A differential game of advertising for national and store brands

Chapter 11 A DIFFERENTIAL GAME OF ADVERTISING FOR NATIONAL AND STORE BRANDS Salma Karray Georges Zaccour Abstract 1. We consider a differential game model for a marketing channel formed by one manufacturer and one retailer. The latter sells the manufacturer’s product and may also introduce a private label at a lower price than the manufacturer’s brand. The aim of this paper is twofold. We first assess in a dynamic context the impact of a private label introduction on the players’ payoffs. If this is beneficial for the retailer to propose his brand to consumers and detrimental to the manufacturer, we wish then to investigate if a cooperative advertising program could help the manufacturer to mitigate the negative impact of the private label. Introduction Private labels (or store brand) are taking increasing shares in the retail market in Europe and North America. National manufacturers are threatened by such private labels that can cannibalize their market shares and steal their consumers, but they can also benefit from the store traffic generated by their presence. In any event, the store brand introduction in a product category affects both retailers and manufacturers marketing decisions and profits. This impact has been studied using static game models with prices as sole decision variables. Mills (1995, 1999) and Narasimhan and Wilcox (1998) showed that for a bilateral monopoly, the presence of a private label gives a bigger bargaining power to the retailer and increases her profit, while the manufacturer gets lower profit. Adding competition at the manufacturing level, Raju et al. (1995) identified favorable factors to the introduction of a private label for the retailer. They showed in a static context that price 214 DYNAMIC GAMES: THEORY AND APPLICATIONS competition between the store and the national brands, and between national brands has considerable impact on the profitability of the private label introduction. Although price competition is important to understand the competitive interactions between national and private labels, the retailer’s promotional decisions do also affect the sales of both product (Dhar and Hoch 1997). Many retailers do indeed accompany the introduction of a private label by heavy store promotions and invest more funds to promote their own brand than to promote the national ones in some product categories (Chintagunta et al. 2002). In this paper, we present a dynamic model for a marketing channel formed by one manufacturer and one retailer. The latter sells the manufacturer’s product (the national brand) and may also introduce a private brand which would be offered to consumers at a lower price than the manufacturer’s brand. The aim of this paper is twofold. We first assess in a dynamic context the impact of a private label introduction on the players’ profits. If we find the same results obtained from static models, i.e., that it is beneficial for the retailer to propose his brand to consumers and detrimental to the manufacturer, we wish then to investigate if a cooperative advertising program could help the manufacturer to mitigate, at least partially, the negative impact of the private label. A cooperative advertising (or promotion) program is a cost sharing mechanism where a manufacturer pays part of the cost incurred by a retailer to promote the manufacturer’s brand. One of the first attempts to study cooperative advertising, using a (static) game model, is Berger (1972). He studied a case where the manufacturer gives an advertising allowance to his retailer as a fixed discount per item purchased and showed that the use of quantitative analysis is a powerful tool to maximize the profits in the channel. Dant and Berger (1996) used a Stackelberg game to demonstrate that advertising allowance increases retailer’s level of local advertising and total channel profits. Bergen and John (1997) examined a static game where they considered two channel structures: A manufacturer with two competing retailers and two manufacturers with two competing retailers. They showed that the participation of the manufacturers in the advertising expenses of their dealers increases with the degree of competition between these dealers, with advertising spillover and with consumer’s willingness to pay. Kim and Staelin (1999) also explored the two-manufacturers, two-retailers channel, where the cooperative strategy is based on advertising allowances. Studies of cooperative advertising as a coordinating mechanism in a dynamic context are of recent vintages (see, e.g., Jørgensen et al. (2000, 2001), Jørgensen and Zaccour (2003), Jørgensen et al. (2003)). 11 A Differential Game of Advertising for National and Store Brands 215 Jørgensen et al. (2000) examine a case where both channel members make both long and short term advertising efforts, to stimulate current sales and build up goodwill. The authors suggest a cooperative advertising program that can take different forms, i.e., a full-support program where the manufacturer contributes to both types of the retailer’s advertising expenditures (long and short term) or a partial-support program where the manufacturer supports only one of the two types of retailer advertising. The authors show that all three cooperative advertising programs are Pareto-improving (profit-wise) and that both players prefer the full support program. The conclusion is thus that a coop advertising program is a coordinating mechanism in also a dynamic setting. Due to the special structure of the game, long term advertising strategies are constant over time. This is less realistic in a dynamic game with an infinite time horizon. A more intuitive strategy is obtained in Jørgensen et al. (2001). This paper reconsiders the issue of cooperative advertising in a two-member channel in which there is, however, only one type of advertising of each player. The manufacturer advertises in national media while the retailer promotes the brand locally. The sales response function is linear in promotion and concave in goodwill. The dynamics are a Nerlove-Arrow-type goodwill evolution equation, depending only on the manufacturer’s national advertising. In this case, one obtains a nondegenerate Markovian advertising strategy, being linearly decreasing in goodwill. In Jørgensen et al. (2000, 2001), it is an assumption that the retailer’s promotion affects positively the brand image (goodwill stock). Jørgensen, et al. (2003) study the case where promotions damage the brand image and ask the question whether a cooperative advertising program is meaningful in such context. The answer is yes if the initial brand image is “weak” or if the initial brand image is at an “intermediate” level and retailer promotions are not “too” damaging to the brand image. Jørgensen and Zaccour (2003) suggest an extension of the setup in Jørgensen et al. (2003). The idea now is that excessive promotions, and not instantaneous action, is harmful to the brand image. To achieve our objective, we shall consider three scenarios or games: 1. Game N : the retailer carries only the N ational brand and no cooperative advertising program is available. The manufacturer and the retailers play a noncooperative game and a feedback Nash equilibrium is found. 2. Game S: the retailer offers a Store brand along with the manufacturer’s product and there is no cooperative advertising program. 216 DYNAMIC GAMES: THEORY AND APPLICATIONS The mode of play is noncooperative and a feedback Nash equilibrium is the solution concept. 3. Game C: the retailer still offers both brands and the manufacturer proposes to the retailer a Cooperative advertising program. The game is played à la Stackelberg with the manufacturer as leader. As in the two other games, we adopt a feedback information structure. Comparing players’ payoffs of the first two games allows to measure the impact of the private label introduction by the retailer. Comparing the players’ payoffs of the last two games permits to see if a cooperative advertising program reduces the harm of the private label for the manufacturer. A necessary condition for the coop plan to be attractive is that it also improves the retailer’s profit, otherwise the will not accept to implement it. The remaining of this paper is organized as follows: In Section 2 we introduce the differential game model and define rigorously the three above games. In Section 3 we derive the equilibria for the three games and compare the results in Section 4. In Section 5 we conclude. 2. Model Let the marketing channel be formed of a manufacturer (player M ) and a retailer (player R). The manufacturer controls the rate of national advertising for his brand A(t), t ∈ [0, ∞). Denote by G(t) the goodwill of the manufacturer’s brand, which dynamics evolve à la Nerlove and Arrow (1962): Ġ(t) = λA(t) − δG(t), G(0) = G0 ≥ 0, (11.1) where λ is a positive scaling parameter and δ > 0 is the decay rate. The retailer controls the promotion efforts for the national brand, denoted by p1 (t), and for the store brand, denoted by p2 (t). We consider that promotions have an immediate impact on sales and do not affect the goodwill of the brand. The demand functions for the national brand (Q1 ) and for the store brand (Q2 ) are as follows: Q1 (p1 , p2 , G) = αp1 (t) − βp2 (t) + θG(t) − µG2 (t) , Q2 (p1 , p2 , G) = αp2 (t) − ψp1 (t) − γG(t), (11.2) (11.3) where α, β, θ, µ, ψ andγ are positive parameters. Thus, the demand for each brand depends on the retailer’s promotions for both brands and on the goodwill of the national brand. Both demands are linear in promotions. 11 A Differential Game of Advertising for National and Store Brands 217 We have assumed for simplicity that the sensitivity of demand to own promotion is the same for both brands considering that the retailer is using usually the same media and methods to promote both brands. However, the cross effect is different allowing for asymmetry in brand substitution. We assume that own brand promotion has a greater impact on sales, in absolute value, than competitive brand promotion, i.e., α > β and α > ψ. This assumption mirrors the one usually made on prices in oligopoly theory. We further suppose that the marginal effect of promoting the national brand on the sales of the store brand is higher than the marginal effect of promoting the store brand on the sales of the national brand, i.e., ψ > β. This actually means that the manufacturer’s brand enjoys a priori a stronger consumer preference than the retailer’s one. Putting together these inequalities leads to the following assumption A1 : α > ψ > β > 0. Finally, the demand for the national brand is concave increasing in 1 its goodwill (i.e., ∂Q ∂G = θ − 2µG > 0, ∀G > 0) and the demand for the store brand is decreasing in that goodwill. Denote by D(t), 0 ≤ D(t) ≤ 1, the coop participation rate of the manufacturer in the retailer’s promotion cost of the national brand. We assume as in, e.g., Jørgensen et al. (2000, 2003), that the players face quadratic advertising and promotion costs. The net cost incurred by the manufacturer and the retailer are as follows CM (A) = CR (p1 , p2 ) = 1 1 uM A2 (t) + uR D (t) p21 (t), 2 2 o  1 n uR 1 − D (t) p21 (t) + p22 (t) , 2 where uR , uM > 0. Denote by m0 the manufacturer’s margin, by m1 the retailer’s margin on the national brand and by m2 her margin on the store brand. Based on empirical observations, we suppose that the retailer has a higher margin on the private label than on the national brand, i.e., m2 > m1 . Ailawadi and Harlam (2004) found indeed that for product categories where national brands are heavily advertised, the percent retail margins are significantly higher for store brands than for national brands. We denote by r the common discount rate and we assume that each player maximizes her stream of discounted profit over an infinite horizon. Omitting the time argument when no ambiguity may arise, the optimization problems of players M and R in the different games are as follows: 218 DYNAMIC GAMES: THEORY AND APPLICATIONS Game C: Both brands are offered and a coop program is available.  Z +∞
C −rt max JM = m0 αp1 − βp2 + θG − µG2 e A,D 0  uM 2 uR 2 − A − Dp1 dt, 2 2  Z +∞
−rt C e m1 αp1 − βp2 + θG − µG2 max JR = p1 ,p2 0 i
1 h
2 2 + m2 αp2 − ψp1 − γG − uR 1 − D p1 + p2 dt. 2 Game S: Both brands are available and there is no coop program.   Z +∞
uM 2 −rt 2 S e m0 αp1 − βp2 + θG − µG − A dt, max JM = A 2 0  Z +∞
S −rt max JR = e m1 αp1 − βp2 + θG − µG2 p1 ,p2 0 
uR  2 2 + m2 αp2 − ψp1 − γG − p1 + p2 dt. 2 Game N : Only manufacturer’s brand is offered and there is no coop program.   Z +∞
uM 2 2 N −rt max JM = m0 αp1 + θG − µG − e A dt, A 2 0   Z +∞
uR 2 N −rt 2 p dt. max JR = e m1 αp1 + θG − µG − p1 2 1 0 3. Equilibria We characterize in this section the equilibria of the three games. In all cases, we assume that the players adopt stationary Markovian strategies, which is rather standard in infinite-horizon differential games. The following proposition gives the result for Game N . Proposition 11.1 When the retailer does not sell a store brand and the manufacturer does not provide any coop support to the retailer, stationary feedback Nash advertising and promotional strategies are given by pN 1 = αm1 , uR 11 A Differential Game of Advertising for National and Store Brands 219 AN (G) = X + Y G, where √ 2m0 θλ r + 2δ − 2 ∆1 √
, X= , Y = 2λ r + 2 ∆1 u M  r 2 2µm0 λ2 ∆1 = δ + + . 2 uM Proof. A sufficient condition for a stationary feedback Nash equilibrium is the following: Suppose there exists a unique and absolutely continuous solution G (t) to the initial value problem and there exist bounded and continuously differentiable functions Vi : ℜ+ → ℜ, i ∈ {M, R}, such that the Hamilton-Jacobi-Bellman (HJB) equations are satisfied for all G ≥ 0: 
(11.4) rVM (G) = max m0 αp1 + θG − µG2 A 
1 ′ − uM A2 + VM (G) λA − δG | A ≥ 0 , 2 rVR (G) = 
max m1 αp1 + θG − µG2 p1 
1 2 ′ − uR p1 + VR (G) λA − δG | p1 ≥ 0 . 2 (11.5) The maximization of the right-hand-side of equations (11.4) and (11.5) yields the following advertising and promotional rates: A (G) = λ ′ V (G) , uM m p1 = αm1 . uR Substituting the above in (11.4) and (11.5) leads to the following expressions  2  2 α m1 λ2  ′ 2 ′ rVM (G) = m0 + θG − µG + VM (G) − δGVM (G) , uR 2uM (11.6)   2   2 λ α m1 2 ′ ′ + θG − µG + VR (G) V (G) − δG . rVR (G) = m1 2uR uM M (11.7) 220 DYNAMIC GAMES: THEORY AND APPLICATIONS It is easy to show that the following quadratic value functions solve the HJB equations; 1 VM (G) = a1 + a2 G + a3 G2 , 2 1 VR (G) = b1 + b2 G + b3 G2 , 2 where a1 , a2 , a3 , b1 , b2 , b3 are constants. Substitute VM (G), VR (G) and their derivatives into equations (11.6) and (11.7) to obtain:  a3  m0 α2 m1 λ2 a22 + r a1 + a2 G + G2 = 2 uR 2uM     λ2 a2 a3 λ2 a23 + m0 θ − δa2 + G − µm0 + δa3 − G2 , uM 2uM   1 α2 m21 λ2 2 r b1 + b2 G + b3 G = + a2 b2 2 2uR uM     λ2 λ2 + m1 θ − δb2 + (a2 b3 + a3 b2 ) G − m1 µ + δb3 − b3 a3 G2 . uM uM By identification, we obtain the following values for the coefficients of the value functions:
√ δ + 2r ± ∆1 m1 µ , b3 = − r a3 = 2 λ2 λ /uM 2 + δ − uM a3 a2 = a1 = where m0 θ r+δ− λ2 uM a3 , λ2 a22 m0 α2 m1 + , ruR 2ruM b2 = b1 = λ2 uM b3 a2 2 r + δ − uλM a3 α2 m21 λ2 a2 b2 m1 θ + 2ruR + ruM  r 2 2µm0 λ2 ∆1 = δ + . + 2 uM To obtain an asymptotically stable steady state, choose the negative solution for a3 . Note that the identified solution must satisfy the con′ (G) = A(G), this assumption is true for straint A(G) > 0. Since uλM VM   G ∈ 0, ḠN , where √ r + 2δ − 2 ∆1 a2 λ ′ 2m0 θλ N √
+ Ḡ = − , A (G) = G. V (G) = a3 uM M 2λ r + 2 ∆1 u M 2 11 A Differential Game of Advertising for National and Store Brands 221 The above proposition shows that the retailer promotes always the manufacturer’s brand at a positive constant rate and that the advertising strategy is decreasing in the goodwill. The next proposition characterizes the feedback Nash equilibrium in Game S. Proposition 11.2 When the retailer does sell a store brand and the manufacturer does not provide any coop support to the retailer, assuming an interior solution, stationary feedback Nash advertising and promotional strategies are given by pS1 = αm1 − ψm2 , uR pS2 = αm2 − βm1 , uR AS (G) = AN (G). Proof. The proof proceeds exactly as the previous one and we therefore print only important steps. The HJB equations are given by: 
rVM (G) = max m0 αp1 − βp2 + θG − µG2 A 
uM 2 ′ A + VM (G) λA − δG | A ≥ 0 , − 2 

rVR (G) = max m1 αp1 − βp2 + θG − µG2 + m2 αp2 − ψp1 − γG p1 ,p2 


uR 2 2 ′ p + p2 + VR (G) λA − δG | p1 , p2 ≥ 0 . − 2 1 The maximization of the right-hand-side of the above equations yields the following advertising and promotional rates: A (G) = λ ′ αm1 − ψm2 Vm (G) , p1 = , uM uR p2 = αm2 − βm1 . uR We next insert the values of A (G), p1 and p2 from above in the HJB equations and assume that the resulting equations are solved by the following quadratic functions: 1 VM (G) = s1 + s2 G + s3 G2 , 2 1 VR (G) = k1 + k2 G + k3 G2 , 2 where k1 , k2 , k3 , s1 , s2 , s3 are constants. Following the same procedure as in the proof of the previous proposition, we obtain
√ δ + 2r ± ∆2 m1 µ , k3 = − r , s3 = 2 λ2 /uM + δ − λ s3 2 uM 222 DYNAMIC GAMES: THEORY AND APPLICATIONS s2 = where m0 θ m1 θ − m2 γ + λ2 uM k3 s2 , k2 = , 2 2 r + δ − uλM s3 r + δ − uλM s3  λ2 2 m0  α (m1 α − m2 ψ) − β (m2 α − m1 β) + s , s1 = ruR 2ruM 2  1  λ2 k1 = (m1 α − m2 ψ)2 + (m2 α − m1 β)2 + k2 s2 , 2ruR ruM  r 2 2µm0 λ2 + . ∆ 2 = ∆1 = δ + 2 uM In order to obtain an asymptotically stable steady state, we choose  for s3 the negative solution. The assumption A(G) > 0 holds for G ∈ 0, ḠS , where ḠS = − ss32 . Note also that s3 = a3 , s2 = a2 and b3 = k3 . Thus AS (G) = AN (G) and ḠS = ḠN . 2 Remark 11.1 Under A1 (α > ψ > β > 0) and the assumption that 1 m2 > m1 , the retailer will always promote his brand, i.e., pS2 = αm2u−βm R 2 > 0. For pS1 = αm1u−ψm to be positive and thus the solution to be inteR rior, it is necessary that (αm1 − ψm2 ) > 0. This means that the retailer will promote the national brand if the marginal revenue from doing so exceeds the marginal loss on the store brand. This condition has thus an important impact on the results and we shall come back to it in the conclusion. In the last game, the manufacturer offers a coop promotion program to her retailer and acts as leader in a Stackelberg game. The results are summarized in the following proposition. Proposition 11.3 When the retailer does sell a store brand and the manufacturer provides a coop support to the retailer, assuming an interior solution, stationary feedback Stackelberg advertising and promotional strategies are given by αm2 − βm1 2αm0 + (αm1 − ψm2 ) , pC , 2 = 2uR uR 2αm0 − (αm1 − ψm2 ) AC (G) = AS (G) , D = . 2αm0 + (αm1 − ψm2 ) pC 1 = Proof. We first obtain the reaction functions of the follower (retailer) to the leader’s announcement of an advertising strategy and a coop support rate. The later HJB equation is the following 11 223 A Differential Game of Advertising for National and Store Brands 

rVR (G) = max m1 αp1 − βp2 + θG − µG2 + m2 αp2 − ψp1 − γG p1 ,p2 − uR 2 (11.8) 


′ 2 2 (1 − D) p1 + p2 + VR (G) λA − δG | p1 , p2 ≥ 0 . Maximization of the right-hand-side of (11.8) yields p1 = αm1 − ψm2 , uR (1 − D) p2 = αm2 − βm1 . uR (11.9) The manufacturer’s HJB equation is: 
uM 2 1 rVM (G) = max m0 αp1 − βp2 + θG − µG2 − A − uR Dp21 A,D 2 2 
′ + VM (G) λA − δG | A ≥ 0, 0 ≤ D ≤ 1 . Substituting for promotion rates from (11.9) into manufacturer’s HJB equation yields 
αm1 − ψm2 αm2 − βm1 rVM (G) = max m0 α −β + θG − µG2 A,D uR (1 − D) uR   
αm1 − ψm2 2 uM 2 uR ′ + VM (G) λA − δG A − D − 2 2 uR (1 − D) Maximizing the right-hand-side leads to A (G) = 2αm0 − (αm1 − ψm2 ) λ ′ VM (G) , D = . uM 2αm0 + (αm1 − ψm2 ) (11.10) Using (11.9) and (11.10) provides the retailer’s promotional strategies p1 = 2αm0 + (αm1 − ψm2 ) , 2uR p2 = αm2 − βm1 . uR Following a similar procedure to the one in the proof of Proposition 11.1, it is easy to check that following quadratic value functions provide unique solutions for the HJB equations, 1 1 VM (G) = n1 + n2 G + n3 G2 , VR (G) = l1 + l2 G + l3 G2 , 2 2 where n1 , n2 , n3 , l1 , l2 , l3 are constants given by:
√ δ + 2r ± ∆3 m1 µ , l3 = − r , n3 = 2 λ2 /uM + δ − λ n3 2 uM 224 DYNAMIC GAMES: THEORY AND APPLICATIONS n2 = m0 θ m1 θ − m2 γ + λ2 uM l 3 n 2 , l2 = , 2 2 r + δ − uλM n3 r + δ − uλM n3

 1 m0  α αm0 + (m1 α − m2 ψ) − β m2 α − m1 β n1 = ruR 2 2
2  1  2 2 1 λ , n22 − α m0 − m1 α − m2 ψ + 2ruM 2ruR 4
 (m1 α − m2 ψ)  1 l1 = αm0 + m1 α − m2 ψ 2ruR 2 2 2 (m2 α − m1 β) λ l2 n2 + + , 2ruR ruM
2 2 where ∆3 = ∆2 = ∆1 = δ + 2r + 2µm0 uλM . To obtain an asymptotically stable steady state, we choose the negative solution for n3 . Note that n3 = s3 = a3 , n2 = s2 = a2 , l3 = k3 = b3 and l2 = k2 . Thus AC (G) = AS (G) = AN (G). 2 Remark 11.2 As in Game S, the retailer will always promote her brand at a positive constant rate. The condition for promoting the manufacturer’s brand is (2αm0 + αm1 − ψm2 ) > 0 (the numerator of pC 1 has to be positive). The condition for an interior solution in Game S was that (αm1 − ψm2 ) > 0. Thus if pS1 is positive, then pC 1 is also positive. Remark 11.3 The support rate is constrained to be between 0 and 1. It is easy to verify that if pC 1 > 0, then a necessary condition for D < 1 is that (αm1 − ψm2 ) > 0, i.e., pS1 > 0. Assuming pC 1 > 0, otherwise there is no reason for the manufacturer to provide a support, the necessary condition for having D > 0 is (2αm0 − αm1 + ψm2 ) > 0. 4. Comparison In making the comparisons, we assume that the solutions in the three games are interior. The following table collects the equilibrium strategies and value functions obtained in the three games. In terms of strategies, it is readily seen that the manufacturer’s advertising strategy (A(G)) is the same in all three games. This is probably a by-product of the structure of the model. Indeed, advertising does not affect sales directly but do it through the goodwill. Although the later has an impact on the sales of the store brand, this does not affect the profits earned by the manufacturer. The retailer adopts the same promotional strategy for the private label in the games where such brand is available, i.e., whether a coop program is offered or not. This is also due to the simple structure of our model. 11 A Differential Game of Advertising for National and Store Brands Table 11.1. p1 p2 A(G) D VM (G) VR (G) 225 Summary of Results Game N Game S Game C αm1 uR αm1 −ψm2 uR αm2 −βm1 uR AN (G) 2αm0 +(αm1 −ψm2 ) 2uR αm2 −βm1 uR AN (G) s1 + a2 G + a23 G2 k1 + k2 G + b23 G2 2αm0 −(αm1 −ψm2 ) 2αm0 +(αm1 −ψm2 ) n1 + a2 G + a23 G2 l1 + k2 G + b23 G2 AN (G) a1 + a2 G + a23 G2 b1 + b2 G + b23 G2 The remaining and most interesting item is how the retailer promotes the manufacturer’s brand in the different games. The introduction of the store brand leads to a reduction in the promotional effort of the manuψm2 S facturer’s brand (pN 1 − p1 = uR > 0). The coop program can however reverse the course of action and increases the promotional  effort for the  2αm0 −αm1 +ψm2 C S > 0 . This result is manufacturer’s brand p1 − p1 = 2uR expected and has also been obtained in the literature cited in the introduction. What is not clear cut is whether the level of promotion could reach
back the one in the game without the store brand. Indeed, C is positive if the condition that (αm + ψm > 2αm ) is satpN − p 1 2 0 1 1 isfied. We now compare the players’ payoffs in the different games and thus answer the questions raised in this paper. Proposition 11.4 The store brand introduction is harmful for the manufacturer for all values of the parameters. Proof. From the results of Propositions 11.1 and 11.2, we have: S N VM (G0 ) − VM (G0 ) = s1 − a1 = − m0 [m2 ψα + β (m2 α − m1 β)] < 0. ruR 2 For the retailer, we cannot state a clear-cut result. Compute, VRS (G0 ) − VRN (G0 ) = k1 − b1 + (k2 − b2 ) G0 i 1 h = (m1 α − m2 ψ)2 + (m2 α − m1 β)2 − α2 m21 2ruR 2m2 γ 4λ2 m0 m2 θγ √
G0 . + √
2 + r r + ∆2 ruM r + ∆2 226 DYNAMIC GAMES: THEORY AND APPLICATIONS Thus for the retailer to benefit from the introduction of a store brand, the following condition must be satisfied √
r + ∆2 2 2 2λ2 m0 θ S N √
VR (G0 ) − VR (G0 ) > 0 ⇔ G0 > α m1 − 4m2 γuR u M r + ∆2 √
i r + ∆2 h − (m1 α − m2 ψ)2 + (m2 α − m1 β)2 . 4m2 γuR The above inequality says that the retailer will benefit from the introduction of a store brand unless the initial goodwill of the national one is “too low”. One conjecture is that in such case the two brands would be too close and no benefit is generated for the retailer from the product variety. The result that the introduction of a private label is not always in the best interest of a retailer has also been obtained by Raju et al. (1995) who considered price competition between two national brands and a private label. Turning now to the question whether a coop advertising program can mitigate, at least partially, the losses for the manufacturer, we have the following result. Proposition 11.5 The cooperative advertising program is profit Paretoimproving for both players. Proof. Recall that k2 = l2 , k3 = l3 = n3 and n2 = s2 . Thus for the manufacturer, we have 3 2 VM (G0 ) − VM (G0 ) = n1 − s1 = 1 [2αm0 − (αm1 − ψm2 )]2 > 0. 8ruR For the retailer VRC (G0 )−VRS (G0 ) = l1 −k1 = 1 (m1 α − m2 ψ) (2αm0 − m1 α + m2 ψ) 4ruR which is positive. Indeed, (m1 α − m2 ψ) = ur pS1 which is positive by the assumption of interior solution and (2αm0 − m1 α + m2 ψ) which is also positive (it is the numerator of D). 2 The above proposition shows that the answer to our question is indeed yes and, importantly, the retailer would be willing to accept a coop program when suggested by the manufacturer. 5. Concluding Remarks The results so far obtained rely heavily on the assumption that the solution of Game S is interior. Indeed, we have assumed that the retailer 11 A Differential Game of Advertising for National and Store Brands 227 will promote the manufacturer’s brand in that game. A natural question is that what would happen if it were not the case? Recall that we required that pS1 = αm1 − ψm2 > 0 ⇔ αm1 > ψm2 . uR If αm1 > ψm2 is not satisfied, then pS1 = 0 and the players’ payoffs should be adjusted accordingly. The crucial point however is that in such event, the constraint on the participation rate in Game C would be impossible to satisfy. Indeed, recall that D= and compute 2αm0 − (αm1 − ψm2 ) , 2αm0 + αm1 − ψm2 2 (αm1 − ψm2 ) . 2αm0 + αm1 − ψm2 Hence, under the condition that (αm1 − ψm2 < 0), the retailer does not invest in any promotions for the national brand after introducing
the private label pS1 = 0 . In this case, the cooperative advertising program can be implemented only if the retailer does promote the national brand and the manufacturer offers the cooperative advertising program i.e., a positive coop participation rate, which is possible only if (2αm0 + αm1 − ψm2 ) > 0. Now, suppose that we are in a situation where the following conditions are true 1−D = αm1 − ψm2 < 0 and 2αm0 + αm1 − ψm2 > 0 (11.11) In this case, the retailer does promote the manufacturer’s product (pC 1 > 0), however we obtain D > 1. This means that the manufacturer has to pay more than the actual cost to get her brand promoted by the retailer in Game C and the constraint D < 1 has to be removed. For pS1 = 0 and when the conditions in (11.11) are satisfied, it is easy to show that the effect of the cooperative advertising program on the profits of retailer and the manufacturer are given by (αm1 − ψm2 ) (2αm0 + m1 α − m2 ψ) < 0 4ur 1 C S (2m0 α + αm1 − ψm2 )2 > 0 VM (G0 ) − VM (G0 ) = 8ur VRC (G0 ) − VRS (G0 ) = In this case, even if the manufacturer is willing to pay the retailer more then the costs incurred by advertising the national brand, the retailer will not implement the cooperative program. 228 DYNAMIC GAMES: THEORY AND APPLICATIONS To wrap up, the message is that the implementability of a coop promotion program depends on the type of competition one assumes between the two brands and the revenues generated from their sales to the retailer. The model we used here is rather simple and some extensions are desirable such as, e.g., letting the margins or prices be endogenous. References Ailawadi, K.L. and Harlam, B.A. (2004). An empirical analysis of the determinants of retail margins: The role of store-brand share. Journal of Marketing, 68(1):147–165. Bergen, M. and John, G. (1997). Understanding cooperative advertising participation rates in conventional channels. Journal of Marketing Research, 34(3):357–369. Berger, P.D. (1972). Vertical cooperative advertising ventures. Journal of Marketing Research, 9(3):309–312. Chintagunta, P.K., Bonfrer , A., and Song, I. (2002). Investigating the effects of store brand introduction on retailer demand and pricing behavior. Management Science, 48(10):1242–1267. Dant, R.P. and Berger, P.D. (1996). Modeling cooperative advertising decisions in franchising. Journal of the Operational Research Society, 47(9):1120–1136. Dhar, S.K. and S.J. Hoch (1997). Why store brands penetration varies by retailer. Marketing Science, 16(3):208–227. Jørgensen, S., Sigué, S.P., and Zaccour, G. (2000). Dynamic cooperative advertising in a channel. Journal of Retailing, 76(1):71–92. Jørgensen, S., Taboubi, S., and Zaccour, G. (2001). Cooperative advertising in a marketing channel. Journal of Optimization Theory and Applications, 110(1):145–158. Jørgensen, S., Taboubi, S., and Zaccour, G. (2003). Retail promotions with negative brand image effects: Is cooperation possible? European Journal of Operational Research, 150(2):395–405. Jørgensen, S. and Zaccour, G. (2003). A differential game of retailer promotions. Automatica, 39(7):1145–1155. Kim, S.Y. and Staelin, R. (1999). Manufacturer allowances and retailer pass-through rates in a competitive environment. Marketing Science, 18(1):59–76. Mills, D.E. (1999). Private labels and manufacturer counterstrategies. European Review of Agricultural Economics, 26:125–145. Mills, D.E. (1995). Why retailers sell private labels. Journal of Economics & Management Strategy, 4(3):509–528. 11 A Differential Game of Advertising for National and Store Brands 229 Narasimhan, C. and Wilcox, R.T. (1998). Private labels and the channel relationship: A cross-category analysis. Journal of Business, 71(4): 573–600. Nerlove, M. and Arrow, K.J. (1962). Optimal advertising policy under dynamic conditions. Economica, 39:129–142. Raju, J.S., Sethuraman, R., and Dhar, S.K. (1995). The introduction and performance of store brands. Management Science, 41(6):957–978. View publication stats