Household Debt and the Heterogeneous Effects of Forward Guidance

K.7 Household Debt and the Heterogeneous Effects of Forward Guidance Ferrante, Francesco and Matthias Paustian Please cite paper as: Ferrante, Francesco and Matthias Paustian (2019). Household Debt and the Heterogeneous Effects of Forward Guidance. International Finance Discussion Papers 1267. https://doi.org/10.17016/IFDP.2019.1267 International Finance Discussion Papers Board of Governors of the Federal Reserve System Number 1267 November 2019

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 1267 November 2019 Household Debt and the Heterogeneous Effects of Forward Guidance Francesco Ferrante and Matthias Paustian NOTE: International Finance Discussion Papers (IFDPs) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the International Finance Discussion Papers Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at www.ssrn.com.

Household Debt and the Heterogeneous Effects of Forward Guidance ∗ FrancescoFerrante†andMatthiasPaustian‡ November2019 Abstract We develop an incomplete-markets heterogeneous agent New-Keynesian (HANK) model in which households are allowed to lend and borrow, subject to a borrowing constraint. We show that, in this framework, forwardguidance, thatisthepromisebythecentralbanktolowerfutureinterestrates, can be a powerful policy tool, especially when the economy is in a liquidity trap. In our model, the power offorwardguidanceisamplifiedbythreeredistributivechannels, absentinarepresentativeagentnew- Keynesianmodel(RANK)orinaHANKmodelwithoutprivatedebt. First,expectedlowerratesimply a future transfer of wealth from savers to borrowers, reducing precautionary motives and stimulating currentdemandandinflation.Second,higherinitialinflationlowersthepathoftherealrateincreasingthe wealthofborrowers,whohaveahighermarginalpropensitytoconsume(MPC).Third,ifdebtisnominal, debtdeflationgeneratesalsoawealthtransfertowardshigh-MPCborrowing-constrainedagents,further increasing aggregate consumption and inflation. These channels amplify each other in a liquidity trap, andcanmakeforwardguidancemorepowerfulinaHANKmodelthaninaRANKframework. These results contrast with previous research on HANK models, which focused on frameworks where agents werenotallowedtoborrow,andwhichfoundnegligibleeffectsofforwardguidance. Keywords: HANKmodel,zerolowerbound,forwardguidance,householddebt JELClassification: E21,E32,E52,E58 *We would like to thank Andrea Prestipino, Nils Gornemann, Gaston Navarro, Florin Bilbiie, Ludwig Straub, Gianluca Violante for their comments, Sebastian Schmidt for his discussion, as well as seminar participants at the Federal Reserve Board, and conference participants at the 2019 SED Conference, at the 2019 Dynare conference and at the Bank of Canada workshop on Frontiers of Monetary Policy and Financial Studies. †FederalReserveBoardofGovernors,francesco.ferrante@frb.gov ‡FederalReserveBoardofGovernors,matthias.o.paustian@frb.gov

1 Introduction In the aftermath of the financial crisis, while short-term interest rates were constrained at the zero lower bound, many central banks have used communication about a future lower path for the policy rate as a key alternative tool. The macroeconomic effects of such forward guidance (FG) in representative agent new Keynesian (RANK) models are particularly large, and are not decreasing in the horizon of the announced cut,seeCarlstrom,Fuerst,andPaustian(2015)orDelNegro,GiannoniandPatterson(2012). Attheheartof this so-called ”forward guidance puzzle” in RANK models is the strong role for intertemporal substitution. The consumption Euler equation prescribes that aggregate demand depends on interest rates in all future periods without discounting. Hence, all else equal, interest rate changes in the very distant future have the sameeffectonconsumptionasinterestratechangestoday. AgrowingrecentliteraturehasexaminedthetransmissionofthistypeofunconventionalmonetarypolicyinNewKeynesianmodelswithheterogeneousagentsandincompletemarkets,widelyknownasHANK models.1 TheseminalpaperbyMcKay,NakamuraandSteinsson(2016)(MNShenceforth)documentsthat HANK models can considerably reduce the power of forward guidance. MNS explain that the key mechanism behind their results is that, when households may become borrowing constrained in the future, the resulting dynamics for aggregate consumption mimic that of a representative agent who discounts future interest rates, a ”discounted Euler equation”. As a result, movements in interest rates in the future have a smaller impact on current consumption in their HANK model than in the comparable RANK model, mitigatingtheforwardguidancepuzzle. However, anumberofauthorshavepointedoutthattheaggregatedynamicsinresponsetofuturemonetary policy shocks in HANK models depend crucially on various auxiliary assumptions, which can either dampenoramplifythepowerofforwardguidance. Forexample,Werning(2015)andHagedornetal. (2019) show that the results in MNS are heavily influenced by assumptions on tax progressivity and on the distributionofdividends,whichdonotbearanyconsequenceinaRANKmodel. Theseassumptionsimplythat, inMNS,forwardguidanceredistributesresourcestowardswealthieragentswithlowermarginalpropensity toconsume(MPC),whichlimitstheeffectonaggregatedemand. Furthermore,severalauthorsusetractable versions of HANK models, with a degenerate wealth distribution, to show analytically that the power of forward guidance depends on the cyclicality of income risk (Werning (2015), Acharya and Dogra (2019), Bilbiie(2019)),ofliquidity(Werning(2015))andofinequality(Bilbiie(2019)). Consequently,quantitative 1SeveralotherpapershavealsostudiedthepropagationofconventionalmonetarypolicyinHANK,see,forexample,Kaplan andViolante(2018),andAuclert(2019). 1

HANKmodels,inwhichthewealthdistributionisarelevantstatevariable,cangivedifferentanswerstothe questionofwhetherforwardguidanceisstillapowerfultoolinheterogeneousagentsframeworks. Hagedorn et al. (2019) introduce wage stickiness and nominal government bonds and taxes in their HANK model, in ordertominimizetheredistributivechannelspresentinMNS,andfindthattheimpactofforwardguidance in a liquidity trap is still negligible, even weaker than in MNS. On the other hand, Werning (2015) shows that in a model with extreme market incompleteness, meaning no borrowing and no public debt, forward guidance has the same impact in RANK and HANK models, if individual income moves proportionally to aggregate income. Furthermore, Farhi and Werning (2019), show that one needs both a HANK framework andboundedrationalityinordertosolvetheforwardguidancepuzzle. Inthispaper,werevisittheeffectsofforwardguidanceinaHANKmodelwherehouseholdsareallowed toborrow. TheroleofhouseholddebtpositionshasreceivedsurprisinglylittleattentioninHANKmodelsin thecontextofforwardguidance.23 Infact,MNS,Hagedornetal. (2019),andFarhiandWerning(2019)use models with a no-borrowing constraint, so that households are savers and the borrower is the government.4 In our framework, households borrow and lend among each others. Furthermore, we make two additional assumptions in order to minimize the impact of the non-monetary redistributive channels operating in the model of MNS. First, we assume there is no government debt, so that bonds are in zero net supply. As a result,interestratesdonotdirectlyaffectthegovernmentbudgetconstraint,andhencetherearenoindirect redistributive effects of monetary policy operating through taxes and transfers. Second, we use a dividend distribution scheme that prevents fluctuations in aggregate profits to have redistributive effects. We believe that this approach allows us to abstract from indirect effects of forward guidance, operating through fiscal assumptionsorunrealisticprofitdistributions,andtofocusontheheterogeneouseffectsofforwardguidance that work through borrowing and lending and through fluctuations in interest rates, which represent more directchannelsofmonetarypolicypropagation. OurHANKframeworkintroducesnovelamplificationmechanismsforforwardguidanceoperatingthrough threeredistributivechannels. First,inourmodel,thenewsoflowerfutureratesattimeT isassociatedwitha futuretransferfromsaverstowardsborrowers,someofwhichareattheborrowingconstraintandbehavelike hand-to-mouth agents. This future transfer from low MPC agents to high MPC ones, immediately reduces 2Theroleofnominalpositionshasbeenstudiedinothercontexts,seeforexampleDoepkeandSchneider(2006). 3Tothebestofourknowledge,theonlyexceptionisWerning(2015),whoderivesanalyticalresultsonlyforasimplifiedmodel withmaximumborrowingproportionaltoaggregateincome,log-utilityandzeroinitialbondholdings.Inanycase,Werning(2015) abstractsfromtheredistributivechannelswestudyinthispaper. 4MNSclaim,inafootnote,thattheirresultsarelittlechangedoncehouseholdsareallowedtoborrow. However,asmentioned above,theresultsinMNSaredrivenbycounterfactualredistributiveeffectsduetotaxesanddividendsdistribution. 2

the precautionary motives of agents who are not at the constraint, but are likely to hit the constraint in the future, stimulating aggregate demand even for t ≤ T. We call this effect a transfer news channel. Second, the standard intertemporal response by unconstained agents to FG causes inflation to rise on impact, and it impliesalowerpathfortherealinterestratewhichgeneratesafurthertransferofwealthtowardshighMPC borrowers. ThischannelisverysimilartowhatAuclert(2019)definesastheinterestrateexposurechannel, whichalsoplaysanimportantroleinaccountingfortheredistributiveeffectsofconventionalmonetarypolicy. Third,whendebtisnominal,unexpectedinflationcreatedbyforwardguidanceerodestherealvalueof borrowers’debt,particularlyboostingtheconsumptionofconstrainedagentsthroughadebtdeflationchannel. Duetotheinteractionbetweenredistribution, aggregatedemand, inflationandtherealrate,thesethree channels amplify each other, igniting a positive feedback loop which can magnify the effects of forward guidanceinourHANKmodelcomparedtotheRANKmodel,especiallyinaliquiditytrap. We quantify the effects of forward guidance in three classes of experiments, which are helpful to study theindividualimpactofthethreechannelsdescribedabove. Thefirsttypeofexperimentconsidersforward guidancespecifiedintermsofaconstantpathfortherealinterestratepriortoalooseningoftherealratein the future (real rate forward guidance). By keeping real rates fixed prior to the forward guidance date, we can focus on the anticipation effect of lower future rates, and we can single out the impact of the transfer newschannelbyusinga”transfernewsshock”,whichonlycapturestheeffectsoftheforeseenredistribution from savers to borrowers. In particular, we show that this channel is particularly strong for unconstrained borrowers, who are more likely to become constrained in the future, and it can result in the effect of FG beinglargerinHANKthaninRANK,especiallyforshorterforwardguidancehorizons. Thesecondtypeof experimentconsidersaconstantpathforthenominalinterestratepriortoalooseningofthenominalratein future (nominal rate forward guidance). In this exercise, higher inflation feeds back into a lower real rate also in the periods preceding the forward guidance date, stimulating consumption through an intertemporal substitutionchannel. Inourmodel,thismechanism,whichisatworkalsointheRANKmodel,isamplified by the interest rate exposure channel, which implies higher wealth for high-MPC borrowers and stronger aggregatedemand. Alltheexperimentsareperformedbothwithrealandnominalbonds,inordertoillustrate the impact of the debt deflation channel, which is particularly relevant for constrained agents. Finally, the third experiment considers a scenario in which adverse demand shocks bring the economy into a liquidity trap,andstudiestheeffectsofthecentralbankextendingthestayattheZLBforlongerthanpredicatedbya standard Taylor rule. The three channels amplify each other in a liquidity trap, making forward guidance a very powerful monetary tool in our framework, even more than in a comparable RANK model. This result isrobusttotheintroductionofwagestickiness,which,bylimitingfluctuationsinrealwages,canpotentially 3

reducetheresponseofhand-to-mouthagents. RelatedLiterature: Thispaperisrelatedtoarecentbodyofworkstudyingtheeffectsofconventionaland unconventional monetary policy in models with heterogeneous agents and incomplete market. Compared to the representative agent canon, such models feature two new elements: i) heterogeneous MPCs and ii) precautionarymotives. As regards standard monetary policy, a number of papers have stressed the importance of MPC heterogeneity in shaping the aggregate response to monetary shocks. Kaplan et al. (2018) show how the transmission of monetary policy differs in a quantitative HANK model, with two assets with different liquidity, compared to a RANK one, because of the presence of a large fraction of wealthy hand-to-mouth agents. The behavior of these agents differs from the one of a representative agent, since they are very sensitive to changes in labor income but do no respond to interest rate changes through intertemporal motives. Auclert (2019) stresses that differences in marginal propensities to consume across agents imply that redistribution mattersfortheeffectofmonetarypolicyonaggregatequantities. Inparticular,heshowsthatthetransmission of monetary policy operates via three redistributive mechanisms. First, an earnings heterogeneity channel reflectingthatagentsareaffecteddifferentlybychangesinlaborandprofitearnings. Second,aFisherchannel whereby unexpected inflation revalues net nominal debt positions, transfering resources from savers to borrowers. And finally, an interest rate exposure channel, since a monetary policy loosening redistributes awayfromagentswithpositiveunhedgedinterestrateexposures(likethesaversinourmodel)andtowards thosewithnegativeinterestrateexposures(liketheborrowersinourmodel). Auclert(2019)showsthatthese re-distributivechannelscanamplifytheeffectsofanexpansionarymonetarypolicyshockifthewinnershave a higher MPC than the losers, as it seems to be the case in the data.5 Our main result of amplification via heterogeneity is consistent with the findings in Auclert (2019), as we show that the Fisher channel and the interest rateexposure channel, togetherwith our noveltransfer news channel, also strengthen theimpact of forwardguidance. When we move to consider the question of whether forward guidance is a powerful monetary tool in HANK models, the literature offers a variety of answers, sometimes contrasting with each other. As we mentioned above, MNS were the first to consider HANK models for a possible solution to the forward guidance puzzle present in the canonical RANK model. Their intuition is that the likelihood of becoming constrainedinthefutureeffectivelyimpliesashorterplanninghorizon,introducingdiscountingintheEuler 5DebortoliandGali(2018),provideadditionalinsightsonthepropagationofmonetaryshocksinHANKmodelsbyemploying amoretractabletwoagentNewKeynesianmodel(TANK),withconstrainedandunconstrainedagents. 4

equation of unconstrained agents and hence dampening the strong intertemporal substitution motive at the heartofthestrongpowerofFGinRANKframeworks. Ontheotherhand,Werning(2015)showsthatforasimpleHANKeconomywithnoborrowingorlendingandwithhouseholdincomeproportionaltoaggregateincome,thereactionofaggregateconsumptionto interest rates can be modeled according to a standard representative-agent Euler equation, ”as if” markets were complete. In addition, Werning (2015) suggests that whether HANK models amplify or dampen the effectofforwardguidancedependsonthecyclicalityofincomerisk,affectingprecautionarymotives,andon thecyclicalityofliquidity,affectinghouseholds’abilitytoself-insure. InMNS,theassumptionsondividend distributionimplythatincomeriskisprocyclical,andtheassumptionsongovernmentbondssupplyandon the borrowing constraint imply that liquidity is countercyclical. As a result, aggregate consumption is less sensitive to interest rates. However, a model with countercyclical unemployment risk, or with procyclical liquidity, could potentially overturn this result. A related point is raised by Bilbiie (2019), in the context of a tractable two-agents HANK model (THANK), who shows that the impact of current and future monetary policy crucially depends on the cyclicality of inequality, that is the income difference between constrained and unconstrained agents, and on the cyclicality of risk, that is the probability of becoming constrained. Acharya and Dogra (2019) build a highly stylized and tractable heterogeneous agent framework, a pseudo representative agent New Keynesian model (PRANK), without differences in marginal propensities to consume, and focus on the role of the cyclicality of uninsurable risk in solving New Keynesian paradoxes. In linewithWerning(2015),theyfindthatforwardguidanceislesspowerfulthanintheRANKmodelonlyif income risk is sufficiently procyclical. If instead, income risk is countercyclical heterogeneity exacerbates theforwardguidancepuzzle. Ourfindingsareconsistentwiththeanalyticalresultsobtainedinthetractable models used in these papers. The redistributive channels introduced by household debt imply that income riskdecreaseswithfutureexpansionarymonetarypolicy, asthegapbetweenthetotalincomeofsaversand borrowers narrows, reducing precautionary motives and strengthening the power of forward guidance. Our framework is directly comparable with the setting in MNS and it allows for a richer quantitative analysis thantheanalyticalsettinginBilbiie(2019)orAcharyaandDogra(2019). Hagedornetal. (2019)useaquantitativeHANKmodelandmaintaintheassumptionofazeroborrowing limit for households. They depart from MNS along two dimensions. First, they introduce wage stickiness in order to reduce the redistributive effect of fluctuations in profits. Second they assume that government debt,andtaxes,arenominal. Intheirframework,theeffectofforwardguidanceisnegligible. Thisresultis partiallyduetothefactthathigherprices,inducedbyforwardguidance,lowertherealvalueofgovernment bonds, which, for precautionary reasons, are considered net wealth by households, and depress aggregate 5

demand. Thischannelcounteractstheattemptsofforwardguidancetoincreasepricesandresultsinrelatively stable real interest rates, thereby preventing a strong intertemporal substitution effect.6 The solution of the forward guidance puzzle proposed by Hagedorn et al. (2019) hence hinges on the fiscal assumptions regarding the supply of government debt, which play no role in our model. Furthermore, in the framework of Hagedorn et al. (2019), in which households are not allowed to borrow, debt deflation has an opposite effectcomparedtoourmodel,sincehigherinflationreducestherealsupplyofgovernmentbonds,whichare used for self-insurance purposes by households.7 In the last part of our paper, we introduce wage rigidities as in Hagedorn et al. (2019), and we find that while the impact of FG in a liquidity trap is reduced both in theRANKandintheHANKmodel,theeffectinHANKisstillsizeableandlargerthaninRANK. AnotherpossiblesolutiontotheforwardguidancepuzzleisprovidedbyFarhiandWerning(2019),who introduceboundedrationality,intheformoflevel-kthinking,inaHANKmodel. Theirmainmessageisthat it is the interaction of market incompleteness and limited rationality which effectively mitigate the forward guidance puzzle, while each factor in isolation would not be enough. Compared to our paper, the work by Farhi and Werning (2019) abstracts from households’ borrowing and does not study the power of FG in a liquiditytrap. To summarize, our paper shows that redistributive effects, linked to private debt holdings, are a key determinant of the aggregate impact of forward guidance in a HANK framework; and that taking into account these channels can make forward guidance in HANK models more powerful than in RANK models, incontrasttosomepreviousliterature. Hence, redistributiveeffectsnotonlyamplifytheimpactofconventionalmonetarypolicy,ashighlightedbyAuclert(2019),buttheyarealsoimportantforthetransmissionof unconventionalpolicyactions. Therestofthepaperisorganizedasfollows. Section2describesthemodel. Section3presentsdifferent typesofforwardguidanceexperiments. Section4concludes. 2 Model OurmodelingframeworkisveryclosetotheoneusedbyMNS,andfeatureshouseholds’uninsurableincome risk,borrowingconstraintsandnominalrigidities. Themaindifferencebetweenourmodelandthesetupof MNS,whereagentsaresubjecttoano-borrowingconstraint,isthatweallowagentstoborrowthroughone periodrisk-freedebt,subjecttoapositiveborrowingconstraint. 6TheimplicationsofthismechanismareexplainedmoreindetailinHagedorn(2018a,2018b). 7Giventhathigherinflationreduceshouseholds’abilitytoselfinsure,onecouldinterprettheresultinHagedornetal.(2019)as aninstanceofcountercyclicalliquidity,assuggestedbyWerning(2015). 6

Furthermore, as we explain more in detail below, we depart from MNS also with respect to their assumptionsregardingtheaggregateavailabilityofliquidityandthedistributionofdividends. Aspointedout by Werning (2015) and Hagedorn et al. (2019), the results in MNS are likely driven by the redistributive effects caused by fluctuations in taxes and profits in response to forward guidance. In particular, MNS assume: i) a fixed stock of government debt, financed by levying taxes only on high productivity agents; ii) profits, arising from monopolistic competition, equally distributed across agents in lump sums. Because of these assumptions, forward guidance produces a redistribution of wealth from high MPC agents towards low MPC households: lower interest rates imply lower returns on savings but also a lower debt servicing costforthegovermentandconsequentlylowertaxesforlowMPCagents;inaddition,thepositiveeffectof forward guidance on real wages implies a decline in dividends that hits disproportionately the agents with lowproductivity. Asaresultoftheseredistributiveforces,forwardguidanceproducesonlysmalleffectson aggregatedemandintheMNSmodel. Inordertoabstractfromtheseconfoundingeffects,wemakeassumptionsaimedatminimizingindirect redistributiveeffectsthatcandrivetherelativepowerofforwardguidancewithinanHANKmodelcompared toitsRANKcounterpart. First,weassumethatthereisnosupplyofpublicdebt,sothatthereisnorolefor fiscalpolicyinourmodel. Second,weassumethatdividendsaredistributedlumpsumbutproportionallyto agents’individuallaborincome.8 Asaresult,fluctuationsinaggregateincome,wagesplusdividends,have minimal impact on the distribution of individual non-financial income. Using the terminology of Werning (2015),wecanthinkofhouseholdnon-financialincomeriskasbeing”acyclical”. Giventheseassumptions, we can focus on the redistributive channels of forward guidance that operate only through fluctuations in interestrates,whichisthemaintoolofmonetarypolicy. 2.1 Households The economy is populated by a continuum of households deriving utility from consumption and leisure, accordingto ∞ ∞ (cid:34) c1−σ l1+ϕ (cid:35) E (cid:88) βtU (c ,l ) = E (cid:88) βt j,t − j,t (1) 0 j,t j,t 0 1−σ 1+ϕ t=0 t=0 wherec isconsumptionofhouseholdj attimetandl islaborsupply. Householdsreceiveidiosyncratic j,t j,t labor productivity z which follows a Markov chain with transition probabilities P (z ,z ). Housej,t j,t+1 j,t holds can borrow and lend using a risk-free one-period bond, b , subject to a borrowing constraint. The j,t+1 8AsimilarassumptionisusedinFahriandWerning(2019). 7

valuefunctionofhouseholdj canbewrittenrecursivelyas    (cid:88)  V (b ,z ) = max U (c ,l )+β Pr(z ,z )E V (b ,z ) t j,t j,t j,t j,t j,t+1 j,t t t+1 j,t+1 j,t+1 cj,t,lj,t,bj,t+1 zj,t+1  subjectto c +b ≤ z l W +Rbb +α D +τ (z ) (2) j,t j,t+1 j,t j,t t t j,t j,t t t j,t b ≥ −¯b with ¯b ≥ 0 (3) j,t+1 where W is the real wage and D are aggregate profits, and Rb is the realized gross real interest rate on t t t bonds. In all our experiments we are going to compare the results obtained with either real bonds, for which Rb = R , the realized real rate; or with nominal bonds, in which case Rb = it−1, where i and t t−1 t πt t π represent the gross nominal rate and inflation rate respectively. Equation (3) represents the borrowing t constraint,whichdependsontheparameter ¯b. Eachhouseholdreceivesashareα ofaggregateprofitsand, j,t aswewilldescribemoreindetailbelow,wewillassumethatthisfractionisproportionaltothehousehold’s laborincome. Finally,τ (z )representlumpsumtaxesleviedbythegovernment,andpotentiallydepending t j,t on households’ individual productivity. As we mentioned above, and unlike the framework used by MNS, wewillassumethatbondsareinzeronetsupply,implyingthattaxeswillplaynoroleinourmodel. Thefirstorderconditionsforbondholdingsandlaborsupplyaregivenby (cid:88) c (b ,z )−σ = µ +βE Rb Pr(z ,z )c (b ,z )−σ (4) j,t j,t j,t j,t t t+1 j,t+1 j,t j,t+1 j,t+1 j,t+1 zj,t+1 z W (c (b ,z ))−σ = (l (b ,z ))ϕ (5) j,t t j,t j,t j,t j,t j,t j,t µ (cid:0) b +¯b (cid:1) = 0 (6) j,t j,t+1 whereµ isthemultiplierontheborrowingconstraint. j,t 2.2 Production ThefinalgoodY isproducedfromacontinuumofintermediateinputs,indexedbyh ∈ [0,1],accordingto t theproductionfunction (cid:20)(cid:90) 1 (cid:21) ε− ε 1 Y = Y (h)(ε−1)/εdh (7) t t 0 8

whereY (h)istheproductionofintermediategoodh. t Asaresultthedemandforintermediategoodhwillbegivenby (cid:18) P (h) (cid:19)−ε t Y (h) = Y (8) t t P t whereP (h)representsthepriceofanindividualvariety,andtheaggregatepricelevelP satisfies t t (cid:20)(cid:90) (cid:21) 1 P = P (h)1−εdz 1−ε (9) t t Intermediate goods are produced by monopolistically competitive producers using only labor as input, accordingto Y (h) = N (h) (10) t t Intermediate good producers are owned by a risk-neutral manager discounting the future at rate 1/R .9 t AsisstandardinNew-Keynesianmodels,theycanresetpricesonlyoccasionally,withprobability(1−γp), asinCalvo(1983). Asaresult,theirproblemwillconsistinchoosingthepriceP∗,inordertosolve t (cid:88) ∞ (cid:34) (cid:89) s (cid:18) 1 (cid:19) (cid:35) (cid:20) P∗ (cid:21) maxE γs t −W Y∗ (h) (11) P t ∗ t s=0 p i=0 R t+i P t+s t t+s subjectto (cid:18) P∗ (cid:19)−ε Y∗ (h) = t Y (12) t+s P t+s t+s Thesolutiontothisproblemsatisfies P∗ ε g π∗ = t = 1,t (13) t P (ε−1)g t 2,t where 1 g = pmY +γ E g (π )ε (14) 1,t t t p R t 1,t+1 t+1 t 1 g = Y +γ E g (π )ε−1 (15) 2,t t p t 2,t+1 t+1 R t Inaddition,inflationπ = P /P canbewrittenas t t t−1 ∗(1−ε) (ε−1) 1 = (1−γ )π +γ π (16) p t p t 9This assumption is also used in Hagedorn et al. (2019). We have also tried an alternative specification in which managers discountatrateβasinMNS.Theresultsareprettysimilar. 9

2.3 Government As in MNS, government is assumed to run a balanced budget to keep a constant level of debt each period, B¯, (cid:90) τ (z)dΓ (b,z) = B¯(R −1) (17) t t t where Γ (b,z) represents the distribution of households over their bond holdings and labor productivity at t time t. As mentioned before, in our baseline calibration we will set B¯ = 0 in order to abstract from the redistributiveeffectsoffiscalassumptions. Unlessotherwisespecified,monetarypolicyfollowsasimpleTaylorrule log(i ) = log(R)+φ log(π ) (18) t π t whereRrepresentsthesteadystaterealrate. Finally,therealinterestratesatisfiestheFisherrelation i t R = (19) t π t+1 2.4 Equilibrium Define Γ (b,z) as the distribution of households over their bond holdings and labor productivity at time t. t Thentheequilibriuminthebondmarketrequires (cid:90) b (b,z)dΓ (b,z) = B¯ (20) t+1 t Aggregateprofitsofintermediatefirmsaregivenby D = Y −W N (21) t t t t whereN representaggregatelabordemand. t AggregatelaborsupplyL isgivenby t (cid:90) L = zl (b,z)dΓ (b,z) (22) t t t andlabormarketclearingrequiresthat N = L (23) t t Aggregateproductionwillbegivenby Y = N = L (24) t t t 10

Finally,aggregatingoverhouseholds’budjetconstraintswecanobtaintheaggregateresourceconstraint C = Y (25) t t whereC = (cid:82) c (b,z)dΓ (b,z),representsaggregateconsumption.10 t t t 3 Forward Guidance Experiments In order to highlight the redistributive channels of forward guidance in our model, we will perform several typesofexperiments. Thefirsttypeofexperiment, whichwecallrealrateforwardguidance, assumesthat thecentralbankannouncesadeclineintherealinterestrateTperiodsinthefuture,whiletherealratestays constant in the preceding periods. The second type of experiment, which we call nominal rate forward guidance, assumes that the central bank announces the same decline in the nominal rate in period t+T, but keepsonlythenominalrateunchangedinprecedingperiods. Finally,thethirdtypeofexperimentwillfocus on the impact of forward guidance in a liquidity trap, when current monetary policy is constrained by the zerolowerbound. Inalltheexperiments, wewillcomparetheresultsobtainedwhenusingaversionofthe modelwithrealbondswithoneusingnominalbonds. 3.1 CalibrationandSolution Apart from our assumptions on the supply of government bonds, on the distribution of dividends, and on the borrowing constraint, we calibrate the model exactly as in MNS, as shown in table 1. We use a value for the risk aversion parameter σ, equal to 2. The Frish elasticity of labor supply, 1/ϕ, is set equal to 0.5. We choose a parameter for the elasticity of substitution of intermediate goods, ε, equal to 6, which impliesasteadystatemarkupof20percent. TheCalvoparameterγp issetto0.85,resultingina15percent probabilityofresettingpriceseveryquarter. Wesetthediscountfactorβ = 0.962inordertoobtainasteady state value of the real interest rate of 2 percent annually. As in MNS, we calibrate the idiosyncratic wage 10Aggregate output could potentially depend also on price dispersion S , according to Y = S L , where S satisfies S = t t t t t t (cid:2) (1−γ )(π∗)−ε+γ πεS (cid:3) . However, sincewewillbesolvingourmodelusinglinearizationtechniques, thissecondorder p t p t t−1 termwillnotaffectaggregatedynamicsandweomititinthemaintext.Wecheckedthatoursolutionapproachdeliversverysimilar resultstotheonesobtainedinMNS,whoemployaperfect-foresightnonlinearsolutionalgorithm,fortheFGexperimentswitha fixedpathfortherealrate. Smalldifferencesmayappearinaliquiditytrap,whenpricedispersioncostscanbecomelargerelative tooutput.However,wefollowHagedornetal.(2019),whoabstractfromtherealimpactofRotembergcosts,inordertoavoidthat largepricemovementsresultinunrealisticresourcecostsinaliquiditytrap. 11

risk to match the wage process estimated in Floden and Linde’ (2001), by using an AR(1) process with an autoregressive coefficient of 0.966 and an innovation variance of 0.17. This process is approximated with a three states Markov chain using the Rouwenhorst (1995) method.11 Unless otherwise specified, we use a standard Taylor rule to determine the nominal interest rate, according to logi = logR +φlogπ , with t t φ = 1.5. Asmentionedabove,wedepartfromMNSwithrespecttothreeassumptions. First,weallowhouseholds to borrow, by setting a value for the borrowing constraint, ¯b = 1.3, that is equal to about 5 times average mothly labor income in the steady state equilibrium. This multiple was suggested by MNS as a reasonable approximation of the evidence provided by Kaplan, Violante and Weidner (2014) on the distribution of households’unsecureddebtintheU.S.Second,weassumethatbondsareinzeronetsupply,thatisB¯ = 0, implying that the government does not need to levy any tax, that is τ (z) = 0 for all z,t. As a result, t we can abstract from redistributive effects linked to assumptions on tax progressivity, which are discussed by Hagedorn et al. (2019). Third, we assume that dividend distribution is proportional to labor income, that is α = z l /L .12 This assumption helps to avoid that fluctuations in aggregate profits might have j,t j,t j,t t unrealistic redistributive effects, possibly overcoming the direct effects of forward guidance, as pointed out byWerning(2015). Infact,substitutingthevaluesforα ,andτ (z ),andusingtheequilibriumvalueof j,t t j,t dividendsimpliedbyeq. (21),wecanrewritethehousehold’sbudgetconstraintofeq. (2)as c +b ≤ z l +Rbb (26) j,t j,t+1 j,t j,t t j,t Equation (26) shows that, in our framework, the only aggregate variable directly affecting households’ wealth is going to be the interest rate on bonds, Rb, while there is no redistributive impact from dividends t or taxes.13 As a consequence, we can abstract from assumptions on households’ portfolio choices or on taxprogressivity,andwecanisolatetheredistributiveeffectsofforwardguidancethatoperateonlythrough fluctuationsininterestrates,themainpolicytoolofthemonetaryauthority. Thiscalibrationresultsinabout 18 percent of agents being constrained in the model’s steady state, a number close to the one obtained by MNS.14 We solve the model using by using the Reiter (2009) method, which combines a global solution, to determine the steady state distribution, with linearization around the steady state in order to determine the 11AspointedoutbyHagedornetal.(2019),MNSusevaluesfortheirMarkovchainthatimplyanaggregateproductivitygreater thanone.Weadjustslightlythevaluessothataggregateproductivityisequalto1asinthecompletemarketmodel. 12Thesharesα arenormalizedbytotallaborsupplyinordertosumuptoone. j,t 13Aggregate wage is still going to affect household wealth through the labor supply decision in eq. (5). Since dividends are assumedtobedistributedinlumpsumfashion,α doesnotaffecthouseholds’laborsupply. j,t 14Inaddition,weobtainthatinsteadystate22%ofagentsareunconstrainedborrowersand60%ofagentsaresavers. 12

dynamics of the aggregate economy. In particular, we approximate the distribution of agents over bond holdings and labor productivity, Γ(b,z), by using a grid with 75 bins, obtained by combining 25 bins for bondswith3binsforproductivities.1516 Table1: Calibration Parameters Description Value Target/Source β DiscountFactor 0.96 2%annualrealrate(MNS) σ RiskAversion 2 MNS ϕ InverseFrischElasticity 2 MNS γ PriceStickiness 0.85 MNS p ε CESElasticity 6 MNS B¯ SupplyGovernmnetBonds 0 NoTaxes ¯b BorrowingLimit 1.3 5timesavg. monthlyincome(MNS) φ Taylorrulecoefficientoninflation 1.5 MNS π 3.2 RealRateForwardGuidance Inourfirstexperimentwefocusontheeffectsofananticipateddeclineintherealinterestrateinperiodt+T, withtherealratebeingatitssteadystatevalueinalltheotherperiods(asimilarexperimentisusedinMNS). For simplicity, we assume that monetary policy is characterized by an exogenous rule for the real interest rate, that is R = i /E π +εr , where εr represents a shock to the real interest rate announced j t t t t+1 t,t−j t,t−j periodsinadvance,aforwardguidanceshock. Weconsideranexperimentinwhichthemonetaryauthority promisesa50basispointsdeclineintherealinterestrateT quartersinthefuture,thatisashocktoεr .17 t,t−T Weperformtheexperimentfirstwithrealbondsandthenconsidernominalmonds. 15We have also tried to use a finer grid, which increased computational time but produced identical results. In addition, we havecheckedthattheReitermethodallowstoreplicatethemainresultsofMNS,whouseaperfect-foresightnonlinearalgorithm. Smalldifferencesmightbepresentintheliquiditytrapexperiments,inwhichnon-linearitiescangeneratevisibleeffectsfromprice dispersionthatarenotcapturedbyoursolutionmethod. However, wefollowHagedornetal. (2019)indisregardingpotentially unrealisticoutputeffectsarisingfrompricedispersion. 16OneadvantageofthismethodisthatitcanbeimplementedinDynare,sensiblyreducingthecomputationaltimecomparedwith non-linearperfectforesightalgorithms. Inaddition,thismethodallowstosolvefortheZLBinarelativelyeasywaybyusing,for example,theOccBintoolboxdevelopedbyGuerrieriandIacoviello(2015). 17Alternatively,itispossibletoreplicatethisexperimentbyusingastandardTaylorrule,andbyfeedingasequenceofanticipated monetarypolicyshocksthatdeliversaconstantpathfortherealinterestrateuntilperiodT−1,andthenproducesa50basispoints declineinperiodT.Wehavecheckedthatthisalternativeapproachdeliversbasicallyidenticalresults. 13

3.2.1 RealBonds Figure1reportstheresponseofourHANKmodel(solidblueline)andofthecorrespondingRANKmodel (reddottedline),toananticipatedshockoneyearinthefuture(T = 4),whenbondsarereal. Asexplainedin detailinMNS,theEulerequationintheRANKmodelimpliesthattheinitialchangeinoutputisproportional, withacoefficientof1/σ = 0.5,totheexpectedsumoffuturerealratechanges. Asaresult,outputis0.25% higherthaninsteadystatesforanyperiodprecedingt+T. ThepathofoutputinourHANKmodel,thesolid blueline,isslightlyhigherthanintheRANKmodel. Tounderstandthisresultitisusefultolookattheblue dashedline,whichrepresentsthecontributionofthe”transfernewsshock”impliedbytheforwardguidance experimentinourHANKframework,thatistheimpliedtransfer,fromsaverstoborrowers,generatedbythe decline in the realized real rate at time t+T +1.18 As we see from figure 1, the transfer from low MPC agentstowardshighMPCagentsinthefifthquartergeneratesaspikeinoutput. Furthermore, suchtransfer reduces agents precautionary motives at time 0, resulting in an increase in output also on impact and in the preceding periods. We call this mechanism amplifying the effects of forward guidance, which is absent in theRANKmodelorinHANKmodelswithouthouseholddebt,thetransfernewschannel. Figure2,presents thesameexperimentforaforwardguidancehorizonofT = 20. InthiscasethepathofoutputofHANKis slightlyabovetheoneforRANKafterperiod10,buttheimpactresponseinHANKisslightlysmaller. The impactofthetransfernewsshockisfairlysimilar. Infigure3,weextendthisexperimenttomultiplehorizons,andwereporttheimpactresponsesofoutput and inflation. As was suggested by figure 1 and 2, the output response is larger for the HANK model at shorterhorizons,butitbecomesweakerthantheRANKwhenforwardguidancepertainshorizonsmorethan 10quartersinthefuture. Ithastobenotedthatforwardguidanceappearstobequitestrongerinourmodel comparedtotheframeworkofMNS(greendottedline),bothbecauseoftheabsenceofredistributiveeffects of dividends and taxes, negatively affecting poor households, and because of the presence of the transfer news channel. On the other hand, given that inflation depends on the sum of future marginal costs in both types of models, we have that its impact response is increasing in the forward guidance horizon. However, inthisexperimentthebehaviorofinflationhasnobearingonoutputdynamics,sincethepathoftherealrate isfixedinalltheexperiments. Inordertoinvestigatefurtherthedeterminantsofthedownwardslopingbluelinegeneratedbyourmodel, figure4presentsthetimezeroimpactontheaggregateconsumptionofconstrainedborrowers,unconstrained 18Toimplementthistypeofshock,wesimplyfeedintothemodelalumpsumtransfer,τ˜ ,proportionaltothevalueofagents’ i,t outstandingdebtinordertocaptureeachagent’sexposuretothechangeintherealizedrealrateattimet+T+1,thatisτ˜ = i,t+T+1 εr b ,whereεr representsthechangeintherealrateimpliedbyforwardguidance. t,t−T i,t+T t,t−T 14

borrowers and unconstrainted savers, when considering either the forward guidance shock or the transfer shock.19 The left panel shows how the three groups react very differently to the FG shock for shorter horizons. UnconstrainedborrowersreactmorethantheRANKmodel. Savers’initialresponseisquiteclose toRANK,whereastheresponseofconstrainedagentsissomewhereinthemiddle. Tounderstandwhatdrives thedifferentialresponseofeachgroup,itisusefultoconsidertheimpactoftheimpliedtransfernewsshock, reportedintherightpaneloffigure4. Inadditiontothestandardintertemporalmotive,inourmodel,lower future interest rates stimulate the demand of unconstrained agents by reducing precautionary motives. In fact,lowerrealratesimplyafuturetransferfromsaverstowardsborrowers(constrainedandunconstrained), who are the agents with a higher likelihood of becoming constrained and with a higher MPC. As a result, borrowersreactmoreonimpact,astheyaremorelikelytobenefitfromthetransfer,puttingupwardpressure on aggregate demand. The impact response of constrained agents is mainly driven by the positive general equilibrium effect on wages, which stimulates their labor supply. The total impact on aggregate output dependsontheaverageMPCofeachgroupandontheirshareofaggregateconsumption,andtherightpanel offigure4showsthatthetransfershockcomponenthasanaggregateeffectonoutputofabout0.01%.20 As the FG horizon extends into the future, the initial response of unconstrained savers and borrowers becomes very similar, both for the FG shock and the transfer news shock. This result is due to the fact that asT → ∞,agentsexpecttoconvergetotheergodicdistribution,wheretheexpectedinitialbondholdingis zero. Consequently,theinitialbondholdingsplaylittleroleforashockoccurringfarinthefuture. However, whatmightbesurprisingisthatwhiletheaggregateeffectofthetransfernewsshockisstillpositiveandstable forlongerhorizons,thetotaleffectofFGintheHANKmodeldeclinesbelowtheeffectintheRANKmodel. One possible explanation for this outcome is that in our framework, and in the MNS model, the amount of borrowing allowed to constrained agents is constant at ¯b. Using the terminology proposed by Werning (2015), we can say that the total amount of liquidity in the economy is countercyclical, since it expands and contracts less than output. As suggested by Werning (2015), countercyclical liquidity can hamper the effect of FG as some agents might not be able to smooth consumption in response to an expected future higher income, due to the future lower rates, because of the constant borrowing limit. Consequently, the expectedprobabilityofbecomingconstrainedmightincrease,reducingthetimezeroconsumptionresponse of unconstrained agents. To check this intuition, we perform the same experiment in an alternative version of the model in which the borrowing constraint scales proportionally to aggregate income, that is where 19Savers and borrowers are classified according to their end of period bond holdings in steady state. Very similar results are obtainedifweusetheinitialbondholdingsinstead. 20Withourcalibration,insteadystateconstrainedborrowersaccountforabout7%oftotalconsumption,unconstrainedborrowers accountforabout18%,andsaversaccountfortheremaining75%. 15

agents can borrow up to ¯b Yt . Quoting again Werning (2015), this setup should correspond to a world in Yss which liquidity is acyclical. The implied impact effects of FG on output and inflation are represented by the blue dash-dotted line in figure 3. In this case, the impact of FG is larger in HANK for all the horizons, and it is slightly increasing with T. These results are due to two new effects. First, when output expands, because of a higher demand by unconstrained agents, it also relaxes the borrowing constraint of high-MPC constrained agents, boosting their consumption for all the periods preceting t + T. Second, the larger T, the longer the expansion in output and the longer the period over which the borrowing constraint is being relaxed, generating an increasing attenuation in agents’ precautionary motives.21 In the rest of the paper, we will maintain the setup with a constant borrowing constraint, as it is standard in most Bewley-Ayagari models. It is beyond the scope of this paper to provide a realistic modeling of the cyclicality of liquidity. Furthermore,asshowninfigure3,eliminatingcountercyclicalliquiditywouldonlyamplifyourresults. The main objective of this section was to highlight the presence, especially at shorter horizons, of a transfer news channel in a HANK model with private debt, and to show that real rate FG is stil powerful in our setup. This new channel could be even more powerful in a model with larger income risk or with procyclicalliquidity.22 3.2.2 NominalBonds WenowperformthesamerealrateFGexperimentassumingthatbondsarenominal. Figure5comparesthe horizon effect of forward guidance for real bonds (as reported in figure 3) and nominal bonds (the dashed blackline). WithnominalbondstheimpactofFGbecomesclearlyincreasinginthehorizonoftheannouncement. ThiseffectisduetothefactthattheincreasingimpactofFGoninflation,afeaturesharedwithMNS, now translates in an increasing transfer towards borrowers at time zero, through a debt deflation channel. Figure6decomposestheimpacteffectbetweensaversandborrowers. Inthisinstance,asitcanbeseenfrom theleftpaneloffigure6,theupwardslopingbehaviorofoutputisdrivenbytheincreasingresponseofconstrainedborrowers,whohavehighMPCandwhoseconsumptionreactsstronglytotheunexpectedtransfer. The right panel of figure 6 shows how also the response of unconstrained borrowers starts increasing with the FG horizon after period 10, as the influence of the transfer news channel wanes (as shown in figure 4). 21Inaddition,thetransfernewschannelisamplifiedaswell,sincethehigheroutputresultingfromtheredistributionofresources feedsbackintoahigherconsumptionforconstrainedagents. 22Wehaveexperimentedwithalternativemodelcalibrationsinwhichincomeriskislargerthaninthebaseline. Inthiscase,the transfernewschannelbecomesstronger,increasingthegapbetweentheoutputeffectintheHANKmodelandtheRANKmodelfor shorthorizons. 16

In fact, in this experiment, the transfer news channel is still active, and it amplifies the impact of the debt deflation channel. The key takeaway of this quantitative exercise is that with nominal debt the response of inflation to FG has positive spillovers for output that increase with the FG horizon, a feature that had not beendocumentedinpreviousliterature. 3.3 NominalRateForwardGuidance Inthepreviousexperiments,therealratewasnotallowedtomoveendogenouslyintheperiodsprecedingthe announcedratedeclineattimeT. Asaconsequence,inflationdidnotplayaroleforoutputmovements,apart from the debt deflation effect obtained with nominal bonds at time zero. We now assume that the central bank holds the nominal interest rate i fixed before time T, decreases the nominal interest rate by 50 basis points at time T, and then it allows i to react according to the Taylor rule specified above for the periods followingT.23 Figure 7 reports the results of this experiment, with T = 12. The output response of output in the RANK model is much larger compared to the previous experiments. This result is due to the fact that the initial spike in inflation causes a decline in the real interest rate, which, unlike the previous experiments, stays below steady state for all periods before T, stimulating aggregate demand. This effect is amplified in our HANK model since the lower real rate implies a wealth transfer from savers to borrowers, who have a higherMPCinthemodel. ThischannelissimilartotheinterestrateexposurechanneldescribedbyAuclert (2019) as one of the main redistributive channels of conventional monetary policy. As we will show in the next exercises, also in the context of forward guidance this mechanism is particularly relevant in HANK frameworks. Figure8looksathowtheeffectsofnominalrateforwardguidancevarywiththehorizonofthedeclinein theinterestrate. IntheRANKmodel,theinitialimpactonoutputincreasesexponentiallywiththehorizon, since,asexplainedabove,theinitialjumpofinflationisincreasinginthehorizonofFG.Furthermore,higher consumptionfeedsbackintohigherinflation,amplifyingthetotaleffect. InourHANKmodel,theeffectis evenstronger,astheredistributionthroughlowerrealratesoperatesforalongertimeperiod. Therightpanel offigure8showsthatunconstrainedborrowersareinfacttheagentsreactingmoreonimpact. Atthesame time, the transfer news channel and the intertemporal amplification, present also in the RANK model, are alsooperating,boostingtheconsumptionofalltheagents. Finally, figure 9 reports also the results for the same type of experiment when we consider nominal 23AsimilarexperimenthasbeenperformedalsobyCalstrometal.(2015)andFarhiandWerning(2019). 17

bonds. As expected, the output response is stronger than the case with real bonds, since the debt deflation channel augments the positive effects of low inflation. The channels discussed in this section are going to be helpful to interpret the results of our third type of experiment, in which the nominal rate is fixed at the zero-lower-bound for a certain number of periods and the central bank promises to keep it low even longer inordertostimulatetheeconomy. 3.4 ForwardGuidanceinaLiquidityTrap Forwardguidanceisakeypolicytoolwhenthecurrentpolicyrateisconstrainedbythezero-lower-bound. In thissection,westudythepowerofforwardguidanceinaliquiditytrap. Figure10considersanexperimentin whichthediscountfactorβincreasesforaknownnumberofperiods,beforerevertingtoitssteadystatevalue, causingthenominalinterestratetohittheZLB.24Inthebaselineexperiment,thenominalrateisgovernedby thestandard”naive”Taylorrule,respondingonlytoinflation,employedinthepreviousexercises. Thepath ofthediscountfactoriscalibratedtoobtainaZLBepisodeofeightquartersanda4%declineinoutputboth intheRANKmodel(solidredline),intheHANKmodelwithrealbonds(solidblueline),andintheHANK modelwithnominalbonds(solidblackline).25 Thedashedlinerepresentsthebehaviorofthethreemodels under an ”extended” monetary policy rule which keeps the nominal rate to zero for four extra periods, as it canbeseeinthetopleftpaneloffigure10. In the RANK model, forward guidance reduces the decline in output and inflation by about two thirds. In our HANK model with real debt, the effect of the four extra periods of low rates is more powerful, with output declining only by about 0.5 percent, and inflation only by 0.25 percent. In our HANK model with nominal debt the recession is completely avoided. Hence, when private debt is introduced, in a liquidity trap forward guidance can be a powerful policy tool also in a HANK model, even more than in a standard RANKframework. Thisresultisduetothethreechannelsoutlinedabove: i)areductionintheprecautionary motives of unconstrained agents, the transfer news channel, ii) a transfer of wealth towards high MPC agents through lower real rates, the interest rate exposure channel, and iii) when debt is nominal, a debt deflationchannel. Asthelowerleftpaneloffigure10shows, akeydriveroftherecessionwiththe”naive” monetarypolicyisthespikeintherealrate,whichdepressesaggregatedemand. Throughtheredistributive channelspresentinourmodel,forwardguidanceisveryeffectiveatavoidingalargedropininflationandat stimulatingdemand. Figure11delvesdeeperintothepropagationofforwardguidancepolicyinourmodel 24Thesameapproachtostudyliquiditytrapsisused,amongothers,byMNSandHagedornetal.(2019). 25WeexploitthelinearsolutionofourmodelandweimplementtheZLBbyusingtheOccBintoolboxdevelopedbyGuerrieri andIacoviello(2015). 18

by reporting the behavior of aggregate consumption for savers and borrowers. As expected, on impact the policyisparticularlybeneficialforborrowers’consumption,butalsosaversrespondstronglybecauseofthe intertemporalchanneloperatingthroughlowerrealrates. Another indirect channel, through which monetary policy operates in a HANK model, is by affecting agents’ wealth through real wages and dividends. In figure 10 we see how real wages experience large fluctuationsinaliquiditytrap. Withflexiblewages,dividendsmoveintheoppositedirection. AsaresultFG cancausealargedeclineindividendsbystimulatingdemandandrealwages. Whendividendsaredistributed lumpsum,asinMNS,thenegativeimpactofFGonprofitscandisproportionatelyaffecthighMPCagents, resulting in a much weaker effect of forward guidance in a liquidity trap (see Werning (2015)). As shown in eq. (26), our assumptions on dividends distribution imply that there is no direct redistributive effect of either profits or wages.26 However, wage fluctuations still affect individual labor supply decisions ( see eq. (5) ). Potentially, the labor supply of constrained agents, who have high MPC, could be more responsive to changes in the real wage.27 In the real rate FG experiments with real bonds, constrained agents benefit from lower future rates only because of the general equilibrium effect on wages. As a result, one might wonder whether the power of forward guidance in our HANK framework would be greatly reduced with the introduction ofsticky wages. Hagedorn et al(2019) introduce sticky wages intheir framework in order to reduce the redistributive effects of labor income and in order to minimize fluctuations in dividends. In particular, they assume that each household provides differentiated labor services to a union, which sells these services to a labor packer. The labor packer combines them into aggregate effective labor using a standard CES technology, with elasticity of substitution εw. Unions set the nominal wage for an effective unitoflaborinordertomaximizetheirprofitssubjecttoquadraticadjustmentcostsalaRotemberg(1982). These costs are given by θw (πw −π¯w)2, where πw is wage inflation, and are proportional to aggregate 2 t t hours. Thesolutiontotheunions’problemdeliversafamiliarwagePhillipscurve28 θw(πw −π¯w)πw = (1−εw)W +εw Lϕ t +θwE 1 (cid:0) πw −π¯w(cid:1) πw L t+1 (27) t t t C−σ t R t+1 t+1 L t t t AkeyassumptioninHagedorn(2019)isthat,whensettingwages,unionstakeintoaccounttheaggregate Lϕ marginal rate of substitution between aggregate labor and aggregate consumption, t , implying that all C−σ t householdssupplythesameamountoflaboratalltimes. Consequently,whenweintroducewagestickiness 26OurassumptionsessentiallyallowustoabstractfromthedirectredistributiveeffectsofwhatAuclert(2019)definestheearnings heterogeneitychannel.AccordingtotheempiricalfindingsinAuclert(2019),thischannelshouldamplifyevenmoretheimpactof monetarypolicyinHANKmodels. 27On the other hand, one must keep in mind that these are also the agents with the lowest idiosyncratic labor productivity realization,potentiallyreducingtheelasticityoflaborsupplytorealwages. 28SeeHagedornetal.(2019)foradetailedsolutionoftheunion’sproblem. 19

inthisfashioninaHANKmodel,wearenotonlyaffectingthedynamicsofthemodelthroughthestandard channels of a New-Keynesian RANK model, but, by eliminating labor supply heterogeneity, we are also eliminating agent’s ability to self-insure by adjusting their labor choice.29 Despite this importante caveat, weintroducewagestickinessinthesamewayasinHagedornetal. (2019),sinceitisasimplewaytolimit wage fluctuations.30 We calibrate εw = 6in order to have the same markup in wages and prices, ad set the adjustment cost parameter θw in order to obtain the same Philips curve slope implied by the Calvo setting forpricestickinessusedinthepreviousexperiments.31 Figure12replicatestheexperimentoffigure11,by targetingthesamedurationsofZLBanddeclineinoutput,whileassumingthatwagesaresticky.32 Forward guidanceisnowweakerinallthethreemodelsconsidered,butitisstillmorepowerfulinourHANKmodel, where it reduces the output decline by more than half, twice as much as in the RANK counterpart.33 One reason why FG is weaker in this experiment is that inflation, and consequently the real rate, move less, dampening the the intertemporal channel of forward guidance. In the HANK model, smaller fluctuations in wages and inflation reduce the redistribution towards constrained agents that we have described in the previous quantitative exercises. The power of the three redistributive channels in our model is still non negligible,giventhatFGnoticeablyreducesthedownturn. Furthermore,wehavetokeepinmindthat,inthe setupusedinfigure12,highMPCagentsarenotallowedtoadjusttheirlaborsupplyinresponsetochanges in their wealth and consumption, introducing an additional limitation to the impacf of FG that is unrelated towagestickiness. Summingup,evenwhenintroducing,inasimplifiedway,frictionsinwageadjustments forwardguidanceremainsaneffectivetoolattheZLBinourHANKframework. 29Asaresult,inadditiontothestandardwedgeintroducedmymonopolisticcompetitionintheunionsmarket,wagestickiness also affects the steady state distribution by equalizing labor supply across agents. For example, with our calibration, once we introducewagestickiness,theshareofconstrainedagentsisslightlysmaller,around16%. 30AsimilarapproachisusedalsoinAuclert,RognlieandStraub(2018). 31TheimpliedPhillipscurveslopeforpricesalaCalvo,isgivenby (1−γp )(1−γp/R) andisabout0.025.Asaresult,theimplyed γp θw isaround200. Inaddition,weassumethatalsothepricePhillipscurveisdeterminedwithaRotemberg(1982)approach,with thesameparameters.AsinHagedornetal.(2019),weassumethatadjustmentcostsarenon-pecuniary. 32Withstickywages,inordertoobtainthesamedeclineinoutputandthesamedurationoftheZLBweneedtouseadiscount factorshockwithashorterdurationthanintheexperimentinfigure10,9periodsinsteadof12. 33Inadditiontothedifferentfiscalassumptions,thecomparisonofourliquiditytrapesperimentwiththeonepresentedinHagedornetal. (2019)ismadedifficultbyotherfactors. First,Hagedornetal. (2019),matchaquitelargerslopeforthePhillipscurve of0.11. Inaddition,fortheirmainliquiditytrapexperimenttheyuseaTaylorrulecoefficientoninflationof0.5,whichmakesa comparisonwiththeRANKmodelimpossiblebecauseofindeterminacy. 20

4 Conclusions The transmission of conventional monetary policy and of forward guidance has been extensively studied in standard New Keynesian models with a representative agent. In these type of models, monetary policy operates mainly through an intertemporal substitution channel, which makes forward guidance particularly potent (Carlstrom et al. (2015), MNS). The transmission of monetary policy in models with heterogeneous agents and incomplete markets is more complex, because of the presence of precautionary motives and of heterogeneity in agents’ marginal propensity to consume. Recent papers have studied how these new elements affect the propagation of conventional (Kaplan et al. (2018), Auclert (2019) ) and unconventional monetary policy. In particular, a set of papers has tried to answer the question of whether HANK models dampenoramplify,withrespecttotheRANKframework,theaggregateeffectsofforwardguidance(MNS, Werning (2015), Hagedorn et al (2019), Bilbiie (2019), Acharya and Dogra (2019)). This debate has so far abstractedfromtheredistrubitionarisingfromtheinteractionbetweenhouseholddebtandforwardguidance. Thispapercontributestotheliteraturebystudyingtheroleofredistributivechannelsforthepropagationof newsonfutureinterestratesinHANKmodelsinwhichhouseholdsborrowandlend. Ourmainfindingisthathouseholddebtintroducesthreeredistributivemechanismsthattendtoamplify the power of forward guidance in a HANK model. First, lower future rates imply a future transfer from savers to borrowers, which decreases precautionary motives. Second, higher initial inflation results in a lower path of the real rate, increasing the wealth of borrowers, who have a higher marginal propensity to consume. Third, if debt is nominal, at the time of the policy announcement debt deflation generates also a wealthtransfertowardshigh-MPCborrowing-constrainedagents,furtherincreasingaggregateconsumption and inflation. These channels amplify each other in a liquidity trap, potentially making forward guidance a powerful monetary tool in a HANK model, even more than in a RANK framework. Auclert (2019), showsthatthesecondandthethirdchannelareimportantforthetransmissionofstandardmonetarypolicy, both from a theoretical and an empirical perspective. We show that redistribution, and its interaction with precautionarymotives,canalsoplayanimportantroleforthetransmissionofforwardguidance. Hence, our work suggests that household debt should be taken into account when evaluating the effectivenessofunconventionalmonetarypolicyinincompletemarketsmodels. Furthermore,ourresultscouldbe evenstrongerifdebtweresubjecttodefault,orifitwerecollateralizedbyadurableassetwhosepricereacts tomonetarystimulus,like,forexample,mortgages34. Theseareallinterestingtopicsforfutureresearch. 34see,forexample,Kaplan,MitmanandViolante(2019) 21

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Figure1: RealRateForwardGuidanceforT=4: RealBonds Output Inflation 0.15 0.25 0.2 0.1 S S S S m m o rf e 0.15 o rf e g g n n a a h h C C e e g g a a tn tn e e c c re 0.1 re P P 0.05 0.05 0 0 0 2 4 6 8 10 0 2 4 6 8 10 Quarter Quarter Forward Guidance HANK Transfer News HANK Forward Guidance RANK 24

Figure2: RealRateForwardGuidanceforT=20: RealBonds Output Inflation 0.6 0.25 0.5 0.2 0.4 S S S S m m o rf e 0.15 o rf e g g n a n a0.3 h h C C e e g g a a tn tn e e c c re 0.1 re P P 0.2 0.05 0.1 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Quarter Quarter Forward Guidance HANK Transfer News HANK Forward Guidance RANK 25

Figure3: RealRateForwardGuidancefordifferenthorizons: RealBonds Output Inflation 0.35 1.2 0.3 1 0.25 0.8 0 0 e e m 0.2 m it ta it ta S S S S m m o o rf e 0.15 rf e 0.6 g g n n a a h h C C e e g a 0.1 g a tn tn e c e c0.4 re re P P 0.05 0.2 0 RANK HANK, Real Bond HANK, Real Bond, Acyclical Liq. HANK MNS -0.05 0 0 10 20 30 40 0 10 20 30 40 Horizon Forward Guidance Horizon Forward Guidance 26

Figure4: RealRateFGfordifferenthorizons: RealBonds,SaversvsBorrowers Forward Guidance Shock Transfer News Shock 0.29 0.04 Output Cons. Const. Borr. Cons. Unconst. Borrowers 0.035 Cons. Savers (HANK) 0.28 0.03 0.27 0.025 00.26 0 e e m m it ta it ta 0.02 S S S S m0.25 m o o rf e rf e 0.015 g g n n a a h h C C .tc 0.24 .tc P P 0.01 0.23 0.005 0.22 0 0.21 -0.005 0 10 20 30 40 0 10 20 30 40 Horizon Forward Guidance Horizon Forward Guidance 27

Figure5: RealRateForwardGuidancefordifferenthorizons: RealvsNominalBonds Output Inflation 0.4 1.2 RANK HANK, Nominal Bond HANK, Real Bond 0.35 HANK MNS 1 0.3 0.25 0.8 0 0 e e m m it ta it ta S S S 0.2 S m m o o rf e rf e 0.6 g g n n a 0.15 a h h C C e e g g a a tn tn e c 0.1 e c0.4 re re P P 0.05 0.2 0 -0.05 0 0 10 20 30 40 0 10 20 30 40 Horizon Forward Guidance Horizon Forward Guidance 28

Figure6: RealRateFGfordifferenthorizons: NominalBonds,SaversvsBorrowers Consumption of constrained agents Consumption of unconstrained agents 2.2 0.31 Constrained Borrowers Unconstrained Borrowers Savers (HANK Nominal Bonds) 2 0.3 1.8 0.29 1.6 0.28 0 0 e1.4 e0.27 m m it ta it ta S S S S m1.2 m0.26 o o rf e rf e g g n n a a h 1 h0.25 C C .tc .tc P P 0.8 0.24 0.6 0.23 0.4 0.22 0.2 0.21 0 10 20 30 40 0 10 20 30 40 Horizon Forward Guidance Horizon Forward Guidance 29

Figure7: NominalRateForwardGuidanceforT=12: RealBonds Nominal Rate Output 1.006 2.5 1.005 2 e 1.004 g 1.5 n a h le v1.003 C e 1 e g L a tn e 1.002 c 0.5 re P 1.001 0 1 -0.5 0 10 20 30 40 0 10 20 30 40 Quarter Quarter Inflation Real Rate 1.4 1.006 1.2 1.004 1 e g 1.002 n 0.8 a h C e 0.6 le v 1 g e a L tn 0.4 e c 0.998 re P 0.2 0.996 0 -0.2 0.994 0 10 20 30 40 0 10 20 30 40 Quarter Quarter HANK RANK 30

Figure8: NominalRateForwardGuidancefordifferenthorizons: RealBonds Forward Guidance shock: output Forward Guidance shock: consumption of different groups 8 8 HANK Aggregate C Const. Borr. (HANK) RANK Aggregate C Unconst. Borrowers (HANK) Aggregate C Savers (HANK) 7 7 6 6 05 05 e e m m it ta it ta S S S S m4 m4 o o rf e rf e g g n n a a h h C C .tc 3 .tc 3 P P 2 2 1 1 0 0 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 Horizon Forward Guidance Horizon Forward Guidance 31

Figure9: NominalRateForwardGuidancefordifferenthorizons: RealvsNominalBonds Output Inflation 8 4.5 RANK HANK, Nominal Bond HANK, Real Bond 4 7 3.5 6 3 0 0 e e m5 m it ta it ta S S S S2.5 m m o o rf e 4 rf e g g n n a a 2 h h C C e e g a3 g a tn tn e c e c1.5 re re P P 2 1 1 0.5 0 0 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 Horizon Forward Guidance Horizon Forward Guidance 32

Figure10: ForwardGuidanceinaLiquidityTrap Nominal Rate Output Inflation 60 1 0.5 50 0 0 S S 40 m-1 -0.5 o s tn io P30 rf e g n a-2 tn e c -1 s is h C re P a B e g 20 a-3 -1.5 tn e c re 10 P-4 -2 0 -5 -2.5 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 Quarter Quarter Real Rate Real Wages 1.8 0 HANK Naive, Real Bonds 1.6 HANK Extended, Real Bonds HANK Naive, Nominal Bonds 1.4 S S -5 HANK Extended, Nominal Bonds 1.2 m RANK Naive tn 1 o rf e g RANK Extended e n c a-10 re P 0.8 h C e g 0.6 a tn 0.4 e c re -15 P 0.2 0 -20 0 5 10 15 20 0 5 10 15 20 Quarter Quarter 33

Figure11: ForwardGuidanceinaLiquidityTrap: SaversvsBorrowers Aggregate C Aggregate C Constrained Borrowers 1 2 0 0 S S S S m m o-1 o rf e rf e -2 g g n a-2 n a h h C C e e -4 g a-3 g a tn tn e e c re-4 c re -6 P P -5 -8 0 5 10 15 20 0 5 10 15 20 Quarter Quarter Aggregate C Unconstrained Borrowers Aggregate C Savers 3 0.5 2 0 S S-0.5 S 1 S m m o o -1 rf e g 0 rf e g-1.5 n a-1 n a h C h C -2 e-2 e g a g a-2.5 tn tn e-3 e c c -3 re re P-4 P -3.5 -5 -4 0 5 10 15 20 0 5 10 15 20 Quarter Quarter HANK Naive (Real Bonds) HANK Extended(Real Bonds) HANK Naive (Nominal Bonds) HANK Extended(Nominal Bonds) 34

Figure12: ForwardGuidanceinaLiquidityTrapwithStickyWages Nominal Rate Output Inflation 60 1 0.1 50 0 0 40 S S m -0.1 s tn io P 30 o rf e g n a -1 tn e c- - 0 0 . . 3 2 s is a B 20 h C e g -2 re P -0.4 a 10 tn e -0.5 c re-3 0 P -0.6 -10 -4 -0.7 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 Quarter Quarter Real Rate Real Wages 0.7 0.5 HANK Naive, Real Bonds 0.6 0 HANK Extended, Real Bonds HANK Naive, Nominal Bonds 0.5 S S m -0.5 H R A A N N K K E N x a t i e ve nded, Nominal Bonds tn 0.4 o rf e g -1 RANK Extended e c 0.3 n a re P h C-1.5 0.2 e g a tn -2 0.1 e c re 0 P-2.5 -0.1 -3 0 5 10 15 20 0 5 10 15 20 Quarter Quarter 35