Taxincidencewhenindividualsaretime-inconsistent:thecaseofcigaretteexci - PDF document

Taxincidencewhenindividualsaretime-inconsistent:thecaseofcigaretteexcisetaxesJonathanGruber,BotondKoDepartmentofEconomics,MITandNBER,E52-355,50MemorialDrive,Cambridge,MA02142-1347,USAUCBerkeley,USAReceived13January2002;receivedinrevisedform18June2003;accepted24June2003Oneofthemostcogentcriticismsofexcisetaxesistheirregressivity,withlowerincomegroupsspendingamuchlargershareoftheirincomeongoodssuchascigarettesthandohigherincomegroups.Wearguethattraditionalquantity-basedmeasuresofincidenceareonlyappropriateunderaveryrestrictive‘‘time-consistent’’modelofconsumptionofsingoods.Amodelthatismuchmoreconsistentwithexistingevidenceonsmokingdecisionsisatime-inconsistentformulationwhereexcisetaxesoncigarettesserveaself-controlfunctionthatisvaluedbysmokerswhowouldliketoquitbutcannot.Thisself-controlfunctionbenefitslowerincomegroupsmore,sincetheyhaveasignificantlyhigherpricesensitivityofsmoking.Calibrationsshowthat,asaresult,cigarettetaxesaremuchlessregressivethanpreviouslyassumed,andareevenprogressiveforawidevarietyofparametervalues.2003ElsevierB.V.Allrightsreserved.Keywords:Cigarettes;Excisetax;Self-controlfunction1.IntroductionOneofthemostcogentcriticismsofexcisetaxesistheirregressivity.Lowerincomegroupsconsumethetypesof‘‘sin’’goodstowhichexcisetaxesapplyatanequalorgreaterratethandohigherincomegroups.Asaresult,thereisastronginverserelationshipbetweenincomeandtheincomesharesdevotedtoconsumptionofgoodssuchascigarettes,alcohol,andgasoline.Thisrelationshipisweakened,butnotreversed,whenmorepermanent0047-2727/$-seefrontmatter2003ElsevierB.V.Allrightsreserved.doi:10.1016/j.jpubeco.2003.06.001 *Correspondingauthor.Tel.:+1-617-253-8892;fax:+1-617-253-1330.E-mailaddress:gruberj@mit.edu(J.Gruber).www.elsevier.com/locate/econbaseJournalofPublicEconomics88(2004)1959–1987 measuresofresourcessuchasconsumptionareusedtodetermineincidence(Poterba,Thepresumptionbehindsuchregressivityargumentsisthattheyreflectthedegreetowhicheachincomegroupis‘‘hurt’’bydifferenttaxpolicies.Foraneconomist,theappropriatemeasureforthisanalysisisutility.Forexample,whenwesaythatanincreaseinthetaxongasolinewouldbebornemostheavilybylow-incomeconsumers,we(should)meanthatifthetaxwasinstituted,theirutilitywouldbemostdetrimentallyaffected.Incidenceistraditionallycomputedintermsofquantitiesconsumed,however,becauseinneoclassicaleconomics,thisisanaccuratemeasurefortheutilityeffectoftaxation.Foramaximizingconsumer,theutilityeffectofasmallpriceincreaseisequaltotheproductofthepriceincrease,thequantityconsumed,andthemarginalutilityofwealth.Thus,wecanmeasurethe‘‘harm’’ofaUS$1taxindirectproportiontotheamountofthatgoodconsumedbyagivenincomegroup.Thissameanalysiscanbeappliedtoaddictivegoodsifconsumersare‘‘rationaladdicts’’,tousethetermofBeckerandMurphy(1988).Theirseminalarticlecodifiedwhathadbecomethestandardapproachamongeconomiststothinkingaboutregulationofaddictivebads.Intheirmodel,consumptionofaddictivebadsisgovernedbythesamedecision-makingprocessasisconsumptionofallothergoods.Consumerstradeofftheutilitygainsfromconsumingthegoodagainstthecostsofdoingso.Asrationalforward-lookingagentstheyrecognizethatthosecostsincludethedamagethattheyaredoingtothemselvesthroughconsumption,aswellastheadditionalfuturedamagetowhichtheyaredrivingthemselvesbyconsumingmoreofanaddictivegood.Insuchamodel,incidenceanalysisiseffectivelythesameforaddictiveasfornon-addictivegoods.Inthispaper,weconsideranalternativeformulationofconsumptionofaddictivebads.OurmodelfollowsthatofBeckerandMurphy,withoneexception:weallowagentstobetime-inconsistentintheirsmokingdecisions.Laboratoryevidenceonpreferencesuniform-lyindicatesthatindividualsuselowerdiscountratesinevaluatingfutureintertemporaltrade-offs,relativetothediscountratethattheyuseinevaluatingsimilartrade-offsbetweentodayandthefuture.Forexample,insmokingdecisions,theagentmightwanttoenjoyhercigarettetoday,butwouldprefertoexerciseself-controltomorrow.Sinceshewillhavesimilarpreferencesforimmediaterewardsinthefuture,thereisaconflictbetweentheintertemporalselves.Thiskindoftimeinconsistencyhasbeenmodeledasquasi-hyperbolicdiscountingbyLaibson(1997)O’DonoghueandRabin(1999a),andithasbeenappliedinthecontextofsavingsdecisions(Laibson,1997;Laibsonetal.,1998;O’DonoghueandRabin,1999b),retirementdecisions(DiamondandKoszegi,1998)andgrowth(Barro,1999)Thegoalofthispaperistoexploretheimplicationsofapplyingquasi-hyperbolicdiscountingtoincidenceanalysis.Althoughourtheoreticalmodelisgeneral,wefocusinparticularonthecaseofsmoking.Wedosobecausetheavailableevidence,reviewedbelow,suggeststhatsmokingdecisionsarebettermodeledinthetime-inconsistentframeworkthaninthetime-consistentone.Webegin,inSection1,withsomebackgroundonsmoking,addictionmodeling,andtimeinconsistency,andSection3introducesthemodel.Wethenturntothefocusofthispaper,incidenceanalysis.Standardincidenceanalysis,whichappliestotheBecker–Murphymodel,isinvalidinourmodel.Taxationalsoaffectshowtheagent’sself-controlJ.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 problemplaysout.Inparticular,aprice-induceddecreaseinconsumptionmaybegoodfortheagent,becauseitsoftenstheoverconsumptionduetothedesireforimmediategratification.Ourmodeladjustsforthistheoretically,andundersimplifyingconditions,yieldsasimplemultiplicativeadjustmenttothestandardincidencemeasure.Section4thenpresentsacalibrationexercisethatshowsthequantitativeimportanceofthispoint.Akeyparameterthatdeterminesincidenceinourmodelisthepricesensitivityofsmoking;ahigherpricesensitivityraisesthevalueoftheself-controladjustmenttoincidence.AswedocumentempiricallyusingdataoncigaretteexpendituresfromtheConsumerExpenditureSurvey(CEX),thepricesensitivityofsmokingismuchlargerforthelowestincomegroupsthanfortheirhigherincomecounterparts,withanelasticityof1forthebottomincomequartile.Asaresult,formanyparametervalues,taxesaremuchlessregressiveorevenprogressivewhenthebenefitsofself-controlareincluded.Section6concludeswiththepolicyimplicationsofourfindings.2.Background2.1.AddictionandthecaseforgovernmentinterventionTherehasbeenalong-standinginterestintheeconomicscommunityinmodelingtheconsumptionofaddictivegoods.Untilthemid-1980s,mostofthisliteraturemodeledaddictionashabitformation,wherebypastconsumptionoftheaddictivegoodincreasestasteforcurrentconsumption.Inapathbreakingarticle,BeckerandMurphy(1988)exploredthedynamicbehavioroftheconsumptionofaddictivegoods,andpointedoutthatmanyphenomenapreviouslythoughttohavebeenirrationalareconsistentwithoptimizationaccordingtostablepreferences.IntheBeckerandMurphymodel,individualsrecognizetheaddictivenatureofchoicesthattheymake,butmaystillmakethembecausethegainsfromtheactivityexceedanycoststhroughfutureaddiction.Inthis‘‘rationaladdiction’’framework,individualsrecognizethefullpriceofaddictiveconsumptiongoods:boththecurrentmonetaryprice,andthecostintermsoffutureharmandaddiction.Rationaladdictionhassubsequentlybecomethestandardapproachtomodelingconsumptionofgoodssuchascigarettes.ThekeynormativeimplicationoftheBeckerandMurphy(1988)modelisthattheoptimalregulatoryroleforgovernmentissolelyafunctionoftheinterpersonalexternalitiesinducedbysmoking.Sincesmoking,likeallotherconsumptiondecisions,isgovernedbyrationalchoice,thefactthatsmokersimposeenormouscostsonthemselvesisirrelevant;itisonlythecoststheyimposeonothersthatgivesrisetoamandateforgovernmentaction.Thereisalargeliteraturethatisdevotedtomeasuringthemagnitudeoftheseexternalities.KeycontributorsincludeManningetal.(1989,1991)Viscusi(1995),andtheliteratureisnicelyreviewedinChaloupkaandWarner(1998)Evansetal.(1999a,b).Estimatesvary,butclusteraround40¢perpack,whichiswellbelowtheaveragelevelofstateandfederalexcisetaxation(76¢perpack,accordingtoGruber,2001Anotherconsiderationforpolicy-makersistheincidenceoftobaccotaxation;givenanoverallgovernmentrevenuerequirement,excisetaxationmustbecomparedtoothertaxesincomposingadistributionallyattractiverevenue-raisingpackage.Infact,however,exciseJ.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 taxesoncigarettesareveryregressive,asdiscussedinEvansetal.(1999a,b)andupdatedinGruberandKoszegi(2002).Cigaretteexpendituresasashareofincomeare3.2%inthebottomquartileoftheincomedistribution,butareonly0.4%ofincomeinthetopquartile.Thisregressivitypersistswhenincidenceiscomputedonalifetimebasis,asshowninPoterba(1989);regressivityislessened,butexpendituresinthebottomquartileoftheconsumptiondistributionarestillmuchhigherasapercentageofconsumptionthaninthetopconsumptionquartile.2.2.ThecasefortimeinconsistencyinsmokingTherearefourtypesofevidencefortimeinconsistency,twoofwhichapplytobehavioringeneral,andtwoofwhichapplytosmokinginparticular.Thefirstislaboratoryexperiments,whichdocumentoverwhelminglythatconsumersaretime-inconsistentAinslie,1992;AinslieandHaslam,1992;Thaler,1981,forexample).Inexperimentalsettings,consumersconsistentlyrevealalowerdiscountratewhenmakingdecisionsovertimeintervalsfurtherawaythanforonesclosertothepresent.Thesecondiscalibratingreal-worldbehavioragainstmodelswithandwithouttimeinconsistency,toassesswhichtypeofmodeldoesthebestjobofexplainingobservedpatterns.Forexample,Angeletosetal.(2001)showthatahyperbolicdiscountingmodelfitsobservedconsumptionandsavingspatternsmuchbetterthananexponentialone;inparticular,thepatternofhighilliquidwealthholdings(e.g.housing)combinedwithhighlevelsofborrowingathighinterestrates(oncreditcards)isconsistentwiththeself-controlproblemsinherentintime-inconsistentmodels.AndDellaVignaandMalmendier(2001)arguethatthebehaviorofhealthclubmembers,suchaspayingaflatfee(ratherthanaperusechargewhichalmostalwaysaddsuptolessexpost),isbestexplainedbytime-inconsistentmodels.ThethirdisaneconometrictestinGruberandMullainathan(2001).Drawingonthemodeldevelopedbelow,theyarguethatonemeansofempiricallydistinguishingtime-inconsistentagentsfromtime-consistentagentsistheimpactofcigarettetaxationontheirmeasuredwell-being.Time-consistentsmokerswillbemadeworseoffbycigarettetaxation,bythestandardargumentsthatunderlietheBecker–Murphymodel.Buttime-inconsistentagents,aswemodelhere,canbemadebetteroffbyhighertaxes,astheyprovidetheself-controldevicetheagentsdemand.GruberandMullainathanusedataonself-reportedwell-beingfromtheGeneralSocialSurvey,matchedtoinformationoncigaretteexcisetaxes,toshowthathigherlevelsofexcisetaxesraisereportedwell-beingamongsmokers,butnotamongothers,whichprovidessomeempiricalsupportforthetime-inconsistentmodel.Fourth,thereisevidenceontwokeyfeaturesthatdistinguishtime-consistentagentsfromtime-inconsistentones.Thefirstistheuseofcommitmentdevicesself-control.Wedistinguishaself-controldevicefromanalternativetechnologyforsmokingcessation,quittingaids:whereasquittingaidsdecreasethedisutilityfromnotsmoking,self-controldeviceslowertheutilityfromsmoking.Time-consistentdecision-makersmightuseaquittingaid,butingeneraltheywillnotuseaself-controldevice—withtimeconsistency,loweringtheutilityofanundesiredalternativeisirrelevantfordecision-making.Butforsometypesoftime-inconsistentagents(whatwelabelbelowsophisticatedagents,whoJ.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 recognizetheirowntimeinconsistency),self-controldevicesarevaluedasameansofcombatingone’sowntime-inconsistenttendencies.Intherelativelysmallmedicalliteratureonself-initiatedattemptsatquittingsmoking,thevoluntaryuseofself-controldevicesfiguresprominently.Peopleregularlysetupsociallymanagedincentivestorefrainfromsmokingbybettingwithothers,tellingthemaboutthedecision,andotherwisemakingitembarrassingtosmoke(Prochaskaetal.,.Variouspunishmentandself-controlstrategiesforquittingarealsostudiedincontrolledexperimentsonsmokingcessation(Miller,1978;MurrayandHobbs,1981,andseeBernstein,1970foravarietyof‘‘aversivestimulus’’techniques),andtheyarerecommendedbybothacademicpublications(GrabowskiandHall,1985)andself-help(CDCvariousyears).Inonestudy,forexample,subjectstoreupadollarbillforeverycigarettetheysmokedabovetheirgivendailylimit,andreducedthatlimitgradually.Presumably,theseexperimentsareincorporatingself-controldevicesbecausetheyareseenasthebestoptionforhelpingindividualsquitsmoking,ascouldbethecaseifindividualsweretime-inconsistent.Asecondfeaturethatdistinguishestime-consistentagentsfromtime-inconsistentagentsisaninabilitytoactualizepredictedordesiredfuturelevelsofsmoking.Theformerphenomenonisspecifictoaclassofhyperbolicdiscounterswhomwelabelbelow,inthattheydonotunderstandthattheycannotmakeconsistentplansthroughtime.Infact,unrealizedintentionstoquitatsomefuturedateareacommonfeatureofstatedsmokerpreferences.AccordingtoBurns(1992),eightoftensmokersinAmericaexpressadesiretoquittheirhabit.Unfortunately,thesedesirescanbeinterpretedinanumberofways,andwearenotawareofanyevidenceforadultsontheirspecificpredictionsorintentionsaboutfuturesmokingbehavior.Foryouths,however,thereisclearevidencethattheyunderestimatethefuturelikelihoodofsmoking.Forexample,amonghighschoolseniorswhosmoke,56%saythattheywillnotbesmoking5yearslater,butonly31%ofthemhaveinfactquit5yearshence.Moreover,amongthosewhosmokemorethanonepack/day,thesmokingrate5yearslateramongthosewhostatedthattheywouldbesmoking(72%)isactuallythanthesmokingrateamongthosewhostatedthattheywouldnotbesmoking(74%)(U.S.DepartmentofHealthandHumanServices,1994)Thisisonlyalimitedsetofevidence,andmuchmoreisneededbeforethetime-inconsistentmodelwillbeacceptedastheappropriateformulationofpreferences.Butitisimportanttonotethatthereisevidence,psychologicalorother,thatsupportstime-consistentpreferencesoverthesetime-inconsistentonesinanydomain.Thissuggeststhatalternativeformulationssuchastheonewedevelopinthispaperbetakenseriously,particularlygiventheradicallydifferentimplicationsforgovernmentpolicyweshowbelow. ThereisalargeempiricalliteraturewhichhastestedandgenerallysupportedakeyempiricalcontentionofBeckerandMurphy(1988)model:thatconsumptionofaddictivegoodstodaywilldependnotonlyonpastconsumption,butonfutureconsumptionaswell.But,asGruberandKoszegi(2001)highlight,forward-lookingbehaviorbysmokersalsoarisesintime-inconsistentmodels,sothatthisevidencedoesnotnecessarilysupporttheBeckerandMurphymodelanditsnormativeimplications.Thatis,thisempiricalliteraturetestsonepremiseoftheBeckerandMurphymodel,showingthatsmokersarenotfullymyopic,butnotasecondkeypremise,timeconsistency.J.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 Muchofthisevidenceisconsistentwithatleasttwootherrecentmodelsofindividualdecision-makingaswell(BernheimandRangel,2001;GulandPesendorfer,2001;PesendorferandGul,2000).Bothofthesepapersruleoutaneffectivetaxpolicyessentiallybyassumption,thereforearrivingatdifferentpolicyimplicationsthanthosedevelopedbelow.Therehasbeenlittleattempttodistinguishthesemodels,althoughtheevidenceinGruberandMullainathan(2001)ismoreconsistentwithourformulationthanwiththesealternatives.Itseemslikelythatbehaviorissomecombinationofthethreemodels,andthebestmodeldependsontheaddictivegoodinquestion.Inthispaper,wefocusourattentionontheconsequencesofhyperbolicdiscounting,acknowledgingthatfutureworkshouldtrytoappropriatelydistinguishandcombinethesealternativeapproaches.3.AconsumptionmodelforaddictivegoodsToaddressthemainquestionsofgovernmentinterventioninthemarketforanaddictivegood,wefirstneedtointroduceamodelofindividualchoiceintheseproducts.Weusethebasicmodelthatweintroducedinourearlierpaper(GruberandKoszegi,2001).Thismodelmarrieswhatareinouropinionthetwomostimportantaspectsoftheconsumptionofaddictivegoods.First,theinstantaneousutilityfromconsumingagoodsuchascigarettesdependsinspecificwaysonpastconsumptionofthesamegood.Wewillbefocusingontwokindsofintertemporallinkages:reinforcement—thetendencyofpastconsumptionofcigarettestoincreasethe‘‘craving’’for(formally,themarginalutilityof)acigarettetoday—andhealthcosts.Bothofthesepropertiesofaddictivesubstancescanbecapturedinaninstantaneousutilityfunctionoftheformarethelevelsofconsumptionoftheaddictiveandordinarygoods,respectively.Reinforcementandhealthcostsareincorporatedintotheutilityfunctionthroughthedependenceof,theso-calledstockofpastconsumption,ameasureoftheamountofpastconsumptionofevolvesaccordingtowhere01isthedepreciationrateofthestock.Reinforcementmeanssimplythat&#x-172;&#x.100;)0,whileadversehealtheffectsareformalizedbyassuming).Thesecondimportantaspectofaddictivegoodsconsumptionconcernsthenatureinwhichinstantaneousutilitiesareintegratedintoaglobalutilityfunction.Supposewearein BernheimandRangel(2001)assumethatinthe‘‘visceral’’statewhentheagentoverconsumesthedrug,sheisnotprice-sensitive.GulandPesendorfer(2001)PesendorferandGul(2000),whosemodelisdrivenbydisutilityfromtemptation,assumethattheagentistemptedequallystronglybythedrugforallpossibleprices,aslongasshehasenoughwealthtopayforit.Forexample,BernheimandRangel’s(2001)modelemphasizesthevisceralfactorsinbehavior,andassumesthatsometimesthedrivetotakedrugsoutpacesallrationaloptimization.Thismaybetruefordrugslikecocaineandheroin,butprobablynotforcigarettes.J.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 -periodmodel.Atime-consistent,exponentialdiscounteragentmakesdecisionsattimeaccordingtothediscountedutilityfunctionWewillcontrastthistypeofdiscountingwiththealternativerecentlypopularizedbyLaibson(1997),quasi-hyperbolicdiscounting.Forquasi-hyperbolicdiscounters,dis-countedutilitybecomesareusuallyassumedtobebetweenzeroandone.Thisformulationisintendedtocapturetheidea(forwhichwearguedabove)thatdecision-makersmighthaveself-controlproblemsregardingtheconsumptionofaddictivegoods.Underthisspecificformofinconsistency,thediscountfactorbetweenconsecutivefutureperiods()islargerthanbetweenthecurrentperiodandthenextone().Thus,theagentis‘‘impatient’’whenfacedwithachoicebetweentodayandtomorrow,butshewouldliketo‘‘becomepatient’’inthefuture.Theself-controlproblemarisesfromthefactthatfutureselveswillagainprefertobeimpatientintheshortrun,creatingaconflictbetweenthecurrentselfandfutureones.Therearetwoextremeassumptionsonecanmakeabouthowthisconflictplaysout.Undernaivete,thedecision-makeriscompletelyunawarethatshewillbeimpatientagaininthefuture.Suchanagentmaximizestheutilityfunction4ineachperiod,andchangesherplansoverandoveragainasshechoosestobemoreimpatientthansheexpected.Attheotherextreme,onecanassumesophistication,wheretheagentrealizesthatshewillchangehermindandshebehavesstrategicallyaccordingtothis.Formally,thesuccessiveintertemporalselvesplaythesubgame-perfectequilibriuminanextensive-formgame,inwhichthechoicevariableofeachselfisconsumptioninthatperiod.Tocompletethesetupofthemodel,letbethepriceoftheaddictivegoodinperiodanddenoteincomeinperiod.Tomakeiteasiertosolveourmodelforsophisticatedquasi-hyperbolicdiscounters,weassumethattherearenosavings:theincomeconsumedineachperiod.Basedonsimpleexamplesandintuition,thereisreasontobelievethatrelaxingourliquidityconstraintassumptionwouldactuallydecreaseincidenceoftaxes.Butwebelieveborrowingtofinanceconsumptionisnotanimportantfactorincigarettesmoking.Inaddition,theagent’suseofsavingsandaddictivegoodsconsumptionasaself-controldeviceconsiderablycomplicatestheanalysis,andwehave Iftheagentwasallowedtoborrowfromfutureincome,shewouldtakeadvantageoftheopportunity,andwoulddosopartlytofinanceconsumptionoftheaddictivegood.Thus,thetaxwouldhavetooffsetthetendencytoborrowinadditiontothetendencytooverconsumethedrug,andthisextraeffectwouldbebeneficialfortheagent.Inparticular,theagentmaywanttodepriveherselfofsavings,sothatthefutureselfcannotaffordtheaddictivegood.Conversely,if,forexample,beingaddicteddecreasestheagent’stendencytospendtoomuch,shemaywantto‘‘overaddict’’herselftoimprovehersavingsbehavior.J.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 notsolvedforthedynamicsofourmodelwithsavings.Forsomeveryexpensiveaddictivegoods,however,itmaybeimportanttoextendourmodelinthisway.GruberandKoszegi(2000),wesolvefortheEulerequationsforthethreetypesofconsumers,time-consistentagents,naivehyperbolicdiscounters,andsophisticatedhyper-bolicdiscounters.Forfuturereference,wequotethesophisticates’first-orderconditionhere,whichisvalidaslongasutilityfunctionsandequilibriumstrategiesaredifferen- 
Asinourearlierpaper(GruberandKoszegi,2001)andsimilarlytoBeckerandMurphy,werestrictattentiontoquadraticutilityfunctions;thisisdoneonlytosimplifytheanalysis.Thus,thefunctionstaketheform aaa2a2taasatSt ,andarepositiveand,andarenegative.Thekeyparameteris,whichmeasurestheeffectofpastconsumptiononthemarginalutilityofcurrentconsumption.Toensurethatfirst-orderconditionsaresufficienttofindtheequilibriuminourmodel,weassumethat)isstrictlyconcave(itsHessianisnegativedefinite).Inthiscase,itisveryeasytoprovebybackwardinductionthatislinearin,whereareconstants.GruberandKoszegi(2001)provethat(1/2),convergestoaconstant,sothatmarginalresponsivenesstopastconsumptionisapproximatelystationaryfarfromtheendofthehorizon.Althoughthisstationarityisnotcrucialforanyofourqualitativepoints,wewilltakeadvantageofitinourcalibrations.Noticethatinordertocarryoutouranalysis,weassumethattherearenoincomeeffects.Thismakestheanalysisconsiderablysimplerwithoutaffectingthebasicconclusionswereach.Withoutincomeeffects,thepriceoftheaddictivegoodhasnoinfluenceon(GruberandKoszegi,2001)Asinanymodelwheredifferentsociallyrelevantactorshavedifferenttastes,adiscussionofgovernmentpolicymuststartwiththesetupofthesocialwelfarefunction.Inthecontextofhyperbolicdiscounting,theseactorsarenotseparateindividuals,butdifferentintertemporalincarnationsofthesameindividual.Wefollowtheliteratureandtaketheagent’slong-runpreferencesasthoserelevantforsocialwelfaremaximization. Foramorethoroughintroductionofthismodel,seeGruberandKoszegi(2001);foracontrastofsophisticationandnaivete,seeO’DonoghueandRabin(1999a)J.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 Sincethediscountfactorappliestoallfutureperiods,thisisalsotheappropriatewelfaremeasureifasophisticatedrepresentativeagentweretovoteinataxchangetodaythatisinstitutedstartinginthenextperiod.Moregenerally,asemphasizedbyO’DonoghueandRabin(2003),theextradiscountfactorthateachselfappliestothefutureisonlyrelevantforthatsingleself,soignoringitwhenevaluatingtheagent’soverallwelfaremaybejustified.Evenmoreimportantly,sincewewillbefocusingonaperiodlengthof1month,anyreasonableinourmodelisverycloseto1.Thus,sincetheagentcaresalotaboutfuturediscountedutility,ourresultswouldbealmostidenticalifweconsideredhercurrentpreferencesatthetimeofthetaxchangeastheappropriatewelfaremeasure.4.IncidenceoftaxesonaddictivegoodsInthissection,wearguethatthetraditionaleconomicmethodsforincidenceanalysisareincompleteforaddictivegoodsinthepresenceoftimeinconsistency.Wealsoproposeanalternativeincidencemeasure,basedonwhatwebelieveisthegoalofincidenceanalysis.Broadlyspeaking,thegoalofincidenceanalysisistodeterminewhois‘‘hurt’’bydifferenttaxpolicies.Foraneconomist,theappropriatemeasureforthisanalysisisutility.Sowhyisincidencetraditionallydoneintermsofpricesandquantitiesconsumed?Thereasonistheenvelopetheorem.Consideraconsumerfacingamaximizationproblemoftheform......WeknowthatthederivativeofthemaximuminthisproblemwithrespecttoistheLagrangemultiplierontheconstraint.Thatis,foramaximizingconsumer, Toseethis,notesimplythatapricechangethatstartstomorrowhasnonegativeincomeeffecttoday,andtheresponseoftoday’sconsumptiontofuturetaxeshasnoeffectondiscountedutilityduetotheenvelopetheorem.Thus,theeffectondiscountedutilitycomesexclusivelyfromfutureperiods,whicharediscountedexponentially.Interestingly,ifthetaxwasinstitutedlaterthaninthenextperiod,theagentwouldactuallyvoteinataxthanwhatisimpliedbyherlong-termdiscountfactor.Thisisduetotheintertemporalsubstitutabilityoftaxes,discussedinGruberandKoszegi(2000)InGruberandKoszegi(2002),wediscusstheimplicationsofthismodelfortheoptimalleveloftaxesoncigarettes.Weprovethat,solongas1,itwillbeoptimaltotaxcigarettes,evenintheabsenceofexternalities.AndweshowthroughcalibrationssimilartothosebelowthattheoptimaltaxoncigarettesisUS$1–2ifconsumersareonlyslightlytime-inconsistent,orUS$5–10ifconsumersareconsiderablytime-inconsistent.Thoseresults,aswellastheincidenceresultsinthispaper,presumesthattherearenotfullyeffectiveprivateself-controldevicesavailable,butthisseemsareasonableassumption.Market-providedself-controlmechanismsareprobablyundercutbythemarketmechanismitself:althoughfirmshaveafinancialincentivetoprovideself-controltoagents,otherfirmshaveafinancialincentivetobreakitdown.Forexample,ifafirmdevelopedaself-controlshotthatcausespainwhentheconsumersmokes,anotherfirmwouldhaveanincentivetodevelopadrugthatrelievestheseeffectsforagentswhotemporarilywanttogetridoftheircommitment.Otherproblemsariseincontractingsetups.Ifthereareexpostgainstobemade,thefutureselfmightwanttorenegotiatetoday’scontract.Buteveniftherearenone,thereisanexpostincentivetocheatonthecontract:smokingishardtoverifyincourt.Thisleavesuswithprivatelyprovidedself-controlmechanismslikebettingwithothersorbecominginvolvedinsituationswhereitisverydifficulttosmoke,butthesemechanismsarelikelytorunintoimportantenforcementproblems.J.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 theutilityeffectofasmallpriceincreaseisequaltotheproductofthepriceincrease,thequantityconsumed,andthemarginalutilityofwealth.Thesameistrueforatime-consistentconsumerofaddictivegoods.Forsimplicity,assumethatthepriceisconstant:.Thenthederivativeofself’sdiscountedutilitywithrespectto...Onceagain,theutilityimpactofapriceincreasedependsonthemarginalutilityofwealthandtheamountsconsumed.Asforanygoodthatisconsumedinmultipleperiods,theutilitycostofacigarettetaxdependsnotonlyoncurrent,butonfutureconsumptionasForquasi-hyperbolicdiscounters,theenvelopetheoremdoesnotholdintheaboveform,sostandardincidenceanalysisfailstocapturetheutilitycostoftaxation.Wefocushereonthemorecomplicatedcaseofsophisticates,andmentiononlybrieflyhowourresultsdifferfornaifs.Toarriveatthecorrectmeasureofincidence,wetakethederivativeoftheagent’slong-runutility(thewelfaremeasureforwhichwehavearguedabove)withrespect Here,eachisafunctionof,whichwewillhavetotakeintoaccountinevaluatingthetotalderivativewithrespectto.LetThusweareinterestedin ddpU1S1
p dpva1
S1cI1
pa1U2S2
pca1 Ba1Bpvaa1
S1acd1
d BBS2U2S2
pd Usingthatself1solves)=0,thiscanberewrittenas 1
bb Ba1Bpvaa1
S1ac J.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 Iteratingthisprocedure,wegetthatthederivativeoftheagent’sdiscountedutilitywithrespecttostandardterm BajBp Thefirstsuminthisexpressionissimilartowhatwehadinthetime-consistentcase:sincetheagentnowhastobuyherconsumptionatahigherprice,herutilityisaffectedbytheextracostofthis.Theadditionalterm,whichwecallthe‘‘self-controladjustment’’toincidence,isnew.Itcapturesthevaluetime-inconsistentagentsattachedtotheself-controltoolprovidedbyahigherprice.Eachselfconsumes‘‘toomuch’’fromthelong-runself’spointofview,sotheprice-induceddecreaseinconsumptionincreasesdiscountedutility.Thistermisabsentinthetime-consistentcasebecauseeachselfconsumes‘‘justtherightamount’’fromthelong-runself’spointofview.Byrepeatedapplicationofthesophisticates’first-ordercondition5,wecanrewriteakeypartoftheself-controladjustment: 
...Wewilluseexpressions(10)and(11)tostudytheself-controladjustedincidenceofcigarettetaxation.Buteventhoughouranalysisismotivatedbycigaretteconsumption,themethodsareapplicabletoothergoodsaswell.Wewillneedthefollowingpreliminarylemmaforouranalysis:Lemma1.isdecreasinginforeacht.isdecreasinginforeacht.isincreasingint.AppendixA.Wearenowreadytodiscussourmainresults.First,foranygoodthatisharmfulandaddictive,thisself-controladjustmentwillbepositive,sothattheincidenceislowerthaninthetime-consistentcase.Next,noticethattheright-handsideofEq.(11)dependsontheconstantsintheagent’sconsumptionfunction.Thus,theself-controladjustmentislargerif�0—whenthegoodisaddictiveasopposedtomerelyharmful.Surprisingly,forapersonsufferingfromself-controlproblems,thetaxationofaddictiveharmfulgoodsimposeslessofaburdenthanthetaxationofgoodsthatareexactlyasharmfulbutnotaddictive.Thereasonisthattheconsumptionofaharmfuladdictivegoodimposestwokindsoffuturecosts;itself-controlJ.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 causesdirectharmandexacerbatesthefutureself-controlproblemaswellbyincreasingtheshort-rundesiretoconsume.Bothoftheseeffectsarebadfromalong-runperspective,andaquasi-hyperbolicdiscounterdoesnottakeeitherofthemsufficientlyintoaccount.Thesamelogicholdsfornaiveconsumers,sotheconclusionthatthetrueburdenoftaxationislowerforaddictiveharmfulgoodsistrueforthemaswell.Onedifficultywithgivingcrispconclusionsaboutincidenceinthismodelisthatthemarginaldamage),throughitsdependenceonconsumptionandthestock,dependsontheotherparametersofthemodel.Our(theoreticalandempirical)knowledgeofthesechangesislimited,soweabstractfromthemintherestofourdiscussionandinthecalibration.Ultimately,ourinterestisinexploringtheimplicationsoftheincidenceadjustmentfortheregressivityofcigarettetaxation.Inthecontextofourmodel,therearefourreasonswhythemagnitudeofthisadjustmentmightdifferforthepoorandtherich.First,theself-controladjustmentinexpression(10)isproportionaltothepriceresponsivenessoftheagent.Thus,aslongas1,morepriceelasticconsumersbearlessofthe‘‘true’’burdenoftaxation.Theintuitionissimple:sincetheagentconsumestoomuchineachperiod,thepricehikeincreasesutilitybyrestrainingtheoverconsumption.Thisself-controltoolismoreeffectiveiftheagentismoreresponsivetopriceincentives.Aswedocumentbelow,thepooraremuchmorepricesensitivethantherichintheirsmokingdecisions.Thus,thisfactorwilltendtoreducetheregressivityofexcisetaxation,allelseequal.Theremainingthreefactorsallrelatetoacriticalquestionthathasbeenunansweredbytheliteratureonsmoking:whydothepoorsmokemore?Onereasonmaybethatlower-incomeindividualssmokemoremostlybecausetheyhavealower.Inthiscase,theself-controlincidenceadjustmentisunambiguouslylargerforthem,andthereforetheregressivityofcigarettetaxesisoverestimatedinstandardanalysis.Thefirst,directeffectofadecreaseinisanincreaseinthemultiplier1inthesecondterminexpression(10).Theintuitionisobvious:ifislower,eachselfisignoringmoreofthefutureharmshecausesbysmokingmore,andthereforeadecreaseinconsumptionismorevaluable.Inaddition,Eq.(11)indicatesthatthereisanindirecteffectaswell,comingfromthefactthatthegoodisaddictive.Aswehavenotedabove,theharmfromcurrentsmokingisnotrestrictedtothedirecthealthcostsofsmoking,butincludesthecostofinducedfuturesmokingaswell.Thiscostishigherifislower,sinceinthatcasethefutureselvesaremakingaworsedecision.Exacerbatingtheproblemisthatisgreaterforagentswitha;therefore,theeffectofcurrentconsumptiononfutureconsumption,whichthecurrentselfisnotsufficientlytakingintoaccount,ishigher.Ontheotherhand,anotherreasonwhythepoormightsmokemoreisthattheyhavethesameleveloftimeinconsistencyastherich(thesamevalueof),butalowerlong-rundiscountfactor.Infact,ifthepoorhavealower,then,allelseequal,theburdenoftaxesmoreheavilyonthem.Loweringhasthedirecteffectofincreasingdiscountingin Thisfeatureisallthemorestrikinggiventheresultinourearlierpaperthataddictivenessdecreasestheoptimaltax,sincefutureselvesexertmoreself-controltoreduceanaddictivebehavior.Thedifferenceisdrivenbytherevenuesideofthegovernment’sproblem.Inducingtheagenttoreducesmokingdecreasesherfutureconsumptionaswell,erodingthegovernment’staxbase.Incidenceanalysisdoesnottakethisintoaccount,whereasoptimaltaxationdoes,creatingtheoppositeimplicationsofaddictiveness.J.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 Eq.(10),anddecreasingtheright-handsideofEq.(11),thusdecreasingtheself-controladjustment.Withthefuture‘‘lessimportant,’’eachintertemporalselfignorespartofalessimportantthing,andsothetaxislessdesirableonself-controlgrounds.Thoughthestandardincidencemeasureisalsolowerwhenislower,theformereffectisproportionallylarger.Loweringalsolowersthedollarvalueofdamagepeopleattachtoconsumingcigarettes,sincethisdamagetendstocomeattheendoflife,furtherraisingincidence.Theconclusionmightbesurprising:ifthepoor’shighersmokingratecomesfromashort-termdiscountfactor(),thenacigarettetaxisnotasregressiveascurrentlybelieved,whileifitcomesfromalowerdiscountfactor(),thensuchataxisevenmoreregressivethanrecentestimates,allelseequal.Iflower-incomeindividualssmokemorebecausetheyattachalowervaluetolife,onceagainwehavetoconcludethattheirself-controladjustmentissmaller.Wemodeladecreaseinthevalueoflifeasanincreasein,whichreducesthemarginalharmfromsmoking.Achangeindoesnotaffectforany,sofromEq.(11),adecreaseinthemarginalharmofacigarettedecreasesthevalueoftheself-controlgainapriceincreaseachieves.Theintuitionissimplythatapersonwhoplacesalowervalueonlifecareslessaboutself-controlaimedatprotectingthatlife,decreasingtheself-controladjustment.Thus,insummary,theimpactoftheself-controladjustmenttostandardincidencemeasuresisunclear.Theadjustmentitselfwilltendtolowerincidence,anditwilldosomoreforgroupswheresmokingismorepricesensitive(whichistrueforthepoor).Butontheotherhand,thisadjustmentisreducedasislowerandthevalueoflifeislower,bothofwhicharealsopotentiallytrueforthepoor.Inthenextsection,weturntoacalibrationexercisewhichcanhelpassesstherelativeimportanceoftheseoffsettinginfluences.5.Whatdifferencedoesitmake?5.1.Calibratingtaxincidence—theoryInordertocarryouttheanalysis,wemakethreesimplifyingassumptionsinadditiontoworkingwithaquadraticutilityfunction.First,weassumethatdecision-makersstartofffromasteady-stateconsumptionlevel:doesnotdependon.Thisisnottoorestrictive,sincesmokerstendtoreachtheirsteady-stateconsumptionlevelbytheirearlytwenties.Second,weignoreend-of-lifeeffectsoperatingthroughandthepriceelasticityof Aneffectactingintheoppositedirectionarisesfromtheaddictivenessofthegood:loweringTherefore,currentconsumptionexertsalargerinfluenceonfutureconsumption,andthefactthattheintertemporalselvesarenottakingthissufficientlyintoaccountmakesthetaxeasiertobear.However,theformer(direct)effectofadecreaseinalwaysoutweighsthelatter(indirect)effect.Theproofisomitted,butisavailableuponrequest.Aswenotebelow,arelatedreasonthatthepoormaysmokemoreisbecausesmokingislessdamagingwhenlifeisshorter,sincefeweryearsoflifearelost.Thisoperatesinaparallelfashiontothelowervalueoflifepoint.Notethatthisformulationdiffersfromthestandardinterpersonalexternalitysetup,fromwhichitmayseemthatthecorrectiveeffectofataxshoulddependonlyonthepriceresponsivenessand1timesthehealthcostofsmoking.Thereasonisthatthereareintertemporallinkagesinourframework,sochangingthebehaviorofoneselfalsochangesthebehaviorofotherselves.J.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 consumption.Third,weassumethatthedisutilityassociatedwithsmoking,),isa,expression(11)becomes 

Usingtheaboveandsetting,expression(10)fortheutility-basedmeasureofincidencebecomes(forself1) 1
dT1
da
b
Tj1dj
1 

whichcanberewrittenas 1
dT1
da
b1
d
vs BaBp 11
dd1
bk*s 1
dT1
dddT
Finally,thisexpressionhasamoreconvenientformas 1
dT1
datimes:1
b1
d
vSacp
BaBp pa   horizon-independenteffect
vSacp
BaBp pa dT1
d1
dT
 length-of-horizoneffectThequantityisthetraditionalmeasureoftaxincidenceinadynamicframework—itissimplytheproductofconsumption,themarginalutilityofincome,andahorizontermdependingonhowlongthetaxaffectstheagent.Therefore,expression(15)givesamultiplicativefactorthatcanbeusedtoadjustourtraditionalmeasuresoftheregressivityofcigarettetaxeswhenwewanttouseautility-basedincidencemeasureforquasi-hyperbolicdiscounters.Thedifferenceinthemagnitudesoftheseadjustmentfactorsacrossincomegroupstellsusthedegreetowhichtheusualincidencemeasuresare‘‘off’’whentheutilitygainsfromextraself-controlaretakenintoaccount. Wheninterpretedliterally,thisassumptionisinappropriateinourframework,sinceitcontradictsourassumptionofreinforcement.Butmorenaturallyinterpreted,theassumptionmeansthatweignoretheovertime,andreplaceitwithanaverage.Since,forthepurposesofwelfareanalysis,thehealthconsequencesofsmokingaremuchmoreimportantthantheeffectofsmokingonthefuturepleasureofsmoking,thisdoesnotseemtobetoomuchofastretch.J.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 Thelatterterminexpression(15)canbedecomposedintotwoterms,onethatdoesnotdependonandonethatdoes.Thisallowsustoimmediatelyassesstherelativeimportanceoflength-of-horizoneffectsofincidencerelativetoothereffectsinthemodel.Notethatthelength-of-horizoneffectislessthantimesthehorizon-independentterm.Supposethattheaveragesmoker’shorizonis25years—thissmokeris40yearsold,andwilllivetoage65.Evenforrelativelylarge’s,theabovetermwillbesmall.Forexample,for=0.9,andassuming=0.6,=0.6,and=0.7—thecombinationinourrangeofparametervaluesthatmaximizestherelativeimportanceofthelength-of-horizoneffect—itisequaltoonlyabout5.6%ofthehorizon-independenteffect.Inaddition,thedifferencesinthistermbetweenincomegroupswillbeevensmaller,sincethedifferenceinlifeexpectancybetweenincomegroupsislow.Therefore,weworkwithaformulathatignoresthelength-of-horizonterm.Inourcalibration,wewillusethemoneyequivalentofthecombineddiscounteddamageofacigaretteinallfutureperiods, 1
dd1
dd .Thisgivesthefinalexpressionfortheadjustmentfactor:
HSp
BaBp pa Withthesimplifyingassumptionswemadeforsophisticates,wecanputtheself-controladjustmentinaconvenientformfornaifsaswell,andwedosoinAppendixB.Asweshowthere,wegetamuchlargeradjustmenttostandardincidencefornaifs.Theintuitionforthisresultderivesfromnaifs’misperceptionabouttheirfuturebehaviorcombinedwiththeaddictivenessofthegood.Sincethegoodisaddictiveandfutureselvesdonotconsumeoptimally,anincreaseincurrentconsumptionnotonlydecreasesfutureutility,butsparksfurtherincreasesinconsumptionthatarealsoharmful.Sincenaifsbelievethattheirfutureselveswillbehaveoptimally,theydonottakethisintoaccount,whereassophisticatesdo(albeitonlypartially).Thiseffectisaggravatedbythefactthatnaifsaremoreresponsivetoincreasesinthestockofpastsmokinganyway(Theorem1inAppendixA).5.2.Calibratingtaxincidence—parametersOnedifficultywithestimatingtheoptimaltaxisparameterizing.Clearly,thereisalotofdisutilityassociatedwithsmokingthatishardtoquantify,suchasthatfromconstantcoughingandincreasedvulnerabilitytovariousillnesses.Wewillignoreallthese,andassumethattheonlydisutilityfromsmokingisintheincreasedchanceofearlydeath.Viscusi(1993)reviewstheliteratureonlifevaluationandsuggestsaconsensusrangeof3–7million1990dollarsforthevalueofaworker’slife;choosingthemidpointvalueand Sincewehavemade)constant,ourcalculationsintheappendixdoignoretheeffectofnaifs’optimismaboutfutureconsumptiononcurrentconsumption.Bytheintertemporalcomplementarityofcon-sumption,naifs’optimismaboutthefuturerestrainscurrentoverconsumption.Thismakesataxlessbeneficial.Althoughwehavenowayofestimatingthis‘‘optimismeffect,’’wecanprovethatincidenceisunambiguouslylowerfornaifsthanforsophisticatesevenifitisaccountedfor.J.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 expressingitincurrentdollarsgivesafigureofUS$6.8million.Presumably,thisisapresentdiscountedvalueforallremainingyears.Weassumethattheaverageworkeris40yearsold,andwouldlivetoage79ifanonsmoker.ViscusiandAldy(2003)showthatstudiesofthevalueoflifebyageimplydiscountratesinthewiderangeof1–17%;theseestimateddiscountrates,ofcourse,reflectbothshortandlong-termdiscounting.Wethereforeconsiderlong-termannualdiscountratesof3%and10%forourestimates.Weusethefactthatsmokersdieonaverageroughly6yearsearlier(Cutleretal.,2001)andcomputeforeachage15–73thePDVofthecostoflosing6yearsattheendoflife.Wethentakeaweightedaverageofthesecostsateachage,wheretheweightsaretheshareofcigarettessmokedateachagefromtheMay1999CurrentPopulationSurveyTobaccoUseSupplement,anationallyrepresentativesurveyofsmokers.Finally,wedividethisweightedaveragebytheaveragenumberofcigarettessmokedoverone’slifetime;thatis,weassumethataverageandmarginaldamageisequal.Atthesefigures,thecostintermsoflifeyearslostperpackofcigarettesisUS$35.64whentheannualdiscountrateis3%.Ofcourse,thisfigurewillvarywiththeagent’sweareconsidering;sincecostsareattheendoflife,theywillfallasfalls.Butitisanenormousfigureforanyreasonable,andisontheorderof100timesaslargeasestimatesoftheinterpersonalexternalitiesfromsmoking.Thereareseveraloffsettingbiasestousingthisfigureasanestimateofthedamageperpack.Thisestimateistoohightotheextentthatsmokersvaluetheirliveslessthannonsmokers.TheevidenceinViscusiandHersch(2001)suggeststhatthisistrue,inthatthecompensatingdifferentialsthatsmokersrequireforriskyjobsareabouthalfthoseofthedifferentialsrequiredbynonsmokers.Ontheotherhand,thisestimateistoolowtotheextentthatwehaveignoredallnon-mortalityrelateddamageduetosmoking.Smokingnotonlyshortenslivesbutlowersqualityofyearsspentaliveaswellthroughreducedhealth.Moreover,ourquasi-hyperbolicframeworkimpliesthatthehedonicvaluationestimatesofalifethatweareusingaretoolow.Underhedonicanalysis,lifevaluationsarebackedoutofrevealedpreferenceinthemarket.Withquasi-hyperbolicdiscounting,thisapproachistheoreticallyunfounded:agentsarenotmaximizingtheirdiscountedutility,somarketbehaviorwillingeneralnotreflecttruelong-runvaluations(whichiswhatweneedforourcalibrations).Unlessthejob-relatedriskfacedbytheworkerisconcentratedinthecurrentperiodandtheworkeriscompletelyliquidityconstrained,quasi-hyperbolicdiscounterswillacceptacompensatingdifferentialthatis‘‘toosmall’’relativetothetruelong-runvalueoftheirlife,becausetheyareexcessivelytemptedbytheshort-runrewardsfromacceptingtherisks.Anothercrucialparameterforourcalibrationsistheshort-termdiscountfactornotonlyhardtoestimateduetothelackofevidenceonimpatienceinsmoking,butalsobecauseitdependsontheperiodlengthchosenforouranalysis.Wethinkofoneperiodin Age79andtheuseof6yearsofreducedlifereflectsanaveragingofeffectsformenandwomen.Theaveragecigarettessmokedoverthelifetime(19,418)isderivedbysubtractingtheaveragestartingageofcurrentsmokers(age17)fromage73,multiplyingby365,andthenmultiplyingbyaveragecigarettesperdaysmokedamongdailysmokers(19).Notealsothatitisunclearwhetherthereferencepointformortalityreductionshouldbethefirstpacksmoked(sothatmortalityreductionscomefromtheperspectiveofage79),thelastpacksmoked(sothattheycomefromtheperspectiveofage73),orinbetween.Weusethesumofdamageoverages73–79asanaverage.J.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 ourmodelas1month.Inlaboratoryexperiments,monthly’stendtoliebetween0.6and0.9,sowecalibrateincidenceforbothoftheseextremes(althoughmostestimatesarecloserto0.6).Acentralvariableforcomputingincidenceisthepriceelasticityofdemandforcigarettes,andinparticularhowitvariesacrossincomegroups.Standardmicro-dataestimatesfortheelasticityofconsumptioncenteraround0.45,andthepreviousliteraturehasindicatedthattheelasticityismuchhigherforlowerthanforhigherincomesmokers.Forexample,Evansetal.(1999a,b)estimatetheelasticityforthosewithmissingincome(whichtheypresumetobeonaverageaverylowincomegroup)is0.73,forthosebelowmedianincomeis0.53,whileitis0.13forthoseabove.WeprovideupdatedestimatesofelasticitiesandtheirdistributionbyusingtheConsumerExpenditureSurvey(CEX),anationallyrepresentativesurveywhichprovidesthehighestqualityandmostcomprehensivemicro-dataonconsumptionintheUS.WematchtotheCEXdatafrom1980through1998informationoncigarettepricesandtaxesineachstateineachsurveymonth.Wethenestimatemodelsofcigaretteexpendituresasafunctionofprice,instrumentedbyexcisetax;asdiscussedbyGruberandKo,suchaninstrumentalvariablesstrategyisrequiredbecausetheremaybestate-specificpricingthatisendogenoustocigarettedemand.Wecontrolinourmodelforasetofdemographiccharacteristics(age,education,sex,andraceofthehouseholdhead;dummiesfornumberofpersonsinthehousehold),andafullsetofstatedummies,yeardummies,andcalendarmonthdummies.TheresultsofthisexerciseareshowninTable1.ThecoefficientestimatesshowtheimpactofaUS$1priceincreaseonexpenditures;thestandarderrorsareinparentheses.Beloweachestimateistheelasticityofthequantityofcigarettesconsumedwithrespecttopriceimpliedbythisconsumptionresponse,atthemeanpriceandquantity.Acrossthefullsample,eachdollarpriceincreaseleadstoanincreaseincigaretteexpendituresofonly16.8¢,foranimpliedelasticityof0.66.Thisislargerthanthetraditionalestimatedelasticityof0.45forcigaretteexpenditures,butitisveryclosetotheestimatesusingmorerecentdatainGruberandKoszegi(2001)Weestimatetheimpactofpricesonconsumptionacrossincomequartiles,consumptionquartiles,andeducationcategories(highschooldropout,highschoolgraduatewithnocollege,somecollegebutnobachelor’sdegree,andbachelor’sdegreeorgreater);thelattertwoareproxiesforpermanentincome.Acrossgroups,weseeaclearpatternofhigherpricesensitivityforlowerincome,consumption,oreducationgroups.Ineverycase,forthebottomgroupexpendituresdeclineaspricerises,implyinganelasticityoflessthan1.Forincomecategories,theelasticitiesdeclinemonotonicallyasincomerises,withatopelasticitythatisroughlyone-thirdthatofthebottomgroup.Forconsumptionandeducationcategories,thedeclineismonotonictothethirdcategory,butelasticitiesthenincreaseagainforthetopgroup. DataontaxesandpricesbystatearefromTobaccoInstitute(1999).Taxesaremeasuredmonthlyusinginformationonstateexcisetaxhistories;pricesaremeasuredannuallyeachNovember,sowetakeaweightedaverageofthepastandfutureNovemberpricesinassigningapricetoeachmonth.Theelasticityimpliedbytheestimatedresponseoftotalconsumptionis(1,whereistheestimatedcoefficientandisthesamplemeanofexpenditure.J.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 Whilewediscussedaboveourcomputationofthehealthdamagefromapackofcigarettes,wenowneedtoassesshowthisvariesbygroup.Thisinturnhastwocomponents:differencesbygroupinthevalueofalife;anddifferencesbygroupinthemarginaldamageofsmoking.TheevidenceontheformerisreviewedinViscusiandAldy,andtheysuggestaconsensusincomeelasticityofthevalueoflifeof0.5,whichweusehere.Thereis,todate,littleevidenceonthelatterquestionofhowthemarginaldamageofsmokingdiffersbyincomegroup.Thedifferentialimpactofsmokingbygroupisunclearexante.Ontheonehand,sincelowerincomegroupsliveforfeweryears,therearefeweryearsoflifelostfromashiftupinthehazardcurveofdeath.Ontheotherhand,totheextentthatthedamagefromsmokinginteractswithotherdisease,therecouldbemoredamagetolowerincomegroupswhoareinworsehealthforotherreasonsaswell.Giventhelackofevidence,wesubsumethispointinourvariationinthevalueoflife.Thereisalsolittleevidenceonthedifferenceintimepreferenceparametersacrossincomegroups.Wethereforeinitiallyassumethatpreferenceparametersarethesame,althoughwediscussbelowtheimplicationsofvaryingtheseparametersacrossgroups.GiventheevidenceinGruberandKoszegi(2001),wewillpresentallcalibrationsfor=0.6and=0.7.Infact,wefindthatthebasicpatternofourresultsisnotsensitivetothevalueschosenfortheseparameters.5.3.Calibratingtaxincidence—basicresultsPuttingtogethertheresultsofthelasttwosubsections,Table2showsourresultsforincidence.Thereisonepanelforeachincomedefinition(currentincome,expenditure,andeducation).Ineachcase,weusetheestimatedelasticitiesfromTable2bygroup.ThefiguresinthetablerepresenttheincidenceofaUS$1taxperpackofcigarettes,asashareofincomeinpanels1and3,andasashareofconsumptioninpanel2.Exanteincidenceisshowninthefirstcolumnofeachtable,ascomputedfromthe1997–1998CEXinGruber Table1PricesensitivityofcigaretteexpenditureQuartileorgroupIncomequartilesConsumptionquartilesEducationgroupsOverall0.168(0.049)(0.049)
0.661]Bottomquartile/HSdropouts0.032(0.078)(0.078)
1.086]
0.016(0.059)(0.059)
1.050]
0.044(0.115)15)
1.080]Secondquartile/HSgraduates0.144(0.087)(0.087)
0.699]0.113(0.083)(0.083)
0.770]0.045(0.096)(0.096)
0.927]Thirdquartile/somecollege0.273(0.113)13)
0.534]0.414(0.104)(0.104)
0.311]0.417(0.100)(0.100)
0.107]Topquartile/collegegraduates0.341(0.116)16)
0.387]0.215(0.137)(0.137)
0.642]0.175(0.082)(0.082)
0.400]Tableshowsestimatesofimpactofcigarettepricesoncigaretteconsumptionlevels.Estimatedimpactonconsumptionisfirstnumberineachcell;standarderrorofthatestimateinparenthesesbelow.Figureinsquarebracketsisimpliedelasticityofquantityofcigarettesconsumed.Firstrowshowsfullsampleestimates;remainingrowsshowestimatesbyincome/expenditure/educationcategories.EstimatesfromCEXdataover1980–1998J.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 andKoszegi(2002).Theremainderoftheresultsshowincidenceafterapplyingourtimeinconsistencyadjustment.Therearetwoimportantpointsforinterpretingtheseresults.First,inmanycasesbelow,wefindthatouradjustmentactuallyreversesincidence,whichwewillshowasanegativeburden;thatis,thecorrectivebenefitsaresolargethatthetaxisabenefit,sothatalargernegativenumberimpliesalargerbenefit.Second,insomecasesbelowwefindthattheratiooftheburdenonthepoorrelativetotherichgrows,buttheabsolutegapbetweenthetwonarrows.Itisnotclearwhetherthisisappropriatelyinterpretedasariseorareductionintheregressivityofthetax.Fortheincomeconceptshowninpanel1ofTable2,taxesarethemostregressiveexante,withtheburdenonthelowestincomegroupalmost10timesthatonthehighest.Inthefirstsetofcolumns,weshowtheadjustedincidencemeasurefor=0.9andanannualdiscountrateof3%.Thishasonlyamodesteffectontheratiomeasureofregressivity,butnarrowstheabsolutegapconsiderably,withthegapbetweenthehighestandlowestincomegroupsfallingbytwo-thirdsinabsoluteterms.Varyinghasanenormouseffect.For=0.6,taxesarenowbeneficialforeverygroup,andmuchmoresoforthelowestincomegroup.Thatis,taxesarenowhighlyprogressive,withverylargebenefitsforthelowestincomegroup,andmuchsmallerbenefitsforthehighestincomegroup.Thislargelyreflectsthemuchhigherpricesensitivityofthisbottom Table2Incidencecalibrations(1)(4)(7)(10)(13)Exante=0.9,=0.97=0.6,=0.97=0.9,=0.9=0.6,=0.9CurrentincomedistributionLow1.690.563.601.12Second0.710.211.620.46Third0.470.131.110.30Top0.180.040.480.11ExpendituredistributionLow0.830.182.220.50Second0.660.141.800.40Third0.490.290.430.390.03Top0.221.050.08EducationcategoriesHSD0.753.060.34HSG0.662.620.31SCL0.40.350.140.370.27CLG0.130.050.250.09Tableshowsexante(column1)andadjusted(columns2–4)incidenceofaUS$1/packriseinthetaxoncigarettes.Allestimatesarefor=0.6and=0.7;correspondingforeachestimateisshownintoprow.Firstpanelshowsincidencecalculationsaccordingtoannualincomequartiles;secondpanelshowscalculationsaccordingtoconsumptionquartiles;finalpanelshowsincidencecalculationsaccordingtoeducationgroups.TableresultsfromevaluationofEq.(17)intext.J.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 Thenexttwocolumnsvary.Alowervalueforsignificantlymitigatestheself-controladjustment,sinceitservestoreducethebenefitoflengtheninglife(sincethoselateryearsoflifearemorehighlydiscounted).=0.9,thereislittlechangeinregressivity.Butfora=0.6,taxesareonceagainprogressive.Theconclusionsaremuchstrongerwhenweusemorepermanentmeasuresofincomeinpanels2and3ofTable2.Panel2usestheexpendituredistributionratherthantheincomedistribution.Here,wefindasignificantnarrowingfor=0.9,withthegapinincidencebetweenthetopandbottomgroupsshrinkingfrom0.61inthefirstcolumnto0.23.With=0.6,onceagain,thetaxbecomeshighlyprogressiveusingthisdistributionalmetric.Forlower,thereisonceagainrelativelylittleeffectwhenwehave=0.9,butthetaxesonceagainbecome(moderately)progressivewith=0.6.Perhapsthestrongestfindingsareobtainedwithadifferentproxyforpermanentincome:education.Theresultsforthisproxyforpermanentincomearemuchmoremodest,asshowninpanel3.Thisisbecausetheincomedifferencesaremuchmoremodestbyeducationgroup,sothattherearemuchmoreuniformlifevaluesbygroupthantheothertwopanels.Inthiscase,taxesareprogressiveinthreeofthefourcases,andmodestlyregressiveinthecasewhere=0.9andanannual10%discountrate.Thus,whilethesecalibrationsaresensitivetoparametervaluesandassumptions,theoverallmessageisclear:cigaretteexcisetaxesaremuchlessregressive,andpotentiallyprogressive,whenweaccountfortheself-controlbenefitsoftaxation.Thatis,thesameforceswhichleadustoalargeoptimaltaxoncigarettesleadaswelltotheconclusionthatcigarettetaxesdonotnecessarilyimposeburdensonsmokers,andthatthebenefitsoftaxationarelargestforthoselowincomegroupsthatsmokethemostandhavethemostpricesensitivity.Asdiscussedearlier,akeysensitivityforthesecalculationsistothesourceofhighersmokingrates,andlowervaluesoflife,forthepoor.Ifthepoorsmokemorebecausetheyhavehigherlong-termdiscountfactor,forexample,thenthiswouldincreasetheregressivityoftheresultsrelativetoTable2.Ontheotherhand,ifthepoorsmokemorebecausetheyhaveahighershort-termdiscountfactor,,thentaxesareevenmoreprogressivethanshowninTable2.Whilewehavenoevidenceontherelativevaluesoftheshort-andlong-termdiscountfactorsacrossincomegroups,itisworthnotingthattheeffectofvaryingismuchstrongerthantheeffectofvarying.Wecanillustratethisbyvaryingthesetwoparameters,holdingallelseconstant,inamannerconsistentwiththerevealeddifferencesinvaluesoflifeacrosstherichandthepoor.Forexample,ifthelong-termannualdiscountrateforthepooris33%whenthediscountratefortherichisonly10%—whichisconsistentwiththedifferencesinvalueoflifeimpliedbyanelasticityof0.5—thentheregressivityoftaxationisverysimilartothatshowninouradjustedfiguresinTable2.Iftheannualdiscountrateofthepooris15%whenthediscountratefortherichis3%—whichisalsoconsistentwiththesevalueoflifedifferences—thentheburdenonthepooristwiceaslargeasthatshownintheseTable2 Wevarydeltabychoosingahighervalueofalife-yearsothatthediscountedvalueoflifestillequalstheUS$6.7billionfigure.But,evenwiththesehigherlife-yearvalues,thebenefitofincreasinglifeby6yearsattheendoflifeisgreatlyreducedwithahigherdiscountrate.J.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 adjustedfigures(sothattheburdenontherichiszero,andtheburdenonthepooris1.1%ofincome).Ontheotherhand,if=0.75forthepoorand=0.9fortherich—whichaccountsforonlyaboutone-tenthofthedifferenceinthevaluesoflife—taxesbecomehighlyprogressive,withaburdenontherichof0.03%ofincome,andaburdenonthepoorofnegative2.3%ofincome.Andif=0.6forthepoorand=0.9fortherich—whichaccountsforaboutone-quarterofthedifferenceinthevaluesoflife—thentheburdenonthepoorrisestonegative7%ofincome.Moreover,iflowerincomegroupsaremorenaive,assomeanalystssuggest,thentheconclusionsherearestrengthened,aswecanprovethatincidenceisevenloweronnaivethanonsophisticatedtime-inconsistentconsumers.ThecalibrationspresentedinTable1alsoimposeaconstanteffectoftaxesbyage.Infact,theincidenceoftaxeswillvarywithage.Inparticular,theself-controlbenefitsoftaxationwillbecomelargerforoldersmokers,forwhomthedamageofsmokingisnearest.Oneparticularlyinterestinggrouptofocusonisteenagers,whoareconsideringstartingsmoking.Wehavenotincludedthisgroupinourcalibrations,sincetheyclearlyviolateoneofourimportantsimplifyingassumptions,thattheagentstartsfromasteady-statelevelofconsumption.Butwecancommentqualitativelyonhowoverallincidence,andtheregressivityoftaxes,wouldbeaffectedbytheself-controladjust-ment.Ontheonehand,thedamageofsmokingismostdistantforthisgroup,sothattheself-controladjustmentwillbethesmallest,andtaxeswillbethemostregressive.Ontheotherhand,thosewhoareinitiatingsmokinghavebeenshowntobethemostpricesensitive,withestimatedelasticitiesontheorderof25–50%higherthantheaverageacrossallages(GruberandZinman,2001).Moreover,ifthetaxpreventsteenagersfromsmokingaltogether,ithasnostandardincidenceatall.Thus,allofitseffectcomesfromtheself-controladjustment,whichislikelytobehigherforlowerincomegroups.Anaturalcomparisonforthinkingaboutthebroaderrangeofpublicpolicyisbetweentaxincreasesandregulations,suchascleanairlawsthatrestrictsmokinginpublicplaces.AswehighlightinGruberandKoszegi(2000),ourmodelsuggeststhatcleanairlawsarejustifiednotsimplyasameansofcontrollingsecond-handsmoke,butasanotherinstrumentofself-control.Thisimpliesthattheyhavelowerincidenceacrossallgroupsthanwouldbecomputedfromatraditionalanalysis.Howmuchlowerdependsontheelasticityofsmokingwithregardstotheseregulations;evidenceonthispointismixed,withGruberandZinman(2001)findinglittleeffectforyouths,butEvansetal.(1999a,b)findinglargeeffectsofspecificworkplacebansonsmoking.Inaddition,modelingtheincidenceofregulationswouldrequireinformationbothontheireffectsthroughouttheincomedistribution(e.g.dotheyimpactplaceswhereparticularlylowincomegroupsworkorcongregate?),andontherelativesensitivityofsmokingtoregulationsthroughouttheincomedistribution.Ifregulationsimposetheirlargestcostsintermsoftimelosttocompliance(e.g.duetomovingoutsideforacigarettebreak),andiflowerincomegroupshavethelowestvalueoftheirtime,thenitispossiblethattheregulationscouldhavethesmallestbehavioralimpactonlowincomegroups,incontrasttotaxes.Inthiscase,regulationsmightbemuchmoreregressivethanaretaxincreases.Thisisclearlyanimportantpointforfutureresearch.J.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 6.ConclusionsAppropriategovernmentpolicytowardsaddictivebadssuchassmokinghasbeen,andwillcontinuetobe,amajorsourceofdebateamongbothpolicy-makersandacademics.Cigaretteexcisetaxeshaverisendramaticallyoverthepastdecaderelativetotherecentpast,yettaxeshaveonlyrecentlyreturnedtotheirreallevelofthemid-1950s,andtheshareoftaxesasapercentageofpriceremainswellbelowitshistoricallevels(Gruber,2001)Andinjustthepast5years,wehaveseenamassivepaymentfromthetobaccoindustrytothestates,andanattemptatacomprehensivetobaccoregulationbillatthefederallevel.Economistsshouldbeimportantparticipantsinthedebateovergovernmentpolicyinthisarena.Theguidanceprovidedbyeconomiststodatehasbeenguidedbythenotionthatsmokingdecisionsaremadeinarational,time-consistentfashionalongthelinesoftheBecker–Murphymodel.Butavailableevidence,albeitquiteweakbyempiricaleconomicsstandards,doesnotsupportthisformulation.Thepurposeofthispaperwastowritedownamodelwhichwasmoreconsistentwiththeavailableevidence.OurmodeldeviatesinonlyonewayfromtheBecker–Murphyformulation,byintroducingspecifictime-inconsistentpreferences.Wefindthatthischangeinthemodelhasradicalimplicationsforgovernmentpolicy,sincegovernmentregulationprovidesacommitmentdevicethatisvaluedbytime-inconsistentconsumers,loweringtheincidenceofthetax.Wedevelopanadjustmenttothestandardmeasureoftaxincidencewhichaccountsforthesebenefits.Wenotethat,whileloweringincidenceoverall,thisadjustmentdoesnotunambiguouslyreduceregressivity;thatdependsontherelativepricesensitivities,valuesoflife,degreeoftimeinconsistency,anddegreeoflong-runimpatienceacrossincomegroups.Butourcalibrationsshowthat,giventhemuchhigherpriceelasticitiesoflowerincomesmokers,inalmostallcasestaxesoncigarettesaremuchlessregressiveandinsomecaseareindeedprogressive.Oneconcernaboutthismodelandthissetofconclusionsisthatitpresentsa‘‘slipperyslope’’towardsjustifyingexcessiveregulationofahostofeconomicbehaviors,rangingfromsmokinganddrinkingtodrivingandfastfoodconsumption.Butthereareatleastthreereasonswhysmokingisamoreappropriateplatformforourmodelthanotherbehaviors.First,thereissignificantcasualevidence(andtheoneeconometricstudyofGruberandMullainathan,2001)tosuggestthatsmokingdecisionsaretakeninatime-inconsistentfashion.Second,smokingisclearlyharmfulatalllevels,andtheharmrisesmonotonicallywiththeamountconsumed;drinking,forexample,issometimesarguedtobebeneficialatlowlevelsofconsumption,andonlyharmfulatveryhighlevels.Finally,theinternalcostsofsmokingdwarfsitsexternalcosts;thevastmajorityofharmdonebyasmokeristohimselforherself.Atstandardvaluesofthevalueofalife/year,weestimateabovethatapackofcigarettescostsUS$35.64intermsoflostlifeexpectancy,roughly100timesthelevelofexternalitiesfromsmoking.Thissuggeststhatsimplyrelyingonexternalitiestodetermineoptimalpolicycanleadtoverylargemistakes;inotherwords,thisisaplacewheregettingtheindividualdecision-makingmodelrightmattersalot.Wethinkthatitisvaluabletoconsidertheimplicationsofmodelssuchasthisinotherarenas;buttheargumentfordoingsointhecontextofsmokingismoststrong.Ofcourse,theempiricalevidencefortimeinconsistencyinsmokingremainsweak,andmuchmoreworkisneededhere.Atthesametime,thefactthatthereisnoempiricalJ.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 support,orevenlaboratorysupport,forexponentialdiscountinginthisorrelatedcontextssuggeststhatalternativemodelsofthetypethatwehavederivedbetakenseriously.Theimportantgeneralpointisthat,whenstandardpublicfinanceanalysessuggestthatthetaxonaddictivebadsissimplyequaltotheirexternalcosts,andthatsuchtaxesarehighlyregressive,thoseanalysesareimplicitlyembracingarationaladdictionmodel.Giventheenormousmagnitudeoftheinternalcoststosmoking,however,alternativemodelssuchasoursmustbeconsideredindesigningregulatorypolicytowardsaddictivegoods.AcknowledgementsWethankCindyPerryforresearchassistance,andAlanAuerbach,PeterDiamond,RogerGordon,DavidLaibson,andSendhilMullainathanforhelpfuldiscussions.AppendixA.ProofsLemma1.isdecreasinginforeacht.isdecreasinginforeacht.isincreasingint.Sincetheutilityfunctionisquadratic,wehavePluggingthisintothesophisticates’first-ordercondition,Eq.(5),andassumingeachperiod: Theabovehastobetrueforall,sothecoefficientofintheexpressionhastobezero.After‘some’manipulation,thisimplies J.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 Definethefunction)accordingtoEq.(20).Wewillprovethat iscontinuousandincreasingon(isdecreasinginThesearesufficienttoestablishallpartsofthelemma.First,noticethatthesecondtermof)isthereciprocalofaquadraticwithanegativecoefficienton.Thenifthistermispositivefortwopointsontherealline,itisalsopositivein-betweenthesetwopoints.Moreover,itiseasytoshowthatontheintervalwherethistermispositive,isstrictlyconvex.Therefore,itissufficienttoshowthat1,andthat0.Thefirsttwoensurethatweareonthecontinuousandstrictlyconvexsectionof,andthelastone(togetherwithconvexity)ensuresthatisincreasingon(Therestisjustcarryingouttheabove.Wehave asboththenumeratorandthedenominatorarepositiveinthesecondterm.Proceeding, aas
aaa aas
aaa
aaad1
d2
baaskT
bass  Thisbeingisequivalentto�0.Butthelattercanberewrittenas,andsinceowingtotheconcavityof)wehavethisinequalityholds.�0alsoimpliesthatthesecondtermispositive,sothatMovingon, whichgives
 aas
aaa
aaad1
d2 1
bkaaakaasask
bass z024 Thesecondderivativeofthereciprocalofaquadraticis(),whichispositiveaslongasispositiveandconcave.J.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 Finally,weprovethat)isdecreasingin.For,thisisquitesimple:thetermmultiplyinginthedenominatorispositivesinceandtheutilityfunctionisstrictlyconcave.Ittakesslightlymoreworktoprovethat)isdecreasingin.ThederivativeofthedenominatorwithrespecttoisapositiveconstanttimesNoticethatthisisequaltowhichisgreaterthanzerosincetheutilityfunction’sHessianisnegativedefinite.Thiscompletestheproof.Theorem1.Toprove,defineexceptwith=1.Thedifferencebetweenthedenominatorsofthesecondtermsof,ignoringthepositivemultiplicativeconstant,isbytheconcavityof).Thisimpliesthatforany),andToprovetheinequality,letusintroducerelevantvaluefunctionsforsophisticatesandtime-consistentagents:wherethesuperscriptsrefertothetwodifferentagents.Sincetheutilityfunctionisquadraticandstrategiesarelinear,bothofthesearequadraticin.Furthermore,itisclearthat)forall,so)hasatleastaslargeaprimecoefficientas).Inaddition,thesecoefficientsarenegative. Theeasiestwaytoseethisistocheckthat)isnegativeforasufficientlyhighJ.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 )solvesleadingtothefirst-orderconditionDifferentiatingthistotallywithrespectto BatBSt vasat
St1
d2ViSSSt1
vaaat
St2uWct1
d2ViSSSt1 ,andarepositiveconstants,and,theaboveimpliesAppendixB.IncidencecalibrationfornaifsWhenconsideringtheeffectofapriceincreaseontheutilityofnaivehyperbolicdiscounters,evenwiththediscountstructuregiven,wecanuseatleasttwomeasuresfortheirutility:whendiscountedutilityisevaluatedaccordingtotheirtrueinstantaneousutilitiesandwhenitisevaluatedaccordingtonaifs’perceivedinstantaneousutilities.Webelievethattheappropriatemeasureforwelfareanalysisisagents’trueutility,sowecalibratetheincidenceofaddictivegoodstaxationonnaifsusingthismeasure.Foraquadraticutilityfunction,andignoringlength-of-horizoneffects(takingthehorizontobeinfinite),thederivativeofanaif’sexponentiallydiscountedutilityfunctionwithrespecttoisthen  Thisexpressionisderivedbyconsideringthetotalutilityeffectofchangingtheagent’sconsumptionineachperiod(throughthechangeinprice),includingtheutilityeffectcomingthroughchanginglaterselves’consumptionlevels.Asbefore,weassumethatthedisutilityfromstock)isconstant.LetthisconstantbeJ.Gruber,B.Koszegi/JournalofPublicEconomics88(2004)1959–1987 .Combiningtime-consistentagents’andnaifs’first-orderconditions,wethen Combiningthiswiththerestoftheaboveexpressionyields BatBp d1
d1
d1
d1k*nvS
b d1
d1
d1
dvS
b d1
d1
d1
d Rearranginggives BatBp1
b Iftheagentstartedoffinasteady-statewithforall,themultiplicativeadjustmenttostandardincidenceis
BaBp pa
hSp Thetotaldiscountedfutureutilitycostofanextracigarette,expressedinmonetaryterms,is.Substitutingthisintheaboveyieldsourfinalexpression
BaBp pa
HSp Thisexpressionisalmostidenticaltotheoneforsophisticatedagents,withtwocrucialdifferences.Naturally,thenaifs’self-controladjustmentfeaturesthenaifs’responsivenesstostock,.Perhapsmoresurprisingly,theconsumptionresponsetoanincreaseinstockismultipliedby1forsophisticates,andnotfornaifs.Weproveintheappendixthat(Theorem1),sobothofthesetendtodecreasethemultiplicativeadjustmentfornaivequasi-hyperbolicdiscountersrelativetotheirsophisticatedcounterparts.Therefore,theincidenceofataxtendstobesmalleronnaifsthanonsophisticates.Ainslie,G.,1992.Picoeconomics:TheStrategicInteractionofSuccessiveMotivationalStateswithinthePerson.CambridgeUniv.Press,Cambridge,UK.Ainslie,G.,Haslam,N.,1992.Hyperbolicdiscounting.In:Loewenstein,G.,Elster,J.(Eds.),ChoiceOverTime.RussellSageFoundation,NewYork,NY,pp.57–92.
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