athey - collusion and price rigidity

7/28/2019 Athey - Collusion and Price Rigidity

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The Review of Economic Studies, Ltd.

Collusion and Price RigidityAuthor(s): Susan Athey, Kyle Bagwell and Chris SanchiricoSource: The Review of Economic Studies, Vol. 71, No. 2 (Apr., 2004), pp. 317-349Published by: Oxford University Press

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ReviewfEconomictudies2004) 1,317-349 0034-6527/04/00140317$02.00? 2004TheReviewf conomictudiesimited

Collusion ndPriceRigiditySUSAN ATHEYStanfordniversitynd NBERKYLE BAGWELLColumbiaUniversityndNBER

andCHRIS SANCHIRICOUniversityfPennsylvania

First ersion eceived ecember 000;final ersioncceptedNovember002 Eds.)Weconsider n infinitelyepeated ertrandame, n which rices republicly bserved nd eachfirmeceives privatelybserved,.i.d.cost hockn eachperiod.Wefocus nsymmetricerfectublicequilibria, hereinny punishments"reborne qually y ll firms. e dentifytradeoffhats associ-atedwith ollusive ricingchemesnwhich heprice obecharged y achfirmsstrictlyncreasingnits ost evel: uch fully orting"chemes fferfficiencyenefits,s theynsure hatheowest-costirmmakes he urrentale,but hey lso imply n informationalost distortedricingnd/orquilibrium-path ricewars), ince higher-costirm ust e deterredrommimickinglower-costirmycharginga lower rice. rigid-pricingcheme,here firm'sollusivericesindependentf ts urrentostposition,acrificesfficiencyenefitsut lso diminisheshe nformationalost.For widerange f set-tings,he ptimal ymmetricollusivecheme equiresi) the bsence f quilibrium-pathricewars nd(ii) a rigid rice.ffirmsre ufficientlympatient,owever,he igid-pricingcheme annot eenforced,and he ollusiverice f ower-costirms ay e distortedownwardn orderodiminishhe ncen-tive ocheat.When hemodel s modifiedo nclude .i.d.publicdemandhocks,hedownwardricingdistortionhatccompaniesfirm'sower-costealizationay ccurnly henurrentemandshigh.

1. INTRODUCTIONIn the tandard odelofcollusion,ymmetricirmsnteractn an infinitelyepeated ertrandgame nwhich astprices republiclybserved. he standard odel ffersnumberf nsights,but tpresumesnunchanging arket nvironment.his s an importantimitation,incethescopefor esting theoryfcollusions greater hen he heoryffersredictionsoncerningthemannernwhich ollusive rices arywith nderlyingarketonditions.This imitationspartiallyddressedn two elebratedxtensions.otembergndSaloner(1986) introduceubliclybserved emand hocks hat re .i.d.over ime.1When hedemandshock s large, he ncentive o cheat undercuthecollusiveprice) s acute, nd collusionbecomesmoredifficulto enforce. ather han orego ollusive ctivityltogether,irms henreducethecollusivepriceand thereby iminish he ncentiveo cheat.Thus,markupsrecountercyclical.ike the tandard odel, heirmodeldoes notpredicthat ctual pricewars"occur nthe quilibriumath; ather,he uccess f collusion aries long he quilibriumathwith hedemand hocks hat reencountered.Followinghe eminalwork fStigler1964),a second iteraturetresses hat firmmaybeunable omonitorerfectlyhebehaviourf tsrivals.Green ndPorter1984) explore hispossibilityn an infinitelyepeated ournotmodel.Theyassume hat firm annot bserve

1. HaltiwangerndHarrington1991) andBagwell ndStaiger1997)considerurtherxtensions.317

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318 REVIEW OF ECONOMIC STUDIEStheoutputhoicesof rivalsbut that ll firmsbserve public ignal themarket rice)thatis influencedothby outputhoices ndan unobservedemand hock.2A colluding irm hatwitnesses low marketrice hen aces ninferenceroblem,s it s unclearwhetherhe ow-price utcomerose s a consequencefa baddemandhock r a secret utputxpansionyarival. he Green ndPorter1984)model hus epresentsollusionn the ontextf a repeatedmoral-hazardhidden-action) odel, nda central eaturef theirnalysiss thatwarsoccuralong he quilibriumath ollowingaddemandhocks.In thispaper,we explore thirdxtensionf the tandardollusionmodel.Weconsideran infinitelyepeated ertrandame, n which achfirms privatelynformedf ts unit ostlevel n eachperiod,heres a continuumfpossible osts, nd he ostrealizations i.i.d. crossfirmsndtime. urrentrice electionsbutnot ostrealizations)republiclybserved eforethe eginningf henext eriod.We thus epresentollusionnthe ontextf repeateddverse-selectionhidden-information)odelwith ubliclybservedctionsprices).3Ourmodel swelldesignedocontributeoa long-standingssue n ndustrialrganizationconcerninghe elationshipetweenollusion ndprice igidityn thepresence fcost hocks.Empiricaltudies yMills 1927),Means 1935) andCarlton1986,1989)conclude hat ricesaremore igidnconcentratedndustries,uggestinghat ollusions associatedwith greatertendencyoward rice igidity.n addition,ver hepast everal ecades, ntitrustnforcementhasuncovered umerousrice-fixinggreementsnwhich irmso-ordinaten a particularriceand enforce tablemarket hares ver time. n these xamples, olluding irmsdjustpriceoccasionallynresponseochangesnoverallmarketonditions,ut heyacrificehe fficiencyadvantageshat ouldbe gained yallowing firm ith temporaryost dvantageo servelargermarkethare.4At the sametime, he ndustrial rganizationiteratureas notprovided satisfactorytheoryhat inkspricerigidity ith ollusion. hebestknown heorys the"kinked emandcurve" heoryfferedy Sweezy 1939) andHall andHitch 1939). As Scherer1980) andTirole1988)discuss, owever,his heoryas mportanthortcomings.ndustrialrganizationeconomistsave hus ravitatedoward hemorenformaliew hat rice igiditysappealingocollusive irms,ecause rigid-priceollusive cheme revents istrustndreduces he isk fa pricewar.Carlton1989)explains:

Thepropertyfthekinked emandurveheoryhat rices unresponsiveosomecost luctuationsspreservednmost iscussionsf ligopolyheoryhetherrnot asedonthe inkedemandurve. hereasonings thatnoligopoliesrices luctuateess nresponseocost hangesespeciallymall nes) hanhey ould therwisen order ottodisturbxistingligopolisticiscipline.nytimepricehangeccursn noligopoly,theres a riskhat pricewar ould reakut.Hence, irmsre eluctanto hange rice(Carlton,989, p.914-915).Wedevelophere rigorousvaluationfthis nformaleasoning.ocusing n the rivate ostfluctuationshat irmsxperience, e explore heextentowhich mistrust"imits olluding

2. While heGreen ndPorter1984)model sdevelopednthe ontextfCournotompetition,hemainnsightscanbecapturedn repeatedertrandetting,sTirole1988)shows. heGreen-Porter1984)model s furtherxtendedbyPorter1983),Abreu, earce nd Stacchetti1986,1990)andFudenberg,evine ndMaskin1994).3. AtheyndBagwell 2001),a sequelto thepresentffort,xplores relatedmodel.We discuss his aper nwhat ollows. urmodel s also related o recentwork hat xtendsheGreen ndPorter1984) model o allowforprivatelybserved emand ignals.Compte1998) andKandori ndMatshushima1998) suppose hat irmsubliclychoose messages" fterrivatelybservingheirespectiveemandignals.nour etting,irmsreprivatelynformedas totheirespectiveosts nd he ublic ctions a (payoff-relevant)rice hoice.4. For xample, Europeanartel fcartonboardroducerset tablemarkethares, ut djustedhe ixed riceevery monthsnresponse ochanging emand onditionsEuropean ommission,994).Additionalxamples rediscussednBusinessWeek 1975, 1998)andScherer1980).



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ATHEY ETAL. COLLUSION AND PRICE RIGIDITY 319firms'bilityorespondotheir espectiveostpositions. he costsofmistrustreformalizedin terms f thepricewarsandpricing istortionshat re required o dissuadefirms rommisrepresentingheir rivatenformation.Webeginwith formalnalysis fthe static ertrandamewith nelastic emand ndprivateost nformation.5hisgameconstituteshe tagegameof ourrepeated-game odel.In theuniqueNashequilibriumf thestagegame, hesymmetricricing trategys strictlyincreasingnthe irm'sost evel.AnadvantagefNashpricings that ales nthe urrenteriodare llocated othe irm ithhe owest ost.This s the fficiencyenefitf "fullyorting"i.e.strictlyncreasing)ricingcheme. fcourse, rom he irms'erspective,ashpricinglsohasan mportantimitation:alesare llocated t owprices.We turn ext othe epeated-gameodel nd xplorewhetherirmsanthenupportetter-than-Nashrofits. efocus nthe lassof ymmetricerfectublic quilibriaSPPE). AnSPPEcollusive cheme t a givenpoint n time an be described y (i) a pricefor achcosttypeand ii) an associated quilibriumontinuationalue for achvector fcurrentrices,wherethe ontinuationalue s symmetriccrossfirms.n an SPPE, therefore,olluding irmsmovesymmetricallyhroughny o-operativerprice-warhases.Weobserve hat collusive chememust atisfywokinds f ncentiveonstraints.irst,for very irmnd cost evel, he hort-termainfromheating ith n off-scheduleeviation(i.e.with price hat s not ssigned oany ost ype nd that hus epresentscleardeviation)must e unattractive,nview ofthe off-the-equilibrium-path)ricewarthat uch a deviationwould mply. s is usual nrepeated-gamereatmentsfcollusion,his onstraints sure o bemetffirmsre ufficientlyatient.econd, he roposedonductmust lso be such hat ofirmis ever ttractedo anon-scheduleeviation, hereby firm f a given ost ypemisrepresentsitsprivatenformationndselects pricentendedor differentost ype.To characterizeheoptimal PPE,we build n thedynamic-programmingechniquesutforthyAbreu t al. (1986, 1990) (APS) andFudenbergtal. (1994). We draw n analogybetween urrepeated idden-informationameandthe taticmechanismesign iterature,nwhich heon-schedulencentiveonstraints analogous o the tandardncentive-compatibilityconstraint,he off-schedulencentive onstrainterves as a counterparto the traditionalparticipationonstraint,nd he ontinuationaluesplay he ole f transfers".owever,nlikea standardmechanismesignproblemnwhich ransfersreunrestricted,hesetoffeasiblecontinuationalues s limited ndendogenouslyetermined.nparticular,emay ssociatepricewarwith transferhatsborne ymmetricallyyall firms.We breakouranalysis foptimal PPE intotwoparts.We supposefirsthat irmsrepatient,o that he ff-scheduleonstraints met.The on-scheduleonstrainthen apturesheinformationalostsof collusion hat onfrontrivatelynformedirms.hecentral roblemsthat he chememust e constructedo that higher-costirm oes nothavean incentiveomisrepresentts osts s lower, y electing lower rice ndtherebyecuringor tself higherexpectedmarkethare. n an SPPE, the nformationalostsofcollusionmaybe manifestedntwoways.First,heprices f ower-costirms aybedistortedosub-monopolyevels.This sa potentiallyffective eans felicitingruthfulost nformation,incehigher-costirms indlower ricesessappealing.econd, ollowinghe election f ower rices,he ollusive chememay ometimesall for futurequilibrium-pathricewar. hecurrent-periodenefitf lowerprice henmaybe ofsufficientagnitudeocompensateor hefutureostofa pricewaronlyifthe irmrulyas ower osts nthe urrenteriod.

5. TheBertrand odel ssumes omogeneousoods,which s a common eaturefcollusivemarketsHayandKelley 1974),Scherer1980,p. 203)). Collusion s alsooften ssociatedwithnelastic emandEckbo,1976).



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320 REVIEWOF ECONOMIC STUDIESA rich rrayfcollusive chemes itwithinhe PPE category.nepossibilitys that irmsincur he nformationalosts fcollusion urelyn terms fdistortedricing. nexamplestheNash-pricingcheme,nwhich irmsepeatedlylaytheNashequilibriumfthe tatic ame.Anotherossibilitys that irmsnitiallychieve ull ortingndadopthigher-than-Nashrices.In this ase, some ofthe nformationalostsof collusionmust e reflectednthefutureostofa pricewar: uch scheme atisfieshe n-scheduleonstraintnly fequilibrium-patharssometimesollow he election fprices ssigned o lower-costirms. furtherossibilitysthat irmsmayneutralizehe nformationalostsof collusion ltogether,yadopting rigid-pricing cheme,nwhich ach firmelects he ameprice neachperiod,whateverts currentcostposition. hedownside ftherigid-pricingcheme s thattsacrificesfficiencyenefits:one firmmayhave ower osts han tsrivals,ndyet hefirmshare hemarket. hese chemeshighlighthecentralradeoffetween fficiencyenefitsnd nformationalosts that ollud-ingfirmsmust econcile.Moregenerally,ollusive ricingchemesmaybe strictlyncreasingoversome ntervalsf costsandrigid verother egions,withwarsfollowingomepricingrealizations.Ourfirst ain indings that irmsare oorly nderny PPE collusive cheme hatnsistsuponfull orting.nfact,onsideringhe ntire etoffullyortingPPE,wefind hat irmsando nobetter han heNash-pricingcheme.We next onsiderhefull lass of SPPE collusionschemes ndreport secondmainfinding:f firms re sufficientlyatient,hen n optimalSPPE collusive cheme anbe achievedwithout ecourseoequilibrium-pathricewars i.e.with tationarytrategies).hisfindingontrastsnterestinglyith hepredictionsftheGreenandPorter1984)model.Armedwithhese indings,e add some dditionaltructurendprovide characterizationofthe ptimal PPE collusive cheme.When irmsrepatientndthedistributionf cost ypesis log-concave, e establish thirdmainfinding:ptimal PPE collusions characterizedya rigid-pricingcheme,nwhich irms elect he ameprice namely,hereservationriceof

consumers)neachperiod, hateverheirost evels.We thus ffernequilibriumnterpretationofthe ssociation etween rice igidityndcollusion escribedbove.We then urn o thesecondpart f ouranalysis nd considermpatientirms.We showthat mpatiencereates n additional isadvantageo pricewars: a schemewithhighpricestoday ustained ywars n thefuturemakes a deviation specially rofitableoday,whilesimultaneouslyeducinghevalue ofco-operationnthefuture.ursecond no-wars) indingthereforeontinues o holdwhenfirms re mpatient.ext,we observe hat he off-scheduleconstraintsparticularlyemandingor ower-costypes.ntuitively,hen firmraws lower-costtype, he emptationocheat nd undercuthe ssigned rice s severe, ince heresultingmarket-shareain sthen speciallyppealing. or mpatientirms,collusive cheme husmustensure hatower-costypes eceive ufficient arkethare nd select ufficientlyowpricesnequilibrium,othat hegains romheatingrenot oogreat.This ogic sreminiscentfthe rgument adebyRotembergndSaloner1986),althoughhere t sprivateost hocksas opposed opublic emandhocks) hat ecessitate odificationofthecollusive cheme.We confirmhis ogicwith ur fourth ainfinding:ffirms re notsufficientlyatientoenforce herigid-pricingcheme,heymay till upport partiallyigidcollusive cheme,nwhich hepriceof lower-costypes s reducedn order omitigateheincentiveocheat.Thisfindinguggestshat ymmetricollusion etweenmpatientirms aybe marked yoccasional andperhaps ubstantial)rice eductionsy ndividual irms. hesedeparturesccurwhen firmeceives favourableostshock, ndthey epresentpermitted"escapeclause" i.e. an opportunityo cutprices nd ncreasemarketharewithoutriggeringretaliation) ithinhe ollusive cheme.Moregenerally,e establishonditions ormpatientfirmsnderwhich,fbetter-than-Nashrofitsan be achieved,hen ptimal PPE collusion s



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ATHEYETAL. COLLUSION AND PRICE RIGIDITY 321characterizedy stationaryricingchemenwhich rices rerigid ver ntervalsfcosts i.e.the ptimal ricingcheme s a weakly ncreasingtep unction).Tofurtherevelop he elationshipetweenur heorynd that fRotembergndSaloner(1986),we next xtend urmodel to includepublic .i.d. demand hocks.The off-scheduleconstraintsthenmost ifficultosatisfy henmarketemandshigh nd firm'sost hockslow.We thus ffer fifth ainfinding:nanextendedmodelwith ublic .i.d.demandhocks,iffirmsrenot ufficientlyatientoenforcehe igid-pricingcheme, ptimal PPE collusionmay echaracterizedy stationaryricingcheme,n which n ndividualirmharges lowerprice nhigh-public-demandndlow-private-costtates. he Rotembergnd Saloner 1986)predictionfcountercyclicalricings thus obust oprivateost nformation.ur modelhasthe urtherredictionhat rices remore ariablewhen oday's emandshigh.Throughout,e restrictttentiono symmetricchemes.This restrictions important.Asymmetricchemes llowone firmoenjoy more rofitableontinuationalue han nother.Such schemes husfacilitate ransfersrom ne firmo another. they ndBagwell 2001)analyse ptimal symmetricchemes.n therepeated amethat hey eature,hestagegameallows hat irmsommunicateost nformationndmakemarket-shareroposals eforeettingprices. nder he ssumptionhat nit osts rehigh r ow, heyonstructnasymmetricerfectpublicequilibriumAPPE) that elivers irst-bestrofits hen hepatience f firmsxceedscriticalfinite)evel.6 n this onstruction,hen firmnnounceshat ts costs rehigh,t sfavoured ith reatermarket harenfutureeriods. heresultingquilibriumlay s highlynon-stationary.The present aperdiffers rom hat fAthey ndBagwell 2001),both n terms f themodelsemployedndthepredictionserived. he present aperconsiders model withcontinuumftypes nd studies ptimal ymmetricchemes,whileallowing hatfirmsmaybe impatientnd that emandmaybe volatile.n contrast, they ndBagwell 2001) use atwo-type odel and study ptimal symmetricchemes,with hecentralfirst-best)indingapplying henfirms resufficientlyatient. t a predictiveevel, hepresent aperprovidesa formalnterpretationor he raditionaliew that tandardollusion ntails ixed rices ndstablemarkethares ver ime.AtheyndBagwell 2001) show hat ophisticatedollusion,nwhich irmsrack ndreward ndividual irmehaviour ver ime, ommunicatendallocatemarkethares,s characterizedymarkethares hat re unstable ver ime.Whilewe do notpropose theoryfhowfirmso-ordinatepon nequilibrium,he wopapers ogetheruggestan intriguingossibility:tationaryPPE areappealing imple ndmaybe descriptivef essformalandperhaps acit) ollusive entures, hileoptimalAPPE arequite ophisticatedndmaybe mostplausiblenthepresence f a smallnumber fwell-organizedonspiratorshatinteractrequentlyndcommunicatexplicitly.inally,PPE maybetheonly vailable ptioniffirmsannot bservendividualirmehaviour. hisoccurs, .g. nprocurementuctions ithmore han wobidders,fthewinningid--but ot hename f hewinner--isnnounced.As ust mplied, specialcase of ourmodel s a repeated rocurementuction.Wemaythus elate urfindingsothosedeveloped yMcAfee ndMcMillan 1992), ntheirnalysisofbidding ings. heydescribe vidence hat ixed-pricechemesi.e. "identical idding")rewidely sed. n a staticmodel, hey how hat fixed rice s theoptimal trategyor iddingcartelsnfirst-priceuctions or single bject,when he artelsre weak" i.e.firmsreunabletomake ransfers).n ouranalysis f theoptimal PPE,we generalizeheweak-cartel odel,since he taticmechanism e analyses directlyerived rom repeated ame nd allowsfora restrictedlass oftransferscorrespondingosymmetricricewars).Ourrigid-pricinginding

6. For familyfrepeated rivate-informationames, udenbergt al. (1994) show hat irst-bestayoffsanbereachednthe imit s players ecomenfinitelyatient.



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322 REVIEWOF ECONOMIC STUDIESthus rovidesdditionalheoreticalupportor hepracticef dentical idding.We alsoextendthe nalysiso ncorporatempatientirms.We describe he tatic ndrepeated ames n Sections and3, respectively.he lattersrelated othemechanism-designpproachnSection . Wepresenturfindingsor PPE amongpatient irmsnSection , andSection considersmpatientinns. hese sections lose withdiscussion f an extendedmodelwith ownward-slopingemand. ection concludes.

2. THE STATICGAMEWebeginwith static ame fBertrandompetitionn which irmsossessprivatenformation.Thisgame llustrateshe mmediateradeoffshat onfrontirmsndeterminingheir ricingpolicies nd serves s a foundationn which ur ubsequentynamicnalysis uilds.We positn ex ante identical irms hat ngage n Bertrandompetitionor sales in ahomogeneous-goodarket.ollowing pulber1995),wemodifyhe tandardertrand odelwith he ssumptionhat ach firmsprivatelynformeds to tsunit ost evel.Cost evels rei.i.d.acrossfirms.nparticular,irm's "type"Oi s determinedyan independentraw romthe upport0, 0] accordingothe ommonlynown istributionunction(0). We assume hatthe orrespondingensity(0) - F'(0) is strictlyositiven [0,0]. After hefirmsearn heirrespectiveosttypes, hey imultaneouslyhooseprices. etpi E R? denote heprice hosenbyfirm, with - (pl..., Pn) then epresentinghe ssociated riceprofile.We assumeunitmassof denticalonsumers,ach ofwhom as an nelastic emand or neunit pto somereservationrice ,where > 0.A price trategyor irm is a functioni 0i) mappingrom he etofcosttypes,0,0],to theset of possibleprices,R+. For simplicity,he functioni is assumedcontinuouslydifferentiable,xcept erhaps t a finite umberfpoints so as to allowfor cheduleswithjumps).A price trategyrofiles thus vector (O) - (pi(Oi), P-i(O -i)), where - (Oi, -i)is thevector fcost ypesndp-i (0-i) is theprofilefrival rice trategies.achfirmhoosesitsprice trategyithhegoalofmaximizingts xpected rofit,ivents ost ype. orepresenta firm'sxpected rofit,erequirewofurtherefinitions.irst, edefiner(p,0) - p - 0 astheprofithat firmeceiveswhen tsets hepricep andhas costtype and"wins" he ntireunitmassof onsumers.econd,wespecifyBertrand arket-share-allocationunction, ip),thatndicates irm's marketharewhen he ector frealized ricessp. Thisfunctionllocatesconsumersvenlymong irmshat elect he owest ricenthemarket.Wemaynowrepresentirm's interimrofit,hich s the xpected rofitor irmwhenit hascosttypeOi, elects hepricepi andanticipateshat ivalpriceswill be determinedythe ival ricing trategyrofile,-i (-i). Withm-pi; P-i) - Eo_i mi pi,P-i(0-i))], firm'sinterimrofitunction aybewrittens

(pi,9Oi; -i) -r(Pi, i)M(Pi; P-i). (2.1)Whenfirmsdopta symmetricricing trategy,(.), we use thenotationmh(pi;) andn(Pi, Oi P).We nowdescribe heessential radeoffhat onfronts firmwhen t sets tsprice.Asillustratedn (2.1),when firmf a given ype onsiderswhethero lower tsprice, t mustweigh he ffectf anincrease n the hance fwinningthroughii pi; p)) againsthedirecteffect fthepricereduction n profit-if-winthrough (pi, 0i)). An importanteature f themodel s that ifferentypes eeldifferentlybout his radeoff.nparticular,he nterimrofitfunctionatisfiessingle-crossingroperty:ower ypes ind he xpected-market-sharencreasethat ccompanies price eductionelatively ore ppealinghan ohigher ypes, ince ower



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ATHEYETAL. COLLUSION AND PRICE RIGIDITY 323types ave ower otal osts nd thus igher rofit-if-win.hesingle-crossingropertympliesthat igher-costirmslways elect igher ricesi.e.pi Oi) isnon-decreasing).Thestage amemaybeanalysed sing tandardechniquesrom he uctioniterature.7

Proposition . The staticgamehas a uniqueNash equilibrium. oreover,heuniqueNashequilibriumricing trategy1) is symmetric:i - pe,Vi; (2) is continuouslyifferen-tiable nd strictlyncreasingver0 e (0, 0); (3) is belowthemonopoly rice: pe(0) < r,VO< 0; and 4) yields ositiventerimrofitor ll types ut hehighest,whoneverwins ndwhose ricepe 0) = 0 wouldyield ero rofitven f tdidwin.Notice hat he ymmetricquilibriumricingtrategye is continuousndstrictlyncreasing.Further,he rice lways alls t orbelow0,no matter owhighsr.Forfutureeference,e et7NE denote firm'sxpected rofitntheNashequilibriumfthe tatic ame.

3. THE REPEATED GAMEIn this ection, edefine he epeatedame.We alsopresent "Factoredrogram"nd stablisha relationshipetweenolutionsothis rogramndoptimal PPE.3.1. ThemodelImagine hat irmsmeetperiod fter eriod oplaythe tagegamedescribedn theprevioussection,ach with he bjectivefmaximizingts xpected iscountedtreamfprofit.ssumefurtherhat, pon ntering period fplay, firmbserves nly hehistoryf: i) itsowncostdraws,ii) itsownpricingchedules,nd iii) the ealized rices f tsrivals. hus,we assumethat firmoes not bserve ival ypes rrival rice chedules.Formally, e describe herepeated ame nthefollowingerms. fullpath fplay s aninfiniteequence Ot, t},with given air nthe equence epresentingvector ftypes ndprice chedules tdate .The nfiniteequence mplies publichistoryfrealized rice ectors,{pt , andpathwise ayoffsor irmmaybe thus efineds

ui(0= St-l'(p, Ot)mii(pt).i lotpt})= . -t=liI

Atthecloseofperiod , firm possesses n informationet,whichmaybe writtens hi -{Of, i, ptit=l1. (The nullhistorys thefirm'snformationetat thebeginningf the firstperiod.)A (pure) trategyor irm,si hi) 0i), associates price chedulewith ach nformationsethi. Each strategyrofile = (s, ... ,s,) induces probabilityistributionverplaypaths{ot, pt} in the usualmanner. he expected iscountedayoffrom is thus heexpectationui s) = E[ui {t, pt )] takenwith especto thismeasurenplaypaths.3.2. AdynamicrogrammingpproachUnder urassumptions,irmypes re .i.d. across ime andfirms),ndso therepeated amehas a recursivetructure.t is thereforeatural o follow udenbergt al. (1994) (FLM) andemploy recursiveolutiononcept:we focusupon equentialquilibrianwhich achfirm'sstrategyonditionsnlyuponthepublicly bserved istoryf realized rices. uchstrategiesarecalledpublicstrategiesndsuchsequential quilibria re calledperfect ublic equilibria(PPE). A public trategyay hus eabbreviateds a mapfrominiteublic istoriespt =l to

7. See alsoSpulber1995),who stablishesropositionfor eneral emand unctions.



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324 REVIEW OF ECONOMIC STUDIESprice chedules.We furtherestrictttentionoSPPE,wherebyollowingvery ublichistory,firms dopt symmetric rice schedules: si(p, . .., p') = sj(p, . .., pt), Vi, , t, p ..., p .Symmetryeans hat ll firmsufferutureunishmentsndrewardsogethern anindustry-widebasis.Drawingon the work of Abreuet al. (1986, 1990),we applythetools of dynamicprogrammingo this etting.etV C R denote he etof SPPE continuationalues ndwriteVs - infVsandVs - supVs.Note,nitially,hatwith continuumfpossible ricingtrategiesthere s no a prioribasis fromwhich oargue hat ither s E Vsor E Vs 8 if s e Vs,thenwe saythatVs is anoptimal PPE value.Following PS, any ymmetricublic trategyprofile = (s,... ,s) canbe factorednto first-periodrice chedule and a continuationpayoffunction : Rn+ --+ R. The continuationayoffunctionescribesherepeated-gamepayoff (p) enjoyed yall firms rom heperspectivefperiod onwardfterachfirst-periodprice ealization = (Pl, . Pn) JRn+.e definev(pi; P-i) - Eo_i (Pi, P-i(0-i))] as theexpected ontinuationayoff hen firm elects i andexpects ther irmsoprice ccordingtop-i. In viewof our ymmetryestriction,emay implywritev(pi; ), andsimilarlyachfirm'sxpected ayoffrom can be writtens Eoi 7r(p(Oi), i;p) + 8V(p(0i); p)].We now onsider he actored rogram,nwhichwe choosefactorizationsirectlynordertomaximize firm'sxpected ayoff,ubjecto i) the easibilityonstrainthathe ontinuationpayoffests lwaysnthe PPE value et nd ii) the ncentiveonstrainthat firmannot ainbydeviatingo analternativericingchedulegiven he ontinuationayoffunctionndunderthe ssumptionhat ther irmsollow hepricingchedule):

The FactoredProgram. Chooseprice chedule andcontinuationayoffunctiontomaximizeEoi[J p(Oi),Oi;p) + 8-(p(Oi); p)]subject o:Vp e R',, v(p) E Vs, andVp, Eoi r(p(Oi), Oi; p) +63-r(p(0i); p) ] > Eoi,7(p(Oi), Oi; p) + 6(PV(0i); p)].

Lemma 1. Consider nysymmetricublicstrategyrofile* = (s*... ,s*) with hecorrespondingactorizationp*, v*). Then,* is an optimalPPE if ndonlyf p*, v*)solvesthe actored rogram.This emma, tandardnthe iterature,stablisheshatwemay haracterizehe etofoptimalSPPE bysolvinghe actored rogram.It s alsopossible oanalyse he etofequilibria.ffirmsan randomize ver ontinuationequilibriae.g. using public andomizationevice), hen he etof SPPE values s convex ndis thus ullyharacterizedhen he est ndworstPPE arefound, here heworstquilibriumvalue s attainedyminimizingatherhanmaximizingheobjectiventheFactored rogram.InSection ,weanalyse heworst PPE; until hen, e focus noptimalPPE.3.3. The nterimrogramorgameswith rivatenformationWe next eformulateheFactored rogramo thatt can be analysed sing xistingoolsfromthe static)mechanismesigniterature. ebegin yobservinghat SPPE ina repeated amewith rivatenformationust e immune otwokinds fcurrent-periodeviations.

8. Wecould ttemptoprove ompactnessf the et ngeneralerms.nstead, e establishompactnessn theprocess fcharacterizingest ndworst PPE values.



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ATHEY ET AL. COLLUSION AND PRICERIGIDITY 325A firmeviatesoff-schedule"hentchooses price ot pecifiedor ny ost ealization(i.e. a pricenot ntherange fp). When firmrices n thismanner,thasunambiguouslydeviated:hedeviantrices "off he quilibriumath".As is standard,uchdeviationsremosteffectivelyeterredhen irmssetheworst vailable unishments a threat.ncontrast,firmdeviates on-schedule" hen tchooses price hats assigned nder tosome ost evel, utnot tsown.Forexample, firmmaybe temptedo choose he ower rice ssigned o a lower-costrealizationnorder o ncreasetschances fwinninghemarket.mportantly,rival annotbe sure hat hedeviatingirm as not rulyfthe osttype hatt s imitating:hedeviantprice s "ontheequilibriumath".Anon-schedule eviation fthiskind anbe preventedfthe ollusive chememposes punishmenthen owprices rechosen. ut such punishmentwouldbecostly,ince twould ccur long he quilibriumath fplay,wheneverirmsctuallyrealize owcosts.With his istinctionthand,wetake he onstraintsftheFactored rogram,ut hemninterimormndparse hemnto wogroups,ndrewritehis rograms follows:The Interim rogram. Chooseprice chedule andcontinuationayoffunction tomaximizeEoi 7 (p(Oi),Oi;p) + 8-(p(Oi); p)]subject o:Off-Scheduleonstraints:p' ? p([0_,]),(IC-offl)VO-i, v(p',P-i(0-i)) E Vs(IC-off2) Oi, f (p(Oi),Oi;p) + 8(p(80); p) > fr(p',Oi; ) + 31(p'; p)On-Schedule onstraints:0i,(IC-onl)VO-_i, v(p(0i), P-i (0-i)) E Vs(IC-on2)VOi, nf(p(Oi), i;p) + 86(p(0i); p) > r(p(O0),Oi;p) +p(5(p(0i); p).Noticehow he n-scheduleonstraintsrewrittenn "direct"orm:or iven andv, IC-on2)requireshat firm ithype idoesbettery announcing"hatts ypesOi han y nnouncingsomeotherype, i,when ther irmsrepresumedo announceruthfully.hissuggestshat(IC-on2)maycorrespondo a "truth-telling"onstraintn an appropriate echanismesignformulation.

4. COLLUSION AMONG PATIENTFIRMSAND MECHANISMDESIGNIn this ection,we buildon this uggestion,howinghatwhenfirmsrepatient,he nterimPrograman berelaxed oyield newprogramhatwecall theMechanismesignProgram. ealso useexistingoolsfrom hemechanismesigniteratureobegin ur haracterizationftheoptimal PPE.4.1. TheMechanismesignProgramOurgeneral pproach as twosteps.First,we relax the nterimrogram,ydroppingff-schedule onstraintsndallowing ontinuationayoffunctionseyond hose hat reactuallyfeasiblenthe epeated ame.With omenotationaldjustment,ethen rrivettheMechanismDesignProgram.econd,weprovideonditionsnderwhich he olutionnthe elaxed ettingcorrespondso a factorizationhat ontinueso satisfyll of the constraintsf the nterimProgram.n thisway,we identifyonditions nderwhich ptimal PPE maybe characterized



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326 REVIEW OF ECONOMIC STUDIESbysolvingheMechanism esignProgram.he Mechanism esignPrograms useful ecausewe canapply xistingools o tdirectly;or xample,n what ollows, emake epeatedse ofthe evenuequivalenceheorem.Our firsttep s to relax theconstraintsf the nterimrograms follows: i) theoff-schedule onstraintsre gnorednd ii) (IC-onl) isreplacedwith relaxed onstraint

V(i , v(p(()0; p) _Vs. (IC-onl')Thefirstelaxations withoutoss ofgeneralityf firmsresufficientlyatient,ince hen ff-schedule eviationsre nyway ot empting.oappreciatehe econd elaxation,e recall hatunderIC-onl),for very n-scheduleector fprices,he ontinuationalue s drawn romheSPPE set,Vs; ncontrast,IC-onl') requires nly hat irm's expectedontinuationaluedoesnot xceed hesupremumfthe PPE set,Vs 9Wenextntroduceirect-formotation.et H 0, 0; p) =- (p((0),0; p) denote he urrent-period rofithat firm ftype would xpectwere t to announcehat ts ypes 0. Wedefineaswella generaltransfer"r"punishment"unction, 0), which firmxpects o ncurwhenitannounces . We nowdefine:The MechanismDesignProgram. Chooseprice chedule and a punishmentunctionT tomaximize

Eo nH(0, ; p) - T(0)]subject o:For ll 9,T(0) > 0;(IC-onM)VO, , H(0, 0; p) - T(0) > H(0, 0; p) - T(0).Suppose p, v) satisfieshe onstraintsfthe nterimrogram. emay hen ranslatep, v) into(p, T), accordingo T 0) --8[Vs iv(p(0);p)]. It sdirecthatp, T) satisfieshe onstraintsftheMechanismesignProgram;urther,sing his ranslation,he bjectivesftheMechanismDesignand nterimrogramsank p, v) pairingsn the ameorder. he Mechanism esignPrograms thus relaxed ersionfthe nterimrogram.ThemeaningfT warrantsmphasis. or given PPE, f firmhat nnounces expectsa continuationaluebelow he upremumf he PPE set i.e. fVs > v-(p((0); )), thenwemayinterprethe PPE as specifyinginexpectation)"war".Theresthen nowar" f he xpectedcontinuationalueequalsthe upremumftheSPPE set i.e. ifVs = v3(p(0); )). Usingthetranslation()) = 8[SVs v p(0); p)], we thusmay ssociateT(0) > 0 (T(O) = 0) withfuturehat ollows firm'snnouncementf0 and n which heres a war no war).Wecome nowto the econd tep nour pproach, hereweprovidehe onditionsnderwhich solutionp*, T*) to theMechanism esignPrograman be translatedack nto afactorizationp*, v*)that atisfiesll ofthe onstraintsf the nterimrogram.hefollowingpropositiondentifiesn mportantetofconditionsfthis ind:10

Proposition (Stationarity). uppose p*, T*) solvestheMechanism esignProgramand T* - 0. Then3S E (0, 1) suchthat,orall 8 > 8, there xists n optimal PPE whichis stationary,hereinirmsdopt p* after ll equilibrium-pathistories,nd p* solves thefollowingrogram:maximizeo (0, 0; p) subject oVO, , (0, 9; p) > H(0, 0; p).9. NoteparticularlyhatIC-onl') allowsv - Vs,even hough e asyethaveno assurance hat s e Vs.10. Inourworkingaper Athey, agwell ndSanchirico,998),we show hat hegeneral pproachlso extendstocaseswhere * canbe strictlyositiveinwhich asethe orrespondingquilibriumsnon-stationary).* > 0 maybethe nique olution hen emandsdownward-sloping,utnot, s we will how, ornelastic emand.



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ATHEYETAL. COLLUSION AND PRICERIGIDITY 327To establishhisresult,we develop wo mplicationsf the maintainedssumptionhat(p*, T* = 0) is optimalntherelaxed nvironment.irst, sing urtranslationbove, t thenfollows hat p*, v* - V,) is a solution o the nterimrogram,f t satisfiesheadditionalconstraintsfthis rogram.nturn,hismplies hatp*,v* - Vs) is (weakly) uperioroanySPPE factorization.ormally,o[lH(0,0; p*) + 8Vs] > Vs.Second,we claim hat ffirmsre ufficientlyatient,tthen ollows hat he epeated layofp* (coupledwith ppropriateff-scheduleunishments)sanSPPE. It s straightforwardhatthispatternfplaysatisfiesheon-schedulencentiveonstraint:incep* satisfiesIC-onM)whenT* = 0, eachfirm illfollow * whenfuturelaydoesnotvarywith heon-scheduleprice. urther,he epeated layofthe taticNashequilibrium,e, is always nequilibriumftherepeated ame;therefore,henfirmsresufficientlyatient,he hreatf Nash reversiondeters noff-scheduleeviationwhere firmhooses pricenot ssigned oany osttype yp*).ll Theclaim s thus stablished.nturn,t mplies hat O HI(0,0; p*)]/(1 - 8) < Vs.Combiningur wo nequalities, e obtain hedesired esult:Vs = Eo H 0,0; p*)]/(l -8). In words,fthe Mechanism esignPrograms solved with p*,T* = 0), and iffirmsare sufficientlyatient,hen n optimal PPE is easilycharacterized:irmsdopt hepricingschedule * ineachperiod, here * is the olution o the tatic rogramtatednProposition.It is nowpossible opreviewhe nalysis hat ollows.n Section .2,we characterizehesetof (p, T) that atisfyIC-onM).Thisputsus in position o solvetheMechanism esignProgram,sing ools standardn themechanismesign iterature.n Section , we establishthat here s always solutionnwhichT* _ 0. Forpatient irms,roposition thenmpliesthat noptimal PPE is characterizedythe tationarydoption f the ccompanyingricingschedule, *. Under he ssumptionhat hedistributionunction(0) is log-concave, efindthat heoptimal ricingchedule akes simple orm: *(0) - r. Wethus eportonditionsunderwhich or ufficientlyatientirmshe ptimalPPE is stationaryndrequiresllfirmsocharge he ameprice ,regardlessf theirosts.Thisrigid-pricingchemes supportedythethreathatf ny ther rice s observed,hefirms illreverto theworst PPE,which eliverscontinuationalueV. WhenF(0) is log-concave, e show n Section thatheworst PPE isattainedhroughashreversioni.e. theNash-pricingchemes used n eachperiod).We alsoestablishhere everaldditionalredictionshat risewhen irmsre esspatient.

4.2. Consequences fon-schedulencentiveompatibilityWebeginouranalysis ftheMechanism esign Program ycharacterizinghe mplicationsofthe n-scheduleonstraintIC-onM).We do this nthefollowingemmawherewe use thenotation1-0(0,0; p) = 01(0, 0; P)I=O):

Lemma2 (Constrainteduction). (p, T) satisfiesIC-onM) f ndonly f(p, T) satisfies:(i) p(O) isweaklyncreasing,nd(ii) fI(0,0; p) - T(0) = I(1, ; p) - T(O) - fIHo0(0~, ; p)dO.

This result s standardn themechanismesign iterature,nd it followsfrom hesingle-crossingroperty.edevelop ext n nterpretationfLemma , n order oprovidehe ntuitivefoundationormany f he indingshat ollow,nd nparticularur esulthat ftenhe ptimalSPPE usesthe igid-pricingcheme.11. We establish elow n Proposition(ii) thatp* yieldshigher xpected rofithandoes the staticNash

equilibrium,e.



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328 REVIEW OF ECONOMIC STUDIESThe situationnalysedherecontrasts ith heusualPrincipal-Agentormulation,incenowthe agents"i.e. firms) esign heir wn schedule,with hegoal ofgeneratings muchprofits possible. heexpressionn ii) thus anbe interpreteds reflectingheprofithat anbe distributedo0,withoutnducingmimicryyotherypes.As (ii) reveals, he nterimrofit(inclusive fwars),H(0, 0; p) - T(O), that s "left" or ype , afterncentive-compatibilityconstraintsreconsidered,onsists f twoterms:he profit-at-the-top"ndthe information"or"efficiency"entsarned ythe ypes etween and0.To interprethese erms,etusconsider typeOk hats ustbelow0. How much anthistype arn,withoutnducingmimicryytype ? TypeOk anearn he ameprofits doestypeplusa bit xtra, here he xtra ortions attributableo thegreaterfficiencyi.e. ower osts)that k ctuallynjoys. imilarly,type k-1 hatsslightlyower han k anearn he ameprofitas doesOk lusa bit xtra. ulling hese oints ogether,enowhavea directnterpretationf(ii): theprofitor ny ype equalstheprofit-at-the-toplusthe ccumulatedfficiencyentsofhigher ypesnote hat1l0 < 0). Animportantmplications thatwhen heprofit-at-the-topis increased,hehighest ypehas less incentiveomisrepresenttself s a lower ype,nd thisrelaxationn the ncentiveonstraintsnturn ermitsower ypes oearnhigherrofits.Whatdetermineshemagnitudefthe fficiencyents? o answer his,etus definehemarketharexpectedya firm hen t announcesas M(0;p) - r(p((0);p), where heexpectations over he nnouncementsfother irmsassumedruthful).e observehat

- o0Fl0(,0;p)dO=

f0M(0;p)dO. (4.1)

Themagnitudef he fficiencyentss thus eterminedy he llocationfmarketharescrosstypes.We note hat hefirmsavetwo nstruments,rices ndwars,withwhich o sort etweentypes. t is useful o considerwhetherheavailabilityfthewar nstrumentxpands hesetof ncentive-compatiblearket-sharellocations. o explore his ssue,we introduce simplerestriction.onsider schemep, T) and etOKdenotehe owest forwhich (0) = p(0). Werestrictttentionoschemesp, T) forwhichH OK,OK;p) - T OK) O0.t sstraightforwardoshow hat nly chemeshatatisfyhe estrictionillbeoptimal,ndmaintaininghe estrictionsimplifieshe xpositionf ourfindings.12ecan nowestablishhat heuse of warsdoesnotexpand he ange f ortinglternativesvailable othe irm.ormally:

Lemma3. Given schemep, T) that atisfiesIC-onM)and T > 0, andassociatedmarket-sharellocationM(O; p), there xists n alternativechemep, T -- ) which lsosatisfiesIC-onM), nd such hatM(O; p) = M(O; j) andH(0, 0; p) - T(O) = H(0, 0; P).In short, iven n originalncentive-compatiblecheme ndmarket-sharellocation, e mayconstructn alternativencentive-compatiblecheme hat elivershe amemarket-sharello-cationwithout singwars,while lso providinghesameprofit-at-the-top.his constructionrequireshat heprices readjustedwayfromheirriginalevels, nd so the emma oesnotdetermine hich irmypesif ny)preferhe lternativecheme.Weexplorehis ssuenext.

12. For ny p, T) that ails he estriction,emaymodifyhe rgumentssociated ith emma in he ollowingway.First, ote that ince IC-onM) implies hat verallprofits decreasingn 0, 1F(0,0; p) - T(0) < 0. Next,ratherhan onstructinghe lternativecheme romp, T), we instead ollow he tepsn theAppendixndconstruct(p, T - 0) fromP, T), where (0O) 0 andT(0) = 0 for e [OK, ] and P, T) = (p, T) elsewhere.heresulting(p, 7 0) satisfiesIC-onM), ven hough-, T7)doesnot, nd t also satisfies (-; p) = M(.; p) = M(-; p), andH j, 0; p) - Ti) = F (0, 0; P) > I 0, 0; p) - T 0). Sinceourgoal s toestablishhatp, 7 - 0) provides igherprofithan oes p, T), the atternequalitytrengthenshe esults hat ollow.



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ATHEYETAL. COLLUSION AND PRICE RIGIDITY 329UsingLemma and quation4.1),we observe hatype 's nterimrofits determineds

11(0, ; p) - T(O) = 11(0, ; p) - T(O) + M(0; p)d0. (4.2)Thenext esult, hich swellknown romuctionheory,ollows irectly:Lemma 4 (Revenue quivalenceheorem). Considerany (p, T) whichsatisfies IC-onM).Then nyotherp, T) whichatisfiesIC-onM),M(0; p) = M(0; p) and 1-(0,0; p) -T(0) = 11(0,0; p) - T(0) must lso satisfy10,0; p) - T(0) = 1-(0,0; p) - T(0) for ll 0.Intuitively,upposethatfirms tartwith he scheme p, T) and that hey hen onsider nalternativecheme p, T) which s on-schedulencentiveompatiblend delivers hesameprofit-at-the-top.f in addition he alternativechememaintainsheoriginalmarket-shareallocation,hen he fficiencyents realsopreservedor very ype.As the lternativechemealters eitherheprofit-at-the-topor he fficiencyents,tfollows rom4.2) that his chememaintainsswellthe riginalnterimrofitor lltypes.At thispoint,we have extractedhree essons.First, fter ccounting or ncentivecompatibility,irms aybe attractedopricingchemes hat aise heprofit-at-the-top.econd,for given mount fsorting,irmsre free o choosewhethero mplementhe orrespondingmarket-sharellocationwith wars.Third, nce theprofit-at-the-topnd the market-shareallocationredetermined,nterimrofits fixed or ll types.With hese essons nplace,wearepreparedow ocharacterizeptimalPPE for atientirms.

5. OPTIMALCOLLUSION AMONGPATIENTFIRMSInthis ection, eanalyse ptimal ymmetricollusionmong irmshatrepatient.Wepresentourcentral ointsn five teps.First,we considerymmetricollusive chemes hat refullysortingi.e.p(0) is strictlyncreasing).econd,weexplorewhethernoptimalPPE for atientfirmsequiresquilibrium-pathars i.e. T(0) > 0 for ome0). Third, e add furthertructureandcharacterizeheoptimal ymmetricricingcheme. ourth, e identifyherelationshipsbetweenurfindingsndthosenthe roaderiterature.inally, ediscuss hemannernwhichourresultsxtendwhen emandsdownward-sloping.5.1. Fully ortingricingchemesTheset ffully ortingchemesncludes varietyfcandidates. nepossibilitys that he irmsemploy he taticNashequilibriumneachperiod.Alternatively,hefirmsmay ttemptosortinthefirsteriodwithhigher rices, erhaps ear hemonopolyevel. Suchschemes atisfyon-schedulencentiveonstraintsnlyf heynclude quilibrium-pathars.Considerwhatprofitsre possible n an SPPE wherethepricing unctions strictlyincreasingn thefirsteriod. incethehighest ypemakesno sales,profit-at-the-tops simply-T (0). Because a firm insthemarketf andonly f all other irmsnnounce igher ypes,efficiencyents reuniquely eterminedor he lass offullyortingchemes, ithM(0; p) =[1 - F(0)]n-1. Thus,byLemma4, anytwofully ortingchemeswhich atisfyIC-onM)differnly f theprofit-at-the-top,T(0), differs. he bestprofit-at-the-tops achievedwhenT(0) = 0. But theNash-pricingunction,e, is fully orting,nd (pe, T - 0) satisfiesIC-onM).Recallthat - 0 correspondso i(p(0); p) = Vs for ll 0. Therefore,hebest ollusivescheme hat s fully ortingn the first eriodyields xpected ayoffs qual to rNE ? 8Vs,



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330 REVIEWOF ECONOMIC STUDIEStheprofitrom hestaticNashequilibrium lusthediscountedrofitromhebestavailablecontinuationPPE.Nowconsidermposing strongerequirement,amelyhat hepricingchemes strictly-FSincreasingnevery eriod fplay. etVFSbe the et f uch PPE and etVs be the upremum-FSof the et. The logic of theprevious aragraphmplies hatVs is attainedy a scheme hatdelivers he tatic ashprofitlus hebest vailable ontinuationalue nthe estrictedetVFS.-FS 7 NE -FS -FS NEI( It follows hat n optimal PPE underThus, s s , r Vs NE/(1 - ). It follows that an optimal SPPE underfull ortings simplyherepeated layof the staticNashequilibrium;urther,hisholdsforanydiscountactor,ince heNash-pricingcheme atisfieslloff-scheduleonstraintss well.Summarizing:

Proposition . Amonghe lassofSPPE where he ricingunctionsstrictlyncreasinginevery eriod, oranydistributionunction anddiscountactor , an optimal PPE is therepeated layof he taticNashequilibriumfterll histories.5.2. Nowarsonthe quilibriumathFor he lass offully ortingricingunctions,he nalysisnthe reviousection hows heresnobenefitnsupportingigher riceswith n-schedule ars.We nowextendhis rgumentorany nitialmarket-sharellocation.Considerny riginalchemep, T) that ntailswars omewhere.emma guaranteesheexistence f analternativeo-war chemep, T - 0) that s incentiveompatible,nduces hesame market-sharellocation,ndgenerateshe ameprofit-at-the-top.emma4 thenmpliesthat healternativeo-war chemegivesthe sameinterimrofitinclusive fwars)as didtheoriginal cheme.As shown ntheAppendixproof fLemma3), the lternativecheduleachieves his rofityexchangingnywar nthe riginalchedule or lower rice.Applyingthis rgument,ogether ith ur tationarityesultProposition),we conclude hat:13

Proposition . Allow oranydistributionunction . If (p*, T*) is a solution o theMechanism esign Program,hen here xists s well a solution(p3, ) with (0O)< p(O)and T 0) -- 0. Thus, f irmsre sufficientlyatient,here hen xists n optimal PPE thatis stationary:irms se the ricing cheme (0) followingvery istorylongthe quilibriumpath, ndE0H(8, 0; 3)/(1- 8) = Vs.We see from roposition thatwarshave no value:for nydistribution, ifthere xists noptimalPPE that seswars, hen herexists swell anoptimalPPE that oes not.5.3. OptimalricingWe arenowpreparedodetermineheoptimal PPE pricingchemewhenfirmsrepatient.Given he no-wars"indingromhepreviousubsection,e seek heprice trategy*(0) thatsolves he rogram resentednour tationarityroposition. ith ome dditionaltructure,hispricingchemeseasily haracterized:14

13. We nclude ere restrictionnthediscountactor,o that he tationarityroposition aybeused. nwhatfollows,nProposition,wedevelop he urtherrgumenthathe ff-scheduleonstraintsrerelaxed nmovingromschemewithwars o a no-warcheme,ndwe are hus ble toremove his estriction.14. Thisresultlso holds n amodelwith iscreteypes,xcept hat n additionalarameterestrictionsrequired,one that epends n thegapbetween andthehighestype. herestrictions satisfied hen he istance etweenypesbecomes mall nough. etails re vailable rom he uthors.



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ATHEYETAL. COLLUSION AND PRICE RIGIDITY 331Proposition . For8 sufficientlyarge:

(i) Ifeithera) F is log-concave,r b) r- 0 issufficientlyarge, hen he quilibriumathofthe ptimalPPE ischaracterizedy ricerigidityp*(0) - r) andnowars T*(0) - 0).(ii) In anyoptimal PPE that s stationary,here xists n open ntervalfcosttypes verwhichricingsrigid,ndper-periodrofitreaterhan hat nthe tatic ashequilibriumisattained: > rNE/(1 - ).

Forpatient irms,fthedistributionunctions log-concave rr is large nough, heoptimalSPPE isdescribeds follows: irmselect heprice in eachperiod,whateverheir rivateostrealizations,o long s allfirms ave elected he rice inallprevious eriods.15urther,irmscanalways xceedNashpayoffs,f heyre ufficientlyatient.Therigid-pricingchemep(O) = r has benefitsnd costs.An importantenefit f thisschemes thatt satisfiesheon-schedulencentiveonstraintithoutecourseoequilibrium-pathwars.Furthermore,heprice s as high s possible.However,n evident ostof therigid-pricingcheme s that tsacrificesfficiencyenefits:tmaybe that nefirm as a lowcostwhile nother irm as a high ost, utunder he igid-pricingcheme ach of these irmsellsto1/n-thfthemarket.hecontentf thepropositionparti)) is that hebenefitsfthe igid-pricingcheme xceedthe osts, rovidedhat hedistributionunctions log-concave,rthereservationrice shigh nough.Asthis ropositionscentralothe aper,we nclude he roofnthe ext. y Proposition,wemay ocus nsolutionsotheMechanism esignProgramorwhichT 0) --0.Using 4.2),wemaynowwrite firm'sxpected rofits

Eo[HI(0,0; p)] = Eo7r(p(O), 0) -M(O; p) + jM(O; p)dO .Next, mployingstandardrickrom he iteraturenoptimaluctionsMyerson1981),BulowandRoberts1989)),wemay ntegrateypartsndrewriteur bjective unctions

Eo[IH(, 0; p)] = Eo 7(p(0), 0)-M(; p) + -(0) -M(O;p) . (5.1)fConsider hefirstermf this xpression.incethe ingle-crossingropertymplies hatisnon-decreasing,rofit-at-the-topshighest hen ll firmset he ameprice,o that hehighesttypesnever nderpriced.hemost rofitableigid-pricingchemes the ne nwhich irmsixtheprice t r.Thus,p(O) --rmaximizes rofit-at-the-top,iven hat ownward-slopingricingschemesrenot n-schedulencentiveompatible.Consider ow he econd erm. og-concavitylays n mportantolehere. ntuitively,he"contribution"f an increasena given ype'sprofito thefirm'sxpected rofits governedby thefractionf typesbelow t (which njoya relaxed ncentiveonstraintndthus arnhigher rofit),onditionalnthe probability"hathe iven ypewill ctuallyrise.We nterpretF (0)/f 0) as ameasure fthe ontributionf n ncreasentype 'sprofitothe irm'sxpectedprofit.og-concavitynsures hat hismeasuresnon-decreasing,o thatllocatingmarketharetohigher ypes, ndawayfrom ower andmore fficient)ypes,mprovesartel rofit.hissuggestshat herigid-pricingcheme (O) - r alsomaximizesheexpected fficiencyentterm,Eo[- (O) .M(O; p)].15. Wehave ssumed > 0. Ifthis ssumptionere elaxed,hen he ptimalchemewould ntail hat irms ith

cost ypes reaterhan sit ut atherhan ndure egative rofits.



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332 REVIEW OF ECONOMIC STUDIESToformalizehis uggestion,bserve irsthat or llp, E0[M(0; p)] = 1/n:before firmlearns ts ype,thasanexpectedmarkethare f1/n.Define

(D(0" ) = nf M(0;p)f(0)dO,and note hat1(.; p) is a probabilityistributionor achp. LetpR be a rigid-pricingcheme.Given hatM(0; p) is non-increasinghenp is non-decreasing,(O; pR) = 1/nmust rossM(O; p) from elowfor ny p. Thus, D(-;pR) dominates (.; p) by first-ordertochasticdominanceFOSD) for ll p. In words, rigid-pricingcheme uts hemostweight nhigh-cost ypes. ythedefinitionfFOSD, it followshatfF(0)/lf(0) isnon-decreasing,

E?[F

(0)"MM(O"pR 1 0 F Ifo F[f' nM(0PR]fJ 7(Od P)d1(0p ())=Eo -(0) -M(0;p)

for ll p non-decreasing.hus,the secondtermn (5.1) is maximizedwith rigid-pricingscheme. inally,nly rigid-pricingcheme anbeoptimal ecause, iven ur ssumptionhatf > 0,0= f(0) < f (0) for > 0. Anynon-rigidchemewould lacemoreweightnSince p(O) - r maximizes rofit-at-the-topnd is theuniquely ptimal igid-pricingscheme hat oesso,we conclude hat igid ricingtr strictlyominatesll other ncentive-compatiblericingchemeswhen is og-concave. nderog-concavity,ffirmsre ufficientlypatient,he ptimalPPE thus ntails igid ricingtr in eachperiod.Theassumptionhat hedistributions log-concaves commonnthe ontractingiterature,andmany istributionsatisfyhis ssumption.utwe can also considerheoptimalmarket-share llocation ven fF/f is decreasingn some ntervals.f r is close enough o 0, theoptimal ricing ulemaximizes hesecondterm f (5.1); clearly,t is desirable o putmoreweightnthehigheralues fF/f,subjectop non-decreasing.hismay ntailntervalsfbothsortingndpooling.Under ur ssumptionhat 0) > 0,strictog-concavitylwaysholds naneighbourhoodf0. Thus,t salways ptimal or ufficientlyatientirmsouserigid ricingtthebottom fthepricingunction,ndfrom his artii) ofthepropositionollows: ufficientlypatientirmsansustainPPE per-periodrofittrictlybove he tatic ash quilibriumwhichentails ortingt thebottom). inally,bserve hatf r - 0 is sufficientlyarge, o that rofit-at-the-tops great nough,hefirsterm f 5.1) dominateshe econd.Then, hebenefitsrompoolingmarketharewith igh-costypes utweighhose romllocatingmarkethare mongsttypeswith hehighest/f, andthe igid-pricingchemes again ptimal.We close this ubsection ith ninstructivexample. uppose hat hereretwofirmsndconsider hefamilyftwo-step ricingchedules. uchschedules re characterizedya lowprice l, a high rice 2, anda breakpoint2, uch hat lltypes elow above)02select he ow(high)price.A rigid-pricingchemes then nextremease, nwhich 2 = 0 or02 = 0. For02 e (0, 0), the ingle-crossingropertynsures hat heon-schedulencentive-compatibilityconstraintsre satisfiedfandonly f a firm ftype 2 is indifferentetween hebottomndtop steps: '(pl, 82)M(0; p) = lr(p2,82)M(0; p), whereM(_; p) = F(02)/2 + [1 - F(02)]andM(8; p) = [1 - F(02)]/2. Givenp2 and02, ncentiveompatibilityhusdetermineslandtherebyxpected rofit.upposefurtherhat hedistributionf costs s uniformver heunit nterval.irect alculationshen ieldE[H(0, 0; p)] = [2p2(1- 02)+ 02+ (02)2- 1]/4.Expected rofits thusmaximizedwhenp2 = r and02 is setat a corner.With > 0, themaximums achievedwhen 2= 0. Thus, rigid-pricingchemewith (O) - P2= r isoptimalwithin he wo-step amily.Wemay lso compareherigid-pricingndNash-pricingchemes.



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ATHEYETAL. COLLUSION AND PRICE RIGIDITY 333Expected rofitnder heNash-pricingchemes 1/6,whichs lessthan hevalue 2r - 1)/4that sgeneratedythe igid-pricingcheme.5.4. Related iteratureWe now ompare urfindingsothose stablishednrelated apers.We recall irstheMcAfeeandMcMillan 1992) study fbidding ings.Working ith staticmodelof procurementauctions,McAfee ndMcMillan how hat iddersna "weakcartel"where ransfersrenotallowed) ollude est ftheygree obid the ameprice, . OurPropositionis closely elated;indeed, heirmodelcan be mapped nto heMechanism esign Program y setting -- 0.The analysishereextends heir esults, owever,yformallyonnectinghe static esults otherepeated-gameontextnddemonstratinghat rigid-pricingchemes optimalvenwhenschemes hat ustainortingsing wasteful"ransfersT > 0) areallowed. nthenext ection,we furthereneralizehe nalysis oconsider hepossibilityf mpatientirms.Ourfindingsre lsorelated owork ncartel esign nder rivatenformation.sCramtonandPalfrey1990)andKihlstromndVives 1992)show,ffirmsandesign mechanism herethey ommunicateheir osttypes ndmake ide-paymentso oneanother,hen hefirmsanachieve ull fficiencyenefits ithoutnypricing istortioni.e.productions allocated othelowest-costirm, ho sets tsmonopoly rice), lthoughheparticipationonstraints ayfailfor omecosttypes.16ntherepeated-gameontext,s AtheyndBagwell 2001) show, irmsmayuseasymmetricontinuationalues omimic ide-paymentsromnefirmo nother.n themodel onsideredere, owever,he PPE restrictionreventsuch ross-firmransfers.ut t sstill ossible or irmsoachieve ull fficiencyenefits,f hey sefullyortingricingchemes.In this ase,howevernformationalosts,manifesteds pricing istortionsnd/or uturericewars,must eexperienced.hecosts xceed he enefits hen is log-concave.Interestingly,ommunicationouldnothaveanyvalue n ourmodel.The realizationffull fficiencyenefitsoesnotrequire ommunication,hile he nformationalostsremain,with rwithoutommunication,o ong s firmsannotmake ide-paymentsoruseasymmetriccontinuationalues). nshort,he ptimal irect evelation echanism ithoutide-paymentsscharacterizedyProposition,andnocommunicationsrequiredo mplementt.Ourfindingsrealso related o thosepresentedn thehidden-actionollusioniterature.A central eatureftheGreen nd Porter1984) and Abreu t al. (1986,1990)papers s thatcollusive onductnvolves eriodic eversionsopricewars.Ourmodelcan be placedwithintheir idden-actionodelling ramework,fwe think f a firm'strategy,(O), as its hiddenaction ndtheresulting rice, = p(O), as thepublic ignal,where hedistributionfthispublic ignal s then eterminedythepricing unctiontselfnd thedistributionalropertiesof 0. The maindifferences thatwe allow for nendogenousupportfthepublic ignal. utdifferently,urmodelmaybeunderstoods a hidden-actionodelwith ndogenousmperfectmonitoring.o seethis, ecall hat ntheGreen ndPorter1984) andAbreu tal. (1986,1990)modellingramework,he upportfthepublicly bservedmarketrice s independentftheprivateutputelectionsmadebyfirms.n ourmodel, ycontrast,he upportf the ignal sitself eterminednequilibrium.nparticular,ffirmsmploy rigid-pricingchedulenwhichthey hooser under ll costrealizations,hennequilibriumhe upportf thepublic ignal sdegenerate,s rival irmsxpect o observe heprice ,no matter hat ostrealizationhefirmexperiences.his nturn nables irmso imitwars ooff-equilibrium-pathvents.

16. Their nalysis uLildsn earlier ork yRoberts1985),who hows hat schemewith ull fficiencyenefitsmaynotbe incentiveompatible, henfirms an communicateut reunable omake ide-payments.ee also theMcAfee ndMcMillan1992) analysis f strongartels",n which idders an make ransferso one another.



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334 REVIEWOF ECONOMIC STUDIESWe assumethat irms' osts are i.i.d. overtime.An alternativessumptions that achfirm's rivatelybserved ost s constantvertime.LaCasse (1999) and Chakrabarti2001)analyse hispossibility ith ownward-slopingemand nddiscreteosttypes. heyprovideconditionsnderwhich here xist quilibriahat re eithereparatingrpooling.Ourresultscanbe usedtocharacterizeptimal quilibria f a gamewith onstantosts,whenfirms repatient.nparticular,f F is log-concaverr is sufficientlyigh, hen ptimalquilibriantailrigid ricingtr.17 nthenext ection, e extend urmodel othe ase ofdownward-slopingdemand.Ourfindingsgaincharacterizeptimal quilibria f a gamewith onstantosts ndpatientirms.ptimal quilibriahen enerallyntail t eastpartial igidity.

5.5. Downward-slopingemandWe consider ow the case ofdownward-slopingemand.Whilecollusive ehaviours oftenassociatedwithnelastic-demandarkets, ost emandurves ave t east ome lasticity,ndso it s importanto nvestigatehe obustnessfourresults.To this nd,wemodifyhemodel s follows.Maintaininghe ssumptionhat oodsareperfectubstitutes,e nowdefine he rofit-if-winunctions7r(p, ) - (p - 0)D(p), whereis a twice-continuouslyifferentiablearketemand unctionhat atisfies > 0 > D' over herelevantange.Weassume hat r(p, ) is strictlyuasiconcavenp,with uniquemaximizer,pm0), where m0) > 0. Themonopoly rice, m0), is strictlyncreasingn0.With hismodification,he nterimrofitunctionontinueso be given yH 0,0; p) =7r(p(0),0)M((0; p). We nextdefine (p; p) as thequantity firmxpects o sell when tsetspricep andopponentsse pricing unction. In thepresentontext,(p; p) = D(p) ?Eo_i[mi(p, p-i(0-i))], and so q(p(O); p) = D(p(O))M(0; p). The function1 thusmaybealternativelyxpresseds F(0, 0; p) = (p(O) - O)q(p(O); p). We note hat emma remainsvalid nd mplies hat

n1(0, ; p)- T(0) = H(0, 0; p)- T((0) +j q(p(0); p)dO. (5.2)Thisexpression ighlightsimilaritiesnddifferencesith he nelastic-demandase.Asinthe ase of nelastic emand,ftype expects o sell a higher uantity,hen ll costtypesbelow0 canenjoyhigher rofit,ince fficiencyentsregreaterthigheruantities.utnowthemagnitudefefficiencyents s determinedytheshapeof thepricing unctioni.e. theallocation fmarkethares cross ypes) ndthe evelofpricesi.e.the ize ofmarketemand).As a consequence,emma no onger olds: f firmsonstructn alternativeollusive chemethroughchange np andT that reservesrofit-at-the-topnd ncentiveompatibilityt theoriginalmarket-sharellocation,he nterimrofitf ower ypess notpreserved.ustike n

an auctionmodelwith isk-averseirms,evenue quivalence reaks ownwhen he evelofpricenteractsirectly ith he irm'srue ost ype. hus,wecannotmmediatelyoncludehatpricewars re notneeded. ndeed, urworking aper Athey tal., 1998)provides detaileddiscussionhowingwhenpricewarsmight e usedandwhen hey an be ruled ut.Here,wefocus ur ttentionnprice igidity.Tounderstandhe enefitsndcosts frigid ricing, e take he xpectationfprofitrom(5.2) and ntegrateyparts,oderive nexpressionnalogous o 5.1):Eo[HI(0, 0; p)- T(0)] = n(0, ; p)- T(0)+ Eo q(p(); p)-(0) . (5.3)

17. Building n the resent aper, theyndBagwell 2002) establishhis inding.



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ATHEYETAL. COLLUSION AND PRICE RIGIDITY 335Recallthat nthecase of nelastic emand, rigid rice tr maximizes othprofit-at-the-topandthe xpectedfficiencyenterm. his s no ongerruewhen emandsdownward-sloping.Instead,he wo erms re nconflictbout he evelof theprice. omaximizeheprofit-at-the-top, twouldbe optimalo have rigid rice, s themarkethare or ype is thereby ade slarge s possible; urthermore,hebestrigid ricewouldbe pm0). Tomaximizehe fficiencyrent erm,twould lsobe optimalo have rigid rice, ollowingnargumentnalogous otheoneweusedfornelastic emand; owever,he est igid ricewouldnowbe as low aspossible,inorder omake he uantityroducedandthus he fficiencyents) s large s possible. heconflictetween he woterms bout heoptimal rice evel mplies hat ricerigiditys notnecessarilyptimal.Atthe ame ime,5.3) also ndicateshathere rerobust orcesnfavourfat eastpartialpricerigidity.o see this, uppose hat is strictlyncreasingn [x,y] C [0,0]. Nowdefine(0O)mplicitlyo thatt greeswith outside f x,y]but s constantn x,y]at a price suchthat

Eo[q(p(O); j) 1 E [x,y]]= q(p; jP)= Eo[q(p(O); p) | 0 e [x,y]]. (5.4)Then,ust s inthe nelastic-demandase,wecan define probabilityistribution

((0; p,x, y) = q(p(O); p)dF(O 0 e [x, y]).SinceD(0; j, x,y) dominates (0; p,x,y) byFOSD, ifF(8)/f(0) isnon-decreasing,hen

Eo q(5(0); )-(0) 0 E x,y] > Eo q(p(0); p)-(O) 0 e [x,y] ,anda force nfavour frigiditys thus llustrated. ith ownward-slopingemand, owever,thenewschemedoes not n general atisfyIC-onM)unlessT is modified.Whilewe canalwaysfind omeT that atisfiesIC-onM), hisT might iolate he onstraint(0) > 0, andit is also possible hatT(0) > T(0), implying reductionnprofit-at-the-top.espite hesecomplications,e establish onditionsnderwhich artial r full igiditysoptimal:

Proposition . Suppose hat emandsdownward-sloping.(i) Amonghe lassofSPPE where he ricing unctionsstrictlyncreasingnevery eriod,foranydistributionunction anddiscountactor , theoptimal PPE is therepeatedplay of he taticNashequilibrium.(ii) For8sufficientlyarge, fF is log-concavend demands sufficientlynelastic,hen heequilibriumathof he ptimalPPE ischaracterizedy ricerigidityndnowars.

To interprethis esult,bserve hat roposition(i) representsstrengtheningfProposi-tion : whendemands downward-sloping,herepeated layofthe taticNashequilibriumsuniquelyptimal mong ullyortingPPE. Downward-slopingemandntroducesewforcesinfavour fschemeswith ow (i.e. staticNash)prices nd no equilibrium-pathricewars, sopposed o schemeswithhigh e.g. monopoly) rices ndequilibrium-pathars.With owerprices,he evelofmarket emandsgreater,ndthe fficiencyents rethus igher.An implicationfProposition(i) is that,f firms reto improve ponthe staticNashequilibrium,hen omerigiditysrequired.With ownward-slopingemand,s Spulber1995)shows, hestaticNash equilibriums fully ortingwithpe 0) = 8. It is straightforwardto showthat t is possibleto achieveprofit reaterhan n the staticNash equilibriumf,forexample, m(0) > 0, by introducingomerigidityt thetopof thepricing chedule.



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336 REVIEW OF ECONOMIC STUDIESFurther,roposition(ii) establishes hat rigid-pricingcheme s optimalwhendemand ssufficientlynelastic. dditionalharacterizationsf he ptimalchemewith ownward-slopingdemand reprovidednAtheytal. (1998).We nowconsider rieflyhepossibilityhat irmsell imperfectubstitutes.n this ase,q(p; p) = E0_,[D(p,p-i(0-i))], andexpected rofits still iven y 5.3). This case ismorecomplex, ecause he xpected emand or ach cost ype epends nthe ntirericingunction.Recallour onstructionf ' as in 5.4). As a result fchanginghepricingunctionrom top, the xpected rofitor ypes V [x,y]may ncrease r decrease, ependingn the hape fthedemandurve.fdemands linear, owever,hen irmsareonly bout he xpected rice fopponents.nthis vent, and/ yield he ameexpected rice ndF-0, 0; p) = H 0,0; ')for 8 [x,y].But t still emains o show hat herexists feasible thatmaintainsIC-onM)and doesnotdecrease rofit-at-the-top.heanalysis fthis roblemsbeyondhe copeofthepresentaper;here,we simply ote hat tcan be shown hatfF and 1 - F are og-concave,anddemands linear,henny cheme hatmprovespon he tatic ashequilibriums at eastpartiallyigid.A similaresult oldswhen irmsompetenquantitiesatherhan rices.18Insummary,he esulthat ully orting ricingchemesrenot ptimalsquite obust,ndtheforces nfavourfpooling emainn more eneralmodels, lthoughewforces hat pposerigidity ay ead tooptimalchemes hat reonly artiallyigid.

6. OPTIMALCOLLUSION AMONG MPATIENT IRMSWe nowconsider ollusion mong mpatientirms.We proceed n twogeneral teps.First,we characterizeheworst PPE (themost evere unishment)ndthen eterminehecriticalpatienceevel bovewhich he igid-pricingcheme an beenforced.econd,we consideresspatient irms, ho areunable o enforce herigid-pricingcheme, ndexplore owthey estcollude.mpatiencereates n additional otivationor he voidance fpricewars.naddition,impatientirms ay sepricingchemes hat ntailn"escape lause",wherebyfirmsallowedtodepartromhe igid rice nd et lower ricewhent xperiencesfavourableost hock.nanextended odel,wefind hat uch departuresespeciallyikelywhen emandstemporarilyhigh. inally, e establishonditionsnderwhich he ptimalollusive ricingcheme or esspatientirms ust e a step unctionpartial igidity).6.1. EnforcingigidityffcheduleWebeginwith hedeterminationf he ritical iscountactor.herigid-pricingcheme atisfiesoff-scheduleonstraintsf firmlways egardshe urrent-periodenefitromndercuttingherigid rice s small ncomparisonothediscountedalueoffutureo-operation.nturn,utureco-operations more aluablewhen irmsremore atientnd he unishmenthatwould ollowa deviations more evere.The critical iscount actors thereforeetermineds the owestdiscount actor t which irmsan enforce herigid-pricingcheme,when deviationeads tothemost evere unishment,s.Formally,etussuppose hat he irmsttemptomaintainrigid rice > 0 inallperiodsfor ll costrealizations.f a firmftype were o cheat ndundercutbyE) thisprice, henthe firmwouldwintheentiremarket,s opposedto ust 1/n-thf themarket,nd so thefirm'sncentiveocheat s 1rr(p, 8). Importantly,his ncentives greatestor firm ith

18. Toseethis, ote hatf firm'sxpected rofitsgiven yfl(O, ; q) = q O)-(Ei [P(q (0) + Ej i q Oj))] -0), thenIT00,0; q) = -q(O), and o the xpression5.3) for xpected rofit,swellas the orcesn favour fpooling,wouldbe the ame.



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ATHEY ETAL. COLLUSION AND PRICE RIGIDITY 337the owest ost evel, , since he rofit-if-wins then ighestnd hegain nmarketharesthusmost aluable.f a firm ere ocheat, owever,t wouldforfeithediscountedalueoffutureco-operation.hisvalue s measurednrelationo the ost hat firmxpectsn the uture,O.Forexample,fa deviations punished yan infiniteeversionothe taticNashequilibrium,then he roposed igid-pricingcheme s off-schedulencentiveompatiblef ndonlyfn-l 1rr(p, 0) < (8/(1 - 8)) -r(p, EO) - NE . (6.1)nnIfthe roposedcheme ields reater-than-Nashrofit,6.1) holdswhen is sufficientlyarge.We note s well hat oth he ncentiveocheat nd he uturealue f o-operationncreasewithhe igid rice, . As long s 8 > n-1, however,he atterffectominates,o that he igidprice hat s easiest osupportasp(O)= r ineachperiod. sing 6.1),we find hat he riticaldiscount actor * above which irmsanuse theNashpunishmenthreatoenforceherigid-pricingcheme s (n - 1)7rr(r,_)" . (6.2)(n - 1)7r(r, ) + r(r,EO) - n NE (It is straightforwardoverifyhat * e (n , 1), ifnr(r,EO) > 7NE. Wemaynow state hefollowingroposition.

Proposition .(i) IfF is log-concave,henor ll discountactors ,Vs= 7rNE/(1 - 8).(ii) IfF is log-concavend8 < 8*,then here oes not xist nSPPE with igid ricing.(iii) IfF is log-concaverr - 0 is large nough,hen * e (n, 1) andfor ll 8 > 8*anyoptimalPPE ischaracterizedyrigid ricingtr inevery eriod.

Thus,whenF is log-concave, ash reversions in fact heworst unishment,ndtherigid-pricingcheme an be enforcedf ndonly f8 > 8*,where * e (n1, 1). To see anexample,supposethere re twofirms,osts are uniformlyistributedver 0, 1] and r = 1. Then,7rNE = 1/6, nd nSPPE with igid ricingxists nd soptimalf ndonlyf8 8*= 6/7.It s strikinghathe owest PPE continuationalue,Vs,correspondsoNashplaywhenis log-concave.his s true espite hefact hat PPE may xistnwhich omefirmypes ricebelow ost.Forexample,heremay xistnon-stationaryPPE, inwhich igher-costypes ricebelow ost nthe irsteriod,ustainedythepromisefa betteruturequilibrium.fcourse,SPPE continuationalues annot e drivenoo ow:the chememust fferhehighest-costypeoverall xpected ayoffsreaterhan8s (orelsethe irm illdeviate ff-schedule),nd ower-costtypes annot edeprivedfthe vailable fficiencyents.n searchingor he owest PPEcontinuationalue,wethus onsider ricingchemes hatminimizefficiencyents. ollowingthe ogic fSection .3,theminimumfficiencyentsattainedsing strictlyncreasingricingschemewhenF is log-concave,ndwith hiswe canestablish hat t s notpossible osustainpunishmentsorse hanNash.Aswe confirmnLemma in theAppendix,owever, hen helog-concavityssumptions relaxed, n-schedulencentiveonstraintsaybe compatible ithbelow-Nashfficiencyents.ffirmsresufficientlyatient,on-stationaryPPE with elow-costpricinganthen econstructedhat ield elow-Nashayoffs.19

19. Thescheme sed ogenerateelow-Nashayoffsequiresomefirmsoprice elow ost, nd uch firm ustbedissuaded romeviatingo higherrice.ndeed, firm ould ndertakeust uch deviation,f hemarket-clearingpricewere ublic ut ndividualriceswere therwiseot.nthis ase,theworstPPE involves he epeated lay f hestaticNashequilibriumfor nyF and8). When ndividualrices republic, owever, firman be induced opricebelow ost, ndwhen isnot og-concavehismaydescribe heworst PPE.



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338 REVIEW OF ECONOMIC STUDIESRecallthatwhen irmsave ccesstoa public andomizationevice, he etofequilibriumvalues sconvex. n that ase,Proposition provides completeharacterizationf he PPE setwhenF is log-concavend3 > 3*.An interestingmplicationf Proposition is that ntertemporalluctuationsn costsdiminishhe bilityf firmsocollude. ntuitively,hen ostsfluctuatehroughime, ollusionrequires reateratience,sthe chememustwithstandhe ncentivembalancehat ccurswhena firmraws lowcurrentost evel(0), and hus aces greatncentiveocheat,while ssessingthe ong-termalueofco-operation ith eferenceo an average ost evel EO). Formally,8* > n-n, where n is thecritical iscount actor or he tandard ertrandupergame,nwhich irms' osts retime-invariant.his mplications broadly onsistent ith he ommonassessmentsee,e.g.Scherer,980, .205)thatollusions more ifficulthen osts revariableacross irms.

6.2. Nowars nthe quilibriumathHow do firmsestcolludewhen hey re unable oenforce herigid-pricingcheme?n thissection,we take firsttep owardnsweringhis uestion. llowing or mpatientirms, eestablishhathe copefor ymmetricollusion annot e enhancedandmay estrictlyeduced)byequilibrium-pathars.Thecentraldea is simple. et us startwith n original PPE collusive cheme.Relyingon Proposition, ifthere s a positive robabilityfan equilibrium-patharassociatedwithsomecosttype, henwe canre-engineern alternativeollusive cheme--byliminatinghewarandreducinghepricefor hat ype correspondingmount-that ields or his ype hesameexpected ayoff. he alternativecheme atisfiesheon-scheduleonstraintgiven hatthe riginal id)and hus onstitutespayoff-equivalentPPE for atientirms. henfirmsreimpatient,owever,he ff-scheduleonstraintsalsoa concern,nd t shere hat he lternativeschedule ffersnactual dvantage:y shiftingrofitrom he urrenteriod price sreduced)to the uturewars re liminated),he ncentiveocheat s reducedwhile he uturealueofco-operations enhanced. heoff-scheduleonstraints thereforeoweasier osatisfyhan nderthe riginalcheme.As the ollowingropositiononfirms,his rgumentsquitegeneral:

Proposition . Allow or anydistributionunction and anydiscountactor . IfanSPPE existswith heoptimal ayoffVs, then here xists stationaryPPE, where he amepricingtrategys used ollowingvery quilibrium-pathistory,ithhe ptimal ayoff s.Thisargumenturtherndicateshat revenuequivalence" oesnot xtendo mpatientirms.A scheme hat sespricewarsmayviolate he ff-scheduleonstrainthen payoff-equivalentschemewithoutricewarsdoesnot.6.3. Partial igidityndcollusionmong mpatientirmsThepropositionsevelopedbove uggesthat ur earch or ollusivechemesmong mpatientfirms hould mphasizewo ngredients:he bsence frigid ricingnd no equilibrium-pathwars.Butexactly ow do impatientirmsrice n an optimal PPE? In this ection,we firstpresentufficientonditionsnderwhich two-step ricing cheme an be enforcednd isoptimal or mpatientirms.econd,we consider n extendedmodel that ncludespubliclyobserved luctuationsn industryemand. inally,we argue hat ptimal PPE for mpatient



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ATHEYETAL. COLLUSION AND PRICE RIGIDITY 339firmsscharacterizedy pricingchedule hats a step unctionpartial igidity),f he ollusiveschemes toofferetter-than-Nashrofits.

6.3.1. Introducing second step: an "Escape Clause". Recall that herigid-pricescheme ails o be enforceable hen3 < 8*,because a firm hat raws he owest-costypeis tootemptedoundercuthe igid rice and ncrease tsmarkethare.A naturalonjecturesthathis roblemmaybeovercome hen two-stepricingcheme s employed, ith rices landP2,wherepl < P2, nda break-point2. nthis ase,the owest-costirm as ess ncentivetocheat. irstly,his irm ow xpects reaterhan 1/n-thhare fthemarket,nd so thegaininmarkethare hat ccompanies price ut s diminished.econdly, nygiven ain nmarketshare snow essprofitable,ince he ower-costirmas a lower rice, nd hus he rofit-if-winitexperiencesnthemarkethare tenjoyss now ower.This,however,s not hewhole tory. alanced gainst hisdiminishedncentiveo cheatis a reductionnexpectedong-termrofit:fthedistributionunctions log-concave, two-step cheme ieldsower xpected rofithan oes a rigid cheme,nd so the irmlso nowhasless tolose inthefuturef tcheats oday. omplicating attersurther,henetresolutionfthese onflictingffects or heoff-schedulencentiveonstraint ayhinge pon henaturefthedistributionunction. two-stepchemewill atisfyhe ff-schedulencentiveonstraintfit owers he ncentivehat he owest-costirm as to cheatwithoutubstantiallylteringheexpected rofithat irmsnticipatenthe uture.ntuitively,hiswillbe the ase f hedensityssmall orower-costypes,o thathese ypes ccurnfrequentlynthe uture.Proposition . IfF is log-concavend

1ir(r,EO) - x7NEf(0) < n (6.3)(n - 1)jr(r, )7rNEthen herexists30 < 8*, uch hat,orevery 8 (8o,8*),therexistsn optimalPPE thatsstationarynduses a two-stepricingcheme, ith 2 = r > Pl and02e (0, 0).When F is log-concave,hetwo-pricecheme s optimal or justbelow8*,sincethen hetwo-pricecheme epartsromhe esired igid-pricingcheme nly t the owest-costypes.Proposition describes situationnwhich herealizationf an unlikelynd low costtype esultsna markedeductionnthefirm'srice, uggestinghat are utpronouncedricecutsmay ccur nderymmetricollusion chemeswhen irmsre mpatient.20notherwords,symmetricollusion mong mpatientirmsmaycall for n "escapeclause"provision,nderwhich firms allowed o select lowerprice nthe vent hat very avourableosttype srealized. heprice eduction ust esubstantial,norder oensure hat he owprices attractiveonlywhen firm'sost ypes low.Thisbehaviours reminiscentfthefindingsfRotembergnd Saloner 1986),but hereare mportantifferences:ntheirase,collusive rices djust cross ll firmsnresponseo apublicdemand hock. roposition alsomaybe usefulwhen nterpretingnapparentpisodeof"cheating"na collusivendustry.magine, or xample, situationn which single irmcharges lowprice ndyetfacesnoretaliation.t is difficulto reconcile uch nobservationwith tandardollusionmodels. n ourprivate-informationetting,owever,ptimal ollusionamong mpatientirms ay llowfor rare xceptions"origidity,n which firmubstantiallycuts tsprice ndfacesnoretaliation.

20. Theproofonstructstwo-stepricing quilibriumor02 closeto0. Inthis quilibrium,he ow-steprice sapproximatelyr+ 0(n - 1)]/n ndicatingdiscrete eductionf mountr- 0) (n- 1)/nfromhehigh-steprice fr.Thetwo-stepricingchemehus allsfor greaterrice eduction henmarketsre ess concentrated.



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340 REVIEW OF ECONOMIC STUDIESThe"small-densityondition"6.3)plays n ntuitiveole, ut he ssumptionsrestrictive.Obviously,t s satisfiedff(0) is closeenough o zero.To seea more ubtle xample,onsiderthe log-concave) istributionunctionamily (O) = 0a, with = 0 < 1 = 0. The small-densityonditions satisfied or ny > 1,but t failswhen < 1. The conditionlsofailswhen = 1 (correspondingo theuniformistribution).hen 6.3) is violated,tmaybe that,for ll8 < 86,notwo-step ricingchedule atisfiesn- ndoff-scheduleonstraints;hiss thecase for heuniformistributionseeAtheytal., 1998).A second xamplehighlightsn interestingrediction:f thesupportfthedistributionincreases,heoptimal ollusive chememay witch rom rigid-pricingoa two-step ricingscheme. onsider distribution(O; It, z), where hemean s constantt u,but he upportsparametrizedyz, so that = i - z and0 = it + z. SupposeF(0; ui, ) is log-concavendsatisfieshe mall-densityondition. nexamplesthe triangle"istribution,here hedensityf(0; z) is symmetricbout/t,andf(8; z) = 1(0 - (/ - z)) on [/t- z, z].21We make woobservations.irst, hile ncreasingleaves he er-periodrofitromigid ricing nchanged,it introducesower-costypes hat reespecially emptedo cheat ndtherebyncreases hecritical iscount actor orrigidpricing,*. Second, onsiderncreasing whileholding hediscount actorixed t n-i < 6 < 1.When is small,nformations approximatelyompleteandrigid ricinganbesupported.t critical,however,he igid-pricingcheme reaks own,andProposition implies hat two-pricechemes then ptimal. hus, ncreasedspread"nthecostdistributioneads to increased ricevariability.n an applicationf thisframework,Simon 2000) argues hat nflationan eadtoan ncreasen the pread f costs ndestablishesthat rices remore ariablewhen nflationshigh.6.3.2. Observable fluctuationsn demand. The intuitionnderlying ropositionsuggestshatny xogenous ariationnthe conomicnvironmenthat eightenshe hort-termincentiveocheat nd/or educes he ong-termalueofco-operation ayresultn lower nd

more ariable rices.A variationfparticularmpiricalelevanceccurswhenndustryemandfluctuatesver ime. ollowingRotembergnd Saloner 1986),we now extend urmodeltoanenvironmentnwhichndustryemand luctuatesnan i.i.d.fashion etween ow andhighstates, E {4L, 4H }, whereOH > OL. Profits proportionalo thedemand tate,which spubliclybserved tthebeginningfeachperiod, efore ost hocks rerealized.In thismodel, he ong-termalue ofco-operations proportionalo the demand hat sexpectednfutureeriods, 0, whichs ndependentf he urrentemandtate. ycontrast,heincentiveocheat sgreatest hen urrentemandshigh. heoff-scheduleonstrainthereforebinds irstor hehigh-demandtate. ormally, emaymodify6.1) to calculate or he igid-pricingcheme critical iscountactor,- r(n - 1)7r(r,_)4Hn - 1)7r(r,

_)PH+ [r(r, EO) - nxrNE]Ef'

where8 > 8*, atwhich heoff-scheduleonstraintindswhencurrent emand s high.Similarly,emaydefinen < 8*as the ritical iscount actoror he igid-pricingchementhe ow-demandtate.When hediscount actor alls lightlyelow H, it sno onger ossibleto enforce rigid rice or ll cost evels n thehigh-demandtate; owever,tremains ossibleto do so when hedemand tate s low.Assuminghat is not oo arge, o that > 6, amodificationfPropositionimplieshe ollowing:21. Note hatf(0_)andf(O)areequalto0,which trictlypeaking iolates urmaintainedssumption;ut t sstraightforwardo show hat ll of our esults xtend s long s f' (0) > 0 andf'(0) < 0.



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ATHEYETAL. COLLUSION AND PRICE RIGIDITY 341Proposition 0. Fortwo irms, hen hemarketize is i.i.d.with e {10L,Hn},fF islog-concavend 6.3) holds, hen here xists 0 E [8, 6*i),suchthat,orevery E (S3, *therexists noptimalPPE thatsstationaryndsatisfies:

(i) inthe owdemandtate 0 = jL), firms sethe igid-pricingcheme fProposition;(ii) inthehigh emandtate(0q H),firms sethe wo-pricecheme fProposition.Thispropositionxtends theme fthepreviousubsection:ymmetricollusion etweenimpatientirmsmaybe marked yoccasional andperhaps ubstantial)ricereductionsyindividual irms.We learnhere hat hesedeparturesre most ikely o occurwhenonefirmreceives favourableost hock ndcurrentemandshigh.One implicationf themodel s that hecountercyclical-pricingindingfRotembergandSaloner 1986) is robust o thepresence fprivateostfluctuations.hismodelcanbegeneralizedn a number f directions.or example, ollowing agwelland Staiger1997),wemay onsider nalternativetochasticrocess,nwhich hedemand rowthatefollowsMarkov rocess,o that ecessionsre haracterizedy lowgrowthndbooms re haracterizedbyfast rowth.n such model, ecessionsrethe imewhen ollusion smost ifficult.hus,givenhe rade

athey - collusion and price rigidity

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