Sequential Entry in a Vertically Differentiated Duopoly

Lambe ini, Luca; Tedeschi, Pie o Wo king Pape Sequen ial En y in a Ve ically Di e en ia ed Duopoly Quade ni - Wo king Pape DSE, No. 492 P o ided in Coope a ion wi h: Uni e si y o Bologna, Depa men o Economics Sugges ed Ci a ion: Lambe ini, Luca; Tedeschi, Pie o (2003) : Sequen ial En y in a Ve ically Di e en ia ed Duopoly, Quade ni - Wo king Pape DSE, No. 492, Alma Ma e S udio um - Uni e si à di Bologna, Dipa imen o di Scienze Economiche (DSE), Bologna, h ps://doi.o g/10.6092/unibo/amsac a/4802 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/159333 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen (insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en, gel en abweichend on diesen Nu zungsbedingungen die in de do genann en Lizenz gewäh en Nu zungs ech e. Te ms o use: Documen s in EconS o may be sa ed and copied o you pe sonal and schola ly pu poses. You a e no o copy documen s o public o comme cial pu poses, o exhibi he documen s publicly, o make hem publicly a ailable on he in e ne , o o dis ibu e o o he wise use he documen s in public. I he documen s ha e been made a ailable unde an Open Con en Licence (especially C ea i e Commons Licences), you may exe cise u he usage igh s as speci ied in he indica ed licence. h ps://c ea i ecommons.o g/licenses/by/3.0/ Sequen ial En y in a Ve ically Diffe en ia ed Duopoly Luca Lambe ini#Pie o Tedeschi§ # Uni e si y o Bologna §Uni e si y o Milano-Bicocca No embe 27, 2003 Abs ac We analyse a model o e ical diffe en ia ion ocusing on he ade-offbe- ween en e ing ea ly and exploi ing monopoly powe wi h a low quali y, e - sus wai ing and enjoying a dominan ma ke posi ion wi h a supe io p oduc . We show ha he e exis s a unique equilib ium whe e he leade en e s wi h a lowe quali y han he ollowe , o low discoun ac o s, o high cos s o quali y and o low consume s’ willingness o pay o quali y. J.E.L. Classi ica ion: L13, O31 Keywo ds: e ical diffe en ia ion, p oduc inno a ion, monopoly en 1In oduc ion An appa en ly well es ablished esul in he heo y o e ically diffe en ia ed oligopoly s a es ha ea lie en an s supply goods o highe quali y han la e en an s, in ha he high-quali y p oduc s ea n highe p o i s han low-quali y al e na i es (see, in e alia, Gabszewicz and Thisse, 1979, 1980; Shaked and Su on, 1982, 1983; Donnen eld and Webe , 1992, 1995). A gene al p oo o his esul o e e y con ex ixed-cos unc ion o quali y imp o emen is p o ided by Lehmann-G ube (1997).1 Two basic assump ions a e a he basis o his esul . The i s is ha consume s’ ma ginal willingness o pay o quali y is uni o mly dis ibu ed o e a gi en suppo . Since he densi y o consume s (i.e., demand) is he same a any income le el, he op-quali y ma ke niche is he mos p o i able. The e o e, in a s a ic game, i ms ob iously p e e o en e wi h a p oduc cha ac e ised by he highes possible quali y. The second assump ion conce ns he ime ho izon conside ed in he abo e men ioned li e a u e. En y in a e ically diffe en ia ed ma ke is usually analyzed wi hin a single-pe iod ex ensi e o m game. Howe e , i one models he en y p oblem in an explici dynamic se up, an ob ious ade-offimme- dia ely appea s, e en main aining he p e ious assump ion. In o de o en e wi h an high quali y p oduc , he i m has o wai o he R&D ac i i y o ake place and consequen ly i looses monopoly p o i s. Howe e , pos pon- ing en y, he i m is able o p oduce a highe quali y good, ob aining hus highe p o i s. A s a ic model does no allow o assess he possibili y ha he e exis s such a ade-offbe ween ea ly inno a ion and he a ainmen o a dominan posi ion in he ma ke . Al hough i is gene ally asse ed ha quali y may esul om i ms’ R&D 1Aoki and P usa (1997) adop a speci ic case o he cos unc ion analysed by Lehmann- G ube (1997), o in es iga e he consequences on p o i s, consume su plus and social wel a e o he iming o in es men in p oduc quali y in a e ically diffe en ia ed duopoly whe e he ma ke s age is played in he p ice space. To his ega d, see also Lambe ini (1999). 1 effo s, his aspec o e ical p oduc diffe en ia ion has ecei ed a ela i ely scan y a en ion, he de elopmen phase being summa ised by a cos unc ion which does no accoun o he ime elapsed be o e he good is p oduced and hen ma ke ed. To ou knowledge, ele an con ibu ions dealing explici ly wi h he R&D ac i i y a e Bea h e al. (1987); Mo a (1992); Rosenk anz (1995, 1997) and Du a e al. (1995). These pape s in es iga e he incen i e owa ds R&D coope a ion (Mo a, 1992; Rosenk anz, 1995) and he ela ion- ship be ween R&D and he pe sis ence o quali y leade ship (Bea h e al., 1987; Rosenk anz, 1997). Du a e al. (1995) analyse s a egic iming in he adop ion o a new echnology leading o p oduc diffe en ia ion and quali y imp o emen s. All o hese pape s main ain ha being he quali y leade (i.e., supplying he highes quali y in he ma ke ) en ails highe p o i s han he i als. We p esen a simple model o e ical diffe en ia ion ocusing upon he ade-offbe ween en e ing ea ly and exploi ing monopoly powe wi h a low quali y, e sus wai ing and enjoying a dominan ma ke posi ion wi h a su- pe io p oduc . We e ain he assump ion o a uni o m income dis ibu ion, ha would make i p o i able o p oduce a high quali y good in a s a ic game, bu elax he assump ion o a s a ic ex ensi e o m game. Namely, in ou model he e exis s a unique equilib ium whe e he leade en e s wi h a lowe quali y han he ollowe , o a la ge se o pa ame e alues.2 This highligh s ha an un a ou able posi ion in duopoly (o oligopoly), due o a lowe quali y han he i als’, may well be mo e han balanced by he monopoly en enjoyed ad in e im wi h lowe de elopmen cos s. The e o e, i appea s ha he es ablished wisdom s a ing ha ea ly en y goes along wi h high quali y (and p o i s) is no obus o a ully ledged in es iga ion o he ole o calenda ime in shaping endogenously i ms’ incen i es. The emainde o he pape is s uc u ed as ollows. The basic model 2F om a diffe en se ing, Du a e al. (1995) also de i e an equilib ium whe e he i s en an p oduces a lowe quali y han he second en an . Howe e , in hei model he la e en an makes mo e p o i s. As i will become clea in he emainde , his conclusion es s upon he shape o he cos unc ion. 2 o e ical diffe en ia ion is laid ou in sec ion 2. Sec ion 3 desc ibes he solu ion o all admissible subgames. The subgame pe ec equilib ium o he whole game is de i ed in sec ion 4. Finally, sec ion 6 p o ides concluding ema ks. 2TheModel Conside a ma ke o e ically diffe en ia ed p oduc s. Le his ma ke exis o e ime ,wi h ∈[0,∞).Two single-p oduc i ms, labelled 1 and 2, p oduce goods o diffe en quali ies, q1and q2∈[0,∞), h ough he same echnology.Wi hou loss o gene ali y we can assume ha i ms p oduc ion cos s a e nough , while de elopmen cos s a e Ci(qi)=cZq+qi q e− d (1) wi h i=1 ,2andq≥0. De elopmen cos s Ci(qi) a e e alua ed a he beginning o he pe iod o in es men , he e o e in 0 o i m 1 and in 1 o i m 2. As usual, hese cos s can be in e p e ed as ixed cos due o he R&D effo needed o p oduce a ce ain quali y. We cha ac e ize he echnology ep esen ed by he abo e cos unc ion as ollows: Assump ion 1 The R&D cos s a e cons an o e ime and equal o c.I i m isea ches o a pe iod o leng h i, hen i can p oduce a good a mos o quali y iand any o he lowe quali y. Once en e ed in o he ma ke he i m canno in es anymo e in R&D. The abo e amoun s o assuming ha any change in he quali y le el implies adjus men cos s i and only i he change akes he o m o a quali y inc ease. Con e sely, once i m ihas bo ne he cos o de eloping a gi en quali y, she may decide o dec ease he quali y o he p oduc cos lessly. Fo he sake o simplici y we assume ha quali y is s ic ly co ela ed wi h he ime o en y. Mo e p ecisely, i i m 1 en e s a ime 1,i smaximum easible quali y is 1=q1. Fi m 2’s cos o imi a ion, howe e , a e exac ly equal 3 o he cos s o inno a ion.3The e o e, i m 2’s ime o en y sa is ies he equali y 2=q1+q2. In he emainde , we shall label he i s en an as he leade . Fi m 2 en e s a da e 2∈[ 1,∞),and we shall e e o he as he ollowe . Assump ion 2 P oduc s a e offe ed on a ma ke whe e consume s ha e uni demands, and buy i and only i he ne su plus de i ed om consump- ion θ(qk,p i(qk)) = θqk−pi(qk)≥0,whe e pi(qk)is he uni p ice cha ged by i m ionagoodo quali yqk, pu chased by a gene ic con- sume whose ma ginal willingness o pay is θ∈[θ,¯ θ],wi h θ=¯ θ−1. We assume ha θis uni o mly dis ibu ed wi h densi y one o e such in e al, so ha he o al mass o consume is one. Th oughou he ollowing analysis, we assume pa ial ma ke co e age. The abo e assump ion is a he common in e ically diffe en ia ed p od- uc models. Mo e ele an a e he assump ions ela i e o he iming o he game. Assump ion 3 Fi m 1 chooses when o en e he ma ke wi h he new p od- uc and simul aneously chooses he quali y and he p ice o be offe ed. Then i m 2 decides whe he o imi a e i m 1 and when o en e he ma ke . Once i m 2 has en e ed, he wo i ms choose simul aneously he quali y le els, which become common knowledge. Finally bo h i ms choose simul aneously he p ice le els. This iming can be jus i ied as ollows. Suppose ha i m 1 has in en ed a new p oduc , bu i has o decide he quali y le el o ha p oduc be o e en y. Since nobody knows he exis ence o his new p oduc , only i m 1 can en e i s . The ea e , o he i ms can imi a e i m 1. Suppose only i m 2 has he necessa y echnology. Howe e , i m 2 has o sus ain he R&D 3The case o e y high imi a ion cos s is suppo ed by empi ical indings (see Mans ield e al., 1981; and Le in e al., 1987). 4 cos s be o e being able o en e and his akes ime and p ecisely he pe iod be ween 1and 2.4 3 Solu ion o he Game As usual we will sol e he game backwa ds. Howe e , i is use ul be o e sol ing he model o in oduce wo de ini ions, conce ning i ms’ beha io . In he emainde , we shall e e o he i s en an ( i m 1) as he leade , and o he second en an ( i m 2) as he ollowe .Wea egoing oexamine wo al e na i e pe spec i es: A. The ollowe en e s a 2wi h a p oduc whose quali y is lowe han he leade ’s. We label his case as high-quali y leade ship. B. The ollowe en e s a 2wi h a p oduc whose quali y is highe han he leade ’s. We label his case as low-quali y leade ship. 3.1 The P ice Game In bo h cases, o e ∈[ 2,∞), i ms compe e in p ices. We bo ow om Aoki and P usa (1997) and Lehmann-G ube (1997) he assump ion ha downs eam Be and compe i ion is simul aneous. Ma ke demands o he high- and low-quali y good a e, espec i ely: xH=¯ θ−pH−pL qH−qL and xL=pH−pL qH−qL −pL qL (2) Duopoly e enue unc ions a e RH=pHxHand RL=pLxL.Sol ing o he equilib ium p ices, we ob ain: pH=2 ¯ θqH qH−qL 4qH−qL ;pL=¯ θqL qH−qL 4qH−qL (3) 4To sol e he game we adop subgame pe ec ion, and we look o simul aneous Nash equilib ia in each s age. Conside ing he S ackelbe g solu ion would make calcula ions mo e cumbe some wi hou affec ing signi ican ly he main esul s. 5 which allow o ew i e he e enue unc ion o i ms in e ms o quali ies only, as ollows:5 RH=4¯ θ2q2 H(qH−qL) (4qH−qL)2(4) RL=¯ θ2qHqL(qH−qL) (4qH−qL)2(5) On he basis o exp essions (4-5), p e ious li e a u e, dealing wi h single- pe iod models, es ablishes ha he i s en an would choose o supply he high-quali y good, gi en ha RH>R L. In he emainde , we label he leade ’s quali y as q1and he ollowe ’s quali y as ei he qHo qL,wi h he unde s anding ha qH≥q1and q1≥qL. 3.2 The Followe ’s Quali y Choice We de e mine he condi ions which induce he ollowe o en e ei he wi h a lowe o wi h a highe quali y han he leade . We will de ine he wo si ua ions en y om below and en y o m abo e, and will be analyzed in a sequel. 3.2.1 En y om below The ollowe ’s p o i s when en e ing om below a e: Π2L=Z∞ q1+qL RLe− d −cZq1+qL q1 e− d = RL e−(q1+qL) −c ¡e−q1 −e−(q1+qL) ¢ which using (5) can be ew i en as: R2L(qL,q 1)=¯ θ2q1qL(q1−qL) (4q1−qL)2 e−(q1+qL) −c ¡e−q1 −e−(q1+qL) ¢= ¯ θ2 e− q1µq1qL(q1−qL) (4q1−qL)2e− qL+c ¯ θ2e− qL−c ¯ θ2¶ 5The p oo is omi ed he e, as i is p o ided by se e al au ho s (Gabszewicz and Thisse, 1979; Choi and Shin, 1992; Mo a, 1993; Aoki and P usa, 1997; Lehmann-G ube, 1997). 6 0 0.02 0.06 5 10 0.04 ПH δ x Figu e 1: P o i o he leade when en e ing om abo e 0 0.01 0.02 0.03 0.04 0.05 0.06 0.1 0.2 0.3 0.4 0.5 x δ Figu e 2: Fi s o de condi ion o he leade when en e ing om abo e 13 En y om below The p o i unc ion o i m1whenen e ing ombelow is: RMRqM+qH qMe− d +RLR∞ qM+qHe− d −cRqM 0e− d = RM e− qM−e−(qM+qH) +RL e−(qM+qH) −c1−e− qM = qM¯ θ2 4 e− qM−e−(qM+qH) +¯ θ2qHqL(qH−q1) (4qH−q1)2 e−(qM+qH) −c(1 −e− qM) . This is equi alen o: ¯ θ2ΠML (qM,q H,q L)= qHqL(qH−qL) (4qH−qL)2e−(qM+qH) +1 4¡1−e−qH ¢e−qM qM−γ¡1−e−qM ¢ whe e γ=c/¯ θ2,asusual. We ha e o dis inguish wo diffe en cases. In he i s one, δ≤7/24 and he e o e P oposi ion 4 holds. In he second one, δ>7/24. Le us s a om he i s case. Case I: δ≤7/24. Hence we can se : q1=γ˜ q1=8 49 7−24 γ ,q 2=γ˜ q2=2 7 7−24 γ which can be subs i u ed in he p o i unc ion o yield: 1 γΠML µγ˜ qM,2 7 7−24 γ ,8 49 7−24 γ ¶= 1 168 Ã(7 −24δ−42δ˜ qM)e(−2+48 7δ) δ+42(˜ qM+4) !e−δ˜qM−1. F om he i s o de condi ion w. . . qM,we ob ain: ˜ qM=1 42 (−49 + 24δ)e−2+ 48 7δ+42(1−4δ) δ³1−e−2+ 48 7δ´ 14 No ice ha abo e exp ession cha ac e izes he leade ’s choice when en e ing wi h he low quali y i qM≥q1o ˜ qM(= qM/γ)≥˜ q1(= q1/γ), ha is, i : 1 42 (−49 + 24δ)e−2+48 7δ+42(1−4δ) δ³1−e−2+ 48 7δ´−8 49 7−24δ δ≥0 which a e some manipula ion is equi alen o: 1 294 7e−2+ 48 7δ+984 e−2+48 7δδ+42+24 δ δ³e−2+48 7δ−1´≥0 Since he nume a o is always posi i e, he abo e condi ion implies ha he denomina o should be posi i e, o equi alen ly ha : δ≥7 24 '0.291 67 and ecalling P oposi ion 4 we know ha he condi ion canno be sa is ied o a posi i e quali y le el. We summa ize he abo e analysis in he ollowing p oposi ion: P oposi ion 5 I espec i e o whe he he leade en e s wi h he low o he high quali y, he quali y o he leade a e he ollowe has en e ed he ma ke is equal o ha o he monopoly phase, i.e., q1=qM,i δ≤7/24. Case II: δ>7/24. Now we analyze he si ua ion whe e he monopolis ’s choice is binding in he duopoly phase. In such a case he leade ’s p o i s, a e i ial ans o ma ion, become: ΠML (qH,q 1)=µµqHq1(qH−q1) (4qH−q1)2−1 4q1¶e−qH +1 4q1+γ¶e−q1 −γ and a e he usual a iable ans o ma ions: ΠML (γ˜ qH,γ˜ q1)=γµµ˜ qH˜ q1 ˜ qH−˜ q1 (4˜ qH−˜ q1)2−1 4˜ q1¶e−δ˜qH+1 4˜ q1+1 ¶e−δ˜q1−γ whe e again δ=γ .De ining: ΠL(˜ qH,˜ q1,δ)=1 γΠML (γ˜ qH,γ˜ q1)+γ 15 0 2 0.2 1 xδ ПL Figu e 3: Leade ’s p o i when en e ing om below and se ing ˜ qH=x˜ q1, we ob ain: ΠL(˜ qH,x˜ qH,δ)=1 4µµ1−12 −4x+x2 (4 −x)2e−δ˜qH¶x˜ qH+4 ¶e−δx˜qH Recalling (12), he monopolis p oblem is equi alen o maximizing he ol- lowing exp ession, wi h espec o x: ΠLÃ4(4−3x+2 x2)−δ(4 −x)3 4δ(4 −x)(1−x),x4(4−3x+2 x2)−δ(4 −x)3 4δ(4 −x)(1−x)!(20) Using es ic ion (13), we can p oduce a g aphical explo a ion o he p oblem in Figu e 3. I shows ha he unc ion has a unique global maximum o each alue o δ. Mo eo e , he i s o de condi ion is: DL(x, δ)=(21) ∂ ∂xΠLÃ4(4−3x+2 x2)−δ(4 −x)3 4δ(4 −x)(1−x),x4(4−3x+2 x2)−δ(4 −x)3 4δ(4 −x)(1−x)!=0 and i is no sol able analy ically. Howe e , i s implici plo is in Figu e 4. 16 x δ 0 0.02 0.04 0.06 0.54 0.55 0.56 0.57 0.58 0.59 0.6 Figu e 4: Leade ’s i s o de condi ion when en e ing om below 4 Isi Con enien oEn e heMa ke wi h a High-quali y P oduc ? Now we can sol e o he subgame pe ec equilib iumo he whole game by de e mining whe he he leade will en e wi h a high o a low quali y. We i s p o e a p elimina y esul . P oposi ion 6 No equilib ium wi h he ollowe en e ing he ma ke wi h a lowe quali y han he leade does exis i δ= c/¯ θ2>0.0625. P oo . I is a di ec consequence o (10). We a e now in he posi ion o p o e he main Lemma o his sec ion. Lemma 7 The e exis s a ¯ δsuch ha , o δ∈£0,¯ δ¢ he e is no equilib ium wi h he ollowe en e ing he ma ke and he leade p oducing he lowe quali y good, while o δ∈¡¯ δ,0.0625¤ he e exis s no equilib ium wi h he ollowe en e ing he ma ke and he leade p oducing he highe quali y good. The alue o ¯ δis app oxima ely: ¯ δ=0 .0203125. 17 0 20 40 60 80 0.02 0.04 0.06 ПL δ ПH Figu e 5: P o i s o i m 1 when en e ing wi h high (solid) and when en e ing wi h low (dash) quali y. P oo . We sol e nume ically equa ions (19) and (21), inding he op imal x o he wop oblems o a ious alueso δ. The compu ed alues a e epo ed in he Table 1-3 o he Appendix in columns deno ed espec i ely by xHL and xLH .Byusing(7)wecancompu e˜ q1and ˜ qL=x˜ q1, heop imal alues o ans o med a iables eplacing qLand q1.Byusing(12)wecan compu e, ins ead, ˜ qHand ˜ q1=x·˜ qH. Gi en he a ious le el o quali ies, he p o i s o he monopolis en e ing om abo e and en e ing om below can be compu ed and a e d awn in Figu e 5. I can be seen ha he p o i o he high quali y monopolis a e highe o lowe le el o δand lowe he ea e . The wo cu es c oss a ¯ δ. The wo le els o he ollowe ’s p o i s, RLwhen i chooses a lowe quali y han he leade ’s and RHwhen i chooses a highe quali y, a e ep esen ed in he ollowing wo igu es. The i s one ep esen s he wo a iables when he leade ies o en e wi h a highe quali y han he ollowe and, as we can see, he bes esponse o he ollowe consis s in choosing a highe quali y. 18 In he second Figu e, ins ead, we ep esen he wo ollowe ’s p o i le els when he leade ies o en e wi h a low quali y. In his case, we see ha he esponse o he ollowe is consis en wi h he leade ’s s a egy. The abo e Lemma allows us o in e ha , in ou model, he leade en e s wi h he high quali y only i i can block he ollowe ’s en y. The e o e, he e we may ha e only wo ypes o equilib ia. In he i s one he leade in es s in R&D in such a way o be able o main ain i s monopoly posi ion. In he second ype o equilib ium, he leade en e s wi h he low quali y and hen he ollowe en e s wi h a highe quali y. The ollowing p oposi ion will exclude he i s ou come, ha whe e he leade can ha e a monopoly powe . P oposi ion 8 Fo δsufficien ly small, he e exis s no equilib ium whe e he leade succeeds in p e-emp ying he ma ke . In pa icula , δ≤1/16 is a sufficien condi ion o he leade no o be able o p e-emp he ma ke . P oo . In o de o p o e he P oposi ion, we mus check ha he ollowe can always en e wi h a lowe quali y o any choice o he leade , making posi i e p o i s. Recall ha he op imal choice o he ollowe is exp essed by (7), which is e-w i en o con enience: ˜ qL=4−7x−δ(4 −x)3 δ(1 −x)(4−x) We know ha xmus sa is y inequali y (8): δ≤4−7x (4 −x)3 Recall also ha p o i s o he ollowe en e ing wi h he low quali y a e: cR2L(γ˜ qL,γ˜ q1)=µµ˜ q1˜ qL(˜ q1−˜ qL) (4˜ q1−˜ qL)2+1 ¶e−δ˜qL−1¶e−δ˜q1 and using again he de ini ion o ˜ qLand he ac ha ˜ qL=x˜ q1,p o i s can be e-w i en as: µ4−7x (4 −x)3δe−4−7x−δ(4−x)3 (1−x)(4−x)−1¶e−δ˜q1 19 No ice ha i (8) is sa is ied as an equali y, hen he ollowe p o i s a e nough , o he wise p o i s a e posi i e o any alue o xand δ. The only possible equilib ia le a e hose wi h he leade en e ing wi h he low quali y and he ollowe esponding wi h a highe one and he o he whe e he opposi e happens, depending on he alue o he composi e pa ame e δ. Howe e , we s ill ha e o asce ain whe he i is op imal o he ollowe o espond wi h a highe (lowe ) quali y i he leade en e s wi h a low (high) one. This is done in he ollowing wo p oposi ions. P oposi ion 9 I δ∈£0,¯ δ¢ heleade en e swi hahighquali yand he ollowe will always espond wi h a lowe one. P oo . This p oo is concep ually simila o he p e ious one. On he basis o Lemma 7, we can compu e q1and qL. Wi h he wo le els o quali y we can compu e nume ically Fi m’s 2 p o i as om equa ion (6). Mo eo e , using he i s o de condi ion o he ollowe when en e ing om abo e (12), we can compu e nume ically he co esponding alue o x, o any gi en q1 and δand hence qH=q1/x.Those alueso xa e epo ed in he ables o he Appendix in he column deno ed as xHH. Finally, we use q1and qH o compu e he ollowe ’s p o i whende ia inganden e ingwi h hehigh quali y using (11). We p o ide he e he g aphical ep esen a ion o he wo le els o p o i o he ollowe showing ha he ollowe ne e de ia es om he low quali y. P oposi ion 10 I δ∈³¯ δ,δi, he leade en e s wi h a low quali y and he ollowe will always espond wi h a highe one, while o ³δ,0.0625i, he e is no equilib ium (in pu e s a egies) since he ollowe has an incen i e o unde cu he leade ’s quali y. P oo . Relying on he p oo o Lemma 7, we can compu e q1and qH.Wi h he wo le els o quali y we can compu e nume ically Fi m’s 2 p o i as om equa ion (11). Mo eo e , using he i s o de condi ion o he ollowe when en e ing om below (7), we can compu e nume ically he app op ia e alue o 20 0 1 2 3 4 5 6 0.01 0.02 0.03 0.04 0.05 0.06 δ R2H R2L Figu e 6: The leade en e s wi h he high quali y. Followe ’s p o i s when choosing he low one (solid) and he high one (do s). x, o any gi en q1and δ, and hence qL=xq1.Those alueso xa e epo ed in he ables o he Appendix in he column deno ed as xLL.Finally,weuse q1and qL o compu e he ollowe ’s p o i when de ia ing and en e ing wi h he low quali y using (6). We p o ide he e he g aphical ep esen a ion o he wo p o i le els o he ollowe showing ha he ollowe ne e de ia es om he high quali y, which shows ha he ollowe ’s p o i a ehighe when en e ing wi h he high quali y, excep o e y high alues o δ. A ew ema ks a e now in o de . Fi s , a i ial one, e e s o δ=¯ δ.Fo ha alue o δbo h equilib ia hold. Second, ecall ha δ= c/¯ θ2.The wo P oposi ions 9 and 10 oge he imply ha in he in e al h0,δi, o lowδ he leade will en e wi h high quali y, while wi h high ones he will choose a low quali y. Tha is, he leade will en e wi h he high quali y o low le els o and wi h he low quali y o high le els o , o gi encand ¯ θ.Sincealow implies a high discoun a e, he esul has a e y in ui i e explana ion: a pa ien monopolis will en e la e in o de o ob ain a be e quli y, while impa ien ones will en e ea lie , e en a he cos o choosing a low quali y. 21 0 0.1 0.2 0.3 0.4 0.03 0.04 0.05 0.06 R2H R2L δ Figu e 7: The leade en e s wi h he low quali y. Followe ’s p o i s when choosing he high one (solid) and he low one (do s). I is also a he in ui i e ha c, he cos o in es ing in quali y, has he same effec s as :ahigh cmakes he monopolis impa ien . On he con a y, he consume s’ willingness o pay o quali y, summa ized by ¯ θ, has opposi e effec s, since he s a egy o wai ing o a highe quali y has highe e u ns. Thi d, we should like o assess ou esul s agains hose o Lehmann-G ube (1997) and Du a e al. (1995), so as o e alua e how diffe en assump ions abou he ime ho izon and he echnology affec he ea u es o he sub- game pe ec equilib ium. Lehmann-G ube (1997) gene alises he analisys conduc ed by Shaked and Su on (1982, 1983) o accoun o a echnology which is con ex in he quali y le el, bu emains in a single-pe iod model whe e he e esis s no monopoly phase. This p oduces he esul ha su plus ex ac ion is maximised when he i m loca es a he op o he a ailable quali y spec um. In Du a e al. (1995), i is assumed ha (i) pe -pe iod ope a i e duopoly p o i s a e p opo ional o ela i e quali y and a e symme ic; (ii) adop ion (en y) da es a e endogenous, while (iii) he g ow h o quali y o e ime is 22 0.035625 0.5702663289 0.2050551901 0.2131944419 0.0883337068 0.036250 0.5702252095 0.2007263935 0.2088696914 0.0859475081 0.036875 0.5701828157 0.1963802531 0.2045194100 0.0835635313 0.037500 0.5701391237 0.1920165513 0.2001432774 0.0811825600 0.038125 0.5700941095 0.1876350669 0.1957409718 0.0788054078 0.038750 0.5700477480 0.1832355747 0.1913121692 0.0764329080 0.039375 0.5700000141 0.1788178457 0.1868565444 0.0740659098 0.040000 0.5699508820 0.1743816469 0.1823737701 0.0717052971 0.040625 0.5699003253 0.1699267414 0.1778635173 0.0693519571 0.041250 0.5698483171 0.1654528876 0.1733254556 0.0670068124 29 Table 3: Nume ical solu ions o high δ’s. En y om below En y om abo e δx LH xLL xHL xHH 0.041875 0.5697948299 0.1609598402 0.1687592523 0.0646707925 0.042500 0.5697398357 0.1564473490 0.1641645734 0.0623448659 0.043125 0.5696833057 0.1519151595 0.1595410828 0.0600300129 0.043750 0.5696252105 0.1473630125 0.1548884429 0.0577272307 0.044375 0.5695655203 0.1427906439 0.1502063139 0.0554375448 0.045000 0.5695042043 0.1381977848 0.1454943546 0.0531619999 0.045625 0.5694412313 0.1335841613 0.1407522217 0.0509016549 0.046250 0.5693765691 0.1289494940 0.1359795701 0.0486575954 0.046875 0.5693101851 0.1242934986 0.1311760528 0.0464309355 0.047500 0.5692420458 0.1196158850 0.1263413209 0.0442227972 0.048125 0.5691721169 0.1149163575 0.1214750236 0.0420343272 0.048750 0.5691003633 0.1101946149 0.1165768081 0.0398666744 0.049375 0.5690267493 0.1054503495 0.1116463198 0.0377210423 0.050000 0.5689512381 0.1006832480 0.1066832018 0.0355986197 0.050625 0.5688737922 0.0958929905 0.1016870953 0.0335006319 0.051250 0.5687943732 0.0910792507 0.0966576396 0.0314283115 0.051875 0.5687129417 0.0862416955 0.0915944716 0.0293829044 0.052500 0.5686294573 0.0813799851 0.0864972263 0.0273656750 0.053125 0.5685438792 0.0764937725 0.0813655364 0.0253779143 0.053750 0.5684561647 0.0715827035 0.0761990324 0.0234209261 0.054375 0.5683662708 0.0666464162 0.0709973427 0.0214960158 0.055000 0.5682741531 0.0616845411 0.0657600932 0.0196044799 0.055625 0.5681797662 0.0566967009 0.0604869077 0.0177476705 30 0.056250 0.5680830636 0.0516825096 0.0551774074 0.0159268892 0.056875 0.5679839975 0.0466415730 0.0498312113 0.0141435302 0.057500 0.5678825190 0.0415734883 0.0444479358 0.0123989271 0.058125 0.5677785781 0.0364778434 0.0390271949 0.0106944486 0.058750 0.5676721233 0.0313542170 0.0335686000 0.0090314093 0.059375 0.5675631020 0.0262021781 0.0280717598 0.0074112368 0.060000 0.5674514598 0.0210212859 0.0225362806 0.0058352248 0.060625 0.5673371415 0.0158110892 0.0169617657 0.0043047257 0.061250 0.5672200899 0.0105711263 0.0113478159 0.0028211172 0.061875 0.5671002466 0.0053009246 0.0056940290 0.0013857253 31