The Macroeconomic Effects of Excess Savings

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) The Macroeconomic Effects of Excess Savings Bence Bardoczy, Jae Sim, and Andreas Tischbirek 2024-062 Please cite this paper as: Bardoczy, Bence, Jae Sim, and Andreas Tischbirek (2024). “The Macroeconomic Effects of Excess Savings,” Finance and Economics Discussion Series 2024-062. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2024.062. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) 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 Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

The Macroeconomic Effects of Excess Savings BenceBardo´czy JaeSim AndreasTischbirek* July2024 Abstract Westudytheconsequencesofshockstothehouseholdwealthdistributionindynamic general equilibrium by characterizing the rate at which excess wealth is depleted. Analytical results link the aggregate decumulation rate to the distribution of the additional balances, microintertemporalmarginalpropensitiestoconsume, andgeneralequilibrium feedback. A quantitative heterogeneous agent New Keynesian model matches the depletionpathoftheexcesssavingsbuiltupduringtheCOVID-19pandemicacrosstheincome distribution. The model predicts a substantial but steadily waning boost to consumption andexplainsupto40percentofthesurgeininflationobservedin2020and2021. Keywords: Excesssavings,heterogeneousagentNewKeynesian(HANK)models,incompletemarkets,householdportfolios,inflationdynamics,COVID-19pandemic JELClassification: E21,E31,E32,E52 *BOARDOFGOVERNORSOFTHEFEDERALRESERVESYSTEM,20thStreetandConstitutionAvenueN.W.,Washington,DC20551,USA WethankStephanieAaronson,AdrienAuclert,HessChung,SebastianGraves,MatteoIacoviello,AntoineLepetit, Gaston Navarro, Matthias Paustian, Rodolfo Rigato (discussant), and participants at the Annual Meeting of the AEAinSanAntonio,theNorthAmericanSummerMeetingoftheEconometricSocietyinNashville,andvarious seminarsforvaluablecommentsandsuggestions. WealsothankAdityaAladangady,DavidCho,LauraFeiveson, andEugenioPintoforsharingtheirupdatedestimatesofexcesssavingswithus. Theanalysisandconclusionsset forthinthispaperarethoseoftheauthorsanddonotindicateconcurrencebyothermembersoftheresearchstaff ortheBoardofGovernors. Bardo´czy:bence.a.bardoczy@frb.gov Sim:jae.w.sim@frb.gov Tischbirek:andreas.j.tischbirek@frb.gov

1 Introduction Householdwealthfluctuatessubstantiallyoverthebusinesscycle,withanannualgrowthrate that is more volatile than that of nominal GDP in the U.S. over the post-war period. While fluctuations in wealth are traditionally viewed as a side effect of business cycle fluctuations, they may, in fact, contribute to generate them. The propensity to build up excess savings— wealth in excess of its level in normal times—to smooth consumption differs substantially across households. The aggregate depletion path similarly masks a vast amount of heterogeneity. Our interest lies in characterizing and quantifying such a depletion of excess savings and its macroeconomic implications. The pandemic-era excess savings provide a good case studybecauseestimatesofthedistributionofexcesssavingsarereadilyavailable.1 During the COVID-19 pandemic, U.S. households drastically reduced spending and receivedfiscaltransfersofunprecedentedsize. Despiteearningslosses,theyaccumulatedexcess savingsestimatedtoamounttoabout10percentofpre-crisisGDPattheirpeak. Thesizeand liquidity of the excess balances raised concerns that inflation could surge if they were spent quickly, shedding light on the inconclusiveness of model predictions about their decumulation. Thelargecontributionoffiscaltransferstothebuildupofexcesssavingsmayprompttwo contrastingassessmentsofthedecumulationrate. First,repeatedlyapplyingasizablemarginal propensitytoconsume(MPC)inthespiritofclassicalKeynesianeconomicsyieldstheprediction that excess savings are spent down rapidly. Second, consulting standard macroeconomic models with a representative agent leads to the conclusion that excess savings may never be depleted, as excess savings are the counterpart of the excess debt used to finance fiscal transfers. In line with Ricardian equivalence, the household refrains from consuming out of excess savings,maintainingthemuntilthegovernmentraisestaxestorepayitsdebt. We analyze the mechanisms underlying the depletion of excess savings by relying on a heterogeneous agent New Keynesian (HANK) model that encompasses both Keynesian and Ricardian household behavior. A key theme of our analysis is that the distribution of excess savingshasastrongimpactontheiraggregatedepletionpathbecausehouseholdswithlessliquid wealth are more Keynesian and households with more liquid wealth are more Ricardian. We clarify that, under general conditions, the partial equilibrium impulse response to a shock to household wealth is fully determined by the households’ intertemporal marginal propensities to consume (iMPCs) and the distribution of the excess wealth.2 The iMPCs also affect the amplification of spending out of excess savings but cease being sufficient to characterize the response in general equilibrium. Thus, we construct a quantitative medium-scale HANK modelthatwecalibrateusingtheseinsightsandapplytoevaluatetheeffectsofpandemic-era 1Aladangadyetal.(2022)estimateexcesssavingsbyincomequartileintheU.S.Theestimateisconstructedby cumulatingtheflowdeviationsofhouseholdsavingintheNationalIncomeandProductAccountsfromapre-crisis trend.Itreflectschangesinincomeandspendingandexcludesvaluationeffects. 2Theterm“iMPC”wasintroducedbyAuclertetal.(2018)todescribethesequence-spaceJacobianofanaggregateconsumptionfunctionwithrespecttoincome. 1

excesssavings.3 Webeginbyanalyzingexcesssavingsdepletionfollowinganarbitraryshocktothewealth distribution in a small-scale HANK model.4 Micro iMPCs reflect a household’s expected consumptiongainagivennumberofperiodsafterreceivingamarginalunitofcashonhand. We demonstrate analytically that in partial equilibrium—that is, conditional on a path of the real interestrate—theresponsesofaggregateconsumptionandsavingarefullydeterminedbythe jointdistributionofinitialexcesssavingsandmicroiMPCs. Asthisresultisindependentfrom the sources of the initial excess wealth, empirical work concerned with measuring iMPCs out of unexpected income is informative for the depletion of excess savings. In general equilibrium, the iMPCs also affect the multiplier on spending out of excess savings but they are not sufficienttocharacterizeaggregateconsumption. Forexample,spendingisinflationary,which results in feedback through adjustments of the real interest rate. The multiplier therefore depends not only on the iMPCs but also on the slope of the Phillips curve, the sensitivity of the real interest rate to inflation, and the intertemporal consumption response to changes in the interestratepath. Our quantitative analysis relies on a medium-scale HANK model of the U.S. economy. It builds on the analytical results regarding the importance of iMPCs for the depletion of excess savingsintwoways. First,wedisciplinethepartialequilibriumconsumption-savingbehavior in the model through a calibration procedure that directly ties the iMPCs to empirical estimates. Second,themodelincludesarichsetofcomponentsthatgivesrisetorealisticgeneralequilibriumforcesbeyondthoseimpliedbytheiMPCsalone. Thesecomponentsincludeinvoluntary unemployment resulting from Diamond–Mortensen–Pissarides searching and matchingfrictions,unionNashbargainingwithwageadjustmentfrictions,model-consistentpricing ofclaimstoallprofitsgeneratedintheeconomyandlong-termgovernmentdebtallowingfor asset revaluation effects, monetary policy implemented through a Taylor-type rule, and the dominanceofdebtfinancingwithslowlyadjustingdistortionarytaxation. With the calibrated quantitative model in hand, we estimate the isolated consequences of excesssavingsinthewakeoftheCOVID-19pandemic. Thebaselinesimulationincludesanaccumulationperiodfollowedbyadecumulationperiod. Theformerrangesfromthebeginning ofthepandemicuntilthethirdquarterof2021,whenthepeakoftheaggregateexcesssavings stock is estimated to have occurred. During the accumulation period, the model households are confronted with shocks that allow the model to replicate realized aggregate consumption and estimates of the excess savings accumulated by each quartile of the income distribution. After the peak in excess savings is reached, we restrict all shocks to zero and study the model predictionsaboutthedecumulationperiod. 3The model described in this paper lays the foundations for the Federal Reserve Board’s HANK framework (“FR-HANK”)usedinquantitativepolicyanalysis. 4A number of contributions extend the standard Bewley–Huggett–Aiyagari incomplete markets model with nominalprice-settingfrictions. EarlyexamplesincludeOhandReis(2012),McKayetal.(2016),McKayandReis (2016),GuerrieriandLorenzoni(2017),Kaplanetal.(2018),andGornemannetal.(2021),amongothers. 2

Key findings from the quantitative exercise include the following. First, the model closely matchesthepathofexcesssavings,onaggregateandforeachincomequartile,overthepartof thedecumulationperiodforwhichempiricalestimatesareavailable. Second,thebottomquartile exhausts its excess savings first and the top quartile last. However, even the top households deplete their excess savings within about three years, implying that they are not fully Ricardian. Third,thecorrespondingincreaseindemandexplainsabout40percentofthesurge in inflation that occurred between the first half of 2020 and the second half of 2021, showing that post-COVID inflation was not exclusively a result of supply constraints.5 Fourth, the excesssavingspathoverthedecumulationperiodiswellpredictedbytheiMPCsandtheinitial allocation of excess savings. General equilibrium feedback mildly accelerates excess saving depletion. Fifth,differencesinthefiscalsupportforhouseholdsmayexplainthesubstantially smallercontractionineconomicactivityintheU.S.thanintheeuroarea. Literature. Our analysis is related to work on the role of household wealth in business cycle fluctuationsandtherecentliteratureonquantitativeHANKmodels. The large drop in household net worth seen in the U.S. during the Global Financial Crisis of2007to2009sparkedaseriesofcontributionstotheliteratureontheimplicationsofchanges in household wealth. Mian et al. (2013) use ZIP code–level data to estimate the elasticity of consumption to housing wealth for the crisis period. They find a sizable average MPC out of housing net worth that declines with household income and leverage. In addition, Mian and Sufi (2014) show that employment contracted more strongly in counties with a larger decline in housing wealth, which is indicative of general equilibrium effects set off by the spending response. FurtherresultsonthewealtheffectsofhousingarecontainedinKaplanetal.(2020a) and Guren et al. (2021). Evidence that a deterioration of not only housing but also financial wealth causes spending to adjust is presented by Christelis et al. (2015), and Heathcote and Perri(2018)linkalowvaluationofhouseholdassetstovolatilityfromequilibriummultiplicity. In contrast to these papers and in line with the excess savings built up during the pandemic, we study spending out of highly liquid assets rather than revaluations of illiquid housing or financial wealth. In addition, our analysis is based on a HANK model, which allows us to evaluatetheinfluenceofdistributionalandgeneralequilibriumeffectsalongthepathofwealth depletion. The focus on excess savings is shared by Auclert et al. (2023b), who use a stylized model to argue that excess savings have prolonged effects on aggregate demand. A large part of a dollar initially held by a poorer household with a higher MPC becomes income for wealthier households with lower MPCs, who spend it again—a “trickling up” that is repeated until the dollarlandsinthehandsoftheultrarich. Wecontributetothisinsightbygivingananalytical 5ThisresultisconsistentwiththefindingsofGiannoneandPrimiceri(2024). Theyargue,basedonstructural vectorautoregressionanalysis,thatstrongdemandcontributedsubstantiallytothepost-pandemicinflation. Our paperprovidesaspecificmechanismbehindstrongdemand. 3

characterization of excess savings depletion based on optimal consumption-saving behavior, constructingamedium-scalemodel,andconductingaquantitativeanalysisoftheimplications ofexcesssavingsintheaftermathofthepandemic,accountingforarichsetofgeneralequilibriumforces. OurquantitativeHANKmodeloftheU.S.economyismostsimilartothemodelsofAuclert et al. (2020) and Bayer et al. (2024). In contrast to both models, ours allows for adjustments at the extensive labor margin by including searching and matching frictions coupled with wage bargaining. Followingtheformer,wemodelafinancialintermediarythatpricesallprofitsgenerated in the economy, but we allow the intermediary to build up net worth. Distributions of retained earnings by the intermediary are comparable to distributions from the illiquid account in their framework—from which we abstract. In the latter, households can optimally adjustanilliquidaccountwithaCalvo-typeprobability,butmonopolyprofitsarecollectedby entrepreneurs,whichprecludesequityrevaluationeffectsthatareimportantforourresults. While they are not concerned with excess savings per se, Carroll et al. (2021) and Bayer et al. (2023) use quantitative models to study the effects of the CARES Act passed in March 2020, which contributed to the buildup of excess savings. We regard these analyses of fiscal measures put in place at the onset of the pandemic as complementary to ours. Finally, a large number of papers seek to answer questions that are specific to the COVID-19 pandemic—for example, by developing models with economic and epidemiological features.6 Our work is alsotangentiallyrelatedtothisstreamoftheliterature,butourinterestliesintheimplications ofexcesswealthdecumulationmorebroadly. Overview. Theremainderofthepaperisorganizedasfollows. Section2drawsonastylized HANKmodeltoprovideanalyticalresultsaboutthedynamicsofexcesssavingsinpartialand general equilibrium. Section 3 lays out the quantitative model. Section 4 explains our calibrationstrategy,whichisinformedbytheanalyticalresults. Section5studiesthemacroeconomic effects of excess savings in the aftermath of the COVID-19 pandemic through the lens of our model. Afinalsectionconcludes. 2 Inspecting Excess Savings Depletion Inthissection,weputexcesssavingsinthecontextofworkhorsemacroeconomicmodels. We make two observations that will guide our quantitative exercise. First, in partial equilibrium, thejointdistributionofiMPCsandinitialexcesssavingsissufficienttocharacterizethespeed of depletion. This result holds irrespective of the original cause of excess savings in a broad and relevant class of consumption-savings models. Second, in general equilibrium, iMPCs also affect the multiplier but are not sufficient anymore. Excess savings decumulation is an 6SeeKaplanetal.(2020b)andEichenbaumetal.(2021),amongothers. 4

aggregatedemandshockthatputsupwardpressureonprices. TheslopeofthePhillipscurve and accommodation on behalf of fiscal and monetary policy makers thus become relevant to determiningtheultimateimpactonthemacroeconomy. 2.1 PartialEquilibrium We start by considering the standard incomplete markets (SIM) model, the workhorse model ofheterogeneousagentmacroeconomics. WeusetheSIMmodeltogiveaformaldefinitionof iMPCsatthemicrolevelandarguethattheyfullycharacterizetheexcesssavingsdecumulation process, conditional on the path of real interest rates.7 This result holds in a broader class of modelsthatincludesthestandardrepresentativeagentandspender-savermodels. SIM Model. There is a unit mass of households indexed by i ∈ [0,1] who face idiosyncratic incomeriskandtradeinasingle,non-statecontingentasset. TheBellmanequationis V t (y i,t ,a i,t−1 ) = maxu(c i,t )+β E t [V t+1 (y i,t+1 ,a i,t )], (1) ci,t,ai,t s.t. c i,t +a i,t = (1+r t−1 )a i,t−1 +y i,t , (2) a ≥ a, (3) i,t where y is idiosyncratic income that follows an exogenous Markov process, a are assets i,t i,t−1 at the end of the last period, u : R → R is a standard period utility function that satisfies the Inadaconditions, β ∈ (0,1)isthediscountfactor,andr istherealreturnonassets. t Although distinguishing income and assets as separate state variables is often useful, it masksawell-knownpropertyoftheSIMmodelthatiscrucialforthinkingaboutexcesssavings. Conditionaloncash-on-handx i,t = (1+r t−1 )a i,t−1 +y i,t anditsforecasts E t [x i,t+h ]forallh ≥ 1, fluctuations in the components of cash on hand are irrelevant for households’ decisions. This implies that entering the period with excess savings a +∆ has the same implications as i,t−1 receiving income y i,t +(1+r t−1 )∆ . This is a key observation that means that the extensive empirical evidence on spending responses to lump-sum transfers is directly applicable to the depletionofexcesssavings,irrespectiveoftheinitialcauseofexcesssavings. ExcessSavingsandiMPCs. Considerhouseholdiwithinitialcash-on-handx whoreceives i,0 a one-time transfer in period 0. Our goal is to compare its consumption-saving choices for all t ≥ 0withthecounterfactualwithnotransfer. Theuseofaone-timetransfertogenerateexcess savings is without loss of generality. It is immediate from the Bellman equation (1)–(3) that thepastonlymattersthroughassets a . Therefore,anyshocksthatgeneratethesameinitial i,t−1 cash-on-hand x willhavethesameimplicationsfort ≥ 0. i,0 7Auclertetal.(2018)defineiMPCsasderivativesoftheaggregateconsumptionfunction. ThesemacroiMPCs areequaltothepopulationaverageofmicroiMPCsaswedefinetheminthissection. 5

Letusfixapathofrealinterestrates{r }∞ ofwhichagentshaveperfectforesight. Solving t t=0 the Bellman equation (1)–(3) yields policy functions c (x ) and a (x ), where the time subt i,t t i,t scripttmarksthedependenceontheinterestratepath. Definition. The micro iMPCs of a household with initial state x are defined as the change in the i,0 expected consumption path for all t ≥ 0 in response to an infinitesimal increase in cash on hand in period0. Formally, E[c (x (∆))|y ]−E[c (x )|y ] m (x ) ≡ lim t i,t i,0 t i,t i,0 , (4) t i,0 ∆→0 ∆ where x (∆)iscashonhandinperiodtfollowingatransfer∆inperiod0. i,t ThefollowingpropositionshowsthatiMPCsareeffectivelyadirectmeasureofexcesssavingsdepletionovertime,takingintoaccountthereturnonsavingsaswellastheidiosyncratic incomeshocksthatmayleadhouseholdstotapintotheirexcesssavings. Proposition. ThemicroiMPCsofhouseholdiwithinitialstate x satisfy i,0 (cid:34) (cid:89) t−1 (cid:16) (cid:17)(cid:12) (cid:35) m (x ) = E c ′(x ) (1+r ) 1−c ′(x ) (cid:12)y . (5) t i,0 t i,t s s i,s (cid:12) i,0 s=0 SeeAppendixA.1foraformalproof. The impact iMPC, m (x ) = c′(x ), is equal to the slope of the consumption function 0 i,0 0 i,0 at the initial state. This is the standard static MPC, which shows the fraction that household i spendsofanincomingtransfer. SubsequentiMPCsaretheproductoftwoterms. Thefirstterm, c′(x ),isthestaticMPCattimet. Thesecondterm, (cid:81)t−1(1+r )[1−c′(x )],isthecumulative t i,t s=0 s s i,s returnontheunspentfractionoftheinitialtransfer—i.e.,excesssavings. Thatis,excesssavings aredeterminedbythehistoriesofinterestratesandofthestaticMPCsofhouseholdi. The proposition shows that one can trace out the paths of consumption and excess savings at the household level by repeatedly applying the static micro MPC. This approach is reminiscent of traditional Keynesian economics. In contrast to traditional Keynesian analysis, however, the static MPC is not a primitive of the aggregate consumption function but an endogenousobjectthatreflectsutilitymaximizationbyindividualhouseholds. MicroMPCsmay evolveovertimebecauseofincomeshocksandplannedsaving. Forexample,householdswho arehitbyalargenegativeincomeshockmayplantorundowntheirsavingstotheborrowing limit. Conversely, households who are hit by a large positive shock may embark on a period of wealth accumulation. The result is that MPCs are heterogeneous across households at any giventime. The key takeaway from the proposition is that MPC heterogeneity carries over into heterogeneity in excess savings decumulation. To consider specific examples, a hand-to-mouth household with m = 1 does not accumulate excess savings; at the other extreme, rich house- 0 holds with small MPCs are expected to deplete excess savings slowly. In the representative 6

agent limit with y ≡ Y and a = −∞ , it can be shown that m = 1−β, a very small number i,t t 0 in typical calibrations. The SIM model is useful because it captures a range of MPCs between thesetwoextremes. Figure 1 visualizes the proposition by plotting m (x ), aggregated to cash-on-hand quart i,0 tiles,fromourquantitativemodel. PanelAshowstheiMPCs,whilepanelBshowstheimplied excesssavingspaths. Wheninterpretingthefigure,itisusefultokeepinmindthatthecumulativeiMPCis1foreverybody. Allhouseholdshaveonedollartospend—thequestionishow quickly they spend it. As expected, cash on hand is a strong predictor of the spending path. The bottom 75 percent of households exhaust most of their excess savings within three years, and virtually all in five years. In contrast, the top 25 percent of households display a much moregradualspendingprofile,retainingmorethanathirdofexcesssavingsafterfiveyears. In sum,thedistributionofexcesssavingsiscrucialforthespeedofdecumulationandtheimplied increaseinaggregatedemand. A. iMPCs B. Excess Savings 1.0 Quartile 1 0.4 Quartile 2 0.8 Quartile 3 0.3 Quartile 4 0.6 0.2 0.4 0.1 0.2 0.0 0.0 0 5 10 15 20 0 5 10 15 20 Quarters Quarters Figure1: DecumulationofExcessSavingsinPartialEquilibrium Beyond the SIM Model. The insight that iMPCs capture excess savings depletion relies on twoassumptionsthatareimplicitintheSIMmodel. Assumption 1. Households make two choices in every period: consumption c > 0 and assets a ∈ A, whereA ⊆ Risacompactsubsetoftherealnumbersthatmaydependonotherexogenousstates. Assumption2. Consumptionisnotastatevariableinindividualhouseholds’dynamicprogram. Prominent models that also satisfy these assumptions include the standard representative agent model, in which y is the same for all households and a = −∞ , and the spender-saver i,t model(CampbellandMankiw,1989),inwhichafraction µ ofhouseholdsarehand-to-mouth, with an asset choice set A = {0}, while 1−µ households behave just like in the representative agent model. When it comes to excess savings, the predictions of these two tractable models are the same. Excess savings have to be concentrated in the hands of permanent income households who, in the absence of changes in prices, use their excess savings to finance 7

asmall,permanentincreaseintheirconsumptionandneverexhausttheirexcesssavings. This predictiondifferssharplyfromthegradualdecumulationimpliedbytheSIMmodel. HowrestrictiveareAssumptions1and2? Assumption1rulesoutmodelsinwhichhouseholdsmakelaborsupplychoices(e.g.,AiyagariandMcGrattan1998)orportfoliochoices(e.g., Kaplan et al. 2018). If households can spend their excess savings on both goods and leisure, income effects on labor supply become relevant for shaping the excess savings decumulation process in addition to iMPCs. We shut down labor supply because there is a consensus that incomeeffectsonlaborsupplyaresmallformosthouseholds.8 Ifhouseholdscansaveinmultiple assets, the distribution of excess savings across asset classes and the marginal propensities to save in those asset classes become relevant. We consider portfolio choice an interesting generalization but abstract from it because during the COVID-19 pandemic U.S households accumulatedexcesssavingsalmostexclusivelyinliquidassets.9 Assumption 2 ensures that we do not have to keep track of the indirect effect of excess savings through lagged consumption. This rules out, for example, models of habit formation. Theassumptionsimplifiestheexpositionofthepropositionbutdoesnotaffectthesufficiency ofiMPCs. 2.2 GeneralEquilibrium Next, we embed the SIM model in a small-scale New Keynesian model to highlight the indirect effects of excess savings decumulation in general equilibrium. We do so by analyzing the intertemporal Keynesian cross (IKC) implied by the model, which allows for a sharp decomposition between the direct and indirect effects of excess savings depletion. The main point is that iMPCs are an important determinant of the general equilibrium channels as well, though notsufficientastheyareforthedirecteffect. Weprovideabriefsummaryofthemodelinthe maintext. ThederivationoftheIKCisrelegatedtoAppendixA.2.10 Model Summary. Households solve (1)–(3) with income being y = (1−τ)Yz , where τ i,t t t i,t t is the tax rate, Y is aggregate income, and z is idiosyncratic productivity. The household t i,t blockcanberepresentedbyanaggregateconsumptionfunction{C }∞ = C({τ,r ,Y}∞ , Γ ) t t=0 t t t t=0 0 that maps the sequences of tax rates, real interest rates, income, and the initial distribution of Γ households into a sequence of aggregate consumption. In this setting, excess savings are 0 Γ equivalent to perturbations of the initial (wealth) distribution . The counterpart of excess 0 savings held by households is government debt. We assume that the fiscal authority implements a path for the tax rate {τ}∞ such that government debt B matches aggregate savings t t=0 t 8Inaddition,Auclertetal.(2023a)pointoutthatmatchingsmallbutpositiveincomeeffectsonlaborsupplyin conjunctionwithhighMPCsisproblematicinNewKeynesianmodels.Solvingtheseproblemsrequireintroducing laborsupplyfrictions,exploringwhichisbeyondthescopeofthispaper. 9WereviewempiricalevidenceonexcesssavingsduringtheCOVID-19pandemicinSection5.1. 10Foracomprehensiveintroductiontomacroeconomicanalysisinsequencespace,seetheseminalpapersAuclert etal.(2018)andAuclertetal.(2021). 8

inperiod0buteventuallyreturnstoitssteady-statevalue. InflationfollowsastandardPhillips curve represented by {π }∞ = K({Y}∞ ). The central bank sets the nominal interest rate i t t=0 t t=0 t accordingtoaninflation-targetingrule. GiventheFisherequation r t = i t −E t [log(Π t+1 )], we can represent the Taylor rule as {r }∞ = R({π }∞ ). In equilibrium, Y = C , and the asset t t=0 t t=0 t t marketclearsbyWalras’law. IntertemporalKeynesianCross. Forthepurposesofthissection,itisconvenienttoabbrevi- Γ ate sequences as bold-faced letters. Conditional on the initial distribution , the equilibrium 0 sequenceofoutputsolves Y−C(τ,R(K(Y)),Y, Γ ) = 0, (6) 0 where we expressed the sequence of real rates entering the household block as a function of outputusingthePhillipscurveandthemonetarypolicyrule. Differentiating(6),weobtain (cid:16) (cid:17) dY = (I−C Y −C R R π K Y )−1 CΓdΓ 0 + C τ dτ . (7) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124)(cid:123)(cid:122)(cid:125) multiplier directeffectofES fiscalreactiontoES As we discussed in the previous section, the direct effect of excess savings on aggregate consumptiondependsontheirdistribution dΓ 0 . TheJacobian CΓ isanaggregatorofmicroiMPCs with an arbitrary distribution. Indirect effects come in two forms. First, the path of taxes that supporttheinitialstockofexcesssavingsandensurethatgovernmentbonds(andthusaggregate savings) eventually return to steady state impacts households via the Jacobian C . Taxes τ eventually have to rise to bring down government debt. Households understand this and may respond by lowering their consumption when, or even before, higher taxes materialize. Second, there is a multiplier that captures the fact that one household’s spending is another household’s income via C and the effect of the endogenous response of the real interest rate Y viaC . R Notably,C Y isalsoanaggregatorofmicroiMPCs. ItdiffersfromCΓinthatitweighsindividualsspecificallybytheirproductivity(reflectingy ∝ Yz )andinthatitincludesresponsesto i,t t i,t anticipatedrisesinfutureincome.11 Ifpricesarecompletelyrigid(K = 0),themultipliersim- Y plifies to (I −C )−1, the dynamic, heterogeneous agent equivalent of 1/(1−m). In this case, Y theiMPCsremainasufficientstatisticeveningeneralequilibrium. However,thisisnolonger the case when prices adjust and monetary policy is active. A rise in demand raises inflation andtriggersamonetarytightening,whichtendstolowerconsumptioninitially.12 11Infact,C istheaggregateiMPCmatrixasdefinedbyAuclertetal.(2018). Y 12Thisisassumingthatthenegativesubstitutioneffectsdominatethepositiveincomeeffectsofhigherrealinterest rates,whichisthecaseinmostcalibratedmodels. 9

3 Quantitative Model In this section, we present a medium-scale HANK model that generates realistic iMPCs and accountsforarichsetofgeneralequilibriumeffects. Theeconomyincludesacontinuumofhouseholdssubjecttouninsurableidiosyncraticrisk, a financial institution that intermediates funds between savers and the productive sector, a capital producer faced with investment adjustment frictions, a labor agency that hires workersundersearchingandmatchingfrictionsandbargainsthewagewitharepresentativelabor union, monopolistically competitive producers of intermediate goods, a representative final goods producer, and fiscal and monetary policy authorities. Time is discrete and one period correspondstoaquarter. 3.1 Households Acontinuumofexanteidenticalhouseholds,indexedbyi ∈ [0,1],aresubjecttoidiosyncratic riskthroughshockstotheirproductivityandtheiremploymentstatus,implyingthattheydiffer ex post. Idiosyncratic productivity in period t = 0,1,... is given by z . Productivity follows i,t aMarkovchainwithaconstanttransitionmatrix Πz andcross-sectionaldistributionπz. Mean productivity (cid:80) πz(z)z is normalized to one. Each household is either employed, e = 1, i,t or unemployed, e = 0. The time-varying mass of households in each employment state i,t πe(e) is determined in general equilibrium. All employed agents spend their entire available t time endowment for work, supplying labor. With the work time endowment normalized to one, their nonfinancial income is given by (1−τ)w z , where τ is the tax rate and w is the t t i,t t t real wage. Unemployed workers search for labor and receive unemployment benefits (1− τ)ωUIwz ,withgrossreplacementrateωUI ∈ [0,1]andsteady-statewagew.13 t i,t The households are limited to a noncontingent short-term asset to insure against idiosyncraticrisk,andborrowingisruledout. Theiroptimalconsumptionplansatisfies (cid:26) (cid:27) V t (e i,t ,z i,t ,a i,t−1 ) = max u(c i,t )+β E t [V t+1 (e i,t+1 ,z i,t+1 ,a i,t )] , (8) ci,t,ai,t c +a = (1+rA )a +(1−τ) (cid:2) y (e ,z )+DFI(a ) (cid:3) +T , (9) i,t i,t t−1 i,t−1 t t i,t i,t t i,t−1 i,t a ≥ 0, (10) i,t whereu(·)istheperiodutilityfunctionandy (e ,z )isidiosyncraticlaborincomegivenby t i,t i,t (cid:104) (cid:105) y (e ,z ) = z 1(e = 1)w +1(e = 0)ωUIw . (11) t i,t i,t i,t i,t t i,t Additionally,c isconsumption, βisthediscountfactor,anda areshort-termdepositsata i,t i,t−1 financialintermediarymadeinperiodt−1thatpayasafereturnrA inthefollowingperiod. t−1 13Below,allvariableswithouttimesubscriptstakeontheirrespectivesteady-statevalue. 10

The households receive additional financial income through dividends paid by the financial intermediaryDFI(·).14 T isalump-sumgovernmenttransfer(ortax)thatistakenasgivenby t i,t thehouseholds.15 Thefirst-orderoptimalityconditionis u ′(c ) ≥ β(1+rA)E (cid:2) u ′(c ) (cid:3) (12) i,t t t i,t+1 alongwithequations(9)and(10),where(12)holdswithequalityiftheborrowingconstraintis notbinding. 3.2 FinancialIntermediary The representative financial intermediary holds equity shares S and government bonds B t t financed by net worth NFI and deposits from the household sector A . Its balance sheet cont t straintis pSS +qBB = NFI +A , (13) t t t t t t where pS is the price of shares issued by the firms S and qB is the price of long-term bonds t t t issued by the government B . Government bonds take the form of perpetuities that pay off a t geometricallydecliningcoupon(δ )s withδ ∈ (0,1]ands = 0,1,2,... startinginperiodt+1, B B following Woodford (2001). The deposits that are intermediated on behalf of the households aresubjecttoaunitintermediationcostξ. Intermediarynetworthisgivenby N t FI = (D t +pS t )S t−1 +(1+δ B q t B)B t−1 −(1+r t A −1 +ξ)A t−1 −D t FI. (14) Equitysharesacquiredint−1earndividendsD inadditiontotheex-dividendprice pS,bond t t holdings in t−1 yield the coupon payment δ qB plus the principal, and a unit of deposits B t is associated with costs of the amount of 1+rA +ξ. Net worth is further reduced by the t−1 distributionspaidtothehouseholdsDFI. t Thedividendpayoutstothehouseholdsareassumedtofollowanadhocdistributionrule: DFI = DFI +ϕ(NFI −NFI). (15) t t−1 ϕparametrizestherateatwhichintermediarynetworthreturnstosteadystate.16 Thesmaller 14Weadoptthecommonassumptionthatownershipofthefinancialintermediaryisnontradablebutallowthe dividendpaymentstodifferalongtheliquidwealthdistribution. 15Transfersaretreatedasexogenousevenifweallowthegovernmenttorelatethemtovariablessuchasincome andassetholdings,aswillbecomeclearbelow. Hence,inoursimulationoftheCOVID-19period,thehouseholds do not anticipate the structure of the discretionary support measures paid by the government beyond standard unemploymentbenefits.Nonetheless,theyfullyinternalizethatthetaxrateτtmustultimatelyadjusttosatisfythe governmentbudgetconstraint. 16Stabilityoftheintermediary’sbalancesheetrequiresDFI =(rA+ξ)NFI andϕ>rA+ξ.Weensurethatthese 11

ϕ is, the stronger household income is insulated from return shocks and the less reactive consumptionistoswingsinequityprices,forexample. The financial intermediary maximizes its expected return on net worth after distributions E (1+rN ) = E (NFI /NFI)subjecttothebalancesheetconstraint(13)andthelawofmotion t t+1 t t+1 t ofnetworth(14),whichyieldsstandardno-arbitragerelationships: E t [(D t+1 +pS t+1 )/pS t ] = E t [(1+δ B q t B +1 )/q t B] = 1+r t A+ξ ≡ 1+r t . (16) Inequilibrium,theexpectedreturnisequatedacrossallfinancialassets. 3.3 CapitalProducer A representative firm maintains the economy’s capital stock K and rents it out to the goodst producing sector. Investment I is subject to convex adjustment costs Φ (·), giving rise to real t I rigidity. Thecapitalstockevolvesaccordingtothefollowinglawofmotion: (cid:20) (cid:18) (cid:19)(cid:21) I K t = (1−δ K )K t−1 + 1−Φ I t I t , (17) I t−1 whereδ K ∈ (0,1)isthedepreciationrateand Φ I (I t /I t−1 ) = (ψ I /2)(I t /I t−1 −1)2. The capital producer chooses investment to maximize the expected present value of the expecteddividendstream. Formally,itsolves (cid:26) (cid:27) 1 (cid:104) (cid:105) p t K(K t−1 ,I t−1 ) = r t KK t−1 +m It, a K x t −I t + 1+r t E t p t K +1 (K t ,I t ) (18) subjecttoequation(17).17 Optimalitynecessitates (cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21) (cid:20) (cid:18) (cid:19)(cid:18) (cid:19)2(cid:21) 1 = Q t 1−Φ I I t I − t 1 −Φ′ I I t I − t 1 I t I − t 1 + 1+ 1 r t E t Q t+1 Φ′ I I t I + t 1 I t I + t 1 , (19) whereQ satisfiestherecursion t (cid:20) (cid:21) 1 Q t = 1+r E t r t K +1 +(1−δ K )Q t+1 . (20) t NotethattheshadowvalueofcapitalQ canbeinterpretedasTobin’smarginalQ. t 3.4 LaborAgencyandUnion There is involuntary unemployment resulting from searching and matching frictions in the Diamond–Mortensen–Pissaridestradition. conditionsholdinourmodelcalibration.ThedetailsareshowninAppendixB.1. 17SeeAppendixB.2forthederivations. 12

A labor agency hires workers by posting vacancies, unemployed workers search for employment,andmatchesareformedstochastically. Theagencysellsthelaborservicessupplied by successfully matched workers to the goods-producing sector. The wage is determined by Nashbargainingbetweentheagencyandarisk-neutralunionthatrepresentsallhouseholds.18 We introduce wage rigidity, as the Diamond–Mortensen–Pissarides model generates insufficientvolatilityinvacancycreationandunemploymentwithflexiblewages(Shimer,2004;Hall, 2005;Shimer,2005). Ratherthanmodelingthedeepsourcesofwagerigidity,ourtractableapproachistoassumethatthelaboragencyincurswageadjustmentcostsinthespiritofRotemberg(1982). Thesecostsdiminishthebargainingsurplusandthusprovideanincentiveforboth sidestoreducefluctuationsinthewagebargained.19 TimingandEmploymentFlows. Ineachperiod,thelabormarketoperatesasfollows. 1. Theagencyinheritsastockofemployedworkers N t−1 fromthepreviousperiod. U t−1 = 1−N t−1 workerswerenotmatchedint−1andstillsearchforemployment. Therealizationsofallaggregateshocksbecomeknown. 2. Existing matches are destroyed with an exogenous probability s ∈ (0,1). The newly separated workers become searchers, implying that the mass of active searchers now is u t = U t−1 +sN t−1 . Theagencycreatesv t newvacanciesatcostκ v foreachvacancy. 3. NewmatchesareformedaccordingtoaCobb–Douglasmatchingtechnology,m(u ,v ) = t t Θ uαmv1−αm with Θ > 0and α ∈ (0,1), andthelaboragencypaysahiringcostκ for m t t m m h each new match. The job-finding rate f and the vacancy-filling rate q are, respectively, t t f ≡ m(u ,v )/u = Θ θ1−αm and q ≡ m(u ,v )/v = Θ θ −αm,where θ = v /u islabor t t t t m t t t t t m t t t t market tightness. Unfilled vacancies are destroyed. From the households’ perspective, thelawofmotionofthelabormarketstatuscanbesummarizedas (cid:34) (cid:35) (cid:34) (cid:35)(cid:34) (cid:35) N t = 1−s(1− f t ) f t N t−1 . (21) U t s(1− f t ) 1− f t U t−1 Fromthelaboragency’sperspective,theevolutionofemploymentis N t = (1−s)N t−1 +q t v t . (22) 4. Wagebargainingandproductiontakeplace. Theagencyreceivesafeeh andpaysawage t w perefficiencyunitoflabor. t 18Theassumptionthataunionbargainsonbehalfofthehouseholdsallowsustoabstractfromwagedispersion, whichwouldariseunderincompletemarketsifeachhouseholdbargainedindividuallywiththelaboragency. 19Differentapproacheshavebeenusedtoachievewagerigidityinthecontextofsearchingandmatching. Hall (2005)analyzessimplewagerules,GertlerandTrigari(2009)applyNashbargainingwithstaggeredmulti-period wagesettingoverafixedhorizon,andChristianoetal.(2016)consideralternatingofferbargaining. 13

ValueofaMatch. Thelaboragencyfacesconvexrealwageadjustmentcosts Φ w (w t ,w t−1 ) = ψ w /2(w t /w t−1 −1)2, which gives rise to real wage rigidity. The profit-maximization problem ofthelaboragencyis (cid:26) (cid:27) 1 J t (N t−1 ) = m Nt a ,v x t (h t −w t )N t −(κ v +κ h q t )v t −Φ w (w t ,w t−1 )N t + 1+r t E t [J t+1 (N t )] (23) subjectto(22). Anoptimumischaracterizedbythefollowing: 1−s J t = h t −w t −Φ w (w t ,w t−1 )+ 1+r E t [J t+1 ], (24) t κ J = v +κ , (25) t h q t where J isthelaboragency’sshadowvalueofamatch—thatis,itsbargainingsurplus.20 Equat tion(24)showsthatthevalueofamatchisthefeereceivedfromthegoodsproducersnetofthe wagepaidandtheadjustmentcostincurredplusthecontinuationvalueobtainedifthematch isnotdestroyed. Equation(25)isazeroprofitconditionforthelaboragencyequatingthevalue ofthemarginalworkertothesumofthecostsassociatedwithvacancypostingandhiring. Weassumethatworkersarerepresentedbyarisk-neutralunion,forwhomthevalueofthe marginalmatchis 1−s H t = w t −wUIw+ 1+r E[(1− f t+1 )H t+1 ]. (26) t Itisthewagenetoftheunemploymentbenefit—aworker’soutsideoptionincasenomatchis formed—plusthecontinuationvalue. WageBargaining. UnderNashbargaining,theequilibriumwagemaximizesthejointsurplus, η 1−η H J , where η parametrizes the union’s bargaining power. The surplus is split such that t t Ω J = (1−Ω )H with t t t t η Ω ≡ , (27) t η+(1−η)(−J /H ) w,t w,t governingthesharereceivedbytheunion. 3.5 GoodsProducers Thegoods-productionsectorincludestwotypesoffirms: arepresentativefinalgoodsproducer andacontinuumofdifferentiatedinputproducers,whicharethesourceofnominalrigidity. 20SeeAppendixB.3forthedetails. 14

FinalGoodsProducer. AperfectlycompetitivefirmproducesY unitsofahomogenousgood t usingaconstantelasticityofsubstitution(CES)technologywithelasticityofsubstitutionϵ : p (cid:32) (cid:33) ϵp (cid:90) ϵp−1 ϵp−1 Y = Y ϵp dj . (28) t j,t Cost minimization implies that the demand for intermediate producer j’s inputY with price j,t P isgivenby j,t (cid:18) P (cid:19)−ϵp j,t Y = Y, (29) j,t t P t where P = (cid:16) (cid:82) P 1−ϵpdj (cid:17)1/(1−ϵp ) isthecorrespondingaggregatepriceindex. t j,t IntermediateGoodsProducers. Monopolisticallycompetitivefirmsindexedbyj ∈ [0,1]produce differentiated inputs using a Cobb–Douglas technology, Y = ΘKα N1−α, where Θ is j,t j,t−1 j,t totalfactorproductivity. FollowingRotemberg(1982),theyfacepriceadjustmentcosts χ (cid:20) (cid:18) P (cid:19) (cid:16) (cid:17) (cid:21)2 Φ p (P j,t ,P j,t−1 , Π t−1 ) = 2 p log P j,t −log Πι t p −1 Π1−ιp (30) j,t−1 with Π t = P t /P t−1 denoting inflation. In an optimum, the parameters χ p > 0 and ι p ∈ [0,1] determine the strength of the price adjustment friction and the degree of indexation, respectively.21 ProfitmaximizationyieldsastandardPhillipscurvewithabackward-andaforwardlookingcomponent: (cid:18) (cid:19) (cid:16) (cid:17) (cid:16) (cid:17) ϵ log Π −log Πιp Π1−ιp = κ p mc −1 t t−1 p ϵ −1 t p (cid:20) (cid:21) + 1+ 1 r E t log (cid:16) Π t+1 (cid:17) −log (cid:16) Πι t pΠ1−ιp (cid:17) Y Y t+1 . (31) t t where the slope κ = (ϵ −1)/χ is the same as in the loglinearized version of (31) and real p p p marginalcostisgivenby 1 (cid:18) rK(cid:19)α(cid:18) h (cid:19)1−α mc = t t (32) t Θ α 1−α andthefactorpricessatisfyr t K = αmc t Y t /K t−1 andh t = (1−α)mc t Y t /N t .22 AggregateProfits. Inadditiontothemonopolisticintermediategoodsproducers,thecapital producerandthelaboragencygenerateprofitsthattheydistributeasdividends. Thedividends 21Ifιp =0andΠ=1,thesecondlogterminequation(30)vanishes,andtheadjustmentcostisminimizedifthe priceremainsfixed.Ifιp =1,thecostisminimizedforfullindexationtoinflationinthepreviousperiod. 22SeeAppendixB.4forallderivations. 15

paidare,respectively, D t F = Y t −r t KK t−1 −h t N t −Φ p (P t ,P t−1 , Π t−1 )−Ψ (33) D t K = r t KK t−1 −I t (34) D t L = (h t −w t )N t −(κ v +κ h q t )V˜ t −Φ w (w t ,w t−1 )N t , (35) where Ψ is a fixed cost of operation in goods production. The equity shares S traded in the t economyareclaimstothestreamofaggregatedividends D = DF+DK+DL. t t t t 3.6 Government Governmentpoliciesareimplementedbyafiscalauthorityandacentralbank. FiscalAuthority. Thegovernment’sbudgetconstraintis (cid:16) (cid:17) G +T +(1+δ qB)BS +ωUIwU = qBBS+τ w N +DFI +ωUIwU . (36) t t B t t−1 t t t t t t t t Itspendson,inorder,outputgoods,aggregatetransfers,debtrepayment,andunemployment benefits. Revenue is generated through debt issuance and proportional taxation. Fiscal policy sets a path for transfers, taxes, and spending, implying that debt supply BS is determined by t thebudgetconstraint. Thepathfortransferswillbeanimportantinputtooursimulation—we returntoitbelow. FollowingAuclertetal.(2020),thetaxrateissetaccordingto BS −BS τ = τ+ϕ qB t−1 (37) t B Y with ϕ > 0. It rises with the debt level, ensuring that government debt is stationary. Finally, B weletspendingbeconstant,G = G. t Monetary Policy. The central bank sets the nominal short-term interest rate according to an inertialTaylor-typerule, (cid:20) (cid:18)Π (cid:19)(cid:21) i t = ϕ i i t−1 +(1−ϕ i ) i+ϕ π log Π t (38) andtherealratesatisfiesthestandardFisherequationr t = i t −E t [log(Π t+1 )]. 3.7 Equilibrium Inanequilibrium,allagentsfollowtheiroptimaldecisionrules,andmarketsclear. 16

Theassetmarketclearingconditionsarethefollowing: (cid:90) (cid:88)(cid:88) A t = a t (e,z,a −1 )Γ t (e,z,a −1 )da −1 , (39) e z S = 1, (40) t B = BS, (41) t t where Γ t (e,z,a −1 )isthemassofhouseholdsinanygivenemployment,productivity,andasset state,and a t (e,z,a −1 )istheoptimalsavingpolicy. Labormarketclearingrequires (cid:88) πe(e)e = N, (42) t t e N +U = 1. (43) t t If the equations (39) to (43) are satisfied, Walras’ law ensures that the goods market clearing conditionholds,whichcanbeexpressedas Y = C +I +G +ξA +(κ +κ q )V˜ t t t t t−1 v h t t +Φ w (w t ,w t−1 )N t +Φ p (P t ,P t−1 , Π t−1 )Y t +Ψ , (44) (cid:90) (cid:88)(cid:88) where C t ≡ c(e,z,a −1 )Γ t (e,z,a −1 )da −1 is aggregate consumption. An equilibrium e z canthenbecharacterizedasfollows. Arecursiveequilibriumisasequenceofprices{Π ,pS,qB,r ,rK,Q ,h ,w },policyfunctions t t t t t t t t {c t (e,z,a −1 ),a t (e,z,a −1 )},thedistribution{Γ t (e,z,a −1 )},aggregates{C t ,A t ,Y t ,N t ,B t ,B t S,K t ,I t , DFI,U ,V,U˜ ,V˜ ,v ,θ ,J ,H , Ω , f ,q ,τ,mc , Φ , Φ , Φ ,DF,DK,DL},andpolicy{i ,G ,T} t t t t t t t t t t t t t t p,t I,t w,t t t t t t t suchthat(1)theevolutionofthewealthdistributionisconsistentwithlabormarketoutcomes, the productivity process, and the households’ policy functions; (2) the households, the financial intermediary, the capital producer, the labor agency, the union, the final goods producer, and the intermediate good producers attain their respective constrained optimum; (3) monetaryandfiscalpolicysatisfytheirrespectiverulesaswellasthegovernmentbudgetconstraint; and(4)allmarketsclear—thatis,equations(39)to(43)hold. 4 Calibration InlinewiththeanalyticalresultspresentedinSection2,ourcalibrationstrategyisparticularly attentivetotwosetsofmoments: iMPCsandfiscalmultipliers. Thecalibrationprocedurecan be summarized as follows. First, we set a large set of parameters based on procedures that arestandardintheliterature. Second, weuseasubsetofparameterstotargetiMPCestimates from micro data. Third, we validate that the iMPCs, in combination with our choice of one 17

additional parameter, give rise to realistic general equilibrium forces by comparing the government spending multipliers of the model with reduced-form estimates. We now fill in the details,startingwiththelasttwopoints. 4.1 DiscipliningConsumption-SavingBehaviorinPartialEquilibrium The mapping between the partial equilibrium iMPCs and excess savings decumulation described in Section 2.1 implies that we can discipline the latter in our model by targeting the iMPCs,statisticsthathavebeenestimatedempirically. iMPCsinData. WeopttofollowAuclertetal.(2020)inbasingourcalibrationontheiMPCs estimatedbyFagerengetal.(2021),whicharecalculatedusingNorwegianlotterywinnings.23 LotteryprizesprovideanidealwayofmeasuringiMPCsinpartialequilibriumbecausetheyare drawn at random and each recipient household is infinitesimal relative to macro aggregates. The empirical evidence gives rise to three stylized facts. First, the average annual MPC on impact is sizable—about 0.5 according to Fagereng et al. (2021). Second, the average iMPCs aredecliningwiththeresponsehorizon. However,theyarestillaround0.1inthesecondyear aftertheimpactyear. Third,theimpactMPCsarestronglynegativelycorrelatedwiththeliquid assetpositionofhouseholds. WhileFagerengetal.(2021)estimateadeclineintheannualMPC from0.62to0.46fromthebottomtothetopquartileofthedistribution,Holmetal.(2021)find asomewhatstrongerfalloffatthetopofthedistributionwiththeirmethodofextractingMPCs fromadecompositionoftheconsumptionresponsetomonetarypolicyshocksinNorway.24 Targeting Procedure. Our incomplete markets set-up allows us to generate a high average impactMPCtogetherwithagraduallydecliningaverageiMPCpath(Auclertetal.,2020). We targetthesetwodatamomentsusingthediscountfactorandasimpletransferfunction,leaving thedistributionofiMPCsuntargeted. Thestructuralformofthetransferfunctionassumedis T = τLS+τaa +ε . (45) i,t i,t−1 i,t Withβ = 0.95,τLS = 0.05,andτa = −0.07,weobtainanannualimpactMPCof0.56andaclose modelfitoftheentireiMPCpath,asillustratedintheleftpanelofFigure2,whichcomparesthe annual data with the corresponding model values. Intuitively, a lower discount factor yields lessprecautionarysavingsinsteadystateand,hence,ahigherimpactMPCandsteeperiMPC path. Similarly,amoreredistributivetransfersystemallowsforhigherconsumptionwhenthe 23To our knowledge, iMPC estimates from U.S. households are unavailable to this date. The benefits of well identified moments that are estimated from high-quality administrative tax data from Norway come at the cost of some uncertainty about cross-country differences in the consumption-saving behavior of households. While quantitativedifferencesarepossible,thequalitativefeatureshighlightedbelowareunlikelytodiffer. 24Holmetal.(2021)estimateasimilarimpactMPCatthebottomoftheliquidassetdistributionbutthepoint estimateforthetop20percentofhouseholdsisabout0.35(SeeFigureE.4intheirSupplementalMaterial). 18

0.5 0.4 0.3 0.2 0.1 0.0 0 1 2 3 4 5 Years CPMi launnA A. Average iMPC Data Model 0 5 10 15 20 Quarters CPMi ylretrauQ B. iMPCs by Cash on Hand Quartile 1 Quartile 2 Quartile 3 Quartile 4 Figure2: iMPCsintheDataandintheModel Notes:TheleftpanelshowstheannualiMPCestimatesbyFagerengetal.(2021)(dots)andthecorrespondingmodel values(line).TherightpanelshowsthequarterlyiMPCpathaveragedacrossthehouseholdsineachquartileofthe cash-on-handdistributioninthemodel. borrowing constraint is binding and therefore deters private insurance, raising the iMPCs in the first quarters. While there is no wealth tax in the U.S., the negative value for τa that we obtain can be interpreted as capturing redistributive features of the tax system that we do not modelexplicitly. Theidiosyncratictermε iszeroexceptforwherestatedotherwisebelow. i,t TherightpanelofFigure2showstheiMPCsbyquartileofthecash-on-handdistribution.25 In line with the third stylized fact described above, the model successfully generates a sizable falloff in the annual impact MPCs from the first to the fourth quartile (0.78 to 0.18), which is, however,somewhatlargerthanwhattheempiricalpointestimatessuggest. 4.2 ValidatingGeneralEquilibriumEffects Giventhemodel’ssizableMPCs,itiscapableofgeneratingsubstantialmultipliers,asshownin Section2.2. Wevalidatethegeneralequilibriumfeedbackinthemodelbycomparingthegovernment spending multipliers implied by it with the estimates by Ramey and Zubairy (2018). Specifically, we consider the cumulative government spending multiplier, the ratio of present value sums (cid:80)h (1+r)−tdY/ (cid:80)h (1+r)−tdG for h = 1,2,...,20, where {dY} and {dG } t=0 t t=0 t t t aretheimpulseresponsesofoutputandgovernmentspending,respectively. Theirresults,depicted in Figure 3, are obtained using local projections with long time series from the U.S., in which government spending is instrumented with exogenously identified military news shocks. Inafirststep,weestimatedtwomodelparameters—theparameterofthefiscalruleϕ and B the persistence parameter ρ of government spending specified as an AR(1) process—using G 25Cashonhandisthesumofallearnings,transfers,andfinancialincome(includingtheprincipal)netoftaxes. 19

Cumulative Government Spending Muliplier 2.00 Data: Military Spending News Model: Match Military Spending 1.75 Model: More Deficit Financing 1.50 1.25 1.00 0.75 0.50 0 4 8 12 16 20 Quarters Figure3: CumulativeGovernmentSpendingMultipliersintheDataandintheModel Notes: ThefigureshowsthecumulativegovernmentspendingmultiplierestimatesbyRameyandZubairy(2018) (“Data: MilitarySpendingNews”)aswellasthecorrespondingsequencesfromthemodelforϕ =1.93(“Model: B Match Military Spending”) and ϕ = 0.025 (“Model: More Deficit Financing”). The shaded region is the 95% B confidenceinterval. the estimates by Ramey and Zubairy (2018) as targets.26 Both parameters influence the government spending multipliers in the model: The former governs the sensitivity of the tax rate to additional government debt, and the latter governs the allocation of government spending overtime. ThebluedashedlineinFigure3showsthatthemodelcloselymatchestheempirical targetswiththeestimatedparametervaluesρ = 0.62andϕ = 1.93. Therelativelyhighvalue G B of ϕ implies thata significant part ofspending is tax financed. While thismay be a plausible B description of the fiscal response to military news shocks, the large fiscal programs that contributedtothebuildupofexcesssavingsduringtheCOVID-19periodwerealmostexclusively debt financed. Therefore, in a second step, we lowered ϕ to the smaller value of 0.025 from B Auclert et al. (2020), which implies a high degree of debt financing while ensuring that debt remainsstationary. Thecorrespondingdash-dottedorangelineshowsthattheresultingmultipliersaresomewhathigherinthefirstquarters,asexpected. However,theyonlylieoutsidethe 95-percent confidence region of the estimates from military news in a few intermediate quarters. Given the strong dominance of debt financing during the pandemic and the acceptable matchwiththeempiricalestimates,weoptforthesmallerofthetwovaluesinourapplication. 4.3 RemainingParameters The above calibration steps are conditional on a large set of parameters that we select based on data from the U.S. as well as findings from the literature. For example, the idiosyncratic income process with n = 33 states is taken from Kaplan et al. (2018); the parametrization of z 26Theestimationwasdoneviathesimulatedmethodofmomentsandincludedthestandarderrorsoftheempiricaltargetsintheweightingmatrix. 20

thesearchingandmatchingfrictionscloselyfollowsChristianoetal.(2016);andourtargetfor theratioofhouseholdwealthtoannualoutput(A+NFI)/4Yis3.82,asinAuclertetal.(2020), giving rise to a deposit ratio that is in line with the 2023 wave of the Survey of Consumer Finances. Othervalues, suchasanelasticityofintertemporalsubstitutionof0.5oranaverage U.S.governmentdebtmaturityoffiveyears,arestandard. SectionCintheAppendixcontains adetaileddiscussionoftheremainingparameterchoices,whicharesummarizedinTableC.1. 5 Excess Savings and the COVID-19 Pandemic DuringtheCOVID-19pandemic,U.S.householdsaccumulatedalargeamountofliquidassets. Weuseourquantitativemodeltostudythemacroeconomiceffectsofthechangesinhousehold wealththattookplaceinthishistoricalepisode. Wedeliberatelyabstractfromotherpandemicspecific influences such as inflationary pressure from international supply constraints, as our interestliesintheeffectsthataredirectlyattributabletothedynamicsofhouseholdsavings. 5.1 HistoricalBackground Twofactorsplayeddominantrolesfortheaccumulationoffundsinhouseholdbalancesheets duringtheCOVID-19pandemic: largefiscalsupportpackagesandsocial-distancingmeasures. ThepanelsAandBofFigure4plotaggregatetransferreceiptsandconsumptionexpenditures between2019and2022,respectively. Inthesecondquarterof2020andthefirstquarterof2021, transfers were nearly twice as high as before the pandemic. The two spikes reflect the “Economic Impact Payments” under the CARES Act and the Tax Relief Act, in combination with the American Rescue Plan.27 Moreover, social-distancing measures contributed to a collapse inconsumptioninmid-2020. Whilethesetwochannelsincreasedhouseholdsavings,declines inlaborearningsduetobusinessclosuresandlayoffsreducedthem. Aladangadyetal.(2022) estimatethatchangesinfiscalsupport,outlays,andincomeresultedinacombinedbuildupof about 2.2 trillion dollars of excess savings at their peak in the third quarter of 2021—about 10 percentofthepre-crisisGDPin2019.28 The bottom panel of Figure 4 shows a decomposition of the peak holdings into the three channels outlined above for each quartile of the income distribution based on estimates by Aladangady et al. (2022).29 The figure reveals several distributional patterns. First, the fiscal 27UndertheCoronavirusAid,Relief,andEconomicSecurity(CARES)Act,thegovernmentpaideligibleadultsup to1,200dollarsandchildrenupto500dollarsasalumpsum.Thesepaymentswerephasedoutforindividualswith agrossannualincomeabove75,000dollars. TheTaxReliefActprovidedfundsofupto600dollarsforbothadults andchildrensubjecttothesameincomethresholds. Inathirdroundofstimuluspayments,theAmericanRescue PlanActauthorizeduptoanadditional1,400dollarssubjecttoincomethresholdsforeachadultanddependent, amongotheremergencyreliefmeasures. 28Outlaysarethesumofpersonalconsumptionexpendituresandinterestpayments. Allvaluesarein2019U.S. dollars. 29Each bar represents the deviation of a variable from its pre-crisis trend cumulated between the onset of the pandemicandthethirdquarterof2021. 21

A. Transfer Receipts B. Consumption 6 16 ) ) s5.5 s15.5 n n o o illir 5 illir 15 T T ( ( D4.5 D14.5 S S U 4 U 14 9 9 1 1 03.5 013.5 2 2 3 13 2019 2020 2021 2022 2019 2020 2021 2022 Quarter Quarter C. Contributions to Excess Savings by Income (2021Q3) 2 Income ) s 1.5 Fiscal Support n o Reduction in Outlays illir 1 Total T ( D 0.5 S U 9 0 1 0 2-0.5 -1 1 2 3 4 Quartile of Income Distribution Figure4: AggregateTransfersandConsumption Notes: PanelsAandBshowrealpersonalcurrenttransferreceiptsandrealpersonalconsumptionexpenditures, respectively(seasonally-adjustedannualrates). PanelCshowscontributionstothestockofexcesssavingsatthe peakin2021:Q3byquartileoftheincomedistribution. support measures increased savings across the entire distribution, but their contribution declined with income. Second, changes in outlays reduced savings at the bottom and increased savings at the top of the distribution. The negative effect at the bottom is consistent with significant spending out of the fiscal stimulus payments received. Third, income losses reduced savingsinallquartiles. Thetotalincomelosseswerelargestforthehighest-incomehouseholds. Fourth,excesssavingswereconcentratedatthetopoftheincomedistribution. Morethanhalf ofthestockatthepeakwasheldbythetop25percent. Finally,adeterminantoftherateatwhichexcesssavingsaredepletedistheirallocationto differentassetclasses. EvidencefromtheDistributionalFinancialAccountsoftheU.S.suggests that almost all excess savings were held as liquid assets (Batty et al., 2023). The vast majority was held in the form of bank deposits, with some rebalancing into money market funds and equities. Ofcourse,therewerealsosizablerevaluationsofassetpositionsduringthepandemic. Whilerevaluationeffectsarenotincludedinourdefinitionofexcesssavings,theyarecaptured byourmodelandcontributetotheresults. 22

5.2 SimulationSet-Up Ourexperimentincludesanaccumulationandadecumulationperiodofexcesssavings. Prior tothepandemic,theeconomyisassumedtobeinsteadystate. Startingwiththefirstpandemic period, the first quarter of 2020, we feed exogenous shocks into the model to match data on aggregateconsumptionandthebuildupofexcesssavingsuntiltheirpeakinthethirdquarter of 2021. From then onward, we restrict the shocks to zero and study the model predictions abouttheeffectsofexcesssavingsonmacroeconomicoutcomes. Shocks. We use series of targeted transfer shocks to replicate the contribution of the fiscal support programs. In addition, shocks to the discount factor approximate the contribution of the voluntary consumption restraints that resulted from social-distancing measures. Figure 4 showsthethreemaincomponentsofexcesssavingsaccumulation: fiscalsupport,reductionin outlays,andincome. Ofthesethreecomponents,weonlytargetthefirsttwoandletincomeadjust endogenously in general equilibrium. Since less affluent households received more fiscal support, we allow the transfer shocks to be specific to the quartiles of the cash-on-hand distribution at the onset of the pandemic.30 In sum, in the buildup stage, there are four series of (cid:8) (cid:9) transfer shocks ε ,ε ,...,ε , with q ∈ {1,2,3,4} together with a sequence q,2020Q1 q,2020Q2 q,2021Q3 ofcommondiscountfactors{β ,β ,...,β }. 2020Q1 2020Q2 2021Q3 Targets. The data targets are the stock of excess savings held in all four quartiles of the income distribution and aggregate consumption over the seven quarters of the buildup period. We choose the shocks such that they minimize the squared distance between the data targets andtheirmodelcounterparts. Sincethefourtransfershockseriesandthediscountfactorshock series are effective determinants of household savings by income quartile and aggregate consumption,respectively,weobtainanearexactmatchofthedatatargetsinthemodel. Validation. Our simulation set-up captures the main contributors to the buildup of excess savings in a parsimonious way. By targeting excess savings directly, we leave transfer payments untargeted. To validate our set-up, we compare the aggregate transfers implied by our estimation with realized data in Figure 5. The fact that the transfers required by the model in thebuildupphaseareoftherightmagnitudeisasuccessofthemodel.31 In contrast to Section 2, we now model the buildup of excess savings. This is not strictly necessary. One could also start the simulation in the third quarter of 2021 and confront the model with a set of one-off shocks to the initial conditions that allow the asset distribution to 30Inourmodelwithonlyliquidassets,thereisanequivalencebetweenhouseholdincomeandcashonhand,as allcomponentsofcashonhandareincomeineachquarter.Weexploitthisequivalencebyproxyingforhousehold incomeusingcashonhand,therelevantstatevariableinourmodel. 31Asizablediscrepancyappearsinthefirstquarterofthesimulation,inwhichtheimportanceoftransfersrelative toconsumptionrestraintsisoverstatedbythemodel. However,experimentingwithconstraintsontransfersinthe firstquarterof2020showedthattheireffectonthedecumulationphaseisnegligible. 23

12 10 8 6 4 2 0 2020 2021 2022 2023 2024 2025 tuptuO fo tnecreP Data Model Figure5: AggregateTransfersintheBuildupofExcessSavings Notes: Thefigureshowstheratioofaggregatetransferstosteady-stateoutputinthemodel(blacksolidline)and theratioofrealizedtransferincomeindeviationsfromitspre-crisistrendrelativetothepre-crisistrendinGDPin theU.S.(bluedashedline).Theverticallinemarksthethirdquarterof2021,theendofthebuildupperiod. lineupwiththedataatthatpointintime. Includingthebuildupinthesimulationhasseveral advantages,though. Duringtheaccumulationphase,generalequilibriumforcesaresetoffthat interactwiththeexcesssavingsdepletion. Inourview,theseinteractionsareanimportantaspectoftheexcesssavingsdynamicsandshouldnotbeexcludedfromtheanalysis. Inaddition, aswediscussinthecontextoftheinflationresponsebelow, theforward-lookingnatureofthe model implies that some of the macroeconomic effects of the excess savings depletion occur before the peak stock is reached. Starting the simulation at the onset of the buildup allows us tocapturetheseeffects. 5.3 ExcessSavingsDynamics Figure 6 compares simulated total excess savings with the corresponding empirical counterpart,andFigure7containsadisaggregationintoincomequartiles,withtheempiricalestimates ontheleftandthesimulationoutputontheright. Aggregate Excess Savings. While the near exact data match until the third quarter of 2021 is achieved by construction, an important test is how the model performs in replicating the behavior of realized excess savings between the end of 2021 and the end of 2022, the part of the decumulation phase for which the empirical estimates are available. Figure 6 shows that the model does well in predicting the realized decumulation rate. The model-implied path is slightlysteeperinitially,whichmayreflectamoregradualresolutionofconsumptionrestraints than our experiment assumes. In the model and in the data, the households have spent a bit more than half of their excess savings by the end of 2022. Almost the entire excess savings stock is depleted at the beginning of 2025. A standard representative agent or spender-saver framework would predict that households hold on to excess savings until taxes rise. Given the slow pace of fiscal consolidation, these models would imply almost no decline in excess 24

10 8 6 4 2 0 2020 2021 2022 2023 2024 2025 tuptuO launnA fo tnecreP Targeted Untargeted Data Model Figure6: AggregateExcessSavings Notes:ThefigurecomparesaggregateexcesssavingsinthemodelwiththeempiricalestimatesofAladangadyetal. (2022).Inthemodel,excesssavingsarethedeviationofaggregateassetsfromthesteadystate,whichwenormalize bysteady-stateoutput. Inthedata,excesssavingsarenormalizedbyapre-pandemictrendinnominalGDP.The verticallinemarksthethirdquarterof2021,theendofthebuildupperiod. 10 8 6 4 2 0 2020 2021 2022 2023 2024 2025 tuptuO launnA fo tnecreP A. Disaggregated Data B. Model Q4 10 Targeted Untargeted Q3 Q2 8 Q1 6 4 2 0 2020 2021 2022 2023 2024 2025 Figure7: DistributionofExcessSavings: Datavs. Model Notes: The figure compares excess savings across income quartiles in the model with the empirical estimates of Aladangady et al. (2022). In the model, excess savings are the deviation of aggregate assets from steady state, whichwenormalizebysteady-stateoutput.Inthedata,excesssavingsarenormalizedbyapre-pandemictrendin nominalGDP.Theverticallinemarksthethirdquarterof2021,theendofthebuildupperiod. savings over the period considered. In contrast, our HANK model successfully predicts the rapiddepletionofexcesssavings. Distribution. Figure7demonstratesthatthedepletionrateismatchedwellbythemodelnot only in the aggregate but also across the income distribution. The households in the highest quartilemaintainthelargestshareofexcesssavingsandtheonesinthelowestquartilerunout of excess savings the fastest. These dynamics may be a result of the concentration of the peak excesssavingsstockatthetopofthedistributioninconjunctionwithinitiallyloweriMPCsof 25

the top quartile. They may also result from “trickling up,” the process by which savings flow from high-MPC to low-MPC households in general equilibrium. We return to a separation of partial and general equilibrium effects in Section 5.5. Note also that even the households in thetopquartilerundowntheirexcesssavingsinlessthanfouryears,whichisconsistentwith thestillnonnegligibleMPCsoftophouseholdsinourmodelandatoddswiththebehaviorof permanentincomeconsumersintheabsenceofstronggeneralequilibriumforces. 5.4 MacroeconomicImplications Wenowturntothebroadermacroeconomicconsequencesofexcesssavings. Figure8portrays the impact of excess savings on output, aggregate consumption, and investment on the left as wellasthenominalshort-terminterestrate,inflation,andtherealrate,allexpressedasannualizedpercentagepoint-deviationsfromtheirsteadystate,ontheright. Aggregateconsumption andhenceoutputinitiallycollapseandthenexpand,overshootingtheirrespectivesteady-state valuesattheendoftheaccumulationphase.32 Becauseinflationdependsontheexpecteddiscountedsumofrealmarginalcostandmarginalcostfollowsthedynamicsofoutput,inflation first declines but rises above its target level as soon as mid-2020. The monetary authority adjusts the nominal interest rate accordingly. Depressed demand during the buildup stage and the anticipation of heightened real rates during the decumulation stage weigh on investment, whichpartlycrowdsoutthepositiveeffectofspendingoutofexcesssavingsonaggregatedemand. Theriseintherealratealsoweakensaggregateconsumption. Allinall,theimplications oftheexcesssavingsdynamicsareakintothoseofstandardfiscalshocks,asexpectedfromthe combinationofdisturbancesthatunderlietheirbuildup. Inflation. Oursimulationfurtherindicatesthatthedecumulationofexcesssavingscontributed to the surge in inflation seen in the U.S. in the wake of the pandemic. Realized inflation and the simulated inflation path are compared in Figure 9. Inflation in the model is now recenteredaroundatargetvalueof2percentforcomparability. Overthefirsttwoquartersof2020, average realized inflation dropped to about 0.5 percent. Subsequently, realized inflation rose steeply, averaging 4.9 percent over the second half of 2021. According to our model, excess savingsareassociatedwithanincreaseintheinflationrateofabout1.8percentagepointsover the same period. We therefore conclude that about 40 percent of the inflation built up by late 32The model implies that aggregate consumption jumps up at the onset of the decumulation phase and then decaysgradually. Personalconsumptionexpenditure(PCE)intheU.S.hasindeedbeenremarkablystrongsince themiddleof2021,asFigureD.1intheAppendixshows.AlthoughPCEdidnotjumpasabruptlyasinthemodel, itscumulativestrengthover2022–2024alignswellwiththatinthemodel,andthemodel-impliedpathforexcess savingsmatchestheempiricalestimateswell. Theabruptconsumptionchangeinthemodelreflectsthatweshut down the exogenous drivers of excess savings accumulation for all households at once starting in 2021Q4. We interpretthesmootherconsumptionpathinthedataasaresultofamoregradual“returntonormalcy”butstill prefer a clean separation between excess savings accumulation and depletion for transparency. With a strongly forward-lookingPhillipscurve,thetimingoftheriseinaggregatedemandmattersmuchlessforinflationthanits overallmagnitude. 26

5 0 5 10 15 2020 2021 2022 2023 2024 2025 SS morf noitaiveD tnecreP A. Output and its Main Components Output 1.0 Consumption Investment 0.5 0.0 0.5 1.0 2020 2021 2022 2023 2024 2025 SS morf noitaiveD PP dezilaunnA B. Interest Rates Inflation Nominal Rate Real Rate Figure8: EffectofExcessSavingsDepletiononMacroeconomicAggregates Notes: Thefigureshowsmodel-basedsimulationsofmacroeconomicaggregatesontheleftandinterestratesand inflationontheright. Shownareeitherpercent-orannualizedpercentagepoint-deviationsfromsteadystate(SS). Theverticallinesmarkthethirdquarterof2021,theendofthebuildupperiod. 6 5 4 3 2 1 0 1 2020 2021 2022 2023 etaR ylretrauQ dezilaunnA Data Model Figure9: SimulatedInflationDuetoExcessSavingsandRealizedInflation Notes:Thefigureshowssimulatedinflationinthemodelandrealizedcorepersonalconsumptionexpenditureprice inflation(annualizedquarterlyrates). Thesimulatedseriesisre-centered,reflectinganinflationtargetof2percent forcomparability.Theverticallinemarksthethirdquarterof2021,theendofthebuildupperiod. 2021mayhaveoriginatedinexcesssavingsdynamics. Sensitivity. The inflation path in our model simulation is sensitive to the value assumed for the parameter governing the slope of the Philips curve κ . It is well known that several p identification issues arise in the estimation of the slope parameter based on macro data alone (Mavroeidis et al., 2014; McLeay and Tenreyro, 2020). Our baseline calibration, κ = 0.05, folp lowsGagliardoneetal.(2023)whoestimatetheslopeofthePhilipscurveusingdetailedmicro paneldataatthefirm-productlevelthatmitigatetheseconcerns. Theirestimateof0.05to0.06 fortheslopeofthemarginalcost-basedPhillipscurvelieswithintherangeofvaluesthatcanbe foundintheDSGEliterature. Forexample,theestimationbySmetsandWouters(2007)yields apointestimateof0.01,whileKaplanetal.(2018)optforavalueof0.1referringtoSchorfheide 27

(2008). RecentestimatesbasedonregionaldatasuggestthatthePhillipscurvewithameasure ofeconomicslackastheforcingvariablewasveryflatintheyearsleadinguptotheCOVID-19 pandemic (e.g., Hazell et al., 2022). Gagliardone et al. (2023) demonstrate that this finding is consistentwiththeirestimateswhenthelowelasticityofmarginalcosttotheoutputgaporthe unemployment gap is taken into account.33 Additionally, Harding et al. (2023) argue that the Phillipscurvemayhavesteepenedinthewakeofthepandemic. Table1: InflationandtheSlopeofthePhillipsCurve 2020H1 2021H2 ∆ FractionofData Data 0.47 4.89 4.42 Model κ =0.01 2.36 2.70 0.34 0.08 p κ =0.025 2.00 2.91 0.92 0.21 p κ =0.05 1.37 3.21 1.83 0.42 p κ =0.075 0.79 3.47 2.68 0.61 p κ =0.10 0.24 3.70 3.46 0.78 p Notes: Shownisaveragequarterlyinflationinthefirsttwoquartersof2020(2020H1)andinthelasttwoquarters of2021(2021H2)inthedataandinthemodel,expressedasannualpercentagerates. Themodelresultsareshown fordifferentvaluesoftheslopeofthePhillipscurveκp andarere-centeredaroundaninflationtargetof2percent. ∆isthechangebetween2021H2and2020H1.Thelastcolumngivestheratioof∆inthemodelandinthedata. Table 1 shows how the simulated inflation dynamics change when the slope parameter κ p is varied away from the baseline value of 0.05. As the the Phillips curve becomes flatter and inflation becomes less responsive to marginal cost and hence aggregate demand, the rise in inflationbetweenthefirsthalfof2020andthesecondhalfof2021declines. Amoderatelyhigh slopeofκ ∈ (0.075,0.1)enablesthemodeltomatchtheinitialdropinrealizedinflationto0.47 p percentattheonsetofthepandemic. Withalowvalueofκ = 0.025,themodelsstillassociates p about20percentoftherealizedsurgeininflationwithexcesssavingsdynamics. 5.5 IsolatedDecumulationorEquilibriumFeedback The depletion rate of aggregate excess savings depends on the joint distribution of the initial excesssavingsandtheiMPCs,aswellasgeneralequilibriumforces,asdemonstratedinSection 2. Wenowinvestigatetherelativeimportanceofthesetwofactors,showingthatthedirecteffect ofexcesssavingsisastrongpredictorofthedepletionpath. Formal Decomposition. A decomposition of the excess savings dynamics into partial and general equilibrium components is facilitated by our solution procedure, the sequence-space Jacobian method laid out in Auclert et al. (2021). According to this method, the model is divided into blocks, subsets of model equations, for which Jacobians are calculated in sequence 33Micro-foundedformulationsofthePhillipscurveintermsoftheoutputgaprelyonassumptionsthatmakethe outputgapproportionaltomarginalcostandhenceagoodproxythereof. 28

12 10 8 6 4 2 0 2 2020 2021 2022 2023 2024 2025 tuptuO launnA fo tnecreP A. PE-GE Decomposition B. Unpacking GE Feedback 4 Total Total GE 3 PE Effect Interest Rate GE Feedback Taxes 2 Labor Income 1 Financial Income 0 1 2 3 2020 2021 2022 2023 2024 2025 Figure10: PartialandGeneralEquilibriumContributionstoExcessSavings Notes: The figure shows total excess savings and their decomposition given by equation (46). The vertical lines markthethirdquarterof2021,theendofthebuildupperiod. space. Theequilibriumconditionspertainingtothehouseholdsarecollectedinablockthatdeterminespathsforaggregatesavingsandconsumptionifitissuppliedwithallblockinputs— (cid:8) (cid:9) sequences for the exogenous variables, {dβ } and dε , and sequences for the endogenous t q,t variables, (cid:8) drA (cid:9) , {dw }, {dτ}, (cid:8) dDFI (cid:9) , {df }, {dU }, and {dN }. Let dA = (dA ,dA ,...)′ t t t t t t t 0 1 beacolumnvectorgivingthegeneralequilibriumresponseofaggregateexcesssavingstothe combinationofshocksfedintooursimulation. Then,toafirst-orderapproximation, (cid:88) (cid:88) dA = JAdς+ JAdX, (46) ς X ς∈Iex X∈Ien (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) PE GE whereIex = {β,ε ,ε ,ε ,ε }isthesetofallexogenousinputstothehouseholdblockandIen = 1 2 3 4 (cid:8) (cid:9) rA,w,τ,DFI, f,U,N collects all endogenous inputs. Further, JA is a square matrix with v (cid:2) (cid:3) elements JA = ∂A /∂v foranyv ∈ Iex∪Ien,dςisavectorspecifyingthevalueofshockς v t,s t s ineachperiod,anddXistheevolutionoftheendogenousinputXingeneralequilibrium. Thus, dA canbedecomposedintotwocomponentslabeled“PE”and“GE,”respectively: thepartial response to the exogenous shocks holding all other model variables fixed and the response to theequilibriumdynamicsoftheendogenousvariables. Excess Savings in PE and GE. Figure 10 provides a visual representation of the decomposition. The left panel shows that the partial equilibrium forces dominate the excess savings response. Byequation(46),besidestheshockseries,thePEcomponentdependsonJA,there- β sponseofsavingstochangesinthediscountfactor,andJA,whichisdeterminedbytheiMPCs εq of the households in quartile q. Figure D.2 in the Appendix breaks down the PE component into the contributions of the discount factor shocks and all transfer shocks. It reveals that the formercontributesomewhatmoretothebuildupofexcesssavingsinpartialequilibriumthan 29

the latter. Both, the excess savings generated through the fiscal transfers and those resulting fromvoluntaryconsumptionrestraintsarespentoutrapidly. ThisisthecasebecausethetransferstargetedbottomhouseholdswithhighMPCsandsocialdistancingpredominantlyaffected thespendingoftophouseholds,whoareresponsivetovariationinthetimepreferencerate. The model’s general equilibrium forces reduce excess savings through four channels, as canbeseenfromtherightpanelofFigure10. First,thecontributionoflaborincomeisnegative initially as weaker aggregate demand lowers the wage rate and the job-finding rate before it graduallyrecovers. Second,theinitialdropintherealinterestratediscouragessavingandthe subsequent rise yields a small positive effect in later quarters. Third, with a higher expected real interest rate path over the decumulation period, government bond and equity prices fall. Thedeclineinassetpricesresultsinsmallerdistributionsfromthefinancialintermediaryand a negative total contribution of financial income despite the higher anticipated return on deposits.34 Fourth,risinggovernmentdebtismetwithincreasesintheincometaxrateand,hence, lossesindisposableincome. Our analysis indicates that the general equilibrium feedback, on net, leads households to accumulate less excess savings and to spend them down faster. This result is in contrast with Auclert et al. (2023b), who emphasize that the multiplier process—one household’s spending from excess savings is another household’s income—prolongs the duration of excess savings. Thereasonforthisdiscrepancyisthattheaggregatedemandmultiplierinourmodelisdampenedbyseveraladditionalfactors. Higherspendinggeneratesinflation,promptingthecentral banktoraiseinterestrates. Higherrealratesreducenotonlytheconsumptionofunconstrained householdsbutalsoinvestmentandvacancycreation. AsweexplaininSection4,ourmodel’s cumulative multiplier fits the evidence from military spending shocks well, assuming a low degreeofdeficitfinancing. Inourmainexcess-savingsexperiment,weallowforamuchhigher degree of deficit financing, which raises the multiplier in the short run. Eventually, taxes rise to consolidate real government debt, reducing the cumulative multiplier in the long run. In contrast,Auclertetal.(2023b)letgovernmentdebtrisepermanently. 5.6 TheStabilizingRoleofFiscalTransferPolicy We conclude our analysis by isolating the stabilizing effect of fiscal support. Below, we first comparetheperformanceoftheU.S.economytothatoftheeuroarea,wherefiscalsupportwas significantlyweaker. Then,weconductacounterfactualexperiment,inwhichweestimatehow macroeconomic aggregates in the U.S. would have evolved in the absence of the emergency transferpaymentstohouseholds. 34WhiletheresponseofdistributionsDFIisnegativeatalltimes,thedistributions’contributiontoexcesssavings t issmallbutpositiveinthefirstquarters,ashouseholdswithhighcashonhandinitiallycutintoconsumptionto maintaintheirtargetstockofsavings. 30

2 0 2 4 6 8 10 12 2018 2019 2020 2021 2022 2023 PDG fo tnecreP A. Primary Deficit 110 105 100 95 90 US EA 85 2018 2019 2020 2021 2022 2023 )001 = 4Q9102( xednI B. Consumption US EA Figure11: PrimaryDeficitandConsumption: U.S.vs. EuroArea Notes:ThedatasourcesfortheprimarydeficitaretheCongressionalBudgetOfficeandtheEuropeanCentralBank fortheU.S.andtheeuroarea,respectively. Theconsumptiondata,chain-linkedvolumeindiceswiththe2019Q4 valuesnormalizedto100,areobtainedfromtheOECD.Consumptioncomprisesthefinalconsumptionexpenditure ofhouseholdsandnon-profitinstitutionsservinghouseholds(NPISH).“EA”standsfor“euroarea”. Comparison with Euro Area. The left panel of Figure 11 shows that the fiscal stance was significantly more accommodative in the U.S. than in the euro area during the pandemic. In 2020,theprimarydeficitasafractionofGDProsetomorethan13percentintheU.S.,whileit reachedonlylessthan6percentintheeuroarea. Asubstantialdiscrepancypersistedinto2021. The difference in fiscal positions was largely driven by the U.S.’s economic impact payments. Correspondingly, aggregate consumption contracted more in the euro area than in the U.S. despite a similar spread of the COVID-19 virus and a similar stance of monetary policy, as illustrated in the right panel of the same figure. By the first quarter of 2021, consumption had risenabout1percentaboveitslevelinthefourthquarterof2019intheU.S.,whileitremained about9percentbelowitspre-pandemiclevelintheeuroarea. FiscalTransfersintheU.S. Wenowturntothecounterfactualanalysis,whichconfirmsthat thefiscalstimuluspaymentsintheU.S.andtheassociatedexcesssavingsdynamicscanexplain differences in macroeconomic outcomes of the magnitude observed between the U.S. and the euroarea. Figure12depictstheconsumptionandinflationpathfromtheexperimentdescribed in Section 5.2 together with the results from a simulation, in which we set the transfer shocks (cid:8) (cid:9) ε ,ε ,...,ε inallquartilestozero. q,2020Q1 q,2020Q2 q,2021Q3 Ourmodelsuggeststhat,withoutthefiscaltransfers,consumptionwouldhavecontracted nearly 10 percentage points more in the second quarter of 2020, a gap of about the same size as the difference between consumption in the U.S. and the euro area in that quarter. Because householdsbuilduplesssavingswithoutthetransfers,consumptionisalsolowerattheonset of the decumulation phase. In line with the stronger drop in demand, inflation falls more at the beginning of the pandemic. However, more accommodative monetary policy and hence 31

10 5 0 5 10 15 20 25 2020 2021 2022 2023 2024 2025 SS morf .veD tnecreP A. Consumption 3 2 1 0 With Transfers Without Transfers 1 2020 2021 2022 2023 2024 2025 tnecreP B. Inflation With Transfers Without Transfers Figure12: RoleofFiscalTransfers Notes: Thefigureshowsthebaselineresults(blacksolidlines)andasimulation, inwhichallfiscaltransfersare removed(bluedashedlines). Theverticallinesmarkthethirdquarterof2021, theendofthebuildupperiodof excesssavings. stronger investment imply that inflation is of similar magnitude as in the baseline simulation from the end of 2021 onward. Overall, although fiscal transfers contributed to the inflation pressureinthefirstquartersofthepandemic,ithelpedstabilizeeconomicactivitysubstantially. 6 Conclusion The permanent income hypothesis (PIH) underpinning standard representative agent models implies that wealth shocks are essentially irrelevant. The representative household is willing to hold any amount of assets indefinitely as long as β = (1+r)−1. Two-agent models of the standardspender-savertypesharethisstarkimplication. Bydefinition,excesssavingsareheld bysavers,whobehaveaccordingtothePIH.AsBilbiieetal.(2021)pointout,savingscannever beexcessiveaccordingtothesemodels. HANKmodelspermitarichertheoryofexcesssavings,allowingustoaccountforthefact that excess savings are distributed across households with different iMPCs. The distribution mattersbecausetheiMPCsgovernthespend-outrateofexcesssavings. Moreover,micro-level iMPCsareinformativefortheconsumptionresponsestochangesintaxes,laborearnings,and financial income, which are associated with excess savings depletion in general equilibrium. Building on these analytical insights, we construct a quantitative HANK model with tightly disciplined iMPCs and general equilibrium multipliers to study the macroeconomic consequences of the excess savings built up during the COVID-19 pandemic. A parsimonious set of transfer and discount factor shocks allows us to replicate the buildup of excess savings between the beginning of the pandemic and their peak in the third quarter of 2021. While they are not targeted, the model closely matches both aggregate and distributional data from the partoftherundownperiodforwhichestimatesofexcesssavingsareavailable. 32

We find that the historical spike in inflation that occurred in the U.S. in 2021 had a significant demand component, which contradicts the prevalent view that the inflation surge was driven almost entirely by supply-side constraints. According to conventional wisdom, while centralbanksmayfacetrade-offsfollowingsupplyshocksthatinducethemtoremaininactive, amonetarycontractioniswarrantedinresponsetoexpansionarydemandshocks. Finally,wefindthatthemacroeconomiceffectsofexcesssavingscanbewellpredictedfrom the joint distribution of initial excess savings and iMPCs. Our analysis exemplifies that models that abstract from distributions cannot generally substitute for models that accommodate distributionaldatadirectly. 33

References Aiyagari,S.R.andMcGrattan,E.R.(1998). Theoptimumquantityofdebt. JournalofMonetary Economics,42(3):447–469. Aladangady,A.,Cho,D.,Feiveson,L.,andPinto,E.(2022). ExcessSavingsduringtheCOVID- 19Pandemic. FEDSNotes2022-10-21,BoardofGovernorsoftheFederalReserveSystem. Auclert,A.,Bardo´czy,B.,andRognlie,M.(2023a). MPCs,MPEs,andMultipliers: ATrilemma forNewKeynesianModels. TheReviewofEconomicsandStatistics,105(3):700–712. Auclert,A.,Bardo´czy,B.,Rognlie,M.,andStraub,L.(2021).UsingtheSequence-SpaceJacobian toSolveandEstimateHeterogeneous-AgentModels. Econometrica,89(5):2375–2408. Auclert, A., Rognlie, M., and Straub, L. (2018). The Intertemporal Keynesian Cross. Working Paper25020,NationalBureauofEconomicResearch. Auclert, A., Rognlie, M., and Straub, L. (2020). Micro jumps, macro humps: Monetary policy andbusinesscyclesinanestimatedhankmodel. CESifoWorkingPaperSeries8051,CESifo. Auclert,A.,Rognlie,M.,andStraub,L.(2023b). Thetricklingupofexcesssavings. AEAPapers andProceedings,113:70–75. Batty, M., Deeken, E., Holmquist, E., andVolz, A.H.(2023). WealthinequalityandCOVID-19 intheU.S.: evidencefromthedistributionalfinancialaccounts. InPost-pandemiclandscapefor centralbankstatistics,volume58ofIFCBulletin.BankforInternationalSettlements. Bayer, C., Born, B., and Luetticke, R. (2024). Shocks, Frictions, and Inequality in US Business Cycles. AmericanEconomicReview,114(5):1211–47. Bayer, C., Born, B., Luetticke, R., and Mu¨ller, G. J. (2023). The Coronavirus Stimulus Package: HowLargeistheTransferMultiplier. TheEconomicJournal,133(652):1318–1347. Bilbiie, F., Eggertsson, G., Primiceri, G., and Tambalotti, A. (2021). ”Excess Savings” Are Not Excessive. LibertyStreetEconomics20210405a,FederalReserveBankofNewYork. Campbell, J. Y. and Mankiw, N. G. (1989). Consumption, income, and interest rates: Reinterpretingthetimeseriesevidence. NBERMacroeconomicsAnnual,4:185–216. Carroll, C. D. (2004). Theoretical Foundations of Buffer Stock Saving. NBER Working Papers 10867,NationalBureauofEconomicResearch,Inc. Carroll, C. D., Crawley, E., Slacalek, J., and White, M. N. (2021). Modeling the Consumption ResponsetotheCARESAct. InternationalJournalofCentralBanking,17(1):107–141. 34

Christelis, D., Georgarakos, D., and Jappelli, T. (2015). Wealth shocks, unemployment shocks andconsumptioninthewakeoftheGreatRecession. JournalofMonetaryEconomics,72:21–41. Christiano,L.,Eichenbaum,M.,andTrabandt,M.(2016). Unemploymentandbusinesscycles. Econometrica,84:1523–1569. Eichenbaum, M. S., Rebelo, S., and Trabandt, M. (2021). The Macroeconomics of Epidemics. TheReviewofFinancialStudies,34(11):5149–5187. Fagereng, A., Holm, M. B., and Natvik, G. J. (2021). MPC Heterogeneity and Household BalanceSheets. AmericanEconomicJournal: Macroeconomics,13(4):1–54. Gagliardone, L., Gertler, M., Lenzu, S., and Tielens, J. (2023). Anatomy of the Phillips Curve: MicroEvidenceandMacroImplications.WorkingPaper31382,NationalBureauofEconomic Research. Gertler, M.andTrigari, A.(2009). Unemploymentfluctuationswithstaggerednashwagebargaining. JournalofPoliticalEconomy,117(1):38–86. Giannone, D. and Primiceri, G. E. (2024). The drivers of post-pandemic inflation. Working paper,EuropeanCentralBank. Gornemann,N.,Kuester,K.,andNakajima,M.(2021). DovesfortheRich,HawksforthePoor? DistributionalConsequencesofMonetaryPolicy. Unpublishedmanuscript. Guerrieri,V.andLorenzoni,G.(2017). CreditCrises,PrecautionarySavings,andtheLiquidity Trap. TheQuarterlyJournalofEconomics,132(3):1427–1467. Guren,A.M.,McKay,A.,Nakamura,E.,andSteinsson,J.(2021). HousingWealthEffects: The LongView. TheReviewofEconomicStudies,88(2):669–707. Hall, R. E. (2005). Employment fluctuations with equilibrium wage stickiness. American EconomicReview,95(1):50–65. Harding,M.,Linde´,J.,andTrabandt,M.(2023). Understandingpost-COVIDinflationdynamics. JournalofMonetaryEconomics,140:S101–S118. Hazell, J., Herren˜o, J., Nakamura, E., andSteinsson, J.(2022). TheSlopeofthePhillipsCurve: EvidencefromU.S.States. TheQuarterlyJournalofEconomics,137(3):1299–1344. He, Z., Liao, G., andWang, B.(2022). Whatgetsmeasuredgetsmanaged: Investmentandthe costofcapital. WorkingPaper29775,NationalBureauofEconomicResearch. Heathcote, J. and Perri, F. (2018). Wealth and Volatility. The Review of Economic Studies, 85(4):2173–2213. 35

Heathcote, J., Storesletten, K., and Violante, G. L. (2017). Optimal Tax Progressivity: An AnalyticalFramework. TheQuarterlyJournalofEconomics,132(4):1693–1754. Holm, M. B., Paul, P., and Tischbirek, A. (2021). The Transmission of Monetary Policy under theMicroscope. JournalofPoliticalEconomy,129(10):2861–2904. Kaplan,G.,Mitman,K.,andViolante,G.L.(2020a). TheHousingBoomandBust: ModelMeets Evidence. JournalofPoliticalEconomy,128(9):3285–3345. Kaplan,G.,Moll,B.,andViolante,G.L.(2018).MonetaryPolicyAccordingtoHANK.American EconomicReview,108(3):697–743. Kaplan, G., Moll, B., and Violante, G. L. (2020b). The Great Lockdown and the Big Stimulus: TracingthePandemicPossibilityFrontierfortheU.S. WorkingPaper27794,NationalBureau ofEconomicResearch. Koby, Y. and Wolf, C. (2020). Aggregation in Heterogeneous-Firm models: Theory and measurement. UnpublishedManuscript. Mavroeidis, S., Plagborg-Møller, M., and Stock, J. H. (2014). Empirical Evidence on Inflation ExpectationsintheNewKeynesianPhillipsCurve. JournalofEconomicLiterature, 52(1):124– 88. McKay,A.,Nakamura,E.,andSteinsson,J.(2016). ThePowerofForwardGuidanceRevisited. AmericanEconomicReview,106(10):3133–3158. McKay, A. and Reis, R. (2016). The Role of Automatic Stabilizers in the U.S. Business Cycle. Econometrica,84(1):141–194. McLeay, M. and Tenreyro, S. (2020). Optimal Inflation and the Identification of the Phillips Curve. NBERMacroeconomicsAnnual,34:199–255. Mian, A., Rao, K., and Sufi, A. (2013). Household Balance Sheets, Consumption, and the EconomicSlump. TheQuarterlyJournalofEconomics,128(4):1687–1726. Mian,A.andSufi,A.(2014). WhatExplainsthe2007-2009DropinEmployment? Econometrica, 82(6):2197–2223. Oh, H. and Reis, R. (2012). Targeted transfers and the fiscal response to the great recession. JournalofMonetaryEconomics,59(S):50–64. Ramey,V.andZubairy,S.(2018). Governmentspendingmultipliersingoodtimesandinbad: Evidencefromushistoricaldata. JournalofPoliticalEconomy,126(2):850–901. Rotemberg,J.(1982). Monopolisticpriceadjustmentandaggregateoutput. ReviewofEconomic Studies,49(4):517–531. 36

Schorfheide, F. (2008). DSGE model-based estimation of the New Keynesian Phillips curve. EconomicQuarterly,94(Fall):397–433. Shimer, R. (2004). The consequences of rigid wages in search models. Journal of the European EconomicAssociation,2(2-3):469–479. Shimer,R.(2005). Thecyclicalbehaviorofequilibriumunemploymentandvacancies. American EconomicReview,95(1):25–49. Smets,F.andWouters,R.(2007). ShocksandFrictionsinUSBusinessCycles: ABayesianDSGE Approach. AmericanEconomicReview,97(3):586–606. Woodford,M.(2001).Fiscalrequirementsforpricestability.JournalofMoney,CreditandBanking, 33(3):669–728. 37

A Illustrative Model A.1 ProofoftheProposition Twocomments. First,wewillusethefactthattheconsumptionfunctionisdifferentiable. This canbeestablishedalongthelinesofCarroll(2004). Second,wewillshowtheresultconditional on a particular sequence of productivity shocks that, starting from an initial x , give rise to a 0 sequenceofcash-on-handvalues x ,x ,...,x . Thisissufficienttoprovetheresultinthemain 1 2 t text,whichaveragestheseidiosyncraticshocks. Finally,weomitisubscriptsforsimplicity. Consider t = 0. Let x 0 = (1+r −1 )a −1 +y 0 denote the default cash on hand in this period. TheMPCfromlump-sumtransfersreceivedinperiod0is c (x +∆)−c (x ) m (x ) = lim 0 0 0 0 = c ′(x ). (A.1) 0 0 ∆→0 ∆ 0 0 Thatis,m (x )issimplytheslopeofthetime-0consumptionfunctionattheoriginalstate x . 0 0 0 Consider t = 1. Let x = (1+r )a (x )+y denote the default cash on hand in period 1. If 1 0 0 0 1 ∆ therewaslump-sumtransfer inperiod0,thencashonhandinperiod1is x (∆) = y +(1+r )a (x +∆) (A.2) 1 1 0 0 0 = y +(1+r )[x +∆−c (x +∆)] (A.3) 1 0 0 0 0 = x +(1+r )[∆−c (x +∆)+c (x )] (A.4) 1 0 0 0 0 0 (cid:20) c (x +∆)−c (x ) (cid:21) = x +(1+r )∆ 1− 0 0 0 0 (A.5) 1 0 ∆ Substitutethisintothedefinitionofm : 1 (cid:16) (cid:104) (cid:105)(cid:17) c x +(1+r )∆ 1− c0 (x0 +∆)−c0 (x0 ) −c (x ) 1 1 0 ∆ 1 1 m (x ) = lim (A.6) 1 0 ∆→0 ∆ Sincec (•)isadifferentiablefunctionforallt,wemaytakethelimitinsideandwrite t (cid:16) (cid:17) c x +(1+r )∆[1−c′(x )] −c (x ) 1 1 0 0 0 1 1 m (x ) = lim (A.7) 1 0 ∆→0 ∆ = lim c 1 (x 1 +∆′)−c 1 (x 1 ) (1+r ) (cid:2) 1−c ′(x ) (cid:3) (A.8) ∆′→0 ∆′ 0 0 0 = c ′(x )·(1+r ) (cid:2) 1−c ′(x ) (cid:3) (A.9) 1 1 0 0 0 Thatis,m (x )istheslopeofthetime-1consumptionfunctionatstatex timestheexcesssavings 1 0 1 thatremainsfromthetransferinperiod0. 38

Consider t = 2. Let x = (1+r )a (x )+y denote the default cash on hand in period 2. If 2 1 1 1 2 ∆ therewaslump-sumtransfer inperiod0,thencashonhandinperiod2is x (∆) = y +(1+r )a (x (∆)) 2 2 1 1 1 = y +(1+r )[x (∆)−c (x (∆))] 2 1 1 1 1 (cid:34) x (∆)−x c (cid:0) x (∆) (cid:1) −c (x ) (cid:35) = x +(1+r )∆ 1 1 − 1 1 1 1 2 1 ∆ ∆ Usingtheresultsfort = 1,wehave lim x 1 (∆)−x 1 = lim(1+r ) (cid:20) 1− c 0 (x 0 +∆)−c 0 (x 0 ) (cid:21) = (1+r ) (cid:2) 1−c ′(x ) (cid:3) ∆→0 ∆ ∆→0 0 ∆ 0 0 0 (A.10) c (cid:0) x (∆) (cid:1) −c (x ) lim 1 1 1 1 = (1+r ) (cid:2) 1−c ′(x ) (cid:3) c ′(x ) (A.11) ∆→0 ∆ 0 0 0 1 1 Asbefore,wesubstitutetheselimitsintothedefinitionofm (x ),invokingthedifferentiability 2 0 oftheconsumptionfunctions,toget c (cid:0) x (∆) (cid:1) −c (x ) (cid:104) (cid:105)(cid:104) (cid:105) m (x ) = lim 2 2 2 2 = c ′(x )·(1+r )(1+r ) 1−c ′(x ) 1−c ′(x ) . (A.12) 2 0 ∆→0 ∆ 2 2 1 0 0 0 1 1 Fromhere,it’seasytoseethepatternforgeneralt ≥ 1: (cid:89) t−1 (cid:104) (cid:105) m (x ) = c ′(x ) (1+r ) 1−c ′(x ) (A.13) t 0 t t s s s s=0 The first term is the static MPC in period t. The second term is the excess savings left from the initialtransfer. A.2 IntertemporalKeynesianCross In this appendix, we specify the details of a small-scale HANK model that gives rise to the intertemporalKeynesiancross(IKC) Y = C(τ,R(K(Y)),Y, Γ ). (A.14) 0 The economy is populated by a unit mass of households, a representative firm, a government,andacentralbank. PhillipsCurveK(Y). WeassumethatthefinalgoodpriceP isflexibleandthenominalwage t W t is sticky. Specifically, wage inflation π t w ≡ W t /W t−1 −1 follows the textbook New Keyne- 39

sianwagePhillipscurve (cid:18) v′(N ) (cid:19) πw(1+πw) = κ µ t −1 +E [πw (1+πw )], (A.15) t t w w u′(C ) t t+1 t+1 t whereκ > 0istheslopeofthePhillipscurve,and µ > 1isthedesiredmarkupoftheseller w w of labor services, v′(N ) is the marginal disutility of labor, and u′(C ) is the marginal utility of t t consumptionforahypotheticalrepresentativeagent. Thatis,theterminparenthesesisthepercentagedeviationofthemarginalrateofsubstitutionbetweenhoursandconsumptionfromits steadystate. UsingtheMRSofahypotheticalrepresentativeagentinsteadoftheaverageMRS oftheheterogeneousagentssimplifiesthederivationoftheintertemporalKeynesiancross. Writing (A.15) as K(Y) requires a few more steps. First, let the final good be produced by a competitive firm with linear technology Y = N. Since the nominal profit of the firm t t is PY −W N, the equilibrium price level must be P = W for all t. This implies that wage t t t t t t inflation equals price inflation π t w = P t /P t−1 −1 = π t . Using these results and goods market clearing Y = C , the Phillips curve (A.15) under perfect foresight can indeed be written as t t π = K(Y). Consumption Function C(τ,r,Y, Γ ). The household sector is an instance of the SIM model 0 (1)-(3) in which income is after-tax labor income y = (1−τ)WtN z with real wage W /P, i,t t Pt t i,t t t hours N, and idiosyncratic productivity z that follows an exogenous Markov process and t i,t has a mean of 1. Given the production function Y = N and the result Wt = 1, the only t t Pt endogenousvariablesenteringthehouseholdblockare{τ,r,Y}. Thisprovesthattheaggregate consumptionfunctioncanbewrittenasC(τ,r,Y, Γ ). 0 Real Rate Function R(π). Follows immediately from combining the Taylor rule i = r + t ss ϕ π π t withtheFisherequationr t = i t −E t [π t+1 ]. 40

B Quantitative Model B.1 FinancialIntermediary The financial intermediary chooses S , B , A and NFI to maximize E (1+rN ) subject to the t t t t t t+1 constraints(13)to(15). Asset Returns. By combining the constraints with the definition of the return on net worth afterdistributionsandtakingtheconditionalexpectation,oneobtains: E (1+rN ) = E (D t+1 +pS t+1 )S t +(1+δ B q t B +1 )B t −(1+r t A+ξ)A t −D t FI (B.1) t t+1 t pSS +qBB −A t t t t t Thefirst-orderconditionsare 0 = E t (D t+1 +pS t+1 )N t FI −E t N t F + I 1 pS t , (B.2) 0 = E (1+δ qB )NFI −ENFI qB, (B.3) t B t+1 t t+1 t 0 = −(1+rA+ξ)NFI +E NFI . (B.4) t t t t+1 Togetherwiththedefinitionof1+rN ,theaboveconditionsimply t+1 D +pS E (1+rN ) = E t+1 t+1 (B.5) t t+1 t pS t 1+δ qB = E B t+1 (B.6) t qB t = 1+rA+ξ, (B.7) t whicharetheno-arbitragerelationshipsshowninSection3.2. StabilityofBalanceSheet. Equation(13)to(15)implythatthelawofmotionofnetworthis (cid:104) (cid:105) N t F + I 1 = D t+1 +pS t+1 −(1+r t A+ξ)pS t S t (cid:104) (cid:105) + 1+δ qB −(1+rA+ξ)qB B B t+1 t t t +(1+rA+ξ)NFI −DFI −ϕ(NFI −NFI). (B.8) t t t Takingconditionalexpectationsandusingtheassetpricingrelationships(B.5)to(B.7)gives E NFI = (1+rA+ξ)NFI −DFI −ϕ(NFI −NFI), (B.9) t t+1 t t t 41

orequivalently E ∆NFI = (rA+ξ)NFI −DFI −ϕ(NFI −NFI). (B.10) t t+1 t t t Thus,asteadystateexistsonlyif DFI = (rA+ξ)NFI. Slightlyexpandingequation(B.10),plugginginforDFI,andcollectingtermsyields E ∆NFI = (rA−rA)NFI +(rA+ξ−ϕ)(NFI −NFI), t t+1 t t t whichshowsthat,wheneverrA = rA,networthconvergestoitssteadystatevaluefromabove t orbelowifϕ > rA+ξ. B.2 CapitalProducer Thecapitalproducer’sproblemcanbestatedas (cid:26) 1 (cid:104) (cid:105) p t K(K t−1 ,I t−1 ) = r t KK t−1 +m Q i t nm It, a K x t −I t + 1+r t E t p t K +1 (K t ,I t ) (cid:20) (cid:20) (cid:18) (cid:19)(cid:21) (cid:21)(cid:27) I +Q t (1−δ K )K t−1 + 1−Φ I t I t −K t , (B.11) I t−1 where Q is the Lagrangian multiplier attached to the law of motion of the capital stock. The t first-orderconditionforinvestmentisgivenby 0 = −1+E pK I,t+1 (K t ,I t ) +Q (cid:20) 1−Φ (cid:18) I t (cid:19) −Φ′ (cid:18) I t (cid:19) I t (cid:21) , (B.12) t 1+r t t I I t−1 I I t−1 I t−1 where (cid:18) (cid:19)(cid:18) (cid:19)2 I I pK I,t (K t−1 ,I t−1 ) = Q t Φ′ I I t I t . (B.13) t−1 t−1 Hencecombining(B.12)and(B.13)yieldsequation(19)inthemaintext. Thefirst-orderconditionofcapitalis 1 (cid:104) (cid:105) Q = E pK (K ,I ) , (B.14) t 1+r t K,t+1 t t t with pK K,t (K t−1 ,I t−1 ) = r t K+(1−δ K )Q t . (B.15) Combining(B.14)and(B.13)yieldsequation(20). 42

B.3 LaborMarket This section derives the bargaining surplus of the labor agency and the union and provides detailsontheNashbargainingsolution. LaborAgency’sProblem. Theprofitmaximizationproblemsolvedbythelaboragencyis (cid:26) (cid:27) 1 J t (N t−1 ) = m Nt a ,v x t (h t −w t )N t −(κ v +κ h q t )v t −Φ w (w t ,w t−1 )N t + 1+r t E t [J t+1 (N t )] (B.16) s.t. N t = (1−s)N t−1 +q t v t , (B.17) (cid:18) (cid:19)2 ψ w Φ w (w t ,w t−1 ) = w t −1 , (B.18) 2 w t−1 Let J be the Lagrange multiplier on the first constraint—the shadow value of an additional t workerwithaverageproductivity. Then,thefirst-orderandenvelopeconditionsare 1 0 = h t −w t −Φ w (w t ,w t−1 )− J t + 1+r E t J N,t+1 (N t ), (B.19) t 0 = −(κ +κ q )+ J q , (B.20) v h t t t J N,t (N t−1 ) = (1−s)J t . (B.21) Combiningtheequations(B.19)to(B.21)yieldsequation(24). Equation(25)thenfollowsfrom equation(B.20). Union Surplus. The union’s valuation of the marginal match depends on the value of an employedworkerW andthevalueofanunemployedworkerU , t t 1 W t = w t + 1+r E t {[1−s(1− f t+1 )]W t+1 +s(1− f t+1 )U t+1 }; (B.22) t 1 U t = ωUIw+ 1+r E t [f t+1 W t+1 +(1− f t+1 )U t+1 ]. (B.23) t Hence, H isgivenby t H = W −U t t t 1−s = w t −ωUIw+ 1+r E t (1− f t+1 )H t+1 . (B.24) t whichisequation(26). 43

NashBargaining. TheNashbargainingsolutionsatisfies Ω J = (1−Ω )H ,where t t t t η Ω ≡ , (B.25) t η+(1−η)(−J /H ) w,t w,t 1−s J w,t = −1−Φ 1 w(w t ,w t−1 )+ 1+r E t J w,t+1 , (B.26) t J w,t+1 = −Φ 2 w(w t+1 ,w t ), (B.27) H = 1. (B.28) w,t Thewageadjustmentcostfunctionimplies (cid:18) (cid:19) w 1 Φ 1 w(w t ,w t−1 ) = ψ w w t −1 w ; (B.29) t−1 t−1 (cid:18) (cid:19) w w Φ 2 w(w t+1 ,w t ) = −ψ w w t+1 −1 w t+ 2 1 . (B.30) t t Inasteadystate,−J = H = 1andtherefore Ω = η. w w B.4 IntermediateGoodProducers ThissectionderivesthePhillipscurveunderRotemberg-pricingallowingforpriceindexation. Theproblemcanbewrittenrecursivelyas (cid:26)(cid:18) P (cid:19)1−ϵp X (P ) = max jt Y −rKK −h N −Ψ t jt−1 Pjt,Kjt−1,Njt P t t t jt−1 t jt (cid:27) − χ 2 p (cid:104) log(P jt /P jt−1 )−log (cid:16) Πι t p −1 Π1−ιp (cid:17)(cid:105)2 Y t + 1+ 1 r E t X t+1 (P jt ) t (B.31) (cid:18) P (cid:19)−ϵp s.t. jt Y = ΘKα N1−α. (B.32) P t jt−1 jt t Thecorrespondingfirst-orderconditionsare: (cid:18) P (cid:19)−ϵpY (cid:104) (cid:16) (cid:17)(cid:105)Y 0 = (1−ϵ p ) P jt P t −χ p log(P jt /P jt−1 )−log Πι t p −1 Π1−ιp P t t t jt +λ ϵ (cid:18) P jt (cid:19)−ϵp −1 Y t + 1 E X ′ (P ); (B.33) jt p P P 1+r t t+1 jt t t t 0 = −rK+λ α ΘKα−1N1−α; (B.34) t jt jt−1 jt 0 = −h +λ (1−α)ΘKα N −α. (B.35) t jt jt−1 jt 44

Intheaboveequations,λ istheLagrangemultiplierassociatedwiththeconstraint. Theenvejt lopeconditionis (cid:104) (cid:16) (cid:17)(cid:105) Y X t ′(P jt−1 ) = χ p log(P jt /P jt−1 )−log Πι t p −1 Π1−ιp P t . (B.36) jt−1 The first-order conditions for capital and labor imply that all firms have the same capital-tolaborratioandhencethesamemultiplierλ ,whichcanbeinterpretedasrealmarginalcost, t (rK)α(h )1−α (cid:18) 1 (cid:19)α(cid:18) 1 (cid:19)1−α mc ≡ λ = t t . (B.37) t t Θ α 1−α InasymmetricequilibriumwithP = P ∀j,combiningthefirst-ordercondition(B.33)withthe jt t envelopecondition(B.36)andrearrangingtermsyieldsthePhillipscurve (cid:16) (cid:17) (cid:16) (cid:17) ϵ (cid:18) ϵ −1 (cid:19) log Π −log Πιp Π1−ιp = p mc − p t t−1 χ t ϵ p p (cid:20) (cid:21) + 1+ 1 r E t log (cid:16) Π t+1 (cid:17) −log (cid:16) Πι t pΠ1−ιp (cid:17) Y Y t+1 . (B.38) t t The last equation involves level deviations of real marginal cost from steady state. It’s useful to rewrite it in terms of percentage deviations for the slope κ to be directly comparable to p standardloglinearizedPhillipscurves. ϵ (cid:18) ϵ −1 (cid:19) ϵ ϵ −1 (cid:18) ϵ (cid:19) p p p p p mc − = mc −1 (B.39) χ t ϵ χ ϵ ϵ −1 t p p p p p (cid:124) (cid:123)(cid:122) (cid:125) κp 45

C Calibration Our calibration of the search and matching block follows Christiano et al. (2016). The calibration of the rest of the model is close to Auclert et al. (2020). Although we do not estimate the transition-specific parameters, we show that our model implies cumulative government spendingmultipliersthatmatchwelltheestimatesofRameyandZubairy(2018). Households. We assume a constant relative risk aversion (CRRA) period utility function, u(c) = c1−1/σ/(1−1/σ) with an elasticity of intertemporal substitution of σ = 0.5. We set the quarterly steady-state real interest rate r to 0.5 percent, implying an annual rate of 2 percent. Thiscorrespondstotherealreturnoncapitalandgovernmentbonds. Therealreturnon deposits is rA = r−ξ and we set the quarterly intermediation spread ξ to 1 percent. That is, the real return on deposits is -2 percent per annum. The time preference rate β is 0.95 which implies a liquid wealth to annual GDP ratio of A/(4Y) = 0.24, very close to the value of 0.26 targeted by Kaplan et al. (2018). For the productivity process, we use the quarterly, discretetimeversionoftheleptokurticprocessestimatedbyKaplanetal.(2018),withonemodification: wescaledownthevarianceofinnovationsby(1−0.181)2,where0.181isthedegreeoftaxprogressivityinHeathcoteetal.(2017). Search and Matching. The calibration of this block follows Christiano et al. (2016). The steady-state unemployment rate is 5 percent. We set the job finding rate to f = 0.6 and the vacancyfillingrateto q = 0.7,informedbyCPS(formenaged25–54)andJOLTSdata,respectively. Thesechoicesimplyaquarterlyseparationrateofs = 0.09whichisalsoconsistentwith the separation rate of prime-age men in the CPS data. We set the elasticity of the matching function to α = 0.5 and back out the matching efficiency as a residual. The unemployment m insurance replacement rate is ωUI = 0.5, the most common value across U.S. states. We calibrate the bargaining power of the union such that the total cost of filling a job (κ +κ q)v is 7 v h percentofthequarterlywageoftheaverageworker. AsinChristianoetal.(2016),thevacancy fillingcostκ accountsfor94percentofthetotalcost. h Supply. ThecalibrationofthisblockfollowsAuclertetal.(2020). Totalfactorproductivity Θ ischosensuchthatquarterlyoutputY isnormalizedto1. Weset Bsuchthatgovernmentdebt is46percentofannualoutput,correspondingtodomesticholdings. Wechoosethecouponrate δ tomatchtheaveragedurationofU.S.governmentdebtof5years. GovernmentspendingG B is set to 16 percent of output, which implies a tax rate of τ = 0.24. We choose a depreciation rate of δ = 0.083 annually and calibrate the capital share α to match a quarterly capital to K output ratio of 8.92. These choices imply a steady-state labor share of 62 percent in line with U.S. data. The fixed cost Ψ is calibrated to let total wealth pS+qBB be equal to 382 percent of annualoutputgivenastandardvaluefortheelasticityofsubstitutionofϵ = 7. p 46

Transition-SpecificParameters. Asdicussedinthemaintext,wesettheslopeofthePhillips curve to κ = 0.05 based on Gagliardone et al. (2023) and allow for a moderate amount of inp dexation, ι = 0.2. We set the parameters of the Taylor rule to conventional values, with an p inflation coefficient of ϕ = 1.5 and an inertia parameter of ϕ = 0.8. We calibrate the invest- π r mentadjustmentcosttoψ = 1.8whichimpliesthatthesemi-elasticityofinvestmenttothereal I rateis−5%atanannualfrequency,inlinewiththeestimatesofKobyandWolf(2020)andHe etal.(2022). Therealwageadjustmentcostψ is100whichimpliesthatwagesaremoderately w stickier than wages. The steady state dividend yield from the perspective of the households DFI/p is about 2.1 percent annually. We set the parameter of the intermediary’s distribution rule ϕ to 0.01, which ensures balance sheet stability and implies that the annualized dividend yield rises by about 0.4 percentage points for each 10 percentage points that intermediary net worth exceeds its steady state. These values are consistent with the average dividend yield of the S&P 500 that lay in the range of about 1 to 3 percent over the last decades. Table C.1 summarizesthecalibration. TableC.1: Calibration Description Parameter/Target Value Elasticityofintertemporalsubstitution σ 0.5 Realinterestrate r 0.005 Intermediationcost ξ 0.01 Timepreferencerate β 0.95 Unemploymentrate U 0.05 Jobfindingrate f 0.6 Vacancyfillingrate q 0.7 Searchers’shareofmatchingfunction α 0.5 m Replacementratio ωUI 0.5 Searchcost-to-wageratio (κ +κ q)v/w 0.07 v h Vacancycostshareofsearchcost κ /(κ +κ q) 0.06 v v h Output Y 1 Governmentdebt-to-GDPratio q B/4Y 0.46 B Maturityofgovernmentbond 1/δ 20 B Shareofgovernmentspending G/Y 0.16 Depreciationrate δ 0.083/4 K Capital-to-outputratio K/4Y 8.92/4 Wealth-to-GDPratio (pS+qBB)/4Y 3.82 Elasticityofsubstitution ϵ 7 p SlopeofthePhillipscurve κ 0.05 p Indexationinpricesetting ι 0.2 p Taylorrulecoefficientoninflation ϕ 1.5 π Taylorrulecoefficientoninertia ϕ 0.8 i Investmentadjustmentcost ψ 1.8 I Realwageadjustmentcost ψ 100 w Payoutrateofretainedearnings ϕ 0.01 47

D Application A. Log PCE B. Percent Deviation of PCE from Trend 10.75 Log PCE 2 Loglinear Trend 10.70 0 10.65 2 4 10.60 6 10.55 8 10 10.50 2010 2012 2014 2016 2018 2020 2022 2024 2010 2012 2014 2016 2018 2020 2022 2024 FigureD.1: PersonalConsumptionExpenditure Notes: Panel A shows log per-capita PCE and its loglinear trend estimated over the 2010-2019 period. Panel B showsthepercentdeviationofper-capitaPCEfromtrend. Theverticallineindicatestheendoftheperiodusedto estimatethetrend. 12 10 8 6 4 2 0 2020 2021 2022 2023 2024 2025 tuptuO launnA fo tnecreP Total PE Transfers Discount Factor FigureD.2: ContributionstoExcessSavingsinPartialEquilibrium Notes:Thefigureshowsthepartial-equilibriumcomponentsofexcesssavingsbasedonequation(46).Thevertical linemarksthethirdquarterof2021,theendofthebuildupperiod. 48