WhyiscreditutilizationstablePrecautionpaymentsandcreditcardsoverthebus - PDF document

1 Whyiscreditutilizationstable?Precaution,
Whyiscreditutilizationstable?Precaution,payments,andcreditcardsoverthebusinesscycleandlifecycleScottL.FulfordandScottSchuhApril2017PreliminaryversionAbstractWeshowthatcreditutilization—thefractionoftheavailablecreditcardlimitused—isremarkablystableoverthebusinesscycle,thelife-cycle,andforindividualsintheshortterm.Tounderstandthisstableutilization,webuildandestimateamodeloflife-cyclecredituseinwhichcreditcardshavebothaprecautionaryandpaymentsvalue,unitingtheirmonetaryandrevolvingdebtuses.Bothusessuggestthatincreasesincreditaccompanyincreasesindebt.Weestimatethepreferencesnecessarytomatchtheobservedlife-cycledebtandconsumption.Themodelsuccessfullypredictsstableutilizationoverthelife-cycle,businesscycle,andforindividuals. ScottFulford:ConsumerFinancialProtectionBurueau;email:scott.fulford@cfpb.gov.IconductedsomeofthisworkwhileIwasonthefacultyatBostonCollegeandavisitingscholarattheConsumerPaymentsResearchCenterattheFederalReserveBankofBoston.IwouldliketothanktheBankandtheCenterfortheirknowledgeandhelp.ScottSchuh:ConsumerPaymentResearchCenter,FederalReserveBankofBoston;email:Scott.Schuh@bos.frb.org.WeboththankDavidZhangforhisexcellentresearchassistance.Thispaperhasbenetedfromthecommentsofpar-ticipantsatthe2015CanadianEconomicsAssociation,the2015BoulderSummerConferenceonConsumerFinancialDecisionMaking,theFDICConsumerResearchSymposium,the2016NBERSummerInstituteAggregateImplica-tionsofMicroBehaviorworkshop,andseminarsattheFederalReserveBankofBoston,theBankofCanada,andtheConsumerFinancialProtectionBureau.Toensureappropriateuseofthedata,Equifaxrequiredthatwepre-clearresultsthatusedEquifaxdatabeforemakingthempublic.Th

2 eviewsexpressedinthispaperaretheauthors'
eviewsexpressedinthispaperaretheauthors'anddonotnecessarilyreecttheofcialpositionoftheFederalReserveBankofBoston,theFederalReserveSystem,theConsumerFinancialProtectionBureau,ortheUnitedStates.1 1IntroductionAsbanksattemptedtorepairtheirbalancesheetsduringthenancialcrisisof2008-2009,theyreducedthecreditcardlimitsofmillionsofAmericans,wipingoutnearlyatrilliondollarsinavailablecredit,andreducingtheaveragelimitbyabout40%.Atthesametime,Americansreducedtheircreditcarddebtbyasimilaramountandsotheaveragecreditutilization—thefractionofavailablecreditused—wasnearlyconstantfrom2000to2015(seegure1).Inaggregate,thedebtreductionswereapproximatelydoublethetaxrebatesfromtheEconomicStimulusAct(Parkeretal.,2013)andtheaveragefallindebtwasmorethan$1,000dollars.WhydidsomanyAmericanspaybacksomuchdebtduringarecession?Whilethefallincreditduringthecrisisisthemostdramaticexample,weshowthatlargechangesincreditaccompaniedbysimilarlylargechangesindebtarecommonoverthelife-cycleandforindividualsintheshortterm.UsingalargepanelfromthecreditbureauEquifaxandcollectedbytheNewYorkFederalReserve,weshowthataveragecreditcardlimitsincreasebymorethan400percentbetweenage20and30,andcontinuetoincreaseafterage30,althoughatasomewhatslowerrate.Yetsincecreditcarddebtincreasesatnearlythesamepace,creditutilizationdeclinesonlyveryslowlyoverthelifecycle.Individualsalsofacesubstantialcreditlimitvolatility,severaltimeslargerthanincomevolatility(Fulford,2015),butindividualcreditutilizationisextremelypersistentaswell,withshocksdyingoutalmostcompletelyafterabouttwoyears.Creditcardscanbeusedbothforpaymentsandtoaccumulaterevolvingdebt,andwesuggestthatthereasonfo

3 rstableutilizationisthatbothusespushtowa
rstableutilizationisthatbothusespushtowardsstability,althoughindifferentways.Therstuseisprimarilymonetary:acreditcardmaybeaconvenientwaytopayforthings,especiallyifitofferssomekindofrewards.Inthisway,creditcardsaresimilartodebitcardsorcheckswhichmaytakeseveraldaystoclearandrequireanintermediarywhopromisestopaythemerchantrstandcollectfromtheconsumerlater.1Convenienceuseofcreditcardsthereforetakes 1Thisaspectofcreditcards,whichinvolvestheinter-relationshipbetweencreditandliquidity,hasbeenstudiedrecentlybyTelyukovaandWright(2008)andTelyukova(2013).2 onsomeoftheaspectsofconsumption.Anincreaseinincomethatincreasesbothconsumptionandthecreditcardlimitwillalsoincreaseconvenienceuse,suggestingthatutilizationshouldbestable.Theseconduseisthat,havingdecidedtopayforsomething,aconsumerhastheoptiontoroll-overherdebtfrommonthtomonth.Creditthereforeplaysaprecautionaryroletosmoothovershocksandalifetimeliquidityroletohelpbringforwardconsumption.Sincesuchprecautionaryorbuffer-stockwealth(Carroll,1997;Deaton,1991)isonlythereforinsurancepurposes,anincreaseincrediteffectivelymakespeoplemorewealthy,allowingthemtospendmore(Fulford,2013).Boththemonetaryandprecautionaryfunctionsofcreditcardspointtostableutilizationsincedebtrisesatthesametimeascredit.Tomakethecasethatthecombinedpaymentandprecautionaryaspectsofcreditareimportantandexplainthebroadfactsofcreditcarddebt,weadaptamodeloflife-cycleconsumptionandsavings(GourinchasandParker,2002;Cagetti,2003)toallowbothfortheuseofcreditcardsforpaymentsandthelargechangesincreditlimitsoverthelife-cyclethatwehavedocumented.Withinthemodel,weallowtheconsumertodecidehowmuchofcurrentconsumptiontopayforwithacreditca

4 rdcard.Somepaymentsareveryconvenienttous
rdcard.Somepaymentsareveryconvenienttouseacreditcardfor,otherslessso.Akeyfeatureofcreditcardcontractsallowsustoidentifytheconveniencevalueofcreditcards:cardholderswhohadrevolvedinthepreviousbillingcyclelosetheoatorgraceperiodtopayoffthenewpurchaseswithoutinterest.Oursimplepaymentsmodelisidentiedoffofthedifferenceinbehaviorbetweenthosewhoarerevolvingdebtandthosewhoarenot.UsingnewdatafromtheDiaryofConsumerPaymentChoice,weestimatethatnon-revolverswouldbewillingtopay0.319percentofallconsumptiontocontinueusingcreditcards.Sincethevaluetakesplaceononlythe17%ofconsumptionpayedforwithcreditcards,itsuggeststhatthisformofpaymentisveryconvenient,giventhecurrentpaymentsinfrastructure.Wethenembedthevalueofpaymentchoiceinthelife-cyclemodelwhichallowstheaccumu-lationofcreditcarddebtandsavingsandestimatethepreferencesnecessarytomatchthelife-cycleproleofconsumptionorcreditcarddebtusingtheMethodofSimulatedMoments(McFadden,1989).Allowingforsubstantialheterogeneityofpreferences,themodelmatchesboththelife-3 cyclemomentsofcreditcarddebtandconsumptionwell.Theestimatessuggestthatabouthalfthepopulationmustbeveryimpatientandcarelittleaboutrisktoholdtheamountofrevolvingdebtweobserve.Aroundhalfofcreditcardusersareprimarilyusingtheircardsforpayments,andsoallowingforbothconvenienceandrevolvinguseisnecessaryforunderstandingoverallcreditcarddebt.BuildingontheworkofGrossandSouleles(2002),recentworkestimatestheresponseofdebttochangesincreditandhasnotedthesubstantialheterogeneityinresponses(Agarwaletal.,2015;Aydin,2015).Theheterogeneitycomesbothfromthedifferencesbetweenrevolversandconve-nienceusersaswellasdifferentresponsesacrossthelifecycleandfraction

5 ofcreditused.Al-lowingforbothtypeofuses,
ofcreditused.Al-lowingforbothtypeofuses,thedynamicsofcreditutilizationfromthemodelcloselymatchthereducedformdynamicsweestimateinthecreditbureaudata,suggestingthatthiskindofhetero-geneityisnecessarytounderstandresponsestocreditandotherliquidityshocks.Addingheterogeneoususesforcredittothelife-cyclemodelalsosuggestsasubtlydifferentexplanationforthehumpshapeoflife-cycleconsumption(Attanasioetal.,1999;AttanasioandBrowning,1995)thanthecombinationofprecautionandlife-cyclesavingssuggestedbyGour-inchasandParker(2002),insteadhearkeningtotheolderheterogeneousapproachofCampbellandMankiw(1989)andCampbellandMankiw(1990).Whileprecautionexplainsthelevelofconsumption,itplaysonlyasmallroleinthelife-cycleshapeinourestimates.Instead,thosewhoholdrevolvingdebtconsumeclosetotheirincomeovertheentirelife-cycle,relyingentirelyoncreditforsmoothingpurposes,andsoalloftheir“savings”comefromcreditlimitincreases.ThepatientconvenienceuserslooksmuchlikeastandardLife-Cycle/PermanentIncomeHypothesisconsumerwhoconsumesnearlythesameamountovertheentirelife-cycle.Theaverageofthesetwopopulationsconsumelessthanincomeoverthelife-cyclebutwiththedistincthumpshapeofconsumptioncomingentirelyfromtheincomeproleoftheimpatienthand-to-mouthpopu-lation.Indeed,ourestimatethatrevolvinghand-to-mouthconsumerscomposeabouthalfofthepopulationisveryclosetotheestimatesinCampbellandMankiw(1990).Usingourmodelestimatedtomatchthelife-cycleofcreditandconsumption,wenallyask4 howwellitexplainsthebusinesscycleofcreditanditsindividualdynamics.Thisapproachfollowsthestandardhypothetico-deductivemethod(Deaton,2010),inthatthebestevidenceinfavorofamodelisifithelpsexplainnewphenomenaout

6 ofsample.Wesimulatealargepopulationthatm
ofsample.Wesimulatealargepopulationthatmatchestheagedistributionofthepopulationofcreditusersandgivethemalarge,unexpected,reductionincreditofthesamesizeasoccurredduringthenancialcrisis.Themodelpredictsthestableutilizationthatobtainedovertheperiodbothinaggregateandattheindividuallevel,despitethelargeriseincreditfrom2000to2008andtheabruptfallover2009.2LiteratureHowconsumersrespondtochangesintheavailabilityofcreditisrarelystudied,despitethecen-tralityofcreditanddebtinthenanciallivesofconsumersandhouseholds.Itisparticularlyhardtostudychangesincreditbecausecredit—unlikedebt,assets,orincome—isonlyoccasionallyre-portedinsurveys.Indeed,Zinman(2015)arguesthathouseholddebtisunderstudiedevenwithintheeldofhouseholdnance,itselfunderstudiedcomparedwithotherareasofnance.Otherthansomeworkonmortgages(IacovielloandPavan,2013),thispaperappearstobethersttostudythelifecycleofcreditlimits.Consequently,weviewoneofourmajorcontributionsasdocumentinghowU.S.consumers'creditanddebtvaryoverthelifecycle.Asmallliteraturehasexaminedchangesincreditcardlimits,buthasconsideredchangesinlimitsonlyforsinglecreditcardaccountsandhasnotconsideredthedifferencebetweenconve-nienceusersandrevolvers.ThepioneeringworkisGrossandSouleles(2002),whouseapanelfromasinglecreditprovidertolookathowhouseholdsrespondtochangesininterestratesandchangesinavailablecredit.Increasesincreditlimitswerefollowedbyincreasesindebt,partic-ularlyforconsumersalreadyclosetotheircreditlimit.GrossandSouleles(2002)alsonotedthe“creditcardpuzzle,”theobservationthatsomehouseholdspayhighinterestoncreditcarddebtswhileatthesametimeearninglowinterestonliquidsavings.Aga

7 rwaletal.(2015)usecreditlimitincreasesth
rwaletal.(2015)usecreditlimitincreasesthatarediscontinuouswithcreditscoretoexaminehowexogenouschangesincreditlim-5 itsmatter.Forconsumerswiththelowestscores,adollarincreaseinlimitsisfollowedby59centsmoredebtwithinayear,whilethosewithhighercreditscoresincuralmostnoincreaseindebt.UsingtheEquifax/CCP,Fulford(2015)demonstratesthatthereissubstantialcreditlimitvariabil-ity.Short-termcreditvolatilityislargerthanmostmeasuresofincomevolatility,andlong-termcreditvolatilityismuchgreaterthanlong-termmeasuresofincomevolatility.Thisvolatilitycanhelptoexplainthecreditcardpuzzle.Ludvigson(1999)examinestheresponseofconsumptiontocreditvolatilityattheaggregatelevel.Leth-Petersen(2010)studiesaDanishreformthatallowedhomeownerstousehousingequityascollateralforthersttime.Thereformincreasedavailablecredit,butitproducedrelativelymoderateconsumptionresponses.Theresponsewasstrongestfortheyoungesthouseholds.Sincelargerchangesincreditaremorelikelywhenthereislessnancialintermediationingeneral,muchoftherecentworkonchangesincredithavestudieddevelopingcountries.Fulford(2013)examinestheshort-andlong-termconsumptionresponsesofconsumersinabuffer-stockmodelfollowingchangesincreditandndsevidenceconsistentwiththemodelinIndiafollowingamassivebankingexpansion.Inanstructuralestimationapproachsimilartoourwork,KaboskiandTownsend(2011)estimateamodelofconsumption,savings,andcreditusinganexpansionofcreditinThailand.KarlanandZinman(2010)examinetheeffectsofexpandingcreditaccesstothemarginalborrowerinSouthAfrica.Whilechangesincreditoverthelifecyclehavebeenlargelyunstudied,muchworkexaminesthedecisiontosaveandconsumeoverthelifecycle.Inmuchofthiswork,creditisintention-

8 allypushedtothebackground.Inthestandardv
allypushedtothebackground.Inthestandardversionsofthelife-cycleorpermanentincomehypothesis,forexample,theassumptionthatyoungconsumerscansmoothreliesdirectlyontheassumptionthatcreditisreadilyavailable(Deaton,1992).Morerecentlife-cyclemodelstakese-riouslythatcreditconstraintsmaybind,butdonotallowcredittovaryoverthelifecycle.Thetypicalassumptionisthateitherthereisnocreditavailable,orthatthereisaxedlimit,sothatonlynetassetsneedtobestudied(see,forexample,GourinchasandParker(2002)).Somerecentworkhasattemptedtoendogenizeborrowingconstraints,andmuchofthiswork6 hasdirectlife-cycleimplications.Coccoetal.(2005)buildamodelofconsumptionandportfoliochoiceoverthelifecycle.TheirworkaddsportfoliochoicetotheapproachofGourinchasandParker(2002).Asanextension,theyintroduceendogenousborrowingconstraints.Thesecon-straintsarebasedontheminimumvaluethatincomecantake,sincewithlimitedenforcement,borrowershavetochoosetopaybackratherthanfacethepenaltyofdefault.Borrowinggreatlyaffectsportfoliochoice,asyoungconsumersborrowiftheendogenousconstraintpermits,andonlystartinvestinginequitylaterinlife.Lopes(2008)introducesasimilarlife-cyclemodelwithdefaultandbankruptcy.Lawrence(1995)appearstohavebeenthersttointroducedefaultinalife-cyclemodel.Athreya(2008)developsalife-cyclemodelwithcreditconstraints,default,andsocialinsurance.Relaxingdefaultpolicycreatesseverecreditconstraintsamongtheyoung.Eliminatingdefaultrelaxescreditconstraintsfortheyoungandreducesconsumptioninequality,butincreasesconsumptioninequalityfortheold,whonowcannolongerdefaultaftersufcientlyseverenegativeshocks.Thispaperalsooverlapswithalargerliteratureexaminingthewaysthatindividualsusethe

9 nancialproductsavailabletothem.Stangoand
nancialproductsavailabletothem.StangoandZinman(2009)examinehowmuchasampleofU.S.consumerspaysinfeesandinterestfornancialproducts.Creditcardinterestisthebiggestnancialexpense,althoughsomehouseholdsalsopaysignicantoverdraftfees.Morethanhalfofhouseholdscouldreducefeessubstantiallybymovingamongproducts.Agarwaletal.(2007)notethatmiddle-agedadultspaytheleastinthesekindsofavoidablefees,withtheminimumpaidatage53.Zinman(2013)examinesthequestionofwhethermarketsover-orunder-supplycreditandconcludesthat,whiletherearemodelsthatsuggestover-supplyandmodelsthatsuggestunder-supply,thereisnotstrongevidenceforeither.7 3Creditcarduseoverthebusinesscycle,life-cycle,andforindividualsBothcreditanddebtchangesubstantiallyoverthebusinesscycle,thelifecycleandforindividualsintheshortterm.ThissectionbrieydiscussesthecontextofconsumercreditintheUnitedStates,introducesourmaindatasources,andpresentssomenon-parametricandreducedformresults.Anearlierworkingpaperversionofthispaper(FulfordandSchuh,2015)providesadditionaldescriptivestatisticsincludingadditionalevidenceonthedistributionofcreditandoncreditcardholdingbyageusingtheSurveyofConsumerPaymentChoice(SchuhandStavins,2014).Wethenturntoamodelinthenextsectiontohelpmakesenseoftheseobservations.3.1ThedataTheEquifax/NYFedConsumerCreditPanel(CCP)containsa5percentsampleofallaccountsreportedtothecreditreportingagencyEquifaxquarterlyfrom1999to2014.Formuchoftheanalysisweuseonlya0.1percentsampleforanalyticaltractability.Onceanaccountisselected,itsentirehistoryisavailable.Thedatasetcontainsacompletepictureofthedebtofanyindividualthatisreportedtothecreditagency:creditcard,auto,mortgage,andstudentloandebts,asw

10 ellassomeother,smaller,categories.Leeand
ellassomeother,smaller,categories.LeeandvanderKlaauw(2010)provideadditionaldetailsonthesamplingmethodologyandhowcloselytheoverallsamplecorrespondstothedemographiccharacteristicsoftheoverallUnitedStatespopulation,andconcludethatthedemographicsmatchtheoverallpopulationveryclosely:thevastmajorityoftheU.S.populationovertheageof18hasacreditbureauaccount.TheCCPalsorecordswhetheranaccountisajointorco-signedaccount.Ratherthancapturingtoofewpeople,themainissueisthatasizablefractionofsampledaccountsrepresenteitherincorrectinformationreportedtoEquifaxorindividualswhoareonlylooselyattachedtothecreditsystem,eitherbychoiceorbecausetheylackthedocumentation.Forexample,theaccountsarebasedonSocialSecuritynumbers,soreportinganincorrectSocialSecuritynumbercancreateaccountsincorrectlyattributedtoanindividual.Forthisreason,we8 limitthesamplethroughouttoaccountsthanincludealistedageandshowanopencreditcardaccountatsomepointfrom1999to2014.Thispopulationhassomeaccesstocreditandthusexcludesthosewhocouldnotobtainacreditcardorchosenotapplyforone.Dependingontheanalysis,wealsolimitthesampletothosewithcurrentopenaccounts,debt,orlimits.Muchofthispaperfocusesoncreditcards,sincetheyhaveexplicitlimitsnotreadilyobservableinmostothermarkets.However,thecreditcardlimitsreportedtocreditreportingagenciesareattimesincomplete.TheEquifax/CCPreportsonlytheaggregatelimitforcardsthatareupdatedinagivenquarter.Cardswithcurrentdebtareupdated,butaccountswithnodebtandnonewchargesmaynotbe.Todealwiththisproblem,wefollowFulford(2015)andcreateanimpliedaggregatelimitbytakingtheaveragelimitofreportedcardstimesthetotalnumberofopencards.Thismethodisexactifcardsthathavenotbeenupdatedhaveth

11 esamelimitasupdatedcards.Estimatingthedi
esamelimitasupdatedcards.Estimatingthedifferencebasedonchangesasnewcardsarereportedandthelimitchanges,Fulford(2015)ndsthatnon-updatedcardstypicallyhavelargerlimits,andsotheoveralllimitisanunderestimateforsomeconsumers.Thisissueisaconcernmostlyforconsumerswhoareusingonlyasmallfractionoftheiravailablecredit,sinceitisonlycardsnotusedatallarenotupdated.Fortheconsumerswhousemoreoftheircreditandsomayactuallybeboundbythelimit,thelimitisaccurate,becausealltheircardsareupdated.3.2CreditanddebtoverthebusinesscycleSince2000overallcreditanddebthavevariedtremendously.Figure1showshowtheaverageU.S.consumer'screditcardlimitanddebthavevariedfrom2000through2014.AlthoughtheEquifaxdatasetstartsin1999,weexcludetherstthreequarters,sincethelimitsinitiallyarenotcomparable(seeAveryetal.(2004)foradiscussionoftheinitialreportingproblems).Thescaleontheleftisinlogarithms,soproportionalchangesindebthaveapproximatelythesameimportanceasproportionalchangesincredit.Between2000and2008theaveragecreditcardlimitincreasedbyapproximately40percentfromaround$10,000onaveragetoapeakof$14,000.Over2009,overalllimitscollapsedrapidlybeforerecoveringslightlyin2012.Creditcarddebtshowsasimilar9 variationovertime.From2000to2008,theaverageU.S.consumer'screditcarddebtincreasedfromjustover$4,000tojustunder$5,000,beforereturningtoaround$4,000over2009and2010.2Utilizationismuchlessvolatilethancreditordebt.Thethicklineinthemiddleofthegureshowscreditutilization,theaveragefractionofavailablecreditused.Creditutilizationfellslightlyfromover35percentin2000toaround30percentin2006beforegenerallyincreasingagainto2010anddecliningslightlysincethen.Althoughitisnotimmediatelyevident,the

12 scaleofgure1forcreditanddebtonthelef
scaleofgure1forcreditanddebtontheleftisverysimilartothescaleforutilizationontheright.A1percentagepointchangeinutilizationhasthesameverticaldistanceasa1percentchangeincreditordebt.Largerchangesarelessexactlycomparable,butthedistancebetween$4,000and$5,000,forexample,isstillalmostexactly20percentagepointsontherightaxis.Thesimilarscalesmeanthatwecandirectlycomparetherelativechangesovertimeinlimits,debt,andcreditutilization.Creditanddebtvarytogetherinwaysthatproduceextremelystableutilizationthathasnoobviousrelationshipwiththeoverallbusinesscycle.Sucharelationshipisnotjustmechanical:whencreditwascutin2009,familieshadthechoicewhethertomaintainorincreasetheirdebt,andthecutincreditcouldhavetranslateddirectlyintoanincreaseinutilizationratherthanadecreaseindebt.3.3CreditanddebtoverthelifecycleWenextexaminehowcredit,debt,andutilizationevolveoverthelife-cycle.Figure2showsthecreditcardlimitanddebtinthetoppanelandcreditutilizationinthebottompanel.Eachlineisforanagecohortthatwefollowovertheentiretimepossible.Thegurethereforemakesnoassumptionsaboutcohort,age,ortimeeffects.Creditlimitsincreaseveryrapidlyearlyinlife,increasingbyaround400percentbetweentheagesof20and30.Theycontinuetoincrease,althoughlessrapidly,afterage40.Life-cyclevariationdominateseverythingelseingure2;whilethereisclearlysomecommonvariationoverthebusinesscycle,cohortsmovenearlyinlinewith 2Thefallindebtisnotbecauseofchargeoffsinwhichthebankwritesoffthedebtfromitsbooksasunrecoverable.Theconsumerstillowesthechargedoffdebt.Banksmayeventuallysellcharge-offdebttoacollectionagency,inwhichcaseitmaynolongerappearascreditcarddebtwithincreditbureauaccounts.Chargeoffsarenotlargeen

13 oughtoexplainthefallindebt,althoughtheyd
oughtoexplainthefallindebt,althoughtheydidincreaseover2009.Seehttps://www.federalreserve.gov/releases/chargeoff/delallsa.htmforcharge-offratesforcreditcards.10 age.Estimatingasimplemodelthatdividesbetweencommontimeandageeffectsmakes,gureA-5intheappendixmakesthispointexplicit.3Despitetheverylargevariationoverthebusinesscycleevidentingure1,thebigchangesarehappeningoverthelife-cycle.Creditlimitsanddebtcombinetogivethefractionofcreditused,showninthebottompanelofgure2.Consumerswithzerodebthavezerocreditutilization,andsoareincludedinutilizationbutareexcludedfrommeancredit,whichincludesonlypositivevalues.4Creditutilizationfallsslowlyfromage20toage80.Onaverage,20-year-oldsareusingmorethan50percentoftheiravailablecredit,and50-year-oldsarestillusing40percentoftheircredit.Creditutilizationfallstobelow20percentonlyaroundage70.Theslowfallincreditutilizationcomesfromtwodifferentsourcesoverthelifecycle.Earlyinlife,creditutilizationishigh,asasubstantialportionofthepopulationusesmuchorallofitsavailablecredit.Creditincreasessomewhatmorerapidlythandebt,however,socreditutilizationfallsslowly.Inmid-life,debtstabilizes,butcreditlimitscontinuetoincreaseslowlyFinally,start-ingaroundage60,averagedebt,conditionalonhavingany,startstodecline,socreditutilizationdeclines.3.4TheindividualevolutionofutilizationTheprevioustwosectionsshowthatdespiteverylargechangesincreditanddebtoverthelifecycleandbusinesscycle,creditutilizationisremarkablystable.Theaggregatedatacouldbehidingsubstantialindividualvolatilityinutilization,however.Inthissectionweshowthatutilization 3Estimatingasimplemodelthatseparatesthevariationbetweenageandyearallowsustomaketheimportan

14 ceoflife-cyclevariationevenclearer.Figur
ceoflife-cyclevariationevenclearer.FigureA-5showstheageandyeareffectsfromestimatingasimpleregressionoftheform:lnDit=+t+a+it;(1)wherelnDitiseitherlogdebt,logcreditlimits,orutilization,andallowsthesetovarybetweenageeffectsaandyeareffectstbutimposescommoncohorteffects.Theexcludedgroupisage20andyear2000,soeachpanelingureA-5startsatzeroatage20andyear2000.Theestimatedeffectisinlogunits,andsothescaleoftheguresuggeststhatvariationoverthelife-cycleincreditisaround9(e2::5=e0:3)timeslargerthanovertime,evenwithamassivecreditcontraction.4Thecalculationingure2aretheaverageofloglimitsandlogdebtstomatchlateranalysisandsoexcludezerosexceptforutilization.FigureA-6intheappendixshowsthefractionineachcohortwhohavepositivecreditanddebt.Includingthezeroswouldlowertheaveragecreditlimitanddebt,butactuallymakesthelife-cyclevariationlarger.11 foranindividualisextremelystable;whiledifferentindividualshavedifferentcreditutilizationratiosthatrepresenttheirindividualsteadystateandtheseratiosmaychangeslowlyastheyage,individualsreturnrapidlytotheirowntypicalratio.Creditutilizationisbestcharacterizedbyxedheterogeneityacrossindividuals,andrelativelysmalldeviationsforanindividualovertime.Wepresentnon-parametricresultsinappendixAandappendixgureA-7andreachalmostidenticalconclusionstotheparametricestimates.Thenon-parametricresultssuggestthatthesimplelineardynamicreducedformmodelweemployissurprisinglyaccurate.Table1showshowutilizationthisperiodisrelatedtoutilizationinthepreviousperiod.Weestimateregressionsoftheform:it=t+a+i+it�1+it;(2)whereit=Dit=Bitisthecreditutilizationgiven

15 thecreditlimitBitandthecurrentdebtDit,co
thecreditlimitBitandthecurrentdebtDit,conditionalonthecreditlimitBit￿0,andage(a)andquarter(t)effectsthatallowutilizationtovarysystematicallybyageandyear.Column1doesnotincludexedeffectsandsoassumesacommonintercept,column2includesquarterandageeffects,whiletheothercolumnsincludeindividualxedeffects,quarter,andageeffects.5Withoutxedeffects,creditutilizationisverypersistentandreturnstoanon-zerosteadystateofapproximately40percentutilization(=(1�)=0:38).Notethatthisutilizationisclosetotheaverageingure1,asitshouldbesincetheyareestimatedfromthesamedataandthenon- 5Thecombinedage,year,andindividualxedeffectsinequation(2)arenotfullyidentied.Asintheage-cohort-periodproblem,itisimpossibletofullyidentifyalleffectssincetherecanbeatrendinanyoneofage,time,orcohort,orsplitamongallthree,andanydivisionisobservationallyequivalent,sincebirthcohortequalstheyearminusage.Putdifferently,ifweestimateallindividualeffects,itisnotpossibletofullyseparatebetweengettingolderandatimeshock.Thesizeofthedatasetmeansthatratherthanestimatingindividualcoefcients—sometimesreferredtoasnuisanceparameters—weinsteadperformthestandardwithintransformationbyremovingthemeanfromallvariablesinequation(2).Thewithintransformationmeansthatanyadditionalrestrictionforidenticationmustbeoneitherthetimeorageeffects.Ratherthanimposingthequestionableassumptionthattwooftheageoryeareffectsareexactlyequal—theimplicationofdroppingmorethanoneoftheageoryeardummies—weinsteadimposetherestrictionthatthereisnotrendintheageeffects.Thisrestrictionisinnocuousinthesensethattherecanstillbeatrendwithage,asindividualeffectsthatare

16 olderwhenweobservethemcanhavelarger
olderwhenweobservethemcanhavelargeri,butthattrendwillappearintheindividualeffectsratherthanintheageeffects.Moreover,theageeffectscanstillallowlife-cyclevariation,butthatvariationmustaverageouttozero.WeimplementthisrestrictionfollowingDeaton(1997,pp.123–126)byrecastingtheagedummiessuchthat^Ia=Ia�[(a�1)I21�(a�2)I20],whereIais1iftheageofpersoniisaandzerootherwise.12 parametricconditionalexpectationfunctionshowninappendixgureA-7isnearlylinear.Intheothercolumns,sincetheage,year,andxedeffectschangethesteadystate,wedonotreportit,butitisimportanttorealizethattheaveragesteady-statecreditutilizationisnotzero.Thenexttwocolumnsreporthowcreditutilizationvariesaroundanindividual-specici,byestimatingusingthewithintransformation.Nearlyhalfoftheoverallvarianceinutilizationcomesfromthesexedeffects.Inotherwords,wecanthinkofthedistributionofutilizationascomingabouthalffromfactorsthatarexedforanindividual,allowingforcommonageandyeartrends,andhalffromrelativelyshort-termdeviationsfromthemean.Adeviationfromthemeanutilizationdiminishesatarateofabout0.353=1-0.647perquarter.Andso,aftera10percentagepointincreaseinutilization,6.47percentagepointsremaininonequarter,1.7percentagepointsinayear,andlessthan0.3ofapercentagepointaftertwoyears.Thelastcolumnsuggeststhatthespeedofreturntoindividualutilizationdependsonage.Fora20-year-old,onlyafraction0.58oftheshockisleftafteronequarter,fora60-year-old0.70ofashockremains.Itisnotobviouswhythespeedofreturnshouldincreasewithage.Onepossibilityisthatcreditrepresentsamuchmoreimportantfactorintheoverallportfolioofyoungpeople,whotypicallyhaveacquiredfewassets.A60-year-oldmayhaves

17 omeotherassetsandismorelikelytobeaconven
omeotherassetsandismorelikelytobeaconvenienceuser,andsomaynottargetcreditutilizationasclosely.TheestimatesinTable1emphasizethatcreditutilizationforanindividualisverystable.Whiletherearedeviationsfromthelong-termmean,thesedissipatequicklyandarelargelygonewithintwoyears.Boththeparametricandnon-parametricevidencesuggeststhatindividualshaveastrongtendencytoreturntotheirowncreditutilizationfollowingshocks.Sincecreditutilizationisnotzeroformostpeople,theresultssuggestastrongtendencytoholdcreditcarddebtforthelongterm.Sincetheparametricestimatesincludeageeffects,thesedeviationsarearoundalife-cycleaverage,whichtheprevioussectionshaveshownisdecliningslowly,butpersistently,withage.Boththeslowdeclineofutilizationwithageandthequickreturntoindividualcreditutilizationsuggestthatthepass-throughfromthecreditcardlimittocreditcarddebtislargeandoccursrelativelyrapidly.Intheworkingpaper(FulfordandSchuh,2015),weshowresultsonhowdebtandcredit13 co-evolve,ratherthanxingtheirrelationshipbycombiningthemintoutilization.Relativelylittleislostbysimplifyingonlytoutilization.Moreover,inaGrangerCausalitysense,thedirectionofcausalitymovesprimarilyfromchangesincredittochangeindebt.4Amodeloflife-cycleconsumptionandcreditcarddebtWehavedemonstratedthatthereisastrongtendencyforindividualstohavestableutilizationwhichfallsslowlyoverthelife-cycle.Tounderstandtheseobservations,thissectiondescribesalife-cycleconsumptionmodelsimilartoGourinchasandParker(2002)orCagetti(2003)towhichweaddapaymentchoiceandchangingcreditoverthelife-cycle.Thenextsectionestimatestheparametersofthemodelbasedontheobservedlife-cycleprolesofdebtandconsumptionandaskshowwellitexplainsthestableu

18 tilization.4.1ThedecisionproblemFromanya
tilization.4.1ThedecisionproblemFromanyagetaconsumerseekstomaximizehisorherutilityforremaininglifegivenhercurrentresourcesandexpectedfutureincome.Withadditivelyseparablepreferences,theconsumerofagetwithcash-at-handWtandcurrentcreditlimitBtmaximizesthediscountedvalueofexpectedfutureutility:maxfCs;sgTs=T(E"TXs=ts�tsu(sCs)+T+1S(AT)#)subjecttoCsWsWs+1=R(As)As+Ys+1+Bs+1As=Ws�Bs�Css=(s;IRs)IRs=f1ifAs�10;0elseg14 whereshegetsperiodutilityu(
)fromconsumptionCtwhichismademorevaluablebysde-pendingonhowshepaysforthatconsumption.Wewilldescribehowshedecideswhatfractions2[0;1]ofherconsumptiontopayforwithacreditcardinmoredetailbelow,butconsiderstobethevalueinconsumptionunitsofherpaymentchoicewhichdependsonwhethershehasrevolvedtheperiodbefore(IRs).Shediscountsthefuturewithaxeddiscountedfactor.ShelivesforTperiodswhereTisarandomnumberthatwewillmatchtotheactuallifetables,butassumethatshedieswithcertaintyatage~T.Ondeath,shereceivesanalutilityS(
)fromleft-overpositiveresources.Inourbaseestimations,wesetthebequestmotivetoallowforanannuitytoheirs.AppendixBdiscussesthespecicfunction.Recentworkhasdisagreedovertheimportanceofabequestmotiveasopposedtootherpossiblemotivesforkeepingassetslateinlifesuchaslong-termcareandmedicalneeds(DeNardietal.,2010).Sincewefocusprimarilyondebts,ourmodelandestimatesarenotwellsituatedtodistinguishbetweenmotives.Removingthebequestmotivedoesnotaffecttheresultsmuch,butrequiresthepatientconsumertoexpectnon-assetincomeafterexpensestofallevenmoresubstantiallylateinlife.EndofperiodassetsAsdeterminetherateofreturnforthenextperiod:R(As)=8￿￿&

19 #x]TJ ;� -2;
.51; Td;&#x [00;&
#x]TJ ;� -2;
.51; Td;&#x [00;&#x]TJ ;� -2;
.51; Td;&#x [00;:RifAs0RBifAs0;withaborrowerfacingahigherratethanasaverRBR.ThepaymentdecisionAccompanyingtheconsumptiondecisionisapaymentdecision.Ineveryperiodtheconsumermustdecidethefractionofherconsumptiontopayforwithacreditcards.Theoptimalfractionthenmakesconsumptionmorevaluablebyanamounts=(s;IRs�1)wheresistheoptimalfractionandIRs�1isanindicatorforwhethersheisrevolvingasofthepreviousperiodandsodoesnotgettheoatthatconvenienceusersget.Sincesappearsintheconsumer'sproblemonlyintheutilityfunction,notintheaccumulationequation,itisnotdirectlyaffecting15 thebudgetconstraint.Instead,wemodelthepaymentdecisionasoneofconvenience,inkeepingwithourdualapproachtocreditcards.Forexample,itisoftenmoreconvenienttobuysomethingonlinewithacreditcard.Itmightbepossibletosetupadirectbankwithdrawal,ormailacheck,orvisitabanktoturncashintoacashier'scheck,butthesemethodsrequireadditionalstepsortakelonger.Theprimaryvalueofcreditcardsaspaymentsisthattheyareawidelyacceptedpaymentmeans.Inaddition,somecreditcardsofferrewardsofvarioustypesthatmakepaymentsusingthemmorevaluableinmanyways.Whilesomeoftheserewards,suchas1%cashback,couldaffectthebudgetconstraintdirectly,others,suchasairlinemilesorpreferentialaccesstoeventtickets,areprimarilyintheformofincreasedvalueofconsumption.Giventhiscomplexity,ourapproachistomakeallpaymentchoicespurelyhedonic,whichsimpliestheproblemtremendouslybyallowingustomaketheconsumer'sproblemsequential.Theconsumerrstdecideshowtopayforwhatshewillconsume.Thengivens,whichmakesconsumptionthisperiodmoreorlessvalu

20 able,shedecideshowmuchtoconsume,takingin
able,shedecideshowmuchtoconsume,takingintoaccounthowherdecisionsthisperiodwillaffectherpaymentandconsumptiondecisionsnextperiod.Inparticular,theoptimalamounttopaywithacreditcardthisperioddirectlydependsonwhethersheleftrevolvingdebtfromlastperiod(IRs�1).Thereisnograceperiodoncreditcardpaymentsforrevolvers.Purchasesstartearninginterestassoonastheyaremadeifaconsumerhasunpaidcreditcarddebtfromthepreviousmonth.Theconsumermaychoosetopayanynewpurchasesoffverysoon,andsonotowemuchinterestonthem,ofcourse,butforrevolversthereisnofreecredit.Thedifferencebetweenthepaymentshabitsofconvenienceusersandrevolversiscrucialhere.Unlikeconvenienceuserswhogetaninterestfreeloanforallpayments,andsomayndcreditcarduseveryconvenient,revolversarepayinginterestontheirpurchasesandsondthemlessconvenient.Sincetheonlystatevariablethatmattersforthepaymentdecisioniswhetherornotsheisrevolving,thisapproachimpliesthatthepaymentdecisiontakesononlytwovaluesR=(R;IR=1)fortheoptimalsharebyarevolverRandC=(C;IR=0)forconvenienceuserswithoptimalshareC.Wewillusethisdistinctionandthedifferentpaymentchoicesofrevolversandconvenienceuserstoidentifyasimplefunctionalformfor(s;IRs�1).16 Tomodelthepaymentdecision,weassumethatallofthegoodsandservicesthataconsumerbuyscanberankedinorderofthevalueshegetsfrompurchasingwithacreditcard.Some,suchasgoodsonline,areveryconvenienttobuywithacreditcard.Others,suchastherentormortgagepayments,areinconvenientormaycomewithsomeadditionalcosts,andsoaconsumerwilltrytoavoidpayingforthemwithacreditcard.Orderedthisway,themarginalvalueofspendingalargerfractiononacreditcardisdeclining,andwillevent

21 uallyreachzerowhenitisinconvenienttospen
uallyreachzerowhenitisinconvenienttospendmoreonacreditcard.Weassumethatthedecreaseinthemarginalvalueofconvenienceislinearatrate1anddenotemarginalconvenienceoftherstconsumptiononacreditcardas0.Figure3showsthismarginalvalue.Themarginalconveniencevalueoftherstsmallshareofconsumptiononacreditcardisveryhigh,butasaconsumerchargesmoreandmore,thevaluedecreases.Theoptimumiswherethemarginalvalueofadditionalconsumptiononacreditcardiszero.Revolversmustpayinterest,inadditiontogettingtheconvenienceofthecard,andsothestart-ingmarginalconvenienceforthemislower.Ifconsumptionisspreadevenlyoverthemonth,thenarevolverwillpayadditionalinterestof(rB=12)=2onherconsumptionthatmonth.6Revolversstartoutatalowerinitialvalue,andsotheiroptimumpaymentshareislowerinFigure3,apredictionweseeinthedataandwilldiscussmoreforwhenweestimatethismodelinsection5.Finally,wendtheconsumptionvalueofthispaymentchoicebytakingtheareaunderthecurvesandnormalizingthevalueofothermeansofpaymentto1sothat(0;
)=1.Exceptforthevalueofbequests,thenormalizationdoesnotaffecttheconsumptiondecisionsinceitaffectsthemarginalvalueofconsumptionthisperiodandinthefuture,leavingthetradeoffbetweenthemunchanged.Althoughitisnotrequiredforthesolutiontotheconsumer'sproblem,itisusefultoconsiderwhatthemodelsaysforconvenienceuseandrevolvingcredit.Aconsumerwithnegativeassetsis 6Thisformulacomesfromthewaythatannualcreditcardratesarereportedandinterestcharged.TheAnnualPercentageRateorAPRisnotacompoundrate,andsoitisappropriatetodivideitby12tondtherateofinterest.Thenancingchargeonacreditcardiscalculatedbasedontheaveragedailybalancewithinamonth,andsothenancingchargeonconsum

22 ptionspreadevenlythroughoutamonthishalft
ptionspreadevenlythroughoutamonthishalftheinterestrate.NotethatwhiletheAPRisnotacompoundrate,interestchargesnotpaidoffeachmonthwillcompoundbothinrealityandinourmodel.17 borrowingandhasdebtsthatrevolvefromperiodtoperiod.Sincethepaymentdecisionispurelyhedonic,itdependsonlyonwhetherornotsomeonewasrevolvinglastperiod,andsothereisanoptimalrevolverfractionofconsumptiononacreditcardRconvenienceuserfractiononacreditcardC:Theobservedcreditcarddebtataget,whichincludesbothnewchargesandpreviousdebtisthen:DRit=RCit�AitifAit0forarevolver(forwhomAi;t�10).Forconvenienceusersontheotherhand,assumingtheyarenottransitioningintobeingrevolvers,theobservedcreditcarddebtisjustafractionoftheirconsumption:DCit=CCitifAit&#x]TJ/;༕ ;.9;Ւ ;&#xTf 1;.21; 0 ;&#xTd [;0:ThecreditlimitThelimitingfactorforconsumptioninanygivenperiodiscash-at-handWswhichisdeterminedbyincomeYs,assetsordebtsfromthepreviousperiodAs,andthecurrentcreditlimitBs:ThecreditlimitBsevolveswithpermanentincome:Bs=bsPs;wherebs0isthepossiblystochasticfractionofthisamountthatcanbeborrowed.ThisapproachmeansthatacrossconsumersBswillbeinproportiontoincomePs;butmayfollowadifferentaveragepathoverthelife-cyclethanincome.7 7Theconsumer'sproblemaswritten,withWsasasufcientperiodbudgetconstraint,impliesthataconsumermustimmediatelyrepayalldebtoverherlimitifhercreditlimitfalls.Toseethis,considerwhathappensifBs�1￿0andtheconsumerborrowsleavingsnegativeassetsattheendofperiodAs�10.IfBs=0;thenassetsattheendofperiodsmustbeweaklypositive(As0),andsoalldebthasbeenrepaidwithinasingleperiod.Acutincreditlimitsimpliesanimmediaterepaymentofdebtinexcessof

23 thelimit.Thisdebtrepaymentwhencreditiscu
thelimit.Thisdebtrepaymentwhencreditiscutbelowdebtdoesnotmatchcreditcardcontracts,whichdonotrequireimmediateandcompletepaymentfollowingafallincredit(Fulford,2015)).Instead,creditcardborrowerscanpayofftheirdebtunderthesameterms,theyjustcannotaddtoit.However,allowingforsuchbehaviormeansthattheremustbeanadditionalcontinuousstatevariablesinceWsandBsnolongerfullydescribetheconsumer'sproblem,whichaddssubstantiallytothenumericalcomplexityofthesolutionthroughthecurseofdimensionality.18 TheincomeprocessIncomeordisposableincomefollowsarandomwalkplusdrift:Ys+1=Ps+1Us+1Ps+1=Gs+1PsNs+1whereGs+1istheknowgrowthratefromperiodtoperiod,the“permanent”orrandomwalkshocksNs+1areindependentlyandidenticallydistributedaslognormalwithmeanone:lnNs+1N(�2N=2;2N);andthetransitoryshocksaredistributedas:Us+18￿￿&#x]TJ ;� -2;
.52;&#x Td ;&#x[000;&#x]TJ ;� -2;
.52;&#x Td ;&#x[000;:ULwithprobabilitypL~Us(1�ULpL)=(1�pL)withprobability1�pL;where~Uisani.i.d.lognormalwithmeanone:ln~Us+1N(�2U=2;2U)andULisunemploymentincomeasafractionaspermanentincome.ThestructureoftheshocksensuresthattheexpectedincomenextperiodisalwaysGs+1Pssincethemeanofbothtransitoryandpermanentshocksisone.Recursiveformulationandiso-elasticpreferences.Weassumethatsub-utilitydisplaysCon-stantRelativeRiskAversion(CRRA):u(C)=C1� 1� :WithCRRApreferences,itispossibletonormalizetheproblemintermsofpermanentincomePtatanygivenage(seeCarroll(2012)foramorecompletediscussion).Usinglowercasetorepresentthenormalizedvalue:ct=Ct=Pt,wt=Wt=Pt,andat=At=Pt,andrewritinginrecursiveform,19 theproblemisequivalentto:vt(wt;bt)=maxctu(t

24 ct)+Et[t+1(Gt+1Nt+1)1� vt+1(wt+1
ct)+Et[t+1(Gt+1Nt+1)1� vt+1(wt+1;bt+1)] subjecttoctwtwt+1=Rt+1(IRt+1)at+Ut+1+bt+1at=wt�bt�ctt=(IRt)=maxt(t;IRt)IRt=f1ifat�10;0elsegwhereRt+1(IRt+1)=R=(Gt+1Nt+1)ifat0andRt+1(IRt+1)=RB=(Gt+1Nt+1)ifat0.TheexpectationattincludesthepossibilityofdeathbeforeTandthecertaintyofdeathat~Tleavingabequestwortht+1s(at)wheres(
)isthebequestfunctionnormalizedbyPt.Theinitialdistributionofincomeandwealth,andlatelifeincomeFinally,thereareseveralimportantdecisionparametersthataffectinitialdistributionsanddecisionslateinlife.Weassumethattheinitialdistributionofthewealth/permanentincomeratioislog-normalwithalocationparameter0andvarianceparameterthatmatchesthevarianceofpermanentincomeshocks.WeallowforafallindisposableincomeinretirementoroldagebyassumingthatafterRperiodspermanentincomefallsto1ofitsvalueatageR�1.Suchafallindisposableincomemayalsocomefromincreasedmedicalorcareneeds,andsoisaexiblewayofallowingforthepossibilitythatwealthneedsmaybehigherlateinlifebecauseincomeafternecessaryexpensesmaybelower.ModelfrequencyBecauseofdataandcomputationalconstraints,muchofthestructuralcon-sumptionliteraturehasbeenlimitedtoexaminingdecisionsmadeatayearlyfrequency.Yetcon-sumptiondecisionsmustbemademorefrequentlythenyearly.Ifsmoothingwithintheyearisperfect,thenthefrequencyshouldnotmatter.However,thelogicofthemodelandthedatasug-gestthatpeopledooccasionallyhittheirbudgetconstraint,whichdirectlyimpliesthatignoring20 decisionsmadewithintheyearmaymissimportantfacetsofconsumerbehavior.Inaddition,ifwewanttounderstandwhetherthemodelcanmatchthequarterlydynamicsofindividualandaggregatecreditutiliza

25 tion,itmustbehaveatleastaquarterlyfreque
tion,itmustbehaveatleastaquarterlyfrequency.4.2NumericalsolutionWiththeproblemwrittenrecursively,weproceedthroughbackwardrecursiontondanumericalapproximationoftheconsumer'sproblem.Foragivensentofparameters,oncevT+1(AT;0)isgiven,itispossibletondanapproximationofvT(w;b),andusetheapproximationofvT(w;b)tondvT�1(w;b).Thesolutiontoeachperiod'svaluefunctionisaconsumptionfunctionct(w;b).Wefollowseveralstandardsteps(seeCarroll(2012)foramoreindepthdiscussionofmanyoftheseapproaches).First,wediscretizethelog-normalshocksusingaGauss-Hermitequadraturewhichturnstheintegrationintheexpectationfunctionintoasummationoverdiscretestates.Sincetheincomeprocessissurelynotexactlylog-normalthereisnogainorlossinaccuracyfromdoingso;wearesimplyreplacingoneapproximationofshockswithanother.Second,wefollowthemethodofendogenousgridpoints(Carroll,2006)tondtheoptimalconsumptionthatleadstoendofperiodassetsatatanumberofgridpointsforatandbt.Veryelegantly,itisthenpossibletondoptimalconsumptionwhichleavesthisamountofassetsct(w;b)attheendogenousgridpointsforwsimplyusingtheaccountingidentityat=wt�bt�ct;andsoavoidacomputationallycostlynumericalroot-ndingapproximationentirely.Moreprecisely,iftheconsumerhasnotconsumedallavailablecash-at-handforthenextperiod,andsoisnotstrictlyconstrainedbythecreditlimit,thenthestandardrstorderconditionsandtheEulerequationimplythat:u0((IRt)ct)=Et[tRt+1(at)(Gt+1Nt+1)1� u0(t+1ct+1(wt+1;bt+1))]wheredespiteitssubscriptt+t=(IRt+1)isdeterminedentirelybythechoiceofwhethertoleavepositiveornegativeassetsforthenextperiod.Giventhenextperiodconsumptionfunction,itis21 straightforwardton

26 dtheoptimalconsumptionthatleavesendofper
dtheoptimalconsumptionthatleavesendofperiodassetsatas:cat(a;b;IRt+1)=(1=t)Eth^t+1Rt+1(a)(Gt+1Nt+1)1� (t+1c(Rt+1(a)a+Ut+1+bt+1;bt+1))� i�1= :(3)Foravectorofendofperiodassets~a,itisnearlycostlesstondtheoptimalconsumptionatavectorofendogenouspointsforcash-at-hand~w=~a+b+cat(~a;b;IRt+1)istheamountatwhichconsumingcat(a;b;IRt+1)andleavingafornextperiodisoptimal.Welinearlyinterpolatebetweenthesepointstondanapproximationoftheconsumptionfunction.Notethatinadditiontothecurrentstateofassetsandthecreditlimit,theconsumptionfunctionisalsoafunctionofwhethertheconsumerisrevolvingbyhavingnegativeassetslastperiod.Whileitisnotacontinuousstate,theadditionofanotherstatevariablecomplicatesthesolutionsincewemustndtheoptimalconsumptionforbothrevolvers,whondconsuminglessvaluablesincetheycannotpayforitinquiteasconvenientway,andconvenienceusers.Forthemostpart,someonewhoisnotrevolvingthisperiodwillnotberevolvingnextperiodandsovt=t+1andthepaymentchoicedoesnotaffecttheconsumptiondecisiondirectly.Itdoes,however,makerevolvingsomewhatmorecostly.Sincetheconsumer'sproblemincludesbothanexternallyimposedcreditlimitaswellasinter-estratesthatdifferdependingonwhetherassetsarepositiveornegative,thereareseveraladditionalcomplications.TherstisthatthestandardEulerequationdoesnotholdwhentheconsumerisagainsthercreditlimitandsospendsallavailableresources,sinceshewouldliketospendmoretodaybutcannot(Deaton,1991).Thisproblemisrelativelyeasytodealwith,however,byinclud-ingtheinectionpoint,orthelastpointatwhichtheEulerequationholdsandsoassetsleftforthenextperiodare�bt.Foranycash-at-handlessthanw=cat(�b;b)

27 ,theconsumerconsumesallcash-at-hand,soct
,theconsumerconsumesallcash-at-hand,soct(w;b)=wifww.Thesecondproblemisthattheinterestratedifferen-tialintroducesastepintheconsumptionfunctionsincetherearetwosolutionstoequation(3)fora=0:One,thelimitwithassetsapproachingzerofrombelow,usestheborrowingrateRBandtheotherusesthesavingrateR.Theeconomicintuitionisthatleavingzeroassetsforthenextperiodisoptimalatahighborrowingratewellbeforeitisoptimalatlowsavingsrate.Forcash-at-handbe-22 tweenthesetwopoints,theconsumerhasamarginalpropensitytoconsumeofonesincethereturnonsavingsisnothighenoughtoinducehertosave,butthecostofborrowingissufcienttokeepherfromborrowingandsoadditionalresourcesgostraighttoconsumption.Todealwiththisissue,theendogenousgridpointsincludetwopointswherea=0:therstcBt=cat(0;b;RB)thesolutiontoequation(3)whena=0usingRBandwBt=0+b+cBt,andthesecondcFt=cat(0;b;R)andwFt.Betweenthepoints(wBt;cBt)and(wFt;cFt)theconsumerhasamarginalpropensitytoconsumeofone.FigureA-2showstheconsumptionfunctionatdifferentageswithaconstantabilitytoborrowoverthelifecycleofonefthofincome.8Severalpointsareworthdiscussing.First,thecon-sumptionfunctiongenerallyfallswithage.Thisoccursastheconsumerplansforretirementwhenhavingaccumulatedalargeamountofsavingsisvaluable.SincewesetbT+1=0,andtheremain-ingwealthisdividedamongthemanyyearsofretirement,consumptionoutofavailableresourcescT+1(wT+1)isquitelow.Planningforthishighdemandforsavingsmeansafterage45,consumersseektoaccumulatebyspendingmuchlessfromavailableresources.Earlyinthelife-cycle,retire-mentisfaraway,andshort-termshocksdominate,andsotheconsumptionfunctionat26isverysimilartotheoneat35and45.Second,forlowcash-at-handbelowwthemarginalp

28 ropensitytoconsumeisone.Betweenwandw
ropensitytoconsumeisone.BetweenwandwBtheconsumerisleavingdebtfornextperiod,andsoispayingahighinterestrateRB.BetweenwBandwFtheconsumerdoesnotwanttoborrow,butthereturnonsavingsisnothighenough,andsosheleaveszeroassetsandhasamarginalpropensitytoconsumeofone.Thiskinkintheconsumptionfunctionimpliesthattherecanbeapositivefractionofconsumerswhoholdexactlyzeroassets.ThedistancebetweenwBandwFdependsontheinterestratedifferential,withawiderdifferentialimplyingalargerdistance. 8TheotherrelevantparametersareasavingsrateR=1:03andborrowingrateRB=1:14,acoefcientofrelativeriskaversionof =2,adiscountparameter=0:9andotherparametersandretirementfunctionsmatchingGourinchasandParker(2002).23 5EstimationThissectiondescribesthedatausedtoestimatethemodel,theestimationprocedure,andthere-sults.Alongwithinformationontheincomeprocessdescribedbelowandthepaymentsparame-ters,thelife-cycleportionofthemodelisdescribedbyonlyfourparameters.Weallowforpref-erenceheterogeneitybyallowingfortwosub-populationswithdifferentpreferences.Ofcourse,additionalpreferenceheterogeneityispossible,butourresultsshowthatthisistheminimumhet-erogeneitynecessaryandwepreferthissimpleheterogeneitysinceitmakesobviousthecontribu-tionofdifferentpopulations.Therearethusthreeformsofheterogeneityintheestimatedmodel:ageaspeoplemakedifferentdecisionsatdifferentages;idiosyncraticheterogeneityaspeoplearehitwithdifferentshocksandsohavedifferentassetsandincomes;andpreferenceheterogeneityasdifferentsub-groupsreactdifferentlytoshocks.Foreachsub-groupweestimatethediscountrate;coefcientofrelativeriskaversion ,andthefallinincomeattheendoflife1.Tocombinegroupsweestimatetheshareofg

29 roupA,fAinthepopulation.Incomeandinitial
roupA,fAinthepopulation.IncomeandinitialwealthmustbejointlydeterminedwithfAtokeepthesimulatedmeanincomeandinitialwealthequaltotheirobservedcounterparts.Toallowonegrouptohaveahigherincomethantheotheronaverage,weestimatethemultipleoftheaveragepermanentincomeearnedbygroupA,A,sothatifgroupAearnshalfofthepopulationaverageincomeA=0:5.TogetherfAandAdirectlydetermineB.Similarly,weestimatetheinitialliquidfundstoincomeratioofgroupAA0,andsofAdeterminesB0.9Finally,weestimatetheconveniencevalueofcreditcardsforpaymentsforrevolversRandconvenienceusersC.Weestimatealloftheotherparametersofthedecisioninaseparaterststage.Wethereforeestimate11parametersjointly=f A;A;A0;A1; B;B;B1;fA;A0;R;Cg.Weestimatetheparametersofthehighlynon-linearmodelthatbestmatchthelife-cycleevo-lutionofdebtandconsumptionusingtheMethodofSimulatedMoments(MSM)ofMcFadden(1989).Foragivensetofparameters2,andrststageparameterssuchastheinterest 9InorderfortheaverageincomeofthecombinedpopulationstoequaltheaverageobservedincomefAA+fBB=1which,sincefB=1�fA,impliesthatB=(1�fAA)=(1�fA).Similarly,sincefAA0+fBB0=InitialliquidfundstoincomethenB0=((Initialliquidfundstoincome)�fAA0)=(1�fA).24 rateandincomeprocessestimatedseparately,wenumericallyndconsumptionfunctionsateachage.Thenwithaninitialdistributionofinitialassets,income,andcreditlimits,wedrawastartingvaluefromthisdistributionforeachi=1toNconsumers.Eachconsumermakesaconsumptiondecision,leavingassetsfornextperiod.Foreachconsumer,wedrawthenextperiodincomeandcreditlimitsfromthe

30 irdistributionsandndhisorherconsumpt
irdistributionsandndhisorherconsumptionandthenassetsforthenextperiod.Thisprocessproceedsuntilthelastperiod,generatingforeveryagetheconsumptionandassetsforallNconsumers.Agivensetofmodelparameterscanthusgeneratealife-cycledistributionofconsumption,debt,andsavings.Withthesedistributions,itispossibletocalculatemomentsthatdescribethedistributionandattempttondtheparametersthatminimizethedifferencebetweenthesemomentsandtheirobservablecounterparts.GourinchasandParker(2002),forexample,calculatetheriskaversionanddiscountparametersthatmakethecalculatedmeanconsumptionoverthelife-cyclematchthehumpshapeobservedintheConsumerExpenditureSurvey.Cagetti(2003)insteadmatchestheaccumulationofwealthoverthelife-cycleasobservedintheSurveyofConsumerFinances.Moreformally,foragiven2,andrststageparametersestimatedbelow,wecandenethedifferencebetweenanempiricalmomentandasimulatedmomentasgj(;)foreachofJtotalmoments.Forexample,oneoftheempiricalmomentsweuseisthemeancreditcarddebtatage30estimatedfromtheEquifax/NYFED,andsoonemomentis:gj(;)=(1=I)IXi=1Di30�(1=N)NXk=1^D(;)k30where^D(;)i30isthesimulateddebtofpersoniatage30givenparameters;andDk30isthedebtofpersonkinEquifax/NYFedCCPatage30.TheseareaveragedovertheIpeopleintheCCPsample,andtheNpeoplewesimulate.TheMSMthenseekstominimizetheweightedsquareofthesedifferences:min2g(;)0Wg(;)(4)25 whereg(;)=(g1(;);:::;gJ(;))andWisaJJweightingmatrix.Wegenerallyusetheoptimalweightingmatrixproportionaltotheinversevarianceoftheempiricalmoments,whichgivesmoreweighttobetterestimatedmoments.5.1Theem

31 piricalmomentsandrststageestimatesWe
piricalmomentsandrststageestimatesWeestimatethemodeltoprovidethebestttotwolife-cyclemoments:(1)theobservedcreditcarddebtoverthelife-cyclefromtheNYFed/EquifaxConsumerCreditPaneldescribedinsection3andthe(2)householdconsumptionoverthelife-cyclefromtheConsumerExpenditureSurveyfrom2000-2014.Sinceourobservedcreditdataisfortheindividualaccountholder,ratherthanthehousehold,weadjusthouseholdconsumptionbydividingbythenumberofadultsinthehousehold.Weallowforsomeunobservedtastechangesoverthelife-cyclebyadjustingconsumptionforthenumberofchildreninthehousehold.10Tobeidentied,themodelrequiresanumberofotherancillaryparametersestimatedelse-where.WeadjustthestandarderrorsoftheMSMfortheserststageestimatesusingtheirvariance-covariancematricesfollowingLaibsonetal.(2007)whoalsoupdateGourinchasandParker(2002)byallowingfortheempiricalmomentstohavedifferentnumbersofobservations.WeusedatafromtheDiaryofConsumerPaymentChoicetondthefractionofconsumptiononacreditcardbyrevolversandconvenienceusers,anddiscusstheidenticationofthepaymentsproblembelow.Weestimatetheaveragelife-cyclegrowth(Gt)ofincomebasedafthorderpoly-nomialbesttofaftertaxincomeperadulthouseholdmemberfromtheConsumerExpenditureSurvey.TherawdataandthettedlinesareinappendixgureA-8.Similarly,wetakeafthdegreepolynomialestimateoftheaveragecreditlimitperaccountfromEquifax/NYFedCCPtoformBs.Notsmoothingthesetwobudgetconstraintsmakeslittledifferencetotheoverallesti- 10Formally,weestimateln(Ci;t=Adultsi;t)=a+t+Childreni;t+i;tandthencalculateaveragehouseholdconsumptionperadultateachafterremovingtheeffectofchildrenatthein-dividuallevel.Theremovingt

32 heimpliedconsumptioneffectofchildrenhasa
heimpliedconsumptioneffectofchildrenhasasurprisinglysmalleffect.FigureA-8showstheunadjustedandadjustedconsumption.Childrenraiseexpendituresperadulthouseholdmemberslightlyfromages35-45,buttheadjustmentissmall.26 mates,butintroducesdistractingjumpsinlife-cycleconsumptionanddebtasconsumersrespondtoshort-termbudgetchanges.WeestimatetheinitialliquidfundstoincomeratioofthepopulationfromtheSCFforhouse-holdsage20-27.Liquidfundsareliquidnancialassets(savings,checking,moneymarket,stocks,bonds)plusavailablecredit(creditlimitminusdebt).Dividedbyonequarternormalincome,wecalculatethatthesehouseholdshaveanaverageof1.28quartersofnormalincomeinavailableliquidfunds.Whilethepopulationaverageincomefollowstheobservedlife-cyclepath,individualincomesvarybasedontheiridiosyncraticshocks.WeusetheestimatesoftheannualincomeprocessfromGourinchasandParker(2002),whichareupdatesofCarrollandSamwick(1997),calculatedfromthePanelStudyofIncomeDynamics.Weadjustthesevolatilitiesforquarterlydynamicssothatfourquarterlyshockscombinetoproducethesamevarianceasoneyearlyshock.Acloseapproximationisthatthequarterlytransitoryvariancemustbefourtimestheannualvariancesincequarterlyshocksaverageout,whilethequarterlypermanentvariancemustbeonefourththeyearlyvariancesincepermanentshocksstack.Weobservetwopricesdirectly,althoughthereislikelygreaterheterogeneityinthepricesactuallypaidthanweincorporateinthemodel.WesettheinterestrateondebtRb=14:11basedontheaveragerevolvinginterestrateovertheperiod,adjustedslightlyforthelackofexplicitdefaultriskinthemodel.11Sincethereisonlyonerisklessliquidassetotherthantheabilitytoborrow,theappropriaterateonsavingsisnotobvioussincewewouldliketocapture

33 thereturnsthatpeopleexpecttoreceiveonthe
thereturnsthatpeopleexpecttoreceiveontheirsavings.Wethereforesetthereturnonsavingsastheaveragereturnonanallbondportfoliofrom1926to2015ascalculatedbythemutualfundcompanyVanguardof5.4percent.Weadjustbothborrowingandsavingpricesforthegeometricaverageinationratefrom2000-2015of2.15percent.Intheexpectedgrowthoverthelife-cyclewealso 11WefollowFulford(2015)inusingtheEdelberg(2006)adjustmentfordefault.TheFederalReserveseriesG19(CommercialBankInterestRateonCreditCardPlansNSA)averageovertheperiodis14.73percent(theaveragecreditcardinterestratereportedintheSCFis14.22percent).BasedoncalculatingtheriskofdefaultfromthePSID,Edelberg(2006)calculatesthezerobankruptcyriskratewouldbe0.62percentagepointslower,asmalleradjustmentthaninAngeletosetal.(2001),whoadjustfordefaultby2percentagepoints.27 includeexpectedrealaggregategrowthof1.5%,theaveragecompoundedratefrom1947to2015fromtheBEA(2009chaineddollarsGDPpercapita).125.2IdenticationandestimationofthepaymentsproblemThestructureoftheconsumer'sproblemmeansthatthepaymentdecisionisonlyinuencedbytheconsumptiondecisionthroughwhetherornottheconsumerwasrevolvingasoftheperiodbefore.Wecanthusndthesolutiontothepaymentsproblemrst,andthenallowthesolutiontothepaymentproblemtoinuencetheconsumptionproblem.ThetwoparametersofthepaymentproblemareexactlyidentiedbytwomomentsweobserveintheDiaryofConsumerPaymentChoice:thefractionofconsumptionthatrevolversandconvenienceuserspayforwithacreditcard,alongwiththeobservablemarketrateonborrowingonacreditcard.Table2showsthefractionofallexpendituresoverathreedayperiodthatthenationallyrepre-sentativesampleofconsumersfromtheDairyofConsumerPaymentChoiceputonac

34 reditcard.Theaverageconsumerpaysfor17.2p
reditcard.Theaverageconsumerpaysfor17.2percentofconsumptionwithacreditcard.Revolverspayforslightlylessat15.6percentandconvenienceuserspayforslightlymoreat18.2percent.Dividingthepopulationbyagesuggeststhattheremaybemoresignicantdifferencesintheusagepatternsfortheyoung,butthoseinmiddleagehaveafairlystablepatternofcreditcarduse,withconve-nienceusersspendingslightlymorethanrevolversatmostages.Interestingly,bothrevolversandconvenienceusersover65tendtospendmoreonacreditcard.Perhapsthischangecomesfromashiftinthetypeofexpenditureafterretirement.Thedifferencebetweenrevolversandconvenienceusersthenexactlyidentiesthepayment 12Whileeachoftheseparametersisvolatileanddifferentagentsmayexperiencedifferentprices,thereisnoesti-mationuncertaintyaboutthemandsowedonotadjusttheMSMvariance-covariancematrixforthem.Putadifferentway:the“population”averageinterestrateisnotanestimateandsodoesnothaveastandarderror(itmaybeadrawfroma“super-population”process,butoureconomicagentsliveinthispopulation,andsotheonlyrelevantpriceistheonetheyface).Instead,theseparametersmaybemiss-specied,whichisamodelingproblem,notanestimationone.Economicagentsmayfaceuncertaintyabouttheirrateofreturn,ination,andeconomicgrowthwhichcanbemodeled,althoughwithinourmodelallagentstaketheseasgiven.Similarly,therearemultiplewaysofsavingwithdifferentratesofreturn,andasinglerateofreturnonlyimperfectlycapturesthisheterogeneity.WethankChrisCarrollforpointingoutthatevenifweremovetrendsfromlife-cycleproles,theeconomicdecisionoftheagentincludesexpectedaggregategrowth.28 modelasgure3illustrates.Sincerevolverslosetheoatonpurchases,theirpaymentsuseofacr

35 editcardisshifteddownbyanamountequaltoth
editcardisshifteddownbyanamountequaltothelossofoat.ThedifferenceintheoptimalfractiononacreditcardRandCidentiestheslopeandinterceptofthetwolines.WeuseasimpleDeltamethodexpansiontocalculatethestandarderrorsforthepaymentparameters0and1.AppendixCshowsthesecalculationsandTable2showstheestimatedcoefcientswithaninterestrateonborrowingof14.11percentadjustedforinationof2.15percent(seediscussioninsection5.1).Themodelthendirectlygivestheconveniencevalueofcreditcreditcards.Atcloseto12percentrealborrowingrate,theconveniencevalueofusingacreditcardforpaymentsoverothermethodsisworth0.319percentofconsumptiontoconvenienceusersand0.235percenttore-volvers.Thiscalculationistheconsumersurplusofcreditcardsasapaymentmechanismandsodoesnotdirectlycalculatewelfare.Theaverageconveniencedependsonthefractionofrevolversandconvenienceusers.Inaddition,thecalculationdoesnottakeintoaccountthecostsofoperatingthepaymentssystemortheproducersurplusfromadditionalsalesmadebecausesomepurchasesaremoreconvenient,orthegainstotheprocessors,networkoperators,andbanks.Nonetheless,a0.319percentmultipleofallconsumptionisasubstantialnumber,andsinceitisproducedfromonlythe18.2percentofconsumptionpaidforwithacreditcard,suggeststhatthevaluethatcon-sumersgetfromusingcreditcardsforthepurchasestheydousethemforisverylarge.Thevalueoftheintercept0,forexample,suggeststhatforthemostvaluablepurchases,usingacreditcardhasaconveniencevalueof4.1percentofallconsumptionforthesepurchases.Forcomparison,ifallconvenienceconsumersreceivedtheequivalentofonepercentcashbackontheirpurchaseswithcreditcards,theimpliedconsumersurplusis0.182percentofconsumption.Thiscalcu

36 lationlikelyoverstatesthedirectvalueofre
lationlikelyoverstatesthedirectvalueofrewardssincenotallcardsofferrewards,butsuggeststhatintheneighborhoodofhalfoftheconveniencevaluefromcreditcardscomesfromdirectrewardsandtheotherhalffromtheirvalueasaconvenientpaymentmechanism.29 5.3Identicationandestimationofthelife-cyclemodelUsingtherststageestimatesofthepaymentsproblemandtheotherparameters,wenextestimatethefulllife-cycleproblemandthendiscussthevariationthathelpsidentifythedifferentparam-eters.13Table3showsthemodelestimateswhilegure4showsthehowdebtandconsumptionvaryoverthelife-cycle.Sincethescaleofthetwopanelsofgure4isinlogs,theMSMises-sentiallyndingtheparameterssothatthesumofthesquareddifferencesbetweenthepredictedconsumptionanddebtlinesisassmallaspossible.Itisclearthatgiventheconstraintsoftheop-timizationmodel,themodelestimatescansuccessfullycapturethelife-cycleprolesofdebtandconsumption.Todoso,themodelsuggeststhatabouthalfthepopulation(fA)mustbeimpatientandalmostriskneutral.Thishalfofthepopulation,whichthegureandtablescallpopulationA,startslifewithasubstantialamountofdebt(A0)andhassubstantialrevolvingdebtthroughoutthelife-cycle.Inordertomatchtheamountofdebtandconsumption,theestimatessuggestthatthispopulationhasaslightlyhigherincomethantheaverage(A)andsoititisconsumingalmostallofitsincomeinanygivenquarter.Thispopulationthuslivesessentiallyhand-to-mouthovertheentirelife-cycle.Itreliesoncreditforallofitssmoothing,althoughitsaverageutilizationishighthroughmuchofthelife-cycle(thethirdpanelingure4),andsoitisnotkeepingmuchcreditforprecautionaryreasonseither.Theestimatessuggestthattheotherhalfofthepopulationisrelativelypatientandriskavers

37 e.Itsriskaversionislowerthanmanyestimate
e.Itsriskaversionislowerthanmanyestimateswhichareoftenaround2,butarehigherthantheestimatesinGourinchasandParker(2002).PopulationBistoopatienttoeverwanttoholdmuchdebtandquicklyrepaystheinitialdebtwhichtheestimatessuggestitholds(A0).Afteraboutage30,thispopulationonlyusescreditcardsforpayments.14Sincetheestimatessuggestthatthispopulationexpectstoreceivelittleincomeafterexpenses(B1)inlatelifeanditisrelatively 13Inrobustnesschecks,itappearsthatthequadraticfunctioninequation(4)hasmultiplelocalminima.Wediscusswhatappearstobethebestlocalminimumhere,butexpectthatinfutureversionswewillprovidegreaterevidencethatitisindeedthebestminimumbasedonstartingtheestimationatagridofstartingpoints.Wealsoplantocharacterizeotherminimaandhowtheyhelpunderstandidentication.14Theaddeddebtfromconvenienceuseofcreditcardsisonemonthworthofconsumption(1/3ofquarterlycon-sumption)timestheestimatedrateofconsumptiononacreditcardforaconvenienceuserfromtable2.30 patient,almosttheentirelife-cycleisspentacquiringassetsforoldage.Consumptionisthereforerelativelyatovertheentirelife-cyclesinceitisattemptingtosmoothoverthisfallinincomeafterexpenses.Sincethisisahighlynon-linearmodel,allmomentsaretypicallyusedtoidentifyallparame-ters,butitisusefultounderstandhowdifferentsourcesofvariationidentifytheparameters.Boththeconsumptionanddebtthatweobserveoverthelife-cycleareaveragesovertheentirepopula-tion,andsothemodelisidentiedoffoftheaverageofthetwopopulations.Theestimatesshowthatforthemodeltoproduceasmuchdebtasinthedata,approximatelyhalfofthepopulationmustberelativelyimpatientandnotmuchconcernedaboutdebt.Sincethispopulationlivesnearlyhand-to-mouth,itsconsumpt

38 ioncloselyfollowsincomeoverthelife-cycle
ioncloselyfollowsincomeoverthelife-cycle.Togetanaverageconsumptionproleinwhichconsumptionisbelowincomeformuchofthelife-cyclethereforerequirestheotherhalfofthepopulationtoberelativelypatientandriskaverse.Thispopulationistoopatienttokeepmuchdebtathighinterestrates,andpaysoffalmostallofcreditcarddebtithasearlyinlifebyitsmid20s.Mostofthelife-cycledynamicsofdebtthereforecomefromtheimpatientpopulation.Whilethelevelofconsumptionanddebtcomesfromtheaverage,thelife-cycleprolesarelargelydeterminedbyonlyoneofthepopulations.Sincethepatientpopulationcarriesalmostnodebt,theproleofcreditcarddebtlargelyidentiesthepreferencesoftheimpatientpopulation,theirinitialwealth,andtheirexpectedresidualincomelateinlife.Thediscountrateofthispopu-lationislargelyidentiedbytherealinterestoncreditcarddebt;ifthepopulationistoopatient,thenitwillnotaccumulateenoughdebt,tooimpatientanditwillacquiretolittledebt.Forthisimpatientpopulation,thehumpshapeofdebtcomesfromincreasesincreditlimitsearlyinlifewhichallowsittoincreasesitsdebts,andthefallinincomelateinlifewhichmakescarryingdebtmorepainful.Theriskaversionisidentiedbyhowmuchcreditthispopulationkeepsasabuffer.Themorepatientandriskaversepopulationcarrieslittledebtpastits20s,soitspreferencesarelargelyidentiedbytheconsumptionprole.TheimpatientpopulationAhasastronghumpinconsumptionasitfollowsincome.Fortheaverageconsumptionproletobebelowaverage31 income,thepatientpopulationBmusthavearelativelyatconsumptionprole.Itspreferencesforrisk( B),discountrate(B),andexpectedlatelifeincomeafterexpensesB)aredeterminedbythisshape.Thediscountrateisalsocloselyidentiedbytherateofret

39 urnonsavings.Theestimatedmodelmatchesoth
urnonsavings.Theestimatedmodelmatchesotherlife-cycleprolessuchastheutilizationandthewealthpathwellasshownintheremainingpanelsofgure4.Creditutilizationstartshighandslowlyfallsoverthelife-cycle.Theestimatessuggestthatutilizationisonaverageslightlylowerthanweobserveinthedata.Toexaminetheevolutionofwealth,whichmaybenegative,wetakethelogofwealthaftergivingeveryone$10,000,whichallowsustoconsiderthefulldistributioninasinglegraph.Themodelestimatespredictslightlylesswealthaccumulationoverthelife-cyclethantheSCF,butpredictsthesametrendincreaseandatteningafterage55.Themodelwasnotestimatedtomatchthesetwoproles,andsobeingabletosuccessfullypredicttheirevolutionsuggeststhatthemodeliscapturingimportantfacetsoflife-cycledecisionmaking.Theheterogeneityinpreferencesiskeyhere.GourinchasandParker(2002)estimateparameterstomatchtheconsumptionproleandvastlyunder-predictwealthaccumulation,whileCagetti(2003)estimatesparameterstomatchthewealthprole,butneedsahighdegreeofrisk-aversethathecannotcapturetheconsumptionprole.Themodelpredictsthatthefractionrevolvingwillbeapproximatelyconstantoverthelife-cycle,whiletheSCFsuggestsitshoulddeclinenearlycontinuously.15Thepopulationaveragefractionrevolvinginthemodelisapproximatelycorrect,butsincetheimpatientpopulationAisalwaysindebt,andthepatientpopulationBalmostneverrevolves,thefractionrevolvingdoesnotchangemuchoverthelife-cycle.Itispossiblethatadditionalheterogeneityacrossthepopulationcouldreplicatethisbehaviorbyallowingpartofthepopulationtobeontheedgebetweenrevolvingandnot,withthebalanceshiftingoverthelife-cycle.Theproblemwiththisapproachisthatthedifferencebetweenthesavingsinterestrateand

40 theborrowingrateissolargethatitisonlyfor
theborrowingrateissolargethatitisonlyfor 15SincetheSCFsometimeshastroublewithdebts(Zinman,2009),andisatahouseholdratherthanindividuallevel,thereisreasontoquestionwhethertheSCFfractionrevolvingisthebestbenchmark.OurestimatesfromtheDairyofConsumerPaymentsChoiceinappendixgureA-1suggestthatthethefractionrevolvingisapproximatelyconstantuntilage50,butthendeclinessteadily.Whilethisproleisclosertothemodelprediction,themodelisstillnotpredictingamovementoutofrevolvinglateinlife.32 veryparticularpreferencesthatsomeonewhoisnotwillingtoborrowlateinlifewouldbewillingtoborrowearlyinlife.Suchparticularpreferencesareverysensitivetochangesininterestrates,whichsuggeststhatcredituseissensitivetointerestrates.Thestabilityofutilizationin1despitelargechangesininterestratesovertheperiodsuggeststhattheanswermustlieelsewhere.Instead,itseemslikelythatthereissomesortofpreferenceheterogeneityoverthelife-cycle.Youngpeoplewhothinkthemselvesinvinciblemayhavelowriskaversion,sincetheyhavenotseenthebadconsequencesofpoorplanning,butbecomeincreasinglypatientandriskaverseastheyageandexperiencebadshocks.Thelasttwocolumnsoftable3showestimatesthatuseadifferentweightingmatrixandturnofftheendogenouspaymentchoicebymakingallconsumersgettherevolvingvaluefromcredituse.Optimalweightsgiveeachmomentaweightproportionaltohowwellitisestimated.Sincethedebtmomentsareestimatedfromlargesampleofadministrativedata,theestimationtendstoputmoreweightonminimizingthedebtresiduals.Analternativeweightingschemeistoallowoptimalweightingwithineachofthecreditanddebtmoments,butrenormalizesothattheaveragemomentfordebtandconsumptiongetsthesameweight.Thisweightmatrixproducesnearlyidenticalesti

41 mates.Similarly,theendogenouspaymentscho
mates.Similarly,theendogenouspaymentschoicedoesnotmuchaffecttheestimates.Thereasonisthatsincesofewpeopleswitchfromrevolvingtoconvenienceuse,thevalueanindividualgetsfromcreditcardconsumptionthisperiodisalmostalwaysthesameasnextperiod.Inmarginalutilitythevalueofconsumptiononacreditcarddoesnotaffectthetradeoffbetweentodayandthefuture.Itmakesnotrevolvingslightlymorevaluable,butsincehardlyanyoneisontheedge,thisvaluedoesnotmatter.Includingcreditcardsasapaymentmechanismisimportant,however.Thedebtsweobserveinthecreditbureaudataincludebothrevolvingandconvenienceuseandnotaccountingfortheconvenienceusewouldmaketheaveragedebttoohigh.5.4ModelpredictionsforthebusinesscycleandindividualsInthissection,wetakethemodelandestimatedparametersandaskhowwellitpredictsphenom-enaoutsidethelife-cyclethatitwasnotestimatedtomatch.Agoodwaytothinkoftheseresults33 isthattheyprovidebothanout-of-sampleexaminationofhowgoodthemodelestimatesareandwhetherthemodelcansuccessfullyexplainotherphenomena.Wetakethisexaminationasdistinctfromourdiscussionintheprevioussectionofwhetherthemodelcanreplicateotherlife-cyclemomentssuchasutilizationandwealthwhichhelpvalidatethemodel.Showingthatthemodelcanhelpunderstandindividualandbusinesscycledynamics,ontheotherhand,providesatestofthemodelpredictions.ToperformthistestwesimulatealargepopulationwiththesameageproleasinEquifaxfromage24to74.Weallowfortheconstantrealgrowthof1.5percenttheconsumersinthemodelassume,andadjustthedollarvaluesfortheaverageinationrate.Finally,tomimicthefallincreditlimitsthatstartedinthelastquarterin2008andwentthrough2009,weintroduceafallincreditof40percenttoonesixthofthepopulationforsixquarters.Thisparti

42 cularexperimentisthesimplestwaytogetthea
cularexperimentisthesimplestwaytogettheapproximately40percentdropincreditlimitsspreadovermorethanayearevidentingure1.Itisnotafullreplicationofthechangingenvironment.Inparticular,itdoesnotincludeafallinincomeoradeclineinthecondenceaboutfutureincomegrowth.Theindividualdynamicsofcreditutilizationfromthesimulationscloselymatchthedynamicsfromthecreditbureaudata.Intable1weshowedthat,oncewecontrolforxedunobservedhet-erogeneitywithxedeffects,shockstoutilizationdisappearquickly,with64.7percentofashocksurvivingeachquarter(thethirdcolumn).Thelastcolumnperformsexactlythesameregressiononthesimulateddata.Thesimulatedconsumersexperiencethelargeunexpectedfallincreditinover2009,andtheexpectedincreaseoverthelife-cycle,buttheonlyunexpectedcreditvolatilitythattheyfacecomesbecausecreditisproportionaltovolatilepermanentincome.Sincevolatilityinincomeismuchlessthanvolatilityincredit(Fulford,2015),theconsumersinthemodelfacelesscreditvolatilitythanactualconsumersoverthetimeperiod.Nonetheless,theirresponsetochangesincreditlimtsareveryclosetoactualconsumers;theestimatedmodelcapturesthedynamicsofcreditutilizationnearlyexactlywith68.3percentofashockpersistingtothenextquarter.Wenallyexaminethebusinesscycledynamicswithwhichwestartedthepaper.Thebot-tompanelofgure1showstheaggregateresponseofthesimulatedconsumerstothe40percent34 fallincreditintroducedoversixquarters.Creditincreasesatthesame1.5percentrealrateasincome,plus2.1percentforaverageination.Modelcreditgrowthisslightlyslowerthantheac-tualcreditgrowthovertheperiodsuggestingthatpeggingcredittoincomedoesnotfullycapturetheaggregategrowth.Sincewealreadycapturethemuchlargerlife-cyclech

43 anges,theremainingunexplainedaggregategr
anges,theremainingunexplainedaggregategrowthduringtheperiodisunimportantforindividualdecisions.Sinceconsumersexpectcreditgrowth,theirdebtsgrowatthesametime,andcreditutilizationisstabledespitethegrowthbeforeandafterthecrisisjustasinthedata.Thesimulatedpopulationhasaslightlylowercreditutilizationoverthelife-cycle(seegure4),andsoalowercreditutilizationinaggregate.Duringthecrisis,debtquicklyadjuststothefallincredit,andsoutilizationismuchsmootherthaneithercreditordebt.Astheindividualdynamicsshow,whileshocksattheindividualleveldisappearquicklyinboththemodelanddata,itstilltakesseveralquartersforconsumerstofullyadjusttheirdebtandsavingstoa40percentfallincredit.Theexcessivesmoothnessofutilizationinthecreditbureaudatasuggeststhattheremaybeadditionalfeaturesoftheperiodnotcapturedbythesimplesimulatedshock,sincethesimulateddatacloselymatchestheindividualresponsestoafallincredit.Thesimulationdoesnotincludeanydeclineinincomewhich,ifexpectedtobepermanent,wouldalsodecreasedebt.Thefallincreditwasalsomoreconcentratedamongsomeconsumers.Whileinitiallythedeclinewasconcentratedamongthosewithhighcreditscores,laterdeclinesweretothosewithlowercreditscores(Fulford,2015),whoseresponsestoafallinlimitsareevenfasterthanaverage(seeourworkingpaper(FulfordandSchuh,2015)formoreevidenceonheterogeneityofresponses).6ConclusionCreditismorevolatilethanincomeoverboththebusinesscycleandlife-cycle.Creditincreasesveryquicklyearlyinlife,yetsincedebtfollowstheseincreases,creditutilizationdecreasesonlyslowlyoverthelife-cycle.Creditutilizationisalsoremarkablystableacrossthebusinesscycle35 despiteanearly40percentdropfollowingthenancialcrisisin2008.Weshowthatthisstabili

44 tycomesfromindividualdecisionsasshocksto
tycomesfromindividualdecisionsasshockstocreditutilizationquicklydisappear.Weintroduceandestimateamodeloflife-cycleconsumption,saving,anddebtthatexplainsthisstability.Creditplaysaprecautionaryroleinthemodel,andsoconsumersmanageitjustastheywouldanybufferofsavings.Creditcardsarealsousedforpaymentsandthisformofdebtissmoothedlikeconsumption.Bothaspectspointstostablecredituse.Weestimatethepaymentsaspectusingthedifferingcreditcarduseforpaymentsofrevolversandconvenienceusersandndthatthepaymentsvalueofcreditcardsisworth0.319percentofconsumptionforconvenienceusers.Estimatingtherestofourmodeloffofthelife-cycleofdebtandconsumption,weshowthatitcanproducethelife-cycleprolesofconsumptionanddebt.Italsopredictsthestableutilizationoverthebusinesscycleandforindividuals.Whiletheestimatedmodeldoescapturetheslowfallinutilizationoverthelifecycle,itpredictsaconstantfractionofthepopulationrevolving,ratherthanasteadydeclineoverthelifecycle.Withfullyforwardlookingbutveryimpatientconsumers,life-cycleconcernsarenotsufcienttogetthosewillingtoborrowinearlylifetostopborrowingatalllateinlife.Instead,thedeclinesuggeststhatsomeadditionalformoflearningorpreferencechangesisneeded.Oneofthestartlingfactsofconsumernanceishowlittlethemedianhouseholdsaves,par-ticularlyinliquidform.Theincreasesincreditoverthelife-cyclehelptoexplainwhyconsumerssavesolittleearlyinlife.Inexpectation,thecreditlimitfortheaveragepersoninher20swillbemuchhighernextyear.Sincecreditisaformofwealthforprecautionarypurposes(Fulford,2013),theyoungareeffectivelybecomingwealthierthroughcreditincreases,reducingtheirneedtosave.Inmiddleage,ontheotherhand,manyhouseholdshavesubstantialdebt.Fo

45 rthesehouse-holds,savingshouldbemostlyab
rthesehouse-holds,savingshouldbemostlyaboutpayingdownpreviousdebt.Payingoffcreditcarddebthasarisklessreturnthataverages14percent,whichnootherassetclasscanmatch.Thelargelife-cyclevariationincreditanddebtsuggestswhytheaveragehouseholdhaslittleinpositiveassetsbeyondasmallemergencyfundandilliquidhousinguntilverylateinlife:itgenerallyhassubstantialandincreasingcredit,andcansavebypayingdowndebt.Nottakingthisvariationintoaccountleads36 toamisunderstandingofthenancialstatusofU.S.households.37 ReferencesAgarwal,Sumit,SouphalaChomsisengphet,NealeMahoney,andJohannesStroebel.2015.“DoBanksPassThroughCreditExpansions?TheMarginalProtabilityofConsumerLendingDur-ingtheGreatRecession.”Technicalreport,SSRN.Agarwal,Sumit,JohnC.Driscoll,XavierGabaix,andDavidLaibson.2007.“TheAgeofReason:FinancialDecisionsOvertheLifecycle.”WorkingPaper13191,NationalBureauofEconomicResearch.Angeletos,George-Marios,DavidLaibson,AndreaRepetto,JeremyTobacman,andStephenWeinberg.2001.“TheHyperbolicConsumptionModel:Calibration,Simulation,andEmpiri-calEvaluation.”TheJournalofEconomicPerspectives,15(3):pp.47–68.Athreya,KartikB.2008.“Default,insurance,anddebtoverthelife-cycle.”JournalofMonetaryEconomics,55(4):752–774.Attanasio,OrazioP.,JamesBanks,CostasMeghir,andGuglielmoWeber.1999.“HumpsandBumpsinLifetimeConsumption.”JournalofBusiness&EconomicStatistics,17(1):pp.22–35.Attanasio,OrazioP.,andMartinBrowning.1995.“ConsumptionovertheLifeCycleandovertheBusinessCycle.”TheAmericanEconomicReview,85(5):pp.1118–1137.Avery,RobertB.,PaulS.Calem,andGlennB.Canner.2004.“CreditReportAccuracyandAccessto

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leCosts.”TheAmericanEconomicReviewPapersandProceedings,99(2):424–429.Stepner,Michael.2013.“BINSCATTER:Statamoduletogeneratebinnedscatterplots.”StatisticalSoftwareComponents,BostonCollegeDepartmentofEconomics,October.Telyukova,IrinaA.2013.“HouseholdNeedforLiquidityandtheCreditCardDebtPuzzle.”Re-viewofEconomicStudies,80(3):1148–1177.Telyukova,IrinaA.,andRandallWright.2008.“AModelofMoneyandCredit,withApplicationtotheCreditCardDebtPuzzle.”ReviewofEconomicStudies,75:629–647.Zinman,Jonathan.2009.“Whereisthemissingcreditcarddebt?Cluesandimplications.”ReviewofIncomeandWealth,55(2):249–265.Zinman,Jonathan.2013.“ConsumerCredit:TooMuchorTooLittle(orJustRight)?”WorkingPaper19682,NationalBureauofEconomicResearch.Zinman,Jonathan.2015.“HouseholdDebt:Facts,Puzzles,Theories,andPolicies.”AnnualReviewofEconomics,7:251–276.40 Figure1:Creditcardlimits,debt,andutilization:2000–2014Observedlimits,debts,andutilization Modelpredictiongivenfallincreditlimits Notes:Theleftaxisshowstheaveragecreditcardlimits(topline)anddebt(bottomline).Notethelogscale.Therightaxisshowsmeancreditutilization(middleline)denedasthecreditcarddebt/creditcardlimitifthelimitisgreaterthanzero.Source:Authors'calculationsfromEquifax/NYFedCCP.41 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1Credit utilization 4000 5000 6000 8000 10000 14000Credit and debt ($ log scale) 2000q1 2005q1 2010q1 2015q1Date Mean credit card limitMean credit card debt Figure2:Creditcardlimits,debt,andcreditutilization Notes:Eachlinerepresentstheaveragecreditcardlimit,debt,andutilizationofonebirthcohort,1999–2014.Source:Aut

50 hors'calculationsfromEquifax/NYFedCCP.42
hors'calculationsfromEquifax/NYFedCCP.42 Credit Limit Credit Debt 1 5 10 20 50 100 redit card limit or debt, thousands (if positive) Credit Utilization 0 .2 .4 .6 .8Credit card debt / Credit Card limit 20 40 60 80Age Figure1:creditlimit,creditdebt,andcreditutilization Figure3:Marginalvalueofadditionalconsumptiononacreditcard Notes:Thisgureillustrateshowthemarginalvalueofadditionalconsumptiononacreditcardchangeswiththesharespentonacreditcard.Thetoplineisforconvenienceuserswhoputasharecofconsumptiononacreditcard.Thebottomlineforrevolversisshifteddownbytheamount�rb=24sincerevolverslosetheoatonpaymentsmadeusingcreditcards.TheythereforeputasmallershareontheircreditcardsR.43 Share of consumption on credit card π 0 1 Marginal value of additional consumption on a credit card ν 0 Slope - ν 1 π C ν 0 - r B /24 π R 0 Figure4:Consumptionanddebtoverthelife-cycle:modelestimatesEstimationmoments:DebtEstimationmoments:Consumption Estimationpredictions:UtilizationEstimationpredictions:Fractionrevolving Estimationpredictions:WealthpathEstimationpredictions:Varianceofdebt Notes:44 Age 30 40 50 60 70 Variance 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Model variance log debt Equifax variance credit card debt Age 30 40 50 60 70 Credit card debt and limit ($, log scale) 200 500 1,000 2,000 5,000 10,000 20,000 Model debt Equifax credit card debt Equifax credit card limit Population A debt Population B debt Age 30 40 50 60 70 Consumption and income ($ quarterly, log scale) 3000 5000 8000 10000 Model consumption CEX consumption CEX income and model income Population A consumption Population B consumption Age 30 40 50 60 70 0 0.2 0.4 0.6

51 0.8 1 Model utilization Equifax utiliza
0.8 1 Model utilization Equifax utilization Population A Population B Age 30 40 50 60 70 Fraction 0 0.2 0.4 0.6 0.8 1 Model fraction revolving SCF fraction revolving Population A Population B Age 30 40 50 60 70 log wealth ($, log scale) 5,000 10,000 20,000 50,000 100,000 Model mean log wealth+10,0000 SCF per person mean log wealth+10,000 Table1:Creditutilization Equifax/NYFedCCPModel Creditutilizationt Creditutilizationt�10.874***0.868***0.647***0.647***0.514***0.683***(0.000876)(0.000892)(0.00131)(0.00139)(0.00441)(0.000329)Creditutilizationt�20.0156***(0.000643)Creditutilt�1Age0.00314***(9.93e-05)Constant0.0479***(0.000461)Observations347,642347,642347,642332,696347,6424,752,000R-squared0.7410.7430.4290.4440.4310.489FixedeffectsNoNoYesYesYesYesAgeandyeareffectsNoYesYesYesYesYesNumberofaccounts10,45110,10310,451102,564Frac.VariancefromFE0.4770.4670.4980.292 Notes:Thesampleincludes0creditutilizationbutexcludesindividualquarterswheretheutilizationisundenedsincethelimitiszeroandwhenutilizationisgreaterthan5(averysmallfraction,seedistributionsofutilizationingure).Allcolumnsincludeageandyeareffects,withageeffectsnormalizedtohavezerotrendwhenxedeffectsareincluded.Source:Authors'calculationsfromEquifax/NYFedCCP. 45 Table2:Fractionofconsumptiononacreditcardandvalueforpayments FractiononStd.Std.Creditcarderrordev. Allconsumers0.1720.00820.310Allrevolvers0.1560.01300.283Allconvenienceusers0.1820.01050.324Revolvers Age250.2340.07600.30125Age350.1570.02870.29935Age450.1340.03440.28245Age550.1420.02130.14255Age650.1440.02070.144Age&#x-250;=650.2020.03990.312ConvenienceUsers Age250.0650.02330.20925Age350.1690.02550.

52 16935Age450.1550.02280.29145Age550.1730.
16935Age450.1550.02280.29145Age550.1730.02150.32755Age650.2160.02420.345Age&#x-250;=650.2430.02710.350 ModelEstimates Level00.0350.0216Slope10.1940.1259 ImpliedvalueofCreditCarduse(inpercentofconsumption) Revolvers0.235%0.1512Convenienceusers0.319%0.0962 Notes:Author'scalculationsfromtheDiaryofConsumerPaymentChoice.Thestandarderrorsarecalculatedbybootstrapping.46 Table3:Modelestimates PopulationACRRA A0.1540.1640.176(0.015)(0.018)(0.019)DiscountA0.8890.8890.889(0.000)(0.000)(0.000)InitialwealthA0-5.320-2.725-2.734(1.031)(0.091)(0.094)Latelifeinc.A10.5150.5220.545(0.009)(0.009)(0.008)PopulationBCRRA A0.9360.8680.958(0.202)(0.217)(0.249)DiscountB0.9610.9630.962(0.003)(0.003)(0.004)InitialwealthB0-0.072-0.090-0.120(0.142)(0.138)(0.155)Latelifeinc.B10.0700.0790.080(0.074)(0.068)(0.072)ShareAfA0.5570.5570.557(0.001)(0.002)(0.002)Inc.mult.AA1.1611.1931.206(0.025)(0.027)(0.027)WeightsOptimalEqualOptimalEndogenouspaymentsYesYesNo Notes:Standarderrorsinparentheses.Equalweightsareoptimalwithineachofthelife-cycledebtandconsumptionmoments,butequalweightforeachsetofmoments.Optimalweightsaretheinverseofthevarianceofeachindividualmoment.Forabriefdescriptionofeachmomentandtheestimationmethodseethebeginningofsection5.47 AppendixForOnlinePublicationAChangesincreditutilization:non-parametricevidenceFigureA-7showsconditionalmeanscatterplotsofcreditutilizationinonequarteragainstcreditutilizationinthenextquarter,inthenextyear,andintwoyears.Thetoprowshowsthemeaninthefuture,conditionalonlyonhavingtheutilizationshownonthex-axisinthatquarter.Thebottomrowinsteadtakesthewithintransformationandallowsforageandyear

53 effects.Itthereforeshowshowfarfromtheind
effects.Itthereforeshowshowfarfromtheindividual'saveragecreditutilizationsheisinthenextquarter,conditionalondifferingfromheraverageutilizationbytheamountonthex-axisthisquarter.Inotherwords,ifanindividualis10percentagepointsabovehertypicalutilizationinonequarter,howfarwillshebeonaverageinthenextquarter,nextyear,andintwoyears?Eachdotcontainsanequalportionofthesample.FigureA-7thuscapturestherelationshipbetweenutilizationtodayandinthefuturewithoutimposinganyparametricassumptions.Eachpanelalsoshowsthebesttlinefortheconditionalmeansandtheestimatedcoefcients.Thetoppanelsshowthatcreditutilizationishighlypersistentanddoesnottendtozeroonaverage.Creditutilizationthisquarteristypicallyveryclosetocreditutilizationnextquarter,sincetheconditionalmeansaretypicallyveryclosetothe45-degreeline.Forexample,onaverageifapersonisusing40percentofhercreditthisquarter,shewillbeusingabout40percentofhercreditnextquarter.Lookingclosely,however,averagecreditutilizationishighernextquarterforthoseusinglessthan20percentoftheircredit,andlowerforthoseusingmorethan80percentoftheircredit.Thebesttlinethroughtheconditionalmeanssuggeststhatcreditutilizationisnottrendingtozero.Instead,thelong-termsteady-stateutilizationis0.39.16Thesameconclusionisevidentfromtheconditionalchangescomparingutilizationthisquartertoayearfromnowandto 16Sincetheconditionalexpectationofutilizationnextquartergiventhisquarterisut+1=0:041+0:896utthesteady-stateutilizationis0.39=0.041/(1-0.896).A-1 twoyearsfromnow.Thoseconsumersusinglessthanapproximately40percentoftheiravailablecreditthisquarterareusingmoreoftheircreditinoneyearandintwoyears,thoseusingmorethan40percentoftheircreditareus

54 inglessoftheircreditonaveragewithinoneye
inglessoftheircreditonaveragewithinoneyearandtwoyears.Thesteady-statecreditutilizationisaround40percent(evidentbyndingwheretheconditionalexpectationfunctioncrossesthe45degreeline),althoughthemovementtowardthesteadystateisfairlyslow.Onaverageindividualsdonottrendtozeroutilization,nortousingalloftheircredit.Con-ditionalonusingzerocreditthisquarter,creditutilizationisnearly5percentwithinonequarterandnearly8percentinayear.Ontheotherhand,theaveragepersonusingallofhercreditinonequarterisusinglessthan90percentofitinayear.ThesecondrowofgureA-7allowsindividualstoreturntotheirownmeanandaddssub-stantialnuance.Thereasoncreditutilizationissopersistentinthetoprowisthatitappearsthatindividualshavetheirownmeantowhichtheyactuallyreturnquiterapidly.Thespeedofthereturnisevidentfromtheslopesofthelines.Onlytwo-thirdsofashocktoutilizationremainsafteronequarterand13percentremainsaftertwoyears.Evenifindividualsreturnveryrapidlytotheirownmeans,itisimportanttonotethatthosemeansarenotzero.CreditutilizationispersistentinthetoprowofgureA-7becauseindividualsaretypicallyquiteclosetotheirownmeancreditutilization.Sincecreditutilizationistheratioofdebtandcredit,thestabilityofcreditutilizationimpliesthatanindividualwithanincreaseincredithasincreasedherdebtsby33percentoftheincreaseincreditwithinonequarter,and87percentoftheincreaseincreditintwoyears.BBequestsInourbaseestimation,wegiveconsumerswhodiewithpositiveresourcessomeextrautilityfrombequests.Wehavealsoexaminedtheconsequencesofallowingconsumerstoplacenovalueonbequests.Sinceourestimationusesdebtandconsumptionmoments,bequestsarenotanimportantA-2 factorintheestimationsincethemedianpersonisleavingveryl

55 ittleinbequests.Wehavefoundthatanyreason
ittleinbequests.Wehavefoundthatanyreasonableapproachtointroducebequestsmakesonlyaveryminordifferenceinourestimates.Thisappendixoutlinesoneexibleapproach,anddiscusseshowtheparameterestimateschangewhennotincludingbequests.Thedifcultywithintroducingbequestsinamodelwithbothdebtandsavingsisthatwhilepositiveassetslefttoheirsmayprovideutility,unsecureddebtsarenotpassedon.Sinceitispossibletodiewithnegativeassets,zerobequestsarepossible.Simplyincludingbequestsinthesub-utilityfunctionwillthereforeproducenegativeinnitevaluefromleavingnobequest,whichconsumerswillcounterfactuallyacttoavoidbyneverbeingindebt.Instead,wemodelthebequestmotiveastheconsumerconsideringthemarginalutilityabequesttoherheirswillbringthemontopoftheirownincomes.TheconsumerdieswithcertaintyatageT,andwithsomeprobabilitybeforethat.Tovaluebequestsreasonably,wetakepositiveassetsasprovidinganannuityvalueontopofsomebaseincomethatheirsalreadygetthatisafunctionoftheconsumer'spermanentincome,andthebequestvalueasthepresentvalueoveralifetimeoftheseadditionalassets.Thevalueofbequestsisthen:S(At)=0@~TXs=0s(Pt+rB(1�IRt+1)At)1� 1� 1A;wheretheconsumeronlyleavesabequestifshehaspositiveassetsattheendoftheperiod(IRt+1is0ifAt0).Theparameterdeterminesthemarginalutilityofbequestsandcanbethoughtofashowmuchmoreorlessincomechildrenhavethanparents.Thisstructureisimportantsincepeoplecandecidetodieindebt,andsoleavenothing.Ifchildrenaremuchbetteroffthenparents,thenthemarginalutilityofleavingabequestislow,andparentswillgenerallyonlyleavealargebequestaccidentally.Inourbaselineestimationwithbequests,weset=2whichimpliesthatattheaverageageofdeathchildren

56 areearningtwiceasmuchastheirelderlyparen
areearningtwiceasmuchastheirelderlyparents,butwehavealsoestimated,andfoundthatwhilevaluesaround2arereasonable,isverypoorlyidentiedgiventhemomentsweuse.Evenverylateinlife,A-3 thelikelihoodofdeathinthenextyearisfairlylow,andsotheexactformofthebequestmotive,includingremovingit,doesnotaffectdecisionsmeaningfullyformostofthedistribution.Thereasontousenitechildren'slifetimeistoallowforpossiblyverypatientparentswithclosetoorgreaterthan1.CIdenticationofthepaymentsmodelThissectionshowshowtoidentifythepaymentmodelparametersandstandarderrorsfromob-servablemoments.Itthencalculatestheconsumersurplusanditsstandarderrors.Weobserve:R=1 NNXi=1i;tjIRi;t�1=1theaverageexpendituresbyrevolversonacreditcard,andsimilarlyCtheaverageforconve-nienceusers.DenoteourestimatesofthestandarderrorsofthesemeansasRandC.ThentheinterceptfortheaverageconvenienceuserisjustC=0=1andforarevolveritisR=(0�rB=24)=1whererBistheAPRinterestchargedonpayments,whichhaveanaveragedailybalanceofhalfofthemonthsconsumption.Solvingfor0and1gives:1=rB=24 C�Rand0=C1=(rB=24)C C�RA-4 .Thepresenceofadifferenceoftworandomvariableswhosesupportsmayoverlapinthede-nominatorofthetransformedvariablesmakescalculatingtheirvariancespotentiallytricky.SinceC�Rismaybecloseto0,then1and0maybeverylarge,whichisadifferentwayofsayingthatthemodelisnotidentiedifthereisnotadifferenceintheaveragebehaviorofconvenienceusersandrevolvers.Wecalculatethestandarderrorsofthetransformedvariablesusingthedeltamethod,whichavoidsthisissuebyexaminingo

57 nlysmallchangesaroundtheoptimum,andsodoe
nlysmallchangesaroundtheoptimum,andsodoesnotconsiderthehighlynon-linearincreasearoundC�R=0.ForsmallchangesCandRaroundCandR:1(rB=24)1 C�R�C�R (C�R)2:SinceRandCareindependent,thevarianceof1isapproximately:Var[1]rB=24 (C�R)22(2C+2R):Takingthesameexpansionfor0includingthecovarianceofthenumeratoranddenominator:Var[0]CrB=24 (C�R)22 2C+R C22R!:Finally,thetotaladditionalconveniencevalueofusingacreditcardoverthealternativesforaconvenienceuserisjusttheareaunderthecurve:C=(C;IRi;t�1=0)=(C0)=2=(rB=48)(C)2 C�Randforarevolversis:R=(R;IRi;t�1=1)=R 2(rB=24)C C�R�rB 24:A-5 TakinganexpansionaroundCandRyields:Var[C]rB 482 2C C�R�C C�R2!22C+C C�R22RandVar[R]rB 482C C�R�CR (C�R)222C+C C�R�CR (C�R)2�122R:A-6 FigureA-1:Fractionofconvenienceusersofcreditcards,conditionalonuse Notes:Eachdotshowsthefractioninthatveyearagegroup(labeledbytheyoungestmember)whodidnotrevolvecreditcarddebtfrommonthtomonth,`conditionalonusingtheircreditcard.95percentcondenceintervalsinbars.Source:Authors'calculationsfromthe2008–2014SurveysofConsumerPaymentChoice.FigureA-2:Consumptionfunctions

58 overthelife-cyclewithborrowing Notes:Sho
overthelife-cyclewithborrowing Notes:Showstheconsumptionfunctionforaconsumeratdifferentpointsofthelife-cycle.Atwtheconsumernolongerconsumesallcash-at-hand(whichincludesallavailablecredit).AtwBtheconsumernolongerborrows,butdoesnotsaveanythingbetweenwBandwFsincetheinterestrateforborrowingishigherthantherateforsaving.AfterwF;theconsumersaves(atthelowerrateofreturn).TheparameterstoproducethesefunctionsareasavingsrateofreturnR=1:03andborrowingrateRB=1:14,acoefcientofrelativeriskaversionof =2,adiscountparameter=0:9andotherparametersandretirementfunctionsmatchingGourinchasandParker(2002).A-7 .3 .4 .5 .6 .7Fraction Non-Revolving 20 30 40 50 60 70Age Group Cash-at-hand w t 0 0.5 1 1.5 2 2.5 3 Consumption c t 0 0.2 0.4 0.6 0.8 1 1.2 A g ! 6 A g 3 $ A g 4 $ A g $ $ A g 6 $ (w * , c * ) (w B , c B ) (w F , c F ) FigureA-3:Creditcardadoptionanduseforpaymentsbyageandyear(A)Creditcardadoptionrate (B)Shareofpaymentsusingcreditcards Source:Authors'calculationsfromthe2013SurveyofConsumerPaymentChoice.A-8 FigureA-4:Creditcardlimit,debt,andcreditutilizationdistributionsbyage(A)CreditCardLimits (B)CreditCardDebt (C)Creditutilization Notes:Eachlineisthepercentileofcreditlimitatthatage,conditionalonhavingapositivecreditlimitonalogscale.Forexample,the90thpercentilelineshowsthat10percentofthepopulation(withapositivecreditlimit)havealimitlargerthanthatline.Source:Authors'calculationsfromEquifax/NYFedCCP.A-9 .1 .2 .5 1 5 10 20 50 100credit limit by pctiles '000s (1, 10, 25, 50, 75, 90, 99) 20 40 60 80Age .2 .5 1 5 10 20 50 100 redit limit ptiles 000s 20 40 60 80Age Figure8:Percentilesoflogcreditcardlimitbycohort(1920,1930,1940,1950,

59 1960,1970,1980). Figure9:Percentilesoflo
1960,1970,1980). Figure9:Percentilesoflogcreditlimitbyage .8 1 1.2 1.4 1.6standard deviation of log credit card limit (if positive) 20 40 60 80Age .1 .2 .5 1 5 10 20 50 100 redit debt ptiles, 000s 20 40 60 80Age Figure10:Percentilesoflogcreditdebtbyage 1 1.5 2 2.5standard deviation of log credit card debt (if positive) 20 40 60 80Age 0 .5 1 1.5 redit utilization percentiles 20 40 60 80Age .1 .2 .3 .4 .5standard deviation of credit card utilization 20 40 60 80Age FigureA-5:Creditcardlimits,debt,andutilization:ageandyeareffectsCreditcardlimits` Creditcarddebt Creditcardutilization Notes:Eachlineshowstheestimatedageoryeareffectsfromequation(1).Notethedifferentscales.Source:Authors'calculationsfromEquifax/NYFedCCP.A-10 0 .5 1 1.5 2 2.5Estimated effect 20 40 60 80Age -.1 0 .1 .2 .3Estimated effect 2000 2005 2010 2015Year 0 .5 1 1.5 2Estimated effect 20 40 60 80Age -.05 0 .05 .1 .15 .2Estimated effect 2000 2005 2010 2015Year -.4 -.3 -.2 -.1 0Estimated effect 20 40 60 80Age -.2 -.1 0 .1 .2Estimated effect 2000 2005 2010 2015Year FigureA-6:FractionwithpositivecreditcardlimitanddebtbycohortandageFractionwithpositivelimitFractionwithpositivedebt Notes:Eachlinerepresentsthefractionwithpositivecreditcardlimitsordebtofonebirthcohort,1999–2014.Source:Authors'calculationsfromEquifax/NYFedCCP. A-11 .2 .4 .6 .8 1Have positive credit card limit 20 40 60 80Age .2 .4 .6 .8Have positive credit card debt 20 40 60 80Age FigureA-7:Changesincreditutilizationinonequarter,oneyear,andtwoyears Notes:Eachpointinthetoprowshowsthemeancreditutilizationinthefutureconditionalonbeinginthebinwithameancreditutilizationonx-axistoday.Thebottomrowshowstheconditi

60 onalrelationshipbetweendeviationsfromthe
onalrelationshipbetweendeviationsfromtheindividualmeanutilizationovertheentiresample,adjustingforageandyear.Source:Authors'calculationsfromEquifax/NYFedCCPusingtheprogrambinscatter(Stepner,2013). A-12 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 1.1Credit utilization inone quarter 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 1.1Credit utilization today E[y|x] = 0.041 + 0.896x 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 1.1Credit utilization inone year 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 1.1Credit utilization today E[y|x] = 0.077 + 0.786x 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 1.1Credit utilization intwo years 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 1.1Credit utilization today E[y|x] = 0.100 + 0.698x -.5 -.4 -.3 -.2 -.1 0 .1 .2 .3 .4 .5Difference in credit utilization inone quarter -.5 -.4 -.3 -.2 -.1 0 .1 .2 .3 .4 .5Difference in credit utilizationfrom own meanE[y|x] = 0.666x -.5 -.4 -.3 -.2 -.1 0 .1 .2 .3 .4 .5Difference in credit utilization inone year -.5 -.4 -.3 -.2 -.1 0 .1 .2 .3 .4 .5Difference in credit utilizationfrom own meanE[y|x] = 0.339x -.5 -.4 -.3 -.2 -.1 0 .1 .2 .3 .4 .5Difference in credit utilization intwo years -.5 -.4 -.3 -.2 -.1 0 .1 .2 .3 .4 .5Difference in credit utilizationfrom own meanE[y|x] = 0.128x FigureA-8:Consumptionandincomeoverthelife-cyclefromtheConsumerExpenditureSurvey Notes:ShowstheaverageconsumptionandincomeateachagefromtheConsumerExpenditureSurvey,poolingallsurveysfrom2000-2014.Consumptionisthetotalhouseholdexpendituresdividedbythenumberofadults.Adjustedconsumptionremovestheestimatedeffectofchildren.Incomeisafter-taxincome,anditssmoothedversionisbasedonaquinticfromage24to81.SincetheCEXpoolsincomeandconsumptionafterage81(or83inlateryears),ages81onth