Research Article Cooperative Advertising in a...
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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 607184, 16 pageshttp://dx.doi.org/10.1155/2013/607184
Research ArticleCooperative Advertising in a Supply Chain withHorizontal Competition
Yi He,1 Qinglong Gou,1 Chunxu Wu,1 and Xiaohang Yue2
1 School of Management, University of Science and Technology of China, Hefei, Anhui 230026, China2 Sheldon B Lubar School of Business, University of Wisconsin-Milwaukee, Milwaukee, WI 53201-0413, USA
Correspondence should be addressed to Qinglong Gou; [email protected]
Received 27 April 2013; Accepted 20 May 2013
Academic Editor: Tsan-Ming Choi
Copyright Β© 2013 Yi He et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Cooperative advertising programs are usually provided by manufacturers to stimulate retailers investing more in local advertisingto increase the sales of their products or services. While previous literature on cooperative advertising mainly focuses on a βsingle-manufacturer single-retailerβ framework, the decision-making framework with βmultiple-manufacturer single-retailerβ becomesmore realistic because of the increasing power of retailers as well as the increased competition among the manufacturers. In view ofthis, in this paper we investigate the cooperative advertising program in a βtwo-manufacturer single-retailerβ supply chain in threedifferent scenarios; that is, (i) each channel membermakes decisions independently; (ii) the retailer is vertically integrated with onemanufacturer; (iii) two manufacturers are horizontally integrated. Utilizing differential game theory, the open-loop equilibrium-advertising strategies of each channel member are obtained and compared. Also, we investigate the effects of competitive intensityon the firmβs profit in three different scenarios by using the numerical analysis.
1. Introduction
To increase the sales of their products or services, somemanufacturers or service providers utilize cooperative adver-tising programs, through which they share a part of theretailerβs advertising cost, to stimulate retailers advertisingmore on their products or services. Generally, advertising canbe divided into national advertising and local advertising.Theformer focuses on building brand image about the productsor services.The latter is often price oriented to stimulate con-sumer to purchase the products or services at once. Supportedby subsidies from a manufacturerβs cooperative advertisingprogram, retailers would always increase their local advertis-ing expenditures and thus improve their profits [1].
Surveys showed that, for many manufacturers or serviceproviders such as General Electric, their advertising budgetsto retailers via cooperative advertising programs are morethree times of that they spent on national advertising [2]. Fur-ther, Dant and Berger found that 25β40% of local advertise-ments are cooperatively funded [3]. Total expenditures on co-operative advertising in 2000 were estimated at $15 billion,compared to $900 million in 1970, nearly a four-fold increase
in real terms [4]. In 2010, about $50 billion was spent on co-operative advertising programs [5].
The tendency toward increased spending on coopera-tive advertising has received significant attention from re-searchers. The cooperative advertising models under studycan be divided into two categories: static models [1, 6β12] anddynamicmodels [13β19]. However, these studiesmainly focuson a βsingle-manufacturer single-retailerβ framework.
Retailers in todayβs market are increasingly more power-ful thanmanufacturers. Useem found that sales throughWal-Mart accounted for 17% of P&Gβs total sales in 2002, 39% ofTandyβs, and over 10% for many other large manufacturers[20]. Tesco is the largest grocer in the United Kingdom, ac-counting for almost 30% of the supermarket sales [21]. HomeDepot and Lowe have more than 50% of the home improve-ment market [22]. As retailers becomemore dominant, man-ufacturers face fierce competition among themselves.Thus, itis necessary to take the competition amongmanufacturers in-to account when studying the cooperative advertising model.
The significant contribution of this paper is that itgeneralizes existing cooperative advertising work on
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2 Mathematical Problems in Engineering
βsingle-manufacturer single-retailerβ framework to the βtwo-manufacturer single-retailerβ framework.This generalizationhas provided new analytical results about how the compe-tition affects the advertising efforts and profit for channelmember. In detail, we study the open-loop equilibrium ad-vertising strategies of each channel member in three differ-ent scenarios, including that (i) each channel member makesdecisions independently; (ii) the retailer is vertically integrat-ed with one manufacturer; (iii) two manufacturers are hori-zontally integrated. Specifically, the following research ques-tions are addressed in this paper. (i) For each scenario, whatare the equilibrium advertising efforts for each channelmem-ber and what is the manufacturerβs optimal participation ratefor the retailerβs local advertising expenditures? (ii)When theretailer integrates with one manufacturer, does the manu-facturer change its decisions about national advertising ex-penditures and participation rates? (iii) How does the hori-zontal integration of two manufacturers affect the decisionsof each channel member?
To answer the above questions, we focus on the coop-erative advertising problem in a βtwo-manufacturer single-retailerβ framework. The dynamic advertising models areproposed based on the Nerlove-Arrow model. Utilizingdifferential game theory, the open-loop equilibrium adver-tising strategies of each channel member are obtained andcompared in three different scenarios.
The remainder of the paper is structured as follows.Previous literature related to our topic is reviewed inSection 2. Section 3 develops the proposed models, and thenthe equilibrium advertising efforts and participation rates inthree different scenarios are discussed. Section 4 offers a nu-merical analysis. Conclusions and suggestions for futureresearch are in Section 5. Proofs for all propositions in thepaper are given in Appendices.
2. Literature Review
Our work is related to several research streams. First is thestream of literature that focuses on cooperative advertising,which can be divided into two main categories: static modelsand dynamic models. A primary static model was proposedby Berger [6], who was the first to analyze cooperativeadvertising. Bergen and Johndeveloped two formalmodels tostudy the effects of the participation rate offered by manu-facturers [7]. By dividing advertising into national and local,Huang et al. were able to study co-op advertising modelsin a static supply chain framework and discuss the channelmembersβ advertising decisions for different relationshipconfigurations between the channel members [1, 8]. Based onthe work of Huang and Li [1], Yue et al. studied the co-opadvertising problem by considering a price discount indemand elasticity market circumstance [9]. Xie and Neyretproposed a more general model, including co-op advertisingand pricing [10]. Further, Seyed Esfahani et al. consideredvertical co-op advertising along with pricing decisions in asupply chain and proved that both the manufacturer andthe retailer reach the highest profits level when they followa cooperation strategy [12]. For the dynamic advertising
models, Chintagunta and Jain extended the work of Nerloveand Arrow [23] to consider the interaction effects of manu-facturer and retailer goodwill on channel sales and developeda dynamic model to study the equilibrium advertising strate-gies in a two-member marketing channel [13]. JΓΈrgensenet al. provided a dynamic model for a cooperative advertisingframework, which allows both channel members to makelong- and short-term advertising efforts to enhance sales andconsumer goodwill [14]. Further, JΓΈrgensen et al. introduceddecreasing marginal returns to goodwill and adopted a moreflexible functional form for the sales dynamics [15]. JΓΈrgensenet al. studied the cooperative advertising program in the casewhere the retailerβs promotions can damage the brand image[16]. Extending the work of JΓΈrgensen et al. [15], Karray andZaccour considered a differential gamemodel for amarketingchannel formed by one manufacturer and one retailer andconcluded that a cooperative advertising program can helpthe manufacturer mitigate the competitive impact of theprivate label [18]. He et al. provided a theoretical analysis ofcooperative advertising plans in a dynamic stochastic supplychain [19].
The above literature is mainly focused on a βsingle-manufacturer single-retailerβ framework. Few studies ad-dress a βmultiple-manufacturer single-retailerβ frameworkor any other framework. Kurtulus and Toktay considered amodel including two competing manufacturers and oneretailer; the result revealed that the retailer can use the formofcategory management and the category shelf space to controlthe intensity of competition between manufacturers to hisbenefit [24]. Adida and DeMiguel studied competition in asupply chain where multiple manufacturers compete inquantities to supply a set of products to multiple risk-averseretailers who compete in quantities to satisfy the uncertainconsumer demand [25]. Cachon and Kok also studied asupply chain system with competing manufacturers and asingle retailer; the results showed that the properties acontractual form exhibits in a one-manufacturer supply chainmay not carry over to the realistic setting in which multiplemanufacturers must compete to sell their goods throughthe same retailer [26]. Further, Lu et al. highlighted theimportance of service frommanufacturers in the interactionsbetween two competing manufacturers and their commonretailer, and their result showed that as themarket base of oneproduct increases, the second manufacturer also benefits butat a lesser amount than the first manufacturer [27]. However,the abovementioned works with multiple manufacturers donot consider cooperative advertising. There are some coop-erative advertising works that focus on a βmultiple-retailerβframework. For example, He et al. [28, 29] extended He et al.[19] by considering the competing retailers, and their resultsshowed that the manufacturerβs support for its retailer ishigher under competition than in its absence.
To our best knowledge, research relating to cooperativeadvertising focused on a βmultiple-manufacturer single-retailerβ framework in the supply chain has not been exploredin literature. In this study, we investigate a cooperative adver-tising model using the βtwo-manufacturer single-retailerβframework.
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Mathematical Problems in Engineering 3
3. Model Description
As shown in Figure 1, we consider a supply chain system con-sisting of two competing manufacturers and one retailer. Thetwo manufacturers produce similar products with differentbrands which are denoted as π, π β {1, 2} that the retailersells simultaneously. The competition is based primarily onthe use of nonprice competitive strategies, namely, the twomanufacturers each advertise their products, and the retaileradvertises two products simultaneously.
We introduce the additional notation in this paper (seeTable 1).
As our goodwill-based model is based upon the modelof Nerlove-Arrow, the changing of the stock of goodwill ofproduct π is given by
β
πΊπ
(π‘) = πππ
β πππ(3βπ)
β πΏπΊπ
, π β {1, 2} , (1)
where 0 < π < 1 is a constant which represents the rivaladvertisingβs negative effect on goodwill, as seen in previousliterature [30]. Next, πΏ > 0 is the diminishing rate of goodwill,which captures the idea that consumers may forget the brandto some extent. National advertising mainly focuses on firmβslong-term objectives such as brand awareness, image, andcredibility [31]. Therefore, we only take the effect of nationaladvertising into the stock of goodwill here. Further, the initialgoodwill of the two products is denoted as
πΊ1(0) = πΊ
10
β₯ 0, πΊ2(0) = πΊ
20
β₯ 0, (2)
and the sales ππ
(π‘) of the two brands along time π‘ satisfy
ππ(π‘) = max {0, πΌπΊ
π
+ πππ π
β πππ (3βπ)
} , π β {1, 2} . (3)
In (3), πΌ, π, π are all positive constants. For the sake ofsimplicity, we suppose that the influence coefficient is identi-cal for these two products. In (3), the item πΌπΊ
π
represents thelong-term effect of national advertising on sales, and the itemπππ π
represents the effect of the retailerβs local advertising onproduct π. As in previous research such as JΓΈrgensen et al. [14],we only take the promotion effect of local advertising into thefunctions of sales herewithout the stock of goodwill.The itemβπππ (3βπ)
illustrates the rival local advertisingβs negative effecton sales. Next, π > π > 0, which implies that the effects ofrival advertising are generally smaller than the effects of oneβsown advertising effect, which is a fairly common assumptionin the relevant literature [30].
The advertising cost functions are quadratic with respectto marketing efforts, namely,
πΆ (πππ(π‘)) =
π2
ππ
(π‘)
2, πΆ (π
π π
) =π2
π π
(π‘)
2, π β {1, 2} .
(4)
This assumption about the advertising cost function iscommonly found in literature [32]. The convex cost functionimplies increasing marginal cost of effort.
Without considering advertising expenditures, themarginal profit of manufacturer π is assumed as π
ππ
β₯ 0,and the marginal profit of the retailer selling the product π is
Manufacturer 1(M1)
Manufacturer 2 (M2)
Customer
Product 1 Product 2
Retailer R
UM1 UM2
UR1 UR2
Figure 1
Table 1: Notation.
π‘ Time π‘, π‘ β₯ 0πΊπ
(π‘) Goodwill of the product π at time π‘, π β {1, 2}πππ
(π‘) Manufacturer πβs national advertising efforts at time π‘ππ π
(π‘) Retailerβs local advertising efforts for product π at time π‘ππ
(π‘) Sales of product π along time π‘π β [0, 1] The national advertisingβs competition coefficientπ β [0, 1] The local advertisingβs competition coefficient
ππ
β [0, 1]Manufacturer πβs participation rate for the retailerβsadvertising cost
πππ
β₯
0Marginal profit of manufacturer π
ππ π
β₯ 0 Marginal profit of the retailer sells the product ππΏ > 0 Diminishing rate of goodwillπ > 0 Discount rate of the manufacturers and the retailerπππ
, ππ
Profit functions forππ and π , respectively
π½ππ
, π½π
Current value of profit functions forππ and π ,respectively
ππ π
β₯ 0. The profit functions of the two manufacturers arethen
πππ(π‘) = π
ππ
ππ(π‘) β
1
2π2
ππ
(π‘) β1
2ππ
π2
π π
(π‘) , π β {1, 2} ,
(5)
and the profit function of the retailer is
ππ (π‘) =
2
β
π=1
(ππ π
ππ(π‘) β
1
2(1 β π
π
) π2
π π
(π‘)) . (6)
In this paper, we assume the participation rate is aconstant over time for the following reasons. (i) Althoughmuch more literature assumes that the participation ratechanges along time [19], a changing participation rate is socomplex that there are no cooperative advertising programsin practice. In the empirical studies of Nagler [4], all the 1470plans explicitly listed a single fixed participation rate. If a firm
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4 Mathematical Problems in Engineering
provides a cooperative advertising program with a changingparticipation rate, the manufacturer would have to knowthe retailerβs daily advertising cost exactly, which is muchmore difficult than learning the whole advertising cost overa certain period of time. (ii) Even, in previous studies whichmodel the participation rate as a function of time, the finaloptimal decision for participation rates were all constant overtime [14, 18, 19, 28].
Please note that πππ
and ππ
are manufacturerβs decisionvariables, and π
π π
is retailerβs decision variable. Then weconsider a two-stage game in this paper. The manufacturersoffer their participation rates for the retailerβs local advertisingexpenditure at stage 1, and then the manufacturers andretailer determine their advertising efforts along time π‘ simul-taneously at stage 2. We firstly keep the participation ratesππ
(π = 1, 2) as fixed, calculate the advertising efforts of themanufacturers and retailer utilizing differential game theory,and then decide the manufacturersβ optimal participationrates.
3.1. Each ChannelMemberMakes Decisions Independently. Inthis scenario, each channel member makes decisions inde-pendently, and the profit functions of all channel membersare given by (5) and (6). Note the profits for all the channelmembers changes along with time π‘. Each channel member,then, strives to maximize the current values of its profit.Witha common discount rate π > 0 and for the sales π
π
β₯ 0, wehave
maxπππβ₯0,1β₯ππβ₯0
π½ππ
= β«
+β
0
πβππ‘
πππ(π‘) ππ‘, π β {1, 2} , (7)
and, for the retailer, we have
maxππ 1β₯0,ππ 2β₯0
π½π
= β«
+β
0
πβππ‘
ππ (π‘) ππ‘. (8)
Taking (1) into account, we get the current value Hamil-tonian of two manufacturers as
π»π1
= ππ1
+ π11
(ππ1
β πππ2
β πΏπΊ1
)
+ π12
(ππ2
β πππ1
β πΏπΊ2
) ,
π»π2
= ππ2
+ π21
(ππ1
β πππ2
β πΏπΊ1
)
+ π22
(ππ2
β πππ1
β πΏπΊ2
) .
(9)
Similarly, we get the retailerβs current value Hamiltonianas
π»π
= ππ
+ π31
(ππ1
β πππ2
β πΏπΊ1
)
+ π32
(ππ2
β πππ1
β πΏπΊ2
) ,
(10)
where ππ1
and ππ2
(π = 1, 2, 3) represent the costate variablesin the firmβs problem corresponding to the firmβs goodwilllevels.
Then using the necessary conditions for equilibrium, weobtain the following results.
Proposition 1. When each channel member makes decisionsindependently and the participation rates π
π
(π = 1, 2) arefixed, the equilibrium advertising efforts for twomanufacturerson their products along time π‘ are all constants; that is,
π(1)
ππ
=πΌπππ
(π + πΏ), π β {1, 2} . (11)
Proposition 1 illustrates the following facts. (i) Whateverthe participation rate that the manufacturer undertakes forthe retailerβs advertising cost, the manufacturerβs equilibriumnational advertising efforts are kept the same and are justlinear with its own marginal profit. The larger the marginalprofit, the more the manufacturer would spend on nationaladvertising. (ii) The manufacturerβs equilibrium advertisingefforts are determined by the item πΌπ
ππ
/(π + πΏ), which isaimed at maintaining the long-term effect of advertising.Specifically, this itemdecreases sharply when the diminishingrate of consumer goodwill becomes very large or the decisionmakers are more short sighted. Therefore, when the decisionmakers do not feel confident in future, or the customerβsgoodwill diminishes quickly, the advertising efforts woulddrop.
Further, we obtain the retailerβs equilibrium advertisingefforts for two brands as follows.
Proposition 2. When each channel member makes decisionsindependently and the participation rates π
π
(π = 1, 2) arefixed, the retailerβs equilibrium advertising efforts for the twobrands along time π‘ are all constants:
π(1)
π π
=
{{
{{
{
πππ π
β πππ (3βπ)
1 β ππ
ππ πππ π
β πππ (3βπ)
β₯ 0,
0 πππ π
π β {1, 2} .
(12)
Proposition 2 holds the following managerial implica-tions. (i) When condition ππ
π π
β πππ ,(3βπ)
β₯ 0, π β {1, 2}is satisfied, a higher participation rate leads the retailer tospend more on local advertising but the advertising effortsare independent of the participation rate which the othermanufacturer provides to the retailer. Therefore, the manu-facturer can use the participation rate to guide the retailerβsadvertising efforts for his product. (ii) The equilibrium localadvertising efforts on product π are increased by the retailerβsmarginal profit of product π.
Furthermore, when condition π > π > 0 is satisfied, weget the following results by (12).
(i) If condition πππ 2
β πππ 1
< 0 holds, we have π(1)π 1
=
(πππ 1
β πππ 2
)/(1 β π1
) and π(1)π 2
= 0. Because themarginal profit which the retailer obtains from prod-uct 2 is extremely small, the benefit ofπ(1)
π 2
from prod-uct 2 does not offset the loss from product 1. Thus,the retailer would not advertise product 2. In otherwords, whatever participation rate manufacturer 2offers, the retailer would never advertise product 2.Under this situation, manufacturer 2 only can changethe situation of no-local-advertising efforts on hisproduct by offering the retailer a higher margin.
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Mathematical Problems in Engineering 5
(ii) If conditionsπππ 1
βπππ 2
β₯ 0 andπππ 2
βπππ 1
β₯ 0hold,we can get π(1)
π 1
= (πππ 1
β πππ 2
)/(1 β π1
) and π(1)π 2
=
(πππ 2
β πππ 1
)/(1 β π2
). In this situation, advertisingfor two brands would lead to so much gain for theretailer that the retailerwould advertise both productsat a certain level.
(iii) If condition πππ 1
β πππ 2
< 0 holds, we have π(1)π 1
= 0
and π(1)π 2
= (πππ 2
β πππ 1
)/(1 β π2
). This situationis similar to the first situation; the retailer wouldadvertise product 2, but not product 1.
When the two manufacturersβ participation rates arefixed, the equilibrium advertising efforts for all channelmem-bers are given by Propositions 1 and 2. Based on these results,we can work out the stock of goodwill for the two productsas well as for the current value of profits for all channel mem-bers, which is given by Proposition 3.
Proposition 3. When each channel member makes decisionsindependently and their advertising efforts are kept as con-stants, that is, π
ππ
(π‘) = π(1)
ππ
and ππ π
(π‘) = π(1)
π π
, π = 1, 2, thenthe accumulated goodwill of two products along time π‘ is
πΊπ(π‘) = π·
π
πβπΏπ‘
+ πΊ(1)
πSS, π β {1, 2} , (13)
where π·π
= πΊπ0
β (π(1)
ππ
β ππ(1)
π,(3βπ)
)/πΏ and πΊ(1)πSS = (π
(1)
ππ
β
ππ(1)
π,(3βπ)
)/πΏ, π = 1, 2. πΊ(1)πSS is the steady-state goodwill for
product π when π‘ β β.
From (13), we obtain the following facts: (i) the steady-state goodwill for product π increases with manufacturer 1βsnational advertising efforts; (ii) the steady-state goodwill forproduct π decreases with the rival manufacturerβs nationaladvertising efforts because of the competitive effect; (iii)steady-state goodwill is only affected by the manufacturerβsadvertising efforts because local advertising has only aninstant promotion effect that has no impact on the stock ofgoodwill; (iv) when the diminishing rate of goodwill becomesvery large, steady-state goodwill decreases.
Substituting (11)β(13) into (7) and (8) and with theparticipation rates π
π
(π = 1, 2) being fixed, we get thecurrent value ofmanufacturer 1βs profit under the equilibriumcondition as follows:
π½(1)
ππ
=π·π
πΌπππ
π + πΏ+
πΌπππ
(π(1)
ππ
β ππ(1)
π,(3βπ)
)
ππΏ
+
πππ
(ππ(1)
π π
β ππ(1)
π ,(3βπ)
)
π
β
(π(1)
ππ
)2
+ ππ
(π(1)
π π
)2
2π, π β {1, 2} .
(14)
The current value of the retailerβs profit is
π½(1)
π
=π·1
πΌππ 1
+ π·2
πΌππ 2
π + πΏ
+
πΌππ 1
(π(1)
π1
β ππ(1)
π2
) + πΌππ 2
(π(1)
π2
β ππ(1)
π1
)
ππΏ
β1 β π1
2(π(1)
π 1
)2
+
ππ 1
(ππ(1)
π 1
β ππ(1)
π 2
)
π
+
ππ 2
(ππ(1)
π 2
β ππ(1)
π 1
)
πβ1 β π2
2(π(1)
π 2
)2
,
(15)
whereπ·π
= πΊπ0
β (π(1)
ππ
β ππ(1)
π,(3βπ)
)/πΏ, π β {1, 2}.Differentiating π½(1)
ππ
with the participation rate ππ
, we getoptimal participation rates, from are given by Proposition 4.
Proposition 4. When all the channel membersmake decisionsindependently, the optimal participation rates that the twomanufacturers provide to the retailer under the equilibriumcondition are
π(1)
π
=
{{
{{
{
π (2πππ
β ππ π
) + πππ (3βπ)
π (2πππ
+ ππ π
) β πππ (3βπ)
ππ πππ
β₯(πππ π
β πππ (3βπ)
)
2π,
0 πππ π
π β {1, 2} .
(16)
For (16), the restraining condition ππ1
β₯ (πππ 1
β
πππ 2
)/2π implies that manufacturer 1 is willing to provide theparticipation rate with the retailer only when he can obtain alarge enough marginal profit. Differentiating π(1)
1
from ππ1
,ππ 1
, and ππ 2
, and knowing that ππ1
β₯ (πππ 1
β πππ 2
)/2π, wefind that ππ(1)
1
/πππ1
> 0; ππ(1)1
/πππ 1
< 0; ππ(1)1
/πππ 2
> 0.The above expressions show that (i) when the manufacturerβsmarginal profit increases, he would offer a high participationrate to the retailer; (ii) when a high marginal profit would beobtained by the retailer, the manufacturer has less incentiveto offer a high participation rate for the cooperative program;(iii) manufacturer 1 would offer a high participation rate if theretailer obtains a larger marginal profit from product 2.
Furthermore, substituting the optimal participation ratesinto (12), we find that the retailerβs equilibrium local advertis-ing efforts on the two brands are all constants, that is,
π(1)
π π
=
{{{{{{{{{{
{{{{{{{{{{
{
ππππ
+1
2πππ π
β1
2πππ (3βπ)
,
if 0 <(πππ π
β πππ (3βπ)
)
2πβ€ πππ
,
πππ π
β πππ (3βπ)
,
if πππ
<(πππ π
β πππ (3βπ)
)
2π,
0 else
π β {1, 2} .
(17)
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6 Mathematical Problems in Engineering
Equation (17) shows that the retailerβs equilibrium adver-tising level π(1)
π π
for a product is not only linear with hisown marginal profit π
π π
, but also linear with the manufac-turerβs marginal profit π
ππ
if the conditions 0 < (πππ π
β
πππ (3βπ)
)/2π β€ πππ
hold. Supposing that πππ
+ ππ π
= ππ
isthe channel marginal profit of one product, the equilibriumadvertising level π(1)
π π
can be rewritten as π(1)π π
= ππππ
/2 +
πππ
/2 β πππ ,(3βπ)
/2 if and only if 0 < πππ π
β πππ ,(3βπ)
β€ 2ππππ
,π = 1, 2 hold. The channel marginal profit of product π isnot changed in the short term; therefore, the above equationsimply that the retailerβs equilibrium advertising efforts areindependent of the marginal profit which the retailer obtainsfrom product π.
3.2. Retailer Integrates with a Manufacturer. In second sce-nario, the retailer integrates with one of the manufacturers.We assume thismanufacturer isπ1.Then, the profit functionof the integration system is π
π1,π
= ππ1
+ ππ
, and theobjective of integration system is
maxππ1β₯0,ππ 1β₯0,ππ 2β₯0
π½π1,π
= β«
+β
0
πβππ‘
(ππ1
+ ππ
) ππ‘. (18)
Further, the objective of manufacturer 2 is
max1β₯π2β₯0,ππ2β₯0
π½π2
= β«
+β
0
πβππ‘
ππ2
ππ‘. (19)
Taking state equation (1) into account, the current valueHamiltonian of the vertical integration system (π1 and π ) is
π»π1,π
= ππ1
+ ππ
+ πΎ11
(ππ1
β πππ2
β πΏπΊ1
)
+ πΎ12
(ππ2
β πππ1
β πΏπΊ2
) ,
(20)
and that of manufacturer 2 is
π»π2
= ππ2
+ πΎ21
(ππ1
β πππ2
β πΏπΊ1
)
+ πΎ22
(ππ2
β πππ1
β πΏπΊ2
) ,
(21)
where πΎπ1
and πΎπ2
(π = 1, 2) represent the costate variables inthe channel memberβs problem corresponding to the firmβsgoodwill level.
Then using (20) and (21), we obtain the following results.
Proposition 5. When the retailer integrates with manufac-turer 1 and the participation rate π
2
provided by manufacturer2 is fixed, the equilibrium advertising efforts of manufacturer 2are constant, that is,
π(2)
π2
=πΌππ2
(π + πΏ). (22)
Compared to Proposition 1, we find that manufacturer 2βsequilibrium advertising level is the same, which implies thatmanufacturer 2βs advertising level does not depend on theintegration between manufacturer 1 and retailer.
Proposition 6. When the retailer integrates with manufac-turer 1 and the participation rate π
2
provided by manufacturer
2 is fixed, manufacturer 1βs equilibrium advertising efforts areconstant, that is,
π(2)
π1
={
{
{
πΌπ1
β ππΌππ 2
π + πΏππ π1
β₯ πππ 2
,
0 πππ π,
(23)
and the retailerβs equilibrium advertising efforts for the twobrands along time π‘ are all constants, that is,
π(2)
π 1
={
{
{
ππ1
β πππ 2
ππ ππ1
β πππ 2
β₯ 0,
0 πππ π,
(24)
π(2)
π 2
=
{{
{{
{
πππ 2
β ππ1
1 β π2
ππ πππ 2
β ππ1
β₯ 0,
0 πππ π,
(25)
where π1
= ππ1
+ ππ 1
.
Proposition 6 shows the following trends. If the conditionπ1
> πππ 2
holds, the national advertising efforts π(2)π1
areaffected by the item (πΌπ
1
β ππΌππ 2
)/(π + πΏ). From this item,we know that the larger the channel marginal profit ofproduct 1 (π
1
), the more the integration system would spendon national advertising for product 1. As opposed to the firstscenario, in this scenario the national advertising efforts arealso affected by the rival productβs marginal profit π
π 2
. Whenππ 2
is increased, the integration system would decreasenational advertising efforts for product 1 and thus decreaseproduct 1βs adverse influence on product 2. Further, if thechannel marginal profit of product 1 is too small, the integra-tion system would not advertise product 1.
If we subtract (17) from (24), we get
Ξππ 1
=
{{{{{{{{
{{{{{{{{
{
πππ1
if 2πππ1
< πππ 1
β πππ 2
,
πππ 1
β πππ 2
2if 0 < ππ
π 1
β πππ 2
β€ 2πππ1
,
ππ1
β πππ 2
if β πππ1
< πππ 1
β πππ 2
β€ 0,
0 else,(26)
where π1
= ππ1
+ ππ 1
.It is easy to prove that Ξπ
π 1
given by (26) is nonnegative,whichmeans that the integration between the retailer and themanufacturer would increase the retailerβs equilibrium localadvertising efforts for product 1.
Furthermore, combining (24) and (25), we obtain similarmanagerial implications as the results of Proposition 2, butwe also find some differences.
(i) When πππ 2
β ππ1
< 0 holds, we have π(2)π 2
= 0 andπ(2)
π 1
= ππ1
β πππ 2
. Note that π1
= ππ1
+ ππ 1
>
ππ 1
, which implies that the integration between theretailer andmanufacturer 1 would lead to the increasein the retailerβs local advertising threshold for product2 and would also increase the retailerβs equilibriumadvertising efforts for product 1.
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Mathematical Problems in Engineering 7
(ii) If conditions ππ1
βπππ 2
β₯ 0 and πππ 2
βππ1
β₯ 0 hold,π(2)
π 2
= ππ1
β πππ 2
and π(2)π 2
= (πππ 2
β ππ1
)/(1 β π2
).In this situation, the retailer increases the equilibriumadvertising efforts for product 1 and decreases effortsfor product 2.
(iii) When ππ1
β πππ 2
< 0 holds, π(2)π 1
= 0 and π(2)π 2
=
(πππ 2
β ππ1
)/(1 β π2
). Note that π1
> ππ 1
, whichimplies that the integration between the retailer andmanufacturer 1 reduces the retailerβs local advertisingthreshold for product 1 and also decreases the retail-erβs equilibrium advertising efforts for product 2.
We can calculate the stock of goodwill for the two pro-ducts and the current value of profits for all channelmembers,which are given by Proposition 7.
Proposition 7. When the retailer integrates with manufac-turer M1, and their advertising efforts are kept constant, thatis, πππ
(π‘) = π(2)
ππ
and ππ π
(π‘) = π(2)
π π
, π = 1, 2, then theaccumulated goodwill of the two products along time π‘ is
πΊπ(π‘) = πΈ
π
πβπΏπ‘
+ πΊ(2)
πSS, π β {1, 2} , (27)
where πΊ(2)πSS = (π
(2)
ππ
β ππ(2)
π,(3βπ)
)/πΏ, πΈπ
= πΊπ0
β (π(2)
ππ
β
ππ(2)
π,(3βπ)
)/πΏ, π = 1, 2. πΊ(2)πSS is the steady-state goodwill for
product π when π‘ β β.
Substituting (22) through (25) and (27) into (18) and (19),and assuming that the participation rates π
π
(π = 1, 2) arefixed, we get the current value of the integration systemβsprofit as follows:
π½(2)
π1,π
=πΈ1
πΌπ1
+ πΈ2
πΌππ 2
π + πΏ
+
πΌπ1
(π(2)
π1
β ππ(2)
π2
) + πΌππ 2
(π(2)
π2
β ππ(2)
π1
)
ππΏ
β
(π(2)
π1
)2
+ (π(2)
π 1
)2
2π+
π1
(ππ(2)
π 1
β ππ(2)
π 2
)
π
+
ππ 2
(ππ(2)
π 2
β ππ(2)
π 1
)
πβ
(1 β π2
) (π(2)
π 2
)2
2π,
(28)
and the profit for manufacturer 2 is
π½(2)
π2
=πΈ2
πΌππ2
π + πΏ+
πΌππ2
(π(2)
π2
β ππ(2)
π1
)
ππΏ
+
ππ2
(ππ(2)
π 2
β ππ(2)
π 1
)
πβ
(π(2)
π2
)2
β π2
(π(2)
π 2
)2
2π,
(29)
where πΈπ
= πΊπ0
β (π(2)
ππ
β ππ(2)
π,(3βπ)
)/πΏ, π = 1, 2.
Differentiating π½(2)π2
from the participation rate π2
, weget the optimal participation rate, which is given byProposition 8.
Proposition 8. When the retailer integrates with manufac-turer 1, and the advertising levels are kept as constants, that is,πππ
(π‘) = π(2)
ππ
,ππ π
(π‘) = π(2)
π π
, π = 1, 2, the optimal participationrate which manufacturer 2 provides is
π(2)
2
=
{{
{{
{
π (2ππ2
β ππ 2
) + ππ1
π (2ππ2
+ ππ 2
) β ππ1
ππ ππ2
β₯(πππ 2
β ππ1
)
2π,
0 πππ π,
(30)
where π1
= ππ1
+ ππ 1
.
Subtracting (16) from (30), we have
Ξπ2
=
{{{{{{{{{{{{{{{{
{{{{{{{{{{{{{{{{
{
4ππππ1
ππ2
πΏ1
πΏ2
if(πππ 2
β πππ 1
)
2πβ€ ππ2
,
π (2ππ2
β ππ 2
) + ππ1
π (2ππ2
+ ππ 2
) β ππ1
if ππ2
<(πππ 2
β πππ 1
)
2πβ€ ππ2
+πππ1
2π,
0 else,
(31)
where πΏ1
= 2πππ2
+πππ 2
βππ1
and πΏ2
= 2πππ2
+πππ 2
βπππ 1
.We can prove that (31) is nonnegative, which implies that
manufacturer 2 would increase his participation rate to theretailer when the retailer integrates with manufacturer 1.
Further, substituting the optimal participation rate into(25), we see that the retailerβs equilibrium local advertisinglevel for product 2 is constant, that is,
π(2)
π 2
=
{{{{{{{
{{{{{{{
{
πππ 2
β ππ1
if ππ2
<(πππ 2
β ππ1
)
2π,
πππ2
+1
2πππ 2
β1
2ππ1
if 0 <(πππ 2
β ππ1
)
2πβ€ ππ2
,
0 else,
(32)
where π1
= ππ1
+ ππ 1
.
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8 Mathematical Problems in Engineering
Subtracting (17) from (32), we have
Ξππ 2
=
{{{{{{{{{{{{{{{{{{{{{{{{{{{{{
{{{{{{{{{{{{{{{{{{{{{{{{{{{{{
{
βπππ1
if 2πππ2
+ πππ1
< πππ 2
β πππ 1
2πππ2
β (πππ 2
β πππ 1
) β πππ1
2
if 2πππ2
< πππ 2
β πππ 1
β€ 2πππ2
+ πππ1
βπππ1
2
if πππ1
< πππ 2
β πππ 1
β€ 2πππ2
βπππ2
β1
2πππ 2
+1
2πππ 1
if 0 < πππ 2
β πππ 1
β€ πππ1
0 else.(33)
Note that the result of (33) is less than zero; the intuitionbehind this can be explained as follows. When the retailerintegrates with manufacturer 1, the retailer would alwaysreduce the local advertising efforts for product 2 to decreasethe competitive influence on product 1.
3.3. The Two Manufacturers Are Horizontally Integrated.When the two manufacturers integrated, it can be seen as asingle firmwith two different brands in the same product cat-egory. Examples in practice include Lenovo. IBMβs personalcomputing division was acquired by Lenovo in 2004, andthe PC of IBM became a subbrand of Lenovo Group namedβThinkpad.βThis is a historical precedent of twomanufactur-ers behaving as a single player, yet, as far as we know, previousresearches on dynamic cooperative advertising programshave never studied such scenario, a single manufacturer withtwo different brands. Most previous research investigateda βsingle-manufacturer single-retailerβ supply chain with asingle brand/product.When themanufacturer advertises twodifferent brands, the result does change; therefore, the thirdscenario must be considered. In this scenario, the integrationsystemβs profit function is π
π1,π2
= ππ1
+ ππ2
, so theobjective is
maxππ1β₯0,ππ2β₯0
1β₯ππβ₯0
π½π1,π2
= β«
+β
0
πβππ‘
(ππ1
+ ππ2
) ππ‘, (34)
and the retailerβs objective is
maxππ 1β₯0,ππ 2β₯0
π½π
= β«
+β
0
πβππ‘
ππ
ππ‘. (35)
The current value Hamiltonian of the integration system(π1 andπ2) is
π»π1,π2
= ππ1
+ ππ2
+ ]11
(ππ1
β πππ2
β πΏπΊ1
)
+ ]12
(ππ2
β πππ1
β πΏπΊ2
) ,
(36)
and that of the retailer isπ»π
= ππ
+ ]21
(ππ1
β πππ2
β πΏπΊ1
)
+ ]22
(ππ2
β πππ1
β πΏπΊ2
) ,
(37)
where ]π1
and ]π2
(π = 1, 2) are the costate variables to thefirmβs goodwill levels.
Using the necessary conditions for equilibrium,we get thefollowing results.
Proposition 9. When the two manufacturers are horizontallyintegrated and the participation rate π
π
(π = 1, 2) is kept fixed,the equilibrium national advertising efforts for the two manu-facturers are all constants, that is,
π(3)
ππ
={
{
{
πΌ (πππ
β πππ(3βπ)
)
π + πΏππ πππ
β₯ πππ(3βπ)
,
0 πππ π
π β {1, 2} ,
(38)
and the retailerβs equilibrium local advertising efforts for thetwo products are all constants, that is,
π(3)
π π
=
{{
{{
{
πππ π
β πππ (3βπ)
1 β ππ
ππ πππ π
β πππ (3βπ)
β₯ 0,
0 πππ π
π β {1, 2} .
(39)
Note that the retailerβs equilibrium local advertisingefforts given by (39) are just the same as Proposition 2. Thisresult implies that whether the two manufacturers integratewith each other or not, the retailer always keeps the same localadvertising efforts for products 1 and 2 only if the participa-tion rates are not changed.
In addition, comparing (38) with the results ofProposition 1, we have
Ξπππ
=
{{
{{
{
βπΌπππ3βπ
π + πΏif πππ
β₯ πππ,3βπ
,
βπΌππ1
π + πΏelse
π β {1, 2} .
(40)
Equation (40) illustrates the following fact. When thetwo manufacturers integrate as a horizontal alliance, theywould decrease their equilibrium advertising efforts to avoidinternal conflict. Specifically, combing the conditions of (38)we have π
ππ
= 0 and ππ,(3βπ)
= (πΌππ3βπ
β πΌππππ
)/(π + πΏ) ifthe condition π
ππ
< πππ,3βπ
holds. This implies that whenthe marginal profit of one product for the manufacturer israther low, the horizontal integration system would invest innational advertising only for the other product.
Proposition 10. When the two manufacturers are horizon-tally integrated and all channel membersβ advertising efforts arekept as constants, that is, π
ππ
= π(3)
ππ
and ππ π
= π(3)
π π
, then theaccumulated goodwill for the two products along time π‘ is
πΊπ(π‘) = πΉ
π
πβπΏπ‘
+ πΊ(3)
πSS, π β {1, 2} , (41)
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Mathematical Problems in Engineering 9
where πΊ(3)πSS = (π
(3)
ππ
β ππ(3)
π,(3βπ)
)/πΏ, πΉπ
= πΊπ0
β (π(3)
ππ
β
ππ(3)
π,(3βπ)
)/πΏ, π = 1, 2. πΊ(3)πSS is the steady-state goodwill for
product π when π‘ β β.
Substituting (38), (39), and (41) into (34) and (35) andwith the participation rates π
π
(π = 1, 2) fixed, we get that thecurrent value of profit for the horizontal integration system is
π½(3)
π1,π2
=πΉ1
πΌππ1
+ πΉ2
πΌππ2
π + πΏ
+
πΌππ1
(π(3)
π1
β ππ(3)
π2
) + πΌππ2
(π(3)
π2
β ππ(3)
π1
)
ππΏ
+
ππ1
(ππ(3)
π 1
β ππ(3)
π 2
)
π+
ππ2
(ππ(3)
π 2
β ππ(3)
π 1
)
π
β
(π(3)
π1
)2
+ (π(3)
π2
)2
+ π1
(π(3)
π 1
)2
+ π2
(π(3)
π 2
)2
2π,
(42)
and the current value of the profit for the retailer is
π½(3)
π
=πΉ1
πΌππ 1
+ πΉ2
πΌππ 2
π + πΏ
+
πΌππ 1
(π(3)
π1
β ππ(3)
π2
) + πΌππ 2
(π(3)
π2
β ππ(3)
π1
)
ππΏ
β1 β π1
2(π(3)
π 1
)2
+
ππ 1
(ππ(3)
π 1
β ππ(3)
π 2
)
π
+
ππ 2
(ππ(3)
π 2
β ππ(3)
π 1
)
πβ1 β π2
2(π(3)
π 2
)2
,
(43)
where πΉπ
= πΊπ0
β (π(3)
ππ
β ππ(3)
π,(3βπ)
)/πΏ, π = 1, 2.Differentiating π½(3)
π1,π2
with the participation rate π1
andπ2
, we get the optimal participation rates:
π(3)
π
=
{{
{{
{
π (2πππ
β ππ π
) + πππ (3βπ)
π (2πππ
+ ππ π
) β πππ (3βπ)
if πππ
β₯(πππ π
β πππ (3βπ)
)
2π
0 else,
π β {1, 2} .
(44)
Note that the above expressions of participation rates areidentical with the results of Proposition 4. Together with theresults of Proposition 9, we find that the equilibrium localadvertising efforts for the two products are identical no mat-ter whether two manufacturers are horizontally integrated ornot.
In this scenario, the equilibrium advertising efforts for thetwo manufacturers become lower, but the equilibrium localadvertising efforts for the two products are not changed.Thiscould lead to the phenomenon that the retailer has so muchpower from advertising the two products that the retailer has
incentive to prevent the horizontal alliance between the twomanufacturers. That is why a successful manufacturerβs hor-izontal integration in a dominant retailer market is very rarein actual practice.
4. Numerical Analysis
In this section, we use numerical analysis to further illustratethe impact of local advertising competition on the profits forall channelmembers and supplement insights from these the-oretical results. In our numerical analysis, we use the follow-ing values to establish ranges for model parameters: πΌ = 12,π = 0.3, πΏ = 0.2, π = 0.2, π = 10, π = 4, π
π1
= 7, ππ2
= 8,ππ 1
= 5, ππ 2
= 4, πΊ10
= 300, and πΊ20
= 320.To obtain qualitative insight regarding how the current
value of each channel memberβs profit varies as competitioncoefficients π and π vary in scenario 1, we keep otherparameters fixed and draw their relationships in Figure 2.
Figure 2 suggests that, in scenario 1, the profit for eachchannel member decreases as competition coefficients π andπ increase. From Figure 2, if competition coefficients π and πequal zero, advertising for one product would not adverselyinfluence the sales of the other product. In this situation, allchannel members would obtain the maximum profits. As thecompetition effects of advertising become intense, the adver-tising effect on sales would scale down, and the profits for allchannel members would decrease.
Figure 3 illustrates the effect of the competition coeffi-cients π and π on the current value of the retailerβs profit inscenario 1 and scenario 3, keeping other parameters fixed.
From Figure 3, we obtain the following facts. (i) Whentwo manufacturers are horizontally integrated, the retailerβsprofit would decrease compared with his profit in scenario 1.Because the retailer would obtain a larger impact on the salesof products in scenario 3, the retailer would have incentive toprevent the horizontal alliance between manufacturers. (ii)Similarly to Figure 2, as competition coefficients π and πincrease, the retailerβs profit would decrease whether the twomanufacturers integrate or not.
Finally, Figure 4 illustrates the impacts of the competitioncoefficients π and π on the current value of the profit ofmanufacturer 2. It suggests the following tendencies: (i)similarly to Figure 2, with competition coefficients π andπ increasing, the profit for manufacturer 2 would decreasewhethermanufacturer 1 integrates with the retailer or not and(ii) whenmanufacturer 1 integrates with the retailer, the profitfor manufacturer 2 would decrease compared to his profit inscenario 1. From Figures 3 and 4, we find that regardless ofwhich two firms (i.e.,π1 and retailer,π2 and retailer, orπ1andπ2) integrate their efforts, the third firm would suffer.
5. Conclusion
Previous research primarily focused on a βsingle-manu-facturer single-retailerβ framework, whereas few studies ad-dress a βmultiple-manufacturer single-retailerβ framework.To fill this gap, this paper investigates the advertising strate-gies for a βtwo-manufacturer single-retailerβ supply chain in
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10 Mathematical Problems in Engineering
Γ103
250
225
200
175
150
125
100
75
50
25
0
The c
urre
nt v
alue
of p
rofit
s
0 0.2 0.4 0.6 0.8
J(1)M1
J(1)M2
J(1)R
The national advertisingβs competition coefficient π
(a)
Γ103
260
240
220
200
180
160
140
120
100
80
600 2 4 6 8 10
J(1)M1
J(1)M2
J(1)R
The c
urre
nt v
alue
of p
rofit
sThe local advertisingβs competition coefficient π
(b)
Figure 2: Relationships between profits and competition coefficients π and π.
Γ103
220
200
180
160
140
120
100
80
600 0.2 0.4 0.6 0.8
J(3)R
J(1)R
The c
urre
nt v
alue
of p
rofit
s
The national advertisingβs competition coefficient π
(a)
Γ103
J(3)R
J(1)R
215
210
205
200
195
190
185
180
175
1700 2 4 6 8 10
The c
urre
nt v
alue
of p
rofit
s
The local advertisingβs competition coefficient π
(b)
Figure 3: Relationships between the retailerβs profit and the competition coefficients π and π.
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Mathematical Problems in Engineering 11
J(2)M2
J(1)M2
0 0.2 0.4 0.6 0.8
140
120
100
80
60
40
0
20
The c
urre
nt v
alue
of p
rofit
s
Γ103
The national advertisingβs competition coefficient π
(a)
J(2)M2
J(1)M2
0 2 4 6 8 10
130
125
120
115
110
105
100
95
90
The c
urre
nt v
alue
of p
rofit
s
Γ103
The local advertisingβs competition coefficient π
(b)
Figure 4: Relationships between the profit ofπ2 and the competition coefficients π and π.
three different scenarios: (i) each channel member makesdecisions independently; (ii) the retailer integrates with oneof themanufacturers; (iii) twomanufacturers are horizontallyintegrated.
Based on the results of the three scenarios, we find the fol-lowing results. (i)Themanufacturerβs equilibrium advertisingefforts are independent of the participation rates that the twomanufacturers offer to the retailer in all three scenarios. (ii)When the retailer integrates with onemanufacturer, the othermanufacturerβs equilibrium advertising efforts would not bechanged. The retailer would enhance the local advertisingefforts for the integrated manufacturer and reduce the localadvertising efforts for the other manufacturer. In response,the other manufacturer would offer a higher (compared toscenario 1) advertising cost participation rate to the retailer.(iii)When the twomanufacturers are horizontally integrated,they would reduce the national advertising efforts to avoidinternal conflict. They also offer the same advertising costparticipation rate to the retailer as in scenario 1. (iv) If any twofirms (i.e.,π1 and retailer,π2 and retailer, orπ1 andπ2)are integrated, the profit of the third firm would decrease. Allthese insights provide important implications and guidelinesfor cooperative advertising program design in supply chainpractice.
It should be noted that our models only consider theeffects of advertising, but this situation may not always hold.In addition, it may be more interesting if we introduce thefactors of pricing and quality to the cooperative advertisingmodel. Additionally, our work on the βtwo-manufacturer
single-retailerβ framework can be extended into a βmultiple-manufacturer single-retailerβ framework.
Appendices
A. Each Channel Member MakesDecisions Independently
Proof of Propositions 1 and 3. When each channel membermakes decisions independently, the current value Hamilto-nian of manufacturer 1 is
π»π1
= ππ1
+ π11
(ππ1
β πππ2
β πΏπΊ1
)
+ π12
(ππ2
β πππ1
β πΏπΊ2
) .
(A.1)
Then we form the Lagrangian:
πΏπ1
= ππ1
+ π11
(ππ1
β πππ2
β πΏπΊ1
)
+ π12
(ππ2
β πππ1
β πΏπΊ2
)
+ π11
(πΌπΊ1
+ πππ 1
β πππ 2
)
+ π12
(πΌπΊ2
+ πππ 2
β πππ 1
) ,
(A.2)
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12 Mathematical Problems in Engineering
the necessary conditions for equilibrium are given by
ππΏπ1
πππ1
= 0, (A.3)
β
π11
= ππ11
βππΏπ1
ππΊ1
, (A.4)
β
π12
= ππ12
βππΏπ1
ππΊ2
, (A.5)
ππΏπ1
ππ1π
> 0, π1π
β₯ 0, π1π
ππΏπ1
ππ1π
= 0, π = 1, 2. (A.6)
Equation (A.3) implies
ππ1
= π11
β ππ12
. (A.7)
Solving (A.4)β(A.6), we getβ
π11
= (π + πΏ) π11
β πΌππ1
β πΌπ1
,
β
π12
= (π + πΏ) π12
β πΌπ2
.
(A.8)
Equation (A.6) implies: π1π
= 0, then substituting π1π
= 0
into (A.8), we getβ
π11
= (π + πΏ) π11
β πΌππ1
,
β
π12
= (π + πΏ) π12
.
(A.9)
Differentiating (A.7) with respect to time and substitutingfor π11
, π12
and their time derivative in (A.9), we getβ
ππ1
= (π + πΏ)ππ1
β πΌππ1
. (A.10)
Solving (A.10) to get the time paths of ππ1
, we find
ππ1(π‘) = πΆ
1
π(π+πΏ)π‘
+πΌππ1
(π + πΏ). (A.11)
Because there is no constraint at π‘ β β, ππ1
shouldsatisfy the free-boundary condition:
limπ‘ββ
ππ1(π‘) < β. (A.12)
Condition (A.12) implies that πΆ1
= 0. Therefore, weobtain the equilibrium advertising effort for manufacturer 1as follows:
π(1)
π1
=πΌππ1
π + πΏ. (A.13)
Similarly consideringmanufacturer 2βs profit maximizingproblem, we obtain the equilibrium advertising level formanufacturer 1 as follows:
π(1)
π2
=πΌππ2
π + πΏ. (A.14)
For (1), we can get the general solutions of (1) as
πΊπ(π‘) = π·
π
πβπΏπ‘
+ πΊπππ
, π β {1, 2} , (A.15)
where πΊπππ
= (πππ
β πππ,(3βπ)
)/πΏ, π = 1, 2.
π·π
is an arbitrary constant. Letting π‘ = 0 in (A.15) andutilizing the initial conditions of (2), we getπ·
π
= πΊπ0
β(πππ
β
πππ,(3βπ)
)/πΏ, π = 1, 2.Substituting (A.13) and (A.14) into (A.15), we find that
πΊπ(π‘) = π·
π
πβπΏπ‘
+ πΊ(1)
πππ
, π β {1, 2} , (A.16)
where π·π
= πΊπ0
β (π(1)
ππ
β ππ(1)
π,(3βπ)
)/πΏ, π = 1, 2 and πΊ(1)πππ
=
(π(1)
ππ
β ππ(1)
π,(3βπ)
)/πΏ, π = 1, 2, when condition 0 β€ π β€
min{ππ1
/ππ2
, ππ2
/ππ1
} holds, the steady-state goodwillπΊπππ
is nonnegative.
Proof of Proposition 2. When each channel member makesdecisions independently, the current value Hamiltonian ofthe retailer is:
π»π
= ππ 1
(πΌπΊ1
+ πππ 1
β πππ 2
)
+ ππ 2
(πΌπΊ2
+ πππ 2
β πππ 1
)
β1
2(1 β π
1
) π2
π 1
β1
2(1 β π
2
) π2
π 2
+ π31
(ππ1
β πππ2
β πΏπΊ1
)
+ π32
(ππ2
β πππ1
β πΏπΊ2
) .
(A.17)
Then we form the Lagrangian
πΏπ
= ππ 1
(πΌπΊ1
+ πππ 1
β πππ 2
)
+ ππ 2
(πΌπΊ2
+ πππ 2
β πππ 1
)
β1
2(1 β π
1
) π2
π 1
β1
2(1 β π
2
) π2
π 2
+ π31
(ππ1
β πππ2
β πΏπΊ1
)
+ π32
(ππ2
β πππ1
β πΏπΊ2
)
+ π31
(πΌπΊ1
+ πππ 1
β πππ 2
)
+ π32
(πΌπΊ2
+ πππ 2
β πππ 1
) .
(A.18)
The necessary conditions for equilibrium are given byππΏπ
πππ 1
= 0, (A.19)
ππΏπ
πππ 2
= 0, (A.20)
β
π31
= ππ31
βππΏπ
ππΊ1
, (A.21)
β
π32
= ππ32
βππΏπ
ππΊ2
, (A.22)
ππΏπ
ππ3π
> 0, π3π
β₯ 0, π3π
ππΏπ
ππ3π
= 0, π = 1, 2. (A.23)
Because ππ π
is constrained to be nonnegative, (A.19)implies that
ππ 1(π‘) = max{0,
(πππ 1
β πππ 2
)
(1 β π1
)} . (A.24)
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Mathematical Problems in Engineering 13
Through a similar proof, we find that
ππ 2(π‘) = max{0,
(πππ 2
β πππ 1
)
(1 β π2
)} . (A.25)
Therefore we can obtain the following results:
π(1)
π π
=
{{
{{
{
πππ π
β πππ (3βπ)
1 β ππ
if πππ π
β πππ (3βπ)
β₯ 0,
0 else,
π β {1, 2} .
(A.26)
Proof of Proposition 4. There are no relationships betweenparticipation rate π
1
and π(1)π 2
or the two manufacturerβsnational advertising efforts. Thus in differentiating π½(1)
π1
fromthe participation rate π
1
we only consider two situations.Situation (1) when ππ
π 1
β πππ 2
β₯ 0 holds, π(1)π 1
= (πππ 1
β
πππ 2
)/(1 β π1
), substituting the above expression into (20)and differentiating π½
π1
with the participation rate π1
, we getmanufacturer 1βs optimal participation rate: π
1
= [π(2ππ1
β
ππ 1
) + πππ 2
]/[π(2ππ1
+ ππ 1
) β πππ 2
]. Since 0 β€ π1
β€ 1,the condition 2ππ
π1
β₯ (πππ 1
β πππ 2
) is required. Situation(2) if condition (ππ
π 1
β πππ 2
) < 0 holds, we get π(1)π 1
= 0.In this situation, whatever participation rate manufacturer1 offers, the retailer would never advertise product 1. Theparticipation rate π
1
is an arbitrary constant. We supposeπ1
= [π(2ππ1
β ππ 1
) + πππ 2
]/[π(2ππ1
+ ππ 1
) β πππ 2
]. Theparticipation rate π
1
is useless in this situation; therefore, theparticipation rate could be negative.
In conclusion, we get the following results:
π(1)
1
=
{{
{{
{
π (2ππ1
β ππ 1
) + πππ 2
π (2ππ1
+ ππ 1
) β πππ 2
if 2πππ1
β₯ πππ 1
β πππ 2
0 else.(A.27)
Through a similar proof, we get manufacturer 2βs optimalshare rate is
π(1)
2
=
{{
{{
{
π (2ππ2
β ππ 2
) + πππ 1
π (2ππ2
+ ππ 2
) β πππ 1
if 2πππ2
β₯ πππ 2
β πππ 1
0 else.(A.28)
B. The Retailer Is Vertically Integrated withOne Manufacturer
Proof of Proposition 5. When the retailer integrates with amanufacturer, the current value Hamiltonian of manufac-turer 2 is
π»π2
= ππ2
(πΌπΊ2
+ πππ 2
β πππ 1
)
β1
2π2
π2
β1
2π2
π2
π 2
+ πΎ21
(ππ1
β πππ2
β πΏπΊ1
)
+ πΎ22
(ππ2
β πππ1
β πΏπΊ2
) .
(B.1)
Then we form the Lagrangian:
πΏπ2
= ππ2
(πΌπΊ2
+ πππ 2
β πππ 1
) β1
2π2
π2
β1
2π2
π2
π 2
+ πΎ21
(ππ1
β πππ2
β πΏπΊ1
)
+ πΎ22
(ππ2
β πππ1
β πΏπΊ2
)
+ π21
(πΌπΊ1
+ πππ 1
β πππ 2
)
+ π22
(πΌπΊ2
+ πππ 2
β πππ 1
) .
(B.2)
At optimality, the necessary conditions are
ππΏπ2
πππ2
= 0,
β
πΎ21
= ππΎ21
βππΏπ1
ππΊ1
,
β
πΎ22
= ππΎ22
βππΏπ1
ππΊ2
,
ππΏπ2
ππ2π
> 0, π2π
β₯ 0, π2π
ππΏπ2
ππ2π
= 0, π = 1, 2.
(B.3)
Proceeding as in the proof for Proposition 1, we get
π(2)
π2
=πΌππ2
(π + πΏ). (B.4)
Proof of Propositions 6 and 7. When the retailer integrateswith a manufacturer, the current value Hamiltonian forintegration system is given by
π»π1,π
= ππ1
+ ππ
+ πΎ11
(ππ1
β πππ2
β πΏπΊ1
)
+ πΎ12
(ππ2
β πππ1
β πΏπΊ2
) .
(B.5)
Then we form the Lagrangian:
πΏπ1,π
= ππ1
+ ππ
+ πΎ11
(ππ1
β πππ2
β πΏπΊ1
)
+ πΎ12
(ππ2
β πππ1
β πΏπΊ2
)
+ π11
(πΌπΊ1
+ πππ 1
β πππ 2
)
+ π12
(πΌπΊ2
+ πππ 2
β πππ 1
) .
(B.6)
Proceeding as in the proof for Proposition 1, and con-straining,as in most cases, the advertising efforts π(π‘) to be
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14 Mathematical Problems in Engineering
nonnegative, we get the following results:
π(2)
π 1
= max {0, π (ππ1
+ ππ 1
) β πππ 2
} ,
π(2)
π 2
= max{0,πππ 2
β π (ππ 1
+ ππ1
)
1 β π2
} .
(B.7)
Thus, the equilibrium advertising levels for manufacturer2 are as follows:
π(2)
π1
={
{
{
πΌ (ππ1
+ ππ 1
)
π + πΏβππΌππ 2
π + πΏif (ππ1
+ ππ 1
) β₯ πππ 2
0 else.(B.8)
Also, we obtain the equilibrium local advertising levels forthe two products:
π(2)
π 1
= {π (ππ1
+ ππ 1
) β πππ 2
if ππ1
β πππ 2
β₯ 0
0 else,
π(2)
π 2
=
{{
{{
{
πππ 2
β π (ππ 1
+ ππ1
)
1 β π2
if πππ 2
β ππ1
β₯ 0
0 else.
(B.9)
Substituting (B.4) and (B.8) into (A.15), we find that
πΊπ(π‘) = πΈ
π
πβπΏπ‘
+ πΊ(2)
πππ
, π β {1, 2} , (B.10)
where πΈπ
= πΊπ0
β (π(2)
ππ
β ππ(2)
π,(3βπ)
)/πΏ, π = 1, 2 and πΊ(2)πππ
=
(π(2)
ππ
β ππ(2)
π,(3βπ)
)/πΏ, π = 1, 2.When condition 0 β€ π β€ min{π
1
/π2
, (π1
β
βπ2
1
β 4ππ2
ππ 2
)/2ππ 2
} holds, the steady-state goodwill πΊ(2)πππ
is nonnegative.
Proof of Proposition 8. There are no relationships betweenparticipation rate π
2
and π(2)π 2
or two manufacturerβs nationaladvertising efforts. Thus in differentiating π½(2)
π2
from theparticipation rate π
2
we only consider two situations. (1)When ππ
π 2
β ππ1
β₯ 0 holds, we get π(2)π 2
= [πππ 2
β π(ππ 1
+
ππ1
)]/(1 β π2
). Substituting the above expression into (38),and differentiating π½
π2
from the participation rate π2
, we getmanufacturer 2βs optimal participation rate. π
2
= [π(2ππ2
β
ππ 2
) + ππ1
]/[π(2ππ2
+ ππ 2
) β ππ1
]. Since 0 β€ π2
β€ 1, theassumption π
π2
β₯ (πππ 2
β ππ1
)/2π is required. (2) Whencondition (ππ
π 2
β πππ 1
) < 0 holds, we get π(2)π 2
= 0. In thissituation, whatever participation rate manufacturer 2 offers,the retailer would never advertise product 2. Thus, π
2
is anarbitrary constant. Here we suppose: π
2
= [π(2ππ2
β ππ 2
) +
ππ1
]/[π(2ππ2
+ππ 2
)βππ1
].The participation rate π1
is uselessin this situation; therefore the participation rate could benegative.
In conclusion, we get the following results:
π(2)
2
=
{{
{{
{
π (2ππ2
β ππ 2
) + ππ1
π (2ππ2
+ ππ 2
) β ππ1
if ππ2
β₯(πππ 2
β ππ1
)
2π
0 else.(B.11)
C. Two Manufacturers AreHorizontally Integrated
Proof of Proposition 9. When the two manufacturers are hor-izontally integrated, the current value Hamiltonian for theintegration system is given by
π»π1,π2
= ππ1
(πΌπΊ1
+ πππ 1
β πππ 2
)
+ ππ2
(πΌπΊ2
+ πππ 2
β πππ 1
)
β1
2π2
π1
β1
2(1 β π
1
) π2
π 1
β1
2π2
π2
β1
2(1 β π
2
) π2
π 2
+ ]11
(ππ1
β πππ2
β πΏπΊ1
)
+ ]12
(ππ2
β πππ1
β πΏπΊ2
) .
(C.1)
Then we form the LagrangianπΏπ1,π2
= ππ1
(πΌπΊ1
+ πππ 1
β πππ 2
)
+ ππ2
(πΌπΊ2
+ πππ 2
β πππ 1
)
β1
2π2
π1
β1
2(1 β π
1
) π2
π 1
β1
2π2
π2
β1
2(1 β π
2
) π2
π 2
+ ]11
(ππ1
β πππ2
β πΏπΊ1
)
+ ]12
(ππ2
β πππ1
β πΏπΊ2
)
+ π11
(πΌπΊ1
+ πππ 1
β πππ 2
)
+ π12
(πΌπΊ2
+ πππ 2
β πππ 1
) .
(C.2)
At optimality, the necessary conditions areππΏπ1,π2
πππ1
= 0, (C.3)
β
]11
= π]11
βππΏπ1,π2
ππΊ1
, (C.4)
β
]12
= π]12
βππΏπ1,π2
ππΊ2
, (C.5)
ππΏπ1,π2
ππ1π
> 0, π1π
β₯ 0, π1π
ππΏπ1,π2
ππ1π
= 0, π = 1, 2.
(C.6)In most case the advertising effort π
π1
is constrained tobe nonnegative. Thus, (C.3) implies
ππ1
= max {0, π½ππ1
+ ]11
β π]12
} . (C.7)Solving (C.4)-(C.5), we get
β
]11
= (π + πΏ) ]11
β πΌππ1
β πΌπ11
,
β
]12
= (π + πΏ) ]12
β πΌππ2
β πΌπ12
.
(C.8)
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Mathematical Problems in Engineering 15
Equation (C.6) implies, π1π
= 0; then substituting π1π
= 0
into (C.8), we getβ
]11
= (π + πΏ) ]11
β πΌππ1
, (C.9)
β
]12
= (π + πΏ) ]12
β πΌππ2
. (C.10)
Differentiating (C.7) with respect to time and substitutingfor ]11
, ]12
and their time derivative in (C.9)-(C.10), we getβ
ππ1
= (π + πΏ)ππ1
β πΌππ1
+ ππΌππ2
. (C.11)
Proceeding as in the proof of Proposition 1, we get
π(3)
π1
= max{0,πΌππ1
(π + πΏ)βππΌππ2
(π + πΏ)} ,
π(3)
π2
= max{0,πΌππ2
(π + πΏ)βππΌππ1
(π + πΏ)} .
(C.12)
Thus, the equilibrium advertising levels for twomanufac-turers are as follows:
π(3)
ππ
={
{
{
πΌ (πππ
β πππ(3βπ)
)
π + πΏif πππ
β₯ πππ,3βπ
,
0 else
π β {1, 2} .
(C.13)
Substituting (C.13) into (A.15), we find that
πΊπ(π‘) = πΉ
π
πβπΏπ‘
+ πΊ(3)
πππ
, π β {1, 2} , (C.14)
where πΉπ
= πΊπ0
β (π(3)
ππ
β ππ(3)
π,(3βπ)
)/πΏ, π = 1, 2 and πΊ(3)πππ
=
(π(3)
ππ
βππ(3)
π,(3βπ)
)/πΏ, π = 1, 2.When condition 0 β€ 2π/(1+π2) β€min{π
π1
/ππ2
, ππ2
/ππ1
} holds, the steady-state goodwillπΊ(3)πππ
is nonnegative.The current value Hamiltonian for the retailer is given by
π»π
= ππ 1
(πΌπΊ1
+ πππ 1
β πππ 2
)
+ ππ 2
(πΌπΊ2
+ πππ 2
β πππ 1
) β1
2(1 β π
1
) π2
π 1
β1
2(1 β π
2
) π2
π 2
+ ]21
(ππ1
β πππ2
β πΏπΊ1
)
+ ]22
(ππ2
β πππ1
β πΏπΊ2
) .
(C.15)
Then we form the Lagrangian:
πΏπ
= ππ 1
(πΌπΊ1
+ πππ 1
β πππ 2
)
+ ππ 2
(πΌπΊ2
+ πππ 2
β πππ 1
)
β1
2(1 β π
1
) π2
π 1
β1
2(1 β π
2
) π2
π 2
+ ]21
(ππ1
β πππ2
β πΏπΊ1
)
+ ]22
(ππ2
β πππ1
β πΏπΊ2
)
+ π21
(πΌπΊ1
+ πππ 1
β πππ 2
)
+ π22
(πΌπΊ2
+ πππ 2
β πππ 1
) .
(C.16)
Proceeding as in the proof of Proposition 2, we get
π(3)
π 1
= max{0,(πππ 1
β πππ 2
)
(1 β π1
)} ,
π(3)
π 2
= max{0,(πππ 2
β πππ 1
)
(1 β π2
)} .
(C.17)
Then we get the following results:
π(3)
π π
=
{{
{{
{
πππ π
β πππ (3βπ)
1 β ππ
if πππ π
β πππ (3βπ)
β₯ 0
0 else,
π β {1, 2} .
(C.18)
Acknowledgment
This work was supported by the National Natural ScienceFoundation of China (Grants nos. 70901068 and 71271198),the Fund for International Cooperation and Exchange of theNational Natural Science Foundation of China (Grant no.71110107024), and the Chinese Universities Scientific Fund(WK2040160008 andWK2040150005). QinglongGouwouldalso like to acknowledge the Science Fund for CreativeResearch Groups of the National Natural Science Foundationof China (Grant no. 71121061) for supporting his research.
References
[1] Z. Huang and S. X. Li, βCo-op advertising models in manu-facturer-retailer supply chains: a game theory approach,β Euro-pean Journal ofOperational Research, vol. 135, no. 3, pp. 527β544,2001.
[2] R. F. Young and S. A. Greyser, βCooperative advertising: prac-tices and problems,β Marketing Science Institute, pp. 82β105,1982.
[3] R. P. Dant and P. D. Berger, βModelling cooperative advertisingdecisions in franchising,β Journal of the Operational ResearchSociety, vol. 47, no. 9, pp. 1120β1136, 1996.
[4] M. G. Nagler, βAn exploratory analysis of the determinants ofcooperative advertising participation rates,β Marketing Letters,vol. 17, no. 2, pp. 91β102, 2006.
[5] R. Yan, βCooperative advertising, pricing strategy and firmperformance in the e-marketing age,β Journal of the Academy ofMarketing Science, vol. 38, no. 4, pp. 510β519, 2010.
[6] P. D. Berger, βVertical cooperative advertising ventures,β JournalMarketing Research, vol. 9, no. 3, pp. 309β312, 1972.
[7] M. Bergen and G. John, βUnderstanding cooperative adver-tising participation rates in conventional channels,β Journal ofMarketing Research, vol. 34, no. 3, pp. 357β369, 1997.
[8] Z. Huang, S. X. Li, and V. Mahajan, βAn analysis of manu-facturer-retailer supply chain coordination in cooperativeadvertising,βDecision Sciences, vol. 33, no. 3, pp. 469β494, 2002.
[9] J. Yue, J. Austin, M. Wang, and Z. Huang, βCoordination ofcooperative advertising in a two-level supply chainwhenmanu-facturer offers discount,β European Journal of OperationalResearch, vol. 168, no. 1, pp. 65β85, 2006.
![Page 16: Research Article Cooperative Advertising in a β¦downloads.hindawi.com/journals/mpe/2013/607184.pdfResearch Article Cooperative Advertising in a Supply Chain with Horizontal Competition](https://reader035.fdocuments.in/reader035/viewer/2022070722/5f01c61c7e708231d400f7f0/html5/thumbnails/16.jpg)
16 Mathematical Problems in Engineering
[10] J. Xie and A. Neyret, βCo-op advertising and pricing models inmanufacturer-retailer supply chains,β Computers and IndustrialEngineering, vol. 56, no. 4, pp. 1375β1385, 2009.
[11] J. Xie and J. C. Wei, βCoordinating advertising and pricing ina manufacturer-retailer channel,β European Journal of Opera-tional Research, vol. 197, no. 2, pp. 785β791, 2009.
[12] M. M. SeyedEsfahani, M. Biazaran, and M. Gharakhani, βAgame theoretic approach to coordinate pricing and vertical co-op advertising in manufacturer-retailer supply chains,β Euro-pean Journal of Operational Research, vol. 211, no. 2, pp. 263β273,2011.
[13] P. K. Chintagunta and D. Jain, βA dynamic model of channelmember strategies for marketing expenditures,β Marketing Sci-ence, vol. 11, no. 2, pp. 168β188, 1992.
[14] S. JΓΈrgensen, S. P. Sigue, and G. Zaccour, βDynamic cooperativeadvertising in a channel,β Journal of Retailing, vol. 76, no. 1, pp.71β92, 2000.
[15] S. JΓΈrgensen, S. Taboubi, and G. Zaccour, βCooperative adver-tising in a marketing channel,β Journal of Optimization Theoryand Applications, vol. 110, no. 1, pp. 145β158, 2001.
[16] S. JΓΈrgensen, S. Taboubi, and G. Zaccour, βRetail promotionswith negative brand image effects: is cooperation possible?βEuropean Journal of Operational Research, vol. 150, no. 2, pp.395β405, 2003.
[17] S. JΓΈrgensen and G. Zaccour, βChannel coordination overtime: incentive equilibria and credibility,β Journal of EconomicDynamics & Control, vol. 27, no. 5, pp. 801β822, 2003.
[18] S. Karray and G. Zaccour, βA differential game of advertisingfor national and store brands,β in Dynamic Games: Theory andApplications, vol. 10, chapter 11, pp. 213β229, Springer, NewYork,NY, USA, 2005.
[19] X. He, A. Prasad, and S. P. Sethi, βCooperative advertising andpricing in a dynamic stochastic supply chain: feedback stack-elberg strategies,β Production and Operations Management, vol.18, no. 1, pp. 78β94, 2009.
[20] J. Useem, βHow retailingβs superpower-and our biggest mostadmired company-is changing the rules for corporate America,βFortune, vol. 65, 2003.
[21] B. Carney, βNot everyone buys Tescoβs strategy,β Business WeekOnline, 2004.
[22] D. Knight, βSurplus of home improvement stores make Indi-anapolis handyman heaven,β Knight Ridder Tribune BusinessNews, vol. 1, 2003.
[23] M. Nerlove and K. J. Arrow, βOptimal advertising policy underdynamic conditions,β Economica, vol. 29, no. 114, pp. 129β142,1962.
[24] M. Kurtulus and L. B. Toktay, βCategory captainship vs. retailercategory management under limited retail shelf space,β Produc-tion and Operations Management, vol. 20, no. 1, pp. 47β56, 2011.
[25] E.Adida andV.DeMiguel, βSupply chain competitionwithmul-tiple manufacturers and retailers,β Operations Research, vol. 59,no. 1, pp. 156β172, 2011.
[26] G. P. Cachon and A. G. Kok, βCompeting manufacturers ina retail supply chain: on contractual form and coordination,βManagement Science, vol. 56, no. 3, pp. 571β589, 2010.
[27] J. Lu, Y. Tsao, and C. Charoensiriwath, βCompetition undermanufacturer service and retail price,β Economic Modelling, vol.28, no. 3, pp. 1256β1264, 2011.
[28] X. He, A. Krishnamoorthy, A. Prasad, and S. P. Sethi, βRetailcompetition and cooperative advertising,β Operations ResearchLetters, vol. 39, no. 1, pp. 11β16, 2011.
[29] X. He, A. Krishnamoorthy, A. Prasad, and S. P. Sethi, βCo-opadvertising in dynamic retail oligopolies,βDecision Sciences, vol.43, no. 1, pp. 73β106, 2012.
[30] A. Nair and R. Narasimhan, βDynamics of competing withquality- and advertising-based goodwill,β European Journal ofOperational Research, vol. 175, no. 1, pp. 462β474, 2006.
[31] C. L. Bovee and W. F. Arrns, Contemporary Advertising, Irwin,Homewood, Ill, USA, 2nd edition, 1986.
[32] P. De Giovanni, βQuality improvement vs. advertising support:which strategy works better for a manufacturer?β EuropeanJournal of Operational Research, vol. 208, no. 2, pp. 119β130, 2011.
![Page 17: Research Article Cooperative Advertising in a β¦downloads.hindawi.com/journals/mpe/2013/607184.pdfResearch Article Cooperative Advertising in a Supply Chain with Horizontal Competition](https://reader035.fdocuments.in/reader035/viewer/2022070722/5f01c61c7e708231d400f7f0/html5/thumbnails/17.jpg)
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