Price competition and store competition: Store brands vs. national brand

European Journal of Operational Research 225 (2013) 166–178

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European Journal of Operational Research

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Price competition and store competition: Store brands vs. national brand

Sungchul Choi a, Karima Fredj b,⇑a School of Business, University of Northern British Columbia, Prince George, Canada V2N 4Z9b Department of Economics, University of Northern British Columbia, Prince George, Canada V2N 4Z9

a r t i c l e i n f o

Article history:Received 24 January 2010Accepted 13 July 2012Available online 26 July 2012

Keywords:Channel leadershipDistribution channelsGame theoryProduct differentiationStore brandsStore differentiation

0377-2217/$ - see front matter Crown Copyright � 2http://dx.doi.org/10.1016/j.ejor.2012.07.016

⇑ Corresponding author. Tel.: +1 2509606686; fax:E-mail address: [email protected] (K. Fredj).

a b s t r a c t

This paper studies pricing strategies in a market channel composed of one national brand manufacturerand two retailers who, each, carry their own store brand and a national brand products. The modelaccounts for product competition between store brands and the national brand products, as well as forstore competition between retailers.

We use a game-theoretic model to examine a variety of channel leadership structures that considerstrategic interactions both at the vertical level (between manufacturer and retailers) and at the horizontallevel (between retailers). This extends the existing literature in main three directions: (a) accounting forboth store competition and product competition in the same model, (b) considering a more comprehen-sive study including all possible interactions between channel members in this context, and (c) introduc-ing horizontal price leadership between the retailers.

Our results suggest that to be more successful and reap higher profits every retailer should pursuegreater store differentiation relative to other competing retailers and less brand differentiation relativeto the national brand to provide relatively homogeneous brands in a well differentiated store. On theother hand, the manufacturer should differentiate his national brand more from the retailers’ storebrands and deal with as many and less strategically powerful retailers as possible. From the consumers’perspective, the manufacturer’s leadership represents the worst scenario as it leads to the highest retailprices for all brands. The lowest prices for the store brand are reached when the retailers have verticalprice leadership and for the national brand when there is no leadership.

Crown Copyright � 2012 Published by Elsevier B.V. All rights reserved.

1. Introduction

Store brands represent more than $65 billion of current retailbusinesses and are achieving new levels of growth every year (Pri-vate Label Manufacturers Association, 2008 Yearbook). One out ofevery five items sold per day in the US supermarkets, drug chainsand mass merchandisers is a store brand product (the UnitedStates Department of Agriculture, 2000). Because of the successof such store brands in several product categories over the pasttwo decades, competition between national brands and storebrands attracted different research interests.

The first stream of research used an empirical approach to inves-tigate the effects of some variables such as advertising, perceivedquality, and industry concentration on competition betweennational brand and store brand, and also explained the variationin store brand market share across different product categories(see for examples: Connor and Peterson, 1992; Hoch and Banerjee,1993; Hoch, 1996; Kim and Parker, 1999; Cotterill et al., 2000).

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+1 2509605545.

The second research stream introduced theoretical models todescribe price competition between national brands and storebrands and to study some related issues. For instance; Raju et al.(1995) developed a game-theoretic model that examines the con-ditions under which it is profitable to introduce store brands;Narasimhan and Wilcox (1998) examined the incentives for storebrand introduction in the loyal/switcher market structure; andSayman et al. (2002), Du et al. (2005), and Choi and Coughlan(2006) investigated positioning issues related to the retailer’s storebrand.

Finally, some research focused directly on the impact of differ-ent leadership structures in various marketing channels on profits,retail pricing and, in a few cases, advertising. Jørgensen et al.(2001), for example, examined the effect of leadership in the singlemanufacturer–single retailer bilateral monopoly channel andfound that manufacturer leadership improved channel efficiencyin terms of pricing and advertising compared to retailer leadership.Choi (1991) dealt with three non-cooperative games between twomanufacturers and one common retailer that covered VerticalNash, Manufacturer Stackelberg and Retailer Stackelberg leader-ships. His results showed that, under a linear duopoly demandfunction, all channel members as well as consumers are better

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S. Choi, K. Fredj / European Journal of Operational Research 225 (2013) 166–178 167

off in absence of a market leader (i.e. under Vertical Nash). Lee andStaelin (1997) used a model with two manufacturers and tworetailers to show that vertical strategic interactions could be a driv-ing key for optimal decisions on channel price leadership and prod-uct line pricing (see also Shugan and Jeuland, 1988; Moorthy andFader, 1989 for more related examples).

In our model, we consider competition between the manufac-turer’s national brand and the two retailers’ store brands in addi-tion to store competition between the two retailers who eachcarry their own store brand and the national brand. In this paperwe aim to address some limitations within the existing literaturein three main directions.

Firstly, in order to guarantee a maximum exposure of a brandand for better consumer convenience, most of consumer goodsare intensively distributed through several independent retailers.Consequently, retailers end up offering different competing brandsof the same product including their own brand in some cases.Therefore, the importance of inter-store competition between largeretailers has been argued to be critical, especially when storebrands are introduced (see, McMaster, 1987). However, most stud-ies of store and national brand competition tend to ignore inter-store competition and focus only on product competition betweena national brand and store brands (e.g. Raju et al., 1995; Narasim-han and Wilcox, 1998; Sayman et al., 2002). On the other hand, pre-vious studies that considered effects of store differentiation in aduopoly common retailers channel, did not consider store brandsin their models (e.g. Choi, 1996; Trivedi, 1998; Basuroy et al.,2001). Finally,the few studies that looked at product competitionwhile considering store brands did not account for store differenti-ation, which make those models different from the present work aswell. For instance; to investigate the effect of retail competition onstore brand introduction strategies, Groznik and Heese (2010)introduced horizontal and vertical product differentiation withina numerical study; while Tarziján (2004) considered a cournotquantity competition between retailers focusing on the impact ofretail concentration; Kurata et al. (2007) considered brand andchannel competition between two retail channels controlledrespectively by one retailer and one manufacturer to study pricingpolicies and brand loyalty building; and finally Corstjens and Lal(2000) used a retail duopoly framework to show that quality storebrands can be used as an implicit and profitable coordination mech-anism when carried by competing retailers. Thus, little is knownabout the role that store brands play in price competition at the re-tail level when introducing, at the same time, store differentiation.One of our contributions is therefore to introduce in the same mod-el store competition between the retailers and product competitionbetween store brands and the national brand. To this purpose, weuse a model framework where the market structure is composedof one national brand manufacturer and two retailers who, each,carry their own store brand and the manufacturer’s national brand.

Secondly, previous studies of store brands investigated eitherthe Manufacturer Stackelberg (MS) channel leadership structure,where manufacturers are modeled as Stackelberg leaders andretailers as followers (see for example: Groznik and Heese, 2010;Raju et al., 1995; Tarziján, 2004; Narasimhan and Wilcox, 1998;Sayman et al., 2002; Yao et al., 2008), or the Vertical Nash (VN)where manufacturers and retailers have the same strategic powerlevel (e.g. Kurata et al., 2007; Corstjens and Lal, 2000). As such,these models have excluded one important leadership structure,the Retailer Stackelberg (RS), where, as opposed to the MS, retailersact as price leaders and manufacturers as price followers. The RS iseven more important to consider in this specific context given thatstore brand introduction has been proven to reduce retailers’dependence on the national brands and to increase their bargain-ing power and store profitability (see for example: Corstjens and

Lal, 2000; Groznik and Heese, 2010; Steiner, 2004; Tarziján,2004). The retailer can then, in this case, control the flow of theproduct and lead the channel, obliging the manufacturer to accord-ingly adjust their decisions (e.g., Choi, 1991, 1996 and Lee andStaelin, 1997). In same line of thought, empirical studies led byNarasimhan and Wilcox (1998) and Morton and Zettelmeyer(2004) examining the strategic role that store brands can play indetermining the manufacturer-retailer relationship, found thatstore brands can be used strategically to affect the wholesale pricesof the manufacturer’ national brand. The RS structure is also sup-ported by some real examples such as Superstore and Sobeys,two major retailers in Canada’s retail food business. These tworetailers earned a significant strategic advantage over nationalbrand manufacturers due to some of their successful store brandssuch as President’s Choice and Compliments (see, Canada Retail FoodSector Report 2009). Henceforth, in this paper we extend our mod-el to study and compare all the possible leadership structures (MS,RS, VN). This would particularly reflect the impact of store brandson the strategic role retailers can play when both product differen-tiation and store differentiation are simultaneously considered.

Finally, all previous studies ignored the possibility of horizontalprice leadership and assumed that channel members interact à-laBertrand–Nash at the horizontal level (e.g. Corstjens and Lal,2000; Groznik and Heese, 2010; Raju et al., 1995; Sayman et al.,2002; Tarziján, 2004). One of our main contributions is, hence, tointroduce a new leadership structure that considers a sequentialinteraction à-la Stackelberg at the horizontal level between thetwo retailers. The latter is mainly justified by the inconsistent re-sults related to the use of the Bertrand–Nash interaction at hori-zontal level, which have been pointed out through severalempirical studies. For instance, Vilcassim et al. (1999) analyzedthe dynamic price and advertising competition among firms in apersonal-care product category and found evidence rejecting Ber-trand–Nash interactions among firms. Kadiyali et al. (2000)pointed out that ‘‘a more general model of interactions is needed’’and Dhar and Ray (2004) proved that grocery supermarkets withstrong store brands can show more strategic interaction betweenretailers in the fluid milk product category (see also Bresnahan,1989 for more related counter-examples). In practice, this channelsetting can be supported by a real example such as Superstore,which is more likely as a major retailer in consumer packagedgoods to have an informational advantage (characteristic of a chan-nel leader) compared to a competing retailer such as Metro. Bothretailers in this example have their own store brand, which givesthem more bargaining power to influence the manufacturers’ pric-ing decisions see Steiner (2004) and illustrates the RDS scenario.

To recapitulate, the primary focus of this paper is to investigateprice competition between a manufacturer’s national brand andtwo retailers’ respective store brands, while simultaneouslyaccounting for store competition between the retailers. This sub-stantially extends the previous literature in three directions: (1)considering both product and store differentiation, (2) applyingdifferent vertical price leadership structures (MS, RS and VN) to astore brand model and (3) applying price leadership at the retailerlevel.

In the following section, we develop a model that examinesprice competition between a national brand and two store brandsin a channel structure with one manufacturer and two retailers. Inthe third section, we derive the analytical equilibrium solutions forthe prices, margins, quantities and profits under the different chan-nel leadership structures. In the fourth section, we study the mainoutcomes and discuss the implications of the different channelleaderships by conducting some sensitivity analysis and comparingresults across different scenarios. Finally, in the last section, weconclude and delineate further research directions.

168 S. Choi, K. Fredj / European Journal of Operational Research 225 (2013) 166–178

2. The model

This section presents the demand and profit functions and pro-vides a description of the different channel leadership structures.We assume that the manufacturer produces a national brand(NB) product and distributes it through two competing retailers.In addition to the national brand, each retailer offers a store brandproduct (SB).1 The manufacturer chooses the wholesale price thatmaximizes his profits and each retailer determines the optimal retailmargin for the national brand and the price of her store brand thatmaximize her combined profits from marketing the two brands.

2.1. The demand structure

Following the established literature, we use linear demandfunctions that capture the main properties: demanded quantityof a product decreases as its price increases or, as the competingproduct’s price decreases, it is also affected by the degree of prod-uct and store differentiation. Accordingly, we extend the demandfunction used in Raju et al. (1995) as follows:

qiM ¼1

2þ 2kð1� piM þ cðpjM � piMÞ þ bðpi � piMÞÞ; ð1Þ

qi ¼1

2þ 2kðk� pi þ bðpiM � piÞÞ; ð2Þ

where M is the index for the manufacturer, i, j = 1, 2 with i – j theindexes for the retailers, qiM the demand of the manufacturer’snational brand at store i, qi the demand of the store brand i, piM

the retail price of the manufacturer’s national brand at store i, pi

the retail price of the store i’s brand, b the cross-price sensitivitybetween the national brand and a store brand at store i, c thecross-price sensitivity between the two stores for the nationalbrand, and k is the store brand’s base level of demand.

b 2 [0,1], with smaller values of b indicating less product substi-tutability (or more product differentiation) between the nationalbrand and a store brand. Hence, a lower value of the cross-pricesensitivity between two products implies that a change in the priceof one of the products will have lower impact on the demand of theother product and vice versa. By considering the same b betweenthe national brand and each of the store brands, we implicitlyassume that the national brand is symmetrically positioned withrespect to the two store brands.

c 2 [0,1], with smaller value indicating less store substitution(or more store differentiation) between the two retailers. Lowervalues of cross-price sensitivity between stores imply that pricedifferences of the national brand between the two stores has lessimpact on the demand the latter will face.

Finally, k indicates the base level of demand of a store brand. k = 0represents the case where the store brand has no base level of de-mand and k = 1 represents the case where the store brand has ex-actly the same base level as the national brand. In general, k cantake any value between 0 and 1, with higher values of k implyinghighly competitive store brands compared to the national brand.

As in Raju et al. (1995), we assume that each retailer procuresher store brand from a manufacturing source. The store brandproducers incur a fixed unit cost for a long-term period (see: Cookand Schutte, 1967; McMaster, 1987). We also assume that the priceat which each retailer procures her store brand is equal to themarginal manufacturing cost. For mathematical tractability andto separate the effects of different channel leadership structuresfrom the effects of cost differences, we assume zero marginal costs.In modeling the demand function, we also assume that price

1 From now on we shall refer to the manufacturer by ‘‘he or him’’ and to eachretailer by ‘‘she or her’’ to avoid confusion.

competition between the two retailers’ store brands is negligible.This assumption is justified by the empirical evidence stating thatthe competition between store brands is very weak compared tothe competition between national brands and between nationalbrand and store brand items (Wedel and Zhang, 2004). Given theseassumptions, the profit function for a national brand manufacturer(M) can be written as:Y

M

¼Xi¼1;2

ðwMqiMÞ ¼ wM

Xi¼1;2

qiM ð3Þ

and the profit function for each retailer Ri, i = 1, 2:YRi

¼ miqiM þ piqi; ð4Þ

where wM is the national brand manufacturer’s wholesale price, andmi = piM � wM is the margin on the national brand for retailer i.2

2.2. The channel leadership structure

We consider strategic interactions both at the vertical level be-tween the manufacturer and the retailers, and the horizontal levelbetween the two retailers. In addition, we consider both simulta-neous and sequential moves at each level of interactions, whichtranslate into the following four pricing games:

� The Manufacturer Stackelberg (MS): refers to a Stackelberg priceleadership at the vertical level with the manufacturer as a lea-der and the retailers as followers, and a simultaneous Nashgame between the two retailers at the horizontal level. Thismeans that the manufacturer, knowing the reaction functionsof both retailers, first announces the wholesale price for hisNB. Then the retailers, simultaneously, choose the margins forthe manufacturer’s NB and their own SB product.� The Vertical Nash (VN): refers to a simultaneous play as per Ber-

trand–Nash between the manufacturer and the two retailers.This means that both the manufacturer and the two competingretailers simultaneously choose the wholesale price and theretailers’ margins, respectively, for the NB and the two storebrands.� The Retailer Stackelberg (RS): as opposed to the MS, under the RS

the manufacturer is a follower and the retailers are leaders. Theretailers, knowing the manufacturer’s reaction function, simul-taneously announce the margins for the manufacturer’s NB andtheir own SB products. The manufacturer would then choosethe wholesale price for his NB product.� The Retailer Double Stackelberg (RDS): is similar to the RS in

terms of the vertical interaction between the manufacturerand the two retailers given that retailers are leaders and themanufacturer is follower in this scenario. However, RDS differsfrom RS at the horizontal level as it considers a sequential inter-action à-la Stackelberg between the two retailers as well. Thatmeans that one of the retailers (Retailer 2 in our model) hasprice leadership over the other (Retailer 1), who then has verti-cal leadership over the manufacturer.

Fig. 1 illustrates the above four configurations of the channelleadership structures. As mentioned in the introduction section,except for the RDS, the different assumptions for the vertical pric-ing game were commonly used in the previous literature (see forexamples, Choi, 1991, 1996; Lee and Staelin, 1997; Kadiyali et al.,2000; Tyagi, 2005). Our main contributions consist of (a) applyingall these assumptions to a model including store brand products

2 Since we assume that the price at which retailers procure their store brand isqual to the marginal costs (set equal to zero), the retail prices and the margins fore store brand are equal.

eth

(a) Manufacturer Stackelberg (MS) (b) Retailer Stackelberg (RS)

(c) Vertical Nash (VN) (d) Retailer Double Stackelberg (RDS)

Retailer 2 Retailer 1

Manufacturer


Manufacturer


Manufacturer

Retailer 2

Retailer 1

Manufacturer

Fig. 1. The channel leadership structure.


and competition between stores and brands, and (b) considering asequential play between the channel members at the horizontal le-vel that constitutes the new channel leadership structure (RDS).

3. The equilibrium outcomes

In this section, we derive the equilibrium results for the differ-ent leadership structures. The optimal results are superscripted byMS, RS, VN, and RDS for the four different leadership assumptionsdescribed above.

3.1. The MS leadership

The MS game is solved backwards. First, we solve for the retail-ers’ reaction functions that represent the optimal margins for thenational brand and own store brands, maximizing retailers’ respec-tive profits while interacting through a simultaneous Nash game.Then, we solve for the manufacturer’s wholesale price of the na-tional brand that maximizes his profits given the retailers’ reactionfunctions.

The retailers’ reaction functions are derived as first orderconditions of maximizing Eq. (4) subject to Eqs. (1) and (2); foreach retailer i = 1, 2:

mMSi ¼

2½ð2bþ 1Þð2bþ 2b2 þ 1Þ þ ðbþ 1Þ3c� þ bc½4ðbþ 1Þð2bþ 1Þ þ ð62½2ðbþ 1Þð2bþ 1Þ þ ð4bþ b2 þ 2Þc�½2ð2bþ 1Þ þ cðbþ

pMSi ¼

2b½ðbþ 1Þð4ð2bþ 1Þ þ c2Þ þ ð2bðbþ 5Þ þ 5Þc� þ c½8ð2bþ 1Þðbþ4½2ðbþ 1Þð2bþ 1Þ þ ð4bþ b2 þ 2Þc�½

mMSi ¼

bþ 1þ bk�wMSM ð2bþ 1Þ

4bþ cþ bcþ 2and

pMSi ¼

2bþ kð2bþ cþ 2Þ þ bcwMSM

2ð4bþ cþ bcþ 2Þ :

It is interesting to notice from these expressions that as the whole-sale price of the national brand increases, each retailer decreasesher margin on the national brand and increases the price of herstore brand. This means that a higher wholesale price for the na-tional brand results in a lower retail margin on his product, towhich the retailers can react by compensating through a higher re-tail margin on their own store brand.

Substituting these reaction functions into the manufacturerprofit function (Eq. (3)) and maximizing it with respect to thewholesale price gives the following optimal value:

wMSM ¼

2½2bþ 1þ cð1þ bÞ� þ bkc2½cð4bþ b2 þ 2Þ þ 2ðbþ 1Þð2bþ 1Þ�

> 0:

Finally, replacing this value of wMSM in the retailers’ reaction func-

tions yields the optimal control variables’ values for each retaileri = 1, 2:

bþ 2b2 þ 3Þc�1Þ�

> 0 and

1Þ2 þ ð3bþ 2Þðbþ 2Þc2 þ 4ð6bþ b2 þ 3Þðbþ 1Þc�ðbþ 1Þcþ 2ð2bþ 1Þ�

> 0:

3 Detailed proofs of all propositions and results discussed in this section areavailable upon request from the authors. Please note that part of the results inpropositions 3 and 4 were proven using numerical analysis and graphs. Numericalanalysis was also used to support and illustrate some of the results that wereanalytically proved in the rest of the article.

4 The calculations supporting these analyses are also available in the appendix.


Finally, replacing the above margins and wholesale price in the de-mand and profit functions, we can easily get the equilibrium quan-tities and profits for each channel member. The full results arepresented in the Appendix.

3.2. No channel leadership (VN)

Under this scenario, all players maximize their respective profitssimultaneously, which results in the following reaction functions:

mVNi ¼

bþ 1þ bk�wVNM ð2bþ 1Þ

4bþ cþ bcþ 2; i ¼ 1;2;

pVNi ¼

2bþ kð2bþ cþ 2Þ þ bcwVNM

2ð4bþ cþ bcþ 2Þ ; i ¼ 1;2 and

wVNM ¼

2þ bðpVN1 þ pVN

2 Þ � ðbþ 1ÞðmVN1 þmVN

2 Þ4ðbþ 1Þ :

The optimal solutions are then obtained by solving the above equa-tions’ system:

mVNi ¼

ð4bþ 3b2 þ 2Þ þ 3ðbþ 1Þbk6ðbþ 1Þð2bþ 1Þ þ cðbþ 2Þð3bþ 2Þ > 0; i ¼ 1;2;

pVNi ¼

bð3bþ cþ 3Þ þ kðbþ 1Þð3bþ 2cþ 3Þ6ðbþ 1Þð2bþ 1Þ þ cðbþ 2Þð3bþ 2Þ > 0; i ¼ 1;2;

wVNM ¼

2ð2bþ 1Þ þ ð2bþ bkþ 2Þc6ðbþ 1Þð2bþ 1Þ þ cðbþ 2Þð3bþ 2Þ > 0:

The equilibrium quantities and profits for each channel member canbe easily derived by substituting these values in the right expres-sions (see appendix).

3.3. The RS leadership

The same as for the MS, this game is solved backwards. Themanufacturer first solves for the wholesale price of the nationalbrand that maximizes his profits. The resulting reaction functionis subsequently used by the retailers while simultaneously maxi-mizing their respective profits.

Solving for the manufacturer optimization problem, we get thisreaction function:

wRSM ¼

2þ bðpRS1 þ pRS

2 Þ � ðbþ 1ÞðmRS1 þmRS

2 Þ4ðbþ 1Þ :

It is obvious from this reaction function that the wholesale price ofthe national brand increases as the retail price of a store brand in-creases (@wM

@p1> 0 and @wM

@p2> 0), which is consistent with the comple-

mentarity property of the two products. Furthermore, the wholesaleprice of the national brand decreases as the retail margin of the na-tional brand increases (@wM

@mRS1< 0 and @wM

@mRS2< 0). This can be explained

by the manufacturer trying in his role as follower to keep compet-itive prices for his brand relative to the store brands.

In the next step, the retailers maximize their profits given thereaction function of the manufacturer and conditionally on eachother’s reaction functions. This yields the following optimalmargins:

mRSi ¼

ð8bþ 5b2 þ 4Þ þ 5ðbþ 1Þbk

10ðbþ 1Þð2bþ 1Þ þ ð16bþ 5b2 þ 8Þc> 0; i ¼ 1;2 and

pRSi ¼

bð5bþ 2cþ 5Þ þ kðbþ 1Þð4cþ 5ðbþ 1ÞÞ10ðbþ 1Þð2bþ 1Þ þ ð16bþ 5b2 þ 8Þc

> 0; i ¼ 1;2:

Finally, substituting these values into the manufacturer reactionfunction solves for the optimal wholesale price:

wRSM ¼

6ðbþ 1Þð2bþ 1Þ þ cð16bþ 7b2 þ 8Þ þ 4ðbþ 1Þbck

2ðbþ 1Þ½10ðbþ 1Þð2bþ 1Þ þ ð16bþ 5b2 þ 8Þc�> 0:

The rest of equilibrium values of the quantities demanded and prof-its for each one of the channel members are then straightforward toderive (see attached Appendix).

3.4. The RDS leadership

Under this channel leadership structure, the two retailers areacting as price leaders with respect to the manufacturer (price fol-lower) at the vertical level. At the horizontal level, one of the retail-ers (Retailer 2) has price leadership over the other (Retailer 1).Solved backward, this means that the manufacturer first chooseshis wholesale price. Then, using the latter’s reaction function; Re-tailer 1 chooses her margins for the national brand and her storebrand. Finally, Retailer 2 uses the information given by the reactionfunctions of the two other channel members to determine the mar-gins for the national brand and her store brand respectively, thatmaximize her profits.

The first maximization problem for the manufacturer gives thesame reaction function as in the RS leadership case. Retailer 1’s(leader at the vertical level and follower at the retailers’ level) reac-tion functions are:

mRDS1 ¼ð4bþ3b2þ2Þþ3ðbþ1Þbkþð3bþ4cþ8bcþ2b2þ3b2cþ1ÞmRDS

2 �ð2bþ1ÞbpRDS2

2ð9bþ4cþ8bcþ6b2þ3b2cþ3Þ;

pRDS1 ¼bð3bþ2cþ3Þþð3bþ4cþ3Þðbþ1Þkþcb2pRDS

2 þð2bþ1ÞbymRDS2

2ð9bþ4cþ8bcþ6b2þ3b2cþ3Þ:

Retailer 2 uses the above information to maximize her profits andsolves for the optimal margins for the national brand and her storebrand. By substituting these values back in the reaction functions,we get the equilibrium retail margins and wholesale price of the na-tional brand and the retail prices of the store brands. These optimalvalues determine the equilibrium solutions of the quantities de-manded of each brand and the profits for each channel member.The complete analytical results for the four channel leadershipstructures are summarized in the Appendix.

4. Discussion

In this section, we conduct some comparisons within and be-tween scenarios to explain some of the impacts the different chan-nel leaderships have on the main equilibrium outcomes. The mainresults are presented in propositions 1– 5.3

We also perform some sensitivity analysis with respect to theprimary parameters of our model to examine their impact on themain results under the different scenarios studied and to comparethe current findings to existing ones, when applicable.

4.1. Sensitivity analysis

In this subsection, we present general results regarding the im-pact of the main parameters of the model; i.e. product differentia-tion, store differentiation, and the base level of store branddemand. In general, the effects of the different model’s parameterson the equilibrium outcomes are consistent through the differentscenarios. They are summarized as follows4:

� Cross-price sensitivity between the national brand and a storebrand (b) has a positive impact on the level of the store branddemand and a negative impact on the manufacturer’s wholesale


price and profits. This implies that more product substitutabil-ity between the store brand and the national brand favoursthe retailers to the detriments of the manufacturer. The positiveimpact on retailers could be explained by the fact that highercross-price sensitivity means that the two brands are close sub-stitutes. Consequently, small changes in the price of thenational brand will have a high impact on the demand of thestore brand. This finding concurs with Raju et al. (1995) con-firming that retailers benefit more from the introduction of astore brand that has high cross-price sensitivity with thenational brand. The negative impact on the manufacturer canbe drawn as logical consequence of the higher competitiveness(substitutability) of the store brand, which will force the manu-facturer to lower his wholesale prices and, consequently, tomake less profit.� Higher cross-price sensitivity between the retailers’ stores (c)

leads to a higher wholesale price but lower margins for theretailers. It has a positive impact on the demand for both prod-ucts and results in higher profits for the manufacturer but lowerprofits for the retailers.The intuition behind this result is that higher store substitut-ability implies more competition between the two stores, whichdirectly leads to lower margins for the retailers on both thenational brand and her store brand and, indirectly, leads to anincrease in their demand level as per the law of demand.This competition tends also to favour the manufacturer vis-à-vis the retailers by allowing him to raise his wholesale price.Henceforth, the manufacturer’s profits increase due to the posi-tive effect on the wholesale price and the quantity demanded ofthe national brand. The negative impact on the retailers’ profits,on the other hand, indicates that the increase in the equilibriumquantities for the national brand and store brand products arenot high enough to compensate for the decrease in the retailmargins of these products.� The retailer’s base level of demand for her store brand (k) has a

positive impact on the wholesale price and on the retail mar-gins. It also has a positive effect on demand for the store brand,but a negative impact on demand for the national brand. Itseffect on profits is negative in the manufacturer’s case andambiguous in the retailers’.These results are consistent with Raju et al. (1995) findings’indicating that a retailer is better-off introducing a store brandfor which the base level of demand is higher. The effect onprices is explained by the fact that an increase in the marketshare of the retailer’s store brand will allow her to increasethe retail prices and margins for the two brands more easilyand would, subsequently, encourage the manufacturer tocharge higher wholesale prices for his national brand, takingadvantage of a higher margin on his national brand product.The impacts on demand for the national and store brands areintuitive as higher values of k indicate stronger demand forthe store brand relative to the national brand. The negativeimpact of k on the manufacturer’s profits, on the other hand,indicates that the decrease in the demand of the national brandoutweighs the increase of the wholesale price. For the retailer,the overall impact could not be determined as it depends onthe relative changes in the manufacturer’s brand demand com-pared to the increase in all retail prices and store branddemand.

4.2. The channel leadership effects

Given the symmetric demand structure in the model, the opti-mal margins, quantities demanded and profits are the same forthe two retailers except in the RDS case. Hence, we only need to

make the distinction between the two retailers’ under the latterscenario.

Proposition 1. The optimal prices of the national brand and storebrand compare as follows for the different scenarios x = VN, MS, RS,and RDS:

pxi ¼ wx

M < pxiM; if k ¼ kx

p;

pxi < wx

M < pxiM; if k < kx

p;

wxM < px

i < pxiM; if k > kx

p;

8><>:where the critical values kx

p are functions of cross-price sensitivities band c and compare as follows: 1 > kMS

p > kVNp > kRS

p > kRDSp1 > kRDS

p2 > 0.In addition to the implicit result of the retailer making a posi-

tive margin on the national brand product, this proposition mainlyshows two different outcomes.

First, it indicates that the retail price of the national brand ateach store, px

iM , is always higher than the store brand’s price. Thisresult proves to be intuitive as proven by the general tendency ofretailers (even the largest ones such as Superstore or Wal-Mart)to charge lower prices for their own store brand compared to theretail price of the national brand’s product they carry. This pricingstrategy is mainly driven by competition factors and aims toencourage consumers to eventually switch their choice from thenational brand to the store brand product.

Secondly, Proposition 1 brings some interesting nuances to theabove intuitive results by providing conditions under which theprice of the store brand can be equal to, or higher than the nationalbrand’s wholesale price, which infers some additional informationwith respect to the margins made by the retailers and the manu-facturer, respectively, on their own brand products. In sum, prop-osition 1 implies that, although the retail prices are alwayshigher for the national brand compared to the store brand, themargin that a retailer makes on her own store brand product com-pares differently to the margin that the manufacturer makes on hisnational brand product depending on the different parameters inthe demand function (b, c and k). This comparison is made by gen-erating critical values of k (the base level demand of the storebrand) that depend on c (the cross-price sensitivities betweenstores) and b (the cross-price sensitivity between products) andthat are specific to each channel leadership structure. When k isgreater than its respective critical value with respect to the specificmarket leadership structure, the price of a retailer’s store brand canbe higher than the wholesale price of the national brand, and viceversa. These findings reinforce the outcomes of the sensitivity anal-ysis, discussed in the last subsection, with respect to k.

Furthermore, by comparing the different critical values ob-tained under the different leadership structures, we find that thehighest critical value of k occurs under the MS leadership case, fol-lowed consecutively by VN, RS, and RDS scenarios. These resultsseem intuitive and consistent with the rest of our findings. Theyshow that as the retailer gains more strategic power in the channel,she requires relatively lower base level of store brand demand thatenables her to charge higher retail prices for her store brand rela-tive to the national brand’s wholesale price.

Finally, it is of interest to note that the critical value is lower un-der the RDS compared to the RS structure even for the retailer act-ing as a follower (Retailer 1 in our model). This implies that bothretailers benefit from a higher strategic power in terms of pricingfor the store brand product when interacting under the RDS sce-nario regardless of the roles they play (leader or follower). This re-sult is going to be reinforced by the next two propositions.

Proposition 2. The national brand’s wholesale price and retailers’margins of the national brand under the different equilibria compare,respectively, as follows:


(i) wMSM > wVN

M > wRSM > wRDS

M ; and

RDS RDS RS VN MS
m2 > m1 > mi > mi > mi ; i ¼ 1;2: (ii)
Proposition 2 mainly implies that Stackelberg channel leadersbenefit from higher unit margins/wholesale prices on the nationalbrand compared to the VN game and that both competing retailersget higher unit margins on the national brand playing a sequentialStackelberg (RDS) rather than interacting simultaneously à-la Ber-trand–Nash. The main implication of this result is that a channelleader can obtain a higher mark-up on the national brand, whichcreates a direct incentive for each channel member to strive forthe informational advantage that would make him the leader ofthe distribution channel.

Proposition 2.i. indicates that the manufacturer gets to chargethe highest wholesale price for the national brand when he is theleader of the channel and the lowest prices when he is acting asthe follower. This part of the results concurs with the previousfindings in the literature (e.g., Choi, 1991; Trivedi, 1998). Thenew finding at this level is the fact that the wholesale price is evenlower under the RDS than under the RS, which is also in agreementwith previous findings in the literature, in the sense that in the caseof RDS the manufacturer loses even more competing power to theadvantage of the retailers (acting as a follower at the third positionrather that second position as in RS).

Proposition 2.ii is coherent with these conclusions and showsthat the retail margins on the national brand follow the exact in-verse order, being the highest under the RDS case followed consec-utively by the RS, VN, and MS scenarios. The ordering of the threelast equilibria is, however, different from previous findings in somestudies that did not consider store competition and/or store brand(for example: Jørgensen et al., 2001), which implies that the intro-duction of store competition and store brand changes the out-comes in that matter.

Finally, an interesting and less expected result emerges fromthe new RDS leadership structure as it indicates that the Stackel-berg price follower among the two retailers (Retailer 1) benefitsfrom higher retail margin on the national brand under the RDScompared to the RS leadership where she plays a simultaneousNash game with her competitor (Retailer 2). This result reinforcesproposition 1 statement relating to the retailers gaining higherstrategic power in terms of pricing when interacting as per Stackel-berg rather than Bertrand–Nash, regardless of their role (leader orfollower).

Proposition 3. Retail prices for the store brand and the nationalbrand products under the different equilibria relate, respectively, asfollow:

(i) pMSiM > pRDS

2M > pRSiM > pRDS

1M > pVNiM and

(ii) pMSi > pVN

i > pRSi and pMS

i > pRDS2 > pRDS

1 > pRS1 :

The first important observation that comes out of Proposition 3is the fact that having the manufacturer as the Stackelberg leadergenerally leads to higher retail prices for both the store and the na-tional brands, which is coherent with the findings of proposition 2.The fact that retailers charge lower retail prices for the nationalbrand and the store brands under RS and RDS leaderships com-pared to the MS leadership can be explained through productand store competition. Product competition would imply that each

retailer sets a relatively high retail prices for her own store brandunder the MS structure because the competing national brandproduct also has higher price in this case (the explanation for high-er retail prices for the national brand being straightforward). At thesame time, store competition between retailers has the effect ofreducing the latitude retailers have in terms of increasing their re-tail prices even when practicing the vertical strategic leadershipunder RDS and RS. This result does, indeed, corroborate with previ-ous findings about the manufacturer’s leadership resulting in high-er retail prices compared to retailers’ leadership when accountingfor store competition (Choi, 1996). This also explains why our re-sults are different than those found in Choi (1991) and Jørgensenet al. (2001) as these previous studies did not include store compe-tition and store brand in their models.

The second important conclusion that Proposition 3 providesis that VN interactions between all chain members lead to thelowest retail prices for the national brand but not necessarilyfor the store brand in which case the RS leadership results incomparatively lower prices. In the latter case, it seems thatretailers practicing vertical strategic interaction under the RSare each using their own SB as leverage to compensate for therelatively higher retail prices of the NB and to control for thestore and brand differentiation they face when competing witheach other. This explains also the higher prices of store brandsunder RDS relative to RS given that the additional horizontalstrategic interaction would provide retailers with an increasedinformational advantage allowing them to compensate for thecompetition effect.

With respect to the two retailers’ Stackelberg scenarios we cansee, as expected, that the Stackelberg price leader (Retailer 2) ben-efits from a higher retail price for both the manufacturer’s NB andhis SB under the RDS compared to the RS. The Stackelberg follower(Retailer 1) has relatively the same strategic advantage for her SBand for the NB. Indeed, even if she charges higher retail price forthe national brand under the RS, according to proposition 2.ii, shemakes larger margin on the NB under the RDS leadership structure.In conclusion both retailers prove to benefit strategically more interms of pricing under RDS compared to RS.

To reach a more conclusive outcome regarding the incentivesretailers have towards the RDS structure we compare the demandfor each brand and the profits they gain under the different scenar-ios in the next propositions.

Proposition 4. Equilibrium quantities for store brand and nationalbrand products under the different equilibria relate, respectively, asfollow:

(i) qVNiM ; q

RDSiM > qRS

iM > qMSiM ;

(ii) qMSi > qRDS

2 > qRSi > qVN

i ; qRDS1 :

Proposition 4 indicates that retailers distribute the largestquantity of the SB (respectively NB) and the lowest quantity ofthe NB (respectively SB) under the MS (respectively VN) leadership.

It also indicates that equilibrium quantities for SBs are larger(respectively lower) under RDS compared to RS equilibrium forthe leading retailer (respectively following retailer). NB quantities,on the other hand, are higher under VN and RDS equilibria (forboth retailers) compared to the RS equilibrium and the lowest un-der the MS equilibrium.

From the consumers’ perspective, propositions 3 and 4 showthat the worst scenario is MS leadership as it leads to the highest


retail prices for both the NB and the SBs and to the lowest equilib-rium quantities. From the same perspective, the lowest retailprices and the highest quantities marketed of the NB product in ab-sence of leadership (under VN) and the lowest prices for the SB arereached under RS. Our findings with respect to the national brandare in agreement with Shugan and Jeuland (1988) and Choi (1996)affirming that consumers are better-off in the absence of channelleadership, but in disagreement with Jørgensen et al. (2001), prov-ing that consumers are better-off under MS leadership. The com-parison with the latter study proves again that introducing storecompetition can change the previous established results. In thiscase we also show that introducing store brand, in addition to storecompetition, can further change consumers’ perspectives as theirlowest prices are reached under the RS and not the VN or the MS.This also brings a new light with respect to channel literature find-ings and to the discussion on who should preferably be the ‘‘chaincaptain’’ (Bowersox et al., 1980; Jørgensen et al., 2001; Stern andEl-Ansary, 1992; Little, 1970).

Proposition 5. Profits earned by the manufacturer and the tworetailers, under the different equilibria, respectively relate as follows:

(i) pMSM > pVN

M > pRSM > pRDS

M ;

RDS RS RDS VN MS
(ii) pR2> pRi
> pR1> pi > pi :

Proposition 5 implies that each channel member obtains higher
profits by acting as a Stackelberg leader rather than a follower orby playing a VN game. This proposition also indicates that the lead-ing retailer (Retailer 2) benefits more from a Stackelberg leadershipat the horizontal level at the detriments of his follower (Retailer 1)who is better off playing a simultaneous game with the competingretailer while keeping leadership with respect to the manufacturer.This implies that, even though the Stackelberg price follower (Re-tailer 1) has strategic power that gives her advantage at the pricinglevel, the disadvantage at the level of the demanded quantities ismore important and leads to lower profits under RDS comparedto RS. However, these profits are still higher under RDS comparedto VN and MS structures, which means that, when leading at thevertical level, Retailer 1 should not fear as much being a followerat the horizontal level, which she should even prefer to no leader-ship at all.
For the manufacturer, on the other hand, unless he is the leader,he is better-off having an equal strategic game with retailers underVN rather than being their follower. The worst case scenario for themanufacturer is indeed when retailers interact as per Stackelbergboth at the horizontal level and the vertical level, in which casehe has the least control and information and makes the lowestprofits.

These results about profits increasing with the level of lead-ership control for each channel member are consistent and inagreement with previous findings in the literature stipulatingthat the leader in the market channel always benefits fromprice leadership regardless of the functional form of the de-mand (e.g. Choi, 1996; Moorthy and Fader, 1989). They are incontradiction with Choi’s (1991) findings showing that the high-est profits are obtained when there is no strategic leadershipand higher when the other channel member has strategic verti-cal leadership. The difference is again attributed to the intro-duction of store and product competition in our model, whichseems to lead to more intuitive results. In this instance Choi(1991) recognized in his study that his own findings aboutthe absence of direct incentive for a channel member to bethe strategic leader was ‘‘puzzling’’.

5. Conclusion

This paper presents a general analytical framework that helpsbetter understand the nature of price competition between na-tional and store brands in the presence of store competition. Infact, we consider both intra-store competition between the na-tional brand and the store brands, and inter-store competition be-tween two stores. In addition, we consider various priceleaderships among the channel members, namely simultaneousinteractions as per Bertrand–Nash and sequential Stackelberg bothat the vertical and horizontal levels of the channel.

Some of our results provide new insights:

� The two competing retailers benefit from price leadership atthe horizontal level (higher prices for the SB and highermargins for the NB) with more advantage for the leaderamong the two retailers.

� Both retailers gain more profits under the RDS compared tothe MS and the VNstructures. But only the leading retailerdoes benefit under RDS compared to RS structure.

� The retail prices of the national brand and store brandsunder the MS case are always higher than those of the tworetailer Stackelberg cases.

Some other results are intuitively appealing:

� Each channel member benefits of higher profits as he/shegains more strategic power. It means that the manufactureris best-off under MS and worst-off under RDS, the retailerfollower is best-off under RS and worst-off under MS, andthe retailer leader is best-off under RDS and worst-off underMS.

� Consumers are always worst-off when the manufacturer hasstrategic leadership in the channel. Retailers’ strategic verti-cal leadership offers them the lowest SB prices. In completeabsence of strategic leadership in the market channel, con-sumers’ benefit of the lowest NB prices and the highestquantities of both brands.

� Demand levels for store brands increase as the cross-pricesensitivity between the national brand and store brandsincreases.

� More competition between retailers’ stores results in lowerretail margins for national and store brands, increasing thedemand levels of both products despite a higher wholesaleprice for the national brand. Accordingly, higher store com-petition only benefits the national brand manufacturer.

� The retailers’ base level of store brand demand has a posi-tive impact on both the wholesale price and the retail mar-gins. Its impacts on the equilibrium quantities of the storebrand are positive and of the national brand are negative,resulting in lower profits for the manufacturer.

In sum, these findings are insightful for practitioners from manyperspectives. First, powerful store brand retailers need to seekabsolute price leadership with respect to other retailers and themanufacturer(s). Retailers should also prefer the position of a fol-lower at the horizontal level as long as they lead with respect tothe manufacturer(s) (RDS leadership structure) to playing a simul-taneous Nash game with the rest of channel members (VN game) orsimultaneous Nash game with other retailers when manufac-turer(s) are leader(s) (MS interaction). Manufacturers on the otherside would have incentive to distribute their national brandthrough as many and strategically less powerful retailers as possi-ble and try to keep a higher leadership in the market structure.


Second, the retailers should offer a store brand that is a close sub-stitute to the national brand, providing relatively homogeneousbrands in the store, in order to guarantee a high level of demand. Thisresult is compatible with Sayman et al. (2002) who found, in a differ-ent channel structure, that closer positioning of a store brand to theleading national brand is the optimal strategy to follow.

Combined, the two managerial implications suggest that to bemore successful, each retailer should pursue greater differentiationof her store from the competitors’ and less differentiation of herstore brand from the national brand by providing relatively homo-geneous brands in the store. On the other hand, the manufacturershould differentiate his national brand more from the retailers’store brands and deal with as many and less strategically powerfulretailers as possible.

To conclude, this study can be extended in many directions:considering more than one manufacturer in the channel, introduc-ing more asymmetry between retailers (for example, different baselevels of demand), allowing for different positioning of the retailersin relation to the manufacturer by using different product differen-

wMSM ¼

2ð2bþ 1þ cð1þ bÞÞ þ bkc2ðcð4bþ b2 þ 2þ 2Þ þ 2ðbþ 1Þð2bþ 1ÞÞ

mMSi ¼

2ð2bþ 1Þð2bþ 2b2 þ 1Þ þ ðbþ 1Þ3cþ bkð4ðbþ 1Þ2bþ 1Þ þ ð6b

2ð2ðbþ 1Þð2bþ 1Þ þ ð4bþ b2 þ 2ÞcÞð2ð2bþ 1Þ þ cðbþ

pMSi ¼

2bððbþ 1Þð4ð2bþ 1Þ þ c2Þ þ ð2bðbþ 5Þ þ 5ÞcÞ þ kð8ð2bþ 1Þðbþ4ð2ðbþ 1Þð2bþ 1Þ þ ð4bþ b2 þ 2ÞcÞ

qMS1M ¼ qMS

2M ¼2ð2bþ 1Þ þ ð2bþ bkþ 2Þc8ð4bþ cþ bcþ 2Þðkþ 1Þ

qMS1 ¼ qMS

2 ¼2bð2bþ cþ bcþ 1Þ þ ð12bþ 4cþ 8bcþ 8b2 þ 3b2cþ 4Þk

8ð2ðbþ 1Þð2bþ 1Þ þ cð4bþ b2 þ 2ÞÞðkþ 1Þ

pMSM ¼

ð2ð2bþ 1Þ þ ð2bþ bkþ 2ÞcÞ2

8ðcð4bþ b2 þ 2Þ þ 2ðbþ 1Þð2bþ 1ÞÞð4bþ cþ bcþ 2Þðkþ 1Þ

pMSRi¼ðbþ1Þk2c2ðð96bþ119b2þ46b3þ4b4þ24Þðbþ1Þcþð208bþ277b2

8ðkþ1Þð4bþcþbcþ2Þ2ð6bþ2cþ4bcþ4b2þ

þð2bþcþbcþ1Þððbþ1Þð4ð2bþ4b2þ1Þð2bþ1Þ2þðbþ1Þb2c3þð18ðkþ1Þð4bþcþbcþ2Þ2ð6

þð32ðbþ1Þ2ð2bþ1Þ2þðbþ1Þðbþ2Þð3bþ2Þc3þ4ð30bþ8b2þ15Þð8ðkþ1Þð4bþcþbcþ2Þ2ð6

þðbþ1Þk2ð64ðbþ1Þ2ð2bþ1Þ3þð3bþ2Þ2ðbþ2Þ2c4þ64ð2bþ1Þ2ð6b

32ðkþ1Þð4bþcþbcþ2Þ2ð6bþ2cþ4bcþ4b2þb2c

tiation parameters between the national brands and each storebrand, considering non-linear demand functions, and consideringother strategic variables than price (such as advertising, qualityand promotions). Finally, it would be of great interest to be ableto validate the results of this model by conducting an empiricalstudy of similar problems if data availability permits.

Acknowledgements

The authors thank the anonymous reviewers, whose commentsand suggestions greatly helped improve this manuscript.

Appendix A. Equilibria outcomes

A.1. The MS scenario

þ 2b2 þ 3Þc1ÞÞ

1Þ2 þ ð3bþ 2Þðbþ 2Þc2 þ 4ð6bþ b2 þ 3Þðbþ 1ÞcÞððbþ 1Þcþ 2ð2bþ 1ÞÞ

þ138b3þ20b4þ52Þð2bþ1ÞÞb2cþ2Þ2

6bþ29b2þ26b3þ4b4þ4Þc2Þþ8ð4bþ8b2þ8b3þ2b4þ1Þð2bþ1ÞcÞbþ2cþ4bcþ4b2þb2cþ2Þ2

bþ1Þð2bþ1Þcþ4ð32bþ41b2þ18b3þ2b4þ8Þc2Þð2bþcþbcþ1Þbk

bþ2cþ4bcþ4b2þb2cþ2Þ2

þ2b2þ3Þðbþ1ÞcÞþ2Þ2

A.2. The VN scenario

wVNM ¼

2ð2bþ 1Þ þ ð2bþ bkþ 2Þc6ðbþ 1Þð2bþ 1Þ þ ð3bþ 2Þðbþ 2Þc

mVNi ¼

ð4bþ 3b2 þ 2Þ þ 3ðbþ 1Þbk6ðbþ 1Þð2bþ 1Þ þ cðbþ 2Þð3bþ 2Þ

pVNi ¼

bð3bþ cþ 3Þ þ kðbþ 1Þð3bþ 2cþ 3Þ6ðbþ 1Þð2bþ 1Þ þ cðbþ 2Þð3bþ 2Þ

qVNiM ¼

ð2ð2bþ 1Þð2bþ bkþ 2ÞcÞðbþ 1Þ2ð6ðbþ 1Þð2bþ 1Þ þ ð3bþ 2Þðbþ 2ÞcÞðkþ 1Þ

qVNi ¼

bð2bþ 1Þ þ ðbþ 1Þbcþ ð3ð2bþ 1Þ þ 2ðbþ 1ÞcÞðbþ 1Þk2ð6ðbþ 1Þð2bþ 1Þ þ ð3bþ 2Þðbþ 2ÞcÞðkþ 1Þ

pVNM ¼

ð2ð2bþ 1Þ þ ð2bþ bkþ 2ÞcÞ2ðbþ 1Þð6ðbþ 1Þð2bþ 1Þ þ ð3bþ 2Þðbþ 2ÞcÞ2ðkþ 1Þ

pVNRi¼ð2bþcþbcþ1Þð12bþ17b2þ9b3þb2cþ4þ2ð9bþ2cþ9Þðbþ1ÞbkÞþð27bþ12cþ24bcþ18b2þ4c2þ4bc2þ9b2cþ9Þðbþ1Þ2k2

2ðkþ1Þð18bþ4cþ8bcþ12b2þ3b2cþ6Þ2

A.3. The RS scenario

wRSM ¼

6ðbþ 1Þð2bþ 1Þ þ cðð16bþ 7b2 þ 8Þ þ 4ðbþ 1ÞbkÞ2ðbþ 1Þð10ðbþ 1Þð2bþ 1Þ þ ð16bþ 5b2 þ 8ÞcÞ

mRSi ¼

ð8bþ 5b2 þ 4Þ þ 5ðbþ 1Þbk

10ðbþ 1Þð2bþ 1Þ þ ð16bþ 5b2 þ 8Þc

pRSi ¼

bð5bþ 2cþ 5Þ þ kðbþ 1Þð4cþ 5ðbþ 1ÞÞ10ðbþ 1Þð2bþ 1Þ þ ð16bþ 5b2 þ 8Þc

pRSiM ¼

ðbþ 1Þðð2bþ 1Þ þ 4bkcþ 2ðð8bþ 5b2 þ 4Þ þ 5ðbþ 1ÞbkÞÞ þ cð16bþ 7b2 þ 8Þ2ðbþ 1Þð10ðbþ 1Þð2bþ 1Þ þ ð16bþ 5b2 þ 8ÞcÞ

qRSiM ¼

6ðbþ 1Þð2bþ 1Þ þ cð4ðbþ 1Þbkþ 16bþ 7b2 þ 8Þ4ðkþ 1Þð10ðbþ 1Þð2bþ 1Þ þ ð16bþ 5b2 þ 8ÞcÞ

qRSi ¼

2ðbþ 1Þð2bþ 1Þð2bþ 5kþ 5bkÞ þ ðbþ 2kþ 2bkÞð3bþ 2Þðbþ 2Þc4ðbþ 1Þðkþ 1Þð10ðbþ 1Þð2bþ 1Þ þ ð16bþ 5b2 þ 8ÞcÞ

pRSM ¼

ððbþ 1Þð2bþ 1Þ þ cðð16bþ 7b2 þ 8Þ þ 4ðbþ 1ÞbkÞÞ2

4ð10ðbþ 1Þð2bþ 1Þ þ ð16bþ 5b2 þ 8ÞcÞ2ðkþ 1Þðbþ 1Þ

pRSRi¼ ððbþ 5b2 þ 4Þ þ 5ðbþ 1ÞbkÞð6ðbþ 1Þð2bþ 1Þ þ cð4ðbþ 1Þbkþ 16bþ 7b2 þ 8ÞÞð10ðbþ 1Þð2bþ 1Þ þ ð16bþ 5b2 þ 8ÞcÞð4ðkþ 1Þð10ðbþ 1Þð2bþ 1Þ þ ð16bþ 5b2 þ 8ÞcÞÞ

þ ðbð5bþ 2cþ 5Þ þ kðbþ 1Þð4cþ 5ðbþ 1ÞÞÞð2ðbþ 1Þð2bþ 1Þð2bþ 5kþ 5bkÞ þ ðbþ 2kþ 2bkÞð3bþ 2Þðbþ 2ÞcÞðbþ 1Þð10ðbþ 1Þð2bþ 1Þ þ ð16bþ 5b2 þ 8ÞcÞð4ðkþ 1Þð10ðbþ 1Þð2bþ 1Þ þ ð16bþ 5b2 þ 8ÞcÞÞ

A.4. The RDS scenario

wRDSM ¼ 242ðbþ1Þ2ð2bþ1Þ3þ2ð2bþ1Þ2ðbþ1Þð1008bþ439b2þ504Þcþð5248bþ7496b2þ4496b3þ957b4þ1312Þ

8ð9bþ4cþ8bcþ6b2þ3b2cþ3Þð34ðbþ1Þð2bþ1Þ2þ4ð40bþ17b2þ20Þðbþ1Þð2bþ1Þcþð128bþ176b2þ96b3þ17b4þ32Þc2Þ

þ ð256bþ360b2þ208b3þ41b4þ64Þð16bþ7b2þ8Þc3

8ðbþ1Þð9bþ4cþ8bcþ6b2þ3b2cþ3Þð34ðbþ1Þ2ð2bþ1Þ2þ4ð40bþ17b2þ20Þðbþ1Þð2bþ1Þcþð128bþ176b2þ96b3þ17b4þ32Þc2Þ

þ bcðð76ðbþ1Þ2ð2bþ1Þ2þ16ð26bþ11b2þ13Þðbþ1Þð2bþ1Þcþð512bþ720b2þ416b3þ83b4þ128Þc2ÞÞ4ð9bþ4cþ8bcþ6b2þ3b2cþ3Þð34ðbþ1Þ2ð2bþ1Þ2þ4ð40bþ17b2þ20Þðbþ1Þð2bþ1Þcþð128bþ176b2þ96b3þ17b4þ32Þc2Þ


mRDS2 ¼ ðbþ 1Þð2ð2bþ 1Þð28bþ 17b2 þ 14Þ þ bkðð112bþ 51b2 þ 56Þcþ 34ðbþ 1Þð2bþ 1ÞÞÞ þ cð192bþ 292b2 þ 200b3 þ 51b4 þ 48Þ

2ððbþ 1Þð2bþ 1Þð4cð40bþ 17b2 þ 20Þ þ 34ðbþ 1Þð2bþ 1ÞÞ þ ð128bþ 176b2 þ 96b3 þ 17b4 þ 32Þc2Þ

pRDS2 ¼ 34bð2bþ 1Þðbþ 1Þ2 þ bcð17ð3bþ 2Þðbþ 2Þðbþ 1Þ þ 2ð16bþ 7b2 þ 8ÞcÞ

2ððbþ 1Þð2bþ 1Þð4cð40bþ 17b2 þ 20Þ þ 34ðbþ 1Þð2bþ 1ÞÞ þ ð128bþ 176b2 þ 96b3 þ 17b4 þ 32Þc2Þ

þ ð34ð2bþ 1Þðbþ 1Þ2 þ 4ð16bþ 7b2 þ 8Þc2 þ ð160bþ 51b2 þ 80Þðbþ 1ÞcÞðbþ 1Þk2ððbþ 1Þð2bþ 1Þð4cð40bþ 17b2 þ 20Þ þ 34ðbþ 1Þð2bþ 1ÞÞ þ ð128bþ 176b2 þ 96b3 þ 17b4 þ 32Þc2Þ

mRDS1 ¼ c2½ð48bþ17b2þ24Þð5bþ4Þð3bþ4Þðbþ1Þbkþð1920bþ4752b2þ6208b3þ4494b4þ1692b5þ255b6þ320Þ�

4ð9bþ4cþ8bcþ6b2þ3b2cþ3Þð34ðbþ1Þ2ð2bþ1Þ2þð128bþ176b2þ96b3þ17b4þ32Þc2þ4ð40bþ17b2þ20Þðbþ1Þð2bþ1ÞcÞ

þðbþ1Þ2ð2bþ1Þ2ð204bkðbþ1Þþ4ð82bþ51b2þ41ÞÞþ2cðbþ1Þð2bþ1Þðð592bþ255b2þ296Þðbþ1Þbkþ960bþ1462b2þ1004b3þ255b4þ240Þ4ð9bþ4cþ8bcþ6b2þ3b2cþ3Þð34ðbþ1Þ2ð2bþ1Þ2þð128bþ176b2þ96b3þ17b4þ32Þc2þ4ð40bþ17b2þ20Þðbþ1Þð2bþ1ÞcÞ

pRDS1 ¼ 102bð7bþ19b2þ25b3þ16b4þ1Þþbð64c3þ408b5þ256bc3þ360b2c3þ208b3c3þ41b4c3Þ


þ ½102ðbþ1Þð2bþ1Þþð752bþ255b2þ376Þc�kðbþ1Þ3ð2bþ1Þ4ð9bþ4cþ8bcþ6b2þ3b2cþ3Þð34ðbþ1Þ2ð2bþ1Þ2þð128bþ176b2þ96b3þ17b4þ32Þc2þ4ð40bþ17b2þ20Þðbþ1Þð2bþ1ÞcÞ

þ ½ð3328bþ4352bþ2048bþ255bþ832Þckþ6ð224bþ85bþ112Þð2bþ1Þb�ðbþ1Þ2c4ð9bþ4cþ8bcþ6b2þ3b2cþ3Þð34ðbþ1Þ2ð2bþ1Þ2þð128bþ176b2þ96b3þ17b4þ32Þc2þ4ð40bþ17b2þ20Þðbþ1Þð2bþ1ÞcÞ

þ ½4ð256bþ360bþ208bþ41bþ64Þckþð2432bþ3260bþ1656bþ255bþ608Þb�ðbþ1Þc2


qRDS1M ¼

ð640bþ 96bkþ 928b2 þ 576b3 þ 129b4 þ 288b2kþ 280b3kþ 88b4kþ 160Þc2

16ðkþ 1Þð34ðbþ 1Þ2ð2bþ 1Þ2 þ ð128bþ 176b2 þ 96b3 þ 17b4 þ 32Þc2 þ 4ð40bþ 17b2 þ 20Þðbþ 1Þð2bþ 1ÞcÞ

þ 4ð120bþ 14bkþ 55b2 þ 14b2kþ 60Þð2bþ 1Þðbþ 1Þcþ 82ðbþ 1Þ2ð2bþ 1Þ2

16ðkþ 1Þð34ðbþ 1Þ2ð2bþ 1Þ2 þ ð128bþ 176b2 þ 96b3 þ 17b4 þ 32Þc2 þ 4ð40bþ 17b2 þ 20Þðbþ 1Þð2bþ 1ÞcÞ

qRDS2M ¼

2ð12cþ7Þð40cþ16c2þ17Þþbðcð136kþ10728Þþc3ð128kþ3712Þþc2ð320kþ12096Þþ2864Þ16ðkþ1Þðbþ1Þð9bþ4cþ8bcþ6b2þ3b2cþ3Þð34ðbþ1Þ2ð2bþ1Þ2þð128bþ176b2þ96b3þ17b4þ32Þc2þ4ð40bþ17b2þ20Þðbþ1Þð2bþ1ÞcÞþ b2ðcð1392kþ48922Þþc3ð1280kþ13984Þþc2ð3072kþ50000Þþ14352Þ

16ðkþ1Þðbþ1Þð9bþ4cþ8bcþ6b2þ3b2cþ3Þð34ðbþ1Þ2ð2bþ1Þ2þð128bþ176b2þ96b3þ17b4þ32Þc2þ4ð40bþ17b2þ20Þðbþ1Þð2bþ1ÞcÞþ b5ðcð13176kþ135374Þþc3ð6084kþ18260Þþc2ð20368kþ96198Þþ56220Þ

16ðkþ1Þðbþ1Þð9bþ4cþ8bcþ6b2þ3b2cþ3Þð34ðbþ1Þ2ð2bþ1Þ2þð128bþ176b2þ96b3þ17b4þ32Þc2þ4ð40bþ17b2þ20Þðbþ1Þð2bþ1ÞcÞþ b6ðcð7584kþ58544Þþc3ð2516kþ5875Þþc2ð10032kþ36189Þþ27896Þ

16ðkþ1Þðbþ1Þð9bþ4cþ8bcþ6b2þ3b2cþ3Þð34ðbþ1Þ2ð2bþ1Þ2þð128bþ176b2þ96b3þ17b4þ32Þc2þ4ð40bþ17b2þ20Þðbþ1Þð2bþ1ÞcÞþ b7cð1760kþ10424Þþc3ð400kþ761Þþc2ð1952kþ5562þ5776Þ

16ðkþ1Þðbþ1Þð9bþ4cþ8bcþ6b2þ3b2cþ3Þð34ðbþ1Þ2ð2bþ1Þ2þð128bþ176b2þ96b3þ17b4þ32Þc2þ4ð40bþ17b2þ20Þðbþ1Þð2bþ1ÞcÞ

qRDS1 ¼8kð40cþ16c2þ17Þþð952kþcð1920kþ136Þþc2ð640kþ64Þþ54Þbþð2584kþcð4444kþ680Þþc2ð1232kþ256Þþ324Þb2

16ðbþ1Þðkþ1Þð34ðbþ1Þ2ð2bþ1Þ2þð128bþ176b2þ96b3þ17b4þ32Þc2þ4ð40bþ17b2þ20Þðbþ1Þð2bþ1ÞcÞ

�ð3400kþcð4976kþ1208Þþcð1136kþ360Þþ702Þbþð2176kþcð2700kþ904Þþcð498kþ208Þþ648Þb4þð544kþcð568kþ240Þþc2ð82kþ41Þþ216Þb5

16ðbþ1Þðkþ1Þð34ðbþ1Þ2ð2bþ1Þ2þð128bþ176b2þ96b3þ17b4þ32Þc2þ4ð40bþ17b2þ20Þðbþ1Þð2bþ1ÞcÞ

qRDS2 ¼ 4kð4cþ 3Þ þ bð48kþ cð48kþ 8Þ þ 5Þ þ b3ð24kþ cð14kþ 7Þ þ 10Þ þ b2ð60kþ cð46kþ 16Þ þ 15Þ

16ðbþ 1Þðkþ 1Þð9bþ 4cþ 8bcþ 6b2 þ 3b2cþ 3Þ

pRDSM ¼ ð2bþ 1Þ

64ðkþ 1Þðbþ 1Þ2ð4cþ bð8cþ 9Þ þ b2ð3cþ 6Þ þ 3Þ2

� 2ð504cþ 656c2 þ 256c3 þ 121Þ þ bðcð152kþ 8064Þ þ c3ð256kþ 3072Þ þ c2ð416kþ 9184Þ þ 2178Þð2ð40cþ 16c2 þ 17Þ þ bð400cþ 128c2 þ 204Þ þ b4ð136cþ 17c2 þ 136Þ þ b3ð524cþ 96c2 þ 408Þ þ b2ð708cþ 176c2 þ 442ÞÞ2

(

þ ½64ðkþ 1Þðbþ 1Þ2ð2ð40cþ 16c2 þ 17Þ þ bð400cþ 128c2 þ 204Þ þ b4ð136cþ 17c2 þ 136Þþ b3ð524cþ 96c2 þ 408Þ þ b2ð708cþ 176c2 þ 442ÞÞ2ð4cþ bð8cþ 9Þ þ b2ð3cþ 6Þ þ 3Þ2��1

� ½b2ðcð1064kþ 26078Þ þ cð1280kþ 7424Þ þ cð2496kþ 25864Þ þ 7986Þ þ b3ðcð2888k

þ 43572Þ þ c3ð2464kþ 9216Þ þ c2ð5760kþ 37480Þ þ 15246Þ þ b4ðcð3800kþ 39638Þþ c3ð2272kþ 6176Þ þ c2ð6400kþ 29437Þ þ 15972Þ þ b5ðcð2432kþ 18600Þ þ c3ð998kþ 2112Þ

þ c2ð3424kþ 11863Þ þ 8712Þ þ b6ðcð608kþ 3512Þ þ c3ð166kþ 287Þ þ c2ð704kþ 1914Þ þ 1936Þ�)


pRDSR1¼½64ðbþ 1Þð9bþ 4cþ 8bcþ 6bþ 3bcþ 3Þðkþ 1Þ��1

� ½34ðbþ 1Þð2bþ 1Þ þ 4ð40bþ 17bþ 20Þðbþ 1Þð2bþ 1Þcþ cð128bþ 176bþ 96bþ 17b4 þ 32Þ��2

� ½8kð40cþ 16c2 þ 17Þ þ bð952kþ cð1920kþ 136Þ þ c2ð640kþ 64Þ þ 54Þ þ b2ð2584kþ cð4444k

þ 680Þ þ c2ð1232kþ 256Þ þ 324Þ þ b3ð3400kþ cð4976kþ 1208Þ þ c2ð1136kþ 360Þ þ 702Þþ b4ð2176kþ cð2700kþ 904Þ þ c2ð498kþ 208Þ þ 648Þ þ b5ð544kþ cð568kþ 240Þ þ c2ð82kþ 41Þ þ 216Þ�� ½4kð4cþ 3Þð40cþ 16c2 þ 17Þ þ bð1632kþ cð5264kþ 672Þ þ c3ð1280kþ 128Þ þ c2ð4992kþ 608Þþ 204Þ þ b2ð5304kþ cð14798kþ 4032Þ þ c3ð2464kþ 512Þ þ c2ð11840kþ 3040Þ þ 1428Þ þ b3ð8976k

þ ð21350kþ 9246Þ þ c3ð2272kþ 720Þ þ c2ð14080kþ 5692Þ þ 3876Þ þ b4ð8364kþ cð16622k

þ 10104Þ þ c3ð996kþ 416Þ þ c2ð8703kþ 4916Þ þ 5100Þ þ b5ð4080kþ cð6578kþ 5238Þþ c3ð164kþ 82Þ þ c2ð2558kþ 1911Þ þ 3264Þ þ b6ð816kþ cð1020kþ 1020Þ þ c2ð255kþ 255Þ þ 816Þ�

pRDSR2¼ ½16ðkþ 1Þð9bþ 4cþ 8bcþ 6b2 þ 3b2cþ 3Þðbþ 1Þ��1

� ½2ð40cþ 16c2 þ 17Þ þ bð400cþ 128c2 þ 204Þ þ b4ð136cþ 17c2 þ 136Þ

þ b3ð524cþ 96c2 þ 408Þ þ b2ð708cþ 176c2 þ 442Þ��2

f4ð1734k2 þ 2048k2c5 þ 833Þ þ cð41888k2 þ 19264Þ þ c4ð47104k2 þ 9216Þ þ c3ð99328k2 þ 33792Þ

þ c2ð94976k2 þ 39808Þ þ b11½353600kþ 110976k2 þ cð933776kþ 318784k2 þ 614992Þ þ c4ð38709k

þ 16269k2 þ 22440Þ þ c3ð330786kþ 130424k2 þ 200362Þ þ c2ð873664kþ 321640k2 þ 552024Þþ

242624�b10½2697152kþ 998784kþ cð7846936kþ 3019744kþ 5293160Þ þ cð3332kþ 3332k2

þ 833Þ þ c4ð464011kþ 214517kþ 268965Þ þ cð3401749kþ 1441636kþ 2103503Þ þ c2ð8086032k

þ 3250768k2 þ 5235856Þ þ 1890176� þ b9½9156064kþ 4050624k2 þ c5ð37380kþ 40712k2 þ 8512Þ

þ c4ð2423206kþ 1265395k2 þ 1406055Þ þ c3ð15411204kþ 7242156k2 þ 9768021Þ þ cð29218220k

þ 12982832k2 þ 20374132Þ þ c2ð33068848kþ 14947730k2 þ 22090358Þ þ 6635616� þ b8½18 208 768k

þ 9765 888k2 þ c5ð179 584kþ 216 964k2 þ 36 384Þ þ c4ð7280 672kþ 4412 987k2 þ 4245 145Þ

þ c3ð40 521 297kþ 21 807 812k2 þ 26 522 188Þcð63 492 370kþ 33 369 096k2 þ 46 354 182Þ

þ c2ð78713524kþ 41213274k2 þ 54803406Þ þ 13856160� þ b7½23484276kþ 15543576k2

þ c5ð487040kþ 666624k2 þ 85376Þ þ c4ð13963312kþ 10105360k2 þ 8246952Þ þ c3ð68500136k

þ 43652556k2 þ 46858840Þ þ cð89175856kþ 56854256k2 þ 69331762Þ

þ c2ð120786604kþ 75541308k2 þ 88954626Þ þ 19121692� þ b6½20518932kþ 17138856k2

þ c5ð824192kþ 1311232k2 þ 120672Þ þ c4ð17883936kþ 15936416k2 þ 10852088Þ þ c3ð77808752k

þ 60832384k2 þ 56656364Þ þ cð84589092kþ 67295656k2 þ 71634422Þ

þ c2ð124868284kþ 96413604k2 þ 99316758Þ þ 18309568� þ b5½12298072kþ 13351 800k2

þ c5ð904 576kþ 1728 768k2 þ 105 472Þ þ c4ð15 505 088kþ 17 638 944k2 þ 9895 872Þ

þ cð54 881 428kþ 56 369 552k2 þ 52 198 178Þ þ c3ð60 151 592kþ 60 055 904k2 þ 47 911 024Þ

þ c2ð88 079 604kþ 87 216 298k2 þ 77 911 352Þ þ 12 410 860�

þ b4½4992424kþ 7345224k2 þ c5ð645120kþ 1549696k2 þ 55808Þ þ c4ð8990720kþ 13686496k2

þ 6269504Þ þ cð24049778kþ 33364472k2 þ 26832640Þ þ c3ð31252384kþ 41890472k2 þ 28375808Þ

þ c2ð41861968kþ 55790866k2 þ 42978036Þ þ 5953924�

þ b3½1313828kþ 2795208kþ cð288768kþ 933888k2 þ 16384Þ þ cð6813232kþ 13657936k2

þ 9537152Þ þ c4ð3338240kþ 7288832k2 þ 2710784Þ þ c3ð10444160kþ 20184320k2 þ 11546752Þþ

þ c2ð12830592kþ 24682368k2 þ 16345488Þ þ 1980612� þ b2½202436kþ 700536k2 þ cð1127024k

þ 3678936k2 þ 2232084Þ þ c5ð73728kþ 362496k2 þ 2048Þ þ c4ð716800kþ 2535424k2 þ 763136Þ

þ c3ð2027264kþ 6385664k2 þ 3076160Þ þ c2ð2290368kþ 7180544k2 þ 4082608Þ þ 435000�

þ b½1 13872kþ 104040k2 þ c5ð8192kþ 81920k2Þ þ cð82688kþ 586432k2 þ 309536Þ þ c4ð67584k

þ 518144k2 þ 125952Þ þ c3ð173568kþ 1191936k2 þ 482816Þ þ c2ð180864kþ 1234688k2 þ 602624Þþ 56756�g



Appendix B. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.ejor.2012.07.016.

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Price competition and store competition: Store brands vs. national brand

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