Pricing and Advertising of Private and National Brands in a Dynamic Marketing Channel

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1 See discussions, stats, and author profiles for this publication at: Pricing and Advertising of Private and National Brands in a Dynamic Marketing Channel Article in Journal of Optimization Theory and Applications June 2008 DOI: /s CITATIONS 25 READS authors: Nawel Amrouche Long Island University 16 PUBLICATIONS 191 CITATIONS SEE PROFILE Guiomar Martín-Herrán Universidad de Valladolid 74 PUBLICATIONS 701 CITATIONS SEE PROFILE Georges Zaccour HEC Montréal - École des Hautes Études commerciales 226 PUBLICATIONS 3,433 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Viability Theory View project Sustainability of cooperation View project All content following this page was uploaded by Georges Zaccour on 29 August The user has requested enhancement of the downloaded file.

2 J Optim Theory Appl (2008) 137: DOI /s Pricing and Advertising of Private and National Brands in a Dynamic Marketing Channel N. Amrouche G. Martín-Herrán G. Zaccour Published online: 5 January 2008 Springer Science+Business Media, LLC 2008 Abstract We consider a marketing channel where a retailer sells, along the manufacturer s brand, its own store brand. We assume that each player invests in advertising in order to build the brand s goodwill. One distinctive feature of this paper is the introduction of the negative effect of own advertising on other player s goodwill stock evolution. We characterize feedback-nash pricing and advertising strategies and assess the impact of the store brand and national brand s goodwill stocks on these strategies in different settings. The main findings suggest first that investing in building up some equity for each brand reduces the price competition between them and propels the market power for both. Second, the retailer will pass to consumer an increase in its purchasing cost of the national brand in all situations as no coordination is taken into account to counter the double marginalization problem. Finally, the higher the brand equity of the store brand, the more the retailer invests in advertising. Keywords Marketing channels Private label Advertising Pricing Differential games Feedback-Nash equilibrium 1 Introduction We consider a marketing channel where a retailer offers to consumers its private label, or store brand, as well as a national manufacturer s brand. The manufacturer N. Amrouche Long Island University, New York, USA G. Martín-Herrán Departamento de Economía Aplicada (Matemáticas), Universidad de Valladolid, Valladolid, Spain G. Zaccour ( ) Chair in Game Theory and Management, GERAD, HEC Montréal, Montréal, Canada georges.zaccour@gerad.ca

3 466 J Optim Theory Appl (2008) 137: controls the wholesale price of its brand to retailer, and invests in advertising in order to build up the brand equity (reputation or goodwill). The retailer decides on the price-to-consumer of the two brands and invests also in advertising to raise its private label equity. We are interested in investigating noncooperative pricing and advertising equilibrium strategies in the context where each player can affect, through advertising, also the other player s brand equity. Store brands are by no means a marginal phenomenon. Indeed, according to the Private Label Manufacturers Association, 1 one in five products sold in U.S. supermarkets in 2004 was a private label. One major determinant of this success is the improved quality of these labels (see e.g. [1 3]). Hoch and Banerji [2] note that this success is however less likely to occur in product categories where the manufacturers have made substantial advertising commitments to brand equity. Although it has been generally the case that retailers spend their communication budgets mainly on features rather than on unaffordable advertising campaigns, some major retailers do advertise their private labels nationally (see e.g. [4] and [5]). We are interested in such context where, indeed, each player s advertising effort has two effects, i.e., raising its own brand equity and decreasing the other s brand one. Intuitively, this context is of interest when the store brand is positioned as one of high quality and actually perceived by consumers as a possible substitute to the manufacturer s brand. 2 In this context of a retailer competing against a manufacturer, we wish to answer the following questions: Assuming that there is a cross-price effect between the two brands, and that each player takes into account the goodwill s dynamics of both brands, how the goodwill stocks affect the players strategies? Assuming that there is no cross-price effect, what is the impact the goodwill stocks on players strategies? Assuming that each player is partially myopic, i.e., takes into account only the goodwill s dynamics of its brand, to what extent do the result may change from the previous situations? Is it a good weapon for the retailer to invest as the manufacturer in advertising its brand? This paper differs from published contributions in many respects. The game theoretic literature on private labels is mainly static. By adopting a differential game framework, our model captures clearly the differences between the advertising investments (flows) and their results in terms of brand equity (stock). The model allows also to analyze the impact of brand equity on strategies. Moreover, the consideration of a store brand building such stock has not been analyzed in the marketing literature, where the assumption has been typically that a store brand derives its revenues by attracting price-sensitive market segments. Finally, it has been seldom the case to fully include in a differential game of marketing channels, as it is done here, both advertising and pricing strategies (see e.g. [6 12]). 1 See 2 For instance, jeans labels as Arizona (J.C. Penney), Canyon River Blues (Sears, Roebuck) and Badge (Federated Department Stores) have become not only acceptable to consumers but even trendy thanks to clever advertising of the product image (apart from the store s image).

4 J Optim Theory Appl (2008) 137: The rest of the paper is structured as follows. In Sect. 2, we develop a model for the channel under study. In Sect. 3, we derive the Feedback-Nash Equilibrium and interpret the results. In Sect. 4, we conclude. 2 Model Consider a marketing channel formed of a manufacturer (player M) of a national brand and a retailer (player R). The retailer offers the national manufacturer s brand (NB) as well as its own store brand (SB) or private label in its outlet. The SB is supplied to the retailer at a constant wholesale price w s by another manufacturer who does not play any strategic role in our setting (as in e.g. [13]). Denote by p n (t) the price-to-consumer of the national brand at time t [0, ) and by p s (t) the retail price of the SB. Both prices are controlled by the retailer. Denote by w n (t) the manufacturer NB s wholesale price to the retailer. Each player advertises its brand in order to increase its brand equity (or market potential). Let A i (t) represent the advertising investment at time t for brand i, i {n, s}, where n stands for national brand and s for store brand. The advertising cost C i (A i ) is assumed convex increasing and taken, for tractability, quadratic as follows: C i (A i ) = c i1 A i + c i2 2 A2 i, i {n, s}, (1) where c i1 and c i2 are positive constants. Denote by G i (t) brand i s market potential whose evolution is governed by the following differential equation: Ġ i (t) = A i (t) δg i (t) k i A j (t), G i (0) = G i0 0, i,j {n, s}, i j, (2) where δ>0 is the decay rate. The above specification modifies the classical Nerlove- Arrow dynamics by adding the term k i A j (t), where k i 0,i {n, s}. This means that there are two sources of depreciation of brand i s equity; the standard one, given by δg i (t) which captures the idea that consumers forget to some extent the brand, and the one due to the advertising by the other brand. Thus, the parameter k i represents the vulnerability of brand i to the advertising made by brand j. We suppose that own advertising effect on brand equity is higher, in absolute terms, to the advertising made by the other brand and hence k i < 1. Reference [14] explained that persuasive advertising, that are more popular in competitive arena, aims to target selective demand for the product. Comparative advertising is one possible format of persuasion. Using these ads, firms try to change the preferences of consumers by increasing their willingness to pay for the sponsoring brand and reducing the one for the comparison brand. Some authors have modeled the spillover effect of advertising assuming either a negative impact of advertising on the competing brand s goodwill ([15] and [16]) or a positive external effects on the market size of all rivals [17]. Following [18], each brand demand depends on its equity and on both prices as follows: D i (t) = α i G i (t) p i (t) + ψ i p j (t), i, j {n, s}, i j, (3)

5 468 J Optim Theory Appl (2008) 137: where ψ i,i {n, s}, is a positive parameter capturing the cross-price effect on demand. This specification deserves some comments. First, it is linear in prices which is a popular assumption in the economics literature, with however a difference that the maximal demand, i.e., the level of demand when the prices are zero, is not a given constant but depends on the brand equity. For simplicity, we set α i = 1, i {n, s}. Second, we normalize in (3) the direct-price effect to 1. Given that the direct-price effect is larger than the cross-price one, see e.g. [19] and [20], we suppose that ψ i < 1, i {n, s}. From now on, we assume G n p n + ψ n p s > 0 and G s p s + ψ s p n > 0 in order to have a positive demand for the national and the store brands. Assuming that both players are profit maximizers, their optimization problems read as follows: max π M = w n,a n max A s, p n,p s π R = 0 0 e ρt {(w n (t) d n )D n (t) C n (A n (t))} dt, (4) e ρt {(p n (t) w n (t))d n (t) + (p s (t) w s )D s (t) C s (A s (t))} dt, (5) where ρ is the common discount rate and d n is the constant unit production cost of the NB. By (1) (5) we have defined a two-player, non-zero sum, infinite-horizon differential game with two state variables (G n,g s ) and five control variables (w n,a n,a s,p n,p s ). We shall from now on eliminate the time argument when no confusion may arise. 3 Feedback Nash Equilibrium We assume that the manufacturer and the retailer play à la Nash and thus decide simultaneously their investments in advertising and the prices. Recall that the manufacturer controls A n and w n, and the retailer A s, p s and p n. As usual in autonomous differential games played over an infinite time horizon, we consider that channel members s strategies are stationary feedback (see, for example, [21]), which means that pricing and advertising strategies as well as value functions are time-independent and only depend on the current levels of the state variables G n and G s. The next proposition characterizes the Nash equilibrium and the equilibrium outcomes. Proposition 3.1 Let denote by V R (G s,g n ) and V M (G s,g n ) the retailer s and the manufacturer s value functions. Let assume that the transfer price is upper bounded by w n. (i) The following expressions satisfy the Hamilton-Jacobi-Bellman equations associated with the retailer s and manufacturer s problems: V R (G s,g n ) = R 1 2 G2 s + R 2 2 G2 n + R 3G n G s + R 4 G s + R 5 G n + R 6, (6) V M (G s,g n ) = M 1 G n + M 2 G s + M 3. (7)

6 J Optim Theory Appl (2008) 137: (ii) The retailer s and the manufacturer s optimal strategies are given by: pn (G s,g n ) = w n(ψ n ψ s + ψn 2 2) [2G n + G s (ψ n + ψ s )] w s (ψ n ψ s ), (ψ n + ψ s 2)(ψ n + ψ s + 2) (8) ps (G s,g n ) = w n(ψ n ψ s ) [2G s + G n (ψ n + ψ s )] w s [2 + (ψ n + ψ s )ψ s ], (ψ n + ψ s 2)(ψ n + ψ s + 2) (9) A s (G s,g n ) = 1 [(R 1 k n R 3 )G s + (R 3 k n R 2 )G n + R 4 k n R 5 c s1 ],(10) c s2 wn (G s,g n ) = w n, (11) A n (G s,g n ) = 1 (M 1 k s M 2 c n1 ), (12) c n2 if there are positive expressions and zero otherwise. The coefficients R i,m j,i = 1,...,6,j = 1, 2, 3 are given in the Appendix. Furthermore, R 1 0, R 2 0 and R 3 has the same sign as the following expression: k n + (ρ + 2δ)( 2k n + ψ n + ψ s )R 1. (13) Moreover, a.1 If R 1 is chosen, then R 1 k n R 3 > 0. a.2 If R 1 is chosen, then R 1 k n R 3 > k 2 n k n(ψ n + ψ s )>0, where R 1,R 1 are given in the Appendix. The sign of R 3 k n R 2 characterizes as follows: b.1 If 1 + kn 2 k n(ψ n + ψ s )<0, then b.1.1 If R 1 is chosen, then R 3 k n R 2 < 0. b.1.2 If R 1 is chosen, then R 3 k n R 2 > 0. b.2 If 2k n ψ n ψ s > 0, then R 3 k n R 2 < 0 4(1 + k 2 n k n(ψ n + ψ s )) + a 1 c s2 (2δ + ρ) 2 > 0. b.3 If 2k n ψ n ψ s < 0 < 1 + k 2 n k n(ψ n + ψ s ), then R 3 k n R 2 > 0 4(1 + k 2 n k n(ψ n + ψ s )) + a 1 c s2 (2δ + ρ) 2 > 0. Proof See the Appendix. Item (i) in the above proposition establishes that the retailer s value function is quadratic in the goodwill levels of the national and store brands, while the manufacturer s value function is linear in these stocks. This is expected given the linearquadratic structure of the retailer s problem and the linear-state structure of the manufacturer s problem. As a consequence, item (ii) shows that the retailer s advertising

7 470 J Optim Theory Appl (2008) 137: strategy is linear in the goodwill levels, while that of the manufacturer is feedback degenerate (i.e., does not depend on the state). The results in item (ii) allow the following remarks. Each pricing strategy depends on both goodwill stocks. Further, the transfer price is set at its upper bound in equilibrium. Note that the higher the latter, the higher is the retail price of the national brand. Indeed, pn ψ n ψ s + ψn 2 (G s,g n ) = 2 w n (ψ n + ψ s + 2)(ψ n + ψ s 2) > 0, by the assumption 0 ψ i < 1, i {n, s}. Therefore, the retail and transfer prices are strategic complements. Put differently, the retailer will pass to consumer an increase in its purchasing cost of the NB. This is the classical double marginalization problem due to the absence of coordination. The relationship between the retail price of the SB and this same transfer price is given by ps ψ n ψ s (G s,g n ) = w n (ψ n + ψ s + 2)(ψ n + ψ s 2) 0 ψ n ψ s. The sign of this derivative depends on the relative magnitude of the cross-price effects ψ n and ψ s.ifψ n >ψ s, then p s w n (G s,g n )<0 and the two variables are strategic substitutes (when the wholesale price increases, the SB s price decreases). If ψ n < ψ s, then we have strategic complementarity (when the wholesale price increases, the SB s price increases too). Ultimately, the actual values of ψ n and ψ s is an empirical matter. The results mean that the retailer will not decrease the SB s price to try to compete against the NB that disfavored it by increasing the wholesale price, unless the cross-price effect of the SB (ψ n ) is high compared the NB s one (ψ s ). Furthermore, the advertising parameters, k n and k s, do not affect the pricing strategies. Indeed, pi (G i,g j ) = 0, for i, j = n, s. k j This result is, most probably, a by-product of the model structure, i.e., the prices do not affect the dynamics of the goodwill stocks. To characterize the relationships between the retailer s pricing strategies and the goodwill stocks, we compute the following derivatives: pn (G s,g n ) = p s 2 (G s,g n ) = G n G s (ψ n + ψ s + 2)(ψ n + ψ s 2) > 0, pn (G s,g n ) = p s ψ n + ψ s (G s,g n ) = G s G n (ψ n + ψ s + 2)(ψ n + ψ s 2) > 0. Thus, increasing any goodwill s level increases both retail prices. An interpretation is that investing in brand equity for any brand enhances their differentiation and reduces the price competition between them, which leads to increase each brand market power. Note also that own goodwill has a higher impact on own price than competing

8 J Optim Theory Appl (2008) 137: brand goodwill, i.e., p i G i (G i,g j )> p i G j (G i,g j ), i, j = s,n, i j. With respect to the relationships between the retailer s and manufacturer s advertising strategies and the goodwill stocks, the following results can be derived from item (ii). If R 1 k n R 3 is positive (always if R 1 is chosen and if R 1 is chosen and condition (3.1) is fulfilled), the retailer s advertising strategy depends positively on the goodwill level of its own brand. This result has been obtained in some papers dealing with advertising competition in horizontal (oligopoly) models (see e.g. [11] and [23 26]). To interpret further the advertising strategies in (10) and (12), we rewrite them as follows: c s1 + c s2 A s (G s,g n ) = V R G s (G s,g n ) k n V R G n (G s,g n ), c n1 + c n2 A n (G s,g n ) = V M G n (G s,g n ) k s V M G s (G s,g n ). Thus, in equilibrium, each brand sets the advertising level such that the marginal cost (left-hand side) is equal to the marginal revenue. The latter is made of the difference between the marginal contributions of own and competitor s stock on value function. Further, from the above expressions we see that the firms advertising decisions interact through the goodwill stocks. Put differently, each advertising strategy cannot be written as function of the other one. In that sense, they are neither strategic substitutes nor complements, but independent. The retailer s reaction to a goodwill variation of the national brand depends on the sign of R 3 k n R 2. If it is positive, then G n has a positive impact on A s.inthe literature on advertising expenditures in dynamic oligopolies, this result is reached when the advertising is of the informative type (see e.g. [22] and [25]). If R 3 k n R 2 is negative, then the retailer s advertising is negatively related to her rival s goodwill stock. In the above references, this case is referred to as predatory advertising. A sufficient condition guaranteeing that the expressions in (6), (7), (10), (12) are retailer s and manufacturer s value functions and advertising strategies is given by lim t e ρt V R (G s (t), G n (t)) = 0, lim t e ρt V M (G s (t), G n (t)) = 0, (14) where (G s (t), G n (t)) is the solution of the closed-loop dynamics obtained after substitution of the advertising strategies (10) and (12) into the goodwill dynamics given by (2). This solution can be written as G n (t) = X n 1 eλ 1t + X n 2 eλ 2t + G ss n, (15) G s (t) = X s 1 eλ 1t + X s 2 eλ 2t + G ss s, (16)

9 472 J Optim Theory Appl (2008) 137: where X1 i,xi 2,i = n, s, are constants, Gss n and Gsss refer to the steady states of the goodwill variables and λ i,i = 1, 2, are the real eigenvalues of the matrix associated to the system of linear differential equations given in (39). The steady-state levels of the goodwill stocks, G ss n, Gss s are given by G ss n = (c n1 M 1 + k s M 2 )[(R 1 k n R 3 )(k n k s 1) + c s2 δ] c n2 k n (c s1 R 4 + k n R 5 )δ c n2 δ(r 1 + kn 2R, 2 2k n R 3 c s2 δ) G ss s = (c n1 M 1 + k s M 2 )[(k n R 2 R 3 )(k n k s 1) c s2 k s δ]+c n2 (c s1 R 4 + k n R 5 )δ c n2 δ(r 1 + kn 2R. 2 2k n R 3 c s2 δ) The values of the goodwill stock at the steady state, (G ss n,gss s ), are obtained after replacing the equilibrium strategies in (2), and solving for G n and G s when G n (t) = 0, G s (t) = 0. The computation of the state variable trajectories allows the identification of the paths for the pricing and advertising strategies, and for the evolution of the demand function for both brands. The results can be obtained after replacing G n (t) and G s (t) by their respective values from (15) and (16) into(8), (9), (10) and (12). The quadratic functional specification in (6) and (7) allows condition (14) tobe satisfied whenever the goodwill stocks are bounded. The next proposition states the conditions for G n,g s to be bounded. Proposition 3.2 The goodwill stocks are bounded if the conditions listed below apply: 1. R 1 + kn 2R 2 2k n R 3 c s2 δ is positive and the initial goodwill stocks are related according to the following expression: G s0 = 2δc s2k n (k n R 2 R 3 ) (G n0 G ss n k n (k n R 3 R 1 ) ) + Gss s, (17) 2. R 1 + k 2 n R 2 2k n R 3 c s2 δ is negative. Proof See the Appendix. Item 1 characterizes the situation where the steady state is stable in the saddle point sense. The saddle point property means that given the initial goodwill level G n0,we can find values of G s0 (that satisfy (17)) such that the closed-loop system converges to the steady state (G ss n,gss s ) as time approaches infinity. Item 2 guarantees a globally asymptotically stable equilibrium. In this case, any initial level of the goodwill stocks converges to the steady-state as time approaches infinity and leads to a bounded time path. It is well known that, in infinite-horizon linear-quadratic optimization models, the time paths of the states (goodwills) and of the controls are exponential functions of time that converge towards the steady state, either from above or below depending on whether the initial condition is higher or lower than the steady state. The initial conditions for the goodwill stocks, and the relationship between the initial values for the state and costate variables, that are required to have a trajectory that converges

10 J Optim Theory Appl (2008) 137: towards the steady state, give us the initial conditions for the firms control variables: advertising investments and prices. To gain some additional qualitative insight into the advertising strategies, we focus on two particular cases. Firstly, we consider that the cross-price effect on demands is zero, that is, ψ n = ψ s = 0(see(3)). Secondly, we consider a situation of partial myopia which means that each player takes into account only the dynamics of its own brand s goodwill when maximizing its profits. 3.1 Zero Cross-Price Effect on Demand When the demand function of each brand is independent of the price of its competitor, the coefficients of the manufacturer s and retailer s value functions particularize as follows: M 1 = w n d n 2(δ + ρ) > 0, M 2 = 0, R 2 = R 3(R 1 k n R 3 c s2 (2δ + ρ)) > 0, R 3 = k n(1 2R 1 (2δ + ρ)), k n (R 1 k n R 3 ) 2(2δ + ρ) R 1 = k2 n (1 + k2 n ) + c s2(2δ + ρ) 2 ± (2δ + ρ) c s2 (c s2 (2δ + ρ) 2 2(1 + kn 2)) > 0. 2(2δ + ρ)(1 + kn 2)2 The manufacturer s advertising is constant and independent of the advertising parameters k n, k s. Some easy algebra allows us to show that for both roots of R 1, expression R 1 k n R 3 is always positive, while expression R 3 k n R 2 is always negative. Since R 1 k n R 3 is always positive, the retailer s advertising strategy depends positively on the goodwill level of its own brand. Since R 3 k n R 2 is always negative, then the retailer s advertising is negatively related to its rival s goodwill stock. Using the terminology previously stated, the retailer s advertising is predatory. 3.2 Partially Myopic Players By partially myopic behavior we mean that each player knows the value of the goodwill stocks for both brands, but it takes into account only the time evolution of its own brand s goodwill and disregards that of its competitor. Such assumption has been madein,e.g.[11] and [22]. The justification of such assumption is that observing the time evolution of the other brand goodwill is costly. Since the price variables do not affect the dynamics of the brands goodwill stocks, the equilibrium pricing strategies remain unchanged under this hypothesis and are given by (8), (9) and (11). Therefore, we compute the advertising equilibrium strategies under the partially myopic behavior. Each channel member decides its advertising investment in order to maximize its profits subject to the dynamics of its own goodwill stock. The manufacturer s and retailer s advertising problems read: max A n π M = 0 e ρt {(w n d n )D n C n (A n )} dt,

11 474 J Optim Theory Appl (2008) 137: s.t. Ġ n = A n δg n k n A s, G n (0) = G n0 ; max π R = e ρt {(p n w n )D n + (p s w s )D s C s (A s )} dt, A s 0 s.t. Ġ s = A s δg s k s A n, G s (0) = G s0. The next proposition fully determines the advertising strategies as well as the equilibrium outcomes when the channel members behave partially myopic. Proposition 3.3 Let denote by V pm R (G s,g n ) and V pm M (G s,g n ) the retailer s and the manufacturer s value functions where the superscript pm stands for partial myopia. (i) The following expressions satisfy the Hamilton-Jacobi-Bellman equations associated with partially myopic retailer s and manufacturer s problems: V pm R (G n,g s ) = R 1 2 G2 s + R 2 2 G2 n + R 3 G n G s + R 4 G s + R 5 G n + R 6, V pm M (G n,g s ) = M 1 G n + M 2 G s + M 3. (ii) Assuming interior solutions, retailer s and manufacturer s advertising strategies at the equilibrium are given by A pm s (G s,g n ) = 1 ( R 1 G s + R 3 G n + R 4 c s1 ), c s2 A pm n (G s,g n ) = 1 ( M 1 c n1 ), c n2 where coefficients R i, M j, i = 1,...,6, j = 1, 2, 3 are given in the Appendix. Furthermore, R 1 0 and 1. If R 1 is chosen, then R 3 has the same sign as the following expression: δ(ρ + δ)c s2 ( 2 + ψ n + ψ s )(2 + ψ n + ψ s ) + 2. (18) 2. If R 1 is chosen, then R 3 0, where R 1, R 1 are given in the Appendix. Proof See the Appendix. Since R 1 is always positive, the retailer s advertising strategy depends positively on the goodwill level of its own brand. The retailer s reaction to a goodwill variation of the national brand depends on the sign of R 3. If it is positive (which is always true if R 1 is chosen and if R 1 is chosen and the expression in (18) is positive), then G n has a positive impact on A s. The advertising is informative. If R 3 is negative (which happens when the root R 1 is chosen and the expression in (18) is negative), then the retailer s advertising is negatively related to its rival s goodwill stock. In this case the advertising is predatory.

12 J Optim Theory Appl (2008) 137: Conclusions We considered a marketing channel where the manufacturer and the retailer invest in advertising in order to increase own brand equity and reduce competitor s one. The main findings are the following: First, investing in building up some equity for each brand reduces the price competition between them and enhances their differentiation which allows more market power for both. Hence, the advertising s option for the SB instead of other usual strategies (features, promotions, etc.) seems to be an interesting weapon to consider. Second, when there is no cross-effect or the players are partially myopic, the higher the brand equity of the SB, the more the retailer invests in advertising. However, the result is ambiguous in the general case. Third, assuming no coordination between both channel s members, the double marginalization problem could not be removed (whatever is the case) and the retailer will pass to consumer an increase in its purchasing cost of the NB. Finally, if we consider that the price of each brand does not have any impact on the demand of the competing brand, then our findings suggest that the retailer will increase its advertising whenever the goodwill of its own brand increases and will decrease it if the goodwill of the NB increases. We made the assumption, as it is generally the case in the literature, that the retail prices do not affect the evolution of the goodwill stocks. It is also of interest to attempt integrating these prices in the dynamics and analyze the interplay between pricing and advertising. This will come, however, at the cost of being able of characterizing the equilibrium only numerically. Finally, we have assumed that the two players are symmetric (information-wise) and implement a Nash equilibrium. Given a certain tradition in marketing channels in assuming a leader-follower information structure, it is relevant to study the resulting Stackelberg equilibrium in this case where the two players invest in advertising. Appendix Proof of Proposition 3.1 The Hamilton-Jacobi-Bellman (HJB) equation associated with the retailer s optimization problem is as follows: { ρv R (G s,g n ) = max (p n w n )[G n p n + ψ n p s ] p n,p s,a s + (p s w s )[G s p s + ψ s p n ] ( c s1 + c ) s2 2 A s A s + V R (G s,g n )(A s δg s k s A n ) G s + V } R (G s,g n )(A n δg n k n A s ). (19) G n

13 476 J Optim Theory Appl (2008) 137: From the necessary conditions for optimality taking the partial derivatives of the RHSin(19) with respect to p n,p s and A s and equating to zero, we obtain G n 2p n + (ψ n + ψ s )p s + w n w s ψ s = 0, (20) G s 2p s + (ψ n + ψ s )p n w n ψ n + w s = 0, (21) c s1 c s2 A s + V R G s (G s,g n ) k n V R G n (G s,g n ) = 0. (22) The HJB equation associated with the manufacturer s optimization problem is as follows: { ( ρv M (G s,g n ) = max (w n d n )[G n p n + ψ n p s ] c n1 + c ) n2 2 A n A n d n w n w n,a n + V M G s (G s,g n )(A s δg s k s A n ) + V M G n (G s,g n )(A n δg n k n A s ) }. (23) Note that the manufacturer s optimization problem is linear in w n. Therefore, since we assume G n p n + ψ n p s > 0 in order to have a positive demand for the national brand, the optimal solution for the transfer price is its upper bound, w n. From the necessary condition for optimality, taking the partial derivative of the RHS of (23) with respect to A n and equating to zero, we obtain c n1 c n2 A n + V M V M (G s,g n ) k s (G s,g n ) = 0. (24) G n G s Solving (20) and (21), conditional on the value of w n which is set to w n,the retailer s optimal pricing strategies are obtained. Moreover, from (22) and (24), the manufacturer s and retailer s optimal advertising strategies are derived as functions of the partial derivatives of the value functions, A s (G s,g n ) = 1 ( ) VR V R (G s,g n ) k n (G s,g n ) c s1, (25) c s2 G s G n ( ) VM R A n (G s,g n ) = 1 c n2 G n (G s,g n ) k s V M G s (G s,g n ) c n1. (26) Inserting the retailer s optimal strategies, (8), (9), (25), and the manufacturer s optimal strategies, (11) and (26) in(19) and (23), the coefficients of the value functions R i, M j, i = 1,...,6, j = 1, 2, 3, are determined by identification and solve the following nine algebraic Riccati equations (the first six correspond to the retailer, whereas the second three correspond to the manufacturer): a 1 (R 1 k n R 3 ) 2 + c s2 (2 + (2δ + ρ)a 1 R 1 ) = 0, (27) a 1 (R 3 k n R 2 ) 2 + c s2 (2 + (2δ + ρ)a 1 R 2 ) = 0, (28)

14 J Optim Theory Appl (2008) 137: k n a 1 (R 1 R 2 + R3 2 ) a 1R 3 (R 1 + kn 2 R 2 (2δ + ρ)c s2 ) + a 6 c s2 = 0, (29) a 1 c s2 [c n2 (δ + ρ)r 5 + (c n1 M 1 + k s M 2 )(R 2 k s R 3 )] + c n2 [a 1 (c s1 + k n R 5 R 4 )(R 3 k n R 2 ) + c s2 (a 2 w s + a 3 w n )]=0, (30) c n2 [a 1 (R 1 k n R 3 )(c s1 + k n R 5 R 4 ) + c s2 (a 7 w s a 2 w n )] a 1 c s2 [(c n1 + k s M 2 M 1 )(k s R 1 R 3 ) c n2 (δ + ρ)r 4 ]=0, (31) 2a1 2 c s2[(c n1 M 1 + k s M 2 )(k s R 4 R 5 ) c n2 ρr 6 ] c n2 {a1 2 (c s1 + k n R 5 R 4 ) 2 2c s2 [a 1 a 4 w n 2 + w2 s (a 1a 5 2ψ s (ψn 3 2ψ3 s ψ n(4 + 3ψs 2 ))) w s w n a 1 a 6 a 4 ]} = 0, (32) a 1 (k n M 1 M 2 )(k n R 2 R 3 ) + c s2 (a 3 (d n w n ) + a 1 (δ + ρ)m 1 ) = 0, (33) a 1 (k n M 1 M 2 )(R 1 k n R 3 ) + c s2 (a 2 (d n w n ) a 1 (δ + ρ)m 2 ) = 0, (34) a 1 {c s2 (c n1 M 1 + k s M 2 ) 2 + 2c n2 [(k n M 1 M 2 )(k n R 5 R 4 + c s1 ) c s2 ρm 3 ]} + 2c n2 c s2 (2 w n a 4 w s a 6 a 5 )( w n d n ) = 0, (35) where a 1 = (ψ n + ψ s + 2)(ψ n + ψ s 2), a 2 = ψ n ψ s, a 3 = 2 + ψ s (ψ n + ψ s ), a 4 = 1 ψ n ψ s, a 5 = 1 + ψ n ψ s, a 6 = ψ n + ψ s, a 7 = 2 + ψ n (ψ n + ψ s ). From (49), we can obtain R 6 as function of the coefficients R 4,R 5,M 1, M 2 : R 6 = where 1 2a1 2c n2c s2 ρ {a2 1 (2c s2 1 (M 1,M 2 )(k s R 4 R 5 ) + c n2 (c s1 R 4 + k n R 5 ) 2 ) c n2 c s2 [2a 1 (a 5 ws 2 + a 4 w n ( w n a 6 w s )) 4ws 2 ψ s(ψn 3 2ψ3 s ψ n (4 + 3ψ 2 s ))]}, 1 (M 1,M 2 ) = c n1 M 1 + k s M 2. From (47), we can derive R 5 as function of the coefficients R 1,R 2,R 3,R 4, M 1, M 2, R 5 = c n2c s2 (a 2 w s + a 3 w n ) + a 1 [c s2 (R 2 k s R 3 ) 1 (M 1,M 2 ) c n2 (k n R 2 R 3 )(c s1 R 4 )] a 1 c n2 (kn 2R. 2 k n R 3 c s2 (δ + ρ)) Replacing this expression in (48), we obtain R 4 as a function of R 1, R 2, R 3, M 1, M 2, R 4 = 1 a 1 c n2 2 (R 1,R 2,R 3 ) {a 1 1 (M 1,M 2 )(k n (k n k s 1)(R 1 R 2 R 2 3 )

15 478 J Optim Theory Appl (2008) 137: where c s2 (k s R 1 R 3 )(δ + ρ)) (R 1 k n R 3 )(c n2 k n (a 2 w s + a 3 w n ) a 1 c n2 c s1 (δ + ρ)) + c n2 (a 2 w n a 7 w s )(k n (k n R 2 R 3 ) c s2 (δ + ρ))}, 2 (R 1,R 2,R 3 ) = (δ + ρ)(r 1 + k n (k n R 2 2R 3 ) c s2 (δ + ρ)). From (52), we can obtain M 3 as function of the coefficients R 4, R 5, M 1, M 2, M 3 = 1 2a 1 c n2 c s2 ρ {a 1[c s2 2 1 (M 1,M 2 ) + 2c n2 (k n M 1 M 2 )(c s1 R 4 + k n R 5 )] + 2c n2 c s2 (d n w n )(a 5 a 6 w s 2a 4 w n )}. Solving simultaneously (50) and (51), we derive M 1 and M 2 as functions of the coefficients R 1,R 2, R 3, M 1 = ( w n d n ) a 2(R 3 k n R 2 ) + a 3 (R 1 k n R 3 c s2 (δ + ρ)), a 1 2 (R 1,R 2,R 3 ) M 2 = ( w n d n ) a 3k n (R 1 k n R 3 ) + a 2 (k n (R 3 k n R 2 ) + c s2 (δ + ρ)). a 1 2 (R 1,R 2,R 3 ) From (46), we can obtain R 2 as a function of R 1 and R 3, R 2 = a 1R 3 (R 1 k n R 3 c s2 (2δ + ρ)) c s2 a 6, (36) a 1 k n (R 1 k n R 3 ) where k n is assumed to be different from zero. 3 Replacing the expression of R 2 in (36) into(45), we can derive R 3 as a function of R 1, R 3 = k n + (2δ + ρ)(ψ n + ψ s 2k n )R 1. (37) (2δ + ρ)(2 k n (ψ n + ψ s )) The factor 2 k n (ψ n + ψ s ) is positive, because 0 k n < 1, 0 <ψ i < 1, i {n, s}. Therefore, the sign of R 3 coincides with the sign of its numerator and condition (55) is derived. Finally, replacing R 3 given in (54) into(44) we obtain a quadratic equation for coefficient R 1 that allows us to obtain R 1 as function of the model parameters, R 1 = a 1[4k 2 n (1 a 6k n + k 2 n ) + c s2[(2δ + ρ)(a 6 k n 2)] 2 ]± 8a 1 (2δ + ρ)(1 a 6 k n + k 2 n )2, (38) where = a 1 [a 1 (4k 2 n (1 a 6k n + k 2 n ) + c s2[(2δ + ρ)(a 6 k n 2)] 2 ) 2 3 If k n = 0, from (46), we can obtain R 1 as a function of R 2 and R 3. Replacing this expression of R 1 into (44), we can derive R 3 as a function of R 2. Finally, replacing this expression of R 3 into (45) we obtain a quadratic equation for coefficient R 2 that allows us to obtain R 2 as function of the model parameters.

16 J Optim Theory Appl (2008) 137: (1 a 6 k n + k 2 n )2 (2c s2 [(2δ + ρ)(a 6 k n 2)] 2 a 1 k 4 n )]. is assumed to be positive in order to have real solutions of (44). Let R 1 be the root with the positive sign affecting the square root, and R 1 the root with the negative sign. Taking into account that a 1 < 0, (44) and (45) can only be satisfied if R 1 and R 2 are positive. From the quadratic equation characterizing coefficient R 1 is easy to conclude that R 1 is positive if and only the following inequality applies: 4(1 + k 2 n k na 6 ) + c s2 [(2δ + ρ)(a 6 2k n )] 2 > 0. In order to determine the dependency of the retailer s advertising investment on its own goodwill stock and on the national brand goodwill stock we have to characterize the sign of the following expressions: R 1 k n R 3 and R 3 k n R 2. After some algebra, these expressions read as follows: R 1 k n R 3 = k2 n 2(2δ + ρ)(1 + k2 n k n(ψ n + ψ s ))R 1, (2δ + ρ)(k n (ψ n + ψ s ) 2) R 3 k n R 2 = c s2(a 1 (2δ + ρ)r 3 + ψ n + ψ s ) a 1 (R 1 k n R 3 ) = c s2(2k n ψ n ψ s )(2 + a 1 (2δ + ρ)r 1 ). a 1 (R 1 k n R 3 )(k n (ψ n + ψ s ) 2) Easy but tedious computations allow us to establish the different possibilities on the sign of R 1 k n R 3 and R 3 k n R 2 as shown in the statement of the proposition. Proof of Proposition 3.2 The eigenvalues λ 1 and λ 2 of the matrix associated to the closed-loop dynamics are λ 1 = δ, λ 2 = R 1 + k 2 n R 2 2k n R 3 c s2 δ c s2. (39) Therefore, if R 1 +kn 2R 2 2k n R 3 c s2 δ is positive, then one eigenvalue is negative, while the other one is positive. Under this assumption, the steady state (G ss n,gss s ) is a saddle point. The initial conditions on the goodwill lying on the stable subspace associated to the negative eigenvalue λ 1, given by (17), allow the system to converge to the steady state as time approaches infinity. If R 1 + kn 2R 2 2k n R 3 c s2 δ is negative, then both eigenvalues λ 1 and λ 2 are negative. Under this assumption, the steady state (G ss n,gss s ) is globally asymptotically stable. Proof of Proposition 3.3 The Hamilton-Jacobi-Bellman (HJB) equation associated with the retailer s optimization problem when it takes into account the dynamics of its own goodwill stock but not that of its competitor, is as follows: { ρv pm R (G s,g n ) = max (p n w)[g n p n + ψ n p s ]+(p s w s )[G s p s + ψ s p n ] A s

17 480 J Optim Theory Appl (2008) 137: ( c s1 + c pm } s2 2 A VR s )A s + (A s δg s k s A n ). (40) G s The HJB equation associated with the partial myopic manufacturer s optimization problem is as follows: { ( ρv pm M (G n,g s ) = max (w n d n )[G n p n + ψ n p s ] c n1 + c ) n2 A n 2 A n A n + V pm M G n (G n,g s )(A n δg n k n A s ) }. (41) From the optimality conditions, the retailer s and the manufacturer s optimal strategies are given by A pm s (G s,g n ) = 1 c s2 ( pm V R G s ) (G s,g n ) c s1, (42) A pm n (G s,g n ) = 1 ( pm ) V M (G s,g n ) c n1, (43) c n2 G n if there are positive expressions, and zero otherwise. Inserting the retailer s optimal strategies, (8), (9), (42), and the manufacturer s optimal strategies, (11) and (43) in(40) and (41), the coefficients of the value functions R i, M j, i = 1,...,6, j = 1, 2, 3, are determined by identification and solve the following nine algebraic Riccati equations (the first six correspond to the retailer, whereas the second three correspond to the manufacturer): a 1 R 1 ( R 1 + 2c s2 (2δ + ρ)) + 2c s2 = 0, (44) a 1 ( R ρc s2 R 2 ) + 2c s2 = 0, (45) a 1 R 3 ( R 1 + (δ + ρ)c s2 ) + (ψ n + ψ s )c s2 = 0, (46) a 1 [ R 3 (c n2 (c s1 R 4 ) c s2 k s (c n1 M 1 )) + c n2 c s2 ρ R 5 ] + c n2 c s2 (w s a 2 + w n a 3 ) = 0, (47) a 1 [ R 1 (c n2 (c s1 R 4 ) c s2 k s (c n1 M 1 )) + c n2 c s2 (δ + ρ) R 4 ] + c n2 c s2 (w s a 7 w n a 2 ) = 0, (48) a 2 1 ( 2c s2k s (c n1 M 1 ) R 4 c n2 (c s1 R 4 ) 2 + 2c n2 c s2 ρ R 6 ) + 2c n2 c s2 a 1 a 4 w 2 n 2c n2 c s2 w s [w s ( a 1 a 5 + 2ψ s (ψn 3 2ψ3 s ψ n(4 + 3ψs 2 ))) + w n a 4 a 6 a 1 ]=0, (49) a 1 M 1 (k n R 3 + c s2 (δ + ρ)) + c s2 a 3 (d n w n ) = 0, (50) a 1 (k n M 1 R 1 + c s2 M 2 ρ) + c s2 a 2 ( w n d n ) = 0, (51) a 1 [ c s2 (c n1 M 1 ) 2 + 2c n2 M 1 (k n R 4 c s1 ) + 2c n2 c s2 ρ M 3 ]

18 J Optim Theory Appl (2008) 137: c n2 c s2 ( w n d n )( 2 w n a 4 + w s a 5 a 6 ) = 0. (52) From (44), we can obtain R 1 as function of the model parameters, R 1 = (2δ + ρ)a 1c s2 ± ((2δ + ρ)a 1 c s2 ) 2 + 8c s2 a 1, (53) 2a 1 where the following condition is assumed to be satisfied in order to have real solutions of (44): c s2 a 1 (2δ + ρ) < 0. Note that a 1 is negative and therefore, both roots in (53) are positive. Let R 1 be the root with the positive sign affecting the square root, and R 1 the root with the negative sign. From (46), we can derive R 3 as a function of R 1, R 3 = c s2 (ψ n + ψ s ) ( R 1 (δ + ρ)c s2 )a 1. (54) Thesignof R 3 is the opposite of the sign of the expression R 1 (δ + ρ)c s2. (55) It can be easily shown that R 3 is positive if R 1 is chosen. If R 1 is selected, the expression (18) is negative (and therefore, R 3 is positive) if and only if δ(ρ + δ)c s2 a > 0. From (45), we can obtain R 2 as a function of R 3, R 2 = a 1 R 3 2 2c s2. a 1 ρc s2 R 2 takes always a positive value. Equation (50) allows us to write M 1 as a function of R 3, a 3 c s2 ( w n d n ) M 1 = a 1 (k n R 3 + c s2 (δ + ρ)). Thesignof M 1 coincides with the sign of k n R 3 + c s2 (δ + ρ). Therefore, if R 3 is positive, then M 1 is also positive. From (48) and (51) we can express the coefficients R 4 and M 2 in terms of the coefficients M 1,the R 1, R 4 = c s2k s (c n1 M 1 ) R 1 a 1 + c n2 [c s1 R 1 a 1 + c s2 (w s a 7 a 2 w n )], c n2 a 1 ( R 1 c s2 (δ + ρ)) M 2 = c s2( w n d n )a 2 + a 1 k n M 1 R 1 c s2 ρa 1.

19 482 J Optim Theory Appl (2008) 137: From (47) we get R 5 as a function of the coefficients R 3, M 1, R 4, R 5 = a 1 R 3 [c s2 k s (c n1 M 1 ) c n2 (c s1 R 4 )] c n2 c s2 (w s a 2 + w n a 3 ). c n2 c s2 a 1 ρ Finally, from (49) and (52) we obtain the expressions of coefficients R 6 and M 3 in terms of coefficients M 1 and R 4, R 6 = M 3 = 1 {a1 2 2c n2 c s2 ρa [c n2(c s1 R 4 ) 2 + 2c s2 k s (c n1 M 1 ) R 4 ] 1 + 2c n2 c s2 [ w n a 4 a 1 ( w n + a 6 w s ) + ws 2 ( a 5 + 2ψ s (ψn 3 2ψ3 s ψ n (4 + 3ψ 2 s )))]}, 1 2c n2 c s2 ρa 1 {2c n2 [k n M 1 a 1 (c s1 R 4 ) + c s2 (d n w n )(w s a 5 a 6 2 w n a 4 )] + c s2 a 1 (c n1 M 1 ) 2 }. Acknowledgements We wish to thank the six anonymous Reviewers for their constructive comments. The second author s research is partially supported by MEC and JCYL under projects SEJ /ECON and VA045A06, co-financed by FEDER funds. The third author s research is supported by NSERC, Canada. References 1. Sethuraman, R.: The effect of marketplace factors on private label penetration in grocery products. Working Papers, Marketing Science Institute, Report No , 1 29 (1992) 2. Hoch, S.J., Banerji, S.: When do private labels succeed?. Sloan Manag. Rev. 34, (1993) 3. Wilensky, D.: Private label success no secret as discount. Discount Store News 33(5), (1994) 4. Anderson, E., Coughlan, A.T., El-Ansary, A., Stern, L.W.: Marketing Channels, 6th edn. Prentice- Hall, New Jersey (2001) 5. Steiner, R.L.: The nature and benefits of national brand and private label competition. Working paper presented at Annual Meeting of the American Economic Association, Atlanta (2002) 6. Jørgensen, S., Sigué, S.P., Zaccour, G.: Stackelberg leadership in a marketing channel. Int. Game Theory Rev. 3, 1 14 (2001) 7. Jørgensen, S., Taboubi, S., Zaccour, G.: Cooperative advertising in a marketing channel. J. Optim. Theory Appl. 110, (2001) 8. Jørgensen, S., Taboubi, S., Zaccour, G.: Retail promotions with negative brand image effects: is cooperation possible?. Eur. J. Oper. Res. 150(2), (2003) 9. Jørgensen, S., Taboubi, S., Zaccour, G.: Incentives for retailer promotion in a marketing channel. In: Haurie, A., Muto, S., Petrosjan, L.A., Raghavan, T.E.S. (eds.) Advances in Dynamic Games. Annals of the International Society of Dynamic Games, vol. 8, pp Birkhäuser, Boston (2006) 10. Jørgensen, S., Zaccour, G.: A differential game of retailer promotions. Automatica 39(7), (2003) 11. Martín-Herrán, G., Taboubi, S.: Shelf-space allocation and advertising decisions in the marketing channel: a differential game approach. Int. Game Theory Rev. 7(3), (2005) 12. Martín-Herrán, G., Taboubi, S., Zaccour, G.: A time-consistent open-loop Stackelberg equilibrium of shelf-space allocation. Automatica 41(6), (2005) 13. Raju, J.S., Sethuraman, R., Dhar, S.K.: The introduction and performance of store brands. Manag. Sci. 41(6), (1995) 14. Kotler, P.: Marketing Management: Analysis, Planning, Implementation and Control, 9th edn. Prentice-Hall, Upper Saddle River (1997). pp. 789

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