast) carbon trading(119890lowast) and green technology investment (120578lowast) strategies satisfy

1205791(119902lowast) = 1205792(120578lowast) = 119896

119901lowast= 119904 + (V minus 119904) 119865 (119902lowast)

119890lowast= (1 minus 120578

lowast) 119902lowastminus 119864

0 lt 120578lowastlt 1

(12)

Proof According to Lemma 1 if (Vminus 119904)119891(119902)119865(119902)119905 minus 1198962 gt 0 let120597120587(119902 119901 120578)120597119902 = 0 120597120587(119902 119901 120578)120597120578 = 0 and combine (9) and(2) we get

(119901 minus 119904) 119865 (119902) minus (119888 minus 119904) minus (1 minus 120578) 119896 = 0

119902119896 minus 119905120578 = 0

119901 = V minus (V minus 119904) 119865 (119902)

119890 = (1 minus 120578) 119902 minus 119864

0 lt 120578 lt 1

(13)

Solving the equations above can derive the optimalproduction pricing carbon trading and green technologyinvestment strategies of themanufacturerThis completes theproof

Proposition 2 shows that considering strategic customerbehavior the manufacturerrsquos optimal production quantitypricing carbon trading volume and green technology invest-ment strategywith cap and trade policy and green technologyinvestment exist and are unique The inherent implicationof the proposition is very intuitive 120579

1(119902) represents the

margin profit of unit carbon emission 1205792(120578) represents the

margin cost of unit carbon emission and 119896 is the unit carbonemission trading price which can be seen as marginal costor profit by carbon emission trading The optimal strategiesof the manufacturer are that the margin profit of unitcarbon emission is equal to the margin cost of unit carbonemission The marginal cost of getting unit carbon emissionfrom different way (green technology investment or buying)is equal The marginal profit of unit carbon emission fordifferent uses (production or sale) is equal

Substitute 119902lowast 119901lowast and 120578lowast into (3) we can obtain the max-imum expected profit of themanufacturer with cap and tradepolicy and green technology investment 120587(119902lowast 119901lowast 120578lowast) =

(119901lowastminus119904)(119902lowastminusint119902lowast

0119865(119909)119889119909)minus(119888minus119904+119896(1minus120578

lowast))119902lowast+119896119864minus(12)119905120578

lowast2

42 Quantity Commitment Scenario In traditional literatureon strategic customer behavior quantity commitment canimprove the manufacturerrsquos expected profit Quantity com-mitment refers to the action in which the manufacturerpromises customers that only a certain number of productsis produced and sold In low carbon supply chain settingcan the maximum expected profit be improved by quantitycommitment We attempt to answer the question in thispaper

We assume that the manufacturer can convince cus-tomers by appropriate means that they only can obtain 119902

units of the product in the wholesales period At this timethe strategic customers no longer need to anticipate theprobability of getting the product at salvage price When 119902

is given the probability of being able to get the product atprice 119904 is 119865(119902) The reservation price is 119901(119902) = Vminus (Vminus 119904)119865(119902)which also is the optimal pricing of the manufacturer Wehave 119890 = (1 minus 120578)119902 minus 119864 then we can obtain the expected profit

6 Discrete Dynamics in Nature and Society

function of themanufacturer with respect to 119902 and 120578 denotedby 120587119902(119902 120578)

120587119902(119902 120578) = (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + 119896 (1 minus 120578)) 119902 + 119896119864 minus1

21199051205782

(14)

So the manufacturerrsquos optimal production quantity andcarbon reduction rate in QC scenario are (119902

119902lowast 120578119902lowast) =

argmax119902ge00lt120578lt1

120587119902(119902 120578) the optimal price is 119901119902lowast = V minus (V minus

119904)119865(119902119902lowast) and the optimal carbon trading volume is 119890119902lowast =

(1 minus 120578119902lowast)119902119902lowastminus 119864

Lemma 3 Given 119902 the optimal green technology investmentstrategy of the manufacturer denoted by 120578119902lowast satisfies

120578119902lowastequiv 120578 (119902) =

119896119902

119905 0 lt 120578

119902lowastlt 1 (15)

Proof Wehave 120597120587119902(119902 120578)120597120578 = 119902119896minus119905120578 1205972120587119902(119902 120578)1205971205782 = minus119905 lt0 Let 120597120587119902(119902 120578)120597120578 = 0 we get 120578119902lowast equiv 120578(119902) = 119896119902119905 In additionaccording to the assumption in Section 3 we know 0 lt 120578

119902lowastlt

1 This completes the proof

Substituting 120578119902lowast = 120578(119902) into (14)

120587119902(119902) equiv 120587

119902(119902 120578 (119902))

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + 119896) 119902 + 119896119864 +11989621199022

2119905

(16)

Then the two-variable optimization problem is simplifiedas a single variable optimization problem

max119902ge0

120587119902(119902) (17)

Lemma4 If 1198962119905minus(Vminus119904)[3119891(119902)119865(119902)+1198911015840(119902)(119902minusint1199020119865(119909)119889119909)] lt

0 120587119902(119902) is a concave function of 119902

Proof The first-order and second-order derivatives of 120587119902(119902)are

119889120587119902(119902)

119889119902= (V minus 119904) [119865

2

(119902) minus 119891 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) +1198962

119905119902

1198892120587119902(119902)

1198891199022=1198962

119905minus (V minus 119904)

sdot [3119891 (119902) 119865 (119902) + 1198911015840(119902) (119902 minus int

119902

0

119865 (119909) 119889119909)] lt 0

(18)

Then we can obtain that 120587119902(119902) is a concave function of 119902This completes the proof

As to the manufacturerrsquos optimal production quantity inQC scenario (denoted by 119902119902lowast) the following proposition isobtained

Proposition 5 If 1198962119905 minus (V minus 119904)[3119891(119902)119865(119902) + 1198911015840(119902)(119902 minus

int119902

0119865(119909)119889119909)] lt 0 the optimal production quantity in QC

scenario (119902119902lowast) satisfies

(V minus 119904) [1198652

(119902) minus 119891 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) +1198962

119905119902 = 0

(19)

Proof It can be directly derived according to Lemma 4 Thiscompletes the proof

Proposition 5 shows that under certain condition theoptimal production quantity of the manufacturer with capand trade and green technology investment exist and areunique

43 The Effect of Quantity Commitment The effect of QCon the optimal strategies and the maximum expected profitof the manufacturer is analyzed by comparing the optimalstrategies and the maximum expected profit of the manufac-turer in RE Equilibrium scenario and QC scenario

Proposition 6 Consider 119902119902lowast lt 119902lowast 119901119902lowast gt 119901lowast 120578119902lowast lt 120578lowast

Proof In order to ensure that the manufacturerrsquos optimalstrategies in the two scenarios exist the condition 119896

2119905 lt

min(Vminus119904)119891(119902)119865(119902) (Vminus119904)[3119891(119902)119865(119902)+1198911015840(119902)(119902minusint1199020119865(119909)119889119909)]

must be held Then 120587119902(119902) is a concave function of 119902According to Proposition 2 we have 120579

1(119902lowast) = 1205792(120578lowast) = 119896

1205792(120578lowast) = 119896 can be written as 120578lowast = 119896119902

lowast119905 Then we also can

rearrange 1205791(119902lowast) = 119896 to (Vminus119904)1198652(119902lowast)minus(119888minus119904+119896)+(1198962119905)119902lowast = 0

We can obtain (119889120587119902(119902)119889119902)|

119902=119902lowast = (V minus 119904)[119865

2

(119902lowast) minus

119891(119902lowast)(119902lowastminus int119902lowast

0119865(119909)119889119909)] minus (119888 minus 119904 + 119896) + (119896

2119905)119902lowast= minus(V minus

119904)119891(119902lowast)(119902lowastminusint119902lowast

0119865(119909)119889119909) lt 0 Then we get 119902119902lowast lt 119902lowast Because

120578 = 119896119902119905 and 119901 = Vminus(Vminus119904)119865(119902) are held in the two scenarioswe can obtain 119901119902lowast gt 119901

lowast and 120578119902lowast lt 120578lowast according to 119902119902lowast lt 119902

lowastThis completes the proof

Proposition 6 shows that compared with RE Equilibriumscenario the manufacturerrsquos optimal production quantity islower the optimal pricing is higher and the optimal carbonreduction rate is lower in QC scenario

Proposition 7 Consider 120587119902(119902119902lowast 120578119902lowast) gt 120587(119902lowast 119901lowast 120578lowast)

Proof In order to ensure that the manufacturerrsquos optimalstrategies in the two scenarios exist the condition 119896

2119905 lt

min(Vminus119904)119891(119902)119865(119902) (Vminus119904)[3119891(119902)119865(119902)+1198911015840(119902)(119902minusint1199020119865(119909)119889119909)]

must be held Then 120587119902(119902) is a concave function of 119902Substitute 119902lowast 119901lowast and 120578

lowast= 119896119902

lowast119905 into (3) we have

120587(119902lowast 119901lowast 120578lowast) = 120587

119902(119902lowast) We know that 120587119902(119902) is a concave

function of 119902 and (119889120587119902(119902)119889119902)|119902=119902119902lowast = 0 Because of 119902119902lowast lt 119902lowast

Discrete Dynamics in Nature and Society 7

120587119902(119902lowast) lt 120587

119902(119902119902lowast) that is 120587(119902lowast 119901lowast 120578lowast) lt 120587

119902(119902119902lowast 120578119902lowast) This

completes the proof

Proposition 7 shows that QC strategy can improve themanufacturerrsquos maximum expected profit when green tech-nology investment and cap and trade policy are taken intoaccount

5 Decentralized Supply Chain Model

In this section the retailer and the manufacturer make deci-sions independently to maximize their own expected profitThe manufacturer is the Stackelberg leader and determinesthe wholesale price and the green technology investmentunder cap and trade policy The retailer is the follower anddetermines the retail price and the order quantity of theproduct

The sequence of events in this section is as follows firstthe manufacturer determines the optimal wholesale pricecarbon trading and green technology investment strategiessecond the retailer forms the belief of customersrsquo reservationprice and then decides the optimal retail price and orderquantity third the manufacture produces products anddelivers to the retailer fourth the customers form the beliefs120585prob of probability of the product sold at salvage price119904 according to the information of market price and thenform the reservation price 119903 fifth the customersrsquo demand issatisfied and the products are sold at full price 119901 finally allremaining products are sold to the external market at salvageprice 119904

51 The Optimal Strategies First we examine the decisionproblem of the retailer Recall that 119908 is the manufacturerrsquoswholesale price The retailerrsquos expected profit denoted by120587119903(119902 119901) is

120587119903(119902 119901) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909) minus (119908 minus 119904) 119902 (20)

Lemma 8 When 119901 is given 120587119903(119902 119901) is a concave function of119902

Proof When 119901 is given according to (20) we have120597120587119903(119902 119901)120597119902 = (119901 minus 119904)119865(119902) minus (119908 minus 119904) and 1205972120587119903(119902 119901)1205971199022 =

minus(119901 minus 119904)119891(119902) lt 0 Then we know that 120587119903(119902 119901) is a concavefunction of 119902 when 119901 is given This completes the proof

As to the retailerrsquos optimal strategies in RE Equilibriumwe can obtain the following proposition

Proposition 9 When the wholesale price is given the retailerrsquosoptimal order strategies (119902119903lowast) and optimal pricing strategies(119901119903lowast) with strategic customer behavior are

119902119903lowast= 119865minus1

(radic119908 minus 119904

V minus 119904) (21)

119901119903lowast= 119904 + radic(119908 minus 119904) (V minus 119904) (22)

Proof In RE Equilibrium (9) still holds Let 120597120587119903(119902 119901)120597119902 = 0and combining with (9) we have

119901 = V minus (V minus 119904) 119865 (119902)

(119901 minus 119904) 119865 (119902) minus (119908 minus 119904) = 0

(23)

According to Lemma 8 we can obtain the optimal orderand pricing strategies of the retailer by solving the equationsabove This completes the proof

Proposition 9 shows that the retailerrsquos optimal order andpricing strategies in REEquilibriumwith cap and trade policyexist and are unique

Second we investigate the manufacturerrsquos decision prob-lem The expected profit function of the manufacturer withgreen technology investment and cap and trade policy is120587119898(119908 119890 120578) = (119908 minus 119888)119902 minus 119896119890 minus (12)119905120578

2According to (21) with green technology investment and

cap and trade policy in wholesale price contract the optimalorder quantity of the retailer 119902119903lowast and the optimal wholesaleprice of the manufacturer 119908119898lowast is a one-to-one relationship

119908 = 119904 + (V minus 119904) 1198652

(119902) (24)

Combined with 119890 = (1 minus 120578)119902 minus 119864 the manufacturerrsquosexpected profit function can be converted into

120587119898(119902 120578) = [(V minus 119904) 119865

2

(119902) minus (119888 minus 119904 + (1 minus 120578) 119896)] 119902

+ 119896119864 minus1

21199051205782

(25)

When using wholesale price contract the total profit ofthe supply chain denoted by 120587sc

(119902 119901 119890 120578) is

120587sc(119902 119901 119890 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909) minus (119888 minus 119904) 119902

minus 119896119890 minus1

21199051205782

(26)

According to (9) and 119890 = (1 minus 120578)119902 minus 119864 the total supplychain profit function in RE Equilibrium can be simplified as

120587sc(119902 120578) = (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578) 119896) 119902 + 119896119864 minus1

21199051205782

(27)

Lemma 10 Given 119902 the optimal green technology investmentstrategy of the manufacturer denoted by 120578119898lowast satisfies

120578119898lowast

equiv 120578 (119902) =119896119902

1199050 lt 120578119898lowast

lt 1 (28)

Proof The first-order and second-order partial derivatives of120587119898(119902 120578) with respect to 120578 are 120597120587119898(119902 120578)120597120578 = 119902119896 minus 119905120578 and

1205972120587119898(119902 120578)120597120578

2= minus119905 lt 0 Let 120597120587119898(119902 120578)120597120578 = 0 we get

120578119898lowast

equiv 120578(119902) = 119896119902119905 In addition according to the assumptionin Section 3 we know 0 lt 120578

119898lowastlt 1 This completes the

proof

8 Discrete Dynamics in Nature and Society

Substituting 120578119898lowast = 120578(119902) into (25)

120587119898(119902) equiv 120587

119898(119902 120578 (119902))

= [(V minus 119904) 1198652

(119902) minus (119888 minus 119904 + 119896)] 119902 + 119896119864

+11989621199022

2119905

(29)

Then the two-variable optimization problem is simplifiedas a single variable optimization problem

max119902ge0

120587119898(119902) (30)

Lemma 11 If 1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus

1199021198912(119902)] lt 0 120587119898(119902) is a concave function of 119902

Proof The first-order and second-order derivatives of 120587119898(119902)with respect to 119902 are 119889120587

119898(119902)119889119902 = (V minus 119904)[119865

2

(119902) minus

2119902119891(119902)119865(119902)] minus (119888 minus 119904 + 119896) + (1198962119905)119902 and 119889

2120587119898(119902)119889119902

2=

1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus 1199021198912(119902)] lt 0 Then

we have that 120587119898(119902) is a concave function of 119902 This completesthe proof

In decentralized supply chain as to the manufacturerrsquosoptimal wholesale price (denoted by 119908

119898lowast) the followingproposition is obtained

Proposition 12 If 1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus1199021198912(119902)] lt 0 the manufacturerrsquos optimal wholesale price in

decentralized supply chain (119908119898lowast) satisfies 119908119898lowast = 119904 + (V minus119904)1198652

(119902119903lowast) where 119902119903lowast represents the retailerrsquos optimal order

quantity and satisfies

(V minus 119904) [1198652

(119902) minus 2119902119891 (119902) 119865 (119902)] minus (119888 minus 119904 + 119896) +1198962

119905119902

= 0

(31)

Proof According to Lemma 11 we can obtain that 119902119903lowastwhichmaximizes themanufacturerrsquos expected profit satisfies(31) Combining with (24) we know that the manufacturerrsquosoptimal wholesale price satisfies 119908119898lowast = 119904 + (V minus 119904)119865

2

(119902119903lowast) at

this time This completes the proof

Proposition 12 shows that the optimal commitment quan-tity of the manufacturer in decentralized supply chain withgreen technology investment and cap and trade policy existand are unique The optimal wholesale price does not relateto the cap allocated by the government but relates to thecarbon trading price This is because the production of themanufacturer is not affected by the cap but is affected bythe margin cost of the product However the margin cost ofthe product varies with the change of carbon trading pricewhen carbon trading is allowed The change results in thatthe optimal commitment quantity is not related to the cap butrelate to the carbon trading price

Substitute 119908119898lowast into (22) we derive the retailerrsquos optimalpricing in decentralized situation119901119903lowast = 119904+radic(119908119898lowast minus 119904)(V minus 119904)

According to Lemma 10 we get the optimal green technologyinvestment of the manufacturer 120578119898lowast = (119896119905)119902119903lowast

Substitute 119902119903lowast 119901119903lowast 119908119898lowast and 120578119898lowast into 120587

119903(119902 119901) 120587119898(119902)

and 120587sc(119902 120578) we obtain the maximum expected profit of

the retailer the manufacturer and the supply chain indecentralized channel

120587119903(119902119903lowast 119901119903lowast) = (119901

119903lowastminus 119904) (119902

119903lowastminus int

119902119903lowast

0

119865 (119909) 119889119909)

minus (119908119898lowast

minus 119904) 119902119903lowast

120587119898(119902119903lowast) = [(V minus 119904) 119865

2

(119902119903lowast) minus (119888 minus 119904 + 119896)] 119902

119903lowast

+ 119896119864 +1198962119902119903lowast2

2119905

120587sc(119902119903lowast 120578119898lowast) = (V minus 119904) 119865 (119902119903lowast) (119902119903lowast minus int

119902119903lowast

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578119898lowast) 119896) 119902119903lowast+ 119896119864

minus1

2119905120578119898lowast2

(32)

52 The Effect of Decentralization We examine the impact ofdecentralization on the supply chain optimal strategies andmaximumexpected profitwhen green technology investmentis taken into consideration by comparing the optimal strate-gies and the maximum expected profit in centralized anddecentralized channel with green technology investment andcap and trade policy

As the effect of decentralization of supply chain we canobtain the following proposition

Proposition 13 Consider 119902119903lowast lt 119902119902lowast

lt 119902lowast 119901119903lowast gt 119901

119902lowastgt 119901lowast

120578119898lowast

lt 120578119902lowastlt 120578lowast

Proof In Proposition 6 119902119902lowast lt 119902lowast 119901119902lowast gt 119901

lowast and 120578119902lowast lt 120578lowast

have been provedTherefore we only need to prove 119902119903lowast lt 119902119902lowast119901119903lowastgt 119901119902lowast and 120578119898lowast lt 120578119902lowast

Define 119867(119902) = 120587119902(119902) minus 120587119898(119902) then

119867(119902)

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

minus (V minus 119904) 1199021198652

(119902)

1198671015840(119902)

= (V minus 119904) 119891 (119902) [2119902119865 (119902) minus (119902 minus int119902

0

119865 (119909) 119889119909)]

(33)

Define 119866(119902) = 2119902119865(119902) minus (119902 minus int119902

0119865(119909)119889119909) then 1198661015840(119902) =

119865(119902) minus 2119902119891(119902) Because 119866(0) = 0 1198661015840(0) = 1 and 119865 satisfiesIFR we know that the value of 119866(119902) starts from 0 increasingfirst and then decreasing in 119902 That is 119866(119902) = 0 has uniquesolution and we denote it as So when 119902 lt 119866(119902) gt 0 that

Discrete Dynamics in Nature and Society 9

is119867(119902) is increasing in 119902 when 119902 gt 119866(119902) lt 0 that is119867(119902)is decreasing in 119902 Then we have that119867(119902) is a quasi-concavefunction of 119902 and 1198661015840 () = 119865() minus 2119891() lt 0 According tothe analysis above we know that1198671015840() = 0 that is 2119865() =( minus int

0119865(119909)119889119909) Substitute into 119889120587119902(119902)119889119902 we have

119889120587119902(119902)

119889119902

100381610038161003816100381610038161003816100381610038161003816119902=

= (V minus 119904) [1198652

() minus 119891 () ( minus int

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) +1198962

119905

= (V minus 119904) 119865 () [119865 () minus 2119891 ()] minus (119888 minus 119904 + 119896)

+1198962

119905lt (V minus 119904) 119865 () [119865 () minus 2119891 ()] lt 0

(34)

According to Lemma 4 we have that 120587119902(119902) is a concavefunction of 119902 so we get 119902119902lowast lt Because 2119865() = ( minus

int

0119865(119909)119889119909) and119867(119902) is a quasi-concave function of 119902 thenwe

get that 119889120587119898(119902)119889119902 lt 119889120587119902(119902)119889119902 if 119902 lt and 119889120587119898(119902)119889119902 gt

119889120587119902(119902)119889119902 if 119902 gt Therefore the only possible orderings for

119902119902lowast 119902119903lowast and are lt 119902

119902lowastlt 119902119903lowast and 119902119903lowast lt 119902

119902lowastlt We

have proved 119902119902lowast lt so we have 119902119903lowast lt 119902119902lowast We get 119901119903lowast gt 119901119902lowastbecause 119901 and 119902 satisfy 119901 = V minus (V minus 119904)119865(119902) in two situationsWe get 120578119898lowast lt 120578

119902lowast because 120578 and 119902 satisfy 120578 = 119896119902119905 in twosituations This completes the proof

Proposition 13 shows that for the decision of a decen-tralized supply chain with green technology investment theoptimal production quantity is increasing the optimal priceis decreasing and the green technology investment (iecarbon reduction rate) is increasing respectively in the threesituations of decentralized supply chain QC scenario and REEquilibrium scenario

The reason is that themanufacturer prefers higher whole-sale price (ie lower production and less green technologyinvestment) to guarantee its ownprofitmaximization thoughthis may harm the retailer even the whole supply chain

In Proposition 7 we have found 120587119902(119902119902lowast 120578119902lowast) gt

120587(119902lowast 119901lowast 120578lowast) that is the manufacturerrsquos maximum profit in

QC scenario is always greater than that in RE EquilibriumTherefore the section analyzes the effect of decentralizationon supply chain performance based on the QC scenario toprovide benchmark for designing of subsequent coordinationcontract

As to the effect of decentralization on themaximumprofitof supply chain we have the following proposition

Proposition 14 Consider 120587sc(119902119903lowast 120578119898lowast) lt 120587119902(119902119902lowast 120578119902lowast)

Proof The optimal green technology investment in decen-tralized supply chain and QC scenario are 120578(119902) = 119896119902119905

according to Lemmas 3 and 10 So the profit function in thetwo situations above can be converted into a same singlevariable function with respect to 119902 that is (16)

According to Lemma 4 we know that 120587119902(119902) is a concavefunction and reaches the maximum value at 119902119902lowast Accordingto Proposition 13 we have 119902119903lowast lt 119902

119902lowast then we get 120587119902(119902119903lowast) lt120587119902(119902119902lowast) that is 120587sc

(119902119903lowast 120578119898lowast) lt 120587

119902(119902119902lowast 120578119902lowast) This completes

the proof

Proposition 14 shows that the maximum profit of decen-tralized supply chain is always lower than that in QC scenariowhich indicates that the profit of the decentralized supplychain can get higher This is the benchmark of supply chaincoordination

6 Supply Chain Coordination

This section coordinates the supply chain with green tech-nology investment and cap and trade policy based on theQC scenario to improve the profit of the decentralized supplychain

61 Coordination Contract Based on Revenue and CostsSharing Contract With revenue sharing contract we assumethat 120601 (0 le 120601 le 1) represents the ratio of revenue kept by theretailer and (1 minus 120601) represents the ratio of revenue deliveredto the manufacturer

The retailerrsquos expected profit function with revenue shar-ing contract denoted by 120587119903

119904(119902 119901 120601) is

120587119903

119904(119902 119901 120601) = 120601 (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119908 minus 120601119904) 119902

(35)

Themanufacturerrsquos expected profit functionwith revenuesharing contract denoted by 120587119898

119904(119908 119890 120578 120601) is

120587119898

119904(119908 119890 120578 120601) = (119908 minus 119888) 119902 + (1 minus 120601)

sdot [int

119902

0

[119901119909 + 119904 (119902 minus 119909)] 119891 (119909) 119889119909

+ int

infin

119902

119901119902119891 (119909) 119889119909] minus 119896119890 minus1

21199051205782

(36)

Substitute 119890 = (1 minus 120578)119902 minus 119864 into the equation above andwe have

120587119898

119904(119908 120578 120601) = (1 minus 120601) (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

+ (119908 minus 119888 minus (1 minus 120578) 119896 + (1 minus 120601) 119904) 119902

+ 119896119864 minus1

21199051205782

(37)

The total expected profit of the whole supply chaindenoted by 120587sc

119904(119902 119901 120578) is

120587sc119904(119902 119901 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578) 119896) 119902 + 119896119864 minus1

21199051205782

(38)

10 Discrete Dynamics in Nature and Society

The subscript 119904 represents the situation of supply chaincoordination using revenue sharing contracts

As to the retailerrsquos optimal order strategy (denoted by119902lowast

119904) and pricing strategy (denoted by 119901lowast

119904) of the supply chain

with revenue sharing contract the following proposition isobtained

Proposition 15 Under revenue sharing contracts the retailerrsquosoptimal order strategy (119902lowast

119904) and optimal pricing strategy (119901lowast

119904)

are

119902lowast

119904= 119865minus1

(radic119908 minus 120601119904

120601 (V minus 119904))

119901lowast

119904= 119904 + radic

(119908 minus 120601119904) ((V minus 119904))120601

(39)

Proof According to the definition of RE Equilibrium wealso have 119901 = V minus (V minus 119904)119865(119902) under revenue sharingcontracts

The first-order and second-order derivative of 120587119903119904(119902 119901 120601)

with respect to 119902 are 120597120587119903

119904(119902 119901 120601)120597119902 = 120601(119901 minus 119904)119865(119902) minus

(119908 minus 120601119904) and 1205972120587119903

119904(119902 119901 120601)120597119902

2= minus120601(119901 minus 119904)119891(119902) lt

0 So given 119901 the retailerrsquos expected profit function is aconcave function of 119902 Let 120597120587119903

119904(119902 119901 120601)120597119902 = 0 and combine

with 119901 = V minus (V minus 119904)119865(119902) we can solve the retailerrsquos optimalstrategies under revenue sharing contractThis completes theproof

Proposition 15 shows that the optimal order and pricingstrategies of the retailer based on revenue sharing contract inRE Equilibrium exist and are uniqueThis is also the retailerrsquosoptimal response function with respect to 119908

Substituting (39) into (43)

120587119898

119904(119902 120578 120601)

= (1 minus 120601) (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ [120601 (V minus 119904) 1198652

(119902) minus (119888 minus 119904 + (1 minus 120578) 119896)] 119902 + 119896119864

minus1

21199051205782

120597120587119898

119904(119902 120578 120601)

120597120578= 119902119896 minus 119905120578

1205972120587119898

119904(119902 120578 120601)

1205971205782= minus119905 lt 0

(40)

Given 119902 and 120601 the manufacturerrsquos green technologyinvestment strategies with revenue sharing contracts exist

and are unique Let 120597120587119898119904(119902 120578 120601)120597120578 = 0 we get 120578119898lowast

119904equiv 120578(119902) =

119902119896119905 and substitute it into 120587119898119904(119902 120578 120601)

120587119898

119904(119902 120601)

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ 120601 (V minus 119904) 119865 (119902) [119902119865 (119902) minus (119902 minus int119902

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) 119902 + 119896119864 +11989621199022

2119905

(41)

Proposition 16 The supply chain with green technologyinvestment and cap and trade policy can not be coordinated byrevenue sharing contract

Proof The derivative of 120587119898119904(119902 120601) with respect to 119902 can be

arranged as follows

120597120587119898

119904(119902 120601)

120597119902=120597120587119902(119902)

120597119902+ 120601(

120597120587119898(119902)

120597119902minus120597120587119902(119902)

120597119902) (42)

Only when 120601 = 0 are the optimal strategies of supplychain under revenue sharing contracts equal to that in QCscenario Let 119908lowast

119904represent the optimal wholesale price of

the manufacture and 119902lowast119904represent the retailerrsquos optimal order

quantity Then 120587119903

119904(119902 119901 120601) = minus119908

lowast

119904119902lowast

119904lt 0 So the revenue

sharing contract cannot realize the supply chain coordinationat this time This completes the proof

Proposition 16 shows that revenue sharing contractscan not realize the supply chain coordination with greentechnology investment and cap and trade policy The rea-son is that the manufacturer undertakes all the costs ofgreen technology investment when using revenue sharingcontracts thus making the manufacturer tend to invest lessin green technology Only when the manufacturer owns allthe revenue will themanufacturerrsquos optimal green technologyinvestment under revenue sharing contracts be equal to thatin QC scenario However the maximum profit of the retaileris negative at this time and it will not participate

62 Coordination Contract Based on Revenue Sharing-CostSharing Contract Proposition 16 states that the supply chainwith green technology investment and cap and trade policycan not be coordinated by revenue sharing contracts becausethe costs of green technology investment are undertaken bythe manufacturer and the game of retailer and customersis characterized by RE Equilibrium So we consider costssharing of green technology investment and carbon tradingand quantity commitment made by the manufacturer basedon revenue sharing contracts Let 120601 (0 lt 120601 le 1) represent theratio of revenue kept by the retailer and let (1minus120601) represent theratio of revenue delivered to the manufacturer 120572 (0 le 120572 le 1)represents the ratio of costs of green technology investmentand carbon trading undertaken by the manufacturer and(1 minus 120572) represents the ratio of costs of green technologyinvestment and carbon trading delivered to the retailer

Discrete Dynamics in Nature and Society 11

The retailerrsquos expected profit function with revenuesharing-cost sharing contract is

120587119903

120572(119902 120601 120572) = 120601 (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119908 minus 120601119904) 119902

minus (1 minus 120572) 119896 [(1 minus 120578) 119902 minus 119864]

minus1

2(1 minus 120572) 119905120578

2

(43)

We have 119890 = (1minus120578)119902minus119864 then themanufacturerrsquos expectedprofit function denoted by 120587119898

120572(119908 120578 120601 120572) is

120587119898

120572(119908 120578 120601 120572)

= (1 minus 120601) (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ (119908 minus 119888 minus 120572 (1 minus 120578) 119896 + (1 minus 120601) 119904) 119902 + 120572119896119864

minus1

21205721199051205782

(44)

The subscript 120572 represents the situation that using rev-enue sharing-cost sharing contract coordinates the supplychain

Proposition 17 If 0 le 120582 le 1 the parameters of revenuesharing-cost sharing (119908 120601 120572) satisfy

119908 = 120582119888 + (1 minus 120578) 119896

120601 = 120582

120572 = 1 minus 120582

(45)

The supply chainwith green technology investment and cap andtrade policy is coordinated

Proof When the parameters of revenue sharing-cost sharingsatisfy (45) the expected profit function of the retailer andthe manufacturer can be written as follows

120587119903

120572(119902 119901 120601 120572) = 120582120587

119902(119902 120578)

120587119898

120572(119908 120578 120601 120572) = (1 minus 120582) 120587

119902(119902 120578)

(46)

The optimal order of decentralized supply chain is equalto that of QC scenario Substitute 119901 = V minus (V minus 119904)119865(119902) into120587sc119888(119902 119901 120578) we get that the profit function of decentralized

supply chain at this time is the same as that of QC scenarioTherefore when the parameters of revenue sharing-costsharing (119908 120601 120572) satisfy (45) the decentralized supply chaincoordination is realizedThemanufacturer earns all the profitif 120582 = 0 and the retailer earns all the profit if 120582 = 1Arbitrary allocation of the supply chain profit between themanufacturer and the retailer is allowed This completes theproof

Proposition 17 shows that revenue sharing-cost sharingcontract can realize the supply chain coordination on condi-tion that the manufacturer makes a quantity commitment to

the customers The optimal production quantity and greentechnology investment in decentralized situation are lowerthan that in QC scenario The quantity commitment of themanufacturer can increase the green technology investmentand production quantity and also reduce the retailer priceThat is the quantity commitment is beneficial for the man-ufacturer the retailer and customers Thus the quantitycommitment is credible to customers now

7 Conclusions and Suggestions for FurtherResearch

This paper investigates the production pricing carbon trad-ing and green technology investment strategies and thecoordination of low carbon supply chain made up of a lowcarbon manufacturer and a retailer The low carbon manu-facturer produces one product under cap and trade policyand performs green technology investment to reduce carbonemissions in production process The retailer orders fromthemanufacturer and distributes the product to homogenouscustomers To the best of our knowledge this is the firststudy on the management of low carbon supply chainswith strategic customer behaviorThis paper provides severalinteresting observations

Observation 1 There are unique optimal production pricingcarbon trading and green technology investment strategiesin centralized (include RE Equilibrium scenario and QCscenario) and decentralized supply chain Therefore byderiving the optimal production pricing carbon trading andgreen technology investment strategies in different situationsthe low carbon manufacturer and the retailer can behaveappropriately based on our findings to maximize their profit

Observation 2 Our findings also show that the centralizedsupply chain can improve its performance by quantity com-mitment However the manufacturerrsquos production and greentechnology investment are reduced while the optimal priceis increased at this time Therefore quantity commitment isbeneficial for the supply chain but is harmful to customers

Observation 3 We also show that the performance of decen-tralized supply chain is lower than that in QC scenario Wefind that the decentralization reduces the manufacturersquos opti-mal production quantity and green technology investmentand increases the retailerrsquos optimal price Besides we find thatthe optimal production quantity is increasing the optimalprice is decreasing and the green technology investment isincreasing respectively in the three situations of decentral-ized supply chain QC scenario andREEquilibrium scenario

Observation 4 We demonstrate that the traditional revenuesharing contracts can not realize the low carbon supply chaincoordination when green technology investment is takenin consideration We also find that the revenue sharing-cost sharing contract realizes the decentralized supply chain

12 Discrete Dynamics in Nature and Society

coordination when the manufacturer makes a quantity com-mitment It is beneficial to the manufacturer the retailerand customers if themanufacturermakes a quantity commit-ment So the quantity commitment is credible to the customerat this time

We assume the customers are homogeneous strategiccustomers But themyopic customers and strategic customersoften coexist and strategic customers also have differenttolerance in real-life Extending this assumption will havea knock-on effect on supply chain decisions So one keyresearch direction is to consider that the customers aremixed customers myopic customers and strategic customersSecond we assume that the supply chain only producesone product While this simplified setting provides someinteresting insights we have to acknowledge that a manufac-turer usually produces two or more productsThese productswill substitute or complement each other So one importantextension of our work is to assume that the manufacturerproduces two or more products which have different unitcarbon emissions in production

Notations

119863 The random demand and119863 ge 0

119891(sdot) Probability density function of119863119865(sdot) Distribution function of119863 We assume

that 119865 satisfy IFR and define 119865 = 1 minus 119865

119888 Unit production cost119904 Unit salvage price of the product which

is an exogenous variableV The customersrsquo utility from consuming

the product119908 Unit wholesale price119901 Unit retail selling price We assume that

it can be observed by the customers119902 The manufacturerrsquos production

quantitythe retailerrsquos order quantityWe assume that it cannot be observedby the customers

119903 The customersrsquo reservation price whichis customersrsquo private information andcannot be observed by the retailer

120585119903 The beliefs of the manufacturer over the

customerrsquo reservation price120585prob The beliefs of customers over their

chances of obtaining the product atphase two

119864 Initial free carbon emission allocated bygovernment

119896 Unit carbon emission trading price119890 Carbon trading volume120578 Carbon emissions reduction rate

Conflict of Interests

The authors declare no conflict of interests

Acknowledgments

This research is partially supported by National Natural Sci-ence Foundation of China (nos 71432003 and 71272128) theProgram forNewCentury Excellent Talents inUniversity (noNCET-12-0087) and Specialized Research Fund for the Doc-toral Program of Higher Education (no 20130185110006)

References

[1] R Alley T Berntsen N L Bindoff et al ldquoSummary forpolicymakersrdquo in Climate Change 2007 The Physical ScienceBasis vol 6 pp 1ndash18 IPCC 2007

[2] X Chen and G Hao ldquoSustainable pricing and productionpolicies for two competing firms with carbon emissions taxrdquoInternational Journal of Production Research vol 53 pp 6408ndash6420 2015

[3] X Chen S Benjaafar and A Elomri ldquoThe carbon-constrainedEOQrdquo Operations Research Letters vol 41 no 2 pp 172ndash1792013

[4] N O Keohane ldquoCap and trade rehabilitated using tradablepermits to control US greenhouse gasesrdquo Review of Environ-mental Economics and Policy vol 3 no 1 pp 42ndash62 2009

[5] B Liu M Holmbom A Segerstedt and W Chen ldquoEffectsof carbon emission regulations on remanufacturing decisionswith limited information of demand distributionrdquo InternationalJournal of Production Research vol 53 no 2 pp 532ndash548 2015

[6] L Zhu and Y Fan ldquoA real options-based CCS investmentevaluation model case study of Chinarsquos power generationsectorrdquo Applied Energy vol 88 no 12 pp 4320ndash4333 2011

[7] XWang and LDu ldquoStudy on carbon capture and storage (CCS)investment decision-making based on real options for Chinarsquoscoal-fired power plantsrdquo Journal of Cleaner Production vol 112part 5 pp 4123ndash4131 2016

[8] W Jiang and X Chen ldquoOptimal strategies for manufacturerwith strategic customer behavior under carbon emissions-sensitive randomdemandrdquo Industrial Management amp Data Sys-tems 2015

[9] Y Levin J McGill and M Nediak ldquoDynamic pricing in thepresence of strategic consumers and oligopolistic competitionrdquoManagement Science vol 55 no 1 pp 32ndash46 2009

[10] A K Parlakturk ldquoThe value of product variety when sellingto strategic consumersrdquo Manufacturing amp Service OperationsManagement vol 14 no 3 pp 371ndash385 2012

[11] X Su and F Zhang ldquoOn the value of commitment andavailability guarantees when selling to strategic consumersrdquoManagement Science vol 55 no 5 pp 713ndash726 2009

[12] W Elmaghraby S A Lippman C S Tang and R Yin ldquoWillmore purchasing options benefit customersrdquo Production andOperations Management vol 18 no 4 pp 381ndash401 2009

[13] W Jiang and X Chen ldquoManufacturersquos production and pricingstrategies with carbon tax policy and strategic customer behav-iorrdquoManagement Science and Engineering vol 9 no 1 pp 30ndash35 2015

[14] I Dobos ldquoThe effects of emission trading on production andinventories in the Arrow-Karlin modelrdquo International Journalof Production Economics vol 93-94 pp 301ndash308 2005

[15] X Chen C K Chan and Y C E Lee ldquoResponsible productionpolicies with substitution and carbon emissions tradingrdquo Jour-nal of Cleaner Production 2015

Discrete Dynamics in Nature and Society 13

[16] X Chen and X Wang ldquoEffects of carbon emission reductionpolicies on transportation mode selections with stochasticdemandrdquo Transportation Research Part E Logistics and Trans-portation Review 2015

[17] X Chang H Xia H Zhu T Fan and H Zhao ldquoProductiondecisions in a hybrid manufacturingndashremanufacturing systemwith carbon cap and trade mechanismrdquo International Journal ofProduction Economics vol 162 pp 160ndash173 2015

[18] J Zhao B F Hobbs and J-S Pang ldquoLong-run equilibriummodeling of emissions allowance allocation systems in electricpowermarketsrdquoOperations Research vol 58 no 3 pp 529ndash5482010

[19] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2 pp519ndash527 2011

[20] A Toptal H Ozlu and D Konur ldquoJoint decisions on inventoryreplenishment and emission reduction investment under dif-ferent emission regulationsrdquo International Journal of ProductionResearch vol 52 no 1 pp 243ndash269 2014

[21] A S Manikas and J R Kroes ldquoA newsvendor approach tocompliance and production under cap and trade emissionsregulationrdquo International Journal of Production Economics vol159 pp 274ndash284 2015

[22] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineering vol10 no 1 pp 99ndash116 2013

[23] X Xu W Zhang P He and X Xu ldquoProduction and pricingproblems in make-to-order supply chain with cap-and-traderegulationrdquo Omega 2015

[24] Y Zhen X Yang and W Jiang ldquoThe impact of transport modeand carbon policy on low-carbon retailerrdquoDiscrete Dynamics inNature and Society vol 2015 Article ID 964305 12 pages 2015

[25] S Swami and J Shah ldquoChannel coordination in green supplychainmanagementrdquo Journal of the Operational Research Societyvol 64 no 3 pp 336ndash351 2013

[26] R H Coase ldquoDurability andmonopolyrdquoThe Journal of Law andEconomics vol 15 no 1 pp 143ndash149 1972

[27] X Su ldquoIntertemporal pricing with strategic customer behaviorrdquoManagement Science vol 53 no 5 pp 726ndash741 2007

[28] D Zhang and W L Cooper ldquoManaging clearance sales in thepresence of strategic customersrdquo Production and OperationsManagement vol 17 no 4 pp 416ndash431 2008

[29] S Whang ldquoDemand uncertainty and the Bayesian effect inmarkdown pricing with strategic customersrdquoManufacturing ampService Operations Management vol 17 no 1 pp 66ndash77 2015

[30] J Du J Zhang and G Hua ldquoPricing and inventory manage-ment in the presence of strategic customers with risk preferenceand decreasing valuerdquo International Journal of Production Eco-nomics vol 164 pp 160ndash166 2015

[31] X Su and F Zhang ldquoStrategic customer behavior commitmentand supply chain performancerdquo Management Science vol 54no 10 pp 1759ndash1773 2008

[32] H Yang ldquoImpact of discounting and competition on benefit ofdecentralization with strategic customersrdquo Operations ResearchLetters vol 40 no 2 pp 123ndash127 2012

[33] D Yang E Qi and Y Li ldquoQuick response and supply chainstructure with strategic consumersrdquo Omega vol 52 pp 1ndash142015

[34] C drsquoAspremont and A Jacquemin ldquoCooperative and nonco-operative R amp D in duopoly with spilloversrdquo The AmericanEconomic Review vol 78 no 5 pp 1133ndash1137 1988

[35] J F Muth ldquoRational expectations and the theory of pricemovementsrdquo Econometrica vol 29 no 3 pp 315ndash335 1961

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

this stream literature from two aspects of centralized anddecentralized supply chain decisions with strategic customerbehavior Su 2007 [27] investigated the dynamic pricingof a monopolist selling a finite inventory over a finite timeperiodThe demand was endogenous and intertemporalThecustomers were heterogeneous and strategic The researchshows that the heterogeneous valuation and patience ofcustomers determine the optimal pricing policies structureZhang and Cooper 2008 [28] considered a firm that sellsa single product over two periods and examined the effectsof strategic customer behavior on the firmrsquos rationing andpricing decisionsTheir research shows that when prices werefixed in advance rationing could improve revenue Whang2015 [29] considered heterogeneous strategic customers withdifferent reservation value and studied the effect of demanduncertainly on the retailerrsquos markdown policy Du et al 2015[30] taking strategic customers into consideration with riskpreference and decreasing value studied the joint stock andpricing decision problem Compared with classical newsboymodels the ordering quantity and total profit got lower whenstrategic customer behavior was taken into considerationAs to the effects of strategic customer behavior on thedecentralized supply chain performance and coordinationstrategy Su and Zhang 2008 [31] examined the effect ofstrategic customer behavior on the performance of supplychain and the valve of commitment They characterizedRational Expected Equilibrium between strategic customersand the retailer They showed that quantity commitmentcould improve the sellerrsquos profit and decentralization couldimprove the supply chain performance Yang 2012 [32]studied the effect of competition and discounting on theperformance of decentralized supply chain with strategiccustomer behavior It showed that the performance of a cen-tralized supply chain was lower than that of a decentralizedsupply chain when strategic customers existed It showed thatretailer competition and the firm and customer discountingwere also driving factors of higher decentralized supply chainperformance except for double marginalization effect Yanget al 2015 [33] addressed the effect of quick response onthe performance of supply chain when strategic customerbehavior was taken into consideration They compared thequick response value in different supply chain structuresThey showed that a decentralized supply chain with revenuesharing contracts could achieve the performance of a central-ized supply chain but allocating the profit arbitrary in supplychain members could not be realized here

Nevertheless despite the increased attention on lowcarbon supply chainmanagement in operationsmanagementliterature very few studies have been carried by consideringgreen technology investments and cap and trade policysimultaneously To the best of our knowledge there are nopublished works that introduce strategic customer behaviorinto low carbon supply chain framework to investigate theoptimal production pricing carbon trading and green tech-nology investment decisions One distinction of our modelis examining the optimal strategies of low carbon supplychain by considering cap and trade policy green technologyinvestment and strategic customer behavior simultaneouslyAnother distinction is systematic consideration of optimizing

low carbon strategies in centralized and decentralized sce-narios The contributions of our work are as follows firstwe obtain the optimal production pricing carbon tradingand green technology investment strategies for companies atall levels of low carbon supply chain in different scenariosSecond we analyze the impact of quantity commitment onthe operation strategies of a centralized supply chain andits performance Third coordination mechanisms for lowcarbon supply chain with strategic customer behavior aredesigned based on revenue sharing-cost sharing contract

3 Model Descriptions and Assumptions

We examine a two-echelon low carbon supply chain madeup of a low carbon manufacturer and a retailer The manu-facturer produces a product and distributes it to customersthrough the retailer We divide the wholesales period of theretailer into two phases The retailer sells the product at fullprice in phase one and at salvage price in phase two If theproduct is sold out in phase one phase two does not existThe customers are homogeneous strategic customers whowill take into account the possibility of buying the productat salvage price to choose buying the product at full price orwait to purchase the product at salvage price to maximize theexpected surplus Each customer purchases one product atmost

Variables and parameters for model developmentadopted in this study are denoted as in Notations section

Because the parameters must meet certain conditions tomake sense we assume the following

(1) 119901 le 119903 Only when the retail price is no more thanthe customer reservation price the customer maypurchase the product at full price

(2) V gt 119901 gt 119908 gt 119888 gt 119904 gt 0 This conditionensures that there is a positive profit margin for themanufacturer retailer and customers when a productis sold to customers In addition the production costis greater than the salvage price which indicates thatthe retailer will lose money when the product fails tosell at full price This prompts the retailer to orderthe products according to the customersrsquo demandbecause the excess inventories generate losses

(3) 119908 gt 119888+119896 This condition states that the manufactureris willing to produce products with cap and tradepolicy Otherwise the manufacturer will not produceproducts but sell carbon emissions to earn profits

(4) The shortage cost of the manufacturer and operatingcost of the retailer are not taken into consideration

(5) We assume that the manufacturer and the retailer areall rational and self-interested that is both of themaim tomaximize their profitWe also assume that theyare risk neutral

(6) The carbon emissions in production is the mainsource of the total carbon emissions So we onlyconsider the carbon emissions of production

4 Discrete Dynamics in Nature and Society

(7) The green technology investment is the function of120591 denoted by 119868(120578) We assume that 119868(120578) ge 01198681015840(120578) gt 0 and 119868

10158401015840(120578) gt 0 This is consistent with

the practice The conditions show that the margingreen technology investment is increasing in carbonemissions reduction rate Referring to drsquoAspremontand Jacquemin 1988 [34] we set 119868(120578) = (12)119905120578

2where 119905 represents the efficiency of green technologyinvestments

Variables and parameters related to low carbon include119890 (a decision variable which represents the carbon emissionstrading policy) 120578 (a decision variable which represents thegreen technology investment policy) and 119864 and 119896 (representthe initial carbon emissions and unit price of carbon emis-sions trading resp)

4 Centralized Supply Chain Model

In this section we assume that themanufacturer is authorizedto determine the production pricing carbon trading andgreen technology investment strategies under cap and tradepolicy to maximize the profit of the centralized supply chainAt the beginning the government allocates a certain amountof free carbon emissions to the manufacturer During theproducing process if the carbon emission is limited themanufacturer should buy extra carbon emissions from theexternalmarket Otherwise if the carbon emission is enoughthe manufacturer could sell extra carbon emissions to gainrevenue At the end of the period carbon emissions of themanufacturer must not exceed the carbon emission rights itholds

The sequence of events in this part is as follows first themanufacturer forms the belief of customersrsquo reservation price120585119903and then decides the retail selling price product quantity

carbon trading volume and green technology investmentsecond the customers form the beliefs 120585prob of probabilityof the product sold at salvage price 119904 according to theinformation of market price and then form the reservationprice 119903 third the customersrsquo demand is satisfied and theproducts are sold at full price119901 finally all remaining productsare sold to the external market at salvage price 119904

41 Rational Expectations Equilibrium Scenario We charac-terize the RE Equilibrium between the manufacturer and thestrategic customers Muth 1961 [35] first proposed rationalexpectations hypothesis which refers to the situation thatthere is no systematic bias between the actual economic resultand peoplersquos expectations Then Su and Zhang 2008 [31]introduced it into operations management to analyze thedecision problems of the enterprises when strategic customerbehavior is taken into account Since then the rationalexpectations hypothesis has been adopted by scholars all overthe world [13 27 30]

First we examine the decision problem of strategiccustomers The customers choose to buy the product imme-diately at price 119901 or wait for markdown to maximize theirexpected surplus The customer surplus is V minus 119901 when thecustomer buys the product at full price and (V minus 119904)120585prob at

salvage price Therefore the maximum expected surplus ofthe customer is max(V minus 119904)120585prob V minus 119901 If and only if (V minus119904)120585prob le Vminus119901 the customer will buy the product at full priceSo given 120585prob we can obtain the customerrsquos reservation price119903(120585prob) = V minus (V minus 119904)120585prob

Then we examine the decision problem of the manufac-turer The profit function of the manufacturer with cap andtrade policy and green technology investment denoted by120587(119902 119901 119890 120578) is

120587 (119902 119901 119890 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909) minus (119888 minus 119904) 119902

minus 119896119890 minus1

21199051205782

(1)

First two items represent the manufacturerrsquos expectedprofit without cap and trade policy and green technologyinvestmentThis is the same with the newsvendormodelThethird item represents the carbon emissions trading costprofitof the manufacturer The forth item represents the cost ofmanufacturer investing in green technology

The carbon trading volume after green technology invest-ment under cap and trade policy is

119890 = (1 minus 120578) 119902 minus 119864 (2)

First term represents the manufacturerrsquos carbon emis-sions in production after green technology investment thesecond term represents initial free carbon emission owned bythe manufacturer

Then 120587(119902 119901 119890 120578) can be transformed into

120587 (119902 119901 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + 119896 (1 minus 120578)) 119902 + 119896119864 minus1

21199051205782

(3)

The beliefs of the manufacturer over the customerrsquosreservation price are 120585

119903 Obviously the manufacturer will

set 119901 = 120585119903 119902 and 120578 are the maximizer max

119902120578120587(119902 119901 120578) when

given 119901 According to the definition of RE Equilibrium theRE Equilibrium solution (119901 119902 120578 119903 120585

119903 120585prob)must meet

119903 = V minus (V minus 119904) 120585prob (4)

119901 = 120585119903 (5)

(119902 120578) = argmax119902120578

120587 (119902 119901 120578) (6)

120585prob = 119865 (119902) (7)

120585119903= 119903 (8)

Conditions (4) (5) and (6) indicate that the manufac-turer and customers will choose the action to maximizetheir own utility Conditions (7) and (8) can ensure that thesolution meets rational expectations hypothesis that is theactual situation of economic operation in line with peoplersquosexpectations

Discrete Dynamics in Nature and Society 5

The Rational Expectations Equilibrium makes

119901 = V minus (V minus 119904) 119865 (119902) (9)

The production pricing and green technology invest-ment decision model of the manufacturer with cap and tradepolicy and green technology investment is

max119902120578

120587 (119902 119901 120578)

st (1 minus 120578) 119902 le 119864

(10)

Defining 1205791(119902) = (1(1 minus 120578))[(119901 minus 119904)119865(119902) minus (119888 minus 119904)]

1205792(120578) = 119905120578119902 120579

1(119902) represents themargin profit of unit carbon

emission that is the profit gained by the manufacturerwith one unit carbon emission input in production 120579

2(120578)

represents the margin cost of unit carbon emission that isthe cost that the manufacturer invests in green technology toget one unit carbon emission reduction

Lemma 1 Given 119901 and (V minus 119904)119891(119902)119865(119902)119905 minus 1198962 gt 0 120587(119902 119901 120578)is a joint concave function of 120578 and 119902 in RE Equilibrium

Proof Given 119901 according to formula (3) 120597120587(119902 119901 120578)120597119902 =

(119901minus119904)119865(119902)minus(119888minus119904)minus(1minus120578)119896 1205972120587(119902 119901 120578)1205971199022 = minus(119901minus119904)119891(119902) lt0 120597120587(119902 119901 120578)120597120578 = 119902119896 minus 119905120578 1205972120587(119902 119901 120578)1205971205782 = minus119905 lt 0 and1205972120587(119902 119901 120578)120597119902120597120578 = 120597

2120587(119902 119901 120578)120597120578120597119902 = 119896

When 119905(V minus 119904)119891(119902)119865(119902) minus 1198962 gt 0 under RE Equilibriumwe get 119905(119901minus119904)119891(119902)minus1198962 gt 0 then we have 1205972120587(119902 119901 120578)120597119902120597120578 =1205972120587(119902 119901 120578)120597120578120597119902 = 119896 and

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205972120587 (119902 119901 120578)

1205971199022

1205972120587 (119902 119901 120578)

120597119902120597120578

1205972120587 (119902 119901 120578)

120597120578120597119902

1205972120587 (119902 119901 120578)

1205971205782

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 119905 (119901 minus 119904) 119891 (119902) minus 1198962

gt 0

(11)

The Hessian Matrix of the problem is negative definite andthen we can prove that 120587(119902 119901 120578) is joint concave function of119902 and 120578 when given 119901 and (V minus 119904)119891(119902)119865(119902)119905 minus 119896

2gt 0 This

completes the proof

As to the manufacturerrsquos optimal production quantity(denoted by 119902

lowast) pricing (denoted by 119901lowast) carbon trading

(denoted by 119890lowast) and green technology investment (denotedby 120578lowast) the following proposition is obtained

Proposition 2 If (Vminus119904)119891(119902)119865(119902)119905minus1198962 gt 0 themanufacturerrsquosoptimal production quantity (119902lowast) pricing (119901lowast) carbon trading(119890lowast) and green technology investment (120578lowast) strategies satisfy

1205791(119902lowast) = 1205792(120578lowast) = 119896

119901lowast= 119904 + (V minus 119904) 119865 (119902lowast)

119890lowast= (1 minus 120578

lowast) 119902lowastminus 119864

0 lt 120578lowastlt 1

(12)

Proof According to Lemma 1 if (Vminus 119904)119891(119902)119865(119902)119905 minus 1198962 gt 0 let120597120587(119902 119901 120578)120597119902 = 0 120597120587(119902 119901 120578)120597120578 = 0 and combine (9) and(2) we get

(119901 minus 119904) 119865 (119902) minus (119888 minus 119904) minus (1 minus 120578) 119896 = 0

119902119896 minus 119905120578 = 0

119901 = V minus (V minus 119904) 119865 (119902)

119890 = (1 minus 120578) 119902 minus 119864

0 lt 120578 lt 1

(13)

Solving the equations above can derive the optimalproduction pricing carbon trading and green technologyinvestment strategies of themanufacturerThis completes theproof

Proposition 2 shows that considering strategic customerbehavior the manufacturerrsquos optimal production quantitypricing carbon trading volume and green technology invest-ment strategywith cap and trade policy and green technologyinvestment exist and are unique The inherent implicationof the proposition is very intuitive 120579

1(119902) represents the

margin profit of unit carbon emission 1205792(120578) represents the

margin cost of unit carbon emission and 119896 is the unit carbonemission trading price which can be seen as marginal costor profit by carbon emission trading The optimal strategiesof the manufacturer are that the margin profit of unitcarbon emission is equal to the margin cost of unit carbonemission The marginal cost of getting unit carbon emissionfrom different way (green technology investment or buying)is equal The marginal profit of unit carbon emission fordifferent uses (production or sale) is equal

Substitute 119902lowast 119901lowast and 120578lowast into (3) we can obtain the max-imum expected profit of themanufacturer with cap and tradepolicy and green technology investment 120587(119902lowast 119901lowast 120578lowast) =

(119901lowastminus119904)(119902lowastminusint119902lowast

0119865(119909)119889119909)minus(119888minus119904+119896(1minus120578

lowast))119902lowast+119896119864minus(12)119905120578

lowast2

42 Quantity Commitment Scenario In traditional literatureon strategic customer behavior quantity commitment canimprove the manufacturerrsquos expected profit Quantity com-mitment refers to the action in which the manufacturerpromises customers that only a certain number of productsis produced and sold In low carbon supply chain settingcan the maximum expected profit be improved by quantitycommitment We attempt to answer the question in thispaper

We assume that the manufacturer can convince cus-tomers by appropriate means that they only can obtain 119902

units of the product in the wholesales period At this timethe strategic customers no longer need to anticipate theprobability of getting the product at salvage price When 119902

is given the probability of being able to get the product atprice 119904 is 119865(119902) The reservation price is 119901(119902) = Vminus (Vminus 119904)119865(119902)which also is the optimal pricing of the manufacturer Wehave 119890 = (1 minus 120578)119902 minus 119864 then we can obtain the expected profit

6 Discrete Dynamics in Nature and Society

function of themanufacturer with respect to 119902 and 120578 denotedby 120587119902(119902 120578)

120587119902(119902 120578) = (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + 119896 (1 minus 120578)) 119902 + 119896119864 minus1

21199051205782

(14)

So the manufacturerrsquos optimal production quantity andcarbon reduction rate in QC scenario are (119902

119902lowast 120578119902lowast) =

argmax119902ge00lt120578lt1

120587119902(119902 120578) the optimal price is 119901119902lowast = V minus (V minus

119904)119865(119902119902lowast) and the optimal carbon trading volume is 119890119902lowast =

(1 minus 120578119902lowast)119902119902lowastminus 119864

Lemma 3 Given 119902 the optimal green technology investmentstrategy of the manufacturer denoted by 120578119902lowast satisfies

120578119902lowastequiv 120578 (119902) =

119896119902

119905 0 lt 120578

119902lowastlt 1 (15)

Proof Wehave 120597120587119902(119902 120578)120597120578 = 119902119896minus119905120578 1205972120587119902(119902 120578)1205971205782 = minus119905 lt0 Let 120597120587119902(119902 120578)120597120578 = 0 we get 120578119902lowast equiv 120578(119902) = 119896119902119905 In additionaccording to the assumption in Section 3 we know 0 lt 120578

119902lowastlt

1 This completes the proof

Substituting 120578119902lowast = 120578(119902) into (14)

120587119902(119902) equiv 120587

119902(119902 120578 (119902))

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + 119896) 119902 + 119896119864 +11989621199022

2119905

(16)

Then the two-variable optimization problem is simplifiedas a single variable optimization problem

max119902ge0

120587119902(119902) (17)

Lemma4 If 1198962119905minus(Vminus119904)[3119891(119902)119865(119902)+1198911015840(119902)(119902minusint1199020119865(119909)119889119909)] lt

0 120587119902(119902) is a concave function of 119902

Proof The first-order and second-order derivatives of 120587119902(119902)are

119889120587119902(119902)

119889119902= (V minus 119904) [119865

2

(119902) minus 119891 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) +1198962

119905119902

1198892120587119902(119902)

1198891199022=1198962

119905minus (V minus 119904)

sdot [3119891 (119902) 119865 (119902) + 1198911015840(119902) (119902 minus int

119902

0

119865 (119909) 119889119909)] lt 0

(18)

Then we can obtain that 120587119902(119902) is a concave function of 119902This completes the proof

As to the manufacturerrsquos optimal production quantity inQC scenario (denoted by 119902119902lowast) the following proposition isobtained

Proposition 5 If 1198962119905 minus (V minus 119904)[3119891(119902)119865(119902) + 1198911015840(119902)(119902 minus

int119902

0119865(119909)119889119909)] lt 0 the optimal production quantity in QC

scenario (119902119902lowast) satisfies

(V minus 119904) [1198652

(119902) minus 119891 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) +1198962

119905119902 = 0

(19)

Proof It can be directly derived according to Lemma 4 Thiscompletes the proof

Proposition 5 shows that under certain condition theoptimal production quantity of the manufacturer with capand trade and green technology investment exist and areunique

43 The Effect of Quantity Commitment The effect of QCon the optimal strategies and the maximum expected profitof the manufacturer is analyzed by comparing the optimalstrategies and the maximum expected profit of the manufac-turer in RE Equilibrium scenario and QC scenario

Proposition 6 Consider 119902119902lowast lt 119902lowast 119901119902lowast gt 119901lowast 120578119902lowast lt 120578lowast

Proof In order to ensure that the manufacturerrsquos optimalstrategies in the two scenarios exist the condition 119896

2119905 lt

min(Vminus119904)119891(119902)119865(119902) (Vminus119904)[3119891(119902)119865(119902)+1198911015840(119902)(119902minusint1199020119865(119909)119889119909)]

must be held Then 120587119902(119902) is a concave function of 119902According to Proposition 2 we have 120579

1(119902lowast) = 1205792(120578lowast) = 119896

1205792(120578lowast) = 119896 can be written as 120578lowast = 119896119902

lowast119905 Then we also can

rearrange 1205791(119902lowast) = 119896 to (Vminus119904)1198652(119902lowast)minus(119888minus119904+119896)+(1198962119905)119902lowast = 0

We can obtain (119889120587119902(119902)119889119902)|

119902=119902lowast = (V minus 119904)[119865

2

(119902lowast) minus

119891(119902lowast)(119902lowastminus int119902lowast

0119865(119909)119889119909)] minus (119888 minus 119904 + 119896) + (119896

2119905)119902lowast= minus(V minus

119904)119891(119902lowast)(119902lowastminusint119902lowast

0119865(119909)119889119909) lt 0 Then we get 119902119902lowast lt 119902lowast Because

120578 = 119896119902119905 and 119901 = Vminus(Vminus119904)119865(119902) are held in the two scenarioswe can obtain 119901119902lowast gt 119901

lowast and 120578119902lowast lt 120578lowast according to 119902119902lowast lt 119902

lowastThis completes the proof

Proposition 6 shows that compared with RE Equilibriumscenario the manufacturerrsquos optimal production quantity islower the optimal pricing is higher and the optimal carbonreduction rate is lower in QC scenario

Proposition 7 Consider 120587119902(119902119902lowast 120578119902lowast) gt 120587(119902lowast 119901lowast 120578lowast)

Proof In order to ensure that the manufacturerrsquos optimalstrategies in the two scenarios exist the condition 119896

2119905 lt

min(Vminus119904)119891(119902)119865(119902) (Vminus119904)[3119891(119902)119865(119902)+1198911015840(119902)(119902minusint1199020119865(119909)119889119909)]

must be held Then 120587119902(119902) is a concave function of 119902Substitute 119902lowast 119901lowast and 120578

lowast= 119896119902

lowast119905 into (3) we have

120587(119902lowast 119901lowast 120578lowast) = 120587

119902(119902lowast) We know that 120587119902(119902) is a concave

function of 119902 and (119889120587119902(119902)119889119902)|119902=119902119902lowast = 0 Because of 119902119902lowast lt 119902lowast

Discrete Dynamics in Nature and Society 7

120587119902(119902lowast) lt 120587

119902(119902119902lowast) that is 120587(119902lowast 119901lowast 120578lowast) lt 120587

119902(119902119902lowast 120578119902lowast) This

completes the proof

Proposition 7 shows that QC strategy can improve themanufacturerrsquos maximum expected profit when green tech-nology investment and cap and trade policy are taken intoaccount

5 Decentralized Supply Chain Model

In this section the retailer and the manufacturer make deci-sions independently to maximize their own expected profitThe manufacturer is the Stackelberg leader and determinesthe wholesale price and the green technology investmentunder cap and trade policy The retailer is the follower anddetermines the retail price and the order quantity of theproduct

The sequence of events in this section is as follows firstthe manufacturer determines the optimal wholesale pricecarbon trading and green technology investment strategiessecond the retailer forms the belief of customersrsquo reservationprice and then decides the optimal retail price and orderquantity third the manufacture produces products anddelivers to the retailer fourth the customers form the beliefs120585prob of probability of the product sold at salvage price119904 according to the information of market price and thenform the reservation price 119903 fifth the customersrsquo demand issatisfied and the products are sold at full price 119901 finally allremaining products are sold to the external market at salvageprice 119904

51 The Optimal Strategies First we examine the decisionproblem of the retailer Recall that 119908 is the manufacturerrsquoswholesale price The retailerrsquos expected profit denoted by120587119903(119902 119901) is

120587119903(119902 119901) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909) minus (119908 minus 119904) 119902 (20)

Lemma 8 When 119901 is given 120587119903(119902 119901) is a concave function of119902

Proof When 119901 is given according to (20) we have120597120587119903(119902 119901)120597119902 = (119901 minus 119904)119865(119902) minus (119908 minus 119904) and 1205972120587119903(119902 119901)1205971199022 =

minus(119901 minus 119904)119891(119902) lt 0 Then we know that 120587119903(119902 119901) is a concavefunction of 119902 when 119901 is given This completes the proof

As to the retailerrsquos optimal strategies in RE Equilibriumwe can obtain the following proposition

Proposition 9 When the wholesale price is given the retailerrsquosoptimal order strategies (119902119903lowast) and optimal pricing strategies(119901119903lowast) with strategic customer behavior are

119902119903lowast= 119865minus1

(radic119908 minus 119904

V minus 119904) (21)

119901119903lowast= 119904 + radic(119908 minus 119904) (V minus 119904) (22)

Proof In RE Equilibrium (9) still holds Let 120597120587119903(119902 119901)120597119902 = 0and combining with (9) we have

119901 = V minus (V minus 119904) 119865 (119902)

(119901 minus 119904) 119865 (119902) minus (119908 minus 119904) = 0

(23)

According to Lemma 8 we can obtain the optimal orderand pricing strategies of the retailer by solving the equationsabove This completes the proof

Proposition 9 shows that the retailerrsquos optimal order andpricing strategies in REEquilibriumwith cap and trade policyexist and are unique

Second we investigate the manufacturerrsquos decision prob-lem The expected profit function of the manufacturer withgreen technology investment and cap and trade policy is120587119898(119908 119890 120578) = (119908 minus 119888)119902 minus 119896119890 minus (12)119905120578

2According to (21) with green technology investment and

cap and trade policy in wholesale price contract the optimalorder quantity of the retailer 119902119903lowast and the optimal wholesaleprice of the manufacturer 119908119898lowast is a one-to-one relationship

119908 = 119904 + (V minus 119904) 1198652

(119902) (24)

Combined with 119890 = (1 minus 120578)119902 minus 119864 the manufacturerrsquosexpected profit function can be converted into

120587119898(119902 120578) = [(V minus 119904) 119865

2

(119902) minus (119888 minus 119904 + (1 minus 120578) 119896)] 119902

+ 119896119864 minus1

21199051205782

(25)

When using wholesale price contract the total profit ofthe supply chain denoted by 120587sc

(119902 119901 119890 120578) is

120587sc(119902 119901 119890 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909) minus (119888 minus 119904) 119902

minus 119896119890 minus1

21199051205782

(26)

According to (9) and 119890 = (1 minus 120578)119902 minus 119864 the total supplychain profit function in RE Equilibrium can be simplified as

120587sc(119902 120578) = (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578) 119896) 119902 + 119896119864 minus1

21199051205782

(27)

Lemma 10 Given 119902 the optimal green technology investmentstrategy of the manufacturer denoted by 120578119898lowast satisfies

120578119898lowast

equiv 120578 (119902) =119896119902

1199050 lt 120578119898lowast

lt 1 (28)

Proof The first-order and second-order partial derivatives of120587119898(119902 120578) with respect to 120578 are 120597120587119898(119902 120578)120597120578 = 119902119896 minus 119905120578 and

1205972120587119898(119902 120578)120597120578

2= minus119905 lt 0 Let 120597120587119898(119902 120578)120597120578 = 0 we get

120578119898lowast

equiv 120578(119902) = 119896119902119905 In addition according to the assumptionin Section 3 we know 0 lt 120578

119898lowastlt 1 This completes the

proof

8 Discrete Dynamics in Nature and Society

Substituting 120578119898lowast = 120578(119902) into (25)

120587119898(119902) equiv 120587

119898(119902 120578 (119902))

= [(V minus 119904) 1198652

(119902) minus (119888 minus 119904 + 119896)] 119902 + 119896119864

+11989621199022

2119905

(29)

Then the two-variable optimization problem is simplifiedas a single variable optimization problem

max119902ge0

120587119898(119902) (30)

Lemma 11 If 1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus

1199021198912(119902)] lt 0 120587119898(119902) is a concave function of 119902

Proof The first-order and second-order derivatives of 120587119898(119902)with respect to 119902 are 119889120587

119898(119902)119889119902 = (V minus 119904)[119865

2

(119902) minus

2119902119891(119902)119865(119902)] minus (119888 minus 119904 + 119896) + (1198962119905)119902 and 119889

2120587119898(119902)119889119902

2=

1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus 1199021198912(119902)] lt 0 Then

we have that 120587119898(119902) is a concave function of 119902 This completesthe proof

In decentralized supply chain as to the manufacturerrsquosoptimal wholesale price (denoted by 119908

119898lowast) the followingproposition is obtained

Proposition 12 If 1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus1199021198912(119902)] lt 0 the manufacturerrsquos optimal wholesale price in

decentralized supply chain (119908119898lowast) satisfies 119908119898lowast = 119904 + (V minus119904)1198652

(119902119903lowast) where 119902119903lowast represents the retailerrsquos optimal order

quantity and satisfies

(V minus 119904) [1198652

(119902) minus 2119902119891 (119902) 119865 (119902)] minus (119888 minus 119904 + 119896) +1198962

119905119902

= 0

(31)

Proof According to Lemma 11 we can obtain that 119902119903lowastwhichmaximizes themanufacturerrsquos expected profit satisfies(31) Combining with (24) we know that the manufacturerrsquosoptimal wholesale price satisfies 119908119898lowast = 119904 + (V minus 119904)119865

2

(119902119903lowast) at

this time This completes the proof

Proposition 12 shows that the optimal commitment quan-tity of the manufacturer in decentralized supply chain withgreen technology investment and cap and trade policy existand are unique The optimal wholesale price does not relateto the cap allocated by the government but relates to thecarbon trading price This is because the production of themanufacturer is not affected by the cap but is affected bythe margin cost of the product However the margin cost ofthe product varies with the change of carbon trading pricewhen carbon trading is allowed The change results in thatthe optimal commitment quantity is not related to the cap butrelate to the carbon trading price

Substitute 119908119898lowast into (22) we derive the retailerrsquos optimalpricing in decentralized situation119901119903lowast = 119904+radic(119908119898lowast minus 119904)(V minus 119904)

According to Lemma 10 we get the optimal green technologyinvestment of the manufacturer 120578119898lowast = (119896119905)119902119903lowast

Substitute 119902119903lowast 119901119903lowast 119908119898lowast and 120578119898lowast into 120587

119903(119902 119901) 120587119898(119902)

and 120587sc(119902 120578) we obtain the maximum expected profit of

the retailer the manufacturer and the supply chain indecentralized channel

120587119903(119902119903lowast 119901119903lowast) = (119901

119903lowastminus 119904) (119902

119903lowastminus int

119902119903lowast

0

119865 (119909) 119889119909)

minus (119908119898lowast

minus 119904) 119902119903lowast

120587119898(119902119903lowast) = [(V minus 119904) 119865

2

(119902119903lowast) minus (119888 minus 119904 + 119896)] 119902

119903lowast

+ 119896119864 +1198962119902119903lowast2

2119905

120587sc(119902119903lowast 120578119898lowast) = (V minus 119904) 119865 (119902119903lowast) (119902119903lowast minus int

119902119903lowast

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578119898lowast) 119896) 119902119903lowast+ 119896119864

minus1

2119905120578119898lowast2

(32)

52 The Effect of Decentralization We examine the impact ofdecentralization on the supply chain optimal strategies andmaximumexpected profitwhen green technology investmentis taken into consideration by comparing the optimal strate-gies and the maximum expected profit in centralized anddecentralized channel with green technology investment andcap and trade policy

As the effect of decentralization of supply chain we canobtain the following proposition

Proposition 13 Consider 119902119903lowast lt 119902119902lowast

lt 119902lowast 119901119903lowast gt 119901

119902lowastgt 119901lowast

120578119898lowast

lt 120578119902lowastlt 120578lowast

Proof In Proposition 6 119902119902lowast lt 119902lowast 119901119902lowast gt 119901

lowast and 120578119902lowast lt 120578lowast

have been provedTherefore we only need to prove 119902119903lowast lt 119902119902lowast119901119903lowastgt 119901119902lowast and 120578119898lowast lt 120578119902lowast

Define 119867(119902) = 120587119902(119902) minus 120587119898(119902) then

119867(119902)

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

minus (V minus 119904) 1199021198652

(119902)

1198671015840(119902)

= (V minus 119904) 119891 (119902) [2119902119865 (119902) minus (119902 minus int119902

0

119865 (119909) 119889119909)]

(33)

Define 119866(119902) = 2119902119865(119902) minus (119902 minus int119902

0119865(119909)119889119909) then 1198661015840(119902) =

119865(119902) minus 2119902119891(119902) Because 119866(0) = 0 1198661015840(0) = 1 and 119865 satisfiesIFR we know that the value of 119866(119902) starts from 0 increasingfirst and then decreasing in 119902 That is 119866(119902) = 0 has uniquesolution and we denote it as So when 119902 lt 119866(119902) gt 0 that

Discrete Dynamics in Nature and Society 9

is119867(119902) is increasing in 119902 when 119902 gt 119866(119902) lt 0 that is119867(119902)is decreasing in 119902 Then we have that119867(119902) is a quasi-concavefunction of 119902 and 1198661015840 () = 119865() minus 2119891() lt 0 According tothe analysis above we know that1198671015840() = 0 that is 2119865() =( minus int

0119865(119909)119889119909) Substitute into 119889120587119902(119902)119889119902 we have

119889120587119902(119902)

119889119902

100381610038161003816100381610038161003816100381610038161003816119902=

= (V minus 119904) [1198652

() minus 119891 () ( minus int

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) +1198962

119905

= (V minus 119904) 119865 () [119865 () minus 2119891 ()] minus (119888 minus 119904 + 119896)

+1198962

119905lt (V minus 119904) 119865 () [119865 () minus 2119891 ()] lt 0

(34)

According to Lemma 4 we have that 120587119902(119902) is a concavefunction of 119902 so we get 119902119902lowast lt Because 2119865() = ( minus

int

0119865(119909)119889119909) and119867(119902) is a quasi-concave function of 119902 thenwe

get that 119889120587119898(119902)119889119902 lt 119889120587119902(119902)119889119902 if 119902 lt and 119889120587119898(119902)119889119902 gt

119889120587119902(119902)119889119902 if 119902 gt Therefore the only possible orderings for

119902119902lowast 119902119903lowast and are lt 119902

119902lowastlt 119902119903lowast and 119902119903lowast lt 119902

119902lowastlt We

have proved 119902119902lowast lt so we have 119902119903lowast lt 119902119902lowast We get 119901119903lowast gt 119901119902lowastbecause 119901 and 119902 satisfy 119901 = V minus (V minus 119904)119865(119902) in two situationsWe get 120578119898lowast lt 120578

119902lowast because 120578 and 119902 satisfy 120578 = 119896119902119905 in twosituations This completes the proof

Proposition 13 shows that for the decision of a decen-tralized supply chain with green technology investment theoptimal production quantity is increasing the optimal priceis decreasing and the green technology investment (iecarbon reduction rate) is increasing respectively in the threesituations of decentralized supply chain QC scenario and REEquilibrium scenario

The reason is that themanufacturer prefers higher whole-sale price (ie lower production and less green technologyinvestment) to guarantee its ownprofitmaximization thoughthis may harm the retailer even the whole supply chain

In Proposition 7 we have found 120587119902(119902119902lowast 120578119902lowast) gt

120587(119902lowast 119901lowast 120578lowast) that is the manufacturerrsquos maximum profit in

QC scenario is always greater than that in RE EquilibriumTherefore the section analyzes the effect of decentralizationon supply chain performance based on the QC scenario toprovide benchmark for designing of subsequent coordinationcontract

As to the effect of decentralization on themaximumprofitof supply chain we have the following proposition

Proposition 14 Consider 120587sc(119902119903lowast 120578119898lowast) lt 120587119902(119902119902lowast 120578119902lowast)

Proof The optimal green technology investment in decen-tralized supply chain and QC scenario are 120578(119902) = 119896119902119905

according to Lemmas 3 and 10 So the profit function in thetwo situations above can be converted into a same singlevariable function with respect to 119902 that is (16)

According to Lemma 4 we know that 120587119902(119902) is a concavefunction and reaches the maximum value at 119902119902lowast Accordingto Proposition 13 we have 119902119903lowast lt 119902

119902lowast then we get 120587119902(119902119903lowast) lt120587119902(119902119902lowast) that is 120587sc

(119902119903lowast 120578119898lowast) lt 120587

119902(119902119902lowast 120578119902lowast) This completes

the proof

Proposition 14 shows that the maximum profit of decen-tralized supply chain is always lower than that in QC scenariowhich indicates that the profit of the decentralized supplychain can get higher This is the benchmark of supply chaincoordination

6 Supply Chain Coordination

This section coordinates the supply chain with green tech-nology investment and cap and trade policy based on theQC scenario to improve the profit of the decentralized supplychain

61 Coordination Contract Based on Revenue and CostsSharing Contract With revenue sharing contract we assumethat 120601 (0 le 120601 le 1) represents the ratio of revenue kept by theretailer and (1 minus 120601) represents the ratio of revenue deliveredto the manufacturer

The retailerrsquos expected profit function with revenue shar-ing contract denoted by 120587119903

119904(119902 119901 120601) is

120587119903

119904(119902 119901 120601) = 120601 (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119908 minus 120601119904) 119902

(35)

Themanufacturerrsquos expected profit functionwith revenuesharing contract denoted by 120587119898

119904(119908 119890 120578 120601) is

120587119898

119904(119908 119890 120578 120601) = (119908 minus 119888) 119902 + (1 minus 120601)

sdot [int

119902

0

[119901119909 + 119904 (119902 minus 119909)] 119891 (119909) 119889119909

+ int

infin

119902

119901119902119891 (119909) 119889119909] minus 119896119890 minus1

21199051205782

(36)

Substitute 119890 = (1 minus 120578)119902 minus 119864 into the equation above andwe have

120587119898

119904(119908 120578 120601) = (1 minus 120601) (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

+ (119908 minus 119888 minus (1 minus 120578) 119896 + (1 minus 120601) 119904) 119902

+ 119896119864 minus1

21199051205782

(37)

The total expected profit of the whole supply chaindenoted by 120587sc

119904(119902 119901 120578) is

120587sc119904(119902 119901 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578) 119896) 119902 + 119896119864 minus1

21199051205782

(38)

10 Discrete Dynamics in Nature and Society

The subscript 119904 represents the situation of supply chaincoordination using revenue sharing contracts

As to the retailerrsquos optimal order strategy (denoted by119902lowast

119904) and pricing strategy (denoted by 119901lowast

119904) of the supply chain

with revenue sharing contract the following proposition isobtained

Proposition 15 Under revenue sharing contracts the retailerrsquosoptimal order strategy (119902lowast

119904) and optimal pricing strategy (119901lowast

119904)

are

119902lowast

119904= 119865minus1

(radic119908 minus 120601119904

120601 (V minus 119904))

119901lowast

119904= 119904 + radic

(119908 minus 120601119904) ((V minus 119904))120601

(39)

Proof According to the definition of RE Equilibrium wealso have 119901 = V minus (V minus 119904)119865(119902) under revenue sharingcontracts

The first-order and second-order derivative of 120587119903119904(119902 119901 120601)

with respect to 119902 are 120597120587119903

119904(119902 119901 120601)120597119902 = 120601(119901 minus 119904)119865(119902) minus

(119908 minus 120601119904) and 1205972120587119903

119904(119902 119901 120601)120597119902

2= minus120601(119901 minus 119904)119891(119902) lt

0 So given 119901 the retailerrsquos expected profit function is aconcave function of 119902 Let 120597120587119903

119904(119902 119901 120601)120597119902 = 0 and combine

with 119901 = V minus (V minus 119904)119865(119902) we can solve the retailerrsquos optimalstrategies under revenue sharing contractThis completes theproof

Proposition 15 shows that the optimal order and pricingstrategies of the retailer based on revenue sharing contract inRE Equilibrium exist and are uniqueThis is also the retailerrsquosoptimal response function with respect to 119908

Substituting (39) into (43)

120587119898

119904(119902 120578 120601)

= (1 minus 120601) (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ [120601 (V minus 119904) 1198652

(119902) minus (119888 minus 119904 + (1 minus 120578) 119896)] 119902 + 119896119864

minus1

21199051205782

120597120587119898

119904(119902 120578 120601)

120597120578= 119902119896 minus 119905120578

1205972120587119898

119904(119902 120578 120601)

1205971205782= minus119905 lt 0

(40)

Given 119902 and 120601 the manufacturerrsquos green technologyinvestment strategies with revenue sharing contracts exist

and are unique Let 120597120587119898119904(119902 120578 120601)120597120578 = 0 we get 120578119898lowast

119904equiv 120578(119902) =

119902119896119905 and substitute it into 120587119898119904(119902 120578 120601)

120587119898

119904(119902 120601)

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ 120601 (V minus 119904) 119865 (119902) [119902119865 (119902) minus (119902 minus int119902

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) 119902 + 119896119864 +11989621199022

2119905

(41)

Proposition 16 The supply chain with green technologyinvestment and cap and trade policy can not be coordinated byrevenue sharing contract

Proof The derivative of 120587119898119904(119902 120601) with respect to 119902 can be

arranged as follows

120597120587119898

119904(119902 120601)

120597119902=120597120587119902(119902)

120597119902+ 120601(

120597120587119898(119902)

120597119902minus120597120587119902(119902)

120597119902) (42)

Only when 120601 = 0 are the optimal strategies of supplychain under revenue sharing contracts equal to that in QCscenario Let 119908lowast

119904represent the optimal wholesale price of

the manufacture and 119902lowast119904represent the retailerrsquos optimal order

quantity Then 120587119903

119904(119902 119901 120601) = minus119908

lowast

119904119902lowast

119904lt 0 So the revenue

sharing contract cannot realize the supply chain coordinationat this time This completes the proof

Proposition 16 shows that revenue sharing contractscan not realize the supply chain coordination with greentechnology investment and cap and trade policy The rea-son is that the manufacturer undertakes all the costs ofgreen technology investment when using revenue sharingcontracts thus making the manufacturer tend to invest lessin green technology Only when the manufacturer owns allthe revenue will themanufacturerrsquos optimal green technologyinvestment under revenue sharing contracts be equal to thatin QC scenario However the maximum profit of the retaileris negative at this time and it will not participate

62 Coordination Contract Based on Revenue Sharing-CostSharing Contract Proposition 16 states that the supply chainwith green technology investment and cap and trade policycan not be coordinated by revenue sharing contracts becausethe costs of green technology investment are undertaken bythe manufacturer and the game of retailer and customersis characterized by RE Equilibrium So we consider costssharing of green technology investment and carbon tradingand quantity commitment made by the manufacturer basedon revenue sharing contracts Let 120601 (0 lt 120601 le 1) represent theratio of revenue kept by the retailer and let (1minus120601) represent theratio of revenue delivered to the manufacturer 120572 (0 le 120572 le 1)represents the ratio of costs of green technology investmentand carbon trading undertaken by the manufacturer and(1 minus 120572) represents the ratio of costs of green technologyinvestment and carbon trading delivered to the retailer

Discrete Dynamics in Nature and Society 11

The retailerrsquos expected profit function with revenuesharing-cost sharing contract is

120587119903

120572(119902 120601 120572) = 120601 (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119908 minus 120601119904) 119902

minus (1 minus 120572) 119896 [(1 minus 120578) 119902 minus 119864]

minus1

2(1 minus 120572) 119905120578

2

(43)

We have 119890 = (1minus120578)119902minus119864 then themanufacturerrsquos expectedprofit function denoted by 120587119898

120572(119908 120578 120601 120572) is

120587119898

120572(119908 120578 120601 120572)

= (1 minus 120601) (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ (119908 minus 119888 minus 120572 (1 minus 120578) 119896 + (1 minus 120601) 119904) 119902 + 120572119896119864

minus1

21205721199051205782

(44)

The subscript 120572 represents the situation that using rev-enue sharing-cost sharing contract coordinates the supplychain

Proposition 17 If 0 le 120582 le 1 the parameters of revenuesharing-cost sharing (119908 120601 120572) satisfy

119908 = 120582119888 + (1 minus 120578) 119896

120601 = 120582

120572 = 1 minus 120582

(45)

The supply chainwith green technology investment and cap andtrade policy is coordinated

Proof When the parameters of revenue sharing-cost sharingsatisfy (45) the expected profit function of the retailer andthe manufacturer can be written as follows

120587119903

120572(119902 119901 120601 120572) = 120582120587

119902(119902 120578)

120587119898

120572(119908 120578 120601 120572) = (1 minus 120582) 120587

119902(119902 120578)

(46)

The optimal order of decentralized supply chain is equalto that of QC scenario Substitute 119901 = V minus (V minus 119904)119865(119902) into120587sc119888(119902 119901 120578) we get that the profit function of decentralized

supply chain at this time is the same as that of QC scenarioTherefore when the parameters of revenue sharing-costsharing (119908 120601 120572) satisfy (45) the decentralized supply chaincoordination is realizedThemanufacturer earns all the profitif 120582 = 0 and the retailer earns all the profit if 120582 = 1Arbitrary allocation of the supply chain profit between themanufacturer and the retailer is allowed This completes theproof

Proposition 17 shows that revenue sharing-cost sharingcontract can realize the supply chain coordination on condi-tion that the manufacturer makes a quantity commitment to

the customers The optimal production quantity and greentechnology investment in decentralized situation are lowerthan that in QC scenario The quantity commitment of themanufacturer can increase the green technology investmentand production quantity and also reduce the retailer priceThat is the quantity commitment is beneficial for the man-ufacturer the retailer and customers Thus the quantitycommitment is credible to customers now

7 Conclusions and Suggestions for FurtherResearch

This paper investigates the production pricing carbon trad-ing and green technology investment strategies and thecoordination of low carbon supply chain made up of a lowcarbon manufacturer and a retailer The low carbon manu-facturer produces one product under cap and trade policyand performs green technology investment to reduce carbonemissions in production process The retailer orders fromthemanufacturer and distributes the product to homogenouscustomers To the best of our knowledge this is the firststudy on the management of low carbon supply chainswith strategic customer behaviorThis paper provides severalinteresting observations

Observation 1 There are unique optimal production pricingcarbon trading and green technology investment strategiesin centralized (include RE Equilibrium scenario and QCscenario) and decentralized supply chain Therefore byderiving the optimal production pricing carbon trading andgreen technology investment strategies in different situationsthe low carbon manufacturer and the retailer can behaveappropriately based on our findings to maximize their profit

Observation 2 Our findings also show that the centralizedsupply chain can improve its performance by quantity com-mitment However the manufacturerrsquos production and greentechnology investment are reduced while the optimal priceis increased at this time Therefore quantity commitment isbeneficial for the supply chain but is harmful to customers

Observation 3 We also show that the performance of decen-tralized supply chain is lower than that in QC scenario Wefind that the decentralization reduces the manufacturersquos opti-mal production quantity and green technology investmentand increases the retailerrsquos optimal price Besides we find thatthe optimal production quantity is increasing the optimalprice is decreasing and the green technology investment isincreasing respectively in the three situations of decentral-ized supply chain QC scenario andREEquilibrium scenario

Observation 4 We demonstrate that the traditional revenuesharing contracts can not realize the low carbon supply chaincoordination when green technology investment is takenin consideration We also find that the revenue sharing-cost sharing contract realizes the decentralized supply chain

12 Discrete Dynamics in Nature and Society

coordination when the manufacturer makes a quantity com-mitment It is beneficial to the manufacturer the retailerand customers if themanufacturermakes a quantity commit-ment So the quantity commitment is credible to the customerat this time

We assume the customers are homogeneous strategiccustomers But themyopic customers and strategic customersoften coexist and strategic customers also have differenttolerance in real-life Extending this assumption will havea knock-on effect on supply chain decisions So one keyresearch direction is to consider that the customers aremixed customers myopic customers and strategic customersSecond we assume that the supply chain only producesone product While this simplified setting provides someinteresting insights we have to acknowledge that a manufac-turer usually produces two or more productsThese productswill substitute or complement each other So one importantextension of our work is to assume that the manufacturerproduces two or more products which have different unitcarbon emissions in production

Notations

119863 The random demand and119863 ge 0

119891(sdot) Probability density function of119863119865(sdot) Distribution function of119863 We assume

that 119865 satisfy IFR and define 119865 = 1 minus 119865

119888 Unit production cost119904 Unit salvage price of the product which

is an exogenous variableV The customersrsquo utility from consuming

the product119908 Unit wholesale price119901 Unit retail selling price We assume that

it can be observed by the customers119902 The manufacturerrsquos production

quantitythe retailerrsquos order quantityWe assume that it cannot be observedby the customers

119903 The customersrsquo reservation price whichis customersrsquo private information andcannot be observed by the retailer

120585119903 The beliefs of the manufacturer over the

customerrsquo reservation price120585prob The beliefs of customers over their

chances of obtaining the product atphase two

119864 Initial free carbon emission allocated bygovernment

119896 Unit carbon emission trading price119890 Carbon trading volume120578 Carbon emissions reduction rate

Conflict of Interests

The authors declare no conflict of interests

Acknowledgments

This research is partially supported by National Natural Sci-ence Foundation of China (nos 71432003 and 71272128) theProgram forNewCentury Excellent Talents inUniversity (noNCET-12-0087) and Specialized Research Fund for the Doc-toral Program of Higher Education (no 20130185110006)

References

[1] R Alley T Berntsen N L Bindoff et al ldquoSummary forpolicymakersrdquo in Climate Change 2007 The Physical ScienceBasis vol 6 pp 1ndash18 IPCC 2007

[2] X Chen and G Hao ldquoSustainable pricing and productionpolicies for two competing firms with carbon emissions taxrdquoInternational Journal of Production Research vol 53 pp 6408ndash6420 2015

[3] X Chen S Benjaafar and A Elomri ldquoThe carbon-constrainedEOQrdquo Operations Research Letters vol 41 no 2 pp 172ndash1792013

[4] N O Keohane ldquoCap and trade rehabilitated using tradablepermits to control US greenhouse gasesrdquo Review of Environ-mental Economics and Policy vol 3 no 1 pp 42ndash62 2009

[5] B Liu M Holmbom A Segerstedt and W Chen ldquoEffectsof carbon emission regulations on remanufacturing decisionswith limited information of demand distributionrdquo InternationalJournal of Production Research vol 53 no 2 pp 532ndash548 2015

[6] L Zhu and Y Fan ldquoA real options-based CCS investmentevaluation model case study of Chinarsquos power generationsectorrdquo Applied Energy vol 88 no 12 pp 4320ndash4333 2011

[7] XWang and LDu ldquoStudy on carbon capture and storage (CCS)investment decision-making based on real options for Chinarsquoscoal-fired power plantsrdquo Journal of Cleaner Production vol 112part 5 pp 4123ndash4131 2016

[8] W Jiang and X Chen ldquoOptimal strategies for manufacturerwith strategic customer behavior under carbon emissions-sensitive randomdemandrdquo Industrial Management amp Data Sys-tems 2015

[9] Y Levin J McGill and M Nediak ldquoDynamic pricing in thepresence of strategic consumers and oligopolistic competitionrdquoManagement Science vol 55 no 1 pp 32ndash46 2009

[10] A K Parlakturk ldquoThe value of product variety when sellingto strategic consumersrdquo Manufacturing amp Service OperationsManagement vol 14 no 3 pp 371ndash385 2012

[11] X Su and F Zhang ldquoOn the value of commitment andavailability guarantees when selling to strategic consumersrdquoManagement Science vol 55 no 5 pp 713ndash726 2009

[12] W Elmaghraby S A Lippman C S Tang and R Yin ldquoWillmore purchasing options benefit customersrdquo Production andOperations Management vol 18 no 4 pp 381ndash401 2009

[13] W Jiang and X Chen ldquoManufacturersquos production and pricingstrategies with carbon tax policy and strategic customer behav-iorrdquoManagement Science and Engineering vol 9 no 1 pp 30ndash35 2015

[14] I Dobos ldquoThe effects of emission trading on production andinventories in the Arrow-Karlin modelrdquo International Journalof Production Economics vol 93-94 pp 301ndash308 2005

[15] X Chen C K Chan and Y C E Lee ldquoResponsible productionpolicies with substitution and carbon emissions tradingrdquo Jour-nal of Cleaner Production 2015

Discrete Dynamics in Nature and Society 13

[16] X Chen and X Wang ldquoEffects of carbon emission reductionpolicies on transportation mode selections with stochasticdemandrdquo Transportation Research Part E Logistics and Trans-portation Review 2015

[17] X Chang H Xia H Zhu T Fan and H Zhao ldquoProductiondecisions in a hybrid manufacturingndashremanufacturing systemwith carbon cap and trade mechanismrdquo International Journal ofProduction Economics vol 162 pp 160ndash173 2015

[18] J Zhao B F Hobbs and J-S Pang ldquoLong-run equilibriummodeling of emissions allowance allocation systems in electricpowermarketsrdquoOperations Research vol 58 no 3 pp 529ndash5482010

[19] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2 pp519ndash527 2011

[20] A Toptal H Ozlu and D Konur ldquoJoint decisions on inventoryreplenishment and emission reduction investment under dif-ferent emission regulationsrdquo International Journal of ProductionResearch vol 52 no 1 pp 243ndash269 2014

[21] A S Manikas and J R Kroes ldquoA newsvendor approach tocompliance and production under cap and trade emissionsregulationrdquo International Journal of Production Economics vol159 pp 274ndash284 2015

[22] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineering vol10 no 1 pp 99ndash116 2013

[23] X Xu W Zhang P He and X Xu ldquoProduction and pricingproblems in make-to-order supply chain with cap-and-traderegulationrdquo Omega 2015

[24] Y Zhen X Yang and W Jiang ldquoThe impact of transport modeand carbon policy on low-carbon retailerrdquoDiscrete Dynamics inNature and Society vol 2015 Article ID 964305 12 pages 2015

[25] S Swami and J Shah ldquoChannel coordination in green supplychainmanagementrdquo Journal of the Operational Research Societyvol 64 no 3 pp 336ndash351 2013

[26] R H Coase ldquoDurability andmonopolyrdquoThe Journal of Law andEconomics vol 15 no 1 pp 143ndash149 1972

[27] X Su ldquoIntertemporal pricing with strategic customer behaviorrdquoManagement Science vol 53 no 5 pp 726ndash741 2007

[28] D Zhang and W L Cooper ldquoManaging clearance sales in thepresence of strategic customersrdquo Production and OperationsManagement vol 17 no 4 pp 416ndash431 2008

[29] S Whang ldquoDemand uncertainty and the Bayesian effect inmarkdown pricing with strategic customersrdquoManufacturing ampService Operations Management vol 17 no 1 pp 66ndash77 2015

[30] J Du J Zhang and G Hua ldquoPricing and inventory manage-ment in the presence of strategic customers with risk preferenceand decreasing valuerdquo International Journal of Production Eco-nomics vol 164 pp 160ndash166 2015

[31] X Su and F Zhang ldquoStrategic customer behavior commitmentand supply chain performancerdquo Management Science vol 54no 10 pp 1759ndash1773 2008

[32] H Yang ldquoImpact of discounting and competition on benefit ofdecentralization with strategic customersrdquo Operations ResearchLetters vol 40 no 2 pp 123ndash127 2012

[33] D Yang E Qi and Y Li ldquoQuick response and supply chainstructure with strategic consumersrdquo Omega vol 52 pp 1ndash142015

[34] C drsquoAspremont and A Jacquemin ldquoCooperative and nonco-operative R amp D in duopoly with spilloversrdquo The AmericanEconomic Review vol 78 no 5 pp 1133ndash1137 1988

[35] J F Muth ldquoRational expectations and the theory of pricemovementsrdquo Econometrica vol 29 no 3 pp 315ndash335 1961

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

(7) The green technology investment is the function of120591 denoted by 119868(120578) We assume that 119868(120578) ge 01198681015840(120578) gt 0 and 119868

10158401015840(120578) gt 0 This is consistent with

the practice The conditions show that the margingreen technology investment is increasing in carbonemissions reduction rate Referring to drsquoAspremontand Jacquemin 1988 [34] we set 119868(120578) = (12)119905120578

2where 119905 represents the efficiency of green technologyinvestments

Variables and parameters related to low carbon include119890 (a decision variable which represents the carbon emissionstrading policy) 120578 (a decision variable which represents thegreen technology investment policy) and 119864 and 119896 (representthe initial carbon emissions and unit price of carbon emis-sions trading resp)

4 Centralized Supply Chain Model

In this section we assume that themanufacturer is authorizedto determine the production pricing carbon trading andgreen technology investment strategies under cap and tradepolicy to maximize the profit of the centralized supply chainAt the beginning the government allocates a certain amountof free carbon emissions to the manufacturer During theproducing process if the carbon emission is limited themanufacturer should buy extra carbon emissions from theexternalmarket Otherwise if the carbon emission is enoughthe manufacturer could sell extra carbon emissions to gainrevenue At the end of the period carbon emissions of themanufacturer must not exceed the carbon emission rights itholds

The sequence of events in this part is as follows first themanufacturer forms the belief of customersrsquo reservation price120585119903and then decides the retail selling price product quantity

carbon trading volume and green technology investmentsecond the customers form the beliefs 120585prob of probabilityof the product sold at salvage price 119904 according to theinformation of market price and then form the reservationprice 119903 third the customersrsquo demand is satisfied and theproducts are sold at full price119901 finally all remaining productsare sold to the external market at salvage price 119904

41 Rational Expectations Equilibrium Scenario We charac-terize the RE Equilibrium between the manufacturer and thestrategic customers Muth 1961 [35] first proposed rationalexpectations hypothesis which refers to the situation thatthere is no systematic bias between the actual economic resultand peoplersquos expectations Then Su and Zhang 2008 [31]introduced it into operations management to analyze thedecision problems of the enterprises when strategic customerbehavior is taken into account Since then the rationalexpectations hypothesis has been adopted by scholars all overthe world [13 27 30]

First we examine the decision problem of strategiccustomers The customers choose to buy the product imme-diately at price 119901 or wait for markdown to maximize theirexpected surplus The customer surplus is V minus 119901 when thecustomer buys the product at full price and (V minus 119904)120585prob at

salvage price Therefore the maximum expected surplus ofthe customer is max(V minus 119904)120585prob V minus 119901 If and only if (V minus119904)120585prob le Vminus119901 the customer will buy the product at full priceSo given 120585prob we can obtain the customerrsquos reservation price119903(120585prob) = V minus (V minus 119904)120585prob

Then we examine the decision problem of the manufac-turer The profit function of the manufacturer with cap andtrade policy and green technology investment denoted by120587(119902 119901 119890 120578) is

120587 (119902 119901 119890 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909) minus (119888 minus 119904) 119902

minus 119896119890 minus1

21199051205782

(1)

First two items represent the manufacturerrsquos expectedprofit without cap and trade policy and green technologyinvestmentThis is the same with the newsvendormodelThethird item represents the carbon emissions trading costprofitof the manufacturer The forth item represents the cost ofmanufacturer investing in green technology

The carbon trading volume after green technology invest-ment under cap and trade policy is

119890 = (1 minus 120578) 119902 minus 119864 (2)

First term represents the manufacturerrsquos carbon emis-sions in production after green technology investment thesecond term represents initial free carbon emission owned bythe manufacturer

Then 120587(119902 119901 119890 120578) can be transformed into

120587 (119902 119901 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + 119896 (1 minus 120578)) 119902 + 119896119864 minus1

21199051205782

(3)

The beliefs of the manufacturer over the customerrsquosreservation price are 120585

119903 Obviously the manufacturer will

set 119901 = 120585119903 119902 and 120578 are the maximizer max

119902120578120587(119902 119901 120578) when

given 119901 According to the definition of RE Equilibrium theRE Equilibrium solution (119901 119902 120578 119903 120585

119903 120585prob)must meet

119903 = V minus (V minus 119904) 120585prob (4)

119901 = 120585119903 (5)

(119902 120578) = argmax119902120578

120587 (119902 119901 120578) (6)

120585prob = 119865 (119902) (7)

120585119903= 119903 (8)

Conditions (4) (5) and (6) indicate that the manufac-turer and customers will choose the action to maximizetheir own utility Conditions (7) and (8) can ensure that thesolution meets rational expectations hypothesis that is theactual situation of economic operation in line with peoplersquosexpectations

Discrete Dynamics in Nature and Society 5

The Rational Expectations Equilibrium makes

119901 = V minus (V minus 119904) 119865 (119902) (9)

The production pricing and green technology invest-ment decision model of the manufacturer with cap and tradepolicy and green technology investment is

max119902120578

120587 (119902 119901 120578)

st (1 minus 120578) 119902 le 119864

(10)

Defining 1205791(119902) = (1(1 minus 120578))[(119901 minus 119904)119865(119902) minus (119888 minus 119904)]

1205792(120578) = 119905120578119902 120579

1(119902) represents themargin profit of unit carbon

emission that is the profit gained by the manufacturerwith one unit carbon emission input in production 120579

2(120578)

represents the margin cost of unit carbon emission that isthe cost that the manufacturer invests in green technology toget one unit carbon emission reduction

Lemma 1 Given 119901 and (V minus 119904)119891(119902)119865(119902)119905 minus 1198962 gt 0 120587(119902 119901 120578)is a joint concave function of 120578 and 119902 in RE Equilibrium

Proof Given 119901 according to formula (3) 120597120587(119902 119901 120578)120597119902 =

(119901minus119904)119865(119902)minus(119888minus119904)minus(1minus120578)119896 1205972120587(119902 119901 120578)1205971199022 = minus(119901minus119904)119891(119902) lt0 120597120587(119902 119901 120578)120597120578 = 119902119896 minus 119905120578 1205972120587(119902 119901 120578)1205971205782 = minus119905 lt 0 and1205972120587(119902 119901 120578)120597119902120597120578 = 120597

2120587(119902 119901 120578)120597120578120597119902 = 119896

When 119905(V minus 119904)119891(119902)119865(119902) minus 1198962 gt 0 under RE Equilibriumwe get 119905(119901minus119904)119891(119902)minus1198962 gt 0 then we have 1205972120587(119902 119901 120578)120597119902120597120578 =1205972120587(119902 119901 120578)120597120578120597119902 = 119896 and

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205972120587 (119902 119901 120578)

1205971199022

1205972120587 (119902 119901 120578)

120597119902120597120578

1205972120587 (119902 119901 120578)

120597120578120597119902

1205972120587 (119902 119901 120578)

1205971205782

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 119905 (119901 minus 119904) 119891 (119902) minus 1198962

gt 0

(11)

The Hessian Matrix of the problem is negative definite andthen we can prove that 120587(119902 119901 120578) is joint concave function of119902 and 120578 when given 119901 and (V minus 119904)119891(119902)119865(119902)119905 minus 119896

2gt 0 This

completes the proof

As to the manufacturerrsquos optimal production quantity(denoted by 119902

lowast) pricing (denoted by 119901lowast) carbon trading

(denoted by 119890lowast) and green technology investment (denotedby 120578lowast) the following proposition is obtained

Proposition 2 If (Vminus119904)119891(119902)119865(119902)119905minus1198962 gt 0 themanufacturerrsquosoptimal production quantity (119902lowast) pricing (119901lowast) carbon trading(119890lowast) and green technology investment (120578lowast) strategies satisfy

1205791(119902lowast) = 1205792(120578lowast) = 119896

119901lowast= 119904 + (V minus 119904) 119865 (119902lowast)

119890lowast= (1 minus 120578

lowast) 119902lowastminus 119864

0 lt 120578lowastlt 1

(12)

Proof According to Lemma 1 if (Vminus 119904)119891(119902)119865(119902)119905 minus 1198962 gt 0 let120597120587(119902 119901 120578)120597119902 = 0 120597120587(119902 119901 120578)120597120578 = 0 and combine (9) and(2) we get

(119901 minus 119904) 119865 (119902) minus (119888 minus 119904) minus (1 minus 120578) 119896 = 0

119902119896 minus 119905120578 = 0

119901 = V minus (V minus 119904) 119865 (119902)

119890 = (1 minus 120578) 119902 minus 119864

0 lt 120578 lt 1

(13)

Solving the equations above can derive the optimalproduction pricing carbon trading and green technologyinvestment strategies of themanufacturerThis completes theproof

Proposition 2 shows that considering strategic customerbehavior the manufacturerrsquos optimal production quantitypricing carbon trading volume and green technology invest-ment strategywith cap and trade policy and green technologyinvestment exist and are unique The inherent implicationof the proposition is very intuitive 120579

1(119902) represents the

margin profit of unit carbon emission 1205792(120578) represents the

margin cost of unit carbon emission and 119896 is the unit carbonemission trading price which can be seen as marginal costor profit by carbon emission trading The optimal strategiesof the manufacturer are that the margin profit of unitcarbon emission is equal to the margin cost of unit carbonemission The marginal cost of getting unit carbon emissionfrom different way (green technology investment or buying)is equal The marginal profit of unit carbon emission fordifferent uses (production or sale) is equal

Substitute 119902lowast 119901lowast and 120578lowast into (3) we can obtain the max-imum expected profit of themanufacturer with cap and tradepolicy and green technology investment 120587(119902lowast 119901lowast 120578lowast) =

(119901lowastminus119904)(119902lowastminusint119902lowast

0119865(119909)119889119909)minus(119888minus119904+119896(1minus120578

lowast))119902lowast+119896119864minus(12)119905120578

lowast2

42 Quantity Commitment Scenario In traditional literatureon strategic customer behavior quantity commitment canimprove the manufacturerrsquos expected profit Quantity com-mitment refers to the action in which the manufacturerpromises customers that only a certain number of productsis produced and sold In low carbon supply chain settingcan the maximum expected profit be improved by quantitycommitment We attempt to answer the question in thispaper

We assume that the manufacturer can convince cus-tomers by appropriate means that they only can obtain 119902

units of the product in the wholesales period At this timethe strategic customers no longer need to anticipate theprobability of getting the product at salvage price When 119902

is given the probability of being able to get the product atprice 119904 is 119865(119902) The reservation price is 119901(119902) = Vminus (Vminus 119904)119865(119902)which also is the optimal pricing of the manufacturer Wehave 119890 = (1 minus 120578)119902 minus 119864 then we can obtain the expected profit

6 Discrete Dynamics in Nature and Society

function of themanufacturer with respect to 119902 and 120578 denotedby 120587119902(119902 120578)

120587119902(119902 120578) = (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + 119896 (1 minus 120578)) 119902 + 119896119864 minus1

21199051205782

(14)

So the manufacturerrsquos optimal production quantity andcarbon reduction rate in QC scenario are (119902

119902lowast 120578119902lowast) =

argmax119902ge00lt120578lt1

120587119902(119902 120578) the optimal price is 119901119902lowast = V minus (V minus

119904)119865(119902119902lowast) and the optimal carbon trading volume is 119890119902lowast =

(1 minus 120578119902lowast)119902119902lowastminus 119864

Lemma 3 Given 119902 the optimal green technology investmentstrategy of the manufacturer denoted by 120578119902lowast satisfies

120578119902lowastequiv 120578 (119902) =

119896119902

119905 0 lt 120578

119902lowastlt 1 (15)

Proof Wehave 120597120587119902(119902 120578)120597120578 = 119902119896minus119905120578 1205972120587119902(119902 120578)1205971205782 = minus119905 lt0 Let 120597120587119902(119902 120578)120597120578 = 0 we get 120578119902lowast equiv 120578(119902) = 119896119902119905 In additionaccording to the assumption in Section 3 we know 0 lt 120578

119902lowastlt

1 This completes the proof

Substituting 120578119902lowast = 120578(119902) into (14)

120587119902(119902) equiv 120587

119902(119902 120578 (119902))

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + 119896) 119902 + 119896119864 +11989621199022

2119905

(16)

Then the two-variable optimization problem is simplifiedas a single variable optimization problem

max119902ge0

120587119902(119902) (17)

Lemma4 If 1198962119905minus(Vminus119904)[3119891(119902)119865(119902)+1198911015840(119902)(119902minusint1199020119865(119909)119889119909)] lt

0 120587119902(119902) is a concave function of 119902

Proof The first-order and second-order derivatives of 120587119902(119902)are

119889120587119902(119902)

119889119902= (V minus 119904) [119865

2

(119902) minus 119891 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) +1198962

119905119902

1198892120587119902(119902)

1198891199022=1198962

119905minus (V minus 119904)

sdot [3119891 (119902) 119865 (119902) + 1198911015840(119902) (119902 minus int

119902

0

119865 (119909) 119889119909)] lt 0

(18)

Then we can obtain that 120587119902(119902) is a concave function of 119902This completes the proof

As to the manufacturerrsquos optimal production quantity inQC scenario (denoted by 119902119902lowast) the following proposition isobtained

Proposition 5 If 1198962119905 minus (V minus 119904)[3119891(119902)119865(119902) + 1198911015840(119902)(119902 minus

int119902

0119865(119909)119889119909)] lt 0 the optimal production quantity in QC

scenario (119902119902lowast) satisfies

(V minus 119904) [1198652

(119902) minus 119891 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) +1198962

119905119902 = 0

(19)

Proof It can be directly derived according to Lemma 4 Thiscompletes the proof

Proposition 5 shows that under certain condition theoptimal production quantity of the manufacturer with capand trade and green technology investment exist and areunique

43 The Effect of Quantity Commitment The effect of QCon the optimal strategies and the maximum expected profitof the manufacturer is analyzed by comparing the optimalstrategies and the maximum expected profit of the manufac-turer in RE Equilibrium scenario and QC scenario

Proposition 6 Consider 119902119902lowast lt 119902lowast 119901119902lowast gt 119901lowast 120578119902lowast lt 120578lowast

Proof In order to ensure that the manufacturerrsquos optimalstrategies in the two scenarios exist the condition 119896

2119905 lt

min(Vminus119904)119891(119902)119865(119902) (Vminus119904)[3119891(119902)119865(119902)+1198911015840(119902)(119902minusint1199020119865(119909)119889119909)]

must be held Then 120587119902(119902) is a concave function of 119902According to Proposition 2 we have 120579

1(119902lowast) = 1205792(120578lowast) = 119896

1205792(120578lowast) = 119896 can be written as 120578lowast = 119896119902

lowast119905 Then we also can

rearrange 1205791(119902lowast) = 119896 to (Vminus119904)1198652(119902lowast)minus(119888minus119904+119896)+(1198962119905)119902lowast = 0

We can obtain (119889120587119902(119902)119889119902)|

119902=119902lowast = (V minus 119904)[119865

2

(119902lowast) minus

119891(119902lowast)(119902lowastminus int119902lowast

0119865(119909)119889119909)] minus (119888 minus 119904 + 119896) + (119896

2119905)119902lowast= minus(V minus

119904)119891(119902lowast)(119902lowastminusint119902lowast

0119865(119909)119889119909) lt 0 Then we get 119902119902lowast lt 119902lowast Because

120578 = 119896119902119905 and 119901 = Vminus(Vminus119904)119865(119902) are held in the two scenarioswe can obtain 119901119902lowast gt 119901

lowast and 120578119902lowast lt 120578lowast according to 119902119902lowast lt 119902

lowastThis completes the proof

Proposition 6 shows that compared with RE Equilibriumscenario the manufacturerrsquos optimal production quantity islower the optimal pricing is higher and the optimal carbonreduction rate is lower in QC scenario

Proposition 7 Consider 120587119902(119902119902lowast 120578119902lowast) gt 120587(119902lowast 119901lowast 120578lowast)

Proof In order to ensure that the manufacturerrsquos optimalstrategies in the two scenarios exist the condition 119896

2119905 lt

min(Vminus119904)119891(119902)119865(119902) (Vminus119904)[3119891(119902)119865(119902)+1198911015840(119902)(119902minusint1199020119865(119909)119889119909)]

must be held Then 120587119902(119902) is a concave function of 119902Substitute 119902lowast 119901lowast and 120578

lowast= 119896119902

lowast119905 into (3) we have

120587(119902lowast 119901lowast 120578lowast) = 120587

119902(119902lowast) We know that 120587119902(119902) is a concave

function of 119902 and (119889120587119902(119902)119889119902)|119902=119902119902lowast = 0 Because of 119902119902lowast lt 119902lowast

Discrete Dynamics in Nature and Society 7

120587119902(119902lowast) lt 120587

119902(119902119902lowast) that is 120587(119902lowast 119901lowast 120578lowast) lt 120587

119902(119902119902lowast 120578119902lowast) This

completes the proof

Proposition 7 shows that QC strategy can improve themanufacturerrsquos maximum expected profit when green tech-nology investment and cap and trade policy are taken intoaccount

5 Decentralized Supply Chain Model

In this section the retailer and the manufacturer make deci-sions independently to maximize their own expected profitThe manufacturer is the Stackelberg leader and determinesthe wholesale price and the green technology investmentunder cap and trade policy The retailer is the follower anddetermines the retail price and the order quantity of theproduct

The sequence of events in this section is as follows firstthe manufacturer determines the optimal wholesale pricecarbon trading and green technology investment strategiessecond the retailer forms the belief of customersrsquo reservationprice and then decides the optimal retail price and orderquantity third the manufacture produces products anddelivers to the retailer fourth the customers form the beliefs120585prob of probability of the product sold at salvage price119904 according to the information of market price and thenform the reservation price 119903 fifth the customersrsquo demand issatisfied and the products are sold at full price 119901 finally allremaining products are sold to the external market at salvageprice 119904

51 The Optimal Strategies First we examine the decisionproblem of the retailer Recall that 119908 is the manufacturerrsquoswholesale price The retailerrsquos expected profit denoted by120587119903(119902 119901) is

120587119903(119902 119901) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909) minus (119908 minus 119904) 119902 (20)

Lemma 8 When 119901 is given 120587119903(119902 119901) is a concave function of119902

Proof When 119901 is given according to (20) we have120597120587119903(119902 119901)120597119902 = (119901 minus 119904)119865(119902) minus (119908 minus 119904) and 1205972120587119903(119902 119901)1205971199022 =

minus(119901 minus 119904)119891(119902) lt 0 Then we know that 120587119903(119902 119901) is a concavefunction of 119902 when 119901 is given This completes the proof

As to the retailerrsquos optimal strategies in RE Equilibriumwe can obtain the following proposition

Proposition 9 When the wholesale price is given the retailerrsquosoptimal order strategies (119902119903lowast) and optimal pricing strategies(119901119903lowast) with strategic customer behavior are

119902119903lowast= 119865minus1

(radic119908 minus 119904

V minus 119904) (21)

119901119903lowast= 119904 + radic(119908 minus 119904) (V minus 119904) (22)

Proof In RE Equilibrium (9) still holds Let 120597120587119903(119902 119901)120597119902 = 0and combining with (9) we have

119901 = V minus (V minus 119904) 119865 (119902)

(119901 minus 119904) 119865 (119902) minus (119908 minus 119904) = 0

(23)

According to Lemma 8 we can obtain the optimal orderand pricing strategies of the retailer by solving the equationsabove This completes the proof

Proposition 9 shows that the retailerrsquos optimal order andpricing strategies in REEquilibriumwith cap and trade policyexist and are unique

Second we investigate the manufacturerrsquos decision prob-lem The expected profit function of the manufacturer withgreen technology investment and cap and trade policy is120587119898(119908 119890 120578) = (119908 minus 119888)119902 minus 119896119890 minus (12)119905120578

2According to (21) with green technology investment and

cap and trade policy in wholesale price contract the optimalorder quantity of the retailer 119902119903lowast and the optimal wholesaleprice of the manufacturer 119908119898lowast is a one-to-one relationship

119908 = 119904 + (V minus 119904) 1198652

(119902) (24)

Combined with 119890 = (1 minus 120578)119902 minus 119864 the manufacturerrsquosexpected profit function can be converted into

120587119898(119902 120578) = [(V minus 119904) 119865

2

(119902) minus (119888 minus 119904 + (1 minus 120578) 119896)] 119902

+ 119896119864 minus1

21199051205782

(25)

When using wholesale price contract the total profit ofthe supply chain denoted by 120587sc

(119902 119901 119890 120578) is

120587sc(119902 119901 119890 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909) minus (119888 minus 119904) 119902

minus 119896119890 minus1

21199051205782

(26)

According to (9) and 119890 = (1 minus 120578)119902 minus 119864 the total supplychain profit function in RE Equilibrium can be simplified as

120587sc(119902 120578) = (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578) 119896) 119902 + 119896119864 minus1

21199051205782

(27)

Lemma 10 Given 119902 the optimal green technology investmentstrategy of the manufacturer denoted by 120578119898lowast satisfies

120578119898lowast

equiv 120578 (119902) =119896119902

1199050 lt 120578119898lowast

lt 1 (28)

Proof The first-order and second-order partial derivatives of120587119898(119902 120578) with respect to 120578 are 120597120587119898(119902 120578)120597120578 = 119902119896 minus 119905120578 and

1205972120587119898(119902 120578)120597120578

2= minus119905 lt 0 Let 120597120587119898(119902 120578)120597120578 = 0 we get

120578119898lowast

equiv 120578(119902) = 119896119902119905 In addition according to the assumptionin Section 3 we know 0 lt 120578

119898lowastlt 1 This completes the

proof

8 Discrete Dynamics in Nature and Society

Substituting 120578119898lowast = 120578(119902) into (25)

120587119898(119902) equiv 120587

119898(119902 120578 (119902))

= [(V minus 119904) 1198652

(119902) minus (119888 minus 119904 + 119896)] 119902 + 119896119864

+11989621199022

2119905

(29)

Then the two-variable optimization problem is simplifiedas a single variable optimization problem

max119902ge0

120587119898(119902) (30)

Lemma 11 If 1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus

1199021198912(119902)] lt 0 120587119898(119902) is a concave function of 119902

Proof The first-order and second-order derivatives of 120587119898(119902)with respect to 119902 are 119889120587

119898(119902)119889119902 = (V minus 119904)[119865

2

(119902) minus

2119902119891(119902)119865(119902)] minus (119888 minus 119904 + 119896) + (1198962119905)119902 and 119889

2120587119898(119902)119889119902

2=

1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus 1199021198912(119902)] lt 0 Then

we have that 120587119898(119902) is a concave function of 119902 This completesthe proof

In decentralized supply chain as to the manufacturerrsquosoptimal wholesale price (denoted by 119908

119898lowast) the followingproposition is obtained

Proposition 12 If 1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus1199021198912(119902)] lt 0 the manufacturerrsquos optimal wholesale price in

decentralized supply chain (119908119898lowast) satisfies 119908119898lowast = 119904 + (V minus119904)1198652

(119902119903lowast) where 119902119903lowast represents the retailerrsquos optimal order

quantity and satisfies

(V minus 119904) [1198652

(119902) minus 2119902119891 (119902) 119865 (119902)] minus (119888 minus 119904 + 119896) +1198962

119905119902

= 0

(31)

Proof According to Lemma 11 we can obtain that 119902119903lowastwhichmaximizes themanufacturerrsquos expected profit satisfies(31) Combining with (24) we know that the manufacturerrsquosoptimal wholesale price satisfies 119908119898lowast = 119904 + (V minus 119904)119865

2

(119902119903lowast) at

this time This completes the proof

Proposition 12 shows that the optimal commitment quan-tity of the manufacturer in decentralized supply chain withgreen technology investment and cap and trade policy existand are unique The optimal wholesale price does not relateto the cap allocated by the government but relates to thecarbon trading price This is because the production of themanufacturer is not affected by the cap but is affected bythe margin cost of the product However the margin cost ofthe product varies with the change of carbon trading pricewhen carbon trading is allowed The change results in thatthe optimal commitment quantity is not related to the cap butrelate to the carbon trading price

Substitute 119908119898lowast into (22) we derive the retailerrsquos optimalpricing in decentralized situation119901119903lowast = 119904+radic(119908119898lowast minus 119904)(V minus 119904)

According to Lemma 10 we get the optimal green technologyinvestment of the manufacturer 120578119898lowast = (119896119905)119902119903lowast

Substitute 119902119903lowast 119901119903lowast 119908119898lowast and 120578119898lowast into 120587

119903(119902 119901) 120587119898(119902)

and 120587sc(119902 120578) we obtain the maximum expected profit of

the retailer the manufacturer and the supply chain indecentralized channel

120587119903(119902119903lowast 119901119903lowast) = (119901

119903lowastminus 119904) (119902

119903lowastminus int

119902119903lowast

0

119865 (119909) 119889119909)

minus (119908119898lowast

minus 119904) 119902119903lowast

120587119898(119902119903lowast) = [(V minus 119904) 119865

2

(119902119903lowast) minus (119888 minus 119904 + 119896)] 119902

119903lowast

+ 119896119864 +1198962119902119903lowast2

2119905

120587sc(119902119903lowast 120578119898lowast) = (V minus 119904) 119865 (119902119903lowast) (119902119903lowast minus int

119902119903lowast

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578119898lowast) 119896) 119902119903lowast+ 119896119864

minus1

2119905120578119898lowast2

(32)

52 The Effect of Decentralization We examine the impact ofdecentralization on the supply chain optimal strategies andmaximumexpected profitwhen green technology investmentis taken into consideration by comparing the optimal strate-gies and the maximum expected profit in centralized anddecentralized channel with green technology investment andcap and trade policy

As the effect of decentralization of supply chain we canobtain the following proposition

Proposition 13 Consider 119902119903lowast lt 119902119902lowast

lt 119902lowast 119901119903lowast gt 119901

119902lowastgt 119901lowast

120578119898lowast

lt 120578119902lowastlt 120578lowast

Proof In Proposition 6 119902119902lowast lt 119902lowast 119901119902lowast gt 119901

lowast and 120578119902lowast lt 120578lowast

have been provedTherefore we only need to prove 119902119903lowast lt 119902119902lowast119901119903lowastgt 119901119902lowast and 120578119898lowast lt 120578119902lowast

Define 119867(119902) = 120587119902(119902) minus 120587119898(119902) then

119867(119902)

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

minus (V minus 119904) 1199021198652

(119902)

1198671015840(119902)

= (V minus 119904) 119891 (119902) [2119902119865 (119902) minus (119902 minus int119902

0

119865 (119909) 119889119909)]

(33)

Define 119866(119902) = 2119902119865(119902) minus (119902 minus int119902

0119865(119909)119889119909) then 1198661015840(119902) =

119865(119902) minus 2119902119891(119902) Because 119866(0) = 0 1198661015840(0) = 1 and 119865 satisfiesIFR we know that the value of 119866(119902) starts from 0 increasingfirst and then decreasing in 119902 That is 119866(119902) = 0 has uniquesolution and we denote it as So when 119902 lt 119866(119902) gt 0 that

Discrete Dynamics in Nature and Society 9

is119867(119902) is increasing in 119902 when 119902 gt 119866(119902) lt 0 that is119867(119902)is decreasing in 119902 Then we have that119867(119902) is a quasi-concavefunction of 119902 and 1198661015840 () = 119865() minus 2119891() lt 0 According tothe analysis above we know that1198671015840() = 0 that is 2119865() =( minus int

0119865(119909)119889119909) Substitute into 119889120587119902(119902)119889119902 we have

119889120587119902(119902)

119889119902

100381610038161003816100381610038161003816100381610038161003816119902=

= (V minus 119904) [1198652

() minus 119891 () ( minus int

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) +1198962

119905

= (V minus 119904) 119865 () [119865 () minus 2119891 ()] minus (119888 minus 119904 + 119896)

+1198962

119905lt (V minus 119904) 119865 () [119865 () minus 2119891 ()] lt 0

(34)

According to Lemma 4 we have that 120587119902(119902) is a concavefunction of 119902 so we get 119902119902lowast lt Because 2119865() = ( minus

int

0119865(119909)119889119909) and119867(119902) is a quasi-concave function of 119902 thenwe

get that 119889120587119898(119902)119889119902 lt 119889120587119902(119902)119889119902 if 119902 lt and 119889120587119898(119902)119889119902 gt

119889120587119902(119902)119889119902 if 119902 gt Therefore the only possible orderings for

119902119902lowast 119902119903lowast and are lt 119902

119902lowastlt 119902119903lowast and 119902119903lowast lt 119902

119902lowastlt We

have proved 119902119902lowast lt so we have 119902119903lowast lt 119902119902lowast We get 119901119903lowast gt 119901119902lowastbecause 119901 and 119902 satisfy 119901 = V minus (V minus 119904)119865(119902) in two situationsWe get 120578119898lowast lt 120578

119902lowast because 120578 and 119902 satisfy 120578 = 119896119902119905 in twosituations This completes the proof

Proposition 13 shows that for the decision of a decen-tralized supply chain with green technology investment theoptimal production quantity is increasing the optimal priceis decreasing and the green technology investment (iecarbon reduction rate) is increasing respectively in the threesituations of decentralized supply chain QC scenario and REEquilibrium scenario

The reason is that themanufacturer prefers higher whole-sale price (ie lower production and less green technologyinvestment) to guarantee its ownprofitmaximization thoughthis may harm the retailer even the whole supply chain

In Proposition 7 we have found 120587119902(119902119902lowast 120578119902lowast) gt

120587(119902lowast 119901lowast 120578lowast) that is the manufacturerrsquos maximum profit in

QC scenario is always greater than that in RE EquilibriumTherefore the section analyzes the effect of decentralizationon supply chain performance based on the QC scenario toprovide benchmark for designing of subsequent coordinationcontract

As to the effect of decentralization on themaximumprofitof supply chain we have the following proposition

Proposition 14 Consider 120587sc(119902119903lowast 120578119898lowast) lt 120587119902(119902119902lowast 120578119902lowast)

Proof The optimal green technology investment in decen-tralized supply chain and QC scenario are 120578(119902) = 119896119902119905

according to Lemmas 3 and 10 So the profit function in thetwo situations above can be converted into a same singlevariable function with respect to 119902 that is (16)

According to Lemma 4 we know that 120587119902(119902) is a concavefunction and reaches the maximum value at 119902119902lowast Accordingto Proposition 13 we have 119902119903lowast lt 119902

119902lowast then we get 120587119902(119902119903lowast) lt120587119902(119902119902lowast) that is 120587sc

(119902119903lowast 120578119898lowast) lt 120587

119902(119902119902lowast 120578119902lowast) This completes

the proof

Proposition 14 shows that the maximum profit of decen-tralized supply chain is always lower than that in QC scenariowhich indicates that the profit of the decentralized supplychain can get higher This is the benchmark of supply chaincoordination

6 Supply Chain Coordination

This section coordinates the supply chain with green tech-nology investment and cap and trade policy based on theQC scenario to improve the profit of the decentralized supplychain

61 Coordination Contract Based on Revenue and CostsSharing Contract With revenue sharing contract we assumethat 120601 (0 le 120601 le 1) represents the ratio of revenue kept by theretailer and (1 minus 120601) represents the ratio of revenue deliveredto the manufacturer

The retailerrsquos expected profit function with revenue shar-ing contract denoted by 120587119903

119904(119902 119901 120601) is

120587119903

119904(119902 119901 120601) = 120601 (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119908 minus 120601119904) 119902

(35)

Themanufacturerrsquos expected profit functionwith revenuesharing contract denoted by 120587119898

119904(119908 119890 120578 120601) is

120587119898

119904(119908 119890 120578 120601) = (119908 minus 119888) 119902 + (1 minus 120601)

sdot [int

119902

0

[119901119909 + 119904 (119902 minus 119909)] 119891 (119909) 119889119909

+ int

infin

119902

119901119902119891 (119909) 119889119909] minus 119896119890 minus1

21199051205782

(36)

Substitute 119890 = (1 minus 120578)119902 minus 119864 into the equation above andwe have

120587119898

119904(119908 120578 120601) = (1 minus 120601) (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

+ (119908 minus 119888 minus (1 minus 120578) 119896 + (1 minus 120601) 119904) 119902

+ 119896119864 minus1

21199051205782

(37)

The total expected profit of the whole supply chaindenoted by 120587sc

119904(119902 119901 120578) is

120587sc119904(119902 119901 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578) 119896) 119902 + 119896119864 minus1

21199051205782

(38)

10 Discrete Dynamics in Nature and Society

The subscript 119904 represents the situation of supply chaincoordination using revenue sharing contracts

As to the retailerrsquos optimal order strategy (denoted by119902lowast

119904) and pricing strategy (denoted by 119901lowast

119904) of the supply chain

with revenue sharing contract the following proposition isobtained

Proposition 15 Under revenue sharing contracts the retailerrsquosoptimal order strategy (119902lowast

119904) and optimal pricing strategy (119901lowast

119904)

are

119902lowast

119904= 119865minus1

(radic119908 minus 120601119904

120601 (V minus 119904))

119901lowast

119904= 119904 + radic

(119908 minus 120601119904) ((V minus 119904))120601

(39)

Proof According to the definition of RE Equilibrium wealso have 119901 = V minus (V minus 119904)119865(119902) under revenue sharingcontracts

The first-order and second-order derivative of 120587119903119904(119902 119901 120601)

with respect to 119902 are 120597120587119903

119904(119902 119901 120601)120597119902 = 120601(119901 minus 119904)119865(119902) minus

(119908 minus 120601119904) and 1205972120587119903

119904(119902 119901 120601)120597119902

2= minus120601(119901 minus 119904)119891(119902) lt

0 So given 119901 the retailerrsquos expected profit function is aconcave function of 119902 Let 120597120587119903

119904(119902 119901 120601)120597119902 = 0 and combine

with 119901 = V minus (V minus 119904)119865(119902) we can solve the retailerrsquos optimalstrategies under revenue sharing contractThis completes theproof

Proposition 15 shows that the optimal order and pricingstrategies of the retailer based on revenue sharing contract inRE Equilibrium exist and are uniqueThis is also the retailerrsquosoptimal response function with respect to 119908

Substituting (39) into (43)

120587119898

119904(119902 120578 120601)

= (1 minus 120601) (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ [120601 (V minus 119904) 1198652

(119902) minus (119888 minus 119904 + (1 minus 120578) 119896)] 119902 + 119896119864

minus1

21199051205782

120597120587119898

119904(119902 120578 120601)

120597120578= 119902119896 minus 119905120578

1205972120587119898

119904(119902 120578 120601)

1205971205782= minus119905 lt 0

(40)

Given 119902 and 120601 the manufacturerrsquos green technologyinvestment strategies with revenue sharing contracts exist

and are unique Let 120597120587119898119904(119902 120578 120601)120597120578 = 0 we get 120578119898lowast

119904equiv 120578(119902) =

119902119896119905 and substitute it into 120587119898119904(119902 120578 120601)

120587119898

119904(119902 120601)

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ 120601 (V minus 119904) 119865 (119902) [119902119865 (119902) minus (119902 minus int119902

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) 119902 + 119896119864 +11989621199022

2119905

(41)

Proposition 16 The supply chain with green technologyinvestment and cap and trade policy can not be coordinated byrevenue sharing contract

Proof The derivative of 120587119898119904(119902 120601) with respect to 119902 can be

arranged as follows

120597120587119898

119904(119902 120601)

120597119902=120597120587119902(119902)

120597119902+ 120601(

120597120587119898(119902)

120597119902minus120597120587119902(119902)

120597119902) (42)

Only when 120601 = 0 are the optimal strategies of supplychain under revenue sharing contracts equal to that in QCscenario Let 119908lowast

119904represent the optimal wholesale price of

the manufacture and 119902lowast119904represent the retailerrsquos optimal order

quantity Then 120587119903

119904(119902 119901 120601) = minus119908

lowast

119904119902lowast

119904lt 0 So the revenue

sharing contract cannot realize the supply chain coordinationat this time This completes the proof

Proposition 16 shows that revenue sharing contractscan not realize the supply chain coordination with greentechnology investment and cap and trade policy The rea-son is that the manufacturer undertakes all the costs ofgreen technology investment when using revenue sharingcontracts thus making the manufacturer tend to invest lessin green technology Only when the manufacturer owns allthe revenue will themanufacturerrsquos optimal green technologyinvestment under revenue sharing contracts be equal to thatin QC scenario However the maximum profit of the retaileris negative at this time and it will not participate

62 Coordination Contract Based on Revenue Sharing-CostSharing Contract Proposition 16 states that the supply chainwith green technology investment and cap and trade policycan not be coordinated by revenue sharing contracts becausethe costs of green technology investment are undertaken bythe manufacturer and the game of retailer and customersis characterized by RE Equilibrium So we consider costssharing of green technology investment and carbon tradingand quantity commitment made by the manufacturer basedon revenue sharing contracts Let 120601 (0 lt 120601 le 1) represent theratio of revenue kept by the retailer and let (1minus120601) represent theratio of revenue delivered to the manufacturer 120572 (0 le 120572 le 1)represents the ratio of costs of green technology investmentand carbon trading undertaken by the manufacturer and(1 minus 120572) represents the ratio of costs of green technologyinvestment and carbon trading delivered to the retailer

Discrete Dynamics in Nature and Society 11

The retailerrsquos expected profit function with revenuesharing-cost sharing contract is

120587119903

120572(119902 120601 120572) = 120601 (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119908 minus 120601119904) 119902

minus (1 minus 120572) 119896 [(1 minus 120578) 119902 minus 119864]

minus1

2(1 minus 120572) 119905120578

2

(43)

We have 119890 = (1minus120578)119902minus119864 then themanufacturerrsquos expectedprofit function denoted by 120587119898

120572(119908 120578 120601 120572) is

120587119898

120572(119908 120578 120601 120572)

= (1 minus 120601) (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ (119908 minus 119888 minus 120572 (1 minus 120578) 119896 + (1 minus 120601) 119904) 119902 + 120572119896119864

minus1

21205721199051205782

(44)

The subscript 120572 represents the situation that using rev-enue sharing-cost sharing contract coordinates the supplychain

Proposition 17 If 0 le 120582 le 1 the parameters of revenuesharing-cost sharing (119908 120601 120572) satisfy

119908 = 120582119888 + (1 minus 120578) 119896

120601 = 120582

120572 = 1 minus 120582

(45)

The supply chainwith green technology investment and cap andtrade policy is coordinated

Proof When the parameters of revenue sharing-cost sharingsatisfy (45) the expected profit function of the retailer andthe manufacturer can be written as follows

120587119903

120572(119902 119901 120601 120572) = 120582120587

119902(119902 120578)

120587119898

120572(119908 120578 120601 120572) = (1 minus 120582) 120587

119902(119902 120578)

(46)

The optimal order of decentralized supply chain is equalto that of QC scenario Substitute 119901 = V minus (V minus 119904)119865(119902) into120587sc119888(119902 119901 120578) we get that the profit function of decentralized

supply chain at this time is the same as that of QC scenarioTherefore when the parameters of revenue sharing-costsharing (119908 120601 120572) satisfy (45) the decentralized supply chaincoordination is realizedThemanufacturer earns all the profitif 120582 = 0 and the retailer earns all the profit if 120582 = 1Arbitrary allocation of the supply chain profit between themanufacturer and the retailer is allowed This completes theproof

Proposition 17 shows that revenue sharing-cost sharingcontract can realize the supply chain coordination on condi-tion that the manufacturer makes a quantity commitment to

the customers The optimal production quantity and greentechnology investment in decentralized situation are lowerthan that in QC scenario The quantity commitment of themanufacturer can increase the green technology investmentand production quantity and also reduce the retailer priceThat is the quantity commitment is beneficial for the man-ufacturer the retailer and customers Thus the quantitycommitment is credible to customers now

7 Conclusions and Suggestions for FurtherResearch

This paper investigates the production pricing carbon trad-ing and green technology investment strategies and thecoordination of low carbon supply chain made up of a lowcarbon manufacturer and a retailer The low carbon manu-facturer produces one product under cap and trade policyand performs green technology investment to reduce carbonemissions in production process The retailer orders fromthemanufacturer and distributes the product to homogenouscustomers To the best of our knowledge this is the firststudy on the management of low carbon supply chainswith strategic customer behaviorThis paper provides severalinteresting observations

Observation 1 There are unique optimal production pricingcarbon trading and green technology investment strategiesin centralized (include RE Equilibrium scenario and QCscenario) and decentralized supply chain Therefore byderiving the optimal production pricing carbon trading andgreen technology investment strategies in different situationsthe low carbon manufacturer and the retailer can behaveappropriately based on our findings to maximize their profit

Observation 2 Our findings also show that the centralizedsupply chain can improve its performance by quantity com-mitment However the manufacturerrsquos production and greentechnology investment are reduced while the optimal priceis increased at this time Therefore quantity commitment isbeneficial for the supply chain but is harmful to customers

Observation 3 We also show that the performance of decen-tralized supply chain is lower than that in QC scenario Wefind that the decentralization reduces the manufacturersquos opti-mal production quantity and green technology investmentand increases the retailerrsquos optimal price Besides we find thatthe optimal production quantity is increasing the optimalprice is decreasing and the green technology investment isincreasing respectively in the three situations of decentral-ized supply chain QC scenario andREEquilibrium scenario

Observation 4 We demonstrate that the traditional revenuesharing contracts can not realize the low carbon supply chaincoordination when green technology investment is takenin consideration We also find that the revenue sharing-cost sharing contract realizes the decentralized supply chain

12 Discrete Dynamics in Nature and Society

coordination when the manufacturer makes a quantity com-mitment It is beneficial to the manufacturer the retailerand customers if themanufacturermakes a quantity commit-ment So the quantity commitment is credible to the customerat this time

We assume the customers are homogeneous strategiccustomers But themyopic customers and strategic customersoften coexist and strategic customers also have differenttolerance in real-life Extending this assumption will havea knock-on effect on supply chain decisions So one keyresearch direction is to consider that the customers aremixed customers myopic customers and strategic customersSecond we assume that the supply chain only producesone product While this simplified setting provides someinteresting insights we have to acknowledge that a manufac-turer usually produces two or more productsThese productswill substitute or complement each other So one importantextension of our work is to assume that the manufacturerproduces two or more products which have different unitcarbon emissions in production

Notations

119863 The random demand and119863 ge 0

119891(sdot) Probability density function of119863119865(sdot) Distribution function of119863 We assume

that 119865 satisfy IFR and define 119865 = 1 minus 119865

119888 Unit production cost119904 Unit salvage price of the product which

is an exogenous variableV The customersrsquo utility from consuming

the product119908 Unit wholesale price119901 Unit retail selling price We assume that

it can be observed by the customers119902 The manufacturerrsquos production

quantitythe retailerrsquos order quantityWe assume that it cannot be observedby the customers

119903 The customersrsquo reservation price whichis customersrsquo private information andcannot be observed by the retailer

120585119903 The beliefs of the manufacturer over the

customerrsquo reservation price120585prob The beliefs of customers over their

chances of obtaining the product atphase two

119864 Initial free carbon emission allocated bygovernment

119896 Unit carbon emission trading price119890 Carbon trading volume120578 Carbon emissions reduction rate

Conflict of Interests

The authors declare no conflict of interests

Acknowledgments

This research is partially supported by National Natural Sci-ence Foundation of China (nos 71432003 and 71272128) theProgram forNewCentury Excellent Talents inUniversity (noNCET-12-0087) and Specialized Research Fund for the Doc-toral Program of Higher Education (no 20130185110006)

References

[1] R Alley T Berntsen N L Bindoff et al ldquoSummary forpolicymakersrdquo in Climate Change 2007 The Physical ScienceBasis vol 6 pp 1ndash18 IPCC 2007

[2] X Chen and G Hao ldquoSustainable pricing and productionpolicies for two competing firms with carbon emissions taxrdquoInternational Journal of Production Research vol 53 pp 6408ndash6420 2015

[3] X Chen S Benjaafar and A Elomri ldquoThe carbon-constrainedEOQrdquo Operations Research Letters vol 41 no 2 pp 172ndash1792013

[4] N O Keohane ldquoCap and trade rehabilitated using tradablepermits to control US greenhouse gasesrdquo Review of Environ-mental Economics and Policy vol 3 no 1 pp 42ndash62 2009

[5] B Liu M Holmbom A Segerstedt and W Chen ldquoEffectsof carbon emission regulations on remanufacturing decisionswith limited information of demand distributionrdquo InternationalJournal of Production Research vol 53 no 2 pp 532ndash548 2015

[6] L Zhu and Y Fan ldquoA real options-based CCS investmentevaluation model case study of Chinarsquos power generationsectorrdquo Applied Energy vol 88 no 12 pp 4320ndash4333 2011

[7] XWang and LDu ldquoStudy on carbon capture and storage (CCS)investment decision-making based on real options for Chinarsquoscoal-fired power plantsrdquo Journal of Cleaner Production vol 112part 5 pp 4123ndash4131 2016

[8] W Jiang and X Chen ldquoOptimal strategies for manufacturerwith strategic customer behavior under carbon emissions-sensitive randomdemandrdquo Industrial Management amp Data Sys-tems 2015

[9] Y Levin J McGill and M Nediak ldquoDynamic pricing in thepresence of strategic consumers and oligopolistic competitionrdquoManagement Science vol 55 no 1 pp 32ndash46 2009

[10] A K Parlakturk ldquoThe value of product variety when sellingto strategic consumersrdquo Manufacturing amp Service OperationsManagement vol 14 no 3 pp 371ndash385 2012

[11] X Su and F Zhang ldquoOn the value of commitment andavailability guarantees when selling to strategic consumersrdquoManagement Science vol 55 no 5 pp 713ndash726 2009

[12] W Elmaghraby S A Lippman C S Tang and R Yin ldquoWillmore purchasing options benefit customersrdquo Production andOperations Management vol 18 no 4 pp 381ndash401 2009

[13] W Jiang and X Chen ldquoManufacturersquos production and pricingstrategies with carbon tax policy and strategic customer behav-iorrdquoManagement Science and Engineering vol 9 no 1 pp 30ndash35 2015

[14] I Dobos ldquoThe effects of emission trading on production andinventories in the Arrow-Karlin modelrdquo International Journalof Production Economics vol 93-94 pp 301ndash308 2005

[15] X Chen C K Chan and Y C E Lee ldquoResponsible productionpolicies with substitution and carbon emissions tradingrdquo Jour-nal of Cleaner Production 2015

Discrete Dynamics in Nature and Society 13

[16] X Chen and X Wang ldquoEffects of carbon emission reductionpolicies on transportation mode selections with stochasticdemandrdquo Transportation Research Part E Logistics and Trans-portation Review 2015

[17] X Chang H Xia H Zhu T Fan and H Zhao ldquoProductiondecisions in a hybrid manufacturingndashremanufacturing systemwith carbon cap and trade mechanismrdquo International Journal ofProduction Economics vol 162 pp 160ndash173 2015

[18] J Zhao B F Hobbs and J-S Pang ldquoLong-run equilibriummodeling of emissions allowance allocation systems in electricpowermarketsrdquoOperations Research vol 58 no 3 pp 529ndash5482010

[19] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2 pp519ndash527 2011

[20] A Toptal H Ozlu and D Konur ldquoJoint decisions on inventoryreplenishment and emission reduction investment under dif-ferent emission regulationsrdquo International Journal of ProductionResearch vol 52 no 1 pp 243ndash269 2014

[21] A S Manikas and J R Kroes ldquoA newsvendor approach tocompliance and production under cap and trade emissionsregulationrdquo International Journal of Production Economics vol159 pp 274ndash284 2015

[22] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineering vol10 no 1 pp 99ndash116 2013

[23] X Xu W Zhang P He and X Xu ldquoProduction and pricingproblems in make-to-order supply chain with cap-and-traderegulationrdquo Omega 2015

[24] Y Zhen X Yang and W Jiang ldquoThe impact of transport modeand carbon policy on low-carbon retailerrdquoDiscrete Dynamics inNature and Society vol 2015 Article ID 964305 12 pages 2015

[25] S Swami and J Shah ldquoChannel coordination in green supplychainmanagementrdquo Journal of the Operational Research Societyvol 64 no 3 pp 336ndash351 2013

[26] R H Coase ldquoDurability andmonopolyrdquoThe Journal of Law andEconomics vol 15 no 1 pp 143ndash149 1972

[27] X Su ldquoIntertemporal pricing with strategic customer behaviorrdquoManagement Science vol 53 no 5 pp 726ndash741 2007

[28] D Zhang and W L Cooper ldquoManaging clearance sales in thepresence of strategic customersrdquo Production and OperationsManagement vol 17 no 4 pp 416ndash431 2008

[29] S Whang ldquoDemand uncertainty and the Bayesian effect inmarkdown pricing with strategic customersrdquoManufacturing ampService Operations Management vol 17 no 1 pp 66ndash77 2015

[30] J Du J Zhang and G Hua ldquoPricing and inventory manage-ment in the presence of strategic customers with risk preferenceand decreasing valuerdquo International Journal of Production Eco-nomics vol 164 pp 160ndash166 2015

[31] X Su and F Zhang ldquoStrategic customer behavior commitmentand supply chain performancerdquo Management Science vol 54no 10 pp 1759ndash1773 2008

[32] H Yang ldquoImpact of discounting and competition on benefit ofdecentralization with strategic customersrdquo Operations ResearchLetters vol 40 no 2 pp 123ndash127 2012

[33] D Yang E Qi and Y Li ldquoQuick response and supply chainstructure with strategic consumersrdquo Omega vol 52 pp 1ndash142015

[34] C drsquoAspremont and A Jacquemin ldquoCooperative and nonco-operative R amp D in duopoly with spilloversrdquo The AmericanEconomic Review vol 78 no 5 pp 1133ndash1137 1988

[35] J F Muth ldquoRational expectations and the theory of pricemovementsrdquo Econometrica vol 29 no 3 pp 315ndash335 1961

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

The Rational Expectations Equilibrium makes

119901 = V minus (V minus 119904) 119865 (119902) (9)

The production pricing and green technology invest-ment decision model of the manufacturer with cap and tradepolicy and green technology investment is

max119902120578

120587 (119902 119901 120578)

st (1 minus 120578) 119902 le 119864

(10)

Defining 1205791(119902) = (1(1 minus 120578))[(119901 minus 119904)119865(119902) minus (119888 minus 119904)]

1205792(120578) = 119905120578119902 120579

1(119902) represents themargin profit of unit carbon

emission that is the profit gained by the manufacturerwith one unit carbon emission input in production 120579

2(120578)

represents the margin cost of unit carbon emission that isthe cost that the manufacturer invests in green technology toget one unit carbon emission reduction

Lemma 1 Given 119901 and (V minus 119904)119891(119902)119865(119902)119905 minus 1198962 gt 0 120587(119902 119901 120578)is a joint concave function of 120578 and 119902 in RE Equilibrium

Proof Given 119901 according to formula (3) 120597120587(119902 119901 120578)120597119902 =

(119901minus119904)119865(119902)minus(119888minus119904)minus(1minus120578)119896 1205972120587(119902 119901 120578)1205971199022 = minus(119901minus119904)119891(119902) lt0 120597120587(119902 119901 120578)120597120578 = 119902119896 minus 119905120578 1205972120587(119902 119901 120578)1205971205782 = minus119905 lt 0 and1205972120587(119902 119901 120578)120597119902120597120578 = 120597

2120587(119902 119901 120578)120597120578120597119902 = 119896

When 119905(V minus 119904)119891(119902)119865(119902) minus 1198962 gt 0 under RE Equilibriumwe get 119905(119901minus119904)119891(119902)minus1198962 gt 0 then we have 1205972120587(119902 119901 120578)120597119902120597120578 =1205972120587(119902 119901 120578)120597120578120597119902 = 119896 and

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205972120587 (119902 119901 120578)

1205971199022

1205972120587 (119902 119901 120578)

120597119902120597120578

1205972120587 (119902 119901 120578)

120597120578120597119902

1205972120587 (119902 119901 120578)

1205971205782

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 119905 (119901 minus 119904) 119891 (119902) minus 1198962

gt 0

(11)

The Hessian Matrix of the problem is negative definite andthen we can prove that 120587(119902 119901 120578) is joint concave function of119902 and 120578 when given 119901 and (V minus 119904)119891(119902)119865(119902)119905 minus 119896

2gt 0 This

completes the proof

As to the manufacturerrsquos optimal production quantity(denoted by 119902

lowast) pricing (denoted by 119901lowast) carbon trading

(denoted by 119890lowast) and green technology investment (denotedby 120578lowast) the following proposition is obtained

Proposition 2 If (Vminus119904)119891(119902)119865(119902)119905minus1198962 gt 0 themanufacturerrsquosoptimal production quantity (119902lowast) pricing (119901lowast) carbon trading(119890lowast) and green technology investment (120578lowast) strategies satisfy

1205791(119902lowast) = 1205792(120578lowast) = 119896

119901lowast= 119904 + (V minus 119904) 119865 (119902lowast)

119890lowast= (1 minus 120578

lowast) 119902lowastminus 119864

0 lt 120578lowastlt 1

(12)

Proof According to Lemma 1 if (Vminus 119904)119891(119902)119865(119902)119905 minus 1198962 gt 0 let120597120587(119902 119901 120578)120597119902 = 0 120597120587(119902 119901 120578)120597120578 = 0 and combine (9) and(2) we get

(119901 minus 119904) 119865 (119902) minus (119888 minus 119904) minus (1 minus 120578) 119896 = 0

119902119896 minus 119905120578 = 0

119901 = V minus (V minus 119904) 119865 (119902)

119890 = (1 minus 120578) 119902 minus 119864

0 lt 120578 lt 1

(13)

Solving the equations above can derive the optimalproduction pricing carbon trading and green technologyinvestment strategies of themanufacturerThis completes theproof

Proposition 2 shows that considering strategic customerbehavior the manufacturerrsquos optimal production quantitypricing carbon trading volume and green technology invest-ment strategywith cap and trade policy and green technologyinvestment exist and are unique The inherent implicationof the proposition is very intuitive 120579

1(119902) represents the

margin profit of unit carbon emission 1205792(120578) represents the

margin cost of unit carbon emission and 119896 is the unit carbonemission trading price which can be seen as marginal costor profit by carbon emission trading The optimal strategiesof the manufacturer are that the margin profit of unitcarbon emission is equal to the margin cost of unit carbonemission The marginal cost of getting unit carbon emissionfrom different way (green technology investment or buying)is equal The marginal profit of unit carbon emission fordifferent uses (production or sale) is equal

Substitute 119902lowast 119901lowast and 120578lowast into (3) we can obtain the max-imum expected profit of themanufacturer with cap and tradepolicy and green technology investment 120587(119902lowast 119901lowast 120578lowast) =

(119901lowastminus119904)(119902lowastminusint119902lowast

0119865(119909)119889119909)minus(119888minus119904+119896(1minus120578

lowast))119902lowast+119896119864minus(12)119905120578

lowast2

42 Quantity Commitment Scenario In traditional literatureon strategic customer behavior quantity commitment canimprove the manufacturerrsquos expected profit Quantity com-mitment refers to the action in which the manufacturerpromises customers that only a certain number of productsis produced and sold In low carbon supply chain settingcan the maximum expected profit be improved by quantitycommitment We attempt to answer the question in thispaper

We assume that the manufacturer can convince cus-tomers by appropriate means that they only can obtain 119902

units of the product in the wholesales period At this timethe strategic customers no longer need to anticipate theprobability of getting the product at salvage price When 119902

is given the probability of being able to get the product atprice 119904 is 119865(119902) The reservation price is 119901(119902) = Vminus (Vminus 119904)119865(119902)which also is the optimal pricing of the manufacturer Wehave 119890 = (1 minus 120578)119902 minus 119864 then we can obtain the expected profit

6 Discrete Dynamics in Nature and Society

function of themanufacturer with respect to 119902 and 120578 denotedby 120587119902(119902 120578)

120587119902(119902 120578) = (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + 119896 (1 minus 120578)) 119902 + 119896119864 minus1

21199051205782

(14)

So the manufacturerrsquos optimal production quantity andcarbon reduction rate in QC scenario are (119902

119902lowast 120578119902lowast) =

argmax119902ge00lt120578lt1

120587119902(119902 120578) the optimal price is 119901119902lowast = V minus (V minus

119904)119865(119902119902lowast) and the optimal carbon trading volume is 119890119902lowast =

(1 minus 120578119902lowast)119902119902lowastminus 119864

Lemma 3 Given 119902 the optimal green technology investmentstrategy of the manufacturer denoted by 120578119902lowast satisfies

120578119902lowastequiv 120578 (119902) =

119896119902

119905 0 lt 120578

119902lowastlt 1 (15)

Proof Wehave 120597120587119902(119902 120578)120597120578 = 119902119896minus119905120578 1205972120587119902(119902 120578)1205971205782 = minus119905 lt0 Let 120597120587119902(119902 120578)120597120578 = 0 we get 120578119902lowast equiv 120578(119902) = 119896119902119905 In additionaccording to the assumption in Section 3 we know 0 lt 120578

119902lowastlt

1 This completes the proof

Substituting 120578119902lowast = 120578(119902) into (14)

120587119902(119902) equiv 120587

119902(119902 120578 (119902))

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + 119896) 119902 + 119896119864 +11989621199022

2119905

(16)

Then the two-variable optimization problem is simplifiedas a single variable optimization problem

max119902ge0

120587119902(119902) (17)

Lemma4 If 1198962119905minus(Vminus119904)[3119891(119902)119865(119902)+1198911015840(119902)(119902minusint1199020119865(119909)119889119909)] lt

0 120587119902(119902) is a concave function of 119902

Proof The first-order and second-order derivatives of 120587119902(119902)are

119889120587119902(119902)

119889119902= (V minus 119904) [119865

2

(119902) minus 119891 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) +1198962

119905119902

1198892120587119902(119902)

1198891199022=1198962

119905minus (V minus 119904)

sdot [3119891 (119902) 119865 (119902) + 1198911015840(119902) (119902 minus int

119902

0

119865 (119909) 119889119909)] lt 0

(18)

Then we can obtain that 120587119902(119902) is a concave function of 119902This completes the proof

As to the manufacturerrsquos optimal production quantity inQC scenario (denoted by 119902119902lowast) the following proposition isobtained

Proposition 5 If 1198962119905 minus (V minus 119904)[3119891(119902)119865(119902) + 1198911015840(119902)(119902 minus

int119902

0119865(119909)119889119909)] lt 0 the optimal production quantity in QC

scenario (119902119902lowast) satisfies

(V minus 119904) [1198652

(119902) minus 119891 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) +1198962

119905119902 = 0

(19)

Proof It can be directly derived according to Lemma 4 Thiscompletes the proof

Proposition 5 shows that under certain condition theoptimal production quantity of the manufacturer with capand trade and green technology investment exist and areunique

43 The Effect of Quantity Commitment The effect of QCon the optimal strategies and the maximum expected profitof the manufacturer is analyzed by comparing the optimalstrategies and the maximum expected profit of the manufac-turer in RE Equilibrium scenario and QC scenario

Proposition 6 Consider 119902119902lowast lt 119902lowast 119901119902lowast gt 119901lowast 120578119902lowast lt 120578lowast

Proof In order to ensure that the manufacturerrsquos optimalstrategies in the two scenarios exist the condition 119896

2119905 lt

min(Vminus119904)119891(119902)119865(119902) (Vminus119904)[3119891(119902)119865(119902)+1198911015840(119902)(119902minusint1199020119865(119909)119889119909)]

must be held Then 120587119902(119902) is a concave function of 119902According to Proposition 2 we have 120579

1(119902lowast) = 1205792(120578lowast) = 119896

1205792(120578lowast) = 119896 can be written as 120578lowast = 119896119902

lowast119905 Then we also can

rearrange 1205791(119902lowast) = 119896 to (Vminus119904)1198652(119902lowast)minus(119888minus119904+119896)+(1198962119905)119902lowast = 0

We can obtain (119889120587119902(119902)119889119902)|

119902=119902lowast = (V minus 119904)[119865

2

(119902lowast) minus

119891(119902lowast)(119902lowastminus int119902lowast

0119865(119909)119889119909)] minus (119888 minus 119904 + 119896) + (119896

2119905)119902lowast= minus(V minus

119904)119891(119902lowast)(119902lowastminusint119902lowast

0119865(119909)119889119909) lt 0 Then we get 119902119902lowast lt 119902lowast Because

120578 = 119896119902119905 and 119901 = Vminus(Vminus119904)119865(119902) are held in the two scenarioswe can obtain 119901119902lowast gt 119901

lowast and 120578119902lowast lt 120578lowast according to 119902119902lowast lt 119902

lowastThis completes the proof

Proposition 6 shows that compared with RE Equilibriumscenario the manufacturerrsquos optimal production quantity islower the optimal pricing is higher and the optimal carbonreduction rate is lower in QC scenario

Proposition 7 Consider 120587119902(119902119902lowast 120578119902lowast) gt 120587(119902lowast 119901lowast 120578lowast)

Proof In order to ensure that the manufacturerrsquos optimalstrategies in the two scenarios exist the condition 119896

2119905 lt

min(Vminus119904)119891(119902)119865(119902) (Vminus119904)[3119891(119902)119865(119902)+1198911015840(119902)(119902minusint1199020119865(119909)119889119909)]

must be held Then 120587119902(119902) is a concave function of 119902Substitute 119902lowast 119901lowast and 120578

lowast= 119896119902

lowast119905 into (3) we have

120587(119902lowast 119901lowast 120578lowast) = 120587

119902(119902lowast) We know that 120587119902(119902) is a concave

function of 119902 and (119889120587119902(119902)119889119902)|119902=119902119902lowast = 0 Because of 119902119902lowast lt 119902lowast

Discrete Dynamics in Nature and Society 7

120587119902(119902lowast) lt 120587

119902(119902119902lowast) that is 120587(119902lowast 119901lowast 120578lowast) lt 120587

119902(119902119902lowast 120578119902lowast) This

completes the proof

Proposition 7 shows that QC strategy can improve themanufacturerrsquos maximum expected profit when green tech-nology investment and cap and trade policy are taken intoaccount

5 Decentralized Supply Chain Model

In this section the retailer and the manufacturer make deci-sions independently to maximize their own expected profitThe manufacturer is the Stackelberg leader and determinesthe wholesale price and the green technology investmentunder cap and trade policy The retailer is the follower anddetermines the retail price and the order quantity of theproduct

The sequence of events in this section is as follows firstthe manufacturer determines the optimal wholesale pricecarbon trading and green technology investment strategiessecond the retailer forms the belief of customersrsquo reservationprice and then decides the optimal retail price and orderquantity third the manufacture produces products anddelivers to the retailer fourth the customers form the beliefs120585prob of probability of the product sold at salvage price119904 according to the information of market price and thenform the reservation price 119903 fifth the customersrsquo demand issatisfied and the products are sold at full price 119901 finally allremaining products are sold to the external market at salvageprice 119904

51 The Optimal Strategies First we examine the decisionproblem of the retailer Recall that 119908 is the manufacturerrsquoswholesale price The retailerrsquos expected profit denoted by120587119903(119902 119901) is

120587119903(119902 119901) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909) minus (119908 minus 119904) 119902 (20)

Lemma 8 When 119901 is given 120587119903(119902 119901) is a concave function of119902

Proof When 119901 is given according to (20) we have120597120587119903(119902 119901)120597119902 = (119901 minus 119904)119865(119902) minus (119908 minus 119904) and 1205972120587119903(119902 119901)1205971199022 =

minus(119901 minus 119904)119891(119902) lt 0 Then we know that 120587119903(119902 119901) is a concavefunction of 119902 when 119901 is given This completes the proof

As to the retailerrsquos optimal strategies in RE Equilibriumwe can obtain the following proposition

Proposition 9 When the wholesale price is given the retailerrsquosoptimal order strategies (119902119903lowast) and optimal pricing strategies(119901119903lowast) with strategic customer behavior are

119902119903lowast= 119865minus1

(radic119908 minus 119904

V minus 119904) (21)

119901119903lowast= 119904 + radic(119908 minus 119904) (V minus 119904) (22)

Proof In RE Equilibrium (9) still holds Let 120597120587119903(119902 119901)120597119902 = 0and combining with (9) we have

119901 = V minus (V minus 119904) 119865 (119902)

(119901 minus 119904) 119865 (119902) minus (119908 minus 119904) = 0

(23)

According to Lemma 8 we can obtain the optimal orderand pricing strategies of the retailer by solving the equationsabove This completes the proof

Proposition 9 shows that the retailerrsquos optimal order andpricing strategies in REEquilibriumwith cap and trade policyexist and are unique

Second we investigate the manufacturerrsquos decision prob-lem The expected profit function of the manufacturer withgreen technology investment and cap and trade policy is120587119898(119908 119890 120578) = (119908 minus 119888)119902 minus 119896119890 minus (12)119905120578

2According to (21) with green technology investment and

cap and trade policy in wholesale price contract the optimalorder quantity of the retailer 119902119903lowast and the optimal wholesaleprice of the manufacturer 119908119898lowast is a one-to-one relationship

119908 = 119904 + (V minus 119904) 1198652

(119902) (24)

Combined with 119890 = (1 minus 120578)119902 minus 119864 the manufacturerrsquosexpected profit function can be converted into

120587119898(119902 120578) = [(V minus 119904) 119865

2

(119902) minus (119888 minus 119904 + (1 minus 120578) 119896)] 119902

+ 119896119864 minus1

21199051205782

(25)

When using wholesale price contract the total profit ofthe supply chain denoted by 120587sc

(119902 119901 119890 120578) is

120587sc(119902 119901 119890 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909) minus (119888 minus 119904) 119902

minus 119896119890 minus1

21199051205782

(26)

According to (9) and 119890 = (1 minus 120578)119902 minus 119864 the total supplychain profit function in RE Equilibrium can be simplified as

120587sc(119902 120578) = (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578) 119896) 119902 + 119896119864 minus1

21199051205782

(27)

Lemma 10 Given 119902 the optimal green technology investmentstrategy of the manufacturer denoted by 120578119898lowast satisfies

120578119898lowast

equiv 120578 (119902) =119896119902

1199050 lt 120578119898lowast

lt 1 (28)

Proof The first-order and second-order partial derivatives of120587119898(119902 120578) with respect to 120578 are 120597120587119898(119902 120578)120597120578 = 119902119896 minus 119905120578 and

1205972120587119898(119902 120578)120597120578

2= minus119905 lt 0 Let 120597120587119898(119902 120578)120597120578 = 0 we get

120578119898lowast

equiv 120578(119902) = 119896119902119905 In addition according to the assumptionin Section 3 we know 0 lt 120578

119898lowastlt 1 This completes the

proof

8 Discrete Dynamics in Nature and Society

Substituting 120578119898lowast = 120578(119902) into (25)

120587119898(119902) equiv 120587

119898(119902 120578 (119902))

= [(V minus 119904) 1198652

(119902) minus (119888 minus 119904 + 119896)] 119902 + 119896119864

+11989621199022

2119905

(29)

Then the two-variable optimization problem is simplifiedas a single variable optimization problem

max119902ge0

120587119898(119902) (30)

Lemma 11 If 1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus

1199021198912(119902)] lt 0 120587119898(119902) is a concave function of 119902

Proof The first-order and second-order derivatives of 120587119898(119902)with respect to 119902 are 119889120587

119898(119902)119889119902 = (V minus 119904)[119865

2

(119902) minus

2119902119891(119902)119865(119902)] minus (119888 minus 119904 + 119896) + (1198962119905)119902 and 119889

2120587119898(119902)119889119902

2=

1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus 1199021198912(119902)] lt 0 Then

we have that 120587119898(119902) is a concave function of 119902 This completesthe proof

In decentralized supply chain as to the manufacturerrsquosoptimal wholesale price (denoted by 119908

119898lowast) the followingproposition is obtained

Proposition 12 If 1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus1199021198912(119902)] lt 0 the manufacturerrsquos optimal wholesale price in

decentralized supply chain (119908119898lowast) satisfies 119908119898lowast = 119904 + (V minus119904)1198652

(119902119903lowast) where 119902119903lowast represents the retailerrsquos optimal order

quantity and satisfies

(V minus 119904) [1198652

(119902) minus 2119902119891 (119902) 119865 (119902)] minus (119888 minus 119904 + 119896) +1198962

119905119902

= 0

(31)

Proof According to Lemma 11 we can obtain that 119902119903lowastwhichmaximizes themanufacturerrsquos expected profit satisfies(31) Combining with (24) we know that the manufacturerrsquosoptimal wholesale price satisfies 119908119898lowast = 119904 + (V minus 119904)119865

2

(119902119903lowast) at

this time This completes the proof

Proposition 12 shows that the optimal commitment quan-tity of the manufacturer in decentralized supply chain withgreen technology investment and cap and trade policy existand are unique The optimal wholesale price does not relateto the cap allocated by the government but relates to thecarbon trading price This is because the production of themanufacturer is not affected by the cap but is affected bythe margin cost of the product However the margin cost ofthe product varies with the change of carbon trading pricewhen carbon trading is allowed The change results in thatthe optimal commitment quantity is not related to the cap butrelate to the carbon trading price

Substitute 119908119898lowast into (22) we derive the retailerrsquos optimalpricing in decentralized situation119901119903lowast = 119904+radic(119908119898lowast minus 119904)(V minus 119904)

According to Lemma 10 we get the optimal green technologyinvestment of the manufacturer 120578119898lowast = (119896119905)119902119903lowast

Substitute 119902119903lowast 119901119903lowast 119908119898lowast and 120578119898lowast into 120587

119903(119902 119901) 120587119898(119902)

and 120587sc(119902 120578) we obtain the maximum expected profit of

the retailer the manufacturer and the supply chain indecentralized channel

120587119903(119902119903lowast 119901119903lowast) = (119901

119903lowastminus 119904) (119902

119903lowastminus int

119902119903lowast

0

119865 (119909) 119889119909)

minus (119908119898lowast

minus 119904) 119902119903lowast

120587119898(119902119903lowast) = [(V minus 119904) 119865

2

(119902119903lowast) minus (119888 minus 119904 + 119896)] 119902

119903lowast

+ 119896119864 +1198962119902119903lowast2

2119905

120587sc(119902119903lowast 120578119898lowast) = (V minus 119904) 119865 (119902119903lowast) (119902119903lowast minus int

119902119903lowast

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578119898lowast) 119896) 119902119903lowast+ 119896119864

minus1

2119905120578119898lowast2

(32)

52 The Effect of Decentralization We examine the impact ofdecentralization on the supply chain optimal strategies andmaximumexpected profitwhen green technology investmentis taken into consideration by comparing the optimal strate-gies and the maximum expected profit in centralized anddecentralized channel with green technology investment andcap and trade policy

As the effect of decentralization of supply chain we canobtain the following proposition

Proposition 13 Consider 119902119903lowast lt 119902119902lowast

lt 119902lowast 119901119903lowast gt 119901

119902lowastgt 119901lowast

120578119898lowast

lt 120578119902lowastlt 120578lowast

Proof In Proposition 6 119902119902lowast lt 119902lowast 119901119902lowast gt 119901

lowast and 120578119902lowast lt 120578lowast

have been provedTherefore we only need to prove 119902119903lowast lt 119902119902lowast119901119903lowastgt 119901119902lowast and 120578119898lowast lt 120578119902lowast

Define 119867(119902) = 120587119902(119902) minus 120587119898(119902) then

119867(119902)

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

minus (V minus 119904) 1199021198652

(119902)

1198671015840(119902)

= (V minus 119904) 119891 (119902) [2119902119865 (119902) minus (119902 minus int119902

0

119865 (119909) 119889119909)]

(33)

Define 119866(119902) = 2119902119865(119902) minus (119902 minus int119902

0119865(119909)119889119909) then 1198661015840(119902) =

119865(119902) minus 2119902119891(119902) Because 119866(0) = 0 1198661015840(0) = 1 and 119865 satisfiesIFR we know that the value of 119866(119902) starts from 0 increasingfirst and then decreasing in 119902 That is 119866(119902) = 0 has uniquesolution and we denote it as So when 119902 lt 119866(119902) gt 0 that

Discrete Dynamics in Nature and Society 9

is119867(119902) is increasing in 119902 when 119902 gt 119866(119902) lt 0 that is119867(119902)is decreasing in 119902 Then we have that119867(119902) is a quasi-concavefunction of 119902 and 1198661015840 () = 119865() minus 2119891() lt 0 According tothe analysis above we know that1198671015840() = 0 that is 2119865() =( minus int

0119865(119909)119889119909) Substitute into 119889120587119902(119902)119889119902 we have

119889120587119902(119902)

119889119902

100381610038161003816100381610038161003816100381610038161003816119902=

= (V minus 119904) [1198652

() minus 119891 () ( minus int

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) +1198962

119905

= (V minus 119904) 119865 () [119865 () minus 2119891 ()] minus (119888 minus 119904 + 119896)

+1198962

119905lt (V minus 119904) 119865 () [119865 () minus 2119891 ()] lt 0

(34)

According to Lemma 4 we have that 120587119902(119902) is a concavefunction of 119902 so we get 119902119902lowast lt Because 2119865() = ( minus

int

0119865(119909)119889119909) and119867(119902) is a quasi-concave function of 119902 thenwe

get that 119889120587119898(119902)119889119902 lt 119889120587119902(119902)119889119902 if 119902 lt and 119889120587119898(119902)119889119902 gt

119889120587119902(119902)119889119902 if 119902 gt Therefore the only possible orderings for

119902119902lowast 119902119903lowast and are lt 119902

119902lowastlt 119902119903lowast and 119902119903lowast lt 119902

119902lowastlt We

have proved 119902119902lowast lt so we have 119902119903lowast lt 119902119902lowast We get 119901119903lowast gt 119901119902lowastbecause 119901 and 119902 satisfy 119901 = V minus (V minus 119904)119865(119902) in two situationsWe get 120578119898lowast lt 120578

119902lowast because 120578 and 119902 satisfy 120578 = 119896119902119905 in twosituations This completes the proof

Proposition 13 shows that for the decision of a decen-tralized supply chain with green technology investment theoptimal production quantity is increasing the optimal priceis decreasing and the green technology investment (iecarbon reduction rate) is increasing respectively in the threesituations of decentralized supply chain QC scenario and REEquilibrium scenario

The reason is that themanufacturer prefers higher whole-sale price (ie lower production and less green technologyinvestment) to guarantee its ownprofitmaximization thoughthis may harm the retailer even the whole supply chain

In Proposition 7 we have found 120587119902(119902119902lowast 120578119902lowast) gt

120587(119902lowast 119901lowast 120578lowast) that is the manufacturerrsquos maximum profit in

QC scenario is always greater than that in RE EquilibriumTherefore the section analyzes the effect of decentralizationon supply chain performance based on the QC scenario toprovide benchmark for designing of subsequent coordinationcontract

As to the effect of decentralization on themaximumprofitof supply chain we have the following proposition

Proposition 14 Consider 120587sc(119902119903lowast 120578119898lowast) lt 120587119902(119902119902lowast 120578119902lowast)

Proof The optimal green technology investment in decen-tralized supply chain and QC scenario are 120578(119902) = 119896119902119905

according to Lemmas 3 and 10 So the profit function in thetwo situations above can be converted into a same singlevariable function with respect to 119902 that is (16)

According to Lemma 4 we know that 120587119902(119902) is a concavefunction and reaches the maximum value at 119902119902lowast Accordingto Proposition 13 we have 119902119903lowast lt 119902

119902lowast then we get 120587119902(119902119903lowast) lt120587119902(119902119902lowast) that is 120587sc

(119902119903lowast 120578119898lowast) lt 120587

119902(119902119902lowast 120578119902lowast) This completes

the proof

Proposition 14 shows that the maximum profit of decen-tralized supply chain is always lower than that in QC scenariowhich indicates that the profit of the decentralized supplychain can get higher This is the benchmark of supply chaincoordination

6 Supply Chain Coordination

This section coordinates the supply chain with green tech-nology investment and cap and trade policy based on theQC scenario to improve the profit of the decentralized supplychain

61 Coordination Contract Based on Revenue and CostsSharing Contract With revenue sharing contract we assumethat 120601 (0 le 120601 le 1) represents the ratio of revenue kept by theretailer and (1 minus 120601) represents the ratio of revenue deliveredto the manufacturer

The retailerrsquos expected profit function with revenue shar-ing contract denoted by 120587119903

119904(119902 119901 120601) is

120587119903

119904(119902 119901 120601) = 120601 (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119908 minus 120601119904) 119902

(35)

Themanufacturerrsquos expected profit functionwith revenuesharing contract denoted by 120587119898

119904(119908 119890 120578 120601) is

120587119898

119904(119908 119890 120578 120601) = (119908 minus 119888) 119902 + (1 minus 120601)

sdot [int

119902

0

[119901119909 + 119904 (119902 minus 119909)] 119891 (119909) 119889119909

+ int

infin

119902

119901119902119891 (119909) 119889119909] minus 119896119890 minus1

21199051205782

(36)

Substitute 119890 = (1 minus 120578)119902 minus 119864 into the equation above andwe have

120587119898

119904(119908 120578 120601) = (1 minus 120601) (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

+ (119908 minus 119888 minus (1 minus 120578) 119896 + (1 minus 120601) 119904) 119902

+ 119896119864 minus1

21199051205782

(37)

The total expected profit of the whole supply chaindenoted by 120587sc

119904(119902 119901 120578) is

120587sc119904(119902 119901 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578) 119896) 119902 + 119896119864 minus1

21199051205782

(38)

10 Discrete Dynamics in Nature and Society

The subscript 119904 represents the situation of supply chaincoordination using revenue sharing contracts

As to the retailerrsquos optimal order strategy (denoted by119902lowast

119904) and pricing strategy (denoted by 119901lowast

119904) of the supply chain

with revenue sharing contract the following proposition isobtained

Proposition 15 Under revenue sharing contracts the retailerrsquosoptimal order strategy (119902lowast

119904) and optimal pricing strategy (119901lowast

119904)

are

119902lowast

119904= 119865minus1

(radic119908 minus 120601119904

120601 (V minus 119904))

119901lowast

119904= 119904 + radic

(119908 minus 120601119904) ((V minus 119904))120601

(39)

Proof According to the definition of RE Equilibrium wealso have 119901 = V minus (V minus 119904)119865(119902) under revenue sharingcontracts

The first-order and second-order derivative of 120587119903119904(119902 119901 120601)

with respect to 119902 are 120597120587119903

119904(119902 119901 120601)120597119902 = 120601(119901 minus 119904)119865(119902) minus

(119908 minus 120601119904) and 1205972120587119903

119904(119902 119901 120601)120597119902

2= minus120601(119901 minus 119904)119891(119902) lt

0 So given 119901 the retailerrsquos expected profit function is aconcave function of 119902 Let 120597120587119903

119904(119902 119901 120601)120597119902 = 0 and combine

with 119901 = V minus (V minus 119904)119865(119902) we can solve the retailerrsquos optimalstrategies under revenue sharing contractThis completes theproof

Proposition 15 shows that the optimal order and pricingstrategies of the retailer based on revenue sharing contract inRE Equilibrium exist and are uniqueThis is also the retailerrsquosoptimal response function with respect to 119908

Substituting (39) into (43)

120587119898

119904(119902 120578 120601)

= (1 minus 120601) (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ [120601 (V minus 119904) 1198652

(119902) minus (119888 minus 119904 + (1 minus 120578) 119896)] 119902 + 119896119864

minus1

21199051205782

120597120587119898

119904(119902 120578 120601)

120597120578= 119902119896 minus 119905120578

1205972120587119898

119904(119902 120578 120601)

1205971205782= minus119905 lt 0

(40)

Given 119902 and 120601 the manufacturerrsquos green technologyinvestment strategies with revenue sharing contracts exist

and are unique Let 120597120587119898119904(119902 120578 120601)120597120578 = 0 we get 120578119898lowast

119904equiv 120578(119902) =

119902119896119905 and substitute it into 120587119898119904(119902 120578 120601)

120587119898

119904(119902 120601)

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ 120601 (V minus 119904) 119865 (119902) [119902119865 (119902) minus (119902 minus int119902

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) 119902 + 119896119864 +11989621199022

2119905

(41)

Proposition 16 The supply chain with green technologyinvestment and cap and trade policy can not be coordinated byrevenue sharing contract

Proof The derivative of 120587119898119904(119902 120601) with respect to 119902 can be

arranged as follows

120597120587119898

119904(119902 120601)

120597119902=120597120587119902(119902)

120597119902+ 120601(

120597120587119898(119902)

120597119902minus120597120587119902(119902)

120597119902) (42)

Only when 120601 = 0 are the optimal strategies of supplychain under revenue sharing contracts equal to that in QCscenario Let 119908lowast

119904represent the optimal wholesale price of

the manufacture and 119902lowast119904represent the retailerrsquos optimal order

quantity Then 120587119903

119904(119902 119901 120601) = minus119908

lowast

119904119902lowast

119904lt 0 So the revenue

sharing contract cannot realize the supply chain coordinationat this time This completes the proof

Proposition 16 shows that revenue sharing contractscan not realize the supply chain coordination with greentechnology investment and cap and trade policy The rea-son is that the manufacturer undertakes all the costs ofgreen technology investment when using revenue sharingcontracts thus making the manufacturer tend to invest lessin green technology Only when the manufacturer owns allthe revenue will themanufacturerrsquos optimal green technologyinvestment under revenue sharing contracts be equal to thatin QC scenario However the maximum profit of the retaileris negative at this time and it will not participate

62 Coordination Contract Based on Revenue Sharing-CostSharing Contract Proposition 16 states that the supply chainwith green technology investment and cap and trade policycan not be coordinated by revenue sharing contracts becausethe costs of green technology investment are undertaken bythe manufacturer and the game of retailer and customersis characterized by RE Equilibrium So we consider costssharing of green technology investment and carbon tradingand quantity commitment made by the manufacturer basedon revenue sharing contracts Let 120601 (0 lt 120601 le 1) represent theratio of revenue kept by the retailer and let (1minus120601) represent theratio of revenue delivered to the manufacturer 120572 (0 le 120572 le 1)represents the ratio of costs of green technology investmentand carbon trading undertaken by the manufacturer and(1 minus 120572) represents the ratio of costs of green technologyinvestment and carbon trading delivered to the retailer

Discrete Dynamics in Nature and Society 11

The retailerrsquos expected profit function with revenuesharing-cost sharing contract is

120587119903

120572(119902 120601 120572) = 120601 (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119908 minus 120601119904) 119902

minus (1 minus 120572) 119896 [(1 minus 120578) 119902 minus 119864]

minus1

2(1 minus 120572) 119905120578

2

(43)

We have 119890 = (1minus120578)119902minus119864 then themanufacturerrsquos expectedprofit function denoted by 120587119898

120572(119908 120578 120601 120572) is

120587119898

120572(119908 120578 120601 120572)

= (1 minus 120601) (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ (119908 minus 119888 minus 120572 (1 minus 120578) 119896 + (1 minus 120601) 119904) 119902 + 120572119896119864

minus1

21205721199051205782

(44)

The subscript 120572 represents the situation that using rev-enue sharing-cost sharing contract coordinates the supplychain

Proposition 17 If 0 le 120582 le 1 the parameters of revenuesharing-cost sharing (119908 120601 120572) satisfy

119908 = 120582119888 + (1 minus 120578) 119896

120601 = 120582

120572 = 1 minus 120582

(45)

The supply chainwith green technology investment and cap andtrade policy is coordinated

Proof When the parameters of revenue sharing-cost sharingsatisfy (45) the expected profit function of the retailer andthe manufacturer can be written as follows

120587119903

120572(119902 119901 120601 120572) = 120582120587

119902(119902 120578)

120587119898

120572(119908 120578 120601 120572) = (1 minus 120582) 120587

119902(119902 120578)

(46)

The optimal order of decentralized supply chain is equalto that of QC scenario Substitute 119901 = V minus (V minus 119904)119865(119902) into120587sc119888(119902 119901 120578) we get that the profit function of decentralized

supply chain at this time is the same as that of QC scenarioTherefore when the parameters of revenue sharing-costsharing (119908 120601 120572) satisfy (45) the decentralized supply chaincoordination is realizedThemanufacturer earns all the profitif 120582 = 0 and the retailer earns all the profit if 120582 = 1Arbitrary allocation of the supply chain profit between themanufacturer and the retailer is allowed This completes theproof

Proposition 17 shows that revenue sharing-cost sharingcontract can realize the supply chain coordination on condi-tion that the manufacturer makes a quantity commitment to

the customers The optimal production quantity and greentechnology investment in decentralized situation are lowerthan that in QC scenario The quantity commitment of themanufacturer can increase the green technology investmentand production quantity and also reduce the retailer priceThat is the quantity commitment is beneficial for the man-ufacturer the retailer and customers Thus the quantitycommitment is credible to customers now

7 Conclusions and Suggestions for FurtherResearch

This paper investigates the production pricing carbon trad-ing and green technology investment strategies and thecoordination of low carbon supply chain made up of a lowcarbon manufacturer and a retailer The low carbon manu-facturer produces one product under cap and trade policyand performs green technology investment to reduce carbonemissions in production process The retailer orders fromthemanufacturer and distributes the product to homogenouscustomers To the best of our knowledge this is the firststudy on the management of low carbon supply chainswith strategic customer behaviorThis paper provides severalinteresting observations

Observation 1 There are unique optimal production pricingcarbon trading and green technology investment strategiesin centralized (include RE Equilibrium scenario and QCscenario) and decentralized supply chain Therefore byderiving the optimal production pricing carbon trading andgreen technology investment strategies in different situationsthe low carbon manufacturer and the retailer can behaveappropriately based on our findings to maximize their profit

Observation 2 Our findings also show that the centralizedsupply chain can improve its performance by quantity com-mitment However the manufacturerrsquos production and greentechnology investment are reduced while the optimal priceis increased at this time Therefore quantity commitment isbeneficial for the supply chain but is harmful to customers

Observation 3 We also show that the performance of decen-tralized supply chain is lower than that in QC scenario Wefind that the decentralization reduces the manufacturersquos opti-mal production quantity and green technology investmentand increases the retailerrsquos optimal price Besides we find thatthe optimal production quantity is increasing the optimalprice is decreasing and the green technology investment isincreasing respectively in the three situations of decentral-ized supply chain QC scenario andREEquilibrium scenario

Observation 4 We demonstrate that the traditional revenuesharing contracts can not realize the low carbon supply chaincoordination when green technology investment is takenin consideration We also find that the revenue sharing-cost sharing contract realizes the decentralized supply chain

12 Discrete Dynamics in Nature and Society

coordination when the manufacturer makes a quantity com-mitment It is beneficial to the manufacturer the retailerand customers if themanufacturermakes a quantity commit-ment So the quantity commitment is credible to the customerat this time

We assume the customers are homogeneous strategiccustomers But themyopic customers and strategic customersoften coexist and strategic customers also have differenttolerance in real-life Extending this assumption will havea knock-on effect on supply chain decisions So one keyresearch direction is to consider that the customers aremixed customers myopic customers and strategic customersSecond we assume that the supply chain only producesone product While this simplified setting provides someinteresting insights we have to acknowledge that a manufac-turer usually produces two or more productsThese productswill substitute or complement each other So one importantextension of our work is to assume that the manufacturerproduces two or more products which have different unitcarbon emissions in production

Notations

119863 The random demand and119863 ge 0

119891(sdot) Probability density function of119863119865(sdot) Distribution function of119863 We assume

that 119865 satisfy IFR and define 119865 = 1 minus 119865

119888 Unit production cost119904 Unit salvage price of the product which

is an exogenous variableV The customersrsquo utility from consuming

the product119908 Unit wholesale price119901 Unit retail selling price We assume that

it can be observed by the customers119902 The manufacturerrsquos production

quantitythe retailerrsquos order quantityWe assume that it cannot be observedby the customers

119903 The customersrsquo reservation price whichis customersrsquo private information andcannot be observed by the retailer

120585119903 The beliefs of the manufacturer over the

customerrsquo reservation price120585prob The beliefs of customers over their

chances of obtaining the product atphase two

119864 Initial free carbon emission allocated bygovernment

119896 Unit carbon emission trading price119890 Carbon trading volume120578 Carbon emissions reduction rate

Conflict of Interests

The authors declare no conflict of interests

Acknowledgments

This research is partially supported by National Natural Sci-ence Foundation of China (nos 71432003 and 71272128) theProgram forNewCentury Excellent Talents inUniversity (noNCET-12-0087) and Specialized Research Fund for the Doc-toral Program of Higher Education (no 20130185110006)

References

[1] R Alley T Berntsen N L Bindoff et al ldquoSummary forpolicymakersrdquo in Climate Change 2007 The Physical ScienceBasis vol 6 pp 1ndash18 IPCC 2007

[2] X Chen and G Hao ldquoSustainable pricing and productionpolicies for two competing firms with carbon emissions taxrdquoInternational Journal of Production Research vol 53 pp 6408ndash6420 2015

[3] X Chen S Benjaafar and A Elomri ldquoThe carbon-constrainedEOQrdquo Operations Research Letters vol 41 no 2 pp 172ndash1792013

[4] N O Keohane ldquoCap and trade rehabilitated using tradablepermits to control US greenhouse gasesrdquo Review of Environ-mental Economics and Policy vol 3 no 1 pp 42ndash62 2009

[5] B Liu M Holmbom A Segerstedt and W Chen ldquoEffectsof carbon emission regulations on remanufacturing decisionswith limited information of demand distributionrdquo InternationalJournal of Production Research vol 53 no 2 pp 532ndash548 2015

[6] L Zhu and Y Fan ldquoA real options-based CCS investmentevaluation model case study of Chinarsquos power generationsectorrdquo Applied Energy vol 88 no 12 pp 4320ndash4333 2011

[7] XWang and LDu ldquoStudy on carbon capture and storage (CCS)investment decision-making based on real options for Chinarsquoscoal-fired power plantsrdquo Journal of Cleaner Production vol 112part 5 pp 4123ndash4131 2016

[8] W Jiang and X Chen ldquoOptimal strategies for manufacturerwith strategic customer behavior under carbon emissions-sensitive randomdemandrdquo Industrial Management amp Data Sys-tems 2015

[9] Y Levin J McGill and M Nediak ldquoDynamic pricing in thepresence of strategic consumers and oligopolistic competitionrdquoManagement Science vol 55 no 1 pp 32ndash46 2009

[10] A K Parlakturk ldquoThe value of product variety when sellingto strategic consumersrdquo Manufacturing amp Service OperationsManagement vol 14 no 3 pp 371ndash385 2012

[11] X Su and F Zhang ldquoOn the value of commitment andavailability guarantees when selling to strategic consumersrdquoManagement Science vol 55 no 5 pp 713ndash726 2009

[12] W Elmaghraby S A Lippman C S Tang and R Yin ldquoWillmore purchasing options benefit customersrdquo Production andOperations Management vol 18 no 4 pp 381ndash401 2009

[13] W Jiang and X Chen ldquoManufacturersquos production and pricingstrategies with carbon tax policy and strategic customer behav-iorrdquoManagement Science and Engineering vol 9 no 1 pp 30ndash35 2015

[14] I Dobos ldquoThe effects of emission trading on production andinventories in the Arrow-Karlin modelrdquo International Journalof Production Economics vol 93-94 pp 301ndash308 2005

[15] X Chen C K Chan and Y C E Lee ldquoResponsible productionpolicies with substitution and carbon emissions tradingrdquo Jour-nal of Cleaner Production 2015

Discrete Dynamics in Nature and Society 13

[16] X Chen and X Wang ldquoEffects of carbon emission reductionpolicies on transportation mode selections with stochasticdemandrdquo Transportation Research Part E Logistics and Trans-portation Review 2015

[17] X Chang H Xia H Zhu T Fan and H Zhao ldquoProductiondecisions in a hybrid manufacturingndashremanufacturing systemwith carbon cap and trade mechanismrdquo International Journal ofProduction Economics vol 162 pp 160ndash173 2015

[18] J Zhao B F Hobbs and J-S Pang ldquoLong-run equilibriummodeling of emissions allowance allocation systems in electricpowermarketsrdquoOperations Research vol 58 no 3 pp 529ndash5482010

[19] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2 pp519ndash527 2011

[20] A Toptal H Ozlu and D Konur ldquoJoint decisions on inventoryreplenishment and emission reduction investment under dif-ferent emission regulationsrdquo International Journal of ProductionResearch vol 52 no 1 pp 243ndash269 2014

[21] A S Manikas and J R Kroes ldquoA newsvendor approach tocompliance and production under cap and trade emissionsregulationrdquo International Journal of Production Economics vol159 pp 274ndash284 2015

[22] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineering vol10 no 1 pp 99ndash116 2013

[23] X Xu W Zhang P He and X Xu ldquoProduction and pricingproblems in make-to-order supply chain with cap-and-traderegulationrdquo Omega 2015

[24] Y Zhen X Yang and W Jiang ldquoThe impact of transport modeand carbon policy on low-carbon retailerrdquoDiscrete Dynamics inNature and Society vol 2015 Article ID 964305 12 pages 2015

[25] S Swami and J Shah ldquoChannel coordination in green supplychainmanagementrdquo Journal of the Operational Research Societyvol 64 no 3 pp 336ndash351 2013

[26] R H Coase ldquoDurability andmonopolyrdquoThe Journal of Law andEconomics vol 15 no 1 pp 143ndash149 1972

[27] X Su ldquoIntertemporal pricing with strategic customer behaviorrdquoManagement Science vol 53 no 5 pp 726ndash741 2007

[28] D Zhang and W L Cooper ldquoManaging clearance sales in thepresence of strategic customersrdquo Production and OperationsManagement vol 17 no 4 pp 416ndash431 2008

[29] S Whang ldquoDemand uncertainty and the Bayesian effect inmarkdown pricing with strategic customersrdquoManufacturing ampService Operations Management vol 17 no 1 pp 66ndash77 2015

[30] J Du J Zhang and G Hua ldquoPricing and inventory manage-ment in the presence of strategic customers with risk preferenceand decreasing valuerdquo International Journal of Production Eco-nomics vol 164 pp 160ndash166 2015

[31] X Su and F Zhang ldquoStrategic customer behavior commitmentand supply chain performancerdquo Management Science vol 54no 10 pp 1759ndash1773 2008

[32] H Yang ldquoImpact of discounting and competition on benefit ofdecentralization with strategic customersrdquo Operations ResearchLetters vol 40 no 2 pp 123ndash127 2012

[33] D Yang E Qi and Y Li ldquoQuick response and supply chainstructure with strategic consumersrdquo Omega vol 52 pp 1ndash142015

[34] C drsquoAspremont and A Jacquemin ldquoCooperative and nonco-operative R amp D in duopoly with spilloversrdquo The AmericanEconomic Review vol 78 no 5 pp 1133ndash1137 1988

[35] J F Muth ldquoRational expectations and the theory of pricemovementsrdquo Econometrica vol 29 no 3 pp 315ndash335 1961

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

function of themanufacturer with respect to 119902 and 120578 denotedby 120587119902(119902 120578)

120587119902(119902 120578) = (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + 119896 (1 minus 120578)) 119902 + 119896119864 minus1

21199051205782

(14)

So the manufacturerrsquos optimal production quantity andcarbon reduction rate in QC scenario are (119902

119902lowast 120578119902lowast) =

argmax119902ge00lt120578lt1

120587119902(119902 120578) the optimal price is 119901119902lowast = V minus (V minus

119904)119865(119902119902lowast) and the optimal carbon trading volume is 119890119902lowast =

(1 minus 120578119902lowast)119902119902lowastminus 119864

Lemma 3 Given 119902 the optimal green technology investmentstrategy of the manufacturer denoted by 120578119902lowast satisfies

120578119902lowastequiv 120578 (119902) =

119896119902

119905 0 lt 120578

119902lowastlt 1 (15)

Proof Wehave 120597120587119902(119902 120578)120597120578 = 119902119896minus119905120578 1205972120587119902(119902 120578)1205971205782 = minus119905 lt0 Let 120597120587119902(119902 120578)120597120578 = 0 we get 120578119902lowast equiv 120578(119902) = 119896119902119905 In additionaccording to the assumption in Section 3 we know 0 lt 120578

119902lowastlt

1 This completes the proof

Substituting 120578119902lowast = 120578(119902) into (14)

120587119902(119902) equiv 120587

119902(119902 120578 (119902))

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + 119896) 119902 + 119896119864 +11989621199022

2119905

(16)

Then the two-variable optimization problem is simplifiedas a single variable optimization problem

max119902ge0

120587119902(119902) (17)

Lemma4 If 1198962119905minus(Vminus119904)[3119891(119902)119865(119902)+1198911015840(119902)(119902minusint1199020119865(119909)119889119909)] lt

0 120587119902(119902) is a concave function of 119902

Proof The first-order and second-order derivatives of 120587119902(119902)are

119889120587119902(119902)

119889119902= (V minus 119904) [119865

2

(119902) minus 119891 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) +1198962

119905119902

1198892120587119902(119902)

1198891199022=1198962

119905minus (V minus 119904)

sdot [3119891 (119902) 119865 (119902) + 1198911015840(119902) (119902 minus int

119902

0

119865 (119909) 119889119909)] lt 0

(18)

Then we can obtain that 120587119902(119902) is a concave function of 119902This completes the proof

As to the manufacturerrsquos optimal production quantity inQC scenario (denoted by 119902119902lowast) the following proposition isobtained

Proposition 5 If 1198962119905 minus (V minus 119904)[3119891(119902)119865(119902) + 1198911015840(119902)(119902 minus

int119902

0119865(119909)119889119909)] lt 0 the optimal production quantity in QC

scenario (119902119902lowast) satisfies

(V minus 119904) [1198652

(119902) minus 119891 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) +1198962

119905119902 = 0

(19)

Proof It can be directly derived according to Lemma 4 Thiscompletes the proof

Proposition 5 shows that under certain condition theoptimal production quantity of the manufacturer with capand trade and green technology investment exist and areunique

43 The Effect of Quantity Commitment The effect of QCon the optimal strategies and the maximum expected profitof the manufacturer is analyzed by comparing the optimalstrategies and the maximum expected profit of the manufac-turer in RE Equilibrium scenario and QC scenario

Proposition 6 Consider 119902119902lowast lt 119902lowast 119901119902lowast gt 119901lowast 120578119902lowast lt 120578lowast

Proof In order to ensure that the manufacturerrsquos optimalstrategies in the two scenarios exist the condition 119896

2119905 lt

min(Vminus119904)119891(119902)119865(119902) (Vminus119904)[3119891(119902)119865(119902)+1198911015840(119902)(119902minusint1199020119865(119909)119889119909)]

must be held Then 120587119902(119902) is a concave function of 119902According to Proposition 2 we have 120579

1(119902lowast) = 1205792(120578lowast) = 119896

1205792(120578lowast) = 119896 can be written as 120578lowast = 119896119902

lowast119905 Then we also can

rearrange 1205791(119902lowast) = 119896 to (Vminus119904)1198652(119902lowast)minus(119888minus119904+119896)+(1198962119905)119902lowast = 0

We can obtain (119889120587119902(119902)119889119902)|

119902=119902lowast = (V minus 119904)[119865

2

(119902lowast) minus

119891(119902lowast)(119902lowastminus int119902lowast

0119865(119909)119889119909)] minus (119888 minus 119904 + 119896) + (119896

2119905)119902lowast= minus(V minus

119904)119891(119902lowast)(119902lowastminusint119902lowast

0119865(119909)119889119909) lt 0 Then we get 119902119902lowast lt 119902lowast Because

120578 = 119896119902119905 and 119901 = Vminus(Vminus119904)119865(119902) are held in the two scenarioswe can obtain 119901119902lowast gt 119901

lowast and 120578119902lowast lt 120578lowast according to 119902119902lowast lt 119902

lowastThis completes the proof

Proposition 6 shows that compared with RE Equilibriumscenario the manufacturerrsquos optimal production quantity islower the optimal pricing is higher and the optimal carbonreduction rate is lower in QC scenario

Proposition 7 Consider 120587119902(119902119902lowast 120578119902lowast) gt 120587(119902lowast 119901lowast 120578lowast)

Proof In order to ensure that the manufacturerrsquos optimalstrategies in the two scenarios exist the condition 119896

2119905 lt

min(Vminus119904)119891(119902)119865(119902) (Vminus119904)[3119891(119902)119865(119902)+1198911015840(119902)(119902minusint1199020119865(119909)119889119909)]

must be held Then 120587119902(119902) is a concave function of 119902Substitute 119902lowast 119901lowast and 120578

lowast= 119896119902

lowast119905 into (3) we have

120587(119902lowast 119901lowast 120578lowast) = 120587

119902(119902lowast) We know that 120587119902(119902) is a concave

function of 119902 and (119889120587119902(119902)119889119902)|119902=119902119902lowast = 0 Because of 119902119902lowast lt 119902lowast

Discrete Dynamics in Nature and Society 7

120587119902(119902lowast) lt 120587

119902(119902119902lowast) that is 120587(119902lowast 119901lowast 120578lowast) lt 120587

119902(119902119902lowast 120578119902lowast) This

completes the proof

Proposition 7 shows that QC strategy can improve themanufacturerrsquos maximum expected profit when green tech-nology investment and cap and trade policy are taken intoaccount

5 Decentralized Supply Chain Model

In this section the retailer and the manufacturer make deci-sions independently to maximize their own expected profitThe manufacturer is the Stackelberg leader and determinesthe wholesale price and the green technology investmentunder cap and trade policy The retailer is the follower anddetermines the retail price and the order quantity of theproduct

The sequence of events in this section is as follows firstthe manufacturer determines the optimal wholesale pricecarbon trading and green technology investment strategiessecond the retailer forms the belief of customersrsquo reservationprice and then decides the optimal retail price and orderquantity third the manufacture produces products anddelivers to the retailer fourth the customers form the beliefs120585prob of probability of the product sold at salvage price119904 according to the information of market price and thenform the reservation price 119903 fifth the customersrsquo demand issatisfied and the products are sold at full price 119901 finally allremaining products are sold to the external market at salvageprice 119904

51 The Optimal Strategies First we examine the decisionproblem of the retailer Recall that 119908 is the manufacturerrsquoswholesale price The retailerrsquos expected profit denoted by120587119903(119902 119901) is

120587119903(119902 119901) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909) minus (119908 minus 119904) 119902 (20)

Lemma 8 When 119901 is given 120587119903(119902 119901) is a concave function of119902

Proof When 119901 is given according to (20) we have120597120587119903(119902 119901)120597119902 = (119901 minus 119904)119865(119902) minus (119908 minus 119904) and 1205972120587119903(119902 119901)1205971199022 =

minus(119901 minus 119904)119891(119902) lt 0 Then we know that 120587119903(119902 119901) is a concavefunction of 119902 when 119901 is given This completes the proof

As to the retailerrsquos optimal strategies in RE Equilibriumwe can obtain the following proposition

Proposition 9 When the wholesale price is given the retailerrsquosoptimal order strategies (119902119903lowast) and optimal pricing strategies(119901119903lowast) with strategic customer behavior are

119902119903lowast= 119865minus1

(radic119908 minus 119904

V minus 119904) (21)

119901119903lowast= 119904 + radic(119908 minus 119904) (V minus 119904) (22)

Proof In RE Equilibrium (9) still holds Let 120597120587119903(119902 119901)120597119902 = 0and combining with (9) we have

119901 = V minus (V minus 119904) 119865 (119902)

(119901 minus 119904) 119865 (119902) minus (119908 minus 119904) = 0

(23)

According to Lemma 8 we can obtain the optimal orderand pricing strategies of the retailer by solving the equationsabove This completes the proof

Proposition 9 shows that the retailerrsquos optimal order andpricing strategies in REEquilibriumwith cap and trade policyexist and are unique

Second we investigate the manufacturerrsquos decision prob-lem The expected profit function of the manufacturer withgreen technology investment and cap and trade policy is120587119898(119908 119890 120578) = (119908 minus 119888)119902 minus 119896119890 minus (12)119905120578

2According to (21) with green technology investment and

cap and trade policy in wholesale price contract the optimalorder quantity of the retailer 119902119903lowast and the optimal wholesaleprice of the manufacturer 119908119898lowast is a one-to-one relationship

119908 = 119904 + (V minus 119904) 1198652

(119902) (24)

Combined with 119890 = (1 minus 120578)119902 minus 119864 the manufacturerrsquosexpected profit function can be converted into

120587119898(119902 120578) = [(V minus 119904) 119865

2

(119902) minus (119888 minus 119904 + (1 minus 120578) 119896)] 119902

+ 119896119864 minus1

21199051205782

(25)

When using wholesale price contract the total profit ofthe supply chain denoted by 120587sc

(119902 119901 119890 120578) is

120587sc(119902 119901 119890 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909) minus (119888 minus 119904) 119902

minus 119896119890 minus1

21199051205782

(26)

According to (9) and 119890 = (1 minus 120578)119902 minus 119864 the total supplychain profit function in RE Equilibrium can be simplified as

120587sc(119902 120578) = (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578) 119896) 119902 + 119896119864 minus1

21199051205782

(27)

Lemma 10 Given 119902 the optimal green technology investmentstrategy of the manufacturer denoted by 120578119898lowast satisfies

120578119898lowast

equiv 120578 (119902) =119896119902

1199050 lt 120578119898lowast

lt 1 (28)

Proof The first-order and second-order partial derivatives of120587119898(119902 120578) with respect to 120578 are 120597120587119898(119902 120578)120597120578 = 119902119896 minus 119905120578 and

1205972120587119898(119902 120578)120597120578

2= minus119905 lt 0 Let 120597120587119898(119902 120578)120597120578 = 0 we get

120578119898lowast

equiv 120578(119902) = 119896119902119905 In addition according to the assumptionin Section 3 we know 0 lt 120578

119898lowastlt 1 This completes the

proof

8 Discrete Dynamics in Nature and Society

Substituting 120578119898lowast = 120578(119902) into (25)

120587119898(119902) equiv 120587

119898(119902 120578 (119902))

= [(V minus 119904) 1198652

(119902) minus (119888 minus 119904 + 119896)] 119902 + 119896119864

+11989621199022

2119905

(29)

Then the two-variable optimization problem is simplifiedas a single variable optimization problem

max119902ge0

120587119898(119902) (30)

Lemma 11 If 1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus

1199021198912(119902)] lt 0 120587119898(119902) is a concave function of 119902

Proof The first-order and second-order derivatives of 120587119898(119902)with respect to 119902 are 119889120587

119898(119902)119889119902 = (V minus 119904)[119865

2

(119902) minus

2119902119891(119902)119865(119902)] minus (119888 minus 119904 + 119896) + (1198962119905)119902 and 119889

2120587119898(119902)119889119902

2=

1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus 1199021198912(119902)] lt 0 Then

we have that 120587119898(119902) is a concave function of 119902 This completesthe proof

In decentralized supply chain as to the manufacturerrsquosoptimal wholesale price (denoted by 119908

119898lowast) the followingproposition is obtained

Proposition 12 If 1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus1199021198912(119902)] lt 0 the manufacturerrsquos optimal wholesale price in

decentralized supply chain (119908119898lowast) satisfies 119908119898lowast = 119904 + (V minus119904)1198652

(119902119903lowast) where 119902119903lowast represents the retailerrsquos optimal order

quantity and satisfies

(V minus 119904) [1198652

(119902) minus 2119902119891 (119902) 119865 (119902)] minus (119888 minus 119904 + 119896) +1198962

119905119902

= 0

(31)

Proof According to Lemma 11 we can obtain that 119902119903lowastwhichmaximizes themanufacturerrsquos expected profit satisfies(31) Combining with (24) we know that the manufacturerrsquosoptimal wholesale price satisfies 119908119898lowast = 119904 + (V minus 119904)119865

2

(119902119903lowast) at

this time This completes the proof

Proposition 12 shows that the optimal commitment quan-tity of the manufacturer in decentralized supply chain withgreen technology investment and cap and trade policy existand are unique The optimal wholesale price does not relateto the cap allocated by the government but relates to thecarbon trading price This is because the production of themanufacturer is not affected by the cap but is affected bythe margin cost of the product However the margin cost ofthe product varies with the change of carbon trading pricewhen carbon trading is allowed The change results in thatthe optimal commitment quantity is not related to the cap butrelate to the carbon trading price

Substitute 119908119898lowast into (22) we derive the retailerrsquos optimalpricing in decentralized situation119901119903lowast = 119904+radic(119908119898lowast minus 119904)(V minus 119904)

According to Lemma 10 we get the optimal green technologyinvestment of the manufacturer 120578119898lowast = (119896119905)119902119903lowast

Substitute 119902119903lowast 119901119903lowast 119908119898lowast and 120578119898lowast into 120587

119903(119902 119901) 120587119898(119902)

and 120587sc(119902 120578) we obtain the maximum expected profit of

the retailer the manufacturer and the supply chain indecentralized channel

120587119903(119902119903lowast 119901119903lowast) = (119901

119903lowastminus 119904) (119902

119903lowastminus int

119902119903lowast

0

119865 (119909) 119889119909)

minus (119908119898lowast

minus 119904) 119902119903lowast

120587119898(119902119903lowast) = [(V minus 119904) 119865

2

(119902119903lowast) minus (119888 minus 119904 + 119896)] 119902

119903lowast

+ 119896119864 +1198962119902119903lowast2

2119905

120587sc(119902119903lowast 120578119898lowast) = (V minus 119904) 119865 (119902119903lowast) (119902119903lowast minus int

119902119903lowast

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578119898lowast) 119896) 119902119903lowast+ 119896119864

minus1

2119905120578119898lowast2

(32)

52 The Effect of Decentralization We examine the impact ofdecentralization on the supply chain optimal strategies andmaximumexpected profitwhen green technology investmentis taken into consideration by comparing the optimal strate-gies and the maximum expected profit in centralized anddecentralized channel with green technology investment andcap and trade policy

As the effect of decentralization of supply chain we canobtain the following proposition

Proposition 13 Consider 119902119903lowast lt 119902119902lowast

lt 119902lowast 119901119903lowast gt 119901

119902lowastgt 119901lowast

120578119898lowast

lt 120578119902lowastlt 120578lowast

Proof In Proposition 6 119902119902lowast lt 119902lowast 119901119902lowast gt 119901

lowast and 120578119902lowast lt 120578lowast

have been provedTherefore we only need to prove 119902119903lowast lt 119902119902lowast119901119903lowastgt 119901119902lowast and 120578119898lowast lt 120578119902lowast

Define 119867(119902) = 120587119902(119902) minus 120587119898(119902) then

119867(119902)

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

minus (V minus 119904) 1199021198652

(119902)

1198671015840(119902)

= (V minus 119904) 119891 (119902) [2119902119865 (119902) minus (119902 minus int119902

0

119865 (119909) 119889119909)]

(33)

Define 119866(119902) = 2119902119865(119902) minus (119902 minus int119902

0119865(119909)119889119909) then 1198661015840(119902) =

119865(119902) minus 2119902119891(119902) Because 119866(0) = 0 1198661015840(0) = 1 and 119865 satisfiesIFR we know that the value of 119866(119902) starts from 0 increasingfirst and then decreasing in 119902 That is 119866(119902) = 0 has uniquesolution and we denote it as So when 119902 lt 119866(119902) gt 0 that

Discrete Dynamics in Nature and Society 9

is119867(119902) is increasing in 119902 when 119902 gt 119866(119902) lt 0 that is119867(119902)is decreasing in 119902 Then we have that119867(119902) is a quasi-concavefunction of 119902 and 1198661015840 () = 119865() minus 2119891() lt 0 According tothe analysis above we know that1198671015840() = 0 that is 2119865() =( minus int

0119865(119909)119889119909) Substitute into 119889120587119902(119902)119889119902 we have

119889120587119902(119902)

119889119902

100381610038161003816100381610038161003816100381610038161003816119902=

= (V minus 119904) [1198652

() minus 119891 () ( minus int

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) +1198962

119905

= (V minus 119904) 119865 () [119865 () minus 2119891 ()] minus (119888 minus 119904 + 119896)

+1198962

119905lt (V minus 119904) 119865 () [119865 () minus 2119891 ()] lt 0

(34)

According to Lemma 4 we have that 120587119902(119902) is a concavefunction of 119902 so we get 119902119902lowast lt Because 2119865() = ( minus

int

0119865(119909)119889119909) and119867(119902) is a quasi-concave function of 119902 thenwe

get that 119889120587119898(119902)119889119902 lt 119889120587119902(119902)119889119902 if 119902 lt and 119889120587119898(119902)119889119902 gt

119889120587119902(119902)119889119902 if 119902 gt Therefore the only possible orderings for

119902119902lowast 119902119903lowast and are lt 119902

119902lowastlt 119902119903lowast and 119902119903lowast lt 119902

119902lowastlt We

have proved 119902119902lowast lt so we have 119902119903lowast lt 119902119902lowast We get 119901119903lowast gt 119901119902lowastbecause 119901 and 119902 satisfy 119901 = V minus (V minus 119904)119865(119902) in two situationsWe get 120578119898lowast lt 120578

119902lowast because 120578 and 119902 satisfy 120578 = 119896119902119905 in twosituations This completes the proof

Proposition 13 shows that for the decision of a decen-tralized supply chain with green technology investment theoptimal production quantity is increasing the optimal priceis decreasing and the green technology investment (iecarbon reduction rate) is increasing respectively in the threesituations of decentralized supply chain QC scenario and REEquilibrium scenario

The reason is that themanufacturer prefers higher whole-sale price (ie lower production and less green technologyinvestment) to guarantee its ownprofitmaximization thoughthis may harm the retailer even the whole supply chain

In Proposition 7 we have found 120587119902(119902119902lowast 120578119902lowast) gt

120587(119902lowast 119901lowast 120578lowast) that is the manufacturerrsquos maximum profit in

QC scenario is always greater than that in RE EquilibriumTherefore the section analyzes the effect of decentralizationon supply chain performance based on the QC scenario toprovide benchmark for designing of subsequent coordinationcontract

As to the effect of decentralization on themaximumprofitof supply chain we have the following proposition

Proposition 14 Consider 120587sc(119902119903lowast 120578119898lowast) lt 120587119902(119902119902lowast 120578119902lowast)

Proof The optimal green technology investment in decen-tralized supply chain and QC scenario are 120578(119902) = 119896119902119905

according to Lemmas 3 and 10 So the profit function in thetwo situations above can be converted into a same singlevariable function with respect to 119902 that is (16)

According to Lemma 4 we know that 120587119902(119902) is a concavefunction and reaches the maximum value at 119902119902lowast Accordingto Proposition 13 we have 119902119903lowast lt 119902

119902lowast then we get 120587119902(119902119903lowast) lt120587119902(119902119902lowast) that is 120587sc

(119902119903lowast 120578119898lowast) lt 120587

119902(119902119902lowast 120578119902lowast) This completes

the proof

Proposition 14 shows that the maximum profit of decen-tralized supply chain is always lower than that in QC scenariowhich indicates that the profit of the decentralized supplychain can get higher This is the benchmark of supply chaincoordination

6 Supply Chain Coordination

This section coordinates the supply chain with green tech-nology investment and cap and trade policy based on theQC scenario to improve the profit of the decentralized supplychain

61 Coordination Contract Based on Revenue and CostsSharing Contract With revenue sharing contract we assumethat 120601 (0 le 120601 le 1) represents the ratio of revenue kept by theretailer and (1 minus 120601) represents the ratio of revenue deliveredto the manufacturer

The retailerrsquos expected profit function with revenue shar-ing contract denoted by 120587119903

119904(119902 119901 120601) is

120587119903

119904(119902 119901 120601) = 120601 (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119908 minus 120601119904) 119902

(35)

Themanufacturerrsquos expected profit functionwith revenuesharing contract denoted by 120587119898

119904(119908 119890 120578 120601) is

120587119898

119904(119908 119890 120578 120601) = (119908 minus 119888) 119902 + (1 minus 120601)

sdot [int

119902

0

[119901119909 + 119904 (119902 minus 119909)] 119891 (119909) 119889119909

+ int

infin

119902

119901119902119891 (119909) 119889119909] minus 119896119890 minus1

21199051205782

(36)

Substitute 119890 = (1 minus 120578)119902 minus 119864 into the equation above andwe have

120587119898

119904(119908 120578 120601) = (1 minus 120601) (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

+ (119908 minus 119888 minus (1 minus 120578) 119896 + (1 minus 120601) 119904) 119902

+ 119896119864 minus1

21199051205782

(37)

The total expected profit of the whole supply chaindenoted by 120587sc

119904(119902 119901 120578) is

120587sc119904(119902 119901 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578) 119896) 119902 + 119896119864 minus1

21199051205782

(38)

10 Discrete Dynamics in Nature and Society

The subscript 119904 represents the situation of supply chaincoordination using revenue sharing contracts

As to the retailerrsquos optimal order strategy (denoted by119902lowast

119904) and pricing strategy (denoted by 119901lowast

119904) of the supply chain

with revenue sharing contract the following proposition isobtained

Proposition 15 Under revenue sharing contracts the retailerrsquosoptimal order strategy (119902lowast

119904) and optimal pricing strategy (119901lowast

119904)

are

119902lowast

119904= 119865minus1

(radic119908 minus 120601119904

120601 (V minus 119904))

119901lowast

119904= 119904 + radic

(119908 minus 120601119904) ((V minus 119904))120601

(39)

Proof According to the definition of RE Equilibrium wealso have 119901 = V minus (V minus 119904)119865(119902) under revenue sharingcontracts

The first-order and second-order derivative of 120587119903119904(119902 119901 120601)

with respect to 119902 are 120597120587119903

119904(119902 119901 120601)120597119902 = 120601(119901 minus 119904)119865(119902) minus

(119908 minus 120601119904) and 1205972120587119903

119904(119902 119901 120601)120597119902

2= minus120601(119901 minus 119904)119891(119902) lt

0 So given 119901 the retailerrsquos expected profit function is aconcave function of 119902 Let 120597120587119903

119904(119902 119901 120601)120597119902 = 0 and combine

with 119901 = V minus (V minus 119904)119865(119902) we can solve the retailerrsquos optimalstrategies under revenue sharing contractThis completes theproof

Proposition 15 shows that the optimal order and pricingstrategies of the retailer based on revenue sharing contract inRE Equilibrium exist and are uniqueThis is also the retailerrsquosoptimal response function with respect to 119908

Substituting (39) into (43)

120587119898

119904(119902 120578 120601)

= (1 minus 120601) (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ [120601 (V minus 119904) 1198652

(119902) minus (119888 minus 119904 + (1 minus 120578) 119896)] 119902 + 119896119864

minus1

21199051205782

120597120587119898

119904(119902 120578 120601)

120597120578= 119902119896 minus 119905120578

1205972120587119898

119904(119902 120578 120601)

1205971205782= minus119905 lt 0

(40)

Given 119902 and 120601 the manufacturerrsquos green technologyinvestment strategies with revenue sharing contracts exist

and are unique Let 120597120587119898119904(119902 120578 120601)120597120578 = 0 we get 120578119898lowast

119904equiv 120578(119902) =

119902119896119905 and substitute it into 120587119898119904(119902 120578 120601)

120587119898

119904(119902 120601)

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ 120601 (V minus 119904) 119865 (119902) [119902119865 (119902) minus (119902 minus int119902

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) 119902 + 119896119864 +11989621199022

2119905

(41)

Proposition 16 The supply chain with green technologyinvestment and cap and trade policy can not be coordinated byrevenue sharing contract

Proof The derivative of 120587119898119904(119902 120601) with respect to 119902 can be

arranged as follows

120597120587119898

119904(119902 120601)

120597119902=120597120587119902(119902)

120597119902+ 120601(

120597120587119898(119902)

120597119902minus120597120587119902(119902)

120597119902) (42)

Only when 120601 = 0 are the optimal strategies of supplychain under revenue sharing contracts equal to that in QCscenario Let 119908lowast

119904represent the optimal wholesale price of

the manufacture and 119902lowast119904represent the retailerrsquos optimal order

quantity Then 120587119903

119904(119902 119901 120601) = minus119908

lowast

119904119902lowast

119904lt 0 So the revenue

sharing contract cannot realize the supply chain coordinationat this time This completes the proof

Proposition 16 shows that revenue sharing contractscan not realize the supply chain coordination with greentechnology investment and cap and trade policy The rea-son is that the manufacturer undertakes all the costs ofgreen technology investment when using revenue sharingcontracts thus making the manufacturer tend to invest lessin green technology Only when the manufacturer owns allthe revenue will themanufacturerrsquos optimal green technologyinvestment under revenue sharing contracts be equal to thatin QC scenario However the maximum profit of the retaileris negative at this time and it will not participate

62 Coordination Contract Based on Revenue Sharing-CostSharing Contract Proposition 16 states that the supply chainwith green technology investment and cap and trade policycan not be coordinated by revenue sharing contracts becausethe costs of green technology investment are undertaken bythe manufacturer and the game of retailer and customersis characterized by RE Equilibrium So we consider costssharing of green technology investment and carbon tradingand quantity commitment made by the manufacturer basedon revenue sharing contracts Let 120601 (0 lt 120601 le 1) represent theratio of revenue kept by the retailer and let (1minus120601) represent theratio of revenue delivered to the manufacturer 120572 (0 le 120572 le 1)represents the ratio of costs of green technology investmentand carbon trading undertaken by the manufacturer and(1 minus 120572) represents the ratio of costs of green technologyinvestment and carbon trading delivered to the retailer

Discrete Dynamics in Nature and Society 11

The retailerrsquos expected profit function with revenuesharing-cost sharing contract is

120587119903

120572(119902 120601 120572) = 120601 (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119908 minus 120601119904) 119902

minus (1 minus 120572) 119896 [(1 minus 120578) 119902 minus 119864]

minus1

2(1 minus 120572) 119905120578

2

(43)

We have 119890 = (1minus120578)119902minus119864 then themanufacturerrsquos expectedprofit function denoted by 120587119898

120572(119908 120578 120601 120572) is

120587119898

120572(119908 120578 120601 120572)

= (1 minus 120601) (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ (119908 minus 119888 minus 120572 (1 minus 120578) 119896 + (1 minus 120601) 119904) 119902 + 120572119896119864

minus1

21205721199051205782

(44)

The subscript 120572 represents the situation that using rev-enue sharing-cost sharing contract coordinates the supplychain

Proposition 17 If 0 le 120582 le 1 the parameters of revenuesharing-cost sharing (119908 120601 120572) satisfy

119908 = 120582119888 + (1 minus 120578) 119896

120601 = 120582

120572 = 1 minus 120582

(45)

The supply chainwith green technology investment and cap andtrade policy is coordinated

Proof When the parameters of revenue sharing-cost sharingsatisfy (45) the expected profit function of the retailer andthe manufacturer can be written as follows

120587119903

120572(119902 119901 120601 120572) = 120582120587

119902(119902 120578)

120587119898

120572(119908 120578 120601 120572) = (1 minus 120582) 120587

119902(119902 120578)

(46)

The optimal order of decentralized supply chain is equalto that of QC scenario Substitute 119901 = V minus (V minus 119904)119865(119902) into120587sc119888(119902 119901 120578) we get that the profit function of decentralized

supply chain at this time is the same as that of QC scenarioTherefore when the parameters of revenue sharing-costsharing (119908 120601 120572) satisfy (45) the decentralized supply chaincoordination is realizedThemanufacturer earns all the profitif 120582 = 0 and the retailer earns all the profit if 120582 = 1Arbitrary allocation of the supply chain profit between themanufacturer and the retailer is allowed This completes theproof

Proposition 17 shows that revenue sharing-cost sharingcontract can realize the supply chain coordination on condi-tion that the manufacturer makes a quantity commitment to

the customers The optimal production quantity and greentechnology investment in decentralized situation are lowerthan that in QC scenario The quantity commitment of themanufacturer can increase the green technology investmentand production quantity and also reduce the retailer priceThat is the quantity commitment is beneficial for the man-ufacturer the retailer and customers Thus the quantitycommitment is credible to customers now

7 Conclusions and Suggestions for FurtherResearch

This paper investigates the production pricing carbon trad-ing and green technology investment strategies and thecoordination of low carbon supply chain made up of a lowcarbon manufacturer and a retailer The low carbon manu-facturer produces one product under cap and trade policyand performs green technology investment to reduce carbonemissions in production process The retailer orders fromthemanufacturer and distributes the product to homogenouscustomers To the best of our knowledge this is the firststudy on the management of low carbon supply chainswith strategic customer behaviorThis paper provides severalinteresting observations

Observation 1 There are unique optimal production pricingcarbon trading and green technology investment strategiesin centralized (include RE Equilibrium scenario and QCscenario) and decentralized supply chain Therefore byderiving the optimal production pricing carbon trading andgreen technology investment strategies in different situationsthe low carbon manufacturer and the retailer can behaveappropriately based on our findings to maximize their profit

Observation 2 Our findings also show that the centralizedsupply chain can improve its performance by quantity com-mitment However the manufacturerrsquos production and greentechnology investment are reduced while the optimal priceis increased at this time Therefore quantity commitment isbeneficial for the supply chain but is harmful to customers

Observation 3 We also show that the performance of decen-tralized supply chain is lower than that in QC scenario Wefind that the decentralization reduces the manufacturersquos opti-mal production quantity and green technology investmentand increases the retailerrsquos optimal price Besides we find thatthe optimal production quantity is increasing the optimalprice is decreasing and the green technology investment isincreasing respectively in the three situations of decentral-ized supply chain QC scenario andREEquilibrium scenario

Observation 4 We demonstrate that the traditional revenuesharing contracts can not realize the low carbon supply chaincoordination when green technology investment is takenin consideration We also find that the revenue sharing-cost sharing contract realizes the decentralized supply chain

12 Discrete Dynamics in Nature and Society

coordination when the manufacturer makes a quantity com-mitment It is beneficial to the manufacturer the retailerand customers if themanufacturermakes a quantity commit-ment So the quantity commitment is credible to the customerat this time

We assume the customers are homogeneous strategiccustomers But themyopic customers and strategic customersoften coexist and strategic customers also have differenttolerance in real-life Extending this assumption will havea knock-on effect on supply chain decisions So one keyresearch direction is to consider that the customers aremixed customers myopic customers and strategic customersSecond we assume that the supply chain only producesone product While this simplified setting provides someinteresting insights we have to acknowledge that a manufac-turer usually produces two or more productsThese productswill substitute or complement each other So one importantextension of our work is to assume that the manufacturerproduces two or more products which have different unitcarbon emissions in production

Notations

119863 The random demand and119863 ge 0

119891(sdot) Probability density function of119863119865(sdot) Distribution function of119863 We assume

that 119865 satisfy IFR and define 119865 = 1 minus 119865

119888 Unit production cost119904 Unit salvage price of the product which

is an exogenous variableV The customersrsquo utility from consuming

the product119908 Unit wholesale price119901 Unit retail selling price We assume that

it can be observed by the customers119902 The manufacturerrsquos production

quantitythe retailerrsquos order quantityWe assume that it cannot be observedby the customers

119903 The customersrsquo reservation price whichis customersrsquo private information andcannot be observed by the retailer

120585119903 The beliefs of the manufacturer over the

customerrsquo reservation price120585prob The beliefs of customers over their

chances of obtaining the product atphase two

119864 Initial free carbon emission allocated bygovernment

119896 Unit carbon emission trading price119890 Carbon trading volume120578 Carbon emissions reduction rate

Conflict of Interests

The authors declare no conflict of interests

Acknowledgments

This research is partially supported by National Natural Sci-ence Foundation of China (nos 71432003 and 71272128) theProgram forNewCentury Excellent Talents inUniversity (noNCET-12-0087) and Specialized Research Fund for the Doc-toral Program of Higher Education (no 20130185110006)

References

[1] R Alley T Berntsen N L Bindoff et al ldquoSummary forpolicymakersrdquo in Climate Change 2007 The Physical ScienceBasis vol 6 pp 1ndash18 IPCC 2007

[2] X Chen and G Hao ldquoSustainable pricing and productionpolicies for two competing firms with carbon emissions taxrdquoInternational Journal of Production Research vol 53 pp 6408ndash6420 2015

[3] X Chen S Benjaafar and A Elomri ldquoThe carbon-constrainedEOQrdquo Operations Research Letters vol 41 no 2 pp 172ndash1792013

[4] N O Keohane ldquoCap and trade rehabilitated using tradablepermits to control US greenhouse gasesrdquo Review of Environ-mental Economics and Policy vol 3 no 1 pp 42ndash62 2009

[5] B Liu M Holmbom A Segerstedt and W Chen ldquoEffectsof carbon emission regulations on remanufacturing decisionswith limited information of demand distributionrdquo InternationalJournal of Production Research vol 53 no 2 pp 532ndash548 2015

[6] L Zhu and Y Fan ldquoA real options-based CCS investmentevaluation model case study of Chinarsquos power generationsectorrdquo Applied Energy vol 88 no 12 pp 4320ndash4333 2011

[7] XWang and LDu ldquoStudy on carbon capture and storage (CCS)investment decision-making based on real options for Chinarsquoscoal-fired power plantsrdquo Journal of Cleaner Production vol 112part 5 pp 4123ndash4131 2016

[8] W Jiang and X Chen ldquoOptimal strategies for manufacturerwith strategic customer behavior under carbon emissions-sensitive randomdemandrdquo Industrial Management amp Data Sys-tems 2015

[9] Y Levin J McGill and M Nediak ldquoDynamic pricing in thepresence of strategic consumers and oligopolistic competitionrdquoManagement Science vol 55 no 1 pp 32ndash46 2009

[10] A K Parlakturk ldquoThe value of product variety when sellingto strategic consumersrdquo Manufacturing amp Service OperationsManagement vol 14 no 3 pp 371ndash385 2012

[11] X Su and F Zhang ldquoOn the value of commitment andavailability guarantees when selling to strategic consumersrdquoManagement Science vol 55 no 5 pp 713ndash726 2009

[12] W Elmaghraby S A Lippman C S Tang and R Yin ldquoWillmore purchasing options benefit customersrdquo Production andOperations Management vol 18 no 4 pp 381ndash401 2009

[13] W Jiang and X Chen ldquoManufacturersquos production and pricingstrategies with carbon tax policy and strategic customer behav-iorrdquoManagement Science and Engineering vol 9 no 1 pp 30ndash35 2015

[14] I Dobos ldquoThe effects of emission trading on production andinventories in the Arrow-Karlin modelrdquo International Journalof Production Economics vol 93-94 pp 301ndash308 2005

[15] X Chen C K Chan and Y C E Lee ldquoResponsible productionpolicies with substitution and carbon emissions tradingrdquo Jour-nal of Cleaner Production 2015

Discrete Dynamics in Nature and Society 13

[16] X Chen and X Wang ldquoEffects of carbon emission reductionpolicies on transportation mode selections with stochasticdemandrdquo Transportation Research Part E Logistics and Trans-portation Review 2015

[17] X Chang H Xia H Zhu T Fan and H Zhao ldquoProductiondecisions in a hybrid manufacturingndashremanufacturing systemwith carbon cap and trade mechanismrdquo International Journal ofProduction Economics vol 162 pp 160ndash173 2015

[18] J Zhao B F Hobbs and J-S Pang ldquoLong-run equilibriummodeling of emissions allowance allocation systems in electricpowermarketsrdquoOperations Research vol 58 no 3 pp 529ndash5482010

[19] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2 pp519ndash527 2011

[20] A Toptal H Ozlu and D Konur ldquoJoint decisions on inventoryreplenishment and emission reduction investment under dif-ferent emission regulationsrdquo International Journal of ProductionResearch vol 52 no 1 pp 243ndash269 2014

[21] A S Manikas and J R Kroes ldquoA newsvendor approach tocompliance and production under cap and trade emissionsregulationrdquo International Journal of Production Economics vol159 pp 274ndash284 2015

[22] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineering vol10 no 1 pp 99ndash116 2013

[23] X Xu W Zhang P He and X Xu ldquoProduction and pricingproblems in make-to-order supply chain with cap-and-traderegulationrdquo Omega 2015

[24] Y Zhen X Yang and W Jiang ldquoThe impact of transport modeand carbon policy on low-carbon retailerrdquoDiscrete Dynamics inNature and Society vol 2015 Article ID 964305 12 pages 2015

[25] S Swami and J Shah ldquoChannel coordination in green supplychainmanagementrdquo Journal of the Operational Research Societyvol 64 no 3 pp 336ndash351 2013

[26] R H Coase ldquoDurability andmonopolyrdquoThe Journal of Law andEconomics vol 15 no 1 pp 143ndash149 1972

[27] X Su ldquoIntertemporal pricing with strategic customer behaviorrdquoManagement Science vol 53 no 5 pp 726ndash741 2007

[28] D Zhang and W L Cooper ldquoManaging clearance sales in thepresence of strategic customersrdquo Production and OperationsManagement vol 17 no 4 pp 416ndash431 2008

[29] S Whang ldquoDemand uncertainty and the Bayesian effect inmarkdown pricing with strategic customersrdquoManufacturing ampService Operations Management vol 17 no 1 pp 66ndash77 2015

[30] J Du J Zhang and G Hua ldquoPricing and inventory manage-ment in the presence of strategic customers with risk preferenceand decreasing valuerdquo International Journal of Production Eco-nomics vol 164 pp 160ndash166 2015

[31] X Su and F Zhang ldquoStrategic customer behavior commitmentand supply chain performancerdquo Management Science vol 54no 10 pp 1759ndash1773 2008

[32] H Yang ldquoImpact of discounting and competition on benefit ofdecentralization with strategic customersrdquo Operations ResearchLetters vol 40 no 2 pp 123ndash127 2012

[33] D Yang E Qi and Y Li ldquoQuick response and supply chainstructure with strategic consumersrdquo Omega vol 52 pp 1ndash142015

[34] C drsquoAspremont and A Jacquemin ldquoCooperative and nonco-operative R amp D in duopoly with spilloversrdquo The AmericanEconomic Review vol 78 no 5 pp 1133ndash1137 1988

[35] J F Muth ldquoRational expectations and the theory of pricemovementsrdquo Econometrica vol 29 no 3 pp 315ndash335 1961

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

120587119902(119902lowast) lt 120587

119902(119902119902lowast) that is 120587(119902lowast 119901lowast 120578lowast) lt 120587

119902(119902119902lowast 120578119902lowast) This

completes the proof

Proposition 7 shows that QC strategy can improve themanufacturerrsquos maximum expected profit when green tech-nology investment and cap and trade policy are taken intoaccount

5 Decentralized Supply Chain Model

In this section the retailer and the manufacturer make deci-sions independently to maximize their own expected profitThe manufacturer is the Stackelberg leader and determinesthe wholesale price and the green technology investmentunder cap and trade policy The retailer is the follower anddetermines the retail price and the order quantity of theproduct

The sequence of events in this section is as follows firstthe manufacturer determines the optimal wholesale pricecarbon trading and green technology investment strategiessecond the retailer forms the belief of customersrsquo reservationprice and then decides the optimal retail price and orderquantity third the manufacture produces products anddelivers to the retailer fourth the customers form the beliefs120585prob of probability of the product sold at salvage price119904 according to the information of market price and thenform the reservation price 119903 fifth the customersrsquo demand issatisfied and the products are sold at full price 119901 finally allremaining products are sold to the external market at salvageprice 119904

51 The Optimal Strategies First we examine the decisionproblem of the retailer Recall that 119908 is the manufacturerrsquoswholesale price The retailerrsquos expected profit denoted by120587119903(119902 119901) is

120587119903(119902 119901) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909) minus (119908 minus 119904) 119902 (20)

Lemma 8 When 119901 is given 120587119903(119902 119901) is a concave function of119902

Proof When 119901 is given according to (20) we have120597120587119903(119902 119901)120597119902 = (119901 minus 119904)119865(119902) minus (119908 minus 119904) and 1205972120587119903(119902 119901)1205971199022 =

minus(119901 minus 119904)119891(119902) lt 0 Then we know that 120587119903(119902 119901) is a concavefunction of 119902 when 119901 is given This completes the proof

As to the retailerrsquos optimal strategies in RE Equilibriumwe can obtain the following proposition

Proposition 9 When the wholesale price is given the retailerrsquosoptimal order strategies (119902119903lowast) and optimal pricing strategies(119901119903lowast) with strategic customer behavior are

119902119903lowast= 119865minus1

(radic119908 minus 119904

V minus 119904) (21)

119901119903lowast= 119904 + radic(119908 minus 119904) (V minus 119904) (22)

Proof In RE Equilibrium (9) still holds Let 120597120587119903(119902 119901)120597119902 = 0and combining with (9) we have

119901 = V minus (V minus 119904) 119865 (119902)

(119901 minus 119904) 119865 (119902) minus (119908 minus 119904) = 0

(23)

According to Lemma 8 we can obtain the optimal orderand pricing strategies of the retailer by solving the equationsabove This completes the proof

Proposition 9 shows that the retailerrsquos optimal order andpricing strategies in REEquilibriumwith cap and trade policyexist and are unique

Second we investigate the manufacturerrsquos decision prob-lem The expected profit function of the manufacturer withgreen technology investment and cap and trade policy is120587119898(119908 119890 120578) = (119908 minus 119888)119902 minus 119896119890 minus (12)119905120578

2According to (21) with green technology investment and

cap and trade policy in wholesale price contract the optimalorder quantity of the retailer 119902119903lowast and the optimal wholesaleprice of the manufacturer 119908119898lowast is a one-to-one relationship

119908 = 119904 + (V minus 119904) 1198652

(119902) (24)

Combined with 119890 = (1 minus 120578)119902 minus 119864 the manufacturerrsquosexpected profit function can be converted into

120587119898(119902 120578) = [(V minus 119904) 119865

2

(119902) minus (119888 minus 119904 + (1 minus 120578) 119896)] 119902

+ 119896119864 minus1

21199051205782

(25)

When using wholesale price contract the total profit ofthe supply chain denoted by 120587sc

(119902 119901 119890 120578) is

120587sc(119902 119901 119890 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909) minus (119888 minus 119904) 119902

minus 119896119890 minus1

21199051205782

(26)

According to (9) and 119890 = (1 minus 120578)119902 minus 119864 the total supplychain profit function in RE Equilibrium can be simplified as

120587sc(119902 120578) = (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578) 119896) 119902 + 119896119864 minus1

21199051205782

(27)

Lemma 10 Given 119902 the optimal green technology investmentstrategy of the manufacturer denoted by 120578119898lowast satisfies

120578119898lowast

equiv 120578 (119902) =119896119902

1199050 lt 120578119898lowast

lt 1 (28)

Proof The first-order and second-order partial derivatives of120587119898(119902 120578) with respect to 120578 are 120597120587119898(119902 120578)120597120578 = 119902119896 minus 119905120578 and

1205972120587119898(119902 120578)120597120578

2= minus119905 lt 0 Let 120597120587119898(119902 120578)120597120578 = 0 we get

120578119898lowast

equiv 120578(119902) = 119896119902119905 In addition according to the assumptionin Section 3 we know 0 lt 120578

119898lowastlt 1 This completes the

proof

8 Discrete Dynamics in Nature and Society

Substituting 120578119898lowast = 120578(119902) into (25)

120587119898(119902) equiv 120587

119898(119902 120578 (119902))

= [(V minus 119904) 1198652

(119902) minus (119888 minus 119904 + 119896)] 119902 + 119896119864

+11989621199022

2119905

(29)

Then the two-variable optimization problem is simplifiedas a single variable optimization problem

max119902ge0

120587119898(119902) (30)

Lemma 11 If 1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus

1199021198912(119902)] lt 0 120587119898(119902) is a concave function of 119902

Proof The first-order and second-order derivatives of 120587119898(119902)with respect to 119902 are 119889120587

119898(119902)119889119902 = (V minus 119904)[119865

2

(119902) minus

2119902119891(119902)119865(119902)] minus (119888 minus 119904 + 119896) + (1198962119905)119902 and 119889

2120587119898(119902)119889119902

2=

1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus 1199021198912(119902)] lt 0 Then

we have that 120587119898(119902) is a concave function of 119902 This completesthe proof

In decentralized supply chain as to the manufacturerrsquosoptimal wholesale price (denoted by 119908

119898lowast) the followingproposition is obtained

Proposition 12 If 1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus1199021198912(119902)] lt 0 the manufacturerrsquos optimal wholesale price in

decentralized supply chain (119908119898lowast) satisfies 119908119898lowast = 119904 + (V minus119904)1198652

(119902119903lowast) where 119902119903lowast represents the retailerrsquos optimal order

quantity and satisfies

(V minus 119904) [1198652

(119902) minus 2119902119891 (119902) 119865 (119902)] minus (119888 minus 119904 + 119896) +1198962

119905119902

= 0

(31)

Proof According to Lemma 11 we can obtain that 119902119903lowastwhichmaximizes themanufacturerrsquos expected profit satisfies(31) Combining with (24) we know that the manufacturerrsquosoptimal wholesale price satisfies 119908119898lowast = 119904 + (V minus 119904)119865

2

(119902119903lowast) at

this time This completes the proof

Proposition 12 shows that the optimal commitment quan-tity of the manufacturer in decentralized supply chain withgreen technology investment and cap and trade policy existand are unique The optimal wholesale price does not relateto the cap allocated by the government but relates to thecarbon trading price This is because the production of themanufacturer is not affected by the cap but is affected bythe margin cost of the product However the margin cost ofthe product varies with the change of carbon trading pricewhen carbon trading is allowed The change results in thatthe optimal commitment quantity is not related to the cap butrelate to the carbon trading price

Substitute 119908119898lowast into (22) we derive the retailerrsquos optimalpricing in decentralized situation119901119903lowast = 119904+radic(119908119898lowast minus 119904)(V minus 119904)

According to Lemma 10 we get the optimal green technologyinvestment of the manufacturer 120578119898lowast = (119896119905)119902119903lowast

Substitute 119902119903lowast 119901119903lowast 119908119898lowast and 120578119898lowast into 120587

119903(119902 119901) 120587119898(119902)

and 120587sc(119902 120578) we obtain the maximum expected profit of

the retailer the manufacturer and the supply chain indecentralized channel

120587119903(119902119903lowast 119901119903lowast) = (119901

119903lowastminus 119904) (119902

119903lowastminus int

119902119903lowast

0

119865 (119909) 119889119909)

minus (119908119898lowast

minus 119904) 119902119903lowast

120587119898(119902119903lowast) = [(V minus 119904) 119865

2

(119902119903lowast) minus (119888 minus 119904 + 119896)] 119902

119903lowast

+ 119896119864 +1198962119902119903lowast2

2119905

120587sc(119902119903lowast 120578119898lowast) = (V minus 119904) 119865 (119902119903lowast) (119902119903lowast minus int

119902119903lowast

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578119898lowast) 119896) 119902119903lowast+ 119896119864

minus1

2119905120578119898lowast2

(32)

52 The Effect of Decentralization We examine the impact ofdecentralization on the supply chain optimal strategies andmaximumexpected profitwhen green technology investmentis taken into consideration by comparing the optimal strate-gies and the maximum expected profit in centralized anddecentralized channel with green technology investment andcap and trade policy

As the effect of decentralization of supply chain we canobtain the following proposition

Proposition 13 Consider 119902119903lowast lt 119902119902lowast

lt 119902lowast 119901119903lowast gt 119901

119902lowastgt 119901lowast

120578119898lowast

lt 120578119902lowastlt 120578lowast

Proof In Proposition 6 119902119902lowast lt 119902lowast 119901119902lowast gt 119901

lowast and 120578119902lowast lt 120578lowast

have been provedTherefore we only need to prove 119902119903lowast lt 119902119902lowast119901119903lowastgt 119901119902lowast and 120578119898lowast lt 120578119902lowast

Define 119867(119902) = 120587119902(119902) minus 120587119898(119902) then

119867(119902)

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

minus (V minus 119904) 1199021198652

(119902)

1198671015840(119902)

= (V minus 119904) 119891 (119902) [2119902119865 (119902) minus (119902 minus int119902

0

119865 (119909) 119889119909)]

(33)

Define 119866(119902) = 2119902119865(119902) minus (119902 minus int119902

0119865(119909)119889119909) then 1198661015840(119902) =

119865(119902) minus 2119902119891(119902) Because 119866(0) = 0 1198661015840(0) = 1 and 119865 satisfiesIFR we know that the value of 119866(119902) starts from 0 increasingfirst and then decreasing in 119902 That is 119866(119902) = 0 has uniquesolution and we denote it as So when 119902 lt 119866(119902) gt 0 that

Discrete Dynamics in Nature and Society 9

is119867(119902) is increasing in 119902 when 119902 gt 119866(119902) lt 0 that is119867(119902)is decreasing in 119902 Then we have that119867(119902) is a quasi-concavefunction of 119902 and 1198661015840 () = 119865() minus 2119891() lt 0 According tothe analysis above we know that1198671015840() = 0 that is 2119865() =( minus int

0119865(119909)119889119909) Substitute into 119889120587119902(119902)119889119902 we have

119889120587119902(119902)

119889119902

100381610038161003816100381610038161003816100381610038161003816119902=

= (V minus 119904) [1198652

() minus 119891 () ( minus int

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) +1198962

119905

= (V minus 119904) 119865 () [119865 () minus 2119891 ()] minus (119888 minus 119904 + 119896)

+1198962

119905lt (V minus 119904) 119865 () [119865 () minus 2119891 ()] lt 0

(34)

According to Lemma 4 we have that 120587119902(119902) is a concavefunction of 119902 so we get 119902119902lowast lt Because 2119865() = ( minus

int

0119865(119909)119889119909) and119867(119902) is a quasi-concave function of 119902 thenwe

get that 119889120587119898(119902)119889119902 lt 119889120587119902(119902)119889119902 if 119902 lt and 119889120587119898(119902)119889119902 gt

119889120587119902(119902)119889119902 if 119902 gt Therefore the only possible orderings for

119902119902lowast 119902119903lowast and are lt 119902

119902lowastlt 119902119903lowast and 119902119903lowast lt 119902

119902lowastlt We

have proved 119902119902lowast lt so we have 119902119903lowast lt 119902119902lowast We get 119901119903lowast gt 119901119902lowastbecause 119901 and 119902 satisfy 119901 = V minus (V minus 119904)119865(119902) in two situationsWe get 120578119898lowast lt 120578

119902lowast because 120578 and 119902 satisfy 120578 = 119896119902119905 in twosituations This completes the proof

Proposition 13 shows that for the decision of a decen-tralized supply chain with green technology investment theoptimal production quantity is increasing the optimal priceis decreasing and the green technology investment (iecarbon reduction rate) is increasing respectively in the threesituations of decentralized supply chain QC scenario and REEquilibrium scenario

The reason is that themanufacturer prefers higher whole-sale price (ie lower production and less green technologyinvestment) to guarantee its ownprofitmaximization thoughthis may harm the retailer even the whole supply chain

In Proposition 7 we have found 120587119902(119902119902lowast 120578119902lowast) gt

120587(119902lowast 119901lowast 120578lowast) that is the manufacturerrsquos maximum profit in

QC scenario is always greater than that in RE EquilibriumTherefore the section analyzes the effect of decentralizationon supply chain performance based on the QC scenario toprovide benchmark for designing of subsequent coordinationcontract

As to the effect of decentralization on themaximumprofitof supply chain we have the following proposition

Proposition 14 Consider 120587sc(119902119903lowast 120578119898lowast) lt 120587119902(119902119902lowast 120578119902lowast)

Proof The optimal green technology investment in decen-tralized supply chain and QC scenario are 120578(119902) = 119896119902119905

according to Lemmas 3 and 10 So the profit function in thetwo situations above can be converted into a same singlevariable function with respect to 119902 that is (16)

According to Lemma 4 we know that 120587119902(119902) is a concavefunction and reaches the maximum value at 119902119902lowast Accordingto Proposition 13 we have 119902119903lowast lt 119902

119902lowast then we get 120587119902(119902119903lowast) lt120587119902(119902119902lowast) that is 120587sc

(119902119903lowast 120578119898lowast) lt 120587

119902(119902119902lowast 120578119902lowast) This completes

the proof

Proposition 14 shows that the maximum profit of decen-tralized supply chain is always lower than that in QC scenariowhich indicates that the profit of the decentralized supplychain can get higher This is the benchmark of supply chaincoordination

6 Supply Chain Coordination

This section coordinates the supply chain with green tech-nology investment and cap and trade policy based on theQC scenario to improve the profit of the decentralized supplychain

61 Coordination Contract Based on Revenue and CostsSharing Contract With revenue sharing contract we assumethat 120601 (0 le 120601 le 1) represents the ratio of revenue kept by theretailer and (1 minus 120601) represents the ratio of revenue deliveredto the manufacturer

The retailerrsquos expected profit function with revenue shar-ing contract denoted by 120587119903

119904(119902 119901 120601) is

120587119903

119904(119902 119901 120601) = 120601 (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119908 minus 120601119904) 119902

(35)

Themanufacturerrsquos expected profit functionwith revenuesharing contract denoted by 120587119898

119904(119908 119890 120578 120601) is

120587119898

119904(119908 119890 120578 120601) = (119908 minus 119888) 119902 + (1 minus 120601)

sdot [int

119902

0

[119901119909 + 119904 (119902 minus 119909)] 119891 (119909) 119889119909

+ int

infin

119902

119901119902119891 (119909) 119889119909] minus 119896119890 minus1

21199051205782

(36)

Substitute 119890 = (1 minus 120578)119902 minus 119864 into the equation above andwe have

120587119898

119904(119908 120578 120601) = (1 minus 120601) (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

+ (119908 minus 119888 minus (1 minus 120578) 119896 + (1 minus 120601) 119904) 119902

+ 119896119864 minus1

21199051205782

(37)

The total expected profit of the whole supply chaindenoted by 120587sc

119904(119902 119901 120578) is

120587sc119904(119902 119901 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578) 119896) 119902 + 119896119864 minus1

21199051205782

(38)

10 Discrete Dynamics in Nature and Society

The subscript 119904 represents the situation of supply chaincoordination using revenue sharing contracts

As to the retailerrsquos optimal order strategy (denoted by119902lowast

119904) and pricing strategy (denoted by 119901lowast

119904) of the supply chain

with revenue sharing contract the following proposition isobtained

Proposition 15 Under revenue sharing contracts the retailerrsquosoptimal order strategy (119902lowast

119904) and optimal pricing strategy (119901lowast

119904)

are

119902lowast

119904= 119865minus1

(radic119908 minus 120601119904

120601 (V minus 119904))

119901lowast

119904= 119904 + radic

(119908 minus 120601119904) ((V minus 119904))120601

(39)

Proof According to the definition of RE Equilibrium wealso have 119901 = V minus (V minus 119904)119865(119902) under revenue sharingcontracts

The first-order and second-order derivative of 120587119903119904(119902 119901 120601)

with respect to 119902 are 120597120587119903

119904(119902 119901 120601)120597119902 = 120601(119901 minus 119904)119865(119902) minus

(119908 minus 120601119904) and 1205972120587119903

119904(119902 119901 120601)120597119902

2= minus120601(119901 minus 119904)119891(119902) lt

0 So given 119901 the retailerrsquos expected profit function is aconcave function of 119902 Let 120597120587119903

119904(119902 119901 120601)120597119902 = 0 and combine

with 119901 = V minus (V minus 119904)119865(119902) we can solve the retailerrsquos optimalstrategies under revenue sharing contractThis completes theproof

Proposition 15 shows that the optimal order and pricingstrategies of the retailer based on revenue sharing contract inRE Equilibrium exist and are uniqueThis is also the retailerrsquosoptimal response function with respect to 119908

Substituting (39) into (43)

120587119898

119904(119902 120578 120601)

= (1 minus 120601) (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ [120601 (V minus 119904) 1198652

(119902) minus (119888 minus 119904 + (1 minus 120578) 119896)] 119902 + 119896119864

minus1

21199051205782

120597120587119898

119904(119902 120578 120601)

120597120578= 119902119896 minus 119905120578

1205972120587119898

119904(119902 120578 120601)

1205971205782= minus119905 lt 0

(40)

Given 119902 and 120601 the manufacturerrsquos green technologyinvestment strategies with revenue sharing contracts exist

and are unique Let 120597120587119898119904(119902 120578 120601)120597120578 = 0 we get 120578119898lowast

119904equiv 120578(119902) =

119902119896119905 and substitute it into 120587119898119904(119902 120578 120601)

120587119898

119904(119902 120601)

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ 120601 (V minus 119904) 119865 (119902) [119902119865 (119902) minus (119902 minus int119902

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) 119902 + 119896119864 +11989621199022

2119905

(41)

Proposition 16 The supply chain with green technologyinvestment and cap and trade policy can not be coordinated byrevenue sharing contract

Proof The derivative of 120587119898119904(119902 120601) with respect to 119902 can be

arranged as follows

120597120587119898

119904(119902 120601)

120597119902=120597120587119902(119902)

120597119902+ 120601(

120597120587119898(119902)

120597119902minus120597120587119902(119902)

120597119902) (42)

Only when 120601 = 0 are the optimal strategies of supplychain under revenue sharing contracts equal to that in QCscenario Let 119908lowast

119904represent the optimal wholesale price of

the manufacture and 119902lowast119904represent the retailerrsquos optimal order

quantity Then 120587119903

119904(119902 119901 120601) = minus119908

lowast

119904119902lowast

119904lt 0 So the revenue

sharing contract cannot realize the supply chain coordinationat this time This completes the proof

Proposition 16 shows that revenue sharing contractscan not realize the supply chain coordination with greentechnology investment and cap and trade policy The rea-son is that the manufacturer undertakes all the costs ofgreen technology investment when using revenue sharingcontracts thus making the manufacturer tend to invest lessin green technology Only when the manufacturer owns allthe revenue will themanufacturerrsquos optimal green technologyinvestment under revenue sharing contracts be equal to thatin QC scenario However the maximum profit of the retaileris negative at this time and it will not participate

62 Coordination Contract Based on Revenue Sharing-CostSharing Contract Proposition 16 states that the supply chainwith green technology investment and cap and trade policycan not be coordinated by revenue sharing contracts becausethe costs of green technology investment are undertaken bythe manufacturer and the game of retailer and customersis characterized by RE Equilibrium So we consider costssharing of green technology investment and carbon tradingand quantity commitment made by the manufacturer basedon revenue sharing contracts Let 120601 (0 lt 120601 le 1) represent theratio of revenue kept by the retailer and let (1minus120601) represent theratio of revenue delivered to the manufacturer 120572 (0 le 120572 le 1)represents the ratio of costs of green technology investmentand carbon trading undertaken by the manufacturer and(1 minus 120572) represents the ratio of costs of green technologyinvestment and carbon trading delivered to the retailer

Discrete Dynamics in Nature and Society 11

The retailerrsquos expected profit function with revenuesharing-cost sharing contract is

120587119903

120572(119902 120601 120572) = 120601 (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119908 minus 120601119904) 119902

minus (1 minus 120572) 119896 [(1 minus 120578) 119902 minus 119864]

minus1

2(1 minus 120572) 119905120578

2

(43)

We have 119890 = (1minus120578)119902minus119864 then themanufacturerrsquos expectedprofit function denoted by 120587119898

120572(119908 120578 120601 120572) is

120587119898

120572(119908 120578 120601 120572)

= (1 minus 120601) (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ (119908 minus 119888 minus 120572 (1 minus 120578) 119896 + (1 minus 120601) 119904) 119902 + 120572119896119864

minus1

21205721199051205782

(44)

The subscript 120572 represents the situation that using rev-enue sharing-cost sharing contract coordinates the supplychain

Proposition 17 If 0 le 120582 le 1 the parameters of revenuesharing-cost sharing (119908 120601 120572) satisfy

119908 = 120582119888 + (1 minus 120578) 119896

120601 = 120582

120572 = 1 minus 120582

(45)

The supply chainwith green technology investment and cap andtrade policy is coordinated

Proof When the parameters of revenue sharing-cost sharingsatisfy (45) the expected profit function of the retailer andthe manufacturer can be written as follows

120587119903

120572(119902 119901 120601 120572) = 120582120587

119902(119902 120578)

120587119898

120572(119908 120578 120601 120572) = (1 minus 120582) 120587

119902(119902 120578)

(46)

The optimal order of decentralized supply chain is equalto that of QC scenario Substitute 119901 = V minus (V minus 119904)119865(119902) into120587sc119888(119902 119901 120578) we get that the profit function of decentralized

supply chain at this time is the same as that of QC scenarioTherefore when the parameters of revenue sharing-costsharing (119908 120601 120572) satisfy (45) the decentralized supply chaincoordination is realizedThemanufacturer earns all the profitif 120582 = 0 and the retailer earns all the profit if 120582 = 1Arbitrary allocation of the supply chain profit between themanufacturer and the retailer is allowed This completes theproof

Proposition 17 shows that revenue sharing-cost sharingcontract can realize the supply chain coordination on condi-tion that the manufacturer makes a quantity commitment to

the customers The optimal production quantity and greentechnology investment in decentralized situation are lowerthan that in QC scenario The quantity commitment of themanufacturer can increase the green technology investmentand production quantity and also reduce the retailer priceThat is the quantity commitment is beneficial for the man-ufacturer the retailer and customers Thus the quantitycommitment is credible to customers now

7 Conclusions and Suggestions for FurtherResearch

This paper investigates the production pricing carbon trad-ing and green technology investment strategies and thecoordination of low carbon supply chain made up of a lowcarbon manufacturer and a retailer The low carbon manu-facturer produces one product under cap and trade policyand performs green technology investment to reduce carbonemissions in production process The retailer orders fromthemanufacturer and distributes the product to homogenouscustomers To the best of our knowledge this is the firststudy on the management of low carbon supply chainswith strategic customer behaviorThis paper provides severalinteresting observations

Observation 1 There are unique optimal production pricingcarbon trading and green technology investment strategiesin centralized (include RE Equilibrium scenario and QCscenario) and decentralized supply chain Therefore byderiving the optimal production pricing carbon trading andgreen technology investment strategies in different situationsthe low carbon manufacturer and the retailer can behaveappropriately based on our findings to maximize their profit

Observation 2 Our findings also show that the centralizedsupply chain can improve its performance by quantity com-mitment However the manufacturerrsquos production and greentechnology investment are reduced while the optimal priceis increased at this time Therefore quantity commitment isbeneficial for the supply chain but is harmful to customers

Observation 3 We also show that the performance of decen-tralized supply chain is lower than that in QC scenario Wefind that the decentralization reduces the manufacturersquos opti-mal production quantity and green technology investmentand increases the retailerrsquos optimal price Besides we find thatthe optimal production quantity is increasing the optimalprice is decreasing and the green technology investment isincreasing respectively in the three situations of decentral-ized supply chain QC scenario andREEquilibrium scenario

Observation 4 We demonstrate that the traditional revenuesharing contracts can not realize the low carbon supply chaincoordination when green technology investment is takenin consideration We also find that the revenue sharing-cost sharing contract realizes the decentralized supply chain

12 Discrete Dynamics in Nature and Society

coordination when the manufacturer makes a quantity com-mitment It is beneficial to the manufacturer the retailerand customers if themanufacturermakes a quantity commit-ment So the quantity commitment is credible to the customerat this time

We assume the customers are homogeneous strategiccustomers But themyopic customers and strategic customersoften coexist and strategic customers also have differenttolerance in real-life Extending this assumption will havea knock-on effect on supply chain decisions So one keyresearch direction is to consider that the customers aremixed customers myopic customers and strategic customersSecond we assume that the supply chain only producesone product While this simplified setting provides someinteresting insights we have to acknowledge that a manufac-turer usually produces two or more productsThese productswill substitute or complement each other So one importantextension of our work is to assume that the manufacturerproduces two or more products which have different unitcarbon emissions in production

Notations

119863 The random demand and119863 ge 0

119891(sdot) Probability density function of119863119865(sdot) Distribution function of119863 We assume

that 119865 satisfy IFR and define 119865 = 1 minus 119865

119888 Unit production cost119904 Unit salvage price of the product which

is an exogenous variableV The customersrsquo utility from consuming

the product119908 Unit wholesale price119901 Unit retail selling price We assume that

it can be observed by the customers119902 The manufacturerrsquos production

quantitythe retailerrsquos order quantityWe assume that it cannot be observedby the customers

119903 The customersrsquo reservation price whichis customersrsquo private information andcannot be observed by the retailer

120585119903 The beliefs of the manufacturer over the

customerrsquo reservation price120585prob The beliefs of customers over their

chances of obtaining the product atphase two

119864 Initial free carbon emission allocated bygovernment

119896 Unit carbon emission trading price119890 Carbon trading volume120578 Carbon emissions reduction rate

Conflict of Interests

The authors declare no conflict of interests

Acknowledgments

This research is partially supported by National Natural Sci-ence Foundation of China (nos 71432003 and 71272128) theProgram forNewCentury Excellent Talents inUniversity (noNCET-12-0087) and Specialized Research Fund for the Doc-toral Program of Higher Education (no 20130185110006)

References

[1] R Alley T Berntsen N L Bindoff et al ldquoSummary forpolicymakersrdquo in Climate Change 2007 The Physical ScienceBasis vol 6 pp 1ndash18 IPCC 2007

[2] X Chen and G Hao ldquoSustainable pricing and productionpolicies for two competing firms with carbon emissions taxrdquoInternational Journal of Production Research vol 53 pp 6408ndash6420 2015

[3] X Chen S Benjaafar and A Elomri ldquoThe carbon-constrainedEOQrdquo Operations Research Letters vol 41 no 2 pp 172ndash1792013

[4] N O Keohane ldquoCap and trade rehabilitated using tradablepermits to control US greenhouse gasesrdquo Review of Environ-mental Economics and Policy vol 3 no 1 pp 42ndash62 2009

[5] B Liu M Holmbom A Segerstedt and W Chen ldquoEffectsof carbon emission regulations on remanufacturing decisionswith limited information of demand distributionrdquo InternationalJournal of Production Research vol 53 no 2 pp 532ndash548 2015

[6] L Zhu and Y Fan ldquoA real options-based CCS investmentevaluation model case study of Chinarsquos power generationsectorrdquo Applied Energy vol 88 no 12 pp 4320ndash4333 2011

[7] XWang and LDu ldquoStudy on carbon capture and storage (CCS)investment decision-making based on real options for Chinarsquoscoal-fired power plantsrdquo Journal of Cleaner Production vol 112part 5 pp 4123ndash4131 2016

[8] W Jiang and X Chen ldquoOptimal strategies for manufacturerwith strategic customer behavior under carbon emissions-sensitive randomdemandrdquo Industrial Management amp Data Sys-tems 2015

[9] Y Levin J McGill and M Nediak ldquoDynamic pricing in thepresence of strategic consumers and oligopolistic competitionrdquoManagement Science vol 55 no 1 pp 32ndash46 2009

[10] A K Parlakturk ldquoThe value of product variety when sellingto strategic consumersrdquo Manufacturing amp Service OperationsManagement vol 14 no 3 pp 371ndash385 2012

[11] X Su and F Zhang ldquoOn the value of commitment andavailability guarantees when selling to strategic consumersrdquoManagement Science vol 55 no 5 pp 713ndash726 2009

[12] W Elmaghraby S A Lippman C S Tang and R Yin ldquoWillmore purchasing options benefit customersrdquo Production andOperations Management vol 18 no 4 pp 381ndash401 2009

[13] W Jiang and X Chen ldquoManufacturersquos production and pricingstrategies with carbon tax policy and strategic customer behav-iorrdquoManagement Science and Engineering vol 9 no 1 pp 30ndash35 2015

[14] I Dobos ldquoThe effects of emission trading on production andinventories in the Arrow-Karlin modelrdquo International Journalof Production Economics vol 93-94 pp 301ndash308 2005

[15] X Chen C K Chan and Y C E Lee ldquoResponsible productionpolicies with substitution and carbon emissions tradingrdquo Jour-nal of Cleaner Production 2015

Discrete Dynamics in Nature and Society 13

[16] X Chen and X Wang ldquoEffects of carbon emission reductionpolicies on transportation mode selections with stochasticdemandrdquo Transportation Research Part E Logistics and Trans-portation Review 2015

[17] X Chang H Xia H Zhu T Fan and H Zhao ldquoProductiondecisions in a hybrid manufacturingndashremanufacturing systemwith carbon cap and trade mechanismrdquo International Journal ofProduction Economics vol 162 pp 160ndash173 2015

[18] J Zhao B F Hobbs and J-S Pang ldquoLong-run equilibriummodeling of emissions allowance allocation systems in electricpowermarketsrdquoOperations Research vol 58 no 3 pp 529ndash5482010

[19] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2 pp519ndash527 2011

[20] A Toptal H Ozlu and D Konur ldquoJoint decisions on inventoryreplenishment and emission reduction investment under dif-ferent emission regulationsrdquo International Journal of ProductionResearch vol 52 no 1 pp 243ndash269 2014

[21] A S Manikas and J R Kroes ldquoA newsvendor approach tocompliance and production under cap and trade emissionsregulationrdquo International Journal of Production Economics vol159 pp 274ndash284 2015

[22] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineering vol10 no 1 pp 99ndash116 2013

[23] X Xu W Zhang P He and X Xu ldquoProduction and pricingproblems in make-to-order supply chain with cap-and-traderegulationrdquo Omega 2015

[24] Y Zhen X Yang and W Jiang ldquoThe impact of transport modeand carbon policy on low-carbon retailerrdquoDiscrete Dynamics inNature and Society vol 2015 Article ID 964305 12 pages 2015

[25] S Swami and J Shah ldquoChannel coordination in green supplychainmanagementrdquo Journal of the Operational Research Societyvol 64 no 3 pp 336ndash351 2013

[26] R H Coase ldquoDurability andmonopolyrdquoThe Journal of Law andEconomics vol 15 no 1 pp 143ndash149 1972

[27] X Su ldquoIntertemporal pricing with strategic customer behaviorrdquoManagement Science vol 53 no 5 pp 726ndash741 2007

[28] D Zhang and W L Cooper ldquoManaging clearance sales in thepresence of strategic customersrdquo Production and OperationsManagement vol 17 no 4 pp 416ndash431 2008

[29] S Whang ldquoDemand uncertainty and the Bayesian effect inmarkdown pricing with strategic customersrdquoManufacturing ampService Operations Management vol 17 no 1 pp 66ndash77 2015

[30] J Du J Zhang and G Hua ldquoPricing and inventory manage-ment in the presence of strategic customers with risk preferenceand decreasing valuerdquo International Journal of Production Eco-nomics vol 164 pp 160ndash166 2015

[31] X Su and F Zhang ldquoStrategic customer behavior commitmentand supply chain performancerdquo Management Science vol 54no 10 pp 1759ndash1773 2008

[32] H Yang ldquoImpact of discounting and competition on benefit ofdecentralization with strategic customersrdquo Operations ResearchLetters vol 40 no 2 pp 123ndash127 2012

[33] D Yang E Qi and Y Li ldquoQuick response and supply chainstructure with strategic consumersrdquo Omega vol 52 pp 1ndash142015

[34] C drsquoAspremont and A Jacquemin ldquoCooperative and nonco-operative R amp D in duopoly with spilloversrdquo The AmericanEconomic Review vol 78 no 5 pp 1133ndash1137 1988

[35] J F Muth ldquoRational expectations and the theory of pricemovementsrdquo Econometrica vol 29 no 3 pp 315ndash335 1961

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

Substituting 120578119898lowast = 120578(119902) into (25)

120587119898(119902) equiv 120587

119898(119902 120578 (119902))

= [(V minus 119904) 1198652

(119902) minus (119888 minus 119904 + 119896)] 119902 + 119896119864

+11989621199022

2119905

(29)

Then the two-variable optimization problem is simplifiedas a single variable optimization problem

max119902ge0

120587119898(119902) (30)

Lemma 11 If 1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus

1199021198912(119902)] lt 0 120587119898(119902) is a concave function of 119902

Proof The first-order and second-order derivatives of 120587119898(119902)with respect to 119902 are 119889120587

119898(119902)119889119902 = (V minus 119904)[119865

2

(119902) minus

2119902119891(119902)119865(119902)] minus (119888 minus 119904 + 119896) + (1198962119905)119902 and 119889

2120587119898(119902)119889119902

2=

1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus 1199021198912(119902)] lt 0 Then

we have that 120587119898(119902) is a concave function of 119902 This completesthe proof

In decentralized supply chain as to the manufacturerrsquosoptimal wholesale price (denoted by 119908

119898lowast) the followingproposition is obtained

Proposition 12 If 1198962119905 minus 2(V minus 119904)[2119891(119902)119865(119902) + 1199021198911015840(119902)119865(119902) minus1199021198912(119902)] lt 0 the manufacturerrsquos optimal wholesale price in

decentralized supply chain (119908119898lowast) satisfies 119908119898lowast = 119904 + (V minus119904)1198652

(119902119903lowast) where 119902119903lowast represents the retailerrsquos optimal order

quantity and satisfies

(V minus 119904) [1198652

(119902) minus 2119902119891 (119902) 119865 (119902)] minus (119888 minus 119904 + 119896) +1198962

119905119902

= 0

(31)

Proof According to Lemma 11 we can obtain that 119902119903lowastwhichmaximizes themanufacturerrsquos expected profit satisfies(31) Combining with (24) we know that the manufacturerrsquosoptimal wholesale price satisfies 119908119898lowast = 119904 + (V minus 119904)119865

2

(119902119903lowast) at

this time This completes the proof

Proposition 12 shows that the optimal commitment quan-tity of the manufacturer in decentralized supply chain withgreen technology investment and cap and trade policy existand are unique The optimal wholesale price does not relateto the cap allocated by the government but relates to thecarbon trading price This is because the production of themanufacturer is not affected by the cap but is affected bythe margin cost of the product However the margin cost ofthe product varies with the change of carbon trading pricewhen carbon trading is allowed The change results in thatthe optimal commitment quantity is not related to the cap butrelate to the carbon trading price

Substitute 119908119898lowast into (22) we derive the retailerrsquos optimalpricing in decentralized situation119901119903lowast = 119904+radic(119908119898lowast minus 119904)(V minus 119904)

According to Lemma 10 we get the optimal green technologyinvestment of the manufacturer 120578119898lowast = (119896119905)119902119903lowast

Substitute 119902119903lowast 119901119903lowast 119908119898lowast and 120578119898lowast into 120587

119903(119902 119901) 120587119898(119902)

and 120587sc(119902 120578) we obtain the maximum expected profit of

the retailer the manufacturer and the supply chain indecentralized channel

120587119903(119902119903lowast 119901119903lowast) = (119901

119903lowastminus 119904) (119902

119903lowastminus int

119902119903lowast

0

119865 (119909) 119889119909)

minus (119908119898lowast

minus 119904) 119902119903lowast

120587119898(119902119903lowast) = [(V minus 119904) 119865

2

(119902119903lowast) minus (119888 minus 119904 + 119896)] 119902

119903lowast

+ 119896119864 +1198962119902119903lowast2

2119905

120587sc(119902119903lowast 120578119898lowast) = (V minus 119904) 119865 (119902119903lowast) (119902119903lowast minus int

119902119903lowast

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578119898lowast) 119896) 119902119903lowast+ 119896119864

minus1

2119905120578119898lowast2

(32)

52 The Effect of Decentralization We examine the impact ofdecentralization on the supply chain optimal strategies andmaximumexpected profitwhen green technology investmentis taken into consideration by comparing the optimal strate-gies and the maximum expected profit in centralized anddecentralized channel with green technology investment andcap and trade policy

As the effect of decentralization of supply chain we canobtain the following proposition

Proposition 13 Consider 119902119903lowast lt 119902119902lowast

lt 119902lowast 119901119903lowast gt 119901

119902lowastgt 119901lowast

120578119898lowast

lt 120578119902lowastlt 120578lowast

Proof In Proposition 6 119902119902lowast lt 119902lowast 119901119902lowast gt 119901

lowast and 120578119902lowast lt 120578lowast

have been provedTherefore we only need to prove 119902119903lowast lt 119902119902lowast119901119903lowastgt 119901119902lowast and 120578119898lowast lt 120578119902lowast

Define 119867(119902) = 120587119902(119902) minus 120587119898(119902) then

119867(119902)

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

minus (V minus 119904) 1199021198652

(119902)

1198671015840(119902)

= (V minus 119904) 119891 (119902) [2119902119865 (119902) minus (119902 minus int119902

0

119865 (119909) 119889119909)]

(33)

Define 119866(119902) = 2119902119865(119902) minus (119902 minus int119902

0119865(119909)119889119909) then 1198661015840(119902) =

119865(119902) minus 2119902119891(119902) Because 119866(0) = 0 1198661015840(0) = 1 and 119865 satisfiesIFR we know that the value of 119866(119902) starts from 0 increasingfirst and then decreasing in 119902 That is 119866(119902) = 0 has uniquesolution and we denote it as So when 119902 lt 119866(119902) gt 0 that

Discrete Dynamics in Nature and Society 9

is119867(119902) is increasing in 119902 when 119902 gt 119866(119902) lt 0 that is119867(119902)is decreasing in 119902 Then we have that119867(119902) is a quasi-concavefunction of 119902 and 1198661015840 () = 119865() minus 2119891() lt 0 According tothe analysis above we know that1198671015840() = 0 that is 2119865() =( minus int

0119865(119909)119889119909) Substitute into 119889120587119902(119902)119889119902 we have

119889120587119902(119902)

119889119902

100381610038161003816100381610038161003816100381610038161003816119902=

= (V minus 119904) [1198652

() minus 119891 () ( minus int

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) +1198962

119905

= (V minus 119904) 119865 () [119865 () minus 2119891 ()] minus (119888 minus 119904 + 119896)

+1198962

119905lt (V minus 119904) 119865 () [119865 () minus 2119891 ()] lt 0

(34)

According to Lemma 4 we have that 120587119902(119902) is a concavefunction of 119902 so we get 119902119902lowast lt Because 2119865() = ( minus

int

0119865(119909)119889119909) and119867(119902) is a quasi-concave function of 119902 thenwe

get that 119889120587119898(119902)119889119902 lt 119889120587119902(119902)119889119902 if 119902 lt and 119889120587119898(119902)119889119902 gt

119889120587119902(119902)119889119902 if 119902 gt Therefore the only possible orderings for

119902119902lowast 119902119903lowast and are lt 119902

119902lowastlt 119902119903lowast and 119902119903lowast lt 119902

119902lowastlt We

have proved 119902119902lowast lt so we have 119902119903lowast lt 119902119902lowast We get 119901119903lowast gt 119901119902lowastbecause 119901 and 119902 satisfy 119901 = V minus (V minus 119904)119865(119902) in two situationsWe get 120578119898lowast lt 120578

119902lowast because 120578 and 119902 satisfy 120578 = 119896119902119905 in twosituations This completes the proof

Proposition 13 shows that for the decision of a decen-tralized supply chain with green technology investment theoptimal production quantity is increasing the optimal priceis decreasing and the green technology investment (iecarbon reduction rate) is increasing respectively in the threesituations of decentralized supply chain QC scenario and REEquilibrium scenario

The reason is that themanufacturer prefers higher whole-sale price (ie lower production and less green technologyinvestment) to guarantee its ownprofitmaximization thoughthis may harm the retailer even the whole supply chain

In Proposition 7 we have found 120587119902(119902119902lowast 120578119902lowast) gt

120587(119902lowast 119901lowast 120578lowast) that is the manufacturerrsquos maximum profit in

QC scenario is always greater than that in RE EquilibriumTherefore the section analyzes the effect of decentralizationon supply chain performance based on the QC scenario toprovide benchmark for designing of subsequent coordinationcontract

As to the effect of decentralization on themaximumprofitof supply chain we have the following proposition

Proposition 14 Consider 120587sc(119902119903lowast 120578119898lowast) lt 120587119902(119902119902lowast 120578119902lowast)

Proof The optimal green technology investment in decen-tralized supply chain and QC scenario are 120578(119902) = 119896119902119905

according to Lemmas 3 and 10 So the profit function in thetwo situations above can be converted into a same singlevariable function with respect to 119902 that is (16)

According to Lemma 4 we know that 120587119902(119902) is a concavefunction and reaches the maximum value at 119902119902lowast Accordingto Proposition 13 we have 119902119903lowast lt 119902

119902lowast then we get 120587119902(119902119903lowast) lt120587119902(119902119902lowast) that is 120587sc

(119902119903lowast 120578119898lowast) lt 120587

119902(119902119902lowast 120578119902lowast) This completes

the proof

Proposition 14 shows that the maximum profit of decen-tralized supply chain is always lower than that in QC scenariowhich indicates that the profit of the decentralized supplychain can get higher This is the benchmark of supply chaincoordination

6 Supply Chain Coordination

This section coordinates the supply chain with green tech-nology investment and cap and trade policy based on theQC scenario to improve the profit of the decentralized supplychain

61 Coordination Contract Based on Revenue and CostsSharing Contract With revenue sharing contract we assumethat 120601 (0 le 120601 le 1) represents the ratio of revenue kept by theretailer and (1 minus 120601) represents the ratio of revenue deliveredto the manufacturer

The retailerrsquos expected profit function with revenue shar-ing contract denoted by 120587119903

119904(119902 119901 120601) is

120587119903

119904(119902 119901 120601) = 120601 (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119908 minus 120601119904) 119902

(35)

Themanufacturerrsquos expected profit functionwith revenuesharing contract denoted by 120587119898

119904(119908 119890 120578 120601) is

120587119898

119904(119908 119890 120578 120601) = (119908 minus 119888) 119902 + (1 minus 120601)

sdot [int

119902

0

[119901119909 + 119904 (119902 minus 119909)] 119891 (119909) 119889119909

+ int

infin

119902

119901119902119891 (119909) 119889119909] minus 119896119890 minus1

21199051205782

(36)

Substitute 119890 = (1 minus 120578)119902 minus 119864 into the equation above andwe have

120587119898

119904(119908 120578 120601) = (1 minus 120601) (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

+ (119908 minus 119888 minus (1 minus 120578) 119896 + (1 minus 120601) 119904) 119902

+ 119896119864 minus1

21199051205782

(37)

The total expected profit of the whole supply chaindenoted by 120587sc

119904(119902 119901 120578) is

120587sc119904(119902 119901 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578) 119896) 119902 + 119896119864 minus1

21199051205782

(38)

10 Discrete Dynamics in Nature and Society

The subscript 119904 represents the situation of supply chaincoordination using revenue sharing contracts

As to the retailerrsquos optimal order strategy (denoted by119902lowast

119904) and pricing strategy (denoted by 119901lowast

119904) of the supply chain

with revenue sharing contract the following proposition isobtained

Proposition 15 Under revenue sharing contracts the retailerrsquosoptimal order strategy (119902lowast

119904) and optimal pricing strategy (119901lowast

119904)

are

119902lowast

119904= 119865minus1

(radic119908 minus 120601119904

120601 (V minus 119904))

119901lowast

119904= 119904 + radic

(119908 minus 120601119904) ((V minus 119904))120601

(39)

Proof According to the definition of RE Equilibrium wealso have 119901 = V minus (V minus 119904)119865(119902) under revenue sharingcontracts

The first-order and second-order derivative of 120587119903119904(119902 119901 120601)

with respect to 119902 are 120597120587119903

119904(119902 119901 120601)120597119902 = 120601(119901 minus 119904)119865(119902) minus

(119908 minus 120601119904) and 1205972120587119903

119904(119902 119901 120601)120597119902

2= minus120601(119901 minus 119904)119891(119902) lt

0 So given 119901 the retailerrsquos expected profit function is aconcave function of 119902 Let 120597120587119903

119904(119902 119901 120601)120597119902 = 0 and combine

with 119901 = V minus (V minus 119904)119865(119902) we can solve the retailerrsquos optimalstrategies under revenue sharing contractThis completes theproof

Proposition 15 shows that the optimal order and pricingstrategies of the retailer based on revenue sharing contract inRE Equilibrium exist and are uniqueThis is also the retailerrsquosoptimal response function with respect to 119908

Substituting (39) into (43)

120587119898

119904(119902 120578 120601)

= (1 minus 120601) (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ [120601 (V minus 119904) 1198652

(119902) minus (119888 minus 119904 + (1 minus 120578) 119896)] 119902 + 119896119864

minus1

21199051205782

120597120587119898

119904(119902 120578 120601)

120597120578= 119902119896 minus 119905120578

1205972120587119898

119904(119902 120578 120601)

1205971205782= minus119905 lt 0

(40)

Given 119902 and 120601 the manufacturerrsquos green technologyinvestment strategies with revenue sharing contracts exist

and are unique Let 120597120587119898119904(119902 120578 120601)120597120578 = 0 we get 120578119898lowast

119904equiv 120578(119902) =

119902119896119905 and substitute it into 120587119898119904(119902 120578 120601)

120587119898

119904(119902 120601)

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ 120601 (V minus 119904) 119865 (119902) [119902119865 (119902) minus (119902 minus int119902

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) 119902 + 119896119864 +11989621199022

2119905

(41)

Proposition 16 The supply chain with green technologyinvestment and cap and trade policy can not be coordinated byrevenue sharing contract

Proof The derivative of 120587119898119904(119902 120601) with respect to 119902 can be

arranged as follows

120597120587119898

119904(119902 120601)

120597119902=120597120587119902(119902)

120597119902+ 120601(

120597120587119898(119902)

120597119902minus120597120587119902(119902)

120597119902) (42)

Only when 120601 = 0 are the optimal strategies of supplychain under revenue sharing contracts equal to that in QCscenario Let 119908lowast

119904represent the optimal wholesale price of

the manufacture and 119902lowast119904represent the retailerrsquos optimal order

quantity Then 120587119903

119904(119902 119901 120601) = minus119908

lowast

119904119902lowast

119904lt 0 So the revenue

sharing contract cannot realize the supply chain coordinationat this time This completes the proof

Proposition 16 shows that revenue sharing contractscan not realize the supply chain coordination with greentechnology investment and cap and trade policy The rea-son is that the manufacturer undertakes all the costs ofgreen technology investment when using revenue sharingcontracts thus making the manufacturer tend to invest lessin green technology Only when the manufacturer owns allthe revenue will themanufacturerrsquos optimal green technologyinvestment under revenue sharing contracts be equal to thatin QC scenario However the maximum profit of the retaileris negative at this time and it will not participate

62 Coordination Contract Based on Revenue Sharing-CostSharing Contract Proposition 16 states that the supply chainwith green technology investment and cap and trade policycan not be coordinated by revenue sharing contracts becausethe costs of green technology investment are undertaken bythe manufacturer and the game of retailer and customersis characterized by RE Equilibrium So we consider costssharing of green technology investment and carbon tradingand quantity commitment made by the manufacturer basedon revenue sharing contracts Let 120601 (0 lt 120601 le 1) represent theratio of revenue kept by the retailer and let (1minus120601) represent theratio of revenue delivered to the manufacturer 120572 (0 le 120572 le 1)represents the ratio of costs of green technology investmentand carbon trading undertaken by the manufacturer and(1 minus 120572) represents the ratio of costs of green technologyinvestment and carbon trading delivered to the retailer

Discrete Dynamics in Nature and Society 11

The retailerrsquos expected profit function with revenuesharing-cost sharing contract is

120587119903

120572(119902 120601 120572) = 120601 (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119908 minus 120601119904) 119902

minus (1 minus 120572) 119896 [(1 minus 120578) 119902 minus 119864]

minus1

2(1 minus 120572) 119905120578

2

(43)

We have 119890 = (1minus120578)119902minus119864 then themanufacturerrsquos expectedprofit function denoted by 120587119898

120572(119908 120578 120601 120572) is

120587119898

120572(119908 120578 120601 120572)

= (1 minus 120601) (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ (119908 minus 119888 minus 120572 (1 minus 120578) 119896 + (1 minus 120601) 119904) 119902 + 120572119896119864

minus1

21205721199051205782

(44)

The subscript 120572 represents the situation that using rev-enue sharing-cost sharing contract coordinates the supplychain

Proposition 17 If 0 le 120582 le 1 the parameters of revenuesharing-cost sharing (119908 120601 120572) satisfy

119908 = 120582119888 + (1 minus 120578) 119896

120601 = 120582

120572 = 1 minus 120582

(45)

The supply chainwith green technology investment and cap andtrade policy is coordinated

Proof When the parameters of revenue sharing-cost sharingsatisfy (45) the expected profit function of the retailer andthe manufacturer can be written as follows

120587119903

120572(119902 119901 120601 120572) = 120582120587

119902(119902 120578)

120587119898

120572(119908 120578 120601 120572) = (1 minus 120582) 120587

119902(119902 120578)

(46)

The optimal order of decentralized supply chain is equalto that of QC scenario Substitute 119901 = V minus (V minus 119904)119865(119902) into120587sc119888(119902 119901 120578) we get that the profit function of decentralized

supply chain at this time is the same as that of QC scenarioTherefore when the parameters of revenue sharing-costsharing (119908 120601 120572) satisfy (45) the decentralized supply chaincoordination is realizedThemanufacturer earns all the profitif 120582 = 0 and the retailer earns all the profit if 120582 = 1Arbitrary allocation of the supply chain profit between themanufacturer and the retailer is allowed This completes theproof

Proposition 17 shows that revenue sharing-cost sharingcontract can realize the supply chain coordination on condi-tion that the manufacturer makes a quantity commitment to

the customers The optimal production quantity and greentechnology investment in decentralized situation are lowerthan that in QC scenario The quantity commitment of themanufacturer can increase the green technology investmentand production quantity and also reduce the retailer priceThat is the quantity commitment is beneficial for the man-ufacturer the retailer and customers Thus the quantitycommitment is credible to customers now

7 Conclusions and Suggestions for FurtherResearch

This paper investigates the production pricing carbon trad-ing and green technology investment strategies and thecoordination of low carbon supply chain made up of a lowcarbon manufacturer and a retailer The low carbon manu-facturer produces one product under cap and trade policyand performs green technology investment to reduce carbonemissions in production process The retailer orders fromthemanufacturer and distributes the product to homogenouscustomers To the best of our knowledge this is the firststudy on the management of low carbon supply chainswith strategic customer behaviorThis paper provides severalinteresting observations

Observation 1 There are unique optimal production pricingcarbon trading and green technology investment strategiesin centralized (include RE Equilibrium scenario and QCscenario) and decentralized supply chain Therefore byderiving the optimal production pricing carbon trading andgreen technology investment strategies in different situationsthe low carbon manufacturer and the retailer can behaveappropriately based on our findings to maximize their profit

Observation 2 Our findings also show that the centralizedsupply chain can improve its performance by quantity com-mitment However the manufacturerrsquos production and greentechnology investment are reduced while the optimal priceis increased at this time Therefore quantity commitment isbeneficial for the supply chain but is harmful to customers

Observation 3 We also show that the performance of decen-tralized supply chain is lower than that in QC scenario Wefind that the decentralization reduces the manufacturersquos opti-mal production quantity and green technology investmentand increases the retailerrsquos optimal price Besides we find thatthe optimal production quantity is increasing the optimalprice is decreasing and the green technology investment isincreasing respectively in the three situations of decentral-ized supply chain QC scenario andREEquilibrium scenario

Observation 4 We demonstrate that the traditional revenuesharing contracts can not realize the low carbon supply chaincoordination when green technology investment is takenin consideration We also find that the revenue sharing-cost sharing contract realizes the decentralized supply chain

12 Discrete Dynamics in Nature and Society

coordination when the manufacturer makes a quantity com-mitment It is beneficial to the manufacturer the retailerand customers if themanufacturermakes a quantity commit-ment So the quantity commitment is credible to the customerat this time

We assume the customers are homogeneous strategiccustomers But themyopic customers and strategic customersoften coexist and strategic customers also have differenttolerance in real-life Extending this assumption will havea knock-on effect on supply chain decisions So one keyresearch direction is to consider that the customers aremixed customers myopic customers and strategic customersSecond we assume that the supply chain only producesone product While this simplified setting provides someinteresting insights we have to acknowledge that a manufac-turer usually produces two or more productsThese productswill substitute or complement each other So one importantextension of our work is to assume that the manufacturerproduces two or more products which have different unitcarbon emissions in production

Notations

119863 The random demand and119863 ge 0

119891(sdot) Probability density function of119863119865(sdot) Distribution function of119863 We assume

that 119865 satisfy IFR and define 119865 = 1 minus 119865

119888 Unit production cost119904 Unit salvage price of the product which

is an exogenous variableV The customersrsquo utility from consuming

the product119908 Unit wholesale price119901 Unit retail selling price We assume that

it can be observed by the customers119902 The manufacturerrsquos production

quantitythe retailerrsquos order quantityWe assume that it cannot be observedby the customers

119903 The customersrsquo reservation price whichis customersrsquo private information andcannot be observed by the retailer

120585119903 The beliefs of the manufacturer over the

customerrsquo reservation price120585prob The beliefs of customers over their

chances of obtaining the product atphase two

119864 Initial free carbon emission allocated bygovernment

119896 Unit carbon emission trading price119890 Carbon trading volume120578 Carbon emissions reduction rate

Conflict of Interests

The authors declare no conflict of interests

Acknowledgments

This research is partially supported by National Natural Sci-ence Foundation of China (nos 71432003 and 71272128) theProgram forNewCentury Excellent Talents inUniversity (noNCET-12-0087) and Specialized Research Fund for the Doc-toral Program of Higher Education (no 20130185110006)

References

[1] R Alley T Berntsen N L Bindoff et al ldquoSummary forpolicymakersrdquo in Climate Change 2007 The Physical ScienceBasis vol 6 pp 1ndash18 IPCC 2007

[2] X Chen and G Hao ldquoSustainable pricing and productionpolicies for two competing firms with carbon emissions taxrdquoInternational Journal of Production Research vol 53 pp 6408ndash6420 2015

[3] X Chen S Benjaafar and A Elomri ldquoThe carbon-constrainedEOQrdquo Operations Research Letters vol 41 no 2 pp 172ndash1792013

[4] N O Keohane ldquoCap and trade rehabilitated using tradablepermits to control US greenhouse gasesrdquo Review of Environ-mental Economics and Policy vol 3 no 1 pp 42ndash62 2009

[5] B Liu M Holmbom A Segerstedt and W Chen ldquoEffectsof carbon emission regulations on remanufacturing decisionswith limited information of demand distributionrdquo InternationalJournal of Production Research vol 53 no 2 pp 532ndash548 2015

[6] L Zhu and Y Fan ldquoA real options-based CCS investmentevaluation model case study of Chinarsquos power generationsectorrdquo Applied Energy vol 88 no 12 pp 4320ndash4333 2011

[7] XWang and LDu ldquoStudy on carbon capture and storage (CCS)investment decision-making based on real options for Chinarsquoscoal-fired power plantsrdquo Journal of Cleaner Production vol 112part 5 pp 4123ndash4131 2016

[8] W Jiang and X Chen ldquoOptimal strategies for manufacturerwith strategic customer behavior under carbon emissions-sensitive randomdemandrdquo Industrial Management amp Data Sys-tems 2015

[9] Y Levin J McGill and M Nediak ldquoDynamic pricing in thepresence of strategic consumers and oligopolistic competitionrdquoManagement Science vol 55 no 1 pp 32ndash46 2009

[10] A K Parlakturk ldquoThe value of product variety when sellingto strategic consumersrdquo Manufacturing amp Service OperationsManagement vol 14 no 3 pp 371ndash385 2012

[11] X Su and F Zhang ldquoOn the value of commitment andavailability guarantees when selling to strategic consumersrdquoManagement Science vol 55 no 5 pp 713ndash726 2009

[12] W Elmaghraby S A Lippman C S Tang and R Yin ldquoWillmore purchasing options benefit customersrdquo Production andOperations Management vol 18 no 4 pp 381ndash401 2009

[13] W Jiang and X Chen ldquoManufacturersquos production and pricingstrategies with carbon tax policy and strategic customer behav-iorrdquoManagement Science and Engineering vol 9 no 1 pp 30ndash35 2015

[14] I Dobos ldquoThe effects of emission trading on production andinventories in the Arrow-Karlin modelrdquo International Journalof Production Economics vol 93-94 pp 301ndash308 2005

[15] X Chen C K Chan and Y C E Lee ldquoResponsible productionpolicies with substitution and carbon emissions tradingrdquo Jour-nal of Cleaner Production 2015

Discrete Dynamics in Nature and Society 13

[16] X Chen and X Wang ldquoEffects of carbon emission reductionpolicies on transportation mode selections with stochasticdemandrdquo Transportation Research Part E Logistics and Trans-portation Review 2015

[17] X Chang H Xia H Zhu T Fan and H Zhao ldquoProductiondecisions in a hybrid manufacturingndashremanufacturing systemwith carbon cap and trade mechanismrdquo International Journal ofProduction Economics vol 162 pp 160ndash173 2015

[18] J Zhao B F Hobbs and J-S Pang ldquoLong-run equilibriummodeling of emissions allowance allocation systems in electricpowermarketsrdquoOperations Research vol 58 no 3 pp 529ndash5482010

[19] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2 pp519ndash527 2011

[20] A Toptal H Ozlu and D Konur ldquoJoint decisions on inventoryreplenishment and emission reduction investment under dif-ferent emission regulationsrdquo International Journal of ProductionResearch vol 52 no 1 pp 243ndash269 2014

[21] A S Manikas and J R Kroes ldquoA newsvendor approach tocompliance and production under cap and trade emissionsregulationrdquo International Journal of Production Economics vol159 pp 274ndash284 2015

[22] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineering vol10 no 1 pp 99ndash116 2013

[23] X Xu W Zhang P He and X Xu ldquoProduction and pricingproblems in make-to-order supply chain with cap-and-traderegulationrdquo Omega 2015

[24] Y Zhen X Yang and W Jiang ldquoThe impact of transport modeand carbon policy on low-carbon retailerrdquoDiscrete Dynamics inNature and Society vol 2015 Article ID 964305 12 pages 2015

[25] S Swami and J Shah ldquoChannel coordination in green supplychainmanagementrdquo Journal of the Operational Research Societyvol 64 no 3 pp 336ndash351 2013

[26] R H Coase ldquoDurability andmonopolyrdquoThe Journal of Law andEconomics vol 15 no 1 pp 143ndash149 1972

[27] X Su ldquoIntertemporal pricing with strategic customer behaviorrdquoManagement Science vol 53 no 5 pp 726ndash741 2007

[28] D Zhang and W L Cooper ldquoManaging clearance sales in thepresence of strategic customersrdquo Production and OperationsManagement vol 17 no 4 pp 416ndash431 2008

[29] S Whang ldquoDemand uncertainty and the Bayesian effect inmarkdown pricing with strategic customersrdquoManufacturing ampService Operations Management vol 17 no 1 pp 66ndash77 2015

[30] J Du J Zhang and G Hua ldquoPricing and inventory manage-ment in the presence of strategic customers with risk preferenceand decreasing valuerdquo International Journal of Production Eco-nomics vol 164 pp 160ndash166 2015

[31] X Su and F Zhang ldquoStrategic customer behavior commitmentand supply chain performancerdquo Management Science vol 54no 10 pp 1759ndash1773 2008

[32] H Yang ldquoImpact of discounting and competition on benefit ofdecentralization with strategic customersrdquo Operations ResearchLetters vol 40 no 2 pp 123ndash127 2012

[33] D Yang E Qi and Y Li ldquoQuick response and supply chainstructure with strategic consumersrdquo Omega vol 52 pp 1ndash142015

[34] C drsquoAspremont and A Jacquemin ldquoCooperative and nonco-operative R amp D in duopoly with spilloversrdquo The AmericanEconomic Review vol 78 no 5 pp 1133ndash1137 1988

[35] J F Muth ldquoRational expectations and the theory of pricemovementsrdquo Econometrica vol 29 no 3 pp 315ndash335 1961

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 9

is119867(119902) is increasing in 119902 when 119902 gt 119866(119902) lt 0 that is119867(119902)is decreasing in 119902 Then we have that119867(119902) is a quasi-concavefunction of 119902 and 1198661015840 () = 119865() minus 2119891() lt 0 According tothe analysis above we know that1198671015840() = 0 that is 2119865() =( minus int

0119865(119909)119889119909) Substitute into 119889120587119902(119902)119889119902 we have

119889120587119902(119902)

119889119902

100381610038161003816100381610038161003816100381610038161003816119902=

= (V minus 119904) [1198652

() minus 119891 () ( minus int

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) +1198962

119905

= (V minus 119904) 119865 () [119865 () minus 2119891 ()] minus (119888 minus 119904 + 119896)

+1198962

119905lt (V minus 119904) 119865 () [119865 () minus 2119891 ()] lt 0

(34)

According to Lemma 4 we have that 120587119902(119902) is a concavefunction of 119902 so we get 119902119902lowast lt Because 2119865() = ( minus

int

0119865(119909)119889119909) and119867(119902) is a quasi-concave function of 119902 thenwe

get that 119889120587119898(119902)119889119902 lt 119889120587119902(119902)119889119902 if 119902 lt and 119889120587119898(119902)119889119902 gt

119889120587119902(119902)119889119902 if 119902 gt Therefore the only possible orderings for

119902119902lowast 119902119903lowast and are lt 119902

119902lowastlt 119902119903lowast and 119902119903lowast lt 119902

119902lowastlt We

have proved 119902119902lowast lt so we have 119902119903lowast lt 119902119902lowast We get 119901119903lowast gt 119901119902lowastbecause 119901 and 119902 satisfy 119901 = V minus (V minus 119904)119865(119902) in two situationsWe get 120578119898lowast lt 120578

119902lowast because 120578 and 119902 satisfy 120578 = 119896119902119905 in twosituations This completes the proof

Proposition 13 shows that for the decision of a decen-tralized supply chain with green technology investment theoptimal production quantity is increasing the optimal priceis decreasing and the green technology investment (iecarbon reduction rate) is increasing respectively in the threesituations of decentralized supply chain QC scenario and REEquilibrium scenario

The reason is that themanufacturer prefers higher whole-sale price (ie lower production and less green technologyinvestment) to guarantee its ownprofitmaximization thoughthis may harm the retailer even the whole supply chain

In Proposition 7 we have found 120587119902(119902119902lowast 120578119902lowast) gt

120587(119902lowast 119901lowast 120578lowast) that is the manufacturerrsquos maximum profit in

QC scenario is always greater than that in RE EquilibriumTherefore the section analyzes the effect of decentralizationon supply chain performance based on the QC scenario toprovide benchmark for designing of subsequent coordinationcontract

As to the effect of decentralization on themaximumprofitof supply chain we have the following proposition

Proposition 14 Consider 120587sc(119902119903lowast 120578119898lowast) lt 120587119902(119902119902lowast 120578119902lowast)

Proof The optimal green technology investment in decen-tralized supply chain and QC scenario are 120578(119902) = 119896119902119905

according to Lemmas 3 and 10 So the profit function in thetwo situations above can be converted into a same singlevariable function with respect to 119902 that is (16)

According to Lemma 4 we know that 120587119902(119902) is a concavefunction and reaches the maximum value at 119902119902lowast Accordingto Proposition 13 we have 119902119903lowast lt 119902

119902lowast then we get 120587119902(119902119903lowast) lt120587119902(119902119902lowast) that is 120587sc

(119902119903lowast 120578119898lowast) lt 120587

119902(119902119902lowast 120578119902lowast) This completes

the proof

Proposition 14 shows that the maximum profit of decen-tralized supply chain is always lower than that in QC scenariowhich indicates that the profit of the decentralized supplychain can get higher This is the benchmark of supply chaincoordination

6 Supply Chain Coordination

This section coordinates the supply chain with green tech-nology investment and cap and trade policy based on theQC scenario to improve the profit of the decentralized supplychain

61 Coordination Contract Based on Revenue and CostsSharing Contract With revenue sharing contract we assumethat 120601 (0 le 120601 le 1) represents the ratio of revenue kept by theretailer and (1 minus 120601) represents the ratio of revenue deliveredto the manufacturer

The retailerrsquos expected profit function with revenue shar-ing contract denoted by 120587119903

119904(119902 119901 120601) is

120587119903

119904(119902 119901 120601) = 120601 (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119908 minus 120601119904) 119902

(35)

Themanufacturerrsquos expected profit functionwith revenuesharing contract denoted by 120587119898

119904(119908 119890 120578 120601) is

120587119898

119904(119908 119890 120578 120601) = (119908 minus 119888) 119902 + (1 minus 120601)

sdot [int

119902

0

[119901119909 + 119904 (119902 minus 119909)] 119891 (119909) 119889119909

+ int

infin

119902

119901119902119891 (119909) 119889119909] minus 119896119890 minus1

21199051205782

(36)

Substitute 119890 = (1 minus 120578)119902 minus 119864 into the equation above andwe have

120587119898

119904(119908 120578 120601) = (1 minus 120601) (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

+ (119908 minus 119888 minus (1 minus 120578) 119896 + (1 minus 120601) 119904) 119902

+ 119896119864 minus1

21199051205782

(37)

The total expected profit of the whole supply chaindenoted by 120587sc

119904(119902 119901 120578) is

120587sc119904(119902 119901 120578) = (119901 minus 119904) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119888 minus 119904 + (1 minus 120578) 119896) 119902 + 119896119864 minus1

21199051205782

(38)

10 Discrete Dynamics in Nature and Society

The subscript 119904 represents the situation of supply chaincoordination using revenue sharing contracts

As to the retailerrsquos optimal order strategy (denoted by119902lowast

119904) and pricing strategy (denoted by 119901lowast

119904) of the supply chain

with revenue sharing contract the following proposition isobtained

Proposition 15 Under revenue sharing contracts the retailerrsquosoptimal order strategy (119902lowast

119904) and optimal pricing strategy (119901lowast

119904)

are

119902lowast

119904= 119865minus1

(radic119908 minus 120601119904

120601 (V minus 119904))

119901lowast

119904= 119904 + radic

(119908 minus 120601119904) ((V minus 119904))120601

(39)

Proof According to the definition of RE Equilibrium wealso have 119901 = V minus (V minus 119904)119865(119902) under revenue sharingcontracts

The first-order and second-order derivative of 120587119903119904(119902 119901 120601)

with respect to 119902 are 120597120587119903

119904(119902 119901 120601)120597119902 = 120601(119901 minus 119904)119865(119902) minus

(119908 minus 120601119904) and 1205972120587119903

119904(119902 119901 120601)120597119902

2= minus120601(119901 minus 119904)119891(119902) lt

0 So given 119901 the retailerrsquos expected profit function is aconcave function of 119902 Let 120597120587119903

119904(119902 119901 120601)120597119902 = 0 and combine

with 119901 = V minus (V minus 119904)119865(119902) we can solve the retailerrsquos optimalstrategies under revenue sharing contractThis completes theproof

Proposition 15 shows that the optimal order and pricingstrategies of the retailer based on revenue sharing contract inRE Equilibrium exist and are uniqueThis is also the retailerrsquosoptimal response function with respect to 119908

Substituting (39) into (43)

120587119898

119904(119902 120578 120601)

= (1 minus 120601) (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ [120601 (V minus 119904) 1198652

(119902) minus (119888 minus 119904 + (1 minus 120578) 119896)] 119902 + 119896119864

minus1

21199051205782

120597120587119898

119904(119902 120578 120601)

120597120578= 119902119896 minus 119905120578

1205972120587119898

119904(119902 120578 120601)

1205971205782= minus119905 lt 0

(40)

Given 119902 and 120601 the manufacturerrsquos green technologyinvestment strategies with revenue sharing contracts exist

and are unique Let 120597120587119898119904(119902 120578 120601)120597120578 = 0 we get 120578119898lowast

119904equiv 120578(119902) =

119902119896119905 and substitute it into 120587119898119904(119902 120578 120601)

120587119898

119904(119902 120601)

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ 120601 (V minus 119904) 119865 (119902) [119902119865 (119902) minus (119902 minus int119902

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) 119902 + 119896119864 +11989621199022

2119905

(41)

Proposition 16 The supply chain with green technologyinvestment and cap and trade policy can not be coordinated byrevenue sharing contract

Proof The derivative of 120587119898119904(119902 120601) with respect to 119902 can be

arranged as follows

120597120587119898

119904(119902 120601)

120597119902=120597120587119902(119902)

120597119902+ 120601(

120597120587119898(119902)

120597119902minus120597120587119902(119902)

120597119902) (42)

Only when 120601 = 0 are the optimal strategies of supplychain under revenue sharing contracts equal to that in QCscenario Let 119908lowast

119904represent the optimal wholesale price of

the manufacture and 119902lowast119904represent the retailerrsquos optimal order

quantity Then 120587119903

119904(119902 119901 120601) = minus119908

lowast

119904119902lowast

119904lt 0 So the revenue

sharing contract cannot realize the supply chain coordinationat this time This completes the proof

Proposition 16 shows that revenue sharing contractscan not realize the supply chain coordination with greentechnology investment and cap and trade policy The rea-son is that the manufacturer undertakes all the costs ofgreen technology investment when using revenue sharingcontracts thus making the manufacturer tend to invest lessin green technology Only when the manufacturer owns allthe revenue will themanufacturerrsquos optimal green technologyinvestment under revenue sharing contracts be equal to thatin QC scenario However the maximum profit of the retaileris negative at this time and it will not participate

62 Coordination Contract Based on Revenue Sharing-CostSharing Contract Proposition 16 states that the supply chainwith green technology investment and cap and trade policycan not be coordinated by revenue sharing contracts becausethe costs of green technology investment are undertaken bythe manufacturer and the game of retailer and customersis characterized by RE Equilibrium So we consider costssharing of green technology investment and carbon tradingand quantity commitment made by the manufacturer basedon revenue sharing contracts Let 120601 (0 lt 120601 le 1) represent theratio of revenue kept by the retailer and let (1minus120601) represent theratio of revenue delivered to the manufacturer 120572 (0 le 120572 le 1)represents the ratio of costs of green technology investmentand carbon trading undertaken by the manufacturer and(1 minus 120572) represents the ratio of costs of green technologyinvestment and carbon trading delivered to the retailer

Discrete Dynamics in Nature and Society 11

The retailerrsquos expected profit function with revenuesharing-cost sharing contract is

120587119903

120572(119902 120601 120572) = 120601 (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119908 minus 120601119904) 119902

minus (1 minus 120572) 119896 [(1 minus 120578) 119902 minus 119864]

minus1

2(1 minus 120572) 119905120578

2

(43)

We have 119890 = (1minus120578)119902minus119864 then themanufacturerrsquos expectedprofit function denoted by 120587119898

120572(119908 120578 120601 120572) is

120587119898

120572(119908 120578 120601 120572)

= (1 minus 120601) (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ (119908 minus 119888 minus 120572 (1 minus 120578) 119896 + (1 minus 120601) 119904) 119902 + 120572119896119864

minus1

21205721199051205782

(44)

The subscript 120572 represents the situation that using rev-enue sharing-cost sharing contract coordinates the supplychain

Proposition 17 If 0 le 120582 le 1 the parameters of revenuesharing-cost sharing (119908 120601 120572) satisfy

119908 = 120582119888 + (1 minus 120578) 119896

120601 = 120582

120572 = 1 minus 120582

(45)

The supply chainwith green technology investment and cap andtrade policy is coordinated

Proof When the parameters of revenue sharing-cost sharingsatisfy (45) the expected profit function of the retailer andthe manufacturer can be written as follows

120587119903

120572(119902 119901 120601 120572) = 120582120587

119902(119902 120578)

120587119898

120572(119908 120578 120601 120572) = (1 minus 120582) 120587

119902(119902 120578)

(46)

The optimal order of decentralized supply chain is equalto that of QC scenario Substitute 119901 = V minus (V minus 119904)119865(119902) into120587sc119888(119902 119901 120578) we get that the profit function of decentralized

supply chain at this time is the same as that of QC scenarioTherefore when the parameters of revenue sharing-costsharing (119908 120601 120572) satisfy (45) the decentralized supply chaincoordination is realizedThemanufacturer earns all the profitif 120582 = 0 and the retailer earns all the profit if 120582 = 1Arbitrary allocation of the supply chain profit between themanufacturer and the retailer is allowed This completes theproof

Proposition 17 shows that revenue sharing-cost sharingcontract can realize the supply chain coordination on condi-tion that the manufacturer makes a quantity commitment to

the customers The optimal production quantity and greentechnology investment in decentralized situation are lowerthan that in QC scenario The quantity commitment of themanufacturer can increase the green technology investmentand production quantity and also reduce the retailer priceThat is the quantity commitment is beneficial for the man-ufacturer the retailer and customers Thus the quantitycommitment is credible to customers now

7 Conclusions and Suggestions for FurtherResearch

This paper investigates the production pricing carbon trad-ing and green technology investment strategies and thecoordination of low carbon supply chain made up of a lowcarbon manufacturer and a retailer The low carbon manu-facturer produces one product under cap and trade policyand performs green technology investment to reduce carbonemissions in production process The retailer orders fromthemanufacturer and distributes the product to homogenouscustomers To the best of our knowledge this is the firststudy on the management of low carbon supply chainswith strategic customer behaviorThis paper provides severalinteresting observations

Observation 1 There are unique optimal production pricingcarbon trading and green technology investment strategiesin centralized (include RE Equilibrium scenario and QCscenario) and decentralized supply chain Therefore byderiving the optimal production pricing carbon trading andgreen technology investment strategies in different situationsthe low carbon manufacturer and the retailer can behaveappropriately based on our findings to maximize their profit

Observation 2 Our findings also show that the centralizedsupply chain can improve its performance by quantity com-mitment However the manufacturerrsquos production and greentechnology investment are reduced while the optimal priceis increased at this time Therefore quantity commitment isbeneficial for the supply chain but is harmful to customers

Observation 3 We also show that the performance of decen-tralized supply chain is lower than that in QC scenario Wefind that the decentralization reduces the manufacturersquos opti-mal production quantity and green technology investmentand increases the retailerrsquos optimal price Besides we find thatthe optimal production quantity is increasing the optimalprice is decreasing and the green technology investment isincreasing respectively in the three situations of decentral-ized supply chain QC scenario andREEquilibrium scenario

Observation 4 We demonstrate that the traditional revenuesharing contracts can not realize the low carbon supply chaincoordination when green technology investment is takenin consideration We also find that the revenue sharing-cost sharing contract realizes the decentralized supply chain

12 Discrete Dynamics in Nature and Society

coordination when the manufacturer makes a quantity com-mitment It is beneficial to the manufacturer the retailerand customers if themanufacturermakes a quantity commit-ment So the quantity commitment is credible to the customerat this time

We assume the customers are homogeneous strategiccustomers But themyopic customers and strategic customersoften coexist and strategic customers also have differenttolerance in real-life Extending this assumption will havea knock-on effect on supply chain decisions So one keyresearch direction is to consider that the customers aremixed customers myopic customers and strategic customersSecond we assume that the supply chain only producesone product While this simplified setting provides someinteresting insights we have to acknowledge that a manufac-turer usually produces two or more productsThese productswill substitute or complement each other So one importantextension of our work is to assume that the manufacturerproduces two or more products which have different unitcarbon emissions in production

Notations

119863 The random demand and119863 ge 0

119891(sdot) Probability density function of119863119865(sdot) Distribution function of119863 We assume

that 119865 satisfy IFR and define 119865 = 1 minus 119865

119888 Unit production cost119904 Unit salvage price of the product which

is an exogenous variableV The customersrsquo utility from consuming

the product119908 Unit wholesale price119901 Unit retail selling price We assume that

it can be observed by the customers119902 The manufacturerrsquos production

quantitythe retailerrsquos order quantityWe assume that it cannot be observedby the customers

119903 The customersrsquo reservation price whichis customersrsquo private information andcannot be observed by the retailer

120585119903 The beliefs of the manufacturer over the

customerrsquo reservation price120585prob The beliefs of customers over their

chances of obtaining the product atphase two

119864 Initial free carbon emission allocated bygovernment

119896 Unit carbon emission trading price119890 Carbon trading volume120578 Carbon emissions reduction rate

Conflict of Interests

The authors declare no conflict of interests

Acknowledgments

This research is partially supported by National Natural Sci-ence Foundation of China (nos 71432003 and 71272128) theProgram forNewCentury Excellent Talents inUniversity (noNCET-12-0087) and Specialized Research Fund for the Doc-toral Program of Higher Education (no 20130185110006)

References

[1] R Alley T Berntsen N L Bindoff et al ldquoSummary forpolicymakersrdquo in Climate Change 2007 The Physical ScienceBasis vol 6 pp 1ndash18 IPCC 2007

[2] X Chen and G Hao ldquoSustainable pricing and productionpolicies for two competing firms with carbon emissions taxrdquoInternational Journal of Production Research vol 53 pp 6408ndash6420 2015

[3] X Chen S Benjaafar and A Elomri ldquoThe carbon-constrainedEOQrdquo Operations Research Letters vol 41 no 2 pp 172ndash1792013

[4] N O Keohane ldquoCap and trade rehabilitated using tradablepermits to control US greenhouse gasesrdquo Review of Environ-mental Economics and Policy vol 3 no 1 pp 42ndash62 2009

[5] B Liu M Holmbom A Segerstedt and W Chen ldquoEffectsof carbon emission regulations on remanufacturing decisionswith limited information of demand distributionrdquo InternationalJournal of Production Research vol 53 no 2 pp 532ndash548 2015

[6] L Zhu and Y Fan ldquoA real options-based CCS investmentevaluation model case study of Chinarsquos power generationsectorrdquo Applied Energy vol 88 no 12 pp 4320ndash4333 2011

[7] XWang and LDu ldquoStudy on carbon capture and storage (CCS)investment decision-making based on real options for Chinarsquoscoal-fired power plantsrdquo Journal of Cleaner Production vol 112part 5 pp 4123ndash4131 2016

[8] W Jiang and X Chen ldquoOptimal strategies for manufacturerwith strategic customer behavior under carbon emissions-sensitive randomdemandrdquo Industrial Management amp Data Sys-tems 2015

[9] Y Levin J McGill and M Nediak ldquoDynamic pricing in thepresence of strategic consumers and oligopolistic competitionrdquoManagement Science vol 55 no 1 pp 32ndash46 2009

[10] A K Parlakturk ldquoThe value of product variety when sellingto strategic consumersrdquo Manufacturing amp Service OperationsManagement vol 14 no 3 pp 371ndash385 2012

[11] X Su and F Zhang ldquoOn the value of commitment andavailability guarantees when selling to strategic consumersrdquoManagement Science vol 55 no 5 pp 713ndash726 2009

[12] W Elmaghraby S A Lippman C S Tang and R Yin ldquoWillmore purchasing options benefit customersrdquo Production andOperations Management vol 18 no 4 pp 381ndash401 2009

[13] W Jiang and X Chen ldquoManufacturersquos production and pricingstrategies with carbon tax policy and strategic customer behav-iorrdquoManagement Science and Engineering vol 9 no 1 pp 30ndash35 2015

[14] I Dobos ldquoThe effects of emission trading on production andinventories in the Arrow-Karlin modelrdquo International Journalof Production Economics vol 93-94 pp 301ndash308 2005

[15] X Chen C K Chan and Y C E Lee ldquoResponsible productionpolicies with substitution and carbon emissions tradingrdquo Jour-nal of Cleaner Production 2015

Discrete Dynamics in Nature and Society 13

[16] X Chen and X Wang ldquoEffects of carbon emission reductionpolicies on transportation mode selections with stochasticdemandrdquo Transportation Research Part E Logistics and Trans-portation Review 2015

[17] X Chang H Xia H Zhu T Fan and H Zhao ldquoProductiondecisions in a hybrid manufacturingndashremanufacturing systemwith carbon cap and trade mechanismrdquo International Journal ofProduction Economics vol 162 pp 160ndash173 2015

[18] J Zhao B F Hobbs and J-S Pang ldquoLong-run equilibriummodeling of emissions allowance allocation systems in electricpowermarketsrdquoOperations Research vol 58 no 3 pp 529ndash5482010

[19] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2 pp519ndash527 2011

[20] A Toptal H Ozlu and D Konur ldquoJoint decisions on inventoryreplenishment and emission reduction investment under dif-ferent emission regulationsrdquo International Journal of ProductionResearch vol 52 no 1 pp 243ndash269 2014

[21] A S Manikas and J R Kroes ldquoA newsvendor approach tocompliance and production under cap and trade emissionsregulationrdquo International Journal of Production Economics vol159 pp 274ndash284 2015

[22] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineering vol10 no 1 pp 99ndash116 2013

[23] X Xu W Zhang P He and X Xu ldquoProduction and pricingproblems in make-to-order supply chain with cap-and-traderegulationrdquo Omega 2015

[24] Y Zhen X Yang and W Jiang ldquoThe impact of transport modeand carbon policy on low-carbon retailerrdquoDiscrete Dynamics inNature and Society vol 2015 Article ID 964305 12 pages 2015

[25] S Swami and J Shah ldquoChannel coordination in green supplychainmanagementrdquo Journal of the Operational Research Societyvol 64 no 3 pp 336ndash351 2013

[26] R H Coase ldquoDurability andmonopolyrdquoThe Journal of Law andEconomics vol 15 no 1 pp 143ndash149 1972

[27] X Su ldquoIntertemporal pricing with strategic customer behaviorrdquoManagement Science vol 53 no 5 pp 726ndash741 2007

[28] D Zhang and W L Cooper ldquoManaging clearance sales in thepresence of strategic customersrdquo Production and OperationsManagement vol 17 no 4 pp 416ndash431 2008

[29] S Whang ldquoDemand uncertainty and the Bayesian effect inmarkdown pricing with strategic customersrdquoManufacturing ampService Operations Management vol 17 no 1 pp 66ndash77 2015

[30] J Du J Zhang and G Hua ldquoPricing and inventory manage-ment in the presence of strategic customers with risk preferenceand decreasing valuerdquo International Journal of Production Eco-nomics vol 164 pp 160ndash166 2015

[31] X Su and F Zhang ldquoStrategic customer behavior commitmentand supply chain performancerdquo Management Science vol 54no 10 pp 1759ndash1773 2008

[32] H Yang ldquoImpact of discounting and competition on benefit ofdecentralization with strategic customersrdquo Operations ResearchLetters vol 40 no 2 pp 123ndash127 2012

[33] D Yang E Qi and Y Li ldquoQuick response and supply chainstructure with strategic consumersrdquo Omega vol 52 pp 1ndash142015

[34] C drsquoAspremont and A Jacquemin ldquoCooperative and nonco-operative R amp D in duopoly with spilloversrdquo The AmericanEconomic Review vol 78 no 5 pp 1133ndash1137 1988

[35] J F Muth ldquoRational expectations and the theory of pricemovementsrdquo Econometrica vol 29 no 3 pp 315ndash335 1961

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Discrete Dynamics in Nature and Society

The subscript 119904 represents the situation of supply chaincoordination using revenue sharing contracts

As to the retailerrsquos optimal order strategy (denoted by119902lowast

119904) and pricing strategy (denoted by 119901lowast

119904) of the supply chain

with revenue sharing contract the following proposition isobtained

Proposition 15 Under revenue sharing contracts the retailerrsquosoptimal order strategy (119902lowast

119904) and optimal pricing strategy (119901lowast

119904)

are

119902lowast

119904= 119865minus1

(radic119908 minus 120601119904

120601 (V minus 119904))

119901lowast

119904= 119904 + radic

(119908 minus 120601119904) ((V minus 119904))120601

(39)

Proof According to the definition of RE Equilibrium wealso have 119901 = V minus (V minus 119904)119865(119902) under revenue sharingcontracts

The first-order and second-order derivative of 120587119903119904(119902 119901 120601)

with respect to 119902 are 120597120587119903

119904(119902 119901 120601)120597119902 = 120601(119901 minus 119904)119865(119902) minus

(119908 minus 120601119904) and 1205972120587119903

119904(119902 119901 120601)120597119902

2= minus120601(119901 minus 119904)119891(119902) lt

0 So given 119901 the retailerrsquos expected profit function is aconcave function of 119902 Let 120597120587119903

119904(119902 119901 120601)120597119902 = 0 and combine

with 119901 = V minus (V minus 119904)119865(119902) we can solve the retailerrsquos optimalstrategies under revenue sharing contractThis completes theproof

Proposition 15 shows that the optimal order and pricingstrategies of the retailer based on revenue sharing contract inRE Equilibrium exist and are uniqueThis is also the retailerrsquosoptimal response function with respect to 119908

Substituting (39) into (43)

120587119898

119904(119902 120578 120601)

= (1 minus 120601) (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ [120601 (V minus 119904) 1198652

(119902) minus (119888 minus 119904 + (1 minus 120578) 119896)] 119902 + 119896119864

minus1

21199051205782

120597120587119898

119904(119902 120578 120601)

120597120578= 119902119896 minus 119905120578

1205972120587119898

119904(119902 120578 120601)

1205971205782= minus119905 lt 0

(40)

Given 119902 and 120601 the manufacturerrsquos green technologyinvestment strategies with revenue sharing contracts exist

and are unique Let 120597120587119898119904(119902 120578 120601)120597120578 = 0 we get 120578119898lowast

119904equiv 120578(119902) =

119902119896119905 and substitute it into 120587119898119904(119902 120578 120601)

120587119898

119904(119902 120601)

= (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ 120601 (V minus 119904) 119865 (119902) [119902119865 (119902) minus (119902 minus int119902

0

119865 (119909) 119889119909)]

minus (119888 minus 119904 + 119896) 119902 + 119896119864 +11989621199022

2119905

(41)

Proposition 16 The supply chain with green technologyinvestment and cap and trade policy can not be coordinated byrevenue sharing contract

Proof The derivative of 120587119898119904(119902 120601) with respect to 119902 can be

arranged as follows

120597120587119898

119904(119902 120601)

120597119902=120597120587119902(119902)

120597119902+ 120601(

120597120587119898(119902)

120597119902minus120597120587119902(119902)

120597119902) (42)

Only when 120601 = 0 are the optimal strategies of supplychain under revenue sharing contracts equal to that in QCscenario Let 119908lowast

119904represent the optimal wholesale price of

the manufacture and 119902lowast119904represent the retailerrsquos optimal order

quantity Then 120587119903

119904(119902 119901 120601) = minus119908

lowast

119904119902lowast

119904lt 0 So the revenue

sharing contract cannot realize the supply chain coordinationat this time This completes the proof

Proposition 16 shows that revenue sharing contractscan not realize the supply chain coordination with greentechnology investment and cap and trade policy The rea-son is that the manufacturer undertakes all the costs ofgreen technology investment when using revenue sharingcontracts thus making the manufacturer tend to invest lessin green technology Only when the manufacturer owns allthe revenue will themanufacturerrsquos optimal green technologyinvestment under revenue sharing contracts be equal to thatin QC scenario However the maximum profit of the retaileris negative at this time and it will not participate

62 Coordination Contract Based on Revenue Sharing-CostSharing Contract Proposition 16 states that the supply chainwith green technology investment and cap and trade policycan not be coordinated by revenue sharing contracts becausethe costs of green technology investment are undertaken bythe manufacturer and the game of retailer and customersis characterized by RE Equilibrium So we consider costssharing of green technology investment and carbon tradingand quantity commitment made by the manufacturer basedon revenue sharing contracts Let 120601 (0 lt 120601 le 1) represent theratio of revenue kept by the retailer and let (1minus120601) represent theratio of revenue delivered to the manufacturer 120572 (0 le 120572 le 1)represents the ratio of costs of green technology investmentand carbon trading undertaken by the manufacturer and(1 minus 120572) represents the ratio of costs of green technologyinvestment and carbon trading delivered to the retailer

Discrete Dynamics in Nature and Society 11

The retailerrsquos expected profit function with revenuesharing-cost sharing contract is

120587119903

120572(119902 120601 120572) = 120601 (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119908 minus 120601119904) 119902

minus (1 minus 120572) 119896 [(1 minus 120578) 119902 minus 119864]

minus1

2(1 minus 120572) 119905120578

2

(43)

We have 119890 = (1minus120578)119902minus119864 then themanufacturerrsquos expectedprofit function denoted by 120587119898

120572(119908 120578 120601 120572) is

120587119898

120572(119908 120578 120601 120572)

= (1 minus 120601) (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ (119908 minus 119888 minus 120572 (1 minus 120578) 119896 + (1 minus 120601) 119904) 119902 + 120572119896119864

minus1

21205721199051205782

(44)

The subscript 120572 represents the situation that using rev-enue sharing-cost sharing contract coordinates the supplychain

Proposition 17 If 0 le 120582 le 1 the parameters of revenuesharing-cost sharing (119908 120601 120572) satisfy

119908 = 120582119888 + (1 minus 120578) 119896

120601 = 120582

120572 = 1 minus 120582

(45)

The supply chainwith green technology investment and cap andtrade policy is coordinated

Proof When the parameters of revenue sharing-cost sharingsatisfy (45) the expected profit function of the retailer andthe manufacturer can be written as follows

120587119903

120572(119902 119901 120601 120572) = 120582120587

119902(119902 120578)

120587119898

120572(119908 120578 120601 120572) = (1 minus 120582) 120587

119902(119902 120578)

(46)

The optimal order of decentralized supply chain is equalto that of QC scenario Substitute 119901 = V minus (V minus 119904)119865(119902) into120587sc119888(119902 119901 120578) we get that the profit function of decentralized

supply chain at this time is the same as that of QC scenarioTherefore when the parameters of revenue sharing-costsharing (119908 120601 120572) satisfy (45) the decentralized supply chaincoordination is realizedThemanufacturer earns all the profitif 120582 = 0 and the retailer earns all the profit if 120582 = 1Arbitrary allocation of the supply chain profit between themanufacturer and the retailer is allowed This completes theproof

Proposition 17 shows that revenue sharing-cost sharingcontract can realize the supply chain coordination on condi-tion that the manufacturer makes a quantity commitment to

the customers The optimal production quantity and greentechnology investment in decentralized situation are lowerthan that in QC scenario The quantity commitment of themanufacturer can increase the green technology investmentand production quantity and also reduce the retailer priceThat is the quantity commitment is beneficial for the man-ufacturer the retailer and customers Thus the quantitycommitment is credible to customers now

7 Conclusions and Suggestions for FurtherResearch

This paper investigates the production pricing carbon trad-ing and green technology investment strategies and thecoordination of low carbon supply chain made up of a lowcarbon manufacturer and a retailer The low carbon manu-facturer produces one product under cap and trade policyand performs green technology investment to reduce carbonemissions in production process The retailer orders fromthemanufacturer and distributes the product to homogenouscustomers To the best of our knowledge this is the firststudy on the management of low carbon supply chainswith strategic customer behaviorThis paper provides severalinteresting observations

Observation 1 There are unique optimal production pricingcarbon trading and green technology investment strategiesin centralized (include RE Equilibrium scenario and QCscenario) and decentralized supply chain Therefore byderiving the optimal production pricing carbon trading andgreen technology investment strategies in different situationsthe low carbon manufacturer and the retailer can behaveappropriately based on our findings to maximize their profit

Observation 2 Our findings also show that the centralizedsupply chain can improve its performance by quantity com-mitment However the manufacturerrsquos production and greentechnology investment are reduced while the optimal priceis increased at this time Therefore quantity commitment isbeneficial for the supply chain but is harmful to customers

Observation 3 We also show that the performance of decen-tralized supply chain is lower than that in QC scenario Wefind that the decentralization reduces the manufacturersquos opti-mal production quantity and green technology investmentand increases the retailerrsquos optimal price Besides we find thatthe optimal production quantity is increasing the optimalprice is decreasing and the green technology investment isincreasing respectively in the three situations of decentral-ized supply chain QC scenario andREEquilibrium scenario

Observation 4 We demonstrate that the traditional revenuesharing contracts can not realize the low carbon supply chaincoordination when green technology investment is takenin consideration We also find that the revenue sharing-cost sharing contract realizes the decentralized supply chain

12 Discrete Dynamics in Nature and Society

coordination when the manufacturer makes a quantity com-mitment It is beneficial to the manufacturer the retailerand customers if themanufacturermakes a quantity commit-ment So the quantity commitment is credible to the customerat this time

We assume the customers are homogeneous strategiccustomers But themyopic customers and strategic customersoften coexist and strategic customers also have differenttolerance in real-life Extending this assumption will havea knock-on effect on supply chain decisions So one keyresearch direction is to consider that the customers aremixed customers myopic customers and strategic customersSecond we assume that the supply chain only producesone product While this simplified setting provides someinteresting insights we have to acknowledge that a manufac-turer usually produces two or more productsThese productswill substitute or complement each other So one importantextension of our work is to assume that the manufacturerproduces two or more products which have different unitcarbon emissions in production

Notations

119863 The random demand and119863 ge 0

119891(sdot) Probability density function of119863119865(sdot) Distribution function of119863 We assume

that 119865 satisfy IFR and define 119865 = 1 minus 119865

119888 Unit production cost119904 Unit salvage price of the product which

is an exogenous variableV The customersrsquo utility from consuming

the product119908 Unit wholesale price119901 Unit retail selling price We assume that

it can be observed by the customers119902 The manufacturerrsquos production

quantitythe retailerrsquos order quantityWe assume that it cannot be observedby the customers

119903 The customersrsquo reservation price whichis customersrsquo private information andcannot be observed by the retailer

120585119903 The beliefs of the manufacturer over the

customerrsquo reservation price120585prob The beliefs of customers over their

chances of obtaining the product atphase two

119864 Initial free carbon emission allocated bygovernment

119896 Unit carbon emission trading price119890 Carbon trading volume120578 Carbon emissions reduction rate

Conflict of Interests

The authors declare no conflict of interests

Acknowledgments

This research is partially supported by National Natural Sci-ence Foundation of China (nos 71432003 and 71272128) theProgram forNewCentury Excellent Talents inUniversity (noNCET-12-0087) and Specialized Research Fund for the Doc-toral Program of Higher Education (no 20130185110006)

References

[1] R Alley T Berntsen N L Bindoff et al ldquoSummary forpolicymakersrdquo in Climate Change 2007 The Physical ScienceBasis vol 6 pp 1ndash18 IPCC 2007

[2] X Chen and G Hao ldquoSustainable pricing and productionpolicies for two competing firms with carbon emissions taxrdquoInternational Journal of Production Research vol 53 pp 6408ndash6420 2015

[3] X Chen S Benjaafar and A Elomri ldquoThe carbon-constrainedEOQrdquo Operations Research Letters vol 41 no 2 pp 172ndash1792013

[4] N O Keohane ldquoCap and trade rehabilitated using tradablepermits to control US greenhouse gasesrdquo Review of Environ-mental Economics and Policy vol 3 no 1 pp 42ndash62 2009

[5] B Liu M Holmbom A Segerstedt and W Chen ldquoEffectsof carbon emission regulations on remanufacturing decisionswith limited information of demand distributionrdquo InternationalJournal of Production Research vol 53 no 2 pp 532ndash548 2015

[6] L Zhu and Y Fan ldquoA real options-based CCS investmentevaluation model case study of Chinarsquos power generationsectorrdquo Applied Energy vol 88 no 12 pp 4320ndash4333 2011

[7] XWang and LDu ldquoStudy on carbon capture and storage (CCS)investment decision-making based on real options for Chinarsquoscoal-fired power plantsrdquo Journal of Cleaner Production vol 112part 5 pp 4123ndash4131 2016

[8] W Jiang and X Chen ldquoOptimal strategies for manufacturerwith strategic customer behavior under carbon emissions-sensitive randomdemandrdquo Industrial Management amp Data Sys-tems 2015

[9] Y Levin J McGill and M Nediak ldquoDynamic pricing in thepresence of strategic consumers and oligopolistic competitionrdquoManagement Science vol 55 no 1 pp 32ndash46 2009

[10] A K Parlakturk ldquoThe value of product variety when sellingto strategic consumersrdquo Manufacturing amp Service OperationsManagement vol 14 no 3 pp 371ndash385 2012

[11] X Su and F Zhang ldquoOn the value of commitment andavailability guarantees when selling to strategic consumersrdquoManagement Science vol 55 no 5 pp 713ndash726 2009

[12] W Elmaghraby S A Lippman C S Tang and R Yin ldquoWillmore purchasing options benefit customersrdquo Production andOperations Management vol 18 no 4 pp 381ndash401 2009

[13] W Jiang and X Chen ldquoManufacturersquos production and pricingstrategies with carbon tax policy and strategic customer behav-iorrdquoManagement Science and Engineering vol 9 no 1 pp 30ndash35 2015

[14] I Dobos ldquoThe effects of emission trading on production andinventories in the Arrow-Karlin modelrdquo International Journalof Production Economics vol 93-94 pp 301ndash308 2005

[15] X Chen C K Chan and Y C E Lee ldquoResponsible productionpolicies with substitution and carbon emissions tradingrdquo Jour-nal of Cleaner Production 2015

Discrete Dynamics in Nature and Society 13

[16] X Chen and X Wang ldquoEffects of carbon emission reductionpolicies on transportation mode selections with stochasticdemandrdquo Transportation Research Part E Logistics and Trans-portation Review 2015

[17] X Chang H Xia H Zhu T Fan and H Zhao ldquoProductiondecisions in a hybrid manufacturingndashremanufacturing systemwith carbon cap and trade mechanismrdquo International Journal ofProduction Economics vol 162 pp 160ndash173 2015

[18] J Zhao B F Hobbs and J-S Pang ldquoLong-run equilibriummodeling of emissions allowance allocation systems in electricpowermarketsrdquoOperations Research vol 58 no 3 pp 529ndash5482010

[19] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2 pp519ndash527 2011

[20] A Toptal H Ozlu and D Konur ldquoJoint decisions on inventoryreplenishment and emission reduction investment under dif-ferent emission regulationsrdquo International Journal of ProductionResearch vol 52 no 1 pp 243ndash269 2014

[21] A S Manikas and J R Kroes ldquoA newsvendor approach tocompliance and production under cap and trade emissionsregulationrdquo International Journal of Production Economics vol159 pp 274ndash284 2015

[22] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineering vol10 no 1 pp 99ndash116 2013

[23] X Xu W Zhang P He and X Xu ldquoProduction and pricingproblems in make-to-order supply chain with cap-and-traderegulationrdquo Omega 2015

[24] Y Zhen X Yang and W Jiang ldquoThe impact of transport modeand carbon policy on low-carbon retailerrdquoDiscrete Dynamics inNature and Society vol 2015 Article ID 964305 12 pages 2015

[25] S Swami and J Shah ldquoChannel coordination in green supplychainmanagementrdquo Journal of the Operational Research Societyvol 64 no 3 pp 336ndash351 2013

[26] R H Coase ldquoDurability andmonopolyrdquoThe Journal of Law andEconomics vol 15 no 1 pp 143ndash149 1972

[27] X Su ldquoIntertemporal pricing with strategic customer behaviorrdquoManagement Science vol 53 no 5 pp 726ndash741 2007

[28] D Zhang and W L Cooper ldquoManaging clearance sales in thepresence of strategic customersrdquo Production and OperationsManagement vol 17 no 4 pp 416ndash431 2008

[29] S Whang ldquoDemand uncertainty and the Bayesian effect inmarkdown pricing with strategic customersrdquoManufacturing ampService Operations Management vol 17 no 1 pp 66ndash77 2015

[30] J Du J Zhang and G Hua ldquoPricing and inventory manage-ment in the presence of strategic customers with risk preferenceand decreasing valuerdquo International Journal of Production Eco-nomics vol 164 pp 160ndash166 2015

[31] X Su and F Zhang ldquoStrategic customer behavior commitmentand supply chain performancerdquo Management Science vol 54no 10 pp 1759ndash1773 2008

[32] H Yang ldquoImpact of discounting and competition on benefit ofdecentralization with strategic customersrdquo Operations ResearchLetters vol 40 no 2 pp 123ndash127 2012

[33] D Yang E Qi and Y Li ldquoQuick response and supply chainstructure with strategic consumersrdquo Omega vol 52 pp 1ndash142015

[34] C drsquoAspremont and A Jacquemin ldquoCooperative and nonco-operative R amp D in duopoly with spilloversrdquo The AmericanEconomic Review vol 78 no 5 pp 1133ndash1137 1988

[35] J F Muth ldquoRational expectations and the theory of pricemovementsrdquo Econometrica vol 29 no 3 pp 315ndash335 1961

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 11

The retailerrsquos expected profit function with revenuesharing-cost sharing contract is

120587119903

120572(119902 120601 120572) = 120601 (V minus 119904) 119865 (119902) (119902 minus int

119902

0

119865 (119909) 119889119909)

minus (119908 minus 120601119904) 119902

minus (1 minus 120572) 119896 [(1 minus 120578) 119902 minus 119864]

minus1

2(1 minus 120572) 119905120578

2

(43)

We have 119890 = (1minus120578)119902minus119864 then themanufacturerrsquos expectedprofit function denoted by 120587119898

120572(119908 120578 120601 120572) is

120587119898

120572(119908 120578 120601 120572)

= (1 minus 120601) (V minus 119904) 119865 (119902) (119902 minus int119902

0

119865 (119909) 119889119909)

+ (119908 minus 119888 minus 120572 (1 minus 120578) 119896 + (1 minus 120601) 119904) 119902 + 120572119896119864

minus1

21205721199051205782

(44)

The subscript 120572 represents the situation that using rev-enue sharing-cost sharing contract coordinates the supplychain

Proposition 17 If 0 le 120582 le 1 the parameters of revenuesharing-cost sharing (119908 120601 120572) satisfy

119908 = 120582119888 + (1 minus 120578) 119896

120601 = 120582

120572 = 1 minus 120582

(45)

The supply chainwith green technology investment and cap andtrade policy is coordinated

Proof When the parameters of revenue sharing-cost sharingsatisfy (45) the expected profit function of the retailer andthe manufacturer can be written as follows

120587119903

120572(119902 119901 120601 120572) = 120582120587

119902(119902 120578)

120587119898

120572(119908 120578 120601 120572) = (1 minus 120582) 120587

119902(119902 120578)

(46)

The optimal order of decentralized supply chain is equalto that of QC scenario Substitute 119901 = V minus (V minus 119904)119865(119902) into120587sc119888(119902 119901 120578) we get that the profit function of decentralized

supply chain at this time is the same as that of QC scenarioTherefore when the parameters of revenue sharing-costsharing (119908 120601 120572) satisfy (45) the decentralized supply chaincoordination is realizedThemanufacturer earns all the profitif 120582 = 0 and the retailer earns all the profit if 120582 = 1Arbitrary allocation of the supply chain profit between themanufacturer and the retailer is allowed This completes theproof

Proposition 17 shows that revenue sharing-cost sharingcontract can realize the supply chain coordination on condi-tion that the manufacturer makes a quantity commitment to

the customers The optimal production quantity and greentechnology investment in decentralized situation are lowerthan that in QC scenario The quantity commitment of themanufacturer can increase the green technology investmentand production quantity and also reduce the retailer priceThat is the quantity commitment is beneficial for the man-ufacturer the retailer and customers Thus the quantitycommitment is credible to customers now

7 Conclusions and Suggestions for FurtherResearch

This paper investigates the production pricing carbon trad-ing and green technology investment strategies and thecoordination of low carbon supply chain made up of a lowcarbon manufacturer and a retailer The low carbon manu-facturer produces one product under cap and trade policyand performs green technology investment to reduce carbonemissions in production process The retailer orders fromthemanufacturer and distributes the product to homogenouscustomers To the best of our knowledge this is the firststudy on the management of low carbon supply chainswith strategic customer behaviorThis paper provides severalinteresting observations

Observation 1 There are unique optimal production pricingcarbon trading and green technology investment strategiesin centralized (include RE Equilibrium scenario and QCscenario) and decentralized supply chain Therefore byderiving the optimal production pricing carbon trading andgreen technology investment strategies in different situationsthe low carbon manufacturer and the retailer can behaveappropriately based on our findings to maximize their profit

Observation 2 Our findings also show that the centralizedsupply chain can improve its performance by quantity com-mitment However the manufacturerrsquos production and greentechnology investment are reduced while the optimal priceis increased at this time Therefore quantity commitment isbeneficial for the supply chain but is harmful to customers

Observation 3 We also show that the performance of decen-tralized supply chain is lower than that in QC scenario Wefind that the decentralization reduces the manufacturersquos opti-mal production quantity and green technology investmentand increases the retailerrsquos optimal price Besides we find thatthe optimal production quantity is increasing the optimalprice is decreasing and the green technology investment isincreasing respectively in the three situations of decentral-ized supply chain QC scenario andREEquilibrium scenario

Observation 4 We demonstrate that the traditional revenuesharing contracts can not realize the low carbon supply chaincoordination when green technology investment is takenin consideration We also find that the revenue sharing-cost sharing contract realizes the decentralized supply chain

12 Discrete Dynamics in Nature and Society

coordination when the manufacturer makes a quantity com-mitment It is beneficial to the manufacturer the retailerand customers if themanufacturermakes a quantity commit-ment So the quantity commitment is credible to the customerat this time

We assume the customers are homogeneous strategiccustomers But themyopic customers and strategic customersoften coexist and strategic customers also have differenttolerance in real-life Extending this assumption will havea knock-on effect on supply chain decisions So one keyresearch direction is to consider that the customers aremixed customers myopic customers and strategic customersSecond we assume that the supply chain only producesone product While this simplified setting provides someinteresting insights we have to acknowledge that a manufac-turer usually produces two or more productsThese productswill substitute or complement each other So one importantextension of our work is to assume that the manufacturerproduces two or more products which have different unitcarbon emissions in production

Notations

119863 The random demand and119863 ge 0

119891(sdot) Probability density function of119863119865(sdot) Distribution function of119863 We assume

that 119865 satisfy IFR and define 119865 = 1 minus 119865

119888 Unit production cost119904 Unit salvage price of the product which

is an exogenous variableV The customersrsquo utility from consuming

the product119908 Unit wholesale price119901 Unit retail selling price We assume that

it can be observed by the customers119902 The manufacturerrsquos production

quantitythe retailerrsquos order quantityWe assume that it cannot be observedby the customers

119903 The customersrsquo reservation price whichis customersrsquo private information andcannot be observed by the retailer

120585119903 The beliefs of the manufacturer over the

customerrsquo reservation price120585prob The beliefs of customers over their

chances of obtaining the product atphase two

119864 Initial free carbon emission allocated bygovernment

119896 Unit carbon emission trading price119890 Carbon trading volume120578 Carbon emissions reduction rate

Conflict of Interests

The authors declare no conflict of interests

Acknowledgments

This research is partially supported by National Natural Sci-ence Foundation of China (nos 71432003 and 71272128) theProgram forNewCentury Excellent Talents inUniversity (noNCET-12-0087) and Specialized Research Fund for the Doc-toral Program of Higher Education (no 20130185110006)

References

[1] R Alley T Berntsen N L Bindoff et al ldquoSummary forpolicymakersrdquo in Climate Change 2007 The Physical ScienceBasis vol 6 pp 1ndash18 IPCC 2007

[2] X Chen and G Hao ldquoSustainable pricing and productionpolicies for two competing firms with carbon emissions taxrdquoInternational Journal of Production Research vol 53 pp 6408ndash6420 2015

[3] X Chen S Benjaafar and A Elomri ldquoThe carbon-constrainedEOQrdquo Operations Research Letters vol 41 no 2 pp 172ndash1792013

[4] N O Keohane ldquoCap and trade rehabilitated using tradablepermits to control US greenhouse gasesrdquo Review of Environ-mental Economics and Policy vol 3 no 1 pp 42ndash62 2009

[5] B Liu M Holmbom A Segerstedt and W Chen ldquoEffectsof carbon emission regulations on remanufacturing decisionswith limited information of demand distributionrdquo InternationalJournal of Production Research vol 53 no 2 pp 532ndash548 2015

[6] L Zhu and Y Fan ldquoA real options-based CCS investmentevaluation model case study of Chinarsquos power generationsectorrdquo Applied Energy vol 88 no 12 pp 4320ndash4333 2011

[7] XWang and LDu ldquoStudy on carbon capture and storage (CCS)investment decision-making based on real options for Chinarsquoscoal-fired power plantsrdquo Journal of Cleaner Production vol 112part 5 pp 4123ndash4131 2016

[8] W Jiang and X Chen ldquoOptimal strategies for manufacturerwith strategic customer behavior under carbon emissions-sensitive randomdemandrdquo Industrial Management amp Data Sys-tems 2015

[9] Y Levin J McGill and M Nediak ldquoDynamic pricing in thepresence of strategic consumers and oligopolistic competitionrdquoManagement Science vol 55 no 1 pp 32ndash46 2009

[10] A K Parlakturk ldquoThe value of product variety when sellingto strategic consumersrdquo Manufacturing amp Service OperationsManagement vol 14 no 3 pp 371ndash385 2012

[11] X Su and F Zhang ldquoOn the value of commitment andavailability guarantees when selling to strategic consumersrdquoManagement Science vol 55 no 5 pp 713ndash726 2009

[12] W Elmaghraby S A Lippman C S Tang and R Yin ldquoWillmore purchasing options benefit customersrdquo Production andOperations Management vol 18 no 4 pp 381ndash401 2009

[13] W Jiang and X Chen ldquoManufacturersquos production and pricingstrategies with carbon tax policy and strategic customer behav-iorrdquoManagement Science and Engineering vol 9 no 1 pp 30ndash35 2015

[14] I Dobos ldquoThe effects of emission trading on production andinventories in the Arrow-Karlin modelrdquo International Journalof Production Economics vol 93-94 pp 301ndash308 2005

[15] X Chen C K Chan and Y C E Lee ldquoResponsible productionpolicies with substitution and carbon emissions tradingrdquo Jour-nal of Cleaner Production 2015

Discrete Dynamics in Nature and Society 13

[16] X Chen and X Wang ldquoEffects of carbon emission reductionpolicies on transportation mode selections with stochasticdemandrdquo Transportation Research Part E Logistics and Trans-portation Review 2015

[17] X Chang H Xia H Zhu T Fan and H Zhao ldquoProductiondecisions in a hybrid manufacturingndashremanufacturing systemwith carbon cap and trade mechanismrdquo International Journal ofProduction Economics vol 162 pp 160ndash173 2015

[18] J Zhao B F Hobbs and J-S Pang ldquoLong-run equilibriummodeling of emissions allowance allocation systems in electricpowermarketsrdquoOperations Research vol 58 no 3 pp 529ndash5482010

[19] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2 pp519ndash527 2011

[20] A Toptal H Ozlu and D Konur ldquoJoint decisions on inventoryreplenishment and emission reduction investment under dif-ferent emission regulationsrdquo International Journal of ProductionResearch vol 52 no 1 pp 243ndash269 2014

[21] A S Manikas and J R Kroes ldquoA newsvendor approach tocompliance and production under cap and trade emissionsregulationrdquo International Journal of Production Economics vol159 pp 274ndash284 2015

[22] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineering vol10 no 1 pp 99ndash116 2013

[23] X Xu W Zhang P He and X Xu ldquoProduction and pricingproblems in make-to-order supply chain with cap-and-traderegulationrdquo Omega 2015

[24] Y Zhen X Yang and W Jiang ldquoThe impact of transport modeand carbon policy on low-carbon retailerrdquoDiscrete Dynamics inNature and Society vol 2015 Article ID 964305 12 pages 2015

[25] S Swami and J Shah ldquoChannel coordination in green supplychainmanagementrdquo Journal of the Operational Research Societyvol 64 no 3 pp 336ndash351 2013

[26] R H Coase ldquoDurability andmonopolyrdquoThe Journal of Law andEconomics vol 15 no 1 pp 143ndash149 1972

[27] X Su ldquoIntertemporal pricing with strategic customer behaviorrdquoManagement Science vol 53 no 5 pp 726ndash741 2007

[28] D Zhang and W L Cooper ldquoManaging clearance sales in thepresence of strategic customersrdquo Production and OperationsManagement vol 17 no 4 pp 416ndash431 2008

[29] S Whang ldquoDemand uncertainty and the Bayesian effect inmarkdown pricing with strategic customersrdquoManufacturing ampService Operations Management vol 17 no 1 pp 66ndash77 2015

[30] J Du J Zhang and G Hua ldquoPricing and inventory manage-ment in the presence of strategic customers with risk preferenceand decreasing valuerdquo International Journal of Production Eco-nomics vol 164 pp 160ndash166 2015

[31] X Su and F Zhang ldquoStrategic customer behavior commitmentand supply chain performancerdquo Management Science vol 54no 10 pp 1759ndash1773 2008

[32] H Yang ldquoImpact of discounting and competition on benefit ofdecentralization with strategic customersrdquo Operations ResearchLetters vol 40 no 2 pp 123ndash127 2012

[33] D Yang E Qi and Y Li ldquoQuick response and supply chainstructure with strategic consumersrdquo Omega vol 52 pp 1ndash142015

[34] C drsquoAspremont and A Jacquemin ldquoCooperative and nonco-operative R amp D in duopoly with spilloversrdquo The AmericanEconomic Review vol 78 no 5 pp 1133ndash1137 1988

[35] J F Muth ldquoRational expectations and the theory of pricemovementsrdquo Econometrica vol 29 no 3 pp 315ndash335 1961

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Discrete Dynamics in Nature and Society

coordination when the manufacturer makes a quantity com-mitment It is beneficial to the manufacturer the retailerand customers if themanufacturermakes a quantity commit-ment So the quantity commitment is credible to the customerat this time

We assume the customers are homogeneous strategiccustomers But themyopic customers and strategic customersoften coexist and strategic customers also have differenttolerance in real-life Extending this assumption will havea knock-on effect on supply chain decisions So one keyresearch direction is to consider that the customers aremixed customers myopic customers and strategic customersSecond we assume that the supply chain only producesone product While this simplified setting provides someinteresting insights we have to acknowledge that a manufac-turer usually produces two or more productsThese productswill substitute or complement each other So one importantextension of our work is to assume that the manufacturerproduces two or more products which have different unitcarbon emissions in production

Notations

119863 The random demand and119863 ge 0

119891(sdot) Probability density function of119863119865(sdot) Distribution function of119863 We assume

that 119865 satisfy IFR and define 119865 = 1 minus 119865

119888 Unit production cost119904 Unit salvage price of the product which

is an exogenous variableV The customersrsquo utility from consuming

the product119908 Unit wholesale price119901 Unit retail selling price We assume that

it can be observed by the customers119902 The manufacturerrsquos production

quantitythe retailerrsquos order quantityWe assume that it cannot be observedby the customers

119903 The customersrsquo reservation price whichis customersrsquo private information andcannot be observed by the retailer

120585119903 The beliefs of the manufacturer over the

customerrsquo reservation price120585prob The beliefs of customers over their

chances of obtaining the product atphase two

119864 Initial free carbon emission allocated bygovernment

119896 Unit carbon emission trading price119890 Carbon trading volume120578 Carbon emissions reduction rate

Conflict of Interests

The authors declare no conflict of interests

Acknowledgments

This research is partially supported by National Natural Sci-ence Foundation of China (nos 71432003 and 71272128) theProgram forNewCentury Excellent Talents inUniversity (noNCET-12-0087) and Specialized Research Fund for the Doc-toral Program of Higher Education (no 20130185110006)

References

[1] R Alley T Berntsen N L Bindoff et al ldquoSummary forpolicymakersrdquo in Climate Change 2007 The Physical ScienceBasis vol 6 pp 1ndash18 IPCC 2007

[2] X Chen and G Hao ldquoSustainable pricing and productionpolicies for two competing firms with carbon emissions taxrdquoInternational Journal of Production Research vol 53 pp 6408ndash6420 2015

[3] X Chen S Benjaafar and A Elomri ldquoThe carbon-constrainedEOQrdquo Operations Research Letters vol 41 no 2 pp 172ndash1792013

[4] N O Keohane ldquoCap and trade rehabilitated using tradablepermits to control US greenhouse gasesrdquo Review of Environ-mental Economics and Policy vol 3 no 1 pp 42ndash62 2009

[5] B Liu M Holmbom A Segerstedt and W Chen ldquoEffectsof carbon emission regulations on remanufacturing decisionswith limited information of demand distributionrdquo InternationalJournal of Production Research vol 53 no 2 pp 532ndash548 2015

[6] L Zhu and Y Fan ldquoA real options-based CCS investmentevaluation model case study of Chinarsquos power generationsectorrdquo Applied Energy vol 88 no 12 pp 4320ndash4333 2011

[7] XWang and LDu ldquoStudy on carbon capture and storage (CCS)investment decision-making based on real options for Chinarsquoscoal-fired power plantsrdquo Journal of Cleaner Production vol 112part 5 pp 4123ndash4131 2016

[8] W Jiang and X Chen ldquoOptimal strategies for manufacturerwith strategic customer behavior under carbon emissions-sensitive randomdemandrdquo Industrial Management amp Data Sys-tems 2015

[9] Y Levin J McGill and M Nediak ldquoDynamic pricing in thepresence of strategic consumers and oligopolistic competitionrdquoManagement Science vol 55 no 1 pp 32ndash46 2009

[10] A K Parlakturk ldquoThe value of product variety when sellingto strategic consumersrdquo Manufacturing amp Service OperationsManagement vol 14 no 3 pp 371ndash385 2012

[11] X Su and F Zhang ldquoOn the value of commitment andavailability guarantees when selling to strategic consumersrdquoManagement Science vol 55 no 5 pp 713ndash726 2009

[12] W Elmaghraby S A Lippman C S Tang and R Yin ldquoWillmore purchasing options benefit customersrdquo Production andOperations Management vol 18 no 4 pp 381ndash401 2009

[13] W Jiang and X Chen ldquoManufacturersquos production and pricingstrategies with carbon tax policy and strategic customer behav-iorrdquoManagement Science and Engineering vol 9 no 1 pp 30ndash35 2015

[14] I Dobos ldquoThe effects of emission trading on production andinventories in the Arrow-Karlin modelrdquo International Journalof Production Economics vol 93-94 pp 301ndash308 2005

[15] X Chen C K Chan and Y C E Lee ldquoResponsible productionpolicies with substitution and carbon emissions tradingrdquo Jour-nal of Cleaner Production 2015

Discrete Dynamics in Nature and Society 13

[16] X Chen and X Wang ldquoEffects of carbon emission reductionpolicies on transportation mode selections with stochasticdemandrdquo Transportation Research Part E Logistics and Trans-portation Review 2015

[17] X Chang H Xia H Zhu T Fan and H Zhao ldquoProductiondecisions in a hybrid manufacturingndashremanufacturing systemwith carbon cap and trade mechanismrdquo International Journal ofProduction Economics vol 162 pp 160ndash173 2015

[18] J Zhao B F Hobbs and J-S Pang ldquoLong-run equilibriummodeling of emissions allowance allocation systems in electricpowermarketsrdquoOperations Research vol 58 no 3 pp 529ndash5482010

[19] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2 pp519ndash527 2011

[20] A Toptal H Ozlu and D Konur ldquoJoint decisions on inventoryreplenishment and emission reduction investment under dif-ferent emission regulationsrdquo International Journal of ProductionResearch vol 52 no 1 pp 243ndash269 2014

[21] A S Manikas and J R Kroes ldquoA newsvendor approach tocompliance and production under cap and trade emissionsregulationrdquo International Journal of Production Economics vol159 pp 274ndash284 2015

[22] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineering vol10 no 1 pp 99ndash116 2013

[23] X Xu W Zhang P He and X Xu ldquoProduction and pricingproblems in make-to-order supply chain with cap-and-traderegulationrdquo Omega 2015

[24] Y Zhen X Yang and W Jiang ldquoThe impact of transport modeand carbon policy on low-carbon retailerrdquoDiscrete Dynamics inNature and Society vol 2015 Article ID 964305 12 pages 2015

[25] S Swami and J Shah ldquoChannel coordination in green supplychainmanagementrdquo Journal of the Operational Research Societyvol 64 no 3 pp 336ndash351 2013

[26] R H Coase ldquoDurability andmonopolyrdquoThe Journal of Law andEconomics vol 15 no 1 pp 143ndash149 1972

[27] X Su ldquoIntertemporal pricing with strategic customer behaviorrdquoManagement Science vol 53 no 5 pp 726ndash741 2007

[28] D Zhang and W L Cooper ldquoManaging clearance sales in thepresence of strategic customersrdquo Production and OperationsManagement vol 17 no 4 pp 416ndash431 2008

[29] S Whang ldquoDemand uncertainty and the Bayesian effect inmarkdown pricing with strategic customersrdquoManufacturing ampService Operations Management vol 17 no 1 pp 66ndash77 2015

[30] J Du J Zhang and G Hua ldquoPricing and inventory manage-ment in the presence of strategic customers with risk preferenceand decreasing valuerdquo International Journal of Production Eco-nomics vol 164 pp 160ndash166 2015

[31] X Su and F Zhang ldquoStrategic customer behavior commitmentand supply chain performancerdquo Management Science vol 54no 10 pp 1759ndash1773 2008

[32] H Yang ldquoImpact of discounting and competition on benefit ofdecentralization with strategic customersrdquo Operations ResearchLetters vol 40 no 2 pp 123ndash127 2012

[33] D Yang E Qi and Y Li ldquoQuick response and supply chainstructure with strategic consumersrdquo Omega vol 52 pp 1ndash142015

[34] C drsquoAspremont and A Jacquemin ldquoCooperative and nonco-operative R amp D in duopoly with spilloversrdquo The AmericanEconomic Review vol 78 no 5 pp 1133ndash1137 1988

[35] J F Muth ldquoRational expectations and the theory of pricemovementsrdquo Econometrica vol 29 no 3 pp 315ndash335 1961

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 13

[16] X Chen and X Wang ldquoEffects of carbon emission reductionpolicies on transportation mode selections with stochasticdemandrdquo Transportation Research Part E Logistics and Trans-portation Review 2015

[17] X Chang H Xia H Zhu T Fan and H Zhao ldquoProductiondecisions in a hybrid manufacturingndashremanufacturing systemwith carbon cap and trade mechanismrdquo International Journal ofProduction Economics vol 162 pp 160ndash173 2015

[18] J Zhao B F Hobbs and J-S Pang ldquoLong-run equilibriummodeling of emissions allowance allocation systems in electricpowermarketsrdquoOperations Research vol 58 no 3 pp 529ndash5482010

[19] B Yalabik and R J Fairchild ldquoCustomer regulatory andcompetitive pressure as drivers of environmental innovationrdquoInternational Journal of Production Economics vol 131 no 2 pp519ndash527 2011

[20] A Toptal H Ozlu and D Konur ldquoJoint decisions on inventoryreplenishment and emission reduction investment under dif-ferent emission regulationsrdquo International Journal of ProductionResearch vol 52 no 1 pp 243ndash269 2014

[21] A S Manikas and J R Kroes ldquoA newsvendor approach tocompliance and production under cap and trade emissionsregulationrdquo International Journal of Production Economics vol159 pp 274ndash284 2015

[22] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineering vol10 no 1 pp 99ndash116 2013

[23] X Xu W Zhang P He and X Xu ldquoProduction and pricingproblems in make-to-order supply chain with cap-and-traderegulationrdquo Omega 2015

[24] Y Zhen X Yang and W Jiang ldquoThe impact of transport modeand carbon policy on low-carbon retailerrdquoDiscrete Dynamics inNature and Society vol 2015 Article ID 964305 12 pages 2015

[25] S Swami and J Shah ldquoChannel coordination in green supplychainmanagementrdquo Journal of the Operational Research Societyvol 64 no 3 pp 336ndash351 2013

[26] R H Coase ldquoDurability andmonopolyrdquoThe Journal of Law andEconomics vol 15 no 1 pp 143ndash149 1972

[27] X Su ldquoIntertemporal pricing with strategic customer behaviorrdquoManagement Science vol 53 no 5 pp 726ndash741 2007

[28] D Zhang and W L Cooper ldquoManaging clearance sales in thepresence of strategic customersrdquo Production and OperationsManagement vol 17 no 4 pp 416ndash431 2008

[29] S Whang ldquoDemand uncertainty and the Bayesian effect inmarkdown pricing with strategic customersrdquoManufacturing ampService Operations Management vol 17 no 1 pp 66ndash77 2015

[30] J Du J Zhang and G Hua ldquoPricing and inventory manage-ment in the presence of strategic customers with risk preferenceand decreasing valuerdquo International Journal of Production Eco-nomics vol 164 pp 160ndash166 2015

[31] X Su and F Zhang ldquoStrategic customer behavior commitmentand supply chain performancerdquo Management Science vol 54no 10 pp 1759ndash1773 2008

[32] H Yang ldquoImpact of discounting and competition on benefit ofdecentralization with strategic customersrdquo Operations ResearchLetters vol 40 no 2 pp 123ndash127 2012

[33] D Yang E Qi and Y Li ldquoQuick response and supply chainstructure with strategic consumersrdquo Omega vol 52 pp 1ndash142015

[34] C drsquoAspremont and A Jacquemin ldquoCooperative and nonco-operative R amp D in duopoly with spilloversrdquo The AmericanEconomic Review vol 78 no 5 pp 1133ndash1137 1988

[35] J F Muth ldquoRational expectations and the theory of pricemovementsrdquo Econometrica vol 29 no 3 pp 315ndash335 1961

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of