This paper presents preliminary findings and is being distributed to e - PDF document

1 This paper presents preliminary findings
This paper presents preliminary findings and is being distributed to economists and other interested readers solely to stimulate discussion and elicit comments. The views expressed in this paper are those of the authors and do not necessarily reflect the position of the Federal Reserve Bank of New York or the Federal Reserve System. Any errors or omissions are the responsibility of the authors. Federal Reserve Bank of New YorkStaff ReportsOptimal Monetary Policy According to HANKSushant AcharyaEdouard ChalleKeshav Dogra Staff Report No. 916 February 2020 Optimal Monetary Policy According to HANKSushant Acharya, Edouard Challe, and Keshav DograFederal Reserve Bank of New York Staff Reports, no. February 2020JEL classification: E21, E30, E52, E62, E63AbstractWe study optimal monetary policy in a eterogenousgent Keynesian economy. A utilitarianplanner seeks to reduce consumptioninequality, in addition to stabilizing output gaps and ination. Theplanner does so both by reducing income risk faced by households, and by reducing the passthrough fromincome to consumption risk, tradingthe benets of lower inequality againstproductive ineciencyand higher ination. When income risk is countercyclical, policy curtails the fall in output in recessionsto mitigate the increase in inequality. We uncover a new form of timeinconsistency of the Ramseyplanthe temptation to exploit households' unhedged interestrate exposure to lower inequality.wordsew Keynesian odel, ncomplete arkets, ptimal onetary olicy_________________Acharya, Dogra: Federal Reserve Bank of New York (emailsushant.acharya@ny.frb.org, keshav.dogra@ny.frb.org). Challe: CREST and Ecole Polytechniquemail: edouard.challe@gmail.com). The authors thank Florin Bilbiie, Christopher Carroll, Russell Cooper, Clodomiro Ferreira, Antoine Lepetit, Galo Nuño, Pedro Teles, Gianluca Violante, and PierreOlivier Weil for helpful discussions. They also received useful comments from seminar participants at HEC Paris, UT Austin, UC3M, EUI, UniversitParisDauphine, UniversitParis 8, Banque de France, and CREST, as well as from conference participants at the Barcelona GSE Summer Forum (Monetary Policy and Centr

2 al Banking), the NBER Summer Institute (
al Banking), the NBER Summer Institute (Micro Data and Macro Models), the Salento Macro Meetings, SED, and T2M. Edouard Challe acknowledges financial support from the French National Research Agency (Labex Ecodec/ANRLABX0047). The views expressed in thpaper are those of the authors and do not necessarily reflect the position of the Federal Reserve Bank of New York or the Federal Reserve System.To view the authors’ disclosure statements, visit https://www.newyorkfed.org/research/stareports/sr.html. 1IntroductionItisincreasinglyrecognizedbyresearchersandpolicymakersthatmonetarypolicycanhaveimportanteectsoninequality.Despitethis,thestudyofhowmonetarypolicyshouldbeconductedoptimallyhasonlyrecentlybeguntodepartfromarepresentativeagentframeworkinwhichconcernsaboutinequalityaretriviallyabsent.WhilethelargerecentheterogeneousagentNewKeynesian(HANK)literaturehasshownthatuninsurableidiosyncraticriskandinequalitycandramaticallychangethepositiveeectsofmonetarypolicyonthemacroeconomy(SeeforexampleRavnandSterk(2017,Forthcoming);Kaplanetal.(2018);denHaanetal.(2018);Auclertetal.(2018);Auclert(2019);Bilbiie(2019a)andmanyothers),thenormativeimplicationsofHANKandthereciprocaleectsofmonetarypolicyonriskandinequality,havebeenlesswellstudied.ThisgapintheliteraturepartlyexistsbecausecharacterizingoptimalpolicyinHANKeconomiesistechnicallydicult.SolvingfortheRamseyoptimalpolicyinvolveschoosingtheevolutionofaninnitedimensionalstatevariable(thewealthdistribution),aswellasinnitedimensionalcontrols(thedistributionofconsumptionandhoursworkedacrossagents)subjecttoaninnitenumberofconstraints(eachhousehold'soptimalityconditionandbudgetconstraints).OneapproachtosolvingoptimalpolicyproblemsforHANKeconomiesiscomputational,andre-searchershaverecentlystarteddevelopingnumericalalgorithmstohandlethem(Bhandarietal.,2018).Weinsteadtakeananalyticalapproach.WestudyastandardNKeconomywithnominalrigiditieswiththeexceptionthathouseholdsfaceuninsurableidiosyncraticrisk.Marketsareincompleteandagentscanonlyself-insurebytradingarisklessbondorbyworkinglongerhours.Weassumethathouseholdshaveconstantabsoluterisk(CARA)utilityandidiosyncraticshocksarenor

3 mallydistributed.AsinAcharyaandDogra(201
mallydistributed.AsinAcharyaandDogra(2018),theseassumptionsimplythattheeconomypermitslinearaggregationwhichinturnmeansthattheinnitedimensionaldistributionsofconsumption,hoursworkedandwealthcanbesumma-rizedbytheircross-sectionalaverages:thepositivebehaviorofmacroeconomicaggregatescanbedescribedindependentlyofthedistributionofthewealthdistribution.Ofcourse,fromanormativeperspective,thedispersionofwealthdoesaectsocialwelfareandhencetheoptimalconductofmonetarypolicy.Crucially,theeectofinequalityonsocialwelfareisalsosummarizedbyanitedimensionalsucientstatistic.Thismakestheoptimalpolicyproblemanalyticallytractable,allowingustodrilldownandidentifyexactlythefeaturesthatmakestheoptimalconductofmonetarypolicydierentinHANKeconomiescomparedtotheRANKbenchmark.Asiscommonlyknown,inRANKtheplannerseekstostabilizepricesandkeepoutputatitsproduc-tivelyecientlevel.InHANK,theplannerhasanadditionalobjective-tousemonetarypolicytoreducethecross-sectionalconsumptiondispersionthatresultsfromthecumulatedeectsofuninsuredidiosyn-craticshocks.ThisincentiveisshutdowninRANKbyconstruction.Weshowthattherearethreewaysinwhichtheplannercanusemonetarypolicytoachievethispurpose.First,thecentralbankmayattempttoreducetheamountofincomeriskthathouseholdsareexposedto(theincomeriskchannel).Howtoachievethisreductionnaturallydependsonthecyclicalityofincomerisk:ifthisriskiscountercyclical,thenthecentralbankhasanincentivetoraiseoutputinordertolowerrisk,whiletheoppositeistrueifriskisprocyclical.Eitherway,thecentralbank'swillingnessandabilitytomanipulatetheamountofidiosyncraticriskthathouseholdsfacegivesitanincentivetomoveoutputawayfromthelevelconsistent1 withstablepricesandproductiveeciency.1Thesecondwayinwhichthecentralbankmayreduceconsumptiondispersion,independentlyofaect-ingthelevelofincomerisk,isbyreducingthepass-throughfromincomerisktoconsumptionrisk,thatis,byloweringthemarginalpropensitytoconsumeoutofachangeinindividualincome(theself-insurancechannel).Thispass-throughultimatelyre ectstheabilityofthehouseholdstoself-insureagainstidiosyn-craticriskthroughborrowingorworkinglongerhours,andisthusaectedbythepathsofrealinterestratesa

4 ndwagesgoingforward.Ontheonehand,lowerin
ndwagesgoingforward.Ontheonehand,lowerinterestratesmakeiteasierforhouseholdstoborrowinresponsetoanunfavorableshock,makingindividualconsumptionlessresponsetochangesinindividualincome;thisultimatelyreducesconsumptiondispersionatanylevelofincomerisk.Ontheotherhand,higherwagesgoingforwardmakeiteasierforhouseholdstobuertheimpactofafallincurrentincomeoncurrentconsumptionbyborrowingtodayandworkinglongerhoursinthefuturetorepaythedebt.Whenfuturewagesarehigh,onlyasmallincreaseinhours(andhencetheincurreddis-utility)workedisrequiredtorepaythisdebt.Thisagainmakesindividualconsumptionlessresponsivetochangesincurrentincome.Itfollowsthatthecentralbankhasanincentivetocommittolowinterestratesandhighwages{i.e.,tobeexpansionary{goingforwardinordertoreducethepass-throughfromincomerisktoconsumptionrisk.ThisincentiveisofcourseabsentinRANK.Finally,thecentralbankcanreduceconsumptiondispersionthroughunanticipatedchangesinthemarginalpropensitytoconsumeoutofwealth.Givenadistributionofwealth,anunexpectedfallininterestratesbenetspoordebtors,reducingtheirinterestpaymentsandincreasingtheirconsumption(theunhedgedinterestrateexposure(URE)channel)(Auclert,2019).Conversely,lowerinterestratesreducetheinterestincomeofrichsavers,reducingtheirconsumption.Overall,lowerratesreducethemarginalpropensitytoconsumeoutofwealth,reducingconsumptioninequality.Importantly,thischannel(unliketheprevioustwo)onlyoperatesforunexpectedchangesininterestrates,asweexplaininmoredetailinSection4.Howdoesthepresenceofthesethreechannels,throughwhichtheplannercanaectinequality,changetheoptimalconductofmonetarypolicy?Asiscommonlyknown,inRANK,optimalmonetarypolicyfeaturesdivinecoincidenceinresponsetoproductivityshocks(BlanchardandGal,2007):itisbothfeasibleandoptimaltostabilizeboththegapbetweenoutputanditsecientlevel(outputgap),andin ation.InourRANKeconomy,intheempiricallyrelevantcasewhereincomeriskiscountercyclical,whileitremainsfeasibletostabilizetheoutputgapandin ation,itisnolongeroptimaltodoso.Buttounderstandthetradeosthatleadtheplannertodeviatefromdivinecoincidenceinthiscase,itisinstructivetostartbyexaminingaHANKeconomywheredivinecoincidenceisoptimal.Thisisthecasewh

5 enriskismildlyprocyclicalandthereisnoini
enriskismildlyprocyclicalandthereisnoinitialwealthinequality.Inthisknife-edgecase,thetwochannelsthroughwhichanticipatedmonetarypolicyaectsconsumptioninequalityexactlyoseteachother:expansionarypolicyraisesoutputandhenceincomerisk,butmakesiteasierforhouseholdstoself-insure,leavingtheconsumptionriskfacedbyhouseholdsunchanged.Inaddition,theabsenceofwealthinequalityatdate0mutestheUREchannel.Thus,whiletheplannerwouldliketoreduceconsumptioninequality,sinceitisnotpossibletodothiswithmonetarypolicy,itremainsoptimaltostabilizeboththe 1EarlierworkhasstressedthatthecyclicalityofincomeriskaectstheresponseofaggregatedemandinHANKeconomies(Werning,2015;AcharyaandDogra,2018;Bilbiie,2019a),andthisinitselfgenericallyrequiresabenevolentcentralbanktoimplementadierentpathofthepolicyratethaninRANK(Challe,2020).Thisdoesnot,however,necessarilywarrantdepartingfrompricestability.2 outputgapandin ation.Awayfromthisknife-edgecase,monetarypolicycan,anddoesexploitthechannelsabovetoaectconsumptioninequality.Supposerstthatriskremainsmildlyprocyclicalsothatincomeriskandself-insurancechannelsoseteachother,butthereiswealthinequalityatdate0.Inthiscase,monetarypolicycannotaectinequalitythroughthersttwochannels,butitcanthroughtheUREchannel.Consequently,theoptimalplanfeaturesanunexpectedcutininterestratesatdate0,whichreducesconsumptionin-equalityatthecostofdeviatingfromtheecientlevelofoutputandin ation.Thatis,optimalpolicyfeatureshigheroutputandin ationatdate0thanwouldbeoptimalinRANK.Whilemonetarypolicycannotaectinequalityfromdate1onwards,outputandin ationcontinuetodeviatefromRANKaspolicyseekstosmooththetransitionbacktosteadystate.IfpolicywassomehowpreventedfromcreatingtheboomtoexploittheUREchannelatdate0,itwouldnotseektodeviatefromRANKatanyotherdateeither.Thisbringsustotheempiricallyrelevantcaseofcountercyclicalrisk.Inthiscaseexpansionarymone-tarypolicycanreduceinequalityatalldatesthroughtheself-insuranceandincomeriskchannels:alowerpathofinterestratesmakesiteasierforhouseholdstoinsurethemselves,andmakesaboominoutputwhichreducesthelevelofincomeriskhouseholdsface.Inaddition,atdate0,acutinratesdeliversafurtherreductioninine

6 qualitythroughtheUREchannel.Consequently
qualitythroughtheUREchannel.Consequently,theplanneralwaystradesothisbenetofmoreexpansionarymonetarypolicy-namely,thatitreducesconsumptioninequality-againstthecostsofinecientlyhighoutputandin ation.Thesetradeoschangetheoptimalresponsetoproduc-tivityshocks,relativetoRANK.Followinganegativeproductivityshock,theplannerletsoutputdeclineasmuchasoutputinthe exiblepricecaseandimplementszeroin ationbyraisingnominalinterestrates.However,inHANK,optimalpolicyimplementsalowerpathofnominalinterestrates,curtailingthefallinoutputinordertomitigatetheincreaseininequality.Eventhoughthisentailshigherin ationandoutputaboveitsecientlevel,thisisoptimalbecauseinequalityisalreadyhigherinrecessionsandsoisthebenetfromareductionininequality.Sofarwehaveonlydiscussedonechannelthroughwhichunexpectedcutsininterestratescanlowerinequality(theUREchannel).Severalauthorshavehighlightedanotherwayinwhichunexpectedchangesinmonetarypolicycanlowerinequality,namelytheFisherchannel:unexpectedin ationredistributesfromsaverswhoholdnominalassetstodebtorswhoholdnominalliabilities.Wedeliberatelyabstractfromthischannelinourbaselinemodel(bylettinghouseholdstradein ation-indexeddebt)inordertoemphasizethattheUREchanneldoesnotdependontheabilityofmonetarypolicytoredistributewealththroughanin ationsurprise.WhileconceptuallydistinctfromtheUREchannel,theFisherchannelprovidesanotheravenuethroughwhichexpansionarymonetarypolicycanreduceinequality.InSection6weshowthatwhenhouseholdstradenominallydenominatedassets,optimalmonetarypolicyismoreexpansionaryinrecessionscomparedtoRANK.RelatedLiteratureThepapermostcloselyrelatedtooursisBhandarietal.(2018),whoalsostudyoptimalmonetarypolicyinaHANKmodel.Themaindierencebetweenourpaperandtheirsismethod-ological.Bhandarietal.(2018)proposeanumericalalgorithmtoderiveoptimalmonetarypolicyinHANKmodels,whilewestudyaHANKeconomywithconstantconstantabsoluteriskaversion(CARA)prefer-3 encesandnormallydistributedshockswhichpermitsclosedformsolutions.2Weseethetwoapproachesasinherentlycomplementary:therstallowsmore exibilityinthestructureofpreferencesandidiosyncraticshocks,whilethesecondbetterisolatesthechannelsbywhichthecentralbankman

7 ipulatesconsumptiondispersionalongtheopt
ipulatesconsumptiondispersionalongtheoptimalpolicyplan.AnothercloselyrelatedpaperisNu~noandThomas(2019)whostudyhowUREandFishereectsaecttheoptimalconductofmonetarypolicyinthepresenceofhet-erogeneity.Unlikeus,theystudyasmallopeneconomyinwhichshort-termrealinterestratesandoutputareunaectedbymonetarypolicy.Thus,theclassicoutputin ationtrade-owhichiscentraltoNewKeynesianeconomiesisabsentintheirsetting.SeveralauthorshavestudiedoptimalmonetarypolicyinsimpleHANKeconomieswithlimitedcross-sectionalheterogeneity{see,e.g.,Bilbiie(2008);BilbiieandRagot(2018);Bilbiie(2019a)andChalle(2020).3Mostofthesepapersachievetractabilitybyimposingthezeroliquiditylimit(householdscan-notborrowandgovernmentdebtisinzeronetsupply).4Thisassumptionrulesouttheself-insurancechannelbecauseinequilibriumhouseholdsdonotborroworlendandhencetheyspendalltheirincomeonconsumption.Ouranalysisshowsthatthisassumptionrulesoutanimportantchannelthroughwhichmonetarypolicyaectsinequality.Moregenerally,ourpaperbelongstothegrowingstrandofliteraturethatrevisitsthetransmissionandoptimalityofvariouseconomicpolicieswithintheHANKframework.Thisincludesnotonlytheworkonconventionalmonetarypolicydiscussedabovebutalsothatonunconventionalmonetarypolicy(McKayetal.,2016;AcharyaandDogra,2018;Bilbiie,2019a;CuiandSterk,2019),onunemployment-insuranceandsocial-insurancepolicies(McKayandReis,2016,2019;denHaanetal.,2018;Kekre,2019),andonscalpolicy(Auclertetal.,2018;Bilbiie,2019b).Therestofthepaperisorganizedasfollows.Section2presentsthemodel.Section3characterizesequilibria,deningtheimplementabilityconstraintstheplannerfaces.Section4showsthattheutilitarianplanner'sobjectivefunctioncanbewrittenintermsofaggregatevariablesandasinglesucientstatisticforthewelfare-relevantmeasureofinequality.Section5characterizesoptimalmonetarypolicy.Section6introducesnominalbondsandshowshowourresultsextendtothatcase.Section7concludes.2Environment2.1HouseholdsWestudyaBewley-Huggetteconomyinwhichhouseholdsfaceuninsurableidiosyncraticshockstotheirdis-utilityfromsupplyinglabor.Weabstractfromaggregateriskbutallowforaonetimeunanticipatedshockatdate0,afterwhichagentshaveperfectforesight.Oureconomyfe

8 aturesaperpetualyouthstructurealaBl
aturesaperpetualyouthstructurealaBlanchard-Yaariinwhicheachindividualfacesaconstantsurvivalprobability#inanyperiod.Populationisxedandnormalizedto1.Consequently,thesizeofanewlyborncohortatanydatetis1�#andthe 2Caballero(1990),Calvet(2001),Wang(2003),AngeletosandCalvet(2006)haveusedsimilarmodelingassumptionsinrealeconomies.Recently,AcharyaandDogra(2018)showsthattheseassumptionsareveryhelpfulinunderstandingthepositivepropertiesofHANKeconomies.3SeealsoNistico(2016),whogeneralizestheTwo-AgentNewKeynesian(TANK)modelofGaletal.(2007)andBilbiie(2008)tothecaseofstochasticasset-marketparticipation,andDebortoliandGal(2018)onthecomparisonbetweentheTANKmodelandaHANKmodelwithhomogeneousborrowing-constrainedhouseholdsandheterogeneousunconstrainedhouseholds.4BilbiieandRagot(2018)isanexceptionasititallowsagentstoholdmoneyinpositiveamountsforself-insurancepurposes.4 datetsizeofacohortbornatstis(1�#)#t�s.Thedatesproblemofanindividualibornatdatesis:maxfcst(i);`st(i);ast(i)g1Xt=s(#)t�sucst(i);`st(i);st(i)s.t.cst(i)+qtast+1(i)=wt`st(i)+ast(i)+Tt(1)ass(i)=0(2)AgentshaveCARApreferencesoverbothconsumptionand(disutilityof)labor:ucst(i);`st(i);st(i)=�1 e� cst(i)�e1 [`st(i)�st(i)](3)Eachagentisavesinrisklessrealactuarialbonds,issuedbynancialintermediaries(describedbelow),whichtradeatapriceofqtatdatetandpayooneunitoftheconsumptiongoodatt+1iftheagentsurvives.5Eachagentcantakeunrestrictedpositiveornegativepositionsinthebondandthesechoicesareonlydisciplinedbythetransversalitycondition.Ttdenoteslump-sumtransfersnetoftaxesanddividendsfromtherms.Forsimplicity,weassumethatdividendsareequallydistributedacrossagents.Ahouseholdsupplieslabor`st(i)atthepre-taxrealwagewt.Theagentfacesuninsurableshocksst(i)N� ;2t
tothedis-utilityofsupplyinglabor.st(i)isindependentacrosstimeandacrossindividuals.Alargerrealizationofst(i)reducesthedis-utilityfromworkand,givenwages,increasesthehousehold'slaborsupply.Equivalently,onemaythinkofst(i)asashocktothehousehold'sendowmentoftimeavailabletosupplylabor.6Toseethis,deneleisureasls

9 t(i)=st(i)�`st(i).Thenonecanrewr
t(i)=st(i)�`st(i).Thenonecanrewritetheperiodutilityfunctional(3)as�e� cst(i)= �e�lst(i)=andthebudgetconstraintas:cst(i)+wtlst(i)+qtast+1(i)=wtst(i)+ast(i)+Tt(4)TheLHSof(4)denotesthepurchasesofconsumption,leisureandbondsbythehouseholdwhiletheRHSdenotesthenotionalcash-on-hand-thevalueofthehousehold'stimeendowmentalongwithsavingsnetoftransfers.Henceforth,wewillsimplyrefertothisascash-on-hand.Weallowforthepossibilitythatthevarianceof,2t,varyendogenouslywiththelevelofeconomicactivityaswediscussbelow.2.2FinancialintermediariesThereisacompetitivenancialintermediationsectorwhichtradesactuarialbondswithhouseholdsandtradesgovernmentdebt.Anintermediaryonlyneedstorepayhouseholdsthatsurvivebetweentandt+1. 5Ineconomieswithadistributionofnominaldebt,unexpectedin ationredistributeswealthbetweencreditorsanddebtors.Bhandarietal.(2018)andNu~noandThomas(2019)discusshowoptimalmonetarypolicytakesthisintoaccount.OurbenchmarkeconomydeliberatelyabstractsfromthischannelinordertohighlighttheotherwaysinwhichoptimalmonetarypolicydiersinHANKandRANKeconomies.InSection6,wereplacerealdebtwithnominaldebt,bringthischannelbackintoplayandshowhowourresultschange.6WethankGianlucaViolanteforsuggestingthisinterpretation.5 Therepresentativeintermediarysolves:maxat+1;Bt+1�#at+1+Bt+1s.t.�qtat+1+t+1Bt+1 1+it0whereBtdenotesgovernmentdebt,atdenotesclaimsheldbyhouseholds,Rt=1+it t+1isthegrossrealreturnongovernmentdebt,itisthenominalinterestratewhichissetbythemonetaryauthorityandt+1denotesgrossin ationbetweentandt+1.Zeroprotsrequirethattheintermediarysells/buybondsfromthehouseholdsatapriceqt=# Rtandthat#at+1=Bt+1.2.3FinalgoodsproducersArepresentativecompetitivenalgoodsrmtransformsthedierentiatedintermediategoodsyjt,j2[0;1]intothenalgoodyaccordingtotheCESaggregatoryt=hR10yt(j)1 dji.Asisstandard,thenalgoodproducer'sdemandforvarietyjis:yt(j)=Pt(j) Pt� �1yt(5)where �1istheelasticityofsubstitutionbetweenvarieties.2.4IntermediategoodsproducersThereisacontinuumofmonopolisticallycompetitiveintermediategoodsrmsindexedbyj2[0;1].Each

10 2;rmfacesaquadraticcostofchangingthepric
2;rmfacesaquadraticcostofchangingthepriceofthevarietyitproduces(Rotemberg,1982).Ifrmjhiresnt(j)unitsoflabor,itcanonlyselltothenalgoodsrmthequantity:yt(j)=ztnt(j)� 2Pt(j) Pt�1(j)�12yt(6)whereztdenotesthelevelofaggregateproductivityatdatet.Firmjsolves:maxfPjt;njt;yjtg1t=01Xt=0Qtj0Pt(j) Ptyt(j)�(1�)wtnt(j)subjectto(5)and(6)whereQtj0=Qts=0R�1s.ThisyieldsthestandardPhillipscurve:(t�1)t= (�1)1�zt (1�)wt+1 Rtyt+1ztwt ytzt+1wt+1(t+1�1)t+1(7)2.5GovernmentThemonetaryauthoritysetstheinterestrateonnominalgovernmentdebt.Thescalauthoritysubsidizesthewagebillofrmsatarateandrebateslumpsumtaxes/transferstoallhouseholdsequally.Thegovernmentbudgetconstraintisgivenby:Bt+1 1+it=PtTt+PtwtZ10nt(j)dj+Bt(8)6 Wefurtherassumethatgovernmentdebtisinzeronetsupply,Bt=0forallt0.2.6MarketclearingInequilibrium,themarketsforthenalgood,laborandassetsmustclear:yt=(1�#)tXs=�1#s�tZicst(i)diZ10nt(j)dj=(1�#)tXs=�1#s�tZi`st(i)di0=at=(1�#)tXs=�1#s�tZiast+1(i)diwherethelastequationholdsbecauseBt=0forallt.2.7ShocksAsmentionedpreviously,weabstractfromaggregateriskbutallowforaonetimeunanticipatedaggregateshocktotheleveloflaborproductivityz0atdate0.Weassumethattheshockdecaysgeometrically:lnzt=%tzlnz0for%z2[0;1).3CharacterizingequilibriaAsinAcharyaandDogra(2018),CARApreferencesandnormallydistributedshocksimplythatthemodelaggregateslinearlyandthedistributionofwealthdoesnotdirectlyaectthedynamicsofaggregatevariables.Webeginbydescribingoptimalhouseholddecisionsinequilibrium.Proposition1(HouseholdDecisionRules).Inequilibrium,theoptimaldatetconsumptionandlaborsupplydecisionsofahouseholdibornatdatesare:cst(i)=Ct+txst(i)(9)`st(i)=lnwt� cst(i)+st(i)(10)wherexst(i)=ast(i)+wt�st(i)� 
isdemeanedcash-on-hand,Ctdenotesaggregateconsumptionandtisthe\marginalpropensitytoconsume"(MPC)outofcash-on-hand.Theseevolveaccordingto:Ct=�1 lnRt+Ct+1� 2t+1w2t+12t+1 2(11)�1t=1+ wt+# Rt�1t+1(12)Proof.SeeAppendixA Tounderstandtheroleo

11 fmarketincompletenessinexplainingthebeha
fmarketincompletenessinexplainingthebehaviorofconsumptionandla-borsupplyitisusefultocompareequations(9)and(10)totheircounterpartsundercompletemarkets.7 Undercompletemarkets,allhouseholdsareinsuredagainstdis-utilityshocks,i.e.themarginalutilityofconsumptione� cst(i)andthemarginaldis-utilityofworke1 (`st(i)�st(i))areequalizedacrossallstatesandso:@cst(i) @st(i)=0and@`st(i) @st(i)=1.Thus,sincehouseholds'consumptionisinsured,ahouseholdwhichdrawsatemporarilyhigherdis-utilityfromworkingcanreducehourswithoutexperiencingadropinconsumption.Instead,whenmarketsareincomplete(9)and(10)implythat:@cst(i) @st(i)=twt�0and@`st(i) @st(i)=1� twt1Forexample,afteranegativeshock(i.e.,greaterdis-utilityfromworking),consumptiondeclinesinsteadofremainingconstantwhilelaborsupplydoesnotfallquiteasmuchasundercompletemarkets.Whilehouseholdsusecreditandlabormarkettoinsurethemselvestosomeextent,thesearenotperfectsubstitutesforArrowsecurities,soagentsareonlyabletopartiallyinsulatethemselvesfromtheshock.Whenthedis-utilityoflaborrises,householdswouldliketoworkless,butreducinghoursasmuchasundercompletemarketswouldcauseconsumptiontodroptoomuch.Theoptimalresponsetotheshockistouselaborsupplyforself-insurance,i.e.toworklongerhoursthanundercompletemarkets.7Proposition1alsostatesthattheMPCoutofcash-on-handisthesameacrossindividuals;(12)describesitsevolutionovertime.Intuitively,considerahouseholdithatreceivesanadditionaldollaratdatet.Theywilloptimallychoosetospenddcst(i)=tofthedollarinthecurrentperiod.Sinceconsumptionandleisurearenormalgoods,theyalsoreducehoursworkedby t,resultingin wttlessincome.Savingtheremaining1�t(1+ wt),theyndthemselveswithdast+1(i)=Rt #[1�t(1+ wt)]nextperiodoutofwhichtheywillconsumedcst+1(i)=t+1dast+1(i).Finally,itisoptimaltosmoothconsumptionsothatdcst(i)=dcst+1(i)whichyieldst=t+1Rt #[1�t(1+ wt)].Rearrangingthisexpressionyieldsequation(12).Iteratingforwardsyields:t=1 P1s=0Qt+sjt(1+ wt+s)TheMPCt,whichmeasuresthepass-throughfromafallincash-on-handtoconsumption,isincreasingincurrentandfutureinteres

12 tratesanddecreasingincurrentandfuturewag
tratesanddecreasingincurrentandfuturewages.Thisisbecauseinterestratesandwagesaecthouseholds'abilitytousethebondandlabormarkets,respectively,forself-insurance.Considerahouseholdwhoreceivesanunfavorableshock0.Thehouseholdrespondsbyworkinglesstoday,borrowinginordertomitigatethedeclineinconsumption,andworkinglongerhoursinthefuture.Alowerpathofinterestratesreducesthecostofborrowing,makingiteasytoselfinsureusingthebondmarketandloweringtheresponsivenessofconsumptiontochangesincash-on-hand.Similarly,higherfuturewagesreducethe(disutility)costofworkingmorehoursinthefuturesinceevenasmallincreaseinhoursworkedsucestorepaythesamedebt.Thistoolowersthesensitivitytofconsumptiontocash-on-hand.8Whilethesensitivityofhouseholdconsumptiontoshocks(t)dependsonthefactorswehavejustdescribed,theaveragelevelofconsumptionintheeconomyCtdependsoninterestratesrelativetoimpa- 7Asalreadymentioned,themodelcanbere-interpretedasonewithanidiosyncratictime-endowmentshockandutilityfromleisuretime.Inthisinterpretation,bothconsumptionandleisuretimestayconstantundercompletemarkets,whilebothco-varywiththeidiosyncraticshockunderincompletemarkets.8WhileAcharyaandDogra(2018)alreadydiscusshowtheMPCrespondstofuturerealinterestrates,thepathofwageshasnoeectontheMPCintheirpaperbecausetheirenvironmentfeaturesinelasticlaborsupply.Inthismodel,however,sincehouseholdscanchoosehowmuchlabortosupply,theyusethisadditionalmargintoself-insure.8 tienceandhouseholds'desireforprecautionarysavings,asshowninequation(11).Absentidiosyncraticrisk,t=0,(11)isastandardEulerEquation;higherrealinterestratesrelativetohouseholdimpatienceraiseconsumptiongrowth.Thelasttermin(11)re ectsprecautionarysavings.Given(9),theconditionalvarianceofdatet+1consumptionofhouseholdiisVt�cst+1(i)
=2t+1w2t+12t+1.Totheextentthatconsumptionriskispositiveandhouseholdsareprudent( �0),householdssavemorethaninarisklesseconomyforthesameinterestrate,i.e.theychooseasteeperpathofconsumptiongrowth.Thevarianceofconsumption,inturn,dependsonboththevarianceofcash-on-handVt�xst+1(i)
=w2t+12t+1,andthepass-throughofcash-on-handriskintoconsumptionriskmeasuredbythe(squared)MPC2t

13 +1.DeterminationofytInasymmetricequilibr
+1.DeterminationofytInasymmetricequilibrium,aggregating(6)acrossrms,wehave:yt=ztnt� 2(t�1)2yt(13)Aggregatinglaborsupply(10)acrosscurrentlyalivehouseholdsandusinggoodsandlabormarketclearing:nt=lnwt� yt+ (14)Combiningthetwoequationsabove,wehave:yt=ztlnwt+  1+ zt+ 2(t�1)2(15)where 2(t�1)2denotestheresourcecostofin ation-higherin ationreducesoutput.DerivingtheaggregateISequationImposinggoodsmarketclearingin(11)yieldstheaggregateISequationwhichdescribestherelationbetweenoutputtodayandtomorrow:yt=yt+1�1 lni+it t+1� 2t+1w2t+12t+1 2(16)TimevaryingtFollowingMcKayandReis(2019),weallowforthevarianceofshockstovaryendogenouslywitheconomicactivitysothatthemodelgeneratescyclicalchangesinthedistributionofearningsrisks.Inparticular,weassumethat2tw2t=2w2e2(yt�y)whereydenotessteadystateoutputand=@lnV(x) @yistheconstantsemi-elasticityofthevarianceofcash-on-handw.r.toutput.This exiblespecicationallowsthevarianceofcash-on-handVt(x)tobeeitherincreasinginyt(procyclicalrisk),when�0;decreasinginyt(countercyclicalrisk),when0;orindependentofthelevelofyt(acyclicalrisk)when=0.3.1SteadystateWenowcharacterizeallocationsinthezeroin ationsteadystate.Wenormalizethelevelofproductivityz=1insteadystate.Imposingt=t+1=1in(7)requiresthatsteadystatewagesw=1 (1�).Given9 thiswage,steadystateoutputisy=lnw+ 1+ .Imposingsteadystatein(16)and(12)yields:R=�1e� 2and=1�e 1+ w(17)where= 22w22denotestheconsumptionriskfacedbyhouseholdsinsteadystate(scaledbythecoecientofprudence)ande=# Risthesteadystatepriceofanactuarialbond.Observethatthepresenceofuninsurablerisk(�0)impliesthattheequilibriumrealinterestrateR�1.Furthermore,thesteadystatedistributionofcash-on-handxinthepopulationisgivenby:F(x)=(1�#)1Xs=0#sx wp s+1(18)where(
)isthecdfofthestandardnormaldistribution.Thisfollowssince,conditionalonsurvival,xisarandomwalkwithnodriftandavarianceofw22insteadystate.3.2LinearizeddemandblockThedemandb

14 lockoftheeconomy,givenapathofinterestrat
lockoftheeconomy,givenapathofinterestrates,canbedescribedbytheISequation(16),theMPCrecursion(12)andthedenitionofGDP(15).Beforeanalyzingoptimalpolicy,itisusefultocomparethedynamicsofthisHANKeconomytoitsRANKcounterpart.Itiseasiesttocomparetherst-orderTaylorexpansionoftheequationsdescribingaggregatedynamicsintheneighborhoodofthezeroin ationsteadystate,whichare:9byt=byt+1�1 (it�t+1)�1 bt+1(19)bt=�(1�e) w 1+ wbwt w+e(bt+1+it�t+1)(20)byt= 1+ bwt w+y 1+ bzt(21)where=1� .InRANK,thereisnoidiosyncraticrisk,i.e.2=0whichimplies=1and=0,sothat(19)becomesthestandardRANKIScurve.AsdiscussedinAcharyaandDogra(2018),uninsurableidiosyncraticriskchangestheISequationintwoways.First,itcanintroduceeitherdiscounting(1)orcompounding(&#x]TJ/;ø 1;�.90;‘ T; 11;&#x.721;&#x 0 T; [0;1)dependingonthecyclicalityofrisk.Ifriskisacyclical,=0,then=1asintheRANKISequation.Ifriskisprocyclical&#x]TJ/;ø 1;�.90;‘ T; 11;&#x.721;&#x 0 T; [0;0then1andthereisdiscountingintheISequation.Thisisbecauseinthissituationlowfutureoutputimplieslowidiosyncraticincomerisk,hence(allelseequal)afallinprecautionarysavingsthatmutesdowntheeectoftheforthcomingrecessiononcurrentaggregatedemand.Theoppositeoccurswhenriskiscountercyclical(0),inwhichcase&#x]TJ/;ø 1;�.90;‘ T; 23;&#x.993;&#x 0 T; [0;1{thatis,wehavecompoundingintheISequation{,becausethentheimpactofafuturerecessiononcurrentaggregatedemandismagniedbytheriseinprecautionarysavings.Second,thestrengthoftheprecautionarymotivedependsnotonlyonthelevelofincomeriskbutalsoonitspass-throughtoconsumptionrisk,i.e.bt+1,whichinturndependsofthefuturepathofrealinterestratesandwages.Bycommittingtoalowerpathofrealratesorahigherpathofwages,monetarypolicycanlowerthestrength 9Welinearizeytandwtinlevelswhileallothervariablesarelog-linearized.10 oftheprecautionarysavingsmotiveatanylevelofincomerisk.Equations(7),(12),(15)and(16)summarizetheoptimalityconditionsoftheprivatesectoranddenetheimplementabilityconstraintsfacedbythecentralbank.

15 Wemaynowturntoitsobjectivefunction.4Obje
Wemaynowturntoitsobjectivefunction.4ObjectivefunctionoftheplannerWeassumethattheplannermaximizesautilitariancriterion;atanydatettheplannerassignsequalweightstothewelfareofallhouseholdscurrentlyaliveandaweightofs�tonthewelfareofcohortswhowillbebornatdatess�t.GiventheseParetoweights,theplanner'sobjectivecanbewrittenasmaximizingP1t=0tUtwhereUt,theperiodtfelicityfunctionoftheplannerissimplytheaverageutilityofallsurvivingagents:10Ut=�(1�#)0Xs=�1#�sZucst(i);`st(i);st(i)di;ThisexpressionforUtfeaturethefullcross-sectionaldistributionofagentsandisingeneralnottractable.Fortunately,itcanbegreatlysimpliedbyexploitingourCARA-Normalstructureagain.Proposition2(SocialWelfareFunction).TheperiodfelicityfunctionUtcanbewrittenasUt=u(yt;nt; )twheret=(1�#)0Xs=�1#�se1 2 22c(t;s)and2c(t;s)isthedate-tcross-sectionaldispersionofconsumptionamongstthesurvivinghouseholdsfromthecohortbornatdatest,i.e.,cst(i)N�yt;2c(t;s)
.Proof.SeeAppendixB.2. Intuitively,u(yt;nt; )isthenotional owutilityoftherepresentativeagent,i.e.,theperiodutilityfunctional(3)evaluatedataggregateconsumptionyt,aggregatelaborsupplynt,andmeanlabordis-utility .tcanbethoughtofasthewelfarecostofinequality,andisincreasinginthewithincohortdispersionofconsumption.Inarisklesseconomy,therewouldbenoconsumptiondispersionandhencet=1atalldates.However,inthepresenceofrisk,t�1,reducingwelfarerelativetothisrepresentativeagentbenchmark.Recallthatu(
)0andsohighertreduceswelfare. 10Notethattheplannerdiscountsfelicityatthesamerateasthehouseholdswouldthemselves.Considerachangeinallocationswhichreducesthedatetfelicityofcohortsbydutandincreasestheirdatet+1felicitybydut+1,whilekeepingthefelicityatallotherdatesandforallotheragentsthesame.Acohortsindividualwillbeindierentregardingthischangeifdut=#dut+1.Fromtheplanner'sperspectivethischangesaggregatewelfareby�#s�tdut+#s+1�tdut+1.Thus,theplannerwillbeindierentaboutthischangeifandonlyiftheindividualsthemselvesareindierent.AsdiscussedbyCalvoandObstfeld(1988),assumingthattheplannerandthehouseholdssharethesam

16 erateoftimepreferenceensuresthatsocialpr
erateoftimepreferenceensuresthatsocialpreferencesaretime-consistent,sothattherst-bestintertemporalallocationofconsumptionacrosscohortsdoesnotchangeovertime.Thisdoesnotpreventotherfromoftimeinconsistenciesfromarisingindecentralizedequilibrium(asshownbelow),buttheseareunrelatedtotheformofsocialpreferences.11 AppendixB.2.1showsthattheevolutionoftfort�0canbewrittenas:lnt=1 2 22tw2t2t+ln[1�#+#t�1](22)withln0=1 2 220w2020+ln[1�#+#�1]+ln0@1�#e 2 1�#e 20 E�1021A| {z }eectofdate0surprise/URE(23)where�1==(1�#)e 2 1�#e 2isthesteadystate.11Intuitively,highercash-on-handriskw2t2tandahigherpass-throughtbothtendtoincreaseconsumptioninequality.Inaddition,consumptioninequalityinheritstheslowmovingdynamicsofwealthinequality,ascanbeseenfromthepresenceoft�1in(22).12Equation(23)showsthattherelationbetween0and0isdierentthantherelationbetweentandtatallotherdates.Thiscanbeexplainedintuitivelyasfollows.Atthebeginningofdate0,thedistributionofwealthaisatitssteadystatelevel:somehouseholdshavepositivenetwealthandsomearedebtors.Sincesaversanddebtorshavedierentunhedgedinterestrateexposures(UREs)inthesenseofAuclert(2019),anunanticipatedchangeininterestratesaectsconsumptioninequality.Supposethatatdate0,thecentralbankchoosesapolicypaththatimplementsatransitorydropintherealinterestrate.Thelowerinterestratesbenetpoordebtors,reducingtheirinterestpaymentsandallowingthemtoincreasetheirconsumption.Bythesametoken,alowerpathofratesreducestheinterestincomeofrichsavers,causingthemtoreduceconsumption.Inotherwords,lowerratesreducetheMPCoutofwealth(0#)whichreducesconsumptioninequalityandhence0.Importantly,ananticipatedcutinrateswouldnotreduceinequalityasmuchasthisunanticipatedcut.Ifwealthyagentsatdatet�1anticipatedlowerratesatdate0,theywouldsavemoreinordertoinsureahigherlevelofconsumptionatdate0.Equally,thepoordebtorswouldborrowmoreatdate-1knowingthattheirdebtwouldbelesscostlytorepay.Forthisreason,whatreduces0throughthischannelisnotafallin0persebutafallin0relativetoitsex

17 pectedvalueE�10,ascanbeseenfromt
pectedvalueE�10,ascanbeseenfromthelasttermin(23).Tobeclear,anticipatedcutsinratesdoreduceinequalityasdiscussedearlier:lowertreducestinequation(22).Butthereisanadditionaleectthatcomesfromasurprisefallinrates.Inourenvironment,sincewedonothaveaggregateshocks(exceptfortheunanticipatedshockatdate0)andthefactthattheRamseyplannerisonlyallowedtore-optimizeatdate0implythatthisadditionalaectofanunanticipatedchangeincanonlyoccuratdate0.Moregenerally,inanenvironmentwithaggregateshocks,surprisechangesinwouldhavethiseectonanydate,forexamplewhenthereisanaggregateproductivityshockandzt6=Et�1zt.Ofcourse,thisone-oredistributionwouldnotoperateintheabsenceofwealthinequalitiesattime0.Iftheeconomywerestartingwithequal(zero)wealthforallhouseholds,insteadofstartingfromtheinvariantdistribution,thenonlythersteectwouldplayoutandtheequilibriumvalueofconsumption 11Weareassumingthattheeconomyisinsteadystateatdate�1.12Notethatwithin-cohortconsumptiondispersion2c(t;s)ingeneralriseswithoutboundsasthecohortages(i.e.,ast�s!1),duetothecumulatedeectofidiosyncraticshocksonthedistributionofcash-on-hand.However,sinceeverycohortgraduallyshrinksinsize,whilenewborncohortshavelittleconsumptiondispersion(i.e.,2c(t;t)=2tw2t2t),tdoesnotnecessarilyblowup.Infact,providedthatthesurvivalrate#e�=2,tisstationary.12 dispersionattimezerowouldsimplybe:ln0=1 2 220w2020:(24)5OptimalmonetarypolicyTheplannerchoosessequencesfwt;t;t;t;it;ntg1t=0tomaximizeP1t=0tu�yt;nt; 
tsubjecttotheaggregateEulerequation(16),agggregatelaborsupply(14),theevolutionoft(12),thePhillipscurve(7),theevolutionoft(22)-(23)andtherelationshipbetweenGDPandwages(15).IntheRANKversionofoureconomy,=0and(22)isreplacedbyt=1forallt.AppendixDpresentstheLagrangianassociatedwiththisproblemalongwiththerstordernecessaryconditionsforoptimality.5.1LongrunoutcomesundertheoptimalRamseyplanandthepayrollsubsidyToproceedfurther,weneedtotakeastandonthevalueofthepayrollsubsidy.InRANK,ifweimposedaproductionsubsidytoeliminatethedistortionscausedbymarketpower,zeroin atio

18 nisoptimalinthelongrunintheabsenceofaggr
nisoptimalinthelongrunintheabsenceofaggregateshocks.ThisneednotbetrueinourHANKeconomywhere�0andsotisendogenous.AsinthestandardNKmodel,thePhillipscurve(7)impliesalong-runtrade-obetweenin ationandeconomicactivity:(�1)= (1�e#�1)(1�) 1�1 (1�)w(25)Thepresenceofalong-runtradeoimpliesthatthepolicymakercanmovewages(orequivalentlyoutput)aboveorbelowits exible-pricelevelbypersistentlydeviatingfrompricestability.WhileitisnotoptimaltodosoinRANK,itmayinfactbeoptimalinHANKbecausethelevelofeconomicactivityaectsboththeamountofincomeriskhouseholdsfacew2t2tandtheirabilitytoself-insureagainstthisrisk,t.Forexample,ifincomeriskiscountercyclical,�1theplannermaywanttocreatein ationtoraisewages(andoutput)abovethelevelconsistentwithproductiveeciency(w�1),andtherebyreduceincomerisk.TomakeourresultsascomparableaspossibletotheclassicNKliteratureonoptimalmonetarypolicy,weeliminatethismotivefordeviatingfrompricestabilitybyintroducinganappropriatelychosenpayrollsubsidy.Toseehowthisworks,considerthecasewithcountercyclicalriskinwhichtheplannerwantstoimplementahighafter-taxwagew�1inthelongruninordertoreduceinequality.From(25)itiseasytoseethatif=�1 (thestandardsubsidyusedintheRANKliterature,suchahighlevelofwagesentailsamarginalcostgreaterthan1whichimpliespositivelongrunin ation�1.However,ifthepayrollsubsidyislarger,thesteadystatemarginalcostcanbebroughtdownto1,consistentwithzeroin ationinthelongrun.Moregenerally,AppendixD.1showsthatinthepresenceofanappropriatelychosenpayrollsubsidytoensurezeroin ation=1,theplanner'srst-orderconditionforwages-whichstatesthatthenet-benet13 ofhighereconomicactivitymustbezeroatanoptimum-becomes:13 |{z}benetfromreductionofinequality (1�e)(1�)| {z }reductionofinequalityduetolowinterestrates+�1 (1�e)(1�)| {z }reductionofinequalityduetohighoutput=w�1 1+ w| {z }costofdeviatingfromproductiveeciency(26)Equation(26)impliesthatthesteadystatewageandpayrollsubsidyconsistentwiththeoptimalityofzerolong-runin ations

19 atisfy:w=1+ 1�  and=&
atisfy:w=1+ 1�  and=�1 +1+   +1(27) summarizesthebenetfromareductioninconsumptioninequalityduetohighereconomicactivity.Theplannerhastheoptiontoreduceinterestratesandraiseoutputabovethe exiblepricelevel.Equation(27)statesthatatanoptimum,themarginalbenetoflowerinequalityduetolowerratesandhigheroutput, ,mustequalthemarginalcostofdistortingproductiveeciencybyraisingoutput(andwages)abovethe exiblepricelevelwhichisproportionaltow�1,thenegativeofthelabor-wedge.Absentuninsurablerisk(=0;=1),thereisnoinequalityandsothereisnobenetfromhighereconomicactivityintermsofreducinginequality, =0.Consequently,inRANK,optimalpolicyequatesthecostofdeviatingfromproductiveeciencyinsteadystateto0andhencew=1inthelongrun.Inthiscase,=1insteadystatecanbeimplementedwiththestandardRANKsubsidy=�1 (fromeq.(27))whichremovesthemonopolisticdistortion.Inthepresenceofrisk(�0),optimalmonetarypolicyseekstoreduceconsumptioninequality.Thiscanbeaccomplishedbothbyreducinglaborincomeriskw2t2tandbymakingiteasierforhouseholdstoself-insureagainstthisrisk(byreducingt).Thusthepolicymakermaywanttodeviatefromproductiveeciencyandpricestability,bothtofacilitateself-insuranceandtoreduceincomerisk;equation(26)showsthatand�1representthestrengthofthesetwomotivesrespectively.Considerrsttheself-insurancechannel.When=1,incomeriskisacyclical:thelevelofeconomicactivitydoesnotaecthouseholdincomerisk.Inthiscase,deviatingfrompricestabilityandproductiveeciency(saybyimplementingawagew�z=1)deliversnobenetsintermsoflowerincomeinequality(secondtermontheLHSof(26)iszero).However,loweringrealinterestratesstillmakesiteasierforhouseholdstosmoothconsumptionbyborrowingandreducesthepass-throughfromincomeshocksintoconsumption,measuredbythersttermoftheLHS,reducingconsumptioninequality.Lowerinterestratesandtheassociatedhigherlevelofeconomicactivityalsohaveacost,sincetheydistortoutputandemploymentabovetheproductivelyecientlevel.Optimalpolicyequatesthesecostsandbenets.So,evenwithacyclicalrisk, �0,andoutputisoptimallyaboveproductivee

20 ciencyinthelongrun.Sincehigherwagesincre
ciencyinthelongrun.Sincehigherwagesincreasethermsmarginalcost,ittakesahigherpayrollsubsidythaninRANK��1 toimplement=1insteadystate.Next,considertheincomeriskchannel.When�1,incomeriskiscountercyclical:highereconomicactivitylowersincomerisk.Inthiscase,stimulatingoutputaboveitsproductivelyecientlevellowersconsumptioninequalityevenforaxed;inaddition,thelowerinterestratesnecessarytoimplement 13Notethatherethereisnotermrepresentingthecostofin ation,preciselybecauseweassumethatwhateverthesteadystatelevelofwages,theappropriatepayrollsubsidyischosensothatrms'marginalcostsareconsistentwithzeroin ationinthelong-run,i.e.,(25)holdswith=1.14 higheroutputreducefurtherreducingconsumptioninequality.Thus,thebenetfromhigheroutputisevenlargerthanif=1-bothLHScomponentsin(26)arepositiveand islarger-andsooutput(andwages)mustbeevenfurtherabovetheproductiveecientlevelthanif=1.Again,ittakesahigherpayrollsubsidy��1 toimplement=1insteadystate.Incontrast,whenriskisprocyclical(1),theeectofhighereconomicactivityandlowerratesonconsumptioninequalityareambiguous.Whilelowerratesdecreasethepassthroughfromincomeshockstoconsumption(&#x]TJ/;ø 1;�.90;‘ T; 11;&#x.615;&#x 0 T; [0;0),theyalsoraiseoutputwhichnowincreasesincomerisk(�10).Forsucientlyprocyclicalrisk,thesecondeectdominates, 0andtheoptimalsteadystatelevelofoutput(andwages)isbelowtheproductivelyecientlevel.Inthiscase,ittakesalowerpayrollsubsidythan�1 toimplement=1insteadystate.Formildlyprocylicalrisk,theself-insurancechanneldominatesand �0withw�1insteadystate.Theself-insurancechannelisperfectlybalancedbytheincomeriskchannelif1�=inwhichcase =0;highereconomicactivityandlowinterestrateshavenorstordereectonconsumptioninequalityandinthiscase,theplannerdoesnotwishtodistortproductiveeciencyinsteadystate,settingw=1.The =0casewillbeausefulbenchmarkinwhatfollows.Remark1(ComparisonwithZero-LiquidityLimits).Theself-insurancechannelisabsentinmodelswhichfeatureincompletemarketsbutimposethezeroliquiditylim

21 itsuchasBilbiie(2019a,b);RavnandSterk(20
itsuchasBilbiie(2019a,b);RavnandSterk(2017);Challe(2020)andothers.Zeroliquidityimpliesthatinequilibrium,thepassthroughfromincomerisktoconsumptionriskisinvarianttopolicysinceitalwaysequals1.Instead,ouranalysisemphasizesthatinHANKeconomiesinterestratepolicyaectswelfarenotjustviaitseectonthelevelofeconomicactivity,asinRANK,butalsobyaectingtheeasewithwhichhouseholdscanself-insure.AsisstandardintheNKliterature,ausefulbenchmarkisthelevelofoutputunder exibleprices.Ina exible-priceversionoftheHANKeconomy,wewouldhavewt=wztatalltimes,andoutputwouldbeynt=zt(lnw+lnzt)+ 1+ ztwhiletheecientlevelofoutputis:yet=ztlnzt+ 1+ ztInRANK,the exible-priceandecientlevelsofoutputcoincide:yet=ynt.ThisisalsotrueinHANKwith =0.Butingeneral,when 6=0,the exible-priceandecientlevelsofoutputnolongercoincide.Withstronglyprocyclicalrisk 0,the exible-pricelevelofoutputyntisalwaysbelowitsecientlevelyet,whilewithmildlyprocyclicalorcountercyclicalrisk &#x]TJ/;ø 1;�.90;‘ T; 12;&#x.411;&#x 0 T; [0;0,yntisalwaysaboveyet.Usingthesedenitions,wecanexpressthelinearizedversionofthePhillipscurve(7)as:t=e #t+1+(byt�bynt)(28)where=1+    �1,byet=+1 1+ bztandbynt=+y 1+ bzt.5.2CalibrationWhileourresultsareprimarilyanalytical,whenplottingIRFsweparameterizethemodelasfollows.Wechoosetonormalizeaggregatesteadystateoutputyto1intheHANKeconomywith =015 (equivalently,intheRANKeconomy).Wecalibratethemodeltoanannualfrequencyandchoosethestandarddeviationofst(i),,sothatthestandarddeviationofincomeinsteadystateequals0:5.14ThisisinlinewithGuvenenetal.(2014)whousingadministrativedatandthestandarddeviationof1yearlogearningsgrowthratetobeslightlyabove0.5.WesettheslopeofthePhillipscurve=0:01,andtheelasticityofsubstitutionbetweenvarieties �1to10,implyinga10percentsteadystatemarkup,=1:1.Wesetthecoecientofrelativeprudenceforthemedianhousehold,�cu000(c) u00(c)= ,tobe3,withintherangeofestimatesintheliterature(seee.g.Cagetti(2003);Fagerengetal.(2017);Christelisetal.(2015)).Wese

22 tr=4%.canbeinterpretedastheFrischel
tr=4%.canbeinterpretedastheFrischelasticityoflaborsupplyforthemedianhousehold;wesetitequalto1=3,withintherangeofestimatesfromthemicroliterature.Wesetthepersistenceoftheshock%z=0:8.FollowingNistico(2016),weset#=0:85.Finally,weconsidertwovaluesforthecyclicalityofincomerisk:= (whichimplies =0),and=�3(whichimplies �0andcountercyclicalrisk).Finally,weset =1+ tonormalizetheecientlevelofoutputinsteadystateto1.5.3DynamicsunderoptimalmonetarypolicyunderRANKAsiscommonintheNKliterature,wecharacterizeoptimalpolicybylinearizingtherstorderconditionsarisingfromtheplanner'sLagrangian(presentedinAppendixE).Itisusefultocomparethischaracteri-zationtooptimalpolicyinaRANKversionofoureconomy.Lemma1(OptimalmonetarypolicyinRANK).InRANK(=0;=1),outputandin ationfbyt;tg1t=0underoptimalpolicysatisfyt=byt�bynt(29)t=t�1� �1t(30)t=e #t+1+(byt�bynt)(31)wheretisthe(normalized)multiplieronthePhillipscurve(7)andisdenedinAppendixE.1.Proof.SeeAppendixE.1. Combining(29)-(30),weseethatoptimalpolicyinRANKsatisesthestandardtargetcriterion:15(byt�bynt)��byt�1�bynt�1
+ �1t=0(32)whereby�1=bye�1=0.Combiningthiswith(31)andusingthefactthatbynt=byetinRANK,weseethattheeconomyfeaturesadivinecoincidence:itisbothfeasibleandoptimalformonetarypolicytosetbyt=byetandt=0atalldatesandstates.Giventheappropriatesteadystatesubsidy=�1 ,the exible-pricelevelofoutput,whichisalsoconsistentwithzeroin ation,maximizessocialwelfare{thereisnotradeobetweenimplementingtheecientlevelofoutputandpricestability. 14Thestandarddeviationofincomeisgivenby(1� w)w.Wecalibrateallparametersexceptthecyclicalityofincomerisktoaneconomywith =0,whichimpliesw=1.15Seeforexample,chapter5inGal(2015).16 5.4DynamicsofmonetarypolicyunderHANKInHANK,theplannerhasanadditionalobjectiverelativetoRANK:inadditiontostabilizingin ationandkeepingoutputclosetoitsecientlevel,shewantstokeepinequalitytaslowaspossible.Theinnovationstoinequalitydependonconsumptionrisk2t2twhichinturnd

23 ependsonbothincomerisk2tandthepass-
ependsonbothincomerisk2tandthepass-through2t.Thiscanbeseenfromthelinearizedversionof(22)whichisgivenby:bt =8���&#x]TJ ;� -1; .63; Td;&#x [00;&#x]TJ ;� -1; .63; Td;&#x [00;&#x]TJ ;� -1; .63; Td;&#x [00;: (1�)byt+bt| {z }consumptionrisk+# Rbt�1 fort�0 (1�)by0+b0+# 1�#b0fort=0(33)Onewaytoreduceconsumptioninequalityistoaectthelevelofoutput:withprocyclicalrisk(1)loweroutputdirectlyreducesincomerisk(&#x]TJ/;ø 1;�.90;‘ T; 12;&#x.243;&#x 0 T; [0;1)facedbyhouseholdswhilewithcountercyclicalrisk,ahigherlevelofoutputisnecessarytoreduceincomerisk.Analternativepathtolowerconsumptioninequalityistocommittoalowerpathofinterestrateswhichreducespass-throughfromincometoconsumptionrisk.However,theplanneronlyhasoneinstrument-thenominalinterestrate.Loweringthenominalinterestratelowersthepass-throughfromincomerisktoconsumptionrisk(measuredbyt)butincreasesoutput.Ifriskiscountercyclical&#x]TJ/;ø 1;�.90;‘ T; 12;&#x.243;&#x 0 T; [0;1,thenthistooreducesconsumptionrisk.However,ifriskisprocyclical1,thenitincreasesincomerisk,leavingtheoveralleectonconsumptionriskunclear.Toseewhatcombinationsoffbyt;btgthattheplannercanimplementwithsomepathofnominalinterestrates,combinetheISequation(19),recursion(20)usingthedenitionofGDP(21)andsolveforwards: h1+1�e ibyt+bt=1�e(1+ ) y y+1Xs=0es(1�)sbynt+s�t(34)where�tisanexogenousprocessdrivenbythesequencefyntg,whichinturndependssolelyonfbztg.5.4.1HANKwith =0Tounderstandthetrade-osfacingtheplanner,itisusefultoconsiderthespecialcaseinwhich =0(orequivalently1�=).Recallfrom(26)thatthisisthecaseinwhichthezeroin ationsteadystatefeaturesproductiveeciency(=1;w=1).Thisbenchmarkfeaturesmildlyprocyclicalrisk:whilethismaynotbetheempiricallyrelevantcase,itisausefulbenchmarkbecauseinthiscase,theconstraintontheplanner'sabilitytoaectconsumptioninequalityisparticularlysevere.Recallthatwhenriskisprocyclical,theeectofexpansionarymone

24 tarypolicyonconsumptioninequalityisgener
tarypolicyonconsumptioninequalityisgenerallyambiguous:higheroutputreducesthelevelofincomerisk,butlowerinterestratesreducesthepassthroughfromincometoconsumptionrisk.When =0,boththeseeectsexactlycanceleachotheroutandconsumptionisinvarianttomonetarypolicytorst-order.Toseethis,notethat(33)becomes:bt =8���&#x]TJ ;� -1; .63; Td;&#x [00;&#x]TJ ;� -1; .63; Td;&#x [00;&#x]TJ ;� -1; .63; Td;&#x [00;:( byt+bt)+# Rbt�1 fort�0( by0+b0)+# 1�#b0fort=0(35)17 while(34)becomes: byt+bt=�t(36)Clearly,inthiscase,theplannercannotaecttheevolutionofconsumptionriskfordatest�0whichissolelydrivenbyexogenousshocksfyntgt0denoted�tin(36).Pluggingin(36)into(35)showsthattheevolutionofinequalityafterdate0isgovernedcompletelybytheexogenoussequencef�tg:bt =8���&#x]TJ ;� -1; .63; Td;&#x [00;&#x]TJ ;� -1; .63; Td;&#x [00;&#x]TJ ;� -1; .63; Td;&#x [00;:�t+# Rbt�1 fort�0�0+# 1�#b0fort=0(37)Again,whileacutininterestrateslowersbt,itincreasesoutputbytandhenceincomerisk,leavingcon-sumptionriskunchanged.Ahigherpathofaggregateproductivityfbztg(whichimpliesahigherpathoffbyntg)increasesconsumptionriskinthiscaseandmonetarypolicycannotdoanythingtopreventit:higherproductivitytendstoincreaseoutputandhencethelevelofincomeriskthathouseholdsfacebuttightermonetarypolicy,whichwouldbeneededtoforestalltheriseinoutput,tendstomakehigherwhichitselfincreasesconsumptionrisk.Eventhoughchangesininterestrates(andhence)cannotaectconsumptioninequalityafterdate0,theplannercanaectconsumptioninequalityatdate0(andthusatallsubsequentdates,becausebtdependsonbt�1).Thisisbecausemonetarypolicychangesafterdate0areanticipatedwhilethedate0changeinmonetarypolicyisunanticipated.Asdescribedinsection4,anunanticipatedcutininterestrates(andhence0)eectivelyredistributesfromsaverstoborrowers.Tobeclear,sincewehaveaneconomywhereagentsholdreal(andnotnominal)claims,thisisnotbecausein ationredistributesdate0realwealthf

25 romsaverstoborrowers:thedate0distributio
romsaverstoborrowers:thedate0distributionofrealwealthisunaected.16Butthedistributionofconsumptionisaected,asrichsaversndthattheyreceivealowerreturnontheirbondholdingsthantheyhadanticipatedwhilepoordebtorsndthattheirinterestpaymentsaresmallerthantheyhadexpected.Insum,whiletheplannerseekstoreduceconsumptioninequalityinadditiontostabilizingpricesandthegapoutputandits exible-pricelevel,thisisnotpossibleafterdate0since(torstorder)theevolutionofconsumptioninequalityisunaectedbypolicy.Eectively,thenafterdate0,theplannerfacesthesametrade-obetweenoutputandin ationasintheRANKeconomy.Atdate0,itispossibletoreduceconsumptioninequalityviaasurprisecutininterestrateswhichexploitshouseholds'unhedgedinterestrateexposure.Thus,theplannerhasanadditionalmotivetocutratesatthisdate.Thisisre ectedintheoptimaldesignofmonetarypolicy,aswenowdemonstrate.Proposition3(Optimalmonetarypolicywith =0).Outputandin ationfbyt;tg1t=0underoptimal 16Section6discussesthecasewherehouseholdsholdnominaldebt.18 policysatisfyt=8:(byt�bynt�)fort=0byt�byntfort1(38)t=1 Rt�1� �1t(39)t=e #t+1+(byt�bynt)(40)where�1;01and&#x]TJ/;ø 1;�.90;‘ T; 31;&#x.322;&#x 0 T; [0;0aredenedinAppendixE.1andtisthe(normalized)multiplieronthePhillipscurve(40).Proof.SeeAppendixE.1. Combining(38)and(39),wegetthefollowingtargetcriterionforalldatest�1:(byt�bynt)�1 R�byt�1�bynt�1
+ �1t=0(41)Comparingequations(32)and(41)showsthatwhen =0thetargetcriteriainHANKandRANKfort�0arealmostidentical(undercompletemarketswehaveR=1sotheformercollapsestothelatter).Thisre ectsthatfactthatmonetarypolicycannotaectconsumptionriskatdatest�0,andthusfacesthesametradeoasRANK.Butthisisnottrueatt=0,wherethetargetcriterionis:byt�bynt+1  �1t=(42)Equation(42)showsthatattime0optimalpolicyinHANKdeviatesfromthatinRANKinthreeways.First,itisoptimaltocreateaboomatdate0.Toseethismostclearly,supposethatproductivityisatitssteady-statevalue,sobynt=b

26 zt=0forallt.Eveninthiscase,itisoptimalto
zt=0forallt.Eveninthiscase,itisoptimaltomoveawayfromby0=0=0andimplementanoutputboomby0�0whichisaccompaniedbyin ation0�0.Thisisbecause,asdiscussedearlier,theevolutionoftisdierentatdate0,comparedtoallotherdates.Asurprisecutininterestratesatdate0reducesconsumptioninequality{asissummarizedbythelasttermof(37)fort=0.Theconstantterm�0inthedate-0targetcriterion(42)re ectsexactlythisbenetfromcuttingratesandreducinginequalityatdate0.Thisisnotbecauseitisinfeasibletosetbyt=byet=0andt=0intheHANKeconomy;thisremainsfeasible,anditremainscostlyfortheplannertodeviatefromthisbenchmark.Butthecostsofdoingsoarebalancedbybenetsofreducingconsumptioninequalityviaasurpriseinterestratecut.Notethatthedesirabilityofexploitinghouseholds'unhedgedinterest-rateexposureforredistribution(theUREchannel)makestheRamseyplantime-inconsistent.SupposethattheplannerhasbeenfollowingaRamseyplansincet=�1andtheeconomyhasconvergedtosteadystate.Giventheopportunitytodeviatefromthisplanatdate0,theplannerwoulddoso,loweringinterestratestemporarily-i.e.thecontinuationofaRamseyplanisnotaRamseyplan.Second,whereasinRANK,optimalpolicyseekstomoveoutputbytone-for-onewithitsecientlevelbyet(=byntinRANK),which uctuatesduetoproductivityshocks,underHANKitisoptimaltotrackthe exible-pricelevelofoutputbyntlessthanone-for-one.Inotherwords,01in(42).Figure1depictstheoptimallevelofdate0outputasafunctionof exible-pricelevelofoutputbyntinHANKandRANK.19 Figure1.Optimallevelofby0inHANK( =0)andRANKabsentmarkupshocksFirst,suppose,byn0=0.InRANK,itisoptimaltotrackthisandtosetby0=byn0=0.ButinHANKthereisarstorderbenetfromcuttinginterestratestoreduceinequality,creatingaboominoutput,untilthemarginalbenetofanadditionalreductioninnominalratesisoutweighedbythecostofdistortingoutputfurtheraboveitsecientlevel(pointAinFigure1).Next,supposethatbyn0�0.Again,theRANKplannersetsby0=byn0(denotedbypointBintheFigure).IftheHANKplannerwerealsotosetby0=byn0�0,thiswouldalreadygenerateasurprisefallininterestrates,whichwouldreduceinequalitytosomeextent.TheHANKplannerstillperceivessomeadditionalbenettoreducinginequalityfurtherbut

27 themarginalbenetissmallersinceinequ
themarginalbenetissmallersinceinequalityhasalreadybeenreduced.Consequently,itisnotoptimaltodeviateasmuchfromproductiveeciencyasinthecasebyn0=0,andsothegapbetweenpointsCandBissmallerthanthatbetweenpointsAandtheorigin.Conversely,whenbyn00,trackingthe exiblepriceallocationwouldentailasurpriseinterestrateincreasewhichwouldincreaseconsumptioninequality.ThebenetofdeviatingfromthisRANKallocationislargerinthiscaseandthereforeitisoptimaltotoleratealargerdeviationfromproductioneciency:thegapbetweenpointsDandEislargerthanthatbetweenpointAandtheorigin.Finally,thedate0targetcriterionputslessweightonin ationrelativetooutput&#x]TJ/;ø 1;�.90;‘ T; 12;&#x.518;&#x 0 T; [0;1.Thisisbecause,giventheconstraintonmonetarypolicyimposedby(36),theevolutionofconsumptioninequalitydependsdirectlyonthesequencesfbyt;byntg1t=0butnotonin ation.Thus,thetargetcriterionputsrelativelylessweightonin ationandmoreonoutput,whichattainsheightenedimportanceduetoitseectsoninequality.Figure2showsthedynamicsofoptimalpolicyabsentshocksintheeconomywith =0.Thepolicymakercutsnominalinterestratesatdate0(paneld),whichgeneratesafallinrealratesandinthepassthroughfromincomerisktoconsumptionriskt(panele).Thefallinconsumptionriskinturnreducesconsumptioninequalityonimpact(panelc),afterwhichitgraduallyreturnstosteadystate.Thefallininterestratesalsogeneratesaboominoutput(panela)andin ation(panelb).Sincetherearenoshockstotheecientlevelofoutputinthisscenario,boththeboominoutputandincreaseinin ationareinthemselvesundesirablefromaneciencyperspective,eveniftheyareapriceworthpayingforapersistentreductionininequality.Inordertoarresttheincreaseinin ation,theplannercommitstotightenpolicy,generatingafallinoutputandde ation,fromdate1onwards.Sincermsareforwardlooking,acommitmenttolowerin ationinthefuturemitigatesthedate0in ationcausedbytheboom20 Figure2.TimeInconsistency:Optimaldynamicsabsentshocks.Allvariablesareplottedaspercentagedeviationsfromtheirsteadystatevalues.inoutput.Next,wediscusstheoptimaldynamicresponseoftheeconomytoanegativeproductivityshock.Figure3showstheimpulseresponsestoanegativeproductivityshockunderoptimalpolicyintheHANKa

28 ndRANKmodels,denedasthedierenc
ndRANKmodels,denedasthedierencebetweenoutcomeswithandwithouttheshock.Thatis,foranyvariableofinterestx,weplotxtjz0=�0:01�xtjz0=0.17IntheRANKeconomy(redlines),theplannerallowsoutputtofall(panela)inlinewiththenaturalrateofoutput,keepingtheoutputgap(panelf)andin ation(panelb)equaltozero.Implementingafallinoutputrequiresanincreaseinnominalinterestrates(paneld),butsinceagentsdonotfaceidiosyncraticrisk,thishasnoeectonconsumptioninequalitywhichisalwayszero.IntheHANKeconomy(bluelines),asharpincreaseininterestrateswouldincreasethepassthroughfromincometoconsumptionriskandwouldpersistentlyincreaseinequality.Toavoidthis,theplanneractuallycutsinterestratesatdate0,dampeningtheincreaseinpassthroughtandtheincreaseininequality.Outputdoesnotfallasmuchasthe exiblepricelevelofoutput(asshownbytheoutputgapinpanelf),andasaresultin ationincreasesatdate0-evenrelativetothescenarioabsentshocks.Thisincreaseinin ationisnotcostless.Tomitigateitsimpact,theplannercommitstoaslightlyhigherpathofinterestrates,andloweroutputandin ation,fromdate1onwards.UnderstandingtheforcesgeneratingtimeinconsistencyThedierenceintheoptimaltargetcriterionthanatallotherdatesarisesonlybecausethereisexistingwealthinequalityatthestartofdate0.Supposethatthereisanequalizationofassetpositionsacrossallhouseholdsatthebeginningofdate0. 17IntheRANKeconomy,theimpulseresponseisidenticaltotheresponseoftheeconomytotheshock,sinceintheabsenceofshocks,optimalpolicykeepsallvariablesattheirsteadystatevaluesandalldeviationsfromsteadystateequalzero.ThisisnottrueintheHANKeconomy,where(asshowninFigure2)theRamseyplannerdeviatesfromsteadystateevenabsentshocks.21 0246810 -0.0200.020.040.060.08 0246810 -2-10123 10-3 0246810 -0.02-0.015-0.01-0.0050 0246810 -0.3-0.2-0.100.1 0246810 -0.25-0.2-0.15-0.1-0.0500.05 Figure3.HANK( =0)vsRANKoptimalresponsetoanegativeproductivityshock.RedlinedenotesRANKwhilebluelinerepresentsHANK( =0).Allvariablesareplottedaspercentagedeviationsfromtheirsteadystatevalues.Inpanelf),weplotoutputgapasthedierencebetweenytandyntnormalizedbysteadystatey:(byt=y)�(bynt=y).Inthisscenario,theUREchannelisinoperative,asdiscussedinsection4andsurprisechangesin0have

29 noadditionaleectrelativetoanticipat
noadditionaleectrelativetoanticipatedchanges.Whenthereisexistingwealthinequality,asurprisecutinthepathofinterestrates(lower0)lowerstheconsumptionofrichsaversandraisestheconsumptionofpoorborrowers,reducingconsumptioninequality.Consequently,att=0,theplannerdeviatesfromthedatet�1targetcriterionbycuttinginterestratesdoengineerpreciselythisoutcome.Absentexistingwealthinequalityatthebeginningofdate0,clearlytherearenoborrowersandsaverstoredistributebetweenandthismotiveisabsent.Infact,with =0,absentinitialwealthinequality,equation(37)becomes:bt =�t+# Rbt�1 8t0(43)withb�1 =�R #ln1�# 1�#e 20.Equation(43)showsthatwithoutinitialwealthinequality,theplannerisunabletoaecttatanydate(uptorst-order).Unsurprisinglythen,theplannerfacesthestandardtrade-obetweenpricestabilityandproductiveineciency.Consequently,theoptimaltargetcriterionatalldatesinthiscasesimpliesto:(byt�bynt)�1 R�byt�1�bynt�1
+ �1t=0(44)withby�1=byn�1=0andsodivinecoincidenceholds-optimalpolicyimplementst=0andbyt=byntatalldatest0inresponsetoproductivityshocks.Remark2(TheinterestrateexposurechannelversustheFisherchannel).Previousresearchhasshown22 0246810 -0.04-0.03-0.02-0.010 0246810 -4-20246 10-4 0246810 -1-0.500.511.52 10-3 0246810 -0.03-0.02-0.0100.010.02 0246810 00.010.020.030.040.050.06 0246810 -5051015 10-3 howtheredistributionofwealththroughin ationwhenassetsandliabilitiesarenominalmaycreateanin ationarybiasonthepartofthecentralbank.This\Fisherchannel"isabsentinourbaselineeconomywherehouseholdsonlyholdrealassets:theplannerisunabletoredistributerealwealthviasurprisein a-tion.Nonetheless,theplannercanredistributeconsumptionviaasurprisecutinthepathofrealratesviatheinterestrateexposurechannel(Auclert,2019).Auclert(2019)studieshowboththesechannelmediatetheeectofinterestratechangesonaggregateconsumptioninanenvironmentwithMPCheterogeneity.Inoureconomy,allagentshavethesameMPC,somonetarypolicydoesnotaectaggregateconsumptionthroughthesechannels.However,itdoesaectthedistributionofconsumptionandinourbasel

30 ineecon-omy,optimalmonetarypolicydeliber
ineecon-omy,optimalmonetarypolicydeliberatelyutilizestheinterestrateexposurechanneltoreduceconsumptioninequality.InSection6,whenweintroducenominalbondsintoourmodel,optimalmonetarypolicyexploitsbothchannelstoachievethesameends.5.4.2OptimalpolicywithcountercyclicalriskWhilethecasewith =0isausefulbenchmarktounderstandtheforcesdrivingoptimalmonetarypolicy,theempiricallyrelevantcaseisoneofcountercyclicalrisk,whichimpliesthat �0.18Inthiscaseacutininterestrates,andtheassociatedboominoutput(andwages),reduceconsumptionriskthroughboththeincomeriskandself-insurancechannels.With =0,riskwasprocyclical,andthebenetoflowerinterestrates(whichlowerandincreasehouseholds'abilitytoselfinsure)wasexactlybalancedbythecostoftheassociatedboominoutputwhichincreasedthelevelofincomeriskfacedbyhouseholds.Incontrast,whenriskiscountercyclical,theincreaseinoutputactuallyreducesthelevelofincomerisk,atthesametimeaslowerinterestratesimprovehouseholds'abilitytoself-insure:expansionarypolicyunambiguouslyreducesconsumptionriskandinequality.Thiscanbeseenbysubstituting(34)into(33):bt =8���&#x]TJ ;� -1; .63; Td;&#x [00;&#x]TJ ;� -1; .63; Td;&#x [00;&#x]TJ ;� -1; .63; Td;&#x [00;:� 1�e byt+�t+# Rbt�1 fort�0� 1�e by0+�0+# 1�#b0fort=0(45)When �0,anincreaseinoutputbytreducestheconsumptionriskfacedbyhouseholdsatanydate,notjustatdate0:@bt @^yt0.Inthisenvironment,theplannerwouldonlyrefrainfromincreasingoutputifdoingsodistortsproductiveeciencyandcreatesin ation.Indeedrecallfrom(26)thatinsteadystatewith &#x]TJ/;ø 1;�.90;‘ T; 12;&#x.178;&#x 0 T; [0;0,themarginalbenetofreducinginequalitybystimulatingdemandisexactlybalancedbythecostofdeviatingfurtherfromproductionineciency.Thereasonamarginalincreaseinoutputhasarst-ordercostintermsofproductiveeciencyisthatthesteadystateinthiscasefeaturesaninecientlyhighlevelofoutputy&#x]TJ/;ø 1;�.90;‘ T; 12;&#x.178;&#x 0 T; [0;1(alsore ectedinw&#x]TJ/;ø 1;�.90;‘ T; 12;&#x.178;&#x 0 T; [0;1).Ou

31 tofsteadystatetheplannerfacesasimilartra
tofsteadystatetheplannerfacesasimilartrade-obetweenthebenetofreducinginequalityviahigheroutputandthecostoffurtherdistortingproductiveeciency,whilealsoseekingtolimitdeviationsfrompricestability.Thistradeoinformstheoptimaldesignofmonetarypolicy,aswediscussnext. 18Moreprecisely,countercyclicalrisk(�1)implies � (1�e)(1�)�0.When0  (1�e)(1�),riskisweaklyprocyclicalsoexpansionarymonetarypolicyreducespassthroughthanitincreaseincomerisk,reducingconsumptionriskinnet.23 Proposition4(Optimalmonetarypolicywithcountercyclicalrisk).Outputandin ationfbyt;tg1t=0underoptimalpolicysatisfyt=8:0( )byt�0( )bynt�( )fort=0( )byt�( )byntfort1(46)t=1 Rt�1� �1yt(47)t=e #t+1+(byt�byet)+ut(48)where( );( );0( );0( )and( )aredenedinAppendixE.1andsatisfy0(0)=,0(0)=,(0)=,(0)=1;(0)=1.Further,whenriskiscountercyclical( );0( )�1and( );0( )1.Proof.SeeAppendixE.1. Bycombining(46)with(48),onecanderivetheoptimaltargetcriterionfort=0:by0�0( )byn0+ �1y 0( )0=( )fort=0(49)andfordatest�1:byt�byt�1�( )byet�byt�1+ �1y ( )t=0(50)Asinthecase =0,optimalpolicydeviatesfromRANKatdate0,creatingaboomevenabsentexogenousshocks.However,unlikeinthecase =0,withcountercyclicalriskoptimalpolicyalsodeviatesfromtheRANKtargetcriterionatsubsequentdatest�1.Thisshouldnotbenotsurprising:with =0,monetarypolicycanonlyaecttheevolutionofconsumptioninequality(torstorder)atdate0.However,withcountercyclicalrisk,anexpansioninoutputreducesinequalityatalldates,somonetarypolicyfacesatradeobetweenthebenetsoflowerinequalityandthecostofdistortingproductiveeciencyandpricestabilityatalldates,anddeviatesfromRANKasaresult.Inparticular,( )1whenriskiscountercyclical:outputtracksthe exiblepricelevelofoutputlessthanoneforoneatalldates.Inaddition,( )&#x]TJ/;

32 8 1;�.90;‘ T; 13;&#x.181;&#x
8 1;�.90;‘ T; 13;&#x.181;&#x 0 T; [0;1:theplannerputsmoreweightonstabilizingoutputrelativetoin ationatalldates.Tounderstandwhyoutputmoveslessthanone-for-onewithits exiblepricelevelatalldate,( )1,recallthedistinctionbetweenthreedierentlevelsofoutput:theproductivelyecientlevelofoutputyet,the exiblepricelevelofoutputynt,andtheequilibriumlevelofoutputyt.When =0,ynt=yet.Inthiscase,thestandardsubsidycorrectsforthedistortionduetomonopolisticcompetition,andproductiveeciencyisobtainedwhenthereiszeroin ationatalldates,orwhenpricesare exible.Inotherwords,the exiblepriceversionoftheeconomywith =0featuresazerolaborwedgeatalldates,Ttlnzt�lnwnt=0wherewntdenotesthewageconsistentwiththe exiblepricelevelofoutput.However,withcountercyclicalrisk, &#x]TJ/;ø 1;�.90;‘ T; 11;&#x.515;&#x 0 T; [0;0,andthe exiblepricelevelofoutputyntishigherthantheecientlevelofoutputyet,whateverthelevelofproductivity.Inthiscase,rmsenjoyasubsidywhichislargerthantheonenecessarytoeliminatethedistortionduetomonopolisticcompetition(seeeq.(27)),andoutputwouldbeinecientlyhighunder exibleprices.The exiblepriceversionofthiseconomyfeaturesanegativelaborwedge-anetsubsidytooutputandemployment-atalltimes.Consequentlykeeping24 z Figure4.HANK( �0)vsRANKoptimalresponsetoanegativeproductivityshock.RedlinedenotesRANK,bluelinerepresentsHANKwithcountercyclicalrisk( �0),andgraylinerepresentsHANKwith =0.Allvariablesareplottedaspercentagedeviationsfromtheirsteadystatevalues.Inpanelf),weplotoutputgapasthedierencebetweenytandyntnormalizedbysteadystatey:(byt=y)�(bynt=y).outputequaltoits exiblepricelevelentailskeepingoutputinecientlyhighatalltimes.Conversely,inordertosetoutputequaltoitsecientlevel,andundotheeectsofthenetsubsidytooutputandemployment,onewouldhavetodriveoutputytdownbelowits exiblepricelevelynt.Absentshocks,theplannercouldalwaysimplementtheecientlevelofoutputbyraisinginterestratesabovetheirsteadystatelevel,reducingoutputbelowitsinecientlyhigh exiblepricelevel,andeliminatingthenegativelaborwedge.Shechoosesnottodothis,however,becausethecostintermsofincreasedconsumptioninequ

33 alityistoohigh-soshesetsoutputequaltoits
alityistoohigh-soshesetsoutputequaltoits exiblepricelevel.Nowconsiderascenarioinwhichproductivityishigherthaninsteadystate(z�1).Inthiscase,theplannercouldcontinuetosetoutputequaltoits exiblepricelevel,keepingthelaborwedgeunchanged(Tt=lnzt�lnwt=�lnw0).Thiswouldentailincreasingytoneforonewithynt.Butitisnolongeroptimaltodoso.Theonlyreasontheplannerdoesnoteliminatetheinecientsubsidyinsteadystateisthatdoingsowouldincreaseinequalitytoomuch.Nowhowever,withhigheroutput,consumptioninequalityisalreadylower,andthebenetoftheinecientsubsidyislower:eveniftheplannerweretoimplementalowerlevelofoutputrelativetothe exiblepricelevel,improvingproductiveeciency,inequalitywouldstillberelativelylow.Optimalpolicyseekstoreducethemagnitudeofthelaborwedgebybringingthelevelofoutputytclosertotheecientlevelofoutputyet.Since1=yeyn=yinsteadystate,thisentailsbytbynt.Conversely,supposeproductivityfallsrelativetosteadystate,z1.Again,theplannercouldtrackthe exiblepricelevelofoutputynt,keepingthelaborwedgeunchanged.Doingsowouldentailreducingoutputoneforonewiththe exiblepricelevelofoutput.Butwhenoutputislower,consumptioninequalityishigher.Inthiscase,therationaleforkeepingoutputaboveitsecientlevelyetisnoteliminated-it25 0246810 -0.04-0.03-0.02-0.010 0246810 -5051015 10-4 0246810 -20246810 10-3 0246810 -0.03-0.02-0.0100.010.02 0246810 00.010.020.030.040.050.06 0246810 -5051015 10-3 isactuallystrengthened.Itisoptimaltodeviateevenfurtherfromproductiveeciency,increasetheeectivesubsidytooutputandemployment,andreduceoutputlessthanoneforonewiththeecientlevelofoutput.Figure4showstheimpulseresponsestoanegativeproductivityshockunderoptimalpolicyinHANKwithcountercyclicalrisk(bluelines),withoutcomesinRANK(redlines)andHANKwith =0(graylines)shownasbenchmarks.Asinthebenchmarkwith =0,theplannerdeviatesfromoptimalpolicyinRANK-whichtracksthe exiblepricelevelofoutput-bycuttingnominalinterestratesatdate0(paneld)andmitigatingthefallinoutput(panela)onimpact,atthecostofcreatinganincreaseinin ation(panelb).Recallthatintheeconomywith =0,theplannerattemptedtodampenthisincreaseinin ationbycommittingtotighterpolicystartingatdate1.Withcountercycl

34 icalrisk,however,anyfallinoutputiscostly
icalrisk,however,anyfallinoutputiscostlybecauseitdirectlyincreasesinequality.Indeed,evenunderoptimalpolicy,thepathofinequalityishigherwithcountercyclicalrisk(panelc).Topreventinequalityfromrisingevenmore,theplannerpostponesthecommitmenttotighterpolicyinthefuture.Whileinterestratesstillriseatdate1(bluelineinpaneld),theincreaseissmallerthanintheRANKeconomy(redline)ortheHANKeconomywith =0(grayline).Asaresult,thefallinin ationandoutput-gaparealsosmoothedoutovertime.Remark3(Theroleofinitialwealthinequalitywithcountercyclicalrisk).Withcountercyclicalrisk( �0),theoptimaltargetcriterionisdierentatdatet=0andatallsubsequentdates,aswasthecasewhen =0.Inparticular,optimalpolicyfeaturesaboomatdate0evenabsentexogenousshocks.Asinthe =0case,suchapolicyisoptimalbecausegivenexistingwealthinequalityatdate0,asurprisecutininterestratesraisestheconsumptionofpoorborrowersandreducestheconsumptionofrichsavers.Tounderstandtheroleofwealthinequalityindrivingthisresult,supposeagainthatthereisanequalizationofhouseholdwealthatthebeginningofdate0.Inthiscase,theoptimaltargetcriterion(50)characterizesoptimalpolicyatalldates(withby�1=by�1=0).Thisimpliesthatabsentshocks,theplannerkeepsinterestratesattheirsteadystatevalue.However,unlikeinthecasewith =0andwealthequalization,itisnolongeroptimaltoimplementzeroin ationandreplicate exiblepriceallocationsinresponsetoproductivityshocks-eventhoughitremainsfeasibletodoso.Figure5plotsdynamicsunderoptimalpolicyinthisscenario.Qualitatively,policyissimilartothecasewithoutwealthequalizationinwhichthereisatimeinconsistencyproblem.RelativetoRANK,theplannerimplementsalowerincreaseininterestratesonimpact(paneld),resultinginalowerincreaseinoutput(panela)andthepassthroughfromincometoconsumptionriskt(panele).Thiscomesatthecostofashort-livedincreaseinin ation.Notethatinequality(showninthepanelc)isbelowitssteadystatelevel,despitetheincreaseinincomeriskcausedbythefallinoutput,becausetheinitialequalizationofwealthreducesconsumptioninequalityatthestartofdate0.6NominaldebtSofarwehaveconsideredaneconomyinwhichhouseholdstradedin ationindexedbonds.AsmentionedinRemark2,wemadethisassumptiontodistinguishbetweentwowaysinwhichmonet

35 arypolicycanaectthedistributionofco
arypolicycanaectthedistributionofconsumptioninHANKeconomies.Therstistheinterestrateexposurechannel:anunanticipatedfallinrealinterestratesincreasestheconsumptionofpoordebtorsandreducesthatofrichsavers.ThesecondistheFishereect:unanticipatedin ationredistributesrealwealthfromsavers26 Figure5.HANK( �0)vsRANKoptimalresponsetoanegativeproductivityshockintheeconomywithwealthequalizationatthebeginningofdate0.RedlinedenotesRANK,bluelinerepresentsHANKwithcountercyclicalrisk( �0).Allvariablesareplottedaspercentagedeviationsfromtheirsteadystatevalues.Inpanelf),weplotoutputgapasthedierencebetweenytandyntnormalizedbysteadystatey:(byt=y)�(bynt=y).whoholdnominalassetstodebtorswithnominalliabilities.Ourbaselinemodelwithin ationindexeddebtabstractsaltogetherfromthesecondeecttofocusontherst.Inthissection,weallowhouseholdstotradenominaldebtandshowhowthethepresenceoftheFishereectchangestheoptimalconductofmonetarypolicy.Financialintermediariesnowtradenominalclaimsatnominalprice# 1+itwhichpayadollartomorrow.Thenthehousehold'sbudgetconstraintcanbewrittenas:Ptcst(i)+# 1+itAst+1(i)=Ptwt`st(i)+Ast(i)+PtTtwhereAst+1(i)isthequantityofnominalclaimspurchasedbythehouseholdatdatet.ThedetailsofthisextensionareinAppendixF.AsbeforetheevolutionofthiseconomyischaracterizedbytheaggregateISequation(11),theevolutionof(12),Phillipscurve(7),thedenitionofGDP(15)andtheevolutionof,replacingRt=1+it t+1.Fouroftheseveequationsareunaectedbytheintroductionofnominalbonds-theexceptionistheevolutionof,whichbecomes:19lnt=1 2 22tw2t2t+ln[1�#+#t�1]+I(t=0)ln0@1�#e 2 1�#e 20 E�10E�10 021A(51)(51)showsthatthepresenceofnominaldebtmeansthatunanticipatedhigherin ationreducesconsump-tioninequality.Sincetherearenoaggregateshocksexceptatdate0andtheRamseyplannercanonlyre-optimizeatdate0,actualandexpectedin ationcoincideinequilibriumexceptatdate0.AsAppendix 19SeeAppendixFforthederivation.27 0246810 -0.04-0.03-0.02-0.010 0246810 -202468 10-4 0246810 -0.1-0.08-0.06-0.04-0.020 0246810 00.0050.010.0150.02 0246810 00.010.020.030.040.050.06 0246810 -202468

36 10-3 Fdetails,thecohortbornatdates0ente
10-3 Fdetails,thecohortbornatdates0entersdate0withwithacross-sectionaldistributionofrealwealthwhichisN0;�sw22E�10 02.Higherthanexpectedin ation0�E�10compressesthedistributionofrealwealth.Thus,bygeneratingin ationatdate0,theplannercanreducewealthandhenceconsumptioninequality.Thisreductioninconsumptioninequalityisinadditiontothereductionachievedbythesurprisecutininterestratesatdate0. Figure6.Timeinconsistencywithnominaldebt:Optimaldynamicsabsentshocksinalternativecalibration(=0:5).BluelinedenotesHANKwithcountercylcicalriskandrealdebt,redlinedenotesHANKwithcountercyclicalriskandnominaldebt.Allvariablesareplottedaspercentagedeviationsfromtheirsteadystatevalues.Qualitatively,thisincreasestheplanner'sincentivetocreateaboomatdate0evenabsentaggregateshocks.Quantitatively,though,thiseectissmallinourbaselinecalibration,asFigure8inAppendixFshows-optimalpolicyinthetwoeconomiesisessentiallyidentical.Thisisprimarilybecauseinourbaselinecalibration,thePhillipscurveisrelatively at(=0:01)whichmeansthatevenalargecutinrealinterestrates,andalargeboominoutput,generatesonlyasmallincreaseinin ation.Inordertogeneratealargeenoughincreaseinin ationtoeectsignicantredistribution,itwouldbenecessarytoengineeramassivedeviationfromtheproductivelyecientlevelofoutput,anditisnotoptimalfortheplannertodothis.Toillustratethequalitativeeectofintroducingin ation-indexeddebt,Figure6showsoptimalpolicyabsentshockswithasteeperPhillipscurve(=0:5).Aspanelbshows,theplannercreatesalargerincreaseinin ationintheeconomywithnominaldebt(redline)thanintheeconomywithin ation-indexeddebt(blueline).Thisresultsinalargerreductionininequality(panelc).Theblackdashedlineinthispanelshowstheeectofimplementingtheoptimalpolicyfromtheeconomywithin ation-indexeddebtintheeconomywithnominaldebt.Eveniftheplannerfollowsthispolicy,anddoesnotactivelyexploittheFishereect,thisalreadyautomaticallygeneratesalargerreductionininequalitythanintheeconomywithin ation-indexeddebt(theblackdashedlineisbelowtheblueline)becausethesameincreaseinin ationnowredistributesrealwealth.Theplanneractuallygeneratesafurtherincre

37 aseinin ation,and28 0246810 -5051015 10-
aseinin ation,and28 0246810 -5051015 10-3 0246810 -0.02-0.0100.010.020.030.04 0246810 -0.015-0.01-0.0050 0246810 -0.08-0.06-0.04-0.0200.02 0246810 -0.06-0.04-0.0200.02 Figure7.Nominalvsin ationindexeddebt:optimalresponsetoanegativeproductivityshockinalternativecalibration(=0:5).BluelinerepresentsHANKwithcountercyclicalriskandrealdebt,redlinedenotesHANKwithcountercyclicalriskandnominaldebt.Allvariablesareplottedaspercentagedeviationsfromtheirsteadystatevalues.Inpanelf),weplotoutputgapasthedierencebetweenytandyntnormalizedbysteadystatey:(byt=y)�(bynt=y).soinequalityfallsevenmore(theredlineisbelowtheblackdashedline).Thepresenceofnominalassetsandliabilitiesalsoaectstheoptimalresponsetoshocks.Figure7plotstheimpulseresponses(denedasintheprevioussection)toanegativeproductivityshockunderoptimalpolicyinthealternativecalibrationwith=0:5.Allocationsunderoptimalpolicyintheeconomywithnominaldebt(redline)aresimilartothosewithin ation-indexeddebt(blueline),evenwithasteepPhillipscurve.However,thelevelofoutputnowdeviatesevenmorefromynt(panelf)inordertogeneratealargersurpriseincreaseinin ation(panelb).Thisisoptimalwhenriskiscountercyclicalbecausenegativeproductivityshocktendstoincreaseinequality(panelc).Creatingmoresurprisein ationpartiallyosetsthis,yieldingalowerincreaseininequality(redline)thanwouldobtainiftheplannerfollowedthesamepolicyasintheeconomywithin ation-indexeddebt(blackdashedline).Asinthepreviousgure,though,evenfollowingthissamepolicywouldautomaticallygeneratealargerreductionininequalitythanintheeconomywithin ation-indexeddebt(blueline).7ConclusionWeuseaanalyticallytractableHANKmodeltostudyhowmonetarypolicyaectsinequality,andtheextenttowhichthiswarrantsachangeintheprinciplesgoverningthedesignofmonetarypolicywhichhavebeendevelopedintheRepresentativeAgentNewKeynesianliterature.Inacompletemarketseconomy,monetarypolicyaectsoutputandin ationbut,trivially,hasnoeectsoninequality(sincehouseholdscanuseArrowsecuritiestoinsurethemselves,eliminatingconsumptioninequality).Inourincompletemarketseconomy,monetarypolicyaectsconsumptionriskandinequalitythroughfourchannels.The29 0246810 -0.04-0.03-0.02-

38 0.010 0246810 -4-202468 10-3 0246810 00.
0.010 0246810 -4-202468 10-3 0246810 00.0050.010.015 0246810 00.0050.010.0150.02 0246810 00.010.020.030.040.050.06 0246810 -2-101234 10-3 rstistheincomeriskchannel:whentheidiosyncraticincomeriskfacedbyhouseholdsiscountercyclical,expansionarymonetarypolicy,bygeneratingaboominoutput,tendstoreduceincomeriskandinequality.Moresubtly,lowerinterestratesmakeiteasierforhouseholdstoself-insureagainstincomeshocks,reducingconsumptionriskforagivenlevelofincomerisk-theself-insurancechannel.Finally,unexpectedcutsininterestratesredistributeconsumptionthroughanunhedgedinterestrateexposurechannel,andunexpectedin ationredistributedrealwealththroughtheFisherchannel.Thus,expansionarymonetarypolicycanreduceconsumptioninequalitythroughallfourofthesechannels.Giventhatmonetarypolicyhasthispowertoreduceinequality,howandwhenshoulditbeused?Autilitarianplannertradesothebenetsoflowerinequalityagainstthecostsofpushingupoutputandin ationabovetheirecientlevels.Inrecessions,inequalityisalreadyhigh-intherelevantcasewithcountercyclicalrisk-sothemarginalbenetofreducinginequalityisparticularlyhigh.Consequently,optimalmonetarypolicyismoreaccommodativeinrecessionsrelativeintoaRANKbenchmark:theplannerpreventsoutputfromfallingasmuchastheecientlevelofoutput,eventhoughthisentailshigherin ation,becausecurtailingthefallinoutputalsocurtailstheriseininequality.ReferencesAcharya,SushantandKeshavDogra,\UnderstandingHANK:insightsfromaPRANK,"StaReports835,FederalReserveBankofNewYorkFebruary2018.Angeletos,George-MariosandLaurentEmmanuelCalvet,\IdiosyncraticProductionRisk,GrowthandtheBusinessCycle,"JournalofMonetaryEconomics,2006,53,1095{1115.Auclert,Adrien,\MonetaryPolicyandtheRedistributionChannel,"AmericanEconomicReview,2019,109(6),2333{2367. ,MattRognlie,andLudwigStraub,\TheIntertemporalKeynesianCross,"WorkingPaper250202018.Bhandari,Anmol,DavidEvans,MikhailGolosov,andThomasJSargent,\Inequality,BusinessCyclesandMonetary-Fiscal-Policy,"WorkingPaper24710,NationalBureauofEconomicResearchJune2018.Bilbiie,FlorinO.,\LimitedAssetMarketsParticipation,MonetaryPolicyand(Inverted)AggregateDemandLogic,"JournalofEconomicTheory,May2008,140(1),162{196. ,\MonetaryPoli

39 cyandHeterogeneity:AnAnalyticalFramework
cyandHeterogeneity:AnAnalyticalFramework,"WorkingPaper,November2019. ,\TheNewKeynesianCross:UnderstandingMonetaryPolicywithHand-to-MouthHouseholds,"Jour-nalofMonetaryEconomics,2019,Forthcoming. andXavierRagot,\OptimalMonetaryPolicyandLiquiditywithHeterogeneousHouseholds,"Work-ingPapers2018.Blanchard,OlivierJeanandJordiGal,\RealWageRigiditiesandtheNewKeynesianModel,"JournalofMoney,CreditandBanking,2007,39,35{65.30 Caballero,RicardoJ.,\ConsumptionPuzzlesandPrecautionarySavings,"JournalofMonetaryEco-nomics,1990,25(1),113{136.Cagetti,Marco,\WealthAccumulationovertheLifeCycleandPrecautionarySavings,"JournalofBusiness&EconomicStatistics,2003,21(3),339{353.Calvet,LaurentEmmanuel,\IncompleteMarketsandVolatility,"JournalofEconomicTheory,2001,98,295{338.Calvo,GuillermoAandMauriceObstfeld,\OptimalTime-ConsistentFiscalPolicywithFiniteLifetimes,"Econometrica,1988,56(2),411{432.Challe,Edouard,\UninsuredUnemploymentRiskandOptimalMonetaryPolicyinaZero-LiquidityEconomy,"AmericanEconomicJournal:Macroeconomics,April2020.Christelis,Dimitris,DimitrisGeorgarakos,TullioJappelli,andMaartenvanRooij,\Con-sumptionUncertaintyandPrecautionarySaving,"CSEFWorkingPapers421,CentreforStudiesinEconomicsandFinance(CSEF),UniversityofNaples,ItalyDecember2015.Cui,WeiandVincentSterk,\QuantitativeEasing,"WorkingPapers,UCL2019.Debortoli,DavideandJordiGal,\MonetaryPolicywithHeterogeneousAgents:InsightsfromTANKmodels,"WorkingPaper2018.denHaan,Wouter,PontusRendahl,andMarkusRiegler,\Unemployment(Fears)andDe ation-arySpirals,"JournaloftheEuropeanEconomicAssocation,2018,16(5),1281{1349.Fagereng,Andreas,LuigiGuiso,andLuigiPistaferri,\Firm-RelatedRiskandPrecautionarySavingResponse,"AmericanEconomicReview,May2017,107(5),393{97.Gal,Jordi,MonetaryPolicy,In ation,andtheBusinessCycle:AnIntroductiontotheNewKeynesianFrameworkandItsApplications,SecondEdition,PrincetonUniversityPress,2015.Gal,Jordi,DavidLopez-Salido,andJavierValles,\UnderstandingtheEectsofGovernmentSpendingonConsumption,"JournaloftheEuropeanEconomicAssociation,2007,5(1),227{270.Guvenen,Fatih,SerdarOzkan,andJaeSong,\TheNatureofCountercyclicalIncomeRisk,"JournalofPoliticalEconomy,2014,

40 122(3),621{660.Kaplan,Greg,BenjaminMoll,
122(3),621{660.Kaplan,Greg,BenjaminMoll,andGiovanniL.Violante,\MonetaryPolicyAccordingtoHANK,"AmericanEconomicReview,March2018,108(3),697{743.Kekre,Rohan,\UnemploymentInsuranceinMacroeconomicStabilization,"WorkingPaper2019.McKay,AlisdairandRicardoReis,\TheRoleofAutomaticStabilizersintheU.S.BusinessCycle,"Econometrica,January2016,84,141{194. and ,\OptimalAutomaticStabilizers,"WorkingPaper2019.31 ,EmiNakamura,andJonSteinsson,\ThePowerofForwardGuidanceRevisited,"AmericanEconomicReview,October2016,106(10),3133.Nistico,Salvatore,\OptimalMonetaryPolicyAndFinancialStabilityInANon-RicardianEconomy,"JournaloftheEuropeanEconomicAssociation,October2016,14(5),1225{1252.Nu~no,GaloandCarlosThomas,\OptimalMonetaryPolicywithHeterogeneousAgents,"WorkingPapers1624,BancodeEspanaNovember2019.Ravn,MortenO.andVincentSterk,\JobUncertaintyandDeepRecessions,"JournalofMonetaryEconomics,2017,90(C),125{141. and ,\MacroeconomicFluctuationswithHANK&SAM:AnAnalyticalApproach,"JournaloftheEuropeanEconomicAssociation,Forthcoming.Rotemberg,JulioJ.,\MonopolisticPriceAdjustmentandAggregateOutput,"ReviewofEconomicStudies,1982,49(4),517{531.Wang,Neng,\CaballeroMeetsBewley:ThePermanent-IncomeHypothesisinGeneralEquilibrium,"AmericanEconomicReview,June2003,93(3),927{936.Werning,Ivan,\IncompleteMarketsandAggregateDemand,"WorkingPaper21448,NationalBureauofEconomicResearchAugust2015.32 AppendixAProofofProposition1Thedatesproblemofanindividualibornatdatescanbewrittenas:maxfcst(i);`st(i);bst+1(i)g�1Xt=s(#)t�s1 e� cst(i)+e1 [`st(i)�st(i)]s.t.cst(i)+qtbst+1(i)=wt`st(i)+bst(i)+Tt(52)wherebss(i)=0andwt=(1�)ewt.Theoptimallaborsupplydecisionsofhousholdiisgivenby:`st(i)=lnwt� cst(i)+st(i)(53)andtheEulerequationisgivenby:e� cst(i)=RtEte� cst+1(i)(54)wherewehaveusedthefactthatqt=# Rt.Next,guessthattheconsumptiondecisionruletakestheform:cst(i)=Ct+txst(i)(55)wherexst(i)=bst�1(i)+wt�st(i)� 
denotes\virtualcash-on-hand".Noticethatxst+1(i)isnormallydistributedandsogiventheguess(55),cst+1(i)isalsonormallydistributedwithmean:Etcst+1(i)=Ct+1+t+1Rt #xst(i)+wt�lnwt+

41 ;
+Tt�(1+ wt)cst(i)andvar
;
+Tt�(1+ wt)cst(i)andvariance:Vt�cst+1(i)
=2t+1w2t+12t+1Takinglogsof(54)andusingthetwoexpressionsabove:cst(i)=�1 lnRt�1 lnEte� cst+1(i)=�1 lnRt+Etcst+1(i)� 2Vt�cst+1(i)
=�1 lnRt+Ct+1+t+1Rt #xst(i)+wt�lnwt+
+Tt�(1+ wt)cst(i)� 2t+1w2t+12t+1 21 Combiningthecst(i)termsandusing(55),theabovecanberewrittenas:t+1Rt #(1+ wt)+# Rt�1t+1fCt+txst(i)g=�1 lnRt+Ct+1+t+1Rt #xst(i)+wt�lnwt+
+Tt� 2t+1w2t+12t+1 2(56)Matchingcoecients:Ct=�#t t+1Rt1 lnRt+#t t+1RtCt+1+twt�lnwt+
+Tt�# Rtt t+1 2t+1w2t+12t+1 2(57)�1t=(1+ wt)+# Rt�1t+1(58)Noticethat(58)isthesameas(12)inthemaintext.Next,inequilibrium,aggregatehoursworkedisgivenby:`t=lnwt� Ct+ andhenceaggregateincomeis:yt=wt`t+Tt=wtlnwt� wtCt+wt +TtUsingthisin(57)andthefactthatCt=ytyieldsequation(11)inthemaintext.BPlanner'sObjectivefunctionB.1ConsumptionisnormallydistributedwithincohortGiventheconsumptionfunction(9)andthenormalityofshocks,toconsumptionofnewlybornin-dividualsatanydatesisnormallydistributedwithmeanysandvariance2c(s;s)=2sw2s2ssincetheyallhavezerowealth.Giventhelinearityofthebudgetconstraint,itfollowsthatnewlybornagents'savingsdecisionsass+1(i)arealsonormallydistributedwithmean0andvariance2a(s+1;s)=�Rs 
2[1�(1+ ws)s]2w2s2s.Byinduction,itfollowsthatforanycohortbornatdates,thecross-sectionaldistributionofconsumptionatanydatet�sisnormalwithmeanytandvariance2c(t;s)=2t2a(t;s)+2tw2t2t(59)whilethedistributionofassetholdingsisnormalwithmean0andvariance2a(t;s)=R2t�1 2[1�(1+ wt�1)t�1]22a(t�1;s)+w2t�12t�1(60)2 B.2ObjectivefunctionofplannerSubstitutinglaborsupply(10)intotheobjectivefunction,wecanwritethedate0expectedutilityofindividualifromthecohortbornatdatesgoingforwardsas:Ws0(i)=�1 E01Xt=0(#)t(1+ wt

42 )e� cst(i)=�1 E01Xt=0(#)t(1+
)e� cst(i)=�1 E01Xt=0(#)t(1+ wt)e� yt� txst(i)wherewehaveusedlaborsupply(10),consumptionfunction(9)andthefactthatinequilibriumCt=yt.Weassumethattheplannerputsaweightof1oneveryindividualaliveatdate0and(#)tonindividuals'lifetimewelfarewhowillbebornatdatet�0.Thenthesocialwelfarefunctioncanbewrittenas:W0=(1�#)0Xs=�1#�sZWs0(i)di| {z }welfareofthosealiveatdate0+1Xs=1(1�#)sZWss(i)di| {z }welfareoftheunbornatdate0UsingthedenitionofWs0(i)andWss(i),noticethatW0canbewrittenas:W0=�1 1Xt=0t(1+ wt)e� yttwheretisdenedas:t=(1�#)tXs=�1#t�sZe� txst(i)di(61)Thus,wecanwriteW0as:W0=1Xt=0tUtwhereUt=�1 (1+ wt)e� yttB.2.1DerivationoftrecursionWrite(61)as:t=(1�#)Ze� txtt(i)di+(1�#)t�1Xs=�1#t�sZe� txst(i)di=(1�#)e 22tw2t2t 2+(1�#)t�1Xs=�1#t�sZe� tfast(i)+wt(st(i)� )gdi=(1�#)e 22tw2t2t 2+(1�#)t�1Xs=�1#t�sZe� tast(i)diZe� twt(st(i)� )di=e 22tw2t2t 2"(1�#)+(1�#)#t�1Xs=�1#t�1�sZe� tRt�1 [�1t�1�(1+ wt�1)]t�1xst�1(i)di#wherewehaveusedthefactthatast(i)=Rt�1 [1�(1+ wt�1)t�1]xst�1(i).Also,atalldatest�0,3 weknowthat�1t�1�(1+ wt�1)= Rt�1�1t.Usingthis,wehaveforallt�0:lnt= 22tw2t2t 2+ln(1�#+#t�1)(62)whichisthesameas(22)inthemaintext.Next,imposingsteadystateuptilldate0,weknowthat�1�(1+ w)=e(e0)�1wheree0denotesthedate�1expectationof0.Thenwecanwrite0as:0=e 220w2020 2"(1�#)+(1�#)#�1Xs=�1#�1�sZe� 0 e0xs�1(i)di#(63)andthevarianceofconsumptionforcohortsbornatdates0is 222x(�1;s)=�s.Therefore:0=e 220w2020 22641�# 1�#e 20 e02375Wealsoknowthatinsteadystate(�1=);[1�#+#�1]1�#e 2=(1�#).Pluggingthisinthepreviousexpressionyields(23)i

43 nthemaintext:ln0= 220w2020
nthemaintext:ln0= 220w2020 2+ln[1�#+#�1]+ln0B@1�#e 2 1�#e 20 e021CACSomeauxiliaryresultsIntheproofsthatfollow,weshallmakeliberaluseofthefollowingassumptionsandresults.Assumption1.Throughoutthepaper,weshallassumethat:1.#1 22.#�e�1 2=0:613.2 =22ln#�1   1+ 2 2(1� )ln#�1 1+2ln#+(1�)!2Lemma2.Giventhat#�e�1 2,wehave1ande1.Proof.Recallthatinsteadystate,= 22w22&#x]TJ/;ø 1;�.90;‘ T; 21;&#x.292;&#x 0 T; [0;0,i.e.:=2 2 w 1+ w21�e 22Rearranging:f() 1�e 22=2 2 w 1+ w2(64)4 Now,f()isincreasingfor�2ln#1givenourassumption,andgoesto1as!.Foranyvaluesofand,wecanndsome0 satisfyingf� 
=2 2.Thus,anysolutionto(64)mustsatisfy 1.Byconstruction,forany,e=e 21. Lemma3.For[0; ),wehave#e 21.Proof.Firstweshowthat#e 2=1impliesthat= .Startingfromtheexpressionsforwagesinsteadystate,using#e 2=1wehave:w�1 1+ w=�1+ (1�)(1�e)=21� ln#�1 (1+2ln#)(1�)Add1tobothsidesandmultiplyby  1+ toget: w 1+ w=242ln#�11�  (1+2ln#)1�e+135  1+ Next,usingtheexpressionaboveinthedenitionof,wehave:2=2ln#�1   1+ 2 �2ln#1�  (1+2ln#)+(1�)!2whichisthesameas denedinAssumption1.Second,notethatwhen2=0,wehave=0and#e 2=#1.Bycontinuityitfollowsthatfor2[0; ),wehave#e 21. Corollary1.Thefollowingistrue:1��1e(1�)�0Proof.1��1e(1�)=1�#e 2(1�)�0 DFirst-orderconditionoftheplanningproblemTheplanningproblemcanbewrittenas:max1Xt=0t�1 (1+ wt)e� ytt5 s.t. yt= yt+1�ln+lnt+1+ln

44 ;�1t�(1+ wt)� 2&#
;�1t�(1+ wt)� 22t+1w22e2(yt+1�y) 2(t�1)t= (�1)1�zt (1�)wt+�1t(t+1�1)t+1lnt= 22tw22e2(yt�y) 2+ln[1�#+#t�1]+I(t=0)ln0B@1�#e 2 1�#e 20 e021CAyt=ztlnwt+  1+ zt+
twheret=1 #�1t�(1+ wt)t+1yt+1ztwt ytzt+1wt+1.TheproblemcanbewrittenasaLagrangian:L=1Xt=0t�1 (1+ wt)e� ytt+1Xt=0tM1;t( yt+1�ln+lnt+1+ln�1t�(1+ wt)� 22t+1w22e2(yt+1�y) 2� yt)+1Xt=0tM2;t (�1)1�zt (1�)wt+t(t+1�1)t+1�(t�1)t+M3;08�&#x]TJ ;� -1; .63; Td;&#x [00;: 220w2020 2+ln[1�#+#�1]+ln0B@1�#e 2 1�#e 20 e021CA�ln09�=�;+1Xt=1tM3;t( 22tw22e2(yt�y) 2+ln[1�#+#t�1]�lnt)+1Xt=0tM4;tyt�ztlnwt+  1+ zt+
tTheoptimaldecisionssatisfy:FOCwrtwtUt wt 1+ wt+�1M2;t�1wtdt�1 dwt(t�1)t�M1;t wt �1t�(1+ wt)+M2;t (�1)zt (1�)wt+wtdt dwt(t+1�1)t+1�M4;tzt 1+ zt+
t=0(65)FOCyt� Ut� M1;t+�1M2;t�1dt�1 dyt(t�1)t+M2;tdt dyt(t+1�1)t+1+�1M1;t�1n � 22t+1w22e2(yt�y)o+M3;t 22tw22e2(yt�y)+M4;t=0(66)6 FOCt�M1;t�1t �1t�1� wt+�1M1;t�1h1� 22w22te2(yt�y)i+�1M2;t�1tdt�1 dt(t�1)t+M2;ttdt dt(t+1�1)t+1+M3;t 22w22te2(yt�y)+I(t=0)M3;0#e 20 e02 1�#e 20 e020 e02=0(67)FOCtUt�M3;t+M3;t+1#t 1�#+#t=0(68)FOCt�1M2;t�1t�1(2t�1)�M2;t(2

45 ;t�1)+M4;tztlnwt+  [1+ 
;t�1)+M4;tztlnwt+  [1+ zt+
t]2 (t�1)=0(69)D.1SteadystateoftheoptimalplanNext,weimposesteadystateandset=1.Thisyieldstheandthemultipliersthatareconsistentwith=1beingoptimalinthelong-run.Using=1in(69):�1e�1M2=0whichimpliesthatM2=0insteadystate.Next,from(68)wehavem3=(1�e)�1wheremi=Mi=Ufori=f1;2;3;4g.Thenequations(65),(66),(67)canbemanipulatedtoyield:m1=e 1�e" 1��1e(1�)#(70)w�1 1+ w= (71)m4= 1��1e(1�) 1��1e(1�)(1+ )(72)where =�(1�) (1�)(1�e)and=1� .Finally,using(71)andsince=1in(25)impliesthat1 (1�)w=1,wecanwrite:w=1+ 1�  and=�1 +1+   +1whicharethesameexpressionsin(27)inthemaintext.7 ELinearizedrstorderconditionsLinearizingtherst-orderconditionsfromtheplannersproblemyieldsthefollowing:FOCw� (1+ )byt+(1+ )bt � 1�e e!(1+ )bm1;t� 1�e e!2  1+ (1+ )2m1bwt w� 1�e e2!(1+ )m1bt+1+  bm2;t�bm4;t +m4 bwt w�m4 1 1+ bzt=0(73)FOCy� (1+ ) 1+ bwt w+ "1+2(1�)2 m3�m1 #byt�bt �bm1;t+ bm1;t�1+2(1�)m3�m1 bt+(1�)bm3;t+bm4;t =0(74)FOC�bm2;t+1 Rbm2;t�1+y 1+ m4t=0(75)FOC(fordatest1)� 1�e e2! (1+ ) 1+ m1bwt w+"2m3�m1 �1�e e2m1#bt+bm3;t+2 (1�)m3�m1 byt�1 e bm1;t�e (1�)bm1;t�1!=0(76)Fort=0� 1�e e2!(1+ )  1+ m1bw0 w+"2m3�m1 �1�e e2m1#b0+2 (1�)m3�m1 by0+# 1�#(m3+bm3;0)+bm3;0+m32+(1�#+#) 1�## 1�#b0�1 e bm1;0�e (1�)bm1;�1!=0(77)FOC w 1+ wbwt w� byt�bm3;t+ebm3;t+1+1��1e2 1�eb&

46 #6;t =0(78)wherebmi=cMi Ufori2f1;2;3
#6;t =0(78)wherebmi=cMi Ufori2f1;2;3;4g.8 E.1DerivingthetargetcriterionCombinetheFOCforbwt(73)andforbyt(74): byt+2 (1�)2 m3�m1 byt+ bt �241+ 1�e e!2(1+ )m135 (1+ ) 1+ bwt w+m4 bwt w+"2(1�)m3�m1 � 1�e e2!(1+ )m1#bt+1+  bm2;t+(1�)bm3;t�m4 1 1+ bzt�1+ 1�e e"bm1;t�e (1�)bm1;t�1#=0(79)Combinewith(76): " +2 (1�)2 m3�m1 �2h1+ 1�ei(1�)m3�m1 #byt+ bt +8:h1+ 1�ei 1�e e2! (1+ ) 1+ m1�241+ 1�e e!2(1+ )m135 (1+ ) 1+ +m4 9=;bwt w+"2(1�)m3�m1 � 1�e e2!(1+ )m1#bt�h1+ 1�ei"2m3�m1 �1�e e2m1#bt+1��h1+ 1�eibm3;t+1+  bm2;t�m4 1 1+ bzt=0Next,usetheGDPdenition(21)tosubstituteoutforbwt w: " +2 (1�)2 m3�m1 �2h1+ 1�ei(1�)m3�m1 #byt+ bt +8:h1+ 1�ei 1�e e2! (1+ ) 1+ m1�241+ 1�e e!2(1+ )m135 (1+ ) 1+ +m4 9=;1+  byt�8:h1+ 1�ei 1�e e2! (1+ ) 1+ m1�241+ 1�e e!2(1+ )m135 (1+ ) 1+ +m4 9=;y bzt+"2(1�)m3�m1 � 1�e e2!(1+ )m1#bt�h1+ 1�ei"2m3�m1 �1�e e2m1#bt+1��h1+ 1�eibm3;t+1+  bm2;t�m4 1 1+ bzt=09 Substituteoutforbtusingbt=�t� h1+1�e ibytandusingthedenitionsofm1;m3andm4,theabovecanbewrittenas: 1��1e(1�) 241�e +1+21��1e1�e 235byt� byt+1��1e(1�) 1��1e(1�)1 

47 6;( +1)byt�+y 1+ bzt&#
6;( +1)byt�+y 1+ bzt� +21��1e 1��1e(1�)�t+1+  bm2;t+ bt �1�e bm3;t=0(80)Guessthat:bm3;t=1 1�ebt + byt+azbzt(81)andusethisin(78)withbwt wsubstitutedoutusingthedenitionofGDP:bt+1 ��1ebt + 1�e byt+1=1�e e (1+ )y 1+ +az1�e%zbztusingthefactthatbzt+1=%zbzt.Usingthedatet+1recursion(45):�t+1=1�e e (1+ )y 1+ +az1�e%zbztUsingthefactthat�t= y 1+ (1�e)(1+ ) 1�e%z(1�)bzt,wehave:%z y 1+ 1�e(1+ ) 1�e%z(1�)bzt=1�e e (1+ )y 1+ +az1�e%zbztwhichimpliesthatazmustsatisfy:az=� (1+ ) 1�ez(1�)y 1+ (82)Usingbm3;t=1 1�ebt + byt+azbztin(80):( )byt�( )bynt=tfort�0(83)wherewehavedenedt=�(1+ ) m4bm2;twhere( )=1�  1+ 241�e +1�21�e (1�)3510 and( )=( )�18:1+ 1�e 1�ez(1�)y +y1+ 1�9=;Similarly,forthedate0targetcriterion,combinethedate0versionof(79)with(77),(81)andthedate1recursion(45)toget:0( )by0�0( )byn0�( )=0(84)where0( )=( )+ m4 2h1+1�e i2 1�e2++2# 1�## 1�#(85)0( )=0( )�124( )( )+ 2( +1) m41+2+2# 1�## 1�#1+1�e 1�ez(1�)y y+35(86)( )=0( )�1  m4"1+(1�e) 1�e## 1�#Insummary,using(84)-(83)andthetargetcriterioncanbewrittenas:t=8:0( )byt�0( )bynt�( )fort=0( )byt�( )byntfort1Thisisthesameas(46)inthemaintext.Next,multiplying(75)by(1+ ) m4yields:t=1 Rt�1�

48 �1ytwhichisthesameasequati
�1ytwhichisthesameasequation(47)inthemaintext.ThisconcludesthederivationoftheexpressionsinProposition4.E.1.1DerivationofexpressionsinProposition3Denethescalars=0(0)and=0(0)and=(0):0(0)=1+ m4 2 1�e2++2# 1�## 1�#�10(0)=�1"1+ m4 2 1�e1+2+2# 1�## 1�#1�e 1�ez(1�)1 1+#=(0)=�1  m4# 1�#�011 Since1�e 1�ez(1�)1 1+1,itisclearthat1.Itisalsoclearbyinspectionthat(0)=(0)=1.Thus,theoptimaltargetcriterionwhen =0canbewrittenas:t=8:byt�bynt�fort=0byt�byntfort1whichisthesameasequation(38).Finally,when =0,y=1andso(47)becomes(39).E.1.2OptimaltargetcriterioninRANKRecallthatinRANKwehave =0and=0.Inthiscase,itisclearfrominspectionthatthecoecientsfurthersimplifyto==1,=0andthetargetcriterioncanbewrittenas:t=byt�byntfort0whichisthesameas(29)inthemaintextE.2Propertiesofcoecients0( );0( );( );( )Claim:( )&#x]TJ ;� -1; .63; Td;&#x [00;1Proof.( )=1+ 1+ 2 (1�)�11�e �1�1+ 1+ 242 (1�)�11�e 1�e(1�)�135wherewehaveusedthefactthat =�1+ (1�e)(1�)andforcountercycicalrisk(�1),wehave � (1�e)(1�).Then,theabovecanbesimpliedto:( )�1+ 1+ 1+ (1�)2�1 Claim:Ifriskiscountercyclical�1,then( )1Proof.Countercyclicalriskor&#x]TJ/;ø 1;�.90;‘ T; 11;&#x.515;&#x 0 T; [0;1impliesthat &#x]TJ/;ø 1;�.90;‘ T; 11;&#x.515;&#x 0 T; [0; (1�e)(1�) Proof.Noticethat( )canbewrittenas:( )=1+ +( + 2) (1�e) 1�ez(1�)y +yh1+ 1�i 1+(1� ) + 22 (1�)�11�e&#

49 12;12 Weneedtoshowthat( )1,i.e
12;12 Weneedtoshowthat( )1,i.e.1+ +( + 2) 1�e 1�ez(1�)y +y1+ 1�1+(1� ) + 22 (1�)�11�eThisexpressioncanbesimpliedtoyield:1+1�e 1�ez(1�)y +y1+ 1� 1�e"2 (1�)�1�1 1�ez(1�)y +y1+ 1�#(87)First,weshowthattheterminthesquarebracketsontheRHSof(87)ispositive,i.e.2�"1�+1+ 1�ez(1�)y +y#Theworstcaseforthistobetrueisifyisverylargeand%z=1.Inthatcase,fortheexpressionabovetobetrue,itmustbethat:e2 2�(1�)whichistruesincee1and2 2�(1�)�1sinceweknowthat01fromAppendixXXX.Thus,theterminthesquarebracketsontheRHSof(87)ispositive.Next,toshowthat(87)holdswithcountercyclicalrisk,itsucestoshowthatitholdsforthelowest consistentwithnon-procyclicalrisk,i.e. = (1�e)(1�).Plugin = (1�e)(1�)into(87),i.e:1+1�e(1+) 1�ez(1�)y +y"2 (1�)�1�1+ 1�ez(1�)y +y 1�#Againtheworstcaseforthisconditiontobesatisedisif%z=1.Supposethatisthecase.Then,theexpressioncanbefurthersimpliedto:y +y1whichistruesincesteadystateoutputispositive. Claim:Ifriskiscountercyclical�1,then0( )�1Proof.Since( )�1,itfollowsfrom(85)that0( )�1. FThemodelwithnominaldebtInthisextension,wechangetheassumptionthatnancialintermediariestraderealclaimswiththehouse-holds.Instead,weassumethatnowtheseintermediariestradenominalclaimsatanominalprice# 1+itwhichpayadollartomorrow.Thenthehousehold'sbudgetconstraintcanbewrittenas:Ptcst(i)+# 1+itAst+1(i)=Ptwt`st(i)+Ast(i)+PtTt13 whereAst+1(i)isthequantityofnominalclaimspurchasedbythehouseholdatdatet.DividingthroughbyPt,weget:cst(i)+# 1+itt+1ast+1(i)=wt`st(i)+ast(i)+Ttwhereast(i)=Ast(i) Ptdenotestherealvalueofwealthheldbythehouseholdatthebeginningofdatet.Giventhese

50 denitions,therestofthemodelisthesam
denitions,therestofthemodelisthesameexceptforthetrecursionwhichcannowbewrittenas:�1t=1+ wt+#t+1 1+it�1t+1thetrecursionwhichwenowderive.Westartwiththebudgetconstraintofahouseholdsandplugintheexpressionforlaborsupply(10)andtheconsumptionfunction(9):ast(i)=1+it�1 #twt�1`st�1(i)+ast�1(i)+Tt�1�cst�1(i)=Rt�1 [1�(1+ wt�1)t�1]xst�1(i)Insteadystate,(12)impliesthatR [1�(1+ w)]=1,sowecanwrite:ast(i)=R [1�(1+ w)]xst�1(i)=xst�1(i)Ifwewereinsteadystateuptilldate0,thedistributionofassetswhichwouldobtainatdate0iftherewasnoin ationbetweent=�1andt=0:~as0(i)=w�1Xk=s�sk(i)� 
and2ea(s;0)=2w2(�s)Butifthereisin ation,thenactualas0(i)is:as0(i)=w�1Xk=s�sk(i)� 
�10Thisimpliesthatforcohortbornatdates0,as0(i)isnormallydistributedwithmean0andvariance�sw22�20.Next,weusethisinformationtoderivetheexpressionfor0.UsingthesameParetoweightsasinthebaselinemodel,wehavefrom(63):0=(1�#)0Xs=�1#�sZe� 0xs0(i)di=(1�#)0Xs=�1#�sZe� 0as0(i)diZe� 0w0(s0� )di=(1�#)e1 2 220w20200Xs=�1#�se� 20 02s=(1�#)e1 2 220w2020 1�#e 20 0214 whereonthelastlinewehaveusedthefactthatforcohortsbornatdatess0;as0(i)N�0;�sw22�20
.Thisexpressioncanbemanipulatedtoyield:ln0=1 2 220w2020+ln[1�#+#]+ln0@1�#e 2 1�#e 20 E�10E�10 021A| {z }eectofdate0surpriseOfcourse,fordatest�0,therecursionstaysthesameasinthebaselinemodelsincetheplannerisnotabletocreateanyin ationsurpriseatdatest1.Inlinearizedterms,therecursioncanbewrittenas:bt =8���&#x]TJ ;� -1; .63; Td;&#x [00;&#x]TJ ;� -1; .63; Td;&#x [00;&#x]TJ ;� -1; .63; Td;&#x [00;:� 1�e byt+�t+# Rbt�1 fort�0� 1�e by0+�0+# 1�#(b0

51 �0)fort=0F.1Optimalmonetarypolic
�0)fort=0F.1OptimalmonetarypolicyThemonetarypolicyproblemisverysimilartothebaselinecase.Theonlydierenceistheexpressionfor0sincenowtheplannercanaectthelevelofconsumptioninequalitybycreatinganin ationsurpriseatdate0.Inlinearizedform,theonlyrstorderconditionsthatdierfromthoseinSectionEisthedate0rst-orderconditionwithrespecttoin ationandwithrespectto.Inlinearizedform,therstorderconditionsforcannowbewrittenas:Fort=0�bm2;0+y 1+  m4+m3# 1�#0�# 1�#bm3;0�m3# 1�#2++# 1�#(b0�0)�m3# 1�#=0(88)andfort1�bm2;t+1 Rbm2;t�1+y 1+  m4t=0(89)andtheFOCsforcanbewrittenas:Fort=0� 1�e e2!(1+ )  1+ m1bw0 w+"2m3�m1 �1�e e2m1#b0+2 (1�)m3�m1 by0+# 1�#(m3+bm3;0)(90)+bm3;0+m32+(1�#+#) 1�## 1�#(b0�0)�1 e bm1;0�e (1�)bm1;�1!=015 andfort1� 1�e e2!(1+ )  1+ m1bwt w+"2m3�m1 �1�e e2m1#bt(91)+2 (1�)m3�m1 byt+bm3;0�1 e bm1;t�e (1�)bm1;t�1!=0F.2DerivingthetargetcriterionItisclearthatthetargetcriterionfordatest1isunchanged.Theonlydierenceisinthedate0targetcriterion,whichwederivenext.Inthiscasewithnominaldebt,thedate0versionofthecombinedFOCforwandy(79)canbewrittenas: by0+2 (1�)2 m3�m1 by0+ b0 �241+ 1�e e!2(1+ )m135 (1+ ) 1+ bw0 w+m4 bw0 w+"2(1�)m3�m1 � 1�e e2!(1+ )m1#b0+1+  bm2;0+(1�)bm3;0�m4 1 1+ bz0�1+ 1�e e"bm1;0�e (1�)bm1;�1#=0Multiply(90)by�h1+ 1�eiandaddtotheequationabovetoget:8: 1��1e(1�) 241�e +1+21��1e1�e 235� +m4  9=;by0�m4  +

52 y 1+ bz0� +21�&
y 1+ bz0� +21��1e 1��1e(1�)�0+1+  bm2;0+ b0 �1�e bm3;0�h1+1�e i# 1�#bm3;0�241+1�e 1�e35# 1�#�241+1�e 1�e352+(1�#+#) 1�## 1�#�0+ h1+1�e i2 1�e2+(1�#+#) 1�## 1�#by0+241+ 1�e 1�e352+(1�#+#) 1�## 1�#0=0Guessbm3;0=1 1�eb0 + by0� (1+ ) 1�ez(1�)y 1+ bz016 Plugthisintotheexpressionabove:8: 1��1e(1�) 241�e +1+21��1e1�e 235+m4  � h1+1�e i9=;by0�m4241�e 1�ez(1�)y y+1+ 1�+1  35byn0+1+  bm2;0+ h1+1�e i# 1�#8:2 +241+1�e 1�e3521+# 1�#+9=;by0�241+1�e 1�ez(1�)35# 1�# (1+ )y y+1+2+# 1�#byn0+241+ 1�e 1�e35# 1�#21+# 1�#+0�241+1�e 1�e35# 1�#=0Thisexpressioncanberearrangedtoyield:0( )[by0�0( )byn0+( )0�( )]=0where0( )( )=  m4241+ 1�e 1�e35# 1�#21+# 1�#+and0( );0( )and( )areasdenedinAppendixE.1.17 Figure8.Timeinconsistencywithnominaldebt:Optimaldynamicsabsentshocksinbaselinecalibration(=0:01).BluelinedenotesHANKwithcountercylcicalriskandrealdebt,redlinedenotesHANKwithcountercyclicalriskandnominaldebt.Allvariablesareplottedaspercentagedeviationsfromtheirsteadystatevalues..18 0246810 -0.0200.020.040.06 0246810 -2-10123 10-3 0246810 -0.04-0.03-0.02-0.010 0246810 -0.2-0.15-0.1-0.0500.05 0246810 -0.2-0