Cournot Vs Stackelberg Equilibria With Entrepreneurial and Labour Managed Firms

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 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. 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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-nc/3.0/ 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. 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