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]
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 “single-manufacturer single-retailer” framework, the decision-making framework with “multiple-manufacturer single-retailer” 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 “two-manufacturer single-retailer” 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 firm’s 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 theretailer’s 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 manufacturer’s 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 25–40% 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, 6–12] anddynamicmodels [13–19]. However, these studiesmainly focuson a “single-manufacturer single-retailer” framework.
Retailers in today’s market are increasingly more power-ful thanmanufacturers. Useem found that sales throughWal-Mart accounted for 17% of P&G’s total sales in 2002, 39% ofTandy’s, 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
2 Mathematical Problems in Engineering
“single-manufacturer single-retailer” framework to the “two-manufacturer single-retailer” 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 manufacturer’s optimal participation ratefor the retailer’s 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 “two-manufacturer single-retailer” 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 channelmembers’ 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]. Jørgensenet 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, Jørgensen et al. introduceddecreasing marginal returns to goodwill and adopted a moreflexible functional form for the sales dynamics [15]. Jørgensenet al. studied the cooperative advertising program in the casewhere the retailer’s promotions can damage the brand image[16]. Extending the work of Jørgensen 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 “single-manufacturer single-retailer” framework. Few studies ad-dress a “multiple-manufacturer single-retailer” 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 “multiple-retailer”framework. For example, He et al. [28, 29] extended He et al.[19] by considering the competing retailers, and their resultsshowed that the manufacturer’s support for its retailer ishigher under competition than in its absence.
To our best knowledge, research relating to cooperativeadvertising focused on a “multiple-manufacturer single-retailer” framework in the supply chain has not been exploredin literature. In this study, we investigate a cooperative adver-tising model using the “two-manufacturer single-retailer”framework.
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 𝑖, 𝑖 ∈ {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 𝑖 is given by
∙
𝐺𝑖
(𝑡) = 𝑈𝑀𝑖
− 𝜃𝑈𝑀(3−𝑖)
− 𝛿𝐺𝑖
, 𝑖 ∈ {1, 2} , (1)
where 0 < 𝜃 < 1 is a constant which represents the rivaladvertising’s negative effect on goodwill, as seen in previousliterature [30]. Next, 𝛿 > 0 is the diminishing rate of goodwill,which captures the idea that consumers may forget the brandto some extent. National advertising mainly focuses on firm’slong-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
𝐺1(0) = 𝐺
10
≥ 0, 𝐺2(0) = 𝐺
20
≥ 0, (2)
and the sales 𝑆𝑖
(𝑡) of the two brands along time 𝑡 satisfy
𝑆𝑖(𝑡) = max {0, 𝛼𝐺
𝑖
+ 𝜆𝑈𝑅𝑖
− 𝜒𝑈𝑅(3−𝑖)
} , 𝑖 ∈ {1, 2} . (3)
In (3), 𝛼, 𝜆, 𝜒 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 𝛼𝐺
𝑖
represents thelong-term effect of national advertising on sales, and the item𝜆𝑈𝑅𝑖
represents the effect of the retailer’s local advertising onproduct 𝑖. As in previous research such as Jørgensen et al. [14],we only take the promotion effect of local advertising into thefunctions of sales herewithout the stock of goodwill.The item−𝜒𝑈𝑅(3−𝑖)
illustrates the rival local advertising’s negative effecton sales. Next, 𝜆 > 𝜒 > 0, which implies that the effects ofrival advertising are generally smaller than the effects of one’sown advertising effect, which is a fairly common assumptionin the relevant literature [30].
The advertising cost functions are quadratic with respectto marketing efforts, namely,
𝐶 (𝑈𝑀𝑖(𝑡)) =
𝑈2
𝑀𝑖
(𝑡)
2, 𝐶 (𝑈
𝑅𝑖
) =𝑈2
𝑅𝑖
(𝑡)
2, 𝑖 ∈ {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 𝑖 is assumed as 𝜌
𝑀𝑖
≥ 0,and the marginal profit of the retailer selling the product 𝑖 is
Manufacturer 1(M1)
Manufacturer 2 (M2)
Customer
Product 1 Product 2
Retailer R
UM1 UM2
UR1 UR2
Figure 1
Table 1: Notation.
𝑡 Time 𝑡, 𝑡 ≥ 0𝐺𝑖
(𝑡) Goodwill of the product 𝑖 at time 𝑡, 𝑖 ∈ {1, 2}𝑈𝑀𝑖
(𝑡) Manufacturer 𝑖’s national advertising efforts at time 𝑡𝑈𝑅𝑖
(𝑡) Retailer’s local advertising efforts for product 𝑖 at time 𝑡𝑆𝑖
(𝑡) Sales of product 𝑖 along time 𝑡𝜃 ∈ [0, 1] The national advertising’s competition coefficient𝜒 ∈ [0, 1] The local advertising’s competition coefficient
𝜙𝑖
∈ [0, 1]Manufacturer 𝑖’s participation rate for the retailer’sadvertising cost
𝜌𝑀𝑖
≥
0Marginal profit of manufacturer 𝑖
𝜌𝑅𝑖
≥ 0 Marginal profit of the retailer sells the product 𝑖𝛿 > 0 Diminishing rate of goodwill𝑟 > 0 Discount rate of the manufacturers and the retailer𝜋𝑀𝑖
, 𝜋𝑅
Profit functions for𝑀𝑖 and 𝑅, respectively
𝐽𝑀𝑖
, 𝐽𝑅
Current value of profit functions for𝑀𝑖 and 𝑅,respectively
𝜌𝑅𝑖
≥ 0. The profit functions of the two manufacturers arethen
𝜋𝑀𝑖(𝑡) = 𝜌
𝑀𝑖
𝑆𝑖(𝑡) −
1
2𝑈2
𝑀𝑖
(𝑡) −1
2𝜙𝑖
𝑈2
𝑅𝑖
(𝑡) , 𝑖 ∈ {1, 2} ,
(5)
and the profit function of the retailer is
𝜋𝑅(𝑡) =
2
∑
𝑖=1
(𝜌𝑅𝑖
𝑆𝑖(𝑡) −
1
2(1 − 𝜙
𝑖
) 𝑈2
𝑅𝑖
(𝑡)) . (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
4 Mathematical Problems in Engineering
provides a cooperative advertising program with a changingparticipation rate, the manufacturer would have to knowthe retailer’s 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 𝑈𝑀𝑖
and 𝜙𝑖
are manufacturer’s decisionvariables, and 𝑈
𝑅𝑖
is retailer’s decision variable. Then weconsider a two-stage game in this paper. The manufacturersoffer their participation rates for the retailer’s local advertisingexpenditure at stage 1, and then the manufacturers andretailer determine their advertising efforts along time 𝑡 simul-taneously at stage 2. We firstly keep the participation rates𝜙𝑖
(𝑖 = 1, 2) as fixed, calculate the advertising efforts of themanufacturers and retailer utilizing differential game theory,and then decide the manufacturers’ 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 𝑡. Each channel member,then, strives to maximize the current values of its profit.Witha common discount rate 𝑟 > 0 and for the sales 𝑆
𝑖
≥ 0, wehave
max𝑈𝑀𝑖≥0,1≥𝜙𝑖≥0
𝐽𝑀𝑖
= ∫
+∞
0
𝑒−𝑟𝑡
𝜋𝑀𝑖(𝑡) 𝑑𝑡, 𝑖 ∈ {1, 2} , (7)
and, for the retailer, we have
max𝑈𝑅1≥0,𝑈𝑅2≥0
𝐽𝑅
= ∫
+∞
0
𝑒−𝑟𝑡
𝜋𝑅(𝑡) 𝑑𝑡. (8)
Taking (1) into account, we get the current value Hamil-tonian of two manufacturers as
𝐻𝑀1
= 𝜋𝑀1
+ 𝜇11
(𝑈𝑀1
− 𝜃𝑈𝑀2
− 𝛿𝐺1
)
+ 𝜇12
(𝑈𝑀2
− 𝜃𝑈𝑀1
− 𝛿𝐺2
) ,
𝐻𝑀2
= 𝜋𝑀2
+ 𝜇21
(𝑈𝑀1
− 𝜃𝑈𝑀2
− 𝛿𝐺1
)
+ 𝜇22
(𝑈𝑀2
− 𝜃𝑈𝑀1
− 𝛿𝐺2
) .
(9)
Similarly, we get the retailer’s current value Hamiltonianas
𝐻𝑅
= 𝜋𝑅
+ 𝜇31
(𝑈𝑀1
− 𝜃𝑈𝑀2
− 𝛿𝐺1
)
+ 𝜇32
(𝑈𝑀2
− 𝜃𝑈𝑀1
− 𝛿𝐺2
) ,
(10)
where 𝜇𝑖1
and 𝜇𝑖2
(𝑖 = 1, 2, 3) represent the costate variablesin the firm’s problem corresponding to the firm’s goodwilllevels.
Then using the necessary conditions for equilibrium, weobtain the following results.
Proposition 1. When each channel member makes decisionsindependently and the participation rates 𝜙
𝑖
(𝑖 = 1, 2) arefixed, the equilibrium advertising efforts for twomanufacturerson their products along time 𝑡 are all constants; that is,
𝑈(1)
𝑀𝑖
=𝛼𝜌𝑀𝑖
(𝑟 + 𝛿), 𝑖 ∈ {1, 2} . (11)
Proposition 1 illustrates the following facts. (i) Whateverthe participation rate that the manufacturer undertakes forthe retailer’s advertising cost, the manufacturer’s 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 manufacturer’s equilibrium advertisingefforts are determined by the item 𝛼𝜌
𝑀𝑖
/(𝑟 + 𝛿), 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 customer’sgoodwill diminishes quickly, the advertising efforts woulddrop.
Further, we obtain the retailer’s equilibrium advertisingefforts for two brands as follows.
Proposition 2. When each channel member makes decisionsindependently and the participation rates 𝜙
𝑖
(𝑖 = 1, 2) arefixed, the retailer’s equilibrium advertising efforts for the twobrands along time 𝑡 are all constants:
𝑈(1)
𝑅𝑖
=
{{
{{
{
𝜆𝜌𝑅𝑖
− 𝜒𝜌𝑅(3−𝑖)
1 − 𝜙𝑖
𝑖𝑓 𝜆𝜌𝑅𝑖
− 𝜒𝜌𝑅(3−𝑖)
≥ 0,
0 𝑒𝑙𝑠𝑒
𝑖 ∈ {1, 2} .
(12)
Proposition 2 holds the following managerial implica-tions. (i) When condition 𝜆𝜌
𝑅𝑖
− 𝜒𝜌𝑅,(3−𝑖)
≥ 0, 𝑖 ∈ {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 retailer’sadvertising efforts for his product. (ii) The equilibrium localadvertising efforts on product 𝑖 are increased by the retailer’smarginal profit of product 𝑖.
Furthermore, when condition 𝜆 > 𝜒 > 0 is satisfied, weget the following results by (12).
(i) If condition 𝜆𝜌𝑅2
− 𝜒𝜌𝑅1
< 0 holds, we have 𝑈(1)𝑅1
=
(𝜆𝜌𝑅1
− 𝜒𝜌𝑅2
)/(1 − 𝜙1
) and 𝑈(1)𝑅2
= 0. Because themarginal profit which the retailer obtains from prod-uct 2 is extremely small, the benefit of𝑈(1)
𝑅2
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.
Mathematical Problems in Engineering 5
(ii) If conditions𝜆𝜌𝑅1
−𝜒𝜌𝑅2
≥ 0 and𝜆𝜌𝑅2
−𝜒𝜌𝑅1
≥ 0hold,we can get 𝑈(1)
𝑅1
= (𝜆𝜌𝑅1
− 𝜒𝜌𝑅2
)/(1 − 𝜙1
) and 𝑈(1)𝑅2
=
(𝜆𝜌𝑅2
− 𝜒𝜌𝑅1
)/(1 − 𝜙2
). 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 𝜆𝜌𝑅1
− 𝜒𝜌𝑅2
< 0 holds, we have 𝑈(1)𝑅1
= 0
and 𝑈(1)𝑅2
= (𝜆𝜌𝑅2
− 𝜒𝜌𝑅1
)/(1 − 𝜙2
). This situationis similar to the first situation; the retailer wouldadvertise product 2, but not product 1.
When the two manufacturers’ 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, 𝑈
𝑀𝑖
(𝑡) = 𝑈(1)
𝑀𝑖
and 𝑈𝑅𝑖
(𝑡) = 𝑈(1)
𝑅𝑖
, 𝑖 = 1, 2, thenthe accumulated goodwill of two products along time 𝑡 is
𝐺𝑖(𝑡) = 𝐷
𝑖
𝑒−𝛿𝑡
+ 𝐺(1)
𝑖SS, 𝑖 ∈ {1, 2} , (13)
where 𝐷𝑖
= 𝐺𝑖0
− (𝑈(1)
𝑀𝑖
− 𝜃𝑈(1)
𝑀,(3−𝑖)
)/𝛿 and 𝐺(1)𝑖SS = (𝑈
(1)
𝑀𝑖
−
𝜃𝑈(1)
𝑀,(3−𝑖)
)/𝛿, 𝑖 = 1, 2. 𝐺(1)𝑖SS is the steady-state goodwill for
product 𝑖 when 𝑡 → ∞.
From (13), we obtain the following facts: (i) the steady-state goodwill for product 𝑖 increases with manufacturer 1’snational advertising efforts; (ii) the steady-state goodwill forproduct 𝑖 decreases with the rival manufacturer’s nationaladvertising efforts because of the competitive effect; (iii)steady-state goodwill is only affected by the manufacturer’sadvertising 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)–(13) into (7) and (8) and with theparticipation rates 𝜙
𝑖
(𝑖 = 1, 2) being fixed, we get thecurrent value ofmanufacturer 1’s profit under the equilibriumcondition as follows:
𝐽(1)
𝑀𝑖
=𝐷𝑖
𝛼𝜌𝑀𝑖
𝑟 + 𝛿+
𝛼𝜌𝑀𝑖
(𝑈(1)
𝑀𝑖
− 𝜃𝑈(1)
𝑀,(3−𝑖)
)
𝑟𝛿
+
𝜌𝑀𝑖
(𝜆𝑈(1)
𝑅𝑖
− 𝜒𝑈(1)
𝑅,(3−𝑖)
)
𝑟
−
(𝑈(1)
𝑀𝑖
)2
+ 𝜙𝑖
(𝑈(1)
𝑅𝑖
)2
2𝑟, 𝑖 ∈ {1, 2} .
(14)
The current value of the retailer’s profit is
𝐽(1)
𝑅
=𝐷1
𝛼𝜌𝑅1
+ 𝐷2
𝛼𝜌𝑅2
𝑟 + 𝛿
+
𝛼𝜌𝑅1
(𝑈(1)
𝑀1
− 𝜃𝑈(1)
𝑀2
) + 𝛼𝜌𝑅2
(𝑈(1)
𝑀2
− 𝜃𝑈(1)
𝑀1
)
𝑟𝛿
−1 − 𝜙1
2(𝑈(1)
𝑅1
)2
+
𝜌𝑅1
(𝜆𝑈(1)
𝑅1
− 𝜒𝑈(1)
𝑅2
)
𝑟
+
𝜌𝑅2
(𝜆𝑈(1)
𝑅2
− 𝜒𝑈(1)
𝑅1
)
𝑟−1 − 𝜙2
2(𝑈(1)
𝑅2
)2
,
(15)
where𝐷𝑖
= 𝐺𝑖0
− (𝑈(1)
𝑀𝑖
− 𝜃𝑈(1)
𝑀,(3−𝑖)
)/𝛿, 𝑖 ∈ {1, 2}.Differentiating 𝐽(1)
𝑀𝑖
with the participation rate 𝜙𝑖
, 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
𝜙(1)
𝑖
=
{{
{{
{
𝜆 (2𝜌𝑀𝑖
− 𝜌𝑅𝑖
) + 𝜒𝜌𝑅(3−𝑖)
𝜆 (2𝜌𝑀𝑖
+ 𝜌𝑅𝑖
) − 𝜒𝜌𝑅(3−𝑖)
𝑖𝑓 𝜌𝑀𝑖
≥(𝜆𝜌𝑅𝑖
− 𝜒𝜌𝑅(3−𝑖)
)
2𝜆,
0 𝑒𝑙𝑠𝑒
𝑖 ∈ {1, 2} .
(16)
For (16), the restraining condition 𝜌𝑀1
≥ (𝜆𝜌𝑅1
−
𝜒𝜌𝑅2
)/2𝜆 implies that manufacturer 1 is willing to provide theparticipation rate with the retailer only when he can obtain alarge enough marginal profit. Differentiating 𝜙(1)
1
from 𝜌𝑀1
,𝜌𝑅1
, and 𝜌𝑅2
, and knowing that 𝜌𝑀1
≥ (𝜆𝜌𝑅1
− 𝜒𝜌𝑅2
)/2𝜆, wefind that 𝜕𝜙(1)
1
/𝜕𝜌𝑀1
> 0; 𝜕𝜙(1)1
/𝜕𝜌𝑅1
< 0; 𝜕𝜙(1)1
/𝜕𝜌𝑅2
> 0.The above expressions show that (i) when the manufacturer’smarginal 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 retailer’s equilibrium local advertis-ing efforts on the two brands are all constants, that is,
𝑈(1)
𝑅𝑖
=
{{{{{{{{{{
{{{{{{{{{{
{
𝜆𝜌𝑀𝑖
+1
2𝜆𝜌𝑅𝑖
−1
2𝜒𝜌𝑅(3−𝑖)
,
if 0 <(𝜆𝜌𝑅𝑖
− 𝜒𝜌𝑅(3−𝑖)
)
2𝜆≤ 𝜌𝑀𝑖
,
𝜆𝜌𝑅𝑖
− 𝜒𝜌𝑅(3−𝑖)
,
if 𝜌𝑀𝑖
<(𝜆𝜌𝑅𝑖
− 𝜒𝜌𝑅(3−𝑖)
)
2𝜆,
0 else
𝑖 ∈ {1, 2} .
(17)
6 Mathematical Problems in Engineering
Equation (17) shows that the retailer’s equilibrium adver-tising level 𝑈(1)
𝑅𝑖
for a product is not only linear with hisown marginal profit 𝜌
𝑅𝑖
, but also linear with the manufac-turer’s marginal profit 𝜌
𝑀𝑖
if the conditions 0 < (𝜆𝜌𝑅𝑖
−
𝜒𝜌𝑅(3−𝑖)
)/2𝜆 ≤ 𝜌𝑀𝑖
hold. Supposing that 𝜌𝑀𝑖
+ 𝜌𝑅𝑖
= 𝜌𝑖
isthe channel marginal profit of one product, the equilibriumadvertising level 𝑈(1)
𝑅𝑖
can be rewritten as 𝑈(1)𝑅𝑖
= 𝜆𝜌𝑀𝑖
/2 +
𝜆𝜌𝑖
/2 − 𝜒𝜌𝑅,(3−𝑖)
/2 if and only if 0 < 𝜆𝜌𝑅𝑖
− 𝜒𝜌𝑅,(3−𝑖)
≤ 2𝜆𝜌𝑀𝑖
,𝑖 = 1, 2 hold. The channel marginal profit of product 𝑖 isnot changed in the short term; therefore, the above equationsimply that the retailer’s equilibrium advertising efforts areindependent of the marginal profit which the retailer obtainsfrom product 𝑖.
3.2. Retailer Integrates with a Manufacturer. In second sce-nario, the retailer integrates with one of the manufacturers.We assume thismanufacturer is𝑀1.Then, the profit functionof the integration system is 𝜋
𝑀1,𝑅
= 𝜋𝑀1
+ 𝜋𝑅
, and theobjective of integration system is
max𝑈𝑀1≥0,𝑈𝑅1≥0,𝑈𝑅2≥0
𝐽𝑀1,𝑅
= ∫
+∞
0
𝑒−𝑟𝑡
(𝜋𝑀1
+ 𝜋𝑅
) 𝑑𝑡. (18)
Further, the objective of manufacturer 2 is
max1≥𝜙2≥0,𝑈𝑀2≥0
𝐽𝑀2
= ∫
+∞
0
𝑒−𝑟𝑡
𝜋𝑀2
𝑑𝑡. (19)
Taking state equation (1) into account, the current valueHamiltonian of the vertical integration system (𝑀1 and 𝑅) is
𝐻𝑀1,𝑅
= 𝜋𝑀1
+ 𝜋𝑅
+ 𝛾11
(𝑈𝑀1
− 𝜃𝑈𝑀2
− 𝛿𝐺1
)
+ 𝛾12
(𝑈𝑀2
− 𝜃𝑈𝑀1
− 𝛿𝐺2
) ,
(20)
and that of manufacturer 2 is
𝐻𝑀2
= 𝜋𝑀2
+ 𝛾21
(𝑈𝑀1
− 𝜃𝑈𝑀2
− 𝛿𝐺1
)
+ 𝛾22
(𝑈𝑀2
− 𝜃𝑈𝑀1
− 𝛿𝐺2
) ,
(21)
where 𝛾𝑖1
and 𝛾𝑖2
(𝑖 = 1, 2) represent the costate variables inthe channel member’s problem corresponding to the firm’sgoodwill 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 𝜙
2
provided by manufacturer2 is fixed, the equilibrium advertising efforts of manufacturer 2are constant, that is,
𝑈(2)
𝑀2
=𝛼𝜌𝑀2
(𝑟 + 𝛿). (22)
Compared to Proposition 1, we find that manufacturer 2’sequilibrium advertising level is the same, which implies thatmanufacturer 2’s 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 𝜙
2
provided by manufacturer
2 is fixed, manufacturer 1’s equilibrium advertising efforts areconstant, that is,
𝑈(2)
𝑀1
={
{
{
𝛼𝜌1
− 𝜃𝛼𝜌𝑅2
𝑟 + 𝛿𝑖𝑓 𝜌1
≥ 𝜃𝜌𝑅2
,
0 𝑒𝑙𝑠𝑒,
(23)
and the retailer’s equilibrium advertising efforts for the twobrands along time 𝑡 are all constants, that is,
𝑈(2)
𝑅1
={
{
{
𝜆𝜌1
− 𝜒𝜌𝑅2
𝑖𝑓 𝜆𝜌1
− 𝜒𝜌𝑅2
≥ 0,
0 𝑒𝑙𝑠𝑒,
(24)
𝑈(2)
𝑅2
=
{{
{{
{
𝜆𝜌𝑅2
− 𝜒𝜌1
1 − 𝜙2
𝑖𝑓 𝜆𝜌𝑅2
− 𝜒𝜌1
≥ 0,
0 𝑒𝑙𝑠𝑒,
(25)
where 𝜌1
= 𝜌𝑀1
+ 𝜌𝑅1
.
Proposition 6 shows the following trends. If the condition𝜌1
> 𝜃𝜌𝑅2
holds, the national advertising efforts 𝑈(2)𝑀1
areaffected by the item (𝛼𝜌
1
− 𝜃𝛼𝜌𝑅2
)/(𝑟 + 𝛿). From this item,we know that the larger the channel marginal profit ofproduct 1 (𝜌
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 product’s marginal profit 𝜌
𝑅2
. When𝜌𝑅2
is increased, the integration system would decreasenational advertising efforts for product 1 and thus decreaseproduct 1’s 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
Δ𝑈𝑅1
=
{{{{{{{{
{{{{{{{{
{
𝜆𝜌𝑀1
if 2𝜆𝜌𝑀1
< 𝜆𝜌𝑅1
− 𝜒𝜌𝑅2
,
𝜆𝜌𝑅1
− 𝜒𝜌𝑅2
2if 0 < 𝜆𝜌
𝑅1
− 𝜒𝜌𝑅2
≤ 2𝜆𝜌𝑀1
,
𝜆𝜌1
− 𝜒𝜌𝑅2
if − 𝜆𝜌𝑀1
< 𝜆𝜌𝑅1
− 𝜒𝜌𝑅2
≤ 0,
0 else,(26)
where 𝜌1
= 𝜌𝑀1
+ 𝜌𝑅1
.It is easy to prove that Δ𝑈
𝑅1
given by (26) is nonnegative,whichmeans that the integration between the retailer and themanufacturer would increase the retailer’s 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 𝜆𝜌𝑅2
− 𝜒𝜌1
< 0 holds, we have 𝑈(2)𝑅2
= 0 and𝑈(2)
𝑅1
= 𝜆𝜌1
− 𝜒𝜌𝑅2
. Note that 𝜌1
= 𝜌𝑀1
+ 𝜌𝑅1
>
𝜌𝑅1
, which implies that the integration between theretailer andmanufacturer 1 would lead to the increasein the retailer’s local advertising threshold for product2 and would also increase the retailer’s equilibriumadvertising efforts for product 1.
Mathematical Problems in Engineering 7
(ii) If conditions 𝜆𝜌1
−𝜒𝜌𝑅2
≥ 0 and 𝜆𝜌𝑅2
−𝜒𝜌1
≥ 0 hold,𝑈(2)
𝑅2
= 𝜆𝜌1
− 𝜒𝜌𝑅2
and 𝑈(2)𝑅2
= (𝜆𝜌𝑅2
− 𝜒𝜌1
)/(1 − 𝜙2
).In this situation, the retailer increases the equilibriumadvertising efforts for product 1 and decreases effortsfor product 2.
(iii) When 𝜆𝜌1
− 𝜒𝜌𝑅2
< 0 holds, 𝑈(2)𝑅1
= 0 and 𝑈(2)𝑅2
=
(𝜆𝜌𝑅2
− 𝜒𝜌1
)/(1 − 𝜙2
). Note that 𝜌1
> 𝜌𝑅1
, whichimplies that the integration between the retailer andmanufacturer 1 reduces the retailer’s local advertisingthreshold for product 1 and also decreases the retail-er’s 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, 𝑈𝑀𝑖
(𝑡) = 𝑈(2)
𝑀𝑖
and 𝑈𝑅𝑖
(𝑡) = 𝑈(2)
𝑅𝑖
, 𝑖 = 1, 2, then theaccumulated goodwill of the two products along time 𝑡 is
𝐺𝑖(𝑡) = 𝐸
𝑖
𝑒−𝛿𝑡
+ 𝐺(2)
𝑖SS, 𝑖 ∈ {1, 2} , (27)
where 𝐺(2)𝑖SS = (𝑈
(2)
𝑀𝑖
− 𝜃𝑈(2)
𝑀,(3−𝑖)
)/𝛿, 𝐸𝑖
= 𝐺𝑖0
− (𝑈(2)
𝑀𝑖
−
𝜃𝑈(2)
𝑀,(3−𝑖)
)/𝛿, 𝑖 = 1, 2. 𝐺(2)𝑖SS is the steady-state goodwill for
product 𝑖 when 𝑡 → ∞.
Substituting (22) through (25) and (27) into (18) and (19),and assuming that the participation rates 𝜙
𝑖
(𝑖 = 1, 2) arefixed, we get the current value of the integration system’sprofit as follows:
𝐽(2)
𝑀1,𝑅
=𝐸1
𝛼𝜌1
+ 𝐸2
𝛼𝜌𝑅2
𝑟 + 𝛿
+
𝛼𝜌1
(𝑈(2)
𝑀1
− 𝜃𝑈(2)
𝑀2
) + 𝛼𝜌𝑅2
(𝑈(2)
𝑀2
− 𝜃𝑈(2)
𝑀1
)
𝑟𝛿
−
(𝑈(2)
𝑀1
)2
+ (𝑈(2)
𝑅1
)2
2𝑟+
𝜌1
(𝜆𝑈(2)
𝑅1
− 𝜒𝑈(2)
𝑅2
)
𝑟
+
𝜌𝑅2
(𝜆𝑈(2)
𝑅2
− 𝜒𝑈(2)
𝑅1
)
𝑟−
(1 − 𝜙2
) (𝑈(2)
𝑅2
)2
2𝑟,
(28)
and the profit for manufacturer 2 is
𝐽(2)
𝑀2
=𝐸2
𝛼𝜌𝑀2
𝑟 + 𝛿+
𝛼𝜌𝑀2
(𝑈(2)
𝑀2
− 𝜃𝑈(2)
𝑀1
)
𝑟𝛿
+
𝜌𝑀2
(𝜆𝑈(2)
𝑅2
− 𝜒𝑈(2)
𝑅1
)
𝑟−
(𝑈(2)
𝑀2
)2
− 𝜙2
(𝑈(2)
𝑅2
)2
2𝑟,
(29)
where 𝐸𝑖
= 𝐺𝑖0
− (𝑈(2)
𝑀𝑖
− 𝜃𝑈(2)
𝑀,(3−𝑖)
)/𝛿, 𝑖 = 1, 2.
Differentiating 𝐽(2)𝑀2
from the participation rate 𝜙2
, 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,𝑈𝑀𝑖
(𝑡) = 𝑈(2)
𝑀𝑖
,𝑈𝑅𝑖
(𝑡) = 𝑈(2)
𝑅𝑖
, 𝑖 = 1, 2, the optimal participationrate which manufacturer 2 provides is
𝜙(2)
2
=
{{
{{
{
𝜆 (2𝜌𝑀2
− 𝜌𝑅2
) + 𝜒𝜌1
𝜆 (2𝜌𝑀2
+ 𝜌𝑅2
) − 𝜒𝜌1
𝑖𝑓 𝜌𝑀2
≥(𝜆𝜌𝑅2
− 𝜒𝜌1
)
2𝜆,
0 𝑒𝑙𝑠𝑒,
(30)
where 𝜌1
= 𝜌𝑀1
+ 𝜌𝑅1
.
Subtracting (16) from (30), we have
Δ𝜙2
=
{{{{{{{{{{{{{{{{
{{{{{{{{{{{{{{{{
{
4𝜆𝜒𝜌𝑀1
𝜌𝑀2
𝐿1
𝐿2
if(𝜆𝜌𝑅2
− 𝜒𝜌𝑅1
)
2𝜆≤ 𝜌𝑀2
,
𝜆 (2𝜌𝑀2
− 𝜌𝑅2
) + 𝜒𝜌1
𝜆 (2𝜌𝑀2
+ 𝜌𝑅2
) − 𝜒𝜌1
if 𝜌𝑀2
<(𝜆𝜌𝑅2
− 𝜒𝜌𝑅1
)
2𝜆≤ 𝜌𝑀2
+𝜒𝜌𝑀1
2𝜆,
0 else,
(31)
where 𝐿1
= 2𝜆𝜌𝑀2
+𝜆𝜌𝑅2
−𝜒𝜌1
and 𝐿2
= 2𝜆𝜌𝑀2
+𝜆𝜌𝑅2
−𝜒𝜌𝑅1
.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 retailer’s equilibrium local advertisinglevel for product 2 is constant, that is,
𝑈(2)
𝑅2
=
{{{{{{{
{{{{{{{
{
𝜆𝜌𝑅2
− 𝜒𝜌1
if 𝜌𝑀2
<(𝜆𝜌𝑅2
− 𝜒𝜌1
)
2𝜆,
𝜆𝜌𝑀2
+1
2𝜆𝜌𝑅2
−1
2𝜒𝜌1
if 0 <(𝜆𝜌𝑅2
− 𝜒𝜌1
)
2𝜆≤ 𝜌𝑀2
,
0 else,
(32)
where 𝜌1
= 𝜌𝑀1
+ 𝜌𝑅1
.
8 Mathematical Problems in Engineering
Subtracting (17) from (32), we have
Δ𝑈𝑅2
=
{{{{{{{{{{{{{{{{{{{{{{{{{{{{{
{{{{{{{{{{{{{{{{{{{{{{{{{{{{{
{
−𝜒𝜌𝑀1
if 2𝜆𝜌𝑀2
+ 𝜒𝜌𝑀1
< 𝜆𝜌𝑅2
− 𝜒𝜌𝑅1
2𝜆𝜌𝑀2
− (𝜆𝜌𝑅2
− 𝜒𝜌𝑅1
) − 𝜒𝜌𝑀1
2
if 2𝜆𝜌𝑀2
< 𝜆𝜌𝑅2
− 𝜒𝜌𝑅1
≤ 2𝜆𝜌𝑀2
+ 𝜒𝜌𝑀1
−𝜒𝜌𝑀1
2
if 𝜒𝜌𝑀1
< 𝜆𝜌𝑅2
− 𝜒𝜌𝑅1
≤ 2𝜆𝜌𝑀2
−𝜆𝜌𝑀2
−1
2𝜆𝜌𝑅2
+1
2𝜒𝜌𝑅1
if 0 < 𝜆𝜌𝑅2
− 𝜒𝜌𝑅1
≤ 𝜒𝜌𝑀1
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. IBM’s personalcomputing division was acquired by Lenovo in 2004, andthe PC of IBM became a subbrand of Lenovo Group named“Thinkpad.”This 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 “single-manufacturer single-retailer” 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 integrationsystem’s profit function is 𝜋
𝑀1,𝑀2
= 𝜋𝑀1
+ 𝜋𝑀2
, so theobjective is
max𝑈𝑀1≥0,𝑈𝑀2≥0
1≥𝜙𝑖≥0
𝐽𝑀1,𝑀2
= ∫
+∞
0
𝑒−𝑟𝑡
(𝜋𝑀1
+ 𝜋𝑀2
) 𝑑𝑡, (34)
and the retailer’s objective is
max𝑈𝑅1≥0,𝑈𝑅2≥0
𝐽𝑅
= ∫
+∞
0
𝑒−𝑟𝑡
𝜋𝑅
𝑑𝑡. (35)
The current value Hamiltonian of the integration system(𝑀1 and𝑀2) is
𝐻𝑀1,𝑀2
= 𝜋𝑀1
+ 𝜋𝑀2
+ ]11
(𝑈𝑀1
− 𝜃𝑈𝑀2
− 𝛿𝐺1
)
+ ]12
(𝑈𝑀2
− 𝜃𝑈𝑀1
− 𝛿𝐺2
) ,
(36)
and that of the retailer is𝐻𝑅
= 𝜋𝑅
+ ]21
(𝑈𝑀1
− 𝜃𝑈𝑀2
− 𝛿𝐺1
)
+ ]22
(𝑈𝑀2
− 𝜃𝑈𝑀1
− 𝛿𝐺2
) ,
(37)
where ]𝑖1
and ]𝑖2
(𝑖 = 1, 2) are the costate variables to thefirm’s goodwill levels.
Using the necessary conditions for equilibrium,we get thefollowing results.
Proposition 9. When the two manufacturers are horizontallyintegrated and the participation rate 𝜙
𝑖
(𝑖 = 1, 2) is kept fixed,the equilibrium national advertising efforts for the two manu-facturers are all constants, that is,
𝑈(3)
𝑀𝑖
={
{
{
𝛼 (𝜌𝑀𝑖
− 𝜃𝜌𝑀(3−𝑖)
)
𝑟 + 𝛿𝑖𝑓 𝜌𝑀𝑖
≥ 𝜃𝜌𝑀(3−𝑖)
,
0 𝑒𝑙𝑠𝑒
𝑖 ∈ {1, 2} ,
(38)
and the retailer’s equilibrium local advertising efforts for thetwo products are all constants, that is,
𝑈(3)
𝑅𝑖
=
{{
{{
{
𝜆𝜌𝑅𝑖
− 𝜒𝜌𝑅(3−𝑖)
1 − 𝜙𝑖
𝑖𝑓 𝜆𝜌𝑅𝑖
− 𝜒𝜌𝑅(3−𝑖)
≥ 0,
0 𝑒𝑙𝑠𝑒
𝑖 ∈ {1, 2} .
(39)
Note that the retailer’s 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
Δ𝑈𝑀𝑖
=
{{
{{
{
−𝛼𝜃𝜌𝑀3−𝑖
𝑟 + 𝛿if 𝜌𝑀𝑖
≥ 𝜃𝜌𝑀,3−𝑖
,
−𝛼𝜌𝑀1
𝑟 + 𝛿else
𝑖 ∈ {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 𝑈
𝑀𝑖
= 0 and 𝑈𝑀,(3−𝑖)
= (𝛼𝜌𝑀3−𝑖
− 𝛼𝜃𝜌𝑀𝑖
)/(𝑟 + 𝛿) ifthe condition 𝜌
𝑀𝑖
< 𝜃𝜌𝑀,3−𝑖
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 members’ advertising efforts arekept as constants, that is, 𝑈
𝑀𝑖
= 𝑈(3)
𝑀𝑖
and 𝑈𝑅𝑖
= 𝑈(3)
𝑅𝑖
, then theaccumulated goodwill for the two products along time 𝑡 is
𝐺𝑖(𝑡) = 𝐹
𝑖
𝑒−𝛿𝑡
+ 𝐺(3)
𝑖SS, 𝑖 ∈ {1, 2} , (41)
Mathematical Problems in Engineering 9
where 𝐺(3)𝑖SS = (𝑈
(3)
𝑀𝑖
− 𝜃𝑈(3)
𝑀,(3−𝑖)
)/𝛿, 𝐹𝑖
= 𝐺𝑖0
− (𝑈(3)
𝑀𝑖
−
𝜃𝑈(3)
𝑀,(3−𝑖)
)/𝛿, 𝑖 = 1, 2. 𝐺(3)𝑖SS is the steady-state goodwill for
product 𝑖 when 𝑡 → ∞.
Substituting (38), (39), and (41) into (34) and (35) andwith the participation rates 𝜙
𝑖
(𝑖 = 1, 2) fixed, we get that thecurrent value of profit for the horizontal integration system is
𝐽(3)
𝑀1,𝑀2
=𝐹1
𝛼𝜌𝑀1
+ 𝐹2
𝛼𝜌𝑀2
𝑟 + 𝛿
+
𝛼𝜌𝑀1
(𝑈(3)
𝑀1
− 𝜃𝑈(3)
𝑀2
) + 𝛼𝜌𝑀2
(𝑈(3)
𝑀2
− 𝜃𝑈(3)
𝑀1
)
𝑟𝛿
+
𝜌𝑀1
(𝜆𝑈(3)
𝑅1
− 𝜒𝑈(3)
𝑅2
)
𝑟+
𝜌𝑀2
(𝜆𝑈(3)
𝑅2
− 𝜒𝑈(3)
𝑅1
)
𝑟
−
(𝑈(3)
𝑀1
)2
+ (𝑈(3)
𝑀2
)2
+ 𝜙1
(𝑈(3)
𝑅1
)2
+ 𝜙2
(𝑈(3)
𝑅2
)2
2𝑟,
(42)
and the current value of the profit for the retailer is
𝐽(3)
𝑅
=𝐹1
𝛼𝜌𝑅1
+ 𝐹2
𝛼𝜌𝑅2
𝑟 + 𝛿
+
𝛼𝜌𝑅1
(𝑈(3)
𝑀1
− 𝜃𝑈(3)
𝑀2
) + 𝛼𝜌𝑅2
(𝑈(3)
𝑀2
− 𝜃𝑈(3)
𝑀1
)
𝑟𝛿
−1 − 𝜙1
2(𝑈(3)
𝑅1
)2
+
𝜌𝑅1
(𝜆𝑈(3)
𝑅1
− 𝜒𝑈(3)
𝑅2
)
𝑟
+
𝜌𝑅2
(𝜆𝑈(3)
𝑅2
− 𝜒𝑈(3)
𝑅1
)
𝑟−1 − 𝜙2
2(𝑈(3)
𝑅2
)2
,
(43)
where 𝐹𝑖
= 𝐺𝑖0
− (𝑈(3)
𝑀𝑖
− 𝜃𝑈(3)
𝑀,(3−𝑖)
)/𝛿, 𝑖 = 1, 2.Differentiating 𝐽(3)
𝑀1,𝑀2
with the participation rate 𝜙1
and𝜙2
, we get the optimal participation rates:
𝜙(3)
𝑖
=
{{
{{
{
𝜆 (2𝜌𝑀𝑖
− 𝜌𝑅𝑖
) + 𝜒𝜌𝑅(3−𝑖)
𝜆 (2𝜌𝑀𝑖
+ 𝜌𝑅𝑖
) − 𝜒𝜌𝑅(3−𝑖)
if 𝜌𝑀𝑖
≥(𝜆𝜌𝑅𝑖
− 𝜒𝜌𝑅(3−𝑖)
)
2𝜆
0 else,
𝑖 ∈ {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 manufacturer’s 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: 𝛼 = 12,𝑟 = 0.3, 𝛿 = 0.2, 𝜃 = 0.2, 𝜆 = 10, 𝜒 = 4, 𝜌
𝑀1
= 7, 𝜌𝑀2
= 8,𝜌𝑅1
= 5, 𝜌𝑅2
= 4, 𝐺10
= 300, and 𝐺20
= 320.To obtain qualitative insight regarding how the current
value of each channel member’s profit varies as competitioncoefficients 𝜃 and 𝜒 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 𝜃 and𝜒 increase. From Figure 2, if competition coefficients 𝜃 and 𝜒equal 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 𝜃 and 𝜒 on the current value of the retailer’s 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 retailer’sprofit 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 𝜃 and 𝜒increase, the retailer’s profit would decrease whether the twomanufacturers integrate or not.
Finally, Figure 4 illustrates the impacts of the competitioncoefficients 𝜃 and 𝜒 on the current value of the profit ofmanufacturer 2. It suggests the following tendencies: (i)similarly to Figure 2, with competition coefficients 𝜃 and𝜒 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.,𝑀1 and retailer,𝑀2 and retailer, or𝑀1and𝑀2) integrate their efforts, the third firm would suffer.
5. Conclusion
Previous research primarily focused on a “single-manu-facturer single-retailer” framework, whereas few studies ad-dress a “multiple-manufacturer single-retailer” framework.To fill this gap, this paper investigates the advertising strate-gies for a “two-manufacturer single-retailer” supply chain in
10 Mathematical Problems in Engineering
×103
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 advertising’s competition coefficient 𝜃
(a)
×103
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 advertising’s competition coefficient 𝜒
(b)
Figure 2: Relationships between profits and competition coefficients 𝜃 and 𝜒.
×103
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 advertising’s competition coefficient 𝜃
(a)
×103
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 advertising’s competition coefficient 𝜒
(b)
Figure 3: Relationships between the retailer’s profit and the competition coefficients 𝜃 and 𝜒.
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
×103
The national advertising’s competition coefficient 𝜃
(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
×103
The local advertising’s competition coefficient 𝜒
(b)
Figure 4: Relationships between the profit of𝑀2 and the competition coefficients 𝜃 and 𝜒.
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)Themanufacturer’s 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 othermanufacturer’s 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.,𝑀1 and retailer,𝑀2 and retailer, or𝑀1 and𝑀2)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 “two-manufacturer
single-retailer” framework can be extended into a “multiple-manufacturer single-retailer” 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
𝐻𝑀1
= 𝜋𝑀1
+ 𝜇11
(𝑈𝑀1
− 𝜃𝑈𝑀2
− 𝛿𝐺1
)
+ 𝜇12
(𝑈𝑀2
− 𝜃𝑈𝑀1
− 𝛿𝐺2
) .
(A.1)
Then we form the Lagrangian:
𝐿𝑀1
= 𝜋𝑀1
+ 𝜇11
(𝑈𝑀1
− 𝜃𝑈𝑀2
− 𝛿𝐺1
)
+ 𝜇12
(𝑈𝑀2
− 𝜃𝑈𝑀1
− 𝛿𝐺2
)
+ 𝜂11
(𝛼𝐺1
+ 𝜆𝑈𝑅1
− 𝜒𝑈𝑅2
)
+ 𝜂12
(𝛼𝐺2
+ 𝜆𝑈𝑅2
− 𝜒𝑈𝑅1
) ,
(A.2)
12 Mathematical Problems in Engineering
the necessary conditions for equilibrium are given by
𝜕𝐿𝑀1
𝜕𝑈𝑀1
= 0, (A.3)
∙
𝜇11
= 𝑟𝜇11
−𝜕𝐿𝑀1
𝜕𝐺1
, (A.4)
∙
𝜇12
= 𝑟𝜇12
−𝜕𝐿𝑀1
𝜕𝐺2
, (A.5)
𝜕𝐿𝑀1
𝜕𝜂1𝑖
> 0, 𝜂1𝑖
≥ 0, 𝜂1𝑖
𝜕𝐿𝑀1
𝜕𝜂1𝑖
= 0, 𝑖 = 1, 2. (A.6)
Equation (A.3) implies
𝑈𝑀1
= 𝜇11
− 𝜃𝜇12
. (A.7)
Solving (A.4)–(A.6), we get∙
𝜇11
= (𝑟 + 𝛿) 𝜇11
− 𝛼𝜌𝑀1
− 𝛼𝜂1
,
∙
𝜇12
= (𝑟 + 𝛿) 𝜇12
− 𝛼𝜂2
.
(A.8)
Equation (A.6) implies: 𝜂1𝑖
= 0, then substituting 𝜂1𝑖
= 0
into (A.8), we get∙
𝜇11
= (𝑟 + 𝛿) 𝜇11
− 𝛼𝜌𝑀1
,
∙
𝜇12
= (𝑟 + 𝛿) 𝜇12
.
(A.9)
Differentiating (A.7) with respect to time and substitutingfor 𝜇11
, 𝜇12
and their time derivative in (A.9), we get∙
𝑈𝑀1
= (𝑟 + 𝛿)𝑈𝑀1
− 𝛼𝜌𝑀1
. (A.10)
Solving (A.10) to get the time paths of 𝑈𝑀1
, we find
𝑈𝑀1(𝑡) = 𝐶
1
𝑒(𝑟+𝛿)𝑡
+𝛼𝜌𝑀1
(𝑟 + 𝛿). (A.11)
Because there is no constraint at 𝑡 → ∞, 𝑈𝑀1
shouldsatisfy the free-boundary condition:
lim𝑡→∞
𝑈𝑀1(𝑡) < ∞. (A.12)
Condition (A.12) implies that 𝐶1
= 0. Therefore, weobtain the equilibrium advertising effort for manufacturer 1as follows:
𝑈(1)
𝑀1
=𝛼𝜌𝑀1
𝑟 + 𝛿. (A.13)
Similarly consideringmanufacturer 2’s profit maximizingproblem, we obtain the equilibrium advertising level formanufacturer 1 as follows:
𝑈(1)
𝑀2
=𝛼𝜌𝑀2
𝑟 + 𝛿. (A.14)
For (1), we can get the general solutions of (1) as
𝐺𝑖(𝑡) = 𝐷
𝑖
𝑒−𝛿𝑡
+ 𝐺𝑖𝑆𝑆
, 𝑖 ∈ {1, 2} , (A.15)
where 𝐺𝑖𝑆𝑆
= (𝑈𝑀𝑖
− 𝜃𝑈𝑀,(3−𝑖)
)/𝛿, 𝑖 = 1, 2.
𝐷𝑖
is an arbitrary constant. Letting 𝑡 = 0 in (A.15) andutilizing the initial conditions of (2), we get𝐷
𝑖
= 𝐺𝑖0
−(𝑈𝑀𝑖
−
𝜃𝑈𝑀,(3−𝑖)
)/𝛿, 𝑖 = 1, 2.Substituting (A.13) and (A.14) into (A.15), we find that
𝐺𝑖(𝑡) = 𝐷
𝑖
𝑒−𝛿𝑡
+ 𝐺(1)
𝑖𝑆𝑆
, 𝑖 ∈ {1, 2} , (A.16)
where 𝐷𝑖
= 𝐺𝑖0
− (𝑈(1)
𝑀𝑖
− 𝜃𝑈(1)
𝑀,(3−𝑖)
)/𝛿, 𝑖 = 1, 2 and 𝐺(1)𝑖𝑆𝑆
=
(𝑈(1)
𝑀𝑖
− 𝜃𝑈(1)
𝑀,(3−𝑖)
)/𝛿, 𝑖 = 1, 2, when condition 0 ≤ 𝜃 ≤
min{𝜌𝑀1
/𝜌𝑀2
, 𝜌𝑀2
/𝜌𝑀1
} holds, the steady-state goodwill𝐺𝑖𝑆𝑆
is nonnegative.
Proof of Proposition 2. When each channel member makesdecisions independently, the current value Hamiltonian ofthe retailer is:
𝐻𝑅
= 𝜌𝑅1
(𝛼𝐺1
+ 𝜆𝑈𝑅1
− 𝜒𝑈𝑅2
)
+ 𝜌𝑅2
(𝛼𝐺2
+ 𝜆𝑈𝑅2
− 𝜒𝑈𝑅1
)
−1
2(1 − 𝜙
1
) 𝑈2
𝑅1
−1
2(1 − 𝜙
2
) 𝑈2
𝑅2
+ 𝜇31
(𝑈𝑀1
− 𝜃𝑈𝑀2
− 𝛿𝐺1
)
+ 𝜇32
(𝑈𝑀2
− 𝜃𝑈𝑀1
− 𝛿𝐺2
) .
(A.17)
Then we form the Lagrangian
𝐿𝑅
= 𝜌𝑅1
(𝛼𝐺1
+ 𝜆𝑈𝑅1
− 𝜒𝑈𝑅2
)
+ 𝜌𝑅2
(𝛼𝐺2
+ 𝜆𝑈𝑅2
− 𝜒𝑈𝑅1
)
−1
2(1 − 𝜙
1
) 𝑈2
𝑅1
−1
2(1 − 𝜙
2
) 𝑈2
𝑅2
+ 𝜇31
(𝑈𝑀1
− 𝜃𝑈𝑀2
− 𝛿𝐺1
)
+ 𝜇32
(𝑈𝑀2
− 𝜃𝑈𝑀1
− 𝛿𝐺2
)
+ 𝜂31
(𝛼𝐺1
+ 𝜆𝑈𝑅1
− 𝜒𝑈𝑅2
)
+ 𝜂32
(𝛼𝐺2
+ 𝜆𝑈𝑅2
− 𝜒𝑈𝑅1
) .
(A.18)
The necessary conditions for equilibrium are given by𝜕𝐿𝑅
𝜕𝑈𝑅1
= 0, (A.19)
𝜕𝐿𝑅
𝜕𝑈𝑅2
= 0, (A.20)
∙
𝜇31
= 𝑟𝜇31
−𝜕𝐿𝑅
𝜕𝐺1
, (A.21)
∙
𝜇32
= 𝑟𝜇32
−𝜕𝐿𝑅
𝜕𝐺2
, (A.22)
𝜕𝐿𝑅
𝜕𝜂3𝑖
> 0, 𝜂3𝑖
≥ 0, 𝜂3𝑖
𝜕𝐿𝑅
𝜕𝜂3𝑖
= 0, 𝑖 = 1, 2. (A.23)
Because 𝑈𝑅𝑖
is constrained to be nonnegative, (A.19)implies that
𝑈𝑅1(𝑡) = max{0,
(𝜆𝜌𝑅1
− 𝜒𝜌𝑅2
)
(1 − 𝜙1
)} . (A.24)
Mathematical Problems in Engineering 13
Through a similar proof, we find that
𝑈𝑅2(𝑡) = max{0,
(𝜆𝜌𝑅2
− 𝜒𝜌𝑅1
)
(1 − 𝜙2
)} . (A.25)
Therefore we can obtain the following results:
𝑈(1)
𝑅𝑖
=
{{
{{
{
𝜆𝜌𝑅𝑖
− 𝜒𝜌𝑅(3−𝑖)
1 − 𝜙𝑖
if 𝜆𝜌𝑅𝑖
− 𝜒𝜌𝑅(3−𝑖)
≥ 0,
0 else,
𝑖 ∈ {1, 2} .
(A.26)
Proof of Proposition 4. There are no relationships betweenparticipation rate 𝜙
1
and 𝑈(1)𝑅2
or the two manufacturer’snational advertising efforts. Thus in differentiating 𝐽(1)
𝑀1
fromthe participation rate 𝜙
1
we only consider two situations.Situation (1) when 𝜆𝜌
𝑅1
− 𝜒𝜌𝑅2
≥ 0 holds, 𝑈(1)𝑅1
= (𝜆𝜌𝑅1
−
𝜒𝜌𝑅2
)/(1 − 𝜙1
), substituting the above expression into (20)and differentiating 𝐽
𝑀1
with the participation rate 𝜙1
, we getmanufacturer 1’s optimal participation rate: 𝜙
1
= [𝜆(2𝜌𝑀1
−
𝜌𝑅1
) + 𝜒𝜌𝑅2
]/[𝜆(2𝜌𝑀1
+ 𝜌𝑅1
) − 𝜒𝜌𝑅2
]. Since 0 ≤ 𝜙1
≤ 1,the condition 2𝜆𝜌
𝑀1
≥ (𝜆𝜌𝑅1
− 𝜒𝜌𝑅2
) is required. Situation(2) if condition (𝜆𝜌
𝑅1
− 𝜒𝜌𝑅2
) < 0 holds, we get 𝑈(1)𝑅1
= 0.In this situation, whatever participation rate manufacturer1 offers, the retailer would never advertise product 1. Theparticipation rate 𝜙
1
is an arbitrary constant. We suppose𝜙1
= [𝜆(2𝜌𝑀1
− 𝜌𝑅1
) + 𝜒𝜌𝑅2
]/[𝜆(2𝜌𝑀1
+ 𝜌𝑅1
) − 𝜒𝜌𝑅2
]. Theparticipation rate 𝜙
1
is useless in this situation; therefore, theparticipation rate could be negative.
In conclusion, we get the following results:
𝜙(1)
1
=
{{
{{
{
𝜆 (2𝜌𝑀1
− 𝜌𝑅1
) + 𝜒𝜌𝑅2
𝜆 (2𝜌𝑀1
+ 𝜌𝑅1
) − 𝜒𝜌𝑅2
if 2𝜆𝜌𝑀1
≥ 𝜆𝜌𝑅1
− 𝜒𝜌𝑅2
0 else.(A.27)
Through a similar proof, we get manufacturer 2’s optimalshare rate is
𝜙(1)
2
=
{{
{{
{
𝜆 (2𝜌𝑀2
− 𝜌𝑅2
) + 𝜒𝜌𝑅1
𝜆 (2𝜌𝑀2
+ 𝜌𝑅2
) − 𝜒𝜌𝑅1
if 2𝜆𝜌𝑀2
≥ 𝜆𝜌𝑅2
− 𝜒𝜌𝑅1
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
𝐻𝑀2
= 𝜌𝑀2
(𝛼𝐺2
+ 𝜆𝑈𝑅2
− 𝜒𝑈𝑅1
)
−1
2𝑈2
𝑀2
−1
2𝜙2
𝑈2
𝑅2
+ 𝛾21
(𝑈𝑀1
− 𝜃𝑈𝑀2
− 𝛿𝐺1
)
+ 𝛾22
(𝑈𝑀2
− 𝜃𝑈𝑀1
− 𝛿𝐺2
) .
(B.1)
Then we form the Lagrangian:
𝐿𝑀2
= 𝜌𝑀2
(𝛼𝐺2
+ 𝜆𝑈𝑅2
− 𝜒𝑈𝑅1
) −1
2𝑈2
𝑀2
−1
2𝜙2
𝑈2
𝑅2
+ 𝛾21
(𝑈𝑀1
− 𝜃𝑈𝑀2
− 𝛿𝐺1
)
+ 𝛾22
(𝑈𝑀2
− 𝜃𝑈𝑀1
− 𝛿𝐺2
)
+ 𝜉21
(𝛼𝐺1
+ 𝜆𝑈𝑅1
− 𝜒𝑈𝑅2
)
+ 𝜉22
(𝛼𝐺2
+ 𝜆𝑈𝑅2
− 𝜒𝑈𝑅1
) .
(B.2)
At optimality, the necessary conditions are
𝜕𝐿𝑀2
𝜕𝑈𝑀2
= 0,
∙
𝛾21
= 𝑟𝛾21
−𝜕𝐿𝑀1
𝜕𝐺1
,
∙
𝛾22
= 𝑟𝛾22
−𝜕𝐿𝑀1
𝜕𝐺2
,
𝜕𝐿𝑀2
𝜕𝜉2𝑖
> 0, 𝜉2𝑖
≥ 0, 𝜉2𝑖
𝜕𝐿𝑀2
𝜕𝜉2𝑖
= 0, 𝑖 = 1, 2.
(B.3)
Proceeding as in the proof for Proposition 1, we get
𝑈(2)
𝑀2
=𝛼𝜌𝑀2
(𝑟 + 𝛿). (B.4)
Proof of Propositions 6 and 7. When the retailer integrateswith a manufacturer, the current value Hamiltonian forintegration system is given by
𝐻𝑀1,𝑅
= 𝜋𝑀1
+ 𝜋𝑅
+ 𝛾11
(𝑈𝑀1
− 𝜃𝑈𝑀2
− 𝛿𝐺1
)
+ 𝛾12
(𝑈𝑀2
− 𝜃𝑈𝑀1
− 𝛿𝐺2
) .
(B.5)
Then we form the Lagrangian:
𝐿𝑀1,𝑅
= 𝜋𝑀1
+ 𝜋𝑅
+ 𝛾11
(𝑈𝑀1
− 𝜃𝑈𝑀2
− 𝛿𝐺1
)
+ 𝛾12
(𝑈𝑀2
− 𝜃𝑈𝑀1
− 𝛿𝐺2
)
+ 𝜉11
(𝛼𝐺1
+ 𝜆𝑈𝑅1
− 𝜒𝑈𝑅2
)
+ 𝜉12
(𝛼𝐺2
+ 𝜆𝑈𝑅2
− 𝜒𝑈𝑅1
) .
(B.6)
Proceeding as in the proof for Proposition 1, and con-straining,as in most cases, the advertising efforts 𝑈(𝑡) to be
14 Mathematical Problems in Engineering
nonnegative, we get the following results:
𝑈(2)
𝑅1
= max {0, 𝜆 (𝜌𝑀1
+ 𝜌𝑅1
) − 𝜒𝜌𝑅2
} ,
𝑈(2)
𝑅2
= max{0,𝜆𝜌𝑅2
− 𝜒 (𝜌𝑅1
+ 𝜌𝑀1
)
1 − 𝜙2
} .
(B.7)
Thus, the equilibrium advertising levels for manufacturer2 are as follows:
𝑈(2)
𝑀1
={
{
{
𝛼 (𝜌𝑀1
+ 𝜌𝑅1
)
𝑟 + 𝛿−𝜃𝛼𝜌𝑅2
𝑟 + 𝛿if (𝜌𝑀1
+ 𝜌𝑅1
) ≥ 𝜃𝜌𝑅2
0 else.(B.8)
Also, we obtain the equilibrium local advertising levels forthe two products:
𝑈(2)
𝑅1
= {𝜆 (𝜌𝑀1
+ 𝜌𝑅1
) − 𝜒𝜌𝑅2
if 𝜆𝜌1
− 𝜒𝜌𝑅2
≥ 0
0 else,
𝑈(2)
𝑅2
=
{{
{{
{
𝜆𝜌𝑅2
− 𝜒 (𝜌𝑅1
+ 𝜌𝑀1
)
1 − 𝜙2
if 𝜆𝜌𝑅2
− 𝜒𝜌1
≥ 0
0 else.
(B.9)
Substituting (B.4) and (B.8) into (A.15), we find that
𝐺𝑖(𝑡) = 𝐸
𝑖
𝑒−𝛿𝑡
+ 𝐺(2)
𝑖𝑆𝑆
, 𝑖 ∈ {1, 2} , (B.10)
where 𝐸𝑖
= 𝐺𝑖0
− (𝑈(2)
𝑀𝑖
− 𝜃𝑈(2)
𝑀,(3−𝑖)
)/𝛿, 𝑖 = 1, 2 and 𝐺(2)𝑖𝑆𝑆
=
(𝑈(2)
𝑀𝑖
− 𝜃𝑈(2)
𝑀,(3−𝑖)
)/𝛿, 𝑖 = 1, 2.When condition 0 ≤ 𝜃 ≤ min{𝜌
1
/𝜌2
, (𝜌1
−
√𝜌2
1
− 4𝜌𝑀2
𝜌𝑅2
)/2𝜌𝑅2
} holds, the steady-state goodwill 𝐺(2)𝑖𝑆𝑆
is nonnegative.
Proof of Proposition 8. There are no relationships betweenparticipation rate 𝜙
2
and 𝑈(2)𝑅2
or two manufacturer’s nationaladvertising efforts. Thus in differentiating 𝐽(2)
𝑀2
from theparticipation rate 𝜙
2
we only consider two situations. (1)When 𝜆𝜌
𝑅2
− 𝜒𝜌1
≥ 0 holds, we get 𝑈(2)𝑅2
= [𝜆𝜌𝑅2
− 𝜒(𝜌𝑅1
+
𝜌𝑀1
)]/(1 − 𝜙2
). Substituting the above expression into (38),and differentiating 𝐽
𝑀2
from the participation rate 𝜙2
, we getmanufacturer 2’s optimal participation rate. 𝜙
2
= [𝜆(2𝜌𝑀2
−
𝜌𝑅2
) + 𝜒𝜌1
]/[𝜆(2𝜌𝑀2
+ 𝜌𝑅2
) − 𝜒𝜌1
]. Since 0 ≤ 𝜙2
≤ 1, theassumption 𝜌
𝑀2
≥ (𝜆𝜌𝑅2
− 𝜒𝜌1
)/2𝜆 is required. (2) Whencondition (𝜆𝜌
𝑅2
− 𝜒𝜌𝑅1
) < 0 holds, we get 𝑈(2)𝑅2
= 0. In thissituation, whatever participation rate manufacturer 2 offers,the retailer would never advertise product 2. Thus, 𝜙
2
is anarbitrary constant. Here we suppose: 𝜙
2
= [𝜆(2𝜌𝑀2
− 𝜌𝑅2
) +
𝜒𝜌1
]/[𝜆(2𝜌𝑀2
+𝜌𝑅2
)−𝜒𝜌1
].The participation rate 𝜙1
is uselessin this situation; therefore the participation rate could benegative.
In conclusion, we get the following results:
𝜙(2)
2
=
{{
{{
{
𝜆 (2𝜌𝑀2
− 𝜌𝑅2
) + 𝜒𝜌1
𝜆 (2𝜌𝑀2
+ 𝜌𝑅2
) − 𝜒𝜌1
if 𝜌𝑀2
≥(𝜆𝜌𝑅2
− 𝜒𝜌1
)
2𝜆
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
𝐻𝑀1,𝑀2
= 𝜌𝑀1
(𝛼𝐺1
+ 𝜆𝑈𝑅1
− 𝜒𝑈𝑅2
)
+ 𝜌𝑀2
(𝛼𝐺2
+ 𝜆𝑈𝑅2
− 𝜒𝑈𝑅1
)
−1
2𝑈2
𝑀1
−1
2(1 − 𝜙
1
) 𝑈2
𝑅1
−1
2𝑈2
𝑀2
−1
2(1 − 𝜙
2
) 𝑈2
𝑅2
+ ]11
(𝑈𝑀1
− 𝜃𝑈𝑀2
− 𝛿𝐺1
)
+ ]12
(𝑈𝑀2
− 𝜃𝑈𝑀1
− 𝛿𝐺2
) .
(C.1)
Then we form the Lagrangian𝐿𝑀1,𝑀2
= 𝜌𝑀1
(𝛼𝐺1
+ 𝜆𝑈𝑅1
− 𝜒𝑈𝑅2
)
+ 𝜌𝑀2
(𝛼𝐺2
+ 𝜆𝑈𝑅2
− 𝜒𝑈𝑅1
)
−1
2𝑈2
𝑀1
−1
2(1 − 𝜙
1
) 𝑈2
𝑅1
−1
2𝑈2
𝑀2
−1
2(1 − 𝜙
2
) 𝑈2
𝑅2
+ ]11
(𝑈𝑀1
− 𝜃𝑈𝑀2
− 𝛿𝐺1
)
+ ]12
(𝑈𝑀2
− 𝜃𝑈𝑀1
− 𝛿𝐺2
)
+ 𝜗11
(𝛼𝐺1
+ 𝜆𝑈𝑅1
− 𝜒𝑈𝑅2
)
+ 𝜗12
(𝛼𝐺2
+ 𝜆𝑈𝑅2
− 𝜒𝑈𝑅1
) .
(C.2)
At optimality, the necessary conditions are𝜕𝐿𝑀1,𝑀2
𝜕𝑈𝑀1
= 0, (C.3)
∙
]11
= 𝑟]11
−𝜕𝐿𝑀1,𝑀2
𝜕𝐺1
, (C.4)
∙
]12
= 𝑟]12
−𝜕𝐿𝑀1,𝑀2
𝜕𝐺2
, (C.5)
𝜕𝐿𝑀1,𝑀2
𝜕𝜗1𝑖
> 0, 𝜗1𝑖
≥ 0, 𝜗1𝑖
𝜕𝐿𝑀1,𝑀2
𝜕𝜗1𝑖
= 0, 𝑖 = 1, 2.
(C.6)In most case the advertising effort 𝑈
𝑀1
is constrained tobe nonnegative. Thus, (C.3) implies
𝑈𝑀1
= max {0, 𝛽𝜌𝑀1
+ ]11
− 𝜃]12
} . (C.7)Solving (C.4)-(C.5), we get
∙
]11
= (𝑟 + 𝛿) ]11
− 𝛼𝜌𝑀1
− 𝛼𝜗11
,
∙
]12
= (𝑟 + 𝛿) ]12
− 𝛼𝜌𝑀2
− 𝛼𝜗12
.
(C.8)
Mathematical Problems in Engineering 15
Equation (C.6) implies, 𝜗1𝑖
= 0; then substituting 𝜗1𝑖
= 0
into (C.8), we get∙
]11
= (𝑟 + 𝛿) ]11
− 𝛼𝜌𝑀1
, (C.9)
∙
]12
= (𝑟 + 𝛿) ]12
− 𝛼𝜌𝑀2
. (C.10)
Differentiating (C.7) with respect to time and substitutingfor ]11
, ]12
and their time derivative in (C.9)-(C.10), we get∙
𝑈𝑀1
= (𝑟 + 𝛿)𝑈𝑀1
− 𝛼𝜌𝑀1
+ 𝜃𝛼𝜌𝑀2
. (C.11)
Proceeding as in the proof of Proposition 1, we get
𝑈(3)
𝑀1
= max{0,𝛼𝜌𝑀1
(𝑟 + 𝛿)−𝜃𝛼𝜌𝑀2
(𝑟 + 𝛿)} ,
𝑈(3)
𝑀2
= max{0,𝛼𝜌𝑀2
(𝑟 + 𝛿)−𝜃𝛼𝜌𝑀1
(𝑟 + 𝛿)} .
(C.12)
Thus, the equilibrium advertising levels for twomanufac-turers are as follows:
𝑈(3)
𝑀𝑖
={
{
{
𝛼 (𝜌𝑀𝑖
− 𝜃𝜌𝑀(3−𝑖)
)
𝑟 + 𝛿if 𝜌𝑀𝑖
≥ 𝜃𝜌𝑀,3−𝑖
,
0 else
𝑖 ∈ {1, 2} .
(C.13)
Substituting (C.13) into (A.15), we find that
𝐺𝑖(𝑡) = 𝐹
𝑖
𝑒−𝛿𝑡
+ 𝐺(3)
𝑖𝑆𝑆
, 𝑖 ∈ {1, 2} , (C.14)
where 𝐹𝑖
= 𝐺𝑖0
− (𝑈(3)
𝑀𝑖
− 𝜃𝑈(3)
𝑀,(3−𝑖)
)/𝛿, 𝑖 = 1, 2 and 𝐺(3)𝑖𝑆𝑆
=
(𝑈(3)
𝑀𝑖
−𝜃𝑈(3)
𝑀,(3−𝑖)
)/𝛿, 𝑖 = 1, 2.When condition 0 ≤ 2𝜃/(1+𝜃2) ≤min{𝜌
𝑀1
/𝜌𝑀2
, 𝜌𝑀2
/𝜌𝑀1
} holds, the steady-state goodwill𝐺(3)𝑖𝑆𝑆
is nonnegative.The current value Hamiltonian for the retailer is given by
𝐻𝑅
= 𝜌𝑅1
(𝛼𝐺1
+ 𝜆𝑈𝑅1
− 𝜒𝑈𝑅2
)
+ 𝜌𝑅2
(𝛼𝐺2
+ 𝜆𝑈𝑅2
− 𝜒𝑈𝑅1
) −1
2(1 − 𝜙
1
) 𝑈2
𝑅1
−1
2(1 − 𝜙
2
) 𝑈2
𝑅2
+ ]21
(𝑈𝑀1
− 𝜃𝑈𝑀2
− 𝛿𝐺1
)
+ ]22
(𝑈𝑀2
− 𝜃𝑈𝑀1
− 𝛿𝐺2
) .
(C.15)
Then we form the Lagrangian:
𝐿𝑅
= 𝜌𝑅1
(𝛼𝐺1
+ 𝜆𝑈𝑅1
− 𝜒𝑈𝑅2
)
+ 𝜌𝑅2
(𝛼𝐺2
+ 𝜆𝑈𝑅2
− 𝜒𝑈𝑅1
)
−1
2(1 − 𝜙
1
) 𝑈2
𝑅1
−1
2(1 − 𝜙
2
) 𝑈2
𝑅2
+ ]21
(𝑈𝑀1
− 𝜃𝑈𝑀2
− 𝛿𝐺1
)
+ ]22
(𝑈𝑀2
− 𝜃𝑈𝑀1
− 𝛿𝐺2
)
+ 𝜗21
(𝛼𝐺1
+ 𝜆𝑈𝑅1
− 𝜒𝑈𝑅2
)
+ 𝜗22
(𝛼𝐺2
+ 𝜆𝑈𝑅2
− 𝜒𝑈𝑅1
) .
(C.16)
Proceeding as in the proof of Proposition 2, we get
𝑈(3)
𝑅1
= max{0,(𝜆𝜌𝑅1
− 𝜒𝜌𝑅2
)
(1 − 𝜙1
)} ,
𝑈(3)
𝑅2
= max{0,(𝜆𝜌𝑅2
− 𝜒𝜌𝑅1
)
(1 − 𝜙2
)} .
(C.17)
Then we get the following results:
𝑈(3)
𝑅𝑖
=
{{
{{
{
𝜆𝜌𝑅𝑖
− 𝜒𝜌𝑅(3−𝑖)
1 − 𝜙𝑖
if 𝜆𝜌𝑅𝑖
− 𝜒𝜌𝑅(3−𝑖)
≥ 0
0 else,
𝑖 ∈ {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.
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