Lambe ini, Luca
Wo king Pape
Cou no Vs S ackelbe g Equilib ia Wi h En ep eneu ial
and Labou Managed Fi ms
Quade ni - Wo king Pape DSE, No. 217
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 (1995) : Cou no Vs S ackelbe g Equilib ia Wi h En ep eneu ial
and Labou Managed Fi ms, Quade ni - Wo king Pape DSE, No. 217, 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/5108
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/159060
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COURNOT VS STACKELBERG EQUILIBRIA
WITH ENTREPRENEURIAL AND LABOUR MANAGED FIRMS
Luca Lambe ini
Dipa imen o di Scienze Economiche#
Uni e si à degli S udi di Bologna
S ada Maggio e 45
40125 Bologna
I aly
el 39-51-6402600
ax 39-51-6402664
e-mail [email p o ec ed]
and
Linac e College
Ox o d OX1 3JA
Uni ed Kingdom
e-mail [email p o ec ed]
Abs ac
The issueo equilib ium selec ion in a duopolygame be ween a p o i maximizing and a labou
managed i m is add essed unde ei he p ice o quan i y compe i ion wi h p oduc
di e en ia ion.I i ms can choose he iming o mo es be o ecompe ing in he ele an ma ke
a iable, he Be and game yields mul iple equilib ia, while he Cou no game has a unique
subgame pe ec equilib ium wi h he p o i maximizing i m in he leade ’s ole and he labou
managed i m in he ollowe ’s ole. Due o a lowe o al ou pu , he Cou no -S ackelbe g
equilib ium yields a lowe le el o social wel a e as compa ed o he simul aneous equilib ium.
This educes he incen i e o ans o m an LM duopoly in o a mixed one.
JEL classi ica ion: D43, D92, L13, L20
Keywo ds: ex ended game, sequen ial play
Acnowledgemen s
I would like o hank Fla io Delbono, Vincenzo Denicolò, Paolo Ga ella and Gianpaolo
Rossini o help ul commen s and discussion. The usual disclaime applies.
#Please send co espondence o he I alian add ess
1. In oduc ion
A la ge body o li e a u e deals wi h he issue o choosing oles in sequen ial duopoly
games.In hecon ex o duopolis iccompe i ionbe weenp o i maximizing(PM) i ms,Gal-O
(1985)andDow ick(1986)show ha ,p o ided i msa esymme ic, heslopeo hei espec i e
eac ion unc ions in he ele an s a egic a iable, i.e., ei he p ice o quan i y, de e mines
whe he hey p e e o ac as a leade o a ollowe . Speci ically, bo h i ms would p e e o be
he leade ( ollowe ) in quan i y (p ice) se ing games i eac ion unc ions a e downwa d
(upwa d) sloping, due o he p esence o s a egic subs i u abili y (complemen a i y) be ween
goods (see Bulow e al., 1985). The esul s eached by he abo e con ibu ions a e ex ended o
he case o di e en ia ed p oduc s by Gal-O (1985) and Boye and Mo eaux (1987).
In a ecen pape ,Okuguchi (1993b) in es iga es he p e e enceso labou managed (LM)
i ms as o he dis ibu ion o oles unde bo h Be and and Cou no compe i ion and p oduc
di e en ia ion, inding ou ha , in sha p con as o wha happens when only en ep eneu ial
i ms a e in ol ed, in he case o a pu e LM duopoly, eac ion unc ions a e upwa d sloping
ega dlesso he kind o compe i ion, be ha inp ices o quan i ies. Hence, bo h LM duopolis s
would p e e o ac as a ollowe , independen ly o he s a egic a iable being se .
E en hough he compa ison be ween he payo s acc uing o duopolis s in simul aneous
and sequen ial games, as well as he conclusions d awn om i , is ele an in i sel , i does no
p o ide any answe o he main ques ion, namely, whe he i ms’ p e e ences would allow o
any o he sequen ial o simul aneous equilib ia o endogenously eme ge as he equilib ium o
he unde lying game one could en isage, i.e., a game whe e i ms a e i s equi ed o announce
he iming o hei espec i e mo es and hen p oceed o se he ele an a iable in o de o
maximize hei own objec i e unc ion in he basic ma ke game. This issue has been ackled
in a e y in luen ial pape by Hamil on and Slu sky (1990). They embed simul aneous and
sequen ial play in o an ex ended game wi h obse able delay whe e playe s mus se bo h he
s a egic a iableo hebasicgameand he ime ose ha a iable.Thela e p ocessisac ually
1
a logical p eplay s age which is no obse ed. I playe s decide o mo e a he same ime, a
simul aneous equilib ium is obse ed, and ice e sa. I is no ewo hy ha he decision o play
ea ly a he han a a la e s age is no su icien o yield S ackelbe g leade ship, since an
analogous decision by he i al de e mines he eme gence o a simul aneous Nash equilib ium.
Thus, a S ackelbe g equilib ium (o sequen ial play) wi h one playe mo ing i s and he i al
second will be he only subgame pe ec equilib ium o he ex ended game i only one o he
wo possible sequen ial play ou comes Pa e o-domina es he simul aneous play ou come
(Hamil on and Slu sky, 1990, Theo em IV, p.37). O he wise, when bo h playe s sha e he same
p e e ences o e he sequence o mo es and he ollowe ’s payo domina es ha associa ed
wi h simul aneous play, hen bo h sequen ial equilib ia (as well as a mixed s a egy one) a e
subgame pe ec equilib ia o he ex ended game, so ha in p inciple i is impossible o know
which o hem will be ac ually obse ed (Hamil on and Slu sky, 1990, Theo em III, p.36).
Applying he ools p o ided by Hamil on and Slus ky (1990), I wan o add ess a ques ion
which so a , o he bes o my knowledge, has emained neglec ed, i.e., which p e e ences
cha ac e ize a mixed duopoly game be ween a p o i maximizing and a labou managed i m,
and consequen ly which kind o equilib ium one can expec o ob ain in such a game i i ms
can decide he iming o mo es be o e p oceeding o compe e in p ices o quan i ies.
The beha iou o LM i ms in mixed oligopolies has been desc ibed by se e al au ho s
(see, in e alia, C eme and C éme , 1992; Delbono and Rossini, 1992; Rossini and Sca pa,
1993; Okuguchi, 1993a). They ha e highligh ed he peculia beha iou o LM i ms unde
quan i y compe i ion, yielding an upwa d sloping eac ion unc ion1ins ead o he usual
downwa d sloping one cha ac e izing he PM i m. Ne e heless, all hese con ibu ions
in es iga e o a ious aims simul aneous play unde ei he quan i y o p ice se ing beha iou .
Iwill show ha , whena p eplays age in he sense o Hamil on andSlu sky (1990) isin oduced,
1. Howe e , he eac ion unc ion o an LM i m is no necessa ily upwa d sloping. See
Miyamo o (1982, p.13).
2
(i) simul aneous play is no o be expec ed unde nei he o m o compe i ion; (ii) Cou no
beha iou yieldsas heuniquesubgamepe ec equilib iumo heex endedgame heS ackelbe g
equilib ium wi h he PM i m mo ing i s , and (iii) Be and beha iou leads o mul iple
equilib ia in which bo h i ms would p e e o mo e la e o play in mixed s a egies.
These esul s ha e some in e es ing implica ions as o he issue o e o ming Eas e n
Eu opeaneconomies.Delbono andRossini (1992)e alua e he easibili yo al e na i e e o ms
o LM ma ke s consis ing in he passage o a mixed oligopoly o a ho izon al me ge whe e he
esul ing i m maximizes an objec i e unc ion in which a posi i e weigh is assigned o ei he
en ep eneu ial p o i o social wel a e. In analysing he case o a mixed duopoly, hey only
conside simul aneous Nash equilib ia. In he p esen pape , i is shown ha only S ackelbe g
equilib ia should be aken in o accoun . Hence, i u ns ou ha a e o m based on ei he he
p i a iza ion o he na ionaliza ion o a labou managed i m implies a smalle social gain han
i could be expec ed on he basis o p e ious li e a u e.
The emainde o he pape is s uc u ed as ollows. Be and compe i ion is desc ibed in
Sec ion 2. Sec ion 3 is de o ed o Cou no compe i ion. Policy implica ions a e discussed in
Sec ion 4. Finally, Sec ion 5 con ains concluding commen s.
2. Be and compe i ion
In o de o sa egua d he compa abili y o wha ollows wi h a leas a pa o he exis ing
li e a u e, I basically adop he same symbology and assump ions as in Okuguchi (1993b). The
magni udes ela ed o he PM and LM i ms a e iden i ied as Pand C, espec i ely.
Bo h i ms p oduce h ough he ollowing echnology:
whe e liis he amoun o labou employed by i m iand xiis he quan i y p oduced by he same
li=hi(xi), i=C,P(1)
3

i m. The echnology is ully cha ac e ized by he ollowing de i a i es:
i.e., hema ginalp oduc i i yo labou isdec easing.Fi msope a einama ke o di e en ia ed
goods, whose demand is
whe e (see Okuguchi, 1993b, pp.2-3):2
The inequali ies in (4.1) s a e ha (i) an inc ease in i m i’s p ice induces a dec ease in he
demand o he own p oduc , (ii) he wo goods a e subs i u es, and (iii) he own p ice e ec is
la ge han he c oss p ice e ec . The inequali ies in (4.2) a e needed o he eac ion unc ion
o he LM i m o be posi i ely sloped.
Since unde he abo e assump ions Okuguchi (1993a,b) has shown ha in a Be and
hi’>0,hi">0,(2)
xi=gi(pi,pj), i,j=C,P,i≠j,(3)
∂gi/∂pi≡gii<0,∂gi/∂pj≡gji>0,−gii>gji;(4.1)
∂2gi
∂pi∂pj≡gij
i≤0,gji+pigij
i>0. (4.2)
2. These assump ions, as well as hose in oduced in he emainde o he pape , hold o
ins ance when linea demand unc ions a e conside ed.
4
se ing he eac ion unc ion o an LM i m is posi i ely sloped i espec i ely o he na u e o
he i al, I can con ine mysel o in es iga e he cha ac e is ics o he en ep eneu ial i m’s
eac ion unc ion. I am going o p o e he ollowing:
LEMMA 1. Unde Be and compe i ion, he eac ion unc ion o he p o i maximizing i m
is upwa d sloping.
PROOF. The objec i e unc ion o he PM i m is he ollowing:
whe e kPde ines he en ep eneu ial i m’s ixed cos . The i s o de condi ion o p o i
maximiza ion w. . . p ice is:
Assume he second o de condi ion is sa is ied. I is known (see Bulow e al., 1985) ha he
slope o he eac ion unc ion has he same sign as he de i a i e o (6) w. . pC:
Acco dingly, i is su icien o de e mine he sign o
πP
B=pPgP(pC,pP)−hP(xP)−kP(5)
∂πP
B
∂pP=gP(pC,pP)+pPgP
P−hP
’gP
P=0. (6)
sign ∂pP
∂pC=sign ∂2πP
B
∂pP∂pC(7)
5
on he basis o he abo e assump ions, i is quickly es ablished ha he sign o (8) is posi i e.
Hence, he eac ion unc ion o he PM i m in he p ice space is upwa d sloping. Q.E.D.
P o ided ha he eac ion unc ion o he PM i m is posi i ely sloped, as claimed in
Lemma 1, and he eac ion unc ion o he LM i m is also inc easing, as shown by Okuguchi
(1993b), I am going o show wha is s a ed in he ollowing:
PROPOSITION1.Theex endedBe andgamebe weenap o i maximizing i mandalabou
managed i m has mul iple equilib ia. None o hem is simul aneous.
PROOF.Since bo h eac ion unc ions a e posi i ely sloped, his se ing is a special case o he
gene al si ua ion depic ed by Hamil on and Slu sky (1990, pp.36-41) in hei Theo ems III,
V(Aii) and VI. Acco ding o hese heo ems, when bo h eac ion unc ions a e inc easing he
ex ended game wi h obse able delay, whe e playe s i s choose he iming o mo es and hen
p oceed o play, has mul iple equilib ia. Namely, bo h sequen ial play a e subgame pe ec
equilib ia; mo eo e , he e exis s a mixed s a egy equilib ium in which i ms andomize o e
he s a egies "mo ing i s " and "mo ing second". This is due o he ac ha bo h eac ion
unc ions in e sec he Pa e o supe io se , i.e., he se o all pai o p ices yielding payo s ha
domina e hose associa ed wi h he simul aneous equilib ium. Q.E.D.
∂2πP
B
∂pP∂pC=gC
P+pPgPC
P−hP
’gPC
P−hP
"gP
PgC
P;(8)
6
3. Cou no compe i ion
In his Sec ion, op imiza ion w. . . quan i y is analised. I he domain o he demand
unc ion (3) is a ec angula egion, i can be in e ed o ob ain:
wi h
Assump ion (10.1) is bo owed om Okuguchi (1993b, p.4). Assump ion (10.2), which is a bi
igh e han he co esponding condi ion in Okuguchi (1993b, p.4), and is bo owed om
Okuguchi (1993a, p.29), implies ha i m i’s ma ginal e enue dec eases as he i al’s ou pu
inc eases. P o ided i m iac s as a p o i maximize , his condi ion also implies ha he own
eac ion unc ion is nega i ely sloped (see No shek, 1985; Dixi , 1986; Okuguchi, 1993a, in e
alia).3P o ided ha he eac ion unc iono heLM i misupwa dsloping(Okuguchi,1993a,b),
he ollowing holds:
PROPOSITION 2. The S ackelbe g equilib ium wi h he p o i maximizing i m mo ing i s
and he labou managed i m mo ing second is he only subgame pe ec equilib ium o he
ex ended Cou no game.
pi= i(xi,xj), i,j=C,P,i≠j,(9)
∂ i/∂xj≡ ji<0; (10.1)
∂2 i
∂xi∂xj≡ ij
i∈[0,− ji
xi[. (10.2)
3.Cou no beha iou mayinduceap o i maximizing i m oconside he i alsass a egic
complemen s.Thishappenswhenala gedominan i mcompe esagains apopula iono smalle
i als. See Bulow e al. (1985, p.500).
7
Appendix
A.1. The leade ’s ou pu in he LM duopoly
The solu ion o he leade ’s p oblem in he pu e LM duopoly is gi en by:
whe e
The ou pu o he ollowe , i m j, can be ob ained h ough he eac ion unc ion (17). Since i
mus be ha he ollowing cons ain is o be sa is ied:
A.2. The leade ’s ou pu in he mixed duopoly
When he PM i m plays he leade ’s ole in he mixed duopoly game, he p oduc ion
amoun s o:
whe e
xi=2
3a−2k
3a+(η+φ)
1/3−a(6k+3a2−(2F/a−2a)2)/3
[27a3(η + φ)]1/3(a.1)
η=−8k3+42a2k2−6a4k−a6
27a3φ= 1
3a√
96a2k3−20k4−24a4k2+2a6k
3(a.2)
a>xi+xj,
a2>k(3+2√2). (a.3)
xP=a
2



3
2+1
ψ1/3+ψ1/3
4


,(a.4)
14

Again, hequan i yp oducedby he ollowe ,in hiscase heLM i m,canbecompu ed eso ing
o he own eac ion unc ion (20). Finally, he condi ion ha mus be me in o de o ma ke
p ice o be posi i e a equilib ium is he ollowing:
ψ=−(16k+a2)
8+a√
8k2+a2k
2.(a.5)
a2>2k.(a.6)
15
Re e ences
Boye , Ma cel and Michel Mo eaux, "On S ackelbe g Equilib ia wi h Di e en ia ed P oduc s:
The C i ical Role o he S a egy Space", Jou nal o Indus ial Economics, XXXVI,
217-30, 1987.
Bulow, Je emy, John Geanakoplos and Paul Klempe e , "Mul ima ke Oligopoly: S a egic
Subs i u es and Complemen s", Jou nal o Poli ical Economy,93, 488-511, 1987.
C eme , Helmu , and Jacques C éme , "Duopoly wi h Employee-Con olled and
P o i -Maximizing Fi ms: Be and s Cou no Compe i ion", Jou nal o Compa a i e
Economics,16, 241-58, 1992.
Delbono,Fla io,andGianpaoloRossini,"Compe i ionPolicy sHo izon alMe ge wi hPublic,
En ep eneu ialandCoope a i eFi ms",Jou nalo Compa a i eEconomics,16,226-40,
1992.
Delbono, Fla io, and Ca lo Sca pa, "Upwa d Sloping Reac ion Func ions unde Quan i y
Compe i ion in Mixed Oligopolies", Dipa imen o di Scienze Economiche, Bologna,
mimeo; o appea in Bulle in o Economic Resea ch, 1995.
Dixi , A inash, "Compa a i e S a ics o Oligopoly", In e na ional Economic Re iew,27,
103-22, 1986.
Dow ick, S e e, " on S ackelbe g and Cou no Duopoly: Choosing Roles", Rand Jou nal o
Economics,17, 251-60, 1986.
Gal-O ,Es he , "Fi s Mo e and Second Mo e Ad an ages", In e na ional Economic Re iew,
26, 649-53, 1985.
16
Hamil on, Jona han H., and S e en M. Slu sky, "Endogenous Timing in Duopoly Games:
S ackelbe g o Cou no Equilib ia", Games and Economic Beha io ,2, 29-46, 1990.
Kahana, Na a, "Do Mul iplan Labo -Managed Monopolies Exis ?", Jou nal o Compa a i e
Economics, 18, 198-201, 1994.
Miyamo o, Yoshina i, "A Labou -Managed Fi m’s Reac ion Func ion Reconside ed",
Economic Resea ch Pape n. 218, Uni e si y o Wa wick, Co en y, 1982.
No shek, William, "On he Exis ence o Cou no Equilib ium", Re iew o Economic S udies,
LII, 85-98, 1985.
Okuguchi,Koji,"Cou no Oligopolywi hP o i -Maximizing andLabo -ManagedFi ms",Keio
Economic S udies,30 (1), 27-38, 1993a.
Okuguchi, Koji, "Followe ’s Ad an age in Labo -Managed Duopoly", Keio Economic S udies,
30 (2), 1-5, 1993b.
Rossini, Gianpaolo, and Ca lo Sca pa, "The T ans o ma ion o a Labou Managed Monopoly:
Compe i ion Policy s Mixed Fi ms", Jou nal o In e na ional and Compa a i e
Economics,2, 191-206, 1993.
Singh, Ni ika , and Xa ie Vi es, "P ice and Quan i y Compe i ion in a Di e en ia ed
Duopoly", Rand Jou nal o Economics,14, 546-54, 1984.
17