Sequential Entry in a Vertically Differentiated Duopoly. Luca Lambertini # Piero Tedeschi # University of Bologna University of Milano-Bicocca

Transcription

1 Sequential Enty in a Vetically Diffeentiated Duopoly Luca Lambetini # Pieo Tedeschi # Univesity of Bologna Univesity of Milano-Bicocca Novembe 27, 2003

2 Abstact We analyse a model of vetical diffeentiation focusing on the tade-off between enteing ealy and exploiting monopoly powe with a low quality, vesus waiting and enjoying a dominant maket position with a supeio poduct. We show that thee exists a unique equilibium whee the leade entes with a lowe quality than the followe, fo low discount factos, fo high costs of quality and fo low consumes willingness to pay fo quality. J.E.L. Classification: L13, O31 Keywods: vetical diffeentiation, poduct innovation, monopoly ent

3 1 Intoduction An appaently well established esult in the theoy of vetically diffeentiated oligopoly states that ealie entants supply goods of highe quality than late entants, in that the high-quality poducts ean highe pofits than low-quality altenatives (see, inte alia, Gabszewicz and Thisse, 1979, 1980; Shaked and Sutton, 1982, 1983; Donnenfeld and Webe, 1992, 1995). A geneal poof of this esult fo evey convex fixed-cost function of quality impovement is povided by Lehmann-Gube (1997). 1 Two basic assumptions ae at the basis of this esult. The fist is that consumes maginal willingness to pay fo quality is unifomly distibuted ove a given suppot. Since the density of consumes (i.e., demand) is the same at any income level, the top-quality maket niche is the most pofitable. Theefoe, in a static game, fims obviously pefe to ente with a poduct chaacteised by the highest possible quality. The second assumption concens the time hoizon consideed in the above mentioned liteatue. Enty in a vetically diffeentiated maket is usually analyzed within a single-peiod extensive fom game. Howeve, if one models the enty poblem in an explicit dynamic setup, an obvious tade-off immediately appeas, even maintaining the pevious assumption. In ode to ente with an high quality poduct, the fim has to wait fo the R&D activity to take place and consequently it looses monopoly pofits. Howeve, postponing enty, the fim is able to poduce a highe quality good, obtaining thus highe pofits. A static model does not allow to assess the possibility that thee exists such a tade-off between ealy innovation and the attainment of a dominant position in the maket. Although it is geneally asseted that quality may esult fom fims R&D 1 Aoki and Pusa (1997) adopt a specific case of the cost function analysed by Lehmann- Gube (1997), to investigate the consequences on pofits, consume suplus and social welfae of the timing of investment in poduct quality in a vetically diffeentiated duopoly whee the maket stage is played in the pice space. To this egad, see also Lambetini (1999). 1

4 effots, this aspect of vetical poduct diffeentiation has eceived a elatively scanty attention, the development phase being summaised by a cost function which does not account fo the time elapsed befoe the good is poduced and then maketed. To ou knowledge, elevant contibutions dealing explicitly with the R&D activity ae Beath et al. (1987); Motta (1992); Rosenkanz (1995, 1997) and Dutta et al. (1995). These papes investigate the incentive towads R&D coopeation (Motta, 1992; Rosenkanz, 1995) and the elationship between R&D and the pesistence of quality leadeship (Beath et al., 1987; Rosenkanz, 1997). Dutta et al. (1995) analyse stategic timing in the adoption of a new technology leading to poduct diffeentiation and quality impovements. All of these papes maintain that being the quality leade (i.e., supplying the highest quality in the maket) entails highe pofits than the ivals. We pesent a simple model of vetical diffeentiation focusing upon the tade-off between enteing ealy and exploiting monopoly powe with a low quality, vesus waiting and enjoying a dominant maket position with a supeio poduct. We etain the assumption of a unifom income distibution, that would make it pofitable to poduce a high quality good in a static game, but elax the assumption of a static extensive fom game. Namely, in ou model thee exists a unique equilibium whee the leade entes with a lowe quality than the followe, fo a lage set of paamete values. 2 This highlights that an unfavouable position in duopoly (o oligopoly), due to a lowe quality than the ivals, may well be moe than balanced by the monopoly ent enjoyed ad inteim with lowe development costs. Theefoe, it appeas that the established wisdom stating that ealy enty goes along with high quality (and pofits) is not obust to a fully fledged investigation of the ole of calenda time in shaping endogenously fims incentives. The emainde of the pape is stuctued as follows. The basic model 2 Fom a diffeent setting, Dutta et al. (1995) also deive an equilibium whee the fist entant poduces a lowe quality than the second entant. Howeve, in thei model the late entant makes moe pofits. As it will become clea in the emainde, this conclusion ests upon the shape of the cost function. 2

5 of vetical diffeentiation is laid out in section 2. Section 3 descibes the solution of all admissible subgames. The subgame pefect equilibium of the whole game is deived in section 4. Finally, section 6 povides concluding emaks. 2 The Model Conside a maket fo vetically diffeentiated poducts. Let this maket existovetimet, witht [0, ). Two single-poduct fims, labelled 1 and 2, poduce goods of diffeent qualities, q 1 and q 2 [0, ), though the same technology. Without loss of geneality we can assume that fims poduction costs ae nought, while development costs ae C i (q i )=c Z q+qi q e t dt (1) with i =1, 2andq 0. Development costs C i (q i ) ae evaluated at the beginning of the peiod of investment, theefoe in 0 fo fim 1 and in t 1 fo fim 2. As usual, these costs can be intepeted as fixed cost due to the R&D effot needed to poduce a cetain quality. We chaacteize the technology epesented by the above cost function as follows: Assumption 1 The R&D costs ae constant ove time and equal to c. If fim i seaches fo a peiod of length t i, then it can poduce a good at most of quality t i and any othe lowe quality. Once enteed into the maket the fim cannot invest anymoe in R&D. The above amounts to assuming that any change in the quality level implies adjustment costs if and only if the change takes the fom of a quality incease. Convesely, once fim i has bone the cost of developing a given quality, she may decide to decease the quality of he poduct costlessly. Fo the sake of simplicity we assume that quality is stictly coelated with the time of enty. Moe pecisely, if fim 1 entes at time t 1,itsmaximumfeasible quality is t 1 = q 1. Fim 2 s cost of imitation, howeve, ae exactly equal 3

6 to the costs of innovation. 3 Theefoe, fim 2 s time of enty satisfies the equality t 2 = q 1 + q 2. In the emainde, we shall label the fist entant as the leade. Fim 2 entes at date t 2 [t 1, ), and we shall efe to he as the followe. Assumption 2 Poducts ae offeed on a maket whee consumes have unit demands, and buy if and only if the net suplus deived fom consumption v θ (q k,p i (q k )) = θq k p i (q k ) 0, whee p i (q k ) is the unit pice chaged by fim i onagoodofqualityq k, puchased by a geneic consume whose maginal willingness to pay is θ [θ, θ], with θ = θ 1. We assume that θ is unifomly distibuted with density one ove such inteval, so that the total mass of consume is one. Thoughout the following analysis, we assume patial maket coveage. The above assumption is athe common in vetically diffeentiated poduct models. Moe elevant ae the assumptions elative to the timing of the game. Assumption 3 Fim 1 chooses when to ente the maket with the new poduct and simultaneously chooses the quality and the pice to be offeed. Then fim 2 decides whethe to imitate fim 1 and when to ente the maket. Once fim 2 has enteed, the two fims choose simultaneously the quality levels, which become common knowledge. Finally both fims choose simultaneously the pice levels. This timing can be justified as follows. Suppose that fim 1 has invented a new poduct, but it has to decide the quality level of that poduct befoe enty. Since nobody knows the existence of this new poduct, only fim 1 can ente fist. Theeafte, othe fims can imitate fim 1. Suppose only fim 2 has the necessay technology. Howeve, fim 2 has to sustain the R&D 3 The case fo vey high imitation costs is suppoted by empiical findings (see Mansfield et al., 1981; and Levin et al., 1987). 4

7 costs befoe being able to ente and this takes time and pecisely the peiod between t 1 and t Solution of the Game As usual we will solve the game backwads. Howeve, it is useful befoe solving the model to intoduce two definitions, concening fims behavio. In the emainde, we shall efe to the fist entant (fim 1) as the leade, and to the second entant (fim 2) as the followe. Weaegoingtoexamine two altenative pespectives: A. The followe entes at t 2 with a poduct whose quality is lowe than the leade s. We label this case as high-quality leadeship. B. The followe entes at t 2 with a poduct whose quality is highe than the leade s. We label this case as low-quality leadeship. 3.1 The Pice Game In both cases, ove t [t 2, ), fims compete in pices. We boow fom Aoki and Pusa (1997) and Lehmann-Gube (1997) the assumption that downsteam Betand competition is simultaneous. Maket demands fo the high- and low-quality good ae, espectively: x H = θ p H p L q H q L and x L = p H p L q H q L p L q L (2) Duopoly evenue functions ae R H = p H x H and R L = p L x L. Solving fo the equilibium pices, we obtain: p H =2 θq H q H q L 4q H q L ; p L = θq L q H q L 4q H q L (3) 4 To solve the game we adopt subgame pefection, and we look fo simultaneous Nash equilibia in each stage. Consideing the Stackelbeg solution would make calculations moe cumbesome without affecting significantly the main esults. 5

8 which allow to ewite the evenue function of fims in tems of qualities only, as follows: 5 R H = 4 θ 2 qh 2 (q H q L ) (4) (4q H q L ) 2 R L = θ 2 q H q L (q H q L ) (5) (4q H q L ) 2 On the basis of expessions (4-5), pevious liteatue, dealing with singlepeiod models, establishes that the fist entant would choose to supply the high-quality good, given that R H >R L. In the emainde, we label the leade s quality as q 1 and the followe s quality as eithe q H o q L,withthe undestanding that q H q 1 and q 1 q L. 3.2 The Followe s Quality Choice We detemine the conditions which induce the followe to ente eithe with a lowe o with a highe quality than the leade. We will define the two situations enty fom below and enty fom above, and will be analyzed in a sequel Enty fom below The followe s pofits when enteing fom below ae: Π 2L = Z which using (5) can be ewitten as: Z q1 +q L R L e t dt c e t dt = q 1 +q L q 1 e (q 1+q L ) R L c e q 1 e (q 1+q L ) R 2L (q L,q 1 )= θ 2 q 1 q L (q 1 q L ) e (q1+ql) (4q 1 q L ) 2 θ 2 µ q1 q L (q 1 q L ) e q 1 c (4q 1 q L ) 2 e q L + c θ2 e q L c θ2 e q 1 e (q 1+q L ) = 5 The poof is omitted hee, as it is povided by seveal authos (Gabszewicz and Thisse, 1979; Choi and Shin, 1992; Motta, 1993; Aoki and Pusa, 1997; Lehmann-Gube, 1997). 6

9 and setting γ c, γ q θ 2 L q L and γ q 1 q 1, and substituting, we obtain: µµ q1 q c R L ( q 1 q L ) 2L (γ q L,γ q 1 )= +1 e δ q L 1 e δ q 1 (6) (4 q 1 q L ) 2 whee δ γ c/ θ 2. Diffeentiating (6) fo q L we obtain the fist ode condition: e δ( q 1+ q L) (7 q L 4 q 1 )( q 1 ) 2 + δ q 1 q L ( q 1 q L )(4 q 1 q L )+δ(4 q 1 q L ) 3 (4 q 1 q L ) 3 =0 If we set q L = x q 1, the numeato becomes: 7x 4+δ q1 x (1 x)(4 x)+δ (4 x) 3 =0 q 3 1 hence: 4 7x δ (4 x)3 q 1 = (7) δx (1 x)(4 x) and: 4 7x δ (4 x)3 q L = x q 1 = δ (1 x)(4 x) Remak 1 The followe enteing with the low quality chooses the following quality level: ag max R 2L (γ q L,γ q 1 )= whee x q L /q 1 and δ c/ θ x δ (4 x)3 δ (1 x)(4 x) q L Notice that in ode to have q L 0wemustimpose δ 4 7x (4 x) 3 (8) whichintunimplies: 0 x , (9) 7 and coespondigly: 0 δ 1 = (10) 16 Moeove: q L x = 8 x 7x (1 x) δ (x +2)(4 x)3 δ (1 x) 2 (4 x) 2 < 0 hence q L is a monotonically deceasing function in the elevant ange. 7

10 3.2.2 Enty fom above Followe s pofits if it entes with the high quality good ae: Z Z q1 +q H Π 2H = R H e t dt c e t dt q 1 +q H q 1 e t 2 R H c e q 1 e (q 1+q H ) which using (4) can be e-witten as follows R 2H (q H,q 1 )= 4 θ 2 qh(q 2 H q 1 ) e (q1+qh) c e q 1 e (q 1+q H ) = (4q H q 1 ) 2 θ 2 µ 4q 2 H (q H q 1 ) (4q H q 1 ) 2 e q H + c θ2 e q H c θ2 e q 1. Then, setting γ c/ θ 2, γ q L q L and γ q 1 q 1 we obtain: µµ 4 q 2 c R 2H (γ q H,γ q 1 )= H ( q H q 1 ) +1 e δ q H 1 e δ q 1 (11) (4 q H q 1 ) 2 whee δ γ c/ θ 2. Diffeentiating (11) fo q H we obtain the fist ode condition: e δ( q 1+ q H) (4 qh q 1 ) 3 +4 q H 2 ( q H q 1 )(4 q H q 1 ) δ 4 q H (4 q H 2 3 q H q 1 +2 q 1) 2 (4 q H q 1 ) 3 =0 If we set q 1 = x q H, the numeato becomes: 4δ (1 x)(4 x) qh 4 4 3x +2x 2 + δ (4 x) 3 q 3 H which is nought if: q H = 4(4 3x +2x2 ) δ (4 x) 3 4δ (4 x)(1 x) Hence, the following holds:. (12) Remak 2 The followe enteing with the high quality chooses the following quality level: ag max R 2H (γ q H,γ q 1 )= 4(4 3x +2x2 ) δ (4 x) 3 q H. 4δ (4 x)(1 x) 8

11 Notice that in ode to have q H 0wemustimpose: Moeove, It is easy to check that: 0 δ 4 (4 3x +2x2 ) (4 x) 3 (13) q H x = 1 4(8+x +7x (1 x)) (x +2)(4 x) 3 δ 4 δ (1 x) 2 (4 x) 2 x q H µ x δ =4 ( 4 3x+2x 2 ) = (4 x) 3 2x (x +5) (4 x) 2 (1 x) δ > 0. Hence, noticing that q H / x is deceasing in δ, wehavethat q H / x > 0in the elevant ange. Theefoe, q H (x) is monotonically inceasing. 3.3 The Leade s Quality Choice The leade has to take two choices on the quality level: one when it entes as a monopolist and the othe when it has to cope with the enty of the competito. On the basis of Assumption 1, the second level of quality cannot exceed the monopoly one. As usual, we stat by analyzing the last quality choice, that when the followe entes The Quality in the Last Stage Game As fo the followe, we detemine the conditions inducing the followe to ente eithe with a lowe o with a highe quality than the leade s. We will define the two situations as enty fom below and enty fom above, and they will be analyzed in a sequel. Enty fom above Fist of all notice that once fim 2 has enteed, fim 1 wishes to poduce at the highest quality level in the poduct space. It is sufficient to compute the deivative of R H with espect to q H and check that it is always positive: R H =4 θ 2 4qH 2 3q H q 2 +2q2 2 q H q H (4q H q 2 ) 3 9

12 which is positive if: 4q 2 H 3q H q 2 +2q 2 2 > 0. Howeve: 4q 2 H 3q H q 2 +2q 2 2 4q 2 2 3q 2 q 2 +2q 2 2 =3q whee the fist inequality is an implication of q H q 2.Sinceq 1 q M,whee q M is the quality level of monopolist s poduct,wecansummaizetheesult in the following poposition. Poposition 3 If the leade entes with the high quality good, then it will poduce a good of the same quality level afte and befoe followe s enty. Enty fom below Afte fim 2 enteed the maket, the leade s optimal quality level is q 1 =4q H /7, if it enteed with a low quality. In fact: R L = q θ 2 q 2 4q H 7q 1 H L (4q H q 1 ) 3 which implies the assetion. Moeove, q 1 =4q H /7implies q 1 =4 q H /7,theefoeifwesubstitutein the followe s fist ode condition we obtain: C 2H µ q H, 4 7 q H = (+7δ q H +48δ 14) q H 3 =0 whose solution is: q H = δ (14) 7 δ which is meaningful if and only if δ = c θ ' (15) Unde the above condition we have: This discussion implies: q 1 = δ δ (16) Poposition 4 If the leade enteed with the low quality good and if the quality chosen afte the followe has enteed is lowe than that chosen in the monopoly phase, the equilibium quality levels when the followes entes ae: q 1 = δ, q 2 = δ (17) 49 δ 7 δ povided that: δ = c/ θ 2 7/24. 10

13 3.3.2 Monopoly Phase Afte having discussed the choices in the competition game, we have to descibe what happens in the monopoly phase. As usual, we stat by descibing the pice policy and then the choice of quality, distinguishing the enty with high and low quality espectively. The Monopolist s Pice In the monopoly phase, evenues ae R M = p θ p/qm ), whee q M is the quality level chosen by fim 1 when monopolist. The fist ode conditions fo the pice is: θqm 2p q M =0 and hence p = θq M /2. Substituting again in the pofits, it yields: R M = 1 4 θ 2 q M Enty fom above. above: The pofit function of fim 1 when enteing fom R M R qm +q L q M e t dt + R H R q M +q L e t dt c R q M 0 e t dt = R M e q M e (q M +q L ) + R H e (q M +q L ) c (1 e q M ) = q M θ2 4 e q M e (q M +q L ) + 4 θ 2 q 2 H(q H q L ) (4q H q L ) 2 e (qm +ql) c (1 e q M ) which is equivalent to: Π 1H (q L,q M )=4 q2 1 (q 1 q L ) θ2 (4q 1 q L ) 2 e (q M +q L ) e q L e q M q M c θ2 1 e q M 4 11

14 We aleady know fom the above analysis that, when the followe entes, the leade will poduce the highest quality and hence we have q 1 = q M. Theefoe, the leade maximizes: Π MH (q L,q 1 )=4 q2 1 (q 1 q L ) (4q 1 q L ) 2 e (q 1+q L ) e q 1 q e (q 1+q L ) q 1 + γe q1 γ with γ = c/ θ 2. Using the usual vaiable tansfomations, we obtain: µ 4 q 2 Π MH (γ q L,γ q 1 )=γ 1 ( q 1 q L ) e δ q L (4 q 1 q L ) q 1e δ q L q 1 +1 e δ q 1 γ whee δ = γ. Then, defining: Π H ( q L, q 1,δ)= 1 γ Π MH (γ q L,γ q 1 )+γ and using q L = x q 1, we obtain: Π H (x q 1, q 1,δ)= 1 µµ 1 8+x xe δx q 4 (4 x) 2 1 q 1 +4 e δ q 1 Using (7) and substituting in the pofit ofthemonopolistweobtainthe following expession: Ã! 4 7x δ (4 x) 3 4 7x δ (4 x)3 Π H, (18) δ (1 x)(4 x) δx (1 x)(4 x) Theefoe, the leade s poblem is fomally equivalent to mazimizing (18) with espect to x. Given estictions (8 10), we can cay out an exploation of the monopolist pofit function in Figue 1, highlighting the existence of a global maximum fo any given value of δ. The monopolist s fist ode condition: Ã! D H (x, δ) = x Π 4 7x δ (4 x) 3 4 7x δ (4 x)3 H, = 0 (19) δ (1 x)(4 x) δx (1 x)(4 x) cannot be solved analytically. Howeve, we can daw the implicit plot in Figue 2. The dotted line plots the locus δ =(4 7x) / (4 x) 3. Accodingly, the only meaningful aea is the one below the dotted line. The continuous line below the dotted one is the locus of the global maxima of the pofit function, as established by compaing Figue 1 and 2. 12

15 П H x δ 0.06 Figue 1: Pofit of the leade when enteing fom above δ x Figue 2: Fist ode condition fo the leade when enteing fom above 13

16 Enty fom below is: The pofit function of fim1whenenteingfombelow R M R qm +q H q M e t dt + R L R q M +q H e t dt c R q M 0 e t dt = R M e q M e (q M +q H ) + R L e (q M +q H ) c 1 e q M = q M θ2 4 e q M e (q M +q H ) + θ 2 q H q L (q H q 1 ) e (qm +qh) (4q H q 1 ) 2 c (1 e q M ). This is equivalent to: θ 2 Π ML (q M,q H,q L )= q H q L (q H q L ) (4q H q L ) 2 e (q M +q H ) e q H e q M q M γ 1 e q M 4 whee γ = c/ θ 2,asusual. We have to distinguish two diffeent cases. In the fist one, δ 7/24 and theefoe Poposition 4 holds. In the second one, δ>7/24. Let us stat fom the fist case. Case I: δ 7/24. Hence we can set: q 1 = γ q 1 = γ, q 2 = γ q 2 = γ 7 which can be substituted in the pofit function to yield: µ 1 γ Π ML γ q M, γ, γ = Ã (7 24δ 42δ q M ) e ( δ)! 7 +42( q M +4) e δ q M δ Fom the fist ode condition w..t. q M, we obtain: q M = ( δ) e 7 δ +42(1 4δ) δ ³1 e δ 14

17 Notice that above expession chaacteizes the leade s choice when enteing with the low quality if q M q 1 o q M (= q M /γ) q 1 (= q 1 /γ), that is, if: 1 42 ( δ) e δ +42(1 4δ) δ ³ δ e δ δ 0 which afte some manipulation is equivalent to: e 7 δ e 7 δ δ δ ³ δ e 7 δ 1 Since the numeato is always positive, the above condition implies that the denominato should be positive, o equivalently that: δ 7 24 ' and ecalling Poposition 4 we know that the condition cannot be satisfied fo a positive quality level. We summaize the above analysis in the following poposition: Poposition 5 Iespective of whethe the leade entes with the low o the high quality, the quality of the leade afte the followe has enteed the maket is equal to that of the monopoly phase, i.e., q 1 = q M,ifδ 7/24. Case II: δ>7/24. Now we analyze the situation whee the monopolist s choice is binding in the duopoly phase. In such a case the leade s pofits, afte tivial tansfomation, become: µµ qh q 1 (q H q 1 ) Π ML (q H,q 1 )= (4q H q 1 ) q 1 e qh q 1 + γ e q1 γ and afte the usual vaiable tansfomations: µµ q H q 1 Π ML (γ q H,γ q 1 )=γ q H q 1 (4 q H q 1 ) q 1 e δ q H q 1 +1 e δ q 1 γ whee again δ = γ. Defining: Π L ( q H, q 1,δ)= 1 γ Π ML (γ q H,γ q 1 )+γ 15

18 П L 2 1 x 0 δ 0.2 Figue 3: Leade s pofit when enteing fom below and setting q H = x q 1, we obtain: Π L ( q H,x q H,δ)= 1 µµ x + x2 (4 x) 2 e δ q H x q H +4 e δx q H Recalling (12), the monopolist poblem is equivalent to maximizing the following expession, with espect to x: Ã! 4(4 3x +2x 2 ) δ (4 x) 3 Π L,x 4(4 3x +2x2 ) δ (4 x) 3 (20) 4δ (4 x)(1 x) 4δ (4 x)(1 x) Using estiction (13), we can poduce a gaphical exploation of the poblem in Figue 3. It shows that the function has a unique global maximum fo each value of δ. Moeove, the fist ode condition is: D L (x, δ) =(21) Ã! x Π 4(4 3x +2x 2 ) δ (4 x) 3 L,x 4(4 3x +2x2 ) δ (4 x) 3 = 0 4δ (4 x)(1 x) 4δ (4 x)(1 x) and it is not solvable analytically. Howeve, its implicit plot is in Figue 4. 16

19 δ x Figue 4: Leade s fist ode condition when enteing fom below 4 IsitConvenienttoEntetheMaketwith a High-quality Poduct? Now we can solve fo the subgame pefect equilibiumof the whole game by detemining whethe the leade will ente with a high o a low quality. We fist pove a peliminay esult. Poposition 6 No equilibium with the followe enteing the maket with a lowe quality than the leade does exist if δ = c/ θ 2 > Poof. It is a diect consequence of (10). We ae now in the position to pove the main Lemma of this section. Lemma 7 Thee exists a δ such that, fo δ 0, δ thee is no equilibium with the followe enteing the maket and the leade poducing the lowe quality good, while fo δ δ, thee exists no equilibium with the followe enteing the maket and the leade poducing the highe quality good. The value of δ is appoximately: δ =

20 80 П H П L δ Figue 5: Pofits of fim 1 when enteing with high (solid) and when enteing with low (dash) quality. Poof. We solve numeically equations (19) and (21), finding the optimal x fothetwopoblemsfovaiousvaluesofδ. The computed values ae epoted in the Table 1-3 of the Appendix in columns denoted espectively by x HL and x LH.Byusing(7)wecancompute q 1 and q L = x q 1,theoptimal values of tansfomed vaiables eplacing q L and q 1. By using (12) we can compute, instead, q H and q 1 = x q H. Given the vaious level of qualities, the pofits of the monopolist enteing fom above and enteing fom below can be computed and ae dawn in Figue 5. It can be seen that the pofit ofthe high quality monopolist ae highe fo lowe level of δ and lowe theeafte. The two cuves coss at δ. The two levels of the followe s pofits, R L when it chooses a lowe quality than the leade s and R H when it chooses a highe quality, ae epesented in the following two figues. The fist one epesents the two vaiables when the leade ties to ente with a highe quality than the followe and, as we can see, the best esponse fo the followe consists in choosing a highe quality. 18

21 In the second Figue, instead, we epesent the two followe s pofit levels when the leade ties to ente with a low quality. In this case, we see that the esponse of the followe is consistent with the leade s stategy. The above Lemma allows us to infe that, in ou model, the leade entes with the high quality only if it can block the followe s enty. Theefoe, hee we may have only two types of equilibia. In the fist one the leade invests in R&D in such a way to be able to maintain its monopoly position. In the second type of equilibium, the leade entes with the low quality and then the followe entes with a highe quality. The following poposition will exclude the fist outcome, that whee the leade can have a monopoly powe. Poposition 8 Fo δ sufficiently small, thee exists no equilibium whee the leade succeeds in pe-emptying the maket. In paticula, δ 1/16 is a sufficient condition fo the leade not to be able to pe-empt the maket. Poof. In ode to pove the Poposition, we must check that he followe can always ente with a lowe quality fo any choice of the leade, making positive pofits. Recall that the optimal choice of the followe is expessed by (7), which is e-witten fo convenience: q L = 4 7x δ (4 x)3 δ (1 x)(4 x) We know that x must satisfy inequality (8): δ 4 7x (4 x) 3 Recall also that pofits fo the followe enteing with the low quality ae: µµ q1 q c R L ( q 1 q L ) 2L (γ q L,γ q 1 )= +1 e δ q L 1 e δ q 1 (4 q 1 q L ) 2 and using again the definition of q L and the fact that q L = x q 1,pofits can be e-witten as: µ 4 7x 4 7x δ(4 x) 3 (4 x) 3 δ e (1 x)(4 x) 1 e δ q 1 19

22 Notice that if (8) is satisfied as an equality, then the followe pofits ae nought, othewise pofits ae positive fo any value of x and δ. The only possible equilibia left ae those with the leade enteing with the low quality and the followe esponding with a highe one and the othe whee the opposite happens, depending on the value of the composite paamete δ. Howeve, we still have to ascetain whethe it is optimal fo the followe to espond with a highe (lowe) quality if the leade entes with a low (high) one. This is done in the following two popositions. Poposition 9 If δ 0, δ theleadeenteswithahighqualityandthe followe will always espond with a lowe one. Poof. This poof is conceptually simila to the pevious one. On the basis of Lemma 7, we can compute q 1 and q L. With the two levels of quality we can compute numeically Fim s 2 pofit as fom equation (6). Moeove, using the fist ode condition of the followe when enteing fom above (12), we can compute numeically the coesponding value of x, fo any given q 1 and δ and hence q H = q 1 /x. Those values of x ae epoted in the tables of the Appendix in the column denoted as x HH. Finally, we use q 1 and q H to compute the followe s pofit whendeviatingandenteingwiththehigh quality using (11). We povide hee the gaphical epesentation of the two levels of pofit of the followe showing that the followe neve deviates fom the low quality. i Poposition 10 If δ ³ δ, δ, the leade entes with a low quality and the ³ i followe will always espond with a highe one, while fo δ, , thee is no equilibium (in pue stategies) since the followe has an incentive to undecut the leade s quality. Poof. Relying on the poof of Lemma 7, we can compute q 1 and q H.With the two levels of quality we can compute numeically Fim s 2 pofit asfom equation (11). Moeove, using the fist ode condition of the followe when enteing fom below (7), we can compute numeically the appopiate value of 20

23 R 2H R 6 2L δ Figue 6: The leade entes with the high quality. Followe s pofits when choosing the low one (solid) and the high one (dots). x, fo any given q 1 and δ, and hence q L = xq 1.Thosevaluesofx ae epoted in the tables of the Appendix in the column denoted as x LL.Finally,weuse q 1 and q L to compute the followe s pofit when deviating and enteing with the low quality using (6). We povide hee the gaphical epesentation of the two pofit levels of the followe showing that the followe neve deviates fom the high quality, which shows that the followe s pofit aehighewhen enteing with the high quality, except fo vey high values of δ. A few emaks ae now in ode. Fist, a tivial one, efes to δ = δ. Fo that value of δ both equilibia hold. Second, ecall that δ = c/ θ 2 h i.thetwo Popositions 9 and 10 togethe imply that in the inteval 0, δ,folowδ the leade will ente with high quality, while with high ones he will choose a low quality. That is, the leade will ente with the high quality fo low levels of and with the low quality fo high levels of, fogivenc and θ. Sincealow implies a high discount ate, the esult has a vey intuitive explanation: a patient monopolist will ente late in ode to obtain a bette qulity, while impatient ones will ente ealie, even at the cost of choosing a low quality. 21

24 R 2H 0.4 R 2L δ Figue 7: The leade entes with the low quality. Followe s pofits when choosing the high one (solid) and the low one (dots). It is also athe intuitive that c, the cost of investing in quality, has the same effects as : ahighc makes the monopolist impatient. On the contay, the consumes willingness to pay fo quality, summaized by θ, has opposite effects, since the stategy of waiting fo a highe quality has highe etuns. Thid, we should like to assess ou esults against those of Lehmann-Gube (1997) and Dutta et al. (1995), so as to evaluate how diffeent assumptions about the time hoizon and the technology affect the featues of the subgame pefect equilibium. Lehmann-Gube (1997) genealises the analisys conducted by Shaked and Sutton (1982, 1983) to account fo a technology which is convex in the quality level, but emains in a single-peiod model whee thee esists no monopoly phase. This poduces the esult that suplus extaction is maximised when the fim locates at the top of the available quality spectum. In Dutta et al. (1995), it is assumed that (i) pe-peiod opeative duopoly pofits ae popotional to elative quality and ae symmetic; (ii) adoption (enty) dates ae endogenous, while (iii) the gowth of quality ove time is 22

25 not endogenously deteminedbyfims; (iv) unit poduction cost is flat w..t. quality; and (v) innovation costs ae summaised by the waiting time befoe the adoption. In this setup, the authos find that a late entant obtains lage pofits than an ealie entant, and no monopoly ent is dissipated at the subgame pefect equilibium. In ou setting, the enty timing is endogenously linked to quality impovement, and the cost bone to supply supeio qualities can be high enough to offset the advantage attached to seving ich customes. The inteplay of these factos may entail that, in some elevant paamete anges, all fims wouldpefetoenteealywithaninfeioqualityathethanlatewitha supeio one. 5 Concluding Remaks We have investigated the beaings of R&D expenditues in continuous time ove the enty pocess in a maket fo vetically diffeentiated goods. We have shown that enteing fist and enjoying an ad inteim monopoly ent may countebalance the incentive towads the supply of high quality goods in duopoly afte the enty of a second innovato. Indeed, we have poved that this is the only subgame pefect equilibium in a lage ange of paametes. The foegoing analysis shows that the established wisdom poduced by pevious liteatue in this field does not popely account fo the ole of time and its inteaction with R&D technology in detemining fims incentives. 23

26 Refeences [1] Aoki, R. and T. Pusa, 1997, Sequential vesus simultaneous choice with endogenous quality, Intenational Jounal of Industial Oganization, 15, [2] Beath, J., Y. Katsoulacos and D. Ulph, 1987, Sequential poduct innovation and industy evolution, Economic Jounal, 97, [3] Choi, C.J. and H.S. Shin, 1992, A comment on a model of vetical poduct diffeentiation, Jounal of Industial Economics, 40, [4] Donnenfeld, S. and S. Webe, 1992, Vetical poduct diffeentiation with enty, Intenational Jounal of Industial Oganization, 10, [5] Donnenfeld, S. and S. Webe, 1995, Limit qualities and enty deteence, RAND Jounal of Economics, 26, [6] Dutta, P.K., S. Lach and A. Rustichini, 1995, Bette late than ealy: vetical diffeentiation in the adoption of a new technology, Jounal of Economics and Management Stategy, 4, [7] Gabszewicz, J.J. and J.-F. Thisse, 1979, Pice competition, quality and income dispaities, Jounal of Economic Theoy, 20, [8] Gabszewicz, J.J. and J.-F. Thisse, 1980, Enty (and exit) in a diffeentiated industy, Jounal of Economic Theoy, 22, [9] Lambetini, L., 1999, Endogenous timing and the choice of quality in a vetically diffeentiated duopoly, Reseach in Economics (Riceche Economiche), 53, [10] Lehmann-Gube, U., 1997, Stategic choice of quality when quality is costly: the pesistence of the high-quality advantage, RAND Jounal of Economics, 28, [11] Levin, R.C., A.K. Klevoick, R.R. Nelson and S.G. Winte, 1987, Appopiating the etuns fom industial eseach and developments, Bookings Papes on Economic Activity,

27 [12] Mansfield, E., M. Schwatz and S. Wagne, 1981, Imitation costs and patents: an empiical study, Economic Jounal, 91, [13] Motta, M., 1992, Coopeative R&D and vetical poduct diffeentiation, Intenational Jounal of Industial Oganization, 10, [14] Motta, M., 1993, Endogenous quality choice: pice vs quantity competition, Jounal of Industial Economics, 41, [15] Rosenkanz, S., 1995, Innovation and Coopeation unde Vetical Poduct Diffeentiation, Intenational Jounal of Industial Oganization, 13, [16] Rosenkanz, S., 1997, Quality impovements and the incentive to leapfog, Intenational Jounal of Industial Oganization, 15, [17] Shaked, A. and J. Sutton, 1982, Relaxing pice competition though poduct diffeentiation, Review of Economic Studies, 49, [18] Shaked, A. and J. Sutton, 1983, Natual oligopolies, Econometica, 51,

28 Appendix Table 1: Numeical solutions fo low δ s. Enty fom below Enty fom above δ x LH x LL x HL x HH

29

30 Table 2: Numeical solutions fo intemediate δ s. Enty fom below Enty fom above δ x LH x LL x HL x HH

31

32 Table 3: Numeical solutions fo high δ s. Enty fom below Enty fom above δ x LH x LL x HL x HH

33