Should Brand Firms Always Take Pioneering Position?

Transcription

1 Shoud Brand Firms Aways Take Pioneering Position? Cong Pan Graduate Schoo o Economics, Osaka University March 15, 2015 Abstract This paper discusses brand irms endogenous timing probem when acing nonbrand irms under quantity competition. We study a market comprising brand and nonbrand products. There exist heterogeneous consumer groups-one group buys ony brand products whie the other one cares itte about the brand. These two consumer groups constitute the high- and ow-end markets respectivey. The brand irms moving order is endogenized, whereas the nonbrand irms are restricted to move in a ater period. We show that i the ow-end market is o an intermediate size, the eader-oower equiibrium outcome occurs, and the eader s proit is ower than the oower s, and the increasing number o nonbrand irms increases the eader s proit but decreases the oower s proit. These resuts oow rom the act that each brand irm s best response unction has an upward ump i the riva s output exceeds a particuar eve. Thus, the eader s proit unction has a downward ump at some particuar point whie the oower s proit does not. The resuts put orward the working o endogenous timing and second mover advantage. Keywords: Marketing strategy; Endogenous timing; Market separation; Second mover advantage; Game theory JEL Cassiication: D21, D43, L10, L13 I woud ike to express my sincere thanks to Noriaki Matsushima or his constructive suggestions and comments on this paper. Besides, I reay appreciate Toshihiro Matsumura, Ikuo Ishibashi, Junichiro Ishida, Tatsuhiko Nariu, Keisuke Hattori s kind hep and suggestions. A errors are mine. Address: 1-7 Machikaneyama, Toyonaka, Osaka , Japan. E-mai: u510590g@ecs.osaka-u.ac.p 1

2 1 Introduction When an estabished irm creates a new product, it gets a chance to choose a pioneering position over its competitor i it presents this new product or sae as eary as possibe. However, there are some rea-word cases in which irms with branded products gave up their pioneering position and intentionay reeased their new product ater their competitors. For exampe, Adidas created Adidas 1, its high-end sports equipment, in However, Adidas postponed its reease and kept sient unti it created its next generation product in 2006, right ater Nike reeased a simiar high-end product, Nike+. 1 In the SmartWatch market, rumors about Appe s iwatch had been spread on the Internet or years. Even the testing machine o iwatch had been spotted aong with its detaied description ong beore Samsung estabished its simiar product-gaaxy Gear-in Berin (September, 2013). Unexpectedy, Appe then reeased their iwatch one year ater. 2 In the above two cases, we see that the brand irms deayed the reease o new, market-ready products. These cases are abnorma because peope conventionay think that a irm shoud take action as eary as possibe to gain advantage. 3 One speciic commonaity in the above two cases is that whie big irms with brand products compete or the high-end market, there aso exist reativey sma irms suppying nonbrand products to the ow-end market. 4 Thereore, whether conventiona thinking ( seeking pioneering position is necessary ) sti makes sense under the above speciicity remains unknown. This paper discusses brand irms endogenous timing probem when they ace nonbrand irms. We consider a three-period game with two brand irms and n nonbrand irms. In the pre-determinate period (period 0), each brand is authorized to choose between moving in period 1 or 2. Each brand irm then decides its production eve according to the timing choices made in period 0. The nonbrand 1 Businessweek (May, 2006) mentioned that Nike and Appe started to suppy their irst ointy produced Nike+ Ipod kit rom that year. 2 Yahoo News (September, 2013); CBS News (September, 2014). 3 Stackeberg (1934) gives one type o expanation to this argument: under duopoy and quantity settings, a irm that moves earier than its competitor can get a arger market share than when it moves simutaneousy with its competitor. It is straightorward that this concusion hods not ony under duopoy but aso oigopoy. Thereore, the pioneering position is advantageous to some extent. Urban et a. (1986) present an empirica anaysis about how pioneering position matters or brand irms. They ind that the earier the brand irm s order o entry, the greater the brand irms ong-term market share. 4 For the irst case, Smith (2010) anayzes how the custer o oca sma irms in China chaenged the might o Nike and Adidas. He describes the sportswear market in China as toe-to-toe, with Adidas and Nike competing with numerous sma domestic irms. For the second case, Forbes Business News (November, 2013) reported that some mobie phone companies in China are heading or the SmartWatch ied. 2

3 irms are restricted to take action ony in a ater stage (period 3). 5 I became inspired by a distinctive market structure introduced by Ishibashi and Matsushima (2009). Here, two heterogeneous consumer groups exist: high-end consumers, who buy ony brand products, and ow-end consumers, who care itte about the brands o products. These two consumer groups respectivey constitute the high- and ow-end markets. There are three types o exogenous parameters: market size, consumer heterogeneity in vauing dierent products, and the number o nonbrand irms. There are two potentia market statuses: brand irms suppy the high-end market at a higher price and nonbrand irms suppy the ow-end market at a ower price separatey; otherwise, brand and nonbrand products are sod at the same price and the high- and ow-end markets integrate. In this setting, two types o symmetric equiibria outcomes can exist under dierent market statuses: when market separation takes pace, both irms choose sma outputs; when the two markets integrate, both irms choose arge outputs. 6 Endogenizing brand irms moving orders enabes the timing equiibrium outcomes to interact with dierent market statuses correspondingy, giving rise to signiicanty dierent resuts compared with the standard endogenous timing game under quantity-setting. I aso ocus on the strategic reation between brand and nonbrand irms. The number o nonbrand irms greaty aects the price in the ow-end market and is the basis on which brand irms decide their outputs. Thus, the activities o nonbrand irms indirecty aect brand irms proits as we. There are two main resuts o this paper. First, i the ow-end market is o an intermediate size, the asymmetric timing outcome (i.e., the eader-oower equiibrium outcome) occurs, and the eader has a ower proit than the oower. Second, the eader s proit increases but the oower s decreases with the increasing number o nonbrand irms. The ogic behind the irst resut is as oows. Due to the existence o the ow-end market, brand irms may exercise their option to overproduce and drive the market price ow enough to enter the 5 Hamiton and Sutsky (1990) insightuy discuss this probem rom two viewpoints: the extended game with observabe deay and that with action commitment. The present paper oows the irst viewpoint. In the extended game with action commitment, instead o simpy announcing the period that each irm woud ike to choose, it chooses an action to which it is then committed. This strand has been widey discussed or quite some time (e.g., Maiath, 1993; Sadanand and Sadanand, 1996; van Damme and Hurkens, 1999). Since it is not the main content o the present paper, I reer to this point ust in sentences. 6 Ishibashi and Matsushima (2009) consider a simiar market structure with the present paper. The brand and nonbrand irms are named as high-end and ow-end irms, respectivey. However, they do not expicity consider the strategic interaction between high-end and ow-end irms and simpy discuss the entry o ow-end irms by assuming that the ow-end market is uy suppied by them and the resuting price in that market is assumed to be zero. 3

4 Figure 1: The Market Structure ow-end market. Thereore, each brand irms best response unction has an upward ump i its riva s output exceeds a certain eve q J. From this property, when brand irms move sequentiay, the eader has to restrict its output to q J in order to maintain the high-end market price at an adequate eve. A tiny excess woud make the oower viciousy raise its output so that the market price coapses rapidy, thus decreasing the eader s proit. However, when brand irms move simutaneousy, because their respective outputs are not restricted by each other, overproducing and ow-end market entrance due to price coapse are more ikey to occur (i one brand irm chooses a arge output, its riva wi aso choose a arge output as a response). When the proitabiity o the ow-end market is ow, sequentia moving enabes brand irms to keep the high-end market price high enough to avoid entering the ow-end market, whereas simutaneous moving eads to a ess proitabe market entrance. I this is the case, the eader-oower equiibrium outcome occurs. At equiibrium, athough the eader restricts output to q J, due to the upward shiting best response unction, the oower need not worry about the price coapsing because it chooses a high output immediatey to compensate or the osses rom the aing price. Thereore, it chooses an output that is arger than the eader s and has a proit higher than the eader. The second resut shows that the competition in the ow-end market aects brand irms proits even when brand irms suppy ony the high-end market. The intuition is as oows. As the number 4

5 o nonbrand irms increases, competition in the ow-end market becomes more intensive, depressing the ow-end market price. Thereore, a price coapse in the high-end market, which eads to market integration, is ess ikey to occur and the eader s output constraint imposed by the oower is aeviated. As a consequence, the eader chooses a higher threshod output q J, and obtains a higher proit. Due to the strategic substitutabiity, the oower chooses ess output and obtains a ower proit. These resuts have two impications or timing strategy. First, new product reease timing is cosey reated to the market s structura eements such as market size. Second, or greater concern, a brand irm may abandon the pioneering position to mitigate head-to-head competition and achieve a higher proit. Some empirica works seem to be consistent with the above arguments. First, Mahaan and Muer (1996) study the IBM case and ind that decisions to introduce a new generation product as soon as possibe or to deay it unti the maturity stage comes are cosey aected by the reative size o the potentia market. 7 Second, Krider and Weinberg (1995) show that in the im industry, some movie companies woud rather deay debut and et their rivas take action irst to avoid head-to-head competition. 8 Athough the main structures o the PC and im industries are not exacty the same as those discussed in the present paper, there is some common ogic to gain with modiication. 9 In reviewing the reevant iterature, I irst discuss the eader-oower equiibrium outcome in the present paper. Hamiton and Sutsky (1990) show that in the strand o observabe deay, the asymmetric timing outcome occurs ony with strategic compementarities. However, this paper provides evidence that the eader-oower outcome can aso exist with strategic substitutabiity. Another reat- 7 As summarized in that study, empirica anaysis inds that IBM postponed the reease o its two mainrame computer generations-360 amiy (integrated circuits) and 370 amiy (siicon chips)-to a very ate stage in the mid-20th century. Athough their main resuts are based on the monopoistic assumption, the authors mention that simiar resuts can aso be deemed under oigopoy settings. 8 Statistica anaysis on the box oice data in the 1990s inds that or avoiding head-to-head competition in the summer and Christmas hoidays, some movie companies et their competitors go irst. 9 Unortunatey, there are imited empirica works investigating the direct reation between irm s pioneering position in reeasing new products and market proitabiity in the market size and consumer heterogeneity in vauating dierent products. For one diicuty, as stated in Ishibashi and Matsushima (2009), it is hard or researchers to access irms proit data; thus, it is hard to measure the magnitude o the proitabiity o a market. For another diicuty, it is ambiguous that to what extent a brand irm postpones its action can be deined as giving up pioneering position. In the present paper, it is the brand irm that chooses to oow its competitor. However, Urban et a. (1986) ind that in some industries, brand irms can deay their actions to dierent extents, i.e., being the irst oower or the second oower means a dierent market share. Despite these diicuties, some indirect empirica works are consistent with the present paper in some reated resuts. Axarogou (2003) anayzes the cycica nature o the timing o new product introductions in U.S. manuacturing. Mukheree and Kadiyai (2008) discusses the reease timing in the DVD market and Engestatter and Ward (2013) study the entry timing in the video games market. 5

6 ed work is by Normann (2002), who discusses the extended game o observabe deay with strategic substitutabiity under asymmetric inormation. Athough eader-oower equiibrium outcomes aso exist in that work, the resut is argey based on the assumption o asymmetric inormation. Thereore, the incentives behind this resut are dierent rom that o the present paper. 10 Our irst resut aso dispays the evidence o second mover advantage. Two reated papers typicay discuss this point. Amir and Stepanova (2006) discusses the endogenous timing probem in a Bertrand-Nash equiibrium case with two irms: one eicient and one ineicient. They ind that the ineicient irm aways has second mover advantage, whereas the eicient one can aso have second mover advantage ony when the eiciency dierence is not that arge. Juien (2011) studies a Cournot sequentia moving case with many eaders and oowers and without considering the endogenous timing probem. When the payers have equa margina costs, each oower has a higher proit than each eader ony i the oower s reaction unction increases in the eader s quantity. In both these works, the increasing reaction unction o the oower is a necessary condition or second mover advantage. However, in this paper, the oower has a reaction unction that is decreasing amost everywhere, except or the upward umping point at equiibrium, athough it obtains a higher proit than the eader. As this paper reers oten to Ishibashi and Matsushima (2009), we share the common concusion that coordination aiure occurs when both brand irms choose arge outputs in order to enter the ow-end market. However, the ocus o Ishibashi and Matsushima is the strategic interaction between brand irms, which is dierent rom the ocus o the present paper. 11 The main ocus here is on brand irms endogenous timing probem and the strategic interaction between brand and nonbrand irms. We extend Ishibashi and Matsushima s (2009) basic mode by adding severa new insights. The remaining paper is organized as oows. In Section 2, we introduce kinked inverse demand unctions. In Section 3, we derive nonbrand irms equiibrium outcomes in period 3 and two cases o 10 There are other works discussing endogenous timing under incompete inormation. Unortunatey, I ind Normann (2002) to be the ony work that beongs to the extended game with observabe deay. Maiath (1993) is the eary work o Normann (2002) that studies amost the same market mode but appies the extended game with action commitment. Severa others consider the situation where market uncertainty vanishes i irms choose to deay actions (e.g., Spencer and Brander, 1992; Sadanand and Sadanand, 1996). For other parae works, Amir and Grio (1999), Yang et a. (2009), and Juien (2011) discuss the strategic rivary between irms; Matsumura and Ogawa (2009) study the reationship between payo and risk dominance. 11 The high-end irms and ow-end irms in Ishibashi and Matsushima (2009) are reerred to as brand irms and nonbrand irms, respectivey, in this paper. 6

7 subgames in which brand irms obtain dierent equiibrium outcomes according to their endogenous timing choices in period 0. In each subgame, I derive parameter ranges within which two types o equiibrium outcomes exist. Then I derive a subgame perect Nash equiibrium in which the eaderoower timing outcome occurs and introduce the reating propositions. Section 4 concudes the paper and the Appendix is presented in Section 5. 2 Theoretica Mode The basic mode takes the spirit o Ishibashi and Matsushima (2009). We consider a demand unction with strategic substitutabe products. There are two brand irms and n nonbrand irms producing brand products and nonbrand products respectivey. A o these products are homogenous or use. For simpicity, we assume the margina cost to be zero, and there is no ixed cost. Here, ony the quantity competition is considered. Let q B i (i = 1, 2) and q N ( = 1,..., n) be the output eve o brand irm and nonbrand irm, respectivey. Denote Q B i qi B, QN q N. We assume that there exists one group o consumers B, who ony buy brand products, and another group o consumers N, who buy whatever product is cheaper. We aso deine the market acing consumer B as the high-end market and that acing consumer N the ow-end market. We assume consumer B s vauation toward brand products to be uniormy distributed on [0, 1] and the size o the high-end market to be 1; consumer N s vauation is uniormy distributed on [0, a], where 0 < a < 1. The ow-end market size is assumed to be b. Let P B be the price o brand products, which is decided by the demand rom the high-end market. Let P N be the price o brand products, which is decided by the demand rom ow-end market. The demand unctions o the high-end and ow-end markets are as oows: D B (P B 0 i P B (1, ) ) = 1 P B i P B [0, 1]. 0 i P D N (P N N (a, ) ) = b(1 PN a ) i PN [0, a]. I P B P N, then brand irms suppy the high-end market and nonbrand irms suppy the ow-end market separatey. I P B < P N, then brand and nonbrand products are sod at the same price, 7

8 which is decided by the aggregate demand rom the high- and ow-end markets. We deine q = (q1 B, qb 2 ; qn 1,..., qn n ). The inverse demand unctions are as oows: 1 Q P B B i 1 Q B a(1 QN b ) (q) = a[1 + b (Q B + Q N )] otherwise. a + b a(1 QN P N b ) i 1 QB a(1 QN b ) (q) = a[1 + b (Q B + Q N )] otherwise. a + b Let πi B(q) and πn (q) denote the proit unction o brand irm i, and nonbrand irm, respectivey. Each irm s proit unction can be expressed as oows: (1 Q πi B (q) = B )qi B i 1 Q B a(1 QN b ) (1) a[1 + b (Q B + Q N )] qi B otherwise, a + b a(1 QN π N (q) = b )qn i 1 Q B a(1 QN b ) (2) a[1 + b (Q B + Q N )] q N otherwise. a + b Let s consider a game with 4 periods. In period 0, each brand irm is authorized strategic options between moving in period 1 or 2. Each irm then decides its production eve according to the timing choices made in period 0. It is noteworthy that when sequentia moving is decided, the brand irm moving in period 2 observes the outcome in period 1. Nevertheess, the remaining nonbrand irms can ony move in period 3. 3 Resut The dierent timing choices give rise to three dierent cases as oows: I, one brand irm moves in period 1 and the other moves in period 2; II, both brand irms move in period 1; III, both brand irms move in period 2. For each case, in equiibrium, both irms can choose either sma outputs to maintain market separation, or arge outputs which cause the high- and ow-end markets to integrate. Case II and III are the same situations, as brand irms engage in the simutaneous moving game and nonbrand irms move aterwards. In the next subsection, we derive the equiibrium outcomes o nonbrand irms. 8

9 We irst introduce the oowing assumption: Assumption 1 In a subgame there coud exist mutipe equiibria within the same parameter range. We appy payo dominance to seect the unique equiibrium outcome. That means that we ony seect the equiibrium outputs that bring brand irms higher proits. 3.1 Nonbrand Firms Equiibrium Outcomes From Eq. (2), we sove each nonbrand irm s maximization probem in the simutaneous moving case. Here, due to kinked inverse demand, we sove the probem by two market statuses: separate or integrated. It is noteworthy that, it is aso possibe that the interior soution we obtain in each market status can not perecty make the corresponding conditiona inequaity hod in the proit unction (2), which means that we do not have either type o interior soution (we have a corner soution). Thereore, we need to consider the threshod vaue when P B is equa to P N, or, the case o a corner soution. We denote the three cases, separate, integrate, or corner soution, by superscripts S, I, and C, respectivey. There are three types o symmetric equiibrium outcomes according to the above three cases respectivey: q NS q NS (Q B ) = b n + 1 ; qni (Q B ) = 1 + b QB n + 1 ; q NC (Q B ) = b n 1 QB (1 ). a (Q B ) (respectivey q NI (Q B )) is the interior soution or the maximization probem o π NS (q) = a(1 Q N /b)q N (respectivey π NI (q) = a[1+b (Q B + Q N )]q N /(a+b)). qnc (Q B ) is the corner soution that makes P B = P N. We need to check the condition under which each type o the above equiibrium outcomes is gobay optima. Ater severa cacuations, we obtain each nonbrand irms equiibrium outcome in period 3 (see Appendix 5.1 or detaied cacuations). q NS (Q B ) i Q B 1 a n + 1, q N (QB ) = q NI (Q B ) i Q B ab > 1 an + bn + b, q NC (Q B ) otherwise. (3) 9

10 3.2 Brand Firms Equiibrium Outcomes In period 3, each nonbrand irm s best response is as derived by Eq. (3). The corresponding price o brand products is as oows: P B (Q B ) = 1 Q B i Q B q K 1 a(1 + b Q B ) (n + 1)(a + b) otherwise. ab an + bn + b, (4) Case I: Either Brand Firm Moves in Period 1 We ca the brand irm moving in period 1 the eader and the one moving in period 2 the oower. The eader chooses its output eve q B and the oower chooses its output eve q B. The subscript ( ) denotes the eader (oower). In period 2, we deine q B (q, q ). By substituting the nonbrand irms best response unction, the oower soves its maximization probem by max π B q B (qb ) = (1 Q B )q B i Q B q K, a(1 + b Q B ) (n + 1)(a + b) qb Otherwise, (5) and obtains two types o the best response unctions. x (y) denotes the type o soution derived rom the irst (second) maximization probem o Eq. (5). Pease see Appendix 5.2 or how to derive the oower s best response unction. q B (qb ) = q Bx (q B ) 1 2 (1 qb ) i qb q J 1 b[a + a(a + b)(n + 1)]/(an + bn + b), q By (q B ) 1 2 (1 + b qb ) Otherwise. (6) The corresponding proit is as oows: π B (qb, qb (qb )) = π Bx (q B, qbx (q B )) 1 4 (1 qb )2 i q B q J, π By (q B, qby (q B )) a(1 + b qb )2 4(n + 1)(a + b) Otherwise, (7) where q J is a threshod vaue o q B that satisies π Bx (q B, qbx (q B )) = πby (q B, qby (q B )). When the eader s output q B reaches q J, the oower raises its own output rom (1 q J )/2 to (1 + b q J )/2, 10

11 which eads to a sudden decrease in the high-end market price. By doing so, the oower enabes ow-end consumers to aord brand products and thereby enters the ow-end market. Athough osing rom the price coapse, the oower maintains the same proit eve by the increase in saes when the eader chooses q J. In period 1, based on the oower s best response unction, the eader s inverse demand unction is as oows: P B (q B, qb (qb )) = P x (q B ) 1 2 (1 qb ) i qb q J, P y (q B ) a(1 + b qb ) 2(a + b)(n + 1) otherwise. (8) The resuting proit unction is as oows: π π B x (qb ) = (qb ) 1 2 (1 qb )qb i q B q J, π y (qb ) a(1 + b qb ) 2(a + b)(n + 1) qb otherwise. (9) We can conirm that P x (q J ) > P y (q J ) or any 0 < a < 1, b > 0, n > 1. The inverse demand curve has a downward ump at q J, which is caused by the oower who viciousy raises its output, thus causing the eader to suer a sudden drop in proit. Since the eader oses in proit whie the oower does not, whether to be a eader or a oower matters a ot to the brand irm s proit when the eader reaches the output eve q B = q J. It is straightorward to ind potentiay three types o eader s equiibrium candidates: two types o ocay optima soutions (q x max q B π Bx (q x 1/2 and q y (1 + b)/2), which are derived rom max q B π Bx (q B) and π By (q B), respectivey, and the corner soution at the ump-point q J. The resuting proits are ) 1/8, π By ) a(1+b) 2 /[8(a+b)(n+1)] and π Bx (q J ) (1 q J )q J /2, respectivey. Thereore, we need to check not ony the goba optimaity at the ocay optima point, but aso compare the eader s ocay optima proits, with the proit gained at q J to obtain the goba optimaity. Ater severa cacuations, we obtain three conditions (A, B, and C), under which each type o equiibrium outcome exists. Pease see Appendix 5.3 or how to derive the equiibrium conditions. Deinition 1 A {(a, b, n) q J {(a, b, n) q J q x and π Bx > q x }; B {(a, b, n) q J q x (q J ) < π By )}. and π Bx (q J ) π By )}; C 11

12 n a b 1 b 1 n a b 1 b 1 n a b 1 b Tabe 1: Numericizing Conditions A, B and C. Condition A can be rearranged to b < b 1 ; condition B can be rearranged to b 1 b < b 1 ; and condition C can be rearranged to b b 1, where b 1 and b 1 are decided given a and n. Lemma 1 When either brand irm moves in period 1, brand irms choose (q x, q x ) (1/2, 1/4) i b < b 1 ; (q J, (1 q J )/2) i b 1 b < b 1 ; or, q y ) ((1 + b)/2, (1 + b)/4) i b b 1. The numericized b 1 and b 1 are depicted in Tabe 1. When either irm moves in period 1, (q x, q x ) is picked up as an equiibrium outcome i the owend market size is sma enough (i.e. b < b 1 ). Under this condition, we aways have π Bx max{π Bx (q J ), π By )}. π Bx (q x ) is arger than π Bx eader wi not expand its output rom q x to q y, or π Bx (q J ) because q x (q x ) π By (q x ) is the ocay optima vaue. The ), because the decrease in price outweighs the increase in quantity. When the ow-end market size is reativey arge (i.e., b b 1 ), i the eader chooses q x, then the high- and ow-end markets integrate and the price is given by P y (q x ). q x wi not be seected by the eader because q y the eader chooses q J or q y is decided by comparing the vaues o π Bx is the ocay optima vaue or π y (qb ). Whether (q J ) and π By ). Due to the downward ump point in the inverse demand curve, the eader has a sudden drop in proit when q B ust exceeds q J, athough its proit rises graduay as it urther increases output unti q y. I b is smaer than b 1, the sudden decrease in proit at q J cannot be compensated by the gradua rise as q B increases unti q y (i.e., π Bx urther (q J ) π By )). Then, q J is picked up by the eader. I the ow-end market size is arge (i.e., b b 1 ), then brand irms aways choose to enter the ow-end market. Thus, q y is picked up by the eader. Case II or Case III: Both Brand Firms Move in Period 1 or Period 2 For simpicity, we ony consider the case when both brand irms move in period 1. In period 1, based on the inverse demand in Eq. (4), each brand irm s best response unction is the same as given 12

13 in Eq. (6) except that we substitute the subscript and with i and i, respectivey. Then, we get qi B (qb i ) = q Bx i (q B i ) 1 2 (1 qb i ) i qb i q J, q By i (q B i ) 1 2 (1 + b qb i ) Otherwise. (10) Based on the inverse demand unction in Eq. (4), brand irm i s proit unction is as oows: πi B (qb ) = π Bx c (q B ) (1 Q B )q B i i Q B q K, π By c (q B ) a(1 + b QB ) (a + b)(n + 1) qb i Otherwise. (11) We use subscript c to denote the Cournot case. Because o the kinked inverse demand unction, brand irms proit unctions are decided by their aggregate output. We have the oowing two types o equiibrium outcome candidates: (q x c, q x c ) (1/3, 1/3) with the equiibrium proit πc Bx (q x c ) 1/9; c, q y c ) ((1 + b)/3, (1 + b)/3) with the equiibrium proit, π By c c ) a(1 + b) 2 /[9(a + b)(n + 1)]. We need to ensure that each type o equiibrium outcome satisies the corresponding conditiona inequaity o best response unction (10). It is noteworthy that mutipe equiibria coud exist within the same parameter range. Then, we use Assumption 1 to seect the unique equiibrium outcome. Pease see Appendix 5.4 or how to derive the equiibrium conditions. Ater severa cacuations, we obtain the oowing condition D in which the equiibrium outcome (q Bx c, q Bx ) exists. c Deinition 2 D {(a, b, n) q J q x c }; E {(a, b, n) q J < q x c }. Condition D can be rearranged to b b 2 ; condition E can be rearranged to b > b 2, where b 2 is decided given a and n. Lemma 2 When both brand irms move in period 1 or period 2, they choose (q x c, q x c ) i b b 2 or c, q y c ) i b > b 2. The numericized b 2 is depicted in Tabe 2. From Lemma 2, we see that when both brand irms move in period 1 or period 2, i the ow-end market size is sma enough, brand irms choose sma equiibrium outputs so as to suppy ony the high-end market. This is the irst part o Lemma 2. Conversey, i the ow-end market size is arge enough, brand irms choose arge equiibrium outputs to enter the ow-end market. 13

14 n a b 2 n a b 2 n a b brand irm 2 brand irm 1 eader oower Tabe 2: Numericizing Conditions D and E. (π Bx eader (πc By c ), πc By c )) (q Bx (π Bx oower (q J ), π Bx (q J, q Bx (q J ), q J ), π Bx (q J )) (π By c c ), π By c (q J )) c )) 3.3 Endogenous Timing Tabe 3: When B and E hod. We ocus on the market condition when both conditions B and E hod. To expicity cariy the condition, we rearrange the inequaities in these two conditions and obtain the oowing simpiied condition: b 2 < b b 1. Under the above condition, when both brand irms move in period 1 or period 2, they choose c, q y c ) and when either o them moves in period 1, they choose (q J, (1 q J )/2). The game in period 0 is depicted in Tabe 3. We discuss brand irm 2 s best timing response given brand irm 1 s timing choice. Given brand irm 1 moving in period 2, we consider the eader s proit π Bx π By ). As π By ) is arger than π By c (q J ). From condition B, π Bx (q J ) c ) or any 0 < a < 1, b > 0 and n > 1, we obtain π Bx (q J ) > πc By c ). (12) Thus, irm 2 reacts by moving in period 1. Given irm 1 moving in period 1, we consider the oower s proit π Bx (q J, q Bx (q J )). From condition E, q J is ess than 1/3. In this inequaity, we see that the eader s output q J is smaer than that o the oower s (1 q J )/2, which impies that the oower obtains a higher proit than the eader, or π Bx (q J, q Bx (q J )) π Bx (q J ). Together with inequaity (12), 14

15 b π Bx (q J ) π Bx (q J, q Bx (q J )) πc By c ) Tabe 4: n = 20, a = 0.6, b < we obtain π Bx (q J, q Bx (q J )) > πc By c ). (13) Thereore, brand irm 2 reacts by moving in period 2. By symmetry, brand irm 1 wi respond the same way given irm 2 s timing choice. Sequentia timing equiibria thus exist. Tabe 4 depicts the corresponding vaues o π Bx (q J ), π Bx (q J, q Bx (q J ), and πc By c ) given n = 20, a = 0.6 and b 2 < b b 1. Next, we show how the number o nonbrand irms aects the proits o brand irms i b 2 < b b 1. As the number o nonbrand irms n increases, q J becomes arger. Since q J q x, the eader s proit, π Bx (q J ) = (1 q J )q J /2 increases in n as we. Conversey, since π Bx (q J, q Bx (q J )) = (1 q J ) 2 /4 deceases in q J, the increasing number o nonbrand irms has a counter eect on the oower s proit. Proposition 1 I b 2 < b b 1, then (i) the eader-oower equiibrium outcome exists, (ii) the eader obtains ess proit than the oower, and (iii) the eader s proit increases but the oower s proit decreases as the number o nonbrand irms increases. Figure 2 depicts brand irms best response unctions and isoproit curves. Due to strategic substitutabiity, each brand irm has a higher proit eve toward the coordinate axis. I b 2 < b b 1, then the brand irms reaction curves ony intersect at c, q y c ). The Pareto superior set reative to the Cournot equiibrium output eve is denoted by the shaded area. Since each brand irm s reaction curve enters the Pareto superior set at q B i q J, either brand irm woud be happy to give the pioneering position to its riva. When brand irm 1 achieves the pioneering advantage over brand irm 2, it chooses to produce at the output eve q J, which brings it the highest proit aong irm 2 s reaction curve inside the Pareto superior set. It is noteworthy that Assumption 1 diminishes the possibiity on which asymmetric timing appears. This is because Assumption 1 seects the type o equiibrium outcome that brings brand irms higher proits. This increases the brand irms proits when they move simutaneousy, 15

16 which enhances the incentives or irms to deviate rom the asymmetric timing situation. However, Proposition 1 shows that asymmetric timing appears even though the brand irms earn higher proits picked up by Assumption 1 under the simutaneous moving situation. Figure 2: Brand Firms Reaction Functions and Pareto Superior Set. The intuition o (ii) is as oows. When brand irms move sequentiay, because o the existence o the ow-end market, the oower has the option o overproducing to make the market price ow enough so that it can enter the ow-end market. Thereore, the oower s best response unction has an upward ump when the eader s output surpasses a certain eve q J. A tiny output excess over q J by the eader woud cause the oower to viciousy raise its output and cause the market price to coapse rapidy. Athough this woud bring the eader demand rom the ow-end market, because the ow-end market is sma (i.e., b b 1 ), the increase in saes woud be outweighed by the decrease in price, which does harm the eader. The eader thus restricts its output to q J to maintain the high-end market price. Conversey, the oower does not need to worry about the price coapsing because it chooses a high output at one stroke to compensate or oses rom the dropping prices. Thereore, it chooses an output that is arger than the eader and has a proit higher than the eader. In (iii), the more intensive competition in the ow-end market increases the eader s proit whie decreasing the oower s, athough brand irms do not suppy the ow-end market directy. The ogic behind this resut is as oows. The increasing number o nonbrand irms makes the ow-end market 16

17 more competitive, which depresses the price P N. Thereore, the ow-end market becomes ess proitabe and the market integration is ess ikey to occur. Thus, the eader s output constraint imposed by the oower is aeviated so that the eader can choose a higher threshod vaue q J and obtain a higher proit. Due to the strategic substitutabiity, the oower obtains a ower proit. 4 Concuding Remarks This paper shows how brand irms endogenous timing decisions are aected by nonbrand irms. I the ow-end market is o an intermediate size, then brand irms may choose to move sequentiay with an outcome that the oower obtains a higher proit than the eader. Furthermore, the nonbrand irms positivey aect the eader s proit but negativey aect the oower s. Due to the existence o the owend market, each brand irm is given an option to choose a arge output and depress the price o brand products such that it can enter the ow-end market. Thus, each brand irm s best response unction has an upward ump i its riva oversuppies beyond a certain eve. This property signiicanty aects the equiibrium outcome when brand irms move sequentiay. The eader has to restrict its output ower than that o the oower, or the oower retaiates by raising its output viciousy to cause a price coapse. Thus, the brand irm who acts as a oower obtains a higher proit than its riva. As the competition in the ow-end market intensiies, it becomes harder or the brand irms to enter the ow-end market, which aeviates the eader s output restriction imposed by the oower, thus, the eader has a higher proit. Due to strategic substitutabiity, the oower s proit decreases. In this paper, the brand irm s discontinuous best response unction pays an important roe in the main resuts. The upward shit o the best response unction gives rise to the sudden drop o price aong the inverse demand unction o brand products, which gives rise to the consequence o second mover advantage. This property is derived rom a particuar kinked inear demand unction, whose price easticity is arger on the right hand side o the kink point. One necessary condition o this property is the existence o a ow-end market. We beieve that other demand unction shapes, rom which the resuting proit unction is gobay nonconcave and has mutipe ocay optima output eves, can aso give rise to this kind o best response unction. The cubic type equation P(Q) = (1.6 Q) 3 is a good exampe. Thereore, this paper may be a good compement to the 17

18 study o second mover advantage. The present paper can be a compementary expanation as to why some brand irms postpone the reease o new products. From our main resuts, a brand irm does this to prevent intensive competition rom ensuing countereit producers (i.e., nonbrand irms) in the ow-end market, because the resuting ow tota outputs give rise to high pricing o brand products and remove ow-end consumers incentives or purchasing brand products. Thereore, when deciding anti-countereiting poicies, a brand irm may consider an indirect method such as postponing reease rather than ace head-to-head competition. From this aspect, this research can aso be extended to countereit deterrence. 5 Appendix 5.1 Equiibrium Outcomes o Nonband Firms: The nonbrand irms inverse demand unction is denoted as oows: a(1 QN P N (Q B, Q N b ) i 1 QB a(1 QN b ), ) = a[1 + b (Q B + Q N )] otherwise. a + b We denote Q N = k qk N. In period 3, we derive nonbrand irm s best response unction taking Q B and Q N as given. We irst consider type S case, P B P N. s best response unction is given as oows: (14) q NS (Q B, Q N ) = arg max π N (q) = a[1 1 b (QN + qn )]qn = 1 2 (b QN ). q N Since q NS (Q B, Q N ) does not aways satisy PB P N, the existence o an interior soution needs to satisy 1 Q B a[1 1 b (qns (Q B, Q N ) + QN )]. Next we consider the type I case, P B < P N. s best response unction is given as oows: q NI (Q B, Q N ) = arg max q N We aso need to make the resut satisy P B < P N. We obtain π N (q) = a(1 + b QB Q N qn ) q N = 1 a + b 2 (1 + b QB Q N ). 1 Q B < a[1 1 b (qni (Q B, Q N ) + QN )]. 18

19 For a[1 (q NI (Q B, Q N )+QN )/b] < 1 QB a[1 (q NS (Q B, Q N )+QN )/b], neither qns (Q B, Q N ) nor q NI (Q B, Q N ) exists. Instead we have corner soutions within the range here. From 1 QB = a[1 (q NC + Q N )/b], we obtain q NC (Q B, Q N ) = b[1 1 a (1 QB )] Q N. Thus, we derive nonbrand irm best response unction as oows: q N (QN B, QN ) = q NS (Q N B, QN ) 1 2 (b QN ) i 1 QB a[1 1 b (qns (Q B, Q N ) + QN )], q NI (Q N B, QN ) 1 2 (1 + b QB Q N ) i 1 QB < a[1 1 b (qni (Q B, Q N ) + QN )], q NC (Q N B, QN ) b[1 1 a (1 QB )] Q N otherwise. (15) We assume that each nonbrand irm has the same equiibrium outcome and wi prove each irm sticks to this outcome ater. Under this assumption, given brand irms aggregate output, we obtain nonbrand irm s equiibrium outcome in period 3 as oows: q N (QB ) = q NS (Q B ) b n + 1 q NI (Q B ) 1 + b QB n + 1 q NC (Q B ) b 1 QB (1 ) otherwise. n a Nonbrand irm s corresponding proit is as oows: π N (QB ) = π NS (Q B ) i Q B 1 a n + 1, i Q B ab > 1 an + bn + b, ab (n + 1) 2 i Q B 1 a n + 1, π NI (Q B ) a(1 + b QB ) 2 (a + b)(n + 1) 2 i Q B ab > 1 an + bn + b, π NC (Q B ) b n (1 QB )(1 1 QB ) otherwise. a We prove the equiibrium outcome denoted by Eq. (16) is gobay optima. We irst derive the condition under which q NS (Q B ) is an equiibrium outcome. Given Q B and Q NS (QB ) k qk NS (Q B ) = (n 1)q NS (Q B ), and based on nonbrand irm s best response unction in (15), i deviates rom q NS (Q B, Q NS (QB )) to q NC (Q B, Q NS (QB )), then it wi choose a deviating outcome q NC (Q B ) b[1 (1 Q B )/a] Q NS (QB ) with a proit π NC (Q B ) (1 Q B )q NC (Q B ). woud not deviate in this manner 19 (16)

20 i it can ony choose q NS (Q B ) or choosing q NS we need either one o the oowing two inequaities to hod: (Q B ) is more proitabe than choosing q NC (Q B ). Then a[1 1 b (qni (Q B ) + +Q NS 1 Q B a[1 1 b (qns (Q B ) + Q NS (QB ))], (17) (QB ))] 1 Q B < a[1 1 b (qns (Q B ) + Q NS (QB ))] and π NC (Q B ) π NS (Q B ). (18) Eq. (17) is the conditiona ormua under which q NS (Q B ) exists. The irst inequaity o (18) is the conditiona ormua under which q NC (Q B ) exists and the second inequaity ensures that gets ess proit when it chooses q NC (Q B ). By arranging Eqs. (17) and (18), we obtain Q B 1 a/(n + 1) and 1 a/(n + 1) < Q B 1 2ab/[(n + 1)(a + 2b)]. Having either o these two inequaities hod, we obtain I deviates rom q NS (Q B, Q NS q NI (Q B ) [1 + 2q NS Q B 1 (QB )) to q NI (Q B ) Q B ]/2 with a proit π NI in this manner i it can ony choose q NS 2ab (n + 1)(a + 2b). (19) (Q B, Q NS (QB )), then it wi choose a deviating outcome (Q B ) a(q NI (Q B )) 2 /(a + b). woud not deviate (Q B ) or choosing q NS (Q B ) is more proitabe than choosing q NI (Q B ). Then we need either one o the oowing two inequaities to hod: 1 Q B a[1 1 b (qns (Q B ) + Q NS (QB ))], (20) 1 Q B < a[1 1 b (qni (Q B ) + Q NS (QB ))] and π NI (Q B ) π NS (Q B ). (21) Arranging Eq. (20), we obtain Q B 1 a/(n + 1). By arranging the irst inequaity o Eq. (21), we obtain Q B > 1 ab/[(n + 1)(a + 2b)]. This contradicts Eq. (19), which ensures that does not deviate rom q NS (Q B ) to q NC (Q B ). Thereore, Eq. (21) is rued out and Q B 1 a/(n + 1) is a necessary condition to ensure that does not deviate rom q NS (Q B ) to q NI (Q B ). Since 1 a/(n + 1) 1 2ab/[(n + 1)(a + 2b)], the condition under which q NS (Q B ) is gobay optima given Q B and Q NS (QB ) is Q B 1 a n + 1. (22) 20

21 Q NI Then, we derive the condition under which q NI (Q B ) is an equiibrium outcome. Given Q B and (QB ) k q NI k Eq. (15), i deviates rom q NI (Q B, Q NI (QB ) = (n 1)q NI (Q B ) and based on nonbrand irm s best response unction in (QB )) to q NC (Q B, Q NI (QB )), then it wi choose a deviating outcome q NC (Q B ) b[1 (1 Q B )/a] Q NI (QB ) with a proit π NC (Q B ) (1 Q B )q NC (Q B ). woud not deviate in this manner i it can ony choose q NI (Q B ) or choosing q NI (Q B ) is more proitabe than choosing q NC (Q B ). Then we need either one o the oowing two inequaities to hod: 1 Q B < a[1 1 b (qni (Q B ) + Q NI (QB ))], (23) a[1 1 b (qni (Q B ) + Q NI (QB ))] 1 Q B < a[1 1 b (qns (Q B ) + Q NI (QB ))] and π NC (Q B ) π NI (Q B ). (24) By arranging Eq. (23), we obtain Q B > 1 ab/(an + bn + b). By arranging Eq. (24), we obtain 1 2ab/( a + 2b + an + 2bn) < Q B 1 ab/(an + bn + b). Having either o theses two inequaities hod, we obtain I deviates rom q NI (Q B, Q NI q NS Q B > 1 (QB )) to q NS 2ab a + 2b + an + 2bn. (25) (Q B, Q NI (QB )), then it wi choose a deviating outcome (Q B ) [b Q NI (QB )]/2 with a proit π NS (Q B ) a(q NS (Q B )) 2 /b. woud not deviate in this manner i it can ony choose q NI (Q B ) or choosing q NI (Q B ) is more proitabe than choosing q NS (Q B ). Then we need either one o the oowing two inequaities to hod: 1 Q B < a[1 1 b (qni (Q B ) + Q NI (QB ))], (26) 1 Q B a[1 1 b (qns (Q B ) + Q NI (QB ))] and π NS (Q B ) π NI (Q B ). (27) By arranging Eq. (26), we obtain Q B > 1 ab/(an + bn + b). By arranging the irst inequaity o Eq. (27), we obtain Q B 1 2ab/( a + 2b + an + 2bn). This contradicts Eq. (25), which ensures that does not deviate rom q NI (Q B ) to q NC (Q B ). Thereore, Eq. (27) is rued out and Q B > 1 ab/(an + bn + b) is a necessary condition to ensure that does not deviate rom q NI (Q B ) to q NS (Q B ). Since 1 ab/(an + bn + b) > 1 2ab/( a + 2b + an + 2bn), the condition under which 21

22 q NI (Q B ) is gobay optima given Q B and Q NI (QB ) is Q NC Q B > 1 ab an + bn + b. (28) Finay, we derive the condition under which q NC (Q B ) is an equiibrium outcome. Given Q B and (QB ) k q NC k Eq. (15), i deviates rom q NC (Q B, Q NC (Q B ) = (n 1)q NC (Q B ) and based on nonbrand irm s best response unction in (QB )) to q NS (Q B, Q NC (QB )), then it wi choose a deviating outcome q NS (Q B ) [b Q NC (QB )]/2 with a proit π NS (Q B ) a(q NS (Q B )) 2 /b. woud not deviate in this manner i it can ony choose q NC (Q B ) or choosing q NC (Q B ) is more proitabe than choosing q NS (Q B ). Then we need either one o the oowing two inequaities to hod: a[1 1 b (qni (Q B ) + Q NC (QB ))] 1 Q B < a[1 1 b (qns (Q B ) + Q NC (QB ))], (29) 1 Q B a[1 1 b (qns (Q B ) + Q NC (QB ))] and π NS (Q B ) π NC (Q B ). (30) By arranging Eq. (29), we obtain 1 a/(n + 1) < Q B 1 ab/(an + bn + b). By arranging Eq. (30), we obtain Q B = 1 a/(n + 1). Having either o these two conditions hod, we obtain I deviates rom q NC (Q B, Q NC 1 a n + 1 ab QB 1 an + bn + b. (31) (QB )) to q NI (Q B, Q NC (QB )), then it wi choose a deviating outcome q NI (Q B ) [1 + b Q B Q NC (QB )]/(n + 1) with a proit π NI (Q B ) a(q NI (Q B )) 2 /(a + b). woud not deviate rom q NC (Q B ) to q NI (Q B ) i it can ony choose q NC (Q B ) or choosing q NC (Q B ) is more proitabe than choosing q NI (Q B ). Then, we need either one o the oowing two inequaities to hod: a[1 1 b (qni (Q B ) + Q NC (QB ))] 1 Q B < a[1 1 b (qns (Q B ) + Q NC (QB ))], (32) 1 Q B < a[1 1 b (qni (Q B ) + Q NC (QB ))] and π NI (Q B ) π NC (Q B ). (33) By arranging Eq. (32), we obtain 1 a/(n + 1) < Q B 1 ab/(an + bn + b). By arranging the irst inequaity o Eq. (33), we obtain Q B > 1 ab/(an + bn + b). This contradicts Eq. (31), which ensures that does not deviate rom q NC (Q B ) to q NS (Q B ). Thereore, Eq. (33) is rued out and 1 a/(n + 1) < Q B 1 ab/(an + bn + b) is a necessary condition to ensure that does not deviate 22

23 rom q NC (Q B ) to q NI (Q B ). Together with Eq. (31), the condition under which q NC (Q B ) is gobay optima given Q B and Q NC (QB ) is 1 a n + 1 < ab QB 1 an + bn + b. (34) 5.2 Foower s Best Response Functions: By soving the proit maximization probem in Eq. (5), we obtain the oower s best response unction as oows: q B (qb ) = q Bx (q B ) i QB q K, q By (q B ) Otherwise. (35) Since q Bx (q B ) does not aways satisy the irst conditiona ormua in Eq. (35), the existence type x outcome needs to satisy q Bx (q B ) + qb q K. By arranging the inequaity, we obtain q B 1 2ab an + bn + b. For the existence o type y outcome, we need q By (q B ) to satisy the second conditiona ormua in Eq. (35): q By (q B ) + qb > q K.By arranging the inequaity, we obtain q B > 1 b 2ab an + bn + b. For 1 b 2ab/(an + bn + b) < q B it a higher proit, or π Bx 1 2ab/(an + bn + b), the oower chooses q Bx (q B ) i this brings (q B, qbx (q B )) πby (q B, qby (q B )). By arranging the inequaity, we obtain12 q B q J. Since 1 b 2ab/(an + bn + b) < q J 1 2ab(an + bn + b) or any 0 < a < 1, b > 0 and n > 1, the oower chooses q Bx (q B ) or qb smaer than q J. Thus, we obtain the oower s best response unction as in Eq. (6). 12 I we simpy derive the threshod vaue o q B, we aso have qb = 1 b[a a(a + b)(n + 1)]/(an + bn + b). However, since q B + q Bx (q B ) = (2 b[a a(a + b)(n + 1)]/(an + bn + b))/2 is arger than the upper bound o Q B, or q K, or any 0 < a < 1, b > 0 and n > 1, q B = 1 b[a a(a + b)(n + 1)]/(an + bn + b) is not an appropriate threshod that π Bx (q B, qbx (q B )) πby (q B, qby (q B )). 23

24 5.3 Equiibrium Conditions When Either Brand Firm Moves in Period 1: We consider three cases: (i)q x < q y First, we consider case (i). For q B i optima soution q x < q J, (ii)q J q x. The equiibrium proit is π Bx < q y, and (iii)q x < q J q y. q J, the optimization probem o π Bx (q B ) has the ocay (q x ). For q B > q J, eader s proit decreases in q B. The optimization probem has the corner soution at qb, which is cose enough to q J. Within this range, brand irm 1 obtains a proit that is stricty ess than π By (q J ). Due to the ump at q J, π Bx (q J ) > π By (q J ). Thereore, π Bx (q x optima soution when the oowing condition hods: ) > π By (q J ), rom which we conirm that q x is the gobay Next we consider case (ii). For q B optima soution q y. The equiibrium proit is π By q By < q J. (36) > q J, the optimization probem o π By (q B ) has the ocay ). For q B q J, brand irm 1 s proit increases in q B. The optimization probem has the corner soution at point q J and the maximized proit is π Bx (q J ). We compare π Bx hods: q y (q J ) with π By ). q J is the gobay optima soution when the oowing condition q x q J and π Bx (q J ) π By ). (37) is the gobay optima soution with a proit when the oowing condition hods: q x q J and π Bx (q J ) < π By ). (38) Finay, we consider case (iii). For q B q J, the optimization probem has the ocay optima soution q x. For q B > q J, the optimization probem has the ocay optima soution q y. Brand irm 1 chooses the soution which brings it a higher proit. Thereore, q x with a proit π Bx q y (q x ) when the oowing condition hods: q x < q J q y and π Bx (q x ) π By is gobay optima with a proit π By ) when the oowing condition hods: q x < q J q y and π Bx (q x ) < π By is the gobay optima soution ). (39) ). (40) 24

25 We arrange the above three situations o q J s vaue and derive the conditions under which each type o equiibrium exists. First, we consider the situation when Eq. (36) or Eq. (39) hods. From the irst inequaity o (39), we arrange the necessary condition o q x < q J q y, q J > q x and obtain b < b 3 [n + 1 4a + (16a 2 8a + n + 1)(n + 1)]/(8a). By arranging the second inequaity o Eq. (39), we obtain b b 4 [n + 1 2a + (4a 2 4a + n + 1)(n + 1)]/(2a). Since b 3 b 4, Eq. (39) can be simpiied as q x < q J q y. Let either Eq. (36) or Eq. (39) hod, we obtain condition A, under which equiibrium outcome (q x, q x ) exists. Next, we consider the situation when Eq. (37) hods: we obtain condition B under which equiibrium outcome (q J, (1 q J )/2) exists. Finay, we consider the situation when Eq. (38) or Eq. (40) hods. It is straight orward to see that the soution o Eq. (40) is an empty set. 13 Thereore, we obtain condition C under which equiibrium outcome, q y ) ((1 + b)/2, (1 + b)/4) exists. 5.4 Equiibrium Conditions When Both Brand Firms Move in Period 1 or Period 2: We derive the condition under which the x and y type equiibrium outcomes exist. For type x e- quiibrium, we irst consider 1/3 q J. Given q B 1 = 1/3, brand irm 2 s best response unction is q Bx 2 (qb 1 ) = 1/3. Then we consider 1/3 > q J. Given q B 1 = 1/3, brand irm 2 s best response unction is q By 2 (qb 1 ) = 1/3 + b/2 > 1/3. Thus, the condition or the existence o type x equiibrium is 1/3 q J. For type y equiibrium, we irst consider (1 + b)/3 > q J. Given q B 1 = (1 + b)/3, brand irm 2 s best response unction is q By 2 (qb 1 ) = (1 + b)/3. Then, we consider (1 + b)/3 q J. Given q B 1 = (1 + b)/3, brand irm 2 s best response unction is q2 Bx(qB 1 ) = 1/3 b/6 < (1 + b)/3. Thus, the condition or the existence o type y equiibrium is (1 + b)/3 > q J. For 1/3 q J < (1 + b)/3, type x and type y equiibria exist. From Assumption 1, brand irms choose a type x equiibrium outcome i πc Bx (qc x ) πc By i ). By arranging the necessary condition o 1/3 q J < (1+b)/3, 1/3 q J, we obtain b 2[(n+1 3a)+ (n + 1)(9a 2 6a + n + 1)]/(9a) b 1. Arranging πc Bx (qc x ) πc By c ), we obtain b [(n + 1 2a) + (n + 1)(4a 2 4a + n + 1)]/(2a) b The irst inequaity o Eq. (40) is the same with that o Eq. (39), rom which we obtain b < b 3. However, the second inequaities o Eq. (40) have an opposite sign to Eq. (39), rom which we obtain b > b 4. Since b 3 b 4, the soution set o Eq. (40) is empty. 25