A goodwill model with predatory advertising Luca Grosseta , Paolo Robertib , Bruno Viscolania,∗ a

Dip. di Matematica Pura e Applicata, University of Padua, Via Trieste 63, I-35121 Padua, Italy b Dip. di Scienze Economiche “M. Fanno”, University of Padua, Via del Santo 22, I-35123 Padua, Italy

Abstract We investigate the dynamic advertising policies of two competing firms in a duopolistic industry, assuming a predatory phenomenon between their advertising campaigns. The resulting model is a differential game which is not linear-quadratic. We show that there exists a Markovian Nash equilibrium, and that it leads to time constant advertising strategies. According to this model, predatory advertising produces a negative externality: the interference between the advertising campaigns decreases the total demand of the market. Keywords: Advertising; Goodwill; Differential games. 1. Introduction We consider a duopolistic industry where two firms are involved in a brand competition using advertising policies with a predatory effect on the opponent’s brand. Predatory advertising is a common effect of advertising in an oligopoly. The advertising of one brand can be deliberately planned against the opponent’s brand, as in a comparative advertising campaign, or it can be an unavoidable effect because of a market structure with few manufacturers. In this case the customers of one firm may change their preferences when they are hit by the advertising of the opponent brand and, as a consequence, the brand goodwill is reduced. ∗

Corresponding author, phone +39 049 827 1397, fax +39 049 827 1479 Email addresses: [email protected] (Luca Grosset), [email protected] (Paolo Roberti), [email protected] (Bruno Viscolani) Preprint submitted to Operations Research Letters

June 13, 2011

We assume that each brand demand is proportional to the same brand goodwill. The goodwill can be identified with the reputation or the brand equity of the product. The goodwill evolution depends positively on the advertising effort of the player. We innovate Nerlove-Arrow dynamics [14] through a nonlinear term which describes the predatory phenomenon. The idea that one player’s advertising effort may hamper the competitor’s sales rate is intuitively appealing, especially when the firms engage in comparative advertising, see e.g. [3], [6]. Moreover, in [7] we can see, in an empirical study, how important the interference phenomenon in advertising is. To the best of our knowledge, the paper [15] extends the Nerlove-Arrow model for the first time introducing an interference phenomenon. In that work, the goodwill evolution of the brand goodwill of the first player is increased by his advertising, but it is decreased by the advertising flow of his competitor. Afterward, this idea is developed in some papers by Amrouche, Mart´ın-Herr´an, and Zaccour [1, 2], where the authors assume that the marketing channel is composed of a manufacturer and a retailer. The retailer sells both a national and a store brand, and the advertising effort for one brand can negatively interfere with the goodwill evolution of the other brand. Unfortunately, under the dynamics proposed by these authors, an important technical feature of the original Nerlove-Arrow model is lost: the goodwill value is no longer necessarily positive. We notice that the same technical problem does not arise in the models where the advertising strategies have a positive spillover effect (see e.g. [4, 5]) because, under these hypotheses, the brand goodwill is always positive for all the advertising effort values chosen by the players. In this paper we propose a model in which the brand goodwill is always positive. The demand function, which is linear with respect to the brand goodwill of the product, is then positive for each possible advertising strategy pair chosen by the competing firms. Moreover, we think that the nonlinear extension of the Nerlove-Arrow dynamics proposed in this paper describes better a comparative advertising campaign. From a technical point of view, the goodwill dynamics used here is a limit case of the model of Leitmann and Schmitendorf [13] (for a deeper analysis of this model we suggest referring to the recent paper [11]). With this work we would like to answer some questions: after modifying a linear quadratic differential game, by changing the state equations of the goodwill variables, is it still possible to find explicit feedback equilibrium strategies? Is the equilibrium advertising strategy of a player affected by the 2

advertising strategy of the competitor? Which are the features of the possible dependence? Are the equilibrium policies in case of predatory advertising more or less expensive than the equilibrium strategies with no interaction? After studying the model and presenting our results, we compare the analysis with the scientific literature on the same topic, showing the similarities and the differences brought by the nonlinear term in the goodwill evolution. The paper is organized as follows: in Section 2 we introduce our model, in Section 3 we characterize the Markovian Nash equilibrium of the differential game, in Section 4 we study the problem under some special assumptions (quadratic advertising costs) which makes the computation simpler and the economic interpretation of the results clearer. 2. Model Let Gi (t) represent the stock of goodwill at time t of the product sold by player i (from now on i, j ∈ {1, 2}, and i 6= j). As justified in the previous section, we refer to the definition of goodwill given by NerloveArrow [14] to describe the variable that summarizes the effects of present and past advertising on demand; in the original model the state variable goodwill needs an advertising effort to increase, but is subject to spontaneous decay. In our model the goodwill value Gi (t) is the joint result of the advertising processes of the two players and we assume that it satisfies the following motion equation G˙ i (t) = γi ui (t) − δGi (t) − εj uj (t) Gi (t) , Gi (0) = αi > 0 ,

(1) (2)

where: • ui (t) ≥ 0 represents the advertising effort controlled by player i; • γi > 0 describes the effectiveness of the advertising effort of player i; • δ > 0 is the goodwill depreciation rate; • uj (t) ≥ 0 represents the advertising effort controlled by player j; • εj ≥ 0 describes the interference of player j’s advertising effort towards player i’s goodwill evolution. 3

The term −εj uj (t) Gi (t) in (1) is the extension of the original model we are proposing in this paper to describe the interference phenomenon. We notice that: • from a technical point of view this term assures that, for all feasible advertising campaigns of the two players, both state variables Gi (t), Gj (t) remain positive; • this term explains the comparative effect of an advertising campaign: the better known the brand of player i, the more effective a comparative advertising effort chosen by player j; • putting together the two decay terms we obtain − (δ + εj uj (t)) Gi (t), hence we are assuming a variable decay factor for the goodwill. In this model we are augmenting the constant decay factor δ (which is present in the Nerlove-Arrow dynamics), by adding to it a variable term εj uj (t), which depends on the advertising flow of the competitor. The assumption that player i is more vulnerable to negative/comparative advertising by the competitor when his goodwill is higher is in line with [13], where the authors argue that: “The adverse effect on sales of a rival’s advertising is proportional to the volume of sales.” Finally, as in the previous literature of this field [12], we assume that each player wants to maximize its own discounted profit Z ∞ e−ρt [βi Gi (t) − ci (ui (t))] dt , (3) 0

where: • ρ > 0 is the discount factor; • βi Gi (t) is the revenue rate of player i and βi > 0 is the marginal revenue of goodwill; • ci (·) is the advertising cost function of player i, a twice continuously differentiable function, increasing, strictly convex, and with ci (0) = 0, c0i (0) = 0, limu7→+∞ c0i (u) = +∞. We remark that the revenue rate βi Gi (t) is linear in the brand goodwill of the product. This form captures the idea that customers demand a product that has a good reputation. We focus on goodwill interactions in the duopoly, whereas we assume that prices do not affect the demand, as in [18]. 4

3. Markovian Nash equilibrium We want to study the noncooperative differential game (2), (1), (3). N Theorem 1. If we denote by (uN 1 , u2 ) a solution of the following system of equations: γi βi , (4) c0i (ui ) = ρ + δ + εj u j N then the couple (uN 1 , u2 ) of constant and positive advertising strategies is a Markovian Nash equilibrium. Using these strategies, the state of the system N ∗ ∗ (GN 1 (t), G2 (t)) converges to the pair (G1 , G2 ), where

G∗i =

γi uN i . δ + εj γj uN j

(5)

Proof of Theorem 1. First of all, we show that the system of equations (4) has a solution. Let ϕi be the inverse of the continuous and strictly increasing function c0i , then ϕi is continuous and strictly increasing too; moreover ϕi (0) = 0. Equation (4) can be rewritten as ui = ϕi (γi βi / (ρ + δ + εj uj )). Now, using the same equation (4) with interchanged indices and substituting the form just obtained for ui , we obtain   −1 γi βi 0 cj (uj ) = γj βj ρ + δ + εi ϕi . (6) ρ + δ + εj u j Both sides of the equation (6) represent continuous and strictly increasing functions in uj ; the left hand side vanishes at uj = 0, and goes to +∞ as uj 7→ +∞, whereas the right hand side has a strictly positive value at uj = 0, and has a finite limit, γj βj /(ρ + δ), as uj 7→ +∞. Hence, there exists a uN j that solves (6). Then, we study the differential game. Let V i (Gi , Gt ) be the player i’s value function, then the Hamilton Jacobi Bellman PDEs associated with the differential game (see [8, Ch.4, Th.4.1, p.92-93]) are: ρV i = maxui {βi Gi − ci (ui ) + VGi i [γi ui − (δ + εj uj ) Gi ] + VGi j [γj uj − (δ + εi ui ) Gj ]

o

,

(7) where we have written V , instead of V (Gi , Gt ), for the sake of simplicity. We guess that the value function is linear in Gi and constant in Gj (the latter i

i

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guess is justified by the fact that Gj is not present in the motion equation, nor in the objective function of player i’s problem): V i (Gi , Gj ) = Ai + B i Gi .

(8)

Then equation (7) becomes ρ(Ai + B i Gi ) = max{βi Gi − ci (ui ) + B i [γi ui − (δ + εj uj ) Gi ]} .

(9)

ui

The control u∗i , which solves the maximum problem (9), is characterized by the equation (10) c0i (u∗i ) = B i γi . Now we check that the guess (8) about the value function is right, by requiring that
ρB i = βi − δ + εj u∗j B i . (11) Using (10), we obtain c0i (u∗i ) =

γi βi , ρ + δ + εj u∗j

(12)

which is the requirement that u∗i satisfies equation (4). After substituting the N unique solution (uN 1 , u2 ) into the state equations (1), we obtain that, for some θi ∈ R, the optimal state variables are N )t

−(δ+εj uj ∗ GN i (t) = Gi + θi e

,

(13)

which, as t 7→ +∞, approach the limit values 5). Finally, we notice that N )T

−ρT lim e−ρT V i (GN (Ai + B i G∗i + B i θi e−(δ+εj uj i (T )) = lim e

T 7→+∞

T 7→+∞

) = 0 (14)

and this ends the proof, because it implies that the infinite horizon transversality conditions hold. 4. Quadratic advertising costs Here we assume that the players’ advertising cost functions are quadratic, in order to obtain an explicit formulation of the advertising strategies: ci (ui ) = κi u2i /2 ,

κi > 0 .

(15)

Using this hypothesis, equation (6) has a unique solution. From an economical point of view it is interesting to distinguish three cases. 6

4.1. No advertising interference The situation of two players who advertise their products without using comparative advertising is the degenerate case of the system under study and is well known (see e.g. [18, Section 3]). If the players do not interfere with each other, i.e. ε1 = ε2 = 0, then there exists a unique equilibrium pair N (uN 1 (t), u2 (t)) of advertising strategies; its components are constant N uN i (t) ≡ ui =

βi γi · , κi (ρ + δ)

(16)

N ∗ ∗ and the associated limit goodwill pair (GN 1 (t), G2 (t)) converges to (G1 , G2 ), where βi γ2 . (17) G∗i = i · κi δ(δ + ρ)

We notice that ui (t) ≡ uN i is the unique dominant strategy of player i. The equilibrium features of this special situation are a natural benchmark for the analysis of more interesting situations where advertising interference is observed. 4.2. Unilateral advertising interference In the special game with unilateral interference from player 1 (i.e. the case where ε1 > 0 and ε2 = 0), we obtain that the unique Markovian Nash N equilibrium (uN 1 , u2 ) has the following analytical form: uN 1 =

γ1 β1 · , κ1 (δ + ρ)

uN 2 =

γ2 β2 · . β1 γ1 κ2 (ρ + δ) + ε1 (ρ+δ)κ 1

(18)

The equilibrium advertising efforts and equilibrium goodwill values for the aggressive player 1 are equal to the values (16) and (17); on the other hand those for the disturbed player 2 are different. In particular, the player 2’s goodwill function GN 2 (t) converges to G∗2 =

γ22 β2  · . β1 γ1 κ2 δ (ρ + δ) + ε1 δ (ρ+δ)κ1 (1 + ε1 G∗1 )

(19)

∗ We notice that the values uN 2 and G2 are both strictly decreasing in ε1 . We can think of this scenario as the situation of a monopolistic market where one incumbent, player 2, has a well established brand and knows that an opposing

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firm, player 1, will enter the market. While the incumbent has a goodwill to defend, the entrant does not have any own brand goodwill, he can then start advertising and predate the incumbent’s brand goodwill. An extensive literature analyzes the optimal strategies for an incumbent and an entrant who compete through advertising. In [16, 17] Schmanlensee suggests that one of the feasible strategies for the incumbent, knowing that the entrant will try to get into the market, is to advertise less than in a monopolistic situation. This is in line with our results: player 2, the incumbent, suffers from the negative advertising of the entrant, and gets both his advertising effort and his goodwill level reduced, compared to the first scenario (compare equation (18) with (16)). This behavior is qualitatively well known and is called the puppy-dog effect [9]. 4.3. Bilateral advertising interference If εi > 0 and εj > 0, then after defining ai = ε i

γi , ki

bi =

βi (aj − ai ) + δ + ρ , δ+ρ

we have uN i

γi bi γi =− + 2ki ai ki

s

bi 2ai

2 +

βi . ai

(20)

(21)

This is the equilibrium characterization in the most general setting and it is interesting to notice that, for each player, the equilibrium advertising flow obtained in this scenario is always less than the equilibrium advertising effort obtained in the first scenario. Here, both the players exhibit the puppy-dog effect (with a different intensity which depends on the magnitude of the interference parameters εi and εj ). 5. Conclusion In this paper we study a goodwill model with a predatory interference between the advertising campaign of a firm and his rival’s brand image. We modify the classical Nerlove-Arrow dynamics introducing an innovative variable decay term, which well models a comparative advertising campaign. Under our model, the goodwill positivity is guaranteed by the system dynamics. Although the studied differential game is not linear quadratic, we are still able to find a Markovian Nash equilibrium. 8

Under the hypotheses of this paper the interference phenomenon between advertising campaigns produces a negative externality. This can be better understood if we focus on the model with quadratic advertising costs and unilateral interference. If player 2 suffers from the predatory advertising of player 1, then the optimal equilibrium of player 2 consists in reducing his advertising effort in order to decrease the negative (comparative) effect of the advertising effort of his opponent. The paper adds to the literature on differential goodwill models. First, it provides an example of a nonlinear quadratic differential game with an explicit Markovian Nash equilibrium. Second, it proposes a solution to the issue of possible negative values of the brand goodwill. Moreover, it can help understand the dynamics between rival brands and advertising policies, when there are interactions in oligopolies. The assumptions of infinite planning horizon and time-independent parameters are strong and precisely these hypotheses lead to value functions that have constant parameters and do not depend on time. The infinite time horizon is an idealization, however there are many economic models formulated over an infinite time horizon because this formulation allows the description of long-lived investment and simplifies the mathematical analysis of the problem [10, p. 155]. Possible extensions of this research regard the study of the finite time horizon version of the same game, in order to understand whether equilibrium strategies with similar characteristics can be found. 6. Acknowledgements We thank Engelbert Dockner, Georges Zaccour and the audience at the ISDG 2010 (Banff) and the SWDGA 2010 (Rimini) for the helpful comments and discussions. The usual disclaimer applies. References [1] N. Amrouche, G. Mart´ın-Herr´an, G. Zaccour, Feedback Stackelberg equilibrium strategies when the private label competes with the national brand, Annals of Operations Research, 164 (2008) 79-95. [2] N. Amrouche, G. Mart´ın-Herr´an, G. Zaccour, Pricing and advertising of private and national brands in a dynamic marketing channel, Journal of Optimization Theory and Applications, 137 (2008) 465-483. 9

[3] F. Barigozzi, P.G. Garella, M. Peitz, With a little help from my enemy: comparative advertising as a signal of quality, Journal of Economics and Management Strategy, 18 (2009) 1071-1094. [4] R. Cellini, L. Lambertini, Advertising with spillover effects in a differential oligopoly game with differentiated goods, Central European Journal of Operations Research, 11 (2003) 409-423. [5] R. Cellini, L. Lambertini, A. Mantovani, Persuasive advertising under Bertrand competition: a differential game, Operations Research Letters, 36 (2008) 381-384. [6] S. Chakrabarti, H. Haller, An analysis of advertising wars, The Manchester School, 79 (2011) 100-124. [7] P.J. Danaher, A. Bonfrer, S. Dhar, The effect of competitive advertising interference on sales for packaged goods, Journal of Marketing Research, 45 (2008) 211-225. [8] E. Dockner, S. Jørgensen, N. Van Long, G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, Cambridge, 2000. [9] D. Fudenberg and J. Tirole, The fat-cat effect, the puppy-dog ploy, and the lean and hungry look, American Economic Review, 75 (1984) 361– 366. [10] D. Grass, G. Feichtinger, J. P. Caulkins, G. Tragler, D. A. Behrens, Optimal Control of Nonlinear Processes, Springer, Berlin, 2008. [11] S. Jørgensen, G. Mart´ın-Herr´an, G. Zaccour, The LeitmannSchmitendorf advertising differential game, Applied Mathematics and Computation, 217 (2010) 1110-1116. [12] S. Jørgensen, G. Zaccour, Differential Games in Marketing, Kluwer, Boston, 2004. [13] G. Leitmann, W.E. Schmitendorf, Profit maximization through advertising: a nonzero sum differential game approach, IEEE Transactions on Automatic Control 23 (1978) 645-650.

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[14] M. Nerlove, K.J. Arrow, Optimal advertising policy under dynamic conditions, Economica, 39 (1962) 129-142. [15] A. Nair, R. Narasimhan, Dynamics of competing with quality- and advertising-based goodwill, European Journal of Operational Research, 175 (2006) 462-474. [16] R. Schmanlensee, Product differentiation advantages of pioneering brands, American Economic Review, 72 (1982) 349-365. [17] R. Schmanlensee, Advertising and entry deterrence: an exploratory model, Journal of Political Economy, 91 (1983) 636-653. [18] B. Viscolani, G. Zaccour, Advertising strategies in a differential game with negative competitor’s interference, Journal of Optimization Theory and Applications, 140 (2009) 153-170.

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