Price vs Quantity in a Repeated Differentiated Duopoly

Lambertini, Luca; Schultz, Christian Working Paper Price vs Quantity in a Repeated Differentiated Duopoly Quaderni - Working Paper DSE, No. 379 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Lambertini, Luca; Schultz, Christian (2000) : Price vs Quantity in a Repeated Differentiated Duopoly, Quaderni - Working Paper DSE, No. 379, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4930 This Version is available at: https://hdl.handle.net/10419/159220 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ Price vsQuantityinaRepeated Di¤erentiatedDuopoly1 LucaLambertini2and Christian Schultz3 23.June2000 1WethankBirgitGrodal, PerBaltzerOvergaardand theaudiencesinCopenhagenand Bologna(XIII Conference onGameTheoryand Applications,4-5June 1999)forveryusefulcommentsand discussion. 2DepartmentofEconomics,UniversityofBologna,StradaMaggiore45,I-40125 Bologna,Italy,fax: +390512092664, [email protected] 3CentreforIndustrialEconomics,InstituteofEconomics,UniversityofCopenhagen,Studiestraede6,DK1455 CopenhagenK, Denmark,fax: +4535323000, e-mail: [email protected] Abstract Weinvestigatethe choice ofmarketvariable,price orquantity,ofanoptimal implicitcartel. Ifthediscountfactorishigh,the cartelcanrealize the monopolypro…tin bothcases. Otherwise, itisoptimalforthe cartelto relyonquantitiesinthe collusivephaseifgoodsaresubstitutesand prices ifgoodsare complements.Thereasonisthat thisminimizesthegainsfrom deviationsfromcollusiveplay. KEYWORDS:partialcollusion,productdi¤erentiation JELClassi…cationCodes:D43,L13 1Introduction Arecurrent themeinindustrialorganizationiswhether…rmschoosequantities,asenvisioned byCournot,orprices,asenvisioned by Bertrand.This lead Singhand Vives(1984) toinvestigatethe equilibriumchoicesofadifferentiated doupoly,where each …rmcanchoosebetweensetting a price ora quantity(butnotboth).Singhand Vives showthat…rmschoosequantities ifgoodsaresubstitutes,whiletheychoosepricesifgoodsare complements. Thepurposeofthepresentpaperistoextend theanalysisofSinghand Vives tothe caseoftacitcollusion. Thequestionweposeiswhether…rmsparticipatinginanoptimizing cartel, whichtrytomaximize pro…tsbuthastorelyontacitcollusion,will usequantitiesorprices.Since themembersofthe cartelcannotwritebinding contractstheyhaveto agree onselfenforcingcontracts, i.e.strategieswhich can besustainedinasubgameperfectequilibrium.Therearetwo…rms producingdi¤erentiatedproducts,butotherwisethe…rmsareidentical. Asis well known,repeatedgameshavemanyandverydivergentequilibria(see e.g. Fudenberg and Maskin,1986).Inoligopolytheory,researchershavetypically focussedonequilibriawhichareundominatedinthesetofequilibria.If …rmsaresymmetric,attention hasbeen driven uponthesymmetricsubgame perfectequilibriumwhichgivesthehighestpro…t to…rms.1Inthispaper 1See,forinstance,Rotemberg and Saloner(1986),Greenand Porter(1984),Abreu, Pearce and Stachetti(1990)and Bernheimand Whinston(1990),orchapter6inTirole (1988)forasurvey. 1 wewill takethesameapproach.Sothequestioncan bereformulatedas follows:do…rms setpricesorquantitiesinthesymmetricsubgameperfect equilibriumwhichgivethemthehighestpro…t? Asiswell knownfromthetheoryofrepeatedgames(see Abreu,1988), anysubgameperfectequilibriumpayo¤can berealizedinasocalledsimple equilibriumconsisting ofanormal(collusive)phaseand apunishmentphase foreachofthe…rms.Westudysuchequilibria.Itisalsowell knownthat theworsethepunishmentphaseis,thehigherpayo¤can berealizedinthe normalphase.Althoughverystrongpunishmentscan bepartofasubgame perfectequilibrium,onemaydoubt theviabilityofsuch punishments(for furtherdiscussionofthis see e.g.Farrell and Maskin,1989).Wetherefore investigatetwokindsofequilibria:equilibriainvolving optimal(verystrong) punishments,and equilibriainvolvingpunishmentsconsisting ofreversionto theone-shotNashequilibrium. Ina one-shotgamethe choice ofprice orquantityis…naland commits the…rmfortherestofthegame.Inarepeatedgame,thisisnotnecessarily so.In principle,the choice ofmarketvariable cancommit the…rmforany numberofperiods.However, itishardtothinkofacommitment technology, whichcancommita…rmtosetaprice (oraquantity)forall future.Inthis paper,therefore,wewill assumethat thechoice ofmarketvariableinaperiod onlycommitsthe…rmforthatperiod,butnotforsubsequentperiods.When setting aprice,the…rmcommitstoselling asmuchasconsumerswill demand at theprice,thiscorrespondsto o¤eringahorizontalsupplycurve.When 2 setting a quantitythe…rmcommitstosellingthisquantityatwhateverprice clearsthemarket, i.e.averticalsupplycurve.Thisisasin Singhand Vives (1984),theyspeakofa “quantitycontract” and “price contract”between a…rmand itscustomers.In principleone couldimagineothercontracts, correspondingtodi¤erentsupplycurvesbutwewill notconsiderthishere. Anoptimizingcartelwill aimat thehighestpossiblepro…t, ideallythe monopolypro…t.Agiven pro…tcan berealized bothwhen …rmschoose pricesand whentheychoosequantities.Therefore,foranoptimizingcartel, the crucialfeatureinthe choice ofmarketvariablesisnot thepro…tina period,but thepro…tabilityofadeviation.Saythat thegoodsareveryclose substitutesand both …rmschoosethemonopolyprice soeach …rmgetshalf ofthemonopolypro…t.Ifa…rmwantstodeviatefromcollusiveplay, it can undercut theother…rmbyasmall amountand gain(almost) thewhole marketand obtain(almost) thewholemonopolypro…t.Whengoodsare closesubstitutes,price settingmakesdeviationsverypro…table,themore sothehigherproductsubstitutabilityis.If,ontheotherhand,…rmseach setaquantityequaltohalfthemonopolyproduction,theyalso obtainthe monopolypro…t.Butnowadeviatorcan nevergainthewholemarket.The cheated …rmwill sell itsquantityregardless oftheprice.Thuswhengoods are closesubstitutes,adeviationisless temptingifthe…rms setquantities thaniftheysetprices. Fora given punishment, it thereforefollowsthatwhengoodsare close substitutesthesmallestdiscountfactorneededtosustainfull collusiononthe 3 monopolyoutcomeis smallerwhen …rmschoosequantitiesinthe collusive phase. Weshowthatwhengoodsaresubstitutesthen,forarangeofintermediatediscountfactors,animplicitcartelcanrealize themonopolypro…tonly ifitreliesonquantities.Similarly, ifthediscountfactoris solowthat the monopolypro…tcannotbesustainedinasubgameperfectequilibrium,we showthat thehighestpro…twhichcan besustainedifthe…rmschoosequantitiesishigherthaniftheychooseprices Thisholdstruewhengoodsaresubstitutes.Whengoodsare complements,thereverseistrue.Ifthediscountfactorisnotveryhigh,thenthe highestpro…twhichcan besustainedinasubgameperfectequilibriumis higherifthe…rmschooseprices.Ifthediscountfactorisveryhigh,the choice ofmarketvariabledoesnotmatter.Thediscountedvalueof future lossesdueto a punishmentisthensohighthat theyaresu¢cient todeter deviationsevenwhenthedeviation pro…tislarge. Hence,formoderatediscountfactors,anoptimizingcartelwill chooseto competeinquantitiesifgoodsaresubstitutesandchoosetocompeteinprices ifgoodsare complements.Thisistrueregardless oftheparticularpunishmentphaseinvolved:optimalor reversiontotheone-shotNashequilibrium. Our resultscould beseenasvindicatingthoseofSinghand Vives.However,themechanismbehind theresultsisdi¤erent.In Singhand Vives’ model, the choice ofmarketvariableismadenon-cooperativelybythe…rms whotrytomaximize shortrunpro…ts.Intherepeatedgame,theoptimal im4 plicitcartelmaximizeslongrun pro…tsrelying ontacitcollusion.The choice ofmarketvariableisthereforeguided bythe consequencesforthedeviation pro…ts:theyshould beminimized. Wealsobrie‡yconsiderthe choice ofmarketvariableinthepunishment phaseoftrigger-strategyequilibriawithNash-punishment.Here,theresults ofSinghand Vivesdirectlygivethat…rmschoosequantitiesinthepunishmentphasewhengoodsaresubstitutesand priceswhengoodsare complements.Withoptimalpunishments,thingsaremoreinvolved,the equilibriumstrategiesarepresumablynon-stationaryand we cannotcharacterize the choice ofmarketvariableinthepunishmentphase. The…rst tostudythe choice ofmarketvariableinarepeated duopoly wasDeneckere(1983,1984).Heanalyzedtrigger-strategyequilibriaàla Friedman(1971),and calculatedthesmallestdiscountfactornecessaryfor sustainingcollusiononthemonopolyoutcomefortwo…rmscommittedto beprice settersinall periodsaswell astwo…rmscommittedtobequantity settersinall periods.Deneckerefound thatwhengoodsaresubstitutesthe crucialdiscountfactorislowerforquantitysetting…rmsthanforprice setting …rms,exceptwhengoodsareveryclosesubstitutes.Theoppositeistrue whengoodsare complements.Deneckereinterpretedthisasacartel ismore stableifitcompetesinquantitieswhengoodsaresubstitutesand morestable ifitcompetesin priceswhengoodsare complements.Majerus(1988)and Rothschild(1992)askedsimilarquestionsinslightlydi¤erentsettings(see alsoAlbækand Lambertini(1998a)foradiscussionofRotschild(1992)). 5 Lambertini(1997)and Albækand Lambertini(1998b)assumethat…rms independentlyand non-cooperativelychoosemarketvariableonce and for all inameta-game,whichtakesplace beforetherepeatedgametakesplace. Thepayo¤tothe…rmsinthemeta-gameisnotpro…t,rathereach …rmis assumed beinterestedinchoosingthemarketvariablewhichminimizesthe discountfactornecessaryforsustainingcollusiononthemonopolypro…tin thesubsequentrepeatedgame.Inorderforthistobeawell speci…edgame, theauthorsalsocalculatethelowestdiscountfactorscompatiblewith …rms realizingmonopolypro…tsinasubgameperfectequilibriumwhenone…rmis aprice setterand theotherisaquantitysetter.Thesepapers showthat the meta-gamemayhavetheformofaprisoners’dilemma,and hence that the non-cooperative choice ofthemarketvariablebeine¢cient-relativetothe payo¤softhemeta-game.Firmschoosetobeprice settersinthemeta-game, althoughcartelstability(inthesenseofDeneckere)ishigheriftheychoseto bequantitysetters. Comparedtothisliterature,ourpaperdi¤ersinseveralaspects.Contrary toLambertiniandAlbæk-Lambertini, weinsist thata…rmspayo¤isthetotal sumofdiscounted pro…ts.Thereisnometa-game constructioninthepaper. Secondly,wedonotassumethat…rmsareabletocommit to a particular marketvariableforall future,the choice ofmarketvariableonlycommits the…rmforoneperiod.Animportantimplicationisthat the choice of marketvariablemaybedi¤erentinthenormaland thepunishmentphase. Thirdly,weareabletosaywhathappenswhen …rmsareunabletocollude 6 2.2Reactionfunctions,prices Suppose…rmsareprice setters.Thepro…tfunctionof…rm1is: ¼1(p1;p2)= 8 > > > > > < > > > > > : ^¼B(p1;p2)=µ1 1+°¡p1 1¡°2+°p2 1¡°2 ¶p1ifp1·1¡°+°p2;p2·1¡°+°p1 e¼B 1(p1;p2)=(1¡p1)p1ifp1·1¡°+°p2;p2¸1¡°+°p1 0ifp1¸1¡°+°p2 (4) Herethe conditionp1·1¡°+°p2ensuresthat…rm1’squantityis non-negativeand p2·1¡°+°p1ensuresthatq2isnon-negative,asisclear from(2).^¼Bisthestandard pro…tfunctionwhentheinvolvedquantitiesare non-negative,~¼Bcorrespondstothe casewhere…rm2’sprice is sohigh,that itsellsnothing(and everythingisasif…rm1wereamonopolist).Thereare similarexpressionsfor…rm2. If…rmtwosetstheprice p;…rm1’sbestreplyistheprice whichsolves max p1¼1(p1;p) Wedenotethisprice RB1(p)and theassociated pro…t¼DB 1(p):Inthesequel, wewill onlybeinterestedin pricesforwhichquantitiesarenon-negative when both …rms set theprice.Using(2),wesee that thisimplythatp·1: Lemma2Considerprice setting(Bertrandbehavior)and assumep2[0;1]: a. Suppose°<0:Then,RB1(p)=1¡°(1¡p) 2and¼DB 1(p)=[1¡°(1¡p)]2 4(1¡°2): 13 b. Suppose°>0: i. Ifp·2¡°¡°2 2¡°2;thenRB1(p)=1¡°(1¡p) 2and¼DB 1(p)=[1¡°(1¡p)]2 4(1¡°2): ii. Ifp2µ2¡°¡°2 2¡°2;1¸;thenRB1(p)=p¡1+° °and¼DB 1(p)= (1¡p)(p¡1+°) °2: Proof.Suppose°<0:From(4)wehavethat^¼Bistherelevantfunction forp·1¡°+°p1,whichisequivalent top1·p¡1+° °:For°<0and p2[0;1];p¡1+° °>1,hence ^¼Bisrelevantforall p12[0;1]:Theresult followsfrom maximizationof^¼Bw.r.t.p1: Nowsuppose°>0:Againfrom(4)wehavethat¼=^¼Bforp·1¡°+°p1 whichisequivalent top1¸p¡1+° °;and ¼=~¼forp1<p¡1+° °: Maximizing^¼Bw.r.t.p1yieldsp1=1¡°(1¡p) 2:If 1¡°(1¡p) 2>p¡1+° °() p<2¡°¡°2 2¡°2 then^¼Bisincreasing atp¡1+° °: Maximizinge¼Bw.r.t.p1yieldsp1=1 2;whichislargerthanp¡1+° °i¤ p<2¡° 2;whichisful…lledsince 2¡° 2>1for°2]0;1[and weassumethat p·1:Hence,~¼Bisincreasing atp¡1+° °:Since ~¼B=^¼Batp¡1+° °; we concludethat theglobaloptimumisattainedintheoptimumof^¼B:This provesb:ioftheLemma. Ifinsteadp>2¡°¡°2 2¡°2;then1¡°(1¡p) 2<p¡1+° °and ^¼Bis decreasingat thecut-o¤pointp¡1+° °:Hence,theoptimalprice islessthan orequaltop¡1+° °;where~¼Bistherelevantpro…tfunction.Maximizing 14 ~¼B;yieldsp1=1=2:However,asweshowedabove,1=2>p¡1+° °for all p·1;so~¼Bisincreasing atp¡1+° °;and theoptimalprice isp1= p¡1+° °:Thisprovesb:ii:oftheLemma.Wealsoneedtocheckthat the pro…tsarenon-negativeat theoptimalsolutions,but thisistrivial. Wemaynotice that,aslong at thereactionfunctionisthe“normal”, wherethequantityoftheother…rmispositive,then pricesarestrategic substituteswhengoodsare complementsand strategic complementswhen goodsaresubstitutes. Usingtheformulafound inaand b:i:oftheLemma,theBertrand equilibriumpriceispBN=1¡° 2¡°;andtheassociatedpro…tis¼BN=1¡° (2¡°)2(1+°). ItiseasilycheckedthatindeedpBN=1¡° 2¡°·2¡°¡°2 2¡°2for°<1: Let¼QPN(¼PQN)betheNashequilibriumpro…t tothequantity(price) setterinthegamewherethe…rmshave chosen di¤erentmarketvariables. Singhand Vives(1984)showthat thefollowingrelationsthen hold: If0<°<1then¼CN>¼QPN>¼BN>¼PQN(5) If¡1<°<0then¼BN>¼PQN>¼CN>¼QPN Theserelationsimplythatifthereisonlyoneperiod,thenthesubgame perfectequilibriumofthetwo-stagegameisunique.If0<°<1;itisa dominantstrategyforboth …rmstochoosequantityasthemarketvariable; if¡1<°<0;itisadominantstrategyforboth …rmstochooseprice asthe marketvariable(Singhand Vives(1984),proposition2). 15 3Deviation pro…ts Wewill beinterestedinsymmetric equilibria,wherethe…rmsget thesame pro…t.Let¼Q(q)bethepro…t toeach …rmiftheybothchoosethequantity q;and ¼B(p)bethepro…tiftheybothchoosetheprice p:Using(1)and (2) theyarerespectively ¼Q(q)=q¡(1+°)q2(6) ¼B(p)=1 1+°p¡µ1 1¡°2¡° 1¡°2 ¶p2 =1 1+°¡p¡p2¢(7) Themonopolyprice,quantityper…rmand pro…tper…rmare pm=1 2;qm=1 21 1+°;¼m=1 41 1+°:(8) Agiven pro…tlevelcan beobtainedeitherbysettingpricesorquantities. Ineachcase,we cancalculatethedeviation pro…tassociatedwiththislevel ofpricesorquantities.Fora given pro…tlevel, ¼;wewouldliketoknow whetherthedeviation pro…t toa…rmis smallerorlargerifthe…rmschoose quantitiesratherthan prices.Iftheyshouldobtainthispro…tlevelbysetting quantities,theyshouldeachchooseaquantity,q(¼);solving ¼=q¡(1+°)q2(9) Thisequation hastworoots q=1+p1¡4(1+°)¼ 2(1+°)and q=1¡p1¡4(1+°)¼ 2(1+°)(10) 16 Thesquarerootiswell de…nedandlessthanone,since ¼·¼m=1 41 1+°.As ¡1<°;thesecondrootisthesmallerofthetwo.Aswewill seeinthesequel, the…rmswill beinterestedinminimizingthedeviation pro…ts.Therefore, therelevantrootistherootwithlowerdeviation pro…t.FromLemma1a and b:i;itisclearthat, if°<0or°>0and q·1+°¡p1¡°2 °(1+°);the deviation pro…tis(1¡°q)2 4:For°<0;thisdeviation pro…tincreasesinq, whiledecreasesinqfor°>0:Hence for°<0;thesecond (lower)rootgives thesmallestdeviation pro…tand isrelevant.For°>0;theoppositeistrue ifindeedq·1+°¡p1¡°2 °(1+°):InsertingtheCournotpro…tinthe…rstroot and evaluating,weget theCournotproduction: q= 1+ s 1¡4(1+°)1 (2+°)2 2(1+°)=1 2+°; whichis smallerthan1+°¡p1¡°2 °(1+°):Since therootisdecreasinginthe pro…tlevel, itislessthan1+°¡p1¡°2 °(1+°)forpro…tlevelsabovetheCournot pro…t.We concludethat,for°>0;the…rstrootgivesrisetothesmaller deviation pro…tsand thereforeistherelevantone.Hence wehave q(¼)= 8 > > < > > : 1¡p1¡4(1+°)¼ 2(1+°)for¡1<°<0 1+p1¡4(1+°)¼ 2(1+°)for0<°<1(11) Infact,theaboveisveryintuitive.When°>0;thereisanegative externalityfromchoosingalargerproductionand theCournotproductionis largerthanthemonopolyproduction.Thelowestdeviation pro…tsobtains when productionishighcorrespondingtothe…rstroot.When°<0,the 17 externalityfromchoosing a largerproductionispositiveand theCournot productionis smallerthanthemonopolyproduction.Thelowestdeviation pro…tsthenobtainwhentheproductionislowcorrespondingtothesecond root. Thedeviationpro…tis¼DC(q(¼)):Wecansummarizetheabovediscussion in Lemma3Considerquantitysetting(Cournotbehavior).Foragivenpro…t level¼thedeviationpro…tisgivenby ¼DC(q(¼)) = 8 > > > > > > > > < > > > > > > > > : à 1¡°1+p1¡4(1+°)¼ 2(1+°) !2 4if °>0 à 1¡°1¡p1¡4(1+°)¼ 2(1+°) !2 4if °<0 (12) Nowconsiderthe casewhere…rms setprices.Theprice whichgivespro…t level¼isp(¼)whichsolves ¼=1 1+°¡p¡p2¢: Therearetworoots p=1¡p1¡4(1+°)¼ 2and p=1+p1¡4(1+°)¼ 2:(13) Again …rmswill chosetheprice level, whichminimizesthedeviation pro…t. FromLemma 2,wesee thatif°<0or°>0and theprice isnot too high µp·2¡°¡°2 2¡°2 ¶;thedeviation pro…tis¼DB 1(p)=(1¡°(1¡p))2 4(1¡°2). 18 For°<0;(1¡°(1¡p))2 4(1¡°2)isdecreasinginp;sothedeviation pro…tis smallestwhenpishigh,and therelevantrootisthesecond (large)root. When°>0;(1¡°(1¡p))2 4(1¡°2)isincreasinginp;soforpbelow2¡°¡°2 2¡°2 the…rst (lower)rootisrelevant.The…rstrootislowerthan2¡°¡°2 2¡°2i¤ 1¡p1¡4(1+°)¼ 2·2¡°¡°2 2¡°2 or 1¡22¡°¡°2 2¡°2·p1¡4(1+°)¼(14) Therighthandsideispositiveforpro…tlevels¼belowthemonopolypro…t¼m= 1 4(1+°):For°·p3¡1;thelefthand sideisnegative.Hence theinequality isful…lledforall relevantpro…tlevelsif°·p3¡1: Forp3¡1·°<1;thelefthand sideof(14)ispositive.Therighthand sideislargerthanthelefthand sideifthepro…tlevel is su¢cientlysmall. Solving(14),wesee thatitisequivalent to ¼·¼¤´1¡µ1¡22¡°¡°2 2¡°2 ¶2 4(1+°):(15) For¼¸¼¤;thesmall rootin(17), i.e., 1¡p1¡4(1+°)¼ 2;islarger than2¡°¡°2 2¡°2:Evidently,soisthelarger root,sofromLemma 2,the deviation pro…tequals(1¡p)(p¡1+°) °2:Notice,thisdeviation pro…tis positiveasp¸2¡°¡°2 2¡°2:We claimthat thedeviation pro…tevaluatedat 19 thesmall rootin(17)is smallerthanevaluatedat thelargeroot.The claim isequivalent to à 1¡1¡p1¡4(1+°)¼ 2 !Ã1¡p1¡4(1+°)¼ 2¡1+° ! °2 · à 1¡1+p1¡4(1+°)¼ 2 !Ã1+p1¡4(1+°)¼ 2¡1+° ! °2 whichisful…lledi¤ à 1¡1¡p1¡4(1+°)¼ 2 ! à 1¡1+p1¡4(1+°)¼ 2 !· Ã1+p1¡4(1+°)¼ 2¡1+° ! Ã1¡p1¡4(1+°)¼ 2¡1+° !(16) (rememberthatall parenthesizesarepositiveasthedeviation pro…tispositiveintherangeweare consideringnow).Condition(16)isclearlyful…lled, thelefthand sideisless thanone,whiletherighthand sideislargerthan one.Hence weknowthat theprice the…rmsuseto obtainthepro…tlevel¼; p(¼);equalsthesmaller root1¡p1¡4(1+°)¼ 2and thedeviation pro…tis given by ¼DB(p(¼)) = à 1¡1¡p1¡4(1+°)¼ 2 !Ã1¡p1¡4(1+°)¼ 2¡1+° ! °2 whenp3¡1·°<1,and ¼¸¼¤:Forlater reference westateour result 20 aboutp(¼): p(¼)= 8 > > > > > < > > > > > : 1+p1¡4(1+°)¼ 2for¡1<°<0 1¡p1¡4(1+°)¼ 2for0<°<1 (17) Wesummarize thisdiscussioninLemma 4 below. Lemma4Considerprice setting(Bertrandbehavior).Foragivenlevelof pro…ts¼;thedeviationpro…tisgivenasfollows 1.If°<0;then ¼DB(p(¼)) = " 1¡° à 1¡1+p1¡4(1+°)¼ 2 !#2 4(1¡°2)(18) 2.If0<°·p3¡1;orp3¡1·°<1,and¼·¼¤;where¼¤isgiven in(15),then ¼DB(p(¼)) = " 1¡° à 1¡1¡p1¡4(1+°)¼ 2 !#2 4(1¡°2)(19) 3.Ifp3¡1·°<1and¼>¼¤;then ¼DB(p(¼)) = à 1¡1¡p1¡4(1+°)¼ 2 !Ã1¡p1¡4(1+°)¼ 2¡1+° ! °2 (20) Asnotedabove,a given pro…t¼can beobtained bysettingpricesand by settingquantities.Ineachcase,therewill beaparticulardeviation pro…t, 21 whichwehavederivedabove.Therefore,wearenowinapositiontocompare thesedeviation pro…ts.AsisclearfromtheavobeLemmata,the comparison dependson°: Ifgoodsare complements¡1<°<0;equations(12)and (18)yield ¼DC(q(¼))¡¼DB(p(¼)) = à 1¡°1¡p1¡4(1+°)¼ 2(1+°) !2 4¡ à 1¡° à 1¡1+p1¡4(1+°)¼ 2 !!2 4(1¡°2) =1 4 ³2¼°+2¼°2¡°¡p(1¡4¼¡4¼°)´°2 (1¡°2)(1+°) Thisexpression hasthesamesignasthesignoftheparenthesis ½(°;¼)´2°(1+°)¼¡°¡p1¡4(1+°)¼(21) Evaluatedat¼=0;½(°;0)=¡°¡1<0:Evaluatedat theBertrand pro…t ¼BN=1¡° (2¡°)2(1+°)weget ½(°;¼BN)=2°(1+°)1¡° (2¡°)2(1+°)¡°¡ s 1¡4(1+°)1¡° (2¡°)2(1+°) =2°1¡° (2¡°)2¡°¡ s 1¡41¡° (2¡°)2 Nowobservethat sign ( 2°1¡° (2¡°)2¡°¡ s 1¡41¡° (2¡°)2 ) =sign ( 2°(1¡°)¡°(2¡°)2¡(2¡°)2s 1¡41¡° (2¡°)2 ) =sign©°2¡°3ª>0for¡1<°<0: 22 subsequentlyplayingtheCournotequilibrium.Ifgoodsare complements (¡1<°<0);thentheychoosepricesand playtheBertrand equilibrium. Let¼Ndenotetheperperiod pro…tofthepunishmentphase. Wedivideintotwocases,…rstwherethediscountfactoris solargethat themonopolypro…tcan berealizedineach period.Secondly,welookat the caseofamoderatediscountfactor,wherethe…rmshavetosettleonapro…t levelsmallerthanthemonopolypro…t.Clearly, ifthediscountfactorisvery closeto one,thenthemonopolypro…tcan besustainedinasubgameperfect equilibrium,regardless ofwhetherthe…rmschoosepricesorquantities.For alowerdiscountfactor,thismaynotbepossible.Foreachcase,quantities and prices,thereisacrucialsmallestdiscountfactor,whichallowsthe…rms tosustainthemonopolypro…tinasubgameperfectequilibrium.Wewill nowderivethese crucialdiscountfactors. Consider…rst the caseofquantities.Thetriggerstrategyequilibrium lookslikethis: I.Ift=0orboth …rmshave chosenQYand qminall previousperiods, chooseQYinthe…rststageand qminthesecond stage. II.Ifthereisanearlierperiodt0<twhereatleastone…rmhaschosen PRinthe…rststageorsomethingdi¤erentfromqminthesecond stage,or ifatleastoneofthe…rmshave chosenPRinthe…rststageofthisperiod, chooseQY;qCN¡PR;pBN¢fromnowonand inall futureif0<°<1; (if¡1<°<0): Since thepunishmentphase(II)consistsofin…niterepetitionoftheone 29 shotNashequilibrium,wejustneedtocheckthatstrategiesareoptimalfor each…rminthenormalphase.Ifa…rmadherestothestrategy, itreceives¼m inall periods.Clearly, ifa…rmwantstodeviate, itshould doitinthesecond stageofaperiod.Ifitdeviatesinthe…rststage, itispunishedalreadyinthe second stage,sothedeviation pro…twill besmaller,thanifitwaitsuntil the second stage.Thebestdeviationconsistsofchoosingthebestreply,which will givethedeviation pro…t¼DC(qm)intheperiodofdeviation,and ¼Nin all future.Hence,the conditionthata…rmwill notdeviateis 1 1¡±¼m¸¼DC(qm)+± 1¡±¼N whichisful…lledifand onlyif ±¸±Q´¼DC(qm)¡¼m ¼DC(qm)¡¼N(30) Notice that,althoughwehavenotexplicitlywrittenit,±Qisafunctionof °.Thenconsiderthe casewhere…rms setpricesinthenormalphase.The non-deviationconstraintbecomes 1 1¡±¼m¸¼DB(pm)+± 1¡±¼N or ±¸±P´¼DB(pm)¡¼m ¼DB(pm)¡¼N(31) As¼m>¼N;thefractionx¡¼m x¡¼Nisincreasinginx:Hence wehavethat ±Q<±P,¼DC(qm)<¼BC(pm)(32) 30 It then directlyfollowsfromProposition5thatifgoodsare complements (¡1<°<0);then±P<±Q;and ifgoodsaresubstitutes(0<°<1);then ±Q<±P: Hence, ±Q<±Pifand onlyif0<°<1 ±Q>±Pifand onlyif¡1<°<0 Wesee thatifgoodsaresubstitutes(0<°<1);thenthereisanonemptyrangeofdiscountfactors, [±Q;±P]wherethe…rmscanrealize the monopolypro…t, iftheychoosequantitieswhilethisisnotpossibleifthey chooseprices.Hence, inthisrangeapro…tmaximizingimplicitcartelwill let the…rmschoosequantities.Whengoodsaresubstitutestheywill alsochoose quantitiesinthepunishmentphase,aswediscussedabove.Whengoodsare complements,ontheotherhand,thereisanon-emptyrangeofdiscountfactors[±P;±Q]forwhich …rmsonlycanrealize themonopolypro…tbychoosing prices,sointhisrangethe cartelchoosesprices.Forveryhigh discountfactors,the…rmscanrealize themonopolypro…twhethertheychooseprices orquantities.TheresultresemblestheresultofDeneckere(1983,1984),but thereisadi¤erence.InDeneckere,…rmsare committedtoeitherpricesor quantitiesinall periodsand phasesoftherepeatedgame,thismeansthat, forgiven°;thediscountfactorforquantitiesiscalculatedwithquantities inthepunishmentphase,whilediscountfactorforpricesiscalculatedwith pricesinthepunishmentphase.Thus,forgiven°;thepunishmentpro…t di¤ersinthetwocases.Inourgame,ontheotherhand,thereisnocommit31 ment,soitisnotnecessarilythe casethat themarketvariableisthesamein thetwophases.Thisimpliesthat,forgiven°;thetwodiscountfactorsare calculatedwiththesamepunishmentpro…t.Hence,althoughtherelative rankingisthesameasifwehad proceededlike Deneckere,the exactvalues ofthediscountfactorsaredi¤erent. Whathappenswhenthediscountfactorisnotsohighthat themonopoly pro…tcan berealized?Recall that, intheoneshotNashequilibrium,…rms choosequantitiesasmarketvariableandthepro…tistheCournotpro…t,¼CN; if0<°<1.Apro…t-maximizingcartelwill atleastget theoneshotNash equilibriumpro…tasaveragepro…t,hence the equilibriumaveragepro…t,¼; ful…ls¼¸¼CNif0<°<1:Similarly, if¡1<°<0;…rmschoosepricesin theoneshotNashequilibriumand thepro…tistheBertrand pro…t¼BN:A cartelwill atleastget thispro…t,hence the equilibriumaveragepro…tful…lls ¼¸¼BNwhen¡1<°<0: Giventhediscountfactor,±;theimplicitcartelwill aimat thehighest averagepro…tlevel, ¼;wherethenon-deviationconstraintisnotviolated. Thus, if…rmscannotget themonopolypro…t,thenthe constraintwill be binding,and thisistrueineach period.Furthermore,thepro…twill bethe sameand equaltotheaveragepro…tineach period.Tosee this,supposethat therearetwoperiodswherethepro…tislowerinthe…rst.Then,theaverage pro…tcan beincreasedinthe…rstperiod bydroppingthe…rstperiodaction and choosingtheactionsprescribedforall subsequentperiodsoneperiod earlier.If,ontheotherhand,the equilibriumpro…tishigherinthe…rst 32 periodthaninthesecond,thepro…tofthesecond periodcan beincreased tothepro…tofthe…rstperiod byrepeatingtheactionsofthe…rstperiod. Thiswill notcause…rmstodeviateinthe…rstperiod,since itwill increase thenormalphasepro…tand thuslessenthedeviationconstraintinthe…rst period. Supposethe…rms setquantities.To obtaintheaveragepro…tlevel¼,the …rmschoosethequantityq(¼)asgiven by(11).Ifa…rmwantstodeviate fromq(¼);itwill receivethedeviation pro…t¼DC(q(¼)).Therefore,thenodeviationconstraintassociatedwiththehighestpro…tlevel, ¼;whichcan be sustained becomes: 1 1¡±¼¸¼DC(q(¼)) +± 1¡±¼N:(33) Thehighestpossiblepro…tlevelattainablewhen …rms setquantities solves thisconditionwithequality.Similarly, if…rms setprices,thebestpro…tlevel, ¼0;isthesolutiontothefollowingnon-deviationconstraint: 1 1¡±¼0¸¼DB(p(¼0))+± 1¡±¼N:(34) Considerthepro…tlevel¼;whichsolvesequation(33)6.Ifgoodsaresubstitutes(0<°<1);thenProposition5,2.directlyimplythatat this¼the righthand sideofequation(34)islargerthanthelefthand side.Hence, if this¼should beobtained bysettingpricesthe…rmswant todeviatefrom collusiveplay.Conversely,becauseofProposition5,2,at thelargest¼0which solves(34)withequality,(33)holdswithstrictinequality.We concludethat, 6Ifthereareseveralsolutions,pickthelargest. 33 ifgoodsaresubstitutes,the cartelcanobtainahigherpro…tbychoosing quantitiesratherthan prices.Whengoodsare complements(¡1<°<0); Proposition5,1,directlygivestheopposite conclusion.If¼0solves(34),then (33)isviolatedat this¼0;sothe cartelcanobtainahigherpro…tbysetting prices. We cansummarize thediscussion: Theorem6Given°:There existdiscountfactors±Qand±Pwhich depend on°where0<±Q;±P<1suchthat thefollowingistruefortheoptimal trigger-strategyequilibriawithNash-punishment. 1.If±>max[±Q;±P];theimplicitcartelisindi¤erentbetweenchoosing pricesorquantitiesinthenormalphase.Firmsreceive themonopoly pro…t. 2.Ifgoodsaresubstitutes(0<°<1);then±Q<±P:If±<±P;…rms setquantitiesinthenormalphase.If±2[±Q;±P];…rmsreceive the monopolypro…t; if ±<±Q;theyreceive less.Forall ±2(0;1),…rms setquantitiesinthepunishmentphase. 3.Ifgoodsare complements(¡1<°<0);then±P<±Q:If±<±Q; …rms setpricesinthenormalphase.If±2[±P;±Q]…rmsreceive the monopolypro…t; if ±<±P;theyreceive less.Forall ±2(0;1),…rms setpricesinthepunishmentphase. Qualitatively,theresultsforthenormalphaseobtainedabove carryover tothe casewherethepunishmentistheoptimalsymmetricpunishment.The 34 argumentsdonotdepend ontheparticularpunishmentphase,thesize of¼N doesnotenterthearguments.Let¼L(±)bethelowestaveragepro…twhich canbesustainedinasymmetricsubgameperfectequilibrium,whenthe…rms’ discountfactoris±:FromtheresultsofAbreu(1986,1988)itisclearthat suchanequilibriumexists.Similarly, let¼H(±)bethehighestaveragepro…t whichcan besustainedinasymmetricsubgameperfectequilibrium.This pro…tcanbeobtainedinasimple equilibrium,wherethepunishmentphaseis as severeaspossible(giventhe equilibriumis symmetric),whichmeansthat itgivesthe…rmsanaveragepro…tof¼L(±):Thesameargumentsasabove showthatifthediscountfactorishigh,themonopolypro…tcan berealized regardless ofthe choice ofmarketvariable.Thereare crucialdiscountfactors ±QO(Oforoptimal)and ±PObelow whichthe choice ofmarketvariableis importantforthepro…t the cartelcanrealize.Withoutfurtherproof,we stateforcompleteness: Theorem7Given°:There existdiscountfactors±QOand±POwhich depend on°where0<±QO;±PO<1suchthat thefollowingistrueforthe optimaltrigger-strategyequilibriawith optimalpunishment. 1.If±>max[±QO;±PO];theimplicitcartelisindi¤erentbetweenchoosing pricesorquantitiesinthenormalphase.Firmsreceive themonopoly pro…t. 2.Ifgoodsaresubstitutes(0<°<1);then±QO<±PO:If±<±PO;…rms setquantitiesinthenormalphase.If±2[±QO;±PO];…rmsreceive the monopolypro…t; if ±<±Q;theyreceive less. 35 3.Ifgoodsare complements(¡1<°<0);then±PO<±QO:If±<±QO; …rms setpricesinthenormalphase.If±2[±PO;±QO]…rmsreceive the monopolypro…t; if ±<±PO;theyreceive less. Aninterestingquestion,whichisnoteasyto answer, iswhichmarketvariablethe…rmsuseintheoptimalpunishmentphase.Unfortunately,wehave notbeenabletosolvethisquestion.Amajorobstacleisthatpresumably theoptimalpunishmentphaseisnon-stationary. 6ConcludingRemarks Wehave consideredthe choice ofmarketvariableofanoptimizingimplicit cartel, which hastorelyontacitcollusion.Theframeworkis similartothe frameworkofSinghand Vives(1984). Our resultspartlycorrespond tothe resultsSinghand Vivesfound fortheoneshotgame.Ifgoodsaresubstitutes …rmscompeteinquantities,ifgoodsarecomplements…rmscompeteinprices. However,themechanismbehind theresultsaredi¤erent.Inthestaticsetting ofSinghand Vives,…rmschoosemarketvariablesnon-cooperativelyinorder tomaximize shortrun pro…ts, intherepeatedgamethe choice ofmarket variableisguidedbydeviationpro…ts,theoptimizingcartelseekstominimize deviation pro…ts. 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