OptimalPricingStrategiesinaProductandServiceSupply ...of retailer’s fairness concerns on the price...

Research ArticleOptimal Pricing Strategies in a Product and Service SupplyChain with Extended Warranty Service Competition consideringRetailer Fairness Concern

Du Zhao 1 Xumei Zhang 1 Tinghai Ren 2 and Hongyong Fu 3

1School of Economics and Business Administration Chongqing University Chongqing 400044 China2School of Business Guizhou University of Finance and Economics Guiyang 550025 China3China Research Institute of Enterprise Governed By Law Southwest University of Political Science and LawChongqing 401120 China

Correspondence should be addressed to Xumei Zhang xmzhang66126com and Hongyong Fu fuhongyongfoxmailcom

Received 25 March 2019 Revised 10 May 2019 Accepted 20 May 2019 Published 27 August 2019

Academic Editor Emilio Gomez-Deniz

Copyright copy 2019DuZhao et alis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

is paper examines optimal pricing in a two-tier product and service supply chain consisting of a manufacturer and a retailer inthe context of vertical competition in extended warranty in two cases one considering the retailerrsquos fairness concerns and onewithout considering the retailerrsquos fairness concerns A manufacturer-dominated product and service supply chain game-theoreticmodel on the Stackelberg model is developed to analyse how the level of vertical competition in extended warranty service and theintensity of a retailerrsquos fairness concerns influence the optimal pricing of products and extended warranties for the manufacturerand retailer is study finds the following (i) Two parties of the supply chain employ differential pricing strategies for extendedwarranties when the retailer has fairness concerns (ii) Compared to the same pricing strategies for extended warranty servicewhen the retailer has no fairness concerns the increase of competition intensity of vertical extended warranty service will enlargethe price difference of extended warranty service Meanwhile it is the intensity of fairness concerns that determines the influencesof retailerrsquos fairness concerns on the price difference of extended warranties (iii) If no fairness concerns are raised an increase inthe level of vertical competition in extended warranty service would benefit both supply chain parties rather than hurting theirprofit If the retailer is fair-minded its fairness utility increases when the intensity of the fairness concerns rises in a reasonablerange and decreases when the intensity exceeds the reasonable range but for the manufacturer its profits will be damaged as longas the retailer raises fairness concerns

1 Introduction

As market competition intensifies the profits generated bymanufacturers and retailers through traditional productproduction and sales are gradually reduced which forcesthem to actively seek new profit drivers A case in point istheir shift to the innovative business model of offeringproducts and related services at the same time [1] Among avariety of services delivered along with products extendedwarranties have become a new profit driver sought after byboth manufacturers and retailers [2] Unlike the traditionalwarranty service an extended warranty service is a pro-longed paid warranty offered to consumers in addition to

the standard warranty on a product [3] Currently bothmanufacturers and retailers have begun to offer extendedwarranties to obtain more profits For example Germany-based Siemens Japan-based Panasonic and other famoushome appliance manufacturers sell extended warranties toconsumers who purchase their products In China SuningGome and other large home appliance retailers also offerextended warranties as a tied service for their products byleveraging their channel advantages As a result manufac-turers and their downstream retailers in the same productand service supply chain compete fiercely with each othervertically for sales and market share of homogeneous ex-tended warranties sales

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 8657463 15 pageshttpsdoiorg10115520198657463

It is worth noting that competition in the extendedwarranty service market is based on products Howeverthe relationship between a manufacturer and a retailerwithin a supply chain is much more about upstream anddownstream partnership regarding production and salesof specific products is requires ensuring that both themanufacturers and retailers within a product and servicesupply chain obtain the optimal benefits e key tosolving this issue lies in making optimal pricing decisionsfor products and services In optimal pricing decisionsthe dominant manufacturer tends to play a powerful rolein setting prices [4 5] while often leads the retailer in aweak position to raise fairness concerns is inevitablyfurther influences optimal pricing decisions [6] To ad-dress the issue this paper discusses how a retailerrsquosfairness concerns influence the optimal pricing decisionsfor products and services in a product and service supplychain with vertical competition in extended warrantyservice

At present the research on extended warranties has gonebeyond extended warranties offered by a single enterprise[7ndash9] to include multiple supply chain members Jiang andZhang [10] studied the impact of a retailerrsquos extendedwarranties on the basic warranties from a manufacturerHeese [11] examined the warranty strategies of manufac-turers and the extended warranty strategies of retailers in thecontext of two competing manufacturers selling productsthrough a common retailer e literature [10 11] puts itsfocus on the impact of a retailerrsquos extended warranties on amanufacturerrsquos basic warranties instead of on the pricing ofextended warranties on products Li et al [12] compared andanalysed the extended warranties provided separately by amanufacturer and a retailer within a two-tier supply chain interms of the length price and profit of the extended war-ranties Cohen and Whang [13] discussed the optimalpricing of products and extended warranties for manufac-turers and third-party service providers in two cases the onewith the extended warranties provided by manufacturersand the other one with the extended warranties provided bythird-party service providers Li et al and Cohen andWhang[12 13] studied the pricing of extended warranties in asupply chain whose members however do not competewith each other Chen et al [14] examined optimal com-peting and collaborating strategy in a complex supply chainconsisting by two manufacturers who manufacture sub-stitutive products and purchase a key component from theother Chen et al [14] studied the competition but it is notwarranty competition In the supply chain consisting of amanufacturer providing warranties and two competingretailers providing extended warranties Bian et al [15]explored the pricing of extended warranties by the tworetailers against the backdrop of horizontal competitionbetween each other is paper extends the work of Bianet al [15] by introducing vertical competition in extendedwarranty between a manufacturer and its downstream re-tailer in a supply chain It also considers how fairnessconcern of the retailer in the weak position in a product andservice supply chain influences the optimal pricing ofproducts and extended warranties

e paper also covers fairness concerns in supply chainsCamerer and aler [16] proved through the ultimatumgame that people punish others for unfair behaviour evenwhen it is costly for them to do so Ho and Zhang [17]confirmed through experimental research that there isfairness concern behaviour in supply chain contracts In thefield of operational research and management science re-search on supply chain fairness concerns generally focuseson two areas e first one is the impact of fairness concernson traditional supply chain contract design Cui et al [6] areamong the first to use the mathematical model to studyfairness concerns in a two-tier supply chain ey find thatwhen both the manufacturer and the retailer have advan-tageous and disadvantageous inequity aversions simulta-neously a simple wholesale price contract can make supplychain coordinated under certain conditions Caliskan-Demirag et al [18] extended the work of Cui et al [6] to anonlinear demand function and found that if both themanufacturer and the retailer have advantageous and dis-advantageous inequity aversions simultaneously in a non-linear demand setting the supply chain members could alsoachieve coordination In addition the exponential demandfunction requires less stringent conditions than the lineardemand to achieve supply chain coordination Wu andNiederhoff [19] extended the analysis of supply chain fair-ness concerns to the random-demand newsvendor model inwhich they studied the impact of fairness concerns on supplychain coordinatione second attractive area of research onsupply chain fairness concerns is the impact of fairnessconcerns on optimal pricing decisions in a supply chain Maet al [20] studied the impact of a retailerrsquos fairness concernson recovery rates sales efforts and profits in a closed-loopsupply chain environment where the product demand issensitive to the retailerrsquos sales effort In the low-carbonsupply chain setting Zhou et al [21] examined the impact ofa retailerrsquos fairness concerns on the supply chain pricingstrategies under both cooperative advertising contracts andcooperative advertising and emission reduction cost-sharingcontracts Ma et al and Zhou et al [20 21] studied theimpact of fairness concerns in supply chains on optimalpricing of products but not of services Liu et al [22] in-vestigated the optimal allocation strategy of logistics serviceorders in a service supply chain composed of a logisticsservice integrator and several logistics service providers(LSPs) based on the LSPsrsquo fairness concern preferences anddemand updating Du and Han [23] analysed optimalpricing strategies under the impact of fairness concerns inthe logistics service supply chain in which the market de-mand is affected by the price and quality defect guarantee oflogistics services at the same time Liu et al and Du and Han[22 23] studied the impact of fairness concerns on theoptimal pricing of services but not of products Wang et al[24] considered a mobile phone supply chain (MPSC)consisting of a service operator and a handset manufacturerand analysed the influence of the nature and level of both thehandset manufacturerrsquos fairness concerns and the operatorrsquosfairness concerns on the pricing decision Li and Li [25]analysed the impacts of a retailerrsquos fairness concerns on thepricing of products and value-added services as well as on

2 Mathematical Problems in Engineering

channel conflict in a dual-channel supply chain in which amanufacturer sells products through a direct channel andalso through a traditional retail channel in which themanufacturerrsquos downstream retailer has fairness concernsand provides value-added services to consumers Wang et al[24] and Li and Li [25] studied the impact of fairnessconcerns on the pricing decisions for both products andservices but without considering service competition isstudy extends the current literature by investigating theoptimal pricing of products and services at the same time ina product and service supply chain with vertical competitionin extended warranty service

Drawing on the extant literatures this paper considersretailersrsquo fairness concern preference and the intensity ofvertical competition in extended warranty service betweenretailers and manufacturers when studying the optimalpricing decisions for products and services in a product andservice supply chain composed of a manufacturer and re-tailer We intend to answer the following questions

What are the optimal pricing decisions for products andservices in a product and service supply chain when theretailer does not have fairness concerns

What are the optimal pricing decisions for products andservices in a product and service supply chain when theretailer has fairness concerns compared with no fairnessconcerns considered

How do the level of the retailerrsquos fairness concerns andthe level of vertical competition in extended warranty serviceaffect the optimal pricing of products and services in aproduct and service supply chain

e rest of this study is organized as follows In Section 2the assumptions and questions related to this paper aredescribed In Section 3 the model used in this paper is solvedin two situations when the retailer has fairness concerns andwhen the retailer has no fairness concerns e optimalpricing strategies for products and services in a product andservice supply chain in both situations are analysed Section4 verifies the main conclusions of the paper using numericalexamples Conclusions and future work are presented in thelast section

2 Model Assumptions

We consider a two-tier product and service supply chainconsisting of a manufacturer and a retailer In the supplyconsumers purchase products from the retailer to satisfytheir needs the retailer orders products from the manu-facturer based on the consumersrsquo needs and the manu-facturer strives to manufacture products to meet theretailerrsquos order requirements is creates a supply-demandflow of products Beyond that the supply chain also has asupply-demand flow of extended warranties on the productsTo better meet consumer demand and improve user expe-rience both the manufacturer and retailer can providehomogeneous extended warranties is means that there isvertical competition in an extended warranty service be-tween the manufacturer and its downstream retailer in thesupply chain

In Figure 1 w and psm are the product wholesale priceand extended warranty service price determined by themanufacturer respectively pp and psr are the product retailprice and extended warranty service price determined by themanufacturer respectively

We assume that the market demand for a specificproduct from the manufacturer and the retailer isDp a minus pp where a (agt 0) is the market size is additivedemand function has been widely used by Chen et al [26]and in other literature Since consumers decide whether topurchase an extended warranty service only after purchasinga product their demand for extended warranties is surelynot higher than their demand for products Without loss ofgenerality the demand for a specific extended warrantyservice from the retailer and the manufacturer can beexpressed as follows

Dsr a minus pp1113872 1113873 minus psr + βpsm

Dsm a minus pp1113872 1113873 minus psm + βpsr(1)

where β (0lt βlt 1) is the level of vertical competition inextended warranty service between the manufacturer andretailer

is paper focuses on analysing how the level of verticalcompetition in extended warranty service and a retailerrsquosfairness concern preference affect the optimal pricing ofproducts and services of the retailer and manufacturer Forease of exposition the manufacturerrsquos cost of productionand the retailerrsquos cost of sales can be set to zero [27] Inaddition the service costs of the manufacturer and retailerdepend primarily on their respective service capabilitieswhich usually remain unchanged within a certain period oftime so the service costs of both parties can also be set tozero [27]

From the above description and assumptions themanufacturerrsquos profit function can be expressed asfollows

πm w a minus pp1113872 1113873 + a minus pp minus psm + βpsr1113872 1113873psm (2)

e retailerrsquos profit function given as follows

πr pp minus w1113872 1113873 a minus pp1113872 1113873 + a minus pp minus psr + βpsm1113872 1113873psr (3)

As discussed in the work of Wu and Niederhoff [19] onfairness concerns if the retailer has fairness concerns itdetermines its optimal product retail price and extendedwarranty service price with the aim of maximizing itsutility Ur and the manufacturer determines its optimalproduct wholesale price and extended warranty serviceprice with the aim of maximizing its own profit Whenthe retailer is fair-minded its utility function isUr πF

r minus λ0πFm where λ0 λ1 + λ λ (λgt 0) is the in-

tensity of the retailerrsquos fairness concerns en the utilityfunction of the fair-minded retailer can be specified asfollows

UFr pp minus w1113872 1113873 a minus pp1113872 1113873 + a minus pp minus psr + βpsm1113872 1113873psr

minus λ0 w a minus pp1113872 1113873 + a minus pp minus psm + βpsr1113872 1113873psm1113960 1113961(4)

Mathematical Problems in Engineering 3

3 Model Solution and Analysis

31 Optimal Decisions of Manufacturer and Retailer withoutFairness Concerns In the partnership between the manu-facturer and the retailer the manufacturer decides thewholesale price w of a specific product the retailer decidesthe product sales price pp and then the manufacturer andretailer decide their extended warranty service price psm andpsr respectively e inverse solution method which hasbeen widely used by Zhang et al [28] and Fu et al [29] isadopted e optimal profit functions of the manufacturerand retailer are given as follows

maxpsm

πm psm( 1113857 w a minus pp1113872 1113873 + a minus pp minus psm + βpsr1113872 1113873psm

(5)

maxpsr

πr psr( 1113857 pp minus w1113872 1113873 a minus pp1113872 1113873

+ a minus pp minus psr + βpsm1113872 1113873psr(6)

e second-order derivative of equation (5) with respectto psm and the second-order derivative of equation (6) withrespect to psr are

d2πm psm( 1113857

dp2sm

minus 2

d2πr psr( 1113857

dp2sr

minus 2

(7)

It can be seen that both πm(psm) with respect to psm andπr(psr) with respect to psr meet the first-order optimalitycondition Hence we have

dπm psm( 1113857

dpsm minus 2psm + a minus pp + βpsr (8)

dπr psr( 1113857

dpsr minus 2psr + a minus pp + βpsm (9)

Based on equations (8) and (9) the optimal extendedwarranty service price for the manufacturer and the retaileris

plowastsm

a minus pp

2 minus β

plowastsr

a minus pp

2 minus β

(10)

Substituting plowastsm and plowastsr into the profit function of theretailer we can have

maxpp

πr pp1113872 1113873 a minus pp1113872 1113873 pp minus w1113872 1113873β2 minus 4 pp minus w1113872 1113873β + a minus 4w + 3pp1113960 1113961

(β minus 2)2

(11)

e second-order derivative of equation (11) with re-spect to pp is

d2πr pp1113872 1113873

dp2p

minus2 β2 minus 4β + 31113872 1113873

(β minus 2)2 (12)

Since 0lt βlt 1 it follows that β2 minus 4β + 3gt 0 en it iseasy to prove d2πr(pp)dp2

p lt 0 us πr(pp) with respect topp meets the following first-order optimality condition

dπr pp1113872 1113873

dpp

a + w minus 2pp1113872 1113873β2 minus 4 a + w minus 2pp1113872 1113873β + 2a + 4w minus 6pp

(β minus 2)2

(13)

Solving equation (13) we can get the retailerrsquos optimalproduct retail price as follows

plowastp

(a + w)β2 minus 4(a + w)β + 2a + 4w

2 β2 minus 4β + 31113872 1113873 (14)

Likewise substituting plowastsm plowastsr and plowastp into the profitfunction of the manufacturer we can have

maxw

πm(w) 2wβ2 minus 8wβ + a + 5w1113872 1113873(β minus 2)2(a minus w)

4 β2 minus 4β + 31113872 11138732

(15)

e second-order derivative of equation (15) with re-spect to w is

d2πm(w)

dw2 minus(β minus 2)2 β2 minus 4β + 521113872 1113873

β2 minus 4β + 31113872 11138732 (16)

Since 0lt βlt 1 it is easy to prove when 0lt βlt2 minus (

6

radic2) the second-order condition d2πm(w)dw2 lt 0

In this case based on the first-order optimality condition wecan get the manufacturerrsquos optimal product wholesale pricewlowast (aβ2 minus 4aβ + 2a)(2β2 minus 8β + 5) Substituting wlowast into

Manufacturer

Productsrsquowholesale

price

Productsrsquoretailprice

w

Retailer

pp psr

psmW

arra

nty

pric

e

War

rant

y pr

ice

Customers

Figure 1 Schematic diagram of a product and service supply chain


the expressions corresponding to plowastp plowastsm and plowastsr we canobtain wlowast a(β2 minus 4β + 2)(2β2 minus 8β + 5) plowastp 3a(β2minus4β + 2)2(2β2 minus 8β + 5) and plowastsm plowastsr a(2 minus β)2(2β2minus8β + 5) From the above analysis we propose the followingproposition

Proposition 1 Without fairness concerns both the manu-facturer and the retailer can arrive at optimal pricing de-cisions e optimal wholesale price and extended warrantyservice price determined by the manufacturer are

wlowast

a β2 minus 4β + 21113872 1113873

2β2 minus 8β + 5

plowastsm

a(2 minus β)

2 2β2 minus 8β + 51113872 1113873

(17)

e optimal retail price and extended warranty serviceprice determined by the retailer are

plowastp

3a β2 minus 4β + 21113872 1113873

2 2β2 minus 8β + 51113872 1113873

plowastsr

a(2 minus β)

2 2β2 minus 8β + 51113872 1113873

(18)

Proposition 1 shows that when the retailer has nofairness concerns both the manufacturer and the retailer canmake their optimal pricing decisions and both parties adoptthe same extended warranty service pricing strategy ismeans that the vertical competition in extended warrantyservice between the two parties without fairness concernsdoes not result in differential pricing of extended warrantiesBased on the analysis of Proposition 1 this paper exploreshow the potential market size of a product and the level ofcompetition in extended warranty service on the productaffect optimal pricing decisions of both the manufacturerand the retailer e first-order partial derivatives of w pppsm and psr with respect to a and β are calculated and thefollowing inference is obtained

Inference 1 If the retailer has no fairness concern themanufacturerrsquos product wholesale price w and extendedwarranty service price psm and the retailerrsquos product retailprice pp and extended warranty service price psr exhibit thefollowing relationships with product market size a and thelevel β of competition in extended warranty service

(i) zwzagt 0 zpsmzagt 0 zwzβlt 0 zpsmzβgt 0(ii) zppzagt 0 zpsrzagt 0 zppzβlt 0 zpsrzβgt 0

Inference 1 indicates that an increase in product marketsize leads both the manufacturer and the retailer choose toincrease the price of their products and extended warrantiesAs the competition in extended warranty service betweenthe two parties intensifies they tend to lower the productprice as optimal decisions is price reduction however isoften accompanied by an increase in the extended warrantyservice price meaning that stiffer service competition

without fairness concerns contributes little to increasingconsumer benefits

Proposition 2 If the retailer has no fairness concerns therelationship between the profits of the manufacturer and theretailer and the level of competition in extended warrantyservice satisfies the following conditions zπmzβgt 0zπrzβgt 0

Proof Substituting wlowast plowastp plowastsm and plowastsr into equation (5) we

can have

πm a2 β2 minus 4β + 41113872 1113873

4 2β2 minus 8β + 51113872 1113873 (19)

Since 0lt βlt 1 β2 minus 4β + 4gt 0 is always true To ensurethat the manufacturerrsquos profit is not less than its retainedprofit namely πm ge 0 the constraint condition 2β2minus8β + 5gt 0 equivalent to 0lt βlt 2 minus (

6

radic2) must be mete

partial derivative of equation (19) with respect to β iszπm

zβ

3a2(2 minus β)

2 2β2 minus 8β + 51113872 11138732 (20)

On the premise of satisfying constraint condition0lt βlt 2 minus (

6

radic2) zπmzβgt 0 can be obtained Similarly

substituting wlowast plowastp plowastsm and plowastsr into equation (6) we canhave

πr a2(1 minus β)(3 minus β)(β minus 2)2

4 2β2 minus 8β + 51113872 11138732 (21)

e partial derivative of equation (21) with respect to β is

zπr

zβ

a2(2 minus β) 4β2 minus 16β + 131113872 1113873

2 2β2 minus 8β + 51113872 11138733 (22)


6

radic2) it is easy to get 2β2 minus 8β + 5gt 0

(2 minus β)(4β2 minus 16β + 13)gt 0 It follows that zπrzβgt 0erefore Proposition 2 is proven

From Inference 1 and Proposition 2 it is clear thatincreasing vertical competition in an extended warrantyservice between a manufacturer and a retailer which has nofairness concerns leads the manufacturer to lower itsproduct wholesale price and raise its extended warrantyservice price as the optimal pricing strategy e same is trueof the retailere reduction by the manufacturer in productwholesale prices can motivate the retailer to order moreproducts from the manufacturer and a lower retail pricemeans a higher demand for a specific product therebyincreasing the sales revenue of the two parties On the otherhand as vertical competition in the extended warrantyservice intensifies the optimal decision of both the manu-facturer and the retailer is to increase the service price withthe aim of increasing the revenue from the service From theabove analysis we can reach a counterintuitive conclusiontougher competition in an extended warranty service in-creases the benefits of both the manufacturer and the


retailer instead of harming their interests However thisinevitably damages the benefits of end users

32 Decisions and Profits of Supply Chain Members with theRetailer Having Fairness Concerns If a retailer is sensitive tofairness it determines its retail price and extended warrantyservice price of a product with the goal of maximizing itsutility Ur and the upstreammanufacturer still determines itswholesale price and extended warranty service price of theproduct with the aim of maximizing its own profiterefore despite the retailerrsquos fairness concern the profitfunction for the manufacturer remains the same as equation(5) and can be rewritten as follows

maxpsm

πFm psm( 1113857 w a minus pp1113872 1113873 + a minus pp minus psm + βpsr1113872 1113873psm

(23)

As discussed in [19 23] the utility function of the fair-minded retailer is

Ur πFr minus λ0π

Fm (24)

where λ0 λ(1 + λ) (λ denotes the retailerrsquos fairnesspreference) Referring to equations (24) and (6) we can getthe following utility function of the fair-minded retailer

maxpsr

UFr psr( 1113857 pp minus w1113872 1113873 a minus pp1113872 1113873 + a minus pp minus psr + βpsm1113872 1113873psr

minus λ0 w a minus pp1113872 1113873 + a minus pp minus psm + βpsr1113872 1113873psm1113960 1113961

(25)

Following the same logic of analysing optimal pricingdecisions without fairness concerns calculate the first-orderand second-order derivatives of equation (23) with respectto psm and calculate the first-order and second-order de-rivatives of equation (25) with respect to psr Combining thetwo first-order optimality conditions we can derive theoptimal extended warranty service price

pFlowastsr

2 minus λ0 minus 1( 1113857β1113858 1113859 a minus pp1113872 1113873

4 + λ0 minus 1( 1113857β2

pFlowastsm

(2 + β) a minus pp1113872 1113873

4 + λ0 minus 1( 1113857β2

(26)

Substituting pFlowastsr and pFlowast

sm into equation (25) we can have

maxpp

UFr pp1113872 1113873 pp minus w1113872 1113873 a minus pp1113872 1113873 + a minus pp minus p

Flowastsr + βp

Flowastsm1113872 1113873p

Flowastsr

minus λ0 w a minus pp1113872 1113873 + a minus pp minus pFlowastsm + βp

Flowastsr1113872 1113873p

Flowastsm1113960 1113961

(27)

Following the same logic of solving plowastp to determine theretailerrsquos optimal retail price for the objective function (27)we can have

pFlowastp

a + λ0w + w( 1113857 λ0 minus 1( 11138572β4 + 2aλ0 λ0 minus 1( 1113857β3 + (2a + 8w)λ20 + 6aλ0 minus 10a minus 8w1113960 1113961β2 + 8a λ0 minus 1( 1113857β + 8 λ0 + 1( 1113857(a + 2w)

2 λ0 minus 1( 11138572β4 + 2λ0 λ0 minus 1( 1113857β3 + 2 λ20 + 7λ0 + 41113872 1113873β2 + 8 λ0 minus 1( 1113857β + 8λ0 + 24

(28)

Substituting pFlowastsr pFlowast

sm and pFlowastp into equation (23) we can

have

maxw

πFm(w) w a minus pFlowastp1113872 1113873 + a minus p

Flowastp minus p

Flowastsm + βp


Flowastsm

(29)

Following the same logic of solving wlowast to determine themanufacturerrsquos optimal wholesale price for the objectivefunction (29) we can have

wFlowast

a λ20 minus 2λ0 + 11113872 1113873β4 + λ20 minus λ01113872 1113873β3 + λ20 + 6λ0 minus 101113872 1113873β2 minus 8β + 81113960 1113961

λ0 + 1( 1113857 2λ20 minus 4λ0 + 21113872 1113873β4 + 2λ20 minus 2λ01113872 1113873β3 + 2λ20 + 13λ0 minus 191113872 1113873β2 + 4λ0 minus 12( 1113857β + 4λ0 + 201113960 1113961 (30)


Substituting wFlowast into the equations corresponding topFlowastsr pFlowast

sm and pFlowastp we can obtain the retailerrsquos and manu-

facturerrsquos optimal prices

pFlowastp

a 3 λ0 minus 1( 11138572β4 + 4λ20 minus 4λ01113872 1113873β3 + 4λ20 + 18λ0 minus 301113872 1113873β2 + 8λ0 minus 24( 1113857β + 8λ0 + 241113960 1113961

4 λ0 minus 1( 11138572β4 + 4λ20 minus 4λ01113872 1113873β3 + 4λ20 + 26λ0 minus 381113872 1113873β2 + 8λ0 minus 24( 1113857β + 8λ0 + 40

pFlowastsm

a(β + 2) 4 + β2 λ0 minus 1( 11138571113960 1113961


pFlowastsr

a 2 minus β λ0 minus 1( 11138571113858 1113859 4 + β2 λ0 minus 1( 11138571113960 1113961


(31)

Based on the above analysis and the analysis of Prop-osition 1 we propose the following proposition

Proposition 3 If the retailer is fair-minded both themanufacturer and the retailer can arrive at optimal pricingdecisions e optimal wholesale price and extended war-ranty service price determined by the manufacturer are

wFlowast


λ0 + 1( 1113857 2λ20 minus 4λ0 + 21113872 1113873β4 + 2λ20 minus 2λ01113872 1113873β3 + 2λ20 + 13λ0 minus 191113872 1113873β2 + 4λ0 minus 12( 1113857β + 4λ0 + 20

pFlowastsm

a(β + 2) 4 + β2 λ0 minus 1( 11138571113960 1113961


(32)

e optimal product retail price and extended warrantyservice price determined by the retailer are

pFlowastp



pFlowastsr



(33)

Proposition 3 shows that when the retailer has fairnessconcerns both the manufacturer and the retailer can makeoptimal pricing decisions and they adopt different extendedwarranty service pricing strategies is indicates that thefairness concerns of the retailer which experiences verticalcompetition in an extended warranty service with themanufacturer lead to differential pricing of the service

Based on the analysis of Proposition 3 this paper in-vestigates how the potential market size of a product thelevel of competition in extended warranty service on theproduct and the retailerrsquos fairness preference influence theoptimal pricing decisions of both the manufacturer and theretailer e first-order partial derivatives of wF pF

p pFsm and

pFsr with respect to a β and λ are calculated and the fol-

lowing inference is obtained

Inference 2 If the retailer has fairness concerns the man-ufacturerrsquos optimal product wholesale price and extendedwarranty service price and the retailerrsquos optimal productretail price and extended warranty service price exhibit thefollowing relationships with product market size a and thelevel β of competition in extended warranty service

(i) zwFagt 0 zwFβlt 0 zwFλlt 0(ii) zpF

pagt 0 zpFpβlt 0 zpF

pλgt 0(iii) zpF

sragt 0 zpFsrβgt 0 zpF

srλlt 0(iv) zpF

smagt 0 zpFsmβgt 0 zpF

smλlt 0

Proof

(i) e partial derivatives of wF with respect to productmarket size a and the level β of competition in


extended warranty service are obtained It is easy toprove zwFagt 0 zwFβlt 0 To determine the sign ofzwFλ the partial derivative of wF with respect to λ0 iscalculated as follows

zwF

λ0

(β + 2)al

m2 (34)

where l 2(λ0 minus 1)2β4 + (9λ20 minus 17λ0 + 8)β3 + 6(λ20minusλ0)β

2 +(4λ20 + 24λ0 minus 32)β minus 32 and m 2(λ0 minus 1)2

β4 +2(λ20 minus λ0)β3 + (2λ20 + 13λ0 minus 19)β2 + 2(2λ0 minus 6)

β + 20 Since 0lt βlt 1 it is easy to prove llt 0 HencezwFλ0 (β + 2)alm2 lt 0 en because λ0 λ(λ+1) it follows that zwFλlt 0

(ii) Solving the partial derivative of retail price pFp with

respect to product market size a we can havezpF

pagt 0 Solving the partial derivative of retail pricepFp with respect to the level β of competition in

extended warranty service we can have

zpFp

β minus

ja

4k2 (35)

where k (λ0 minus 1)2β4 + (λ20 minus λ0)β3 + (λ20 + (132)λ0 minus

(192))β2 + (2λ0 minus 6)β + 2λ0 + 10 and j ((λ0 minus 1)β2+ 4)(λ0(λ0 minus 1)2β4 + (2λ30 minus 5λ20 + 3)β3 + (minus 6λ20 minus 12λ0+ 18)β2 + (minus 2λ0 + 36)β minus 8λ0 + 24) Since 0lt βlt 1λ0 λ(λ + 1) it is easy to prove jgt 0 HencezpF

pβ minus ja4k2 lt 0 To determine the sign of zpFpλ

the partial derivative of pFp with respect to λ0 is cal-

culated as follows

zpFp

λ0 minus

ϕ(β + 2)a

4φ2 (36)

where φ (λ0 minus 1)2β4 + (λ20 minus λ0)β3 + (λ20 + (132)

λ0 minus (192))β2 + (2λ0 minus 1)β + 2λ0 + 10 and ϕ

(λ0minus 1)β4 minus (32λ0 + 52)β3 minus (3λ0 + 1)β2 minus 2β minus 4Since 0lt βlt 1 and λ0 λ(λ + 1) it is easy to proveϕlt 0 Hence zpF

pλgt 0(iii) Solving the partial derivative of the retailerrsquos optimal

extended warranty service price pFsr with respect to

product market size a we can have zpFsragt 0

Following the proof logic in (ii) we can derivezpF

srβgt 0 zpFsrλlt 0

(iv) Likewise we can derive zpFsmagt 0 zpF

smβgt 0zpF

smλlt 0

e results from the analysis of Inference 1 and In-ference 2 show that when the retailer has fairness concernsmarket size and the level of vertical competition in ex-tended warranty service influence the optimal pricingdecisions of the manufacturer in a way similar to what theydo for the retailer is indicates that the retailerrsquos fairness

concerns do not change the direction in which market sizeand vertical competition in extended warranty serviceinfluence optimal pricing decisions in the product andservice supply chain Further analysis reveals that in theface of increasing intensity of the retailerrsquos fairness con-cerns the manufacturer can offset the concerns and in-directly receive compensation by lowering its productwholesale price and the retailer can improve its profits byraising the retail price In addition both the manufacturerand the retailer can decrease the price of extended war-ranties to attract consumers to purchase the servicethereby increasing their revenue

e findings from the analysis of Proposition 3 and In-ference 2 suggest that the retailerrsquos fairness concerns can leadto differential pricing of extended warranties though they donot change the direction in which market size and verticalcompetition in extended warranty service affect the optimalpricing decisions of the retailer and manufacturer It can beseen that the manufacturer and retailer may determine dif-ferent extended warranty service price if the retailer is fair-minded en how does the retailerrsquos fairness concern affectthis price difference And is this price difference connectedwith the level of vertical competition in extended warrantyservice in the context of the retailerrsquos fairness concerns efollowing proposition will answer these questions

Proposition 4

(i) When the retailer has fairness concerns the dif-ference in extended warranty service price betweenthe retailer and manufacturer meets pFlowast

sm gtpFlowastsr

(ii) e relationship between the difference pFlowastsm minus pFlowast

srin extended warranty service price and the level β ofcompetition in extended warranty service meetsz(pFlowast

sm minus pFlowastsr )zβgt 0

(iii) e relationship between the difference pFlowastsm minus pFlowast

srin extended warranty service price and the intensityof the retailerrsquos fairness concerns is as follows if theintensity of the fairness concerns meets 0lt λlt (1 minus τ)τ then z(pFlowast

sm minus pFlowastsr )zλlt 0 if

the concern intensity meets λgt (1 minus τ)τ thenz(pFlowast

smminus pFlowastsr )zλgt 0 where τ (2β3 minus 8β2 + 2

minus 2β4 + 8β3 minus 3β2 minus 8β + 51113969

+ 5β)(β + 2)(2β3 minus 4β2 + β + 2)β

Proof

(i) Referring to the expressions of pFlowastsm and pFlowast

sr inProposition 3 which share the same denominatorwe can tell if pFlowast

sm is larger or smaller than pFlowastsr only by

comparing (β + 2) against [2 minus β(λ0 minus 1)] Since0lt βlt 1 pFlowast

sm gtpFlowastsr is proven

(ii) In the discussion on the relationship between thedifference in extended warranty service price and thelevel of competition in extended warranty servicethe price difference is expressed as follows


pFlowastsm minus p

Flowastsr

aλ0β 4 + β2 λ0 minus 1( 11138571113960 1113961


(37)

e partial derivative of pFlowastsm minus pFlowast

sr with respect to β is

z pFlowastsm minus pFlowast

sr( 1113857

zβ

minus aλ0c4η2

(38)

where c (λ0 minus 1)3β6 minus (λ30 minus (132)λ20 + 8λ0 minus (52))β4 + (4λ20 + 8λ0 minus 12)β3 minus (2λ20 minus 2λ0 + 8)β2 minus 8λ0minus 40 and η (β4 + β3 + β2)λ20 minus (2β4 +β3 minus (132)β2minus 2β minus 2)λ0 + β4 minus (192)β2 minus 6β + 10 Since 0lt βlt 1 and 0lt λ0 lt 1 we have clt 0 Hencez(pFlowast

sm minus pFlowastsr )zβ minus aλ0c4η2 gt 0

(iii) e partial derivative of pFlowastsm minus pFlowast

sr pFlowastsm minus pFlowast

sr withrespect to λ0 is


sr( 1113857

zλ0 minus

aβ(β + 2)]4η2

(39)

where ] (λ0 minus 1)2β5 minus 2(λ0 minus 1)β4 + (12λ20 + 11λ0 minus (19

2))β3 + (λ20 minus 10λ0 + 13)β2 + 22β minus 20 Let τ (2β3 minus 8β2 +

2


+ 5β)(β + 2)(2β3 minus 4β2 + β+

2)β If 0lt λ0 lt τ namely 0lt λlt (1 minus τ)τ we have vgt 0Hence z(pFlowast

sm minus pFlowastsr )zλ minus aβ(β + 2)]4η2 lt 0 If τ lt λ0 lt 1

namely 0lt (1 minus τ)τ lt λ we have vlt 0 Hencez(pFlowast

sm minus pFlowastsr )zλ minus aβ(β + 2)]4η2 gt 0 erefore Propo-

sition 4(iii) is provenProposition 4(i) shows that when the retailer has fairness

concerns the manufacturer tends to develop a differentialpricing strategy in which it offers a specific extended war-ranty service at a price above that of the retailer therebyshifting some of the consumer demand for the service to theretailer to compensate the retailer Proposition 4(ii) impliesthat generally the higher the level of vertical competition inan extended warranty service between the manufacturer andretailer the greater the price difference in the extendedwarranty service provided by the two parties is is mainlybecause increasing competition between the two partiesmeans a greater negotiation power of the retailer whichforces the manufacturer to surrender part of its profits to theretailer by increasing its own extended warranty serviceprice From Proposition 4(iii) it can be seen that the in-tensity of the retailerrsquos fairness concerns directly affects the

differential pricing strategy for an extended warranty servicesubjected to vertical competition with the manufacturer Ifthe intensity of the retailerrsquos fairness concerns is less than acertain threshold (0lt λlt (1 minus τ)τ) they will not drawconsiderable attention from the manufacturer Instead anincrease in the intensity of the fairness concerns will narrowthe price gap in the extended warranty service between themanufacturer and retaileris means that a higher intensityof the retailerrsquos fairness concerns will reduce its own revenuefrom the extended warranty service If the intensity of re-tailerrsquos fairness concerns is greater than the threshold(λgt (1 minus τ)τ gt 0) increasing intensity of the retailerrsquosfairness concerns implies a greater difference in extendedwarranty service price between the manufacturer and re-tailer and the greater the price difference the more likely theretailer is to improve its revenue from the extended warrantyservice

Proposition 5 When the retailer has fairness concerns

(i) e relationship between the manufacturerrsquos profitfunction and the level of vertical competition inextended warranty service as well as the intensity ofthe retailerrsquos fairness concerns meets zπF

mzβgt 0zπF

mzλlt 0(ii) e relationship between the retailerrsquos utility func-

tion and the level of vertical competition in extendedwarranty service as well as the intensity of the re-tailerrsquos fairness concerns is as follows if the intensityof the fairness concernsmeets 0lt λltω(1 minus ω) thenzUF

r zβlt 0 zUFr zλgt 0 if the concern intensity

meets ω(1 minus ω)lt λ then zUFr zβgt 0 zUF

r zλlt 0where ω (2β4 + β3 minus (132)β2 minus 2β minus 2) +

Δ

radic2

(β4 +β3 + β2) and Δ 9β6 + 41β5 + 2094β4 + 6β3 minus

10β2 +8β + 4

Proof

(i) Substituting optimal product prices wFlowast and pFlowastsr and

optimal extended warranty service prices pFlowastsm and pFlowast

pinto the manufacturerrsquos profit function (23) we canhave

πFm a2 β4λ20 + 8β2 minus 2β41113872 1113873λ0 + β4 minus 8β2 + 161113960 1113961

4 2 β4 + β3 + β21113872 1113873λ20 + 13β2 minus 4β4 minus 2β31113872 1113873λ0 + 2β4 minus 19β2 + 201113960 1113961 1 + λ0( 1113857 (40)


e first-order partial derivative of equation (40) withrespect to the level β of vertical competition in extendedwarranty service is

zπFmzβ

a2 4 + λ0 minus 1( 1113857β21113960 1113961

8 1 + λ0( 1113857η2Υ (41)

where Υ [(β4 + 2β3)λ30 minus (2β4 + 5β3 + 6β2)λ20 + (β4minus 12β2 minus 20β minus 8)λ0 + 3β3 + 18β2 + 36β + 24] Since 0ltβlt 1 0lt λ0 lt 1 it is easy to prove 0lt (1 + λ0)0lt 4 + (λ0 minus 1)β2 Υgt 0 erefore zπF

mzβgt 0 isprovenLikewise the first-order partial derivative of equation(40) with respect to λ0 is

zπFm

zλ0 minus

a2 λ0 minus 1( 1113857β2 + 41113960 1113961ψ

8 λ0 + 1( 11138572η2

(42)

where ψ (β6 + β5 + β4)λ30 minus (3β6 + 3β5 minus 9β4 minus 12β3 minus 12β2)λ20 + (3β6 + β5 minus 20β4 + 44β2 + 16β + 16)

λ0 minus β6 + β5 + 18β4 + 12β3 minus 44β2 minus 16β + 48 Since0lt βlt 1 0lt λ0 lt 1 it is easy to prove that 0ltψ isalways true Hence zπF

mzλ0 lt 0 en sinceλ0 λ(λ + 1) zπF

mzλlt 0 is proven(ii) Substituting wFlowast pFlowast

sr pFlowastsm and pFlowast

p into the retailerrsquosutility function (25) we can have

UFr

a2 λ0 minus 1( 1113857β2 + 41113960 11139612

λ0 minus 1( 11138572β4 + λ20 minus λ01113872 1113873β3 + λ20 + 7λ0 minus 91113872 1113873β2 + 4λ0 minus 4( 1113857β + 4λ0 + 121113960 1113961

16η2 (43)

where η (β4 + β3 + β2)λ20 minus (2β4 + β3 minus (132)β2 minus 2β minus 2)

λ0 + β4 minus (192)β2 minus 6β + 10To analyse the relationship between the retailerrsquos utility

function and the intensity of its fairness concerns the first-order partial derivative of equation (43) with respect to λ0 iscalculated as follows

zUFr

zλ0

a2(β + 2) λ0 minus 1( 1113857β2 + 41113960 1113961σ16η3

(44)

where σ (β8 minus β7 minus 5β6 minus 6β5 minus 4β4)λ30 minus (3β8 + 2β7minus (172)β6 + (252)β5 + 37β4 + 30β3 + 20β2)λ20 + (3β8 + 7β8minus 8β6 minus 15β5 + 4β4 minus 46β3 minus 76β2 minus 24β minus 16)λ0 minus β8 minus 4β7 +

(92) β6 + (152)β5 + 37β4 minus 56β2 minus 24β minus 16 Since 0lt βlt1 0lt λ0 lt 1 it is easy to prove that σ lt 0 is always true Itfollows that the numerator in equation (44) is negative Todetermine the sign of zUF

r zλ0 only the sign of η needs to beconsidered Since the root on the left side of equation η 0 is

2β4 + β3 minus (132)β2 minus 2β minus 21113872 1113873 minusΔ

radic

2 β4 + β3 + β21113872 1113873lt 0 (45)

where Δ 9β6 + 41β5 + (2094)β4 + 6β3 minus 10β2 + 8β +4gt 0we assume that the root on the right side isω (2β4 + β3 minus (132)β2 minus 2β minus 2) +

Δ

radic2(β4 + β3 + β2) It

follows that if 0lt λ0 ltω then ηlt 0 It is easy to provezUF

r zλ0 gt 0 Furthermore since λ0 λ(λ + 1) equivalentto zUF

r zλgt 0 we have 0lt λltω(1 minus ω) Likewise whenωlt λ0 equivalent to ω(1 minus ω)lt λ we have zUF

r zλlt 0Following the logic of analysing the relationship between

the retailerrsquos utility function and the intensity λ of its fairnessconcerns we can determine the relationship between theretailerrsquos utility function and the level β of vertical com-petition as follows when 0lt λltω(1 minus ω) we havezUF

r zβlt 0 when ω(1 minus ω)lt λ we have zUFr zβgt 0

Proposition 5(i) shows that when the retailer has fairnessconcerns the manufacturerrsquos profit increases with the level ofvertical competition in extended warranty service Moreovergiven Proposition 2 it can be seen that the retailerrsquos fairnessconcerns do not change the way that the competition levelinfluences themanufacturerrsquos profit On the other hand as theretailerrsquos fairness concern intensifies the profits obtained bythe manufacturer decrease Considering Inference 2 the mainreason for this situation is that in the face of increasing in-tensity of the retailerrsquos fairness concerns the manufacturertends to lower product wholesale price so that the retailerreceives indirect compensation Another ideal solution for themanufacturer is to reduce extended warranty service price toattract consumers to purchase the service In this case aretailer that has a strong sense of fairness of concern may alsochoose to reduce extended warranty service price Howeverthe findings from the analysis of Proposition 4(i) show thatretailers always offer extended warranties at a lower price thanthat of manufacturers In summary retailersrsquo fairness con-cerns negatively affect the profit of manufacturerse greaterthe intensity of the fairness concerns the lower the profit ofmanufacturers

Proposition 5(ii) shows that compared to the impact ofvertical competition level on a retailerrsquos performancewithout fairness concerns an increase in vertical competi-tion level in the context of the retailerrsquos fairness concernsdoes not always enhance the utility of the retailer When theintensity of the fairness concerns is lower than a certainthreshold ω(1 minus ω) the utility of the retailer decreases withincreasing vertical competition level but the retailerrsquos overallbenefits increase with the intensity of the fairness concernsis implies that a retailer can secure benefits by increasingthe intensity of its fairness concerns When the intensity ofthe retailerrsquos fairness concerns is greater than the thresholdλgtω(1 minus ω) a counterintuitive conclusion is reached the


utility of the retailer is reduced if the intensity of its fairnessconcerns increasesis can be explained by the results fromthe analysis of Inference 2 and Proposition 4(iii) Superfi-cially increasing intensity of the fairness concerns can leadto a bigger difference in extended warranty service priceenabling the retailer to earn more from the extended war-ranty However an increase in the intensity of the fairnessconcerns beyond the threshold brings about a higherproduct retail price is is not conducive to increasing theretailerrsquos revenue from the product Since product sales arethe source of the revenue from extended warranties theutility of the fair-minded retailer will decrease

4 Analysis with Numerical Examples

To verify the propositions and inferences in Section 3numerical examples are used to further analyse how the levelβ of competition in extended warranty service between amanufacturer and a retailer and the intensity of the retailersfairness concerns λ influence the pricing strategies of bothparties for products and extended warranties First let a 1λ 1 λ0 λ(1 + λ) 05 e relationships between themanufacturerrsquos product wholesale prices w and wF and thelevel β of competition in extended warranty service can bedetermined in two situations when the retailer has fairnessconcerns and when the retailer has no fairness concernsese relationships are shown in Figure 2

It can be seen from Figure 2 that the greater verticalcompetition in extended warranty service between themanufacturer and the retailer the lower the productwholesale price of the manufacturer If the vertical com-petition is weak the wholesale price wF when the retailer hasfairness concerns is lower than the wholesale price w whenthe retailer has no fairness concerns If the vertical com-petition is strong the wholesale price wF when the retailerhas fairness concerns is higher than the wholesale price w

when the retailer has no fairness concerns As shown inFigure 2 when the retailer has fairness concerns themanufacturer can determine product wholesale price basedon the level of competition in extended warranty servicebetween the two parties e stronger the competition thegreater the wholesale price discount given to the retailer

Figure 3 shows the relationship between the extendedwarranty service prices psm and psr of the manufacturer andretailer and the level β of vertical competition in the ex-tended warranty service without fairness concerns and therelationship between extended warranty service prices pF

smand pF

sr of the manufacturer and retailer and the level β ofvertical competition in the extended warranty service whenthe retailer is fair-minded based on the above parameters

As can be seen from Figure 3 regardless of whether theretailer is fair-minded as the competition in extendedwarranty service between the manufacturer and retailerintensifies both parties tend to raise the price of the serviceAnd the greater the competition the more obvious the risein extended warranty service for both parties is isconsistent with Inferences 1 and 2 When the retailer is notconcerned about fairness the manufacturer and retaileradopt the same extended warranty service pricing strategy

When the retailer has fairness concerns the two partiesadopt differential extended warranty service pricingstrategies In this case the manufacturerrsquos extended war-ranty service price is greater than that of the retailer Inaddition the extended warranty service price of bothparties when the retailer has no fairness concerns is higherthan the corresponding extended warranty service pricewhen the retailer has fairness concerns is result verifiesProposition 3 According to Figure 3 when the retailer hasfairness concerns the two parties can determine extendedwarranty service price based on the level of competition inthe service e stronger the competition the greater theprice difference in extended warranty service between thetwo parties

00

01

01

02

02

03

03

05

05

04

04

06 07β

wF

w

Figure 2 Relationships between wholesale prices w and wF and thelevel β of vertical competition

00

0201

02

03 0504

04

06

06

08

1

07β

psmF

psm (psr)Fpsr

Figure 3 Relationships between extended warranty service pricespsm psr pF

sm and pFsr and β


In the same way let a 1 β 05 e relationshipsbetween the manufacturerrsquos product wholesale price wF andextended warranty service price pF

sm and the retailerrsquosproduct sales price pF

p and extended warranty service priceand pF

sr and the intensity λ of the retailerrsquos fairness concernscan be determined as shown in Figure 4

As can be seen from Figure 4 an increase in the intensityof the retailerrsquos fairness concerns means a lower productwholesale price of the manufacturer a higher sales price ofthe retailer and a smaller difference in extended warrantyservice price between the two parties Moreover a decreasein the intensity of the retailerrsquos fairness concerns will widenthe gap between the wholesale price and sales price andbetween the extended warranty service price of the manu-facturer and retailer ese refsults prove Inference 2 andProposition 4 From Figure 4 it is clear that when the in-tensity of the retailerrsquos fairness concerns increases the re-tailer can make a larger profit on each item sold and enjoy agreater price advantage in the extended warranty servicemarket is price advantage also helps the retailer increaseits revenue

Following the above method of assigning values to ex-ogenous parameters let a 1 λ 1 λ0 λ(1 + λ) 05e relationships between the two partiesrsquo profits or utilitiesπr Ur πm and πF

m and the level β of their competition inextended warranty service can be determined in two situ-ations when the retailer has fairness concerns and when theretailer has no fairness concerns ese relationships areshown in Figures 5(a) and 5(b)

As can be seen from Figures 5(a) and 5(b) regardless ofwhether the retailer has fairness concerns the profits orutilities of the manufacturer and retailer increase with thelevel of competition in an extended warranty service Inaddition the greater the competition is the faster the profitsor utilities of the manufacturer and retailer grow issuggests that the competition in an extended warrantyservice between the two parties is beneficial to both partiese result proves Proposition 2 and Proposition 5(i)

Let a 05 and β 09e revenue of the manufacturerwhen the retailer has fairness concerns and when the retailerhas no fairness concerns is shown in Figure 6

It can be seen from Figure 6 that when the retailer hasfairness concern preference its greater fairness concernsmeans a lower revenue of the manufacturer leading to awider income gap compared with when the retailer has nofairness concern preferenceis verifies Proposition 5(i) Ascan be seen from Figure 6 the retailerrsquos fairness concerns arealways unfavourable to the manufacturer and the strongerthe intensity of the retailerrsquos fairness concerns the lower themanufacturerrsquos revenue

In the above parameter settings the relationship betweenthe utility Ur of the retailer and the intensity λ of its fairnessconcerns can be further obtained as shown in Figure 7

As can be seen from Figure 7(a) when the intensity ofthe retailerrsquos fairness concerns is low and increases theutility of the retailer gradually increases and the greater theintensity is the faster the utility growsis indicates that theretailer can benefit itself from increasing the intensity of itsfairness concerns As shown in Figure 7(b) when the

intensity of the retailerrsquos fairness concerns is high and risesthe utility of the retailer gradually decreases Moreover thegreater the intensity the slower the decrease in the retailerrsquosutility is means that as the retailer increases the intensityof its fairness concerns beyond a certain threshold its benefitis reduced is result provides evidence supporting Prop-osition 5(ii) From Figure 7 it is clear that it is not alwaysbeneficial for a retailer to increase the intensity of its fairnessconcerns Only when the intensity increases are within acertain threshold can they bring a higher utility for theretailer

5 Conclusions

As the product service market grows rapidly both manu-facturers and retailers have begun to provide extendedwarranties Consumers can choose to purchase an extendedwarranty service from either the manufacturer or the re-tailer Since themanufacturers and retailers in a supply chainprovide homogeneous extended warranties services theyface competition with each other At the same time besidesconsidering its own earnings a retailer pays attention to thedistribution of profits in the supply chain A lower-than-expected profit can lead the retailer to have fairness concernpreferences is paper proposes a manufacturer-ledStackelberg game model to investigate optimal pricingstrategies of manufacturers and retailers for their productsand extended warranty services when the retailerrsquos fairnessconcern preference is paper further analyses how thelevel of vertical competition in extended warranty serviceand the intensity of a retailerrsquos fairness concerns influencethe optimal pricing of products and extended warranties andprofits for the manufacturer and retailer

00

02

01

03

04

05

06

1 2 3 4λ

psmF

wF

FpsrpF

Figure 4 Relationships between product wholesale price wFproduct sales price pF

p extended warranty service price pFsm pF

srand λ


is paper presents the following findings Firstly whenthe retailer has no fairness concerns the retailer and themanufacturer share the same optimal extended warrantyservice pricing strategies And increasing competition inextended warranty service between the two parties leads toa higher consistent extended warranty service price and alower retail price In contrast when the retailer has fairnessconcerns the retailer and the manufacturer develop dif-ferential pricing strategies for an extended warranty service

as their optimal choice In this case the manufacturer tendsto set an extended warranty service price higher than that ofthe retailer to surrender part of its profits to the retailer andoffset the retailerrsquos concern about unfairness Secondly thedifference in extended warranty service price between thetwo parties is affected by the level of competition in theextended warranty service and the intensity of the retailerrsquosfairness concerns Increase in vertical competition in anextended warranty service will always widen the price gap

00

02

01

03

04

05

05 1 215

07

08

06

λπm

F

πm

Figure 6 Relationships between πm πFm and λ

0

02

01

03

04

05

06

0 0201 03 04 05 06 07

πrπm

β

(a)

0

02

01

03

04

05

06

0 0201 03 04 05 06 07β

πmF

Ur

(b)

Figure 5 (a) Relationships between πr πm and β (b) Relationships between Ur πFm and β


in the extended warranty service between the manufacturerand retailer e impact of the intensity of the retailerrsquosfairness concerns on this price gap depends on the intensityitself When the intensity of the retailerrsquos fairness concernsis less than a certain threshold an increase in the intensityof the fairness concerns will narrow the price gap in theextended warranty service between the manufacturer andretailer When the intensity of the retailerrsquos fairness con-cerns is greater than the threshold increasing intensity ofthe fairness concerns implies a greater difference in ex-tended warranty service price between the two partiesirdly the results from further analysis of the profit ofboth parties show that when the retailer has no fairnessconcerns its competition with the manufacturer in ex-tended warranty service will increase the income of bothparties rather than harming their interests In comparisona higher intensity of the retailerrsquos fairness concerns alwaysharms the interests of the manufacturer For the retaileronly when the intensity increases are within a certainthreshold can it improve its utility

is paper considers vertical competition in extendedwarranty service in a product and service supply chainconsisting of only one retailer and one manufacturer Inreality a manufacturer may also distribute the same productthrough multiple retailers which may experience horizontalcompetition in an extended warranty service with eachother is issue can be further examined through vectorequalization [30ndash32] Future research can be done on pricingstrategies in a product and service supply chain involvingmultiple retailers competing with each other horizontally inan extended warranty service in the context of fairnessconcerns

Data Availability

e data generated andor analysed during the current studyare available from the corresponding author on reasonablerequest

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is research was funded by the National Key Research andDevelopment Program (grant no 2018YFB1701502) theNational Natural Science Foundation of China (grant nos71572020 and 71501162) the Chongqing Basic Science andFrontier Technology Research Project (grant nocstc2016jcyjA0528) the Scientific and Technological Re-search Program of Chongqing Municipal Education Com-mission (no KJQN201800715) the China PostdoctoralScience Foundation (grant no 2015M580770) the Hu-manities and Social Science Research Program of ChongqingMunicipal Education Commission (grant no 19SKGH061)and the Chongqing Postdoctoral Science Foundation (grantno Xm2015044)

References

[1] J-C Lu Y-C Tsao and C Charoensiriwath ldquoCompetitionunder manufacturer service and retail pricerdquo EconomicModelling vol 28 no 3 pp 1256ndash1264 2011

00 01 02 03 04

05

1

2

15

λ

Ur

(a)

0

05

1

2

15

05 1 215λ

Ur

(b)

Figure 7 Relationship between the retailerrsquos utility Ur and the intensity λ of fairness concerns (a) e influence of the intensity of fairnessconcerns on utility when the intensity is lower (b) e influence of the intensity of fairness concerns on utility when the intensity is higher


[2] J C Hartman and K Laksana ldquoDesigning and pricing menusof extended warranty contractsrdquo Naval Research Logisticsvol 56 no 3 pp 199ndash214 2009

[3] T Chen A Kalra and B Sun ldquoWhy do consumers buyextended service contractsrdquo Journal of Consumer Researchvol 36 no 4 pp 611ndash623 2009

[4] X Chen and X Wang ldquoFree or bundled channel selectiondecisions under different power structuresrdquo Omega vol 53pp 11ndash20 2015

[5] X Chen X Wang and H K Chan ldquoManufacturer and re-tailer coordination for environmental and economic com-petitiveness a power perspectiverdquo Transportation ResearchPart E Logistics and Transportation Review vol 97pp 268ndash281 2017

[6] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquo Management Science vol 53 no 8pp 1303ndash1314 2007

[7] N Jack and D N P Murthy ldquoA flexible extended warrantyand related optimal strategiesrdquo Journal of the OperationalResearch Society vol 58 no 12 pp 1612ndash1620 2007

[8] S Bouguerra A Chelbi and N Rezg ldquoA decision model foradopting an extended warranty under different maintenancepoliciesrdquo International Journal of Production Economicsvol 135 no 2 pp 840ndash849 2012

[9] K Shahanaghi R Noorossana S G Jalali-Naini andM Heydari ldquoFailure modeling and optimizing preventivemaintenance strategy during two-dimensional extendedwarranty contractsrdquo Engineering Failure Analysis vol 28pp 90ndash102 2013

[10] B Jiang and X Zhang ldquoHow does a retailerrsquos service planaffect a manufacturerrsquos warrantyrdquo Management Sciencevol 57 no 4 pp 727ndash740 2011

[11] H S Heese ldquoRetail strategies for extended warranty sales andimpact on manufacturer base warrantiesrdquo Decision Sciencesvol 43 no 2 pp 341ndash367 2012

[12] K Li S Mallik and D Chhajed ldquoDesign of extended war-ranties in supply chains under additive demandrdquo Productionand Operations Management vol 21 no 4 pp 730ndash746 2012

[13] M A Cohen and S Whang ldquoCompeting in product andservice a product life-cycle modelrdquo Management Sciencevol 43 no 4 pp 535ndash545 1997

[14] X Chen X Wang and Y Xia ldquoProduction coopetitionstrategies for competing manufacturers that produce partiallysubstitutable productsrdquo Production and Operations Man-agement vol 28 no 6 pp 1446ndash1464 2019

[15] Y Bian S Yan W Zhang and H Xu ldquoWarranty strategy in asupply chain when two retailerrsquos extended warranties bundledwith the productsrdquo Journal of Systems Science and SystemsEngineering vol 24 no 3 pp 364ndash389 2015

[16] C Camerer and R H aler ldquoAnomalies ultimatums dic-tators and mannersrdquo Journal of Economic Perspectives vol 9no 2 pp 209ndash219 1995

[17] T-H Ho and J Zhang ldquoDesigning pricing contracts forboundedly rational customers does the framing of the fixedfee matterrdquoManagement Science vol 54 no 4 pp 686ndash7002008

[18] O Caliskan-Demirag Y Chen and J Li ldquoChannel co-ordination under fairness concerns and nonlinear demandrdquoEuropean Journal of Operational Research vol 207 no 3pp 1321ndash1326 2010

[19] X Wu and J Niederhoff ldquoFairness in selling to the news-vendorrdquo Production and Operations Management vol 23no 11 pp 2002ndash2022 2014

[20] P Ma K W Li and Z-J Wang ldquoPricing decisions in closed-loop supply chains with marketing effort and fairness con-cernsrdquo International Journal of Production Research vol 55no 22 pp 6710ndash6731 2017

[21] Y Zhou M Bao X Chen and X Xu ldquoCo-op advertising andemission reduction cost sharing contracts and coordination inlow-carbon supply chain based on fairness concernsrdquo Journalof Cleaner Production vol 133 pp 402ndash413 2016

[22] W Liu S Wang D Zhu D Wang and X Shen ldquoOrderallocation of logistics service supply chain with fairnessconcern and demand updating model analysis and empiricalexaminationrdquoAnnals of Operations Research vol 268 no 1-2pp 177ndash213 2018

[23] N Du and Q L Han ldquoPricing and service quality guaranteedecisions in logistics service supply chain with fairnessconcernrdquo Asia-Pacific Journal of Operational Researchvol 35 no 5 article 1850036 2018

[24] N Wang Z-P Fan and X Chen ldquoEffect of fairness onchannel choice of the mobile phone supply chainrdquo In-ternational Transactions in Operational Research 2019 Inpress

[25] Q H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling vol 40 no 15-16pp 7349ndash7367 2016

[26] X Chen L Li and M Zhou ldquoManufacturerrsquos pricing strategyfor supply chain with warranty period-dependent demandrdquoOmega vol 40 no 6 pp 807ndash816 2012

[27] B Dan S Zhang and M Zhou ldquoStrategies for warrantyservice in a dual-channel supply chain with value-addedservice competitionrdquo International Journal of ProductionResearch vol 56 no 17 pp 5677ndash5699 2018

[28] X Zhang X Han X Liu R Liu and J Leng ldquoe pricing ofproduct and value-added service under information asym-metry a product life cycle perspectiverdquo International Journalof Production Research vol 53 no 1 pp 25ndash40 2015

[29] H Fu K L Teo Y Li and L Wang ldquoWeather riskndashrewardcontract for sustainable agri-food supply chain with loss-averse farmerrdquo Sustainability vol 10 no 12 p 4540 2018

[30] X Sun H Fu and J Zeng ldquoRobust approximate optimalityconditions for uncertain nonsmooth optimization withinfinite number of constraintsrdquo Mathematics vol 7 no 1p 12 2019

[31] X-K Sun X-J Long H-Y Fu and X-B Li ldquoSome char-acterizations of robust optimal solutions for uncertain frac-tional optimization and applicationsrdquo Journal of Industrial ampManagement Optimization vol 13 no 2 pp 803ndash824 2017

[32] X Sun K L Teo and L Tang ldquoDual approaches to char-acterize robust optimal solution sets for a class of uncertainoptimization problemsrdquo Journal of Optimization Aeory andApplications vol 182 no 3 pp 984ndash1000 2019


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Submit your manuscripts atwwwhindawicom

It is worth noting that competition in the extendedwarranty service market is based on products Howeverthe relationship between a manufacturer and a retailerwithin a supply chain is much more about upstream anddownstream partnership regarding production and salesof specific products is requires ensuring that both themanufacturers and retailers within a product and servicesupply chain obtain the optimal benefits e key tosolving this issue lies in making optimal pricing decisionsfor products and services In optimal pricing decisionsthe dominant manufacturer tends to play a powerful rolein setting prices [4 5] while often leads the retailer in aweak position to raise fairness concerns is inevitablyfurther influences optimal pricing decisions [6] To ad-dress the issue this paper discusses how a retailerrsquosfairness concerns influence the optimal pricing decisionsfor products and services in a product and service supplychain with vertical competition in extended warrantyservice

At present the research on extended warranties has gonebeyond extended warranties offered by a single enterprise[7ndash9] to include multiple supply chain members Jiang andZhang [10] studied the impact of a retailerrsquos extendedwarranties on the basic warranties from a manufacturerHeese [11] examined the warranty strategies of manufac-turers and the extended warranty strategies of retailers in thecontext of two competing manufacturers selling productsthrough a common retailer e literature [10 11] puts itsfocus on the impact of a retailerrsquos extended warranties on amanufacturerrsquos basic warranties instead of on the pricing ofextended warranties on products Li et al [12] compared andanalysed the extended warranties provided separately by amanufacturer and a retailer within a two-tier supply chain interms of the length price and profit of the extended war-ranties Cohen and Whang [13] discussed the optimalpricing of products and extended warranties for manufac-turers and third-party service providers in two cases the onewith the extended warranties provided by manufacturersand the other one with the extended warranties provided bythird-party service providers Li et al and Cohen andWhang[12 13] studied the pricing of extended warranties in asupply chain whose members however do not competewith each other Chen et al [14] examined optimal com-peting and collaborating strategy in a complex supply chainconsisting by two manufacturers who manufacture sub-stitutive products and purchase a key component from theother Chen et al [14] studied the competition but it is notwarranty competition In the supply chain consisting of amanufacturer providing warranties and two competingretailers providing extended warranties Bian et al [15]explored the pricing of extended warranties by the tworetailers against the backdrop of horizontal competitionbetween each other is paper extends the work of Bianet al [15] by introducing vertical competition in extendedwarranty between a manufacturer and its downstream re-tailer in a supply chain It also considers how fairnessconcern of the retailer in the weak position in a product andservice supply chain influences the optimal pricing ofproducts and extended warranties

e paper also covers fairness concerns in supply chainsCamerer and aler [16] proved through the ultimatumgame that people punish others for unfair behaviour evenwhen it is costly for them to do so Ho and Zhang [17]confirmed through experimental research that there isfairness concern behaviour in supply chain contracts In thefield of operational research and management science re-search on supply chain fairness concerns generally focuseson two areas e first one is the impact of fairness concernson traditional supply chain contract design Cui et al [6] areamong the first to use the mathematical model to studyfairness concerns in a two-tier supply chain ey find thatwhen both the manufacturer and the retailer have advan-tageous and disadvantageous inequity aversions simulta-neously a simple wholesale price contract can make supplychain coordinated under certain conditions Caliskan-Demirag et al [18] extended the work of Cui et al [6] to anonlinear demand function and found that if both themanufacturer and the retailer have advantageous and dis-advantageous inequity aversions simultaneously in a non-linear demand setting the supply chain members could alsoachieve coordination In addition the exponential demandfunction requires less stringent conditions than the lineardemand to achieve supply chain coordination Wu andNiederhoff [19] extended the analysis of supply chain fair-ness concerns to the random-demand newsvendor model inwhich they studied the impact of fairness concerns on supplychain coordinatione second attractive area of research onsupply chain fairness concerns is the impact of fairnessconcerns on optimal pricing decisions in a supply chain Maet al [20] studied the impact of a retailerrsquos fairness concernson recovery rates sales efforts and profits in a closed-loopsupply chain environment where the product demand issensitive to the retailerrsquos sales effort In the low-carbonsupply chain setting Zhou et al [21] examined the impact ofa retailerrsquos fairness concerns on the supply chain pricingstrategies under both cooperative advertising contracts andcooperative advertising and emission reduction cost-sharingcontracts Ma et al and Zhou et al [20 21] studied theimpact of fairness concerns in supply chains on optimalpricing of products but not of services Liu et al [22] in-vestigated the optimal allocation strategy of logistics serviceorders in a service supply chain composed of a logisticsservice integrator and several logistics service providers(LSPs) based on the LSPsrsquo fairness concern preferences anddemand updating Du and Han [23] analysed optimalpricing strategies under the impact of fairness concerns inthe logistics service supply chain in which the market de-mand is affected by the price and quality defect guarantee oflogistics services at the same time Liu et al and Du and Han[22 23] studied the impact of fairness concerns on theoptimal pricing of services but not of products Wang et al[24] considered a mobile phone supply chain (MPSC)consisting of a service operator and a handset manufacturerand analysed the influence of the nature and level of both thehandset manufacturerrsquos fairness concerns and the operatorrsquosfairness concerns on the pricing decision Li and Li [25]analysed the impacts of a retailerrsquos fairness concerns on thepricing of products and value-added services as well as on








2 Model Assumptions




















maxpsm


(5)

maxpsr




d2πm psm( 1113857

dp2sm

minus 2

d2πr psr( 1113857

dp2sr

minus 2

(7)


dπm psm( 1113857


dπr psr( 1113857



plowastsm

a minus pp

2 minus β

plowastsr

a minus pp

2 minus β

(10)


maxpp


(β minus 2)2

(11)


d2πr pp1113872 1113873

dp2p

minus2 β2 minus 4β + 31113872 1113873

(β minus 2)2 (12)



dπr pp1113872 1113873

dpp


(β minus 2)2

(13)


plowastp

(a + w)β2 minus 4(a + w)β + 2a + 4w

2 β2 minus 4β + 31113872 1113873 (14)


maxw


4 β2 minus 4β + 31113872 11138732

(15)


d2πm(w)


β2 minus 4β + 31113872 11138732 (16)


6



Manufacturer


price


w

Retailer

pp psr

psmW

arra

nty

pric

e

War

rant

y pr

ice

Customers





wlowast

a β2 minus 4β + 21113872 1113873

2β2 minus 8β + 5

plowastsm

a(2 minus β)

2 2β2 minus 8β + 51113872 1113873

(17)


plowastp

3a β2 minus 4β + 21113872 1113873

2 2β2 minus 8β + 51113872 1113873

plowastsr

a(2 minus β)

2 2β2 minus 8β + 51113872 1113873

(18)








can have

πm a2 β2 minus 4β + 41113872 1113873

4 2β2 minus 8β + 51113872 1113873 (19)


6



zβ

3a2(2 minus β)

2 2β2 minus 8β + 51113872 11138732 (20)


6




4 2β2 minus 8β + 51113872 11138732 (21)


zπr

zβ

a2(2 minus β) 4β2 minus 16β + 131113872 1113873

2 2β2 minus 8β + 51113872 11138733 (22)


6







maxpsm


(23)


Ur πFr minus λ0π

Fm (24)


maxpsr



(25)


pFlowastsr


4 + λ0 minus 1( 1113857β2

pFlowastsm

(2 + β) a minus pp1113872 1113873

4 + λ0 minus 1( 1113857β2

(26)



maxpp


Flowastsr + βp


Flowastsr



Flowastsm1113960 1113961

(27)


pFlowastp



(28)



have

maxw


Flowastp minus p

Flowastsm + βp


Flowastsm

(29)


wFlowast







pFlowastp



pFlowastsm

a(β + 2) 4 + β2 λ0 minus 1( 11138571113960 1113961


pFlowastsr



(31)



wFlowast



pFlowastsm

a(β + 2) 4 + β2 λ0 minus 1( 11138571113960 1113961


(32)


pFlowastp



pFlowastsr



(33)



p pFsm and






pλgt 0(iii) zpF


srλlt 0(iv) zpF


smλlt 0

Proof




zwF

λ0

(β + 2)al

m2 (34)









zpFp

β minus

ja

4k2 (35)





culated as follows

zpFp

λ0 minus

ϕ(β + 2)a

4φ2 (36)










smβgt 0zpF

smλlt 0




Proposition 4


sm gtpFlowastsr










+ 5β)(β + 2)(2β3 minus 4β2 + β + 2)β

Proof








pFlowastsm minus p

Flowastsr

aλ0β 4 + β2 λ0 minus 1( 11138571113960 1113961


(37)




sr( 1113857

zβ

minus aλ0c4η2

(38)







sr( 1113857

zλ0 minus

aβ(β + 2)]4η2

(39)



2


+ 5β)(β + 2)(2β3 minus 4β2 + β+










mzβgt 0zπF






Δ

radic2


10β2 +8β + 4

Proof








zπFmzβ

a2 4 + λ0 minus 1( 1113857β21113960 1113961

8 1 + λ0( 1113857η2Υ (41)



zπFm

zλ0 minus

a2 λ0 minus 1( 1113857β2 + 41113960 1113961ψ

8 λ0 + 1( 11138572η2

(42)







UFr

a2 λ0 minus 1( 1113857β2 + 41113960 11139612


16η2 (43)




zUFr

zλ0

a2(β + 2) λ0 minus 1( 1113857β2 + 41113960 1113961σ16η3

(44)





radic

2 β4 + β3 + β21113872 1113873lt 0 (45)


Δ

















smand pF




00

01

01

02

02

03

03

05

05

04

04

06 07β

wF

w


00

0201

02

03 0504

04

06

06

08

1

07β

psmF

psm (psr)Fpsr


sm and pFsr and β















5 Conclusions


00

02

01

03

04

05

06

1 2 3 4λ

psmF

wF

FpsrpF



srand λ




00

02

01

03

04

05

05 1 215

07

08

06

λπm

F

πm


0

02

01

03

04

05

06

0 0201 03 04 05 06 07

πrπm

β

(a)

0

02

01

03

04

05

06

0 0201 03 04 05 06 07β

πmF

Ur

(b)





Data Availability




Acknowledgments


References


00 01 02 03 04

05

1

2

15

λ

Ur

(a)

0

05

1

2

15

05 1 215λ

Ur

(b)










































Journal of












Journal of







Volume 2018






Volume 2018














2 Model Assumptions




















maxpsm


(5)

maxpsr




d2πm psm( 1113857

dp2sm

minus 2

d2πr psr( 1113857

dp2sr

minus 2

(7)


dπm psm( 1113857


dπr psr( 1113857



plowastsm

a minus pp

2 minus β

plowastsr

a minus pp

2 minus β

(10)


maxpp


(β minus 2)2

(11)


d2πr pp1113872 1113873

dp2p

minus2 β2 minus 4β + 31113872 1113873

(β minus 2)2 (12)



dπr pp1113872 1113873

dpp


(β minus 2)2

(13)


plowastp

(a + w)β2 minus 4(a + w)β + 2a + 4w

2 β2 minus 4β + 31113872 1113873 (14)


maxw


4 β2 minus 4β + 31113872 11138732

(15)


d2πm(w)


β2 minus 4β + 31113872 11138732 (16)


6



Manufacturer


price


w

Retailer

pp psr

psmW

arra

nty

pric

e

War

rant

y pr

ice

Customers





wlowast

a β2 minus 4β + 21113872 1113873

2β2 minus 8β + 5

plowastsm

a(2 minus β)

2 2β2 minus 8β + 51113872 1113873

(17)


plowastp

3a β2 minus 4β + 21113872 1113873

2 2β2 minus 8β + 51113872 1113873

plowastsr

a(2 minus β)

2 2β2 minus 8β + 51113872 1113873

(18)








can have

πm a2 β2 minus 4β + 41113872 1113873

4 2β2 minus 8β + 51113872 1113873 (19)


6



zβ

3a2(2 minus β)

2 2β2 minus 8β + 51113872 11138732 (20)


6




4 2β2 minus 8β + 51113872 11138732 (21)


zπr

zβ

a2(2 minus β) 4β2 minus 16β + 131113872 1113873

2 2β2 minus 8β + 51113872 11138733 (22)


6







maxpsm


(23)


Ur πFr minus λ0π

Fm (24)


maxpsr



(25)


pFlowastsr


4 + λ0 minus 1( 1113857β2

pFlowastsm

(2 + β) a minus pp1113872 1113873

4 + λ0 minus 1( 1113857β2

(26)



maxpp


Flowastsr + βp


Flowastsr



Flowastsm1113960 1113961

(27)


pFlowastp



(28)



have

maxw


Flowastp minus p

Flowastsm + βp


Flowastsm

(29)


wFlowast







pFlowastp



pFlowastsm

a(β + 2) 4 + β2 λ0 minus 1( 11138571113960 1113961


pFlowastsr



(31)



wFlowast



pFlowastsm

a(β + 2) 4 + β2 λ0 minus 1( 11138571113960 1113961


(32)


pFlowastp



pFlowastsr



(33)



p pFsm and






pλgt 0(iii) zpF


srλlt 0(iv) zpF


smλlt 0

Proof




zwF

λ0

(β + 2)al

m2 (34)









zpFp

β minus

ja

4k2 (35)





culated as follows

zpFp

λ0 minus

ϕ(β + 2)a

4φ2 (36)










smβgt 0zpF

smλlt 0




Proposition 4


sm gtpFlowastsr










+ 5β)(β + 2)(2β3 minus 4β2 + β + 2)β

Proof








pFlowastsm minus p

Flowastsr

aλ0β 4 + β2 λ0 minus 1( 11138571113960 1113961


(37)




sr( 1113857

zβ

minus aλ0c4η2

(38)







sr( 1113857

zλ0 minus

aβ(β + 2)]4η2

(39)



2


+ 5β)(β + 2)(2β3 minus 4β2 + β+










mzβgt 0zπF






Δ

radic2


10β2 +8β + 4

Proof








zπFmzβ

a2 4 + λ0 minus 1( 1113857β21113960 1113961

8 1 + λ0( 1113857η2Υ (41)



zπFm

zλ0 minus

a2 λ0 minus 1( 1113857β2 + 41113960 1113961ψ

8 λ0 + 1( 11138572η2

(42)







UFr

a2 λ0 minus 1( 1113857β2 + 41113960 11139612


16η2 (43)




zUFr

zλ0

a2(β + 2) λ0 minus 1( 1113857β2 + 41113960 1113961σ16η3

(44)





radic

2 β4 + β3 + β21113872 1113873lt 0 (45)


Δ

















smand pF




00

01

01

02

02

03

03

05

05

04

04

06 07β

wF

w


00

0201

02

03 0504

04

06

06

08

1

07β

psmF

psm (psr)Fpsr


sm and pFsr and β















5 Conclusions


00

02

01

03

04

05

06

1 2 3 4λ

psmF

wF

FpsrpF



srand λ




00

02

01

03

04

05

05 1 215

07

08

06

λπm

F

πm


0

02

01

03

04

05

06

0 0201 03 04 05 06 07

πrπm

β

(a)

0

02

01

03

04

05

06

0 0201 03 04 05 06 07β

πmF

Ur

(b)





Data Availability




Acknowledgments


References


00 01 02 03 04

05

1

2

15

λ

Ur

(a)

0

05

1

2

15

05 1 215λ

Ur

(b)










































Journal of












Journal of







Volume 2018






Volume 2018










maxpsm


(5)

maxpsr




d2πm psm( 1113857

dp2sm

minus 2

d2πr psr( 1113857

dp2sr

minus 2

(7)


dπm psm( 1113857


dπr psr( 1113857



plowastsm

a minus pp

2 minus β

plowastsr

a minus pp

2 minus β

(10)


maxpp


(β minus 2)2

(11)


d2πr pp1113872 1113873

dp2p

minus2 β2 minus 4β + 31113872 1113873

(β minus 2)2 (12)



dπr pp1113872 1113873

dpp


(β minus 2)2

(13)


plowastp

(a + w)β2 minus 4(a + w)β + 2a + 4w

2 β2 minus 4β + 31113872 1113873 (14)


maxw


4 β2 minus 4β + 31113872 11138732

(15)


d2πm(w)


β2 minus 4β + 31113872 11138732 (16)


6



Manufacturer


price


w

Retailer

pp psr

psmW

arra

nty

pric

e

War

rant

y pr

ice

Customers





wlowast

a β2 minus 4β + 21113872 1113873

2β2 minus 8β + 5

plowastsm

a(2 minus β)

2 2β2 minus 8β + 51113872 1113873

(17)


plowastp

3a β2 minus 4β + 21113872 1113873

2 2β2 minus 8β + 51113872 1113873

plowastsr

a(2 minus β)

2 2β2 minus 8β + 51113872 1113873

(18)








can have

πm a2 β2 minus 4β + 41113872 1113873

4 2β2 minus 8β + 51113872 1113873 (19)


6



zβ

3a2(2 minus β)

2 2β2 minus 8β + 51113872 11138732 (20)


6




4 2β2 minus 8β + 51113872 11138732 (21)


zπr

zβ

a2(2 minus β) 4β2 minus 16β + 131113872 1113873

2 2β2 minus 8β + 51113872 11138733 (22)


6







maxpsm


(23)


Ur πFr minus λ0π

Fm (24)


maxpsr



(25)


pFlowastsr


4 + λ0 minus 1( 1113857β2

pFlowastsm

(2 + β) a minus pp1113872 1113873

4 + λ0 minus 1( 1113857β2

(26)



maxpp


Flowastsr + βp


Flowastsr



Flowastsm1113960 1113961

(27)


pFlowastp



(28)



have

maxw


Flowastp minus p

Flowastsm + βp


Flowastsm

(29)


wFlowast







pFlowastp



pFlowastsm

a(β + 2) 4 + β2 λ0 minus 1( 11138571113960 1113961


pFlowastsr



(31)



wFlowast



pFlowastsm

a(β + 2) 4 + β2 λ0 minus 1( 11138571113960 1113961


(32)


pFlowastp



pFlowastsr



(33)



p pFsm and






pλgt 0(iii) zpF


srλlt 0(iv) zpF


smλlt 0

Proof




zwF

λ0

(β + 2)al

m2 (34)









zpFp

β minus

ja

4k2 (35)





culated as follows

zpFp

λ0 minus

ϕ(β + 2)a

4φ2 (36)










smβgt 0zpF

smλlt 0




Proposition 4


sm gtpFlowastsr










+ 5β)(β + 2)(2β3 minus 4β2 + β + 2)β

Proof








pFlowastsm minus p

Flowastsr

aλ0β 4 + β2 λ0 minus 1( 11138571113960 1113961


(37)




sr( 1113857

zβ

minus aλ0c4η2

(38)







sr( 1113857

zλ0 minus

aβ(β + 2)]4η2

(39)



2


+ 5β)(β + 2)(2β3 minus 4β2 + β+










mzβgt 0zπF






Δ

radic2


10β2 +8β + 4

Proof








zπFmzβ

a2 4 + λ0 minus 1( 1113857β21113960 1113961

8 1 + λ0( 1113857η2Υ (41)



zπFm

zλ0 minus

a2 λ0 minus 1( 1113857β2 + 41113960 1113961ψ

8 λ0 + 1( 11138572η2

(42)







UFr

a2 λ0 minus 1( 1113857β2 + 41113960 11139612


16η2 (43)




zUFr

zλ0

a2(β + 2) λ0 minus 1( 1113857β2 + 41113960 1113961σ16η3

(44)





radic

2 β4 + β3 + β21113872 1113873lt 0 (45)


Δ

















smand pF




00

01

01

02

02

03

03

05

05

04

04

06 07β

wF

w


00

0201

02

03 0504

04

06

06

08

1

07β

psmF

psm (psr)Fpsr


sm and pFsr and β















5 Conclusions


00

02

01

03

04

05

06

1 2 3 4λ

psmF

wF

FpsrpF



srand λ




00

02

01

03

04

05

05 1 215

07

08

06

λπm

F

πm


0

02

01

03

04

05

06

0 0201 03 04 05 06 07

πrπm

β

(a)

0

02

01

03

04

05

06

0 0201 03 04 05 06 07β

πmF

Ur

(b)





Data Availability




Acknowledgments


References


00 01 02 03 04

05

1

2

15

λ

Ur

(a)

0

05

1

2

15

05 1 215λ

Ur

(b)










































Journal of












Journal of







Volume 2018






Volume 2018










wlowast

a β2 minus 4β + 21113872 1113873

2β2 minus 8β + 5

plowastsm

a(2 minus β)

2 2β2 minus 8β + 51113872 1113873

(17)


plowastp

3a β2 minus 4β + 21113872 1113873

2 2β2 minus 8β + 51113872 1113873

plowastsr

a(2 minus β)

2 2β2 minus 8β + 51113872 1113873

(18)








can have

πm a2 β2 minus 4β + 41113872 1113873

4 2β2 minus 8β + 51113872 1113873 (19)


6



zβ

3a2(2 minus β)

2 2β2 minus 8β + 51113872 11138732 (20)


6




4 2β2 minus 8β + 51113872 11138732 (21)


zπr

zβ

a2(2 minus β) 4β2 minus 16β + 131113872 1113873

2 2β2 minus 8β + 51113872 11138733 (22)


6







maxpsm


(23)


Ur πFr minus λ0π

Fm (24)


maxpsr



(25)


pFlowastsr


4 + λ0 minus 1( 1113857β2

pFlowastsm

(2 + β) a minus pp1113872 1113873

4 + λ0 minus 1( 1113857β2

(26)



maxpp


Flowastsr + βp


Flowastsr



Flowastsm1113960 1113961

(27)


pFlowastp



(28)



have

maxw


Flowastp minus p

Flowastsm + βp


Flowastsm

(29)


wFlowast







pFlowastp



pFlowastsm

a(β + 2) 4 + β2 λ0 minus 1( 11138571113960 1113961


pFlowastsr



(31)



wFlowast



pFlowastsm

a(β + 2) 4 + β2 λ0 minus 1( 11138571113960 1113961


(32)


pFlowastp



pFlowastsr



(33)



p pFsm and






pλgt 0(iii) zpF


srλlt 0(iv) zpF


smλlt 0

Proof




zwF

λ0

(β + 2)al

m2 (34)









zpFp

β minus

ja

4k2 (35)





culated as follows

zpFp

λ0 minus

ϕ(β + 2)a

4φ2 (36)










smβgt 0zpF

smλlt 0




Proposition 4


sm gtpFlowastsr










+ 5β)(β + 2)(2β3 minus 4β2 + β + 2)β

Proof








pFlowastsm minus p

Flowastsr

aλ0β 4 + β2 λ0 minus 1( 11138571113960 1113961


(37)




sr( 1113857

zβ

minus aλ0c4η2

(38)







sr( 1113857

zλ0 minus

aβ(β + 2)]4η2

(39)



2


+ 5β)(β + 2)(2β3 minus 4β2 + β+










mzβgt 0zπF






Δ

radic2


10β2 +8β + 4

Proof








zπFmzβ

a2 4 + λ0 minus 1( 1113857β21113960 1113961

8 1 + λ0( 1113857η2Υ (41)



zπFm

zλ0 minus

a2 λ0 minus 1( 1113857β2 + 41113960 1113961ψ

8 λ0 + 1( 11138572η2

(42)







UFr

a2 λ0 minus 1( 1113857β2 + 41113960 11139612


16η2 (43)




zUFr

zλ0

a2(β + 2) λ0 minus 1( 1113857β2 + 41113960 1113961σ16η3

(44)





radic

2 β4 + β3 + β21113872 1113873lt 0 (45)


Δ

















smand pF




00

01

01

02

02

03

03

05

05

04

04

06 07β

wF

w


00

0201

02

03 0504

04

06

06

08

1

07β

psmF

psm (psr)Fpsr


sm and pFsr and β















5 Conclusions


00

02

01

03

04

05

06

1 2 3 4λ

psmF

wF

FpsrpF



srand λ




00

02

01

03

04

05

05 1 215

07

08

06

λπm

F

πm


0

02

01

03

04

05

06

0 0201 03 04 05 06 07

πrπm

β

(a)

0

02

01

03

04

05

06

0 0201 03 04 05 06 07β

πmF

Ur

(b)





Data Availability




Acknowledgments


References


00 01 02 03 04

05

1

2

15

λ

Ur

(a)

0

05

1

2

15

05 1 215λ

Ur

(b)










































Journal of












Journal of







Volume 2018






Volume 2018










maxpsm


(23)


Ur πFr minus λ0π

Fm (24)


maxpsr



(25)


pFlowastsr


4 + λ0 minus 1( 1113857β2

pFlowastsm

(2 + β) a minus pp1113872 1113873

4 + λ0 minus 1( 1113857β2

(26)



maxpp


Flowastsr + βp


Flowastsr



Flowastsm1113960 1113961

(27)


pFlowastp



(28)



have

maxw


Flowastp minus p

Flowastsm + βp


Flowastsm

(29)


wFlowast







pFlowastp



pFlowastsm

a(β + 2) 4 + β2 λ0 minus 1( 11138571113960 1113961


pFlowastsr



(31)



wFlowast



pFlowastsm

a(β + 2) 4 + β2 λ0 minus 1( 11138571113960 1113961


(32)


pFlowastp



pFlowastsr



(33)



p pFsm and






pλgt 0(iii) zpF


srλlt 0(iv) zpF


smλlt 0

Proof




zwF

λ0

(β + 2)al

m2 (34)









zpFp

β minus

ja

4k2 (35)





culated as follows

zpFp

λ0 minus

ϕ(β + 2)a

4φ2 (36)










smβgt 0zpF

smλlt 0




Proposition 4


sm gtpFlowastsr










+ 5β)(β + 2)(2β3 minus 4β2 + β + 2)β

Proof








pFlowastsm minus p

Flowastsr

aλ0β 4 + β2 λ0 minus 1( 11138571113960 1113961


(37)




sr( 1113857

zβ

minus aλ0c4η2

(38)







sr( 1113857

zλ0 minus

aβ(β + 2)]4η2

(39)



2


+ 5β)(β + 2)(2β3 minus 4β2 + β+










mzβgt 0zπF






Δ

radic2


10β2 +8β + 4

Proof








zπFmzβ

a2 4 + λ0 minus 1( 1113857β21113960 1113961

8 1 + λ0( 1113857η2Υ (41)



zπFm

zλ0 minus

a2 λ0 minus 1( 1113857β2 + 41113960 1113961ψ

8 λ0 + 1( 11138572η2

(42)







UFr

a2 λ0 minus 1( 1113857β2 + 41113960 11139612


16η2 (43)




zUFr

zλ0

a2(β + 2) λ0 minus 1( 1113857β2 + 41113960 1113961σ16η3

(44)





radic

2 β4 + β3 + β21113872 1113873lt 0 (45)


Δ

















smand pF




00

01

01

02

02

03

03

05

05

04

04

06 07β

wF

w


00

0201

02

03 0504

04

06

06

08

1

07β

psmF

psm (psr)Fpsr


sm and pFsr and β















5 Conclusions


00

02

01

03

04

05

06

1 2 3 4λ

psmF

wF

FpsrpF



srand λ




00

02

01

03

04

05

05 1 215

07

08

06

λπm

F

πm


0

02

01

03

04

05

06

0 0201 03 04 05 06 07

πrπm

β

(a)

0

02

01

03

04

05

06

0 0201 03 04 05 06 07β

πmF

Ur

(b)





Data Availability




Acknowledgments


References


00 01 02 03 04

05

1

2

15

λ

Ur

(a)

0

05

1

2

15

05 1 215λ

Ur

(b)










































Journal of












Journal of







Volume 2018






Volume 2018











pFlowastp



pFlowastsm

a(β + 2) 4 + β2 λ0 minus 1( 11138571113960 1113961


pFlowastsr



(31)



wFlowast



pFlowastsm

a(β + 2) 4 + β2 λ0 minus 1( 11138571113960 1113961


(32)


pFlowastp



pFlowastsr



(33)



p pFsm and






pλgt 0(iii) zpF


srλlt 0(iv) zpF


smλlt 0

Proof




zwF

λ0

(β + 2)al

m2 (34)









zpFp

β minus

ja

4k2 (35)





culated as follows

zpFp

λ0 minus

ϕ(β + 2)a

4φ2 (36)










smβgt 0zpF

smλlt 0




Proposition 4


sm gtpFlowastsr










+ 5β)(β + 2)(2β3 minus 4β2 + β + 2)β

Proof








pFlowastsm minus p

Flowastsr

aλ0β 4 + β2 λ0 minus 1( 11138571113960 1113961


(37)




sr( 1113857

zβ

minus aλ0c4η2

(38)







sr( 1113857

zλ0 minus

aβ(β + 2)]4η2

(39)



2


+ 5β)(β + 2)(2β3 minus 4β2 + β+










mzβgt 0zπF






Δ

radic2


10β2 +8β + 4

Proof








zπFmzβ

a2 4 + λ0 minus 1( 1113857β21113960 1113961

8 1 + λ0( 1113857η2Υ (41)



zπFm

zλ0 minus

a2 λ0 minus 1( 1113857β2 + 41113960 1113961ψ

8 λ0 + 1( 11138572η2

(42)







UFr

a2 λ0 minus 1( 1113857β2 + 41113960 11139612


16η2 (43)




zUFr

zλ0

a2(β + 2) λ0 minus 1( 1113857β2 + 41113960 1113961σ16η3

(44)





radic

2 β4 + β3 + β21113872 1113873lt 0 (45)


Δ

















smand pF




00

01

01

02

02

03

03

05

05

04

04

06 07β

wF

w


00

0201

02

03 0504

04

06

06

08

1

07β

psmF

psm (psr)Fpsr


sm and pFsr and β















5 Conclusions


00

02

01

03

04

05

06

1 2 3 4λ

psmF

wF

FpsrpF



srand λ




00

02

01

03

04

05

05 1 215

07

08

06

λπm

F

πm


0

02

01

03

04

05

06

0 0201 03 04 05 06 07

πrπm

β

(a)

0

02

01

03

04

05

06

0 0201 03 04 05 06 07β

πmF

Ur

(b)





Data Availability




Acknowledgments


References


00 01 02 03 04

05

1

2

15

λ

Ur

(a)

0

05

1

2

15

05 1 215λ

Ur

(b)










































Journal of












Journal of







Volume 2018






Volume 2018









zwF

λ0

(β + 2)al

m2 (34)









zpFp

β minus

ja

4k2 (35)





culated as follows

zpFp

λ0 minus

ϕ(β + 2)a

4φ2 (36)










smβgt 0zpF

smλlt 0




Proposition 4


sm gtpFlowastsr










+ 5β)(β + 2)(2β3 minus 4β2 + β + 2)β

Proof








pFlowastsm minus p

Flowastsr

aλ0β 4 + β2 λ0 minus 1( 11138571113960 1113961


(37)




sr( 1113857

zβ

minus aλ0c4η2

(38)







sr( 1113857

zλ0 minus

aβ(β + 2)]4η2

(39)



2


+ 5β)(β + 2)(2β3 minus 4β2 + β+










mzβgt 0zπF






Δ

radic2


10β2 +8β + 4

Proof








zπFmzβ

a2 4 + λ0 minus 1( 1113857β21113960 1113961

8 1 + λ0( 1113857η2Υ (41)



zπFm

zλ0 minus

a2 λ0 minus 1( 1113857β2 + 41113960 1113961ψ

8 λ0 + 1( 11138572η2

(42)







UFr

a2 λ0 minus 1( 1113857β2 + 41113960 11139612


16η2 (43)




zUFr

zλ0

a2(β + 2) λ0 minus 1( 1113857β2 + 41113960 1113961σ16η3

(44)





radic

2 β4 + β3 + β21113872 1113873lt 0 (45)


Δ

















smand pF




00

01

01

02

02

03

03

05

05

04

04

06 07β

wF

w


00

0201

02

03 0504

04

06

06

08

1

07β

psmF

psm (psr)Fpsr


sm and pFsr and β















5 Conclusions


00

02

01

03

04

05

06

1 2 3 4λ

psmF

wF

FpsrpF



srand λ




00

02

01

03

04

05

05 1 215

07

08

06

λπm

F

πm


0

02

01

03

04

05

06

0 0201 03 04 05 06 07

πrπm

β

(a)

0

02

01

03

04

05

06

0 0201 03 04 05 06 07β

πmF

Ur

(b)





Data Availability




Acknowledgments


References


00 01 02 03 04

05

1

2

15

λ

Ur

(a)

0

05

1

2

15

05 1 215λ

Ur

(b)










































Journal of












Journal of







Volume 2018






Volume 2018








pFlowastsm minus p

Flowastsr

aλ0β 4 + β2 λ0 minus 1( 11138571113960 1113961


(37)




sr( 1113857

zβ

minus aλ0c4η2

(38)







sr( 1113857

zλ0 minus

aβ(β + 2)]4η2

(39)



2


+ 5β)(β + 2)(2β3 minus 4β2 + β+










mzβgt 0zπF






Δ

radic2


10β2 +8β + 4

Proof








zπFmzβ

a2 4 + λ0 minus 1( 1113857β21113960 1113961

8 1 + λ0( 1113857η2Υ (41)



zπFm

zλ0 minus

a2 λ0 minus 1( 1113857β2 + 41113960 1113961ψ

8 λ0 + 1( 11138572η2

(42)







UFr

a2 λ0 minus 1( 1113857β2 + 41113960 11139612


16η2 (43)




zUFr

zλ0

a2(β + 2) λ0 minus 1( 1113857β2 + 41113960 1113961σ16η3

(44)





radic

2 β4 + β3 + β21113872 1113873lt 0 (45)


Δ

















smand pF




00

01

01

02

02

03

03

05

05

04

04

06 07β

wF

w


00

0201

02

03 0504

04

06

06

08

1

07β

psmF

psm (psr)Fpsr


sm and pFsr and β















5 Conclusions


00

02

01

03

04

05

06

1 2 3 4λ

psmF

wF

FpsrpF



srand λ




00

02

01

03

04

05

05 1 215

07

08

06

λπm

F

πm


0

02

01

03

04

05

06

0 0201 03 04 05 06 07

πrπm

β

(a)

0

02

01

03

04

05

06

0 0201 03 04 05 06 07β

πmF

Ur

(b)





Data Availability




Acknowledgments


References


00 01 02 03 04

05

1

2

15

λ

Ur

(a)

0

05

1

2

15

05 1 215λ

Ur

(b)










































Journal of












Journal of







Volume 2018






Volume 2018









zπFmzβ

a2 4 + λ0 minus 1( 1113857β21113960 1113961

8 1 + λ0( 1113857η2Υ (41)



zπFm

zλ0 minus

a2 λ0 minus 1( 1113857β2 + 41113960 1113961ψ

8 λ0 + 1( 11138572η2

(42)







UFr

a2 λ0 minus 1( 1113857β2 + 41113960 11139612


16η2 (43)




zUFr

zλ0

a2(β + 2) λ0 minus 1( 1113857β2 + 41113960 1113961σ16η3

(44)





radic

2 β4 + β3 + β21113872 1113873lt 0 (45)


Δ

















smand pF




00

01

01

02

02

03

03

05

05

04

04

06 07β

wF

w


00

0201

02

03 0504

04

06

06

08

1

07β

psmF

psm (psr)Fpsr


sm and pFsr and β















5 Conclusions


00

02

01

03

04

05

06

1 2 3 4λ

psmF

wF

FpsrpF



srand λ




00

02

01

03

04

05

05 1 215

07

08

06

λπm

F

πm


0

02

01

03

04

05

06

0 0201 03 04 05 06 07

πrπm

β

(a)

0

02

01

03

04

05

06

0 0201 03 04 05 06 07β

πmF

Ur

(b)





Data Availability




Acknowledgments


References


00 01 02 03 04

05

1

2

15

λ

Ur

(a)

0

05

1

2

15

05 1 215λ

Ur

(b)










































Journal of












Journal of







Volume 2018






Volume 2018














smand pF




00

01

01

02

02

03

03

05

05

04

04

06 07β

wF

w


00

0201

02

03 0504

04

06

06

08

1

07β

psmF

psm (psr)Fpsr


sm and pFsr and β















5 Conclusions


00

02

01

03

04

05

06

1 2 3 4λ

psmF

wF

FpsrpF



srand λ




00

02

01

03

04

05

05 1 215

07

08

06

λπm

F

πm


0

02

01

03

04

05

06

0 0201 03 04 05 06 07

πrπm

β

(a)

0

02

01

03

04

05

06

0 0201 03 04 05 06 07β

πmF

Ur

(b)





Data Availability




Acknowledgments


References


00 01 02 03 04

05

1

2

15

λ

Ur

(a)

0

05

1

2

15

05 1 215λ

Ur

(b)










































Journal of












Journal of







Volume 2018






Volume 2018





















5 Conclusions


00

02

01

03

04

05

06

1 2 3 4λ

psmF

wF

FpsrpF



srand λ




00

02

01

03

04

05

05 1 215

07

08

06

λπm

F

πm


0

02

01

03

04

05

06

0 0201 03 04 05 06 07

πrπm

β

(a)

0

02

01

03

04

05

06

0 0201 03 04 05 06 07β

πmF

Ur

(b)





Data Availability




Acknowledgments


References


00 01 02 03 04

05

1

2

15

λ

Ur

(a)

0

05

1

2

15

05 1 215λ

Ur

(b)










































Journal of












Journal of







Volume 2018






Volume 2018










00

02

01

03

04

05

05 1 215

07

08

06

λπm

F

πm


0

02

01

03

04

05

06

0 0201 03 04 05 06 07

πrπm

β

(a)

0

02

01

03

04

05

06

0 0201 03 04 05 06 07β

πmF

Ur

(b)





Data Availability




Acknowledgments


References


00 01 02 03 04

05

1

2

15

λ

Ur

(a)

0

05

1

2

15

05 1 215λ

Ur

(b)










































Journal of












Journal of







Volume 2018






Volume 2018










Data Availability




Acknowledgments


References


00 01 02 03 04

05

1

2

15

λ

Ur

(a)

0

05

1

2

15

05 1 215λ

Ur

(b)










































Journal of












Journal of







Volume 2018






Volume 2018















































Journal of












Journal of







Volume 2018






Volume 2018








OptimalPricingStrategiesinaProductandServiceSupply ...of retailer’s fairness concerns on the price...

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Transcript of OptimalPricingStrategiesinaProductandServiceSupply ...of retailer’s fairness concerns on the price...