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1LévydistributionandlongcorrelationtimesinsupermarketsalesRobertD.GrootUnileverResearchVlaardingenPOBox114,3130ACVlaardingen,TheNetherlandsSalesdatainacommoditymarket(supermarketsalestoconsumers)hasbeenanalysedbystudyingthefluctuationspectrumandnoisecorrelations.Threerelatedproducts(ketchup,mayonnaiseandcurrysauce)havebeenanalysed.Mostnoiseinsalesiscausedbypromotions,butherewefocusonthefluctuationsinbaselinesales.Thesecharacterisethedynamicsofthemarket.Fourhithertounnoticedeffectshavebeenfoundthataredifficulttoexplainfromsimpleeconometricmodels.Theseeffectsare:(1)thenoiselevelinbaselinesalesismuchhigherthancanbeexpectedforuncorrelatedsalesevents;(2)weeklybaselinesalesdifferencesaredistributedaccordingtoabroadnon-Gaussianfunctionwithfattails;(3)thesefluctuationsfollowaLévydistributionofexponentα=1.4,similartofinancialexchangemarketsandinstockmarkets;and(4)thisnoiseiscorrelatedoveraperiodof10to11weeks,orshowsanapparentpowerlawspectrum.Thesimilaritytostockmarketssuggeststhatmodelsdevelopedtodescribethesemarketsmaybeappliedtodescribethecollectivebehaviourofconsumers.PACS:89.65.Gh,05.40.Ca,05.40.Fb,05.45.TpKeywords:markets,minoritygame,econophysics,consumergoods,baselinesales,noise,detrendedtimeseriesanalysisEmail:[e-mail],Lévydistributionandlongcorrelationtimesinsupermarketsales21IntroductionThesearchforgeneraltrendsinmarketshasalonghistory.Nevertheless,some70%ofnewproductsputinthemarketfails.InmarketingsciencetheadoptionofnewtechnologiesistraditionallystudiedusingtheBassmodel[1],whichwasfirstposedin1969,andwhichrelatesthefractionofconsumersadoptinganewtechnologyperunitoftime,tothefractionpresentatthattime.Thismodelignoresthevariationamongconsumersanditignorestheirnetworkofsocialrelations.Oneimportantrecentinsightinthedescriptionofhumaneconomicbehaviouristhatthesenetworksmaybeveryimportantforthesuccessorfailureofnewproducts.Arthur[2]arguedthattheamountofinformationhumanstypicallyhavedealwithistoolargetohandle,andmoreoverindividualsusuallyhavetobasetheirdecisionsonpartialinformation.Thereforetheyturntorulesofthumb,orstrategies,todeterminetheiractions.Thesestrategiesareoftenjustcopiedfrompeers;thisisthefirstreasonwhysocialnetworksareimportant.Thesecondreasonwhynetworksareimportant,isthatpartoftheperceivedvalue(orutility)ofaproductdependsonitsacceptanceinthesocialnetworkoftheconsumer.Theutilityofaproductrepresentsmorethanitsdirectapplication;itsvalueinusecanbeincreasedbecausetheconsumerfeelsthatitattributestohis/hersocialstatus[3].Forexample,clothingcanbevaluabletoexpressthemembershipofapeer-group,orotherwiseexpensivecarsmaycuestatustothepeer-group.Thissocialeffectisnotlimitedtoproductslikecars,butmayalsoholdforcommoditymarkets.RecentlyStauffer[4],WeisbuchandSolomonhavestudiedtheimpactofsocialnetworksandpercolationphenomenatotheadoptionofnewproducts.Solomonetal.[5]discussthepossibilityofobservingself-organisedcriticality.Thequestionaddressedis:whydoesoneobservethatsomenewlylaunchedproductsaredoomedtofailurewhileothersbecomegreathits,ratherthanafeaturelessdistributionofpartialsuccess?Thegeneralexplanationforsuchabimodaldistributionisdiscussedintermsofboundedrationality.Decisionsarenottakenuponcompleteinformation,butconsumerscopythebehaviouroftheirpeers[6].Thus,thespreadofanewtechnologyisoftencomparedtothespreadofavirusoverthepopulation,andisdeterminedbyapercolationtransition.Goldenbergetal.[7]bringtheseideasfurtherandcomparethemtomarketingofconsumergoods,namelytheintroductionofcarsandLCDcolortelevisionsets.However,themodelsmentionedareallbasedontheoreticalideasofhowconsumerschoose,thelinktoempiricaldataisweakormissing.BeforeattemptingtodescribetheRDGroot,Lévydistributionandlongcorrelationtimesinsupermarketsales3dynamicsofnewmarketintroductions,oneshouldatleastbeabletodescribethedynamicsofastationarymarket.Thefirststeptodeveloparealisticmodelofthecollectivebehaviourofconsumers,whichtakesintoaccountpsychologicalandsocialfactors,shouldbetocollectempiricalfactsthatcanbeusedasastandardtotestthemodels.Thedevelopmentofmodelsforfinancialandstockmarketshasbenefitedgreatlyfromtheanalysisofpricefluctuations[8,9].Sincethiswasthecaseforourunderstandingofstockmarkets,theobviousfirststeptowardsbuildingamodelofconsumersales,istoanalysethefluctuationsinthismarket.Thisshouldunearththeempirical–stylised–factsofconsumersales.Suchananalysisseemstobemissinginthisfield.Oncethestylisedfactsareestablishedwecananalysetheircauses,andfinallybuildmodelsbasedontheseinsights.Asafirstattemptinthisdirectionthefluctuationsinactualsalesdataofanumberofconsumergoodsareanalysed.Herewerestricttopresentingtheempiricaldataanditsimplicationsformodelling.2TheconsumergoodsmarketTheavailableobservationsareactualsalesdataofproductsinsupermarkets.TimeseriesofsalescanbeobtainedfromACNielsen[10].Thelongestseriesavailableisoveraperiodof120weeks.Asanexampleofsalesinacommoditymarket,thesalesdataofketchupintheNetherlandswillbediscussedfirst.IntheNetherlandsthereareabout16x106consumers,dividedoversome7x106households.Thesehouseholdsshopatsome5200su