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
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h ps://hdl.handle.ne /10419/159333
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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