Macroeconomic Effects of Capital Tax Rate Changes

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Macroeconomic Effects of Capital Tax Rate Changes Saroj Bhattarai, Jae Won Lee, Woong Yong Park, and Choongryul Yang 2022-027 Please cite this paper as: Bhattarai, Saroj, Jae Won Lee, Woong Yong Park, and Choongryul Yang (2022). “Macroeconomic Effects of Capital Tax Rate Changes,” Finance and Economics Discussion Series 2022-027. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2022.027. 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.

Macroeconomic Effects of Capital Tax Rate Changes* SarojBhattarai† JaeWonLee‡ WoongYongPark§ ChoongryulYang¶ UTAustin SeoulNationalUniv. SeoulNationalUniv. FederalReserveBoard Abstract We study aggregate, distributional, and welfare effects of a permanent reduction in the capital tax rate in a quantitative model with capital-skill complementarity and household heterogeneity. Suchataxreformleadstoexpansionarylong-runaggregateoutputandinvestment effects,butthosearecoupledwithincreasesinwage,consumption,andincomeinequality. The taxreformisnotself-financinganditseffectsdependcruciallyonwhetherthegovernmentcuts lump-sum transfers or raises distortionary labor or consumption tax rates for financing. The formerresultsinalargeraggregateexpansion,butattheexpenseofagreaterriseininequality. As a result, the latter is relatively more beneficial for unskilled households. We find that the taxreform, whentheconsumptiontaxrateadjusts, leadstoaParetoimprovementintermsof life-time welfare. For transition dynamics, monetary policy, in addition to the fiscal adjustments,matters. Inparticular,ifmonetarypolicyinflatesawayaportionofthepublicdebt,the economycanavoidtheshort-runcontractionthatwouldariseotherwise. JELClassification: E62,E63,E52,E58,E31 Keywords: Capital tax rate; Distortionary financing; Capital-skill complementarity; Inequality;Welfareimplications *WearegratefultoScottBaker,MartinBeraja,OliCoibion,KeremCos¸ar,ChrisHouse,HashmatKhan,Minjoon Lee, Philipp Lieberknecht, Karel Mertens, Ezra Oberfield, Valerie Ramey, Eric Young, and seminar and conference participantsattheUTAustin, CarletonUniversity, FrontiersofMonetaryPolicyandFinancialStudiesWorkshopat Bank of Canada, Midwest Macro conference, North American Econometric Society Summer meetings, MacroeconomicModelingandModelComparisonNetworkconferenceatStanfordUniversity,andRichmondFed–University ofVirginiaworkshop,forhelpfulcommentsandsuggestions. Theviewsexpressedherearethoseoftheauthorsand donotnecessarilyreflectthoseoftheFederalReserveBoardortheFederalReserveSystem. Firstversion: Jan2018; Thisversion: Jan2022. †DepartmentofEconomics,UniversityofTexasatAustinandCAMA,2225Speedway,StopC3100,Austin,TX 78712,U.S.A.Email: saroj.bhattarai@austin.utexas.edu. ‡Department of Economics, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, South Korea. Email: jwlee7@snu.ac.kr. §Department of Economics, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, South Korea. Email: woongyong.park@snu.ac.kr. ¶FederalReserveBoardofGovernors,20thStreetandConstitutionAvenueNW,Washington,DC20551,U.S.A. Email: choongryul.yang@frb.gov. 1

1 Introduction Themacroeconomicanddistributionaleffectsofpermanentcapitaltaxcutshaverecentlybecomea subjectofwidespreaddiscussion,spurredbytherecentU.S.taxreformthatreducedthecorporate tax rate from 35% to 21%. Several questions have been raised. What are the long-run and the short-run effects on output and investment? What are the distributional consequences in terms of wage and consumption inequality? What are the welfare implications? Will such a large tax cut be self-financing? If not, what are the positive and normative implications of alternative ways to finance the reform? How does the monetary policy response matter for the short-run effects of a capitaltaxcut? Given the nature of these questions, it is useful to pursue an analysis through the lens of a quantitative dynamic model that can capture both aggregate and distributional effects. Moreover, since the tax reform is large-scale, it is imperative to consider general equilibrium effects as well as the sources of financing. This paper therefore addresses these questions using a quantitative, dynamic general equilibrium model with skill and consumption heterogeneity.1 We present the long-runequilibrium,aswellasfull(nonlinear)transitiondynamicsandwelfareevaluation. Compared to existing studies on the effects of capital tax changes, our analysis is integrative in terms of the model used to address them. We consider several relevant features, such as (equipment) capital-skill complementarity, household heterogeneity, different types of long-run fiscal adjustments, as well as short-run monetary and fiscal policy interactions, in a unified framework. In terms of research questions, this paper can also be distinguished from the existing studies in that we pay special attention to the source of financing. We show how the government finances capitaltaxcuts–thatis,howtheresultingincreaseinpublicdebtisultimatelyretired–hasimportant aggregate,distributional,andwelfareconsequencesinthelong-runaswellasduringthetransition periods. The paper starts with a long-run analysis. We show analytically in a simplified model and numericallyinthequantitativemodelwithcapital-skillcomplementarityandhouseholdheterogeneity that capital tax cuts, as expected, have expansionary long-run aggregate effects on the economy. For instance, with a permanent reduction of the capital tax rate from 35% to 21%, output in the new steady state, compared to the initial steady state, is greater by 4.24%, structure investment by 20.24%, and equipment investment by 6.27%, in our baseline calibration. The mechanism for aggregate effects is well understood. A reduction in the capital tax rate leads to a decrease in the rental rate of capital, raising demand for capital by firms. This stimulates investment and capital accumulation. A larger amount of capital stock, in turn, makes workers more productive, raising wagesandhours. Finally,giventheincreaseinthefactorsofproduction,outputexpands. 1Inparticular,unskilledhouseholdsinourmodelcannotsmoothconsumptionovertime. 2

Thisaggregateexpansionhowever,iscoupledwithworseningwageandconsumptioninequality in our model. For instance, skilled wages increase by 4.66% while unskilled wages increases byonly0.56%,drivenbycapital-skillcomplementarity. Inaddition,thereisariseinconsumption inequality. The major reason is that the aggregate effects above are obtained in our baseline scenario where the government finances the capital tax cuts by cutting back lump-sum transfers on the unskilled household.2 In this scenario, consumption of the unskilled falls with the capital tax ratereform,by3.86%. Incontrast,consumptionoftheskilledrisesby4.82%. Importantly, in this baseline scenario, although aggregate labor hours increase as in a standard representative-agent framework, skilled hours and unskilled hours move in opposite directions. A capital tax rate cut financed by appropriate transfer adjustments leads to an increase in unskilled hours not only due to the aforementioned standard mechanism that works through the labor demandchannel,butalsoduetothewealtheffectsthatencouragetheunskilledhouseholdstosupply more hours. By contrast, the wealth effects on labor supply and the standard labor demand channel work in opposite directions for the skilled households: the former now discourages the skilled householdstosupplyhourswhereasthelattercontinuestoworkasaforcetoincreaseskilledhours. In the baseline case, the former dominates the latter. Consequently, there is a slight decrease in skilled hours. Notice that because of these two-way wealth effects, the rise in consumption inequality will lead to a further increase in wage inequality. Thus, in our model with heterogeneous households, consumption inequality and wage inequality interact and reinforce each other to amplifythedistributionaleffects.3 The baseline scenario above, while it may serve as a useful benchmark, is unrealistic because in our calibration, the lump-sum transfer cuts in fact imply imposing lump-sum taxes on the unskilled households. We thus consider two other financing options in which the government relies ondistortionarylabororconsumptiontaxes. Thethreefinancingschemesunderconsideration–the lump-sum transfer adjustment, the labor tax rate adjustment, and the consumption tax rate adjustment–generally produce different effects on aggregate output because each scheme influences workers’ labor supply decisions differently. First, in our baseline scenario above, as mentioned, lump-sum transfer cuts generate a (negative) wealth effect on the unskilled labor supply, which boosts unskilled hours and in turn, contributes to greater aggregate output (again, in addition to the standard mechanism through firms’ demand for labor). In comparison, a rise in the labor or consumption tax rate decreases the effective wage rate (as is well-understood) and additionally, 2Regardingthedistributionofunearnedincome,inourbaselinecase,skilledhouseholdsreceiveprofitsfromfirms andunskilledhouseholdsreceivetransfersfromthegovernment. Wehowever,parameterizeourmodelinawaysuch thatotherdistributionpossibilitiescanbeeasilyexplored. Weprovidesomesensitivityanalysis. 3Thisadditionalfeedbackchannelisabsentinarepresentative-householdversionofourmodel.Inaddition,household heterogeneity amplifies the aggregate effects, too. The reason is that the wealth-effect-induced increase in unskilled hours is only partially offset by the wealth-effect-induced decrease in skilled hours in this baseline scenario. Wedonotshowthisresultforbrevity,butisavailableuponrequest. 3

weakens the wealth effect for the unskilled household. These two mechanisms work together to generate a smaller aggregate expansion under the distortionary tax adjustments. We nevertheless find that aggregate expansion is greater under the consumption tax rate adjustment than under the the labor tax adjustment. We show this result again both analytically in a simplified model and numericallyinthequantitativemodel. Specifically, in our baseline calibration, a permanent reduction of the capital tax rate from 35% to 21% requires an increase in the labor tax rate from 22.7% to 25.4%.4 Then, output in the new steady state, compared to the initial steady state, is greater by 2.08%, equipment investment by 17.75%, and structures investment by 4.81%. The reason for the smaller boost in aggregate variables,comparedtothelump-sumtransferadjustmentcase,isafallinhoursofboththeskilled and the unskilled. Importantly, the behavior of unskilled hours is different even qualitatively, comparedtothelump-sumtransferadjustmentcase,fortworeasons. First,theafter-taxwagerate for unskilled hours in fact declines. Second, the wealth effect on unskilled labor supply is not as strong because the massive reduction in transfer income for the unskilled household is absent. Overall, the decrease in skilled and unskilled hours dampens the expansionary aggregate effect of capitaltaxcuts. On distributional implications, both wage and consumption inequality increase, as they did before with the transfer adjustment case. In comparison to the transfer adjustment case however, these distributional effects are smaller. The main reason is that the burden from the labor tax increaseissharedbybothtypesofhouseholds,whereastransferreductiononlyaffectstheunskilled. Therefore, consumption of the unskilled does not fall as much while consumption of the skilled doesnotincreaseasmuchnow. Thereducedconsumptioninequalityinturndiminishestheroleof thewealtheffectonlaborsupply,leadingtoasmallerincreaseintheskillpremium. For consumption tax rate adjustment, the third source that can finance the reduction in the capital tax rate from 35% to 21%, consumption tax rates have to increase from 1.29% to 3.49%. Generally, the effects are qualitatively similar to those in the labor tax rate adjustment case. However,quantitatively,theaggregateeffectsarebiggercomparedtolabortaxrateadjustment,aslabor supplygetsdistortedless.5 Oneimportantqualitativedifference,comparedtobothtransferandlabortaxrateadjustment,isinconsumptionoftheunskilledhouseholds,whichincreasesslightlyin thelong-run. Thisisdrivenbythelargeroutputeffectswhentheconsumptiontaxrateadjustsand hasimportantimplicationsforourwelfareresults. We then move to an analysis of transition dynamics as the economy evolves from the initial steady-state to the new steady-state. During the transition, the economy experiences a decline in 4We keep debt-GDP ratio the same between the initial and the new steady-state. Debt-GDP ratio, however, is allowedtodeviatefromthesteady-statelevelalongthetransitionpath,whenwestudyshort-runeffects. 5Thishappensasincomeandsubstitutioneffectsoffsetunderconsumptiontaxrateincrease. 4

not only consumption of both types of households, but also output. This holds even if lump-sum transfers finance the capital tax rate cut.6 Consumption and output falls are more severe under the distortionary tax rate adjustments. The short-run contraction may be viewed as another side effect ofapermanentcapitaltaxcutbesidestheincreaseininequalityinthelong-runthatwehighlighted above. Another important aspect of transition dynamics that we highlight is on the need to analyze monetary and fiscal policy adjustments jointly. This is because the short-run effects depend critically on the monetary policy response. In particular, when the government has access only to distortionary labor taxes, we consider a case where the central bank accommodates inflation so that nominal government debt is partially inflated away along the transition after the capital tax ratecut. Inthisinterestingscenario,thegovernmentdoesnotraiselabortaxratesasmuch,andthe rise of inflation in the short-run completely negates any short-run contraction in output as well as consumption. Next, while our paper does not study optimal policy, we analyze welfare consequences of the permanent capital tax rate cut, given the various financing possibilities we consider. We show that long-term social or aggregate welfare gains contrast with short-term (but, still prolonged) aggregatewelfarelosses,regardlessofhowthecapitaltaxratecutisfinanced. Moreover,thesame resultmostlyholdsatthehouseholdlevelaswell: Eventheskilledhousehold,whoalwaysbenefits from the capital tax reform in the long-run, suffers significant short-run welfare losses except for the case when lump-sum transfers adjust. The unskilled households experience welfare losses in theshort-rununderallfinancingoptions. Nowfocusingonthe(long-term)life-timewelfare,weshowthattheskilledgainsattheexpense oftheunskilledundertransferadjustment,andthusthecapitaltaxreformisnotParetoimproving. Thesameresultholdswhenlabortaxrateadjusts. Infact,financingacapitaltaxcutbylump-sum transfer adjustment, while leading to a higher level of aggregate output, is not necessarily better than financing it by labor tax adjustment. Intuitively, when some agents’ income relies relatively more on transfers (the unskilled households), a reduction in transfer to offset tax revenue losses from capital tax cuts can decrease their welfare. In contrast, labor tax adjustment works better for the unskilled as the labor tax increase burden is shared by both types, whereas transfer reduction onlyaffectstheunskilled. Finally, to underscore yet again the importance of considering and modeling different sources of financing the capital tax reform, we show that compared to the labor tax rate adjustment, the welfare results are different for consumption tax rate adjustment. In this case, there is in fact a Pareto improvement as not only the skilled households, but also the unskilled households, gain 6Theshort-runfallinoutputisaresultofinvestmentadjustmentcost,nominalrigidities,andourempiricalmonetarypolicyrule. 5

in terms of lifetime welfare.7 The main reason for this result is that compared to labor tax rate adjustment, consumption of the unskilled goes up in the long-run, as we discussed above, due to lowerdistortionsonlaborsupply,leadingtoalargeraggregateoutputeffect. Related literature This paper is related to several strands of the literature, some of which have been developed without much interaction with each other. While we focus mostly on a positive analysis, our paper is related to classic normative analysis of Chamley (1986) and Judd (1985), whichwasre-addressedrecentlyinStraubandWerning(2020).8 Wedonotanalyzeoptimalmonetary and fiscal policy issues, but do compute welfare implications given the capital tax rate cut and various financing rules we consider. While doing so, we find that increasing the consumption tax rate to finance the capital tax rate cut leads to a Pareto improvement in our heterogeneous householdmodel. Our analysis of the central bank allowing inflation to directly facilitate debt stabilization, through passive monetary policy, when the government has access to only distortionary taxes relatesthispapertotheliteratureonmonetaryandfiscalpolicyinteractions–inparticular,thenormative analysis in Sims (2001). We implement this scenario using a rules-based positive description of interest rate policy, as in Leeper (1991), Sims (1994), and Woodford (1994) for instance.9 Relatedly, our work is also motivated by the study of effects of government spending and how that depends on the monetary policy response, as highlighted recently by Christiano, Eichenbaum and Rebelo(2011),Woodford(2011),andLeeper,TraumandWalker(2017). In terms of analyzing the long-run effects of changes in the capital tax rate in an equilibrium macroeconomic model, our paper is close to Trabandt and Uhlig (2011) and the more recent work ofBarroandFurman(2018)thatanalyzestheU.S.taxreform. Comparedtothisliterature,onekey differenceisthatourbaselinemodelfeaturescapital-skillcomplementarity,followingKruselletal. (2000),aswellashouseholdheterogeneity,suchthatbothwageandconsumptioninequalityissues canbeanalyzed. TrabandtandUhlig’s(2011)mainfocusisontheimportantissueofthepresence ofLaffercurvesforlaborandcapitaltax,undereithertransferorgovernmentspendingadjustment. We show, both analytically and numerically, how the macroeconomic effects of a given capital tax rate change are different depending on whether non-distortionary or distortionary sources of 7Whilelife-timewelfareishigherfortheunskilledhouseholds,thereiswelfarelossintheshort-andmedium-run, aswementionedabove. 8Theliteratureonoptimalcapitaltaxationinadynamicsettingisextensive. Earlierworktypicallyfindssignificant welfare gains from eliminating capital income taxes in representative-household infinite-horizon frameworks. More recentstudiesmoveawayfromthestandardsetup,featuringheterogeneoushouseholdsandoverlappinggenerations, and find optimality of non-zero capital taxes (e.g. Aiyagari 1995, Erosa and Gervais 2002, and Conesa, Kitao and Krueger2009.) 9In this case, the central bank does not follow the Taylor principle. Bhattarai, Lee and Park (2014) and Ascari, FlorioandGobbi(2020)analyticallycharacterizetheeffectsofsuchacaseinamodelwithstickyprices. Canzoneri, CumbyandDiba(2010),LeeperandLeith(2016),andCochrane(2019)provideexcellentsurveysoftheliterature. 6

government financing are available as well as how different types of distortionary financing can have qualitatively different welfare implications. In addition, we study transition dynamics in detail, highlighting that it is imperative to model monetary and fiscal policy adjustments jointly for determining short-run effects, and explore aggregate, distributional, and welfare implications takingdynamicsfullyintoaccount. BarroandFurman’s(2018)recentimportantcontributionstudiesmacroeconomicimplications ofagivencapitaltaxratechange,likewedo,inamodelwithmoredetailsofthetaxcodeandfive types of capital. Our baseline model is simpler in that respect, but features endogenous labor supplysuchthatdistortionarysourcesofcapitaltaxreformfinancingcanbeexploredcarefully. Moreover,asmentionedabove,wealsostudywageandconsumptioninequalityimplications,transition dynamics, and welfare properties. Furthermore, our model with endogenous labor supply and household heterogeneity allows us to highlight how wage inequality and consumption inequality interactandreinforceeachother,therebyamplifyingdistributionaleffects. Another closely related paper is Domeij and Heathcote (2004). Similar to our study, they explicitly take into account both transition dynamics and steady-state change after a tax reform in the welfare analysis, and show that a capital tax rate reduction is not Pareto improving. We also find such a result in this paper for labor tax rate adjustment. For consumption tax rate adjustment however, we do find that capital tax rate reduction is Pareto improving in our baseline calibration. Domeij and Heathcote’s (2004) model, in addition, abstracts from capital-skill complementarity. ThemorerecentworkofSlavíkandYazici(2019)usesamodelwithcapital-skillcomplementarity and analyzes the effects of a tax reform that eliminates tax differentials between equipment and structurecapital. Besides the different research questions we aim to address, our paper is different from these two contributions in modeling choices. Their models feature a richer form of household heterogeneity, as in Aiyagari (1994), while our model is more stylized in that dimension, focusing on a particulartypeofheterogeneityasinthetraditionoftheTwoAgentNewKeynesian(TANK)literature. Ouranalysisthusmissespotentiallyimportantimplicationsofarealisticwealthdistribution. It however, allows us to include a richer set of model elements that enable us to conduct a more realistic analysis of transition dynamics. Moreover, we do find new results related to distortionary financingusingconsumptiontaxes. Furthermore,ourempiricallymotivatedspecificationsofmonetaryandfiscalpolicy,coupledwithmodelelementsthatmatterfortransition,allowustoconsider positiveandwelfareimplicationsofmonetaryandfiscalpolicyinteractions. As mentioned above, the way we introduce household heterogeneity connects our paper to the growingTANKliterature. Thisliteraturehasanalyzedextensivelyvariousissuesonmonetarypolicy(e.g. Bilbiie2008,Bhattarai,LeeandPark2015,CúrdiaandWoodford2016,andDebortoliand Galí2017.) Onthefiscalside,Galí,López-SalidoandVallés(2007),Bilbiie,MonacelliandPerotti 7

(2013) and Eggertsson and Krugman (2012) have considered the effects of government spending and (lump-sum) transfers. Much of the literature, however, abstracts from capital accumulation (withtheexceptionofGalí,López-SalidoandVallés2007),therebyprecludingananalysisofcapital income taxes. Existing papers also do not consider several sources of distortionary financing. Moreover, our model importantly also features capital-skill complementarity, which allows us to analyzetheeffectofapolicychangeonthewagedistribution.10 There is by now a fairly large dynamic stochastic general equilibrium modeling literature that assesses the effects of distortionary tax rate changes and of fiscal policy generally. For instance, among others, Forni, Monteforte and Sessa (2009) study transmission of various fiscal policies, including government spending and transfer changes in a quantitative model. Sims and Wolff (2018) additionally study state-dependent effects of tax rate changes. These papers often study effectsoftransitoryandsmallchangesinthetaxratewhileourmainfocusisonthelong-runeffects of a permanent reduction in the capital tax rate under various sources of financing, and then on an analysisoffull(nonlinear)transitiondynamicsfollowingafairlylargereduction. Additionally,we provide analytical results on comparison of different sources of financing that help illustrate the key mechanisms on the long-run effects, while in the quantitative part, we use a model that can assessdistributionalconsequences. While we are motivated by the particular recent U.S. episode of a permanent tax rate change, generally,ourpaperisinfluencedalsobyalargeliteraturethatempiricallyassessesthemacroeconomiceffectsoftaxpolicy. Inparticular,variousidentificationstrategies,suchasnarrative(Romer and Romer 2010) and statistical (Blanchard and Perotti 2002, Mountford and Uhlig 2009) have beenusedtoassessequilibriumeffectsoftaxchanges. Relatedly,HouseandShapiro(2008)study a particular case of change in investment tax incentive. The effects on aggregate variables that we find using a calibrated equilibrium model is consistent with this work, although these papers have generally focused either explicitly on temporary tax policies or do not explicitly separate out permanent changes from transitory ones. Perhaps most importantly, we also use our model to assess distributionalandwelfareeffectsfollowingapermanentcapitaltaxratecut. 2 Model We now present the baseline model, which is a quantitative equilibrium framework augmented with two types of workers (skilled and unskilled) and two types of capital (structures and equipment). We introduce equipment capital-skill complementarity following Krusell et al. (2000), and a skill premium arises endogenously in the model. Moreover, the households are heterogeneous: 10Onatechnicalside,theexistingliteraturetypicallyfocusesonlineardynamicsaroundasteadystate. Wepresent exact,nonlineartransitiondynamicsinaTANKmodelwithcapital-skillcomplementarity. 8

theskilledhouseholdmakesoptimalconsumption/savingsdecisionswhiletheunskilledhousehold is “hand-to-mouth”. This framework allows us to study both aggregate as well as wage and consumption inequality implications of a capital tax rate change in a unified way. The model also featuresadjustmentcostsininvestment,variablecapacityutilization,andnominalpricingfrictions to enable a realistic study of transition dynamics. Pricing frictions additionally allow an analysis oftheroleofmonetarypolicyforthetransitiondynamics. 2.1 Private sector Westartbydescribingthemaximizationproblemsoftheprivatesector. 2.1.1 Households There are two types of households who supply skilled labor (type s) and unskilled labor (type u), respectively. Themeasureoftype-ihouseholdfori∈{s,u}isdenotedbyNi. Theskilledhouseholds are“Ricardian”,andtheirproblemisto   max E  (cid:88)∞ βtU (cid:0) Cs,Hs(cid:1)  (cid:110) C t s,H t s,B t s,I b s ,t ,I e s ,t ,Kˆ b s ,t+1 ,Kˆ e s ,t+1 ,u e s ,t ,u b s ,t (cid:111) 0 t=0 t t  subjecttoasequenceofflowbudgetconstraints (cid:16) (cid:17) 1+τC PCs+P Is +P Is +Bs t t t t b,t t e,t t (cid:16) (cid:17) = 1−τH WsHs+R Bs +λ τKP Is +λ τKP Is t t t t−1 t−1 b t t b,t e t t e,t (cid:16) (cid:17) (cid:16) (cid:17) + 1−τK RK,bus Kˆs + 1−τK RK,eus Kˆs t t b,t b,t t t e,t e,t (cid:16) (cid:17) (cid:16) (cid:17) P (cid:16) (cid:17) (cid:16) (cid:17) χs χs −P 1−λ τK A us Kˆs − t 1−λ τK A us Kˆs +P ΦΦ +P S S , t b t b b,t b,t q e t e e,t e,t t Ns t t Ns t t where E isthemathematicalexpectationoperator,Cs isconsumption, Hs ishours,and Is and Is t t t b,t e,t are investment in the capital stock of structures and equipment denoted by Kˆs and Kˆs , respecb,t e,t tively. Similarly, Ks ≡us Kˆs and Ks ≡us Kˆs are the effective structure and equipment capital b,t b,t b,t e,t e,t e,t and us and us are the respective variable capacity utilization rates. A (us ) and A (us ) are the b,t e,t b b,t e e,t costsofvariablecapitalutilization. The skilled Ricardian households trade nominal risk-less one-period government bonds Bs. t They are paid a fraction χs of the aggregate profits Φ from the firms and a fraction χs of the Φ t S aggregate lump-sum transfers S from the government. The aggregate price level is P , Ws is the t t t K,b K,e nominal wage for skilled households, R is the nominal one-period interest rate, and R and R t t t are the rental rate of capital structures and equipment, respectively. The government levies taxes onconsumption,laborincome,andcapitalincomewithtaxratesτC,τH,andτK,respectively. The t t t 9

parameters λ and λ are the rates of expensing of capital investment in structures and equipment, b e respectively. Thediscountfactorisβ. Theevolutionsofthetwotypesofcapitalstockaredescribedby    Kˆ b s ,t+1 =(1−d b )Kˆ b s ,t +  1−S I I s b s ,t   I b s ,t , b,t−1    Kˆ e s ,t+1 =(1−d e )Kˆ e s ,t +  1−S I I s e s ,t   I e s ,t q t , e,t−1 where q is the relative price between investment in capital structures and equipment, and d and t b d are the rates of depreciation of the capital stock invested in structures and equipment, respece (cid:16) (cid:17) tively.11 S It representinvestmentadjustcost. I t−1 Theunskilledhouseholdsare“hand-to-mouth”(HTM),andtheirproblemisto max U (cid:0) Cu,Hu(cid:1) t t {Cu,Hu} t t subjecttoaflowbudgetconstraint (cid:16) (cid:17) (cid:16) (cid:17) χu χu 1+τC PCu= 1−τH WuHu+P ΦΦ +P S S , t t t t t t t Nu t t Nu t whereCu istheirconsumptionand Hu ishours. TheunskilledHTMhouseholdsarepaidafraction t t χu oftheaggregateprofitsΦ fromthefirmsandafractionχu oftheaggregatelump-sumtransfers Φ t S S fromthegovernment. Wu isthenominalwageforunskilledhouseholds. t t (cid:16) (cid:17) The period utility U(C ,H ), investment adjustment cost S It , and variable capacity utilizat t I t−1 tioncostA(u )havestandardproperties,whicharedetailedlater. t 2.1.2 Firms The model has final goods firms and intermediate goods firms. Perfectly competitive final goods firmsproduceaggregateoutputY bycombiningacontinuumofdifferentiatedintermediategoods, t (cid:18) (cid:82) (cid:19) θ indexedbyi∈[0,1],usingtheCESaggregatorgivenbyY t = 1 Y t (i) θ− θ 1 di θ−1 ,whereθ>1isthe 0 elasticity of substitution between intermediate goods. The corresponding optimal price index P t (cid:18) (cid:82) (cid:19) 1 forthefinalgoodis P = 1 P (i)1−θdi 1−θ ,where P (i)isthepriceofintermediategoodiandthe t t t 0 optimaldemandforgoodiisY (i)= (cid:16) Pt(i) (cid:17)−θ Y . Thefinalgoodisusedforprivateandgovernment t Pt t consumptionaswellasinvestmentincapitalstructuresandequipment. Monopolistically competitive intermediate goods firms, indexed by i, produce output using a 11Aswedescribeindetaillater,thisrelativepriceisexogenoustoensurebalancedgrowthinthemodel. 10

CRSproductionfunction F(.) Y (i)=F (cid:0) A,K (i),K (i),L (i),L (i) (cid:1) , (2.1) t t b,t e,t s,t u,t where A is an exogenous stochastic process that represents technological progress, with its gross t growthrategivenbya ≡ At =a¯.12 Aswedescribeindetaillater,wefollowKruselletal.(2000) t A t−1 in functional form assumptions on F(.), a nested CES formulation, and parameterizations of the elasticitiesofsubstitutionacrossfactors,suchthatitfeatures(equipment)capital-skillcomplementarity. Firms rent the two types of capital and hire the two types of labor in perfectly competitive factormarkets. Intermediate goods firms face nominal rigidity. As in Calvo (1983), a firm resets its price optimally with probability 1−α every period. Firms that do not optimize adjust their price ac- P cording to the indexation rule P t (i)= P t−1 (i)π γ t− P 1 π¯1−γP, where γ P measures the extent of dynamic indexationandπ¯ isthesteady-statevalueofthegrossinflationrateπ ≡P /P . t t t−1 Optimizingfirmschooseacommonprice P∗ tosolvetheirproblem t   {P∗ t ,Yt+k(i) m ,H a t+ x k(i),Kt+k(i)} E t   (cid:88) k ∞ =0 (cid:16) α p β (cid:17)k Λ Λ t+ t k P t+k Φ t+k (i)   , subjectto(2.1),whereΛ isthemarginalutilityofnominalincomeoftheskilled household13 and t flowprofitΦ (i)isgivenby t P∗ Wu Ws RK,b RK,e Φ t+k (i)= t X P,t,k Y t,t+k (i)− t+kL u,t+k (i)− t+kL s,t+k (i)− t+kK b,t+k (i)− t+kK e,t+k (i), P t+k P t+k P t+k P t+k P t+k where  X =  (π t π t+1 ···π t+k−1 )γPπ¯(1−γP)k, k≥1 P,t,k 1 k=0 and (cid:32) P∗X (cid:33)−θ Y t,t+k (i)= t P,t,k Y t+k . P t+k Note that there is an endogenous skill premium in the model, which we define as the wage Ws of skilled labor relative to that of unskilled labor, t . Given the CRS production function and the Wu t assumptionofperfectlycompetitivefactormarkets,thefactorpricesareequaltomarginalproducts 12Steady-stateofavariablexisdenotedbyxthroughout.Aswediscusslater,werestrictpreferencesandtechnology suchthatthemodelisconsistentwithbalancedgrowth. 13Whilewehavewrittenthemodelgenerallyaboveintermsofprofitdistributiontohouseholds,weassumethatfirm profitsgototheskilledhouseholdsinourbaselineparameterization. Therefore,thefirmsusethestochasticdiscount factoroftheskilledhouseholdintheirdynamicprofitmaximizationproblem.Forlong-runanalysis,aswewillassume thatthemarginalutilityisequatedacrosshouseholds(only)insteady-state,itisirrelevantwhichstochasticdiscount factorisusedbythefirms. 11

ofeach factormultipliedby firms’marginalcosts. Moreover, asweshow indetaillater, ifcapitalskillcomplementarityexists,theskillpremiumincreasesintheamountofequipmentcapitalwhen the quantities of the two types of labor inputs are held fixed. It is also increasing in the ratio of unskilledtoskilledlabor. 2.2 Government Wenowdescribetheconstraintonthegovernmentanddeterminationofmonetaryandfiscalpolicy. 2.2.1 Governmentbudgetconstraint Thegovernmentflowbudgetconstraint,writtenbyexpressingfiscalvariablesasratioofoutput,is P B Y t +τC t C Y t +τ t H (cid:32) P W Y t s L s,t + P W Y t u L u,t (cid:33) +τ t K (cid:88)  P R t K Y ,i K i,t −λ i (cid:0) I i,t +AC i,t (cid:1)   t t t t t t t t t i∈{b,e} B 1 Y G S =R t−1 t−1 + t + t , t−1 P Y π Y Y Y t−1 t−1 t t t t where B =NsBs, S =(cid:80) NiSi, Si = χi SS , AC =NsA (us )Kˆs , AC =NsA (us )Kˆs , andG is t t t i∈{s,u} t t Ni t b,t b b,t b,t e,t e e,t e,t t governmentspendingonthefinalgood.14 2.2.2 Monetarypolicy MonetarypolicyisgivenbyafeedbackruleasinCoibionandGorodnichenko(2011), R R¯ t = (cid:20)R R t ¯ −1 (cid:21)ρR 1 (cid:20)R R t ¯ −2 (cid:21)ρR 2   (cid:18)π π¯ t (cid:19)ϕπ (cid:32) Y Y t− t 1 (cid:33)ϕ∆y (cid:32) Y Y t n t (cid:33)ϕx   (1−ρR 1 −ρR 2 ) , (2.2) whereρRandρRgoverninterestratesmoothing,ϕ ≥0isthefeedbackparameteroninflation,Ynis 1 2 π t the natural (that is, flexible price) level of output, ϕ∆y is the feedback parameter on output growth, ϕ isthefeedbackparameteronoutputgap,andR¯ isthesteady-statevalueofR . Forlargeenough x t feedback coefficients (a combination of ϕ π , ϕ∆y , and ϕ x ), the Taylor principle is satisfied.15 We will also consider a case, described in more detail next, where the Taylor principle is not satisfied, andinflationresponsewillplayadirectroleingovernmentdebtstabilizationalongthetransition. 14We introduce government spending in the model for a realistic calibration. As we discuss later, government spending-to-GDPratioisheldfixedthroughoutinthemodel. 15In the textbook linearized sticky price model, this condition is ϕ >1. In the model here, as there are several π endogenouspropagationmechanismsandanempiricallygroundedinterestraterule,suchaconditionhastobedeterminednumerically,althoughϕ >1isoftenagoodbenchmark. π 12

2.2.3 Fiscalpolicy Weconsideraone-timepermanentchangeinthecapitaltaxrateτK inperiod0,whentheeconomy t is in the initial steady-state. In order to isolate the effects of the capital tax rate cut, Gt is kept Yt unchanged from its initial steady-state value in all periods. The debt-to-GDP ratio, Bt , may PtYt deviate from the initial steady-state in the short-run but will converge back to the initial steadystateinthelong-run,throughappropriatechangesinfiscalinstruments. Weconsiderthefollowing fourpolicyadjustments. First,onlylump-sumtransfersadjustfollowingafeedbackrulesimilarto themonetarypolicyrulespecificationinCoibionandGorodnichenko(2011), (cid:32) (cid:33) (cid:32) (cid:33) S S¯ S S¯ S S¯ t − new =ρS t−1 − new +ρS t−2 − new (2.3) Y t Y¯ new 1 Y t−1 Y¯ new 2 Y t−2 Y¯ new    (cid:32) (cid:33) (cid:32) (cid:33) + (cid:16) 1−ρS 1 −ρS 2 (cid:17)  ψS B  P B t− Y 1 − P B Y  +ψS ∆y Y Y t +ψS x Y Y n t   , t−1 t−1 t−1 t where 0 ≤ ρS +ρS < 1 governs labor tax rate smoothing, ψS ≥ 0 is the feedback parameter on 1 2 B outstanding debt, ψS is the feedback parameter on output growth, ψS is the feedback parameter ∆y x on the output gap, S¯ is the new steady-state value of S , and B is the (initial and new) steadynew t PY statevalueof Bt . Alargeenoughfeedbackcoefficientondebt(highψS)ensuresthatfiscalpolicy PtYt B leadstostationarydebtdynamics. Second,onlylabortaxratesτH adjustfollowingafeedbackrulesimilartothemonetarypolicy t rulespecificationinCoibionandGorodnichenko(2011), (cid:16) (cid:17) (cid:16) (cid:17) τH−τ¯H =ρH τH −τ¯H +ρH τH −τ¯H (2.4) t new 1 t−1 new 2 t−2 new    (cid:32) (cid:33) (cid:32) (cid:33) + (cid:16) 1−ρ 1 H−ρ 2 H (cid:17)  ψH B  P B t− Y 1 − P B Y  +ψ ∆ H y Y Y t +ψH x Y Y n t   , t−1 t−1 t−1 t where 0 ≤ ρH +ρH < 1 governs labor tax rate smoothing, ψH ≥ 0 is the feedback parameter on 1 2 B outstanding debt, ψH is the feedback parameter on output growth, ψH is the feedback parameter ∆y x on output gap, τ¯H is the new steady-state value of τH.16 A large enough feedback coefficient on new t debt(highψH)ensuresthatfiscalpolicyleadstostationarydebtdynamics.17 B 16FeedbackrulesforfiscalpolicywereestimatedinanearlycontributionbyBohn(1998). Bhattarai,LeeandPark (2016)estimateslightlysimplerversionsthanaboveinageneralequilibriummodelwithlump-sumtaxes. Noteimportantlythatin(2.4),distortionarytaxratesadjustsmoothlyduringthetransition.Thisismotivatedbythetheoretical analysisofBarro(1979), butalso, byourempiricalestimatesoftaxrulesthattakethisform. Forcompletenessand comparison,wewillalsoconsideracasewherelabortaxratesadjustasnecessarytoensureaconstantdebt-to-GDP ratiothroughoutthetransition. 17Inthetextbooklinearizedmodelwithonesourceoftaxes,thisconditionisψH >β−1−1. Inthemodelhere,we B solvefornon-lineardynamicsandadditionally,asthereareseveralsourcesoftaxesandanempiricallygroundedtax rule,suchaconditionhastobedeterminednumerically. 13

Third,onlyconsumptiontaxratesτC adjustfollowingthesimplefeedbackrule t (cid:16) (cid:17) (cid:16) (cid:17) τC−τ¯C =ρC τC −τ¯C +ρC τC −τ¯C (2.5) t new 1 t−1 new 2 t−2 new    (cid:32) (cid:33) (cid:32) (cid:33) + (cid:16) 1−ρC 1 −ρC 2 (cid:17)  ψC B  P B t− Y 1 − P B Y  +ψC ∆y Y Y t +ψC x Y Y n t   , t−1 t−1 t−1 t where0≤ρC+ρC <1governsconsumptiontaxratesmoothing,ψC ≥0isthefeedbackparameter 1 2 B onoutstandingdebt,ψC isthefeedbackparameteronoutputgrowth,ψC isthefeedbackparameter ∆y x ontheoutputgap,andτ¯C isthenewsteady-statevalueofτC. Alargeenoughfeedbackcoefficient new t ondebt(highψC)ensuresthatfiscalpolicyleadstostationarydebtdynamics. B For transition dynamics, the behavior of the monetary authority generally matters. In the three fiscal policies described above, the monetary policy rule (2.2) satisfies the Taylor principle, which thereby, implies that inflation plays no direct role in government debt stabilization. Moreover, given our restrictions that ψS, ψH and ψC are high enough, taxes respond strongly enough to B B B ensurethatdebtdynamicsaremean-reverting. We consider a fourth case to highlight the role of monetary policy response to inflation for transition dynamics. In this case, labor taxes adjust, but not sufficiently, as the tax rule response coefficients are not large enough, and inflation partly plays a direct role in government debt stabilization, as the monetary rule response coefficients are not large enough. The monetary and labor tax rules are still given by (2.2) and (2.4), but now with these appropriate restrictions on the feedback parameters. Thus, in this fourth case, we allow debt stabilization, (only) along the transition, tooccurpartlythroughdistortionarylabortaxesandpartlythroughinflation.18 2.3 Equilibrium and functional forms Equilibrium definition is standard, given the maximization problems of the private sector and the monetaryandfiscalpolicyrulesdescribedabove. Goods,asset,andfactormarketsclearinequilibrium. Theeconomyfeaturesbalancedgrowth. Aswedescribebelow,weusestandardassumptions onpreferencesthatensurebalancedgrowth. Moreover,sinceourproductionfunctionfeaturestwo types of capital and capital-skill complementarity, we impose an additional assumption on the growth rate of q , the exogenous relative price between investment in capital structures and equipt ment. Generally, we normalize variables growing along the balanced growth path by the level of technology. Fiscal variables, as mentioned above, are normalized by output. We use the notation, for instance, Y˜ ≡ Yt and b˜ ≡ Bt to denote these stationary variables where γ is the growth rate t γt t PtYt ofoutput. We alsousethenotationTC, TH, andTK todenote (real)consumption,labor, andcapt t t 18An analogous consumption tax rule, with low enough response to debt, generates similar results and is thus omittedhere. Moreover,whileweconsiderthesevariousfiscal/monetaryadjustmentscenariostoinvestigatehowboth positiveandnormativeresultsdependonpolicychoices,ouranalysisisnotintheRamseypolicytradition. 14

ital tax revenues. Nominal variables are denoted in real terms in small case letters, for instance, w = Wt. AlltheequilibriumconditionsarederivedandgivenindetailintheAppendixA.1. t Pt Weusethefollowingfunctionalformsforpreferencesandtechnology (cid:16) (cid:17)1+φ Hi U(Ci,Hi)≡logCi−ω¯i t , t t t 1+φ F (cid:0) A t ,K b,t ,K e,t ,L s,t ,L u,t (cid:1) ≡A t (cid:0) K b,t (cid:1)α (cid:104) µL u σ ,t +(1−µ) (cid:0) λ (cid:0) K e,t (cid:1)ρ+(1−λ) (cid:0) L s,t (cid:1)ρ(cid:1)σ ρ (cid:105)1− σ α , andstandardfunctionalformsfortheinvestmentadjustmentandvariablecapacityutilizationcosts (cid:32) I (cid:33) ξ (cid:32) I (cid:33)2 χ2 S t ≡ t −γ , A(u)≡χ (u −1)+ 2(u −1)2. t 1 t t I 2 I 2 t−1 t−1 The utility function is standard and consistent with balanced growth. The production function F(·)isanestedCESstructureusedinKruselletal.(2000). Thisimpliesthatequipmentcapitaland skilled labor have the same elasticity of substitution against unskilled labor, given by 1/(1−σ). Theelasticityofsubstitutionbetweenequipmentcapitalandskilledlaboris1/(1−ρ). Capital-skill complementarityexistswhenσ>ρ. Theparametersµandλgovernincomeshares. Notethatwhen ρ → 0, the production function reduces to a standard Cobb-Douglas formulation, which we will use for analytical results. Suppose that the gross growth rate of A is a¯ = γ1−α. We then assume t that the exogenous relative price between consumption (structures) and equipment investment, q , t grows at rate γ =1/γ, which leads to balanced growth in the model. It follows that all growing q variablesexcept A andequipmentcapitalgrowatrateγ.19 t 3 Long-run results This section presents our results on the long-run effects of changes in the capital tax rate. Our key focus is on the source of financing the capital tax rate cut. We consider the three different fiscalpolicyadjustmentspresentedaboveinSection2.2.3,allofwhichensurethatthegovernment debt-to-GDP ratio is at the same level in the long-run. The first is by (non-distortionary) transfer adjustment,whichwetakeasthestartingpoint. Wethenlookathowadistortionaryadjustmentof labortaxrateandconsumptiontaxratealterstheresults. 19King,PlosserandRebelo(2002)describestherequiredrestrictionsonpreferencesandtechnologyinthestandard neoclassical model. Balanced growth with capital-skill complementarity in the production function was shown in MaliarandMaliar(2011),whopointedouttheneedtohaveanexogenouspathforrelativepricebetweenconsumption (structures)andequipmentandrestrictionsonthegrowthrate. 15

3.1 Analytical results of a simplified model We start with analytical results that help clarify the mechanisms regarding the long-run aggregate effectsunderthedifferentwaysoffinancing. Forthis,wesimplifyourmodelsuchthatitconverges to a textbook business cycle model. In particular, we first assume ρ → 0 to get a nested version of the model with a Cobb-Douglas production function. It is also assumed that the two share parametersarezero,µ=α=0,andthatthefractionofskilledhouseholdsis1,NS =1. Inthiscase, themodelnowhasonetypeofcapital K andonetypeoflabor L andastandardCobb-Douglas e,t s,t production function that implies a unit elasticity of substitution between capital and labor. In the analytical results below, we then drop subscripts e and s for variables. For simplicity, there is no expensingofthetaxrate. Themechanismfortheaggregateeffectsofacapitaltaxratecutwhennon-distortionarytransfers adjust in such a textbook model is well-understood. A reduction in the capital tax rate leads to a decrease in the rental rate of capital, raising firms’ demand for capital. This stimulates investment, and the capital-to-labor ratio increases as a result. A larger amount of capital stock, in turn, makes workers more productive, raising wages and hours. Given the increase in the factorsofproduction,outputincreases,whichalsoraisesconsumptionunlessthesteady-stateratioof governmentspending-to-GDPisunrealisticallyveryhigh.20 How do these effects change when distortionary labor or consumption tax rates increase to financethecapitaltaxreform? Proposition3.1belowpresentstheresults. (cid:16) (cid:17) (cid:16) (cid:17) Proposition 3.1. Let τ¯K =τ¯K+∆ τ¯K denote a new capital tax rate with a small change ∆ τ¯K . new Also,denotethenewlaborandconsumptiontaxraterequiredtofinancethecapitaltaxratechange (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) by τ¯H = τ¯H +∆ τ¯H and τ¯C = τ¯C +∆ τ¯C , respectively, with respective changes ∆ τ¯H and new new (cid:16) (cid:17) ∆ τ¯C . The new steady-state values of a variable X¯ in transfer adjustment case, in labor tax new rate adjustment case, and in consumption tax rate adjustment case are denoted by X¯T , X¯L and new new (cid:110) (cid:111) X¯C ,respectively. For X ∈ C˜,K˜,I˜,Y˜,H new (cid:32) (cid:33) (cid:32) (cid:33) X¯T (cid:16) (cid:17) 1 1 (cid:16) (cid:17) ln new =−ΘT∆ τ¯K = ∆ τ¯H , X¯L L 1+φ 1−τ¯H new (cid:32) (cid:33) (cid:16) (cid:17) whereΘT = 1 1 λ 1+ a¯−(1−d) τ¯C >0,and L 1+φ 1−τ¯H 1−λ a¯−(1−d) β (cid:32) (cid:33) (cid:32) (cid:33) X¯T (cid:16) (cid:17) 1 1 (cid:16) (cid:17) ln new =−ΘT∆ τ¯K = ∆ τ¯C , X¯C C 1+φ 1+τ¯C new 20In such a case, government consumption or investment crowds out private consumption. See Appendix B.3, PropositionB.1fordetails,whichformallypresentstheresultsdiscussedabove. 16

(cid:16) (cid:17) whereΘT >0ifG ¯˜ <1−λθ−1 a¯−(1−d) 1−τ¯K . Moreover, C θ a¯−(1−d) new β (cid:32) (cid:33) (cid:32)(cid:32) (cid:33) (cid:32) (cid:33) (cid:33) X¯L (cid:16) (cid:17) 1 1 (cid:16) (cid:17) 1 (cid:16) (cid:17) ln new =ΘL∆ τ¯K = ∆ τ¯C − ∆ τ¯H , X¯C C 1+φ 1+τ¯C 1−τ¯H new (cid:16) (cid:17) where ΘL > 0 if G ¯˜ < 1−λθ−1 a¯−(1−d) 1−τ¯K −(1−λ)θ−11−τ¯H . If follows that under the mild C θ a¯−(1−d) new θ 1+τ¯C β (cid:16) (cid:17) conditionongovernmentspending,forasmallchange∆ τ¯K <0, X¯L <X¯C <X¯T . new new new Proof. SeeAppendixB.5. □ Proposition 3.1 implies that, generally, when the capital tax rate is cut, output, capital, investment, consumption and hours increase by more in the transfer adjustment case than in either the labor tax rate adjustment case or the consumption tax adjustment case. Moreover, generally, the consumption tax rate adjustment leads to bigger expansionary effects than the labor tax rate adjustment. Comparedtothecaseoflump-sumtransferadjustment,themacroeconomiceffectsaresmaller becauseofdistortionscreatedbythelabororconsumptiontaxrateincreasesthataffectlaborsupply negatively. Moreover, Proposition 3.1 states that the differences in the changes in output, investment, consumption, and hours across the three financing options are given by the same amount. Thisconstantdifferencedependsintuitivelyandpreciselyonthelaborsupplyparameterforagiven change in the tax rates. A higher Frisch elasticity (1) makes workers more responsive to labor tax φ ratesorconsumptiontaxrates,therebygeneratinggreaterdistortions,whichinturn,magnifiesthe difference. The three fiscal adjustments produce the same outcomes when labor supply is completely inelastic (1 =0). Moreover, as is intuitive, higher is the initial level of the labor tax rate, φ bigger is the difference. Thus, for the same change in the labor tax rate, if the initial labor tax rate ishigher,theincreaseinoutput,investment,consumption,andhourswillberelativelysmaller. When it comes to comparing the labor tax rate to the consumption tax rate adjustment, the latterislessdistortionaryforlaborhours,therebygeneratingbiggeraggregateeffects. Infact,with β=1, steady-state hours do not depend at all on the consumption tax rate, but depend negatively on the labor tax rate.21 This interesting finding will play a critical role when we later conduct a welfareanalysisinSection5. 21SeeAppendixB.5fordetails. Thisresultarisesbecauseofthecompleteoffsetofsubstitutionandincomeeffects. Moreover,ifthereisnocapitaltaxrateinthemodel,thentherestrictionthatβ=1isnotneeded. Thiscasehowever, isnotrelevantforourpaperasthefocusisontheresponseoftheeconomytochangesinthecapitaltaxrate. 17

3.2 Numerical results of the quantitative model Wenowpresentthenumericalresultsemployingourbaselinequantitativemodelwithheterogeneityandskillpremium. 3.2.1 Parameterization The frequency of the model is a quarter. Table 1 contains numerical values we use for the parameters that are relevant for long-run effects. The parameterization is standard, and we provide detailed justification and references in Table 1. As given above, we use separable preferences that imply log utility in consumption and then calibrate a modest, unit Frisch elasticity of labor supply (1=1) based on Smets and Wouters (2007). For the production function elasticity of substitution φ parameters, we use the estimates in Krusell et al. (2000) (σ = 0.401,ρ = −0.495). This parameterization implies (equipment) capital-skill complementarity. We also follow Krusell et al. (2000) in matching the income share of structure (α=0.117) as well as the depreciation rates of the two types of capital. For the income share of equipment, we pick parameter value to get a steady-state aggregate labor income share of 0.56, which is consistent with estimates in Elsby, Hobijn and S¸ahin(2013)andOhanian,OrakandShen(2021). We use the Monthly Outgoing Rotation Group (MORG) of the US Census Bureau’s Current Population Survey data to calculate both the fraction of unskilled households and the skill premium. The skill premium is defined as the ratio of the hourly wage of workers with 14 or more years of schooling to the hourly wage of workers with less than 14 years of schooling. The share of workers with 14 or more years of schooling is 0.505 and the skill premium for median wage is 55%. Wethensettheincomeshareofunskilledhouseholds(µ=0.359)tomatchtheskillpremium observedindata. Additionally, across various fiscal adjustment scenarios and preference and technology functionsspecifications,wenormalizehoursforskilledlabortobe0.330andhoursforunskilledlabor to be 0.307 in the initial steady-state by appropriately adjusting the scaling parameters ω¯s and ω¯u. We follow the calibration of Lindquist (2004) for this choice of steady-state hours as well as the fractionofskilledlabor(Ns=0.5). Thesteadystateofthefiscalvariablessuchasthedebt-to-GDPratio,thegovernmentspendingto-GDPratio,andthetaxes-to-GDPratio,ismatchedtotheirrespectivelong-runvaluesinthedata. AppendixCdescribesthisdataindetail. Wethencalibratethesteady-statemarkuptoobtaina35% capital tax rate initially. The implied initial levels of labor tax rate and consumption tax rate are 12.8% and 0.9% respectively. For the effective expensing rates of the two types of capital, we use the estimates in Barro and Furman (2018), which imply lower expensing of structure investment. We assume that the profit shares for skilled labor (χs ) is 1 and the transfers share for unskilled Φ 18

Table1: Calibrationforlong-runanalysis Value Description References Households β 0.9975 Timepreference SmetsandWouters(2007) φ 1.0 InverseofFrischelasticityoflaborsupply SmetsandWouters(2007) ω¯s 4.67 Laborsupplydisutilityparametersforskilledhouseholds Steady-stateH¯s=0.33 ω¯u 9.90 Laborsupplydisutilityparametersforunskilledhouseholds Steady-stateH¯u=0.31 Ns 0.5 Fractionofskilledlabor Data(SeeAppendixC) de 0.031 Equipmentcapitaldepreciation Kruselletal.(2000) db 0.014 Structurescapitaldepreciation Kruselletal.(2000) Firms σ 0.401 Elasticityofsubstitutionbetweenunskilledlaborandequipment Kruselletal.(2000) ρ -0.495 Elasticityofsubstitutionbetweenskilledlaborandequipment Kruselletal.(2000) α 0.117 StructurescapitalIncomeshare Kruselletal.(2000) Steady-statelaborshare:56% λ 0.35 Equipmentcapitalincomeshare (Ohanian,OrakandShen,2021) Steady-stateskillpremium:55% µ 0.359 Unskilledlaborincomeshare Data(SeeAppendixC) γ 1.0054 Long-rungrowthrateofoutput SmetsandWouters(2007) π¯ 1.0078 Steady-stateinflationrate SmetsandWouters(2007) Steady-statemarkup:33% θ 4.0 Elasticityofsubstitutionbetweengoods (Hall,2018) q 0 0.95 Relativepriceofstructuretoequipmentcapital MaliarandMaliar(2011) Government b ¯˜ 0.363 Steady-statedebttoGDPratio Data(SeeAppendixC) G ¯˜ 0.161 Steady-stategovernmentspendingtoGDPratio Data(SeeAppendixC) T ¯˜C 0.009 Steady-stateconsumptiontaxrevenuetoGDPratio Data(SeeAppendixC) T ¯˜H 0.128 Steady-statelabortaxrevenuetoGDPratio Data(SeeAppendixC) χs 1 Fractionofprofitdistributiontoskilledworker Assigned Φ χs 0 Fractionoftransfertoskilledworker Assigned S λb 0.338 Effectiveexpensingrateofstructureinvestment BarroandFurman(2018) λe 0.812 Effectiveexpensingrateofequipmentinvestment BarroandFurman(2018) labor(χu)is1.22 S WepresentdetailedsensitivityanalysisofourbaselineparameterizationinSection6. 3.2.2 Numericalresults The results for long-run effects under various financing policies are in Figure 1. While our focus is on a reduction of the capital tax rate from 35% to 21%, which are clearly shown with colored dotsintheFigure,weshowtheentirerangeoftaxratechangesforcompleteness. Let us first look at the case of transfer adjustment presented by the blue lines in Figure 1. A decrease in the capital tax rate reduces total (tax) revenues-to-GDP ratio (driven by decrease in capital tax revenue-to-GDP ratio). The reduction in tax revenues is then financed by a decline in 22WediscusshowresultsmightchangewithalternateassumptionslaterinasensitivityanalysisinSection6. 19

10 40 0 5 0 20 0 -10 0 -20 -20 -20 -40 -5 -40 -60 -30 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0.34 0.34 10 0 0.32 0 0.335 -10 0.3 -10 0.33 0.28 -20 -20 0.325 0.26 -30 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 50 10 50 0 0 0 0 -10 -10 -20 -50 -50 -30 -100 -20 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 7 40 8 0 20 6 0 5 6 -20 -20 -40 4 4 -40 -60 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 4 10 15 2 10 0 -2 10 5 -4 5 -6 5 0 -8 0 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 Figure1: Long-runeffectsofpermanentcapitaltaxratechanges 20

transfers-to-GDP ratio from 1.0% to -0.49% as shown in the last panel of the Figure 1.23 This need to engage in lump-sum tax to finance the capital tax reform is unrealistic, and motivates our analysisofdistortionaryfinancinglater. In terms of aggregate implications, for a reduction of the capital tax rate from 35% to 21%, output increases by 4.24% relative to the initial steady state, structure investment by 20.24%, and equipment investment by 6.27%.24 These effects are illustrated respectively in the fifth, fourth, and third panels of Figure 1. The aggregate effects are driven by the same mechanism as in the simple model we discussed earlier, where the rental rate of capital falls, which leads to a boost in thecapital-laborratioandinvestment. Let us now look at distributional implications. First, the skill premium or wage inequality rises following a capital tax rate cut as skilled wages increase by more than unskilled wages. In particular, the former increases by 4.66% while the latter increases by only 0.56%. Accordingly, theskillpremiumgoesupby6.32%points(shownintheeleventhpaneloftheFigure1).25 Second, income inequality, measured by the the ratio of after-tax capital-to-labor income (shown in the tenthpanel),unambiguouslyincreases—althoughbothtypesofincomeincrease. Third,theserises in wage and income inequality, coupled with the fall in transfers which are all distributed to the unskilled in our baseline calibration, result in an increase in consumption inequality. Specifically, consumptionoftheunskilledfallsby3.86%whileconsumptionoftheskilledrisesby4.82%. Asa result,therelativeconsumptionoftheskilledvs. theunskilled,increasesinthelong-runby27.7% (shown in the twelfth panel). Notice that the rise in consumption inequality generates differential wealth effects on labor supply. Consequently, skilled hours decreases slightly while unskilled hours increases (shown in the sixth and seventh panels).26 The increases in wage, consumption, and income inequality can be considered as caveats to the effectiveness of the capital tax rate cut inourmodel,evenwhenlump-sumtransfersareallowedtofinancethetaxcut. To understand the mechanisms behind these increases in inequality, we can express the skill 23Note that this result is obtained not only because output (i.e. the denominator) increases. In fact, the total tax revenuesalsodecline. Inparticular,thereisasignificantdecreaseincapitaltaxrevenues(about44%declinerelative to the initial steady state), which is only partially offset by an increase in consumption and labor tax revenues. The governmentthereforefinancessuchadeficitbytakingresourcesawayfromthehousehold:transfersdeclinebyroughly 149%oftheinitialsteady-state. Thereisa“Laffercurve”forcapitaltaxrevenuesbutthecapitaltaxrevenuestartsto declineatveryhighandempiricallyirrelevantrange,suchasabove90%inourbaselinecalibration. 24Forcomparison,BarroandFurman(2018)predictthatthelong-runincreaseinoutputwillbe3.1%forapermanent capital tax rate cut from 38% to 26%, assuming that the employment-to-population ratio is fixed. Unlike their analysis,wemodelanendogenouslaborsupplydecision,whichisespeciallyimportantwhenconsideringdistortionary financing. 25InAppendixB.2,weshowanalyticallyhowtheskillpremiumincreaseswiththecapitaltaxratecutinourmodel. 26Additionally,wefindthatstructureinvestmentincreasesbymorethanequipmentinvestment. Quantitatively,the majorreasonforthisfindingisthelowerexpensingrateonstructureinvestmentinourcalibration.Qualitatively,arole isalsoplayedbythefactthatintheproductionfunction,theelasticityofsubstitutionbetweenequipmentinvestment andskilledhoursmakesthemcomplements. 21

premiuminourmodelas Ws (1−µ)(1−λ) (cid:32) (cid:32) K (cid:33)ρ (cid:33)σ ρ −ρ (cid:32) L (cid:33)1−σ t = λ e,t +(1−λ) u,t . Wu µ L L t s,t s,t Thus,ifacapital-skillcomplementarity(σ>ρ)exists,theskillpremiumincreasesintheamountof equipment capital when the quantities of the two types of labor inputs are held fixed. This mechanism—whichisoperativeeveninamodelwitharepresentativehouseholdorperfectconsumption insurance—drives our result on the skill premium. On top of this well-known mechanism, in our model with heterogeneous households, the rise in consumption inequality affects labor hours in a way such that wage inequality increases even further. To see this, first notice that the skill premiumisincreasingintheratioofunskilled-to-skilledlaborintheexpression. Asstatedabove,the unskilled-to-skilled labor ratio increases in response to a capital tax cut because the rise in consumptioninequalityproducesdifferentialwealtheffectsonlaborsupply. Furthermore,thedecrease in skilled hours raises the equipment capital-to-skilled labor ratio, which in turn generates a further increase in wage inequality. The two sources of heterogeneity we introduce into an otherwise textbookmacroeconomicmodel(skillheterogeneityandhouseholdheterogeneity)thusinteractin economicallymeaningfulwaystogeneratenewdistributionaleffects. Now, we contrast the results when labor tax rate, instead of transfers, adjusts to finance the capital tax rate cut. These results are illustrated by the red lines in Figure 1. In this case, labor tax rates,inthelong-run,havetoincreasefrom22.7%to25.4%tofinancethereductionofthecapital tax rate from 35% to 21%. While the capital tax cut continues to be expansionary, the increase in output and investment is now less under labor tax rate adjustment—as is consistent with what we showed in Proposition 3.1 in the simplified model. In particular, for the baseline experiment of a reduction of the capital tax rate from 35% to 21%, output increases by 2.08%, equipment investment by 17.75%, and structures investment by 4.81%. The reason for the smaller boost in aggregate variables is fall in hours of both the skilled and the unskilled. There is now a smaller increaseintheafter-taxwagesforskilledworkers,whichleadstoabiggerdecreaseinlaborhours in the long-run, from 0.330 to 0.327. For unskilled workers, there is in fact a decline of the aftertax wages, by 1.45%, which leads to a smaller increase in labor hours in the long-run. The labor supply response of the unskilled is qualitatively different from the lump-sum transfer adjustment case,asnowthewealtheffectonlaborsupplyduetodecreaseintransfersisnolongerinoperation. Overall,thedecreaseinskilledandunskilledhoursdampenstheexpansionaryeffectofcapitaltax cutsonoutputandinvestment. Turning to distributional effects, both wage and consumption inequality increase, as they did in the transfer adjustment case. In comparison to the transfer adjustment case however, these distributional effects are smaller. The main reason is that the burden from the labor tax increase is 22

shared by both types of households, whereas transfer reduction only affects the unskilled. Therefore, consumption of the unskilled does not fall as much while consumption of the skilled does not increase as much now. The reduced consumption inequality in turn diminishes the role of the wealtheffectonlaborsupply,leadingtoasmallerincreaseintheskillpremium.27 We finally analyze the case when consumption tax rate increases in the long-run to finance the capital tax rate cut. In this case, to finance the reduction of the capital tax rate from 35% to 21%, consumption tax rates have to increase from 1.29% to 3.49%. Generally, as we emphasized before in Proposition 3.1 for the simple model, the effects are qualitatively similar to labor tax rate adjustment, with the main distortion again coming in labor supply decisions, which leads to a smallerexpansionaryeffectonoutputcomparedtotransferadjustment. One important qualitative difference compared to both transfer and labor tax rate adjustment can be seen in consumption of the unskilled household, which now increases slightly (shown in thesecondpanelofFigure1). Asinthelabortaxrateadjustmentcase,theburdenonconsumption tax rate increase is shared by both types of households, and thus the unskilled household benefits compared to transfer adjustment case. Compared to labor tax rate adjustment, consumption tax rateadjustmentismorebeneficialfortheunskilledastheaggregateoutputeffectsarebigger. This numerical result is consistent with what we showed in Proposition 3.1 in the simplified model in which consumption taxes are relatively less distortionary for labor supply. As transfers-to-GDP is fixed in the long-run, a greater output expansion implies that transfers (in level) will increase by morenow,whichallowstheunskilledtosustainahigherlevelofconsumptioninthelong-run. 4 Transition dynamics Wenowdiscusstransitiondynamicsassociatedwithapermanentcapitaltaxratecut,from35%to 21%, by tracing out the evolution of the economy from the initial steady-state to the new steadystate. Studyingtransitiondynamicsisimportantaswefindthatittypicallytakesaquitelongtime, around 156 quarters, for consumption to converge to a new steady-state following a permanent reductioninthecapitaltaxrate. Thisallowsusinparticulartoanalyzeshort-runeffects,whichare thefocushere. Asinthelong-runanalysisintheprevioussection,wepresentthebaselinemodelunderdifferent policy adjustments. Compared to the long-run analysis, we pay special attention to the role of monetary policy, which can be potentially important due to nominal rigidities in the short-run. A newthemewehighlightthusinthissectionishowajointanalysisofmonetaryandfiscalpolicyis importanttounderstandtheshort-runeffectsofapermanentcapitaltaxratechange. 27Finally, our measure of income inequality also continues to increases, but by more here compared to transfer adjustment,asafter-taxlaborincomenowdecreases. 23

Table2: Calibrationfortransitiondynamics Value Description References Households ξ 4.0 Investmentadjustmentcost SmetsandWouters(2007) ′′ A 0.85 Elasticityofcostofcapitalutilization SmetsandWouters(2007) A′ Firms α 0.65 Calvostickypriceparameter SmetsandWouters(2007) P γ 0.22 Degreeofpriceindexation SmetsandWouters(2007) P Government(Fiscal/MonetaryPolicy):TransferorLaborTaxRateAdjustment ρR 1.12 Interestratesmoothingparameterlag1 CoibionandGorodnichenko(2011) 1 ρR -0.18 Interestratesmoothingparameterlag2 CoibionandGorodnichenko(2011) 2 ϕ 1.58 InflationfeedbackparameterunderTaylorrule CoibionandGorodnichenko(2011) π ϕ 0.11 OutputgapfeedbackparameterunderTaylorrule CoibionandGorodnichenko(2011) x ϕ∆y 2.21 OutputgrowthfeedbackparameterunderTaylorrule CoibionandGorodnichenko(2011) ρH 0.869 Transfer/Labortaxratesmoothingparameterlag1 Estimated(SeeAppendixD) 1 ρH 0.0 Transfer/Labortaxratesmoothingparameterlag2 Estimated(SeeAppendixD) 2 ψH 0.111 Transfer/Labortaxrateresponsetodebt Estimated(SeeAppendixD) B ψH 0.831 Transfer/Labortaxrateresponsetooutputgrowth Estimated(SeeAppendixD) ∆y ψH 0.0 Transfer/Labortaxrateresponsetooutputgap Estimated(SeeAppendixD) x Government(Fiscal/MonetaryPolicy):LaborTaxRateandInflationAdjustment ρR 1.12 Interestratesmoothingparameterlag1 CoibionandGorodnichenko(2011) 1 ρR -0.18 Interestratesmoothingparameterlag2 CoibionandGorodnichenko(2011) 2 ϕ 0.95 InflationfeedbackparameterunderTaylorrule Assigned π ϕ 0.0 OutputgapfeedbackparameterunderTaylorrule Assigned x ϕ∆y 0.0 OutputgrowthfeedbackparameterunderTaylorrule Assigned ρH 0.785 Labortaxratesmoothingparameterlag1 Estimated(SeeAppendixD) 1 ρH 0.107 Labortaxratesmoothingparameterlag2 Estimated(SeeAppendixD) 2 ψH 0.001 Labortaxrateresponsetodebt Estimated(SeeAppendixD) B ψH 1.821 Labortaxrateresponsetooutputgrowth Estimated(SeeAppendixD) ∆y ψH 0.0 Labortaxrateresponsetooutputgap Estimated(SeeAppendixD) x 4.1 Parameterization For transition dynamics, the parameterization of policy rules, investment adjustment costs, nominal rigidities, and capacity utilization costs matters. The parameterization is in Table 2. We use estimates from Smets and Wouters (2007) for investment adjustment costs, capacity utilization costs,theprobabilityofresettingprices,anddegreeofinflationindexation. We now discuss how we parameterize the policy rules, which govern the associated fiscal and monetary adjustments along the transition. We use estimates from Coibion and Gorodnichenko (2011) for monetary policy rule parameters from the post-Volcker period (1983-2002), in which the Taylor principle is satisfied. For the transfer and tax rule parameters, we use our estimates of thepolicyrule(2.4)usingUSdata. ThedetailsoftheestimationarepresentedinAppendixDand 24

TableF.1showstheestimationresults. We point out here that the transfer and tax rule parameters we use for the case where transfer and taxes respond sufficiently to debt are based on estimates obtained using post-Volcker period dataforthesameperiodasinCoibionandGorodnichenko(2011)(1983-2002). Thiscombination, using data and estimates for the exact time period, then describes the regime where labor tax rates adjust to ensure stationary debt dynamics while monetary policy stabilizes inflation. Next, the tax ruleparametersweuseforthecasewheretaxesdonotrespondsufficientlytodebt,whileinflation plays a role in debt stabilization in the short-run, are based on estimates obtained using data from a later period (2001-2019). In this latter case, the monetary policy rule does not satisfy the Taylor principle and we parameterize it accordingly.28 In this regime, inflation plays a partial, but direct role,indebtstabilization. 4.2 Four different fiscal/monetary adjustments We now consider four different fiscal/monetary policy adjustments, as described in Section 2.2.3. Inparticular,anewpolicyresponsethatweconsiderhereisonewhereinflationplaysapartial,but direct,roleindebtstabilization. 4.2.1 Lump-sumtransferadjustment Once again, the starting point is the case of transfer adjustment, shown by the blue lines in Figure 2.29 What makes the short-run distinct from the long-run is that in principle, capital tax cuts can nowgenerateacontractionaryaggregateeffectduringthetransitionperiods. The results can be best understood as depicting transition dynamics when the capital stock is initially below the new steady-state. As mentioned before, a reduction in the capital tax rate leads to a decrease in the rental rate of capital, thereby facilitating capital accumulation via more investment. In the short-run, to finance this increase of investment, consumption of skilled declines initially. Moreover, there is a large decline along the transition in consumption of the unskilled. This is because of the large dynamic decline in transfers. Given this postponement of consumption,combinedwithstickyprices,outputalsofallstemporarily,beforerisingtowardsthehighnew steady-state. Thetemporarycontractioninoutputisaresultofstickypricesandinvestmentadjustment costs, which renders output (partially) demand-determined and markups countercyclical in 28In particular, we use ϕ π <1, ϕ∆y =0, andϕ x =0. We do not formally estimate the monetary policy rule for this sub-period due to the binding ZLB for much of the sample and the lack of enough observations. Passive monetary policyarguablycharacterizesthisperiodwell. Ourresultsarerobusttothepreciseparameterizationofϕ aslongasit π isbelow1,asweshowinSection6. 29ThedashedstraightlinesintheFigure2representthenewsteadystaterelativetotheinitialsteadystate. 25

0 4 6 30 -2 2 -4 4 20 0 -6 2 10 -8 0 0 -2 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0.33 4 0.33 0.325 4 2 0.325 0.32 2 0 0.32 0.315 0 0.31 -2 0.315 -2 -4 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 30 2 6 30 0 20 4 20 -2 10 2 10 -4 0 0 0 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 3.9 3 6 2.9 15 3.8 2.8 4 10 2.7 2 5 3.7 2.6 0 0 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 1 3.5 15 44 0 0.5 3 42 -0.5 2.5 0 14 40 -1 -0.5 38 2 13 -1.5 -1 36 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 Figure2: Transitiondynamicsofapermanentcapitaltaxratedecreaseunderalternatefinancing 26

the model.30 The temporary fall in output (which is coupled with increased capital stock), in turn leads to fall in hours of the skilled. For the unskilled workers however, because of the effects on marginal utility, they work more along the transition. This increased labor supply by the unskilled mitigatesthefallinoutputthatwouldoccurotherwise. Inflationalsodeclines. Itisdeterminedbyforwardlookingbehavioroffirms,andthusdepends oncurrentandfuturerealmarginalcosts. Asrealmarginalcostsareafunctionofwagesandcapital rental rate, the path of inflation roughly follows that of these factor prices. The decrease in wages is driven by both supply and demand forces. The drop in consumption and the rise in marginal utility of consumption raise the supply of hours for a given wage rate, which plays an important role for unskilled labor response. On the other hand, labor demand declines as firms produce a smalleramountofoutputasdiscussedabove. In terms of inequality, Figure 2 shows that the skill premium and consumption inequality increase in the short-run and slowly converge to the new steady state. The capital-to-labor income ratioalsoincreasesintheshort-run,abovethenewlong-runlevel. Moreover, the long-run positive effects of capital tax cuts come at the expense of short-run decline of labor tax revenue—even under lump-sum transfer adjustments. Furthermore, the decrease in labor income requires a larger adjustment of transfers. Transfers fall sharply and in fact, have to go negative after a few periods (due to the smoothing feature of our transfer policy). This needstoengageinlump-sumtaxesisarguablyunrealistic,andmotivatesourstudyofdistortionary financingnext. 4.2.2 Labortaxrateadjustment Next, we analyze the case of labor tax rate increase, which is shown by the red lines in Figure 2. Here, labor tax rate evolves according to the tax rate rule, (Equation 2.4), given in Section 2.2.3. Overall,modeldynamicsarequalitativelysimilartothoseinthetransferadjustmentcase. Westill see capital accumulation, achieved by increased investment and postponement of consumption, whichinturnalsocausesoutputtofallwithstickyprices. Quantitatively, however, the drop in consumption and output is larger in this case compared to the lump-sum transfer adjustment case. As in the lump-sum transfer adjustment case, delayed consumption decreases hours by lowering firm’s labor demand. In addition, increased labor tax rate decreases hours even further by discouraging workers from supplying labor. Consequently, hours in equilibrium fall by more, of both the skilled and the unskilled. In fact, in contrast to the transfer adjustment case, unskilled hours now decreases. This is due to the same mechanism as in the long-run we discussed above. The wealth effect on labor supply is relatively weak for the 30Themonetarypolicyrulespecificationalsoplaysarolequantitatively. 27

unskilled because there is no reduction in transfers and the burden of labor tax rate increases is shared by both types of households. Therefore, the substitution effects dominate. Overall, the fall inhoursofbothtypesinturnamplifiestheshort-runcontractioninconsumptionandoutput. In terms of distributional implications, consumption inequality increase is less pronounced compared to the transfer adjustment case. This fiscal adjustment is relatively more beneficial for theunskilledasitdoesnotfeatureadeclineintransfers.31 Thelabortaxincreaseburdenisshared bybothtypes,whereastransferreductiononlyaffectstheunskilled. Note that the dynamics associated with labor tax rate adjustment are fairly close, especially in the very short-run, compared to lump-sum transfer adjustment. This is driven by our use of an empirically driven tax rule where tax rates adjust smoothly, allowing higher-than-normal debt-to- GDP ratio. To highlight this, in Figure 2 we additionally consider a case where labor tax rates adjust as necessary to maintain a constant debt-to-GDP ratio throughout the transition. In that case, the short-run contraction is clearly more severe compared to lump-sum transfer adjustment, driven by more rapid increases in the labor tax rates that has a strong negative effect on labor hours—especiallyoftheskilledhouseholds. 4.2.3 Labortaxrateandinflationadjustment The results are quite different, even qualitatively, in the case where labor tax rate increases, but not sufficiently, and inflation partly plays a role in government debt stabilization, as described in Section 2.2.3.32 Figure 3 shows these results, where for comparison we also show the pure labor taxrateadjustmentcasediscussedjustabove. The main difference now, compared to the pure labor tax adjustment analysis, is that there is a short-run burst of inflation to help stabilize debt. This increase in inflation, as the model has nominal rigidities, helps counteract the short-run contractionary effects. Output, consumption, investment, hours, and wages, in fact, all increase in the short-run and the differences are quite pronounced for all variables other than investment. After 8 quarters or so, the transition dynamics become very similar to the pure labor tax rate adjustment case. Interestingly, debt-to-GDP ratio, in sharp contrast to other fiscal adjustment cases, decreases for extended periods due to the rise in outputandthepricelevel. On the distribution side, a new result emerges. As we discussed above, labor tax rate adjustment,comparedtotransferadjustment,isrelativelymorebeneficialfortheunskilledasitdoesnot feature a decline in transfers and the labor tax increase burden is shared by both types. Generat- 31Labortaxrateschangesmoothlyhere,asdotransfers. 32Notethatinthiscase,themonetarypolicyrule(2.2)doesnotsatisfytheTaylorprinciple,whichiscoupledwith a weak response of the tax rate in the tax rule (2.4). Clearly, we can analyze a similar fiscal adjustment case where inflationplaysaroleindebtstabilizationevenwithlump-sumtransferadjustment. Whennon-distortionarysourcesof revenueispossible,allowinginflationtoplayaroleindebtstabilizationmightnotbeaveryinsightfulexperiment. 28

80 30 1 60 4 20 0 40 2 -1 20 10 0 -2 0 0 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 30 0.311 0.45 40 20 0.31 0.4 20 10 0.309 0.35 0 0.308 0 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 80 0 0 60 20 40 -10 -50 10 20 -20 -100 0 0 -30 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 6 60 5 3.5 4 40 4.5 3 2 20 4 0 0 2.5 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 5 22 8 20 40 6 4 18 4 30 3 16 2 14 0 2 20 12 -2 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 Figure 3: Transition dynamics of a permanent capital tax rate decrease under labor tax rate and inflation adjustment 29

ing inflation, on top of that, favors the unskilled even more as government debt, whose real value deteriorates during the transition, is owned only by the skilled. Thus, while both the skilled and unskilled household’s consumption rises due to a smaller increase in the labor tax rate with rising inflation, the latter rises relatively more. The decrease in consumption inequality in turn leads to a decline in skill premium in the short-run due to the wealth effect on labor supply. These results onconsumptionandwageinequalityareincontrastwiththoseobtainedunderothertypesoffiscal adjustments. 4.2.4 Consumptiontaxrateadjustment For completeness, we next study transition dynamics for the case of consumption tax rate adjustment. We show the results in Appendix Figure F.1 in Appendix F, where we use the same policy rule parameters as for the labor tax rate adjustment. The transition dynamics associated with the labortaxrateandconsumptiontaxrateadjustmentareverysimilar. 5 Welfare implications While the focus of this paper is not necessarily on normative policy issues, we can nevertheless evaluate welfare implications of the permanent capital tax rate cut from 35% to 21%. Our results in the previous sections suggest that a reduction of the capital tax rate has different welfare implications depending on time horizon, household types, and policy adjustments. In this section, we formally calculate a measure of welfare gain that can be achieved through a permanent capital tax cut,takingintoaccounttransitiondynamicsaswellasthelong-runeffect. 5.1 Welfare measure Our measure of welfare gain for type-i agent, µi , is implicitly defined by the sequence of values k,t satisfying (cid:88)t (cid:16) (cid:17) (cid:88)t (cid:16)(cid:16) (cid:17) (cid:17) βjU C˜i,Hi = βjU 1+µi C ¯˜i,H¯i , j j k,t j j j=0 j=0 (cid:110) (cid:111) (cid:110) (cid:111) where C˜i,Hi and C ¯˜i,H¯i arerespectivelythetimepathoftype-iagent’snormalizedconsumption j j j j andhourswithandwithoutacapitaltaxcutunderthevariousfiscal/monetaryadjustments(indexed by k) we have considered above. We denote by µi the case of transfer adjustment, by µi the T,t H,t case of labor tax rate adjustment, and by µi the case of consumption tax rate adjustment. Thus, C,t µi measureswelfaregainsfromperiod0,whenthetaxreforminitiates,till(arbitrary)periodt,in k,t 30

units of a percentage of the level of normalized initial consumption.33 The lifetime (total) welfare gain is then measured by lim µi , which is of interest in the business cycle literature (Lucas t→∞ k,t 1987). 5.2 Welfare results Panel (a) of Figure 4 shows welfare results for a reduction in the capital tax rate from 35% to 21% when transfers adjust. It is clear that in this case, the tax reform does not lead to a Pareto improvement: theskilledgainattheexpenseoftheunskilledbecausethelattertype,aswepointed outearlier,consumesless,whichinturnalsoforcestheagenttoworkmorethroughwealtheffects on labor supply. Specifically, the lifetime welfare gains under transfer adjustment amounts to 4.59% of the initial consumption level for the skilled and negative 8.24% for the unskilled.34 Turningtotheshort-run,theskilledworker’swelfareneverbecomenegativewhileincontrast,for theunskilled,µu neverbecomepositive. T,t ThetaxreformnotleadingtoaParetoimprovementisalsotrueunderlabortaxrateadjustment, as shown in panel (b) of Figure 4. Moreover, when labor tax rates adjust, compared to transfer adjustment, while welfare gains are smaller for the skilled in the long-run and µ for the skilled H,t becomes positive only 80 quarters after the onset of the capital tax reform, welfare losses are in fact smaller for the unskilled. Labor tax adjustment works better for the unskilled as the labor tax increase burden is shared by both types, whereas transfer reduction only affects the unskilled. We discussed the same mechanism for the consumption effects in the long-run in Section 3.2.2. This finding implies that lump-sum transfer adjustment, while leading to a higher level of aggregate output,isnotnecessarilyabetterpolicyresponsethanlabortaxrateadjustmentinourmodel. Compared to the labor tax rate adjustment, the results are different for consumption tax rate adjustment, as shown in panel (b) of Figure 4. In this case, the capital tax reform leads to a Pareto improvement as the unskilled workers also gain in terms of lifetime welfare. The main reason for this result is that the consumption of the unskilled goes up in the long-run, as we discussed in Section3.2.2,duetoalargeraggregateoutputeffectcomparedtolabortaxrateadjustment. While life-time welfare is higher for the unskilled workers, there is welfare loss in the short-run: µ for C,t theunskilledbecomespositiveonly264quartersaftertheonsetofthecapitaltaxreform. 33Itthusmeasureswelfaregainsatthepointwhentheagentsaretquartersold. 34ThelifetimewelfaregainsarepresentedbythedottedlinesintheFigure4. 31

(a)TransferAdjustment 4.59 0 0 -8.24 -10 -10 0 50 100 150 200 0 50 100 150 200 (b)LaborTaxRateandConsumptionTaxRateAdjustment 1.24 1.05 0 0.14 -1.37 -4 -4 0 50 100 150 200 0 50 100 150 200 Figure4: Welfareimplicationsofapermanentcapitaltaxratedecrease 6 Sensitivity analysis Before concluding the paper, we present some important sensitivity analyses. All the results from thissectionaredescribedindetailinAppendixEandwediscussthemsuccinctlybelow. 6.1 Long-run results We start with long-run effects. First, we present comparative statics results with respect to Frisch elasticity of labor supply. This is an important parameter, given that different sources of financing imply different labor supply responses. We show how a higher Frisch elasticity leads to larger outputeffectsundertransferadjustmentswhilethereverseholdsunderlabortaxadjustments. Next, 32

wecomparethetwofiscaladjustmentcasesvaryingthevaluesofFrischelasticity. Consistentwith Proposition3.1,thedifferencebetweenthetwocasesisbiggerforahigherFrischelasticity. As there are heterogeneous agents in our model, clearly the assumptions made on how profits and transfers are distributed across the two types of households makes a non-trivial difference for distributional variables. We show long-run results under various combinations of these distributions. For instance, if the skilled workers receive both the profits and (cut in) transfers, it leads to a decline in consumption inequality, in sharp contrast to the baseline case. The results also show that aggregate effects on output and investment however, are relatively similar across the various possibilitiesforprofitsandtransferdistributions. Finally, we do a sensitivity analysis on the equipment capital share parameter, λ. Our calibrationstrategyforthisparameteristomatchthelaborshareandnowwevarythetargetedlaborshare to both higher and lower values than the baseline. We find that the results are robust qualitatively overall and for aggregate variables, the quantitative differences are small. For consumption, there aresomequantitativedifference,asasmallerλismorebeneficialfortheunskilled. 6.2 Transition dynamics Wenextturntotheshort-runanalysis. Wecomparethetransitiondynamicsunderthelabortaxand inflation adjustment finance scheme for different inflation feedback parameters in the Taylor rule (the inflation feedback parameter has to be below 1 in this regime). Our results are very robust. Thedifferencesacrosstheparameterizationsshowupmostclearlyininflationanddebtdynamics, withastrongerTaylorrulecoefficientinfactleadingtoabiggereffectoninflationdynamically. 35 We then consider transition dynamics under transfer adjustment with two different rules for profit and transfer distributions, one the baseline and the other where the skilled workers get both the profits and (cut in) transfers. Again, as in the long-run, the differences are less prominent in outputeffects,butshowupmoreprominentlyindistributionalvariables. Forinstance,consumption of the unskilled falls for a short-period only, whereas consumption of the skilled falls persistently, thereby leading consumption inequality to actually fall after a few periods. Moreover, the same dynamicpatternholdsforwageinequality,whichfallsafterafewperiods. 6.3 Welfare results We end with welfare results under various values of Frisch elasticity of labor supply. We show that our main finding that transfer and labor tax adjustments do not lead to a Pareto improvement, but consumption tax adjustment in fact does, is robust to both a higher and lower Frisch elasticity than our baseline value. Next, we conduct a sensitivity analysis on the equipment capital share 35ThisisconsistentwiththeanalyticalresultsforthesimplestickypricemodelinBhattarai,LeeandPark(2014). 33

parameter, λ, and again find that our main finding that transfer and labor tax adjustments do not leadtoaParetoimprovement,butconsumptiontaxadjustmentinfactdoes,isrobust. 7 Conclusion We study aggregate, distributional, and welfare effects of a permanent reduction in the capital tax rate in a quantitative equilibrium model with capital-skill complementarity and household heterogeneity. Suchataxreformleadstoexpansionarylong-runaggregateoutputandinvestmenteffects, but those are coupled with increases in wage, consumption, and income inequality. The expansionary aggregate effects in the long-run are smaller when distortionary labor or consumption tax rateshavetoincreasetofinancethecapitaltaxratecut. We study transition dynamics and show that there are contractionary effects in the short-run, withafallnotjustinconsumptionofboththeskilledandunskilled,butalsoaggregateoutput,and which is in turn coupled with increases in both wage and consumption inequality. The short-run contraction in consumption and output are more severe under distortionary tax rate adjustment. Importantly, we show that joint modeling of monetary and fiscal policy response is critical for analyzing short-run effects. In particular, when the government has access only to distortionary labortaxes,weconsiderthecentralbankdirectlyaccommodatinginflationtofacilitategovernment debt stabilization along the transition. In this interesting scenario, the government does not raise labor tax rates as much, and the rise of inflation in the short-run completely negates any short-run contractioninoutputaswellasconsumption. Finally, we show that the capital tax cut has different welfare implications for each type of household depending on time horizon and policy adjustments. In the long-run, the tax reform increases the life-time utility of the skilled households under all the financing schemes considered, whereas it decreases the life-time utility of the unskilled households under lump-sum transfer and labor tax rate adjustments. The tax reform benefits the skilled households the most when transfers adjust, whereas the unskilled households prefer distortionary financing to avoid a significant reduction in transfer incomes. Importantly, we find that increasing consumption tax rate leads to a Pareto improvement. In the short-run, distortionary financing leads to short-term welfare losses forboththeskilledandtheunskilled. We could extend our analysis in future work in a couple of directions. Our analysis of the short-run and the long-run suggests that the proposed tax reform will have heterogeneous effects on different generations. Thus, exploring generational heterogeneity is a particularly interesting avenueforfutureresearch. Introducingfirmheterogeneityandfinancingconstraints,inadditionto household and skill heterogeneity we already incorporate, might also be an interesting avenue for research. 34

References Aiyagari, S. Rao. 1994. “Uninsured Idiosyncratic Risk and Aggregate Saving.” The Quarterly Journal of Economics,109(3):659–84. Aiyagari, S. Rao. 1995. “Optimal Capital Income Taxation with Incomplete Markets, Borrowing Constraints,andConstantDiscounting.”JournalofPoliticalEconomy,103(6):1158–75. Ascari, Guido, Anna Florio, and Alessandro Gobbi. 2020. “Monetary-Fiscal Interactions under Price LevelTargeting.”mimeo. Barro, Robert, and Jason Furman. 2018. “Macroeconomic Effects of the 2017 Tax Reform.” Brookings PapersonEconomicActivity,49(1):257–345. Barro, Robert J. 1979. “On the Determination of the Public Debt.” Journal of Political Economy, 87(5):940–71. Bhattarai, Saroj, Jae Won Lee, and Woong Yong Park. 2014. “Inflation dynamics: The Role of Public DebtandPolicyRegimes.”JournalofMonetaryEconomics,67:93–108. Bhattarai, Saroj, Jae Won Lee, and Woong Yong Park. 2015. “Optimal Monetary Policy in a Currency UnionwithInterestRateSpreads.”JournalofInternationalEconomics,96(2):375–97. Bhattarai, Saroj, Jae Won Lee, and Woong Yong Park. 2016. “Policy Regimes, Policy Shifts, and U.S. BusinessCycles.”TheReviewofEconomicsandStatistics,98(5):968–83. Bilbiie, Florin O. 2008. “Limited Asset Markets Participation, Monetary Policy and (Inverted) Aggregate DemandLogic.”JournalofEconomicTheory,140(1):162–96. Bilbiie,FlorinO.,TommasoMonacelli,andRobertoPerotti.2013.“PublicDebtandRedistributionwith BorrowingConstraints.”TheEconomicJournal,123(566):F64–F98. Blanchard, Olivier, and Roberto Perotti. 2002. “An Empirical Characterization of the Dynamic Effects of Changes in Government Spending and Taxes on Output.” The Quarterly Journal of Economics, 117(4):1329–68. Bohn, Henning. 1998. “The Behavior of U.S. Public Debt and Deficits.” The Quarterly Journal of Economics,113(3):949–63. Calvo, Guillermo A. 1983. “Staggered Prices in a Utility-Maximizing Framework.” Journal of Monetary Economics,12(3):383–98. Canzoneri, Matthew, Robert Cumby, and Behzad Diba. 2010. “The Interaction Between Monetary and Fiscal Policy.” In Handbook of Monetary Economics. Vol. 3, ed. Benjamin M. Friedman and Michael Woodford,Chapter17,935–99.Elsevier. Chamley, Christophe. 1986. “Optimal Taxation of Capital Income in General Equilibrium with Infinite Lives.”Econometrica,54(3):607–22. Christiano,Lawrence,MartinEichenbaum,andSergioRebelo.2011.“WhenIstheGovernmentSpendingMultiplierLarge?” JournalofPoliticalEconomy,119(1):78–121. 35

Cochrane,John.2019.“TheFiscalTheoryofthePriceLevel.”unpublished. Coibion,Olivier,andYuriyGorodnichenko.2011.“MonetaryPolicy,TrendInflation,andtheGreatModeration: AnAlternativeInterpretation.”AmericanEconomicReview,101(1):341–70. Conesa,JuanCarlos,SagiriKitao,andDirkKrueger.2009.“TaxingCapital? NotaBadIdeaafterAll!” AmericanEconomicReview,99(1):25–48. Cúrdia, Vasco, and Michael Woodford. 2016. “Credit Frictions and Optimal Monetary Policy.” Journal ofMonetaryEconomics,84:30–65. Debortoli, Davide, and Jordi Galí. 2017. “Monetary Policy with Heterogeneous Agents: Insights from TANKModels.”DepartmentofEconomicsandBusiness,UniversitatPompeuFabraEconomicsWorking Papers1686. Domeij, David, and Jonathan Heathcote. 2004. “On The Distributional Effects Of Reducing Capital Taxes.”InternationalEconomicReview,45(2):523–54. Eggertsson,GautiB.,andPaulKrugman.2012.“Debt,Deleveraging,andtheLiquidityTrap: AFisher- Minsky-KooApproach.”TheQuarterlyJournalofEconomics,127(3):1469–1513. Elsby,Michael,BartHobijn,andAys¸egülS¸ahin.2013.“TheDeclineoftheU.S.LaborShare.”Brookings PapersonEconomicActivity,44(2):1–63. Erosa,Andres,andMartinGervais.2002.“OptimalTaxationinLife-CycleEconomies.”JournalofEconomicTheory,105(2):338–69. Forni, Lorenzo, Libero Monteforte, and Luca Sessa. 2009. “The General Equilibrium Effects of Fiscal Policy: EstimatesfortheEuroArea.”JournalofPublicEconomics,93(3):559–85. Galí,Jordi,J.DavidLópez-Salido,andJavierVallés.2007.“UnderstandingTheEffectsofGovernment SpendingonConsumption.”JournaloftheEuropeanEconomicAssociation,5(1):227–70. Hall,RobertE.2018.“NewEvidenceontheMarkupofPricesoverMarginalCostsandtheRoleofMega- FirmsintheUSEconomy.”NationalBureauofEconomicResearchWorkingPaper24574. House, Christopher L., and Matthew D. Shapiro. 2008. “Temporary Investment Tax Incentives: Theory withEvidencefromBonusDepreciation.”AmericanEconomicReview,98(3):737–68. Jones, John Bailey. 2002. “Has Fiscal Policy Helped Stabilize the Postwar U.S. Economy?” Journal of MonetaryEconomics,49(4):709–46. Judd, KennethL.1985.“RedistributiveTaxationinaSimplePerfectForesightModel.”JournalofPublic Economics,28(1):59–83. King, Robert, Charles Plosser, and Sergio Rebelo. 2002. “Production, Growth and Business Cycles: TechnicalAppendix.”ComputationalEconomics,20(1-2):87–116. Krusell, Per, Lee E. Ohanian, José-Víctor Ríos-Rull, and Giovanni L. Violante. 2000. “Capital-Skill ComplementarityandInequality: AMacroeconomicAnalysis.”Econometrica,68(5):1029–53. Leeper, Eric M. 1991. “Equilibria under ’Active’ and ’Passive’ Monetary and Fiscal Policies.” Journal of MonetaryEconomics,27(1):129–47. 36

Leeper, Eric M., and Campbell Leith. 2016. “Understanding Inflation as a Joint Monetary-Fiscal Phenomenon.” In Handbook of Macroeconomics. Vol. 2, ed. John B. Taylor and Harald Uhlig, Chapter 30, 2305–2415.Elsevier. Leeper, Eric M., Nora Traum, and Todd B. Walker. 2017. “Clearing Up the Fiscal Multiplier Morass.” AmericanEconomicReview,107(8):2409–54. Lindquist, Matthew J. 2004. “Capital-Skill Complementarity and Inequality Over the Business Cycle.” ReviewofEconomicDynamics,7(3):519–40. Lucas,RobertE.1987.ModelsofBusinessCycles.Oxford[Oxfordshire]:B.Blackwell. Maliar,Lilia,andSergueiMaliar.2011.“Capital-SkillComplementarityandBalancedGrowth.”Economica,78(310):240–59. Mountford,Andrew,andHaraldUhlig.2009.“WhataretheEffectsofFiscalPolicyShocks?” Journalof AppliedEconometrics,24(6):960–92. Ohanian, Lee E, Musa Orak, and Shihan Shen. 2021. “Revisiting Capital-Skill Complementarity, Inequality,andLaborShare.”NationalBureauofEconomicResearchWorkingPaper28747. Romer,ChristinaD.,andDavidH.Romer.2010.“TheMacroeconomicEffectsofTaxChanges: Estimates BasedonaNewMeasureofFiscalShocks.”AmericanEconomicReview,100(3):763–801. Sims, Christopher A. 1994. “A Simple Model for Study of the Determination of the Price Level and the InteractionofMonetaryandFiscalPolicy.”EconomicTheory,4(3):381–99. Sims,ChristopherA.2001.“FiscalConsequencesforMexicoofAdoptingtheDollar.”JournalofMoney, CreditandBanking,33(2):597–616. Sims,Eric,andJonathanWolff.2018.“TheState-DependentEffectsofTaxShocks.”EuropeanEconomic Review,107:57–85. Slavík,Ctirad,andHakkiYazici.2019.“OntheConsequencesofEliminatingCapitalTaxDifferentials.” CanadianJournalofEconomics,52(1):225–52. Smets, Frank, and Rafael Wouters. 2007. “Shocks and Frictions in US Business Cycles: A Bayesian DSGEApproach.”AmericanEconomicReview,97(3):586–606. Straub, Ludwig, and Iván Werning. 2020. “Positive Long-Run Capital Taxation: Chamley-Judd Revisited.”AmericanEconomicReview,110(1):86–119. Trabandt, Mathias, and Harald Uhlig. 2011. “The Laffer Curve Revisited.” Journal of Monetary Economics,58(4):305–27. Woodford, Michael. 1994. “Monetary Policy and Price Level Determinacy in a Cash-in-Advance Economy.”EconomicTheory,4(3):345–80. Woodford, Michael.2011.“SimpleAnalyticsoftheGovernmentExpenditureMultiplier.”AmericanEconomicJournal: Macroeconomics,3(1):1–35. 37

Appendix (not for publication) A Model with household heterogeneity A.1 Stationary equilibrium A.1.1 Notations A balanced growth path can be achieved if At =a¯ =γ1−αand qt =γ = 1, i.e. γ γ =1. In this A t−1 q t−1 q γ q case,thegrowthrateofoutput Yt =γ. Y t−1 Cs Cu Is Is Ks Ks Quantities:C˜s= t ,C˜u= t , I˜s = b,t , I˜s = e,t , K˜s = b,t , K˜s = e,t t γt t γt b,t γt e,t γt b,t γt e,t (cid:16) (cid:17)t γ γ q Y˜ = Y t , K˜ = K b,t , K˜ = K e,t , K ˜ˆ = Kˆ b,t , K ˜ˆ = Kˆ e,t t γt b,t γt e,t (cid:16) (cid:17)t b,t γt e,t (cid:16) (cid:17)t γ γ γ γ q q Ws Wu RK,b RK,e Prices: w˜s= t , w˜u= t , rK,b= t , rK,e= t t Pγt t Pγt t P t Pγt t t t t Φ Φi P P∗ MC Φ˜ = t , Φ˜i= t, π = t , p˜∗= t , mc = t t γt t γt t P t P t P t−1 t t Z Z Z˜ = 1,t , Z˜ = 2,t 1,t (P)θ+1γt 2,t (P)θγt t t B G TC TH Fiscalvariables: b˜ = t , G˜ = t , T˜C = t , T˜H = t t PY t Y t Y t Y t t t t t TK,b TK,e S Si T˜K,b= t , T˜K,e= t , S˜ = t , S˜i= t t Y t Y t Y t Y t t t t Multipliers: Λ˜i=γtPΛi, Ψ˜i =γtΨi , Ψ˜i =Ψi t t t b,t b,t e,t e,t A.1.2 Stationaryequilibriumconditions We considera symmetric equilibrium acrossfirms, where all firms set the same priceand produce the same amount of output. Given nonlinear equilibrium conditions, we detrended variables to specify stationary equilibrium conditions. All the notations are the same as before. We now state allthestationaryequilibriumequationsunderincompletemarkets. • Productionfunction:(Let A =1) 0 Y˜A=K˜α (cid:20) µLσ +(1−µ) (cid:16) λK˜ρ +(1−λ)Lρ (cid:17)σ ρ (cid:21)1− σ α t b,t u,t e,t s,t • Aggregateoutput Y˜A=Y˜ Ξ t t t 38

• Costminimization Y˜A rK,b=αmc t t t K˜ b,t r t K,e=(1−α)mc t K Y ˜ ˜ e t A ,t  µLσ (1 + − ( µ 1 ) − (cid:16) λ µ K ) ˜ (cid:16) e ρ λ ,t K + ˜ρ (1 + − ( λ 1 ) − L λ ρ s, ) t (cid:17) L σ ρ ρ (cid:17)σ ρ    λK˜ e ρ ,t + λ ( K 1 ˜ e ρ − ,t λ)Lρ s,t   u,t e,t s,t   w˜u t =(1−α)mc t L Y˜ u t A ,t µLσ +(1−µ) (cid:16) λ µ K L ˜ u σ ρ ,t +(1−λ)Lρ (cid:17)σ ρ  u,t e,t s,t w˜ t s=(1−α)mc t L Y˜ s t A ,t  µLσ (1 + − ( µ 1 ) − (cid:16) λ µ K ) ˜ (cid:16) e ρ λ ,t K + ˜ρ (1 + − ( λ 1 ) − L λ ρ s, ) t (cid:17) L σ ρ ρ (cid:17)σ ρ    λK˜ e ρ ( ,t 1 + − ( λ 1 ) − L λ ρ s, ) t Lρ s,t   u,t e,t s,t • Skill-premium w˜s (1−µ)(1−λ) (cid:32) (cid:32) K˜ (cid:33)ρ (cid:33)σ ρ −ρ (cid:32) L (cid:33)1−σ t = λ e,t +(1−λ) u,t w˜u µ L L t s,t s,t • Firms’maximization: Z˜ 1,t =mc t Y˜ t +α p β (cid:16) (π t )γPπ¯(1−γP) (cid:17)−θ E t (cid:40)Λ˜ Λ˜ t+1 Z˜ 1,t+1 (π t+1 )θ (cid:41) (A.1) t Z˜ 2,t =Y˜ t +α p β (cid:16) (π t )γPπ¯(1−γP) (cid:17)1−θ E t (cid:40)Λ˜ Λ˜ t+1 Z˜ 2,t+1 (π t+1 )θ−1 (cid:41) (A.2) t • Pricedispersion Ξ t =(1−α P ) (cid:0) p˜∗ t (cid:1)−θ+α P πθ t (cid:16) πγ t− P 1 π¯1−γP (cid:17)−θΞ t−1 (A.3) where θ Z˜ p˜∗= 1,t . t θ−1Z˜ 2,t • Aggregatepriceindex π1−θ=(1−α ) (cid:0) π p˜∗(cid:1)1−θ+α (cid:16) πγP π¯1−γP (cid:17)1−θ t P t t P t−1 • Profit Φ˜ =Y˜ −w˜uL −w˜sL −rK,bK˜ −rK,eK˜ t t t u,t t s,t t b,t t e,t 39

• Hand-to-mouthhouseholds w˜u 1−τ t H =ω¯uC˜u(cid:0) Hu(cid:1)φ t 1+τC t t t (cid:16) (cid:17) (cid:16) (cid:17) 1+τC C˜u= 1−τH w˜uHu+Φ˜u+S˜uY˜ t t t t t t t t • Skilledhouseholds – Marginalutilities: (cid:16) (cid:17) 1 Λ˜s 1+τC = t t C˜s t Λ˜s (cid:16) 1−τH (cid:17) w˜s=ω¯s(cid:0) Hs(cid:1)φ t t t t – Capacityutilizationcosts χ2 A (cid:0) u (cid:1)=χ (cid:0) u −1 (cid:1)+ b,2(cid:0) u −1 (cid:1)2 b e,t b,1 b,t b,t 2 χ2 A (cid:0) u (cid:1)=χ (cid:0) u −1 (cid:1)+ e,2(cid:0) u −1 (cid:1)2 e e,t e,1 e,t e,t 2 – FOCsandCapitalAccumulation w˜s 1−τ t H =ω¯sC˜s(cid:0) Hs(cid:1)φ t 1+τC t t t (cid:40) (cid:41) β 1 Λ˜s= RE Λ˜s (A.4) t γ t t t+1π t+1 Ψ˜ b s ,t = γ β E t (cid:110) (1−d b )Ψ˜ b s ,t+1 + (cid:104)(cid:16) 1−τ t K +1 (cid:17) r t K + , 1 bu b,t+1 − (cid:16) 1−λ b τ t K +1 (cid:17) A b (cid:0) u b,t+1 (cid:1)(cid:105) Λ˜ t s +1 (cid:111) (A.5) (cid:16) 1−λ b τ t K (cid:17) Λ˜ t s=Ψ˜ b s ,t   1−S  I˜ I s ˜ b s ,t γ   −S′  I˜ I s ˜ b s ,t γ  I˜ I s ˜ b s ,t γ   b,t−1 b,t−1 b,t−1   + γ β E t   Ψ˜ b s ,t+1   I˜ b I s ˜ , s t+1 γ   2 S′   I˜ b I s ˜ , s t+1 γ     b,t b,t (cid:40) (cid:34) (cid:35) (cid:41) Ψ˜ e s ,t =βE t (1−d e )Ψ˜ e s ,t+1 + (cid:16) 1−τ t K +1 (cid:17) r t K + , 1 eu e,t+1 − q 1 (cid:16) 1−λ e τ t K +1 (cid:17) A e (cid:0) u e,t+1 (cid:1) Λ˜ t s +1 (A.6) 0       (cid:16) 1−λ e τ t K (cid:17) q 1 0 Λ˜ t s=Ψ˜ e s ,t  1−S I˜ e I s ˜ , e s t− ,t 1 γ  −S′ I˜ e I s ˜ , e s t− ,t 1 γ I˜ e I s ˜ , e s t− ,t 1 γ    + γ β E t   Ψ˜ e s ,t+1   I˜ e I s ˜ , e s t+ ,t 1 γ   2 S′   I˜ e I s ˜ , e s t+ ,t 1 γ     40

γK ˜ˆ b s ,t+1 =(1−d b )K ˜ˆ b s ,t +   1−S  I˜ I s ˜ b s ,t γ     I˜ b s ,t b,t−1    K ˜ˆ e s ,t+1 =(1−d e )K ˜ˆ e s ,t +  1−S I˜ I s ˜ e s ,t γ   I˜ e s ,t q 0 e,t−1 (cid:16) 1−τK (cid:17) rK,b= (cid:16) 1−λ τK (cid:17) A ′ (cid:0) u (cid:1) t t b t b b,t (cid:16) 1−τK (cid:17) rK,e= 1 (cid:16) 1−λ τK (cid:17) A ′(cid:0) u (cid:1) t t q e t e e,t 0 • Resourceconstraint (cid:32) (cid:33) (cid:16) 1−G˜ (cid:17) Y˜ =Ns C˜s+I˜s +I˜s +A (cid:0) u (cid:1) K ˜ˆs + 1 A (cid:0) u (cid:1) K ˜ˆs +NuC˜u t t t b,t e,t b b,t b,t q e e,t e,t t 0 =C˜ +I˜ +I˜ +A (cid:0) u (cid:1) K ˜ˆ + 1 A (cid:0) u (cid:1) K ˜ˆ (A.7) t b,t e,t b b,t b,t e e,t e,t q 0 • Marketclearing C˜ =NsC˜s+NuC˜u t t t L =NuHu,L =NsHs u,t t s,t t K˜ =NsK˜s ,K˜ =NsK˜s b,t b,t e,t e,t I˜ =NsI˜s ,I˜ =NsI˜s b,t b,t e,t e,t K˜s =us K ˜ˆs ,K˜s =us K ˜ˆs b,t b,t b,t e,t e,t e,t • Governmentbudgetconstraint 1 Y˜ b˜ +T˜C+T˜H+T˜K,b+T˜K,e=R b˜ t−1 +G˜ +S˜ (A.8) t t t t t t−1 t−1 π t γ Y˜ t t t where (cid:32) (cid:33) C˜ (cid:88) L T˜C =τC t , T˜H =τH w˜i i,t , t Y˜ t t t Y˜ t i∈s,u t T˜ t K,b=τ t K   r t K,b K ˜ˆ Y˜ b,t u b,t −λ b   I˜ Y˜ b,t +A b (cid:0) u b,t (cid:1)K ˜ˆ Y˜ b,t     , t t t T˜ t K,e=τ t K   r t K,e K ˜ˆ Y˜ e t ,t u e,t −λ e   I˜ Y˜ e, t t + q 1 0 A e (cid:0) u e,t (cid:1)K ˜ˆ Y˜ e t ,t     41

• FiscalPolicyRules: (cid:32) (cid:33) (cid:32) (cid:33) S S¯ S S¯ S S¯ t − new =ρS t−1 − new +ρS t−2 − new (A.9) Y t Y¯ new 1 Y t−1 Y¯ new 2 Y t−2 Y¯ new    (cid:32) (cid:33) (cid:32) (cid:33) + (cid:16) 1−ρS 1 −ρS 2 (cid:17)  ψS B  P B t− Y 1 − P B Y  +ψS ∆y Y Y t +ψS x Y Y n t   , t−1 t−1 t−1 t (cid:16) (cid:17) (cid:16) (cid:17) τH−τ¯H =ρH τH −τ¯H +ρH τH −τ¯H (A.10) t new 1 t−1 new 2 t−2 new    (cid:32) (cid:33) (cid:32) (cid:33) + (cid:16) 1−ρ 1 H−ρ 2 H (cid:17)  ψH B  P B t− Y 1 − P B Y  +ψ ∆ H y Y Y t +ψH x Y Y n t   , t−1 t−1 t−1 t (cid:16) (cid:17) (cid:16) (cid:17) τC−τ¯C =ρC τC −τ¯C +ρC τC −τ¯C = (A.11) t new 1 t−1 new 2 t−2 new    (cid:32) (cid:33) (cid:32) (cid:33) + (cid:16) 1−ρC 1 −ρC 2 (cid:17)  ψC B  P B t− Y 1 − P B Y  +ψC ∆y Y Y t +ψC x Y Y n t   t−1 t−1 t−1 t  τK =  τ¯K ift=0 (A.12) t τ¯K ift>0 New G˜ =G ¯˜ (A.13) t • Monetarypolicy (cid:16) (cid:17) R R¯ t = (cid:20)R R t ¯ −1 (cid:21)ρR 1 (cid:20)R R t ¯ −2 (cid:21)ρR 2   (cid:18)π π¯ t (cid:19)ϕπ (cid:32) Y Y t− t 1 (cid:33)ϕ∆y (cid:32) Y Y t n t (cid:33)ϕx   1−ρR 1 −ρR 2 (A.14) • GovernmentdebttoGDP(ortransfers): S˜ =S ¯˜ifadjustinglabortaxrate (A.15) t A.2 Steady state Recallthatinsteady-state,S(γ)=S′(γ)=0: From(A.1)–(A.3),weget θ−1 m¯c= . θ Fromskilledworkers’optimalcapitalinvestmentdecisions,weget (cid:32) (cid:33) γ 1−λ τ¯K r¯K,b= −(1−d ) b β b 1−τ¯K (cid:32) (cid:33) 1 1 1−λ τ¯K r¯K,e= −(1−d ) e q β e 1−τ¯K 0 Fromtheproductionfunction,weget Y L ¯˜ ¯ A = L¯ Y ¯˜ =   K L¯ ¯˜ b   α   µ (cid:32) L L ¯ ¯ u (cid:33)σ +(1−µ)   λ   K L¯ ¯˜ e   ρ +(1−λ)   σ ρ   1− σ α s s s s s 42

and Ξ¯ =1. Fromfirms’optimalityconditions,weget w¯˜s=(1−λ)(1−µ)(1−α)m¯c (cid:32) α r¯K m¯ ,b c (cid:33)1− 1− σ α   K L¯ ¯˜ b   1−σ  λ   K L¯ ¯˜ e   ρ +(1−λ)   σ ρ −1 s s w¯˜u (cid:32) L L ¯ ¯ u (cid:33)1−σ =µ(1−α)m¯c (cid:32) α r¯K m¯ ,b c (cid:33)1− 1 α − − α σ  K L¯ ¯˜ b   1−σ s s r¯K,e   K L¯ ¯˜ e   1−ρ =λ(1−µ)(1−α)m¯c (cid:32) α r¯K m¯ ,b c (cid:33)1− 1− σ α   K L¯ ¯˜ b   1−σ  λ   K L¯ ¯˜ e   ρ +(1−λ)   σ ρ −1 s s s (cid:32) α r¯K m¯ ,b c (cid:33) = (cid:32) K L¯ ˜ b (cid:33)α−1   µ (cid:32) L L ¯ ¯ u (cid:33)σ +(1−µ)   λ   K L¯ ¯˜ e   ρ +(1−λ)   σ ρ   1− σ α s s s Also,fromskilled-household’sbudgetconstraint,wehavebudgetconstraint (cid:16) (cid:17)C ¯˜s (cid:16) (cid:17) H¯s 1+τ¯C = 1−λs τ¯H w¯˜s Y ¯˜ τH Y ¯˜ (cid:16)(cid:16) (cid:17) (cid:16) (cid:17) (cid:17)K ¯˜ˆs + 1−τ¯K r¯K,b− 1−λ τ¯K (γ−(1−d )) b b b Y ¯˜ (cid:32) (cid:16) (cid:17) (cid:16) (cid:17)d (cid:33) K ¯˜ˆs + 1−τ¯K r¯K,e− 1−λ τ¯K e e e q 0 Y ¯˜ + (cid:32) 1 −1 (cid:33) b ¯˜s+ χ Φ s Φ˜ t + χ S s S˜ β Ns Y ¯˜ Ns t wheretheprofitis Φ¯˜ L¯ L¯ K ¯˜ K ¯˜ =1−w¯˜u u −w¯˜s s −r¯K,b b −r¯K,e e Y ¯˜ Y ¯˜ Y ¯˜ Y ¯˜ Y ¯˜ andthetransferis S ¯˜ =τ¯C C Y ¯˜ ¯˜ +τ¯H   (cid:88) λi τH w¯˜i L Y ¯ ¯˜ i   +τ¯Kr¯K,b   K Y ¯˜ˆ ¯˜ b −λ b I Y ¯˜ ¯˜ b   +τ¯Kr¯K,e   K Y ¯˜ˆ ¯˜ e −λ e I Y ¯˜ ¯˜ e   − (cid:32)(cid:32) β 1 −1 (cid:33) b ¯˜+G ¯˜ (cid:33) . i∈{s,u} From(A.5)and(A.6)forbothtypesofhouseholds,weget K ¯˜ˆs I ¯˜s (γ−(1−d )) b = b b H¯s H¯s K ¯˜ˆs I ¯˜s d e = e q e H¯s H¯s 0 K ¯˜ˆu I ¯˜u (γ−(1−d )) b = b b H¯u H¯u K ¯˜ˆu I ¯˜u d e = e q e H¯u H¯u 0 43

Fromhousehold’sintra-temporalEulerequations,wehave w¯˜u 1−τ¯H =ω¯uC ¯˜u (cid:16) H¯u (cid:17)φ 1+τ¯C (cid:16) (cid:17) (cid:16) (cid:17) 1+τ¯C C ¯˜u= 1−τ¯H w¯˜uH¯u+Φ¯˜u+S ¯˜uY ¯˜ 1−τ¯H w¯˜s=ω¯sC ¯˜s (cid:16) H¯s (cid:17)φ 1+τ¯C Λ¯˜s C ¯˜u Λ = = s,u Λ¯˜u C ¯˜s (cid:32) (cid:33) C ¯˜s= 1 C ¯˜ NuΛ +Ns s,u (cid:32) Λ (cid:33) C ¯˜u= s,u C ¯˜ NuΛ +Ns s,u Fromthemarketclearingconditions,wehave K ¯˜ K ¯˜s L¯ K ¯˜u b = b + u b L¯ H¯s L¯ H¯u s s K ¯˜ K ¯˜s L¯ K ¯˜u e = e + u e L¯ H¯s L¯ H¯u s s From(A.7),weget C ¯˜ K ¯˜ d K ¯˜ (cid:16) (cid:17) Y ¯˜ +(γ−(1−d )) b + e e = 1−G˜ L¯ s b L¯ s q 0 L¯ s L s ThenominalinterestrateisobtainedfromEulerequation(A.4) γπ¯ R¯ = . β We fix steady-state hours for skilled labor H¯s =0.33 by assuming the skilled works 40 hours perweekand H¯u =0.93∗H¯s(Skilledworkerswork7%morethanlow-skilledworker). 44

B Additional Analytical Results and Proofs of Propositions B.1 Steady-state equilibrium equations for a nested version of the model We assume µ = 0, α = 0, and ρ → 0 to get a nested version of the model with a Cobb-Douglas production function. In this case, to have balanced growth, the growth rate of output is the same withthegrowthrateoftechnology,γ=a¯,andthegrowthrateofrelativeprice(γ )is1. Letq =1. q 0 Also, let the fraction of skilled workers NS =1 and set χs =1 and χs =1 for profit and transfers Φ S distributions. Then,inthiseconomy,wehaveonetypeofcapitalK andonetypeoflaborL . We e,t s,t derive steady-state equilibrium equations for this nested version of the model and drop subscripts eand s. • Marginalcost θ−1 m¯c= . θ • Capitalrentalrate a¯ −(1−d) r¯K = β (B.1) 1−τ¯K • Productionfunction H Y ¯˜ ¯ =  H K ¯˜ ¯   λ (B.2) • Wagesandcapital-to-laborratio K ¯˜ (cid:32) r¯K (cid:33) λ− 1 1 = (B.3) H¯ λm¯c w¯˜ =(1−λ)m¯c  H K ¯˜ ¯   λ =(1−λ)(λ)1− λ λ(m¯c)1− 1 λ (cid:16) r¯K (cid:17) λ− λ 1 (B.4) • Resourceconstraint C ¯˜ (cid:16) (cid:17) Y ¯˜ I ¯˜ = 1−G ¯˜ − . (B.5) H¯ H¯ H¯ • Profit Φ¯˜ H¯ K ¯˜ =1−w¯˜ −r¯K Y ¯˜ Y ¯˜ Y ¯˜ • Transfer (cid:32) (cid:33) S ¯˜ = 1− R¯ b ¯˜−G ¯˜+T ¯˜C+T ¯˜H+T ¯˜K. (B.6) π¯a¯ 45

• Theconsumption,laborincomeandcapitalincometaxratesarerespectivelygivenas: T ¯˜C 1 T ¯˜H 1 T ¯˜K τ¯C = , τ¯H = , τ¯K = . C¯˜ w¯˜ H¯ r¯K K˜ Y¯˜ Y¯˜ Y˜ • Intra-temporalEulerequation   1 H¯ = ω¯ (cid:18) 1+τ¯ 1 C C¯˜ 1 (cid:19)  1+φ . (B.7) 1−τ¯H H¯ w¯˜ • Investment I ¯˜ K ¯˜ = (a¯−(1−d)). (B.8) H¯ H¯ • Nominalinterestrate a¯π¯ R¯ = . β B.2 Analytical results with capital-skill complementarity Theskillpremiumcanbeexpressedasfollows: (cid:18) λ (cid:18) K¯˜ e (cid:19)ρ +(1−λ) (cid:19)σ ρ −1 w¯˜s = (1−λ)(1−µ) L¯ s w¯˜u µ (cid:16) L¯ u (cid:17)σ−1 L¯ s (1−λ)(1−µ) 1−σ σ−ρ(cid:16) φ (cid:17) = χ σ (F) ρ 1−σ+φ µ c  (cid:18) (cid:19) σ whereχ c =  (1−λ) µ (1−µ) ω ω ¯ ¯ u s Λ˜ (cid:16) N N s,u u s (cid:17)φ(cid:18) 1 1 − − λ λ u τ τ s H H τ τ ¯ ¯ H H (cid:19)  σ−φ−1 >0and F (cid:18) K L¯ ¯˜ s e (cid:19) =λ (cid:18) K L¯ ¯˜ s e (cid:19)ρ +(1−λ)>0. Then, ∂τ ∂ ¯K (cid:32) w w ¯˜ ¯˜ u s (cid:33) =(σ−ρ) (cid:32) 1−σ φ +φ (cid:33) w w ¯˜ ¯˜ u s F 1 λ   K L¯ ¯˜ e   ρ−1 ∂τ ∂ ¯K   K L¯ ¯˜ e   s s <0. B.3 Lump-sum transfer adjustment We start with the case where lump-sum transfers adjust to finance the capital tax rate cut. It is usefultostateamildrestrictionongovernmentspendinginsteady-state.36 Assumption1. G ¯˜ <1−θ−1 (cid:32) a¯−(1−d) (cid:33) (cid:16) 1−τ¯K (cid:17) =1−1 (cid:18) I ¯˜ (cid:19) intheinitialsteady-state. θ a¯−(1−d) λ Y ¯˜ β 36This restriction is very mild, and is just to ensure that government spending in steady-state is not very high. For instance, except for a case of an unrealistically high markup, this holds for any reasonable parameterization of governmentspendinginsteady-state. 46

Then, we can formally show that a permanent capital tax rate cut leads to an increase in output, consumption, investment, and wages, and a decline in the rental rate of capital, as given in PropositionB.1. PropositionB.1. Fixτ¯H andb ¯˜. Withlump-sumtransferadjustment, 1. Rental rate of capital is increasing, while capital to hours ratio, wage, hours, capital, investment,andoutputaredecreasinginτ¯K. 2. UnderAssumption1,consumptionisalsodecreasinginτ¯K. Proof. SeeAppendixB.4. □ B.4 Proof of Proposition B.1 Proof. From(B.1)and(B.4),weget ∂r¯K r¯K = >0 ∂τ¯K 1−τ¯K ∂w¯˜ (cid:32) w¯˜ (cid:33)(cid:18) λ (cid:19)∂r¯K =− <0. ∂τ¯K r¯K 1−λ ∂τ¯K Letk ¯˜ = K ¯˜ andy¯˜ = Y ¯˜ . From(B.2)and(B.3),weget H¯ H¯ ∂k ¯˜ k ¯˜ 1 ∂r¯K =− <0 ∂τ¯K r¯K 1−λ∂τ¯K ∂y¯˜ (cid:32) y¯˜ (cid:33)1 ε ∂k ¯˜ =λ <0 ∂τ¯K k ¯˜ ∂τ¯K Combining(B.4)and(B.5)with(B.7),werewritethesteady-statehoursas H¯ =   ω¯  1− 1 λ1 1 − + τ τ ¯ ¯ H C   1 m − ¯c G ¯˜ −λ a a ¯ ¯ − − ( ( 1 1 − − d d ) ) (cid:16) 1−τ¯K (cid:17)       − 1+ 1 φ . β Then,thepartialderivativewithrespecttocapitaltaxrateis   ∂ ∂ τ¯ H¯ K =− H 1 ¯ + 2+ φ φ ω¯ 1− λ λ1 1 − + τ τ ¯ ¯ H C a a ¯ ¯ − − ( ( 1 1 − − d d ) )  <0. β Now, we find the partial derivatives of levels of variables. For capital, investment and output, we caneasilyverifythat ∂K ¯˜ =H¯ ∂k ¯˜ +k ¯˜ ∂H¯ <0 ∂τ¯K ∂τ¯K ∂τ¯K ∂I ¯˜ ∂K ¯˜ = (a¯−(1−d))<0 ∂τ¯K ∂τ¯K ∂Y ¯˜ ∂y¯˜ ∂H¯ =H¯ +y¯˜ <0 ∂τ¯K ∂τ¯K ∂τ¯K 47

Forconsumption,combining(B.2)and(B.8)with(B.5),weget C ¯˜=   (cid:16) 1−G ¯˜ (cid:17) H Y ¯˜ ¯ − H I ¯˜ ¯   H¯  (cid:16) (cid:17) λ   =  m¯c a λ ¯ − 1 ( − 1− τ¯K d)  1−λ (cid:16) 1−G ¯˜ (cid:17) −λm¯c (a a ¯ ¯ − − ( ( 1 1 − − d d ) ) )(cid:16) 1−τ¯K (cid:17) H¯. β β Then,thepartialderivativeofconsumptionwithrespecttocapitaltaxrateis ∂ ∂ τ¯ C ¯˜ K =− (cid:32) 1− λ λ1− 1 τ¯K (cid:33)  λm a¯ ¯ − c (cid:16) ( 1 1 − − τ¯ d K ) (cid:17)  1− λ λ   (cid:16) 1−G ¯˜ (cid:17) −m¯c (a a ¯ ¯ − − ( ( 1 1 − − d d ) ) )(cid:16) 1−τ¯K (cid:17)   H¯+ H C ¯ ¯˜ ∂ ∂ τ¯ H¯ K . β β UnderAssumption1,wefind ∂C ¯˜ <0. □ ∂τ¯K B.5 Proof of Proposition 3.1 Proof. From(B.1),(B.3),and(B.4),weget K ¯˜ w¯˜ λ = H¯ r¯K 1−λ   1 = a¯ − λ ( m 1 ¯c −d) (cid:16) 1−τ¯K (cid:17) 1−λ . β Theamountofchangesincapitaltohoursratiotothecapitaltaxcutisthesameinbothlump-sum transfers adjustment case and labor tax rate case. In a similar way, we know that output to hours ratio,investmenttohoursratio,andconsumptiontohoursratiochangebythesameamountinboth cases. Thus,allthemagnitudesofchangesinmacroquantitiestocapitaltaxcutsaredeterminedby the hours responses. Now, we compare the changes in hours to capital tax rate changes under the transfersadjustmentcasewiththechangesunderthelabortaxrateadjustmentcase. Noticethatthe initial steady-states are the same in both cases. Let H¯T and H¯L denote the steady-state hours new new after the capital tax changes in the transfers adjustment case and in the labor tax rate adjustment case,respectively. Then,from(B.7),weget H H ¯ ¯ n n T L e e w w =   ( ( 1 1 − − ω λ ¯ ω λ ¯ ) ) m m ¯c ¯c 1 1 1 1 − − + + τ¯ τ τ τ ¯ ¯ ¯ n H H C C ew (cid:32) (cid:32) 1 1 − − G G ¯˜ ¯˜ − − λ λ m m ¯ β a c ¯ ¯ β a c ¯ ( − ( a¯ − a ( ¯ − ( 1 − 1 ( − ( 1 − 1 d − d − ) d ) d ) ) ) ) (cid:16) (cid:16) 1 1 − − τ¯ τ¯ n K n K e e w w (cid:17) (cid:17) (cid:33) (cid:33)   − 1+ 1 φ  (cid:16) (cid:17)− 1 =  1− ∆ 1− τ¯ τ¯ H H  1+φ =   1+ 1− λ λ (cid:32) 1− 1 τ¯H (cid:33)  1+τ¯C   a a ¯ ¯ − − ( ( 1 1 − − d d ) )     ∆ (cid:16) τ¯K (cid:17)   − 1+ 1 φ . β 48

(cid:16) (cid:17) Forsmallchangesincapitaltaxrate∆ τ¯K ,weget ln (cid:32) H H ¯ ¯ n T L ew (cid:33) =− 1+ 1 φ1− λ λ (cid:32) 1− 1 τ¯H (cid:33)  1+τ¯C   a a ¯ ¯ − − ( ( 1 1 − − d d ) )     ∆ (cid:16) τ¯K (cid:17) new β (cid:16) (cid:17) =−Θ∆ τ¯K (cid:32) (cid:33) (cid:16) (cid:17) whereΘ= 1 λ 1 1+τ¯C(a¯−(1−d)) >0. Then,forthelevelsofoutput,consumption,capital 1+φ1−λ 1−τ¯H a¯−(1−d) β andinvestment,thedifferencesarethesame: thatis, ln   Y Y ¯˜ ¯˜ n n T L e e w w   =ln   C C ¯˜ ¯˜ n n T L e e w w   =ln   K K ¯˜ ¯˜ n n T L e e w w   =ln   I I ¯˜ ¯˜ n n T L e e w w   =ln (cid:32) H H ¯ ¯ n n T L e e w w (cid:33) =−Θ∆ (cid:16) τ¯K (cid:17) . (cid:32) (cid:33) Notice that ∂ln Y Y ¯˜ ¯˜ n n T L e e w w = − Θ ∆ (cid:16) τ¯K (cid:17) implying that the gap is increasing in the initial level of τ¯H ∂τ¯H 1−τ¯H whenthereisacapitaltaxcut. Now,let H¯T and H¯C denotethesteady-statehoursafterthecapitaltaxchangesinthetransnew new fersadjustmentcaseandintheconsumptiontaxrateadjustmentcase,respectively. Then,weget H H ¯ ¯ n n C T e e w w =   ( ( 1 1 − − ω λ ¯ ω λ ¯ ) ) m m ¯c ¯c 1 1 1 1 + − − + τ¯ τ τ τ ¯ ¯ C ¯ n H H C ew (cid:32) (cid:32) 1 1 − − G G ¯˜ ¯˜ − − λ λ m m ¯ β a c ¯ ¯ a c ¯ ( − ( a¯ − a ( ¯ − ( 1 − 1 ( − ( 1 − 1 d − d − ) d ) d ) ) ) ) (cid:16) (cid:16) 1 1 − − τ¯ τ¯ n K n K e e w w (cid:17) (cid:17) (cid:33) (cid:33)   − 1+ 1 φ β  (cid:16) (cid:17) 1 =  1+ ∆ 1+ τ¯ τ¯ C C  1+φ =   1−   1+ (cid:32) 1 a β a ¯ ¯ − − + ( ( 1 1 τ¯ − − C d d ) ) (cid:33) τ¯C  1+ (cid:32) a¯− ∆˜ (1 (cid:16) − τ¯ d K ) (cid:33) (cid:17) ∆˜(cid:0) τ¯K (cid:1)   1+ 1 φ a¯−(1−d) β (cid:16) (cid:17) Forsmallchangesincapitaltaxrate∆ τ¯K ,weget  (cid:32) (cid:33)  ln (cid:32) H H ¯ ¯ n n C T e e w w (cid:33) =− 1+ 1 φ  1+ 1 a β a ¯ ¯ − − + ( ( 1 1 τ¯ − − C d d ) ) τ¯C   (cid:16) 1−G ¯˜ (cid:17) − a a ¯ ¯ − − ( ( 1 1 λ − − m d d ) ¯ ) c λm¯c (cid:16) 1−τ¯ n K ew (cid:17)   ∆ (cid:16) τ¯K (cid:17) β Then, for the levels of output, consumption, capital and investment, the differences are the same: 49

thatis, ln   Y Y ¯˜ ¯˜ n n C T e e w w   =ln   C C ¯˜ ¯˜ n n C T e e w w   =ln   K K ¯˜ ¯˜ n n C T e e w w   =ln   I I ¯˜ ¯˜ n n C T e e w w   =ln (cid:32) H H ¯ ¯ n n C T e e w w (cid:33) (cid:16) (cid:17) 1 ∆ τ¯C = 1+φ1+τ¯C  (cid:32) (cid:33)  =− 1+ 1 φ  1+ 1 a β a ¯ ¯ − − + ( ( 1 1 τ¯ − − C d d ) ) τ¯C   (cid:16) 1−G ¯˜ (cid:17) − a a ¯ ¯ − − ( ( 1 1 λ − − m d d ) ¯ ) c λm¯c (cid:16) 1−τ¯ n K ew (cid:17)   ∆ (cid:16) τ¯K (cid:17) β (cid:16) (cid:17) =−MT∆ τ¯K . C     Then,M C T = 1+ 1 φ  1+ β a a ¯ ¯ 1 − − + ( ( 1 1 τ¯ − − C d d ) )  τ¯C   (cid:16) 1−G ¯˜ (cid:17) − a a ¯ ¯− − ( ( 1 1 − λ − d d m ) ) ¯c λm¯c(1−τ¯ n K ew )   >0if β G ¯˜ <1−λ θ−1a¯−(1−d)(cid:16) 1−τ¯K (cid:17) θ a¯ −(1−d) new β ≂1−0.15=0.85 inourcalibration. Inthiscase,capitaltaxcutismoreexpansionaryinthetransferadjustmentcase thanintheconsumptiontaxrateadjustmentcase. Noticethat ∂ln ∂ (cid:32) τ¯ Y Y ¯˜ ¯˜ C n n C T e e w w (cid:33) = 1+ 1 φ   1− (   1+ β a a ¯ ¯ − − τ¯ ( ( C 1 1 − ) − d 2 d ) )       (cid:16) 1−G ¯˜ (cid:17) −a a ¯ ¯ − − ( ( 1 1 − λ − d d m ) ) ¯c λm¯c(1−τ¯ n K ew )   ∆ (cid:16) τ¯K (cid:17) implyingthatthegapis β decreasingintheinitiallevelofτ¯C whenthereisacapitaltaxcut. Lastly,letH¯L andH¯C denotethesteady-statehoursafterthecapitaltaxchangesinthelabor new new taxadjustmentcaseandintheconsumptiontaxadjustmentcase,respectively. Then,weget H H ¯ ¯ n n C L e e w w =   ( ( 1 1 − − ω ω λ λ ¯ ¯ ) ) m m ¯ ¯ c c 1 1 1 1 − + − + τ τ ¯ ¯ τ τ ¯ C ¯ n n H H C e e w w (cid:32) (cid:32) 1 1 − − G G ¯˜ ¯˜ − − λ λ m m ¯ ¯ β a a c c ¯ ¯ ( ( − − a a ¯ ¯ ( ( − − 1 1 ( ( − − 1 1 d d − − ) ) d d ) ) ) ) (cid:16) (cid:16) 1 1 − − τ τ ¯ ¯ n n K K e e w w (cid:17) (cid:17) (cid:33) (cid:33)   − 1+ 1 φ β (cid:32) 1+τ¯C 1−τ¯H (cid:33) 1+ 1 φ = new new 1+τ¯C 1−τ¯H  (cid:16) (cid:17) (cid:16) (cid:17) 1 =   1+ ∆ 1+ τ¯ τ¯ C C   1− ∆ 1− τ¯ τ¯ H H   1+φ =     1− 1+ 1 a β a ¯ ¯ − − + ( ( 1 1 τ¯ − − C d d ) ) τ¯C (cid:16) 1−G ¯˜ (cid:17) − a a ¯ ¯ − − ( ( 1 1 λ − − m d d ) ¯ ) c λm¯c (cid:16) 1−τ¯ n K ew (cid:17) ∆ (cid:16) τ¯K (cid:17)     1+ 1+ 1 a β a ¯ ¯ − − − ( ( 1 1 τ¯ − − H d d ) ) τ¯C 1− λ λ ∆ (cid:16) τ¯K (cid:17)     1+ 1 φ β 50

(cid:16) (cid:17) Forsmallchangesincapitaltaxrate∆ τ¯K ,weget ln (cid:32) H H ¯ ¯ n C L ew (cid:33) = 1+ 1 φ   ∆ 1+ (cid:16) τ¯ τ¯ C C (cid:17) − ∆ 1− (cid:16) τ¯ τ¯ H H (cid:17)  new   = 1+ 1 φ   1+ a β a ¯ ¯ − − ( ( 1 1 − − d d ) ) τ¯C    1− 1 τ¯H 1− λ λ − 1+ 1 τ¯C (cid:16) 1−G ¯˜ (cid:17) − a a ¯ ¯ − − ( ( 1 1 λ − − m d d ) ¯ ) c λm¯c (cid:16) 1−τ¯ n K ew (cid:17)  ∆ (cid:16) τ¯K (cid:17) β = 1+ 1 φ (cid:18) 1− λ λ (cid:19)   1+ 1 a β a ¯ ¯ − − − ( ( 1 1 τ¯ − − H d d ) ) τ¯C     1− 1 1 − + τ τ ¯ ¯ H C (cid:16) 1−G ¯˜ (cid:17) − a a ¯ ¯ ( − − 1 ( ( 1 1 − − − d d λ ) ) ) λ m m ¯ ¯ c c (cid:16) 1−τ¯ n K ew (cid:17)   ∆ (cid:16) τ¯K (cid:17) β Then, for the levels of output, consumption, capital and investment, the differences are the same: thatis, ln   Y Y ¯˜ ¯˜ n n C L e e w w   =ln   C C ¯˜ ¯˜ n n C L e e w w   =ln   K K ¯˜ ¯˜ n n C L e e w w   =ln   I I ¯˜ ¯˜ n n C L e e w w   =ln (cid:32) H H ¯ ¯ n n C L e e w w (cid:33) = 1+ 1 φ   ∆ 1+ (cid:16) τ¯ τ¯ C C (cid:17) − ∆ 1− (cid:16) τ¯ τ¯ H H (cid:17)  = 1+ 1 φ (cid:18) 1− λ λ (cid:19)   1+ 1 a β a ¯ ¯ − − − ( ( 1 1 τ¯ − − H d d ) ) τ¯C     1− 1 1 − + τ τ ¯ ¯ H C (cid:16) 1−G ¯˜ (cid:17) − a a ¯ ¯ ( − − 1 ( ( 1 1 − − − d d λ ) ) ) λ m m ¯ ¯ c c (cid:16) 1−τ¯ n K ew (cid:17)   ∆ (cid:16) τ¯K (cid:17) β (cid:16) (cid:17) =ML∆ τ¯K . C    NoticethatM C L = 1+ 1 φ (cid:16) 1− λ λ (cid:17) 1+ β a a ¯ ¯ 1 − − − ( ( 1 1 τ¯ − − H d d ) ) τ¯C   1−1 1 − + τ τ ¯ ¯ H C (cid:16) 1−G ¯˜ (cid:17) −a¯− ( ( 1 1− − d λ ) ) λ m m ¯c ¯c(1−τ¯K )  >0if a¯−(1−d) new β G ¯˜ <1−λ θ−1a¯−(1−d)(cid:16) 1−τ¯K (cid:17) −(1−λ) θ−11−τ¯H θ a¯ −(1−d) new θ 1+τ¯C β ≂1−0.4976=0.5023 inourbaselinecalibration. To see that labor tax adjustment is more distortionary than consumption tax rate adjustment case,wecancombineEquations(B.5),(B.6),and(B.7)toderive H¯ =   ω¯  1 1 − + τ τ ¯ ¯ H C H C ¯ ¯˜ w 1 ¯˜     − 1+ 1 φ =   ω¯  1− 1 τ¯H   (cid:32) 1− (cid:32) 1− β 1 (cid:33) b ¯˜+S ¯˜ (cid:33) H Y ¯˜ ¯ −τ¯Hw¯˜ −τ¯Kr¯K H K˜ ¯ − H I ¯˜ ¯  w 1 ¯˜     − 1+ 1 φ =   ω¯  1− 1 τ¯H (1− 1 λ)   1− (cid:16) 1− m¯ β 1 c (cid:17) b ¯˜+S ¯˜ −(1−λ)τ¯H−λτ¯K− λ( a¯ a¯ − − ( ( 1 1 − − d d ) ))(cid:16) 1−τ¯K (cid:17)       − 1+ 1 φ . β 51

Whenβ=1,thenweget H¯ =   ω¯   1−m m ¯ ¯ c c +S ¯˜ (1−λ) 1 (cid:0) 1−τ¯H (cid:1) +1     − 1+ 1 φ . □ C Data appendix We calibrate the steady-state fiscal variables using US quarterly data for the post-Volcker period from1982:Q4to2008:Q2. C.1 Debt and spending data Weusethefollowingdefinitionsforourdebtandspendingvariables: • Governmentdebt=marketvalueofprivatelyheldgrossfederaldebt; • Governmentexpenditures=governmentconsumption; Notethatweuseasinglepricelevel,GDPdeflator,forbothvariables. ThemarketvalueofprivatelyheldgrossfederaldebtserieswasobtainedfromFederalReserve Bank of Dallas and the government consumption data series was taken from National Income and ProductAccounts(NIPA)tables. C.2 Tax data WefollowamethodoriginallybasedonJones(2002). Additionally,weusethetaxrevenuesofthe federalgovernmentandlocalpropertytaxes. We use federal taxes on production and imports (lines 4 of NIPA Table 3.2) for consumption taxrevenues. LetthisbeTC. The average personal income tax rate is computed to get both capital tax revenues and labor taxrevenues. Wefirstcomputetheaveragepersonalincometaxrateas IT τP = W+PRI/2+CI where IT is the personal current tax revenues (line 3 of NIPA Table 3.2), W is wage and salary accruals (line 3 of NIPA Table 1.12), PRI is proprietor’s income (line 9 of NIPA Table 1.12), and CI is capital income, which is the sum of rental income (line 12 of NIPA Table 1.12), corporate 52

profits(line13ofNIPATable1.12),interestincome(line18ofNIPATable1.12),and PRI/2. We hereregardhalfofproprietor’sincomeaswagelaborincomeandtheotherhalfascapitalincome. Thenthecapitaltaxrevenueis TK =τPCI+CT +PT whereCT is taxes on corporate income (line 7 of NIPA Table 3.2), and PT is property taxes (line 8 of NIPA Table 3.3). In NIPA, home owners are thought of as renting their houses to themselves and thus property taxes are included as taxes on rental income or capital income. The labor tax revenueiscomputed TH =τP(W+PRI/2)+CSI whereCSI iscontributionsforgovernmentsocialinsurance(line11ofNIPATable3.2). C.3 Skill Premium in the CPS Data FollowingLindquist2004,IusetheMonthlyOutgoingRotationGroup(MORG)oftheUSCensus Bureau’s Current Population Survey data and check the share of the skilled and the unskilled and alsotheskillpremium. Theskillpremiumisdefinedastheratioofthehourlywageofworkerswith 14 or more years of schooling to the hourly wage of workers with less than 14 years of schooling. Werestrictthesamplewithagebetween20and70. Theshareofworkerswith14ormoreyearsof schooling is 0.505 and the skill premium for mean wage is 66% and the skill premium for median wageis55%. D Estimation of labor tax adjustment rule Thelabortaxrateadjustmentruleisspecifiedasthefollowing: (cid:16) (cid:17) (cid:16) (cid:17) τH−τ¯H =ρH τH −τ¯H +ρH τH −τ¯H t new 1 t−1 new 2 t−2 new    (cid:32) (cid:33) (cid:32) (cid:33) + (cid:16) 1−ρ 1 H−ρ 2 H (cid:17)  ψH B  P B t− Y 1 − P B Y  +ψ ∆ H Y Y Y t +ψH x Y Y n t   (D.1) t−1 t−1 t−1 t where Yn is the natural level of output. We estimate an empirical version of this rule by OLS. We t (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) first estimate the composite coefficients 1−ρH−ρH ψH, 1−ρH−ρH ψH , and 1−ρH−ρH ψH and 1 2 B 1 2 ∆Y 1 2 x then recover ψH, ψH , and ψH using the estimate of ρH and ρH. Quarterly US data is used for B ∆Y x 1 2 estimation: tax revenues-to-output ratio, market value of government debt-to-output ratio, output growth and the gap between actual output and potential output. The rule is estimated on two sub-periods. The first sub-sample covers the period from 1983Q1 through 2002Q4 as in Coibion 53

and Gorodnichenko (2011) and the second sub-sample covers the period from 2001Q1 through 2019Q3. Forthefirstsub-sample,wedropthesecondlagofthetaxrevenuestoensurestationarity of the tax rule. The data is taken from FRED of the Federal Reserve Bank of St. Louis. Potential output is real potential gross domestic product estimated by US Congressional Budget Office. TableF.1showstheestimationresults. E Sensitivity analysis Wepresentsomeadditionalresultsonsensitivityanalysisandextensions. E.1 Long-run results We start with long-run effects. First, we present comparative statics results with respect to Frisch elasticity of labor supply. This is an important parameter, given that different source of financing imply different labor supply response. Appendix Figures F.2 and F.3 show how with transfer adjustment higher Frisch elasticity leads to larger output effects while the reverse holds for labor tax adjustment. Next, in Appendix Figures F.4 and F.5, we compare across the three fiscal adjustmentsforagivenFrischelasticity. ConsistentwithProposition3.1,thedifferencebetweentransfer adjustmentcaseandlabortaxadjustmentcaseisbiggerforahigherFrischelasticity. As there are heterogeneous agents in our model, clearly the assumptions made on how profits and transfers are distributed across the two types of households makes a non-trivial difference for distributional variables. Appendix Figure F.6 shows long-run results under various combinations ofthesedistributions. Forinstance,iftheskilledworkersgetboththeprofitsand(cutin)transfers, it leads to a decline in consumption inequality, in sharp contrast to the baseline case. The results alsoshowthataggregateeffectsonoutputandinvestmenthowever,arerelativelysimilaracrossthe variouspossibilitiesforprofitsandtransferdistributions. We also present results on the equipment capital share parameter, λ. in Appendix Figures F.7 and F.8. Our calibration strategy for this parameter is to match the labor share and now we vary the targeted labor share to both higher and lower values than baseline. We find that the results are robust qualitatively overall and for aggregate variables, the quantitative differences are small. For consumption, there are some quantitative difference, as a smaller λ is more beneficial for the unskilled. E.2 Transition dynamics Wenextmovetotransitiondynamics. AppendixFigureF.9comparesthetransitiondynamicsinthe baseline model under the labor tax and inflation adjustment finance scheme for different inflation 54

feedback parameters in the Taylor rule (the inflation feedback parameter has to be below 1 in this regime). Our results are very robust. The differences across the parameterizations show up most clearly in inflation and debt dynamics, with a stronger Taylor rule coefficient in fact leading to a bigger effect on inflation dynamically. This is consistent with the analytical results for the simple stickypricemodelinBhattarai,LeeandPark(2014). Wethenshowtransitiondynamicsundertransfersadjustmentwithtwodifferentrulesforprofit and transfer distributions, one the baseline and the other where the skilled workers get both the profits and (cut in) transfers. Again, like with the long-run, Appendix Figure F.10 shows that the differences are less prominent in output effects, but show up more prominently in distributional variables. For instance, consumption of unskilled falls for a short-period only, whereas skilled consumption falls persistently, thereby leading consumption inequality to actually fall after a few periods. Moreover, the same dynamic pattern holds for wage inequality, which falls after a few periods. E.3 Welfare results Finally, we end with welfare results under various values of Frisch elasticity of labor supply. Appendix Figure F.11 shows that our main finding that transfer and labor tax adjustment do not lead to a Pareto improvement, but consumption tax adjustment in fact does is robust to both a higher and lower Frisch elasticity than our baseline parameterization. Next, we do a sensitivity analysis on the equipment capital share parameter, λ. As shown in Appendix Figure F.12, we find that our main finding that transfer and labor tax adjustments do not lead to a Pareto improvement, but consumptiontaxadjustmentinfactdoes,isrobusttodifferentvaluesofλ. 55

F Appendix tables and figures AppendixTableF.1: Estimationresultsforlabortaxrateadjustmentrules (1)Sample(1983Q1-2002Q4) (2)Sample(2001Q1-2019Q3) 0.869 0.785 ρH 1 (0.075) (0.171) 0.107 ρH 2 (0.168) 0.111 0.007 ψH B (0.107) (0.069) 0.831 1.821 ψH ∆Y (0.633) (1.473) 0.032 0.040 ψH x (0.035) (0.099) 0.001 -0.002 constant (0.003) (0.001) R2 0.832 0.853 Observations 79 75 Notes: ThetableshowsOLSestimatesofthelabortaxrateadjustmentrule(D.1). QuarterlyUSdataisusedfor estimation: tax revenues-to-output ratio, market value of government debt-to-output ratio, output growth and the gap between actual output and potential output. Column (1) shows the estimation results using the sample from from1983Q1through2002Q4andcolumn(2)showstheestimationresultsusingthesamplefrom2001Q1through 2019Q3. Forthecolumn(1),wedropthesecondlagofthetaxrevenuestoensurestationarityofthetaxrule. See AppendixDfordetails. 56

6 0 30 1 4 20 -2 0 2 10 -1 -4 0 0 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 3 4 0.3084 2 0.33 0.3083 2 1 0.3082 0 0.325 0 0.3081 -2 -1 0.308 0.32 -4 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 3 15 2 20 10 0 2 -2 10 5 1 -4 0 0 0 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 3.9 3 6 20 2.9 15 4 3.8 2.8 10 2.7 2 5 3.7 2.6 0 0 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 3.5 14.5 44 2.5 0 3 14 42 2 -0.5 13.5 40 2.5 13 1.5 38 -1 2 12.5 1 36 -1.5 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 Appendix Figure F.1: Transition dynamics of a permanent capital tax rate decrease under labor tax rate andconsumptiontaxrateadjustment 57

10 50 10 0 0 5 0 -10 0 -20 -20 -5 -50 -30 -40 -10 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 10 10 0.35 0.33 0 0 -10 0.3 -10 0.328 -20 -20 0.25 -30 -30 0.326 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 40 20 0 50 20 0 -5 0 0 -20 -20 -50 -10 -40 -60 -40 -100 -15 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 7 20 8 0 6 0 5 6 -20 -20 -40 4 4 -40 -60 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0.98 10 0.96 12.8 10 0.94 12.6 5 0.92 12.4 5 0.9 12.2 0 0.88 0 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 Appendix Figure F.2: Long-run effects of permanent capital tax rate changes under transfer adjustment withdifferentFrischelasticities 58

2 40 0 0 20 -5 0 0 -20 -20 -10 -2 -40 -15 -40 -60 -4 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0.311 0 0 0.34 -5 0.31 -10 -10 0.33 0.309 -15 -20 0.308 -20 0.32 -30 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 50 0 0 0 -5 0 -10 -20 -10 -15 -50 -40 -20 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 7 40 8 0 20 6 0 5 6 -20 -20 -40 4 4 -40 -60 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0.98 2 15 0.96 10 1.5 0.94 10 1 0.92 5 0.9 0.5 5 0.88 0 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 Appendix Figure F.3: Long-runeffectsofpermanentcapitaltaxratechangesunderlabortaxrateadjustmentwithdifferentFrischelasticities 59

10 50 0 0 0 0 -10 -20 -20 -5 -50 -30 -40 -10 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 10 10 0.35 0 0.34 0 -10 0.3 -10 0.33 -20 -20 0.25 -30 -30 0.32 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 50 20 50 0 0 0 0 -10 -20 -50 -50 -40 -100 -20 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 7 40 8 0 20 6 0 5 6 -20 -20 -40 4 4 -40 -60 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 4 10 15 2 10 0 -2 10 5 -4 5 -6 5 0 -8 0 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 AppendixFigureF.4: Long-runeffectsofpermanentcapitaltaxratechangesunderdifferentfiscaladjust- (cid:16) (cid:17) ments 1 =4.0 φ 60

40 10 0 0 20 5 0 -10 0 -20 -20 -20 -5 -40 -40 -60 -30 -10 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0.336 0.32 0 0.334 0 -10 0.332 0.3 -10 0.33 -20 -20 0.328 0.28 0.326 -30 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 10 50 50 0 0 0 0 -10 -10 -50 -20 -50 -100 -20 -30 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 7 40 8 0 20 6 0 5 6 -20 -20 -40 4 4 -40 -60 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 4 10 15 2 10 0 5 -2 10 -4 5 -6 5 0 -8 0 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 AppendixFigureF.5: Long-runeffectsofpermanentcapitaltaxratechangesunderdifferentfiscaladjust- (cid:16) (cid:17) ments 1 =0.5 φ 61

10 10 40 4.8 6.3 0 5 0 20.2 0 -10 0 -20 -20 -4 -40 -20 -40 -60 -30 -10 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0.34 4.2 0.34 4.7 0 0.32 0 0.32 0.3 -10 -5 0.28 -10 -20 0.3 0.26 -15 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 40 10 50 0 19.1 6.67 29.2 0 0 0 -10 -20 -10 -50 -40 -20 -20 -60 -100 -30 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 7 10 40 0 20 8 6 0 6 -20 -20 5 -40 4 4 -40 -60 -80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0.98 10 12.8 0.96 10 0.94 12.6 5 0.92 12.4 5 0.9 0 0.88 12.2 0 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 Appendix Figure F.6: Long-run effects of permanent capital tax rate changes under transfer adjustment (differentprofitsandtransferdistributionrules) 62

50 10 10 0 0 5 0 -10 0 -20 -20 -5 -30 -40 -50 -10 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 10 0.35 0.34 0 0 0.335 -10 0.3 0.33 -20 -20 0.325 -30 0.25 -40 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 20 50 0 50 0 0 0 -10 -20 -50 -50 -100 -20 -40 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 7 40 8 0 20 6 0 6 -20 -20 5 -40 4 4 -40 -60 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 4 15 15 2 10 0 10 -2 10 5 -4 5 -6 5 -8 0 0 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 AppendixFigureF.7: Long-runeffectsofpermanentcapitaltaxratechangesunderdifferentfiscaladjustments(λ=0.5,laborshare=0.53) 63

4 40 2 0 20 0 0 0 -2 -10 -4 -20 -20 -40 -6 -20 -8 -40 -60 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0.32 0 0.335 0 0.3 -10 0.33 -10 0.28 -20 0.325 -20 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 50 40 5 0 20 0 0 -5 0 -5 -20 -10 -10 -40 -50 -15 -60 -15 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 7 40 8 0 20 6 0 6 -20 5 -20 -40 4 4 -60 -40 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 6 8 2 14 4 6 0 12 2 4 -2 10 0 2 -4 8 -2 0 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 0 21 35 50 65 80 AppendixFigureF.8: Long-runeffectsofpermanentcapitaltaxratechangesunderdifferentfiscaladjustments(λ=0.07,laborshare=0.63) 64

80 1 4 60 20 3 0 40 2 10 -1 20 1 0 0 0 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 30 0.311 0.45 40 20 0.31 0.4 20 10 0.309 0.35 0 0 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 80 0 0 20 60 -10 -50 40 10 -20 20 -100 0 0 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 3.8 5 3.6 4 40 3.4 4.5 3.2 3 2 20 4 2.8 2.6 0 0 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 5 22 35 20 6 4 30 18 4 25 3 16 2 20 14 2 0 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 Appendix Figure F.9: Transition dynamics of a permanent capital tax rate decrease under labor tax rate andinflationadjustmentwithdifferentinflationfeedbackparameters 65

4 2 6 30 0 2 4 20 -2 -4 2 10 0 -6 0 0 -2 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0.34 0.33 5 4 0.335 0.325 2 0.33 0.32 0 0 0.325 0.315 0.32 0.31 -2 -5 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 2 20 6 30 0 15 4 20 -2 10 2 10 -4 5 0 0 -6 -10 0 -2 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 3.9 3 20 6 2.9 15 3.8 4 2.8 10 3.7 2.7 2 5 2.6 0 0 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 3.5 1 44 0 3 0.5 42 -0.5 0 40 -1 2.5 -1.5 -0.5 38 2 -2 -1 36 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 Appendix Figure F.10: Transition dynamics of a permanent capital tax rate decrease under transfer adjustment(differentprofitsandtransferdistributionrules) 66

(a)TransferAdjustment 5.35 4.59 4.16 0 0 -7.74 -8.24 -9.13 -10 -10 0 50 100 150 200 0 50 100 150 200 (b)LaborTaxRateAdjustment 111..12.324 0 0 --- 111 ... 532 578 -2 -4 -6 -8 0 50 100 150 200 0 50 100 150 200 (c)ConsumptionTaxRateAdjustment 11..0056 0 000...111 0246 -2 -4 0 50 100 150 200 0 50 100 150 200 Transferadjustment Labortaxadjustment Consumptiontaxadjustment 1 =4.0 1 =1.0 1 =0.5 1 =4.0 1 =1.0 1 =0.5 1 =4.0 1 =1.0 1 =0.5 φ φ φ φ φ φ φ φ φ Skilled 5.352 4.589 4.156 1.118 1.236 1.301 1.056 1.052 1.050 Unskilled -9.128 -8.242 -7.742 -1.546 -1.370 -1.276 0.157 0.136 0.124 AppendixFigureF.11: WelfarewithdifferentFrischelasticities 67

(a)TransferAdjustment 5.37 4.59 3.25 0 0 -5.73 -8.24 -10 -10-.1170 0 50 100 150 200 0 50 100 150 200 (b)LaborTaxRateAdjustment 11..2348 0.9 0 0 -0.65 -1.37 -1.78 -2 -4 0 50 100 150 200 0 50 100 150 200 (c)ConsumptionTaxRateAdjustment 1.24 1.05 0.72 0.36 0 00..01 044 -2 0 50 100 150 200 0 50 100 150 200 Transferadjustment Labortaxadjustment Consumptiontaxadjustment λ=0.5 λ=0.35 λ=0.07 λ=0.5 λ=0.35 λ=0.07 λ=0.5 λ=0.35 λ=0.07 Skilled 5.368 4.589 3.248 1.376 1.236 0.895 1.238 1.052 0.725 Unskilled -10.173 -8.242 -5.729 -1.776 -1.370 -0.655 0.039 0.136 0.36 AppendixFigureF.12: Welfarewithdifferentequipmentcapitalincomeshare(λ) 68

60 60 40 40 20 20 0 0 -20 -20 0 50 100 150 200 0 50 100 150 200 AppendixFigureF.13: Welfareimplicationsunderlabortaxrateandinflationadjustment 69