We are grateful to all seminar participants for comments and suggestions Any errors are our own Email fnavalseacuk pschiraldilseacuk Abstract Sales are a widespread and wellknown phenome ID: 483742
Download Pdf The PPT/PDF document "Storage Francesco NavaDiscussio..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Storageïª Francesco NavaDiscussion Paper No. TE/2011/549 October 2006 The Suntory-Toyota International Centres for Economics and Related Disciplines London School of Economics and Political Science Houghton Street London WC2A 2AE Tel.: 020-7955 6674 We are grateful to all seminar participants for comments and suggestions. Any errors are our own. Email: f.nava@lse.ac.uk, p.schiraldi@lse.ac.uk. Abstract Sales are a widespread and well-known phenomenon that has been documented in several product markets. Regularities in such periodic price reductions appear to suggest that the phenomenon cannot be entirely attributed to random variations in supply, demand, or the aggregate price level. Certain sales are traditional and so well publicized that it is difficult to justify them as devices to separate informed from uninformed consumers. This paper presents a model in which sellers want to reduce prices periodically in order to improve their ability to collude over time. In particular, the study shows that if buyers have heterogeneous storage technologies, periodic sales may facilitate collusion by magnifying intertemporal linking in consumers decisions. The stability and the profitability of different sale strategies is then explored. The optimal sales discount and timing of sales are characterized. A trade-off between cartel size and aggregate profits arises. Keywords: Storage, sales, collusion, cartel size, repeated games. JEL classification: L11, L12, L13, L41. The authors. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source. 1IntroductionTheoccurrenceofperiodicpricereductions,orsales,onavarietyofitemsisapervasiveandwell-knownmicroeconomicphenomenonthathasbeendocumentedinseveralproductmarkets.Typically,ahighpricesarechargedinmostperiods,butoccasionallypricesarecuttosupplymoreunitstoapotentiallylargergroupofconsumers.Theregularoccurrenceofsuchphenom-enonappearstosuggestthatsalescannotbeentirelyexplainedbyrandomvariationsinsupply,demand,ortheaggregatepricelevel.Moreover,certainsaleperiodsaretraditionalandsowellpublicizedthatitisdi¢culttojustifythemasdevicestoseparateinformedfromuninformedconsumers.Agrowingempiricalliteraturealso,appearstosuggestthatthemajorityofperi-odicsalestakeplaceforproductsthatarefairlystorable,andthatstoragecapacityexplainsinparttheresponsivenessofconsumerstochangesinprices(BellandHilber2006,HendelandNevo2006&2010,Erdemetal2003,Seiler2010).Suchevidencehighlightstheprimaryrolethatstorageconstraintsmayplayindeterminingconsumers’purchasingbehaviorandthus,retailers’pricingdecisions.Thepresentpaperstudieshowsalestrategiesmayfostercollusioninamarketinwhichgoodscanbestored.Thustheaimofthestudyistoprovideanadditionalmotivefor…rmstoengageinsalestrategiesandtoshedlightontheoptimalsalesdiscountandtiming.Whilefactorssuchasinformationaldi¤erencesandheterogeneityinwillingnesstosearchorpayhavereceivednotableattention,thetheoreticalliteratureonstorageconstraintsremainsscarcedespiteempiricalrelevancedocumentedbyagrowingliterature.NotableexceptionsareSalopandStiglitz1982,Hong,McAfeeandNayyar,2002,andDudine,HendelandLizzeri2006.Weconsideranindustryinwhichineveryperiod,n…rmsproduceahomogeneousstorablegood,andsellittoamassofheterogeneousconsumerswithunitdemand.Consumersdi¤eronlybytheiraccesstostorage.Inparticular,werestrictattentiontoeconomieswithtwotypesofconsumers:thosewithoutstoragecapacity,andthosewithstoragecapacityS.1Inthiscontext,weexaminethee¤ectsofheterogeneousstoragetechnologieson…rms’incentivestoholdperiodicsalestosupportagreaterdegreeofcooperationamong…rmsinarepeatedcompetitionsetting.Salestrategieswillbecharacterizedbyaregularprice,byapricemark- 1Thepaperassumesthatconsumersaresmallandunabletoestablishreputations.SeeNolkeandWhite2007forthecaseofstrategicbuyerswithinterdependentdemands.2 down(ontheregularprice)andbythefrequencyofsales.Weshowthatperiodicsalestrategiessustainedbygrimtriggerpunishmentsallow…rmstocolludeonsigni…cantpro…tlevelsevenwhenstandardno-salestrategiescannotsustaincollusionatsuchpro…tlevels.Insuchanenvironmentsalestrategiesmaystrengthencollusion,asthestoragetechnologyintertemporallylinksconsumerdemandandthus,reducestheshort-rungainsfromadeviation.Inparticular,inanysuchequilibrium…rmswillcharge,inanyperiodofsales,abigenoughdiscounttoinduceallconsumerswithopenstoragecapacitytostockpileaquantitysu¢cienttosatisfyalltheirdemanduntilthenextsale.Suchbehaviorreducesincentivestodeviateinregularpriceperiods,asonlyconsumerswithoutstoragepurchaseunitsinsuchperiods.Moreover,theincentivestodeviatealso,declineinperiodswithsalesbothbecausealowerpriceischargedinsuchperiods,andbecauseconsumerswithstoragereducetheirdemandwhenadeviationisobservedinthewakeoftheimminentpricewar(impliedbythegrimtriggerpunishments).Oftenwewillrefertotheincentivesnottodeviatefromanequilibriumstrategyasthestabilityofastrategy.The…rstpartoftheanalysis:characterizesconsumerdemandfortheproposedenvironment;characterizesthesetEofsalestrategieswhicharemorestablethananystrategywithoutsales;andprovidesnecessaryandsu¢cientconditionsforthenon-emptinessofsuchaset.Thesecondpartoftheanalysisrestrictsattentiontothecaseofsingle-unitstorageandcharacterizesthemoststableandthemostpro…tablesalestrategiesinthesetE.Withinsuchsetatrade-o¤isproventoemergebetweencollusivepro…tsandstability(whichismeasuredbythemaximalnumberof…rmsthatcancolludeongivenpro…ts).Suchtrade-o¤isexplicitlycharacterized.Thesecondpartoftheanalysisconcludesbystudyinghowsuchtrade-o¤isa¤ectedbychangesintheenvironment.Inparticular,increasesinpatience(i.e.thefrequencyofinteraction)willleadbothtolargercartelsatanypro…tlevelandtothepersistenceofthepro…t-stabilitytrade-o¤foralargerrangeofpro…ts.Anincreaseinthepro…tabilityofamarketinstead,willincreasebothcartelsizeatintermediatepro…tlevelsandtherangeofpro…tsforwhichthepro…t-stabilitytrade-o¤persists.Alargerfractionofconsumerswithoutstoragewillhaveambiguouse¤ectsoncartelsizeathighpro…tlevelsandwillreducethemaximalcartelsizeatlowpro…tlevels.Ingeneral,thee¤ectofsuchachangewillremainambiguous,asalargefractionofconsumerwithstoragecouldreduceequilibriumpro…tsduetothecostofanticipatingproduction.However,therangeofpro…tsforwhichthe3 pro…t-stabilitytrade-o¤existswillbeproventodeclineinthefractionofconsumerswithoutstorage.Relationshipbetweentheoptimalsalemarkdownandtheenvironmentwillalsobeexplored.The…nalpartoftheanalysisconsistsoftwoextensions.The…rstextensionconsidersthemulti-unitstoragescenario,characterizestheoptimaltimingofsales,andhighlightsthataccesstomulti-unitstoragetechnologiescanreducethepro…t-stabilitytrade-o¤.Thesecondextensionshowsthatsalesdonotneedtobesynchronizedwhen…rmscompeteinmultiplemarkets.Notetheintheproposedmodelcollusionisstrengthenedattheexpenseofaggregatepro…ts,sincedeviationpro…tsdeclinemorethanequilibriumpro…tswhensalestrategiesareemployed.Aggregatepro…tsmustdeclineintheproposedsetup,sinceallconsumersarehomogeneousintheirwillingnesstopay.Notehowever,thatifconsumerswithhigherstoragecapacityhadalowerwillingnesstopay,asalestrategymayachievehigherpro…tsthantheno-salestrategybypricediscriminatingamongdi¤erenttypesofconsumers,andthusfostercollusionevenfurther.Wehaveelectedtokeepvaluationshomogeneousacrossconsumersinordertodisplaymoreexplicitlythee¤ectsoftheintertemporallinkinginconsumerdemand.LiteratureReview:Oneofthe…rsttheoreticalexplanationsforsalesrelatesconsumersearchbehaviortopricediscrimination.TwoprominentexamplesinthisliteratureareVarian1980andSalopandStiglitz1982.Varian1980arguesthatinthepresenceheterogeneouslyinformedconsumers,retailpricevariationscanariseasanaturaloutcomeofmixedstrategyequilibriuminwhich…rmspricediscriminateconsumerswithdi¤erentinformation.SalopandStiglitz1982instead,considersamodelwithsearchcostsinwhichconsumersareimperfectlyinformedaboutthepriceschargedbystoresanddi¤erintheirabilitytostockpile.Insuchframeworktheauthorsshowthatstoreshaveincentivestoholdunannouncedsalestoinduceconsumerstopurchasefutureconsumption.Bothmodelshowever,areessentiallystaticmodelsandcannotaccountforcorrelationinprices.Eventhoughtherandomsalesfeatureremainsacompellingexplanationforsomeerraticpricebehavior,itappearslesssuitedtoaccountformanyofthedocumentedretailmarkdownsthatarepredictable,publiclyknow,andtakeplaceinmoststoressimultaneously(Pesendofer2002andWarnerandBarsky1995).Theappealingfashion/clearanceparadigmforsales(Lazear1986,Pashigian1988,andPashigianandBowen1991)canalso,hardlybeappliedtoawidevarietyofretailitemsfor4 whichthefashionhypothesisappearsapriorilessappropriate(eitherbecausetheitemsarehomogeneous,orbecausestyleschangelittleovertime).A…nalrelevantliteraturehasmotivatedsalesasaformofintertemporalpricediscrimi-nation(Conlisk,GerstnerandSobel1984,HendelandNevo2010,Hong,McAfeeandNayyar2002,NarasimhanandJeuland1985,Sobel1984).Conlisk,GerstnerandSobel1984,andinparticularSobel1984studytheincentivestoholdcyclicalsimultaneoussalesasameansofpricediscriminationinadurable-goodenvironment.Inmostperiods,pricesarekepthightoextractsurplusfromhighvalueconsumers,butperiodicallypricesaredecreasedinordertoselltoalargergroupofconsumerswithlowerreservationvalues.Akeyassumptiontogen-eratesuchpricecyclesistheconstantin‡owofnewheterogeneousconsumersinthemarket.HendelandNevo2010,Hong,McAfeeandNayyar2002,NarasimhanandJeuland1985,studytheincentivestoholdperiodicsalesinamarketwithstorablegoodsandheterogeneouscon-sumers.Inthissetup,theincentivestopricediscriminateconsumersovertimewithsalesfullyexplainedbythepositivecorrelationbetweenstoragecostsandconsumers’willingnesstopay.Wecomplementthesepapersbyo¤eringanewexplanationfortheexistenceofsalestrategiesinadynamicstorablegoodsmodelinwhichtheincentivestoholdperiodicsalesarisesevenintheabsenceofsuchacorrelation.Moreover,asinSobel1984,wecharacterizetheoptimaltimingforsales.Thepaperisalso,closelyrelatedtoseveralstudieswhichpointedoutimportanceoftheintertemporallinkingindecisionstoexplaincollusivebehavior(AusubelandDeneckere1987,DanaandFong2010,Gul1987,SchiraldiandNava2010).Roadmap:Section2introducesthemodel,de…nestherelevantclassofsalestrategies,andpresentsseveralpreliminaryresultscomparingdi¤erentclassesofsalestrategiesintermsoftheirstability.Section3restrictsattentiontoamodelwithunit-storage.Forthatsetuptworelevantsalestrategiesarecharacterized.Namely,themoststablesalestrategyandthemostpro…tablesalestrategyinE.Thesectionconcludeswiththefullcharacterizationofthepro…t-stabilitytrade-o¤thatdi¤erentsalestrategiesentail.Comparativestaticresultsshowhowthetrade-o¤isa¤ectedbychangesinpatience,inpro…tabilityandinthefractionofconsumerswithaccesstostorage.Section4extendsthebaselinemodelintwodirections.The…rstconsidersthegeneralmodelwithmulti-unitstorageandagaincharacterizesthemoststableandmost5 pro…tablesalestrategiesinsuchamoregeneralframework.Itwillbeshownthatsalestrategieswithinfrequentsalesmaybeincreasebothstabilityandpro…ts,andmaythus,alleviatethetrade-o¤.Thesecondextensionshowswhysynchronizationinsalesisnotnecessarywhen…rmsoperateinseveralmarkets.Section5concludes.AllproofsarerelegatedtoappendixA.AppendixBcontainsseveralusefulderivationsomittedfromthemaintextforclarity.2AModelwithStorableGoodsThissection…rstintroducesasimpleeconomywithstorage,de…nestheclassofsalestrategiesthatwillbeanalyzedthroughoutthepaper,anddevelopsseveralpreliminaryresultsonstability.TheSimpleEconomywithStorageConsideranin…nite-horizondiscrete-timemodelwithin…nitelylivedproducesandconsumers.Supposethattwogoodsaretradedinthemodelwhichweshallrefertoasconsumptionqandmoneym.Ineachperiod,allconsumersareendowedwithalargeamountofmoneyMandwithnoconsumption.Thepreferencesofaconsumerareseparableoverthetwogoodsandsatisfy:u(q;m)=8:v+mifq1mifq1Hence,themarginalvalueofconsumptionisvforthe…rstunitconsumedand0foranyadditionalunitconsumed.Thebudgetconstraintfacedbyeachconsumerineveryperiodrequiresthat:m=M�pwherepdenotestheamountofmoneyspentonconsumptiongood.Allconsumersdiscountthefutureatacommonfactor,andtheirtime-preferencesoverutilitysequencesfutg1t=0satisfy:(1�)X1t=0tutThereisaunitmeasureofconsumers.Consumersdi¤eronlyintheirabilitytostoretheconsumptiongood.Inparticular,assumethatafraction 0oftheconsumersisunabletostoregoods,whileafraction ScanstoreuptoSadditionalunitsofconsumption.Suchunitsdo6 notdepreciate,whenstoredandcanbeconsumedinanyfutureperiod.A…nitesetof…rms,Nwithcardinalityn,suppliesconsumptiongoodtothismarket.All…rmshaveacommonconstantmarginalcostofproducingconsumptiongood,c.Ifp=(p1;:::;pn)arethepricessetbyeach…rmontheunitsofconsumptionsold,andifd(p)denotestheaggregatedemandatsuchprices,theindividualdemandfacedby…rmisatis…es:di(p)=8:1 jargminj2Npjjd(p)ifpiminj2Npj0ifpi�minj2NpjThestagepro…tof…rmi2Ngivenapricevectorpsatis…es:vi(p)=(pi�c)di(p)All…rmsdiscountthefutureatacommonfactor,andtheirtime-preferencesoverpro…tsequencesfvtg1t=0satisfy:(1�)X1t=0tvtSaleStrategiesandEquilibriumFirmsandconsumersobserveallthepricesquotedinthemarketinallpreviousperiods.Thus,thesetofpossiblehistoriesintheinducedgamecanbede…nedas:H=f;g[f[1t=1ts=1Rn+gA…rm’sstrategymapshistoriesintoapricequotedatagivendate.Consumersusetheinformationaboutpastquotedpricesandtheequilibriumstrategyofthe…rmstoformbeliefsaboutfuturepricesintheeconomy.Sinceconsumersaresmallweshallassumethattheirindividualdecisionsareunobservabletoanyotherindividual.Consumerswillthus,decidehowmanyunitstopurchasefrom…rmsquotingthelowestprice,justsotomaximizetheirindividualpayo¤.Inparticular,consideranysequenceoffuturepricespt=fpzg1z=t.Let7 pz=mini2Nfpizgdenotethemarketpriceinperiodz,andlet:T(pt)=minzsubjectto pt�z pt+zdenotehowlongaconsumerhastowaitbeforethecurrentmarketpriceexceedsthefuturediscountedprice.Thenextpropositionpinsdownconsumerdemandfortheproposedenviron-ment.Remark1If pzvinanyperiodz0,thedemandforconsumptiongoodattimet:(1)byconsumerswithoutstoragetechnologysatis…esd0(pt)=1;(2)byconsumerswithstoragetechnologyandwithsunitsalreadyinstoragesatis…es:dS(s;pt)=maxminT(pt);S+1 �s;0 Thisisthecasesinceconsumerswithaccesstostoragepurchasemultipleunitsonlyiftheyperceivethestoragecosttobesmallerthanthecostoffuturepriceincreases.2Sincebuyerscannotbuildreputations,allconsumersofthesametypepurchasethesamenumberofunitsineachperiod.Hence,theaggregatedemandinaperiodinwhichallconsumerswithstoragehavethesamenumberofunitsssatis…es:d(s;pt)= 0+ SdS(s;pt)Theequilibriumstrategiesthatwillbeanalyzedthroughoutthepaperdisciplinedeviationsastrigger-strategieswould.However,equilibriumpriceswillvaryalongtheequilibriumpath.Inparticular,consideranystrategyinwhichall…rmssetpricesalongtheequilibriumpathsothatforsome{2f2;3;:::g: pt=8:(1+)cifmod(t;{)6=0(1+)cifmod(t;{)=0wheremod(t;{)6=0denotesthe{moduloofthetimeperiodt.Suchstrategymaybeinterpretedasacyclicalsalespolicyinwhichall…rmsjointlyreducepricesevery{periods,wheredenotesthemarkupinperiodswithoutsales,andwhere2[0;1]denotesthefractionof 2Thecostofstoragecoincideswiththediscountfactor,astherateoftimepreferencesrepresentstheoppor-tunitycostofspendingmoneysoonertostockunits.8 suchamarkupchargedduringsaleperiods.Deviationsfromtheequilibriumpatharepunishedviareversiontocompetitivepricingineachfuturetimeperiod.Inparticular,foranyhistoryht2Hoflengtht,allthesalestrategiesthatwillbeconsideredwillsatisfy:(ht)=8: ptifpiz= pzforanyiandanyztcifotherwiseTheequilibriumpunishmentstrategyisNashinanysubgameinwhichadeviationhasalreadyoccurred,sinceno…rmcanbene…tfromadeviationwhenalltheother…rmsarepricingcom-petitively.Thus,theincentivestocomplywithasalestrategywillbepinneddownbylookingonlyatdeviationsfromtheequilibriumpath.Lett()denotetheaggregatepayo¤ontheequilibriumpath,lett()denotethepayo¤ofthemostpro…tabledeviationfromtheequi-libriumpath,andlettheratioofequilibriumtodeviationpro…ts,t=t,bedenotedbyRt().Thefollowingresultcharacterizestheupper-boundoncartelsizeforwhichtheproposedsalestrategyconstitutesaSubgamePerfectequilibrium.Remark2AsalestrategyisaSPEofthein…niterepetitionofthegameifandonlyif:n1 1�Rt()foranyt0(1)Theupper-boundonthepopulationsizedisciplinesthelargestnumberof…rmsthatcansustainstrategyasasubgameperfectequilibrium.Notethatforanystrategywithoutsales(andthuswithoutstorage)suchconditionwouldsimplifytothecommonrequirementn1 1�.Throughout,wewillrefertothestrategiesde…nedinthissectionassalestrategies.Anyoneofthesestrategieswillbecompletelypinneddownbythethreeparameters:thesalesdiscount,theregularmarkup,andthefrequencyofsales{.ThesetofpossiblesalestrategieswillbedenotedbyS=�0;v�c c[0;1]f2;3;:::g.Theupper-boundimposedonthemarkuprequiresconsumerswithoutstoragetobewillingtotradeinperiodswithoutsales.Suchrestrictionisimposed,sinceanystrategywithahighermarkupcouldnotbeoptimalforpro…tsandforstability.SimilarconsiderationsshowwhyrelaxinganyoneoftheotherboundsonSwouldnota¤ectanyofresultsdevelopedinwhatfollows.9 PreliminaryResultsonStability&Pro…tsThissub-sectiondevelopsseveralpreliminaryresultsthatcomparedi¤erentsalestrategiesinSintermsofpro…tsandstability.Necessaryandsu¢cientconditionforastrategywithsalestobemorestablethanastrategywithoutsalesarepresented.Necessaryandsu¢cientfortheexistenceofsuchsalestrategiesarealsopresented.Tothisend,letusbeginwithatwode…nitionsclarifyingtheintentofourcomparison.De…nition1Asalestrategy2Sissaidtobemorestablethanstrategy0if:mint0Rt()mint0Rt(0)De…nition2Asalestrategy2Sissaidtoraisemorepro…tsthanstrategy0if:0()0(0)Notethatfora…xeddiscountfactor,thede…nitionofstabilitycomparesanytwostrategiesbylookingatthelargestnumberof…rmsn()thatcancolludeonstrategy:n()=mint0Rt() 1�Analternative,butsimilar,de…nitionofstabilitymayinvolvethe‡oorofthemapn().Butsimilarconclusionswouldhold.Alternatively,onecouldfocusonthethelowestpossibledis-countfactor()neededtocolludeonstrategyinapopulationof…xedsizen.Todosoa…xedpointargumentwouldhavetobeemployedtosolvefor:()=1�mint0Rt(j()) nWerefrainfromdoingsoandinstead,focusonn(),sincetheanalysissimpli…esconsiderablywhilethequalitativeresultscoincide.Inordertocomparethestabilityoftwosalestrategies,theequilibriumanddeviationpro…tsmustbeexpressedintermsoftheparametersofthestrategy.Recursivelyde…netheequilibrium10 demanddtandstoragestineachperiodt0asfollows:d0=d(0;p0)&dS0=dS(0;p0)&s1=dS0�1dt=d(st;pt)&dSt=dS(st;pt)&st+1=st+dSt�1Thenextremarkshowsthatanysalestrategyiscyclical,andthatitiswithoutlosstoconsideronlythe…rst{periodstocharacterizetheentirestreamofpayo¤s.Remark3Ifmod(t;{)=0,st=0.Ifmod(t;{)=mod(z;{),dt=dz.TheclaimfollowsfromthepropertiesoftheequilibriumpricingpathpandofthemapdS(st;pt).Itrequiresconsumersnottohaveanyunitsstoredinperiodsofsaleandthus,tohavethesamedemandatcongruentdatesinthecycle.Thus,for:S(t)=8:0ifmod(t;{)=0{�mod(t;{)ifmod(t;{)6=0equilibriumpayo¤smustsatisfyfort=f1;:::;{�1g:t()=(1�) 1�{P{�1z=0zvt+z(pt+z)==(1�) 1�{hP{�1z=0zdt+z�(1�)S(t)d0icSinceadeviationtopricey6=ptatstagetleadstoaconjecturedpricepathyt=�(y;p�it);c;c;c;:::,foryptthedeviationpayo¤sateachstagemustsatisfy:t(y;)=(y�c)d(st;yt)Forconvenience,letusde…nefourclassesofsalestrategiesforwhichallpreliminaryresultswillbedeveloped:De…nition3LetNSdenotethosestrategiessuchthat=1.LetVSdenotethosestrategiesforwhichdt&#x-277; 0andmod(t;{)6=0forsomet.LetCSdenotethosestrategiesforwhich{S+1and(1+){�1(1+).LetECdenotethosestrategiesforwhich 0 { S+ 0.11 ThesetNconsistsofallthosesalestrategiesforwhichnodiscountisevero¤eredalongtheequilibriumpath.ThesetVcomprisesallthosestrategiesforwhichthereisaperiodalongtheequilibriumpathwithnosalesandunitdemand.ThesetCinstead,willbeproventoconsistsofallthosestrategiesinwhichconsumerswithaccesstostoragepurchaseonlyinperiodswithsales.TherestofthesectionprovesthatEcomprisesallthesalestrategieswhicharemorestablethanastrategywithoutsales,andprovidesnecessaryandsu¢cientconditionsforthenon-emptinessofE.Thenextpropositionpresentsseveralintroductoryresultsonthestabilityandtheprof-itabilityofdi¤erentsalestrategies.Inparticular,itshowsthatVandCpartitionSandthatanysalestrategyinwhichconsumerswithstoragepurchaseunitsduringano-salesperiodisdominatedbothintermsofpro…tsandintermsofstabilitybyapolicyinwhichnosalesevertakeplace.Sucharesultconsiderablysimpli…estheanalysisofthepro…t-stabilitytrade-o¤.Thelatterobservationwillbeexploitedtocharacterizethesetofsalestrategiesthatcanbemorestablethantherevenuemaximizingno-salespolicy,=v�c cand=1.Proposition4Thefollowingclaimsmusthold:(1)SnC=V(2)NV(3)anystrategyinNismorestablethananystrategyinV;(4)anystrategythatsets=v�c cand=1ispro…tmaximizingwithinS;(5)foranystrategyinC,st=S(t)and:dt=8: 0+{ Sifmod(t;{)=0 0ifmod(t;{)6=0ThepropositionprovesthatVandCpartitionthesetofsalestrategies.Moreover,itimpliesthatallstrategieswithnosales(i.e.strategiesinN)areequallystableandmorestablethananyothersalestrategyinwhichconsumerswithstoragepurchaseunitsinperiodsofsales(i.e.strategiesinV).Thisshouldalso,clarifywhysaleswhereassumedtotakeplaceintheinitialperiod,sinceanysalestrategyviolatingsucharequirementwouldbelongtoV.Thelastpartofthepropositionconsiderablysimpli…estheexpressionoftheequilibriumpayo¤sforanystrategy12 inC.Inparticular,forany2Cweget:t()= 0+1� 1�{S(t)[(x S+ 0)� 0]cThelemmaalso,pinsdowntherevenuemaximizingdeviationsforsuchcandidatestrategies:t()=maxyt(y;)=8:cifmod(t;{)=0 0cifmod(t;{)6=0andthecorrespondingpro…tratios:Rt()=8: 0 +1� 1�x({ S+ 0)� 0 ifmod(t;{)=01+1� 1�{S(t)h({ S+ 0) 0�1iifmod(t;{)6=0Thenexttabledisplaysthee¤ectsofmarginalchangesinthesalestrategyonequilibriumanddeviationpro…tsandtheirratio: ddd{ dt() �0�0?dt() 000dRt() 0??ThesignofdRt()=disunspeci…ed,sinceitisnegativeifmod(t;{)=0andpositiveoth-erwise.Thesignsofd1()=d{anddR1()=d{coincide.Inparticular,suchderivativesarepositiveeitherifmod(t;{)=0,orifmod(t;{)6=0andthefractionofconsumerswithstor-ageissu¢cientlysmall,andnegativeotherwise.AppendixBcontainsthederivationofthesederivativesandtheirsigns.Asproveninthenextproposition,wheneverastrategybelongsCitiswithoutlosstoignoreall,butthe…rsttwoperiods,inordertocharacterizeitsstability.SuchconclusioncoupledwiththeobservationthatmintRt()isindependentofandsinglepeakedinforanygiven{,impliesallstrategiesinEmustbemorestablethanastrategywithnosales.Proposition5AnystrategyinEismorestablethananystrategyinSnE.Therefore,wheneverthesetEisnon-emptysalestrategiesexistthataremorestablethan13 strategieswithoutsales.ItiseasytoobservethatastrategyinEwillbestrictlymorestablethanastrategywithoutsalesifandonlyif:2 0 S{+ 0;+1 {�1�1 Suchexpressionandthepreviousresultcanbeexploitedtoderivenecessaryandsu¢cientconditionsforthesetEtobenon-empty.Corollary6Econtainsastrategywithacycleoflength{2f2;:::;S+1gifandonlyif:{�1v�c v 0 {� 0({�1)+c vThus,ifv�c v 0 2� 0+c v,thenEisnon-empty.Suchconditionsjointlydisciplinetheallfreeparametersofthemodel,namely:thefractionofconsumerswithstorage 0,thepro…tabilityofthemarketv�c,andthediscountfactor.Thecomparativestaticsresults,developedbelow,discussindetailhowthesizeofEthedependsonsuchfreeparameters.Thenewboundimposedonthediscountratearisesfromtheconsumers’demand.Thus,suchconditionwouldonlydisciplinethetimepreferencesoftheconsumers,ifthosecoulddi¤erfromtimepreferencesofthe…rms.However,evenwhenthetwocoincide,therestrictionimposedonremainsindependentofthenumberof…rmsinthemarket.Beforeproceedingtothenextsection,fornotationalconvenience,de…ne({)astheuniquepositiverootofthefollowingquadraticequation:R1(({);{)=R0(({);{)ifsuchasolutionexistsin[0;1],andset({)=1otherwise.Thedetailsofthederivationof({)andtheproofofuniquenessaredeferredtoappendixB.Further,de…ne({)asthesmallestdiscountforwhichconsumerswithaccesstostoragewouldpurchase{unitsinperiodsofsaleswhentheregularmarkupissetatthemonopolylevel=(v�c)=c:({)=v v�c{�1�c v�c14 3Single-UnitStorageandthePro…t-StabilityTrade-O¤Forsakeoftractabilitythissectionconsidersenvironmentsinwhichconsumerscanstoreatmostasingleunit,S=1.Wedoso,sincemostqualitativeresultsareuna¤ectedbythisassumption.Partoftheextensionssectionisdevotedtogeneralizingtoarbitrarystoragecapacities.ThesectioncomparessalestrategiesinSintermsofpro…tsandstability,andshowsthatatrade-o¤canemergebetweenthetwo.ThenexttwopropositionscharacterizetwostrategiesparticularstrategiesinE.Theformerwillbethemoststablesalestrategy,whilethelatterwillbethemostpro…tableofallthesalestrategiesinE.Wheneverthetwostrategiesdonotcoincide,atrade-o¤betweencartelpro…tsandstabilitywillemerge.Thesectionproceedswithtoexplicitcharacterizationofthepro…t-stabilitytrade-o¤andtoseveralcomparativestatics.Tosimplifynotationinthissectionlet = 0and=c=(v�c).TriviallyobservethatS=1impliesthat{=2foranypolicyinE.Notethatgiventhestatedassumptionsandde…nitions,(2)and(2)respectivelysatisfy:(2)=minf1;g(2)=(1+)�whereistheuniquepositiverootofthefollowingquadraticequation:2(2� )� (1� )� 2=0IfEisnon-empty,themoststablesalestrategyischaracterizedbythefollowingresult.Proposition7IfE6=;,nostrategyinSisstrictlymorestablethanstrategy2E: { 1= minf(2);(2)g 2 Moreover,isthemostpro…tableofallthestrategiesinSwithequalstability.Suchastrategyrequires…rmstosetcollusivemarkupsinperiodswithoutsalesanduniquelypinsdowntheoptimaldiscountfortheremainingperiods.Themonopolymarkupcanbecharged15 inperiodswithoutsalessincehasnoe¤ectonthestability.TheoptimalsalesdiscountischosentominimizeminfR0();R1()gwithinthefeasibleset2[ =(2� );(1+)�]andcruciallydependsonthefractionofconsumerswithstorageandonthemonopolymarkupintheeconomy.Thecomparativestaticssectiondiscussessuchdependenceindetail.Noticethattheoptimalsalestrategymaydependonalltheparametersofthemodelexceptforn,sinceboth(2)and(2)areindependentofn.Thus,thelargestnumberof…rmsn()willingtocolludeonanystrategyinScanbefoundbylookingat:n()=R1() 1�Eventhoughsuchstrategyisoptimalintermsofstability,morepro…tablepoliciesexistinE.Thenextpropositioncharacterizes,themostpro…tablesalestrategyinE.Proposition8IfE6=;,nostrategyinEisstrictlymorepro…tablethanstrategy+2E: + + {+ 1= (2) 2 Asinthepreviouspropositionthestrategyrequires…rmstosetcollusivemarkupsinperiodswithoutsales.However,thepro…tmaximizingsalediscountistheuniquelypinneddownbytheconsumer’sstorageconstraint.Obviously,suchadiscountmaybesmallerthanthatofthemoststablepolicy,andnolongerdependsonthefractionofconsumerswithstorageintheeconomy.Hence,theupper-boundonthenumberof…rmsneededtoacolludeon+maybesmallerthanfor:n(+)=minfR1(+);R0(+)g 1�n()Theprevioustwopropositionsweremeanttohighlightthetrade-o¤thatmayarisebetweenpro…tsandstabilityinsuchenvironments.The…rstresultinfact,showedthatsalestrategiescouldbeusedimprovestabilityattheexpenseofpro…ts.Strategywasproventobemorestablethananyotherstrategywithoutsales,butlesspro…tablethanfullcollusion.Similarlywasclearlymorepro…tableandlessstablethanthecompetitiveoutcome(i.e.theNashequilibriumofthestagegame).Thesecondresultinstead,showedthatevenwithinEpro…t-stabilitytrade-o¤swouldarisewhenever6=+.16 SinceSwas…xedto1,amorestringentcharacterizationofthepro…t-stabilitytrade-o¤withinScanbederived.Inparticular,notethatthepreviouspropositionsrequirethat:(1)increasingcanonlybene…tpro…tsandcannotharmthestability;(2)increasingcanonlybene…tpro…ts;(3)increasingcanharmstabilityifandonlyif2[(2);(2)].Then,for…xedvaluesof ,,and,considerastrategy(1=;;2).LetR()=mintRt(1=;;2)denoteitsstability,let()=0(1=;;2)denoteitspro…tsandlet:n()=1 1�R()denotethemaximalnumberof…rmsthatcansustainsuchastrategyinequilibrium.Notethat()isstrictlyincreasing,andthusinvertible,in.Finally,foranypro…tlevel2R+letN()=n(�1())denotethemaximalnumberof…rmsthatcancolludeonsuchapro…tlevelwhileemployingasalestrategy.Ifso,notethatatrade-o¤emergesbetweenpro…tsandstabilityfor2[(2);(2)]:dN() d=dn=d d=d(�1())=1 1�dR0=d d0=d(1=;�1();2)0sincethedenominatorispositive,whilethenumeratorisnegative(detailsinAppendixB).Inparticular,thelastexpressionimpliesthatmaximalnumberof…rmsthatcansustainacollusivesalestrategy(1=;;2)mustdeclineaspro…tsincrease.Furthernoticethatanysalestrategythatraisesmorepro…tsbysetting�(2)cannotbeoptimalintermsofstability,sinceapolicyinNexiststhatisbothmorestableandmorepro…table,as:1 1��n()Similarly,nopolicysetting(2),couldeverbeoptimal,sinceitwouldsimultaneouslyreducestabilityandpro…ts.Infact,bycontinuityasalestrategy(;(2);2)couldbeproventoexistthatisequallypro…tableas(1=;;2),butmorestable.Forconvenienceletm=v�cdenotethemonopolypro…tandletand+denotethepro…tsrespectivelyofthemoststable17 andthemostpro…tablestrategiesinE:= 1++2� 1+(v�c)+= 1++2� 1++(v�c)Noteimmediatelythat+.Thenextpropositionhighlightsthespeci…cnatureofthepro…t-stabilitytrade-o¤foreconomiesinwhichS=1.Proposition9IfE6=;,foranypro…tlevel2(0;m]themaximalnumber…rmsthatcancolludeonwhileemployingasalestrategysatis…es:N()=8���&#x]TJ ;� -2;.51; Td;&#x [00;&#x]TJ ;� -2;.51; Td;&#x [00;&#x]TJ ;� -2;.51; Td;&#x [00;:1 1�if2(+;m]1 1�(2� ) (1+)� (v�c)if2(;+]n()if2(0;]Intheinterval(;+],N()isdecreasingandconvex,andbyconstructionsatis…esN()�1=(1�).Thus,propositionexactlyquanti…esthetrade-o¤betweenpro…tsandcartelsize(sta-bility)thatdi¤erentsalestrategiesimply.Suchatrade-o¤emergessincelargersalesdiscountsmayfavorstability,butcertainlyhurtpro…ts. Figure1:TheleftplotdepictsR1(;{)andR0(;{)asfunctionsof,where ({)= 0=( S{+ 0).Therightplotdepictsthepro…t-stabilitytrade-o¤N().Theleftplotof…gure(1)depictsR1(;{)andR0(;{)andshowsthatforanyvalueof{18 auniquediscount({)existswhichmaximizesthesizeofthecartel.Therightplotinstead,depictspro…t-stabilitytrade-o¤N()derivedinthepreviousproposition.Tohighlightthetrade-o¤moreexplicitlyconsideraneconomyinwhich=0:95, =0:15,v=10,andc=1.Noticethatthemaximalcartelsizegrowsfrom20to29when…rmscolludeonthemoststablesalestrategyinsteadofthemonopolystrategym.Pro…tshowever,declinesigni…cantlyfrom9to2as…rmsneedtopriceveryaggressivelyduringsalesinordertosustaincollusion.Themostpro…tablesalestrategyinstead,marginallyimprovesthecartelsize,butraisesalmostasmuchpro…tasthemonopolypolicy.Thefollowingtablereportsalltherelevantvariablesfortheexamplediscussed: n{ m 20:09:001:0098+ 20:58:720:9492 28:61:960:1592Thenextsectiondevelopscomparativestaticstohighlighthowchangesintheenvironmentmaya¤ectsuchatrade-o¤.ComparativeStaticsAllthecomparativestaticsaredevelopedforthefourrelevantfreeparametersinthemodel,, ,vandc.The…rstpreliminaryresultcharacterizeshowthesizeofthesetEchangesassuchparametersvary.Asexpected,moresalestrategiesarestablebothwhenmanyconsumerscanstoreunits,andwhenagentsarepatient.Increasesinthepro…tabilityofthemarketfurtherimprovetheabilitytocolludeonagivensalestrategy.Proposition10ThesizeofthesetEdecreaseswithcand ,andincreaseswithvand.TheresultisprovenbystudyinghowtheboundscharacterizingthestablestrategysetEvarywiththefreeparameters.AlargerfractionofconsumerswithstorageincreasesthesizeofE,sincemoresalediscountsarestableatanyfrequency{.Similarlypatience,andpro…tabilityv�c,increasethesizeofE,sincetheconsumerdemandconstraint({)isrelaxedwhensuchvariablesgrow.19 Amorecompellingresultcharacterizeshowthepro…t-stabilitytrade-o¤isa¤ectedbychangesinthefreeparameters.Thenextpropositionexplicitlycharacterizesthisdependence,andshowshowequilibriumstrategiesarea¤ectedbychangesintheenvironment.Proposition11IfE6=;,foranypro…tlevel2(0;m]themaximalnumber…rmsthatcancolludeonwhileemployingasalestrategysatis…es:dN() dd dvdc 2(+;m] +0002(;+] +?+�2(0;]\6=+ +�002(0;]\=+ +�+�wheredN()=d �0ifandonlyif�=(2m�).Moreover,thecut-o¤pro…tlevelsand+andthemapsandevaluatedat{=2and(2)(2)furthersatisfy: dd dvdc d+ +++�d �++�d �+00d +0+�Thepropositionshowsthatincreasesinpatience(i.e.thefrequencyofinteraction)mayleadbothtolargerequilibriumcartelsatanypro…tlevelandtothepersistenceofthepro…t-stabilitytrade-o¤onalargerrangeofpro…ts.Increaseinthepro…tabilityofamarket(i.e.v�c)instead,wereshowntoincreasethemaximalequilibriumcartelsize,butonlyforintermediatepro…tlevels,asthestabilitybothofthemoststablestrategyandofthemonopolystrategymwereproventobeindependentofvaluesandcosts.Increasesinthepro…tabilityofamarketwerealsoproventoincreasetherangeofpro…tsforwhichthepro…t-stabilitytrade-o¤persists.Finally,increasingthefractionofconsumerswithoutstorage(i.e. )wasproventoreducethestabilityofthemoststablesalestrategyasintertemporallinkingbetweendecisionswoulddecline.Thee¤ectofsuchachangeonthemaximalcartelsizeatintermediatepro…tvalueswasinstead,provenambiguous,asalargefractionofconsumerwithstoragecouldleadtoa20 declineinequilibriumpro…tsduetothecostofanticipatingproduction.Clearlysuchachangewouldhavenoe¤ectonthestabilityofstrategieswithoutsales.However,therangeofpro…tsforwhichthepro…t-stabilitytrade-o¤existswoulddeclinein andwouldeventuallyvanishatsomevalue 1.Notethatthediscounto¤eredduringaperiodwithsales(i.e.1�)inthemostpro…tablepolicy+declineswithpatienceandpro…tabilityandisuna¤ectedbythefractionofconsumerswithstorage.Thediscounto¤eredinthemoststablestrategy(whensuchstrategydoesnotcoincidewith+)instead,growswithpatienceandthefractionofconsumerswithstorageandisuna¤ectedbypro…tability.Figure2belowprovidesavisualcharacterizationofthecomparativestaticsresultspresentedinthepreviousproposition. Figure2:Comparativestaticsonthetrade-o¤N()withrespecttoanincrease:in(topleft),in (topright),inv(bottomleft),andinc(bottomright).21 4Multi-UnitStorageandAsynchronizedSalesThissectionconsiderstwoextensionsofthebaselinemodel.Inthe…rstconsumershaveaccesstomulti-unitstoragetechnologies,S�1,whileinthesecond…rmscompeteinmultiplemarkets.GeneralStorageS�1The…rstextensioncharacterizestheoptimaltimingofsales,andhighlightsthataccesstomulti-unitstoragetechnologiescanreducethepro…t-stabilitytrade-o¤.Thiswillbethecase:sincethemoststablesalestrategymaybothdisplayinfrequentsalesandbebothmorepro…table,thanthemoststablepolicywithfrequency2;andbecausethemostpro…tablesalestrategywillnotbea¤ectedbythechangeinstorageconstraints.Asintheprevioussection,thenexttwopropositionscharacterizethemoststableandthemostpro…tablesalestrategiesinE.Forconvenience,letusidentifytwoparticularsalefrequencieswhichwillbeemployedinthecharacterizationofthemoststablesalestrategies:{=argmax{2f2;:::;S+1gR0(({);{)s.t.({)({){=argmin{2f2;:::;S+1g{s.t.({)({)Proposition12AssumethatE6=;.If{isde…nedandifR0(({);{)�R1(({);{)when-ever{isalsode…ned,thennostrategyinSisstrictlymorestablethanstrategy2E: { 1= ({) { otherwise,nostrategyinSisstrictlymorestablethanstrategy2E: { 1= ({) { Moreover,isthemostpro…tableofallthestrategiesinSwithequalstability.Suchastrategyrequires…rmstosetcollusivemarkupsinperiodswithoutsalesanduniquelypinsdowntheoptimaldiscountfortheremainingperiods.Comparedtothepolicyfoundintheprevioussection,themoststablestrategymaydisplaylessfrequentsales,{�2.Inparticular,22 anecessaryconditionforthisphenomenontooccuristhatthecriticalratio,R0(({);{),beincreasingwith{at{=2.Theoptimalsalesdiscountandtheoptimalfrequencyofsalescruciallydependonthefractionofconsumerswithstorage,onthediscountrate,andonthemonopolymarkupintheeconomy.Inmanycommonscenarios,thesalestrategycanbesimpli…edto(1=;({�1);{�1)if{�2,andto(1=;(2);2)otherwise.Themostpro…tablesalestrategyinEremainsuna¤ectedwhenconsumerscanstoremultipleunitsandthus,thepropositioncharacterizingsuchstrategycoincideswiththeonepresentedintheprevioussection.Proposition13IfE6=;,nostrategyinEisstrictlymorepro…tablethanstrategy+2E: + + {+ 1= (2) 2 Thepro…tmaximizingsalestrategycannotchangeevenifconsumerscanstoremultipleunits,sincealowersalesfrequencywouldleadtoabiggerdiscountandthustolowerpro…ts.Again,theoptimalsalesdiscountistheuniquelypinneddownbytheconsumer’sstorageconstraint.Asintheprevioussection,suchpropositionswoulddisplaythepro…t-stabilitytrade-o¤forthismoregeneralenvironment.The…rstresultinfact,showedthatsalestrategieswithinfrequentsalescouldbeusedtofurtherimprovestabilityattheexpenseofpro…ts.Infact,strategywasproventobemorestablethananyotherstrategywithoutsales,butlesspro…tablethanfullcollusion.Similarlywasclearlymorepro…tableandlessstablethanthecompetitiveoutcome(i.e.theNashequilibriumofthestagegame).Thesecondresultinstead,showedthatthepro…t-stabilitytrade-o¤wouldpersistwhenwouldentailinfrequentsales.ThecomparativestaticresultsonthesizeofthesetEdevelopedintheprevioussectionalsoholdwithoutfurthermodi…cationsinthismorecomplexenvironment.Asexpected,moresalestrategieswillbestable:whenconsumersaremorepatient;whenmoreconsumershaveaccesstostorage;orwhenthemarketbecomesmorepro…table.Proposition14ThesizeofthesetEdecreaseswithcand ,andincreaseswithvand.Obviously,allthecomparativestaticsdevelopedintheprevioussectiononthemostpro…tablepolicy+holdwithoutanyfurthermodi…cation.Asforthemoststablepolicy,resultswould23 havetobeadjustedtoallowoptimalfrequency{torespondtochangesintheenvironment.However,resultswitha‡avorsimilartotheonesdevelopedintheprevioussectionwouldholdatanygivenfrequency{.Werefrainfromdevelopingsuchcomparativestaticsinfullgeneralityastheadditionalintuitiongainedislimited.Toconcludethispartoftheanalysisconsideragaintheeconomyinwhich=0:95, =0:15,v=10,andc=1.Supposethatconsumerswithaccesstostoragecanstoreuptothirtyunits,S=30.Noticethatthemaximalcartelsizeisachievedwithinfrequentsaleswhichtakeplaceevery21periods.Maximalcartelsizegrowsfromto37when…rmscolludeonthemoststablesalestrategy(S).Sincesalesoccurlessfrequentlysmallerdiscountsarenecessarytosustainthemaximalcartelsize.Hence,inthisenvironmentthepro…tsofthemoststablestrategy(S)canbelargerthanthoseassociatedtothemoststablestrategy(1)ofaneconomyinwhichatmostasingleunitcanbestoredS=1,asisthecaseintheexamplereportedbelow.Thus,thepro…t-stabilitytrade-o¤candecreasewhenconsumersgainaccesstomoree¢cientstoragetechnologies.Thefollowingtablereportsalltherelevantvariablesfortheexamplediscussed: n{ m 20:09:001:0098+ 20:58:720:9492(1) 28:61:960:1592(S) 37:44:620:27921MultipleMarketsandAsynchronizedSalesFromthepreviousdiscussion,itmayappearthatcoordinationinsalesisnecessarytoachieveanystabilitygain.Incontrast,weprovideasimpleexampletoarguethatsalestrategiesdonotneedtosynchronized.Inparticular,wewillarguethatwhen…rmsoperateinmultiplemarkets,salesdonotneedtobesimultaneousandsymmetriceitherwithinoracrossmarkets.ConsideravariationonthepreviouslydescribedeconomyinwhichtherearetwoidenticalmarketsAandB,eachwithamass1=2ofconsumers,andanevennumbern4of…rmsoperatinginbothmarkets.3Corollary9providessu¢cientconditionsfortheexistenceofastablesalestrategy 3Noticethatsincemarketsaresymmetric,since…rms’objectivefunctionsarenotstrictlyconcave,andsincereturnstoscaleareconstant,therearenostabilitygainsduetothemulti-marketsetup(BernheimandWhinston24 k2Eineachmarketk2fA;Bg.Eachofthesestrategiesstillrequires…rmstochargethe…xedmarkup,k,inalmosteveryperiod,andtoperiodicallyholdsalesbyreducingthemarkuptokkevery{kperiods.Themoststablesalestrategyinsuchanenvironmentstillprescribessetk=ineachmarketk2fA;Bg.Suchasalestrategysustainscollusionifinanyperiodt2f0;1g:n1 1�t(A)+t(B) t(A)+t(B)=1 1�Rt()(2)sincet(k)=t()andt(k)=t()foranyk2fA;Bg.Nowconsiderastrategyinwhichthemarkupineachmarketis…xedtok=,butdi¤erent…rmsholdsalesindi¤erentmarketsevery{periods.Inparticular,considerastrategyinwhichsalesoccurringalongtheequilibriumpathsatisfyineveryperiodt:(1)ifmod(t;2{)=0,…rmsf1;2;:::;n=2gsetadiscountA=inmarketAandB�inmarketB,whilealltheremaining…rmssetB=inmarketBandA�inmarketA;(2)ifmod(t;2{)={,…rmsfn=2+1;:::;ngsetadiscountA=inmarketAandB�inmarketB,whilealltheremaining…rmssetB=inmarketBandA�inmarketA;(3)ifmod(t;2{)6=0;{,A=B=1forevery…rmineverymarket.Notethatany…rmchargingk�inmarketkdoesnotcollectpro…tsinthatmarketduringasalesperiod.Also,observethatthetotalpro…tacrossmarketsisconstantforeach…rmandequaltothetotalpro…tachievedincaseofsimultaneoussales.Withoutlossconsiderperiod0anda…rmi2f1;2;:::;n=2gandnotethat:0i(A)=1 2�x 1�x 0+1� 1�2x[2({ S+ 0)]c0i(B)=1 2�x 1�x 0+x�x+1 1�2x[2({ S+ 0)]c)0i(A)+0i(B)=0(A)+0(B)Moreover,notethatinanyperiodtthedeviationpro…tsofeachplayercoincideineachmarketkwiththoseofthemoststablesalestrategysincet(k)=t(k).Thefewlastobservationsinturnimplythatstrategyisasstableasmoststablestrategy.Withoutlossofgeneralityconsidertheincentivestodeviateinperiod0ofafora…rmi2f1;2;:::;n=2gholdingasales 1990,andSpagnolo1999).25 inmarketA:n1 1�0(A)+0(B) 0(A)+0(B)=1 1�R0()whichisequivalenttocondition(2).Similarly,incentivestocomplywiththeequilibriumstrategyremainuna¤ectedinperiodswithoutsales.Thus,maximalcartelsizeunderwhichasalestrategysustainscollusionremainsuna¤ectedevenwithasynchronizedsales.Hence,anasynchronizedsalestrategywouldstrictlydominateasimultaneoussalestrategyforanyarbitrarilysmallmenucostincurredby…rmswhilechangingprices.Thepreviousargumentrequiredthenumberof…rmsoperatingineachmarkettoexceedfour.Thiswasnecessary,sincethedeviationpayo¤0(k)wouldincrease,ifasingle…rmheldsalesinmarketk,ask�forany…rmnotholdingsales.Ifso,thelargestsustainablecartelwithasynchronizedsaleswouldsmallerthanwithsynchronizedsales,asstabilityisinverselyrelatedtothelowestpricechargedbyacompeting…rm.Notethatthestraightforwardextensionofthemulti-marketmodeltoasymmetricmarketswouldgeneratesalestrategieswhicharenotsynchronizedacrossmarketsaswellaswithinmarkets.Let1denotethevariantofstrategyinwhichasingle…rmhassalesinmarketAinperiodsmod(t;2{)=0andinmarketBinperiodsmod(t;2{)={.AgainconsideraneconomyinwhichS=1,=0:95, =0:15,v=10,andc=1.Fixthethreatdiscountofallthe…rmsnotsellinginamarketkduringasalesperiodtok=0:2.Asexpected,thestabilityofstrategy1issmallercomparedtoascartelsizedeclineswheneverdeviationpro…tsgrow: n{ m 20:09:001:0098+ 20:58:720:9492 28:61:960:1592 28:61:960:15921 24:71:960:159226 5ConclusionTheanalysispresentedanovelrationaleforsalesinanindustryinwhichahomogeneousstorablegoodisproducedbyn…rms,andsoldtoconsumerswithaccesstoheterogeneousstoragetechnologies.Inthiscontext,thepaperexaminedthee¤ectsofheterogeneityinstorageon…rms’incentivestoholdperiodicsalestosupportagreaterdegreeofcollusion.Insuchanenvironmentsalestrategieswereproventostrengthentheincentivestocollude,asstoragewouldintertemporallylinkconsumerdemandandthus,reducetheshort-rungainsfromadeviation.Inparticular,inanystableequilibrium…rmswouldchargeinanyperiodofsalesabigenoughdiscounttoinduceallconsumerswithopenstoragecapacitytostockpileaquantitysu¢cienttosatisfyalltheirdemanduntilthenextsale.Suchbehaviorwasshowntoreduceincentivestodeviatebothinregularpriceperiods(asonlyconsumerswithoutstoragewouldpurchaseunitsinsuchperiods)andinperiodswithsales(bothbecausealowerpricewouldbechargedinsuchperiods,andbecauseconsumerswithstoragewouldreducetheirdemandifadeviationwereobservedinthewakeofanimminentpricewar).The…rstpartoftheanalysis:characterizedconsumerdemand;characterizedthesetEofsalestrategieswhicharemorestablethananystrategywithoutsales;andpresentednecessaryandsu¢cientconditionsforthenon-emptinessofsuchaset.Thesecondpartoftheanalysisfocusedonsingle-unitstorageandcharacterizedthemoststableandthemostpro…tablesalestrategiesinthesetE.Withinsuchsetatrade-o¤wasproventoemergebetweencollusivepro…tsandcartelsize.Suchtrade-o¤anditsdependenceontheenvironmentwereexplicitlycharacterized.Relationshipbetweentheoptimalsalemarkdownandtheenvironmentwasalsoexplored.The…nalpartoftheanalysisextendedthebaselinemodelintwodirections.The…rstextensionconsideredthemulti-unitstoragescenario,characterizedtheoptimaltimingofsales,andshowedthataccesstomulti-unitstoragetechnologiescouldreducethepro…t-stabilitytrade-o¤.Thesecondextensioninstead,provedthatsynchronizationinsaleswouldnotbenecessaryinmulti-marketsetup.Intheproposedmodelcollusionwasstrengthenedattheexpenseofaggregatepro…ts,sincedeviationpro…tswoulddeclinemorethanequilibriumpro…tsifsalestrategieswereemployed.Aggregatepro…tshowever,hadtodeclinewhen…rmsheldsales,asconsumerswerehomogeneousintheirwillingnesstopay.Notehowever,thatifconsumerswithhigherstoragecapacityhada27 lowerwillingnesstopay,asalestrategycouldachievehigherpro…tsthantheno-salestrategybypricediscriminatingamongdi¤erenttypesofconsumers,andcouldthusfostercollusionevenfurther.Valuationswerekepthomogeneousacrossconsumersonlytodisplaymoreexplicitlythee¤ectsoftheintertemporallinkinginconsumerdemand.Finally,notethattherationalityimposedonconsumerdemandrequiredthatallbuyerswouldunderstandtheconsequencesofadeviationonfutureprices.Thisassumptioncouldeasilyberelaxedbyintroducingbehavioralbuyerswhocanonlyimperfectlyforecastfutureprices.Allresultswouldstillholdqualitatively,eventhoughthestabilityofanygivenstrategymaydecline.References[1]Ausubel,L.andR.Deneckere,“Oneisalmostenoughformonopoly”,RANDJournalofEconomics,1987,18:2,pp.255-274.[2]Bell,D.R.andC.A.L.Hilber,“AnEmpiricalTestoftheTheoryofSales:DoHouseholdStorageConstraintsA¤ectConsumerandStoreBehavior?”,QuantitativeMarketingandEconomics,2006,4(2),pp.87-117[3]BernheimD.B.andM.D.Whinston.“Multimarketcontactandcollusivebehavior”,RANDJournalofEconomics,1990,21,pp.1-26.[4]Dana,J.D.andY.F.Fong,“Long-LivedConsumers,IntertemporalBundling,andTacitCollusion”,Mimeo,2006.[5]Dudine,P.,Hendel,I.andLizzeri,A.,“StorableGoodMonopoly:TheRoleofCommit-ment”,AmericanEconomicReview,2006,96(5),1706–1719.[6]Erdem,T.,M.KeaneandS.Imai,“ConsumerPriceandPromotionExpectations:Cap-turingConsumerBrandandQuantityChoiceDynamicsunderPriceUncertainty”,Quan-titativeMarketingandEconomics,2003,1,pp.5-64.[7]Gul,F.,“NoncooperativeCollusioninDurableGoodsOligopoly”,RANDJournalofEco-nomics,1987,18:2,pp248-254.28 [8]Hendel,I.andA.Nevo,“MeasuringtheImplicationsofSalesandConsumerInventoryBehavior”,Econometrica,2006.[9]Hendel,I.andA.Nevo,“IntertemporalPriceDiscriminationinStorableGoodsMarkets”,Mimeo,NorthwesternUniversity.[10]HongP.,P.McAfee,andA.Nayyar,“EquilibriumPriceDispersionwithConsumerInven-tories”,JournalofEconomicTheory,2002,105(2),pp.503-517.[11]Pesendorfer,M.,“RetailSales.AStudyofPricingBehaviorinSupermarkets”,JournalofBusiness,2002,75(1),pp.33-66.[12]Salop,S.andJ.E.Stiglitz,“Bargainsandripo¤s:Amodelofmonopolisticallycompetitivepricedispersion”,ReviewofEconomicStudies,1977,44(3),pp.493-510.[13]Salop,S.andJ.E.Stiglitz,“Thetheoryofsales:Asimplemodelofequilibriumpricedispersionwithidenticalagents”,AmericanEconomicReview,1982,72(5),pp.1121-1130.[14]SeilerS.,“TheImpactofSearchCostsonConsumerBehavior:aDynamicApproach”,2010,Mimeo,LondonSchoolofEconomics.[15]Schiraldi,P.andF.Nava,“ResaleandCollusioninaDynamicMarketforSemidurableGoods”,Mimeo,LondonSchoolofEconomics,2010.[16]Sobel,J.“Thetimingofsales”,ReviewofEconomicStudies,51(3),pp.353-368(1984).[17]SpagnoloG.“OnInterdependentSupergames:MultimarketContact,Concavity,andCol-lusion”,JournalofEconomicTheory,1999,89,pp.127-139.[18]Varian,H.“Amodelofsales”,AmericanEconomicReview,1980,70(4),pp.651-659.[19]VolkerN.andL.White,“DoVerticalMergersFacilitateUpstreamCollusion?”,AmericanEconomicReview,2007,97(4),pp.1321-1339.6AppendixPartA:Proofs29 Remark1If pzvinanyperiodz0,thedemandforconsumptiongoodattimet:(1)byconsumerswithoutstoragetechnologysatis…esd0(pt)=1;(2)byconsumerswithstoragetechnologyandwithsunitsalreadyinstoragesatis…es:dS(s;pt)=maxminT(pt);S+1 �s;0 Proof.Part(1)oftheclaimistrivial.Toprove(2)noticethatbyconstructiondS(s;pt)2[0;S�1+s].Theupper-boundmusthold,sincenoconsumercanstoremorethanSunits.Thelower-boundmustholdbecausenoplayercanbene…tbydisposingalreadypurchasedunitsgiventhat pz0foranyz.Alsonoticethatonlypro…lesofdemandsuchthatguaranteeaconsumptionstreamofaunitineveryperiodcanbeoptimal,sincepricessatisfy pzv.Thus,payo¤streamcanbecomparedbylookingonlyatthetotalexpenditureonconsumptiongood.Then,considerthecaseinwhichT(pt)S+1.Bycontradictionconsiderapro…leofdemandforthesuccessiveT(pt)periods,fdt+zgT(pt)z=0andsupposethatdt6=maxfT(pt)�s;0g.Ifso,thereexistsapro…leofdemandsd0t+z T(pt)z=0thatcostslessandthatleavestheconsumerwithexactlyasmanyunitsstoredinperiodt+T(pt).Infact,consider:d0t=maxT(pt)�s;0 d0t+z=0ifz2�0;T(pt)d0t+T(pt)=hPT(pt)z=0dzi�d0tbyconstructionthepro…leleavestheconsumerwithexactlyasmanyunitsstoredinperiodt+T(pt).Moreoverd0costsless,since:PT(pt)z=0z pzdz=PT(pt)�1z=0z pzdz+T(pt)pt+T(pt)dt+T(pt) ptPT(pt)�1z=0dz+T(pt)pt+T(pt)dt+T(pt)== ptd0t+ pthPT(pt)�1z=0dz�d0ti+T(pt)pt+T(pt)dt+T(pt)== ptd0t+ ptd0t+T(pt)�dt+T(pt)+T(pt)pt+T(pt)dt+T(pt)==ptd0t+T(pt)pt+T(pt)d0t+T(pt)+hT(pt)pt+T(pt)� ptidt+T(pt)�d0t+T(pt)ptd0t+T(pt)pt+T(pt)d0t+T(pt)30 giventhat:(i)PT(pt)�1z=0dzd0t&dt+T(pt)d0t+T(pt),sincethatconsumersconsumeoneunitineveryperiod;and(ii)T(pt)pt+T(pt) ptz pt+zforanyz2(0;T(pt)).Thusacontradictionisestablished.AverysimilarandomittedargumentworksalsoforthecaseinwhichT(pt)�S+1andestablishestheclaim. Remark2StrategyisaSPEofthein…niterepetitionofthegameifandonlyif:1�1 nt tProof.Theproofoftheresultistrivial.Noplayerbene…tsfromadeviationalongtheequilibriumpathif:t n(1�)twheretdenotesthemostpro…tabledeviation.Suchconditionisexploitedtopindowntherequirementonthecriticaldiscountrate.Moreovernodeviationcanbepro…tableo¤theequilibriumpath,sinceallplayersmakeatmostzeropro…tswhenallcompetitorsquotepricesatmarginalcost. Remark3Ifmod(t;{)=0,st=0.Ifmod(t;{)=mod(z;{),dt=dz.Proof.The…rstclaimisprovenbyinduction.Notethatmod(0;{)=0ands0=0.WeshowthatiftheclaimistrueforanytTsuchthatmod(t;{)=0itistrueforanytT+{suchthatmod(t;{)=0.InfactconsiderthelargestdatetsuchthattTandmod(t;{)=0.Suchdateexistsbytheinitialconditionandtheinductionhypothesis.Atsuchdatethedemandofanindividualwithstoragesatis…es:dS(0;pt)=minT(pt);S+1 MoreoverT(pt){,since(1+)�{(1+).Hence,dS(0;pt){andst{giventhatoneunitwillbeconsumed.Moreover,inanyperiodz2ft+1;:::;t+{g,since(1+)&#x]TJ/;༴ ;.9;Ւ ;&#xTf 1;.80; 0 ;&#xTd [;(1+)&#x]TJ/;༴ ;.9;Ւ ;&#xTf 1;.80; 0 ;&#xTd [;(1+),wehavethatT(pt)=1andconsequently:dS(st;pt)=8:0ifst&#x]TJ ;� -2;.52;&#x Td ;&#x[000;01ifst=0)st+1=8:st�1ifst&#x]TJ ;� -2;.52;&#x Td ;&#x[000;00ifst=031 Whichestablishesthatst+{=0,sincest{.Thesecondclaimfollowsimmediately,sincefromthepreviouspartoftheproofitisstraightforwardtoobservethat:dS(st;pt)=8&#x]TJ/;༴ ;.9;Ւ ;&#xTf 1;.69; 0 ;&#xTd [;&#x]TJ/;༴ ;.9;Ւ ;&#xTf 1;.69; 0 ;&#xTd [;&#x]TJ/;༴ ;.9;Ւ ;&#xTf 1;.69; 0 ;&#xTd [;&#x]TJ ;� -2;.52;&#x Td ;&#x[000;&#x]TJ ;� -2;.52;&#x Td ;&#x[000;&#x]TJ ;� -2;.52;&#x Td ;&#x[000;:dS(0;p0)ifmod(t;{)=00ifst&#x]TJ ;� -2;.52;&#x Td ;&#x[000;0&mod(t;{)6=01ifst=0&mod(t;{)6=0(3) Proposition4Thefollowingclaimsmusthold:(1)SnC=V(2)NV(3)anystrategyinNismorestablethananystrategyinV;(4)anystrategythatsets=v�c cand=1ispro…tmaximizingwithinS;(5)foranystrategyinC,st=S(t)and:dt=8: 0+{ Sifmod(t;{)=0 0ifmod(t;{)6=0Proof.Toprovepart(1),webeginbyarguingthatSnCV.Considerastrategy2SnCforwhichtheconstraint(1+){�1(1+)isviolated.Bythedemandstructureestablishedincondition(3)aperiodexistsinwhichmod(t;{)6=0anddSt&#x]TJ ;� -2;.51; Td;&#x [00;1,sinced0S{.Therefore,everystrategyviolating(1+){�1(1+)mustbelongtoV.Similarly,if{&#x]TJ/;༴ ;.9;Ւ ;&#xTf 1;.44; 0 ;&#xTd [;S+1aperiodwouldexistsinwhichdSt&#x]TJ/;༴ ;.9;Ւ ;&#xTf 1;.44; 0 ;&#xTd [;1andmod(t;{)6=0,andthestrategywouldagainbelongtoV.Thus,SnCV.Furthernotethatcondition(5)requiresC\V=;,andthusestablishes(1).Notethat(2)isimmediate,because=1impliesdS(st;pt)=1foranytas pt+1= ptrequires pt� pt+1.Toprove(3),…rstobservethatallstrategiesinNareequallystable.Notethat,bytheproofof(2),foranystrategy2Nequilibriumpayo¤ssimplifytot()=c.Thus,adeviatingplayercancaptureatmostsuchapro…tbyundercuttingthepricemarginally.Anydeviationtoapricey2(c;(1+)c),mustsatisfyd(st;yt)1,since pt�candthereforet(y;)(y�c).Hence,t()=candRt()=1foranyt2f0;1;:::gandany2N.Now,considerastrategy2Vandaperiodtinwhichdt� 0and pt=(1+)c.Notesuchconditionsimplythat32 st=0anddt1.Ifso,by pt=(1+)c,wegetthatt(y;)=(y�c)d(st;yt)=(y�c),andt()=c.Moreover,ifsuchaperiodexists,itmustbethat:dS0=minT(p0);S+1 {becauseoftheevolutionofsavingsanddemanddiscussedintheperviouslemma(condition3).Inturnthisrequiresthatt�1(1+)(1+)&#x]TJ/;༴ ;.9;Ւ ;&#xTf 1;.42; 0 ;&#xTd [;t(1+)forsomet2f1;:::;{�1g.Ifso,pickthesmallesttforwhich(1+)&#x]TJ/;༴ ;.9;Ւ ;&#xTf 1;.42; 0 ;&#xTd [;t(1+)andnoticethat:t()=(1�) 1�{hP{�1z=0zdt+z�(1�)S(t)d0ic==(1�) 1�{hhPS(t)�1z=0z+ 0P{�1z=S(t)+1zi+S(t)( 0+ St)ic== 00t()+ SSt()cwherethelastinequalitymustholdsince:0t()=1�S(t)+S(t)+1�{+(S(t)�S(t)+1) 1�{ccSt()=1�S(t)+t(S(t)�S(t)+1) 1�{c==�1�{�t+t({�t�{�t+1) 1�{cc,�1�t t(1�)Theinequalitybounding0t()musthold,sinceitcannotbepro…tabletocutpricesoncon-sumersthatdonotaltertheirdemand.TheinequalityboundingSt()mustholdinstead,since…rmsprefertodelayproductioncostsandbecauset�1(1+)(1+)requires:1+ 1+t�1Pt�1z=0z t=�1�t t(1�)Hence,astrategy2VcannotbemorestablethanastrategyinN,sinceRt()1.Theproofof(4)istrivial.Theproposedstrategyraisesapro…tofv�c,sincedt=1foranyt.Nostrategyinwhichdt=1foranytcandobetter,sincevisthehighestpricethatabuyerwillingpayforaunitofconsumption.Butanyotherstrategysuchthatdt6=1forsometmustsatisfydS0�1,bythepropertiesofthedemandfunctionderivedincondition(3).In33 turn,ifdS0�1,itmustbethat(1+)(1+).Thusby(3),wegetthatpro…tscanbeexpressedasfollowsforsomedS02(1;{]:0()=(1�) 1�{P{�1z=0zdz�(1�)d0c==(1�) 1�{hh 0PdS0�1z=1z+P{�1z=d0Szi+( 0+ SdS0)icAnargumentsimilartotheonedevelopedinthepreviouspartoftheproofshowsthat0()c.Inparticular,writepro…tsas0()= 000()+ SS0()andnoticethatforthesamereasondescribedinpart(3)00()c.Thenlett=d0snoticethat:S0()=�t�{+t(1�) 1�{cc,�1�t t(1�)wheretheinequalityboundingS0()isestablishedbyt�1(1+)(1+)asinpart(3).Thisestablishes(4),sincecv�cisnecessaryforpro…tstobemaximalbythepropertiesofthedemandfunction.Part(5)followstriviallyfromcondition(3)andthedemandfunctionsofbothtypesofconsumersdiscussedinthetext. Proposition5AnystrategyinEismorestablethananystrategyinSnE.Proof.FirstweestablishthatastrategyEinismorestablethanastrategyinN.Considerastrategy2E.Byde…nitionofE, 0 { S+ 0,andthereforewegetthatR1()1andthatforanyt2f1;2;:::;{�2g:Rt()=1+(1�) 1�{{�t{ S+ 0 0�1Rt+1()Hence,thestabilityofastrategyin2EwillbepinneddownbytheminimumbetweenR0()andR1().Moreover,R0()�1sinceforanyandforany{�1:R0()= 0 +1� 1�xh({ S+ 0)� 0 i 0+(1�){ 1�x S= 0+{ 1++:::+x�1 S�134 wherethe…rstinequalityholdssincedR0()=d0.WhichestablishesthatifastrategybelongstoEthenitmustbemorestablethananystrategyinN,sincemint0Rt()1.SinceanystrategyinNismorestablethananystrategyinV,whatremainstobeprovenisthatanystrategyinEismorestablethanstrategiesinCnE.Butthisisimmediatesince2CnEimplies 0 { S+ 0,andthusR1()1. Corollary6Econtainsastrategywithacycleoflength{2f2;:::;S+1gifandonlyif:{�1v�c v 0 {� 0({�1)+c v(4)Thus,ifv�c v 0 2� 0+c v,thenEisnon-empty.Proof.First,weestablishthat4impliestheexistenceofastrategywithacycleoflength{2f2;:::;S+1ginE.Letconstraint4holdforsome{2f2;:::;S+1g.Takeanystrategythatsets=v c�1and2 0 S{+ 0;{�11+1 �1 (5)atthegivenvalue{.ThestrategyobviouslybelongstoE.Moreover,suchastrategyexistssincetheintervalinwhichwaschosenisnon-empty,whenever4holdsat{.Nextweestablishthenecessityof4.AnystrategyinEmustsatisfy5byconstruction.Consideranyoneofthesestrategies,andnoticethat: 0 S{+ 0;{�11+1 �1 0 S{+ 0;{�1v v�c�c v�cSincethenon-emptinessofthebiggerintervalisequivalentto4,wegetthat4beingviolatedpreventtheexistenceofapolicywithcyclelength{inE.Thisestablishesthenecessity.Thelastobservationisatrivialcorollary. Proposition7IfE6=;,nostrategyinSisstrictlymorestablethanstrategy2E: { 1= minf(2);(2)g 2 Moreover,isthemostpro…tableofallthestrategiesinSwithequalstability.35 Proof.Thisfollowsfromtheproofofproposition(12). Proposition8IfE6=;,nostrategyinEisstrictlymorepro…tablethanstrategy+2E: + + {+ v=c�1 (2) 2 Proof.Thisfollowsfromtheproofofproposition(13). Proposition9IfE6=;,foranypro…tlevel2(0;m]themaximalnumber…rmsthatcancolludeonwhileemployingasalestrategysatis…es:N()=8���&#x]TJ ;� -2;.52;&#x Td ;&#x[000;&#x]TJ ;� -2;.52;&#x Td ;&#x[000;&#x]TJ ;� -2;.52;&#x Td ;&#x[000;:1 1�if2(+;m]1 1�(2� ) (1+)� (v�c)if2(;+]n()if2(0;]Proof.Firstnotethatif�+nostrategyinEismorepro…tablethan+.Thus,nosuchpro…tlevelcanbesustainedbyasalestrategybelongingtoE.Ifso,themoststablestrategyisonewithoutsales.However,allstrategiesinNareequallystablebyproposition(7)andthus,N()=1 1�foranysuchstrategy.Thensupposethatandconsideranystrategy(;;2)withpro…ts.Notethatimpliesthateitheror.Alsonotethatadi¤erentstrategy(;;2)existswhichraisesexactlythesamepro…ts,sinceanypro…tlevelcanbeobtainedbypicking2(0;].Thusobservethatstrategy(;;2)isequallystabletostrategy=(;;2)andthusmorestablethan(;;2).Finallyconsiderthecaseinwhich2(;+].Notethatforthistobethecaseitmustbethat+,whichinturnrequires=(2)(2)=+Notethatsetting=isalwaysoptimalforbothpro…tsandstability.Thus,foranypro…tlevel2(;+]acorrespondingsalesdiscountexists()2(;+]whichsustainspro…t36 level.Suchadiscountisfoundbysolvingthefollowingequalitywithrespectto:= 1++(2� )1 1+c)()=1 2� v�c(1+)� ThevalueofN()insuchintervalcanthenbefoundbycomputingn(()):N()=n(())=1 1�R0(;();2)=1 1� ()c==1 1�(2� ) (1+)� (v�c)whichestablishesthedesiredresult. Proposition10ThesizeofthesetEdecreaseswithcand ,andincreaseswithvand.Proof.Firstnotethatbycorollary9thesu¢cientconditionfortheexistenceofasalestrategywithperiod2requires: 2� +c v1� 2� =h(6)Furthernoticethatsuchconditioniseasiertosatisfywhentherighthandsideof(6)issmaller.Thisinturnimpliesthedesiredresultssince:dh d =v�c v2 (2� )2�0dh dv=�c v21� 2� 0dh dc=1 v1� 2� �0The…nalobservationonistrivialthelefthandsideof(6)increasesin. Proposition11IfE6=;,foranypro…tlevel2(0;m]themaximalnumber…rmsthatcan37 colludeonwhileemployingasalestrategysatis…es:dN() dd dvdc 2(+;m] +0002(;+] +?+�2(0;]\6=+ +�002(0;]\=+ +�+�wheredN()=d �0ifandonlyif�=(2m�).Moreover,thecut-o¤pro…tlevelsand+andthemapsandevaluatedat{=2and(2)(2)furthersatisfy: dd dvdc d+ +++�d �++�d �+00d +0+�Proof.Firstnotewhen2(+;m]thesignofallthederivativesofN()=1=(1�)istrivial.Next,considerthecaseinwhich2(;+].Notethatwithinsuchinterval2(;+]and:dN() d =1 1�[2(v�c)�(1+)] ((1+)� (v�c))2�0,2(v�c)�(1+)dN() d=1 (1�)2(2� )[2(� (v�c))+ (v�c)] ((1+)� (v�c))2�0dN() dv=�dN() dc=1 1�(2� ) ((1+)� (v�c))2�0Thesecondinequalityholds,since�(v�c)and� togetherimply� (v�c)(wherethe…rstconditionholdssince0�0forthestrategytobelongtoE,andwherethesecondconditionholdssince�andsincetheonlypositiverootof(2)satis…es� ,asexplainedinappendixB).Beforeweproceed…nalscenario2(0;],letusprovealltheremainingresults.First,observethatd(2)=dv=d(2)=dc=0,sincebothR0andR1areindependentofvaluesand38 costs(seeappendixB).Furthernotethatbytheimplicitfunctiontheoremappliedtothemap(2):=d(2) d=�R1�R0 R1�R0=�(1� ) [2(2� )�(1� ) ] =d(2) d =�R1 �R0 R1�R0=(+1)+2 (�) [2(2� )�(1� ) ]Moreover,notethatd(2)=d0,since2(2� )&#x]TJ/;༕ ;.9;Ւ ;&#xTf 3;.31; 0 ;&#xTd [;(1� ) byde…nitionof;andthatintheonlyrelevantscenario(i.e.(2)&#x]TJ/;༕ ;.9;Ւ ;&#xTf 3;.31; 0 ;&#xTd [;(2))d(2)=d &#x]TJ/;༕ ;.9;Ւ ;&#xTf 3;.31; 0 ;&#xTd [;0,since&#x]TJ/;༕ ;.9;Ւ ;&#xTf 3;.31; 0 ;&#xTd [;(2)&#x]TJ/;༕ ;.9;Ւ ;&#xTf 3;.31; 0 ;&#xTd [;(2)=.Also,notethatR1&#x]TJ/;༕ ;.9;Ւ ;&#xTf 3;.31; 0 ;&#xTd [;0,R00,R1 0andR0 0.Thesignofthederivativesofthemap(2)followtriviallyfromitsde…nition.Thennotethat+anditsderivativeswithrespecttotherelevantparameterssatisfy:+= 1+(v�c)+2� 1+(v�c)d+ d=2v+2(1� )c (1+)2�0&d+ d =c(1�) 1+�0d+ dv=2 1+�0&d+ dc=�2� (1�) 1+0Tocomputethederivativesof,considerthecaseinwhich(2)&#x]TJ/;༕ ;.9;Ւ ;&#xTf 1;.42; 0 ;&#xTd [;(2).Orelse,and+andtheirrespectivederivativeswouldcoincide.Ifso:= 1++2� 1+(2)(v�c)d d=1 (1+)2 �(2� )(2)+(2� )(1+)d(2) d(v�c)0d d =�(2) 1++2� 1+d(2) d (v�c)�0d dv=�d dc= 1++2� 1+(2)�0wherethe…rstinequalityholdssinceE6=;implies �(2� )(2)0,andthesecondinequalityholdssince(2)�(2)implies�(2).Atlast,considerthecaseinwhich39 2(0;].Supposethat6=+.Ifso,(2)�(2)andtherefore:dN() d=1 (1�)2[R0+(1�)[R0+R0]]�0(7)dN() d =1 1�[R0 +R0 ]0(8)dN() dv=dN() dc=0(9)where(7)ispositivesinceR0�0andbecause:R0+(1�)R0= +2 (1+)2h(2� )� i�0where(8)isnegativesince:R0 +R0 =R1R0 �R0R1 R1�R0==�1 R1�R0 (1+)2[ +(2� )]0andwhere(9)holdstrivially.Finally,considerthecaseinwhich2(0;]and=+.Ifso,(2)(2)and:dN() d=1 �1�222+2� (1+2)+(1�2)�0dN() d =� 1�22 20dN() dv= 1�22� v�0&dN() dc= 1�22� c0whichconcludestheproof. Proposition12AssumethatE6=;.If{isde…nedandifR0(({);{)�R1(({);{)when-ever{isalsode…ned,thennostrategyinSisstrictlymorestablethanstrategy2E: { 1= ({) { 40 otherwise,nostrategyinSisstrictlymorestablethanstrategy2E: { 1= ({) { Moreover,isthemostpro…tableofallthestrategiesinSwithequalstability.Proof.Toprovetheclaimitsu¢cestoshowthatismorestablethananyotherstrategyinE.Consideranyotherstrategy=(;;{)2S.Firstletusestablishthatif2E,thenthesalestrategy({)=(1=;minf({);({)g;{)alsobelongstoEandismorestablethan.Notethat: ({) 0 S{+ 0{�11+1 �1 ({)wherethe…rstinequalityholdsby2E,andthesecondsince1=.Moreover,minf({);({)g2[ ({);({)]since({) ({)giventhatR1(;{)1R0(;{)forany({).Thus,({)2E.Toprovethat({)ismorestablethan,…rstnotethatthemarkupdoesnota¤ectmint0Rt()andincreases({),whichinturnimpliesthatsettingtoitsupper-boundcannotreducethestability.Thennotethatmint0Rt(;{)issinglepeakedin2[0;1],sinceR0(;{)decreasesin,sinceR1(;{)increasesin,andsinceR0(0;{)&#x-286;R1(0;{)(seeappendixpartBfordetail).Alsonoticethatthepeakmint0Rt(;{)withrespecttoisachievedexactlyat=({).Thus,if({)({),nostrategywiththesamecyclelengthcanbemorestablethan({).Ifhowever,({)&#x-286;({),themoststablestrategymustsatisfy=({),sincemint0Rt(;{)increasesinfor({).Nextobservethatbytheimplicitfunctiontheoremwegetthat:0({)=�R1{�R0{ R1�R0041 Notethatthedenominatoristriviallypositive(seeappendixB),andthatthenumeratorisnegativesinceat=({):R0{�R1{=1� 1�{ S1�{�1 0+{�1log 1�{ 0 �1 ({)�10wherethe…rsttermispositivesince({) 0={�1,andwherethesecondtermispositivesince({)minf ({); 0g(seeappendixBfordetails).Hence,since0({)0,thereexistsauniquevalue{suchthat({)=({).Letf({)=(({)= ({))�1.Notethatforasalestrategytobemorestablethanastrategywithoutsalesitmustbethatf({)0.Furthermore:@ @{R1(({);{)={�1�{ 1�{f0({)+log 1�{f({){�1�{ 1�{f0({)�1� 1�{f({){�1�{ (1�{){[{f0({)�f({)]0wherethe…rstinequalityholdssincelog�1,andthesecondsince1�{{(1�).Thethirdinequalityinstead,holdssince:{f0({)�f({)=1 ({){0({)�{ 0({)({) ({)�({)+ ({)==1 ({)[log{(({)+1=)+({)(1� ({))�({)+ ({)]==1 ({)log{({)+1 + ({)(1�({))==1 ({)1+1 log{�{�1+ ({)�1�{�11 ({)1+1 ({�1){�1+ ({)�1�{�11 ({)1+1 ( ({)�{�1)�1�{�1042 wherethe…rstinequalityholds,sincelog{{�1,wherethesecondholdstrivially,andwherethelastinequalityholdssince({) ({)isequivalentto:{�1 ({)++1 (1� ({))� ({)Thelastfewobservationstogetherestablishedthatif({)({)forsome{,thenincreasingthecyclelengthwouldonlyreducethestabilityofthesalestrategy({).Inturnthisestablishesthatsetting{�{cannotimprovestability.Finally,notethat,if{exits,nostrategywithperiod{{canbemorestablethan(1=;({);{)byde…nitionof{.Thus,themoststablesalestrategywillbeeither(1=;({);{)or(1=;({);{)dependingontherelativestabilityofthetwo.Theobservationaboutpro…tsfollowstrivially,sincechangingand{wouldnecessarilyreducestabilitybyconstructionofandbecause=v c�1raisesthehighestpro…tandcannotlowerstability. Proposition13IfE6=;,nostrategyinEisstrictlymorepro…tablethanstrategy+2E: + + {+ v=c�1 (2) 2 Proof.Bythepropertiesofthetimezeropro…tfunctiondiscussedinappendixB,pro…tsattime0increasein,,and{.Thus,themostpro…tablestrategyinEwithacycleoflength{musttriviallysatisfy=v c�1and=({),since{�11+1 �1 increasesin.Thus,(2)followsimmediatelysince{ischosenbyde…nitionsotomaximizepro…tsinEandsince:@0(;({);{) @{= 0c(1�)2{�1(1+) (1�{)2log+ Sc(1�) (1�{)2(1�{)({�1(1+)�1)+{�1((1+)�)logx Sc(1�) (1�{)2({�1(1+)�1)(1�{)+{�1((1+)�)logx Sc(1�) (1�{)2({�1(1+)�1)�{�1((1+)�)(1�{)==� Sc(1�)2 (1�{)043 wherethesecondinequalityholdslogxx�1andtherestissimplealgebra. Proposition14ThesizeofthesetEdecreaseswithcand ,andincreaseswithvand.Proof.Firstnotethatbycorollary9thesu¢cientconditionfortheexistenceofasalestrategywithperiod{requires:{�11 1+ {� ({�1)+ 1+=h( ;)(10)Furthernoticethatsuchconditionishardertosatisfywheneitheror increasesince:dh( ;) d =1 1+{ ({� ({�1))2�0dh( ;) d=1 (1+)2(1� ){ {� ({�1)�0Thus,thesizeofthesetEdecreaseswithbothand .Toestablishthecomparativestaticsoncandv,simplynotethatd=dv0andthatd=dc&#x]TJ/;༕ ;.9;Ւ ;&#xTf 4;.99; 0 ;&#xTd [;0.The…nalobservationonistrivialthelefthandsideof(10)increasesin. PartB:DerivativesandSignsRecallthatforanystrategy2Cequilibriumanddeviationpro…tsinthetwocriticalperiodsrespectivelysatisfy:0()= 0+1� 1�x[({ S+ 0)� 0]c1()= 0+{�1�{ 1�{[({ S+ 0)� 0]c0()=c1()= 0c44 Derivativesatt=0:d0() d= 0+1� 1�x[({ S+ 0)� 0]c�0d0() d=1� 1�{({ S+ 0)c�0d0() d{=1� 1�{ S+log{�{+1 (1�{)2[({ S+ 0)� 0]c�0d0() d=c0&d0() d=c�0&d0() d{=0Tosignd0()=d{considerhardercase,namely S=1.Ifso:d0() d{=1� 1�{1+{log{ 1�{c0whichispositive,since:xlog=logx1�1 {Similarly,derivativesatt=1,satisfy:d1() d= 0+{�1�{ 1�{[({ S+ 0)� 0]c�0d1() d={�1�{ 1�{({ S+ 0)c�0d1() d{={�1�{ 1�{ S+log{�1�{ (1�{)2[({ S+ 0)� 0]cd1() d= 0c&d1() d=d1() d{=0Again,tosignd1()=d{considerhardercase,namely S=1.Ifso:d1() dx={�1�{ 1�{1+{log1 1�{c0whichisnegative,since:log{{�145 Moreoverd1()=d{�0,when 0=1.Thus,thesignofd1()=d{dependsonthefractionofconsumerswithstorageintheeconomy.Noticethattheresultingcriticalratiosareindependentof:R0()= 0 +1� 1�xh({ S+ 0)� 0 iR1()=1+{�1�{ 1�{({ S+ 0) 0�1Derivativesatt=0:dR0() d=��x 1�x 0 20dR1() d={�1�{ 1�{{ S+ 0 0�0dR0() d{=1� 1�{ S+log{�{+1 (1�{)2h{ S+ 0� 0 i0dR1() d{={�1�{ 1�{ S 0+log{�1�{ (1�{)2{ S+ 0 0�1wherethesignofdR1()=d{coincideswiththatofd1()=d{.NoticethatR1()�R0()=0requires:2x�1 S{+ 0 0+1�{�1 1��[ S{+ 0]� 0�{ 1�=0Suchconditionalwayshasuniquepositivesolutionwhichsatis…es2[ 0; 0={�1],sinceitisnegativebothat=0andat= 0,andpositiveat= 0={�1.However,thesolutioncouldinprinciplerequire�1.Ifso,thesolutiontothegeneralprogrammax2[0;1]mintRt()=R1()willsatisfy({)=minf1;g.46