Advertising and Endogenous Exit in a Differentiated Duopoly

Mantovani, Andrea; Mion, Giordano Working Paper Advertising and Endogenous Exit in a Differentiated Duopoly Quaderni - Working Paper DSE, No. 455 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Mantovani, Andrea; Mion, Giordano (2002) : Advertising and Endogenous Exit in a Differentiated Duopoly, Quaderni - Working Paper DSE, No. 455, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4840 This Version is available at: https://hdl.handle.net/10419/159296 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ Advertising and endogenous exit in a differentiated duopoly∗ Andrea Mantovani†and Giordano Mion‡ December, 2002 Abstract In this paper we consider a duopoly two-stage duopoly where firms first decide whether to invest in advertising and then compete in prices. Advertising has two effects: a market enlargement for both firms and a predatory gain for the investing firm only. Both symmetric and asymmetric equilibria may arise. The two most interesting cases are a coordination game where both firms investing and non-investing are equilibria, and a chicken game where only one firm invests while the other is possibly driven (endogenously) out of the market. Our results suggest that product differentiation has an ambiguous impact on investment in advertising and that strong product substitutability may induce a coordination problem. Keywords: Advertising, product differentiation, endogenous exit, asymmetric equilibria, coordination games. JEL classification: C72, L11, L13, M37. ∗We thank Rabah Amir and Jacques Thisse for useful comments and suggestions. Both authors gratefully acknowledge financial support from CORE. The usual disclaimer applies. †CORE, Université Catholique de Louvain, 34 Voie du Roman Pays, B-1348, Louvain-la-Neuve, Belgium, [email protected] and Department of Economics, University of Bologna, Strada Maggiore 45, I-40125 Bologna, Italy, [email protected] ‡CORE, Université Catholique de Louvain, 34 Voie du Roman Pays, B-1348, Louvain-la-Neuve, Belgium, [email protected] and Department of Economics, University of Bari, Italy 1 1Introduction The role of advertising in the competition among firms has always represented an interesting issue. Advertising has been studied following different aspects of its nature. Advertising can be informative,giventhat it provides informations to consumers about the potential quality of a brand. Furthermore, by advertising, afirm reveals the features of its product and this tends to increase product differentiation. As Kaldor acknowledged: Advertising is a method of differentiating, in the eyes of the consumer, the products of one firm from those of its competitor; it is a method, therefore, of reducing the scope and effectiveness of price-competition by attaching a strong element of “goodwill” to each firm (Kaldor, 1950, p.14). But advertising is also persuasive, given that the investing firm could aim at convincing the consumer that what he really wants is its particular variety. The dual role of advertising becomes then evident. On the one hand, advertising acts to shift firms’ demand curves; while on the other hand, it makes a good more differentiated from the one produced by rivals. The study of this tension will be one of the main subject of this paper. The economic literature has initially dealt with the negative impact of advertising with respect to welfare considerations. Advertising has in fact been considered as socially harmful. Kaldor (1950) himself recognized that advertising could have a “manipulative” effect that reduces competition by convincing consumers that two identical products are differentiated. On the other hand, Nelson (1970) and Demsetz (1979) acknowledged the beneficial function of advertising when it conveys the right information to consumers, whose searching costs then tend to decline. Moreover, advertising could give rise to barriers to entry for newcomers that would need to spend a substantial amount of money to overcome the reputation of the incumbents. Many authors focused on the issue of strategic advertising as an instrument to deter entry (Bagwell and Ramey, 1988 and 1990). Schmalensee (1983) considered a duopoly two-stage Cournot model where an initial investment in advertising was able to deter the entry of new rivals. More recently, Ishigaki (2000) found that Schmalensee’s results did not hold in a similar Bertrand setting. In our study of advertising we will be particularly concerned with two aspects: first, we analyze the impact of advertising in the enlargement of a market for a ‘non well-known’ product; second, we explicitly deal with the predatory interaction that could characterize advertising games. The first consideration comes from the fact that consumers might not be fully aware of the presence of certain types of products in the market. This is especially true for products belonging to the hi-tech sector. A firm that develops a ‘novelty’ must invest resources to explain which kind of product has become available. The creation of a new market, or the enlargement of an existing one, could represent nonetheless an advantage for a potential rival, which would benefit from an information spill-over that shifts the demand 2 curve upward for all those kinds of goods. In the literature this has been often referred to as cooperative advertising (Friedman, 1983; Martin, 1993, ch. 6), even if firms do not necessarily cooperate in the profit maximization stage. The issue of advertising that increases the size of the market has been analyzed also in a dynamic setting (see Jorgensen, 1982 and Dockner et al., 2000, ch.11 for exhaustive surveys). Cellini and Lambertini (2002) consider a differential oligopoly game with differentiated goods where firms compete à la Cournot in the market phase and may finance advertising to enlarge their market shares. Furthermore, each firm’s advertising effort produces a positive spill-over for the rival in terms of market enlargement. The second consideration mentioned above refers to the conventional view that advertising also creates “brand loyalty” and “goodwill” that sticks to a determined brand. In particular, we focus on the predatory nature of advertising (Friedman, 1983; Martin, 1993, ch. 6). In fact, by engaging in advertising, a firm increases its own demand while at the same time it reduces the demand of the rival. An example is given by the use of comparative advertising, through which a firm compare the characteristics of its product with those of the competitors.1Crucially, and that is why we decided to deal with price competition and product differentiation, this is more likely to happen the higher the substitutability among the products. Thedegreeofdifferentiation on the product market has a direct impact on advertising decisions, and this interaction could not be properly modeled in the standard quantity competition framework. Grossman and Shapiro (1984), for example, considered a differentiation duopoly model with price competition and showed that advertising is positively related to the degree of product differentiation. Other models dealing with such a relationship can be found in Butters (1977), Wolinsky (1984) and Von der Fehr and Stevik (1998). As a consequence, in our analysis an investment in advertising will have two effects, that we will denote as a “market enlargement effect” and a “predatory effect”.2The former captures the expansion of the market and represents an advantage for every operating firm, while the latter accounts for the individual incentive for each single firm to spend resources on advertising. As we will see, the relative strength of these two components will determine which outcome represent an equilibrium. Depending on the parameter values, both symmetric and asymmetric equilibria may possibly arise. Among them, two outcomes are of particular interest: a coordination game in which both investing and non-investing are simultaneously equilibria; a chicken game in which only one firm invests in equilibrium with the second one possibly driven (endogenously) out of the market. Furthermore, we will also provide some insight about the Pareto optimality (from firms’ standpoint) of market outcomes that will enable us to identify prisoner dilemma situations. Particular attention will be paid to the role of product differentiation in determining the equilibrium level of advertising, as well as to shed some light on the problem of coordination. This paper is organized as follows. In the following section we will introduce the analytic features of the model. Section 3 analyzes the second stage price game while in Section 4 we solve (backward) the first 1The use of comparative advertising progressively increased both in the United States and, more recently, also in the European Union. According to Muehling et al. (1990),in the United States around 40% of all advertising is comparative. 2For an alternative (dynamic) framework in which advertising is both cooperative and predatory, see Piga (1998). 3 stage advertising game. Section 5 provides a complete characterization of the equilibria of the game in terms of parameters and then turns to their economic interpretation. Section 6 finally provides conclusions and directions for further research. 2 The Model Consider an industry composed of two symmetric firms that produce a differentiated good. They are engaged in the following two-stage game. In the first stage, each firm decides whether to devote resources to advertising or not, while in the second stage they compete in prices. In particular, firms can only decide to advertise or not, while the strategy set of each firm for the second stage price game is the entire <+.3 Marginal costs are supposed to be zero and there are no fixed costs in production. When a firm engages in advertising it modifies its own demand, as well as that of its rival, while incurring a fixedcostthatwe normalize to one. We restrict our attention to subgame perfect Nash equilibria. The demand structure turns out to be extremely important in our analysis. As we want to deal with both product differentiation and price competition, a natural starting point is the linear demand function:4 qi=a−bpi+c(p−i−pi)=a−(b+c)pi+cp −i(1) The parameter astands the market size, while bis meant to represent the surplus of the own price over the cross price effect. The parameter cis an (inverse) measure of product differentiation; the higher c,the higher the substitutability between the products, given the stronger impact of a price difference. As we mentioned in the introduction, the kind of advertising we are interested in is such that demand curves shift outward (informative advertising), thus enlarging the market for that product. Suppose that a new kind of product becomes available, or that such product is not very well-known. When a firm advertises its own good, it provides also general information about that kind of product. This turns out to be beneficial also for a potential rival that produces a similar good. A good example can be traced in the DVD market expansion boosted by Sony’s massive advertising campaign. This positive spill-over, that we call “market enlargement effect”, gives then an advantage to all firms as sellers of that type of product, and could be modeled with a symmetric shift of the parameter ain the demand function of our two firms. On the other hand, by advertising, each firm also creates “brand loyalty” and “goodwill” for its own product, thus drawing consumers away from rivals. This leads to a kind of ‘stealing’ process that we call, coherently with the existing literature, “predatory effect”. Crucially, the lower is the degree of product 3Clearly, it would be preferable to use a more sophisticated set of alternatives for advertising decisions. However, as many other forms of investment, advertising can have a discrete nature in the sense that it is sometimes more important to decide whether to invest or not rather than the exact amount to be spent on it. Furthermore, as we will see afterwards, our simple binary assumption will allow for a complete characterization of all possible equilibria of the game. 4The proposed linear demand function is consistent with utility maximizing consumers with quadratic utility functions (see Shubik and Levitan 1980). 4 differentiation (high values of c), the higher will be the impact of this second effect. In fact, as long as products are perceived as highly substitutes, firms have a strong incentive to attach an element of differentiation on their own good through advertising. In order to capture this “predatory effect”, we make the hypothesis that a firm that does advertising receives a demand gain cα, while imposing at the same time an equivalent demand cut −cαon its rival. Although our two firms are aprioriidentical, the game could have asymmetric equilibria. In particular, we have to deal with the possibility that firm isets a price lower or equal to a limit price pl i, pushing the other firm (endogenously) out of the market. This clearly raises the problem of defining the demand received by the remaining firm. Starting from equation (1), the solution we adopt is to define demand in the limit pricing domain in such a way that continuity is preserved for all admissible price strategies. This leads to the following demand system:5 qi(pi,p −i,I i,I −i)=   max {a(Ii,I −i)−bp i+c[p−i−pi+αi(Ii)−α−i(I−i)],0},ifp i>p l i max {2a(Ii,I −i)−bp i−bϕ(pi),0},ifp i≤pl i   (2) where: Ii={0,1}for i =1,2 a(Ii,I −i)=a(Ii+I−i)=        aif Ii+I−i=0 a+γif Ii+I−i=1 a+3γ/2if Ii+I−i=2        αi=   0if Ii=0 αif Ii=1   for i =1,2 ϕ(pi)=max½a(Ii+I−i)+c[α−i(I−i)−αi(Ii)] b+c+c b+cpi,0¾ a,b,α,γ,c>0 The binary variable Iirepresents advertising strategies: firm icould either advertise (Ii=1)ornot (Ii=0). The “market enlargement” effect induced by advertising is captured by a(I1,I 2)and depends upon total investment: I1+I2.If no firm advertises, then a(I1+I2)is stuck to a basic level a.Ifonly 5See Appendix A.1 for further details. It is important to stress that the requirement of continuity for all admissible prices leads to a ‘unique’ definition of demand in the limit price domain. 5 Figure 1 : The demand function 6 - qi pi 0 pl i HHHHHHHHHHHHHHHHH HZZZZZZZ Z r one firm advertises, then a(I1+I2)increases to a+γ, while if the other does the same the new marginal increase is just (1/2)γ. This series of diminishing increments accounts for the fact that there cannot be unlimited expansion of the market.6The “predatory effect” is instead parameterized by αi,thatcould be either zero or α. If only one firm advertises then, as long as the other one is actually on the market (pi>p l i), its demand increases by cαwhile the demand of the rival decreases by the same amount. As argued in the previous section, the magnitude of the “predatory effect” is in fact positively related to product substitutability. On the other hand, if prices and advertising strategies are such that only firm imakes positive sells (pi≤pl i), its demand depends only on piin such a way that continuity in prices is guaranteed. 3 The second stage price game In equilibrium, there can obviously be just two possibilities: either the two firms sell a positive amount of goods, or just one of them receives a positive demand while the other has a zero output. As we already pointed out, our demand system (2) is continuous for all p1,p 2∈[0,∞)and it is clearly monotone decreasing (increasing) in each firm own (cross) price whenever a firm’s demand is positive. Anyway, firstorder conditions alone do not suffice to characterize Nash Equilibria in the price game because demand functions have kinks. In fact, demands are just piece-wise linear in both own and cross price, and their slope changes in view of limit pricing. However, in Appendix A.1 we show that this change is “well-behaved” in the sense that the slope is lower (in absolute value) when only one firm sells on the market. Figure 1 shows the demand of firm iin case a positive limit price pl iexists. This change in price responsiveness comes from the fact that, when both firms are active on the market, 6In particular, we adopt the geometric series a(n)=a+Pn i 1 iγ. 6 Figure 2 : The profit function 6 - pi πi 0pl i r a price reduction by one firm induces not only new customers to buy the product, but also usual clients of the rival to drift to the price-reducing firm. This has a very useful implication for firms’ profit functions which, by linearity of demands and absence of variable costs, turn out to be strictly concave whenever quantities are positive. Therefore, we can state the following: Claim 1: If a NE in the second stage price game with both firms making positive sells exists, then first-order conditions are necessary and sufficient to identify it. In Figure 2 we represent an example of how the profit function of firm ilooks like when a positive limit price pl iexists. As we will see afterwards, for equilibria in which only one firm is active on the market, first-order conditions will be of a little help since one typically faces corner solutions. Let us now analyze in details the 3 possible outcomes arising as a consequence of different advertising decisions. 3.1 Case A: None invests We start by considering the symmetric case where no firm invests in advertising. If both firms receive positive demands, then the demand curves of each firm are (with I1=I2=0): q1=a−bp 1+c(p2−p1)and q2=a−bp 2+c(p1−p2). Since we assume that marginal costs are zero, and there are no fixed costs in production, profits are: π1=p1q1=p1[a−bp 1+c(p2−p1)] and π2=p2q2=p2[a−bp 2+c(p1−p2)] . 7 Profits are quadratic in each firm’s own price, and by first-order conditions we get equilibrium prices: pA 1=pA 2=a 2b+c>0. The demands corresponding to these prices are always positive (so that this NE is always acceptable) and the equilibrium profits (obtained using equilibrium prices pA 1and pA 2)are: πA 1=π1(pA 1,p A 2)=πA 2=π2(pA 1,p A 2)=a2(b+c) (2b+c)2>0(3) According to Claim 1, the pair {pA 1,p A 2}thus represents the unique “SPE” characterized by both firms making positive sells. On the other hand, it is straightforward to check that each firm can always find here, whatever the other does, a strictly positive price such that it receives some demand and makes strictly positive profits. This clearly means that there is no room for equilibria with just one active firm, i.e.: Lemma 1: In the subgame where no firm invests, there is a unique Nash equilibrium in pure strategies given by {pA 1,p A 2}. 3.2 Case B: Only one firm invests We now examine the case where only one firm invests in advertising. Without loss of generality, we assume that firm 1 invests while firm 2 does not: I1=1and I2=0.In case of positive sells for both firms, the demand curves are given by: q1=a+γ−bp 1+c(p2−p1+α)and q2=a+γ−bp 2+c(p1−p2−α). Compared to the previous case, where none of them invested in advertising, both firms enjoy here an increase in demand equal to γdue to the market enlargement effect. However, due to the predatory effect, firm 1 receives an additional gain cα, while imposing a penalty −cαto the rival. Profits are now given by: π1=p1q1−1=p1[a+γ−bp 1+c(p2−p1+α)] −1and π2=p2q2=p2[a+γ−bp 2+c(p1−p2−α)] By first-order conditions we get equilibrium prices: pB 1Ac =(a+γ)(2b+3c)+cα(2b+c) (2b+c)(2b+3c)>0(4) pB 2Ac =   (a+γ)(2b+3c)−cα(2b+c) (2b+c)(2b+3c)if α<αa 0otherwise (5) 8 1. When only πA i>πB 1holds, then (0,0) is the unique “SPE” of the game. The two firms do not invest in advertising and, depending on the value of γ, we could possibly obtain a prisoner dilemma game. 2. When only πC i>πB 2holds, then (1,1) turns out to be the unique “SPE” of the game. Again, depending on γ, we may have or not a prisoner dilemma. 3. If both conditions hold together, we obtain a coordination game with two “SPE” along the principal diagonal. 4. Lastly, if both these conditions are not satisfied, we get a chicken game with two asymmetric “SPE” along the secondary diagonal characterized by only one firm investing in advertising. To link equilibrium profits with the parameters of the model, we have to consider three different expressions associated to πB 1, depending on the value taken by α. We can give necessary and sufficient conditions on (α,γ)for (0,0) to be a “SPE”: Proposition 1 (0,0) is a “SPE” for sufficiently low combinations between the values of αand the ones of γ.Inparticular:(i)whenα≥αb, we need γ≤γ2; (ii) when αa≤α<αb, we need either γ≤γ2or, if γ>γ2,thenα≤αc(<αb); (iii) when 0<α<αa, we need either γ≤γ3(>γ2)or, if γ>γ3,then α≤αd(<αa).When γ≥γ4(>γ3),(0,0) is never an equilibrium, independently of α. Moreover, this Nash Equilibrium, when it exists, turns out to be Pareto dominant from firms’ standpoint for sufficiently low values of γ(γ<γ1), otherwise the game is of a prisoner dilemma type. Proof. see Appendix A.2. The dashed area in Figure 4 indicates those values that sustain (0,0) as a “SPE” in the (α,γ)space.9 As one can see, both firms decide not to invest when the combination of the market size effect and the strategic predatory effect is weak enough. There is, indeed, a certain degree of substitution in the two effects. A strong predatory gain αcould be compensated with a weakening of the market enlargement in order for (0,0) to be a “SPE”. However, if γis big enough (γ≥γ4), then, whatever αis, (0,0) is never a “SPE”. Furthermore, when it exists as an equilibrium, (0,0) is Pareto dominant for firms only when the market size effect is sufficiently weak (γ<γ1). Interestingly, a prisoner dilemma (indicated by the portion of the dashed area on the right of γ1)arises when there are quite good perspectives of enlarging the market, but the predatory gain is limited. One firm alone has no advantage to invest since it does not steal that much from the other which can, by contrast, enjoy the enlargement of the market due to advertising without paying any cost for it. Although both firms would be better offby investing, they thus refrain from doing so. Let us now consider the equilibrium (1,1). By evaluating πC ivs πB 2, and taking into account the restrictions on both profit functions, we can conclude that in the parameter space (α,γ): 9Figure 4 has been depicted using c=1,b=1,anda=0.3. 15 Figura 4 : The equilibrium (0,0) 6 - γ α 0γ4 γ1 γ3 γ2 αd αc αa αb ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯   T T T T T T T T T T T T T ³³³ ³ ³³³ ³ ³³³ ³ ³³³ ³ ³³³ ³ ³³³ ³ ³³³ ³ ³³³ ³ ³³³ ³ ³³³³ ³ ³³³³ ³ ³³³³³ ³ ³³³³³³ ³³³³³³ ³ ³³³³³³³ ³ ³³³³³³³³ ³ ³³³³³³³³ ³ ³³³³³³³³³ ³ ³³³³³³³³³ ³ ³³³³³³ ³³ ³ Proposition 2 (1,1) is a “SPE” for sufficiently high combinations between the values of αand those of γ.In particular, we need at least that γ≥γ5and either α≥αa,or, when α<αa,α≥αe(<αa). When γ≤γ5,(1,1) is never an equilibrium, independently of α.Onthecontrary,ifγ≥γ6(>γ5),(1,1) is always a “SPE”. Such a solution represents a Pareto dominant strategy for firms for sufficiently high values of γ(γ>γ1), otherwise it gives rise to a prisoner dilemma. Proof. see Appendix A.3. The dotted area in Figure 5 describes our equilibrium conditions in the (α,γ)space.10 Contrary to before, both firms invest in equilibrium when the combination of the two effects is strong enough. There is, again, a certain degree of substitution between αand γ. When the predatory effect is weak, (1,1) constitutes a “SPE” of the game only if the market expansion translates into a considerable increase in firms’ profits. However, if γis big enough (γ>γ6), then, whatever is α,(1,1) is always a “SPE”. Furthermore, (1,1) is Pareto dominant from firms’ standpoint whenever γ>γ1.Conformingtointuition, aprisoner dilemma situation (indicated by the portion of the dotted area on the left of γ1) still arises, but its nature is the mirror image of the previous case. Crucially, the predatory gain here needs to be strong enough. Firm would in fact be better offwithout doing advertising because the market expansions possibilities are quite limited (γ>γ1). However, they advertise in equilibrium because they are fully aware of the substantial gain (loss) of being the only advertising (non-advertising) firm. 10Figure 5 has also been drawn using c=1,b=1,anda=0.3. 16 Figura 5 : The equilibrium (1,1) 6 - γ α 0γ5γ1γ6 αa αe   Combining Proposition 1 and 2, we can fully characterize the four possible outcomes of the model in terms of the parameters. In the next section we will give some insights on how these equilibria configurations react to changes in parameters as well as their underlying economic interpretation. 5 Further results and economic interpretations In the previous section we gave necessary and sufficient conditions on the two-dimensional parametric space (α,γ)such that (0,0) and (1,1) are “SPE”. Following Propositions 1 and 2, we reasonably expect to find that, when there are small incentives for firms to advertise (i.e. low values of αand γ), (0,0) is the only equilibrium of the game. By contrast, for sufficiently high values of αand γ, we expect (1,1) to be the only outcome. Now, what is not clear is what happens in intermediate situations. Both a coordination and a chicken game will be possible, but conditions leading to each outcome still remain unknown at this stage. In order to shed some light on the forces underpinning the game, we resort to comparative statics analysis. This task turns out to be extremely difficult because equilibria are characterized in the twodimensional space (α,γ)and we need to consider simultaneously all the different threshold values appearing in Propositions 1 and 2. After tedious calculations, one can show that: γ6>γ1>γ5and γ4>γ1>γ3>γ2. 17 Figure 6 : Analysis of equilibria: First case 6 - γ α 0γ5γ1γ6γ4 γ3 γ2 αd αc αe αa   αb ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ T T T T T T T T T T T T T ³³³ ³ ³³³ ³ ³³³ ³ ³³³ ³ ³³³ ³ ³³³ ³ ³³³ ³ ³³³ ³ ³³³ ³ ³³³³ ³ ³³³³ ³ ³³³³³ ³ ³³³³³³ ³³³³³³ ³ ³³³³³³³ ³ ³³³³³³³³ ³ ³³³³³³³³³ ³ ³³³³³³³³³³ ³ ³³³³³³³³³³ ³ ³³³³³³ ³ ³³³ Unfortunately, we cannot directly order γ6vs γ4and γ5vs γ3,γ2,but we use the initial size of the market, parameterized by a, to discriminate between such threshold values of γ. This will allow for a complete characterization of the equilibria. In particular, depending on the relative position of awith respect to two critical values, a1and a2,with0<a 1<a 2, we get the following results:11 Lemma 6: (i) When a<a 1,thenγ4>γ6and γ5>γ3;(ii)whena1<a<a 2,thenγ4<γ6and γ5>γ3; (iii) when a>a 2,thenγ4<γ6and γ5,γ2,γ3<0. Using the above results, we can sketch a complete rank of the threshold values of γ.Depending on the value taken by a, three different situations will appear: 1. a<a 1=⇒γ4>γ6>γ1>γ5>γ3>γ2>0; 2. a1<a<a 2=⇒γ6>γ4>γ1>γ5>γ3>γ2>0; 3. a2<a=⇒γ6>γ4>γ1>0>γ5,γ2,γ3. Figure 6 describes the whole situation in the first case (i.e. a<a 1)12. We can now clearly identify both a chicken game and a coordination game. In the white area, neither the conditions of Proposition 1 nor those of Proposition 2 hold, and so (1,0) and (0,1) are the (unique) equilibria. This scenario appears 11Calculations are available upon request. 12Figure 6 merges figures 4 and 5. It has in fact been drawn using c=1,b=1,anda=0.3<a 1=0.43. 18 when the possibility of enlarging the market is neither too limited nor too excessive (otherwise either (0,0) or (1,1) would respectively be the only outcomes), and the strategic predatory gain is sufficient for the investing firm to profitably cover the investment costs. Moreover, as we know from Lemma 3, if α<αa, then the firm which is not investing is still selling a positive amount of product, while for α>αait is endogenously driven out of the market. In particular, when αa≤α<αbthe investing firm finds it convenient to set a limit price, while above αbit has such a big advantage that it can charge a kind of monopoly price. On the other hand, where the dotted and dashed areas overlap, we have an interesting coordination game. The simple structure of the game allows us to treat the (relatively) unaddressed issue of coordination in advertising decisions in a fairly straightforward way. Both (0,0) and (1,1) can be simultaneously equilibria and this happens again for intermediate values of γ, but now coupled with a weak strategic effect. When αis small, a firm that invests alone must bear all costs of advertising, while its gain comes almost entirely from the enlargement of the market. At the same time, the other firm gains more or less the same, without paying anything for it. If the return on advertising is quite good (in terms of γ), then the non-investing firm could find it profitable to devote resources to adverting too. On the contrary, if this return is not too high, then the other firm can reasonably reconsider its investment decision. Therefore, for intermediate values of γand low α, both kinds of deviations are plausible and we have a problem of coordination. Interestingly, as the figure shows, when such a problem arises the Pareto optimal equilibrium for our firmsistheonewithinvestment. Figure 7 and 8 depict respectively the all set of equilibrium conditions for the cases where a1<a<a 2 and a2<a.13 There are two main differences with respect to Figure 6. First, the dashed area shrinks indicating that the equilibrium (0,0) is less and less likely to occur. This is due the fact that, when the initial size of the market aincreases, then firms are, ceteris paribus, more capable to cover the fixed costs of advertising. This reasonably makes firms more willing to invest in advertising. Second, the coordination game disappears. This happens for the same reasons that cause the dashed area to reduce. Rising a,itis less likely that a firm cannot cover the fixed cost of advertising, even if it invests alone. Further intuitions could be drawn by the relative dimension of c, the parameter which measures the degree of substitutability between the products of the two firms. Actually, we can alternatively rephrase Lemma6intermofc. As a function of c,botha1and a2vary from values close to zero to infinity and their first derivatives are strictly positive. Therefore, for a given a, we can always find values of csufficiently high to have a<a 1, and then decreasing cwe pass to the other two situations a1<a<a 2and a2<a. For high values of c, we are then more likely to happen in a situation like the one depicted in Figure 6, where the dashed area expands while the dotted one shrinks with respect to Figures 7 and 8. Intuitively, when products are close substitutes (high c), then competition in prices turns out to be very fierce. Consequently, equilibrium profits decrease (and at the limit they tend to zero) and firms are then more 13Figure7and8havebeendrawnstilltakingc=1,andb=1, but while the former refers to a=1(which is in between a1=0.43 and a2=2.12), the second uses a=2.5>a 2=2.12. 19 Figure 7 : Analysis of equilibria: Second case 6 - γ α 0γ2γ3γ5γ1γ4γ6 αa ¢¢¢¢¢¢¢¢¢¢¢¢ αb ¯¯¯¯¯¯¯¯ αc αd αe ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ ³³ ³³ ³ ³³ ³ ³³³ ³³³ ³ ³³³ ³ ³³³³ ³³³³ ³ ³³³³ ³ ³³³³³ ³ ³³³³³ ³ ³³³³³ ³ ³³³³ ³ ³³ ³ ³³ Figure 8 : Analysis of equilibria: Third case 6 - γ α 0γ1γ4γ6 αdαe T T T T T T T ³³ ³ ³ ³ ³ ³ ³ ³³ ³ ³ ³ 20 reluctant to advertise given that such an activity requires a fixed cost. This behavior is certainly consistent with the findings of Grossman and Shapiro (1984) inter alia. Turning to the asymmetric situation where just one firm advertises, the net effect of a change in cis instead quite ambiguous. Indeed, the impact of the strategic effect on demands, given in equation (2) by cα, would be stronger, giving a relative advantage to the investing firm. On the other hand, the increase in competition in the goods market lowers profits, dampening the incentive to advertise. Consequently, the size of the white area may either increase or decrease. In our simulations, it actually increases from Figure 8 to 7, while decreasing when moving from 7 to 6. As products become more differentiated, it is not necessarily the case that a firm finds it profitable to invest in advertising if the other does not. Taking into account asymmetric outcomes thus leads to discover this somehow counter-intuitive relation between the equilibrium level of advertising and the degree of product differentiation. A similar ambiguous relation has been highlighted, although in a different framework, by von der Fehr and Stevik (1998). Finally, coming back to the coordination game, we can observe that it arises only for high values of c. In other words, coordination becomes an issue when the degree of interdependence among agents, captured here by price competition, is strong enough. This indeed conforms with intuition, but we can further prove that in such a case investing is the best choice: Proposition 3 The coordination game could arise only for low values of a(a<a 1) or, equivalently, when product are highly substitutes (big c). Furthermore, when both (0,0) and (1,1) are “SPE”, then the latter is always Pareto dominant from firms’ standpoint. Proof. see Appendix A.4. 6Conclusions In this paper we considered a two-stage duopoly model with differentiated products where firms decide whether to invest in advertising or not and then they compete in prices. We focused on two main effects of advertising: a market enlargement effect and a predatory effect. Particular attention has been paid to the specificroleofproductdifferentiation in advertising decisions as well as to the assessment of their Pareto optimality from firms’ standpoint. Depending on the values taken by the parameters, both symmetric and asymmetric equilibria appeared. Among them, two outcomes are of particular interest: a coordination game in which both investing and non-investing are simultaneously equilibria; a chicken game in which only one firm invests in equilibrium with the second one possibly driven (endogenously) out of the market. The coordination game arises only when products are strongly substitutes, suggesting that coordination matters when the degree of interdependence among agents, captured by price competition, is sufficiently high. Interestingly, when such a problem of coordination appears, the investment strategy leads in our model to the Pareto optimum. 21 Turningtothechicken game, we actually found two very interesting results. First, there exists a parameter region that supports a limit pricing behavior by the investor with the rival being endogenously squeezed out of the market. Furthermore, if the predatory advantage for the investing firm is strong enough, then it is able to take the entire market just by setting a kind of monopoly price. Second, in this asymmetric case the impact of product substitutability on the advertising efforts is ambiguous. This result contradicts the common view of a positive relationship between product differentiation and the equilibrium level of advertising and comes from the interplay between two opposite forces at work in the asymmetric equilibrium: the predatory gain that depends negatively on differentiation, and the strength of price competition that is instead relaxed by a decrease in product substitutability. Starting from a very simple framework, we obtained quite interesting results. However, one of the main limitations of our approach stands in the use of a binary strategy set for investment decisions. Ideally, it would be better to use a more sophisticated relationship between investment in advertising and demand changes. On the other hand, as many other forms of investment, advertising has a strong discrete nature in the sense that it is sometimes more important to decide whether to invest or not rather than the exact amount to be spent on it. Furthermore, our plain structure turns out to be extremely flexible in the sense that it allows us to treat a large number of parameters (without resorting to normalization) and to completely characterize the game. Situations like the coordination game or the chicken game would in fact have been hardly treated in a continuous framework. The same forces at work in the present paper could also be translated into a dynamic setting. Every firm could in fact be endowed with a stock of advertising that summarizes the effects of past advertising efforts. As a consequence, apart form current advertising, both the enlargement of the market and the consumers’ shift from one firm to the other would depend on the stock of accumulated goodwill. References [1] Amir, R. (2000), R&D Returns, Market Structure, and Research Joint Venture, Journal of Institutional and Theoretical Economics 4, 583-598. [2] Bagwell, K. and G. Ramey (1988), Advertising and Limit Pricing, Rand Journal of Economics 19, 59-71. [3] Bagwell, K. and G. Ramey (1990), Advertising and Pricing to Deter or Accommodate Entry When Demand is Unknown, International Journal of Industrial Organization 8, 93-113. [4] Cellini, R. and L. Lambertini (2002), Advertising with Spillover Effects in a Differential Oligopoly Game with Differentiated Goods, in L.A. Petrosjan and N.A. Zenkevich (eds.), Proceedings of the X International Symposium on Dynamic Games and Applications, International Society of Dynamic Games and St. Petersburg State University, vol. I, 189-96. 22 [5] Butters, G. (1977), Equilibrium distribution of prices and advertising, Review of Economic Studies 44, 465-491. [6] Demsetz, H. (1979), Accounting for Advertising as a Barrier to Entry, Journal of Business 52, 345-360. [7] Dockner, E. J., S. Jørgensen, N. Van Long and G. Sorger (2000), Differential Games in Economics and Management Science, Cambridge, cambridge University Press. [8] Friedman, J.W. (1983), Advertising and Oligopolistic Equilibrium, Bell Journal of Economics 14, 464-473. [9] Grossman, G. M. and C. Shapiro (1984), Informative advertising with differentiated products, Review of Economic Studies 51, 63-81. [10] Ishigaki, H. (2000), Informative advertising and entry deterrence: a Bertrand model, Economics Letters 67, 337-343. [11] Jørgensen, S. (1982), A Survey of Some Differential Games in Advertising, Journal of Economic Dynamics and Control 4, 341-369. [12] Kaldor, N. (1950), The Economic Aspects of Advertising, Review of Economic Studies 18, 1-27. [13] Martin, S. (1993), Advanced Industrial Economics, Oxford, Blackwell. [14] Muehling, D., J. Stoltman and S. Grossbart (1990), The Impact of Comparative Advertising on Levels of Message Involvement, Journal of Advertising 19, 41-50. [15] Nelson, P. (1974), Advertising as Information, Journal of Political Economy 82, 729-754. [16] Piga, C. (1998), A Dynamic Model of Advertising and Product Differentiation, Review of Industrial Organization 13, 509-522. [17] Schmalensee, R. (1983), Advertising and entry deterrence: an exploratory model. Journal of Political Economy 91, 636-653. [18] Shubik, M., and R. Levitan (1980), Market Structure and Behavior, Harvard University Press, Cambridge, Mass. [19] von der Fehr, N.-H. and K. Stevik (1998), Persuasive Advertising and product Differentiation, Southern Economic Journal 65(1), 113-126. [20] Wolinsky, A. (1984), Product differentiation with imperfect information, Review of Economic Studies 51, 53-61. 23 AAppendix A.1 The demand structure Let’s start by considering demand functions qiand q−ifor our firms as given by equation (1). This analytic formulation is clearly meaningful as long as price strategies are such that the implied qiand q−iare non negative. Our goal here is to show how the demand system (2) can be obtained from equation (1) using continuity arguments. Consider the limit case in which piand p−iare such that q−i,ascomputedfrom (1), exactly equals zero. Solving the equation q−i=a(Ii,I −i)−bp −i+c[pi−p−i+α−i(I−i)−αi(Ii)] = 0 for p−iand plugging the solution into the equation of qione gets (after rearranging terms): qi=2a(Ii,I −i)−bp i−bϕ(pi)(A1) which, as we argued, is precisely the demand of firm iwhen the other firm gets zero sells (pi≤pl i). Equation (A1) is certainly correct for any couple of prices piand p−isuch that q−i=0in (1). What remains to prove is that this is true for all prices piand p−ithat leads firm −ito be out of the market, that is for lower piand greater p−i. Consider for example a higher p−i.Sincefirm −iis already out of the market, it cannot certainly hope to ameliorate its position by increasing the price. Demands should thus be invariant to this increases in p−iand, by continuity, qiequals (A1), which is in fact a function of pionly. On the other hand, if firm i charges a price lower then before, then firm −iis again out of the market, and we will actually have q−i<0 in (1). Following the above reasoning, demand of firm ishould not, as long as q−icomputed with (1) is non-positive, depend on p−i. Everything thus works as if firm −iwas charging a new price p−ithat would make q−iexactly equal to zero, leading us back to formulation (A1). Finally, since the price p−isolution to the equation q−i=0cannot be negative, we have ϕ(pi)=maxna(Ii+I−i)+c[α−i(I−i)−αi(Ii)] b+c+c b+cpi,0o. Now let’s turn to price responsiveness of our demand system (2). As long as firm −iis on the market, the appropriate demand curve is qi=a(Ii,I −i)−bp i+c[p−i−pi+αi(Ii)−α−i(I−i)] and its derivative with respect to piis simply −(b+c).Ifpricepinow goes below the limit price pl i, then the right demand function is (A1) and its slope can be either −b(b+2c) b+c>−(b+c)or, if piis so low to hit the constrain ϕ(pi)=0, it amounts to −b>−b(b+2c) b+c. As a conclusion, when pidecreases demand qibecomes less and less sensitive to price changes. Finally, the limit price pl iis simply the non-negative solution (if it exist) to the equation q−i= a(Ii,I −i)−bp −i+c[pi−p−i+α−i(I−i)−αi(Ii)] = 0 with respect to pi. This solution to turns out to be: pl i=αi(Ii)−α−i(I−i)+−a(Ii,I −i)+(b+c)p−i c, 24