美联储-收入冲击及其向消费的传导（英）

FinanceandEconomicsDiscussionSeries

FederalReserveBoard,Washington,D.C.

ISSN1936-2854(Print)

ISSN2767-3898(Online)

IncomeShocksandTheirTransmissionintoConsumption

EdmundCrawleyandAlexandrosTheloudis

2024-038

Pleasecitethispaperas:

Crawley,Edmund,andAlexandrosTheloudis(2024).“IncomeShocksandTheirTransmissionintoConsumption,”FinanceandEconomicsDiscussionSeries2024-038.Washington:BoardofGovernorsoftheFederalReserveSystem,

/10.17016/FEDS.2024.038

.

NOTE:StafworkingpapersintheFinanceandEconomicsDiscussionSeries(FEDS)arepreliminarymaterialscirculatedtostimulatediscussionandcriticalcomment.TheanalysisandconclusionssetfortharethoseoftheauthorsanddonotindicateconcurrencebyothermembersoftheresearchstafortheBoardofGovernors.ReferencesinpublicationstotheFinanceandEconomicsDiscussionSeries(otherthanacknowledgement)shouldbeclearedwiththeauthor(s)toprotectthetentativecharacterofthesepapers.

1

IncomeShocks

andTheirTransmissionintoConsumption*

EdmundCrawleyAlexandrosTheloudisMay29,2024

Measuringhowhouseholdconsumptionrespondstoincomeshocksisimportantforun-derstandinghowfamiliescopewithadverseevents,fordesigninggovernmentinsuranceorotherincomesupportpolicies,andforunderstandingthetransmissionofbusinesscyclesandmonetarypolicy.Itisalsoimportantforevaluatingtheeffectsoffiscalorlabormar-ketreformsonconsumerwelfareandforexaminingthewaythesereformsmayaffectthemacroeconomygiventhatconsumptionisalargeshareofgrossdomesticproduct.

Thispaperreviewstheeconomicsliteratureof,primarily,thepast20yearsthatstud-iesthelinkbetweenincomeshocksandconsumptionfluctuationsatthehouseholdlevel.Weidentifythreebroadapproachesthroughwhichresearchersestimatetheconsumptionresponsetoincomeshocks:(1)structuralmethodsinwhichafullyorpartiallyspecifiedmodelhelpsidentifytheconsumptionresponsetoincomeshocksfromthedata,(2)naturalexperimentsinwhichtheconsumptionresponseofonegroupthatreceivesanincomeshockiscomparedwithanothergroupthatdoesnot,and(3)elicitationsurveysinwhichconsumersareaskedhowtheyexpecttoreacttovarioushypotheticalevents.Noneoftheseapproachesareexclusivetoasinglefieldwithineconomics;studiesthatuseanyofthesemethodsareordinarilyclassified,dependingontheirspecificfocus,inmacroeconomics,laboreconomics,orpublicfinance—tonameonlyafewfields.

Ouraiminthisshortpaperistosurveythisincreasinglybusyliteratureandprovideanaccessiblesummaryofthevariousestimatesoftheconsumptionresponsetoincomeshocks.Weconcentrateonthesimilaritiesanddifferencesbetweenthevariousstudies,inparticularwithrespecttothemethod,data,consumptionnotion,andtypeofincomeshockanalyzed,

*Crawley:FederalReserveBoard;email:

edmund.s.crawley@.

Theloudis:DepartmentofEcono-metrics&OR,TilburgUniversity;email:

a.theloudis@.

ViewpointsandconclusionsstatedinthispaperaretheresponsibilityoftheauthorsaloneanddonotnecessarilyreflecttheviewpointsoftheFederalReserveBoard.Thisreviewwillappearintheforthcoming“ElgarEncyclopediaofConsumption,”editedbyJos´eM.LabeagaandJos´eAlbertoMolina.

2

andthusalsowithrespecttothetypeofconsumptionresponseeachworkidentifies.Ourfocusisonresponsestoshocks,thatis,unanticipatedincomechanges.

JappelliandPistaferri

(2010)reviewtheearlierevidenceonresponsesto

anticipatedincomechanges.

Thesurveyproceedsasfollows.Section

1

introducesabrieftheoreticalframeworkthathelpsfixideasforthesubsequentdiscussion.Thenextsectionsaredevotedtothedifferentapproachestotheestimationoftheconsumptionresponse.Section

2

surveysthestudiesthatemploystructuralmethods,section

3

reviewstheevidencefromnaturalexperiments,andsection

4

focusesontheelicitationsurveys.Section

5

concludes.Twotablessummarizesomeoftheresults:Table

1

providesasummaryofestimatesfromstructuralmodels,whileTable

2

providesasummaryofestimatesfromnaturalexperimentsandelicitationsurveys.

1TheoreticalBackground

AhouseholdichoosesconsumptionCit,savingsAit+1,andperhapsotherbehaviorscapturedinLit(possiblyavector)—forexample,laborsupply,tomaximizeitsexpectedlifetimeutility

T

forwhichweassumeseparabilityovertimeandgeometricdiscounting.1ExpectationsaboutfuturestatesoftheworldarecapturedbyE0.TheutilityfunctiondependsonconsumptionandonLit,whichmaybeachoicevariable(forexample,endogenouslaborsupply)ortakenasgiven(forexample,exogenouslaborsupply).Utilityissubscriptedbyitoreflectpreferenceheterogeneityacrosshouseholds.Wehaveemployedafinite-horizonsettingwithTasitsterminalperiod,whichisnaturalgivenourfocusonhouseholds.2

Theproblemissubjecttothesequentialbudgetconstraint

(1+r)Ait+Ti(Yi;Lit)=Cit+Ait+1,(2)

whichlinksresourcesovertimeundertheassumptionthatconsumerscanborrowandsave

ataninterestrater.3TimapsgrossincomebeforetaxesandtransfersYitodisposable

householdincomeYi.Yimaybeavector;forinstance,withtwoearnersinthehousehold,

Yi=(Yi1,Yi2)′,whereYejindicatestheearningsofmemberj.Timaydependonchoices

1Strictly,thehouseholdchoosesaplanforCit,Ait+1,andLitcontingentonfuturestatesoftheworld,including,forexample,futurestatesofincome.Wedonotexplicitlyshowthecontingentstatestoeasethenotation.

2Theextensiontoinfinitehorizonistrivialandmostlyinconsequential(

JappelliandPistaferri,

2010)

.

3Theextensiontomultipleorriskyassetsisalsostraightforward.

3

Lit;forexample,withendogenouslaborsupply,theprimitivesourceofgrossincomeisthe

hourlywage—thatis,Yi=(Yi1,Yi2)′,whereYwjindicatesthewageofmemberj.Ti

issubscriptedbyitoreflectheterogeneityintaxes,welfarebenefits,contingenttransfers,orexternalsourcesofincome(oftencalledexternalinsurance).TheremaybeaborrowingconstraintinsomeperiodssuchthatAit≥Bit,whereBitdenotestheapplicableborrowinglimit.Finally,thereisaterminalconditiononAiT+1,whichreflectsthathouseholdsrundowntheirassetsbeforedeathorbequeaththemtotheiroffspring.

Therearemanyalternativespecificationsfortheprocessthatgovernsincome;ageneralformulationis

Yi=fik(Xit,Yi−1,v,u,...),(3)

wherek={d,g,e1,e2,w1,w2,...}indicatesthetypeofincomeconsidered—thatis,dispos-able,gross,andsoon.fikreflectsthepreciseprocess,whichallowsforheterogeneityacrosshouseholdsanddependsonobservablesXit(forexample,age,time,andeducation),past

incomeYi−1,andidiosyncraticshocksvandutologincome,suchaspermanentandtran-

sitoryshocks.Itmayalsodependonolderincome,othershocks(includingaggregateones),andpastshocks,dependingonthespecificprocessthatisbeingconsidered.

Meghirand

Pistaferri

(2011)offerareviewofthevastincomedynamicsliteratureanddiscusspopular

cases.4

Solving(

1

)subjectto(

2

),theborrowingconstraint,andtheterminalconditionyieldsaconsumptionpolicyrulewhoseexactformulationdependsonthepreferencespecification,theincomeprocess(

3

),thetightnessoftheborrowingconstraint,andthemarketenvironmentinwhichthehouseholdoperates—forexample,theextenttowhichithasaccesstocontingenttransfersorexternalinsurance.Ageneralformulationfortheconsumptionruleis

Cit=gi(Ait,Xit,Yi−1,v,u,...),(4)

whichallowsforheterogeneityacrosshouseholds(reflecting,amongotherthings,heterogene-ityinUi)andgenerallydependsonassetsandthevariouscomponentsofincome.Policyrule

(4)subsumesseveralpopularsettingsintheliterature,and

JappelliandPistaferri

(2010)

offerspecificexamples.Itisnonethelessnonexhaustiveofallpossiblesettings.Forinstance,durablegoodsnecessitateaccountingfortheirstockandpossibleadjustmentcosts.Herewesimplysee(

4

)asageneralorganizationaldeviceratherthanthesolutiontoanygivenmodel.

Interestliesinhowandbyhowmuchtheincomeshocksv,u,andsoon,affectconsump-

4Therearemultiplegeneralizationsof(

3

).Forexample,onemayallowfiktodependontime(age),thusenablingtheeffectofshockstobetimevarying.Weview(

3

)asasimpleorganizingdeviceratherthanasanexhaustiverepresentationofeverypossibleincomeprocess.

4

tion.Tomeasurethisconsumptionresponse,theliteraturefocusesontwomainparameters:themarginalpropensitytoconsume(MPC),broadlydefinedasthederivativeofconsump-

tionwithrespecttoincome—thatis,dCit/dYi—andthepass-throughrate,broadlydefined

asthederivativeofconsumptiongrowthwithrespecttotheshock—thatis,d∆lnCit/dv.

Wenowturntothethreebroadapproachestoestimatingtheseparameters.

2StructuralMethods

Earlypapers.Aseriesofinfluentialpapersinthe1980sand‘90stestthepredictionsofthepermanentincomehypothesisandthecomplete-marketsmodel,thethenbenchmarkmodelsintheliterature.Thesetestsaredonethroughformingappropriatehypothesesonthelinkbetweenconsumptionandincomefluctuationsthatemanatefromthesemodels.WhilethisearlyworkdoesnotstrictlymeasureMPCsorpass-throughrates,itprovidesmotivatingevidenceforthesubsequentworkthatexplicitlymeasurestheconsumptionresponsetoshocks.

HallandMishkin

(1982),oneofthefirststudies,investigatethesensitivityoffoodcon

-sumptiontoincomeusingmicrodatafromthePanelStudyofIncomeDynamics(PSID).Consumptionvariesmorecloselywithpermanentthanwithtransitoryshocks,whichisanimplicationofthepermanentincomehypothesisinwhichhouseholdssmoothconsumptionthroughself-insurance(savingandborrowing).Yet,thesensitivityofconsumptiontotran-sitoryincomeismuchstrongerthantheorypredicts,whichleadstorejectionofthemodel.5

Thepermanentincomehypothesispostulatesthatinequalityinconsumptiongrowsoverthelifecycle.Thisresultisthemotivatingobservationfor

DeatonandPaxson

(

1994

)who,usingmicrodatafrommultiplecountries,confirmthatthevarianceofconsumption(andincome)growswithage.Whiletheycannotrejectthepermanentincomehypothesis,theyadmitthattheevidenceisalsoconsistentwithothermodelsofintertemporalchoice,suchasmodelsthatpermitsomeexternalinsurancetoidiosyncraticincomeshocks.

Ontheoppositeendoftheory,

Cochrane

(

1991

)testsforfullinsurancebyassessingthesensitivityofconsumptiontoincomegrowthandtoeventssuchasillnessandjobloss.Undercompletemarkets,consumershaveaccesstocontingenttransfers,sohouseholdconsumptiongrowthshouldbeunrelatedtoidiosyncraticevents.FocusingonfoodconsumptioninthePSID,thehypothesisisrejectedfollowinglongillnessorjobloss,butnotrejectedinresponse

toshortunemploymentspells,thusprovidingearlyevidenceforpartialinsurance.6

5Otherearlypapersthattestthepermanentincomehypothesisare

Hall

(1978),whodoesnotrejectit;

Sargent

(1978),whorejectsit;and

Flavin

(1981),whoalsorejectsit

.Theyallusetime-seriesdata.

6

AltugandMiller

(1990)modelacomplete-marketsenvironmentandallowfornon-separabilitywith

laborsupply;usingfoodconsumptioninthePSID,theycannotrejectfullinsurancetowagefluctuations.

5

AttanasioandDavis

(1996)testforfullinsuranceacrossbirthcohortsandeducation

groups,accountingforconsumption-workcomplementarityandcommondemographicsdriv-ingconsumptionandwages.Undercompletemarketsandintheabsenceofaggregateshocks,consumptiongrowthshouldnotco-movewithwagegrowth.Drawingsyntheticpan-elsfromtheConsumerExpenditureSurvey(CEX)andCurrentPopulationSurvey(CPS),theysharplyrejectfullinsurancetolow-frequencywageshifts.Inasimilarmodel,

Hayashi,

Altonji,andKotlikoff

(1996)testforfullinsuranceacrossandwithinextendedfamilies,which

theyrejectbasedonthestrongcorrelationbetweenwageandfoodconsumptiongrowthinthePSID.Usingsimilardatainasimplersetting,

Altonji,Hayashi,andKotlikoff

(1992)also

rejectfullinsurance.Bycontrast,

Mace

(

1991

)and,inparticular,

Townsend

(

1994

),whostudiesvillageinsuranceinIndia,findmixedevidence.

Insum,theseearlyworksfrequentlyrejectthebenchmarkmodelsofpermanentincome(self-insurance)andcomplete-markets(fullinsurance).Yet,thedataconsistentlyrevealthathouseholdshaveaccesstosomeinsurancetoincomeshocks.7Theliteratureinthebusy2000sthrough2010sattemptstomeasurethedegreeofinsuranceandidentifyitssources.

Covariancerestrictions.

Blundell,Pistaferri,andPreston

(2008),abbreviatedasBPP,

introducetheseminalmethodologytomeasuretheconsumptionresponsetoincomeshocks.Theirideaisthattheextenttowhichconsumptiongrowthvarieswithincomegrowth—thelatterbeingdrivenbyvariousincomeshocks—reflectsthedegreeoftransmissionofthoseshocksintoconsumption.8Thisideaismotivatedbyaconsumptionprocessthatislog-linearinincomeshocks—namely,

∆cit=ξit+ϕtv+ψtu,(5)

where∆citisconsumptiongrowth∆lnCitnetofobservablesXit.9vanduare,respec-

tively,apermanentandatransitoryshocktodisposablehouseholdincome(sok=dhere);ϕtandψtaretheirtransmissionparameters;andξitisapreferenceshockunrelatedtoincome.If

incomefollowsthecanonicalpermanent-transitoryprocess—namely∆y=v+∆u,where

∆yisincomegrowth∆lnYinetofXit—thetransmissionparametersareidentifiedthrough

Cov(∆cit,Zit)/Cov(∆y,Zit)—namely,aregressionof∆citon∆yusingappropriateinstru-

7Amongthefirststudiestoquantifytheextentofconsumptioninsurance,

Gruber

(1997)measureshow

unemploymentinsurancereducesthefallinconsumptionuponunemployment.

8Inapredecessorpaper,

BlundellandPreston

(1998)assumethatthattheriseinconsumptioninequality

observedby

DeatonandPaxson

(1994)isdrivenbypermanentbutnottransitoryshocks,towhichconsumers

canfullyselfinsure.Thisassumptionallowsthemtouseincomeandconsumptionmomentstoidentifythevariancesofpermanentandtransitoryincomeshocks.

Attanasioetal.

(2002)extendthisideatoasettingof

twoearnerswithseparateincomestreams,while

PrimiceriandvanRens

(2009)extendittoheterogeneous

incomeprocesses.

9Weuse∆todenotethefirstdifferenceoperator;so∆Xt=Xt−Xt−1.

6

mentsZforincome.Inthecaseofpermanentshocks,permanentincomeZit=

nets∆yfromthetransitoryshockatt,soitscovariancewith∆citidentifiesϕt.Inthecase

oftransitoryshocks,futureincomeZit=∆y+1shifts∆ybecauseofmeanreversionofthe

transitoryshock,soitscovariancewith∆citidentifiesψt.Thisstrandofliteraturehastakenitsnamefromthesecovariancerestrictions.

Theconsumptionequation(

5

)canbeobtainedthroughalog-linearizationofthepolicyrule(

4

)inalife-cyclepermanentincomemodelwithCRRAutility,apermanent-transitoryincomeprocess,andslackborrowingconstraints.Inthiscase,ϕtreflectstheshareofcon-sumptionthatisfundedbyfuturelaborincome(asopposedtoassetswhosevalueremains

unchangedbytheshocktoincome)and,dependingonthemeasureofYi,featuresofthetax

andbenefitssystem;ψtissimilaruptoanannuitizationfactorforthehousehold’sremaininghorizon.Yet,(

5

)isalsoconsistentwithothersettingsinwhichthetransmissionofshocksdependsontheirpersistence—forexample,environmentswithmoralhazard(

Attanasioand

Pavoni

,

2011)orexternalinsurance(

Blundell,Pistaferri,andSaporta-Eksten,

2016)

.Thelinearinsuranceequation(

5

)is,therefore,thereducedformofmultipleenvironmentsthatmaydifferintheirinsurancecontent.Assuch,ϕtandψtarecalledpartialinsuranceparam-eters;theymeasuretheoverallpass-throughof(or,ontheflipside,insuranceto)permanentandtransitoryincomeshocks,regardlessoftheprecisemechanismsthatgivegroundsforsuchinsurance.

Empirically,BPPfocusonnondurableconsumptionanddisposablehouseholdincomefrom1980to1992.WhilethePSIDisidealforitsincomedata,itsconsumptiondata,includingmostlyfooditems,arelimiteduntil1999.BPPthusimputeconsumptionfromtheCEXintothePSID.Theyestimateϕt=0.64(a10percentpermanentincomecutre-ducesconsumptionbyonly6.4percent—householdsarethuspartiallyinsuredtopermanentshocks)andψt=0.05(statisticallynotdifferentfromzero,soconsumptionisfullyinsuredagainsttransitoryshocks).

BPPsparkedmultipleextensions.

Blundell,Low,andPreston

(2013)derive(5)under

anautoregressiveincomeprocessandgeneralpreferences.Theyassesstheapproximationvis-`a-visthetruepolicyruleandshowthatitperformswellwhenliquidityconstraintsdonotbind.

Blundell,Pistaferri,andSaporta-Eksten

(2016)modelendogenouslaborsupply

tomeasuretheconsumptionresponsetowageshocks.UsingincomeandconsumptiondatainthePSIDafter1999,theyestimateϕt=0.32/0.19(male/femalewages)andfindlittleexternalinsurancewhenaccountingforfamilylaborsupply,assets,andthetaxandbenefitssystem.10

Theloudis

(2017)extendsthismodeltoanintra-householdbargaining(collective)

10

Hyslop

(2001)usescovariancerestrictionstomeasurethepass-throughofwageshockstohousehold

earnings.

JessenandK¨onig

(2023)usearelatedapproach

.UsinghoursandearningsdatainthePSIDover

7

setting,inwhichlackofcommitmenttolifetimemarriagelimitstheinsuranceroleoffamilylaborsupply.

Chopra

(2023)arguesthattheinsuranceroleoffamilylaborsupplyincreases

duringrecessions.

Mostoftheseworkspermitlimitedheterogeneity.

Arellano,Blundell,andBonhomme

(2017),ABBinshort,relaxlinearityintheincomeprocessandletshocksfeature

nonlinearpersistencedependingontheirsignandsize.Theconsumptionresponsetoshocksvariesflexiblywiththeirlevelandpasthistory.Usingaquantile-basedestimationmethodandPSIDdataover1999to2011,theymeasuretheconsumptionresponsetologincome—atypeofMPC—at0.2to0.4,onaverage(specificationwithhouseholdheterogeneity),thoughtheresponsevariesoverthedistributionofshocks.Althoughallowingforheterogeneitymakesinterpretingtheirparametersimilartoapass-throughrate,theyfindtheresponseismarkedlydifferentfromBPP.ThisdifferencemaybeduetoABB’sflexibleincomeprocessorthenewconsumptiondatainthePSID.11

GhoshandTheloudis

(2023)relaxlinearityintheconsumptionprocessbywriting∆c

itasaquadraticpolynomialinincomeshocks.Thisspecificationstemsfromasecond-orderapproximationtothepolicyrule(

4

)inamodelsimilartoBPP.Thepass-throughofshocksnowdependsontheirsignandsize,sothismethodallowssmallversuslarge,orgoodversusbadshocks,tohaveanasymmetriceffectonconsumption.Identificationrequiressecond,third,andfourthmomentsofincomeandsecondmomentsofconsumption,incontrasttoABBwhorequireobservingtheirentiredistribution.UsingPSIDdataafter1999,theyestimatethepass-throughoftheaveragepermanentshockat0.13;badorlargepermanentshockshaveamuchbiggereffectonconsumption,andtheirpass-throughincreaseswiththeirseverity.12

Alan,Browning,andEjrnæs

(2018)estimatetheextentofinsurancetoincomeshocks

allowingforflexiblejointheterogeneityintheconsumptionandincomeprocesses.TheyusePSIDdataover1968to2009andfindpervasivecross-householdheterogeneityinthepass-through,rangingfrom0.05to0.69.

Theloudis

(

2021

)allowsforunobservedprefer-enceheterogeneityinamodelwithfamilylaborsupply.Heexploreshigher-ordermomentsofincomeandconsumptiontoidentifythecontributionofheterogeneitytoconsumptioninequality.

Aconsistentempiricalfindinginthisliteratureisthat,onaverage,consumptionisfully

1970to1997,theyfindthatwageandtasteshockshaveacomparablecontributiontothetotalvarianceofearnings.

11ABBisnotacovariancemethod,buttheirframeworkfallsfirmlyinthiscategorybecausetheymeasuretheoveralleffectofshocks,asinBPP.

Arellanoetal.

(2024)advancethismethodtounbalancedpanelsand

flexibleheterogeneity.Theymeasuretheconsumptionresponsetologincomeat0.2onaverage.

12Theaveragepass-throughisclosetoABBbutmarkedlydifferentfromBPP.ThediscrepancyisduetotheconsumptionimputationinBPPandthebiennial(versusannual)frequencyofthemoderndata.

8

insuredagainsttransitoryshocks.Severalpaperschallengethisresult.

Crawley

(2020)argues

thatBPPneglecttimeaggregationinthePSID.Timeaggregationoccurswhenincomeisobservedlessfrequently(annually)thantheunderlyingtruedata(forexample,monthlypayments);incomegrowthisthenmechanicallypositivelycorrelated,whichchangesthecovariancerestrictionsusedtoidentifythevarianceofshocksandtheirtransmissionintoconsumption.

Crawley

(2020)addressesthispointandestimatesthepass-throughofshocks

atϕt=0.34andψt=0.24(statisticallysignificant).

Commault

(2022)extendsthelinearinsurancemodel(

5

)toallowconsumptiontorespondtopasttransitoryshocks.Severalunderlyingstructuresjustifythisapproach.Theexactsolutionto∆citinalife-cyclepermanentincomemodelincludeshigher-ordertermsthatdependonpastvariables(forinstance,wealth);modelswithlimitedcommitmentareanotherexample.

Commault

(2022)proposesanewestimatorfor

ψt,onethatselectsasinstrument

theonlyfuture∆y+κ+1thatisuncorrelatedwithpastshocks.Sheestimatesψt=0.6(MPC

at0.32),whichhelpsbridgethediscrepancyinψtbetweenstudiesthatemploycovariancerestrictionsandthosethatrelyonnaturalexperiments.13

CrawleyandKuchler

(2023)addressneglectedtimeaggregation

andallowconsumptiontodependonpasttransitoryshocks.Theyestimateheterogeneousϕtandψtoverwell-definedsubpopulationsusingDanishadministrativedataandrelatetheirestimatestoliquidwealthandotherhouseholdbalance-sheetcharacteristics.

HryshkoandManovskii

(2022)identifyhouseholdsinthePSIDwithvastlydifferent

degreesofinsurance.TheyshowthatthesonsoffamiliesoriginallysurveyedbythePSIDin1968exhibitalmostnoinsurance,whilethedaughtershavesubstantialpartialinsurance.Theydoathoroughjobexplainingthisdiscrepancybydifferentialincomepersistenceacrossthetwogroups,whichisfurtherexplainedbydifferentialattritionfromthesurvey.

Fullyspecifiedmodels.Thestudiesemployingcovariancerestrictionsdonotfullyspecifypreferences,expectations,andthebudgetset,sotheytakenostanceontheexactmechanismsthatgiverisetopartialinsurance.Duetotheirsemi-parametricnature,theseworksareoflimiteduseforpolicycounterfactuals.Anotherapproachistofullyspecifythechannelsthroughwhichconsumerssmoothincomeshocks,andtakesuchmodelstothedata.

Attanasio,Low,andS´anchez-Marcos

(2005)quantifytheinsuranceroleoffemalela

-borsupply(measuredintermsofwelfarecostsofincomeuncertainty)throughastructural

13Aswereviewsubsequently,naturalexperimentsoftenimplyalargerψtthantypicallyestimatedthroughcovariancerestrictions.Itisunclear,however,ifsurveydatasuchasthePSID,onwhichthelatterstudiesrely,reflectlargertransitoryshocksthataretypicalofexperiments.Studiesthatallowthepass-throughtodependonthesizeoftheshocktypicallyfindthatlargershockshavelargerpass-through,thusalsohelpingbridgethegapinestimatesbetweennaturalexperimentsandcovariancerestrictions.

9

modelofconsumptionandlaborsupplywithearningsrisk.

KruegerandPerri

(2006)show

thatamodelwithacompletesetofArrowsecuritiesbutlimitedenforceabilityofcontractsreproducesincomeandconsumptioninequalityintheU.S.,suggestingthathouseholdspos-sessmoreinsurancethanself-insurance.

Storesletten,Telmer,andYaron

(2004)showthat

alife-cyclepermanentincomemodelcanalsoproduceempiricallyconsistentincomeandconsumptioninequalityifthetaxandbenefitssystemandtheaggregatelevelofwealtharetakenintoaccount.

Heathcote,Storesletten,andViolante

(2008)calibratealife-cycle

modelofconsumptionandlaborsupplywithpartialinsurancetowageshockstomeasurethewelfarecostsofriskandmarketincompleteness.Partialinsuranceisfixedbytheauthors:Permanentshocksareuninsured,whiletransitoryshocksarefullyinsured.

Low,Meghir,and

Pistaferri

(2010)calibratealife-cyclemodelofconsumption,laborsupply,andjobmobility

withincomeandemploymentrisk.Self-insuranceaside,themodelallowsforthreechannelsofpartialinsurance:unemploymentbenefits,disabilityinsurance,andfoodstamps.

Theseearlierquantitativemodelsmeasurethewelfareimplicationsofrisk(orofcertaininsurancemechanisms)butnotthedegreeofconsumptioninsuranceperse.14

Kaplanand

Violante

(2010)explicitlymeasurethedegreeofinsuranceinacalibratedlife-cycleper

-manentincomemodelwithincomerisk.ThisisthemodelwhoseconsumptionruleBPPlog-linearize.Whiletheyfindalmostfullinsurancetotransitoryshocks,asinBPP,theyesti-mateφt=0.78,larger