Strategic Brand Proliferation: Monopoly vs. Duopoly

Strategic Brand Proliferation: Monopoly vs. Duopoly Kentaro Inomata March 15, 2014 Abstract This paper ascertains whether (i) the existence of competition encourages existing firms to proliferate their brands, and (ii) intensifying competition brings about increases in the profits of the firms. To show these, we compare a Cournot duopoly with a monopoly. We show that in symmetric-firms setting, there exist asymmetric equilibria in that as competition intensifies (product substitutability increases), only one of the firms has an incentive to become a multi-product firm (MPF) and therefore increases its profit. This is because the brands of the MPF act as complements in duopoly whereas those always work as substitutes in monopoly. JEL classification: L13; D43 Keywords: brand proliferation, multi-product firm (MPF), product substitution, Cournot competition, substitutes and complements This research was supported by a Grani-in-Aid for Japan Society for the Promotion of Science (JSPS) Fellows (Grant number 25 5882). I would like to thank Junichiro Ishida, Shingo Ishiguro and Noriaki Matsushima for detailed their comments and many suggestions. I am also indebted to Takanori Adachi, Satoshi Fukuda, Jun-ichi Itaya, Akio Kawasaki, Keisuke Kawata, Hiroshi Kitamura, Akira Miyaoka, Tadashi Morita, Tatsuhiko Nariu, Tetsuya Shinkai, Tadanobu Tanno, Yoshihiro Yoshida and the seminar participants at Keio, Kyoto, Kyoto Sangyo, Kyushu Sangyo, and Osaka University. Graduate School of Economics, Osaka University. Research Fellow (DC2), Japan Society for the Promotion of Science. E-mail: kge001ik@mail2.econ.osaka-u.ac.jp 1

1 Introduction In the literature on industrial organization, brand proliferation which means an increase in the number of differentiated goods in an industry has been broadly discussed since Schmalensee (1978) s seminal paper. This is because brand proliferation is a type of important strategic tools for firms in that it may bring about any radical change in the strategic interactions among competitive firms. In this meaning, even though brand proliferation is similar to the variety expansion, those are different. This is why brand proliferation is one of central topics in the literature on industrial organization. So in this paper we would like to focus on the relation between the incentives of firms for brand proliferations and competitive pressure. The purposes of this paper are to ascertain whether (i) the existence of competition encourages existing firms to proliferate their brands, and (ii) intensifying competition brings about increases in the profits of the firms. To consider these purposes, we construct a simple model comparing existing firms incentives to proliferate their brands in Cournot duopoly with the one in monopoly. Of course, we cannot directly observe such investments in two cases, monopoly and duopoly. But by this thought experiment, we can shed light on the incentives of competitive firms to obtain their larger market shares which do not exist in monopoly. This comparison is the significant feature of the paper because there are a few studies comparing the incentives under monopoly and the one under competition in spite of the importance. The first, (i) is motivated from a simple viewpoint that competitive pressure encourages each firm to invest to supply a new commodity. In our daily lives, we often see the examples of brand proliferation, new releases of mobile phones, cars, and beverages. Those are usually provided not by a firm but by several firms. So this fact impresses us competitive pressure encourages the firms to proliferate their brands. However, in theoretical perspectives, it tends to opposite. That is, since firms yield lower profits under competition than under monopoly, each investment level is likely to be lower. This is the negative aspect of competition for brand proliferation. Meanwhile, the other effect peculiar to competition, the business-stealing effect, exists. Because this effect works positive for the advantageous firm, competitive firms try to invest aggressively. Thus the problem here is which effect dominates the other. On the topic of brand proliferation as a strategic tool, one of the most cited studies, Judd (1985), which analyzes an entry and exit problem by using a spatial model, refers that market competition discourages the firms from proliferating their brands. This theoretical expectation has been widely supported in the literatures on brand proliferation and entry deterrence. 12 1 For instance, Belleflamme and Peitz (2010, p. 417), one of the latest textbooks on industrial organization, states, If the incumbent can withdraw its product at sufficiently low cost from a segment in which it faces a direct competitor, brand proliferation is not a credible strategy for entry deterrence. This is the essence of Judd (1985). 2 In this point, Ishibashi (2003) implies the possibility that competitive pressure stimulates a monopolist to proliferate their brand for entry deterrence. Ishibashi (2003) supports 2

Denicolò et al. (2007) compare the incentive of an incumbent to release a low type with a high type in monopoly and the one in duopoly, and derive the fact that a monopolist does not provide a low type good in addition to a high type good, in contrast with a duopoly where an efficient firm provides for predation. In this meaning, Denicolò et al. (2007) imply the fact that competition encourages an existing firm to proliferate its brand. This result is just what we derive in the paper. However, Denicolò et al. (2007) assume two key points that i) there is a cost difference between the incumbent to provide two brands and the one of the entrant, ii) only the efficient firm has the option to provide an additional good. In Denicolò et al. (2007) s setting, it is apparent that these assumptions are definitively important in the result. Thus, the pure effect of brand proliferation on the strategic interaction disappears when the cost difference is zero, so any firm no longer proliferates its brand because of the cannibalization effect. 3 The second, (ii) is purely a theoretical question whether intensifying competition (increasing in the degree of product differentiation) always makes firms less profitable. This type of question was firstly brought up by Zanchettin (2006) where the efficient firm may increase its profit by providing a closer substitute. The paper follows the argument and extends to a symmetric but multi-product case. In other words, the paper conducts the proliferating strategies of identical firms instead of asymmetric costs. The structure of our study is so simple: first deriving the incentive of a monopolist to provide an additional variety and that of duopolists, then comparing those incentives. The specification of the model is following. i) Monopoly case: in the first stage, a monopolist decides whether to proliferate its brand and become a multi-product firm (MPF) by paying the fixed cost, then the firm sets quantity to maximize its profit in the second stage. 4 Similarly, ii) duopoly case: another identical firm exists. At stage one, firms make their decisions of brand proliferation simultaneously. At stage two, firms compete with each other in Cournot fashion (sub-game perfect Nash equilibrium). The only difference between two cases is whether an identical rival firm exists. By setting so, we can identify the pure effect of the existence of the opponent. The characteristic of the model is on the inverse demands. Sometimes in researches on differentiated oligopoly (including duopoly), all the relation between any two differentiated goods are represented by a single parameter of product differentiation. Such an assumption helps the models be tractable. But at the same time, it restricts the situation tightly in the case more than three Schmalensee (1978) s suggest and indicates a counter example using a strategic interaction between two entrants. However, still, once a rival firm succeeds to enter the market, the incumbent no longer has an incentive to proliferate brands. From this point of view, Ishibashi (2003) supports the fact that for existing firms, the existence of the rivals works as a disincentive to proliferate brands. 3 The cannibalization effect is the only characteristic of the multi-product firm (MPF) which provides multiple differentiated goods. Since an MPF provides similar goods, it faces share reductions in each product it provides even if the MPF is a monopolist. 4 The representative models of MPFs in oligopoly are, for example, Lin (2004), Grossmann (2007), Lin and Zhou (2013). 3

goods are provided. Under the restriction, we cannot focus on the effect by the relativity between two factors. In this point, our model allows heterogeneity that the degree of product differentiation between firm 1 s good A and B differs from the one between firm 1 s good A and 2 s A. We call the former the cannibalization effect, and the latter the competition effect. relative magnitudes of all the parameters themselves may radically change each interaction between any two goods. This type of demand system has been well-known by Dobson and Waterson (1996, 2007) and Zhao and Xing (2008) etc. After all, we show that (i) if the competition effect is sufficiently high and the cannibalization effect is much small, there exist asymmetric equilibria in which only one of the competitive firms proliferates and the other does not. In this case, the proliferating firm one-sidedly increases its profit whereas the rival loses its market share widely. Also, ii) the MPF s incentive for brand proliferation is stronger in duopoly than in monopoly. Such a result implies the possibility that given the productivity, the existence of the rival firm accelerates brand proliferation. In addition, iii) we find that intensifying competition, that is, an increase in the competition effect yields the MPF and decreases the profit of the rival firm if the cannibalization effect relatively small. The intuition behind the results is that the goods provided by a firm work as if those are complements each other when the rival exists. In contrast, two goods provided by a monopolist always act as substitutes. Besides, the reason why there is a possibility that only one firm proliferates its brand, even though the firms are identical in the setting, derives from the fixed cost and the concept of strategic substitute. In the first stage, given the strategy of its rival, each firm chooses whether to proliferate its brand or not. So in addition to strategic substitute, if the fixed cost is not too low and not so high, brand proliferation is no longer than the best response to the rival s strategic brand proliferation, and hence the above asymmetric equilibria can emerge. The situation that this result implies is here: in Japan, at a large station, it is usual to see multiple cafe brands, like Starbucks, Tully s, around it. Especially, one is near the south entrance (call this S) and the other next to the north entrance (N 1 ). In this situation, when searching a coffee shop to take a rest, consumer faces a choice between N 1 or S to come in. Besides this, we often see that another branch of a brand (N 2 ) whose former branch S is at south opens at the north entrance where the rival N 1 exists. For consumers of coffee shops, such a situation may make them firstly choose a cafe brand (e.g. Starbucks or Tully s) and then the branch of the brand they chose, that is, S or N 2. In this case, it is natural to interpret that the subjective distances between two cafe brands are same and the cannibalization effect between two goods of an MPF can be weaker than the competition effect between two goods of firms. That is, this proliferation strategy should be harmful for the firm (because of the cannibalization effect) but maybe more for the rivals (because of the competition effect). So the total effect becomes positive and the MPF obtains higher profit because of the strong business-stealing effect. Of course, the cafe market around a station is perhaps so large and therefore a new entry may not hurt any shop. However, cases matching the logic we mentioned above seem 4

quite common in our daily lives. There are further studies related to our paper. Tabuchi (2012) investigates the incentives of the firms under Hotelling s situation. By using the feature of Hotelling s model, Tabuchi (2012) indicates that under the case three or more firms exist, firms proliferate their brands because of the consumers costs of moving, although there is no incentive to proliferate their brands under one or two firms. Tabuchi (2012) is similar to the paper in that competition encourages the firms to proliferate their brands. An empirical study by Kadiyali et al. (1998) shows that brand proliferation by a duopolist in the yogurt industry yields market expansion and therefore both firms raise their demands. Similarly, Draganska and Jain (2005) extend the argument of Kadiyali et al. (1998) and indicate that brand proliferation by oligopolists in the yogurt industry may mitigate price competition. These findings highlight the fact that brand proliferation acts as a strategic tool to yield the proliferating firm a profit, although the cannibalization effect hurts the firm. The remainder of this paper is as follows. In Section 2, we present the basic model and analyze the equilibrium in monopoly and duopoly. In Section 3, we compare the firm s incentives to provide another good in monopoly and duopoly. In Section 4, we extend the basic model to a symmetric option case, and then generalize the main result. Finally, we conclude this paper. 2 The Basic Model In this section, we present a model to determine sure whether a firm s incentive to provide another differentiated good is stronger in duopoly than in monopoly. First, we set up a two-stage monopoly model in which the monopolist, Firm 1, chooses whether to become a multi-product firm (MPF) by paying fixed cost F or remain a single-product firm (SPF) in the first stage, and then decides the quantity level in the second stage. Second, we provide a differentiated duopoly model where there exists a rival firm, Firm 2. Then, by comparing the former with the latter, we derive the condition to accomplish the purpose of the analysis. For simplicity, we assume that only Firm 1 has the option to provide a new variety of good and become an MPF in this section (this assumption will be relaxed in Section 4). This assumption can be interpreted as a situation in which Firm 2 faces some financial restriction or has no technology to create a new variety. Firm 1 provides goods A and B. Let q A and q B denote the quantities of good A and B respectively. Similarly, q 2 indicates the quantity of good 2 provided by Firm 2. The inverse demand functions are as follows. 5 p A =a q A γ AB q B γ A2 q 2, (1) p B =a q B γ AB q A γ B2 q 2, (2) p 2 =a q 2 γ A2 q A γ B2 q B, (3) 5 This type of inverse demand function is also used in Zhao and Xing (2006, 2008). 5

where a is the intercept of the inverse demand for each demand. γ ij (0, 1) indicates the degree of product substitutability between goods i and j for i, j {A, B, 2}, i j. If there exists no rival, q 2 = 0. Similarly, q B = 0 in the single-product firm case. 2.1 Monopoly Case First, this paper examines the monopoly case. After solving two profit maximization problems of a monopolist (second stage), Firm 1 compares both profits and chooses the more profitable one (first stage). 2.1.1 Second Stage: Quantity Decision In case of the single-product firm, the profit function of Firm 1 is π S m(q A ) = (a q A c)q A, (4) where c > 0 is the constant marginal cost. The equilibrium quantity and profit of single-product firm are q S m = a c 2, πs m = ( a c 2 )2. (5) On the other hand, in case of the multi-product firm, the profit function is π M m (q A, q B ) = (a q A γ AB q B c)q A + (a q B γ AB q A c)q B F, (6) where F is the fixed cost for the entry of good B. γ AB implies the degree of product substitutability between goods A and B, which measures the cannibalization effect. The MPF s equilibrium quantities and profit are q A = q B = π M m (γ AB ) = a c 2(1 + γ AB ), (7) 2 ( a c 1 + γ AB 2 )2 F. (8) By eq. (8), it is clear that the MPF s quantities and profit are the decreasing function of γ AB. 2.1.2 First Stage: MPF or SPF? In the first stage, Firm 1 chooses whether to create an additional differentiated good: SP F if πm M (γ AB ) πm S < 0, indifferent if πm M (γ AB ) πm S = 0, MP F if πm M (γ AB ) πm S > 0. 6

These conditions can be rewritten as SP F if F m (γ AB ) < F, SP F ormp F if F m (γ AB ) = F, MP F if F m (γ AB ) > F, (9) where F m (γ AB ) 1 γ AB 1+γ AB ( a c 2 )2. F m (γ AB ) also decreases in γ AB. 2.2 Duopoly Case Next, the case where there is a rival firm, Firm 2, is presented. For simplicity, we assume that only Firm 1 has the option to become an MPF. 2.2.1 Second Stage: Cournot Competition The profit functions are π S 1 = (a q A γ A2 q 2 c)q A, (10) π S 2 = (a q 2 γ A2 q A c)q 2, (11) where γ A2 denotes the degree of product substitutability between goods A, provided by Firm 1, and good 2, provided by Firm 2. 6 γ A2 measures the competition effect between the firms. The equilibrium quantities and profits are q A = q 2 = a c 2 + γ A2, (12) π S 1 = π S 2 = ( a c 2 + γ A2 ) 2. (13) Similarly, if Firm 1 becomes a multi-product firm, the profit functions are π M 1 =(a q A γ AB q B γ A2 q 2 c)q A + (a q B γ AB q A γ B2 q 2 c)q B F, (14) π M 2 =(a q 2 γ A2 q A γ B2 q B c)q 2, (15) where γ B2 indicates the degree of product substitutability between good B and good 2. In the following, to focus on the relation between the degree of the competition effect (γ A2 and γ B2 ) and that of the cannibalization effect (γ AB ), we assume γ A2 = γ B2 and redefine these parameters as γ 12. That is, γ A2 = γ B2 γ 12. 7 This redefined parameter implies the degree of product substitutability between the goods of Firm 1 and those of Firm 2. 6 If γ A2 = 0, these two firms are the different market monopolists. In addition, if γ A2 = 1, the firms provide homogeneous goods. 7 In the sense that both goods A and B are produced by Firm 1 and only good 2 is produced by firm 2, γ A2 and γ B2 can be summarized in a competition effect. 7

The equilibrium quantities and profits are qa M = qb M = (2 γ 12)(a c) 2(2 + 2γ AB γ12 2 ), qm 2 = (1 + γ AB γ 12 )(a c) 2 + 2γ AB γ12 2, (16) π M 1 = (1 + γ AB)(2 γ 12 ) 2 (a c) 2 2(2 + 2γ AB γ 2 12 )2 F, π M 2 = (1 + γ AB γ 12 ) 2 (a c) 2 (2 + 2γ AB γ 2 12 )2. 2.2.2 First Stage: MPF or SPF? (17) In the first stage, Firm 1 in the differentiated duopoly chooses whether to enter an additional differentiated good: SP F if π1 M (γ 12, γ AB ) π1 S (γ 12 ) < 0, indifferent if π1 M (γ 12, γ AB ) π1 S (γ 12 ) = 0, MP F if π1 M (γ 12, γ AB ) π1 S (γ 12 ) > 0. These conditions can be rewritten as SP F if F d (γ 12, γ AB ) < F, SP F ormp F if F d (γ 12, γ AB ) = F, MP F if F d (γ 12, γ AB ) > F, (18) where F d (γ 12, γ AB ) (1 γ AB)(8 + 8γ AB γ 4 12) 2(2 + 2γ AB γ 2 12 )2 (2 + γ 12 ) 2 (a c)2. 3 Does Quantity Competition encourage a Duopolist to become an MPF? In this section, we compare Firm 1 s incentive to provide another variety between the monopoly case and the duopoly case. Since the purpose of this paper is to determine whether there exists a case where quantity competition encourages Firm 1 to provide another differentiated good, we combine two conditions (11) and (20). The condition where a duopolist, Firm 1 provides another differentiated good and becomes an MPF but monopolistic Firm 1 does not is F d (γ 12, γ AB ) F F m (γ AB ). (19) This inequality implies the parameter condition that Firm 1 s incentive to provide differentiated good is larger in duopoly than in monopoly. Likewise, since F d (0, γ AB ) = F m (γ AB ), the above condition can be rewritten as F d (γ 12, γ AB ) F F d (0, γ AB ). (20) Hence, what we investigate hereafter is summarized as follows: whether the existence of quantity competition enhances the profitability of MPF. To verify 8

this, we first partially differentiate F d (γ 12, γ AB ) with respect to γ 12, then find the conditions of γ 12 and γ AB where the derivative is positive: F d (γ 12, γ AB ) > 0 (21) As a consequence, we obtain Proposition 1 and Corollary 1. Proposition 1 (i) For γ 12 (2 2(1 γ AB ), 1) and γ AB (0, 1 2 ), the profit of an MPF is monotonically increasing in the competition effect. In addition, (ii) the profit difference between MPF and SPF, F d (γ 12, γ AB ), is also monotonically increasing in the competition effect. Proof of Proposition 1 F d (γ 12, γ AB ) = πm 1 πs 1. (22) Apparently, the second term of RHS ( πs 1 ) is positive. Thus, the remaining problem is only the sign of the first term: π1 M = πm 1 (q 1, q 2, γ 12, γ AB ) q 2 (γ 12, γ AB ) (23) q 2 + πm 1 (q 1, q 2, γ 12, γ AB ) =2q 1 ( γ 12 q 2 (γ 12, γ AB ) q 2 ), (24) where the first term is positive (Business-stealing effect) and the second term is negative (price down effect or market shrink effect). If the former effect dominates the latter, the sign of πm 1 becomes positive. It is a function of γ 12 and γ AB : 2(γ 12 q 2 + q 2 ) > 0 γ 12 (γ 12 4) 2(1 + γ AB ) > 0. (25) Q.E.D. The implications of Proposition 1 are as follows. (i) As the competition between the firms becomes more intense, MPF yields more profit. This result is similar to that of Zanchettin (2006), who indicates that an intense competition may benefit an efficient firm. However, our result holds even if firms costs are symmetric. (ii) There exists the possibility that the existence of quantity competition gives more benefit to a duopolist that provides another differentiated good. Corollary 1 The cross-derivative of the incentive in duopoly, F d (γ 12, γ AB ), is negative: 2 F d (γ 12,γ AB ) γ AB < 0. 9

Corollary 1 implies that the benefit from a marginal decrease in the degree of the cannibalization effect, γ AB, is larger in duopoly than in monopoly. Accordingly, Proposition 1 and Corollary 1 yield a remarkable consequence: if the cannibalization effect is sufficiently low, a higher competition effect makes the MPF more profitable. Based on these, Proposition 2 is derived. Proposition 2 One of the duopolists wants to provide a differentiated good more aggressively in duopoly than in monopoly if the competition effect, γ 12, is sufficiently large but the cannibalization effect, γ AB, is sufficiently small. Proof of Proposition 2 Take the limits γ 12 1 and γ AB 0. Then, eq. (22) is satisfied in the neighborhood of (γ 12, γ AB ) = (1, 0) since F d (γ 12, γ AB ) and F m (γ AB ) are continuous. Further, Corollary 2 is derived from Proposition 2. Q.E.D. Corollary 2 The brands provided by an MPF competing with a rival that provides a close substitute act as complements as long as Proposition 2 holds: nevertheless, they act as substitutes in monopoly. Proof of Corollary 2 Transform the inverse demand function of good A (B) into the demand function and differentiate it with respect to the price of good q B (A). Then, in duopoly, i p j < 0, i, j {A, B}, i j, if Proposition 2 holds. In contrast, in monopoly, the sign is always opposite. Q.E.D. The intuition behind Proposition 1 is as follows. If γ 12 is sufficiently large, the driving out power becomes stronger. On the one hand, if γ AB is small enough, Firm 1 s goods A and B do not cannibalize each other too much. As a result, these two effects improve only Firm 1 s position, and yield it higher profit (and reduce Firm 2 s profit). Such an effect never occurs in monopoly. This implies that the brands provided by the MPF do not act as substitutes, but as complements to each other. This can explain the case in which a firm that competes with its rivals intensively attempts to provide multiple similar brands. In addition, the following result is also derived from Proposition 2. Corollary 3 If the degree of the competition effect equals that of the cannibalization effect (γ 12 = γ AB ), there exists no case that a duopolist wants to add another good more aggressively than a monopolist. This indicates the importance of the separation of the measure of product substitutability into the measures of two effects, the competition effect and the cannibalization effect. 10

4 Symmetric option case In this section, we generalize the basic model to a symmetric option case. The assumption in which only one of the two firms has the option to provide another good is relaxed in this subsection. That is, each firm has the option. In this case, each firm chooses its strategy in the first stage (Table 1). Firm 2 Firm 1 MPF SPF MPF SPF π1 MM, π2 MM π1 MS, π2 SM π1 SM, π2 MS π1 SS, π2 SS Table 1: MPF or SPF? Note that the consequence of the asymmetric case is shown in the previous section. Therefore, the remaining problem in this subsection is the result of MPF vs. MPF (π1 MM vs. π2 MM ). If both firms choose to provide good A with good B in the first stage, the inverse demand functions that the two firms face in the second stage are given by The profit functions are p 1A = a q 1A γ AB q 1B γ 12 (q 2A + q 2B ), (26) p 1B = a q 1B γ AB q 1A γ 12 (q 2A + q 2B ), (27) p 2A = a q 2A γ AB q 2B γ 12 (q 1A + q 1B ), (28) p 2B = a q 2B γ AB q 2A γ 12 (q 1A + q 1B ). (29) π1 MM = (p 1A c)q 1A + (p 1B c)q 1B F, (30) π2 MM = (p 2A c)q 2A + (p 2B c)q 2B F. (31) The equilibrium quantity and profit are, respectively, q 1A = q 1B = q 2A = q 2B = a c 2(1 + γ AB + γ 12 ), (32) π1 MM = π2 MM = (1 + γ AB)(a c) 2 F. (33) 2(1 + γ 12 + γ AB ) 2 After that, in the first stage, each firm chooses its strategy (MPF or SPF) given the opponent s strategy. Then, we likewise compare the incentive in duopoly with a rival MPF to the one in monopoly. The condition is described by an inequality as follows. F MM (γ 12, γ AB ) F F MM (0, γ AB ) (34) where F MM (γ 12, γ AB ) = π MM 1 π SM 1 + F and F MM (0, γ AB ) = F m (γ AB ). As a consequence, we obtain the following result. 11

Proposition 3 If the rival firm also provides multiple goods (MPF vs. MPF), then the incentive to provide another variety is always less in duopoly than in monopoly. Proof of Proposition 3 Suppose F MM (γ 12, γ AB ) F F MM (0, γ AB ). However, for γ 12, γ AB [0, 1], the maximum value of F MM (γ 12, γ AB ) F MM (0, γ AB ) is 0 if γ 12 = 0 or γ AB = 1. Therefore, γ 12, γ AB (0, 1), F MM (0, γ AB ) > F MM (γ 12, γ AB ). This contradicts the above inequality. Q.E.D. Then, we show that asymmetric equilibria can exist in the first stage. In the previous section, we suppose that only one of the firms has the option of brand proliferation. Nevertheless, in this subsection, because of the symmetry of the option, a symmetric equilibrium in which both firms proliferate their brands can be shown. However, at the same time, there can also exist two asymmetric equilibria because of the fixed costs. Thus, we now need to ascertain the parameter domain of the fixed cost that holds the asymmetric equilibria. If it is non-empty, we can see the generality of the consequence of the previous section. As supposed in the previous section, we assume anew that F MS (γ 12, γ AB ) F where F d (γ 12, γ AB ) = F MS (γ 12, γ AB ) = π1 MS π1 SS + F. If so, the thing to be shown is whether an inequality π MS 1 π SS 1 π SM 1 π MM 1 (35) holds under F MS (γ 12, γ AB ) F F m (γ AB ). The first half of the inequality, π1 MS π1 SS, is the incentive to proliferate the brand given the opponent s strategy for an SPF. Likewise, the second half of the inequality, π1 SM π1 MM, implies the disincentive to provide another variety B given Firm 2 s strategy for an MPF. Then, π1 SS π1 SM. We must now prove that the incentive condition (F MS (γ 12, γ AB ) F ) is consistent with the disincentive condition (F F MM (γ 12, γ AB )). By Proposition 2 and Proposition 3, we obtain Proposition 4. Proposition 4 If the competition effect is sufficiently large and the cannibalization effect is sufficiently small, the asymmetric equilibria where one of the firms proliferates its brand and the other remains an SPF exist even if the rival firm also has the option of brand proliferation. Proof of Proposition 4 By Proposition 3, γ 12, γ AB (0, 1), F m (γ AB ) > F MM (γ 12, γ AB ). Hence, F MS (γ 12, γ AB ) F F m (γ AB ) > F MM (γ 12, γ AB ) holds in the neighborhood of (γ12, γab ) = (1, 0). Q.E.D. In contrast, if the fixed cost is relatively low, the disincentive condition may not hold. In this case, the MPF strategy is the dominant strategy for 12

both players, and hence (MPF, MPF) is the unique Nash equilibrium. In other words, if the cost is relatively high, eq. (35) holds and quantity competition may encourage one of the firms to provide an additional variety in the case of a symmetric option in addition to the asymmetric option in Section 3. Thus, from the above argument, we can conclude that our main proposition (Proposition 2) is persuasive if the fixed costs are relatively high. 5 Concluding Remarks In this paper, we investigate the strategic effect of the brand proliferation. Since the competition effect and the cannibalization effect have been described by an identical parameter in most previous works, an increase in the degree of product substitutability among the goods tends to mean that the incentive to provide another variety and become an MPF is less in duopoly than in monopoly. However, in this paper, it is seen that the effects are separated. As a result, we find that if the competition effect is sufficiently high and the cannibalization effect is sufficiently low, the power of predation increases since the two brands act as complements in duopoly but as substitutes in monopoly. Thus, in this case, the existence of a rival firm encourages one of the firms to provide another differentiated good and become an MPF. This paper constrains the game of brand proliferation to be one-shot. However, in reality, the strategic interaction is continuous: the rival SPF ought to respond to the strategy. Thus expansion to a sequential proliferation model is a subject for future research. References [1] Belleflamme, Paul and Peitz, Martin, Industrial Organization: Markets and Strategies Cambridge University Press (2010) [2] Denicolò, Vincenzo., Polo, Michele., and Zanchettin, Piercarlo., Entry, Product Line Expansion, and Predation Journal of Competition Law and Economics, 2007, 3(4): pp. 609-624 [3] Dixit, Avinash, A Model of Duopoly Suggesting a Theory of Entry Barriers The Bell Journal of Economics Vol. 10 (1979): pp. 20-32 [4] Dobson, Paul W., and Michael Waterson. Product range and interfirm competition. Journal of Economics & Management Strategy 5.3 (1996): 317-341. [5] Dobson, Paul W., and Michael Waterson. The competition effects of industry-wide vertical price fixing in bilateral oligopoly. International Journal of Industrial Organization 25.5 (2007): 935-962. 13

[6] Eckel, Carsten and Neary, Peter J., Multi-Product Firms and Flexible Manufacturing in the Global Economy The Review of Economic Studies Vol. 77 (2010): pp. 188-217 [7] Grossmann, Volker, Firm Size and Diversification: Multiproduct Firms in Asymmetric Oligopoly International Journal of Industrial Organization Vol. 25 (2007): pp. 51-67 [8] Ishibashi, Ikuo A Note on Credible Spatial Entry Deterrence International Journal of Industrial Organization Vol. 21 (2003): pp. 283-289 [9] Jing, Bing and Zhang, John Z., Product line competition and price promotions Quantitative Marketing and Economics Vol. 9 (2011): pp. 275-299 [10] Judd, Kenneth L, Credible Spatial Preemption The Rand Journal of Economics Vol. 16 (1985): pp. 153-166 [11] Kadiyali, Vrinda, Vilcassim, Naufel, and Chintagunta, Pradeep, Product Line Extensions and Competitive Market Interactions: An Empirical Analysis Journal of Econometrics Vol. 89 (1998): pp. 339-363 [12] Klemperer, Paul, Equilibrium Product Lines: Competing Head-to-Head May be Less Competitive The American Economic Review Vol. 82 (1992): pp. 740-755 [13] Klemperer, Paul and Padilla, Jorge A., Do Firms Product Lines Include Too Many Varieties? The Rand Journal of Economics Vol. 28 (1997): pp. 472-488 [14] Lin, Ping Process and Product R&D by a Multiproduct Monopolist Oxford Economic Papers Vol. 56 (2004): pp. 735-743 [15] Lin, Ping, and Wen Zhou. The effects of competition on the R&D portfolios of multiproduct firms International Journal of Industrial Organization (2013). [16] Schmalensee, Richard, Entry Deterrence in the Ready-to-Eat Breakfast Cereal Industry The Bell Journal of Economics Vol. 9 (1978): pp. 275-289 [17] Tabuchi, Takatoshi, Multiproduct Firms in Hotelling s Spatial Competition Journal of Economics & Management Strategy Vol. 21 (2012): 445 467 [18] Xing, Yuqing, and Zhao, Laixun, Reverse Imports, Foreign Direct Investment and Exchange Rates Japan and the World Economy Vol. 20 (2008): pp. 305-327 [19] Zanchettin, Piercarlo, Differentiated Duopoly with Asymmetric Costs Journal of Economics and Management Strategy Vol. 15 (2006): 999-1015 14

[20] Zhao, Laixun and Xing, Yuqing Global Production and Currency Devaluation Review of International Economics Vol. 14 (2006): pp. 202-211 15