Size Distribution and Anti-trust

Lambertini, Luca; Sasaki, Dan Working Paper Size Distribution and Anti-trust Quaderni - Working Paper DSE, No. 390 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Lambertini, Luca; Sasaki, Dan (2000) : Size Distribution and Anti-trust, Quaderni - Working Paper DSE, No. 390, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4911 This Version is available at: https://hdl.handle.net/10419/159231 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/ Sizedistributionand anti-trust LucaLambertiniDan Sasaki DepartmentofEconomics DepartmentofEconomics UniversityofBologna UniversityofExeter StradaMaggiore45 Exeter,DevonEX4 4PUEngland I-40125 Bologna,ItalyUnitedKingdom [email protected]@exeter.ac.uk August2000 Sizedistributionand anti-trust Abstract:Extensiveliteraturenotwithstanding,e¤ectsofthesize distributionof…rms onconsumers’surplusand onsocialwelfareleavesroomforfurtherexploration.In thispaperwediscoverthatsize distributionimposestwocounter-balancinge¤ects onaggregatesurplusoftheindustry: [i] even distributionof…rmsizestypically facilitatestacitcollusioncomparedtoslightlyuneven distribution,whilst[ii] very uneven distributionresemblesmonopoly.Thetrade-o¤betweenthesetwocounterforcescanmaketheoverall welfare e¤ectof…rms’size distribution(givena…xed numberof…rms)non-monotoneinthedegree ofconcentration. Keywords:Competitive equilibrium,predatoryequilibrium,collusivesustainability, concentrationindeces. JEL classi…cation : L11,D43, K22. 1 1Introduction Concentrationindicesare commonlyused by variousantitrustauthoritiesall overthe worldinordertomeasurehowanti-competitivethemarketis.Game-theoreticmicroeconomicslargelysupportsthepredictiongiven bytheseindicesasfarasthenumberof…rms isconcerned.Thatis,themore…rmscoexistinamarket, [1]theless marketpowereach …rmcanexerciseinstaticoligopolistic equilibria,and [2]theharderitistosustaintacit collusionasasubgameperfectequilibriumwhenthemarketisrepeated,beitCournotor Bertrand oranythingin between. Infact,however,concentrationindicesmeasuretwothingsinseparably.Theyre‡ect notonlythenumberof…rms,butalsotheirsizedistributionaswell. Namely, ifthenumber of…rmsisthesame,themoreuneventheirsizesare,thehigherthe concentrationindices are.However,fromatheoreticalviewpoint, itisless straightforwardwhetheruneven size distributionsnecessarilymakethemarketless competitiveasopposedtomore evenly sizedoligopoly. Existingstudiesanalysingthe exerciseofmarketpowerbya prioriheterogeneous …rmsincludeHarrington(1989;1991),Lambson(1994;1995),and Rothschild(1999), interalia.Harrington(1989)considers…rmswith heterogeneousdiscountfactorsand relatestheirdi¤erentdegreesofmyopiatothewell-knowne¤ectsintermsofbargaining power.Lambson(1994;1995)characterises…rmswith unequallysizedcapacityand the e¤ectsoftheirsize distributiononthesubgame-perfectsustainabilityoftacitcollusion withoptimalpunishmentàlaAbreu(1986;1988)and Abreu,Pearce and Stacchetti (1986).InHarrington(1991)and Rothschild(1999),…rmheterogeneityismodelledin termsofheterogeneousproductioncosts. Technically,ouranalysisinthispaperis somewherein betweenLambson’scapacity approach,andHarrington’sandRothschild’scostapproach.Morephilosophically,however, wehaveaslightlydi¤erentviewfrom mostofthe existingcontributionsinthefollowing sense.Inourcomparativestatics,we…xtheindustry’saggregateproductioncapacity (de…nedintermsoftheindustry-widemarginalcostfunction)and divideitbetween …rms atvariousparametrised proportions.Thiscontrastswiththeso-farmorepopularview ofcomparingeach …rm’scost (orcapacityasaspecialcasethereof)parametersdirectly, 2 withoutfocusingcentralattentionontheindustry’saggregateproductioncapacity. Thereasonwhywetakethis somewhatunconventionalapproachisdueto ourinterest inassociating ourtheoreticalanalysistopracticalpolicyimplications.Namely,when discussingconcentrationand variouswaysofitsindexation,whatinterestsus(orpolicy makersingeneral) themostisthedistributionofproductioncapacityacross …rms,noless thaneach …rm’sindividualcapacityorcostlevels.Forthispurpose,we…nd ourapproach tobethemostnaturaland alsotechnicallythemoststraightforward. Weshowthatsize distribution producestwo opposite e¤ectsonaggregatesurplusof theindustry: [i]evendistributionof…rmsizestypicallyfacilitatestacitcollusioncompared toslightlyuneven distribution,whilst[ii] veryuneven distributionresemblesmonopoly. Fora given numberof…rms,thetrade-o¤betweenthesetwoforcescanmaketheoverall welfare e¤ectof…rms’size distributionnon-monotoneinthedegree ofconcentrationof theindustry. Theremainderofthepaperisorganisedasfollows.Insection2welayoutand analyse ourbasicduopolymodel, wherewemainlyconcentrateonBertrand duopolywhichturns out tobethesimplestalthoughthegistofouranalysisisapplicableto otherforms ofoligopolisticmarkets.Theninsection3wepresentasimpleillustrative exampleto developconcreteintuitionastothenon-monotonicityofthe e¤ectsofsize distributionon industry-aggregatewelfare.Finally,section4concludesthepaper. 2Bertrand duopoly 2.1Stagegame Consideranindustrywheretheindustry-wideaggregatemarginalcostfunctionisM[Q], whereM[Q]>0and M0[Q]>0forall Q¸0.Therearetwo…rmscompetinginthis industry,referredto as…rm1 and …rm2henceforth.Theircapacityratioisk:(1¡k), thatis,their respectivemarginalcostfunctionsare m1[q1]=M·q1 k ¸;m2[q2]=M·q2 1¡k ¸; 3 where0<k<1.Thereareno…xedcostsirrespectiveofk. Assumeforsimplicitythat thesetwo…rmsareperfectlysubstitutablesuppliersfacing the common, industry-wideinversedemand functionD¡1[Q],suchthatD¡1[0]>M[0] and (D¡1)0[Q]<0forall Q¸0.These…rmsaresimultaneous-moveprice-setters. 2.2Full collusioninaBertrand supergame Nowsupposethat thestagegamede…nedin2.1isin…nitelyrepeatedwiththediscount factor±whichiscommon betweenthetwo…rms. Obviously,wheneverthere existsaQF2 argmax q Zq Q=0 ³D¡1[Q]¡M[Q]´dQ, ifthetwo…rmsareabletosustaintacitcollusionat themonopolylevel p1=p2=PF wherePF=D¡1[QF],selling q1=kQF;q2=(1¡k)QF; thenthisisthemostpro…tableoutcomefromthetwo…rms’pointofview,exercisingtheir marketpowerinfull, earningnett pro…ts(perstagegame) ¼1=k¦F;¼2=(1¡k)¦F where¦F=ZQF Q=0 ³D¡1[Q]¡M[Q]´dQ. Wenowexaminewhetherthismonopolypricingis sustainablebytriggerstrategies. Asaninstrumentforsustainingtacitcollusion,considertheone-shotNashreversionwith thefollowingstaticBertrand-Nashequilibriumasathreatpoint.Notethat there can bemultiplestaticBertrand-Nashequilibria,but that thereisonewithzeronett payo¤s foreither…rm.1This static equilibrium,denoted byp1=p2=PBNhereinafter,satis…es ZQBN Q=0 ³D¡1[Q]¡M[Q]´dQ=0,wherePBN=D¡1[QBN]. Let¦[k;PF]denote…rm1’sdeviation pro…t, i.e., themaximumone-shotnett pro…t attainablefor…rm1 givenp2=PF.Providedthat¦[k;PF]¸k¦F,…rm1hasan incentivetoremainintacitcollusionifand onlyif ±¸1¡k¦F ¦[k;PF]: 1Thispenalcodeautomaticallyensuresthe“securitylevelpunishment”,asde…ned byLambson(1987). 4 Likewise, let¦[1¡k;PF]denote…rm2’sdeviation pro…tgivenp1=PF.Providedthat ¦[1¡k;PF]¸(1¡k)¦F…rm2hasanincentivetoremainintacitcollusionifand only if ±¸1¡(1¡k)¦F ¦[1¡k;PF]: Theminimumadmissiblediscountfactorinordertosustaintacitcollusionistherefore ±=max(1¡k¦F ¦[k;PF];1¡(1¡k)¦F ¦[1¡k;PF] ) because,tosustaintacitcollusion,theincentivestoremainincollusionshould besatis…ed forboth …rms.Figure1illustratesthetwofunctions1¡k¦F ¦[k;PF]and 1¡(1¡k)¦F ¦[1¡k;PF], whichobviouslyaremirrorimagesofeachotherwithrespect tok=1 2. Proposition1:²If1¡k¦F ¦[k;PF](andthus1¡(1¡k)¦F ¦[1¡k;PF]aswell)happenstohave alocalmaximumatk=1 2,then±alsohasalocalmaximumatk=1 2.However, insofarasthesetwofunctionsaresmooth,theyhaveazeroslopeatk=1 2and so does±,whichimpliesthatasmall departureofkawayfrom1 2entailsno…rst-order e¤ecton±. ²Inall othercases,±hasalocalminimumatk=1 2.Furthermore,whenever1¡ k¦F ¦[k;PF](andthus1¡(1¡k)¦F ¦[1¡k;PF]aswell)hasanon-zeroslopeatk=1 2,thelocal minimumof±at thispointisadownward-dippingkink,hence anylocaldeparture ofkawayfrom1 2entailsa…rst-orderincreasein±. Obviously,thelatternot theformeristhegeneric case,asdrawninFigure1(thelocal slopesofthetwofunctionsmaybeoppositefromthoseinthediagram; insofarasthey arenon-zero ourgenericobservation upholds). 5 Figure1:Minimumcollusivediscountfactorneark=1 2. -k 6 ±(thickenedlocus) 1 2 1¡k¦F ¦[k;PF]1¡(1¡k)¦F ¦[1¡k;PF] Ineconomictermsthisproposition(itssecondhalf,thegenericcase)suggeststhatunequalising thesize distributionofthe…rmsawayfromthe50 -50 splitcan help destabilisetacit collusionand therebycontributetowelfare, inspiteofthefact thatitincreasesmostof the commonlyusedconcentrationindices.Theintuitiontothispropositioncan bebest obtained bymeansofanexample,astheoneinsection3. 2.3PartialcollusioninaBertrand supergame Whenthediscountfactor±istoo lowtosustainfull collusionasdescribedin2.2,there canstill besustainedsometacitprice collusionthatisnotaspro…table.Thistypeof collusionisoftenreferredto aspartialcollusion.Inthespiritof“tacitcollusion”where …rmsarenotunderanyexplicitlycollusiveagreementsuchas sidepaymentsand hence aretosharepro…tsaccordingtotheirsales share,wefocusonthoseoutcomeswherethe …rms sharethemarketaccordingtotheircapacityratiok:(1¡k).2Inorderforthetwo …rmstocolludeataprice PT2(PBN;PF)withquantitiesq1=kQTand q2=(1¡k)QT 2Size asymmetrybetweenthetwo…rmsinourmodelcouldgenerallygiverisetotwodimensions inwhichtacitprice collusioncan be“partial.”Oneisthelevelofthe collusiveprice,whichisthe dimensionordinarilyconsideredwhen discussingpartialcollusion.Theotherdimensionisthereshu-ing ofquantitiesbetweenthetwo…rms,givenanycollusiveprice level; betweenequallysized …rmsthe equal quantityshareswouldalwaysbethebestsustainableand themostpro…table collusive con…guration, whilstbetween unequallysized …rmsthequantitysharesproportionaltotheirsizesmaynotbethebest sustainablealbeitunambiguouslythemostpro…tablewheneversustainable.Fixingthequantityratio at k:(1¡k)isforustoconcentrateontheformerdimension not thelatter. 6 wherePT=D¡1[QT],analogouslytosection2.2,thediscountfactormustsatisfy ±¸max(1¡k¦¤[PT] ¦[k;PT];1¡(1¡k)¦¤[PT] ¦[1¡k;PT] ) where¦¤[PT]=ZQT Q=0 ³D¡1[Q]¡M[Q]´dQ,and ¦[k;PT]and ¦[1¡k;PT]denotethe deviation pro…tsfor…rm1 and …rm2,respectively,giventheother…rmconformingto PT. Weareinterestedinthemostpro…tablepartialcollusiongiven±2(0;±)and k. Letting PT[±;k]=arg PT (1¡k¦¤[PT] ¦[k;PT]=±) ;PT[±;1¡k]=arg PT (1¡(1¡k)¦¤[PT] ¦[1¡k;PT]=±) ; themostpro…tablepartiallycollusiveprice can bede…nedas P¤[±;k]=minfPT[±;k];PT[±;1¡k]g: The“dual” ofProposition1isherebyasfollows. Proposition2:²IfPT[±;k](and thusPT[±;1¡k]aswell)happenstohavealocal minimumatk=1 2,thenP¤[±;k]alsohasalocalminimumatk=1 2.However, insofarasthetwofunctionsPT[±;k]and PT[±;1¡k]aresmooth,theyhaveazero slopeatk=1 2and sodoesP¤[±;k],whichimpliesthatasmall departureofkaway from1 2entailsno…rst-ordere¤ectonP¤[±;k]. ²Inall othercases,P¤[±;k]hasalocalmaximumatk=1 2.Furthermore,whenever PT[±;k](and thusPT[±;1¡k]aswell)hasanon-zeroslopeatk=1 2,thelocal maximumofP¤[±;k]atthispointisaupward-kinkedridge,henceanylocaldeparture ofkawayfrom1 2entailsa…rst-orderdecreaseinP¤[±;k]. Thegeneric case,thesecond itemofProposition2, isillustratedinFigure2(asinFigure 1,thelocalslopesofthetwofunctionsmaybeoppositefromthoseinthediagram; insofar astheyarenon-zero ourgenericobservation upholds). 7