Non-Linear Employment Effects of Tax Policy

Board of Governors of the Federal Reserve System International Finance Discussion Papers ISSN 1073-2500 (Print) ISSN 2767-4509 (Online) Number 1333 December 2021 Non-Linear Employment Effects of Tax Policy Domenico Ferraro and Giuseppe Fiori Please cite this paper as: Ferraro, Domenico and Giuseppe Fiori (2021). “Non-Linear Employment Effects of Tax Policy,” International Finance Discussion Papers 1333. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/IFDP.2021.1333. NOTE: International Finance Discussion Papers (IFDPs) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the International Finance Discussion Papers Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at www.ssrn.com.

∗ Non-linear Employment Effects of Tax Policy Domenico Ferraro Giuseppe Fiori November 16, 2021 Abstract We study the non-linear propagation mechanism of tax policy in a heterogeneousagent equilibrium business cycle model with search frictions in the labor market and an extensive margin of employment adjustment. The model exhibits endogenous job destruction and endogenous hiring standards in the form of occasionally-binding zero-surplus constraints. After parameterizing the model using U.S. data, we find that the dynamic response of employment to a temporary change in the labor income tax is highly non-linear, displaying sizable asymmetries and state-dependence. Notably, the response to a tax rate cut is at least twice as large in a recession as in an expansion. JEL Classification: E12, E24, E32, E62 Keywords: Search frictions, job destruction, heterogeneity, aggregation, tax policy ∗Ferraro: Department of Economics, W. P. Carey School of Business, Arizona State University, PO Box 879801, Tempe, AZ85287-9801, UnitedStates(e-mail: domenico.ferraro@asu.edu). Fiori: BoardofGovernorsoftheFederalReserveSystem,DivisionofInternationalFinance,20thandConstitutionAvenueN.W., Washington,DC20551,UnitedStates(e-mail: giuseppe.fiori@frb.gov). Thefirstversionofthispaperwas presented on November 3, 2017, at the Junior Macroeconomics Conference hosted by CASEE at Arizona StateUniversity,withthetitle“TheCountercyclicalTaxMultiplier.” WethankBartHobijn,NirJaimovich, ValerieRamey,RichardRogerson,andAys¸egülS¸ahinaswellasconferenceandseminarparticipantsatthe Junior Macroeconomics Conference at Arizona State University and North Carolina State University, for valuablecommentsandsuggestions. Allerrorsareourown.

1 Introduction This paper studies the non-linear propagation mechanism of discretionary tax policy in the context of a heterogeneous-agent equilibrium business cycle model featuring search frictions in the labor market, and an extensive margin of employment adjustment, that operates through endogenous job destruction and hiring. We ask three questions related totheeffectsofshockstoflat-ratelabortaxesontheaggregateemploymentrate: (i) Istheeffectofataxratehikelargerthantheeffectofataxratecut? (ii) Does the marginal effect of a tax rate change depend on the size of the tax rate increaseordecrease? (iii) Istheeffectofataxratecutlargerinarecessionthaninanexpansion? Using a quantitative version of our model, we find that the answer to these questions is “yes.” Overall, the interaction of search frictions and worker heterogeneity in skills produces significant non-linearities in the propagation mechanism of tax policy to the aggregate employment rate. Three main results stand out. First, we find that a cut in tax rates increases the employment rate by less than a tax rate hike reduces it, and that the largerthesizeoftheshock,thelargerthesignasymmetrybetweennegativeandpositive taxshocks. Second,themarginaleffectofataxshockisdecreasinginthesizeoftheshock. Third, the effects of tax changes are state dependent. Notably, the effect of a tax rate cut isatleasttwiceaslargeinarecessionasinanexpansion. Ourresultshaveimportantimplicationsforanumberoffiscalpolicyissues,including, butnotlimitedto,countercyclicaltaxpolicy: totheextentthatthemarginaleffectofatax rate cut is decreasing in its size, tax policy is less effective as a countercyclical policy instrument than the estimates based on linear structural vector autoregressions (SVARs) wouldimply. Our theoretical analysis builds on two premises. First, a typical U.S. recession is “Lshaped,”asopposedto“V-shaped,”featuringasharpdropintheaggregateemployment rate, followed by a recovery phase in which the employment rate slowly reverts back to its pre-recession level. These patterns clearly point to the presence of significant nonlinearities in aggregate dynamics (McKay and Reis, 2008; McQueen and Thorley, 1993; Morley and Piger, 2012; Neftçi, 1984; Sichel, 1993). The Great Recession of 2008-09 and 1

the contraction led by the spread of the new coronavirus (COVID-19) in early 2020 are the two most recent episodes of this phenomenon. We view endogenous job destruction as a key feature of the propagation mechanism of macroeconomic shocks, including tax shocks. Second,congestioneffectsduetorandomsearchinthelabormarket,exacerbatedbya disproportionate inflow of workers in the unemployment pool at the onset of recessions, can induce convexity in hiring costs and thereby a high degree of curvature in the cost of producing output. This curvature leads to interesting non-linearities, implying that the responsiveness of the economy to discretionary tax policy may considerably vary over thebusinesscycle,anditsefficacydependsonthemagnitudeofthetaxratechangeitself. Prominentfeaturesofourmodelare(i)searchfrictions,whichgiverisetoequilibrium unemployment, as in the Diamond-Mortensen-Pissarides (DMP) framework (Diamond, 1982; Mortensen, 1982; Pissarides, 1985), and (ii) worker heterogeneity in productivity or skills,whichdeliversanextensivemarginofemploymentadjustment.1 Taxratesimpinge onequilibriumallocationsviatwochannels. First,theyaltertherelativereturnofmarket to non-market activity, as in the standard model of labor supply with home production. The second channel operates through the effective bargaining power of the worker. That is, the higher the tax rate on labor income, the lower the share of the surplus generated byamatchaccruingtotheworker. The model features endogenous separation and hiring in the form of occasionally binding zero-surplus constraints: given a value for the tax rate and a level of aggregate productivity, a zero-surplus constraint defines a cutoff on worker productivity such that existing matches with workers whose productivity is below the cutoff are endogenously destroyed. Analogously, meetings between employers posting vacancies and workers whose productivity is below the cutoff are not converted into jobs. In this sense, the model exhibits endogenous hiring standards in that who gets hired depends on tax rates aswellasaggregateproductivity. Giventhesefeatures,themodelembedstwomechanismswhoseinteractioncanyield highly non-linear effects of tax policy. First, absent search frictions, the model collapses to a frictionless setting with indivisible labor and heterogeneous workers. A reservation wage rule determines the extent to which available labor services are fully used or left 1Recent empirical work points to the importance of worker heterogeneity for aggregate labor-market dynamics(AhnandHamilton,2019;BarnichonandFigura,2015). 2

idle. More specifically, if the prevailing wage is above the reservation wage, the individual is employed and producing output—otherwise, he or she is unemployed. This mechanism generates V-shaped responses to tax rate hikes: sharp contractions followed by quick recoveries. Second, search frictions, the extent of which varies over the business cycle, impede the instantaneous creation of jobs, leading to gradual recoveries. This mechanisminducesL-shaped-typeresponsestotaxratechanges. The dynamic response of the aggregate employment rate to changes in tax rates is drivenbytwoforces. First,theextentofsearchfrictions,asmeasuredbythemarkettightness ratio, varies in response to tax shocks. Specifically, the tightness ratio, and therefore theprobabilitythatanunemployedworkerbumpsintoajobvacancy,fallsinresponseto ataxratehike. Further,becausethejob-meetingprobabilityisconcaveinthetightnessratio,itdropsmoreinresponsetoataxratehikethanitrisesinresponsetoanequally-sized taxratecut. Notealsothatthestimulative,positiveeffectofataxratecutonemployment isdampedbythepresenceofsearchfrictions. Intheabsenceoffrictions,theemployment ratejumpstoitsnewsteady-statelevelinsofarasthetaxratecutislargeenoughtomake workerswillingtosupplylabor. Second,ataxratehikemakesthezero-surplusconstraintbind,implyinganimmediate adjustment in employment through endogenous job destruction. Further, the larger the tax rate increase, the larger the fraction of workers hitting the zero-surplus constraint and the more important is the active margin of job destruction. This mechanism is a key driver of state-dependence in tax policy. Specifically, during recessions, a larger fraction oflow-productivityworkersisatthemargin,soevenasmalltaxratecutcaninducelarge responsesintheaggregateemploymentrate. To quantify these mechanisms, we parametrize the model using U.S. data, including impulse response functions (IRFs) from proxy SVARs. Along with other data moments, the model reproduces the “peak” response of the U.S. employment rate (one minus the unemployment rate) to a narratively identified shock to the average marginal tax rate (AMTR).Inthemodel,consistentlywiththeIRFfromproxy-SVARs,a1percentagepoint (pp), temporary reduction in the tax rate leads to about a 0.65 pp increase in the employmentrate.2 2FollowingthevastliteratureonSVARs,hereweconfrontthemodelwiththeevidenceonunanticipated AMTRshocks. SeeMertensandRavn(2012)forempiricalevidenceontheeffectsofanticipatedtaxpolicy shocksintheUnitedStates. 3

We use the quantitative model to gauge the extent of non-linearities in tax policy: non-linearitiesarepervasive,rangingfromsignificantsignandsizeasymmetriestostatedependence. It is worth emphasizing that such non-linear responses arise from shocks whosemagnitudesarecomparablewiththoseobservedhistoricallyintheUnitedStates. First, sign asymmetry is sizable. In response to a 1.5 pp tax rate cut, the employment rate rises by nearly 0.45 pp, whereas in response to an equally sized tax rate hike, the employment rate falls by 0.8 pp. Such asymmetry increases with the size of the shock: in response to a 3 pp tax rate change, the drop in the employment rate is nearly three times aslargeasthepeakresponsetoataxratecut. A tax rate hike induces endogenous job destruction on impact, followed by a period of depressed vacancy posting. So the pool of unemployment rises, and at the same time, the probability of bumping into a job vacancy and finding a job falls. In this sense, tax rate hikes exacerbate the extent of search frictions. In response to a tax rate cut, instead, the adjustment brings about an increase in vacancy posting, leading to a sluggish and hump-shapedincreaseintheemploymentrate. Second,sizeasymmetryisalsoquantitativelyimportant. Forexample,atthepeak,the employment rate increase in response to a 3 pp tax rate cut is approximately one-third of two times the increase in the employment rate in response to a 1.5 pp tax rate cut. In this sense, the marginal effect of a tax rate cut decreases sharply with the size of the tax cut. While size asymmetry is present for tax rate increases, too, it is much less pronounced relativetotaxratecuts. Third, the stimulative effect on the aggregate employment rate of a tax rate cut is at least twice as large in a recession as in an expansion. In the model, during expansions, unemployment is of the frictional type. This implies that tax rate changes work through changes in the extent of frictions, as measured by the speed at which an unemployed workerbumpsintoajobopportunity. Ataxratecutcausesanincreaseinvacancyposting thatinturnleadstoahigherprobabilityoffindingajob. However,duringanexpansion, vacancy posting is already high and the labor market is tight, so the increase in vacancy postinginducedbythetaxcutissmall. During recessions, instead, tax rate changes operate through changes in the extent of frictions, as well as changes in the fraction of zero-surplus workers. A tax rate cut that is large enough can then make a large fraction of unemployed workers viable for hiring. In 4

addition, vacancy posting, too, is more responsive to tax rate cuts during recessions than expansions. The interaction of these two mechanisms also implies that the extent of state dependence rises with the depth of the recession. Notably, the deeper the recession, the moreeffectivetaxratereductionsareinstimulatingaggregateemployment. The rest of the paper is organized as follows. In Section 2, we discuss the related literature. In Sections 3 and 4, we present the model and illustrate its main qualitative properties. In Sections 5 and 6, we parametrize the model and study some of its basic quantitative properties. In Section 7, we show non-linear IRFs to gauge the extent of non-linearities in the propagation mechanism of tax shocks. Finally, Section 8 concludes. AppendixAcontainsdetailsondatasources,variables’construction,andestimation. 2 Related Literature Thispapercontributestotheimportantandgrowingliteraturestudyingnon-linearitiesin thepropagationmechanismoffiscalpolicy. Empirically,astrandofthisliteratureaimsat identifying(i)thecausaleffectsofshockstogovernmentspendingandtaxratesonmacro variables and (ii) the extent to which the magnitude of these effects depends on the sign and size of the fiscal policy shock as well as on the state of the business cycle. Given the limited number of time-series observations on aggregate macro variables, and the high frequency of countercyclical fiscal policy changes, it is well known that the identification of the effects of fiscal policy remains a challenge in linear econometric models, let alone in the context of non-linear models. Perhaps not surprisingly, then, there is no consensus abouttheextentofasymmetryandstatedependenceinthetransmissionoffiscalpolicy. Early empirical studies find that government spending multipliers are significantly larger in recessions than in expansions (see, e.g., Auerbach and Gorodnichenko, 2012, 2013;BachmannandSims,2012;Fazzari,MorleyandPanovska,2015).3 Usingahistorical dataset of U.S. government spending, Ramey and Zubairy (2018) find that the spending multiplier is acyclical, thus calling into question the early empirical findings. Recently, Barnichon, Debortoli and Matthes (2019) find that the sign of the change in government spendingmattersforthesizeofthemultiplier. Notably,areductioningovernmentspending has contractionary effects on economic activity with a multiplier above one, which 3A“multiplier”isdefinedasthechangeinoutputcausedbya$1changeingovernmentpurchases. 5

are largest during recessions, whereas an increase in government spending is associated with a multiplier below one, which is virtually the same in recessions and in expansions. These findings establish the presence of both asymmetric and state-dependent effects of contractionaryfiscalshocks. Onthetheoryside,thereisrelativelylittlework. Michaillat(2014)proposesatheoryof a countercyclical government spending multiplier, using the “rationing unemployment” framework developed in Michaillat (2012). In Michaillat’s model, job rationing emerges in equilibrium because firms face decreasing returns to labor and wages are rigid: after a negative technology shock, the marginal product of labor falls, but rigid wages adjust downward only partially. Thus, if the adverse shock is sufficiently large, the marginal productfallsbelowthewagesothatitisunprofitableforfirmstohireworkersregardless ofthecostofpostingajobvacancy. Pizzinelli, Theodoridis and Zanetti (2020) show that a DMP model with endogenous job separation and on-the-job search replicates the empirical findings from a threshold vector autoregression model that the unemployment rate, job-separation rate, and jobfinding rate exhibit a larger response to productivity shocks during periods when aggregate productivity is low. Ghassibe and Zanetti (2020) study the implications of search frictions in the product market for state-dependence in fiscal policy. They find that the magnitude and the variation of the fiscal multiplier over the business cycle depend on the source of economic fluctuations—that is, on whether the economy is hit by demand or supply shocks. Finally, Fernández-Villaverde et al. (2020) develop a DMP model with searchcomplementaritiesintheinter-firmmatchingprocessinwhichanequilibriumwith high output and low unemployment coexists with another equilibrium with low output and high unemployment. In their model, fiscal policy is considerably more potent in the badequilibriumcharacterizedbylowoutputandhighunemployment. Wecontributetothetheoreticalstrandoftheliteratureintwoways. First,wepropose anewmechanismforthenon-lineareffectsoffiscalpolicy,basedonworkerheterogeneity and search frictions. More specifically, we stress the role of endogenous job destruction, aswellasendogenoushiringstandards,intheformofoccasionallybindingzero-surplus constraints. The resulting extensive margin of employment adjustment interacts with search frictions, magnifying the extent of non-linearities. To the best of our knowledge, theimportanceofanactiveextensivemarginsubjecttofrictionshasnotbeenunderscored bytheliterature. 6

Second, we analyze asymmetry and state dependence in a unified framework, which encompassesdynamicresponsestotaxratechangesofdifferentshapesdependingonthe relative importance of search frictions vis-à-vis the composition of the unemployment pool. Importantly, we discipline our quantitative model with narratively identified IRFs so that the model reproduces the peak response of the employment rate to an identified taxrateshock. 3 Model Inthissection,wepresenttheheterogeneous-agentequilibriumbusinesscyclemodelthat welaterusetostudythenon-linearpropagationmechanismoftaxshocks. Themodelretains two main ingredients of the framework developed by Ferraro (2018): (i) worker heterogeneity in market productivity and (ii) search frictions in the labor market. We beginbypresentingtheeconomicenvironment: preferences,budgetconstraints,technology, market structure, and the stochastic process for tax shocks. We then set up the value functions,discusswagedetermination,and,finally,turntotheequilibriumofthemodel. 3.1 Environment Time is discrete and continues forever, indexed by t = 0,1,..., ∞ . The model economy is populated by a continuum of measure one of workers as well as by a continuum of employers or firms whose mass is determined in free-entry equilibrium. Workers and employers are infinitely-lived and risk-neutral, and they discount future values at the same rate 0 < β < 1. Workers are endowed with an indivisible unit of time per period, which is supplied as labor or used for job search. We adopt a two-state representation of the labor market such that workers are either employed and producing output, or unemployed and searching for a job. Employers are either matched with a worker, and soproducingoutput,orpostingjobvacanciestohireunemployedworkers. Heterogeneity Workersareheterogeneousintermsoftheirmarketproductivity,which, for lack of a better term, we refer to as “skills.” There are M ≥ 2 skill types, indexed by x ∈ X = {x ,x ,...,x }, where x < x < ... < x . We view the skill type x as 1 2 M 1 2 M aninnatecharacteristicoftheworker,whichisknownbyemployerswithnouncertainty. 7

Themassofworkersoftypexisdenotedby f(x),andthelaborforceisnormalizedtoone, i.e. ∑ f(x) = 1. Identical employers have access to the same production technology, x∈X withlaborservicesastheonlyinput. Preferencesandbudgetconstraints Workers’preferencesoverconsumptionc ≥ 0and t timespentatwork n ∈ {0,1} aredescribedby t ∞ E ∑ βt(c −Bn ), (1) 0 t t t=0 where E is the expectation operator conditional on information available at t = 0 and 0 B ≥ 0 is the utility cost of working. Workers’ budget constraint is c ≤ (1−τ)w n + t t t t (1−n )Γ (w ),where(1−τ)w istheafter-taxwage,0 ≤ τ < 1isaflat-ratetaxonlabor t t t t t t income, and Γ(w ) ≥ 0 denotes government transfers to unemployed workers, such as t unemploymentinsurance(UI)benefits. Technology Productionrequiresamatchbetweenoneemployerandoneworker. When a match (or, equivalently, a “job”) is created, output y is produced with a linear production function, y = Ax, where A denotes aggregate productivity, which we later use to parametrizethestateofthebusinesscycle,and x isthelevelofskillsasdefinedearlier. To capture the presence of search frictions, we assume that the number of meetings between unemployed workers and job vacancies is determined by a constant-returns-toscale(CRS)meetingtechnology. Specifically,let θ ≡ v/u denotetheratioofjobvacancies to unemployed workers, then an unemployed worker meets a job vacancy with probability p(θ) : R + → [0,1]. We further assume that p(θ) is strictly increasing and twice continuouslydifferentiable,with p(0) = 0and p(∞) = 1. Conversely,theprobabilitythat avacancymeetsanunemployedworkeris q(θ) = p(θ)/θ,with q(0) = 1and q(∞) = 0. Market structure In the spirit of the “directed search” framework of Moen (1997), we consider a market structure featuring segmentation by skill types so that the aggregate labor market consists of a collection of M submarkets indexed by x. Specifically, workers areassignedtodifferentsubmarkets,andemployerscanpostvacanciesinonesubmarket at a time, directing their search by choosing the submarket with the highest expected payoff. In each submarket, then, the rate at which unemployed workers meet employers 8

depends on the tightness ratio θ(x) ≡ v(x)/u(x), where v(x) and u(x) are job vacancies andunemployedworkersinsubmarket x,respectively. Beyond its theoretical appeal, this formulation of the market structure also captures a salientfeatureoftherecruitingprocess. Tostreamlinetheapplicationprocess,jobvacancy postings typically include job qualifications as well as job requirements, specifying the skills,experience,andotherattributesthattheemployerisseekingtofindintheapplicant who is hired for the job. Hence, in the model, as to a large extent in actual labor markets, thesearchprocessis,infact,directedthroughvacancyposting. Exogenous stochastic process for tax rates The flat-rate tax on labor income follows a first-orderautoregression(AR(1))processinlogs: log(τ ) = (1−ρ)log(τ¯)+ρlog(τ)+σ(cid:101) , (2) t+1 t t+1 whereτ¯ istheunconditionalmeanofthetaxrateandtheparameters0 < ρ < 1andσ > 0 iid govern the persistence of the shock and the volatility of the innovations (cid:101) ∼ N(0,1), t respectively. Tobesure,intheUnitedStates,labortaxrateshavechangedovertimeforanumberof reasons,including,butnotlimitedto,theresponseofthegovernmenttocurrentbusiness cycle conditions (see Romer and Romer, 2010, for a narrative account of postwar U.S. tax policy). Here, instead, we model the stochastic process for tax rates to capture the “exogenous” component of tax policy, akin to the structural shock approach in SVARs. Specifically, an innovation to the tax rate meets three requirements: (i) it is unpredictable given current and past information; (ii) it is uncorrelated with other potential structural shocks hitting the economy; and (iii) it is unanticipated—namely, it is not news about futurechangesintaxrates. Timeline Withinaperiod,eventsunfoldasfollows. Atthebeginningoftheperiod,the aggregate shock is realized. After this realization, the period has two stages. First, jobseparation, job-creation, and search decisions are made simultaneously. Second, at the productionstage,outputisproducedandwagesarepaid. In the setting here, there is a separation between the time at which a meeting occurs and the formation of a match. During the period, job search and job vacancies determine 9

meetings between workers and employers. At the beginning of the next period, bilateral Nash bargaining takes place, and, if profitable, a meeting is converted into a job. This formulationofthehiringprocessisconsistentwiththe“non-sequential”searchparadigm (seeVanOursandRidder,1992;AbbringandVanOurs,1994;OmmerenandRusso,2014; DavisanddelaParra,2017).4 3.2 Value Functions Weformulatetheproblemofworkersandemployersinrecursiveform. Notably,wewrite the Bellman equations at the production stage, when the current decisions of continuing, destroying, or creating a match have already been made. All of the information needed for optimal decision making is the realization of the tax rate τ, which is the state variable ofthedynamicprogram. Henceforth, we omit time subscripts and use primes to denote next-period variables. Also, abusing notation slightly, we denote the probability that an unemployed worker meets a job vacancy by p(x) ≡ p(θ(τ;x)), and the probability that a job vacancy meets anunemployedby q(x) ≡ q(θ(τ;x)). Employer Bellman equations At the production stage, the value of a job in submarket x satisfiestheBellmanequation (cid:20) (cid:26) (cid:27)(cid:21) J(τ;x) = Ax−w(x)+β E max (cid:0) 1−d (cid:48)(x) (cid:1) J(τ (cid:48) ,x)+d (cid:48)(x)maxV(τ (cid:48) ;x˜) , (3) d(cid:48)(x) x˜ where Axistheoutputofamatch;w(x)isthewagerate,whosedeterminationwediscuss later;and x˜ indicatestheemployer’schoiceaboutwheretopostajobvacancy. Theemployer’sdecisiontodestroythematchissubsumedintheindicatorfunction (cid:40) δ if J(τ;x) > max V(τ;x˜) x˜ d(x) = (4) 1 if J(τ;x) ≤ max V(τ;x˜), x˜ 4SeeStigler(1961), Gal, LandsbergerandLevykson(1981), Morgan(1983), andMorganandManning (1985) for early theoretical work on non-sequential search and Gautier (2002), Wolthoff (2014), and Albrecht,GautierandVroman(2006)forrecentworkontheefficiencypropertiesofenvironmentswithnonsequentialsearch. 10

where0 < δ < 1isanexogenousrateofjobdestruction. Notethatthemaxoperatorin(4) capturestheemployer’sdecisiontopostajobvacancyinthesubmarketwiththehighest expectedvalue. Inthissense,searchisdirected. At the search stage, the value of a job vacancy posted in submarket x satisfies the Bellmanequation (cid:20) (cid:26) (cid:27) V(τ;x) = −k+β E q(x)max h (cid:48)(x)J(τ (cid:48) ;x)+ (cid:0) 1−h (cid:48)(x) (cid:1) maxV(τ (cid:48) ;x˜) h(cid:48)(x) x˜ (cid:21) +(1−q(x))maxV(τ (cid:48) ;x˜) , (5) x˜ wherek ≥ 0istheunitcostofpostingandmaintainingajobvacancy,andtheemployer’s job-creationdecisionissubsumedintheindicatorfunction (cid:40) 1 if J(τ;x) > max V(τ;x˜) x˜ h(x) = (6) 0 if J(τ;x) ≤ max V(τ;x˜). x˜ Worker Bellman equations At the production stage, the value of being employed in submarket x satisfiestheBellmanequation (cid:20) (cid:26) (cid:27)(cid:21) W(τ;x) = (1−τ)w(x)−B+β E max (cid:0) 1−s (cid:48)(x) (cid:1) W(τ (cid:48) ;x)+s (cid:48)(x)U(τ (cid:48) ;x) , (7) s(cid:48)(x) where the worker’s instantaneous return from working (1−τ)w(x)− B is the after-tax wagenetofthedisutilityofwork. Theworker’sjob-separationdecisionissubsumedintheindicatorfunction (cid:40) δ if W(τ;x) > U(τ;x) s(x) = (8) 1 if W(τ;x) ≤ U(τ;x). At the search stage, the value of being unemployed in submarket x satisfies the Bell- 11

manequation (cid:20) (cid:26) (cid:27) U(τ;x) = Γ(w)+β E p(x)max a (cid:48)(x)W(τ (cid:48) ;x)+ (cid:0) 1−a (cid:48)(x) (cid:1) U(τ (cid:48) ;x) a(cid:48)(x) (cid:21) +(1− p(x))U(τ (cid:48) ;x) , (9) where Γ(w) is government transfers, and the worker’s job acceptance decision is subsumedintheindicatorfunction (cid:40) 1 if W(τ;x) > U(τ;x) a(x) = (10) 0 if W(τ;x) ≤ U(τ;x). 3.3 Equilibrium Asisstandardintheliterature,weconsiderafree-entryequilibriuminwhichthevalueof postingajobvacancyequalszeroinallsubmarketsatalltimes. Thisequilibriumconcept, coupled with a directed search process, greatly simplifies the computation of the model, as the Bellman equations, and thereby the total match surplus, depend solely on the tax shock τ,whichcapturesalloftherelevantinformationforoptimaldecisions. Asaresult, individual agents’ decision rules and tightness ratios do not depend on the distribution ofworkers overwages andunemployment acrosssubmarkets. The literaturerefers toan equilibrium with such properties as a Block Recursive Equilibrium (BRE) (see Shi, 2009; MenzioandShi,2010a,b,2011).5 In addition, as in the standard DMP model, the equilibrium within each submarket is block-recursive,too. Thatis,wecansolvefortheequilibriumtightnessratioanddecision rulesindependentlyofthestocksofemploymentorunemployment. Giventhesedecision rules and tightness ratio, we then compute the evolution of the stocks of employment by calculatingtheflowsofhiresandseparations. Tosummarize,theequilibriumineachsubmarketischaracterizedbythreeequations: (i) a Bellman equation for the total match surplus generated by the match; (ii) a free- 5ABREgivesaconsiderableadvantageoveranequilibriumwithrandomsearch. Toseethis,consider analternativemarketstructureinwhichmarketsarenotsegmentedsothatindividualswithdifferentlevels of skills search in the same market. In such an environment, under Nash bargaining, vacancy-posting firms would need to keep track of the distribution of unemployed workers over skills. In a free-entry equilibrium, this implies that job-finding probabilities become a function of the distribution of skills, an infinitelydimensionalobjectthatcannotbecharacterizedanalytically. 12

entry condition for vacancy posting; and (iii) a wage equation that implicitly determines the split of the surplus between workers and employers. We solve for the equilibrium match surplus and tightness in each submarket separately, and then aggregate across submarkets. 3.3.1 WageSetting The wage is determined via bilateral Nash bargaining. When a worker and an employer meet, they bargain over how to split the surplus generated by the match. As is standard in the literature, we assume that bargaining resumes every period so that workers in old and new matches receive the same wage. Specifically, in each submarket, the wage is determined to maximize the weighted product of the net match surplus accruing to the worker,W(τ;x)−U(τ;x),andthenetsurplusaccruingtothefirm, J(τ;x),i.e., w(x) = argmax[W(τ;x)−U(τ;x)]ηJ(τ;x)1−η, (11) where 0 ≤ η ≤ 1 is a parameter reflecting the worker “bargaining power.” In the case in whichthebargainingsetisempty,weset w(x) = 0. The solution to the Nash-bargaining problem specified in (11) satisfies a modified sharingrulethattakesintoaccountthepresenceoftheflat-ratetaxonlaborincome: η(1−τ)J(τ;x) = (1−η)[W(τ;x)−U(τ;x)]. (12) This equation specifies that shifts in the labor tax rate act like changes in the effective bargaining power of the worker. Specifically, the higher the tax rate, the smaller is the shareofthetotalmatchsurplusthatgoestotheworker. To see this clearly, let S(τ;x) ≡ [W(τ;x)−U(τ;x)]/(1−τ)+ J(τ;x) denote the total match surplus generated by a match. Using the sharing rule (12), we obtain (after some algebra) that the value of a job and the net value of working are proportional to the total matchsurplus: J(τ;x) = (1−η)S(τ;x), (13) W(τ;x)−U(τ;x) = η(1−τ)S(τ;x). (14) 13

Note that equations (13) and (14) imply “bilateral efficiency.” That is, employers and workersalwaysagreeonthedecisiontodestroyanexistingmatch,ortocreateanewone, implying that d(x) = s(x) and h(x) = a(x) for all possible realizations of the tax shock. In this sense, changes in tax rates do not induce inefficient separations. Yet, tax shocks will affect equilibrium outcomes by changing the total surplus generated by the match, aswellasthesplitofthissurplusbetweenworkersandemployers. 3.3.2 ThreeEquationsthatDeterminetheEquilibrium We now turn to describing the three equations that determine the equilibrium of the model. Total match surplus At the production stage, the total match surplus generated by a matchinsubmarket x satisfiestheBellmanequation S(τ;x) = Ax− B+Γ(w(x)) + ηβ [1−δ− p(θ(x))]E(cid:2) (1−τ (cid:48))max (cid:8) S(τ (cid:48) ;x),0 (cid:9)(cid:3) 1−τ 1−τ +β(1−δ)(1−η)E(cid:2) max (cid:8) S(τ (cid:48) ;x),0 (cid:9)(cid:3) . (15) First, worker productivity Ax net of the tax-adjusted opportunity cost of employment, B+Γ(w(x))/(1−τ)—henceforth, “net productivity”—enters total match surplus as the instantaneous return to the match such that shocks to τ act like exogenous shifts in the relative return to non-market work. Everything else being equal, the higher the tax rate, the higher is the effective opportunity cost of employment or the lower is the relative returntomarketwork. Second,ashocktothecurrenttaxrateτ changestheconditionalexpectationofthetax (cid:48) ratenextperiod τ . Suchachangeinexpectationsmattersforcurrentdecisionsinsofaras a higher (lower) tax rate next period (i) reduces (raises) the expected match surplus, and (ii)lowers(raises)theexpectedshareofthetotalsurplusaccruingtotheworker. This latter effect interacts with the level of slack in the labor market. The value of the market tightness ratio θ(x) affects, through the job-meeting probability p(θ(x)), the rate atwhichtheindividualworkerandemployerdiscounttheexpectedmatchsurplus. This effect captures the option value of job search that the worker forgoes by accepting a job 14

offer. Theexistenceofthisoptionvaluereducesthetotalmatchsurplustotheextentthat the worker has some bargaining power, i.e. η > 0. Specifically, the lower the tightness ratio, the more important is the expectation of future tax rates for current decisions to destroyorcreateamatch. Free-entry condition Employers post job vacancies up to the point where the expected costequalstheexpectedbenefitofopeningandmaintainingajobvacancy: k ≥ β(1−η)q(θ(x))E(cid:2) max (cid:8) S(τ (cid:48) ;x),0 (cid:9)(cid:3) , (16) withequalityinthecaseofaninteriorsolutionforjobvacancies. Thisfree-entrycondition determines the tightness ratio in submarket x independently of the other submarkets. In this setting, the distribution of unemployed workers over skills across submarkets plays noroleinthedeterminationofjobvacancies. (Attheaggregatelevel,theskilldistribution ofunemploymentremainsakeydeterminantoftotalhires.) Persistentshockstotaxrates affect tightness and so the degree of slack in each submarket, through changes in the expected surplus, on the right-hand side of the free-entry condition (16). The higher the tax rate next period, the lower the expected surplus, which mandates a lower tightness ratiointhecurrentperiod.6 Nash-bargainedwages AsinthestandardDMPmodel,ifgovernmenttransfersdonot depend on wages—i.e., Γ(w(x)) = 0—we can compute the equilibrium of the model by solvingtheBellmanequationforthetotalmatchsurplus(15)andthefree-entrycondition (16). If,instead, Γ(w(x)) > 0,asinourcase,thenawageequationhastobeincludedand jointlysolvedwiththesurplusequationandthefree-entrycondition. Usingthesharingrule(13)andtheequationforthevalueofajob(3),weobtain(after somealgebra)aforward-lookingwageequation: w(x) = Ax−(1−η)S(τ;x)+β(1−η)(1−δ)E(cid:2) max (cid:8) S(τ (cid:48) ;x),0 (cid:9)(cid:3) . (17) Next, using the free-entry condition (16) and the Bellman equation for the total match 6Notethataslongastheexpectedsurplusontheright-handsideofthefree-entryconditionispositive, theequilibriumfeaturesaninteriorsolutionforvacancyposting. 15

surplus(15),weadjustequation(17)sothatitbecomes (1−δ)k w(x) = Ax+ −(1−η)S(τ;x). (18) q(θ(x)) 3.3.3 IndividualDecisionRulesandAggregation In each submarket, the joint employer-worker decision of creating a job or destroying an existing one satisfies a cutoff rule. The zero-surplus constraint S(τ¯;x) = 0 yields cutoffs on tax rates τ¯(x) that vary by skills such that if τ ≥ τ¯(x) a match with a worker of skill level x is destroyed, or, similarly, a meeting between a vacancy and an unemployed of skill level x is not converted into a job. Notably, the higher the skill level, the higher the cutoff on the tax rate that makes workers and employers indifferent between continuing ordestroyinganexistingemploymentrelationship,i.e. τ¯(x ) < ... < τ¯(x ). 1 M Individual decision rules More formally, in each submarket, endogenous job creation satisfiesthedecisionrule (cid:40) 1 if S(τ;x) > 0 h(τ;x) = a(τ;x) = (19) 0 if S(τ;x) = 0. Equivalently,endogenousjobdestructionsatisfiesthedecisionrule (cid:40) δ if S(τ;x) > 0 d(τ;x) = s(τ;x) = (20) 1 if S(τ;x) = 0. Aggregation Ineachsubmarket,theemploymentrateis e (cid:48)(x) = e(x)+h (cid:48)(x)p(θ(x))u(x)−d (cid:48)(x)e(x). (21) Summing across submarkets, we find that the aggregate employment rate evolves Φ(cid:48) ∆(cid:48) overtimebasedonthetotalflowsofhires, U,andseparations, E,suchthat E (cid:48) = E+Φ(cid:48) U−∆(cid:48) E, (22) 16

where E = ∑ f(x)e(x), and the aggregate job-finding rate ( Φ ) and job-separation rate x ∆ ( )nextperiodarecalculatedas Φ(cid:48) = (1/U) ∑ h (cid:48)(x)p(θ(x))h(x)u(x), (23) x ∆(cid:48) = (1/E) ∑ d (cid:48)(x)h(x)e(x). (24) x 4 Inspecting the Propagation Mechanism Before proceeding to the quantitative part of the paper, here, to build intuition, we study the qualitative properties of the model. We begin by studying the deterministic steadystate equilibrium of the model, in which the tax rate is constant over time, and the flows intounemployment(separations)equaltheflowsoutofunemployment(hires). Then,we turntoperfectforesighttransitiondynamicstoprovideinsightintotheeconomy’sdynamic responsetoshocks. Inthemodel,twomechanismsarerelevantfortaxpolicy: (i) Search frictions. At the individual level, job-meeting probabilities driven by the market tightness ratios determine the speed at which unemployed individuals find job opportunities. In this sense, the faster an unemployed individual bumps into a vacancy, the lesser is the extent of search frictions impinging on equilibrium allocations. (ii) Extensive margin of employment adjustment. At the individual level, the zerosurplus constraint determines whether a worker-employer meeting is converted into a job or an existing job is destroyed. At the aggregate level, this mechanism determines the mass of “marginal” or “zero surplus” workers, a key determinant of thesensitivityoftheeconomytotaxshocks. Tosummarize, thedynamicresponseof theaggregateemployment ratetochangesin tax rates is driven by two mechanisms. First, the extent of search frictions, as measured bythetightnessratio,variesinresponsetotaxshocks. Notably,thetightnessratioineach submarket, and so the probability that an unemployed worker meets a job vacancy, falls in response to a tax rate hike. Further, because the job-meeting probability is concave in 17

the tightness ratio, it drops more in response to a tax rate hike than to an equally sized taxratecut. Second,shiftsintaxratesleadtooccasionally-bindingzero-surplusconstraints,which is a key source of non-linearity in the propagation mechanism of tax shocks. Indeed, the magnitude and persistence of tax shocks hitting the economy impinge on the frequency at which the zero-surplus constraints bind. For example, a tax rate hike can make the zero-surplusconstraintbind,implyinganimmediateadjustmentinemploymentthrough endogenousjobdestruction. Further,thelargerthetaxrateincrease,thelargerthefraction ofsubmarketshittingthezero-surplusconstraintandthemoreimportantistheextensive marginofemploymentadjustment. 4.1 Deterministic Steady State Frictional equilibrium From equation (21), the steady-state employment rate in submarket x is h(x)p(θ(x)) e(x) = . (25) h(x)p(θ(x))+d(x) If the steady-state total match surplus is positive, then h(x) = 1 and d(x) = δ. In this case, e(x) = p(θ(x))/(p(θ(x)) + δ), as in the standard DMP model. Otherwise, if the steady-statesurplusiszero, h(x) = 0and d(x) = 1,suchthat e(x) = 0. Setting Γ(w(x)) = 0 to simplify calculations, we specify that equations (15)-(17) yield the steady-state equations determining the total match surplus (26), tightness ratio (27), andwagerate(28),respectively: Ax−B/(1−τ) S(x) = , (26) 1−β[1−δ−ηp(θ(x))] k ≥ β(1−η)q(θ(x))S(x), (27) B w(x) = (1−η) +η[Ax+kθ(x)]. (28) 1−τ The zero-surplus constraint S(x) = 0 gives the cutoff on the tax rate τ¯ = 1− B/Ax such that for τ ≥ τ¯ the match with a worker of skill level x is not viable. Again, the taxratecutoffisincreasingintheskilllevel,implyingthathigherskilledworkersremain 18

viable for hiring for higher values of the tax rate. (In the stochastic economy, this implies that matches with higher skilled workers are less likely to be destroyed.) Alternatively, one can think in terms of a cutoff on the skill level x¯ = B/(1−τ)A, which is increasing inthetaxrate,suchthatworkersofskilllevel x ≤ x¯ arenotviableforemployment. Inthesteadystatewithsearchfrictions,theaggregateemploymentrateis ∑ ∑ f(x)p(θ(x)) E = f(x)e(x) = , (29) p(θ(x))+δ x>x¯ x>x¯ wheretheextensivemarginofemploymentadjustmentiscapturedbytheconditionx > x¯ in the summation over submarkets, which determines the fraction of the labor force that isviableforemployment. Frictionlessequilibrium Inthefrictionlesslimit,theeconomycollapsestoacollectionof spotlabormarketswithindivisiblelaborandskillheterogeneity. Ask (cid:38) 0,lim θ(x) = k(cid:38)0 ∞ , such that, abusing notation slightly, p(∞) = 1 and q(∞) = 0. In this frictionless economy,thewageinsubmarketxequalsthemarginalproductoflabor,suchthatw(x) = Ax. A reservation wage rule then determines whether an individual is at work or idle. Notably, the reservation wage is w¯ = B/(1−τ) such that if w(x) > w¯, workers of skill level x areemployed;otherwise,if w(x) ≤ w¯,theyareunemployedornonemployed. Finally,ifweset δ = 0,theaggregateemploymentrateinthefrictionlesseconomyis ∑ E = f(x) = 1−F(x¯), (30) x>x¯ where F(x¯) isthecumulativedistributionfunctionofworkerskillsevaluatedatthezerosurpluscutoff x¯. 4.2 Convex Hiring Costs Using the surplus equation (26), the free-entry condition (27), at the interior solution for vacancy posting, and the time discount rate r ≡ 1/β−1, we obtain (after some algebra) arelationbetweennetproductivityandthetightnessratio,whichdeterminesthesteady- 19

stateequilibrium: (cid:20) (cid:21) B 1 k Ax− = (r+δ) +ηkθ(x) . (31) 1−τ 1−η q(θ(x)) On theleft-hand side, net productivity, which isequal to the marginalproduct of labor Ax minus the opportunity cost of employment B/(1−τ), represents the instantaneous return to a match or the relative return to market versus non-market activity. Permanent shifts in tax rates directly affect this margin by impinging on work incentives for a given tightnessratio. Thetwotermsontheright-handsidecapturetheeffectofthetightnessratioonhiring costs,akeyequilibriumrelationshipduetothepresenceofsearchfrictions. Thefirstterm, (r+δ)k/q(θ(x)),isthediscountedvalueoftheexpectedcostofpostingandmaintaininga vacancy. Thistermisincreasingandconcaveinmarkettightness: thehigherthetightness ratio, the lower the probability that a vacancy meets an unemployed worker and the higherthedurationofanunfilledvacancy. Thesecondterm,ηkθ(x),capturesthepositive effect that a tight labor market exerts on the wage. The higher the tightness ratio, the higher the probability an unemployed meets a vacancy—a mechanism that strengthens theworker’sbargainingposition,whichputsupwardpressureonwages. Importantly, note that the steady-state employment rate implied by (25) is increasing and concave in tightness, which yields a convex relationship between the employment rate and hiring costs. In this sense, search frictions can induce a highdegree of curvature inhiringcostsandtherebyinthecostofproducingoutput. 4.2.1 RoleofSearchFrictions To isolate the role of search frictions from the extensive margin of employment adjustment, here we focus on the case of positive match surplus, in which workers’ skill levels are above the skill cutoff—i.e., x > x¯—for all values of the tax rate. Figure 1 depicts the steady-stateequilibriumimpliedbyequation(31),beforeandafterataxrateincrease,for high-skilled workers generating “large surplus” (panels A and B), and for low-skilled or “smallsurplus”workers(panelsCandD). Panel A shows the steady-state equilibrium for high-skilled workers determined as the intersection of the net productivity curve (NPC), the left-hand side of equation (31), 20

andthehiringcostcurve(HCC),theright-handsideofequation(31). PanelBshowstwo different equilibria: (i) the initial one depicted in panel A; and (ii) the new equilibrium that results from an increase in the tax rate. The HCC remains unchanged, whereas the NPC shifts downward, implying a lower employment rate. Similarly, panels C and D illustrate the two equilibria before and after an equally sized tax rate increase for lowskilled,small-surplusworkers. Twomaintheoreticalinsightsemerge. First, for large-surplus workers, the effect of a tax rate change is generally small and approximately linear. Second, for small-surplus workers, instead, a tax rate increase of the same size has a bigger negative effect on the employment rate. Further, the effects of tax rate changes on small-surplus workers are potentially asymmetric. Because of the curvature in HCC, the effect of a tax rate increase can be considerably larger than that of a tax rate cut. Hence, the overall effect of a tax rate hike or cut on the aggregate employment rate critically depends on the share of low-skilled, small-surplus workers in the unemployment pool. As the composition of the unemployment pool varies over the businesscycle,taxratechangescanhavestate-dependenteffects. 4.2.2 RoleofZero-SurplusConstraint As in Figure 1, panels A and B of Figure 2 depict the equilibrium before and after the increase in the tax rate for high-skilled workers whose surplus remains positive in spite of the tax rate hike. For these workers, the drop in the employment rate induced by the tax rate hike comes entirely from search frictions, as in the DMP framework. A tax rate increasedepressesvacancyposting,whichinturnlowerstheprobabilityoffindingajob. Panels C and D of Figure 2 illustrate the effect of the tax rate increase for low-skilled workers, whose match surplus hits the zero-surplus constraint. In this latter case, the employment rate drops to zero. Importantly, the larger the size of the tax rate increase, the larger the share of workers hitting the zero-surplus constraint. This mechanism is an important driver of state-dependence in tax policy. To see this, think of a scenario where the economy is in a recessionary state in which aggregate productivity A < A¯ is below normal—say, A¯. In such a recession, NPC has shifted downward (as in the case of a tax rate increase), such that the zero-surplus constraint could be binding for a large share of workers. In this situation, a tax rate cut has a larger effect than it would have otherwise hadifaggregateproductivityhadbeenabovenormal. 21

Notealsothatthestimulative,positiveeffectofataxratecutonemploymentisdamped bythepresenceofsearchfrictions. Intheabsenceoffrictions,thesubmarketwouldjump back up to full employment insofar as the tax rate cut is large enough to make workers viableforhiringagain. 4.3 Transition Dynamics Theequilibriumofthemodelfeaturestransitiondynamicsofdifferentshapesdepending on the importance of search frictions vis-à-vis the extensive margin of employment. In the frictionless case, there is no internal propagation of shocks. For example, in response toataxrateincreasefollowedbyanequallysizeddecrease,theemploymentratefallson impact and then instantaneously jumps back up to its initial level before the shock. By contrast, in the presence of search frictions and occasionally binding zero-surplus constraints, transition dynamics are sluggish. In general, employment rate responses to tax rateshockscantakedifferentshapesinbetweentheV-andL-shapedcases. Before proceeding, we note that in the stochastic version of the model, the transition dynamicsoftheemploymentrateinsubmarket x aregovernedbyastochasticdifference equation: e (cid:48)(x) = (cid:2) 1−h(τ (cid:48) ;x)p(θ(τ;x))−d(τ (cid:48) ;x) (cid:3) e(x)+h(τ (cid:48) ;x)p(θ(τ;x)). (32) In Section 7, we leverage the full stochastic structure of the model—i.e., model solution and the stochastic difference equation (32) for each submarket—to simulate non-linear IRFs. Here,however,toprovideanalyticalinsight,weconsiderthecaseofperfectforesight transitiondynamicswithpermanentandtemporarytaxrateshocks. Permanent tax rate shocks For the case of permanent shocks, think of the following scenario. At time t = 0, submarket x rests at the steady-state employment rate e (x) ≥ 0 0 associated with the tax rate τ . Unexpectedly, a permanent tax rate shock realizes. After 0 the shock, the agents expect the tax rate to be τ ∗ (cid:54)= τ for all t ≥ 1. Note that since 0 the tightness ratio is a “jump” variable, and the indicator variables for endogenous job creation and destruction depend solely on the value of the tax rate, they jump to their newsteady-statevaluesonimpactandremainconstantthereafter. 22

Temporarytaxrateshocks Forthecaseoftemporaryshocks,weconsiderasequenceof “MIT shocks.” Again, think of submarket x at the steady state with e (x) ≥ 0. Unexpect- 0 edly,ashockrealizessuchthatagentsexpectthetaxratetobeτ ∗ (cid:54)= τ fort ≥ 1. However, 0 after a number of periods, unexpectedly, at time t = T ≥ 2, a new shock realizes, such thatthetaxratetakesonanewvalue—say,τ —whichisexpectedtolastforeverfort ≥ T. 0 In a nutshell, the agents in the model perceive every shock as permanent, akin to a randomwalk. However,differentlyfromarandomwalk,theagentsalsoexpectinnovations to the tax rate to be zero at all times. In this sense, in the case of MIT shocks, agents are systematically“surprised”bychangesintaxrates. 4.3.1 SearchFrictionsversusExtensiveMarginofEmploymentAdjustment Inthecaseofperfectforesight,theemploymentrateineachsubmarketevolvesovertime accordingtoadeterministicdifferenceequation: e (x) = [1−h(τ ∗ ;x)p(θ(τ ∗ ;x))−d(τ ∗ ;x)]e (x)+h(τ ∗ ;x)p(θ(τ ∗ ;x)). (33) t+1 t Thesolutionto(33)is e (x) = e ∗(x)+λt(τ ∗ ;x)[e (x)−e ∗(x)], (34) t 0 where e ∗(x) is the new steady-state level of the employment rate associated with τ ∗ and λ(τ ∗ ;x) governsthespeedoftransitiontothenewsteadystate: λ(τ ∗ ;x) ≡ [1−h(τ ∗ ;x)p(θ(τ ∗ ;x))−d(τ ∗ ;x)] ≤ 1. (35) As is evident from (35), the speed of adjustment to the shock, λ(τ ∗ ;x), depends on the ∗ specificvalueofthenewtaxrate τ anditvariesacrosssubmarkets. Two polar cases Two cases are of special interest. First, absent labor-market frictions (i.e.,δ = 0and p(∞) = 1),λ(τ ∗ ;x) = 0. Thisimpliesthattheemploymentratebecomesa jumpvariable: itfallsorrisesonimpacttoitsnewsteady-statelevel,implyingnointernal propagationoftaxshocks. Second, in the presence of frictions, the model displays a fundamental asymmetry in 23

thedynamicadjustmenttotaxratehikesvis-à-vistaxratecuts. Specifically,ifthenewtax rate is larger than the tax rate cutoff for zero surplus (τ ∗ ≥ τ¯), then λ(τ ∗ ;x) = 0, so the employment rate drops on impact to the new lower level, as in the frictionless case. By contrast,inthecaseofataxratecut(withτ ∗ < τ¯),thenλ(τ ∗ ;x) = [1− p(θ(τ ∗ ;x))−δ] ∈ (0,1),implyingaslowadjustmenttowardahigherleveloftheemploymentrate. Figure 3 illustrates the variety of responses to tax rate changes implied by the model. In sum, the sign, as well as the size, of tax rate changes and the mix of frictionless, as opposed to frictional, adjustments to shocks allow for substantial non-linearities and a richsetofIRFshapes. 4.3.2 AggregateDynamics At the aggregate level, the speed of adjustment of the economy to tax rate shocks is a complexequilibriumobject. Specifically,theeconomy-widecounterpartof(34)is E = E ∗ + ∑ λt(τ ∗ ;x)f(x) (cid:20) e 0 (x)−e ∗(x) (cid:21) (E −E ∗), (36) t E −E∗ 0 x 0 (cid:124) (cid:123)(cid:122) (cid:125) Λ t where E ≡ ∑ f(x)e (x) and E ∗ ≡ ∑ f(x)e ∗(x) are the initial and the new steady- 0 x 0 x state levels of the aggregate employment rate. Thus, the aggregate speed of adjustment Λ depends on (i) heterogeneous speeds of adjustment across submarket λ(τ ∗ ;x), (ii) t different initial employment rates across submarkets e (x), and (iii) the new steady-state 0 employmentrates e ∗(x) thatvaryacrosssubmarkets. 5 Bringing the Model to the Data In this section, we specify the functional form for the meeting technology and discuss parameter values. The parametrization of the model consists of three steps. First, we estimate the parameters governing the stochastic process for tax shocks. Second, to ease comparison with previous work, we calibrate a subset of parameters based on common values in the literature. Third, we estimate the remaining parameters to match a select number of empirical targets, including the peak response of the employment rate to a 1 percentagepointcutintheaveragemarginaltaxrate(AMTR),estimatedinthecontextof 24

proxy SVARs.7 See Appendix A for data sources, variables’ construction, and estimation details. 5.1 Functional Form for the Meeting Technology WeconsideraCRSmeetingfunction,thatdeterminesthenumberofmeetings,m,interms ofunemployedworkersseekingjobs,u,andjobvacancies,v. BasedonDenHaan,Ramey andWatson(2000),weadoptthefollowingmeetingfunction: vu m(v,u) = , (37) (vξ +uξ)1/ξ where ξ > 0isaparametertobecalibrated. With this specification, the probability that an unemployed worker meets a job vacancy is p(θ) = m(v,u)/u = (cid:0) 1+θ −ξ (cid:1)−1/ξ , and the probability that a job vacancy meets an unemployed worker is q(θ) = m(v,u)/v = (cid:0) 1+θξ (cid:1)−1/ξ . The appealing property of specification (37) over a Cobb-Douglas is that meeting probabilities are guaranteed to be between 0 and 1. Further, the elasticity of the job-meeting probability with respect to the tightnessratiois1/(1+θξ). DifferentlyfromaCobb-Douglasspecification,thiselasticity isnotconstant;rather,itisdecreasingintightness. 5.2 Parametrization We now turn to the calibration of the parameters for preferences, technology, and the stochasticprocessforthetaxrate. Asisstandardinadynamicgeneralequilibriummodel like ours, none of the parameters have a one-to-one relationship to a moment. Nonetheless,itisusefultodescribethecalibrationprocedureinafewdistinctsteps. Tosummarize,themodelhas14parameters(τ¯,ρ,σ,β,γ,δ,η,B,k,ξ,x ,x ,µ ,and min max x σ ). A model period is taken to be a “month.” Table 1 reports the baseline parameter x values. Next, we discuss in detail key steps of the parametrization: data sources, interpretationofmodelparameters,choiceofdatamoments,andestimation. 7TheAMTRistheweightedaverageofthemarginaltaxratesfacedbyindividualtaxpayers. Weights areconstructedasindividualincomedividedbytotalincomeinagivenyear.Thenotionofincomeincludes wages,self-employment,partnership,andS-corporationincome.SeeBarroandSahasakul(1983)andBarro andRedlick(2011)foradditionaldetails. 25

5.2.1 StochasticProcessforTaxShocks Parametrization of the stochastic process for tax shocks is an important element of our calibration. OurapproachhereistochoosetheparametersoftheAR(1)processforthetax rate (τ¯,ρ,σ) such that the tax rates in the model reproduce salient properties of AMTRs intheUnitedStates. Wedosointwosteps. First, we use annual data on AMTRs constructed by Mertens and Montiel Olea (2018) for the period from 1946 to 2012. Then, we estimate an AR(1) process by regressing the AMTRonaconstantanditslaggedvalue: AMTR = 0.0183+0.9407·AMTR +εAMTR, (38) t+1 t t+1 where t = 1946,...,2012 and εAMTR is a residual error term. Our estimates imply a residt+1 ual standard deviation of 0.0131 and a long-run mean τ¯ of 0.0183/(1−0.9407) = 0.3086. Next, using standard conversion formulas, we find that an autoregressive coefficient of 0.9407 at the annual frequency corresponds to 0.94071/12 = 0.9949 at the monthly fre- (cid:113) (cid:14) quency,andthatanannualstandarddeviationof0.0131correspondsto0.0131 ∑12 0.94072(j−1) = j=1 0.005atthemonthlyfrequency. Second,wediscretizetheAR(1)processfortaxrateshocks(2)usingtheRouwenhorst method.8 Specifically, we use 51 grid points and set the standard deviation σ to 0.005, consistentwiththemonthlystandarddeviationcalculatedearlier. Settingthepersistence parameter ρ to 0.9949 produces, however, an IRF of the tax rate that is too persistent comparedwiththatestimatedfromaSVAR.Toovercomethisissue,weselectthevalueof ρsuchthattheIRFofthetaxratetoa1percentagepointshockinthemodelreplicatesthe IRFestimatedusingnarrativelyidentifiedshockstoAMTRsinFerraroandFiori(2020). This IRF matching approach is based on an iterative procedure that consists of five steps: (i)giventhevalueofσ = 0.005,guessavalueforρ—say,0.9949—andapproximate the stochastic process for tax rates with the Rouwenhorst method; (ii) simulate artificial time series for tax rates using the Markov chain approximation of the AR(1) process; (iii) fitanAR(1)processtotheartificial“monthly”data;(iv)usetheestimatedautoregressive coefficient to compute an IRF up to four years after the shock; and (v) iterate until the simulatedIRFisclosetotheestimatedIRF.Thisproceduregives ρ = 0.98. 8KopeckyandSuen(2010)showthattheRouwenhorstmethodismoreaccuratethantheotheravailable methodsinapproximatinghighlypersistentprocesses. 26

5.2.2 ExogenouslySetParameters We set the time discount factor β to (1/1.04)1/12 = 0.9967 such that the model generates an annual real interest rate of 4%, a value that is commonly used in the business cycle literature(see,e.g.,McGrattanandPrescott,2003;Gomme,RavikumarandRupert,2011). WeassumethatUIbenefits Γ(w) = γwrepresent40%ofwagesasinShimer(2005),sowe setthereplacementrate γ to0.4. Next, we turn to three parameters related to labor market frictions (δ,η,k). We set the exogenous job-separation rate δ to 0.02, consistent with the average of the monthly quits rate in the U.S. nonfarm business sector for 2000:M12 through 2020:M1. We set the worker bargaining weight η to 0.5. In the median state—i.e., τ = τ¯ = 0.31—the worker’s share of surplus is η(1−τ¯) = 0.5(1−0.31) = 0.345, which lies between the value of 0.5 in Mortensen and Pissarides (1994) and Hall and Milgrom (2008) and of 0.19 implied by thecalibrationinHagedornandManovskii(2008). Finally, we allow for the parameter governing the unit vacancy cost k to vary across submarkets. Similarly to the calibration in Shimer (2005), we set k such that the tightness ratioineachsubmarketequalsonewhenthetaxrateisatthemedianstate. Thisapproach yields that the unit cost of posting a vacancy is increasing in skills.9 And that the ratio of the unit vacancy cost to skills is increasing and concave in the level of skills. Figure 4 shows the distribution of skills, and the relation between vacancy costs and skills in the calibrated model. Notably, for the highest-skilled workers, vacancy costs represent 35% of the monthly output of a job, whereas for the lowest-skilled workers in the labor force, unit vacancy costs are a trivial fraction of output. In this sense, search frictions are relativelymoreimportantforthehigh-thanthelow-skilledworkers. 5.2.3 JointlyEstimatedParameters Given the exogenously set parameters, and the parametrized stochastic process for the tax rate, we are left to determine seven parameter values (B,k,ξ,x ,x ,µ ,andσ ). min max x x Here, we begin by discussing key model statistics, and then we turn to the choice of data 9This is broadly consistent with the empirical evidence discussed in Hamermesh and Pfann (1996, p. 1268): “Theaveragecostof[employment]adjustmentrisesveryrapidlywiththeskilloftheworker. Thus while external costs may be very low in jobs filled by high-turnover, low-skilled workers, they are very largeforhigh-skilledjobsthatareusuallyoccupiedbylong-tenureworkers.” 27

momentsandestimation. Opportunitycostofemployment ThedisutilityofworkBisakeyparametergoverning willingness to work. At the individual level, the flow opportunity cost of moving from unemployment to employment consists of forgone UI benefits plus the forgone value of leisure. Attheaggregatelevel,theaverageflowopportunitycostofemploymentis 1 ∑ z ≡ [B+γw(x)]u(x), (39) U x where u(x) is the mass of unemployed in submarket x and U = ∑ u(x) is aggregate x unemployment. Similarly,theaggregatewagerateiscalculatedas 1 ∑ w = w(x)e(x), (40) E x wheree(x)isthemassofemployedworkersinsubmarket xandE = ∑ e(x)isaggregate x employment. Theratioofztotheafter-taxwage(1−τ)wisakintoa“replacementratio,” whichcapturestherelativereturnofnon-markettomarketwork. Skilldistribution Workerskills x ∈ {x ,x ,...,x }aredistributedaccordingtoaLog- 1 2 M Normal (µ , σ2), where the parameters −∞ < µ < +∞ and σ > 0 govern the scale and x x x x the shape of the distribution, respectively.10 To proceed, we discretize the support of this distributionwitha200-pointapproximation,wheretheskilltypesareevenlyspacedover a grid ranging from the lowest type x to the highest type x . We normalize µ = 0, min max x set x = 10, and calibrate x and σ jointly with the utility parameter B and the max min x matchingfunctionparameterξ usingasimulatedmethod-of-momentsprocedure,whose implementationwedescribelater. Choiceofdatamoments Todeterminetheremainingfourparameters(B,ξ,x ,andσ ), min x we estimate their values such that model-implied moments match their corresponding datamoments: 1. averageunemploymentrate(5.5%), 10Thisassumptionisinformedbytheevidenceonthelog-normalityofempiricalwagedistributions(see, e.g.,Moscarini,2005;Pizzinelli,TheodoridisandZanetti,2018). 28

2. averagejob-findingrate(40%), 3. averageopportunitycostofemploymentrelativetoafter-taxwages(0.7),and 4. peak response of the aggregate employment rate to a 1 pp cut in the AMTR (0.65 pp). Afewremarksaboutthechoiceofthedatamomentsjustdescribedareinorder. First, the two targets for the average unemployment rate (5.5%) and job-finding rate (40%) are valuescommonlyusedintheliterature. Dependingondifferentdatasourcesandestimation techniques, available estimates for the average job-finding rate in the United States rangefrom35%to45%(seeShimer,2005,2012;FujitaandRamey,2009). Herewechoose themidpointoftheavailablerangeofestimates. Second, there is an ongoing discussion in the literature about the calibrated value of the opportunity cost of employment, or the flow value of non-market work. In the standard DMP model, the value of non-market work is a key parameter determining the elasticity of the tightness ratio to productivity as well as to tax shocks. For productivity shocks, Shimer (2005) shows that a standard DMP model, calibrated to a 40% UI replacement rate, fails to generate a realistic elasticity of the unemployment rate with respect totheobservedmovementsinlaborproductivity. HagedornandManovskii(2008)argue thattheflowvalueofnon-marketworkincludesnotonlyUIbenefits,butalsotheforgone values of home production and leisure, and that a calibration based on a replacement ratioashighas90%improvesthemodel’sabilitytoyieldfluctuationsintheunemployment rateofamagnitudecomparablewiththedata. Here,wechoosetotargetanintermediate valueof0.7whichisconsistentwiththeestimateofHallandMilgrom(2008). Third, a peak employment rate response of 0.65 percentage points is within the range of available empirical estimates in the literature. Specifically, while the point estimate is taken from Ferraro and Fiori (2020), the magnitude of such peak response is consistent withexistingempiricalstudies—e.g.,MertensandMontielOlea(2018). Indeed,estimates based on narratively-identified tax shocks have stood the test of time, by consistently producingIRFsofsimilarmagnitude(seeRamey,2016,forasurveyoftheliterature). Simulated method of moments Because of non-linearities, the ergodic distribution of the stochastic economy is not centered around the deterministic steady state. As a result, the average values of endogenous variables differ from their corresponding steady-state 29

values implied in a version of the model with perfect foresight. Moreover, the larger the extentofnon-linearities,thelargerthisdiscrepancy. Implementing our method-of-moments approach requires, then, an iterative procedure. Givenaninitialguessforparametervalues,wesolvetheequilibriumofthestochastic model, compute the theoretical means of the endogenous variables using the ergodic probability distribution for the tax rate, and compare those theoretical means with the corresponding sample averages. This procedure gives B = 0.1843, ξ = 0.89, x = 0.64, min and σ = 0.15. x 5.3 Linear IRFs: Model versus Data Figure5showslinearIRFstoa1percentagepointcutinthetaxrateestimatedonartificial data generated from the model (dotted line with triangles) and from actual data in a proxy-SVARframework(solidlinewithcircles),uptofouryearsaftertheshock. In the model, as in the data, tax rate shocks are persistent with a half-life of roughly three years. In fact, the persistence of tax shocks is an important factor determining forward-lookingbehaviorinthemodel,suchasvacancypostingthroughentrydecisions, as well as job destruction and job acceptance decisions. Overall, the model does a good jobofaccountingfortheempiricalIRFtothenarratively-identifiedtaxrateshock. While, as per our calibration strategy, the model matches the peak response of the employment rate,itgeneratesanIRFthatexhibitsmorepersistencerelativetotheempiricalIRF. Since the IRFs in Figure 5 are linear IRFs, they are mirrors image of the IRFs to an equally sized tax rate increase. Further, as a byproduct of the linearity assumption, the magnitude of the responses—as measured by, say, the peak or trough, as well as the shape—isthesameinrecessionsandexpansions. Tobesure,linearIRFsarenottypically viewed as an exact representation of macroeconomic variables, however, a long tradition intheliteraturetakesthemasa“goodapproximation”ofaggregatedynamics. Inthatview, linear IRFs are interpreted as weighted averages of the responses to positive and negative shocks and to small and big shocks, as well as of the responses in recessions and expansions. InSections6and7,weshowthattheinteractionofsearchfrictionswiththeextensive margin of employment adjustment induces a high degree of non-linearity: a linear IRF 30

would mask quantitatively sizable differences in the aggregate response of the economy to tax shocks based on the state of the business cycle and on the sign and size of the shock. Forexample,theaverageresponsetoashockisnotinformativeabouttheexpected responseoftheeconomytoataxpolicychangeduringarecession. 6 Basic Properties of the Calibrated Economy In this section, we examine basic properties of the calibrated economy, relating tax rates to key aggregate variables, such as the unemployment rate, output, and job-finding and job-separationrates. 6.1 Stochastic Steady State Weconsiderthestochasticsteadystateofthemodel,definedastheequilibriuminwhich endogenous variables remain constant in the presence of expected future shocks, when theinnovationstotheseshocksturnouttobezero. In Figure 6, panels A through D plot the aggregate unemployment rate, output, and job-finding and job-separation rates against the tax rate.11 The unemployment rate is highly non-linear and convex in the tax rate. As a result of this convexity, the equilibrium of the model features positive skewness in the unemployment rate, so the mean of the unemployment rate is above the median. Specifically, in our calibrated economy, the medianunemploymentrateis4.4%,asopposedtoameanof5.5%. Job-finding and job-separation rates are, respectively, concave and convex in the tax rate. Again, tax rates can affect job-finding rates through changes in tightness ratios, and thereby in the extent of search frictions, and changes in the fraction of marginal or zerosurplus workers that are viable for hiring. In the calibrated economy, median and mean job-finding rates are 0.46 and 0.4, respectively. Similarly, the median job-separation rate is2.2%,anditsmeanis2.6%,whichisconsistentwiththeestimateinShimer(2005). Aggregate output is highly non-linear and concave in the tax rate. For a large range of 11The values of these endogenous variables are constructed assuming that each submarket rests at its ownsteadystateforagivenvalueofthetaxrate,andthatsuchtaxratewhileexpectedtochangeovertime, takesonthesamevalueonrealizationforever. 31

tax rates, output is indeed flat. However, for tax rates above 35%, aggregate output falls sharply. Inthissense,themodelheredisplaysapropertysimilartothe“pluckingmodel” offluctuationsproposedbyMiltonFriedman(Friedman,1993). Thatis,outputfeaturesa ceiling,punctuatedbysharpcontractionsinducedbysizableincreasesintaxrates. 6.2 Long-Run Behavior of the Economy Figures 7 and 8 show (i) the ergodic distribution of the unemployment rate and of the tax rate computed by using 1 million monthly periods simulated from the model and (ii) the relation between the unemployment rate and the tax rate implied by the ergodic distributionofthemodel. Two main results stand out. First, in panel A of Figure 7, the ergodic distribution of theunemploymentrateisskewedtotheright. Notably,theunemploymentratefluctuates most of the time in the range of 4% to 6%, with infrequent spikes above the 10% level. In panel B of the same figure, the distribution of tax rates is instead symmetric around the mean. Thisimpliesthattheskewnessinthedistributionofunemploymentratesisdueto the internal propagation mechanism of tax rate changes, not to the stochastic process of taxshocks. Second, Figure 8 shows that the relation between the unemployment rate and the tax rate from the model’s ergodic distribution is convex, which confirms the convexity of the unemployment rate in the stochastic steady state shown in panel A of Figure 6. A new result emerges, though: the level of the tax rate determines not only the average unemployment rate, but also the variability of unemployment rates. Notably, the higher the tax rate, the larger the responsiveness of the unemployment rate to tax rate changes. Thus, the history of past realizations of tax rate shocks, too, is a factor determining the dynamic response to a current tax shock. This is yet another form of non-linearity, which cannotbetakenintoaccountinthecontextoflinearIRFs. 7 Non-linearities in the Propagation of Tax Shocks To quantify the non-linearities in the propagation mechanism of tax shocks embodied in the model, we rely on non-linear IRFs. We focus on tax shocks whose magnitudes 32

are comparable with those observed historically in the United States. In Section 7.1, we brieflydiscussissuesrelatedtothekeypropertiesandcomputationofnon-linearIRFs. In Sections7.2and7.3wereportresultsforasymmetryandstatedependence. 7.1 Non-Linear IRFs PropertiesofIRFs Incontrasttothemorecommonlyused,linearIRFs,non-linearIRFs aregenerallynotinvarianttothesignandthesizeofshocks,nortotherealizedsequence ofpastshocks. Asaresult,onecannotinfertheshapeoftheIRFtoanegativeshockfrom thattoapositiveshockbysimplyflippingthesignoftheresponse,norcanonethinkofan IRF to a small shock as a scaled-down version of the response to a big shock. In fact, the magnitude of the marginal effect of a given shock, as well as its dynamic implications, critically depends on whether the shock is positive or negative, whether it is large or small,and whetheritwaspreceded byahistoryof positiveornegativerealizations. This createsawell-knownreportingproblem,whichwetacklebyproducinganumberofIRFs underalternativescenarios. Computation of IRFs In the model, shocks to the tax rate are symmetric, so the asymmetry in outcomes is solely the result of the internal propagation mechanism at play. Also,thetaxrateistheonlystatevariableofthemodel. Thisimpliesthatgeneratingtime series from the model requires simulating sequences of tax rates and then using model solutions for individual decision rules and tightness ratios to compute the equilibrium dynamicsoftheemploymentrateasimpliedbythesimulatedsequencesoftaxrates. We compute IRFs by simulating the equilibrium paths of two alternative economies, whichwerefertoas“benchmark”and“counterfactual.” Forthebenchmarkeconomy,we simulate 50,000 paths for, say, the employment rate over 60 model periods (“months”). The sequences of tax shocks are simulated as follows. At t = 0, the initial tax rate across all simulations rests at the median state, i.e., τ = τ¯. From t = 0 onward, tax rates 0 are simulated according to the Markov chain implied by the Rouwenhorst method. This procedure yields time series of realized tax rates (cid:8) τi (cid:9)60 , where i = 1,2,...,50,000 and t t=0 τi = τ¯ for all replications. Associated with these sequences of tax rates, the model gen- 0 erates 50,000 time series of employment rates (cid:8) ei (cid:9)60 . For the counterfactual economy, t t=0 we implement the same steps, but with a different initial tax rate τ = τ¯ ±∆ , where ∆ 0 τ τ 33

parametrizes the size of the initial shock to the tax rate. The average difference between thesimulatedpathsofthebenchmarkeconomyandthoseofthecounterfactualeconomy istheIRFtoa ±∆ taxrateshock. Toquantifytheextentofstatedependence,wecompareIRFsacrossdifferentstatesof the business cycle, as captured by the level of aggregate productivity A. To achieve this goal, we keep the same parameter values as in the baseline calibration, and re-compute the equilibrium of the model with A = A¯ ± ∆ , where ±∆ captures the state of the A A business cycle, relative to a benchmark economy in which aggregate productivity is normalized to A = A¯ ≡ 1. For example, to compare the IRF to a −∆ tax rate cut in “good” τ versus“bad”times,wecomputetheIRFtoataxcutinaneconomywith A = 1+∆ and A theIRFtoanequallysizedtaxcutinadepressedeconomyinwhichaggregateproductivity A = 1−∆ is ∆ percentbelownormal. A A 7.2 Asymmetry Signasymmetry Figure9showsthenon-linearIRFssimulatedfromthemodeltoa0.75 percentage points tax rate cut (solid line) and to an equally-sized tax rate hike (dotted line). The response of the employment rate to symmetric shocks is asymmetric. Notably, while both responses are hump-shaped, the trough of the IRF to a tax rate hike is larger thanthepeakresponsetotheequally-sizedtaxratecut. Figures10and11showtheIRFsto1.5and3percentagepointstaxshocks,respectively. In a nutshell, the larger the size of the tax shock, the larger the asymmetry between the response to a tax rate hike and that to a tax rate cut. Next, compare the non-linear IRFs to a tax rate cut in Figures 9-11 with the linear IRF in Figure 5. According to the linear IRF estimated on artificial data generated from the model, at the peak of the response, the employment rate is approximately 0.65 percentage points above the long-run mean. According to the non-linear IRFs simulated from the model, instead, tax rate cuts have much smaller effects on the employment rate. Specifically, a peak response close to 0.65 percentage points can come only from a tax cut of 3, as opposed to 1, percentage points. In sum, the marginal effects implied by the linear IRF in Figure 5 are somewhat similar to those implied by the non-linear IRF to a tax rate hike but are significantly larger than thoseimpliedbythenon-linearIRFtoataxratecut. 34

Size asymmetry Figure 12 shows the IRF to a 3 percentage points tax rate cut (solid line) and two times the IRF to a 1.5 percentage points tax rate cut (dotted line). Similarly, Figure 13 shows IRFs to tax rate hikes. In the context of linear IRFs, the two responses equal each other. Here, instead, IRFs do not scale with the size of the tax shock. Notably, the marginal effect of a tax rate change at each time horizon falls sharply in the size of theshock. Thus,notonlythesign,butalsothesizeofthetaxshockmattersfortheshape of the employment rate response. In a nutshell, the larger the size of the tax rate cut, the lessertheeffectivenessoftaxcutsatincreasingtheemploymentrate. 7.3 State Dependence Inevaluatingtheextentofstatedependenceinthepropagationoftaxshocks,weconsider two scenarios in which aggregate productivity is 4% below (bad times) and 4% above (good times) the level of productivity in normal times A¯ = 1; thus Agood = 1+0.04 and Abad = 1−0.04. Figure 14 shows the IRFs to a small tax rate cut in bad times (solid line) and in good times (dotted line). The difference in the magnitude of the responses is sizable. Notably, the increase in the employment rate in bad times is twice as large as that in good times. Importantly,thisdifferenceintheresponsivenessoftheemploymentratetoataxratecut becomeslargerasweincreasethesizeofthetaxratecut,asshownbyFigure15. In good times, all unemployment is frictional. Thus, changes in tax rates impinge on equilibriumallocationssolelythroughchangesintheextentofsearchfrictionsinthelabor market. Specifically, a temporary tax rate cut induces an increase in vacancy posting and thereby a higher probability that an unemployed worker meets a posted vacancy. Since in good times all meetings generate a positive surplus, a lower tax rate leads to a higher job-finding probability and so an expansion in employment and output. In bad times, instead,theeconomyfeaturesamixoffrictionalandzero-surplusunemployment. Inthis scenario, a tax rate cut boosts vacancy posting, as in good times; further, it makes zerosurplus workers viable for hiring. Importantly, the deeper the recession, the larger the fractionofzero-surplusworkers,andthelargertheexpansionaryeffectofataxcut. 35

8 Conclusion We study the propagation mechanism of tax policy in the context of a heterogeneousagent equilibrium business cycle model with search frictions in the labor market and an active extensive margin of employment adjustment. The model is estimated to match a select number of data moments for the U.S. economy, including the peak response of the aggregateemploymentratetoaonepercentagepointcutintheaveragemarginaltaxrate, asestimatedinthecontextofproxy-SVARsbasedonnarratively-identifiedtaxshocks. In the absence of search frictions, the model exhibits a propagation mechanism of tax rate changes that is “approximately linear” such that linear IRFs would accurately capture aggregate equilibrium dynamics. However, in the presence of realistic search frictions and occasionally binding zero-surplus constraints, we find that the response of the aggregateemploymentratetoataxratechangeishighlynon-linear,displayingasymmetry in the sign and size of the tax rate shock, as well as state dependence. Three main results stand out. First, the response to a tax rate cut is considerably smaller than that to anequallysizedtaxratehike. Second,themarginaleffectofataxratecutdecreaseswith thesizeofthetaxcut. Third,theresponseoftheemploymentratetoataxratecutismuch largerinarecessionthaninanexpansion. Overall, the results in this paper suggest that heterogeneity in the composition of the unemployment pool, which varies over the business cycle, and the interaction of search frictions with an extensive margin of employment adjustment can produce significant non-linearities in the propagation mechanism of tax policy. Accounting for these nonlinearitiesseemsparamountfortheestimationoftheeffectsofcountercyclicalfiscalpolicy as well as for the design of automatic stabilizers. While of great importance, these issues havebeenleftforfutureresearch. 36

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0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.7 0.75 0.8 0.85 0.9 0.95 1 Employment rate stsoc gnirih ,ytivitcudorp teN A. Steady-State Equilibrium: Large Surplus 0.8 0.7 0.6 0.5 Hiring costs Net productivity 0.4 Equilibrium 0.3 0.2 0.1 0 0.7 0.75 0.8 0.85 0.9 0.95 1 Employment rate stsoc gnirih ,ytivitcudorp teN B. Steady-State Equilibrium: Large Surplus Hiring costs Old net productivity New net productivity: Tax hike Old equilibrium New equilibrium: Tax hike 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.7 0.75 0.8 0.85 0.9 0.95 1 Employment rate stsoc gnirih ,ytivitcudorp teN C. Steady-State Equilibrium: Small Surplus 0.8 0.7 0.6 0.5 Net productivity Hiring costs 0.4 Equilibrium 0.3 0.2 0.1 0 0.7 0.75 0.8 0.85 0.9 0.95 1 Employment rate stsoc gnirih ,ytivitcudorp teN D. Steady-State Equilibrium: Small Surplus Old net productivity New net productivity: Tax hike Hiring costs Old equilibrium New equilibrium: Tax hike Figure1: RoleofSearchFrictions Notes:Thefigureillustratesthesteady-stateequilibriumofthemodelbeforeandafterataxrateincrease. Onthex-axis,employmentissteady-stateemploymentinequation(25). Onthey-axis,netproductivity andhiringcostsaretheleft-andtheright-handsidesofequation(31),respectively. PanelsAandBrefer tohigh-skilled(largesurplus)workers. PanelsCandDrefertolow-skilled(smallsurplus)workers. In allpanels,workerskilllevelsareabovetheskillcutoffforzerosurplus. 42

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.7 0.75 0.8 0.85 0.9 0.95 1 Employment rate stsoc gnirih ,ytivitcudorp teN A. Steady-State Equilibrium: High-Skilled 0.8 0.7 0.6 0.5 Hiring costs Net productivity 0.4 Equilibrium 0.3 0.2 0.1 0 0.7 0.75 0.8 0.85 0.9 0.95 1 Employment rate stsoc gnirih ,ytivitcudorp teN B. Steady-State Equilibrium: High-Skilled Hiring costs Old net productivity New net productivity: Tax hike Old equilibrium New equilibrium: Tax hike 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.7 0.75 0.8 0.85 0.9 0.95 1 Employment rate stsoc gnirih ,ytivitcudorp teN C. Steady-State Equilibrium: Low-Skilled 0.8 0.7 0.6 0.5 Net productivity Hiring costs 0.4 Equilibrium 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Employment rate stsoc gnirih ,ytivitcudorp teN D. Steady-State Equilibrium: Low-Skilled Old net productivity New net productivity: Tax hike Hiring costs Old equilibrium New equilibrium: Tax hike Figure2: RoleofZero-SurplusConstraint Notes:Thefigureillustratesthesteady-stateequilibriumofthemodelbeforeandafterataxrateincrease. Onthex-axis,employmentissteady-stateemploymentinequation(25). Onthey-axis,netproductivity and hiring costs are the left- and the right-hand sides of equation (31), respectively. High- and lowskilledworkersaredefinedasworkerswhoseskilllevelsareaboveandbelowtheskillcutoffforzero surplusafterthetaxrateincrease,respectively. 43

0.34 0.32 0.3 0 5 10 15 20 Periods after shock etar xaT A. Permanent Tax Rate Cut 1 Perfect foresight 0.5 0 0 5 10 15 20 Periods after shock etar tnemyolpmE B. IRF to Permanent Shock Frictional Frictionless 0.34 0.32 0.3 0 5 10 15 20 Periods after shock etar xaT C. Temporary Tax Rate Cut 1 MIT shocks 0.5 0 0 5 10 15 20 Periods after shock etar tnemyolpmE D. IRF to Temporary Shock Frictional Frictionless Figure3: FrictionalversusFrictionlessTransitionDynamics Notes:Thefigureillustratesthedynamicresponseoftheemploymentrateinthecaseswith frictions (solid line with circles) and without frictions (dotted line with diamonds) to a permanenttaxratecut(panelsAandB)andtoatemporarytaxratecut(panelsCandD). 44

0.15 0.1 0.05 0 0 1 2 3 4 5 6 7 8 9 10 Skill level ytisneD A. Skill Distribution B. Vacancy Cost C. Vacancy Cost / Skills 4 0.4 3 0.3 2 0.2 1 0.1 0 0 0 2.5 5 7.5 10 0 2.5 5 7.5 10 Skill level Skill level Figure4: WorkerHeterogeneityandVacancyCosts Notes: Panel A shows the distribution of skills implied by the calibration of themodel: x = 0.64, x = 10,µ = 0,andσ = 0.15. PanelBshowsthe min max x x valuesofthevacancycostparameterkbyskilllevels. PanelCshowstheratio oftheunitvacancycosttotheskilllevel,i.e.,k/x. 45

1.2 1 0.8 0.6 0.4 0.2 0 -0.2 0 1 2 3 4 Years after shock )pp( noitaiveD A. Employment Rate 0.2 Data Model 0 -0.2 -0.4 -0.6 -0.8 -1 0 1 2 3 4 Years after shock )pp( noitaiveD B. Tax Rate Data Model Figure5: LinearIRFtoaTaxShock-ModelversusData Notes: ThefigureshowsthelinearIRFestimatedonactualdatawithaproxy SVAR (solid line with circles) and that estimated on artificial data simulated fromthemodel(dottedlinewithtriangles). 46

A. Unemployment Rate B. Output 0.4 1.4 0.3 1.3 0.2 1.2 0.1 0 1.1 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 Tax rate Tax rate C. Job-Finding Rate D. Job-Separation Rate 0.8 0.2 0.6 0.15 0.4 0.1 0.2 0.05 0 0 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 Tax rate Tax rate Figure6: StochasticSteadyStates Notes: The figure shows the stochastic steady state implied by the calibrated model. Foreachvalueofthetaxrateonthe x-axis, panelsAthroughDplot the corresponding values of the aggregate unemployment rate, output, and job-findingandjobseparationratesonthey-axis. 47

A. Histogram of Unemployment Rates 6 4 2 0 0 0.05 0.1 0.15 0.2 0.25 0.3 Unemployment rate ycneuqerF 5 10 B. Histogram of Tax Rates 15 10 5 0 0.2 0.25 0.3 0.35 0.4 0.45 Tax rate ycneuqerF 4 10 Figure7: ErgodicDistribution Notes: The figure shows the histograms of the unemployment rate and tax rate using 1 millionmonthlyperiodssimulatedfromthemodel’sergodicdistribution. 48

Figure8: ConvexRelationbetweenUnemploymentandTaxRates Notes: The figure shows the unemployment rate against the tax rate using 1 million monthlyperiodssimulatedfromthemodel’sergodicdistribution. 49

0.6 0.4 0.3 0.2 0 -0.2 -0.3 -0.4 -0.6 0 20 40 60 Months after shock )pp( noitaiveD A. Employment Rate 0.8 0.6 0.4 0.2 0 -0.2 -0.4 Tax cut -0.6 Tax hike -0.8 0 20 40 60 Months after shock )pp( noitaiveD B. Tax Rate Tax cut Tax hike Figure9: SignAsymmetry-SmallTaxShock 50

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 20 40 60 Months after shock )pp( noitaiveD A. Employment Rate 1.5 Tax cut Tax hike 1 0.5 0 -0.5 -1 -1.5 0 20 40 60 Months after shock )pp( noitaiveD B. Tax Rate Tax cut Tax hike Figure10: SignAsymmetry-MediumTaxShock 51

3 2 1 0 -1 -2 -3 0 20 40 60 Months after shock )pp( noitaiveD A. Employment Rate 3 2 1 0 -1 -2 Tax cut Tax hike -3 0 20 40 60 Months after shock )pp( noitaiveD B. Tax Rate Tax cut Tax hike Figure11: SignAsymmetry-LargeTaxShock 52

4 3 2 1 0 -1 -2 0 20 40 60 Months after shock )pp( noitaiveD A. Employment Rate 1 0.5 0 -0.5 -1 -1.5 -2 IRF to 3 pp tax cut -2.5 2 IRF to 1.5 pp tax cut -3 0 20 40 60 Months after shock )pp( noitaiveD B. Tax Rate Tax cut Figure12: SizeAsymmetry-LargeTaxCut 53

3 2 1 0 -1 -2 -3 -4 -5 0 20 40 60 Months after shock )pp( noitaiveD A. Employment Rate 3 IRF to 3 pp tax hike 2 IRF to 1.5 pp tax hike 2.5 2 1.5 1 0.5 0 -0.5 -1 0 20 40 60 Months after shock )pp( noitaiveD B. Tax Rate Tax hike Figure13: SizeAsymmetry-LargeTaxHike 54

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 0 20 40 60 Months after shock )pp( noitaiveD A. Employment Rate 0.2 0 -0.2 -0.4 -0.6 Tax cut in "bad" times Tax cut in "good" times -0.8 0 20 40 60 Months after shock )pp( noitaiveD B. Tax Rate Tax cut Figure14: StateDependence-SmallTaxShock 55

0.8 0.6 0.4 0.2 0 -0.2 -0.4 0 20 40 60 Months after shock )pp( noitaiveD A. Employment Rate 0.5 0 -0.5 -1 Tax cut in "bad" times Tax cut in "good" times -1.5 0 20 40 60 Months after shock )pp( noitaiveD B. Tax Rate Tax cut Figure15: StateDependence-MediumTaxShock 56

Appendix A Empirics In this appendix, we describe data sources, variables’ construction, and the details of the procedureadoptedtoestimatethedynamiceffectsofchangesintheaveragemarginaltax rate (AMTR), using a structural vector autoregressions (SVARs) approach, as in Mertens andMontielOlea(2018)andFerraroandFiori(2020). Average marginal tax rate (AMTR) To construct AMTR, we follow Barro and Redlick (2011) and consider a notion of “labor income” that includes wages, self-employment, partnership, S-corporation income. The data are taken from the CPS March Supplement. AMTRisthesumofthefederalindividualincometaxandthepayroll(FICA)tax. Weuse the NBER-TAXSIM program to simulate marginal income tax rates and marginal payroll taxratesattheindividuallevel. WethenconstructAMTRasthesumofaveragemarginal individual income tax rate (AMIITR) and average marginal payroll tax rate (AMPTR), usingadjustedgrossincome(AGI)sharesasweights. Identification of tax shocks Tax shocks are identified in the context of SVARs using proxies for exogenous variation in tax rates as external instruments (Mertens and Ravn, 2013). We use the proxies constructed in Ferraro and Fiori (2020) for exogenous changes inAMTRs. Toselectinstancesofexogenousvariationintaxrates,FerraroandFiori(2020) followthenarrativeapproachproposedbyRomerandRomer(2010): changesintotaltax liabilities are classified as “exogenous” based on the motivation for the legislative action beingeitherlong-runconsiderations,thatareunrelatedtothebusinesscycle,orinherited budgetdeficits. To account for potential “anticipation effects,” only individual income tax liability changeslegislatedandimplementedwithintheyearareincluded,thisapproachisinline with Mertens and Montiel Olea (2018). According to this criterion, seven tax reforms are identified as exogenous: (1) Revenue Act of 1964; (2) Revenue Act of 1978; (3) Economic Recovery Tax Act 1981; (4) Tax Reform Act of 1986; (5) Omnibus Budget Reconciliation Act of 1990; (6) Omnibus Budget Reconciliation Act of 1993; (7) Jobs and Growth Tax 57

ReliefReconciliationActof2003. The impact of a reform is measured as the difference between two counterfactual tax rates. The first counterfactual tax rate is calculated using year t−1 income distribution and year t statutory tax rates and brackets. The second is calculated based on the year t−1 income distribution and year t−1 statutory tax rates and brackets. The difference between the two isolates then the impact that a tax reform implemented in year t had on the AMTR. An issue that arises with these type of calculations is the indexing of the federal tax system starting in 1985. To address this concern, we rescale incomes by the automaticadjustmentsinbracketwidthsembeddedinthefederaltaxcode. SVAR specification The baseline reduced-form VAR specification includes the average marginal tax rate, the unemployment rate, the participation rate, and a set of aggregate control variables for the sample of annual observations for the period 1961-2012. Control variablesincludethelogofrealGDPpercapita,thelogoftheS&Pindex,andthefederal funds rate, which allows us to capture business cycle dynamics, the monetary policy stance, as well as the effects of bracket creep. To explicitly allow for the feedback from debt to taxes and spending, the log of real government spending per capita (purchases and net transfers), the average tax rate and the change in log real federal government debtpercapitaarealsoincluded. 58