1IntroductionSimpletextbookmodelsofcompetitivemarketsforhomogeneouspro - PDF document

1IntroductionSimpletextbookmodelsofcompe
1IntroductionSimpletextbookmodelsofcompetitivemarketsforhomogeneousproductssuggestthatall-outcompetitionamong…rmswillleadtotheso-called“lawofoneprice.”Yet,empiricalstudiesspanningmorethanfourdecades(seeTables1aand1b)revealthatpricedispersionistheruleratherthantheexceptioninmanyhomogeneousproductmarkets.Theobservationthatthepricesdi¤erent…rmschargeforthesameproductoftendi¤erby30percentormoreledHalVariantosuggestthat“the‘lawofoneprice’isnolawatall”(Varian,1980,p.651).Thischapterprovidesauni…edtreatmentofseveraltheoreticalmodelsthathavebeendevelopedtoexplainthepricedispersionobservedinhomogeneousproductmarkets,andsurveystheburgeoningempiricalliterature(includingthestudiessummarizedinTables1aand1b)whichdocumentsubiquitouspricedispersion.AkeymotivationforthischapteristodispeltheerroneousviewthattheInternet—throughitsfacilitationofdramaticdeclinesinconsumersearchcosts—willultimatelyleadtothe“lawofoneprice.”Whenconfrontedwithevidenceofpricedispersion,manyarequicktopointoutthateveninmarketsforseeminglyhomogeneousproducts,subtledi¤erencesamongthe“services”o¤eredbycompeting…rmsmightleadthemtochargedi¤erentpricesforthesameproduct.NobelLaureateGeorgeStigler’sinitialresponsetowagsmakingthispointwasphilosophical:“...[While]aportionoftheobserveddispersionispresumablyattributabletosuchdi¤erence[s]...itwouldbemetaphys-ical,andfruitless,toassertthatalldispersionisduetoheterogeneity”(Stigler,1961,p.215).Thirty-…veyearslater,theliteraturehasamassedconsiderablesupportforStigler’sposition.AsweshallseeinSections2and3,thereisstrongtheoreticalandempiricalevidencethatmuch(andinsomemarkets,most)oftheobserveddispersionstemsfrominformationcosts—consumers’costsofacquiringinformationabout…rms,and/or…rms’costsoftransmittinginformationtoconsumers.AsFigure1reveals,researchoninformation,search,andpricedispersionhasbecomeincreas-inglyimportantsincethepublicationofStigler’sseminalarticleontheEconomicsofInformation.Untilabout1998,moststudiesfocusedonenvironmentswhereconsumersincurapositivecostofobtainingeachadditionalpricequote.Searchcostsinthesestudiesconsistofconsumers’oppor-tunitycostoftimeinsearchingforlowerprices(so-called“shoe-leather”costs),plusothercostsassociatedwithobtainingpricequotesfromcompeting…rms(suchastheincrementalcostofthepostagestampsorphonecallsusedinacquiringpriceinformationfrom…rms).Consumersintheseenvironmentsweighthecostofobtaininganadditionalpricequoteagainsttheexpectedbene…tsofsearchinganadditional…rm.AswediscussinSection2.1,equilibriumpricedispersioncanarisein2Asecondapproachdeemphasizesthemarginalsearchcostasasourceforpricedispersion.Instead,consumersaccesspriceinformationbyconsultingan“infor

mationclearinghouse”(e.g.,anewspape
mationclearinghouse”(e.g.,anewspaperoranInternetpricecomparisonsite);e.g.SalopandStiglitz(1977),Shilony(1977),Rosenthal(1980),Varian(1980),Narasimhan(1988),Spulber(1995),BayeandMorgan(2001),Baye,Morgan,andScholten(2004a).1Thedistinguishingfeatureof“clearinghousemodels”isthatasubsetofconsumersgainaccesstoalistofpriceschargedbyall…rmsandpurchaseatthelowestlistedprice.Intheearliestofthesemodels,equilibriumpricedispersionstemsfromexanteheterogeneitiesinconsumersor…rms.Forexample,intheVarianandSalop-Stiglitzmodels,someconsumerschoosetoaccesstheclearinghousetoobtainpriceinformation,whileothersdonot.InShilony,Rosenthal,andNarasimhan,someconsumersareloyaltoaparticular…rm(andthuswillbuyfromitevenifitdoesnotchargethelowestprice),whileotherconsumersare“shoppers”andonlypurchasefromthe…rmchargingthelowestprice.Spulber(1995)showsthatequilibriumpricedispersionarisesevenwhenallconsumerscancostlesslyaccesstheclearinghouse—providedeach…rmisprivatelyinformedaboutitsmarginalcost.BayeandMorgan(2001)o¤eraclearinghousemodelthatendogenizesnotonlythedecisionsof…rmsandconsumerstoutilizetheinformationclearinghouse(inthepreviousclearinghousemodels,…rms’listingdecisionsareexogenous),butalsothefeeschargedbytheowneroftheclearinghouse(the“informationgatekeeper”)toconsumersand…rmswhowishtoaccessortransmitpriceinformation.Theyshowthatadispersedpriceequilibriumexistsevenintheabsenceofanyexanteheterogeneitiesinconsumersor…rms.Inthissection,weprovideanoverviewofthekeyfeaturesandideasunderlyingtheseliteratures.2.1Search-TheoreticModelsofPriceDispersionWebeginwithanoverviewofsearch-theoreticapproachestoequilibriumpricedispersion.Theearlyliteraturestressestheideathat,whenconsumerssearchforpriceinformationandsearchiscostly,…rmswillchargedi¤erentpricesinthemarket.Therearetwobasicsortsofmodelsused:Modelswith…xedsamplesizesearchandmodelswheresearchissequential.Wewilldiscusseachoftheseinturn.1Athirdapproachdeemphasizesconsumersearchandmainlyfocusesonwhetherpricedispersioncanarisewhenconsumers“passively”obtainpriceinformationdirectlyfrom…rms(asindirectmailadvertisements);cf.Butters(1977),GrossmanandShapiro(1984),Stegeman(1991),RobertandStahl(1993),McAfee(1994),andStahl(1994).Arelatedmarketingliteratureexaminessimilarissues,rangingfromloyaltyandpricepromotionstrategiestochannelcon‡ictsandtheInternet;seeLalandVillas-Boas(1998),LalandSarvary(1999),Raju,Srinivasan,andLal(1990),andRao,ArjunjiandMurthi(1995).4the…rmo¤eringthelowestprice;and3.Thedistributionof…rms’pricesisgivenbyanexogenousnon-degeneratecdfF(p)onp;p.Stiglerassumesthataconsumerchoosesa…xedsamplesize,n,tominimizetheexpectedtotalcost(expectedpurchasecostplussearchcost)ofpurchasingKunitsoftheproduct

:E[C]=KEhp(n)mini+cnwhereEhp(n)mini=E[mi
:E[C]=KEhp(n)mini+cnwhereEhp(n)mini=E[minfp1;p2;:::;png];thatis,theexpectedlowestpricequoteobtainedfromndrawsfromF:SincethedistributionofthelowestofndrawsisF(n)min(p)=1�[1�F(p)]n;E[C]=KZpppdF(n)min(p)+cn=K"p+Zpp[1�F(p)]ndp#+cnwherethesecondequalityobtainsfromintegrationbyparts.Noticethattheterminsquarebracketsre‡ectstheexpectedpurchaseprice,whichisadecreasingfunctionofthesamplesize,n.However,sinceeachadditionalpriceobservationcostsc�0toobtain,anoptimizingconsumerwillchoosetosearcha…nitenumberoftimes,n,andthuswillgenerallystopshortofobtainingthebestprice�p
inthemarket.ThedistributionoftransactionpricesisthedistributionofthelowestofndrawsfromF;thatis,F(n)min(p)=1�(1�F(p))nFromthis,Stiglerconcludesthatdispersioninbothpostedpricesandtransactionspricesarisesasaconsequenceofcostlysearch.Howdotransactionspricesandsearchintensityrelatetothequantityoftheitembeingpur-chased(orequivalently,tothefrequencyofpurchases)?4Stigler’smodelo¤erssharppredictionsinthisdimension.Notethattheexpectedbene…ttoaconsumerwhoincreaseshersamplesizefromn�1tonisEhB(n)i=Ehp(n�1)mini�Ehp(n)miniK;(1)4Kmayberelatedtopurchasefrequencyasfollows.Supposepricesare“valid”forTperiods,andtheconsumerwishestobuyoneuniteverytTperiods;thatis,trepresentsaconsumer’spurchasefrequency.ThenthetotalnumberofunitspurchasedduringtheTperiodsisKT=t:Thus,anincreaseinpurchasefrequency(t)isformallyequivalenttoanincreaseinKinthemodelabove.6SinceGisameanpreservingspreadofF;thereexistsauniqueinteriorpointu=F(EF[P])suchthatF�1(u)=G�1(u):Further,foralluu;F�1(u)�G�1(u)&#x-427;0andforallu&#x-427;u;F�1(u)�G�1(u)0:Thus
=n Zu0�F�1(u)�G�1(u)
(1�u)n�1du+Z1u�F�1(u)�G�1(u)
(1�u)n�1du!Next,noticethat(1�u)n�1isstrictlydecreasinginu;hence
&#x]TJ/;༕ ;.9; T; 11;&#x.515;&#x 0 T; [00;n Zu0�F�1(u)�G�1(u)
(1�u)n�1du+Z1u�F�1(u)�G�1(u)
(1�u)n�1du!=n(1�u)n�1Z10�F�1(u)�G�1(u)
du=0wherethelastequalityfollowsfromthefactthatFandGhavethesamemean.Proposition2SupposethatanoptimizingconsumerobtainsmorethanonepricequotewhenpricesaredistributedaccordingtoF,andthatpricedistributionGisameanpreservingspreadofF.Thentheconsumer’sexpectedtotalcostsunderGarestrictlylessthanthoseunderF:Proof.Supposethat,underF;theoptimalnumberofsearchesisn:Thentheconsumer’sexpectedtotalcostunderFisE[CF]=EFhp(n)miniK�cn�EGhp(n)miniK�cnE[CG]wherethestrictinequalityfollowsfromProposition1,andtheweakinequalityfollowsfromthefactthatnsearchesmaynotbeoptimalunderthedistributionG:At…rstblush,itmightse

emsurprisingthatconsumersengagedin…
emsurprisingthatconsumersengagedin…xedsamplesearchpayloweraveragepricesandhavelowerexpectedtotalcostsinenvironmentswherepricesaremoredispersed.Theintuition,however,isclear:Inenvironmentswherepricesaremoredispersed,theprospectsforpriceimprovementfromsearcharehigherbecausethelefttailofthepricedistribution—thepartofthedistributionwhere“bargains”aretobefound—becomesthickeraspricesbecomemoredispersed.84.Aconsumerwhoischargedthemonopolypriceearnssurplussu¢cienttocoverthecostofobtainingasinglepricequote;thatisv(p)�c:Inthisenvironment,all…rmspostthemonopolypriceandconsumersvisitonlyonestore,purchaseatthepostedpricep;andobtainsurplusv(p)�c�0.Giventhestoppingruleofconsumers,each…rm’sbestresponseistochargethemonopolyprice;giventhatall…rmschargep,itisoptimalforeachconsumertosearchonlyonce.Toseethatthisistheuniqueequilibriuminundominatedstrategies,supposetothecontrarythatthereisanequilibriuminwhichsome…rmpostedapricebelowthemonopolyprice(clearly,pricingabovethemonopolypriceisadominatedstrategy).Letp0bethelowestsuchpostedprice.A…rmpostingthelowestpricecouldpro…tablydeviatebyraisingitspricetothelowerofporp0+c:Anyconsumervisitingthat…rmwouldstillrationallybuyfromitsincethemarginalbene…tofanadditionalsearchissmallerthanc—themarginalcostofanadditionalsearch.Thus,sucha…rmwillnotloseanycustomersbythisstrategyandwillraiseitsearningsoneachofthesecustomers.TheDiamondparadoxisstriking:eventhoughthereisacontinuumofidentical…rmscompetinginthemodel—atextbookconditionforperfectcompetition—inthepresenceofanysearchfrictionswhatsoeverthemonopolypriceistheequilibrium.Rothschild’scriticismoftheStiglermodel,alongwiththeDiamondparadox,spawnedseveraldecadesofresearchintowhethercostlysearchcouldpossiblygenerateequilibriumpricedispersion—asituationwhereconsumersareoptimallygatheringinformationgivenadistributionofprices,andwherethedistributionofpricesoverwhichconsumersaresearchingisgeneratedbyoptimal(pro…t-maximizing)decisionsof…rms.2.1.3TheReinganumModelandOptimalSequentialSearchReinganum(1979)wasamongthe…rsttoshowthatequilibriumpricedispersioncanariseinasequentialsearchsettingwithoptimizingconsumersand…rms.Reinganum’sresultmaybeseeninthefollowingspecialcaseofourenvironmentwhere:1.Consumershaveidenticaldemandsgivenby�v0(p)=q(p)=Kp",where"�1andK&#x]TJ/;༣ ;.9; T; 19;&#x.633;&#x 0 T; [00;0;2.Consumersengageinoptimalsequentialsearch;3.FirmshaveheterogeneousmarginalcostsdescribedbytheatomlessdistributionG(m)on[m;m];10Case3.h(p)0:Thentheconsumer’soptimalstrategyistosearchuntilsheobtainsapricequoteatorbelowthereservationprice,r;wherersolvesh(r)=Zrp(v(p)�v(r))dF(p)�c=0(4)Equation(4)representsa

priceatwhichaconsumerisexactlyindi¤eren
priceatwhichaconsumerisexactlyindi¤erentbetweenbuyingandmakinganadditionalsearch.Toseethatsuchapriceisuniquelyde…nedbythisequation,noticethath�p
=�c0,h(p)0,andh0(z)=B0(z)�0:Aconsumerwhoobservesapricethatexceedsrwilloptimally“reject”thatpriceinfavorofcontinuedsearch,whileaconsumerwhoobservesapricebelowrwilloptimally“accept”thatpriceandstopsearching.Case1isclearlynoteconomicallyinterestingasitleadstotheabsenceofanymarketfortheproductinthe…rstplace.Case2ariseswhentheexpectedutilityofpurchasingtheproductexceedsthecostofaninitialsearch,butthedistributionofpricesissu¢ciently“tight”relativetosearchcoststomakeadditionalsearchessuboptimal.Mostoftheexistingsearchliterature,includingReinganum,restrictsattentiontoCase3,asweshalldohereafter.Thereservationpricede…nedinequation(4)hasseveralinterestingcomparativestaticprop-erties.Totallydi¤erentiatingequation(4)withrespecttorandc;andusingequation(3)revealsthatdrdc=1q(r)F(r)=1Kr"F(r)�0Thus,anincreaseinsearchcostsleadstoahigherreservationprice:Otherthingsequal,therangeof“acceptable”pricesisgreaterforproductswithhighersearchcosts.Notethat,forthespecialcasewhenq(r)=1;dr=dc=1=F(r)�1:Inthiscase,aoneunitincreaseinsearchcostsincreasestherangeofacceptablepricesbymorethanoneunit—thatis,thereisa“magni…catione¤ect”ofincreasesinsearchcosts.6ReinganumavoidsRothschild’scriticismandthe“Diamondparadox”byintroducing…rmcostheterogeneities.Sinceeach…rmjdi¤ersinitsmarginalcosts,mj;priceswilldi¤eracross…rmsevenwhentheypriceasmonopolists.Supposethatafraction01of…rmspriceaboverandrecallthatthereareconsumersper…rm.Arepresentative…rm’sexpectedpro…twhenitpricesatpjis:Ej=8:(pj�mj)q(pj)1�ifpjr0ifpj�r6Ingeneral,theremaybeeitheramagni…cationoranattenuatione¤ectofaoneunitincreaseinthecostofsearch.12wherethelastequalityfollowsfromthefactthatristheoptimalreservationpricewhenconsumersfacethepricedistribution^F:Inshort,Reinganum’sassumptionsofdownwardslopingdemandandcostheterogeneitygiverisetoanequilibriumofpricedispersionwithoptimizingconsumersand…rms.Notethatdownwardslopingdemandandcostheterogeneitiestogetherplayacriticalroleingeneratingequilibriumpricedispersioninthisenvironment.Toseethatbothassumptionsarerequired,suppose…rstthatcostsareheterogeneousbutthateachconsumerwishedtopurchaseoneunitoftheproduct,valuedatv.Inthiscase,givenareservationpriceofrv,all…rmswould…nditoptimaltopriceatr;andthedistributionofpriceswouldbedegenerate.Ofcourse,areservationpriceofrvisinconsistentwithoptimizingbehavioronthepartofconsumers.Toseethis,supposethataconsumerwasunexpectedlypresentedwithapricep0=r+;where

c:Accordingtothesearchstrategy,such
c:Accordingtothesearchstrategy,suchaconsumerissupposedtorejectthispriceandcontinuesearching;however,thebene…tfromthisadditionalsearchislessthanthecost.Thus,aconsumershouldoptimallyacceptapricep0ratherthancontinuingtosearch.Theupshotofthisisthattheonlyequilibriumreservationpriceisr=v:However,thesearepreciselytheconditionsgiveninCase1;hencetheonlyequilibriumiswherenoconsumersshopatall.8Ifdemandweredownwardslopingbut…rmshadidenticalmarginalcostsofm,each…rmwouldhaveanincentivetosetthesameprice,p=minfr;m"=(1+")g;giventhereservationprice.ThisleadsbacktoCase2andoneobtainstheDiamondparadox:All…rmschargethemonopolyprice,p=m"=(1+").Indeed,intheenvironmentabove,alimitingcasewherethedistributionofmarginalcostsconvergestoapointisexactlytheDiamondmodel.Finally,weexaminehowthevarianceinthedistributionofposted(andtransactions)pricesvarieswithsearchcosts.Notethat,inequilibrium,thevarianceinpricesisgivenby2=Ep2�(E[p])2=Zrpp2dF(p)� ZrppdF(p)!2=Zrpp2^f(p)dp+1�^F(r)r2� Zrpp^f(p)dp+1�^F(r)r!28CarlsonandMcAfee(1983)showthatifoneintroducesheterogeneitiesinconsumersearchcosts,adispersedpriceequilibriummayexistprovidedthatindividualconsumershaveperfectlyinelastic(incontrasttodownwardsloping)demand.142.1.5TheMacMinnModelInlightofthefactthatthereareinstancesinwhich…xedsamplesizesearchisoptimal,onemaywonderwhetherequilibriumpricedispersioncanariseinsuchasetting.MacMinn(1980)providesana¢rmativeanswertothisquestion.MacMinn’sresultmaybeseeninthefollowingspecialcaseofourenvironmentwhere:1.Consumershaveunitdemandwithvaluationv;2.Consumersengageinoptimal…xedsamplesearch;and93.FirmshaveprivatelyobservedmarginalcostsdescribedbytheatomlessdistributionG(m)on[m;m],wheremv:Atthetime,MacMinnderivedequilibriumpricingbysolvingasetofdi¤erentialequationsunderthespecialcasewhereGisuniformlydistributed.However,subsequenttohispaper,akey…ndingofauctiontheory,theRevenueEquivalenceTheorem(Myerson,1981)wasdeveloped.10Usingtherevenueequivalencetheorem,wecangeneralizeMacMinn’sresultstoarbitrarycostdistributions.Toseethis,noticethatwhenconsumersoptimallyengageina…xedsamplesearchconsistingofn…rms,each…rme¤ectivelycompeteswithn�1other…rmstoselloneunitoftheproduct.Ofthesen…rms,the…rmpostingthelowestpricewinsthe“auction”.Usingtherevenueequivalencetheorem,onecanshowthattheexpectedrevenuestoa…rmwithmarginalcostminany“auction”wherethe…rmchargingthelowestpricealwayswinsandthe…rmwiththehighestmarginalcostearnszerosurplusisR(m)=m(1�G(m))n�1+Zmm(1�G(t))n�1dt(5)IntheMacMinnmodel,expectedrevenuesaresimplya…rm’spostedprice,p(m),multipliedbytheprobabilityitchargesthelowestprice,which,inequilibrium,is(1�G

(m))n�1:UsingthefactthatR(m)=p(m)
(m))n�1:UsingthefactthatR(m)=p(m)(1�G(m))n�1,substitutingintoequation(5);andsolvingforp(m)yieldstheequilibriumpricingstrategyofa…rmwithmarginalcostmwhenconsumerssamplen…rms:p(m)=m+Zmm1�G(t)1�G(m)n�1dt(6)9MacMinnalsoprovidesaversionofthemodelthatisvalidforoptimalsequentialsearch.10SeeKlemperer(1999)foranon-technicalsurveyofauctiontheoryincludingtherevenueequivalencetheorem.McAfeeandMcMillan(1988)establishesanequivalencebetweensearchandauctionsinamechanismdesigncontext.16notethatasthesamplesizeincreases,thevarianceinequilibriumpricesincreases.Thisimpliesthat,takingintoaccounttheinteractionbetweenconsumersand…rmsinthis…xed-samplesizesearchmodel,dispersionvariesinverselywithsearchcosts.Conclusion2IntheMacMinnmodel,areductioninsearchcostsincreasesthevarianceofequi-libriumprices.ThisconclusionisincontrasttoConclusion1,wherepreciselytheoppositeimplicationisobtainedintheReinganumsequentialsearchmodel.Thishighlightsanimportantfeatureofsearch-theoreticmodelsofpricedispersion:Dependingonthemodel,areductioninsearchcostsmaybeassociatedwithhigherorlowerlevelsofpricedispersion.IntheReinganummodel,areductioninsearchcostsreducesthereservationpriceofconsumersandthusinducesmarginal“high-cost”…rmstoreducetheirpricesfromtheirmonopolypricetothereservationprice.Sincethemonopolypricesoflow-cost…rmsarebelowthereservationprice,theirpricesremainunchanged;lowersearchcoststhusreducetherangeofprices.IntheMacMinnmodel,lowersearchcostsinduceconsumerstosamplemore…rmsbeforepurchasing—ine¤ect,each…rmcompeteswithmorerivals.Asaconsequence,theoptimalamountof“bidshading”(pricingabovemarginalcost)isreduced,thusincreasingthelevelofpricedispersion.2.1.6TheBurdettandJuddModelBurdettandJudd(1983)werethe…rsttoshowthatequilibriumpricedispersioncanariseinasearch-theoreticmodelwithexanteidenticalconsumersand…rms.11BurdettandJudd’smainresultmaybeseeninthefollowingspecialcaseofourenvironmentwhere:1.Consumershaveunitdemanduptoapricev;2.Consumersengageinoptimal…xedsamplesearch;123.Each…rmhasconstantmarginalcost,m;andwouldoptimallychargeallconsumerstheuniquemonopolyprice,p=v;and4.Aconsumerwhoischargedthemonopolypriceearnssurplussu¢cienttocoverthecostof11JanssenandMoraga-González(2004)provideanoligopolisticversionoftheBurdettandJuddmodel.12BurdettandJuddalsoprovideaversionofthemodelthatisvalidunderoptimalsequentialsearch.18followsthataconsumermustbeindi¤erentbetweenobtainingonepricequoteandobtainingtwopricequotes.Thatis,inanydispersedpriceequilibriumEhB(1)i�EhB(2)i=c�EhB(3)i�:::�EhB(n)i:Thus,inanydispersedpriceequilibrium,1;2�0whilei=0foralli�2:Let1=and2=1�:Wearenowinapositiontocharacterizea

natomlessdispersedpriceequilibrium.First
natomlessdispersedpriceequilibrium.First,notethatsince2(0;1);thereisapositiveprobabilitythata…rmfacesnocompetitionwhenitsetsitsprice.Thus,if…rmichargesthemonopolyprice,itearnsexpectedpro…tsofE[ijpi=v]=(v�m)Incontrast,a…rmchoosingsomelowerprice“wins”whenitspriceisbelowthatoftheother…rmaconsumerhassampled.Thus,if…rmichargesapricepiv;itearnsexpectedpro…tsofE[ijpiv]=(pi�m)(+(1�)(1�F(pi)))Thus,foragivendistributionofsearches,equilibriumpricedispersionrequiresthatthedistributionof…rmprices,F(
);satis…es+(1�)(1�F(p))=(v�m)(p�m)orF(p)=1�(v�p)(p�m)1�(10)whichisawell-behavedatomlesscumulativedistributionhavingsupport[m+(v�m);v].Finally,itremainstodetermineanequilibriumvalueof:Sinceeachconsumermustbeindif-ferentbetweensearchingoneortwo…rms,EhB(2)i=cNoticethat,when=0or=1;EB(2)=0whileEB(2)�0forall2(0;1):BurdettandJuddshowthatEB(2)isquasi-concave;thus,whencissu¢cientlylow,therearegenericallytwodispersedpriceequilibria—oneinvolvingarelativelyhighfractionofconsumersmakingtwosearches,theotherwitharelativelylowfractionofconsumers.1414Thereisanon-dispersedpriceequilibriumwhereallconsumerssearchonceandall…rmschargethemonopolyprice.20withamaximalwillingnesstopayofv�m:15Ofthese,amass,S�0,oftheconsumersareprice-sensitive“shoppers.”Theseconsumers…rstconsulttheclearinghouseandbuyatthelowestpricelistedthereprovidedthispricedoesnotexceedv.Ifnopricesareadvertisedattheclearinghouseoralllistedpricesexceedv,thena“shopper”visitsoneofthe…rmsatrandomandpurchasesifitspricedoesnotexceedv.AmassL0ofconsumersper…rmpurchasefromthat…rmifitspricedoesnotexceedv.Otherwise,theydonotbuytheproductatall.ItcanbeshownthatifL�0or�0,equilibriumpricedispersionarisesinthegeneralmodel—providedofcoursethatisnotsolargethat…rmsrefusetolistpricesattheclearinghouse.Moreprecisely,Proposition3Let0n�1n(v�m)S.Then,inasymmetricequilibriumofthegeneralclearinghousemodel:1.Each…rmlistsitspriceattheclearinghousewithprobability=1�nn�1(v�m)S1n�1:2.Ifa…rmlistsitspriceattheclearinghouse,itchargesapricedrawnfromthedistributionF(p)=10@1�nn�1+(v�p)L(p�m)S1n�11Aon[p0;v];wherep0=m+(v�m)LL+S+nn�1L+S:3.Ifa…rmdoesnotlistitspriceattheclearinghouse,itchargesapriceequaltov:4.Each…rmearnsequilibriumexpectedpro…tsequaltoE=(v�m)L+1n�1Proof.First,observethatifa…rmdoesnotlistitspriceattheclearinghouse,itisadominantstrategytochargeapriceofv:Next,noticethat&#

11;2(0;1]whenevern(n�1)(v�m)
11;2(0;1]whenevern(n�1)(v�m)S1:15BayeandMorgan(2001)consideranenvironmentwithdownwardslopingdemand.221.Itiscostlessfor…rmstolistpricesontheclearinghouse:=0and;2.Each…rmhasapositivemassofloyalconsumers:L�0:Since=0;itfollowsfromProposition3that=1;thatis,allofthen…rmsadvertisetheirpriceswithprobabilityone.UsingthisfactandProposition3,theequilibriumdistributionofpricesisF(p)=1�(v�p)(p�m)LS1n�1on[p0;v](11)wherep0=m+(v�m)LL+SThepricedispersionarisingintheRosenthalmodelstemsfromexogenousdi¤erencesintheprefer-encesofconsumers.Whileshoppersviewallproductsasidenticalandpurchaseatthelowestlistedprice,each…rmisendowedwithastockofLloyals.Theequilibriumpricedispersionarisesoutofthetensioncreatedbythesetwotypesofconsumers.Firmswishtochargevtoextractmaximalpro…tsfromtheloyalsegment,butifall…rmsdidsoa…rmcouldslightlyundercutthispriceandgainalloftheshoppers.Onemightimaginethatthis“undercutting”argumentwouldleadtotheBertrandoutcome.However,oncepricesgetsu¢cientlylow,a…rmisbettero¤simplychargingvandgivinguponattractingshoppers.Thus,theonlyequilibriumisinmixedstrategies—…rmsrandomizetheirprices,sometimespricingrelativelylowtoattractshoppersandothertimespricingfairlyhightomaintainmarginsonloyals.Itisinterestingtoexaminetheequilibriumtransactionspricesinthemarket.Loyalcustomersexpecttopaytheaveragepricechargedby…rms:E[p]=Zvp0pdF(p)whileshoppersexpecttopaythelowestofndrawsfromF(p);thatis,theexpectedtransactionpricepaidbyshoppersisEhp(n)mini=Zvp0pdF(n)min(p)whereF(n)min(p)isthecdfassociatedwiththelowestofndrawsfromF:Howdotransactionspricesvarywiththenumberofcompeting…rms?Rosenthal’sstrikingresultisthat,asthenumberofcompeting…rmsincreases,theexpectedtransactionspricespaidbyallconsumersgoup.Asweshallseebelow,theresulthingesonRosenthal’sassumptionthatentry242.Thetotalmeasureof“uninformed”consumerslackingaccesstotheclearinghouseisU�0;hence,each…rmisvisitedbyL=Unoftheseconsumers.Again,since=0;itfollowsthat=1andhencealln…rmsadvertisetheirpricesattheclearinghouse:UsingthisfactandsettingL=U=ninProposition3,theequilibriumdistributionofpricesisF(p)=1� (v�p)(p�m)UnS!1n�1on[p0;v]wherep0=m+(v�m)UnUn+SThefactthatthisatomlessdistributionofpricesexistswheneverthereisanexogenousfractionofconsumerswhodonotutilizetheclearinghouseraisestheobviousquestion:Canthisequilibriumpersistwhenconsumersaremakingoptimaldecisions?Varianshowsthattheanswertothisquestionisyes—provideddi¤erentconsumershavedi¤erentcostsofaccessingtheclearinghouse.Theeasiestwaytoseethisistonotethatthevalueofinformationprovidedbytheclearinghouseisthedi¤erenceintheexpectedpricepaidbythoseaccessingtheclearinghouse,Ehp(n)mini;a

ndthosenot,E[p];thatis;VOI(n)=E[p]�Eh
ndthosenot,E[p];thatis;VOI(n)=E[p]�Ehp(n)mini(12)whereVOIdenotesthevalueof(price)informationcontainedattheclearinghouse.Supposeconsumersfaceacostofaccessingtheinformationprovidedbytheclearinghouse.Notethatthiscostisessentiallya…xedcostofgainingaccesstotheentirelistofprices,notaperpricecostasinthesearch-theoreticmodelsconsideredabove.VarianassumesthatthecosttotypeSandLconsumersofaccessingtheclearinghouseisSandL,withSL.ThenprovidedSVOI(n)LtypeSconsumerswilloptimallyutilizetheclearinghousewhilethetypeLconsumerswillnot.Inshort,ifdi¤erentconsumershavedi¤erentcostsofaccessingtheclearinghouse,thereexistsanequilibriumofpricedispersionwithoptimizingconsumersand…rms.Insuchanequilibrium,informedconsumerspayloweraveragepricesthanuninformedconsumers.Itisimportanttoemphasizethat,whenoneendogenizesconsumers’decisionstobecomein-formedintheVarianmodel,thelevelofpricedispersionisnotamonotonicfunctionofconsumers’informationcosts.Wheninformationcostsaresu¢cientlyhigh,noconsumerschoosetobecomeinformed,andall…rmschargethe“monopolyprice,”v.Whenconsumers’informationcostsarezero,allconsumerschoosetobecomeinformed,andall…rmspriceatmarginalcostinasymmetric26advertisetheirpricesandforconsumerstogainaccesstothelistofpricespostedattheclear-inghouse.Forexample,newspaperscharge…rmsfeestoadvertisetheirpricesandmaychoosetochargeconsumerssubscriptionfeestoaccessanypostedinformation.Thesameistrueofmanyonlineenvironments.Moreover,theclearinghouseisitselfaneconomicagent,andpresumablyhasanincentivetoendogenouslychooseadvertisingandsubscriptionfeestomaximizeitsownexpectedpro…ts.Thus,BayeandMorganexaminetheexistenceofdispersedpriceequilibriainanenviron-mentwithoptimizingconsumers,…rms,andamonopoly“gatekeeper”whocontrolsaccesstotheclearinghouse.Speci…cally,BayeandMorganconsiderahomogeneousproductenvironmentwherenidentical,butgeographicallydistinct,marketsareeachservedbya(single)local…rm.Distanceorothertransactioncostscreatebarrierssu¢cienttoprecludeconsumersinonemarketfrombuyingthisproductinanothermarket;thuseach…rminalocalmarketisamonopolist.Nowimaginethatanentrepreneurcreatesaclearinghousetoserveallmarkets.IntheInternetage,onecanviewtheclearinghouseasavirtualmarketplace–throughitscreation,thegatekeeperexpandsbothconsumers’and…rms’opportunitiesforcommerce.Eachlocal…rmnowhastheoptiontopaythegatekeeperanamounttopostapriceontheclearinghouseinordertogainaccesstogeographicallydisparateconsumers.Eachconsumernowhastheoptiontopaythegatekeeperanamounttoshopattheclearinghouseandtherebypurchasefrom…rmsoutsidethelocalmarket.Themonopolygatekeeper…rstsetsandtomaximizeitsownexpectedpro…ts.Giventhesefees,pro…tmaximizing…rms

makepricingdecisionsanddeterminewhethero
makepricingdecisionsanddeterminewhetherornottoadvertisethemattheclearinghouse.Similarly,consumersoptimallydecidewhethertopaytoaccesstheclearinghouse.Followingthis,aconsumercansimplyclickhermousetoresearchpricesattheclearinghouse(ifsheisasubscriber),visitthelocal…rm,orboth.Withthisinformationinhand,aconsumerdecideswhetherandfromwhomtopurchasethegood.BayeandMorganshowthatthegatekeepermaximizesitsexpectedpro…tsbysettingsu¢-cientlylowthatallconsumerssubscribe,andcharging…rmsstrictlypositivefeestoadvertisetheirprices.Thus,BayeandMorgan’smainresultsmaybeseeninthefollowingspecialcaseofthegeneralclearinghousemodel:1.Thegatekeeperoptimallysetspositiveadvertisingfees:�0and;2.Thegatekeeperoptimallysetssubscriptionfeessu¢cientlylowsuchthatallconsumersaccesstheclearinghouse;thatis,L=0:28Whydoesthegatekeeper…nditoptimaltosetlow(possiblyzero)feesforconsumerswishingtoaccessinformation,butstrictlypositivefeesto…rmswhowishtotransmitpriceinformation?BayeandMorganpointoutthatthisresultstemsfroma“freerider”problemontheconsumersideofthemarketthatisnotpresentonthe…rmside.Recallthatthegatekeepercanonlyextractrentsequaltothevalueoftheoutsideoptionof…rmsandconsumers.Foreachsideofthemarket,theoutsideoptionconsistsofthesurplusobtainablebynotutilizingtheclearinghouse.Asmoreconsumersaccessthesite,thenumberofconsumersstillshoppinglocallydwindlesandtheoutsideoptionfor…rmsiseroded.Incontrast,asmore…rmsutilizetheclearinghouse,vigorouspricecompetitionamongthese…rmsreduceslistedpricesandleadstoamorevaluableoutsideoptiontoconsumersnotusingtheclearinghouse.Thus,tomaximizepro…ts,thegatekeeperoptimallysubsidizesconsumerstoovercomethis“freeriderproblem”whilecapturingrentsfromthe…rmsideofthemarket.Noanalogous“freeriderproblem”arisesonthe…rmside;indeedgreaterconsumerparticipationattheclearinghouseincreasesthefrequencywithwhich…rmsparticipate(increases)andhencepermitsgreaterrentextractionfrom…rms.2.2.4ModelswithAsymmetricConsumersIngeneral,littleisknownaboutthegeneralclearinghousemodelwithasymmetricconsumers.17However,forthespecialcaseoftwo…rms,resultsareavailable.Hereweshowhowonecanadaptthegeneralclearinghousemodeltoaccountforasymmetriesinduopolymarkets.Supposetherearetwo…rms(i=1;2)competinginthemarket.AmassL1ofcustomersareloyalto…rm1whileL2customersareloyalto…rm2whereL1L2.Proposition4Let012(v�m)S.Then,inanasymmetricdispersedpriceequilibrium:1.Each…rmlistsitspriceattheclearinghousewithprobability=1�2S(v�m):2.Ifa…rmlistsitspriceattheclearinghouse,itchargesapricedrawnfromthedistributionFi(p)=11�2+(v�p)Lj(p�m)Son[p0;1;v];wherep0;1=m+(v�m)L1L1+S+2L1+S:17F

orspeci…cclearinghousemodels,somere
orspeci…cclearinghousemodels,someresultsareavailable.Forinstance,Baye,Kovenock,anddeVries(1992)characterizeallequilibriainaversionoftheVarianmodelinwhich…rmshaveasymmetricnumbersofconsumers.30forintoequation(13)andsolvingforFj;oneobtainstheexpressionforthedistributionsofadvertisedpricesgivenintheproposition.Itisstraightforwardtoverifythatthisisawell-de…nedcdfon[p0;1;v];andthatneither…rmcangainbychargingapricepi=2[p0;1;v]:ThereareseveralnoteworthyfeaturesoftheequilibriumpricingandadvertisingstrategiesgiveninProposition4.First,when=0;both…rmsadvertisepricesontheclearinghousewithprobabilityone.Narasimhan(1988)analyzesduopolycompetitionwhere…rmshaveanasymmetricnumberofloyalcustomersundertheassumptionthatboth…rmslistpricesattheclearinghousewithcertainty.ThemodelpresentedabovethustakesthemainpartofNarasimhan’sanalysisasaspecialcase.When�0;itisinterestingtonotethatthepropensitytoadvertiseislessthanunityand,moresurprisingly,itisexactlythesameforboth…rms.Thus,asymmetriesinthecustomerbaseof…rmsneednotleadtoasymmetriesin…rmpropensitiestoadvertise.Incontrast,the…rms’distributionsofadvertisedpricesdodependontheircustomerbases.Comparingthedistributionsofpricesforthetwo…rms,one…ndsthat:F1(p)�F2(p)=1v�p(p�m)S[L1�L2]�0Thatis,the…rmwithfewerloyalsisactuallylessaggressiveinitspricingstrategythanthe…rmwithmoreloyals.Thelargertheasymmetryinloyals,thelargerthedi¤erenceintheaveragepriceschargedbythetwo…rms—butinanunexpecteddirection.Indeed,the…rmwithmoreloyals(…rm1)o¤ersthelowestprice,p0;1inthemarketwithstrictlypositiveprobability.18AsinBayeandMorgan(2001),onemayendogenizetheadvertisingfeebyallowingapro…t-maximizinggatekeepertodeterminethelevelofthatmaximizesitsexpectedpro…ts.19Forexample,iftheonlycostsare…xedcosts(FC);theexpectedpro…tsoftheclearinghousearesimplyitsexpectedadvertisingrevenuesminuscosts:E[]=2�FC=21�2S(v�m)�FCEquatingthegatekeeper’sexpectedmarginalpro…tstozeroandsolvingfortheoptimaladvertisingfeeyields=S(v�m)4(14)18Since…rm2’sdistributionisatomless,atieatpricep0;1isazeroprobabilityevent.19Forsimplicity,weassumetheclearinghousemustsetthefeechargedtoconsumersforaccessat=0toinducethemalltoparticipate.Thisistypicallythecase,forexample,atonlinepricecomparisonsites.32ItisinterestingtocomparetheSpulbermodel,whichoccursintheclearinghouseframework,withthesearch-theoreticframeworkofMacMinn.Noticethat,whenthenumberofcompeting…rmsinSpulber,n,isequaltotheoptimal…xedsamplesizeforconsumersintheMacMinnmodel,n;theequilibriumdistributionofprices,equations(15)and(7),areidenticalinthetwomodels.Thatis,costhete

rogeneitiesaresu¢cienttogeneratepricedi
rogeneitiesaresu¢cienttogeneratepricedispersioninoligopolymodelswhereallconsumersobtaincompletepriceinformation,aswellasinmodelswhereacontinuumof…rmscompetebuteachconsumeronlyobtainspricequotesfroma…nitenumbernofthese…rms.2.3BoundedRationalityModelsofPriceDispersionSeveralrecentpapershaveemphasizedthatboundedrationalitycanalsoleadtopricedispersion.TheideaistorelaxtheNashequilibriumassumption–whichrequiresthateachdecisionmakerinthemarketischoosinganaction(beitapriceorasearchstrategy)thatisabestresponsetogivenactionsofothermarketparticipants.Twoequilibriumconcepts–quantalresponseequilib-rium(McKelveyandPalfrey,1995)andepsilonequilibrium(Radner,1980)–areparticularlyusefulbecausetheynestthestandardNashequilibriumconceptasaspecialcase.Inaquantalresponseequilibrium(QRE),thelikelihoodthataparticular…rmsetsaspeci…cpricedependsontheexpectedpro…tsarisingfromthatprice(seeLopez-Acevedo,1997).A…rm’spriceisdeterminedbyastochasticdecisionrule,butpricesleadingtohigherexpectedpro…tsaremorelikelytobecharged.Ofcourse,each…rm’sexpectedpro…tsfromdi¤erentpricingdecisionsdependontheprobabilitydistributionsofotherplayers’prices.AQRErequiresthatall…rmsholdcorrectbeliefsabouttheprobabilitydistributionsofotherplayers’actions.ThenondegeneratedistributionsofpricesresultinginaQREmaybeviewedasshocksto…rms’pro…tfunctions.Alternatively,nondegeneratepricedistributionsmightstemfromdecisionerrorsby…rms.Sucherrorsmayarisefromlimitationsinmanagers’cognitiveprocessingabilitiesor“bugs”indynamicpricingalgorithmsusedbyInternetretailers.Inan"-equilibrium,thepriceschargedbyeach…rmaresuchthatno…rmcangainmorethan"inadditionalpro…tsbychangingitsprice.Suchanequilibriummayarisebecauseofcognitiveormotivationalconstraintsonthepartof…rms.Forexample,ifitiscostlytoreprogramdynamicpricingalgorithms,managersmaynotbewillingtoincurtheseeconomicorpsychiccostswhentheresultinggainissmall(lessthan").Recently,BayeandMorgan(2004)appliedQREand"-equilibriumconceptstopricinggamesandshowedthatonlyalittleboundedrationalityisneededtogeneratethepatternsofprice34inge¤orts,di¤erentialservicequalities,orreputations)cancontributetoequilibriumpricedispersion,suchdi¤erencesarenotnecessaryforequilibriumpricedispersion.6.ThankstotheInternet,informationgatekeepersareplayinganincreasinglyimportantroleintheeconomy.Intheirattempttomaximizepro…tsandenhancethevalueofinformationprovidedbytheirsites,informationgatekeepershaveanincentivetochargefeesfortheirservicesthatinduceequilibriumpricedispersion.7.Alittleboundedrationalitygoesalongwayinexplainingpricedispersion.3EmpiricalAnalysisofPriceDispersionWenowturntotheempiricalliteratureonpricedispersion.InSection3.1,wediscusssomeofthestrengthsandwe

aknessesofcommonlyusedmetricsformeasurin
aknessesofcommonlyusedmetricsformeasuringpricedispersioninonlineando­inemarkets.Section3.2providesanoverviewoftheempiricalliterature,andhighlightsempiricalevidencesuggestingthatinformationcosts(eitherontheconsumeror…rmsideofthemarket)contributetopricedispersion;thatis,dispersionisnotpurelyanartifactofsubtleproductheterogeneities.3.1MeasuringPriceDispersionTheequilibriummodelsofpricedispersionpresentedaboveeachimplynon-degeneratedistributionsofprices,F(p);onsomeintervalp;p.Givensuchadistribution,astandardmeasureofdispersionisthevarianceinprices.Foreachmodelofequilibriumpricedispersion,thismeasurecanbedirectlycomputed.Forinstance,intheMacMinnmodel,if…rmshaveuniformlydistributedmarginalcosts,thevarianceinpricesis2p=n�1n2(m�m)212Noticethatoneistheninapositiontotestcomparativestaticpredictionsofthemodelusingthismeasure.Inasimilarmanner,expressionsforthevarianceinpricesmaybederivedfromtheothermodelspreviouslypresented.Anumberofauthorsusethesamplevariancetomeasurepricedispersion(e.g.,Pratt,Wise,andZeckhauser(1979)andAncaraniandShankar(2004)).Theobviousadvantageisthatitusesallavailabledata.Adrawbackofthismeasureisapparentwhencomparingdispersionacrossproductsorovertime.Forinstance,supposethat,duringanin‡ationaryperiod,themarginal36Understandardconditionstherewillexistauniquesymmetricequilibriumwhereall…rmspriceatmarginalcost.Butinaddition,thereareasymmetricequilibriawheretwo…rmspriceatmarginalcostandtheremainingn�2…rmspricestrictlyabovemarginalcost.Thus,pricedispersioncanariseinaclassicalBertrandenvironment.Yet,theapparentpricedispersionisarguablynoteconomicallyrelevantbecausetheuniquetransactionspriceismarginalcost.Toremedythistheoreticaldefect,Baye,MorganandScholten(2004a)proposeameasurecalled“thegap,”whichtheyde…netobethedi¤erencebetweenthetwolowestpricesinthemarket.Lettingp(n)2denotethesecondlowestpricerealizationfromndrawsfromF;the(sample)gapisde…nedas21G(n)=p(n)2�p(n)minTheclassicalBertrandmodel(aswellastextbookmodelsofperfectcompetition)impliesthatthegapbetweenthetwolowestpricesiszeroinanyequilibrium(symmetricorotherwise).Alloftheoligopolymodelsofpricedispersiondiscussedabove,incontrast,implyapositivegap.Anadditionalpropertyofthegapisthatitgivesgreaterweighttolowprices,which,intheabsenceofquantitydata,onemightreasonablyassumeleadtomoresalesthanhigherprices.Thekeydisadvantage,sharedbytherange,isthatitreliespurelyonextremevaluesofthedata.Hence,therangeandgaparemoresensitivetooutliersandotherformsof“noise”thanmeasuresthatusealltheavailabledata,suchasthesamplevarianceandcoe¢cientofvariation.Inadditiontothesemeasures,thevalueofinformation(VOI)de…nedearlierinequation(12)canalsobeusedasagaugeofdispersion.Thismeasure,whichissimplythedi¤erencebetweentheave

rageobservedpriceandthelowestobservedpri
rageobservedpriceandthelowestobservedprice,iszerointheabsenceofanypricedispersionbutotherwisepositive.Theprincipaladvantageofthismeasureofdispersionisthatithasaveryintuitiveinterpretation:Itsvalueindicatestheamountofmoneyaconsumersavesbypurchasingatthebestpriceratherthanfromarandomlyselected…rminthemarket.3.2PriceDispersionintheFieldIfpricedispersionstemsfromfrictionsrelatedtotheacquisitionandtransmissionofinformation(asimpliedbythemodelsinSection2)ratherthansubtledi¤erencesin…rms’servicelevels,observedlevelsofdispersionshouldsystematicallydependon“environmentalfactors”presentinthemodels.21Aswiththerange,onecanperformcomparativestaticanalysesforanyofthetheoreticalmodelsusingtheexpectedgap,anditissometimesusefultonormalizethegapbydividingbythelowestprice.Inthisformulation,thegaprepresentsthedi¤erencebetweenthetwolowestpricesexpressedasapercentageofthelowestpricerealization.38iteminaconsumer’soverallbudget,andthefrequencywithwhichanitemispurchased,aregoodproxiesforK.Dispersionfor“Cheap”versus“Expensive”ItemsStigler(1961)providescasualevidenceinsupportofhis…rsthypothesis—thatdispersionislowerforitemsthataccountforalargeexpenditureshareofasearcher’sconsumptionbundle(“expensiveitems”)thanthosethataccountforasmallerexpenditureshare(“cheapitems”).Governmentcoalpurchasesareasmallpercentageoftheoverallgovernmentbudget,whileahousehold’sexpendituresonanautomobilecomprise(in1961aswellastoday)amuchlargerpercentageofitsoverallbudget.Stiglerobtaineddi¤erentsellers’pricesfortwohomogeneousproducts—anthracite-gradecoaltobesoldtothegovernment,andanautomobiletobesoldtoahousehold.Pricesforanthracitecoalrangedfrom$15.46to$18.92,withanaveragepriceof$16.90andastandarddeviationof$1.15.Pricesfortheautomobile(basedonwhatStiglercalled“anaverageamountofhiggling”)rangedfrom$2,350to$2,515,withanaveragepriceof$2,436andstandarddeviationof$42.Stigler’sdatathustendtosupporthis…rstconjecture:Ifonecalculatestheimpliedcoe¢cientofvariationbasedonStigler’s…gures,thecoe¢cientofvariationforcoal(whichmakesupasmallpercentageofthegovernment’sbudget)is14.7percent,whilethatforanautomobile(whichmakesupalargepercentageofahousehold’sbudget)is1.7percent.Pratt,WiseandZeckhauser(1979)observeasimilarpatterninacross-sectionofconsumerproductssoldinBostoninthe1970s.Theyobtainthefollowingregressionresultregressingthesample(log)standarddeviationofpricesforagivenitemonthesample(log)meanpriceforthesameitem.ln=�1:517+0:892lnE[p](16)Straightforwardmanipulationofequation(16)revealsthata1percentincreaseinthemeanpriceofanitemdecreasesthecoe¢cientofvariationby10.8percent.Thus,thePratt,Wise,andZeckhauserdataalsosuggestthat,empirically,thecoe¢cientofvariationislowerformoreexp

ensiveitemsthancheaperitems.However,equa
ensiveitemsthancheaperitems.However,equation(16)alsohighlightsthattherelationshipdependscruciallyonthemeasureofpricedispersionused:Ifoneweretousethestandarddeviationtomeasurepricedispersion,equation(16)impliesthataonepercentincreaseinthemeanpriceofaproductleadstoa0.892percentincreaseindispersion,asmeasuredbythestandarddeviation.Anumberofotherauthorshavereportedsimilarpatternsinonlineando­inemarkets,bothintheUSandinEuropeforproductsrangingfromconsumersundries,electronicproducts,andgasoline;cf.Marvel(1976),CarlsonandPescatrice(1980),ClayandTay(2001),Scholtenand40dealingwiththeseproblems.OneimportantexampleisBrownandGoolsbee(2002).TheirstartingpointistheStahl(1989)modelofequilibriumpricedispersion,whichaswenotedinSection2,predictsthatpricedispersionisinitiallyanincreasingfunctionofthefractionof“shoppers”whoenjoyzerosearchcosts,butafterathreshold,isadecreasingfunctionofthefractionofshoppers.BrownandGoolsbeepointoutthattheStahlmodelcloselymatchesthemarketforterm-lifeinsuranceduringthe1992-1997period.ConsumerswhodidnothaveanInternetconnectionarguablyhadtosearchsequentiallytoobtainpricequotesfromdi¤erentinsuranceagents,whilethosewithInternetaccesscouldusewebsitessuchasQuickquote.comto“costlessly”identifythecompanyo¤eringthelowestannualpremium.Intheirdata,variationinthefractionof“shoppers”(thosewhoresearchinsuranceonline)stemsnotonlyfromthegeneralriseinInternetpenetrationduringthe1990s,butmoreimportantly,fromvariationinthegrowthratesinInternetusageacrossdi¤erentgroupsofpolicyholders.BrownandGoolsbeeregressthestandarddeviationinresiduals(obtainedfromapriceregressionthatcontrolsforobservablecharacteristicsofpeopleandpolicytypes)onacubicfunctionoftheirproxyforthefractionof“shoppers.”ConsistentwiththepredictionoftheStahlmodel,pricedispersioninitiallyrisesasthefractionofshoppersincreases,butstartstodeclineoncethefractionofconsumersresearchinginsuranceonlineexceedsabout5percent.Asimilarapproachisimplicitinanumberofpapersthathavecomparedlevelsofdispersioninonlineversuso­inemarkets(cf.BrynjolfssonandSmith,2000;CarltonandChevalier,2001;AncaraniandShankar,2004;andScholtenandSmith,2002.)Thebasicpremiseisthatsearchcostsarelowerinonline(searchentailsclicks)versuso­inemarkets(searchentailstravelcosts).22Ingeneral,sincedi¤erentsearchmodelsmakedi¤erentpredictionsabouttheimpactofreductionsinsearchcostsonlevelsofpricedispersion,itisnottoosurprisingthatthe…ndingsofthisliteraturearedecidedlymixed;forsomeproducts,dispersionislowerinonlinemarkets;forotherproducts,dispersionisactuallyhigheronline.2322Theviewthatonlinesearchiseithermoreprevalentorcheaperthano­inesearchisamatterofsomedebate;see,forinstance,AdamicandHuberman(2001),Johnson,Moe,Fader,Bellman,andLohse(2004).Bakos(1997)wasamongthe…rsttoadvanceatheoreticalargumentthatwhenth

ecostofpriceinformationisclosetozero,equ
ecostofpriceinformationisclosetozero,equilibriumpriceisclosetomarginalcost.Morerecently,however,Harrington(2001)hasarguedthatBakos’resultsare‡awed.Finally,theInternetitselfalsoo¤ersopportunitiesforobfuscation(seeEllisonandEllison(2004))orunobservedlackofinventories(seeArnold&Saliba(2002))thatcanraisesearchand/ortransactionscostsrelativetoo­inemarkets.23Onemayspeculatethatonceshippingcostsareaccountedfor,pricedispersiononlinevanishes.Thisisnotthecase;cf.Pan,RatchfordandShankar(2002);AncaraniandShankar(2004);Brynjolfsson,DickandSmith(2004);BrynjolfssonandSmith(2000);SmithandBrynjolfsson(2001);DinlersozandLi(2005).42aswesawintheVarianandRosenthalmodels.Thus,examiningtherelationshipbetweenthepricedispersionandthenumberofcompetingsellersnotonlyprovidesatestofwhetherinformationalfactorsplayaroleingeneratingobservedpricedispersion,butalsoinmakingdistinctionsamongthevarioustheorymodels.Forinstance,Baye,MorganandScholten(2004a)examinethetheoreticalandempiricalrela-tionshipbetweenthenumberofcompetitorsandlevelsofpricedispersioninclearinghousemodels.Theyshowthatthetheoreticalrelationshipbetweennumberofcompetitorsandthelevelofpricedispersioninclearinghousemodelsis,ingeneral,ambiguous,duetocompeting“orderstatistic”and“strategic”e¤ects.ThroughacalibrationdisplayedinFigure2,theyshowthattheimpactofthenumberofsellersonpricedispersiondependsonthevariantofthemodel.Asthe…gureshows,intheVarianmodel(where…rms’informationtransmissioncostsdonotdrivepricedispersion),theexpectedgapbetweenthetwolowestpricesisinitiallyincreasinginthenumberofsellers,andthendeclines.Incontrast,intheBayeandMorganmodel(where…rms’informationtransmissioncostsarethemaindriverofpricedispersion),theexpectedgapismonotonicallydecreasinginthenumberof…rms.Basedononlinedatafromapopularpricecomparisonsiteforconsumerelec-tronicsproducts,andcontrollingforotherfactorscontributingtopricedispersion,they…ndaninverserelationbetweenthegapandthenumberofonlinesellers.Thisrelationshipisdepictedasthedotted“observed”lineinFigure2.Asthe…gurereveals,thenon-monotonicitypredictedbytheVarianmodel,aswellastherelatively‡atrelationshipbetweenthegapandnumberof…rmspredictedinthecalibratedversionoftheRosenthalmodel,isabsentinthedata.Speci…cally,inmarketsservedbybetweentwoandfour…rms,theaveragegap(asapercentageofthelowestprice)isabout14percent.Theaveragepercentagegapfallstoabout3percentinmarketswith…vetoten…rms,andislessthanonepercentinmarketswithmorethan10…rms.Morebroadly,severalempiricalpapershavesuggestedthattheamountofpricedispersionobservedinthemarketdependsonvariousmeasuresofthenumbersofcompetitors.Marvel(1976)reportsthatanincreaseinthenumberofcompetitors(measuredbytheln(HHI))reducestherangeinthepriceofgasoline.Barron,Tayl

orandUmbeck(2004)studythestructuraldeter
orandUmbeck(2004)studythestructuraldeterminantsofpricedispersionintheretailgasolineindustryinfourgeographiclocations,andprovideempiricalevidencethat,controllingforstation-levelcharacteristics,anincreaseinstationdensitydecreasesbothpricelevelsandpricedispersion.24BorensteinandRose(1994)investigatetherelationshipbetweendispersionamongairfaresandthenumberofcompetitorsor‡ightdensity.They…ndthat24SeealsoPngandReitman(1994).44alsosuggestthatheterogeneitieseitheracross…rmsoracrossmarketsimpactpricedispersioninonlinemarkets(cf.Smith,BaileyandBrynjolfsson(1999);Clay,KrishnanandWol¤(2001);SmithandBrynjolfsson(2001);ChenandHitt(2002);ResnickandZeckhauser(2002);andBrynjolfsson,DickandSmith(2004)).Inallcases,however,evenaftercontrollingforvariousheterogeneities,economicallysigni…cantlevelsofpricedispersionremain.Thereisalsoevidencethatonlinepricesexhibittemporalpricedispersion.Forinstance,Baye,MorganandScholten(2004b)examineturnoveroftheidentityofthelow-priceandhigh-price…rmsusingadatasetconsistingof36popularconsumerelectronicsproductssoldovera19-monthperiod.They…ndconsiderableevidenceformonth-to-monthchangesintheidentityofthelow-price…rms,butsomeevidenceofpersistenceintheidentityofhigh-priced…rms.Similarly,IyerandPazgal(2003)collectbi-weeklypricedataonmusicCDs,movievideosandbooksfrom…vepricecomparisonsites:MySimon,BottomDollar,EvenBetter,BsillyandPricescanduringtheperiodApril-October2000and…ndempiricalresultssuggestingthatnosingle…rmconsistentlychargesthelowprice.Finally,Baye,MorganandScholten(2004a)examinetheconvergencehypothesisofpricedis-persionusingadatasetconsistingofoverfourmilliondailypriceobservationsforoveronethousandconsumerelectronicsproductssoldonapopularInternetpricecomparisonsiteoveraneightmonthperiod.Evenallowingforanonlinearrelationshipbetweenobservedpricedispersionandtime,they…ndnoevidencefortheconvergencehypothesisinthismarket—thelevelofpricedispersionre-mainedstableovertheperiod.3.3ConcludingRemarks:EmpiricsWeconcludewithfoursimpleobservations.1.AsisevidentfromthestudieshighlightedinTable1,pricedispersionisubiquitousandpersistent.Regardlessoftheparticularproduct(tinplatecansorPDAs),thevenueinwhichtheyaresold(onlineoro­ine,intheUSorabroad),orthetimeperiod(1901or2005),theinescapableconclusionfromtheempiricalliteratureisavalidationofStigler’sandVarian’sinitialobservations:Informationremainsavaluableresource,andthelawofonepriceisstillnolawatall.2.Theoryisusefulforunderstandingdispersiondata,anddispersiondataisusefulfordiscrim-inatingamongalternativetheoreticalmodels.46ReferencesAdamic,L.A.andB.A.Huberman.2001.“TheWeb’sHiddenOrder.”Comm.ACM,44(9),55-59.Ancarani,F.andV.Shankar.2004.“PriceLevelsandPriceDispersionWithinandAcrossMultipleRetailerTypes:FurtherEvidence

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lEconomics,50(3),351-367.Clemons,E.I.HannandL.Hitt.2002.“PriceDispersionandDi¤erentiationinOnlineTravel:AnEmpiricalInvestigation.”ManagementScience,48,534-549.Dana,J.D.1994.“LearninginanEquilibriumSearchModel.”InternationalEconomicReview,35,745-771.Daughety,A.1992.“AModelofSearchandShoppingbyHomogeneousConsumerswithoutPricePrecommitmentbyFirms,”JournalofEconomicsandManagementStrategy1(3),455-473.Diamond,P.1971.“AModelofPriceAdjustment.”JournalofEconomicTheory,3,156-168.50Johnson,E.J.,W.Moe,P.Fader,S.BellmanandJ.Lohse.2004.“OntheDepthandDynamicsofWorldWideWebShoppingBehavior.”ManagementScience,50(3),299-308.Johnson,R.N.2002.“SearchCosts,LagsandPricesatthePump.”ReviewofIndustrialOrganiza-tion,20,33-50.Klemperer,P.1999.“AuctionTheory:AGuidetotheLiterature.”JournalofEconomicSurveys,13,227-286.Lach,S.2002.“ExistenceandPersistenceofPriceDispersion:AnEmpiricalAnalysis.”ReviewofEconomicsandStatistics,84(3),433-444.Lal,R.andM.Villas-Boas.1998.“PricePromotionsandTradeDealswithMultiproductRetailers.”ManagementScience,44(7),935-949.Lal,R.andM.Sarvary.1999.“WhenandHowistheInternetLikelytoDecreasePriceCompeti-tion.”MarketingScience,18(4),485-503.Lopez-Acevedo,G.1997.“QuantalResponseEquilibriaforPostedO¤erMarkets.”EstudiosEconómicos,12,95-131.MacMinn,R.D.1980.“SearchandMarketEquilibrium.”JournalofPoliticalEconomy,88(2),308-327.Marvel,H.P.1976.“TheEconomicsofInformationandRetailGasolinePriceBehavior:AnEm-piricalAnalysis.”JournalofPoliticalEconomy,84,1033-1080.McAfee,R.P.1994.“EndogenousAvailability,Cartels,andMergerinanEquilibriumPriceDisper-sion.”JournalofEconomicTheory,62,24-47.McAfee,R.P.1995.“MultiproductEquilibriumPriceDispersion.”JournalofEconomicTheory,67,83-105.McAfee,R.P.andJ.McMillan.1988.“SearchMechanisms.”JournalofEconomicTheory,44,99-123.McKelvey,R.andT.Palfrey1995.“QuantalResponseEquilibriaforNormalFormGames.”GamesandEconomicBehavior,10,6-38.52Reinganum,J.F.1979.“ASimpleModelofEquilibriumPriceDispersion.”JournalofPoliticalEconomy,87,851-858.Resnick,P.andR.Zeckhauser.2002.“TrustAmongStrangersinInternetTransactions:EmpiricalAnalysisofeBay’sReputationSystem.”AdvancesinAppliedMicroeconomics,11,127-158.Robert,J.andD.O.Stahl.1993.“InformativePriceAdvertisinginaSequentialSearchModel.”Econometrica,61,657-686.Roberts,M.J.andD.Supina.2000.“OutputPriceandMarkupDispersioninMicroData:TheRolesofProducerHeterogeneityandNoise.”AdvancesinAppliedMicroeconomics,9,1-36..Rosenthal,R.W.1980.“AModelinWhichanIncreaseintheNumberofSellersLeadstoaHigherPrice.”Econometrica,48(6),1575-1580.Rothschild,M.1973.“ModelsofMarketOrganizationwithImperfectInformation:ASu

rvey.”JournalofPoliticalEconomy,81(
rvey.”JournalofPoliticalEconomy,81(6)1283-1308.Rothschild,M.1974.“SearchingfortheLowestPriceWhentheDistributionofPricesisUnknown.”JournalofPoliticalEconomy,82(4),689-711Salop,S.C.andJ.E.Stiglitz.1977.“BargainsandRipo¤s:AModelofMonopolisticallyCompetitivePriceDispersion,”ReviewofEconomicStudies,44,493-510.Scholten,P.andA.Smith.2002.“PriceDispersionThenandNow:EvidencefromRetailandE-tailMarkets.”inMichaelR.Baye(ed.)inTheEconomicsoftheInternetandE-commerce,AdvancesinAppliedMicroeconomics,11,2002.Scott-Morton,F.,F.ZettelmeyerandJ.Silva-Risso.2001.“InternetCarRetailing.”JournalofIndustrialEconomics,49(4),501-519.Shilony,Y.1977.“MixedPricinginOligopoly.”JournalofEconomicTheory,14,373-388.Smith,M.D.,J.BaileyandE.Brynjolfsson.1999.“UnderstandingDigitalMarkets:ReviewsandAssessments”inUnderstandingtheDigitalEconomy,BrynjolfssonandKahin,eds.,MITPress,Cambridge,MA.Smith,M.D.andE.Brynjol¤son.2001.“ConsumerDecision-MakingatanInternetShopbot:BrandStillMatters.”JournalofIndustrialEconomics,49(4),541-558.54*2005 data through third quarter.Figure 1: Percentage of Articles Published in the American Economic Review, Journal of Political Economy, and Econometrica on Information, Search or Price DispersionSource: Social Science Citation Index, Keyword search for "Information OR Price Dispersion OR Search," and Authors' Calculations.5.00%10.00%1961-19651966-19701971-19751976-19801981-19851986-19901991-19951996-20002001-2005*0102030405060Percentage GapVarianRosenthalBaye-MorganTable 1a: Measures of Price Dispersion RepStudyData PeriodProduct MarketBailey (1998)1997Books 13.2%Standard DeviationBooks 10.4%Standard DeviationCompact Discs 17.6%Standard DeviationCompact Discs 11.0%Standard DeviationSoftware 7.1%Standard DeviationSoftware 8.1%Standard DeviationBorenstein and Rose (1994)1986U.S. Airline 0.018 - 0.416Gini coefficientCarlson and Pescatrice (1980)1976Consumer Sundries3.3% - 41.4%Coefficient of VariationEckard (2004)1901 - 2001Baking Powder, Sugar, Salt -- 1901 3.1% - 10.1%Coefficient of VariationBaking Powder, Sugar, Salt -- 2001 0.0% - 13.4%Coefficient of Variation1999Books $54.00 - $122.00RangeBooks $21.94 - $76.20Standard DeviationCompact Discs $20.00 - $40.00RangeCompact Discs $12.91 - $23.86Standard DeviationBooks (Sweden)$19.00 - $58.00RangeCompact Discs (Sweden)$21.00 - $46.00RangeLach (2002)1993 - 1996Refrigerator (Israel)4.9%Coefficient of VariationChicken, Flour, Coffee (Israel)11.4% - 19.7%Coefficient of VariationMarvel (1976)1964 - 1971Regular Gasoline $0.048RangeRegular Gasoline $0.015Standard DeviationPremium Gasoline $0.048RangePremium Gasoline $0.017Standard Deviation1975Various Products and Services 4.4% - 71.4%Coefficient of VariationVarious Products and Services 11.0% - 567.

0%RangeVarious Products and Services 7.2
0%RangeVarious Products and Services 7.2% - 200.0%Value of InformationRoberts and Supina (2000)1963 - 1987Wood Products 13.8% - 90.2%Coefficient of VariationFabrics 18.8% - 78.1%Coefficient of VariationCoffee 14.3% - 25.1%Coefficient of VariationReady-Mixed Concrete 13.2% - 37.2%Coefficient of VariationNewsprint 4.5% - 8.2%Coefficient of VariationGasoline 6.2% - 11.8%Coefficient of VariationTinplate Steel Cans 25.0% - 31.0%Coefficient of VariationPan Bread 26.0% - 49.6%Coefficient of VariationCorrugated Shipping Containers 21.8% - 39.6%Coefficient of VariationScholten and Smith (2002)1976 - 2000Consumer Sundries -- 1976 3.3% - 41.4%Coefficient of VariationConsumer Sundries -- 2000 1.6% - 42.0%Coefficient of VariationConsumer Sundries -- 2000 5.7% - 28.4%Coefficient of VariationSorensen (2000)1998Prescription Drugs $13.17RangePrescription Drugs 22.0%Coefficient of VariationStigler (1961)1953Anthracite Coal $3.46RangeAnthracite Coal $1.15Standard Deviation1959Identical Automobiles $165.00RangeIdentical Automobiles $42.00Standard DeviationVillas-Boas (1995)1985 - 1987Coffee 21.5%Coefficient of VariationConsumer Electronics (UK)7.0% - 54.9%RangeConsumer Electronics (Denmark)12.8% - 42.9%RangeConsumer Electronics (France)1.6% - 16.1%GapConsumer Electronics (Italy)3.6% - 13.7%GapConsumer Electronics (Netherlands)8.9% - 34.6%GapConsumer Electronics (Spain)3.7% - 18.0%GapConsumer Electronics (Sweden)5.9% - 15.6%GapConsumer Electronics (UK)2.5% - 14.5%GapConsumer Electronics (Denmark)3.6% - 31.9%GapHong and Shum (Forthcoming)2002Books $8.19 - 27.05RangeBooks 6.2% - 8.5%Coefficient of Variation2004Market for Keyboards $6.50 - $91.67RangeMarket for Keyboards 8.0% - 52.0%Coefficient of Variation2000Books 15.0%Coefficient of VariationCompact Discs 15.4%Coefficient of VariationDVDs 12.7%Coefficient of VariationPDAs 11.8%Coefficient of VariationSoftware 11.7%Coefficient of VariationConsumer Electronics 9.6%Coefficient of Variation2000 - 2003Consumer Electronics and Books 9.8% - 11.7%Coefficient of VariationBooks 33.3% - 48.9%RangeCompact Discs 22.2% - 51.0%RangeDVDs 30.7% - 43.7%RangeComputers 15.0% - 34.4%RangeSoftware 19.0% - 35.6%RangeConsumer Electronics 22.1% - 45.7%RangeSmith and Brynjolfsson (2001)1999Books 28.0% - 33.0%Value of InformationBooks $6.29 - $10.51Standard DeviationInformation,Search,andPriceDispersionMichaelR.BayeKelleySchoolofBusinessIndianaUniversityJohnMorganHaasSchoolofBusinessandDepartmentofEconomicsUniversityofCalifornia,BerkeleyPatrickScholtenBentleyCollegeThisDraft:October31,2005.Forthcoming,HandbookonEconomicsandInformationSystems(Elsevier,T.Hendershott,ed.).AbstractWeprovideauni…edtreatmentofalternativemodelsofinformationacquisition/transmissionthathavebeenadvancedtorationalizepricedispersioninonlineando­inemarketsforhomo-geneousprodu

cts.Thesedi¤erentframeworks—whichi
cts.Thesedi¤erentframeworks—whichincludesequentialsearch,…xedsam-plesearch,andclearinghousemodels—revealthatreductionsin(ortheeliminationof)con-sumersearchcostsneednotreduce(oreliminate)pricedispersion.Ourtreatmenthighlightsa“duality”betweensearch-theoreticandclearinghousemodelsofdispersion,andshowshowauction-theoretictoolsmaybeusedtosimplify(andevengeneralize)existingtheoreticalre-sults.Weconcludewithanoverviewoftheburgeoningempiricalliterature.Theempiricalevidencesuggeststhatpricedispersioninbothonlineando­inemarketsissizeable,pervasive,andpersistent—anddoesnotpurelystemfromsubtledi¤erencesin…rms’productsorservices.WeoweaspecialdebttoMichaelRauhandFelixVárdyforprovidinguswithdetailedhand-writtencommentsonearlierdrafts.WealsothankVilleAalto-Setälä,FabioAncarani,MariaArbatskaya,JudyChevalier,KarenClay,WoodyEckard,SaraFisherEllison,XianjunGeng,RupertGatti,JoseMoragaGonzalez,JoeHarrington,TerryHendershott,GaneshIyer,MaartenJanssen,RonaldJohnson,KenJudd,RamayyaKrishnan,SolLach,RajivLal,PrestonMcAfee,XingPan,Je¤Perlo¤,IvanPng,RamRao,JenniferReinganum,NancyRose,VenkyShankar,JorgeSilva-Risso,MichaelSmith,AlanSorensen,DanSpulber,MarkStegeman,BeckTaylor,MiguelVillas-Boas,XiaolinXing,andRichardZeckhauserforencouragement,helpfulcomments,andsuggestions.Ofcourse,weareresponsibleforanyshortcomingsthatremaininthiso¤ering.1Information,Search,andPriceDispersionMichaelR.BayeKelleySchoolofBusinessIndianaUniversityJohnMorganHaasSchoolofBusinessandDepartmentofEconomicsUniversityofCalifornia,BerkeleyPatrickScholtenBentleyCollegeThisDraft:October31,2005.Forthcoming,HandbookonEconomicsandInformationSystems(Elsevier,T.Hendershott,ed.).AbstractWeprovideauni…edtreatmentofalternativemodelsofinformationacquisition/transmissionthathavebeenadvancedtorationalizepricedispersioninonlineando­inemarketsforhomo-geneousproducts.Thesedi¤erentframeworks—whichincludesequentialsearch,…xedsam-plesearch,andclearinghousemodels—revealthatreductionsin(ortheeliminationof)con-sumersearchcostsneednotreduce(oreliminate)pricedispersion.Ourtreatmenthighlightsa“duality”betweensearch-theoreticandclearinghousemodelsofdispersion,andshowshowauction-theoretictoolsmaybeusedtosimplify(andevengeneralize)existingtheoreticalre-sults.Weconcludewithanoverviewoftheburgeoningempiricalliterature.Theempiricalevidencesuggeststhatpricedispersioninbothonlineando­inemarketsissizeable,pervasive,andpersistent—anddoesnotpurelystemfromsubtledi¤erencesin…rms’productsorservices.WeoweaspecialdebttoMichaelRauhandFelixVárdyforprovidinguswithdetailedhand-writtencommentsonearlierdrafts.WealsothankVilleAalto-Setälä,FabioAncarani,MariaArbatskaya,JudyChevalier,KarenClay,WoodyEckard,SaraFisherEllison,XianjunGeng,RupertGatti,JoseMoragaGonzalez

,JoeHarrington,TerryHendershott,GaneshIy
,JoeHarrington,TerryHendershott,GaneshIyer,MaartenJanssen,RonaldJohnson,KenJudd,RamayyaKrishnan,SolLach,RajivLal,PrestonMcAfee,XingPan,Je¤Perlo¤,IvanPng,RamRao,JenniferReinganum,NancyRose,VenkyShankar,JorgeSilva-Risso,MichaelSmith,AlanSorensen,DanSpulber,MarkStegeman,BeckTaylor,MiguelVillas-Boas,XiaolinXing,andRichardZeckhauserforencouragement,helpfulcomments,andsuggestions.Ofcourse,weareresponsibleforanyshortcomingsthatremaininthiso¤ering.theseenvironmentsunderavarietyofmarketconditionsandsearchstrategies(includingsequentialand…xedsamplesearch).Whilemarginalsearchcostsareusefulinexplainingpricedispersioninsomemarkets,inmanyonlinemarketsincrementalsearchcostsareverylow—andinsomecases,zero.Forexample,pricecomparisonsitesandshopbottechnologiescreateenvironmentswhereconsumersmayobtainalistofthepricesthatdi¤erentsellerschargeforthesameproduct.Despitethefactthatthisinformationisavailabletoconsumersinseconds,ultimatelyatthecostofasingle“mouseclick,”theoverwhelmingempirical…ndingisthatevenintheseenvironments,pricedispersionispervasiveandsigni…cant—thelawofonepriceisegregiouslyviolatedonline.InSection2.2,weexamineanalternativelineoftheoreticalresearchwheremarginalsearchcostsarenotthekeydriverforpricedispersion.OurtheoreticalanalysisconcludesinSection2.3withadiscussionofalternativebehavioralrationalesforpricedispersion(includingboundedrationalityonthepartof…rmsand/orconsumers).Section3providesamoredetailedoverviewofthegrowingempiricalliterature.AsonemightsuspectbasedonthetrendinFigure1andtheresearchsummarizedinTables1aand1b,mostempiricalstudiesofpricedispersionpost-datetheInternetandrelyononlinedata.Ourviewisthatthisismoreanartifactoftherelativeeasewithwhichdatamaybecollectedinonlinemarkets—notanindicationthatpricedispersionismoreimportant(ormoreprevalent)inonlinethano­inemarkets.Forthisreason,wehaveattemptedtoprovideabalancedtreatmentoftheliteraturesononlineando­inepricedispersion.Asweshallargue,theoverwhelmingconclusionofbothliteraturesisthatpricedispersionisnotpurelyanartifactofproductheterogeneities.2TheoreticalModelsofPriceDispersionThissectionpresentsalternativemodelsthathavebeenusedtorationalizethepricedispersionob-servedinbotho­ineandonlinemarkets.Oneapproachistoassumethatitiscostlyforconsumerstogatherinformationaboutprices.Inthese“search-theoretic”models,consumerssearchingforthebestpriceincurapositivecostofobtainingeachadditionalpricequote.RepresentativeexamplesincludeStigler(1961),Rothschild(1973),Reinganum(1979),MacMinn(1980),Braverman(1980),BurdettandJudd(1983),CarlsonandMcAfee(1983),Rob(1985),Stahl(1989,1996),Dana(1994),McAfee(1995),JanssenandMoraga-González(2004),aswellasJanssen,Moraga-González,andWildenbeest(2005).3theseenvironmentsunderavarietyofmarketconditionsandsearchstrategies(including

sequentialand…xedsamplesearch).Whil
sequentialand…xedsamplesearch).Whilemarginalsearchcostsareusefulinexplainingpricedispersioninsomemarkets,inmanyonlinemarketsincrementalsearchcostsareverylow—andinsomecases,zero.Forexample,pricecomparisonsitesandshopbottechnologiescreateenvironmentswhereconsumersmayobtainalistofthepricesthatdi¤erentsellerschargeforthesameproduct.Despitethefactthatthisinformationisavailabletoconsumersinseconds,ultimatelyatthecostofasingle“mouseclick,”theoverwhelmingempirical…ndingisthatevenintheseenvironments,pricedispersionispervasiveandsigni…cant—thelawofonepriceisegregiouslyviolatedonline.InSection2.2,weexamineanalternativelineoftheoreticalresearchwheremarginalsearchcostsarenotthekeydriverforpricedispersion.OurtheoreticalanalysisconcludesinSection2.3withadiscussionofalternativebehavioralrationalesforpricedispersion(includingboundedrationalityonthepartof…rmsand/orconsumers).Section3providesamoredetailedoverviewofthegrowingempiricalliterature.AsonemightsuspectbasedonthetrendinFigure1andtheresearchsummarizedinTables1aand1b,mostempiricalstudiesofpricedispersionpost-datetheInternetandrelyononlinedata.Ourviewisthatthisismoreanartifactoftherelativeeasewithwhichdatamaybecollectedinonlinemarkets—notanindicationthatpricedispersionismoreimportant(ormoreprevalent)inonlinethano­inemarkets.Forthisreason,wehaveattemptedtoprovideabalancedtreatmentoftheliteraturesononlineando­inepricedispersion.Asweshallargue,theoverwhelmingconclusionofbothliteraturesisthatpricedispersionisnotpurelyanartifactofproductheterogeneities.2TheoreticalModelsofPriceDispersionThissectionpresentsalternativemodelsthathavebeenusedtorationalizethepricedispersionob-servedinbotho­ineandonlinemarkets.Oneapproachistoassumethatitiscostlyforconsumerstogatherinformationaboutprices.Inthese“search-theoretic”models,consumerssearchingforthebestpriceincurapositivecostofobtainingeachadditionalpricequote.RepresentativeexamplesincludeStigler(1961),Rothschild(1973),Reinganum(1979),MacMinn(1980),Braverman(1980),BurdettandJudd(1983),CarlsonandMcAfee(1983),Rob(1985),Stahl(1989,1996),Dana(1994),McAfee(1995),JanssenandMoraga-González(2004),aswellasJanssen,Moraga-González,andWildenbeest(2005).Thesearchmodelsconsideredinthissubsectionareallbasedonthefollowinggeneralenvi-ronment.Acontinuumofprice-setting…rms(withunitmeasure)competeinamarketsellinganidentical(homogeneous)product.Firmshaveunlimitedcapacitytosupplythisproductatacon-stantmarginalcost,m:Acontinuumofconsumersisinterestedinpurchasingtheproduct.Letthemassofconsumersinthemarketbe;sothatthenumberofcustomersper…rmis:Eachconsumerhasaquasi-linearutilityfunction,u(q)+y,whereqisthequantityofthehomogeneousproductandyisthequantityofsomenumerairegoodwhosepriceisnormalizedtobeunity.Thisimpliesthattheindirectutilityofac

onsumerwhopaysapricepperunitoftheproduct
onsumerwhopaysapricepperunitoftheproductandwhohasanincomeofMisV(p;M)=v(p)+Mwherev(
)isnonincreasinginp:ByRoy’sidentity,notethatthedemandfortheproductofrelevanceisq(p)�v0(p).Toacquiretheproduct,aconsumermust…rstobtainapricequotefromastoreo¤eringtheproductforsale.Supposethatthereisasearchcost,c,perpricequote.2If,afterobtainingnpricequotes,aconsumerpurchasesq(p)unitsoftheproductfromoneofthe…rmsatpricepperunit,theconsumer’s(indirect)utilityisV=v(p)+M�cnTheanalysisthatfollowsfocusesonpostedpricemarketswhereconsumersknowthedistributionofpricesbutdonotknowthepriceschargedbyparticularstores.32.1.1TheStiglerModelStigler(1961)considersthespecialcaseofthisenvironmentwhere:1.EachconsumerwishestopurchaseK1unitsoftheproduct;thatis,q(p)=�v0(p)=K;2.Theconsumer’ssearchprocessis…xedsamplesearch—priortosearching,consumersdeter-minea…xedsamplesize,n;of…rmsfromwhomtoobtainpricequotesandthenbuyfrom2Inwhatfollows,weassumethatconsumershaveidenticalsearchcosts.Axell(1977)o¤ersamodelofpricedispersionwithheterogeneoussearchcosts.3ThisassumptionisrelaxedinRothschild(1974),BenabouandGertner(1993)andDana(1994),wherebuyerslearnaboutthedistributionofpricesastheysearch,andinRauh(1997),wherebuyers’searchstrategiesdependononly…nitelymanymomentsofthedistributionofprices.Daughety(1992)o¤ersanalternativesearch-theoreticmodelofequilibriumpricedispersionthatresultsfrominformationalasymmetriesandalackofpriceprecommitmentonthepartof…rms.5Thesearchmodelsconsideredinthissubsectionareallbasedonthefollowinggeneralenvi-ronment.Acontinuumofprice-setting…rms(withunitmeasure)competeinamarketsellinganidentical(homogeneous)product.Firmshaveunlimitedcapacitytosupplythisproductatacon-stantmarginalcost,Acontinuumofconsumersisinterestedinpurchasingtheproduct.Letthemassofconsumersinthemarketbesothatthenumberofcustomersper…rmisEachconsumerhasaquasi-linearutilityfunction,)+,whereisthequantityofthehomogeneousproductandisthequantityofsomenumerairegoodwhosepriceisnormalizedtobeunity.Thisimpliesthattheindirectutilityofaconsumerwhopaysapriceperunitoftheproductandwhohasanincomeofp;M)=)+whereisnonincreasinginByRoy’sidentity,notethatthedemandfortheproductofrelevanceis�Toacquiretheproduct,aconsumermust…rstobtainapricequotefromastoreo¤eringtheproductforsale.Supposethatthereisasearchcost,,perpricequote.If,afterobtainingquotes,aconsumerpurchasesunitsoftheproductfromoneofthe…rmsatpriceperunit,theconsumer’s(indirect)utilityis)+Theanalysisthatfollowsfocusesonpostedpricemarketswhereconsumersknowthedistributionofpricesbutdonotknowthepriceschargedbyparticularstores.2.1.1TheStiglerModelStigler(1961)considersthespecialcaseofthisenvironmentwhere:1.Eachconsumerwishestopurchaseunitsoftheproduct;thatis,)=)=2.Theconsumer

’ssearchprocessis…xedsamplesea
’ssearchprocessis…xedsamplesearch—priortosearching,consumersdeter-minea…xedsamplesize,of…rmsfromwhomtoobtainpricequotesandthenbuyfromInwhatfollows,weassumethatconsumershaveidenticalsearchcosts.Axell(1977)o¤ersamodelofpricedispersionwithheterogeneoussearchcosts.ThisassumptionisrelaxedinRothschild(1974),BenabouandGertner(1993)andDana(1994),wherebuyerslearnaboutthedistributionofpricesastheysearch,andinRauh(1997),wherebuyers’searchstrategiesdependononly…nitelymanymomentsofthedistributionofprices.Daughety(1992)o¤ersanalternativesearch-theoreticmodelofequilibriumpricedispersionthatresultsfrominformationalasymmetriesandalackofpriceprecommitmentonthepartof…rms.whichisdecreasinginn.Furthermore,theexpectedbene…tfromsearcharegreaterforproductsboughtingreaterquantitiesormorefrequently;thatis,equation(1)isincreasinginK:SincethecostofthenthsearchisindependentofKwhiletheexpectedbene…tisincreasinginK;itimmediatelyfollowsthattheequilibriumsearchintensity,n;isincreasinginK:Thatis,consumersobtainmorepricequotesforproductstheybuyingreaterquantities(orfrequencies).DespitethefactthattheStiglermodelassumeseachindividualinelasticallypurchasesKunitsoftheproduct,aversionofthe“lawofdemand”holds:Each…rm’sexpecteddemandisanonincreasingfunctionofitsprice.Toseethis,notethata…rmchargingpricepisvisitedbynconsumersando¤ersthelowestpricewithprobability(1�F(p))n�1:Thus,arepresentative…rm’sexpecteddemandwhenitchargesapriceofpisQ(p)=nK(1�F(p))n�1(2)whichisdecreasinginp:TheStiglermodelimpliesthatboththeexpectedtransactionsprice(Proposition1)aswellastheexpectedtotalcostsinclusiveofsearchcosts(Proposition2)arelowerwhenpricesaremoredispersed(inthesenseofameanpreservingspread).5Proposition1SupposethatapricedistributionGisameanpreservingspreadofapricedistri-butionF.Thentheexpectedtransactionspriceofaconsumerwhoobtainsn�1pricequotesisstrictlylowerunderpricedistributionGthanunderF:Proof.Let
=EFhp(n)mini�EGhp(n)minibethedi¤erenceintheexpectedtransactionspriceunderFcomparedtoG:Wewillshowthatforalln�1;
�0:Usingthede…nitionofEhp(n)mini;
=Z1�1pn(1�F(p))n�1dF(p)�Z1�1tn(1�G(t))n�1dG(t)Letu=F(p)andv=G(p);sothatdu=dF(p),dv=dG(p);p=F�1(u);andt=G�1(v):Then
=nZ10F�1(u)(1�u)n�1du�nZ10G�1(v)(1�v)n�1dv=nZ10�F�1(u)�G�1(u)
(1�u)n�1du5GisameanpreservingspreadofFif(a)R1�1[G(p)�F(p)]dp=0and(b)Rz�1[G(p)�F(p)]dp0;withstrictinequalityforsomez:Notethat(a)isequivalenttothefactthatthemeansofFandGareequal.Together,thetwoconditionsimplythatFandGcrossexactlyonce(atthemean)ontheinteriorofthesupport.7whichisdecreasingin.Furthermore,theexpectedbene…tfromsearcharegreaterforprod

uctsboughtingreaterquantitiesormorefrequ
uctsboughtingreaterquantitiesormorefrequently;thatis,equationisincreasinginK:Sincethecostofthethsearchisindependentofwhiletheexpectedbene…tisincreasinginK;immediatelyfollowsthattheequilibriumsearchintensity,isincreasinginK:Thatis,consumersobtainmorepricequotesforproductstheybuyingreaterquantities(orfrequencies).DespitethefactthattheStiglermodelassumeseachindividualinelasticallypurchasesunitsoftheproduct,aversionofthe“lawofdemand”holds:Each…rm’sexpecteddemandisanonincreasingfunctionofitsprice.Toseethis,notethata…rmchargingpriceisvisitedbyconsumersando¤ersthelowestpricewithprobabilityThus,arepresentative…rm’sexpecteddemandwhenitchargesapriceof)=whichisdecreasinginTheStiglermodelimpliesthatboththeexpectedtransactionsprice(Proposition1)aswellastheexpectedtotalcostsinclusiveofsearchcosts(Proposition2)arelowerwhenpricesaremoredispersed(inthesenseofameanpreservingspread).Proposition1Supposethatapricedistributionisameanpreservingspreadofapricedistri-bution.Thentheexpectedtransactionspriceofaconsumerwhoobtainsn�pricequotesisstrictlylowerunderpricedistributionthanunderF:Proof.
=minminbethedi¤erenceintheexpectedtransactionspriceundercomparedtoWewillshowthatforalln�Usingthede…nitionofmin
=dFsothatdudFdG;pThen
=)(1)(1dvisameanpreservingspreadofiffG(p)�F(p)]dp=0anddG(p)�F(p)]dp0;withstrictinequalityforsomez:Notethatisequivalenttothefactthatthemeansofandareequal.Together,thetwoconditionsimplythatandcrossexactlyonce(atthemean)ontheinteriorofthesupport.2.1.2TheRothschildCritiqueandDiamond’sParadoxWhileStiglero¤eredthe…rstsearch-theoreticrationaleforpricedispersion,themodelhasbeencriticizedfortworeasons.First,aspointedoutinRothschild(1973),thesearchprocedureassumedinStigler’smodelmaynotbeoptimal.In…xedsamplesearch,consumerscommittoa…xednumber,n,ofstorestosearchandthenbuyatthelowestpriceattheconclusionofthatsearch.Acleardrawbacktosuchastrategyisthatitfailstoincorporatenewinformationobtainedduringsearch,suchasanexceptionallylowpricefromanearlysearch.Indeed,oncethebestpricequoteobtainedissu¢cientlylow,thebene…tintheformofpriceimprovementdropsbelowthemarginalcostoftheadditionalsearch.Aswewillseebelow,sequentialsearchresultsinanoptimalstoppingrulesuchthataconsumersearchesuntilshelocatesapricebelowsomethreshold,calledthereservationprice.Second,thedistributionofprices,F;isexogenouslyspeci…edandisnotbasedonoptimizing…rmbehavior.Infact,inlightofequation(2),arepresentative…rmwithconstantmarginalcostofmenjoysexpectedpro…tsof(p)=(p�m)Q(p):Thatis,absentanycostheterogeneities,each…rmfacesexactlythesameexpectedpro…tfunc-tion.Whythen,would…rmsnotchoosethesamepro…t-maximizingpriceor,moregenerally,howcouldthedistributionofpricesgenerated

bypro…t-maximizing…rmsbeconsis
bypro…t-maximizing…rmsbeconsistentwiththepricedistributionoverwhichconsumersweresearching?Inshort,Rothschildpointedoutthatitisfarfromclearthatinformationcostsgiverisetoanequilibriumofpricedispersionwithoptimizingconsumersand…rms;inStigler’smodel,onlyonesideofonemarket,theconsumers,areactinginanoptimizingfashionconsistentwithequilibrium.Forthisreason,Rothschildcriticizedtheearlyliteratureforits“partial-partialequilibrium”approach.Diamond(1971)advancedthisargumentevenfurther—heessentiallyidenti…edconditionsundercostlysearchwheretheuniqueequilibriuminundominatedstrategiesinvolvesall…rmschargingthesameprice—themonopolyprice.Diamond’sresultmaybereadilyseeninthefollowingspecialcaseofourenvironmentwhere:1.Consumershaveidenticaldownwardslopingdemand,i.e.�v00(p)=q0(p)0;2.Consumersengageinoptimalsequentialsearch;3.A…rmactingasamonopolywouldoptimallychargeallconsumerstheuniquemonopolyprice,p;and92.1.2TheRothschildCritiqueandDiamond’sParadoxWhileStiglero¤eredthe…rstsearch-theoreticrationaleforpricedispersion,themodelhasbeencriticizedfortworeasons.First,aspointedoutinRothschild(1973),thesearchprocedureassumedinStigler’smodelmaynotbeoptimal.In…xedsamplesearch,consumerscommittoa…xednumber,,ofstorestosearchandthenbuyatthelowestpriceattheconclusionofthatsearch.Acleardrawbacktosuchastrategyisthatitfailstoincorporatenewinformationobtainedduringsearch,suchasanexceptionallylowpricefromanearlysearch.Indeed,oncethebestpricequoteobtainedissu¢cientlylow,thebene…tintheformofpriceimprovementdropsbelowthemarginalcostoftheadditionalsearch.Aswewillseebelow,sequentialsearchresultsinanoptimalstoppingrulesuchthataconsumersearchesuntilshelocatesapricebelowsomethreshold,calledthereservationprice.Second,thedistributionofprices,F;isexogenouslyspeci…edandisnotbasedonoptimizing…rmbehavior.Infact,inlightofequation,arepresentative…rmwithconstantmarginalcostenjoysexpectedpro…tsof)=(Thatis,absentanycostheterogeneities,each…rmfacesexactlythesameexpectedpro…tfunc-tion.Whythen,would…rmsnotchoosethesamepro…t-maximizingpriceor,moregenerally,howcouldthedistributionofpricesgeneratedbypro…t-maximizing…rmsbeconsistentwiththepricedistributionoverwhichconsumersweresearching?Inshort,Rothschildpointedoutthatitisfarfromclearthatinformationcostsgiverisetoanequilibriumofpricedispersionwithoptimizingconsumersand…rms;inStigler’smodel,onlyonesideofonemarket,theconsumers,areactinginanoptimizingfashionconsistentwithequilibrium.Forthisreason,Rothschildcriticizedtheearlyliteratureforits“partial-partialequilibrium”approach.Diamond(1971)advancedthisargumentevenfurther—heessentiallyidenti…edconditionsundercostlysearchwheretheuniqueequilibriuminundominat

edstrategiesinvolvesall…rmscharging
edstrategiesinvolvesall…rmschargingthesameprice—themonopolyprice.Diamond’sresultmaybereadilyseeninthefollowingspecialcaseofourenvironmentwhere:1.Consumershaveidenticaldownwardslopingdemand,i.e.)=2.Consumersengageinoptimalsequentialsearch;3.A…rmactingasamonopolywouldoptimallychargeallconsumerstheuniquemonopoly;and4.Aconsumerwhoischargedthemonopolypricebya…rmwiththehighestmarginalcost,m;earnssurplussu¢cienttocoverthecostofobtainingasinglepricequote;thatisv"1+"m�c:Reinganumshowsthat,undertheseassumptions,thereexistsadispersedpriceequilibriuminwhich…rmsoptimallysetpricesandeachconsumerengagesinoptimalsequentialsearch.Toestablishthis,we…rstshowhowonederivestheoptimalreservationpriceinasequentialsearchsetting.SupposeconsumersareconfrontedwithanondegeneratedistributionofpricesF(p)onp;pthatisatomless,exceptpossiblyatp:Consumersengageinoptimalsequentialsearchwithfreerecall.If,followingthenthsearch,aconsumerhasalreadyfoundabestpricezmin(p1;p2;:::;pn);then,bymakinganadditionalsearch,suchaconsumerexpectstogainbene…tsofB(z)=Zzp(v(p)�v(z))dF(p)=Zzp�v0(p)F(p)dp;wherethesecondequalityobtainsthroughintegrationbyparts.UsingLeibnitz’rule,wehaveB0(z)=�v0(z)F(z)=Kz"F(z)�0(3)Thus,theexpectedbene…tsfromanadditionalsearcharelowerwhentheconsumerhasalreadyidenti…edarelativelylowprice.Sincesearchiscostly(c�0),consumersmustweightheexpectedbene…tsagainstthecostofanadditionalsearch.Theexpectednetbene…tsofanadditionalsearchareh(z)B(z)�cIftheexpectedbene…tsfromanadditionalsearchexceedtheadditionalcost,h(z)�0;itisoptimalfortheconsumertoobtainanadditionalpricequote.Ifh(z)0,theconsumerisbettero¤purchasingatthepricezthanobtaininganadditionalpricequote.Aconsumer’soptimalsequentialsearchstrategymaybesummarizedasfollows:Case1.h(p)0andRppv(p)dF(p)c:Thentheconsumer’soptimalstrategyistonotsearch.Case2.h(p)0andRppv(p)dF(p)dpc:Thentheconsumer’soptimalstrategyistosearchuntilsheobtainsapricequoteatorbelowthereservationprice,r=p:114.Aconsumerwhoischargedthemonopolypricebya…rmwiththehighestmarginalcost,earnssurplussu¢cienttocoverthecostofobtainingasinglepricequote;thatis1+"Reinganumshowsthat,undertheseassumptions,thereexistsadispersedpriceequilibriuminwhich…rmsoptimallysetpricesandeachconsumerengagesinoptimalsequentialsearch.Toestablishthis,we…rstshowhowonederivestheoptimalreservationpriceinasequentialsearchsetting.Supposeconsumersareconfrontedwithanondegeneratedistributionofprices;thatisatomless,exceptpossiblyatConsumersengageinoptimalsequentialsearchwithfreerecall.If,followingthethsearch,aconsumerhasalreadyfoundabestpricemin(;p;:::;pthen,bymakinganadditionalsearch,suchaconsumerexpectstogainbene…tsof)=(v(p)

�v(z))dF(p)=Zzpdp;wherethesecondequa
�v(z))dF(p)=Zzpdp;wherethesecondequalityobtainsthroughintegrationbyparts.UsingLeibnitz’rule,wehave)=KzThus,theexpectedbene…tsfromanadditionalsearcharelowerwhentheconsumerhasalreadyidenti…edarelativelylowprice.Sincesearchiscostlyc�,consumersmustweightheexpectedbene…tsagainstthecostofanadditionalsearch.Theexpectednetbene…tsofanadditionalsearchareIftheexpectedbene…tsfromanadditionalsearchexceedtheadditionalcost,itisoptimalfortheconsumertoobtainanadditionalpricequote.If,theconsumerisbettero¤purchasingatthepricethanobtaininganadditionalpricequote.Aconsumer’soptimalsequentialsearchstrategymaybesummarizedasfollows:Case1.p)0andRppdFc:Thentheconsumer’soptimalstrategyistonotsearch.Case2.p)0andRppdFdpThentheconsumer’soptimalstrategyistosearchuntilsheobtainsapricequoteatorbelowthereservationprice,p:11Ignoringforamomentthefactthata…rm’sdemandiszeroifitpricesabover;notethatpro…t-maximizationimpliesthe…rst-ordercondition(pj�mj)q0(pj)+q(pj)1�=0:Standardmanipulationofthe…rst-orderconditionforpro…t-maximizationimpliesthat…rmj’s(unconstrained)pro…t-maximizingpriceisaconstantmarkupoveritscost:pj="1+"mj:Supposethat…rmssimplyignoretheconsumer’sreservationprice,r;andpriceatthismarkup.Thiswouldimplythatconsumersfaceadistributionofpostedprices^F(p)=G(p(1+")=")ontheinterval[m"=(1+");m"=(1+")].Giventhisdistributionofprices,optimizingconsumerswouldsetareservationprice,r;suchthath(r)=Zrp(v(p)�v(r))d^F(p)�c=0Furthermore,ifrm"=(1+");…rmschargingpricesintheinterval(r;m"=(1+")]wouldenjoynosales.Sincetheelasticityofdemandisconstant,…rmsthatwouldmaximizepro…tsbypricingaboverintheabsenceofconsumersearch…nditoptimaltosettheirpricesatrwhenconsumerssearch.7Thus,thedistributionofprices,^F(p);isinconsistentwithoptimizingbehavioronthepartof…rms.Infact,giventhereservationpricer;optimizingbehavioronthepartof…rmswouldimplyadistributionofpricesF(p)=8:^F(p)ifpr1ifp=rToestablishthatthisis,infact,anequilibriumdistributionofpricesonemustverifythatconsumersfacingthis“truncated”distributionofpriceshavenoincentivetochangetheirreservationprice.Giventhistruncateddistributionofprices,thenetexpectedbene…tsofsearchareh(r)=Zrp(v(p)�v(r))dF(p)�c=Zrp(v(p)�v(r))d^F(p)+h1�^F(r)i[v(r)�v(r)]�c=Zrp(v(p)�v(r))d^F(p)�c=07Reinganumassumesthatmm"=(1+"),whichguaranteesthat…rmswhowouldotherwisepriceabover…nditpro…tabletopriceatr.13Ignoringforamomentthefactthata…rm’sdemandiszeroifitpricesabover;notethatpro…t-maximizationimpliesthe…rst-ordercondition)+=0Standardmanipulationofthe…rst-orderconditionforpro…t-maximizationimpliesthat…rm

’s(unconstrained)pro…t-maximiz
’s(unconstrained)pro…t-maximizingpriceisaconstantmarkupoveritscost:1+Supposethat…rmssimplyignoretheconsumer’sreservationprice,r;andpriceatthismarkup.Thiswouldimplythatconsumersfaceadistributionofpostedprices)=(1+ontheinterval(1+(1+.Giventhisdistributionofprices,optimizingconsumerswouldsetareservationprice,r;suchthat)==0Furthermore,ifr(1+…rmschargingpricesintheintervalr;(1+wouldenjoynosales.Sincetheelasticityofdemandisconstant,…rmsthatwouldmaximizepro…tsbypricingaboveintheabsenceofconsumersearch…nditoptimaltosettheirpricesatwhenconsumerssearch.Thus,thedistributionofprices,isinconsistentwithoptimizingbehavioronthepartof…rms.Infact,giventhereservationpricer;optimizingbehavioronthepartof…rmswouldimplyadistributionofprices)=prToestablishthatthisis,infact,anequilibriumdistributionofpricesonemustverifythatconsumersfacingthis“truncated”distributionofpriceshavenoincentivetochangetheirreservationprice.Giventhistruncateddistributionofprices,thenetexpectedbene…tsofsearchare)=dF)+=0Reinganumassumesthatmm(1+,whichguaranteesthat…rmswhowouldotherwisepriceabove…nditpro…tabletopriceatwhere^f(p)isthedensityof^F(p):Hence,d2dr=2r1�^F(r)�2 Zrpp^f(p)dp+1�^F(r)r!1�^F(r)=2h1�^F(r)i(r�E[p])0withstrictinequalityifrm"=(1+"):Thus,wehave:Conclusion1IntheReinganummodel,areductioninsearchcostsdecreasesthevarianceofequilibriumprices.Aswewillseebelow,however,thisisnotageneralpropertyofsearch-theoreticmodelsofpricedispersion.2.1.4RemarksonFixedversusSequentialSearchItisusefultohighlightsomekeydi¤erencesbetweensequentialand…xedsamplesizesearch.Withsequentialsearch,thenumberofsearchesisarandomvariablefromageometricdistribution,andtheexpectednumberofsearches,givenadistributionofpricesF(p)andreservationpricer,isE[n]=1F(r)Incontrast,with…xedsamplesizesearch,consumerscommitupfronttonsearches.Bothtypesofsearchhaveadvantagesanddisadvantages,andindeedMorganandManning(1985)haveshownthatbothtypesofsearchcanbeoptimalindi¤erentcircumstances.Thekeyadvantageofsequentialsearchisthatitallowsasearchertoeconomizeoninformationcosts–thedecision-makerweighstheexpectedbene…tsandcostsofgatheringadditionalpriceinformationaftereachnewpricequoteisobtained.Ifanacceptablepriceisobtainedearlyon,theexpectedgainsfromadditionalsearchesaresmallandthereisnoneedtopaythecostofadditionalsearches.Theprimaryadvantageof…xed-samplesizesearchisthatitallowsonetogatherinformationquickly.Consider,forinstance,a…rmthatrequiresrawmaterialsbytheendoftheweek.Ifittakesaweekforarawmaterialsvendortoprovideapricequote,sequentialsearchwouldpermitthe…rmtoobtainonlyapricequotefromasinglevendor.Inthiscase,…xedsamplesizesearchisoptimal—the…rmcommi

tstoobtainquotesfromnvendors,wherenischo
tstoobtainquotesfromnvendors,wherenischosenbythe…rmtominimizeexpectedcostsasoutlinedaboveinourdiscussionoftheStiglermodel.15whereisthedensityofHence,d=2=22p])0withstrictinequalityifr(1+Thus,wehave:Conclusion1IntheReinganummodel,areductioninsearchcostsdecreasesthevarianceofequilibriumprices.Aswewillseebelow,however,thisisnotageneralpropertyofsearch-theoreticmodelsofpricedispersion.2.1.4RemarksonFixedversusSequentialSearchItisusefultohighlightsomekeydi¤erencesbetweensequentialand…xedsamplesizesearch.Withsequentialsearch,thenumberofsearchesisarandomvariablefromageometricdistribution,andtheexpectednumberofsearches,givenadistributionofpricesandreservationprice,isisn]=Incontrast,with…xedsamplesizesearch,consumerscommitupfronttosearches.Bothtypesofsearchhaveadvantagesanddisadvantages,andindeedMorganandManning(1985)haveshownthatbothtypesofsearchcanbeoptimalindi¤erentcircumstances.Thekeyadvantageofsequentialsearchisthatitallowsasearchertoeconomizeoninformationcosts–thedecision-makerweighstheexpectedbene…tsandcostsofgatheringadditionalpriceinformationaftereachnewpricequoteisobtained.Ifanacceptablepriceisobtainedearlyon,theexpectedgainsfromadditionalsearchesaresmallandthereisnoneedtopaythecostofadditionalsearches.Theprimaryadvantageof…xed-samplesizesearchisthatitallowsonetogatherinformationquickly.Consider,forinstance,a…rmthatrequiresrawmaterialsbytheendoftheweek.Ifittakesaweekforarawmaterialsvendortoprovideapricequote,sequentialsearchwouldpermitthe…rmtoobtainonlyapricequotefromasinglevendor.Inthiscase,…xedsamplesizesearchisoptimal—the…rmcommitstoobtainquotesfromvendors,whereischosenbythe…rmtominimizeexpectedcostsasoutlinedaboveinourdiscussionoftheStiglermodel.Noticethat,afterintegrationbyparts,wecanrewriteequation(6)toobtainthefamiliarformulaforequilibriumbiddinginreverse…rst-priceauctionsp(m)=Ehm(n�1)minjm(n�1)minmi(7)wherem(n�1)ministhelowestofn�1drawsfromthedistributionG:ForthespecialcasewhereGisuniformlydistributed,theequilibriumpricingstrategysimpli…estop(m)=n�1nm+1nm:(8)Noticethattheequilibriumpricingstrategygivesrisetoadistributionofpostedprices,F(p),inducedbythedistributionofcosts;thatisF(p)=G(p(m))Forthistobeanequilibriumdistributionofprices,itmustbeoptimalforconsumerstosamplen…rms.Thatis,EhB(n+1)icEhB(n)iwheretheexpressionEB(n),aspreviouslyde…nedinequation(1)whenK=1,istheexpectedbene…tfromincreasingthenumberofpricequotesobtainedfromn�1ton:AsintheStiglermodel,areductioninsearchcostsincreasestheoptimalsamplesizen(sothatconsumersoptimallysamplemore…rms).Thus,MacMinnshowsthat,providedsearchcostsarelowenough,adispersedpriceequilibriumexists.Thisnotonlyleadstoexpostdi¤erencesinco

nsumers’informationsets(di¤erentco
nsumers’informationsets(di¤erentconsumerssampledi¤erent…rmsandsoobservedi¤erentprices),butinducesadegreeofcompetitionamong…rms(sincetheyarecompetingagainstatleastoneother…rm,whosecosttheydonotknow).AsintheReinganummodel,thelevelofpricedispersiondependsonthedispersionin…rms’costs.Forthespecialcasewherecostsareuniformlydistributed,thevarianceinequilibriumprices�2p
isgivenby2p=n�1n22m(9)wherenistheoptimalnumberofsearchesbyconsumersand2misthevariancein…rm’scosts.Twointerestingresultsemergefromthemodel.First,thevarianceinpricesincreasesasthevariancein…rms’marginalcostsincreases.Thisresultisintuitive.Somewhatcounterintuitively,17Noticethat,afterintegrationbyparts,wecanrewriteequationtoobtainthefamiliarformulaforequilibriumbiddinginreverse…rst-priceauctions)=minminwhereministhelowestofdrawsfromthedistributionForthespecialcasewhereisuniformlydistributed,theequilibriumpricingstrategysimpli…es)=nm+1nNoticethattheequilibriumpricingstrategygivesrisetoadistributionofpostedprices,inducedbythedistributionofcosts;thatis)=Forthistobeanequilibriumdistributionofprices,itmustbeoptimalforconsumerstosample…rms.Thatis,+1)cwheretheexpression,aspreviouslyde…nedinequationwhen=1,istheexpectedbene…tfromincreasingthenumberofpricequotesobtainedfromAsintheStiglermodel,areductioninsearchcostsincreasestheoptimalsamplesize(sothatconsumersoptimallysamplemore…rms).Thus,MacMinnshowsthat,providedsearchcostsarelowenough,adispersedpriceequilibriumexists.Thisnotonlyleadstoexpostdi¤erencesinconsumers’informationsets(di¤erentconsumerssampledi¤erent…rmsandsoobservedi¤erentprices),butinducesadegreeofcompetitionamong…rms(sincetheyarecompetingagainstatleastoneother…rm,whosecosttheydonotknow).AsintheReinganummodel,thelevelofpricedispersiondependsonthedispersionin…rms’costs.Forthespecialcasewherecostsareuniformlydistributed,thevarianceinequilibriumpricesisgivenbywhereistheoptimalnumberofsearchesbyconsumersandisthevariancein…rm’scosts.Twointerestingresultsemergefromthemodel.First,thevarianceinpricesincreasesasthevariancein…rms’marginalcostsincreases.Thisresultisintuitive.Somewhatcounterintuitively,obtainingasinglepricequote:13IntheBurdettandJuddmodel,anequilibriumconsistsofapricedistributionF(p)(basedonoptimalpricingdecisionsby…rms)andanoptimalsearchdistributionn&#x-278;1n=1,wheren&#x-278;1n=1isthedistributionofthenumberoftimesaconsumersearchesinthepopulation.Thus,iistheprobabilitythataconsumersearches(oralternatively,thefractionofconsumersthatsearch)exactlyi…rms.If1=1;thenallconsumerssampleonlyone…rm.If1=0;thenallconsumerssampleatleasttwo…rms,andsoon.Consumerspurchas

efromthe…rmsampledthato¤ersthelowe
efromthe…rmsampledthato¤ersthelowestprice.WebeginbystudyingoptimalsearchonthepartofconsumersgivenapricedistributionF(p):Recallthattheexpectedbene…ttoaconsumerwhoincreaseshersamplesizefromn�1tonisEhB(n)i=Ehp(n�1)mini�Ehp(n)miniasintheStiglermodel.Moreover,theexpectedbene…tscheduleisstrictlydecreasinginn:Thus,anoptimalnumberofpricequotes,n;satis…esEhB(n+1)icEhB(n)iFirstconsiderthecasewhereallconsumersobtaintwoormorepricequotes;thatis,where1=0:Inthiscase,theoptimalpricingstrategyonthepartof…rmsistopriceatmarginalcost(theBertrandparadox)sinceeach…rmisfacingpurepricecompetitionwithatleastoneother…rmandall…rmsareidentical.Ofcourse,ifall…rmsarepricingatmarginalcost,thenitwouldbeoptimalforaconsumertosampleonlyone…rm,whichcontradictsthehypothesisthat1=0.Thus,wemayconcludethat,inanyequilibrium1&#x-278;0:Next,considerthecasewhereconsumersallobtainexactlyonepricequote.Inthatcase,each…rmwouldoptimallychargethemonopolyprice,p=v:Hence,16=1inanydispersedpriceequilibrium.Fromthesetwoargumentsitfollowsthat,inanydispersedpriceequilibrium,12(0;1).Inlightofthefactthatconsumers’expectedbene…tsfromsearcharedecreasinginthesamplesize,it13Theseassumptionsaresatis…ed,forexample,whenq(p)=8���&#x]TJ ;� -1;.13; Td;&#x[000;&#x]TJ ;� -1;.13; Td;&#x[000;&#x]TJ ;� -1;.13; Td;&#x[000;:1ifpv1�p�vifvpv+0ifp�v+and�c=2:19obtainingasinglepricequote13IntheBurdettandJuddmodel,anequilibriumconsistsofapricedistribution(basedonoptimalpricingdecisionsby…rms)andanoptimalsearchdistribution,whereisthedistributionofthenumberoftimesaconsumersearchesinthepopulation.Thus,istheprobabilitythataconsumersearches(oralternatively,thefractionofconsumersthatsearch)exactly…rms.If=1thenallconsumerssampleonlyone…rm.If=0thenallconsumerssampleatleasttwo…rms,andsoon.Consumerspurchasefromthe…rmsampledthato¤ersthelowestprice.WebeginbystudyingoptimalsearchonthepartofconsumersgivenapricedistributionRecallthattheexpectedbene…ttoaconsumerwhoincreaseshersamplesizefromminminasintheStiglermodel.Moreover,theexpectedbene…tscheduleisstrictlydecreasinginThus,anoptimalnumberofpricequotes,satis…es+1)cFirstconsiderthecasewhereallconsumersobtaintwoormorepricequotes;thatis,where=0Inthiscase,theoptimalpricingstrategyonthepartof…rmsistopriceatmarginalcost(theBertrandparadox)sinceeach…rmisfacingpurepricecompetitionwithatleastoneother…rmandall…rmsareidentical.Ofcourse,ifall…rmsarepricingatmarginalcost,thenitwouldbeoptimalforaconsumertosampleonly…rm,whichcontradictsthehypothesisthat=0Thus,wemayconcludethat,inanyequilibriumNext,considerthecasewhereconsume

rsallobtainexactlyonepricequote.Inthatca
rsallobtainexactlyonepricequote.Inthatcase,each…rmwouldoptimallychargethemonopolyprice,v:Hence,=1inanydispersedpriceequilibrium.Fromthesetwoargumentsitfollowsthat,inanydispersedpriceequilibrium,.Inlightofthefactthatconsumers’expectedbene…tsfromsearcharedecreasinginthesamplesize,it13Theseassumptionsaresatis…ed,forexample,when)=ifpvififp�vand�c=Tosummarize,BurdettandJuddshowthatequilibriumpricedispersioncanariseevenwhenall…rmsandconsumersareexanteidentical.Intheequilibriumpricedistribution,all…rmschargepositivemarkups.Afractionofconsumersdonotcomparisonshop—theysimplysearchatonestoreandpurchase.Theremainingfractionofconsumersare“shoppers”—theseconsumerssearchattwostoresandbuyfromwhichevero¤ersthelowerprice.2.2Modelswithan“InformationClearinghouse”Insearch-theoreticmodels,consumerspayanincrementalcostforeachadditionalpricequotetheyobtain.Thesemodelsarerelevant,forexample,whenconsumersmustvisitorphonetraditionalsellersinordertogatherinformationaboutprices.Theyarealsorelevantinonlineenvironmentswhereconsumersmustsearchthewebsitesofindividualretailerstogatherinformationaboutthepricestheycharge.Analternativeclassofmodelsisrelevantwhenathirdparty—aninformationclearinghouse—providesasubsetofconsumerswithalistofpriceschargedbydi¤erent…rmsinthemarket.Examplesofthisenvironmentincludenewspaperswhichdisplaypricesdi¤erentstoreschargeforthesameproductorserviceandonlinepricecomparisonsites.Inthissectionweprovideageneraltreatmentofclearinghousemodels,andshowthatthesemodelsaresurprisinglysimilartothosethatariseunder…xedsamplesizesearch.Oneofthekeymodelingdi¤erencesisthatclearinghousemodelstendtobeoligopolymodels;thus,thereisnotacontinuumof…rmsinsuchsettings.Wherepossible,weshallusethesamenotationasintheprevioussection;however,forreasonsthatwillbecomeclearwhenwecompareclearinghousemodelswiththesearchmodelspresentedabove,wenowletndenotethenumberof…rmsinthemarket.ThegeneraltreatmentthatfollowsreliesheavilyonBayeandMorgan(2001)andBaye,MorganandScholten(2004a).Considerthefollowinggeneralenvironment(whichwewillspecializetocoveravarietyofdi¤er-entmodels).Thereisa…nitenumber,n�1;ofprice-setting…rmscompetinginamarketsellinganidentical(homogeneous)product.Firmshaveunlimitedcapacitytosupplythisproductataconstantmarginalcost,m:Acontinuumofconsumersisinterestedinpurchasingtheproduct.Thismarketisservedbyapriceinformationclearinghouse.Firmsmustdecidewhatpricetochargefortheproductandwhethertolistthispriceattheclearinghouse.Letpidenotethepricechargedby…rmi:Itcostsa…rmanamount0ifitchoosestolistitsprice.Allconsumershaveunitdemand21Tosummarize,BurdettandJuddshowthatequilibriumpricedispersioncanariseevenwhenall…rmsandconsumersareexanteiden

tical.Intheequilibriumpricedistribution,
tical.Intheequilibriumpricedistribution,all…rmschargepositivemarkups.Afractionofconsumersdonotcomparisonshop—theysimplysearchatonestoreandpurchase.Theremainingfractionofconsumersare“shoppers”—theseconsumerssearchattwostoresandbuyfromwhichevero¤ersthelowerprice.2.2Modelswithan“InformationClearinghouse”Insearch-theoreticmodels,consumerspayanincrementalcostforeachadditionalpricequotetheyobtain.Thesemodelsarerelevant,forexample,whenconsumersmustvisitorphonetraditionalsellersinordertogatherinformationaboutprices.Theyarealsorelevantinonlineenvironmentswhereconsumersmustsearchthewebsitesofindividualretailerstogatherinformationaboutthepricestheycharge.Analternativeclassofmodelsisrelevantwhenathirdparty—aninformationclearinghouse—providesasubsetofconsumerswithalistofpriceschargedbydi¤erent…rmsinthemarket.Examplesofthisenvironmentincludenewspaperswhichdisplaypricesdi¤erentstoreschargeforthesameproductorserviceandonlinepricecomparisonsites.Inthissectionweprovideageneraltreatmentofclearinghousemodels,andshowthatthesemodelsaresurprisinglysimilartothosethatariseunder…xedsamplesizesearch.Oneofthekeymodelingdi¤erencesisthatclearinghousemodelstendtobeoligopolymodels;thus,thereisnotacontinuumof…rmsinsuchsettings.Wherepossible,weshallusethesamenotationasintheprevioussection;however,forreasonsthatwillbecomeclearwhenwecompareclearinghousemodelswiththesearchmodelspresentedabove,wenowletdenotethenumberof…rmsinthemarket.ThegeneraltreatmentthatfollowsreliesheavilyonBayeandMorgan(2001)andBaye,MorganandScholten(2004a).Considerthefollowinggeneralenvironment(whichwewillspecializetocoveravarietyofdi¤er-entmodels).Thereisa…nitenumber,n�ofprice-setting…rmscompetinginamarketsellinganidentical(homogeneous)product.Firmshaveunlimitedcapacitytosupplythisproductataconstantmarginalcost,Acontinuumofconsumersisinterestedinpurchasingtheproduct.Thismarketisservedbyapriceinformationclearinghouse.Firmsmustdecidewhatpricetochargefortheproductandwhethertolistthispriceattheclearinghouse.Letdenotethepricechargedby…rmItcostsa…rmanamountifitchoosestolistitsprice.AllconsumershaveunitdemandThisconditionholds,sincen�1n(v�m)S:Noticethatp0�m;providedthatL�0or�0:Inthiscase,itcanbeshownthatFisawell-de…ned,atomlesscdfon[p0;v].WhenL=0and=0,noticethatp0=m.Inthiscase,thesymmetricequilibriumdistributionofpricesisdegenerate,withall…rmspricingatmarginalcost(theBertrandparadoxoutcome).Next,weshowthat,conditionalonlistingaprice,a…rmcandonobetterthanpricingaccordingtoF:ItisobviousthatchoosingapriceaboveorbelowthesupportofFisdominatedbychoosingapriceinthesupportofF:A…rmchoosingapricepinthesupportofFearnsexpectedpro…tsofE(p)=(p�m) L+ n�1Xi=0n�1i

;i(1�)n�1�i(1�F(p)
;i(1�)n�1�i(1�F(p))i!S!�:Usingthebinomialtheorem,wecanrewritethisas:E(p)=(p�m)L+(1�F(p))n�1S�=(v�m)L+n�1;wherewehavesubstitutedforFtoobtainthesecondequality.Sincea…rm’sexpectedpro…tsareconstanton[p0;v],itfollowsthatthemixedpricingstrategy,F;isabestresponsetotheothern�1…rmspricingbasedonF:When=0;itisaweaklydominantstrategytolist.Itremainstoshowthatwhen�0and2(0;1),a…rmearnsthesameexpectedpro…tsregardlessofwhetheritlistsitsprice.Buta…rmthatdoesnotlistearnsexpectedpro…tsofE=(v�m)L+Sn(1�)n�1=(v�m)L+n�1;whichequalstheexpectedpro…tsearnedbylistinganypricep2[p0;v].Wearenowinapositiontoexaminethemanywell-knownclearinghousemodelsthatemergeasspecialcasesofthisgeneralenvironment.2.2.1TheRosenthalModelRosenthal(1980)wasamongthe…rsttoshowthatequilibriumpricedispersioncanariseinaclearinghouseenvironmentwhensomeconsumershaveapreferenceforaparticular…rm.Underhisinterpretation,each…rmenjoysamassLof“loyal”consumers.Rosenthal’smainresultsmaybeseeninthefollowingspecialcaseofthegeneralclearinghousemodel:23Thisconditionholds,sinceS:Noticethat�m;providedthatL��Inthiscase,itcanbeshownthatisawell-de…ned,atomlesscdfononp0;v.When=0=0,noticethat.Inthiscase,thesymmetricequilibriumdistributionofpricesisdegenerate,withall…rmspricingatmarginalcost(theBertrandparadoxoutcome).Next,weshowthat,conditionalonlistingaprice,a…rmcandonobetterthanpricingaccordingF:ItisobviousthatchoosingapriceaboveorbelowthesupportofisdominatedbychoosingapriceinthesupportofF:A…rmchoosingapriceinthesupportofearnsexpectedpro…tsofE)=(Usingthebinomialtheorem,wecanrewritethisas:E)=(F=(wherewehavesubstitutedfortoobtainthesecondequality.Sincea…rm’sexpectedpro…tsareconstantononp0;v,itfollowsthatthemixedpricingstrategy,F;isabestresponsetotheother…rmspricingbasedonF:=0itisaweaklydominantstrategytolist.Itremainstoshowthatwhen�,a…rmearnsthesameexpectedpro…tsregardlessofwhetheritlistsitsprice.Buta…rmthatdoesnotlistearnsexpectedpro…tsofE=(=(whichequalstheexpectedpro…tsearnedbylistinganypriceicep0;vWearenowinapositiontoexaminethemanywell-knownclearinghousemodelsthatemergeasspecialcasesofthisgeneralenvironment.2.2.1TheRosenthalModelRosenthal(1980)wasamongthe…rsttoshowthatequilibriumpricedispersioncanariseinaclearinghouseenvironmentwhensomeconsumershaveapreferenceforaparticular…rm.Underhisinterpretation,each…rmenjoysamassof“loyal”consumers.Rosenthal’smainresultsmaybeseeninthefollowingspecialcaseofthegeneralclearinghousemodel:bringsmoreloyals

intothemarket.Indeed,thefractionofshoppe
intothemarket.Indeed,thefractionofshoppersinthemarketisS=(S+nL)anditmayreadilybeseenthatasnbecomeslarge,shoppersaccountforanincreasinglysmallfractionofthecustomerbaseof…rms.Asaconsequence,theincentivestocompeteforthesecustomersisattenuatedandpricesriseasaresult.Thekeyistorecognizethatincreasesinnchangethedistributionofprices,andthise¤ectaswellasanyorderstatistice¤ectassociatedwithanincreaseinnmustbetakenintoaccount.Formally,noticethattheequilibriumdistributionofprices,F;isstochasticallyorderedinn:Thatis,thedistributionofpriceswhentherearen+1…rmscompeting…rst-orderstochasticallydominatesthedistributionofpriceswheretherearen…rmscompeting.Thisimpliesthatthetransactionspricespaidbyloyalsincreaseinn.Toshowthatthetransactionspricespaidbyshoppersalsoincreaseinnrequiresabitmorework;however,onecanshowthatthesamestochasticorderingobtainsforthecdfF(n)min(p):Finally,itisusefultonotethesimilaritybetweentheRosenthalversionoftheclearinghousemodelandthesearch-theoreticmodelofBurdettandJudd.InBurdettandJudd,eventhoughthereisacontinuumof…rms,eachconsumeronlysamplesa…nitenumberof…rms(oneortwo).Further,inBurdettandJudd,a…xedfractionofconsumersper…rm,;sampleonlyasingle…rm.Ine¤ect,theseconsumersare“loyal”tothesingle…rmsampledwhilethefraction(1�)ofcustomerssamplingtwo…rmsare“shoppers”—theychoosethelowerofthetwoprices.Forthisreason,whenn=2intheRosenthalmodel,theequilibriumpricedistributiongiveninequation(11)isidenticaltoequation(10)intheBurdettandJuddmodel(modulorelabelingthevariablesforloyalsandshoppers).2.2.2TheVarianModelVarian(1980)wasamongthe…rsttoshowthatequilibriumpricedispersioncanariseinaclear-inghouseenvironmentwhenconsumershavedi¤erentexanteinformationsets.16VarianinterpretstheSconsumersas“informedconsumers”andtheLconsumersas“uninformed”consumers.Thusamass,S;ofconsumerschoosetoaccesstheclearinghousewhileothers,themassLper…rm,donot.Varian’smainresultmaybeseeninthefollowingspecialcaseofthegeneralclearinghousemodel:1.Itiscostlessfor…rmstolistpricesontheclearinghouse:=0;and16PngandHirshleifer(1987),aswellasBayeandKovenock(1994),extendtheVarianmodelbyallowing…rmstoalsoengageinpricematchingor“beatorpay”advertisements.25bringsmoreloyalsintothemarket.Indeed,thefractionofshoppersinthemarketisS=anditmayreadilybeseenthatasbecomeslarge,shoppersaccountforanincreasinglysmallfractionofthecustomerbaseof…rms.Asaconsequence,theincentivestocompeteforthesecustomersisattenuatedandpricesriseasaresult.Thekeyistorecognizethatincreasesinchangethedistributionofprices,andthise¤ectaswellasanyorderstatistice¤ectassociatedwithanincreaseinmustbetakenintoaccount.Formally,noticethattheequilibriumdistributionofprices,F;isstochasti

callyorderedinThatis,thedistributionofpr
callyorderedinThatis,thedistributionofpriceswhenthereare+1…rmscompeting…rst-orderstochasticallydominatesthedistributionofpriceswherethereare…rmscompeting.Thisimpliesthatthetransactionspricespaidbyloyalsincreasein.Toshowthatthetransactionspricespaidbyshoppersalsoincreaseinrequiresabitmorework;however,onecanshowthatthesamestochasticorderingobtainsforthecdfminFinally,itisusefultonotethesimilaritybetweentheRosenthalversionoftheclearinghousemodelandthesearch-theoreticmodelofBurdettandJudd.InBurdettandJudd,eventhoughthereisacontinuumof…rms,eachconsumeronlysamplesa…nitenumberof…rms(oneortwo).Further,inBurdettandJudd,a…xedfractionofconsumersper…rm,;sampleonlyasingle…rm.Ine¤ect,theseconsumersare“loyal”tothesingle…rmsampledwhilethefractionofcustomerssamplingtwo…rmsare“shoppers”—theychoosethelowerofthetwoprices.Forthisreason,when=2intheRosenthalmodel,theequilibriumpricedistributiongiveninequationidenticaltoequationintheBurdettandJuddmodel(modulorelabelingthevariablesforloyalsandshoppers).2.2.2TheVarianModelVarian(1980)wasamongthe…rsttoshowthatequilibriumpricedispersioncanariseinaclear-inghouseenvironmentwhenconsumershavedi¤erentexanteinformationsets.16Varianinterpretsconsumersas“informedconsumers”andtheconsumersas“uninformed”consumers.Thusamass,S;ofconsumerschoosetoaccesstheclearinghousewhileothers,themassper…rm,donot.Varian’smainresultmaybeseeninthefollowingspecialcaseofthegeneralclearinghousemodel:1.Itiscostlessfor…rmstolistpricesontheclearinghouse:=0;and16PngandHirshleifer(1987),aswellasBayeandKovenock(1994),extendtheVarianmodelbyallowing…rmstoalsoengageinpricematchingor“beatorpay”advertisements.equilibrium—theBertrandparadox.Thus,forsu¢cientlyhighorlowinformationcosts,thereisnopricedispersion;formoderateinformationcosts,pricesaredispersedonthenondegenerateinterval[p0;v].AsimilarresultobtainsinStahl(1989),whichisrelatedtoVarianasfollows.Stahlassumesafractionofconsumershavezerosearchcostsand,asaconsequence,viewall…rms’pricesandpurchaseatthelowestpriceinthemarket.TheseconsumersplaytheroleofSinVarian’smodel(informedconsumers).TheremainingfractionofconsumerscorrespondtotheL’sintheVarianmodel,butratherthanremainingentirelyuninformed,theseconsumersengageinoptimalsequen-tialsearchinpresenceofpositiveincrementalsearchcosts.Stahlshowsthatwhenallconsumersareshoppers,theidentical…rmspriceatmarginalcostandthereisnopricedispersion.Whennoconsumersareshoppers,Diamond’sparadoxobtainsandall…rmschargethemonopolyprice.Asthefractionofshoppersvariesfromzerotoone,thelevelofdispersionvariescontinuously—fromzerotopositivelevels,andbackdowntozero.Conclusion3Ingeneral,pricedispersionisnotamonotonicfuncti

onofconsumers’informationcostsorthe
onofconsumers’informationcostsorthefractionof“shoppers”inthemarket.Howdoesthenumberofcompeting…rmsa¤ecttransactionsprices?IntheRosenthalmodel,wesawthatincreased“competition”ledtohigherexpectedtransactionspricesforallconsumers.IntheVarianmodel,incontrast,thee¤ectofcompetitiononconsumerwelfaredependsonwhetherornottheconsumerchoosestoaccesstheclearinghouse.Morgan,Orzen,andSefton(forthcoming)showthatasnincreases,thecompetitivee¤ectpredictablyleadstoloweraveragetransactionpricesbeingpaidbyinformedconsumers.However,theoppositeistrueforuninformedconsumers—asthenumberofcompeting…rmsincreases,…rmsfacereducedincentivestocutpricesinhopesofattractingthe“shoppers”and,asaconsequence,theaveragepricechargedbya…rm,whichisalsotheaveragepricepaidbyanuninformedconsumer,increases.IfoneviewstheclearinghouseasrepresentingaccesstopriceinformationontheInternet,thenonecaninterpretthepricee¤ectasoneconsequenceoftheso-called“digitaldivide;”seeBaye,Morgan,andScholten(2003).ConsumerswithInternetaccessaremadebettero¤bysharperonlinecompetitionwhilethosewithoutsuchaccessaremadeworseo¤.2.2.3TheBayeandMorganModelAlloftheabovemodelsassumethatitiscostlessfor…rmstoadvertisetheirpricesattheclear-inghouse.BayeandMorgan(2001)pointoutthat,inpractice,itisgenerallycostlyfor…rmsto27equilibrium—theBertrandparadox.Thus,forsu¢cientlyhighorlowinformationcosts,thereisnopricedispersion;formoderateinformationcosts,pricesaredispersedonthenondegenerateintervalalp0;v.AsimilarresultobtainsinStahl(1989),whichisrelatedtoVarianasfollows.Stahlassumesafractionofconsumershavezerosearchcostsand,asaconsequence,viewall…rms’pricesandpurchaseatthelowestpriceinthemarket.TheseconsumersplaytheroleofinVarian’smodel(informedconsumers).Theremainingfractionofconsumerscorrespondtothe’sintheVarianmodel,butratherthanremainingentirelyuninformed,theseconsumersengageinoptimalsequen-tialsearchinpresenceofpositiveincrementalsearchcosts.Stahlshowsthatwhenallconsumersareshoppers,theidentical…rmspriceatmarginalcostandthereisnopricedispersion.Whennoconsumersareshoppers,Diamond’sparadoxobtainsandall…rmschargethemonopolyprice.Asthefractionofshoppersvariesfromzerotoone,thelevelofdispersionvariescontinuously—fromzerotopositivelevels,andbackdowntozero.Conclusion3Ingeneral,pricedispersionisnotamonotonicfunctionofconsumers’informationcostsorthefractionof“shoppers”inthemarket.Howdoesthenumberofcompeting…rmsa¤ecttransactionsprices?IntheRosenthalmodel,wesawthatincreased“competition”ledtohigherexpectedtransactionspricesforallconsumers.IntheVarianmodel,incontrast,thee¤ectofcompetitiononconsumerwelfaredependsonwhetherornottheconsumerchoosestoaccesstheclearinghouse.Morgan,Orzen,andSefton(forthcomin

g)showthatasincreases,thecompetitivee¤e
g)showthatasincreases,thecompetitivee¤ectpredictablyleadstoloweraveragetransactionpricesbeingpaidbyinformedconsumers.However,theoppositeistrueforuninformedconsumers—asthenumberofcompeting…rmsincreases,…rmsfacereducedincentivestocutpricesinhopesofattractingthe“shoppers”and,asaconsequence,theaveragepricechargedbya…rm,whichisalsotheaveragepricepaidbyanuninformedconsumer,increases.IfoneviewstheclearinghouseasrepresentingaccesstopriceinformationontheInternet,thenonecaninterpretthepricee¤ectasoneconsequenceoftheso-called“digitaldivide;”seeBaye,Morgan,andScholten(2003).ConsumerswithInternetaccessaremadebettero¤bysharperonlinecompetitionwhilethosewithoutsuchaccessaremadeworseo¤.2.2.3TheBayeandMorganModelAlloftheabovemodelsassumethatitiscostlessfor…rmstoadvertisetheirpricesattheclear-inghouse.BayeandMorgan(2001)pointoutthat,inpractice,itisgenerallycostlyfor…rmstoUndertheseconditions,usingProposition3,weobtainthefollowingcharacterizationofequilib-rium…rmpricingandlistingdecisions:Each…rmlistsitspriceattheclearinghousewithprobability=1�nn�1(v�m)S1n�12(0;1)Whena…rmlistsattheclearinghouse,itchargesapricedrawnfromthedistributionF(p)=1 1�nn�1(p�m)S1n�1!on[p0;v];wherep0=m+nn�1S:Whena…rmdoesnotlistitsprice,itchargesapriceequaltov;andeach…rmearnsequilibriumexpectedpro…tsequaltoE=1n�1Noticethatnrepresentstheaggregatedemandby…rmsforadvertisingandisadecreasingfunctionofthefeechargedbythegatekeeper.Pricesadvertisedattheclearinghousearedispersedandstrictlylowerthanunadvertisedprices(v).Severalfeaturesofthisequilibriumareworthnoting.First,equilibriumpricedispersionariseswithfullyoptimizingconsumers,…rms,andendogenousfee-settingdecisionsonthepartoftheclearinghouse–despitethefactthattherearenoconsumeror…rmheterogeneitiesandallconsumersare“fullyinformed”inthesensethat,inequilibrium,theyalwayspurchasefroma…rmchargingthelowestpriceintheglobalmarket.Second,whileequilibriumpricedispersionintheVarianmodelisdrivenbythefactthatdi¤erentconsumershavedi¤erentcostsofaccessingtheclearinghouse,BayeandMorganshowthatanoptimizingclearinghousewillsetitsfeessu¢cientlylowthatallconsumerswillrationallyaccesstheclearinghouse.Equilibriumpricedispersionarisesbecauseofthegatekeeper’sincentivestosetstrictlypositiveadvertisingfees.Strikingly,despitethefactthatallconsumersusethegatekeeper’ssiteandthuspurchaseatthelowestglobalprice,…rmsstillearnpositivepro…tsinequilibrium.Inexpectation,thesepro…tsareproportionaltothecost,;ofaccessingtheclearinghouse.Conclusion4IntheBayeandMorganmodel,equilibriumpricedispersionpersistsevenwhenitiscostlessforallconsumerstoaccesstheinfor

mationpostedatthegatekeeper’ssite.I
mationpostedatthegatekeeper’ssite.Indeed,pricedispersionexistsbecauseitiscostlyfor…rmstotransmitpriceinformation(advertiseprices)atthegatekeeper’ssite.29Undertheseconditions,usingProposition3,weobtainthefollowingcharacterizationofequilib-rium…rmpricingandlistingdecisions:Each…rmlistsitspriceattheclearinghousewithprobability=1n�1(v�m)S1Whena…rmlistsattheclearinghouse,itchargesapricedrawnfromthedistribution)= 1�nn�1(p�m)S1p0;vwheren�1Whena…rmdoesnotlistitsprice,itchargesapriceequaltov;andeach…rmearnsequilibriumexpectedpro…tsequaltoENoticethatrepresentstheaggregatedemandby…rmsforadvertisingandisadecreasingfunctionofthefeechargedbythegatekeeper.PricesadvertisedattheclearinghousearedispersedandstrictlylowerthanunadvertisedpricesSeveralfeaturesofthisequilibriumareworthnoting.First,equilibriumpricedispersionariseswithfullyoptimizingconsumers,…rms,andendogenousfee-settingdecisionsonthepartoftheclearinghouse–despitethefactthattherearenoconsumeror…rmheterogeneitiesandallconsumersare“fullyinformed”inthesensethat,inequilibrium,theyalwayspurchasefroma…rmchargingthelowestpriceintheglobalmarket.Second,whileequilibriumpricedispersionintheVarianmodelisdrivenbythefactthatdi¤erentconsumershavedi¤erentcostsofaccessingtheclearinghouse,BayeandMorganshowthatanoptimizingclearinghousewillsetitsfeessu¢cientlylowthatallconsumerswillrationallyaccesstheclearinghouse.Equilibriumpricedispersionarisesbecauseofthegatekeeper’sincentivestosetstrictlypositiveadvertisingfees.Strikingly,despitethefactthatallconsumersusethegatekeeper’ssiteandthuspurchaseatthelowestglobalprice,…rmsstillearnpositivepro…tsinequilibrium.Inexpectation,thesepro…tsareproportionaltothecost,accessingtheclearinghouse.Conclusion4IntheBayeandMorganmodel,equilibriumpricedispersionpersistsevenwhenitiscostlessforallconsumerstoaccesstheinformationpostedatthegatekeeper’ssite.Indeed,pricedispersionexistsbecauseitiscostlyfor…rmstotransmitpriceinformation(advertiseprices)atthegatekeeper’ssite.3.Ifa…rmdoesnotlistitspriceattheclearinghouse,itchargesapriceequaltov:4.Firmiearnsequilibriumexpectedpro…tsequaltoEi=(v�m)Li+Proof.LetAi2f0;1gdenoteanindicatorvariableforwhether…rmiadvertisesitspriceattheclearinghouse.LetidenotetheprobabilitythatAi=1:Thus,…rmi’sexpectedpro…tsfromeachofthetwolistingdecisionsgiventhestrategy(j;Fj)ofitsrivalareE[i(pjAi=0)]=(Li+(1�j)S2)(p�m)andE[i(pjAi=1)]=(Li+S(1�jFj(p)))(p�m)�Fromthe…rstequation,itisclearthatthedominantstrategyofa…rmnotlistingitspriceistochargethemonopolyprice,v:Furthermore,i2(0;1)requiresthatE[i(vj

Ai=0)]=E[i(pjAi=1)](13)forpinthesup
Ai=0)]=E[i(pjAi=1)](13)forpinthesupportofFi:Supposethatthemonopolyprice,v;istheupperboundofthesupportofprices.Inthiscase,theaboveequalityreducesto:(Li+(1�j)S2)(v�m)=(Li+S(1�j))(v�m)�Theuniquesolutiontothisequationis1=2=;whereisde…nedinProposition4.Letp0;idenotethepricesuchthat,if…rmichargedthelowestpriceinthemarketandattractedallshoppers,itsexpectedpro…tsatpricep0;iwouldexactlyequalthepro…tsgainedbysimplypricingatv.Thispricesatis…es(Li+(1�)S2)(v�m)=(Li+S)(p0;i�m)�Substitutingforandsolvingyieldsp0;i=m+Li(v�m)+2Li+SNoticethatp0;1�p0;2;thatis,thelowest“sale”priceo¤eredtoattractshoppersishigherforthelarge…rmthanforthesmall…rm.Sinceequilibriumpricedistributionsmusthaveidenticalsupports,itfollowsthatFi(p)hassupport[p0;1;v]foralli:Finally,substitutingtheexpression313.Ifa…rmdoesnotlistitspriceattheclearinghouse,itchargesapriceequaltov:4.Firmearnsequilibriumexpectedpro…tsequaltoE=(Proof.2fdenoteanindicatorvariableforwhether…rmadvertisesitspriceattheclearinghouse.Letdenotetheprobabilitythat=1Thus,…rm’sexpectedpro…tsfromeachofthetwolistingdecisionsgiventhestrategy;Fofitsrivalareei(pjAi=0)]=(+(1i(pjAi=1)]=()))(Fromthe…rstequation,itisclearthatthedominantstrategyofa…rmnotlistingitspriceistochargethemonopolyprice,v:Furthermore,requiresthatthati(vjAi=0)]==i(pjAi=1)]inthesupportofSupposethatthemonopolyprice,v;istheupperboundofthesupportofprices.Inthiscase,theaboveequalityreducesto:+(1)=(Theuniquesolutiontothisequationis;whereisde…nedinProposition4.denotethepricesuchthat,if…rmchargedthelowestpriceinthemarketandattractedallshoppers,itsexpectedpro…tsatpricewouldexactlyequalthepro…tsgainedbysimplypricing.Thispricesatis…es+(1)=()(Substitutingforandsolvingyields)+2Noticethat�p;thatis,thelowest“sale”priceo¤eredtoattractshoppersishigherforthelarge…rmthanforthesmall…rm.Sinceequilibriumpricedistributionsmusthaveidenticalsupports,itfollowsthathassupportrtp0;1;vforallFinally,substitutingtheexpressionEquation(14)showsthatthegatekeeperoptimallychargesafeeto…rmsadvertisingpricesthatisproportionaltotheconsumertra¢c(S)onitssite.Thisfeeinducesequilibriumpricedispersion;indeed,as…rstnotedbyBayeandMorgan,pricedispersionisnecessaryinorderforthegatekeepertopro…tablymaintainaclearinghouse.2.2.5CostHeterogeneitiesandtheSpulberModelSpulber(1995)considersasituationwhereconsumershaveaccesstothecompletelistofpricesandbuyfromthe…rmo¤eringthelowestprice.Ofcourse,insuchasetting,if…rmswereidenticalonewouldimmediatelyobtaintheBertrandoutcome.Togeneratepricedispersion,Spulberexaminesthesituationwhere…rmshavehe

terogeneouscostsandconsumershavedownward
terogeneouscostsandconsumershavedownwardslopingdemand.However,themaineconomicintuitionunderlyingthemodelmaybeseenthroughthefollowingadaptationofourgeneralclearinghouseframeworkfortheunitdemandcase:1.Allconsumersareshoppers:S�0andL=0;2.Thereisnocosttoadvertisepricesontheclearinghouse:=0;and3.FirmshaveprivatelyobservedmarginalcostsdescribedbytheatomlessdistributionG(m)on[m;m]:Sincetherearenocoststoadvertiseprices,all…rmslistpricesontheclearinghouse.Each…rmfacescompetitionfromn�1other…rmswithrandommarginalcosts.Sincethe…rmchargingthelowestpricewinstheentiremarket,…rmsaree¤ectivelycompetinginanauctioninwhichtheirowncostsareprivateinformation.Forthespecialcaseofunitdemand,theequilibriumpricefora…rmisagainthefamiliarexpressionfroma…rst-priceauction:p(m)=Ehm(n�1)minjm(n�1)minmi(15)wherem(n�1)ministhelowestofn�1drawsfromthedistributionG:Thereareseveralnoteworthyfeaturesofthisequilibrium.First,equilibrium…rmpricingentailspositivemarkupsdespitethefactthatallconsumersare“shoppers”andhaveacompletelistofprices.Intuitively,thereisatrade-o¤betweenloweringone’spricetoattractshoppersandthepro…tabilityofthisprice.Inequilibrium,thisresultsinamarkupwhichdependsonthenumberofcompeting…rms.Asthenumberof…rmsgrowslarge,theequilibriummarkupbecomessmall.Second,noticethatcostheterogeneityleadstoequilibriumpricedispersiondespitethefactconsumersareidenticalandallconsumersarepurchasingatthelowestprice.33showsthatthegatekeeperoptimallychargesafeeto…rmsadvertisingpricesthatisproportionaltotheconsumertra¢conitssite.Thisfeeinducesequilibriumpricedispersion;indeed,as…rstnotedbyBayeandMorgan,pricedispersionisnecessaryinorderforthegatekeepertopro…tablymaintainaclearinghouse.2.2.5CostHeterogeneitiesandtheSpulberModelSpulber(1995)considersasituationwhereconsumershaveaccesstothecompletelistofpricesandbuyfromthe…rmo¤eringthelowestprice.Ofcourse,insuchasetting,if…rmswereidenticalonewouldimmediatelyobtaintheBertrandoutcome.Togeneratepricedispersion,Spulberexaminesthesituationwhere…rmshaveheterogeneouscostsandconsumershavedownwardslopingdemand.However,themaineconomicintuitionunderlyingthemodelmaybeseenthroughthefollowingadaptationofourgeneralclearinghouseframeworkfortheunitdemandcase:1.Allconsumersareshoppers:S�=02.Thereisnocosttoadvertisepricesontheclearinghouse:=0;and3.Firmshaveprivatelyobservedmarginalcostsdescribedbytheatomlessdistribution;Sincetherearenocoststoadvertiseprices,all…rmslistpricesontheclearinghouse.Each…rmfacescompetitionfromother…rmswithrandommarginalcosts.Sincethe…rmchargingthelowestpricewinstheentiremarket,…rmsaree¤ectivelycompetinginanauctioninwhichtheirowncostsareprivateinformation.Forthespecialcaseofunitdemand,t

heequilibriumpricefora…rmisagainthe
heequilibriumpricefora…rmisagainthefamiliarexpressionfroma…rst-priceauction:)=minminwhereministhelowestofdrawsfromthedistributionThereareseveralnoteworthyfeaturesofthisequilibrium.First,equilibrium…rmpricingentailspositivemarkupsdespitethefactthatallconsumersare“shoppers”andhaveacompletelistofprices.Intuitively,thereisatrade-o¤betweenloweringone’spricetoattractshoppersandthepro…tabilityofthisprice.Inequilibrium,thisresultsinamarkupwhichdependsonthenumberofcompeting…rms.Asthenumberof…rmsgrowslarge,theequilibriummarkupbecomessmall.Second,noticethatcostheterogeneityleadstoequilibriumpricedispersiondespitethefactconsumersareidenticalandallconsumersarepurchasingatthelowestprice.dispersiondocumentedinlaboratoryexperimentsaswellasobservedonInternetpricecomparisonsites.Inasimilarvein,Rauh(2001)showsthatpricedispersioncanarisewhenmarketparticipantsmakesmallbutheterogeneousmistakesintheirbeliefsaboutthedistributionofprices.Ellison(2005)providesamoredetailedtreatmentofrecentadvancesalongtheselines.2.4ConcludingRemarks:TheoryDespiteaslowstart,therearenowavarietyofmodelsthatcanbeusedtorationalizeequilibriumpricedispersioninonlineando­inemarkets.Weconcludeourtheoreticaldiscussionwiththefollowinggeneralobservations:1.Thereisnota“one-size-…ts-all”modelofequilibriumpricedispersion;di¤erentmodelsareappropriateforanalyzingdi¤erentmarketenvironments.Forinstance,search-theoreticmod-elsaremostappropriateforanalyzingenvironmentswhereconsumersmustvisitdi¤erentstoresor…rms’websitestogatherpriceinformation.Clearinghousemodelsareappropriatewhenconsumersareabletoaccessalistofprices(forexample,inanewspaperoratapricecomparisonsite).2.Thedistributionofpricesisdeterminedbytheinteractionofallmarketparticipants—…rms,consumersand,inthecaseofclearinghousemodels,informationgatekeepers.Asaconse-quence,thelevelofpricedispersiondependsonthestructureofthemarket–thenumberofsellers,thedistributionofcosts,consumers’elasticitiesofdemand,andsoon.3.Reductionsinsearchcostsmayleadtoeithermoreorlesspricedispersion,dependingonthemarketenvironment.Furthermore,theeliminationofconsumersearchcostsneednoteliminatepricedispersion.4.Dependingonthemarketenvironment,heightenedcompetition(increasesinthenumberof…rms)canincreaseordecreasethelevelofdispersion.Moreover,insomemodels,heightenedcompetitionofthisformleadstohighertransactionspricespaidbyallconsumers.Inothermodels,thee¤ectofincreasedcompetitiononthewelfareofconsumersdependsonwhichsideofthe“digitaldivide”aconsumerresides.5.Pricedispersionisnotpurelyanartifactofexanteheterogeneitiesin…rmsorconsumers.Whiledi¤erencesin…rms’costsorbaseofloyalconsumers(stemmingfrom…rms’brand-35dispersiondocumentedinlaboratoryexperimentsaswellasobserve

donInternetpricecomparisonsites.Inasimil
donInternetpricecomparisonsites.Inasimilarvein,Rauh(2001)showsthatpricedispersioncanarisewhenmarketparticipantsmakesmallbutheterogeneousmistakesintheirbeliefsaboutthedistributionofprices.Ellison(2005)providesamoredetailedtreatmentofrecentadvancesalongtheselines.2.4ConcludingRemarks:TheoryDespiteaslowstart,therearenowavarietyofmodelsthatcanbeusedtorationalizeequilibriumpricedispersioninonlineando­inemarkets.Weconcludeourtheoreticaldiscussionwiththefollowinggeneralobservations:1.Thereisnota“one-size-…ts-all”modelofequilibriumpricedispersion;di¤erentmodelsareappropriateforanalyzingdi¤erentmarketenvironments.Forinstance,search-theoreticmod-elsaremostappropriateforanalyzingenvironmentswhereconsumersmustvisitdi¤erentstoresor…rms’websitestogatherpriceinformation.Clearinghousemodelsareappropriatewhenconsumersareabletoaccessalistofprices(forexample,inanewspaperoratapricecomparisonsite).2.Thedistributionofpricesisdeterminedbytheinteractionofallmarketparticipants—…rms,consumersand,inthecaseofclearinghousemodels,informationgatekeepers.Asaconse-quence,thelevelofpricedispersiondependsonthestructureofthemarket–thenumberofsellers,thedistributionofcosts,consumers’elasticitiesofdemand,andsoon.3.Reductionsinsearchcostsmayleadtoeithermoreorlesspricedispersion,dependingonthemarketenvironment.Furthermore,theeliminationofconsumersearchcostsneednoteliminatepricedispersion.4.Dependingonthemarketenvironment,heightenedcompetition(increasesinthenumberof…rms)canincreaseordecreasethelevelofdispersion.Moreover,insomemodels,heightenedcompetitionofthisformleadstohighertransactionspricespaidbyallconsumers.Inothermodels,thee¤ectofincreasedcompetitiononthewelfareofconsumersdependsonwhichsideofthe“digitaldivide”aconsumerresides.5.Pricedispersionisnotpurelyanartifactofexanteheterogeneitiesin…rmsorconsumers.Whiledi¤erencesin…rms’costsorbaseofloyalconsumers(stemmingfrom…rms’brand-costsofall…rmsintheMacMinnmodelincreasedbyafactor �1:Inthatcase,thenewvariancewouldsimplyscaleuptheoriginalvariancebyafactor 2:Thus,thismeasureofpricedispersionwouldchangeeventhoughtheunderlyingrealeconomicsofthesituationarethesameafterthein‡ationaryperiod.Forthisreason,ifonewishestocomparelevelsofpricedispersioneitheracrossdi¤erentproductsoracrosstime,onemuststandardizethedatainsomefashion.Analternativeistousethecoe¢cientofvariation,CV=p=E[p](oritssampleanalogue),whichishomogenousofdegreezerointhelevelofprices.TheCVisparticularlyusefulwhencomparinglevelsofpricedispersionoverlongperiodsoftime(e.g.,ScholtenandSmith(2002)andEckard(2004))oracrossdi¤erentproducts(e.g.,CarlsonandPescatrice(1980);Sorensen(2000);Aalto-Setälä(2003);andBaye,MorganandScholten(2004a,b)).Anaddedadvantageisthat,unlikesomemethodsofstandard

ization,thecoe¢cientofvariationmayprese
ization,thecoe¢cientofvariationmaypreservethecomparativestaticpredictionsofthemodelofinterest.Forinstance,intheMacMinnmodel,equation(8)impliesthattheexpectedpriceisE[p]=n�1nm+m2+mn,andthusthecoe¢cientofvariationisCV=1p3(n�1)(m�m)(n�1)(m+m)+2mOnemayverifythatthisstatisticis,likethevariance,decreasinginsearchcosts,but,unlikethevariance,thisstatisticdoesnotchangewithamultiplicativeshiftin…rms’costs.Anotherwidelyusedmeasureofpricedispersionisthe(sample)range;see,forinstance,Pratt,Wise,andZeckhauser(1979)andBrynjolfssonandSmith(2000).Lettingp(n)minandp(n)maxdenote,respectively,thelowestandhighestofnobservedpricesdrawnfromF;thentherangeisR(n)=p(n)max�p(n)minGiventheequilibriumdistributionofpricesimpliedbyaparticulartheoreticalmodel,comparativestaticpredictionsaboutchangesintherangearepossiblebasedonthebehaviorofthehighestandlowestorderstatistics.Thatis,onecanperformcomparativestaticanalysisontheexpectedrange:20EhR(n)i=Ehp(n)maxi�Ehp(n)miniUnfortunately,alloftheabovemeasuresofpricedispersionsu¤erfromapotentialtheoreticaldefect.Supposethatn�2…rmscompeteinaclassicalhomogeneousproductBertrandsetting.20Tofacilitatecomparisonsacrossdi¤erentproductsorovertime,itissometimesusefultonormalizetherangebydividingitbytheminimumoraverageprice;seeBaye,Morgan,andScholten(2004b)andBrynjolfssonandSmith(2000).37costsofall…rmsintheMacMinnmodelincreasedbyafactor\r�Inthatcase,thenewvariancewouldsimplyscaleuptheoriginalvariancebyafactorThus,thismeasureofpricedispersionwouldchangeeventhoughtheunderlyingrealeconomicsofthesituationarethesameafterthein‡ationaryperiod.Forthisreason,ifonewishestocomparelevelsofpricedispersioneitheracrossdi¤erentproductsoracrosstime,onemuststandardizethedatainsomefashion.Analternativeistousethecoe¢cientofvariation,CVVp](oritssampleanalogue),whichishomogenousofdegreezerointhelevelofprices.TheCVisparticularlyusefulwhencomparinglevelsofpricedispersionoverlongperiodsoftime(e.g.,ScholtenandSmith(2002)andEckard(2004))oracrossdi¤erentproducts(e.g.,CarlsonandPescatrice(1980);Sorensen(2000);Aalto-Setälä(2003);andBaye,MorganandScholten(2004a,b)).Anaddedadvantageisthat,unlikesomemethodsofstandardization,thecoe¢cientofvariationmaypreservethecomparativestaticpredictionsofthemodelofinterest.Forinstance,intheMacMinnmodel,equationimpliesthattheexpectedpriceisisp]=nm+m2+m,andthusthecoe¢cientofvariationisCVp1)(m�m)1)(m+m)+2Onemayverifythatthisstatisticis,likethevariance,decreasinginsearchcosts,but,unlikethevariance,thisstatisticdoesnotchangewithamultiplicativeshiftin…rms’costs.Anotherwidelyusedmeasureofpricedispersionisthe(sample)range;see,forinstance,Pratt,Wise,andZeckhauser(1979)andBrynjolfssonandSmith(2000).Lettingminmaxdenote,re

spectively,thelowestandhighestofobserved
spectively,thelowestandhighestofobservedpricesdrawnfromF;thentherangeismaxminGiventheequilibriumdistributionofpricesimpliedbyaparticulartheoreticalmodel,comparativestaticpredictionsaboutchangesintherangearepossiblebasedonthebehaviorofthehighestandlowestorderstatistics.Thatis,onecanperformcomparativestaticanalysisontheexpectedrange:20maxminUnfortunately,alloftheabovemeasuresofpricedispersionsu¤erfromapotentialtheoreticaldefect.Supposethatn�…rmscompeteinaclassicalhomogeneousproductBertrandsetting.20Tofacilitatecomparisonsacrossdi¤erentproductsorovertime,itissometimesusefultonormalizetherangebydividingitbytheminimumoraverageprice;seeBaye,Morgan,andScholten(2004b)andBrynjolfssonandSmith(2000).Forexample,inhisseminalarticleontheeconomicsofinformation,GeorgeStigler,advancedthefollowinghypotheses:“...dispersionitselfisafunctionoftheaverageamountofsearch,andthisinturnisafunctionofthenatureofthecommodity:1.Thelargerthefractionofthebuyer’sexpendituresonthecommodity,thegreaterthesavingsfromsearchandhencethegreatertheamountofsearch.2.Thelargerthefractionofrepetitive(experienced)buyersinthemarket,thegreaterthee¤ectiveamountofsearch(withpositivecorrelationofsuccessiveprices).3.Thelargerthefractionofrepetitivesellers,thehigherthecorrelationbetweensuccessiveprices,andhence,thelargertheamountofaccumulatedsearch.4.Thecostofsearchwillbelarger,thelargerthegeographicsizeofthemarket.”Stigler(1961,p.219).Stigler’shypotheseso¤erausefulguideforunderstandingtheempiricalliteratureonpricedispersion.MuchofthisliteraturetestsStigler’shypothesesbyexaminingwhethersearchintensity(proxiedbyvariablesthata¤ectthebene…tsandcostsofsearch)iscorrelatedwithlevelsofpricedispersion.Aswehaveseen,however,whenonetakesRothschild’scriticismintoaccount,anincreaseinsearchintensitycanleadtoincreasesordecreasesinthelevelofequilibriumpricedispersion,dependingonthemodel.Thus,onechallengeforempiricalresearchersischoosingamodelthatcloselyapproximatesthe“datagenerating”environment.Asecondchallengeistocontrolforfactorsoutsideofthemodelthatmightin‡uencelevelsofdispersion.Athirdchallengearisesbecause…rmoptimizationisabsentinStigler’smodel,butisclearlypresentinthedata.Forthisreason,anumberofempiricalstudieslookbeyondStigler’shypothesestotesthypothesesderivedfromspeci…csearch-theoreticorclearinghousemodelsofequilibriumpricedispersion.Weprovideabroadoverviewoftheseandrelatedstrandsoftheliteraturebelow.3.2.1Dispersionandthe“Bene…ts”ofSearchThesearch-theoreticmodelspresentedinSection2implythatsearchintensitydepends,inpart,ontheconsumer’sdemandforaproduct.IntheStiglermodel,demandisrepresentedbytheparameter,K.ThegreaterisK;thegreatertheexpectedbene…tsofsearchandhencethegreaterthesearchintensity.Stigler’s…r

sttwohypothesesarebasedonthenotionthatth
sttwohypothesesarebasedonthenotionthattheshareofan39Forexample,inhisseminalarticleontheeconomicsofinformation,GeorgeStigler,advancedthefollowinghypotheses:“...dispersionitselfisafunctionoftheaverageamountofsearch,andthisinturnisafunctionofthenatureofthecommodity:1.Thelargerthefractionofthebuyer’sexpendituresonthecommodity,thegreaterthesavingsfromsearchandhencethegreatertheamountofsearch.2.Thelargerthefractionofrepetitive(experienced)buyersinthemarket,thegreaterthee¤ectiveamountofsearch(withpositivecorrelationofsuccessiveprices).3.Thelargerthefractionofrepetitivesellers,thehigherthecorrelationbetweensuccessiveprices,andhence,thelargertheamountofaccumulatedsearch.4.Thecostofsearchwillbelarger,thelargerthegeographicsizeofthemarket.”Stigler(1961,p.219).Stigler’shypotheseso¤erausefulguideforunderstandingtheempiricalliteratureonpricedispersion.MuchofthisliteraturetestsStigler’shypothesesbyexaminingwhethersearchintensity(proxiedbyvariablesthata¤ectthebene…tsandcostsofsearch)iscorrelatedwithlevelsofpricedispersion.Aswehaveseen,however,whenonetakesRothschild’scriticismintoaccount,anincreaseinsearchintensitycanleadtoincreasesordecreasesinthelevelofequilibriumpricedispersion,dependingonthemodel.Thus,onechallengeforempiricalresearchersischoosingamodelthatcloselyapproximatesthe“datagenerating”environment.Asecondchallengeistocontrolforfactorsoutsideofthemodelthatmightin‡uencelevelsofdispersion.Athirdchallengearisesbecause…rmoptimizationisabsentinStigler’smodel,butisclearlypresentinthedata.Forthisreason,anumberofempiricalstudieslookbeyondStigler’shypothesestotesthypothesesderivedfromspeci…csearch-theoreticorclearinghousemodelsofequilibriumpricedispersion.Weprovideabroadoverviewoftheseandrelatedstrandsoftheliteraturebelow.3.2.1Dispersionandthe“Bene…ts”ofSearchThesearch-theoreticmodelspresentedinSection2implythatsearchintensitydepends,inpart,ontheconsumer’sdemandforaproduct.IntheStiglermodel,demandisrepresentedbythe.ThegreaterisK;thegreatertheexpectedbene…tsofsearchandhencethegreaterthesearchintensity.Stigler’s…rsttwohypothesesarebasedonthenotionthattheshareofanSmith(2002),Johnson(2002),GattiandKattuman(2003),andAalto-Setälä(2003).Morere-cently,Eckard(2004)comparespricedispersionforstapleproductsin1901and2001,andreportscoe¢cientsofvariationin2001thatarealmosttwicethosebasedondatafrom1901.Eckardarguesthatonereasonfortheincreaseddispersionisthathissampleconsistsofstapleitems(suchassugarandbakingpowder)thataccountedforamuchlargershareofhouseholdbudgetsin1901thanin2001.DispersionandPurchaseFrequencyInhissecondhypothesis,Stiglerarguesthatinmarketswheretherearemorerepetitiveorexperiencedbuyers,thegreateristheamountofe¤ectivesearch.Unfortunately,itisdi¢culttodirectlytestth

ishypotheses,sinceinmostmarketsthereisno
ishypotheses,sinceinmostmarketsthereisnotadirect(objective)measureof“buyerexperience”or“purchasefrequency”touseinexaminingitsimpactonlevelsofpricedispersion.Anumberofthestudiesmentionedabove,however,providecasualevidencethatpurchasefrequencyimpactsthelevelofpricedispersion(cf.CarlsonandPescatrice,1980;Pratt,Wise,andZeckhauser,1979).Sorensen(2000),however,hasprovidedavery“clean”andeleganttestofStigler’ssecondhypothesis.Hisanalysisisbasedondatafromthemarketforprescriptiondrugs.Theuniqueaspectofthismarketisthatpurchasefrequency—thetypicaldosageanddurationoftherapyforagivenprescriptiondrug—maybeobjectivelymeasured.Aconsumers’bene…tpersearchisclearlyhighestforfrequentlypurchaseddrugs,and,Sorensenargues,thisshouldleadtogreatersearchandlowerpricedispersion.Hisempiricalanalysisidenti…esastronginverserelationshipbetweenpurchasefrequencyandpricedispersion.Forexample,aftercontrollingotherfactors(whichtogetherexplainaboutone-thirdofthevariationinprices),Sorensen…ndsthatthepricerangeforadrugthatmustbepurchasedmonthlyisabout30percentlowerthanifitwereaone-timetherapy.Importantly,Sorensenshowsthattheresultsarequalitativelysimilarwhenalternativemeasuresofpricedispersion(suchasthestandarddeviation)areused.3.2.2Dispersionandthe“Cost”ofSearchResearchersstudyingtheempiricalrelationshipbetweensearchcostsandpricedispersionhavefacedobstaclessimilartothoseofresearchersfocusingonthebene…tsideofthesearchequation.First,thepredictedimpactofsearchcostsonlevelsofdispersiondependsnotonlyonthemodel,butalsoonthemetricusedformeasuringdispersion.Second,searchcostsaregenerallyunobservable.Someofthemorein‡uentialpapersintheareaareonesthathavedevisedinnovativemethodsof41Smith(2002),Johnson(2002),GattiandKattuman(2003),andAalto-Setälä(2003).Morere-cently,Eckard(2004)comparespricedispersionforstapleproductsin1901and2001,andreportscoe¢cientsofvariationin2001thatarealmosttwicethosebasedondatafrom1901.Eckardarguesthatonereasonfortheincreaseddispersionisthathissampleconsistsofstapleitems(suchassugarandbakingpowder)thataccountedforamuchlargershareofhouseholdbudgetsin1901thaninDispersionandPurchaseFrequencyInhissecondhypothesis,Stiglerarguesthatinmarketswheretherearemorerepetitiveorexperiencedbuyers,thegreateristheamountofe¤ectivesearch.Unfortunately,itisdi¢culttodirectlytestthishypotheses,sinceinmostmarketsthereisnotadirect(objective)measureof“buyerexperience”or“purchasefrequency”touseinexaminingitsimpactonlevelsofpricedispersion.Anumberofthestudiesmentionedabove,however,providecasualevidencethatpurchasefrequencyimpactsthelevelofpricedispersion(cf.CarlsonandPescatrice,1980;Pratt,Wise,andZeckhauser,1979).Sorensen(2000),however,hasprovidedavery“clean”andeleganttestofStigler’ssecondhypo

thesis.Hisanalysisisbasedondatafromthema
thesis.Hisanalysisisbasedondatafromthemarketforprescriptiondrugs.Theuniqueaspectofthismarketisthatpurchasefrequency—thetypicaldosageanddurationoftherapyforagivenprescriptiondrug—maybeobjectivelymeasured.Aconsumers’bene…tpersearchisclearlyhighestforfrequentlypurchaseddrugs,and,Sorensenargues,thisshouldleadtogreatersearchandlowerpricedispersion.Hisempiricalanalysisidenti…esastronginverserelationshipbetweenpurchasefrequencyandpricedispersion.Forexample,aftercontrollingotherfactors(whichtogetherexplainaboutone-thirdofthevariationinprices),Sorensen…ndsthatthepricerangeforadrugthatmustbepurchasedmonthlyisabout30percentlowerthanifitwereaone-timetherapy.Importantly,Sorensenshowsthattheresultsarequalitativelysimilarwhenalternativemeasuresofpricedispersion(suchasthestandarddeviation)areused.3.2.2Dispersionandthe“Cost”ofSearchResearchersstudyingtheempiricalrelationshipbetweensearchcostsandpricedispersionhavefacedobstaclessimilartothoseofresearchersfocusingonthebene…tsideofthesearchequation.First,thepredictedimpactofsearchcostsonlevelsofdispersiondependsnotonlyonthemodel,butalsoonthemetricusedformeasuringdispersion.Second,searchcostsaregenerallyunobservable.Someofthemorein‡uentialpapersintheareaareonesthathavedevisedinnovativemethodsofAlongthesesamelines,anumberofstudiescompareaveragepricesinonlineversuso­inemarkets.Theideaisthatsearchcostsareloweronline,thusa¤ectingnotonlytherangeorvarianceinprices,butalsothemeanprice(andhencethecoe¢cientofvariationthroughboththemeanandvariance).Scott-Morton,ZettelmeyerandSilva-Risso(2001)…ndthatpricesarelowerinonlinemarketsforautomobiles.ConsumerswhopurchaseacarthroughtheInternetreferralserviceAutobytel.comreducetheirpurchasepricebyapproximately2.2percent.Apotentiallyconfoundingexplanationforthispricedi¤erenceisthattheconsumerswhochoosetoshoponlinemayalsobeskilled“higglers,”touseStigler’sphraseandthusthepricedi¤erencemightpurelyre‡ectadi¤erenceinthenegotiatingskillsofconsumersacrossthetwochannels.Interestingly,Zettelmeyer,Scott-MortonandSilva-Risso(2004)provideevidencethatthisisnotthecase:consumerswhopurchaseautomobilesonlinearenottypicallythosewhonegotiatewellinthetraditionalchannel.Thereareanumberofotherstudies,however,that…ndequalorhigherpricesonline(cf.Clemons,HannandHitt(2002);Bailey(1998);Goolsbee(2001);Clay,etal.(2003);Erevelles,RollandandSrinivasan(2001)).Furtherstudiesdistinguishpricelevelsdependingonwhethertheretailerisasolelyonlineor“multichannel”(cf.ChevalierandGoolsbee(2003)and;TangandXing(2001)).Analternativeapproachisto“recover”searchcostsusingstructuralparametersfromapar-ticularmodelofpricedispersion.Forexample,HongandShum(forthcoming)obtainsearchcostsestimatesusingrestrictionsimposedbytheoreticalsearchmodelsandassumi

ngthatobservedpricedispersionisanequilib
ngthatobservedpricedispersionisanequilibriumphenomenonarisingfromheterogeneousconsumersearchcosts.Theirestimationtechniqueisappliedtoonlinepricedataonfoureconomicsandstatisticstextbooks.Theyobtainsearchcostestimatesrangingfrom$1.31to$29.40fortheseitems.Asimilarapproachcanbeusedinclearinghousemodels.Villas-Boas(1995)usesthetheoreticaldensityfunctionim-pliedbytheVarian(1980)clearinghousemodeltoobtainestimatesofthenumberofshoppersintheo­ineco¤eeandsaltinecrackermarkets.Morerecently,Baye,Gatti,Kattuman,andMorgan(2005)usedatheoreticalclearinghousemodelasthebasisforestimatingthefractionof“shop-pers”inanonlinemarketforPDAsintheUK.Theirresultssuggestthatabout13percentoftheconsumersinthismarketareshoppers.3.2.3DispersionandtheNumberofSellersTheoligopolymodelspresentedinSection2revealthatequilibriumdistributionsofprices,andhencelevelsofdispersion,varywiththenumberofsellerscompetinginthemarket.Thedirectioninwhichpricesmoveasaconsequenceofachangeinthenumberofsellersis,however,modelspeci…c,43Alongthesesamelines,anumberofstudiescompareaveragepricesinonlineversuso­inemarkets.Theideaisthatsearchcostsareloweronline,thusa¤ectingnotonlytherangeorvarianceinprices,butalsothemeanprice(andhencethecoe¢cientofvariationthroughboththemeanandvariance).Scott-Morton,ZettelmeyerandSilva-Risso(2001)…ndthatpricesarelowerinonlinemarketsforautomobiles.ConsumerswhopurchaseacarthroughtheInternetreferralserviceAutobytel.comreducetheirpurchasepricebyapproximately2.2percent.Apotentiallyconfoundingexplanationforthispricedi¤erenceisthattheconsumerswhochoosetoshoponlinemayalsobeskilled“higglers,”touseStigler’sphraseandthusthepricedi¤erencemightpurelyre‡ectadi¤erenceinthenegotiatingskillsofconsumersacrossthetwochannels.Interestingly,Zettelmeyer,Scott-MortonandSilva-Risso(2004)provideevidencethatthisisnotthecase:consumerswhopurchaseautomobilesonlinearenottypicallythosewhonegotiatewellinthetraditionalchannel.Thereareanumberofotherstudies,however,that…ndequalorhigherpricesonline(cf.Clemons,HannandHitt(2002);Bailey(1998);Goolsbee(2001);Clay,etal.(2003);Erevelles,RollandandSrinivasan(2001)).Furtherstudiesdistinguishpricelevelsdependingonwhethertheretailerisasolelyonlineor“multichannel”(cf.ChevalierandGoolsbee(2003)and;TangandXing(2001)).Analternativeapproachisto“recover”searchcostsusingstructuralparametersfromapar-ticularmodelofpricedispersion.Forexample,HongandShum(forthcoming)obtainsearchcostsestimatesusingrestrictionsimposedbytheoreticalsearchmodelsandassumingthatobservedpricedispersionisanequilibriumphenomenonarisingfromheterogeneousconsumersearchcosts.Theirestimationtechniqueisappliedtoonlinepricedataonfoureconomicsandstatisticstextbooks.Theyobtainsearchcostestimatesrangingfrom$1.31to$29.40fortheseitems.Asimilarapproachcanbeusedincl

earinghousemodels.Villas-Boas(1995)usest
earinghousemodels.Villas-Boas(1995)usesthetheoreticaldensityfunctionim-pliedbytheVarian(1980)clearinghousemodeltoobtainestimatesofthenumberofshoppersintheo­ineco¤eeandsaltinecrackermarkets.Morerecently,Baye,Gatti,Kattuman,andMorgan(2005)usedatheoreticalclearinghousemodelasthebasisforestimatingthefractionof“shop-pers”inanonlinemarketforPDAsintheUK.Theirresultssuggestthatabout13percentoftheconsumersinthismarketareshoppers.3.2.3DispersionandtheNumberofSellersTheoligopolymodelspresentedinSection2revealthatequilibriumdistributionsofprices,andhencelevelsofdispersion,varywiththenumberofsellerscompetinginthemarket.Thedirectioninwhichpricesmoveasaconsequenceofachangeinthenumberofsellersis,however,modelspeci…c,dispersioninfaresincreasesonrouteswithlower‡ightdensityormorecompetition.Thus,thereisevidencethatthenumberofsellersmattersforpricedispersion.3.2.4DispersionandPricePersistenceVarian(1980)wasthe…rsttodistinguishbetweenwhathereferredtoas“spatial”and“temporal”pricedispersion.Underspatialpricedispersion,di¤erent…rmschargedi¤erentpricesatanypointintime,buta…rm’spositioninthedistributionofpricesdoesnotchangeovertime.Absentrandomcostshocks,spatialpricedispersionarisesintheReinganum,MacMinn,andSpulbermodels.Incontrast,withtemporalpricedispersion,…rmschargedi¤erentpricesateachpointintime,buttheirpositioninthedistributionofpriceschangesovertime.Temporalpricedispersionarisesinthegeneralclearinghousemodel(andvariousspecialcases)aswellasintheBurdettandJuddmodel.Variancritiquesmodelsofspatialpricedispersion,arguingthatifconsumerscanlearnfromexperiencethatsome…rmspersistentlyo¤erlowerpricesthanother…rms,thenmodelsofspatialpricedispersionsuggesta“convergencehypothesis”:pricedispersionshoulddiminishovertimeduetothepositivecorrelationinsuccessiveprices(touseStigler’sterminology)andcumulativesearchinformation.Thishasledtoanumberofstudiesthatexaminewhetherthereisanyevidencefortheconvergencehypothesisandwhetherthetemporalpricedispersionpredictedbytheclearinghousemodelsis,infact,presentinthedata.Usingmonthlystore-levelpricedatafromIsrael,andaftercontrollingforobservedandun-observedproductheterogeneities,Lach(2002)…ndssomeevidenceoftemporalpricedispersion.Lachestimatesmonth-to-monthtransitionsamongquartilesby…rms;thatis,theprobabilitythata…rmo¤eringapriceinagivenquartileatthestartofthemonthisstillo¤eringapriceinthesamequartileattheendofthemonth.Hisestimatessuggestthattheprobabilityofremaininginthesamequartileis78percentfor…rmssellingrefrigeratorsand71percentfor…rmsselling‡our.Theseprobabilitiesaresomewhatlowerfor…rmssellingchicken(51percent)andco¤ee(43per-cent).Whenthetransitionperiodisextendedtosixmonthsinsteadofonemonth,theprobabilityofremaininginthesame

quartileisconsiderablylower—falling
quartileisconsiderablylower—fallingtoaround30-35percent.RobertsandSupina(2000)suggestthatstructuraldi¤erencesin…rms’costsaccountforaconsiderableportionofpricedispersionintheo­inesector—aspredictedbyavarietyofsearch-theoreticmodels.Usingplant-levelUSCensusdata,they…ndsomeevidenceforpricepersistence.Theevidenceisstrongestinthetailsofthedistribution:high-price…rmstendtopersistentlychargehighprices,andlow-price…rmstendstopersistentlychargelowprices.Avarietyofotherstudies45dispersioninfaresincreasesonrouteswithlower‡ightdensityormorecompetition.Thus,thereisevidencethatthenumberofsellersmattersforpricedispersion.3.2.4DispersionandPricePersistenceVarian(1980)wasthe…rsttodistinguishbetweenwhathereferredtoas“spatial”and“temporal”pricedispersion.Underspatialpricedispersion,di¤erent…rmschargedi¤erentpricesatanypointintime,buta…rm’spositioninthedistributionofpricesdoesnotchangeovertime.Absentrandomcostshocks,spatialpricedispersionarisesintheReinganum,MacMinn,andSpulbermodels.Incontrast,withtemporalpricedispersion,…rmschargedi¤erentpricesateachpointintime,buttheirpositioninthedistributionofpriceschangesovertime.Temporalpricedispersionarisesinthegeneralclearinghousemodel(andvariousspecialcases)aswellasintheBurdettandJuddmodel.Variancritiquesmodelsofspatialpricedispersion,arguingthatifconsumerscanlearnfromexperiencethatsome…rmspersistentlyo¤erlowerpricesthanother…rms,thenmodelsofspatialpricedispersionsuggesta“convergencehypothesis”:pricedispersionshoulddiminishovertimeduetothepositivecorrelationinsuccessiveprices(touseStigler’sterminology)andcumulativesearchinformation.Thishasledtoanumberofstudiesthatexaminewhetherthereisanyevidencefortheconvergencehypothesisandwhetherthetemporalpricedispersionpredictedbytheclearinghousemodelsis,infact,presentinthedata.Usingmonthlystore-levelpricedatafromIsrael,andaftercontrollingforobservedandun-observedproductheterogeneities,Lach(2002)…ndssomeevidenceoftemporalpricedispersion.Lachestimatesmonth-to-monthtransitionsamongquartilesby…rms;thatis,theprobabilitythata…rmo¤eringapriceinagivenquartileatthestartofthemonthisstillo¤eringapriceinthesamequartileattheendofthemonth.Hisestimatessuggestthattheprobabilityofremaininginthesamequartileis78percentfor…rmssellingrefrigeratorsand71percentfor…rmsselling‡our.Theseprobabilitiesaresomewhatlowerfor…rmssellingchicken(51percent)andco¤ee(43per-cent).Whenthetransitionperiodisextendedtosixmonthsinsteadofonemonth,theprobabilityofremaininginthesamequartileisconsiderablylower—fallingtoaround30-35percent.RobertsandSupina(2000)suggestthatstructuraldi¤erencesin…rms’costsaccountforaconsiderableportionofpricedispersionintheo­inesector—aspredicte

dbyavarietyofsearch-theoreticmodels.Usin
dbyavarietyofsearch-theoreticmodels.Usingplant-levelUSCensusdata,they…ndsomeevidenceforpricepersistence.Theevidenceisstrongestinthetailsofthedistribution:high-price…rmstendtopersistentlychargehighprices,andlow-price…rmstendstopersistentlychargelowprices.Avarietyofotherstudies3.Therelationshipbetweenpricedispersionandeconomicprimitivesisoftensensitivetothemeasureofpricedispersionused.4.Despitethewidespreadadoptionofinventionssuchastheautomobile,thetelephone,televi-sion,andtheInternet,pricedispersionisstilltheruleratherthantheexceptioninhomoge-neousproductmarkets.Reductionsininformationcostsoverthepastcenturyhaveneitherreducednoreliminatedthelevelsofpricedispersionobservedinhomogeneousproductmar-kets.473.Therelationshipbetweenpricedispersionandeconomicprimitivesisoftensensitivetothemeasureofpricedispersionused.4.Despitethewidespreadadoptionofinventionssuchastheautomobile,thetelephone,televi-sion,andtheInternet,pricedispersionisstilltheruleratherthantheexceptioninhomoge-neousproductmarkets.Reductionsininformationcostsoverthepastcenturyhaveneitherreducednoreliminatedthelevelsofpricedispersionobservedinhomogeneousproductmar-kets.Baye,M.R.andJ.Morgan.2004.“PriceDispersionintheLabandontheInternet:TheoryandEvidence.”RandJournalofEconomics,35(3),449-466.Baye,M.R.,J.MorganandP.Scholten.2003.“TheValueofInformationinanOnlineConsumerElectronicsMarket.”JournalofPublicPolicyandMarketing,22(1),17-25.Baye,M.R.,J.MorganandP.Scholten.2004a.“PriceDispersionintheSmallandintheLarge:EvidencefromanInternetPriceComparisonSite.”JournalofIndustrialEconomics,52(4),463-496.Baye,M.R.,J.MorganandP.Scholten.2004b.“TemporalPriceDispersion:EvidencefromanOnlineConsumerElectronicsMarket.”JournalofInteractiveMarketing,18(4),101-115.Baylis,K.andJ.M.Perlo¤.2002.“PriceDispersionontheInternet:GoodFirmsandBadFirms.”ReviewofIndustrialOrganization,21,305-324.Benabou,R.andR.Gertner.1993.“SearchwithLearningfromPrices:DoesIncreasedUncertaintyLeadtoHigherMarkups?”ReviewofEconomicStudies,60(202),69-94.Borenstein,S.andN.L.Rose.1994.“CompetitionandPriceDispersionintheU.S.AirlineIndus-try.”JournalofPoliticalEconomy,102(4),653-683.Braverman,A.1980.“ConsumerSearchandAlternativeMarketEquilibria.”ReviewofEconomicStudies47,487-502.Brown,J.R.andA.Goolsbee.2002.“DoestheInternetMakeMarketsMoreCompetitive?EvidencefromtheLifeInsuranceIndustry.”JournalofPoliticalEconomy,110(3),481-507.Brynjolfsson,E.andM.D.Smith.2000.“FrictionlessCommerce?AComparisonofInternetandConventionalRetailers.”ManagementScience,46(4),563-585.Brynjolfsson,E.,A.A.DickandM.D.Smith.2004.“SearchandProductDi¤erentiationatanInternetShopbot.”WorkingPaper.Burdett,K.andK.L.Judd.1983.“EquilibriumPriceDispersion.”Econometri

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