Temporary Partial Expensing in a General-Equilibrium Model

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Temporary Partial Expensing in a General-Equilibrium Model Rochelle M. Edge and Jeremy B. Rudd 2005-19 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.

Temporary Partial Expensing in a General-Equilibrium Model Rochelle M. Edge Jeremy B. Rudd Federal Reserve Board∗ Federal Reserve Board∗∗ First Draft: July 22, 2003 This Draft: April 12, 2005 Abstract This paper uses a dynamic general-equilibrium model with a nominal tax system to consider the effects of temporary partial expensing allowances on investment and other macroeconomic aggregates. ∗ Corresponding author. Mailing address: Mail Stop 61, 20th and C Streets NW, Washington, DC 20551. E-mail: rochelle.m.edge@frb.gov. ∗∗ E-mail: jeremy.b.rudd@frb.gov. We thank Alan Viard, David Reifschneider, and seminar participants at the Federal Reserve Board and Atlanta Fed System Conference for useful comments, andDarrelCohenforhelpfulconversations. Theviewsexpressedareourownanddonotnecessarily reflect the views of the Board of Governors or the staff of the Federal Reserve System.

1 Introduction In recent years, the use of forward-looking general-equilibrium models to analyze the conduct of monetary policy has become a commonplace of the macroeconomics literature. Bycontrast,considerablylessprogresshasbeenmadeinemployingthese models to examine questions related to fiscal policy.1 In this paper, we incorporate a nominal tax system into an otherwise standard sticky-pricemonetarybusinesscycle(MBC)model,andusetheresultingframework to examine the effect of a temporary partial expensing allowance on investment expenditures,realactivity,andgovernmentrevenues.2 Fromatechnicalstandpoint, temporary expensing allowances provide an excellent candidate for this kind of approach: There is significant scope for the general-equilibrium effects of these policies to differ from what a partial-equilibrium analysis would predict; moreover, the fact that these tax changes are temporary requires us to explicitly consider how agents’ behavior today is affected by their expectations of future events.3 In addition, expensing allowances figured prominently in the fiscal stimulus packages that were enacted in the wake of the most recent recession; hence, an analysis of the effects of these policies has important topical relevance. Besides analyzing the general-equilibrium effects of investment incentives, a broadergoalofthispaperistocontributetoourunderstandingofhowthecanonical MBC model responds to fiscal policy changes. Previous research has provided us 1There is an irony here inasmuch as one of the earliest calls for a structural approachto policy modelling—Lucas’s 1976 paper “Econometric Policy Evaluation: A Critique”—invoked a fiscal policy example (the establishment of an investment tax credit) to make its point. 2Partial expensing allowances permit firms to deduct a fraction of the cost of newly purchased capital goods from their taxable income. An expensing allowance is therefore similar to an investment tax credit (ITC) in that it allows a firm to raise its posttax income through purchases of capital goods; importantly, however, a firm is not allowed to claim any future depreciation allowances for its expensed capital (under an ITC, such a restriction is partly or wholly absent). 3Previous analyses of investment tax policies have not typically employed a framework that permitsthesimultaneoustreatmentoftheseissues(recentworkbyHouseandShapiro,2005,isan importantexception). Forexample,ElmendorfandReifschneider(2003)usearational-expectations macromodel (the Federal Reserve Board’s FRB/US model) to examine a permanent change in an investment tax credit, but are unable to treat the effect of a temporary credit. Similarly, Cohen,Hansen,andHassett(2002)provideacarefulstudyofthepartial-equilibriumimpactofan expensing allowance on user costs, while Abel (1982) examines the partial-equilibrium effects of temporary and permanent changes in tax incentives for investment. 1

with a relatively broad understanding of the model’s strengths and shortcomings as a tool for monetary policy evaluation. However, the model’s successes (or failures) in illuminating monetary policy issues need not translate to a corresponding degree of success in the fiscal policy context. In particular, this focus on monetary policy (as well as these models’ inherent complexity) has often led researchers to place less emphasis on capturing features of the economy—such as the capital formation process—that are likely to matter much more when fiscal policy concerns are paramount. We therefore provide a relatively detailed description of how the modelrespondstotheparticularfiscalpolicychangesweconsider,andidentifythose components of the model’s structure that most profoundly influence our results. In the remainder of the paper, we derive our theoretical model under various assumptions as to the type of costs faced by firms in adjusting their capital stock and labor inputs, the nature of the economy’s aggregate supply relation, and the way in which household saving is determined. Our motivation for considering a number of alternative investment specifications stems from the fact that the fiscal policies we consider involve current and prospective changes to the tax system; given the forward-looking nature of the problem, then, the nature of the investment adjustment costs faced by firms will have an important effect on their capital expenditures. Similarly, we demonstrate that the presence of a nominal tax system implies that real and nominal interest rates will have an important influence on the model’s real responses; hence, it is important to consider the degree to which the model’s predicted effect of changes in tax-based investment incentives depends on its implied dynamics for real interest rates and expected inflation. We also use the model to explore a practical question concerning the relative effects of two types of tax-based investment incentives; specifically, we examine whathappenstooutput,governmentrevenue,andcapitalformationwhenexpensing allowancesareincreasedwiththecorrespondingresponseofthesevariablesfollowing a reduction in capital taxes. This type of “bang-for-the-buck” calculation is similar to that discussed by Abel (1978) for permanent tax changes in a partial-equilibrium setting; however, our own treatment represents the first time this topic has been addressed within the context of a fully specified microfounded dynamic generalequilibrium framework. 2

2 A Sticky-Price Model with Nominal Taxation Our model economy is characterized by three sets of agents: households, firms, and the government. Households consume output, supply (homogeneous) labor, and purchase goods that are then transformed into capital and rented to firms. There are two classes of firms: a continuum of monopolistically competitive intermediategoods producers, each of whom hires labor and capital to produce a differentiated good, and a single final-good producer who aggregates the intermediate goods to produce output for final demand. Finally, the government consists of a fiscal authority, who levies taxes that are rebated to households as lump-sum transfers, and a monetary authority who sets interest rates according to a Taylor rule. With the exception of our treatment of taxation and investment, our theoretical setup is quite similar to the sticky-price monetary business cycle models used by Woodford (2003) and others to analyze monetary policy. We therefore devote most of this section to a detailed examination of those features of the model that are affected by the introduction of a nominal tax system, and relegate a more complete description of the model to the Appendix. 2.1 Households The preferences of household i (where i ∈ [0,1]) are represented by the utility function ∞ 1 1−σ 1 1+s U0 = E0 ( X t=0 δt (cid:20) 1−σ (cid:16) C t i (cid:17) − 1+s (cid:16) H t i (cid:17) (cid:21)) , (1) where Ci is defined as household i’s consumption, Hi is its labor supply, and δ and t t s denote the household’s discount factor and labor supply elasticity, respectively. Thehousehold’sbudgetconstraint—whichreflectsitsroleinaccumulatingphysical capital—is given by ∞ Ai t+1 /R t f =Ai t +R t kK t i−F t k R t kK t i−X t P t I t i− κ(1−κ)v−1 (1−X t−v )P t−v I t i −v v=1 ! X + 1−Fh W Hi+Profitsi +Ti−P Ci−P Ii, (2) t t t t t t t t t (cid:16) (cid:17)(cid:16) (cid:17) where R f = R −Fh(R −1). (3) t t t t 3

The variable Ai denotes the nominal value of household i’s bond holdings at the t beginning of period t; W is the nominal wage paid on labor; Rk is the rental t t rate paid to household i for the use of its capital stock Ki (where Ki depreciates t t geometrically at the rate κ); Profitsi represents the profits disbursed (as dividends) to households from the monopolistically competitive intermediate-goods producers; Ti are lump-sum transfers from the fiscal authority; P is the price of final output; t t Ii denotes the household’s current-period purchases of investment goods; and R is t t the gross pretax nominal interest rate between periods t and t+1. The fiscal system that we assume taxes all forms of nominal personal income (that is, income from financial assets, dividends, and labor) at the rate Fh, and t taxes capital income at the rate Fk.4 Hence, households receive an after-tax ret turn R f on their financial assets that is given by equation (3).5 In addition, two t types of deductions are permitted against capital income: depreciation charges and expensing allowances. The presence of depreciation allowances reflects the fiscal authority’s recognition that part of the payment capital owners receive from renting out their capital stock merely reflects compensation for the depreciation of the stock from its use in production. An expensing allowance, meanwhile, represents a (partial) rebate of the purchase price of a new capital good. Unlike a pure subsidy or credit, however, future depreciation of the portion of the new investment good that is expensed may not later be deducted from taxable income. Thus, an expensing allowance can be loosely thought of as a completely “front-loaded” depreciation allowance. We make the standard simplifying assumption that households directly own all capital in the economy and rent it out to firms; hence, tax provisions on investment are directly reflected in the budget constraint (2), as follows. First, an expensing allowance X is applied to household i’s time-t nominal expenditure on new capital t 4Wearemakinganarbitrary(butultimatelyunimportant)distinctionherebetweenthe“profits” that appear in equation (2)—which represent a pure surplus over the payments to the factors of production that is distributed as a dividend to firm owners—and payments to households in their capacityasownersofthecapitalstock,whichserveasthebaseofthecorporateincometax. While it is somewhat artificial to assume that the former payments are not considered profits by the tax code, thisassumptionhasnosubstantiveeffect onouranalysisbecause monopolyprofitshavethe sameeffectonhouseholdbudgetconstraintsasalump-sumpayment(andarezeroinequilibrium). 5Note that the form of this expression reflects the fact that only interest—not principal—is subject to taxation. 4

goods, P Ii. Second, the dollar value of depreciation at time-t from all previous t t purchases of capital is given as ∞ v=1 κ(1−κ)v−1P t−v I t i −v . However, because previouslyexpensedcapitalmaynot P receiveadepreciationallowance,eachtermP t−v I t i −v in the sum in equation (2) must be multiplied by (1−X t−v ). In addition, under the U.S. tax code depreciation is computed using historical cost; as a result, the investment price in the depreciation term is written with a t−v subscript.6 In practice, depreciation allowances are based on a legislated schedule of depreciation rates, not the true (economic) depreciation rate κ. In our model, using legislated depreciation rates to compute depreciation allowances would merely involve replacing ∞ v=1 κ(1−κ)v−1P t−v I t i −v in equation (2) with V v=1 κi v rsP t−v I t i −v , whereV denotesthetax-lifeofthecapitalstock—whichaveragesaround5-1/2years P P (22 quarters) for equipment investment—and κirs denotes the rate of depreciation v for tax purposes (specified by the tax code) in the vth period of the capital stock’s life. However, this extension significantly increases the number of state variables in the model, and complicates our interpretation of the resulting first-order conditions for investment. In addition, it turns out that few of the model’s qualitative results are affected by our equating tax depreciation with economic depreciation.7 We therefore assume that κirs = κ(1−κ)v−1 throughout. v Intheabsenceofadjustmentcostsoncapitalorinvestmentspending,thecapital accumulation process is given by Ki = (1−κ)Ki+Ii. (4) t+1 t t For our baseline model, we assume that is it costly to adjust firms’ capital stocks, with adjustment costs taking a quadratic form. This yields the following capital 6The difference between a partial expensing allowance and a pure investment subsidy can be easily described in the context of equation (2). Under partial expensing, when the household deducts its allowed proportion of current investment spending from current capital income future depreciationallowancesarescaledbackaccordingly(hencetheterm1−X tmultiplyingthedepreciationallowanceterms). Bycontrast,underaninvestmentsubsidytheallowancetodaywouldleave future depreciation allowances unaffected, so that allowable deductions to taxable income would be given by X t P t I t i− ∞ v=1 κ(1−κ)v−1P t−v I t i −v . 7Intuitively,reasonablechangestotheassumedpatternofcapitaldepreciationhaveaverysmall P effect on the cost of capital relative to the effect that obtains from the presence or absence of an expensing allowance. Hence, it is this latter factor that is the dominant influence on the contour of the model’s impulse response function for investment. 5

evolution equation: 2 χk Ki Ki = (1−κ)Ki+Iiexp − t+1 −1 , (5) t+1 t t  2 K t i !    where the parameter χk controls the curvature of the adjustment-cost function. In the baseline model, then, the household takes as given its initial bond stock Ai, the expected path of the gross nominal interest rate R , the price level P , the 0 t t wage rate W , the rental rate Rk, profits income, and the legislated personal income t t ∞ taxratesandexpensingallowances(Fh,Fk,andX ),andchooses Ci,Hi,Ii,Ki t t t t t t t+1 t=0 so as to maximize equation (1) subject to the budget constraint ((cid:8)2) and the capita(cid:9)l evolution equation (5). 2.2 Intermediate- and Final-Goods Producers j j The monopolistically competitive firm j chooses labor H and capital K to minit t j mize its cost of producing output Y , taking as given the wage rate W , the rental t t rate Rk, and the production function. Specifically, firm j solves: t 1−α α min W H j +RkK j such that H j K j −FC ≥ Y j , (6) {Hj,Kj} ∞ t t t t t t t t t t=0 (cid:16) (cid:17) (cid:16) (cid:17) whereα istheelasticityofoutputwithrespecttocapitalandandFC isafixedcost (set equal to FC = Y∗ ) that is assumed in order to preclude positive steady-state θ−1 profits. Thecost-minimizationproblemimplieslabor-andcapital-demandschedules j for each firm as well as an expression for the firm’s marginal cost MC . We bring t sticky prices into the model by assuming that intermediate-goods producers are Calvo price-setters: In any period, a fraction (1−η) of firms can reset their price, while the remaining fraction η are constrained to charge their existing price (which is indexed to the steady-state inflation rate). We also assume a representative final-good producing firm who takes as given the prices {P j }1 that are set by each intermediate-good producer, and chooses t j=0 intermediate inputs {Y j }1 to minimize its cost of producing aggregate output Y t j=0 t subject to a Dixit-Stiglitz production function: min 1 P j Y j dj s.t. Y ≤ 1 Y j θ− θ 1 dj θ− θ 1 . (7) {Y t j} ∞ t=0Z 0 t t t (cid:18)Z 0 t (cid:19) 6

Thiscost-minimizationproblemyieldsdemandfunctionsforeachintermediategood that are given by Y j = Y (P j /P ) −θ, where P , the price of final output, is defined t t t t t as P t = ( 0 1 (P t j )1−θdz)1− 1 θ. R 2.3 The Monetary Authority The central bank sets the nominal interest rate according to a Taylor-style feedback rule. Specifically, the target nominal interest rate R¯ is assumed to respond to t deviations of output and the (gross) inflation rate from their respective target levels Π¯ and Y¯: R¯ t = Π t /Π¯ β Y t /Y¯ γ R∗, (8) (cid:0) (cid:1) (cid:0) (cid:1) where R∗ denotes the economy’s steady-state (equilibrium) interest rate. For simplicity,wewillassumethatthecentralbanktargetstheeconomy’ssteady-statelevel of output, implying that Y¯ = Y∗. Policymakers smoothly adjust the actual interest rate to its target level: R t = (R t−1)ρ R¯ t 1−ρ exp[ξ t r], (9) (cid:0) (cid:1) where ξr represents a policy shock. t 2.4 The Fiscal Authority To keep the number of fiscal distortions in the model to a minimum, we assume a role for government that is as simple as possible; namely, one in which the fiscal authority merely raises revenues via taxation and then rebates these revenues as lump-sum transfers Ti to households. Hence, the government faces the following t budget constraint: 1 1 1 1 Tidi = Revenue = FhW Hidi+ FkRkKidi+ FhProfitsidi (10) t t t t t t t t t 0 0 0 0 Z Z Z Z 1 1 1 + F t h(R t−1−1) Ai t /R t−1 di− F t kX t P t I t idi− F t kLiabi t ,κ di. 0 0 0 Z (cid:16) (cid:17) Z Z Thegovernment’sdepreciationallowanceliabilitytohouseholdiinperiodt,Liabi,κ , t is given by: ∞ Liabi t ,κ = κ(1−κ)v−1 (1−X t−v )P t−v I t i −v =κ(1−X t−1)P t−1I t i −1 +(1−κ)Liabi t , − κ 1 v=1 X 7

under our assumption that depreciation allowances equal true economic depreciation.8 Note that if the net stock of bonds in the economy is zero (as it will be when all bonds are domestic and privately issued), then the first term in the second line of equation (11) drops out. An additional variable that we define here (since it will prove useful when we attempt to score different tax policies) is the present discounted value of revenues. This is given as: ∞ PDVr t ev = E t " v=0 δvM M U U t+ t / v / P P t t+v Rev t+v # = Rev t +E t (cid:20) δM M U t U + t 1 / / P P t t+v PDVr t+ ev 1 (cid:21) (11) X where the dependence on the marginal utility of consumption, MU , reflects the use t of a stochastic discount factor to value future income. Finally, we note in passing that changes in tax policy in our framework can be equated with shocks to suitably specified exogenous processes for the fiscal variables. For example, the introduction of a permanent partial expensing allowance is captured by a one-time shock to X , where the expensing allowance is assumed to t follow an AR(1) process with a unit autoregressive root: X t = X t−1+ǫx t . (12) Similarly, a temporary (n-period) partial expensing allowance can be treated as an innovation to X under the assumption that the allowance follows an MA(n−1) t process: X t = ǫx t +ǫx t−1 +···+ǫx t−n+1 . (13) Naturally, shocks to other fiscal variables (such as Fk) can be treated in a parallel t fashion. 2.5 The Model’s First-Order Conditions Weonlyconsiderthefirst-orderconditionsthataredirectlyaffectedbythepresence of nominal taxation; other first-order conditions are described in the Appendix. The household’s utility-maximization problem yields an intertemporal Euler equation along with a supply schedule for labor: f 1 R = δE t (14) C t σP t t "C t σ +1 P t+1# 8With legislated depreciation rates, this liability equals V v=1 κi v rs(1−X t−v)P t−v I t i −v . P 8

and W (1−Fh) t t = HsCσ. (15) P t t t The solution to the household’s maximization problem also yields a capital supply condition; however, when adjustment costs are present, this expression is relatively complicated. WethereforerelegateittotheAppendix(asequation45), andinstead give the capital supply equation that obtains when there are no adjustment costs for capital or investment, namely: Rk (1−Fk ) R f E t+1 t+1 =E t 1−FkX −PDVκ(1−X ) t " P t+1 # t "Π t+1 t t t t # (cid:16) (cid:17) −E t (1−κ) 1−F t k +1 X t+1−PDV t κ +1 (1−X t+1) , (16) h (cid:16) (cid:17)i where the variable R f is defined by equation (3). The variable PDVκ in equat t tion (16) is the present discounted value of future depreciation allowances that households can deduct from their tax liability; when depreciation allowances for tax purposes are equal to true economic depreciation, this is given by ∞ PDVκ = E δvMU t+v /P t+v κ(1−κ)v−1 Fk , (17) t t ( v=1 MU t /P t t+v ) X where we again use a stochastic discount factor to value future income streams.9 In addition, factor demand schedules (in which labor and capital demand is expressed as a function of output and factor-price ratios) are obtained from the intermediate-goods producers’ problem, while the final-goods producer’s problem yields demand functions for intermediate goods and an expression for the aggregate price level. These relations (along with the economy’s market-clearing condition) are described in detail in the Appendix. 2.6 The Log-Linearized Model Equations We obtain a linear model by log-linearizing the model equations about a deterministic steady state. Again, we mainly focus on describing and interpreting those equations that are directly affected by the presence of a nominal tax system; other log-linearized model equations are presented in the Appendix. 9When allowances are based on legislated depreciation rates, the κ(1−κ)v−1 term in equation (17) is replaced by κirs. v 9

The household’s Euler equation (14) becomes 1 f c t = E t c t+1− σ r t −E t π t+1 , (18) (cid:16) (cid:17) with π defined as the log-difference of the price level (here and elsewhere, we use lower-case letters to denote log deviations of variables from their steady-state values). As is clearly evident from this equation, consumption growth is a function of the real posttax interest rate. The log-linearized posttax nominal interest rate is given by r f = Π¯ δ −F∗ h r − Π¯ δ −1 · F∗ h E fh , (19) t Π¯ t Π¯ 1−F∗ h t t+1 δ δ where an asterisk in lieu of a time subscript denotes a variable’s steady-state value. Finally, the household’s labor supply condition log-linearizes to w = F∗ h fh+σ·c +s·h . (20) t 1−F∗ h t t t When capital adjustment costs are present, the capital supply condition yields the following log-linear expression for the user cost: E t r t k +1 = "1− F∗ k F∗ k # f t k +1 + (cid:20) 1−δ( 1 1−κ) (cid:21)(cid:16) r t f −E t π t+1 (cid:17) − 1 · PDV∗ κ pdvκ−δ(1−κ)E pdvκ (cid:20) 1−δ(1−κ) 1−PDV∗ κ (cid:21) t t t+1 (cid:0) (cid:1) 1 F∗ k−PDV∗ κ − "1−δ(1−κ) · 1−PDV∗ κ # (X t −δ(1−κ)E t X t+1) χk ·κ 1 − "1−δ(1−κ) · 1−PDV∗ κ # (δE t k t+2−(1+δ)k t+1+k t ), (21) with pdvκ = δ/Π¯ (1−κ)E pdvκ + 1− δ/Π¯ (1−κ) E fk −r f . (22) t t t+1 t t+1 t (cid:0) (cid:1) (cid:0) (cid:0) (cid:1) (cid:1) (Note that equation 21 is the log-linearized version of equation 45 from the Appendix.) As can be seen from these equations, there are two important ways in which the presence of a nominal tax system affects aggregate demand determination. First, consumptiongrowthandtheusercostarebothfunctionsoftherealposttax interest 10

rate, which will not move one-for-one with changes in the nominal interest rate whenincometaxesarenonzero. Second, becausedepreciationallowancesarevalued at historic cost, they will be worth less in current-dollar terms when inflation is positive—putdifferently, thenominalnatureofdepreciationallowancesimpliesthat nominal interest rates determine their discounted present value. Hence, an increase in nominal interest rates raises the user cost of capital in two ways: first by raising the posttax real interest rate, and second by lowering the expected present value of depreciation allowances.10 The other components of the log-linearized model are quite standard. Capital and labor demand are given by θ−1 k = y +(1−α)w −(1−α)rk (23) t θ t t t (cid:18) (cid:19) and θ−1 h = y −αw +αrk, (24) t θ t t t (cid:18) (cid:19) respectively, while the log-linearized aggregate supply relation is a new-Keynesian Phillips curve of the form (1−η)(1−ηδ) π t = δE t π t+1+ mc t . (25) η Finally, the log-linearized monetary policy rule is r t = ρr t−1+(1−ρ)(βE t π t+1+γy t )+ξ t r, (26) which combines equations (8) and (9).11 2.7 Calibration Thestructuralparametervaluesthatweuseinordertocalibratethebaselinemodel are summarized in the table below. The values for α, σ −1, and θ are set so as to match Kimball’s (1995) preferred calibration; δ is taken from Clarida, Gal´ı, and Gertler (2000, p. 170); and κ is computed from the depreciation rates and nominal 10Theseintersectionsofthetaxsystemwithaggregatedemanddeterminationchangetheconditions required for the existence of a determinate and stable rational-expectations equilibrium; see Edge and Rudd (2002) for a discussion. 11The Appendix details the model’s remaining log-linearized equations. 11

stocks in Katz and Herman (1997). None of these is particularly controversial.12 For χk, we choose a value that gives our capital adjustment cost function the same curvature properties as Kimball’s specification; more concretely, the adjustment costs under this calibration are such that, following a permanent shock (and in partial equilibrium), the capital stock adjusts 30 percent of the way to its desired level after one year.13 Finally, our assumed value for η implies that firms’ prices are fixed for one year on average, which is again standard; conditional on this value for η, our assumed (inverse) labor supply elasticity s is then chosen so as to yield an elasticity of inflation with respect to output that is similar to what Clarida, et al. employ in their work.14 Calibrated Values of Common Structural Parameters Parameter Description Value α Elasticity of output with respect to capital 0.30 σ −1 Intertemporal elasticity of substitution 0.20 θ Elasticity of substitution of intermediates 11 κ Depreciation rate 0.034 δ Households’ discount factor 0.99 χk Curvature parameter in adjustment cost function 500 (1−η) Probability firm can reset price 0.25 s Inverse labor supply elasticity 2.75 F∗ h Steady-state tax rate on noncapital income 0.30 F∗ k Steady-state tax rate on capital income 0.48 Π¯ Inflation target 1.00 12Note that our assumed value of θ implies an equilibrium markup of 10 percent. In addition, the depreciation rate κ and discount factor δ are expressed at a quarterly—not annual—rate; for example, our assumed value for depreciation equals 13 percent per year. 13Kimball’scalibrationisparticularlyrelevantforourpurposessinceitisinformedbytheresults of Cummins, Hassett, and Hubbard’s (1994) study, which uses variation in business tax rates (includingITCprovisionsanddepreciationallowances)toidentifyandestimatestructuralinvestment equations. 14With this value of s, a 2.75 percent increase in wages is required to raise hours supplied by one percent (all else equal). While this implies a labor supply curve that is steeper than what is commonlyemployedbyRBCmodellers,itisquiteconsistentwiththerangeofvaluesfoundinthe micro-labor literature (see, for example, Abowd and Card, 1989, table 10); it also yields a much more realistic implication for the representative consumer’s marginal expenditure share of leisure (c.f. the discussion in Kimball, 1995, pp. 1267-69). 12

For the policy-related parameters, the assumed values for F∗ h and F∗ k are intended to capture the average marginal tax rates on noncapital and capital income that are implied by the current U.S. tax code; a detailed description of how these valueswerechosen(togetherwithadiscussionofhowsensitiveourresultsaretodifferent assumptions about F∗ h and F∗ k) is provided in the Appendix. The Π¯ value we specify implies an inflation target of zero—which is the assumed steady-state value of inflation in the model—while the parameter values we set in our Taylor rule are β = 1.80, γ = 0.0675, and ρ = 0.79, which are the post-1979 values estimated by Orphanides (2001) using real-time data. 3 Effects of Partial Expensing Allowances In this section, we use the baseline model to examine the effects of permanent and temporary changes in the expensing allowance on capital investment, with a particular focus on the way in which the general-equilibrium character of the model influences its response to fiscal shocks. We then discuss how the model’s basic predictions change when the firm faces alternative adjustment-cost specifications for its capital stock or investment spending. Toprovideausefulbenchmark,wefirstpresentresultsfromapartial-equilibrium model that uses the same neoclassical investment specification that underpins the general-equilibrium model. Hence, any difference in results that obtains under the general-equilibrium framework arises because of the effects that changes in investment demand have on output, real interest rates, and consumption demand. In addition, when we compare the results from our general-equilibrium setup to those that obtain in a partial-equilibrium analysis, we use a version of the baseline model in which prices are assumed to be fully flexible (since aggregate price rigidities are irrelevant when output is exogenous). Later, this will permit us to separately identify the role played by sticky prices in our framework. 3.1 Effect of a Permanent Partial Expensing Allowance We first consider the effects of a permanent 30 percent expensing allowance.15 Figure 1 shows the predicted responses of the capital stock, gross investment, and 15We choose 30 percent for our example because it corresponds to the size of the (temporary) expensing allowance that was instituted under the 2002 Job Creation and Worker Assistance Act. 13

the real rental rate from the partial-equilibrium model, while Figure 2 gives the corresponding responses from the flexible-price version of the baseline model. (As consumption is an endogenous variable in the general-equilibrium model, we also plot its response in Figure 2.)16 In both models, the presence of the expensing allowance makes new capital a more attractive investment. Households therefore immediately begin to purchase capital goods, which raises the economy’s capital stock; the aggregate rental rate then falls as this new capital is added to the economy. Aclosercomparisonofthetwosetsofresultsrevealssomeimportantdifferences, however. Inthepartial-equilibriummodel, theonlyconstraintagentsfaceinadding to the capital stock is the presence of adjustment costs. By contrast, in a generalequilibriumframework, additional capital spending canonly occurif moreoutput is produced and/or a greater share of output is devoted to investment. In the model, this process is mediated by higher real interest rates (not shown), which induce households both to give up some of their consumption and to supply more labor (thus raising output).17 Itisalsoimportanttonotethatessentiallyallofthesluggishnessoftheresponse ofthecapitalstockinthegeneral-equilibriummodelreflectstheendogenousreaction of the other variables in the model. This can be most clearly seen by comparing the path of the capital-output ratio in the baseline general-equilibrium model to its path in a version of the model in which adjustment costs are completely absent, which we do in Figure 3. As is evident from this plot, capital adjustment costs have a relatively small incremental effect on the path of the capital-output ratio that obtains in the general-equilibrium model. This point can also be illustrated by noting that the capital-output ratio eventually rises about five percent above its baseline level as a result of the expensing allowance. In the partial-equilibrium model, therefore, the capital stock has moved roughly three-fourths of the way to its long-run value after twenty quarters. In the general-equilibrium setup, however, 16All variables are expressed as percentage deviations from their steady-state values, with the exception of the rental rate, which is given as a percentage-point deviation at a quarterly rate. 17Note that the rise in real rates actually pushes the economywide real rental rate above its baseline level for several periods after the expensing allowance comes into effect. (Intuitively, the rise in aggregate demand that results from the increased demand for investment goods makes installed capital more valuable.) Even so, there is still an incentive to invest, since the expensing allowance implies that new capital remains attractive even with the rise in real rates. 14

the capital-output ratio has moved about a third of the way to its long-run level after the same period of time has elapsed. 3.2 Effect of a Temporary Partial Expensing Allowance Wenowturntoanexaminationoftheeffectsofpartialexpensingallowancesthatare put into place for a limited period of time. This adds an important forward-looking aspecttothemodel,sincefirms’currentbehaviorwillanticipatetheexpectedfuture change in tax policy. As a result, the model’s dynamic responses will be richer, and will further highlight how the general-equilibrium nature of the analysis influences the results. The specific experiment we consider is the introduction of a 30 percent expensing allowance that lasts for three years; all agents are assumed to fully understand and believe the temporary nature of the allowance. Panel A of Figure 4 plots the predicted responses of capital, investment, and the rental rate from the partial-equilibrium model (with capital adjustment costs) following the introduction of the temporary expensing allowance. As before, the newallowancesmakenewcapitalinvestment(temporarily)moreattractive, thereby leading to a gradual increase in the capital stock and an immediate jump in investmentexpenditure; overtime,asmorecapitalisaddedtotheeconomy,theaggregate rental rate declines. Interestingly, however, in this case the temporary nature of the allowancesinducesfirmsto“pullforward”theirinvestmentspending(ascanbeseen from the figure, the path of the capital stock following a permanent increase in the expensing allowance—plotted here as a dotted line—lies below the response from the temporary-allowance case for the first four years). Later, when the expensing allowance expires, the capital stock lies above its steady-state level. Disinvestment is costly, however (there are adjustment costs), and so takes place over an extended period. The result is a persistent investment “pothole,” as the level of investment falls below its steady-state level. The responses of these variables (and consumption) in the flexible-price generalequilibrium model are plotted in panel B of Figure 4. As is apparent from a comparison with the partial-equilibrium case, the responses of capital and investment are smaller in the general-equilibrium model; in addition, there is no longer an investment pothole inasmuch as investment remains above its steady-state level even after the expensing allowance comes off (though we still obtain a sharp drop in the 15

level of investment—and thus a reduction in its growth rate—in the period that the allowance expires). Once again, the source of this more muted response of investment is the endogenous response of real interest rates and consumption to changes in investment demand.18 In general equilibrium, higher aggregate demand pushes up real interest rates (this is needed in order to call forth more saving), which acts to attenuate the increase in investment and the capital stock. Then, when the expensing allowance comes off, the resulting decline in aggregate demand is partly buffered by a reduction in real rates. Both of these factors imply that the resulting overcapacity (and desire to disinvest) is not as severe. Itisworthnotingthatverylittleinvestmentispulledforwardunderatemporary allowance in the general-equilibrium case (this can be seen from the leftmost plot in panel B of the figure, which also plots the response of capital following a permanent expensing allowance). Put differently, the usual conclusion that a temporary investment tax incentive will have a greater (short-term) effect on investment than a permanenttaxchange—aninsightthatisreadilydrawnfromthepartial-equilibrium framework—need not be correct once general-equilibrium considerations are taken into account.19 3.3 Expensing Allowances When Prices are Sticky Up to this point, we have examined a version of the general-equilibrium model in which prices were assumed to be fully flexible (this was done in order to permit a direct comparison with the partial-equilibrium setup). We now assume that prices arestickybyincorporatingthelog-linearizedaggregatesupplyrelation(equation25) into the model with costly capital adjustment. Panel A of Figure 5 plots the responses of capital, consumption and investment, and the real rental rate from this model following the introduction of a three-year, 30 percent expensing allowance. Comparison with panel B of Figure 4 reveals that 18Note that, under our calibration, the contribution of consumption to output growth (which here is analogous to nonfarm business output) is a little less than four times as great as that for investment. 19This is not, of course, a completely general result: As Auerbach (1989) demonstrates, the differentialeffectsoninvestmentoftemporaryandpermanenttaxchanges(oranychangetothecost ofcapital)dependsonthenatureoftheadjustment-costfunction. However,ourresultobtainsfor alloftheadjustment-costspecificationsthatweconsiderunderreasonablecalibrations. Moreover, thereisinvariablyapronounceddifferencebetweenthepartial-andgeneral-equilibriumpredictions of the model, which is the point that we are seeking to establish. 16

adding sticky prices changes the model’s predictions in several ways. First, the response of investment is greater than in the flexible-price case (though it is still smaller than the investment response from the partial-equilibrium model). Second, investment now temporarily falls below its steady-state level after the expensing allowance expires (so in this sense, we once again obtain an investment pothole). The intuition for both of these findings is relatively straightforward. Under sticky prices, firms commit to meeting all demand for their output at their fixed, posted price. Since output is partly demand-determined, there is less need for consumption to be crowded out through an increase in real interest rates, since a positiveaggregatedemandshockispartlymetbyincreasedsupply. Inaddition,this makes firms more concerned with their capacity (now and in the future), since an increaseindemandwillcauseasharpriseintheirrealmarginalcosts—and,hence,a declineintheirrealprofits—unlesstheyincreasetheircapitalstock. Likewise,under sticky prices there is an incentive to disinvest more rapidly in the face of a slump in demand, since firms are not able to make up for a demand shortfall by cutting prices. Finally, the movements in the rental rate for capital reflect the interaction of these swings in capital demand with currently available capital supply.20 3.4 Alternative Adjustment Cost Specifications In the preceding analysis, the presence of costly capital adjustment added an important forward-looking element to firms’ and households’ decisionmaking. However, recent work on investment dynamics has moved away from using this type of adjustment-costmechanismtomodelthefrictionsfacingfirmswhentheyseektoadjust their inputs. In this section, therefore, we consider two other specifications for factor-adjustment costs: one in which it is costly to adjust capital and the capitaloutput ratio (which can be thought of as a convex approximation to a putty-clay technology), and one in which it is costly to change the level of investment (which captures elements of time-to-build).21 20Sticky prices also affect the model’s response to a permanent change in expensing allowances (notshown). Asnotedearlier,thefactthatdepreciationallowancesarecalculatedusinghistorical costs implies that the nominal interest rate has an independent influence on the cost of capital (by determining the present value of future depreciation allowances). A permanent expensing allowanceyieldsapermanentlyhigherlevelofthecapitalstock,whichinturnimpliespermanently lower marginal costs and persistently lower inflation. As a result, nominal interest rates and the cost of capital both decline, which generates a larger eventual response of the capital stock. 21See section A.3 of the Appendix for the log-linearized versions of these equations. 17

Costly factor-ratio adjustment: Under costly factor-ratio adjustment, the production technology in the intermediate-goods sector becomes 2 1−α α χf K j /H j Y j = H j K j exp − t t −1 −FC (27) t (cid:16) t (cid:17) (cid:16) t (cid:17)  2 K t j −1 /H t j −1 !    (again, we also assume that it is costly to adjust the capital stock as well as the factor mix).22 Panel B of Figure 5 plots the results from this version of the model. The additional source of inflexibility implies that there is now less benefit from adjusting the capital stock independently; as a result, the investment response in this version of the model is more muted than what we obtained in the model with capital-adjustmentcostsonly(inparticular, investmentnowreachesitspeakalittle earlier, and also remains above its steady-state level after the expensing allowance comes off). However, we also now see a much larger swing in the rental rate for capital than before. Previously, firms facing changes in demand for their output were able to change their production by altering the amount of labor they hired. Here, though, this avenue is partly closed off, as it is now also costly to adjust labor inputs; theresultisamorepronouncedswingindemandforinstalledcapital, which shows up as a relatively larger change in the rental rate.23 Investment adjustment costs: Both the capital and factor-ratio adjustmentcostspecificationsimplyasharpinitialjumpininvestmentspendingwhenexpensing allowancesarefirstintroduced. Inpractice,however,firms’abilitytorapidlychange their capital expenditure plans is likely to be hampered by the presence of timeto-plan and time-to-build considerations. One straightforward way to capture this is to assume that investment—rather than the capital stock—is costly to adjust.24 We do this by assuming the following form for the capital evolution equation: 2 χa Ii Ki = (1−κ)Ki+Iiexp − t −1 , (28) t+1 t t  2 I t i −1 !    22We set χf in equation (27) equal to 25, which roughly halves the swing in the capital-labor ratio that occurs around the expiration date of the temporary expensing allowance. 23Note, however, that the responses of pre- and posttax nominal interest rates (not shown) are considerably smoother than the response of the rental rate, which reflects swings in the marginal product of installed capital and the markup of prices over marginal costs. 24Investment adjustment costs have been used by a number of authors; see Christiano, Eichenbaum, and Evans (2001) for a recent example. 18

where the adjustment cost parameter χa is set equal to 1.75 in order to match the estimated value found in Edge, Laubach, and Williams (2003). Panel C of Figure 5 plots results from this version of the model. The investment response in this case is smootherandshowsamorepronouncedhump(withthepeakofthehumpoccurring about a year before the expensing allowance expires); in addition, the decline in investment that occurs immediately after the expiration of the expensing allowance is not as sharp and is spread over a longer period of time. Similarly, the swings in the rental rate are more muted as well. 4 Additional Extensions to the Baseline Model In addition to the specific form of adjustment costs we assume, several other features of our model can conceivably affect the predicted response of investment to a change in tax policy. For example, the responses of saving and hours worked to changes in the real interest rate will obviously influence the response of investment spending to a tax shock; similarly, the independent role of the nominal interest rate on capital demand (which arises as a result of the nominal character of depreciation allowances) yields an additional way in which our characterization of the model economy’s aggregate supply relation affect the model’s predicted responses. Finally, a less obvious aspect of the model’s specification that turns out to have an interesting effect on our results is our implicit assumption that labor and capital can be used to produce either consumption or investment goods (which in turn reflects the single-good nature of our baseline theoretical framework). In this section, therefore, we consider how our results are affected by employing alternative specifications for household consumption and aggregate supply, and also extend the model to incorporate sector-specific factor inputs. 4.1 Modelling Habit Persistence in Consumption Within the macromodelling literature, it has become increasingly common to advocate a specification for household preferences that yields habit persistence in consumption.25 We add “external” habit persistence to our model by assuming the 25See Fuhrer (2000) for a representative example. 19

following alternative representation of preferences: ∞ 1 1−σ 1 1+s U0 = E0 ( X t=0 δt (cid:20) 1−σ (cid:16) C t i−bC t−1 (cid:17) − 1+s (cid:16) H t i (cid:17) (cid:21)) , (29) in the sticky-price model with investment adjustment costs.26 (We use the model with investment adjustment costs as our jumping-off point because we view its real responses—specifically, the hump-shaped response of output and investment that obtains following a shock—to be the most realistic.) The new first-order conditions for the household’s consumption and labor supply decisions are given in the Appendix; in log-linearized form, the consumption Euler equation is given by 1 1 1 f 1−b (c t −bc t−1) = 1−b (E t c t+1−bc t )− σ r t −E t π t+1 (30) (cid:18) (cid:19) (cid:18) (cid:19) (cid:16) (cid:17) and the household’s labor supply curve is w t = 1− F∗ h F∗ h f t h+ 1− σ b (c t −bc t−1)+s·h t . (31) Panel A of Figure 6 plots the responses to a three-year, 30 percent temporary expensing allowance from this alternative version of the model against the baseline model’s responses. As is evident from the figure, incorporating habit persistence has essentially no effect on the predicted path for investment. The source of this result can be readily seen if we re-write the log-linearized labor supply curve in the following equivalent fashion: w = F∗ h fh+ σ ((1−b)c +b∆c )+s·h (32) t 1−F∗ h t 1−b t t t (note that we are making use here of the identity c ≡ (1 − b)c + bc ). In the t t t baseline model without habit persistence, the response of consumption is already relativelysmooth(seepanelCofFigure5). Asaresult, ∆c ≈ 0, whichimpliesthat t equation (32) approximately reduces to the labor supply curve from the baseline model. Hence,thereislittlescopeforhabitpersistencetoinfluencethelaborsupply decision—and, thus, the response of the real economy—in this context. 26This particular specification of habit formation is taken from Christiano, Eichenbaum, and Evans (2001). We set b in equation (29) equal to 0.8, which implies a relatively large degree of habit persistence. 20

4.2 More Inertial Price-Setting The new-Keynesian Phillips curve that we employ in the baseline model has been criticized on the grounds that it implies a too-rapid response of inflation to real shocks. We therefore gauge the influence of this assumption on our results by consideringa“hybrid”Phillipscurve(duetoChristiano,Eichenbaum,andEvans,2001) in which inflation is partially indexed to its own lag.27 In log-linearized form, this aggregate supply relation is given as: 1 δ (1−η)(1−ηδ) π t = π t−1+ E t π t+1+ mc t . (33) 1+δ 1+δ (1+δ)η PanelBofthefiguregivestheresultsfromthisversionofthemodel. Inbroadterms, thepredictedresponsesforinvestmentarequitesimilaracrossthetwospecifications: Although the hybrid inflation equation does in fact yield a smaller initial response of inflation (not shown), after a few quarters the path of the inflation rate is similar to what is obtained in the baseline model (in addition, because we assume that the monetaryauthoritytriestosmoothitspolicyrate,thepathofnominalinterestrates is also quite similar across the two models). However, because the hybrid Phillips curve imparts more inertia to price setting, the path of inflation (and nominal interest rates) remains higher over a longer period in the alternative model. As a result, the response of investment is attenuated slightly relative to the baseline case.28 4.3 Multisector Production with Limited Factor Mobility Up to this point, the models that we have examined have all implicitly assumed a one-sector production structure in which labor and (existing) capital can be instantaneously and costlessly allocated to the production of either consumption or capital goods. As a result, a large portion of any increase in investment demand in our baseline model is accommodated by an increase in output, as households supply more hours to the economy’s single production sector. 27We would not want to leave the impression that we are advocating this model of price-setting behavior,sincethereiscompellingevidencethatittoohasdifficultyincapturingobservedinflation dynamics (see Rudd and Whelan, 2003, for a discussion). Rather, our motivation for employing this specification stems from its representing the most commonly cited alternative to a purely forward-looking inflation equation. 28Recall that nominal interest rates have an independent influence on investment through their effect on the present discounted value of depreciation allowances. 21

A more realistic production structure would involve separate sectors for the production of consumption and investment goods and would take into account the fact that capital and labor inputs tend to be sector specific (particularly over short horizons). In such an economy, it will be more difficult to rapidly increase production in a given sector; in particular, we would expect the rise in investment demand that results from the introduction of a temporary expensing allowance to be only partiallymet. Asaresult, wewilltendtoseeaslowerresponseofaggregateinvestment to a change in tax policy. Wemodelsector-specificlaborinputsbyassumingthathouseholdsincura(convex) adjustment cost whenever they change the number of hours that they supply to the consumption or investment sector; this implies the following alternative representation for household preferences: U0 = E0 X t ∞ =0 δt 1− 1 σ (cid:16) C t i (cid:17) 1−σ − 1+ 1 s (H t c,i +H t k,i ) 1+sexp  χ 2 l H H t c − t c, 1 i / / H H t t k k − ,i 1 −1 ! 2  ,   (34) where the k and c superscripts on hours index the investment- and consumptiongoods sectors, respectively. We set the adjustment-cost parameter χl equal to a relatively low value (namely unity).29 We allow for sector-specific capital by assuming distinct accumulation processes for the capital stocks employed in the consumption- and investment-goods sectors. Specifically, household i’s holdings of sector-n capital evolve according to 2 χa I i,n K i,n = (1−κ)K i,n +I i,n exp − t −1 (n = k,c), (35) t+1 t t  2 I t i − ,n 1 !    which,exceptforthensuperscript,hasanidenticalformtotheinvestmentadjustmentcost specification from the one-sector model (equation 28).30 PanelCofFigure6plotsthemodel’sresponsetoathree-year,30percentexpensing allowance. As is evident from this panel, the presence of sector-specific factor 29This calibration, along with the other parameter values we assume, implies that wages in the consumption-goods sector must move 1-3/4 percent above wages in the capital-goods sector in order to have one percent of the economy’s aggregate labor supply shift into the production of consumption goods (and vice-versa). 30The household faces a slightly different budget constraint in this version of the model; we describe it in detail in the Appendix. In addition, note that firms’ production functions will now reflect the sector-specific nature of factor inputs. 22

supplies further mutes investment’s response to a temporary expensing allowance relative to the baseline model. 5 Effects of Changes in the Capital Income Tax Rate An alternative policy that is often suggested as a means of stimulating investment spending involves reducing the tax rate on capital income. We next examine the effect of this policy on investment and output, and compare it with the effect of an expensing allowance that has the same impact on government revenues. Model responses: Figure 7 plots the usual set of model responses (under several different adjustment-cost specifications) following a three-year, 20 percentage point reduction in the capital tax rate Fk. Qualitatively, a temporary cut in the capital tax rate yields a path for investment spending that is much more front-loaded than the path that obtains under an expensingallowance. The reasonis that the benefits from a reduction in capital taxes are received for as long as the policy is in place; as a result, purchasing and holding a unit of capital for the full three-year period yieldsthegreatestgains. Bycontrast, anexpensingallowancerepresentsaone-time boon (in the quarter that the capital is purchased) that is worth roughly as much at the start of the three-year period as toward the end. Revenue Impact of Alternative Tax Policies: One of the most useful features of our model is its ability to assess the revenue consequences of alternative tax policies—in particular, we can compare the investment responses induced by a capital tax cut and an expensing allowance, where each policy is constrained to have an identical impact on government revenue. In Figure 8, we compare the effect of a temporary capital tax cut with that of a temporary expensing allowance, where each policy is set so as to yield the same change in the present value of government revenues.31 (We focus on the variant of the baseline model of Section 3 that incorporates investment adjustment costs; results for the other specifications are similar.) The present value is computed over a ten-year (or 40-quarter) period—this corresponds to the width of the “budget window” that is typically used to score the revenue effects of fiscal policy changes— 31Specifically, we compare a 30 percent temporary (three-year) expensing allowance with a 19.5 percentage point three-year reduction in the capital income tax. 23

using the following expression, 39 v−1 1−δ pdv rev (10) t = 1−δ40 E t  rev t − δv  rev t+v − (r t f +j −π t+1+j )  , (36) (cid:18) (cid:19) v=1 j=0 X X    which corresponds to a log-linearized, finite-period version of equation (11). As can be seen from the figure, the expensing allowance yields a uniformly higher response of investment (and output). Intuitively, since a capital tax applies to the incomefromall capitalwhileanexpensingallowanceappliestoexpendituresonnew capitalonly, theformerrepresentsarelativelyexpensivewaytocallforthadditional investment spending.32 6 Conclusions and Directions for Future Work This paper has attempted to analyze tax-based investment incentives in the context of a fully specified general-equilibrium model. Our analysis revealed two rather surprisingresults,andconfirmedanotherthathadbeenwell-establishedbyprevious researchers working in a partial-equilibrium framework. • First, our findings highlight the need to take general-equilibrium considerations seriously when thinking about the impact of tax incentives on investment. Forexample, thestandard(andintuitive)viewthattemporarychanges in expensing allowances induce a larger response of investment spending than do permanent changes turns out to depend on whether saving is endogenous. • Second, while the development of models that incorporate habit persistence and alternative specifications for price-setting behavior has led to important improvements in our ability to credibly assess monetary policy, we find that these extensions carry considerably less importance in the context of fiscal policy evaluation. By contrast, whether we model the multisector nature of production—which receives almost no attention in the monetary policy literature—turns out to have a relatively important effect on the model’s response to a tax change. In intuitive terms, the fiscal shocks that we are 32Note also that the capital stock is much higher at the end of the three-year period under the expensingallowance. Thus,iftherevenueconsequencesofeachpolicywereconsideredoveralonger (or infinite) period, expensing allowances would appear even more attractive. 24

considering here are more like real disturbances, so the model’s responses— and the determinants of its dynamics—are often more akin to those of a real (as opposed to monetary) business cycle model. • Finally, our analysis allowed us to confirm the result—previously only considered in a partial-equilibrium setup—that for policies with the same effect on the present value of government revenues, a change in the rate of capital taxation represents a relatively less efficacious way of stimulating investment than does a change in the expensing rate for newly purchased capital. A natural next step is to refine the framework developed here into one that can be used for quantitative simulations. Attaining this goal would require advancing our analysis along at least three fronts. First, any serious quantitative assessment of expensing allowances must recognize the fact that these tax provisions pertain to equipment investment only. Constructing a fully specified model in which different types of capital are used in production would require making difficult decisions about the degree of substitutability across capital types. However, for the purposes of short-term analysis, it might be sufficient to consider a model in which the stock of structures is assumed to be fixed; this would permit the model to generate more realisticpredictedresponsesofoutputtotax-inducedchangesin(equipment)capital without requiring us to explicitly model the investment decision for structures. Extending the baseline model to an open-economy setting would also represent an important refinement. With no external sector, the endogenous response of the real interest rate is larger following a tax-induced change in investment, since only domestically produced output can be used to meet the additional demand for physical capital. For the U.S. economy, this might be a reasonable first approximation (thoughseeAuerbach,1989,foracontraryview),asonlyaboutathirdoftheequipment purchased for investment in the U.S. is produced abroad. Nevertheless, an explicit treatment of external considerations in this context—while difficult given the current state of open-economy dynamic general-equilibrium modelling—would yield a framework with even greater practical relevance. Morefundamentally, anymodelthatpurportstoinformreal-worlddecisionmaking should be able to demonstrate a reasonable degree of empirical validity. For the applicationconsideredhere,formalempiricaljustificationislikelytobecomplicated 25

by the fact that the effect of tax changes on investment—let alone on interest rates, consumption, and inflation—is probably very difficult to parse out; moreover, relatively few historical examples of these sorts of tax changes exist. This suggests that considering tax changes alone will not allow us to identify all of the model’s parameters (though it might be possible to estimate these parameters by examining the model’s predicted response to other shocks). Finally, an additional useful extension would involve constructing an apparatus thatwouldpermittheassessmentofuncertainfuturepolicies. Inpractice,thelikelihood and/or length of proposed tax policies are typically not known with certainty, and this should act to attenuate the economy’s response to announced changes in tax policy. Whether the effects of such uncertainty could be quantified in a linear framework, however, is far from clear. 26

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[11] Edge, Rochelle M., Thomas Laubach, and John C. Williams (2003). “The Responses of Wages and Prices to Technology Shocks.” Finance and Economics Discussion Series Paper No. 2003-65. [12] Edge, Rochelle M., and Jeremy B. Rudd (2002). “Taxation and the Taylor Principle.” Federal Reserve Board Finance and Economics Discussion Series Paper No. 2002-51. [13] Fuhrer, Jeffrey C. (2000). “Habit Formation in Consumption and Its Implications for Monetary-Policy Models.” American Economic Review, 90, 367-390. [14] House,ChristopherL.andMatthewD.Shapiro(2005).“TemporaryInvestment TaxIncentives: TheorywithEvidencefromBonusDepreciation.”Unpublished manuscript (January 13th draft). [15] Katz, Arnold J. and Shelby W. Herman (1997). “Improved Estimates of Fixed Reproducible Tangible Wealth, 1929-95.” Survey of Current Business, May, 69-92. [16] Kimball, Miles S. (1995). “The Quantitative Analytics of the Basic Neomonetarist Model.” Journal of Money, Credit, and Banking, 27, 1241-1277. [17] Orphanides, Athanasios (2001). “Monetary Policy Rules, Macroeconomic Stability, and Inflation: A View from the Trenches.” Federal Reserve Board Finance and Economics Discussion Series Paper No. 2001-62. [18] Rudd, Jeremy and Karl Whelan (2003). “Can Rational Expectations Sticky- Price Models Explain Inflation Dynamics?” Federal Reserve Board Finance and Economics Discussion Series Paper No. 2003-46. [19] Woodford, Michael (2003). Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton, N.J.: Princeton University Press. 28

A Detailed Model Derivations This section of the Appendix gathers together the first-order conditions from the baseline model of section 2 (and its later versions) that are not discussed in the text, and explicitly describes the model’s equilibrium and steady-state solution. A.1 Omitted First-Order Conditions Theintermediate-goods producers’cost-minimizationproblem(6)yieldsfactor demand schedules for each firm; these have the form: 1−α α Rk/P α H j = Y j +FC t t and (37) t (cid:18) α (cid:19) (cid:16) t (cid:17) W t /P t! α 1−α W /P 1−α j j t t K = Y +FC . (38) t (cid:18) 1−α (cid:19) (cid:16) t (cid:17) R t k/P t! In addition, this problem implies a marginal cost function (which is identical for all firms) that is given by: MC j W /P 1−α Rk/P α t = t t t t . (39) P t (cid:18) 1−α (cid:19) α ! An intermediate-goods producing firm that is able to reset its price in period t j takes as given its nominal marginal cost MC , the aggregate price level P , and t t aggregate output Y and solves: t ∞ max ηkE δkMU t+k /P t+k P j −MC j Y j −P FC (40) {P t j} k X =0 t " MU t /P t (cid:16)(cid:16) t t+k (cid:17) t+k t (cid:17) # such that j −θ P Y t j +k = Y t+k P t+ t k! , (41) where MU denotes the marginal utility of consumption. This implicitly defines an t j optimal price P for firms who do change their prices in periodt, which is expressed t as: j ∞ k=0 ηkE t ((δkMU t+k /P t+k )/(MU t /P t ))MC t j +k θY t+k P = . (42) t P ∞ k=0 ηkE t h[((δkMU t+k /P t+k )/(MU t /P t ))(θ−1)Y t+k ]i Thefinal-gooPd producing firm’s cost-minimization problem(equation7) yields a demand function for each of the intermediate goods: −θ j j Y = Y P /P . (43) t t t t (cid:16) (cid:17) 29

The demand functions for the intermediate goods imply that the competitive price P for the final (actual) good is defined implicitly as: t 1 P = 1 (P j ) 1−θdz 1−θ . (44) t t 0 (cid:18)Z (cid:19) The economy’s goods-market clearing condition implies that C +I = Y , t t t where I denotes actual spending on capital goods. t The first-order condition for capital supply for the model with capital adjustment costs is given by: Rk 1−Fk t+1 t+1 E (45) t  (cid:16)P t+1 (cid:17) = E  R t f 1−exp −  χk K t+1 −1 2 FkX +PDVκ(1−X ) t "Π t+1 " 2 (cid:18) K t (cid:19) # (cid:16) t t t t (cid:17) ! ×exp χk K t+1 −1 2 1+χk K t+1 −(1−κ) K t+1 −1 " 2 (cid:18) K t (cid:19) #(cid:18) (cid:18) K t (cid:19)(cid:18) K t (cid:19)(cid:19)# f +E R t FkX +PDVκ(1−X ) χk K t+1 −(1−κ) K t+1 −1 t "Π t+1 (cid:16) t t t t (cid:17) (cid:18) K t (cid:19)(cid:18) K t (cid:19)# −E t " 1−exp " − χ 2 k (cid:18) K K t t + + 1 2 −1 (cid:19) 2 # (cid:16) F t k +1 X t+1+PDV t κ +1 (1−X t+1) (cid:17) ! ×exp χk K t+2 −1 2 (1−κ)+χk K t+2 −(1−κ) K t+2 −1 K t+2 " 2 (cid:18) K t+1 (cid:19) #(cid:18) (cid:18) K t+1 (cid:19)(cid:18) K t+1 (cid:19) K t+1 (cid:19)# −E t (cid:20)(cid:16) F t k +1 X t+1+PDV t κ +1 (1−X t+1) (cid:17) χk (cid:18) K K t t + + 1 2 −(1−κ) (cid:19)(cid:18) K K t t + + 1 2 −1 (cid:19) K K t t + + 1 2 (cid:21) while the corresponding expression for capital supply under investment adjustment costs is given by: Rk (1−Fk ) R f Q E t+1 t+1 = E t t −FkX −PDVκ(1−X ) t " P t+1 # t "Π t+1 (cid:18) P t t t t t (cid:19)# −E t (cid:20) (1−κ) (cid:18) Q P t t + + 1 1 −F t k +1 X t+1−PDV t κ +1 (1−X t+1) (cid:19)(cid:21) where E t Q t+1 I t+1exp − χi I t+1 −1 2 χi I t+1 I t+1 −1 "P t+1 " 2 (cid:18) I t (cid:19) # (cid:18) I t (cid:19)(cid:18) I t (cid:19)# = E R t f I − Q t I exp − χi I t+1 −1 2 1−χi I t+1 I t+1 −1 . t t t "Π t+1 P t " 2 (cid:18) I t (cid:19) #(cid:18) (cid:18) I t (cid:19)(cid:18) I t (cid:19)(cid:19)!# 30

Finally, when it is costly to adjust factor ratios (i.e., the production function takestheformofequation27),thefirst-orderconditionsfromtheintermediate-good producing firm’s cost-minimization problem become: j (W /P )H t t t −(1−α) j (MC /P )(Y +FC) t t t j j j j K /H K /H = χf t t −1 t t j j j j K t−1 /H t−1 ! K t−1 /H t−1! j j j j j −χfE Π t+1 · (MC t+1/P t+1)(Y t+1 +FC) K t+1 /H t+1 −1 K t+1 /H t+1 t " R t f (MC t /P t )(Y t j +FC) K t j /H t j ! K t j /H t j !# and (Rk/P )K j t t t −α j (MC /P )(Y +FC) t t t j j j j K /H K /H = −χf t t −1 t t j j j j K t−1 /H t−1 ! K t−1 /H t−1! j j j j +χfE Π t+1 · (MC t+1/P t+1)(Y t+1+FC) K t+1 /H t+1 −1 K t+1 /H t+1 . t " R t f (MC t /P t )(Y t +FC) K t j /H t j ! K t j /H t j !# A.2 Steady-State Equilibrium In deriving the model’s steady-state equilibrium, we first note that the steady-state valueoftheinflationrate,Π∗,isassumedtoequalthecentralbank’sinflationtarget, Π¯. The steady-state values of all other variables in the model are functions of the model’s parameters as well as of the steady-state inflation rate and the steady-state value of the tax variables (F∗ h, F∗ k, and X∗). From equations (3) and (14), the steady-state pretax and posttax nominal interest rates are given by: Π¯ 1 R∗ = −F∗ h and (46) δ ! 1−F∗ h Π¯ R∗ f = . (47) δ Thesteady-statevalueofrealmarginalcostisgivenbytheinverseofthemarkup, while equations (16) and (39) imply that the steady-state values of the factor prices 31

are given by: j MC∗ MC∗ θ−1 = = , (48) P∗ P∗ θ R∗ k 1−PDV∗ κ 1 = −(1−κ) , and (49) P∗ (cid:18) 1−F∗ k (cid:19)(cid:18) δ (cid:19) 1 α W∗ MC∗ 1−α α 1−α =(1−α) . (50) P∗ (cid:18) P∗ (cid:19) (cid:18) R∗ k/P∗ (cid:19) The variable PDV∗ κ is equal to either κ V δ v PDV∗ κ = F∗ k · Π¯ −(1−κ) or PDV t κirs = F∗ k Π¯ κi v rs, δ v X =1(cid:18) (cid:19) dependingonwhetherweuseeconomicdepreciationorthelegislatedtaxschedulefor depreciation allowances. Implicit in the definition of R∗ k/P∗ is the assumption that there are no expensing allowance provisions in the steady-state (which characterizes the U.S. tax code since 1986); as a result, X∗ = 0 and X t −X∗ = X t . The steady-state ratios H∗ j = H∗, K∗, I∗, and C∗ can be derived from equa- Y∗ Y∗ Y∗ Y∗ Y∗ tions (4), (37), (38), and the market-clearing condition. This yields: H∗ j H∗ θ 1−α α R∗ k/P∗ α = = , (51) Y∗ Y∗ (cid:18) θ−1 (cid:19)(cid:18) α (cid:19) W∗/P∗! K∗ θ α 1−α W∗/P∗ 1−α = , (52) Y∗ (cid:18) θ−1 (cid:19)(cid:18) 1−α (cid:19) (cid:18) R∗ k/P∗ (cid:19) I∗ K∗ θ α 1−α W∗/P∗ 1−α =κ· = κ , (53) Y∗ Y∗ (cid:18) θ−1 (cid:19)(cid:18) 1−α (cid:19) (cid:18) R∗ k/P∗ (cid:19) C∗ I∗ θ α 1−α W∗/P∗ 1−α =1− = 1−κ . (54) Y∗ Y∗ (cid:18) θ−1 (cid:19)(cid:18) 1−α (cid:19) (cid:18) R∗ k/P∗ (cid:19) Equations (51) and (54), together with the steady-state version of equation (15), yield the steady-state solution for real output: 1 1 (W∗/P∗)σ+s (1−F∗)σ+s Y∗ = s σ (55) (H∗/Y∗)σ+s (C∗/Y∗)σ+s 1 1 (W∗/P∗)σ+s (1−F∗)σ+s = . θ σ+ s s 1−α σ s + α s R∗ k/P∗ σ s + α s 1−κ θ α 1−α W∗/P∗ 1−α σ+ σ s θ−1 α W∗/P∗ θ−1 1−α R∗ k/P∗ (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:18) (cid:16) (cid:17)(cid:16) (cid:17) (cid:16) (cid:17) (cid:19) Together with equations (51) through (54), equation (56) yields solutions for the steady-state values of H∗, K∗, I∗, and C∗. 32

Finally, in the steady state real revenue is: Rev∗ = F∗ hY∗+(F∗ k −F∗ h) R∗ k K∗−F∗ k Liab ∗ κ , (56) P∗ P∗ P∗ where real depreciation allowance liabilities are Liabκ ∗ κ P∗ = Π¯ −(1−κ) I∗, (57) when we assume that firms deduct true economic depreciation and Li P a ∗ bκ ∗ = V κi v rs Π 1 ¯ v I∗, (58) v=1 (cid:18) (cid:19) X when deductions follow the legislated schedule of allowances. The steady-state present discounted value of real revenues is given by: PDVr ∗ ev 1 Rev∗ = . P∗ 1−δ P∗ (cid:18) (cid:19) A.3 Additional Log-Linearized Model Equations The log-linear expression for economy-wide marginal cost is mc = (1−α)w +αrk. (59) t t t The log-linear government revenue expression is given by rev = F∗ hY∗ fh+y + (F∗ k−F∗ h)(R∗ k/P∗)K∗ rk+k + F∗ k(R∗ k/P∗)K∗ fk t Rev∗/P∗ t t Rev∗/P∗ t t Rev∗/P∗ t (cid:16) (cid:17) (cid:16) (cid:17) − F∗ h(R∗ k/P∗)K∗ fh− F∗ kI∗ X − F∗ k(Liabκ ∗/P∗) fk +liabκ , Rev∗/P∗ t Rev∗/P∗ t Rev∗/P∗ t t (cid:16) (cid:17) where Π¯ −(1−κ) Π¯ −(1−κ) liabκ t = Π¯ ! I∗(i t−1−π t −X t−1)+ Π¯ ! liabκ t−1 −π t , (cid:0) (cid:1) for the case where economic depreciation is used to compute firms’ depreciation allowances, and 1 V 1 v liabκ t irs = V v=1 Π 1 ¯ v κi v rs v X =1(cid:18) Π¯ (cid:19) κi v rs(i t−v −X t−v ) P (cid:16) 1 (cid:17) V V 1 v + V v=1 Π¯ 1 v κi v rs w X =1 v X =w(cid:18) Π¯ (cid:19) κi v rs ! π t k +w−1 (60) (cid:16) (cid:17) P 33

for the case where legislated depreciation rates are used. We can then log-linearize equation (11), which yields: pdvr t ev = (1−δ)rev t −δ(r t −E t π t+1)+δE t pdvr t+ ev 1 . (61) In addition, note that if depreciation rates for tax purposes are given by a legislated schedule (and, so, not equivalent to economic depreciation), then the expressionforthelog-linearizedpresentvalueofdepreciationallowances(equation22) becomes 1 V δ v pdvκirs = κirsE fk (62) t V v=1 Π¯ δ v κi v rs v X =1(cid:18) Π¯ (cid:19) v t t+v P (cid:16) 1 (cid:17) V V δ v + V v=1 Π¯ δ v κi v rs w X =1 v X =w(cid:18) Π¯ (cid:19) κi v rs ! E t r t k +w−1 . (63) (cid:16) (cid:17) P Forthemodelwithoutcapitalorinvestmentadjustmentcosts,thecapitalsupply condition yields the following log-linear expression for the user cost: E t r t k +1 = "1− F∗ k F∗ k # f t k +1 + (cid:20) 1−δ( 1 1−κ) (cid:21)(cid:16) r t f −E t π t+1 (cid:17) − 1 · PDV∗ κ pdvκ−δ(1−κ)E pdvκ (cid:20) 1−δ(1−κ) 1−PDV∗ κ (cid:21) t t t+1 (cid:0) (cid:1) 1 F∗ k−PDV∗ κ − "1−δ(1−κ) · 1−PDV∗ κ # (X t −δ(1−κ)E t X t+1). (64) When investment adjustment costs are present, the corresponding expression for the user cost is given by: E t r t k +1 = "1− F∗ k F∗ k # f t k +1 + (cid:20) 1−δ( 1 1−κ) (cid:21)(cid:16) r t f −E t π t+1 (cid:17) − 1 · PDV∗ κ pdvκ−δ(1−κ)E pdvκ (cid:20) 1−δ(1−κ) 1−PDV∗ κ (cid:21) t t t+1 (cid:0) (cid:1) 1 F∗ k−PDV∗ κ − "1−δ(1−κ) · 1−PDV∗ κ # (X t −δ(1−κ)E t X t+1) 1 1 + (cid:20) 1−δ(1−κ) · 1−PDV∗ κ (cid:21) (q t −δ(1−κ)E t q t+1), and (65) q t =χa(1+δ)i t −χa·i t−1−χa·E t δi t+1. (66) 34

while the log-linearized first-order conditions under costly factor-ratio adjustment are: θ−1 χf θ−1 h t − θ y t = 1+(1+δ)χf h t−1− θ y t−1 (cid:18) (cid:18) (cid:19) (cid:19) (cid:18) (cid:18) (cid:19) (cid:19) δχf θ−1 + 1+(1+δ)χf E t h t+1− θ y t+1 (cid:18) (cid:18) (cid:19) (cid:19) α − (w −rk), and 1+(1+δ)χf t t θ−1 χf θ−1 k t − θ y t = 1+(1+δ)χf k t−1− θ y t−1 (cid:18) (cid:18) (cid:19) (cid:19) (cid:18) (cid:18) (cid:19) (cid:19) δχf θ−1 + 1+(1+δ)χf E t k t+1− θ y t+1 (cid:18) (cid:18) (cid:19) (cid:19) 1−α + (w −rk). 1+(1+δ)χf t t Finally, the economy’s goods-market clearing condition log-linearizes to C∗ I∗ y = c + i . (67) t t t Y∗ Y∗ A.4 Additional Equations for the Habit-Persistence Model With habit-persistence the consumption Euler equation and labor supply curve become: 1 Rt = δE t (C t −bC t−1)σP t t "(C t+1−bC t )σP t+1# and W (1−Fh) t P t = H t s(C t −bC t−1)σ. t A.5 Additional Equation for the Hybrid New-Keynesian Phillips Curve Under the assumption that firms who cannot reset their prices index to lagged inflation equation (42) becomes: j 2−θ ∞ j k 2−θ −1 P P MU t t (θ−1)Y t + γk+1 βk+1 E0 MU t+k+1 t Π t+l (θ−1)Y t+k+1  P t! k=0  P t+k+1 l=0 !  X Y  j j 1−θ   MC P × MU t t θY t t  P t P t!  35

∞ MC j P j k 1−θ + γk+1 βk+1 E0 MU t+k+1 t+k+1 t Π t+l θY t+k+1 = 1. k=0  P t+k+1 P t+k+1 l=0 !  X Y   (68) When we log-linearize this, we obtain the hybrid new-Keynesian Phillips curve (equation 33). A.6 Additional Equations for the Sector-Specific Factors Model With two sectors, the labor and capital demand schedule, marginal cost function, and pricing equation all generalize in a straightforward manner (there is now one for each sector). The labor supply schedules are more complicated, however. For c,i H the first-order condition is given by: t 2 χl H c,i /H k,i exp t t −1 (69)  2 H t c −1 /H t k −1 !   s  1 1+s H c,i /H k,i 1/H k,i × H c,i +H k,i + H c,i +H k,i χl t t −1 t , t t 1+s t t H t c −1 /H t k −1 ! H t c −1 /H t k −1! (cid:16) (cid:17) (cid:16) (cid:17) k,i while the first-order condition for H is: t 2 χl H c,i /H k,i exp t t −1 (70)  2 H t c −1 /H t k −1 !   s  1 1+s H c,i /H k,i H c,i −1/H k,i × H c,i +H k,i + H c,i +H k,i χl t t −1 t · t . (cid:16) t t (cid:17) 1+s (cid:16) t t (cid:17) H t c −1 /H t k −1 ! H t k,i H t c −1 /H t k −1! These log-linearize as follows: wc = F∗ h fh+σ·c +s H∗ c ·hc+ H∗ k ·hk (71) t 1−F∗ h t t H∗ c+H∗ k t H∗ c+H∗ k t ! + 1+ 1 s H∗ c H + ∗ c H∗ k ! χl hc t −hk t − hc t−1 −hk t−1 (cid:16)(cid:16) (cid:17) (cid:16) (cid:17)(cid:17) wk = F∗ h fh+σ·c +s H∗ c ·hc+ H∗ k ·hk t 1−F∗ h t t H∗ c+H∗ k t H∗ c+H∗ k t ! + 1+ 1 s H∗ c H + ∗ k H∗ k ! χl hk t −hc t − hk t−1 −hc t−1 . (cid:16)(cid:16) (cid:17) (cid:16) (cid:17)(cid:17) 36

B Calibrating the Steady-State Tax Rates This portion of the Appendix describes how the effective tax rates on income are calibrated, and discusses how our main results are affected by different assumed values for the capital tax rate. B.1 Calibration of Fh ∗ We follow Edge and Rudd (2002) in using tabulations from the Statistics of Income (Table 3.4) to compute average marginal Federal tax rates on earned income. For 2001 (the most recent year for which these data are available), we obtain an average marginal rate that is a little more than 25 percent. We then adjust this figure to reflect income taxation by state and local governments; specifically, data from the National Income and Product Accounts (NIPAs) indicate that state and local personal income taxes represented about 2-1/2 percent of overall personal income in 2001. As this is an average (not marginal) rate, we double it to capture the progressive nature of most state and local tax systems. The sum of these two rates yields the 30 percent average marginal tax rate that we assume. B.2 Calibration of Fk ∗ We require an estimate of the average marginal tax rate on capital income. Excluding depreciation, net capital income can be divided into three categories: dividends, retained earnings, and interest payments. If the corporate income tax rate is given by F∗ c, and if dividends (and capital gains) are taxed at the rate F∗ d, then the effective tax rate on capital income F∗ k is implicitly defined by 1−F∗ k = (1−ω)(1−F∗ d)(1−F∗ c)+ω(1−F∗ h), (72) where ω denotes the share of net interest payments in overall capital income. Under current law, the Federal corporate income tax rate is 35 percent, while the Federal tax rate on dividends and capital gains is 15 percent. (We add an additional 5 percentage points to these rates to reflect taxation at the state and local level.) Using NIPA data, we estimate that 17.5 percent of the capital income share is paid out as net interest. All together, these figures imply a capital tax rate of 48 percent, which is the value we assume for F∗ k in our baseline model. 37

The preceding assumes that the double taxation of dividends (at the corporate and personal level) matters in determining the cost of capital. Under the so-called “new view” of dividend taxation, however, the taxation of dividend income at the personal level is immaterial as far as the cost of capital is concerned.33 In this case, the first tax term in parentheses on the right-hand side of equation (72) equals one, implying that the effective tax rate on capital income is 38 percent. Finally, the simplest possible case arises when firms are financed exclusively through debt (in which case taxable corporate income is zero). This implies that all capital income is taxed at the personal tax rate, or that F∗ k = F∗ h = 30 percent. To assess how sensitive our results are to alternative assumptions about F∗ k, the table below gives the long-run change in the real rental rate of capital (expressed as a percent deviation from its steady-state level) following a permanent 30 percent expensing allowance for various assumed values of F∗ k.34 Based on the figures in the table (and given the log-linear structure of the model), assuming a value of F∗ k consistent with dividend taxation’s having no effect on the cost of capital would reduce the model’s responses by about a third, while assuming that firms are purely debt-financed would scale them down by about a half. Long-Run Percent Change in Real Rental Rate Tax rate F∗ k Description Change 48 percent Baseline assumption −5.21 38 percent “New view” of dividend taxation −3.68 30 percent Fully debt-financed firms −2.67 33See Auerbach (1979) and Bradford (1981) for discussions of this issue. 34Thefiguresinthetablegivethedirect effectontherentalratethatobtainsfromachangeinthe expensing allowance under the specified tax rate; they do not incorporate any general-equilibrium effects. 38

FIGURE 1.: Permanent Expensing in a Partial-Equilibrium Model with Capital Adjustment Costs A.: 20 Quarters Capital Investment Rental Rate Percent Percent Percentage points 8 10 0.1 8 6 0.0 6 4 -0.1 4 2 -0.2 2 0 0 -0.3 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 B.: 200 Quarters Capital Investment Rental Rate Percent Percent Percentage points 8 10 0.1 8 6 0.0 6 4 -0.1 4 2 -0.2 2 0 0 -0.3 0 40 80 120 160 200 0 40 80 120 160 200 0 40 80 120 160 200

FIGURE 2.: Permanent Expensing in a General-Equilibrium Flexible-Price Model with Capital Adjustment Costs A.: 20 Quarters Capital Investment and Consumption Rental Rate Percent Percent Percentage points 8 8 0.1 6 6 0.0 4 4 -0.1 2 2 -0.2 0 I C 0 -2 -0.3 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 B.: 200 Quarters Capital Investment and Consumption Rental Rate Percent Percent Percentage points 8 8 0.1 6 6 0.0 4 4 -0.1 2 2 -0.2 0 I C 0 -2 -0.3 0 40 80 120 160 200 0 40 80 120 160 200 0 40 80 120 160 200

FIGURE 3.: Effect of Permanent Expensing on the Capital-Output Ratio in a General-Equilibrium Flexible-Price Model Percent 5 4 3 2 1 0 With capital adjustment costs Without capital adjustment costs -1 0 40 80 120 160 200

FIGURE 4.: Temporary Expensing with Capital Adjustment Costs A.: Partial-Equilibrium Model Capital Investment Rental Rate Percent Percent Percentage points 6 15 0.00 -0.05 10 4 -0.10 5 -0.15 2 0 -0.20 Temp. Perm. 0 -5 -0.25 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 B.: General-Equilibrium Flexible-Price Model Capital Investment and Consumption Rental Rate Percent Percent Percentage points 3 6 0.06 0.03 4 2 0.00 2 -0.03 1 0 -0.06 Temp. Perm. I C 0 -2 -0.09 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20

FIGURE 5: Temporary Partial Expensing, Alternative Adjustment-Cost Specifications A.: Capital Adjustment Costs Capital Investment and Consumption Rental Rate Percent Percent Percentage points 4 12 0.4 3 8 0.2 2 4 0.0 1 0 -0.2 I C 0 -4 -0.4 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 B.: Factor-Ratio Adjustment Costs Capital Investment and Consumption Rental Rate Percent Percent Percentage points 4 12 2 1 3 8 0 2 4 -1 1 0 -2 I C 0 -4 -3 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 C.: Investment Adjustment Costs Capital Investment and Consumption Rental Rate Percent Percent Percentage points 4 12 0.10 0.05 3 8 0.00 2 4 -0.05 1 0 -0.10 I C 0 -4 -0.15 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20

FIGURE 6.: Temporary Expensing Allowance, Alternative Models A.: Model with Habit-Persistence in Consumption Capital Investment Rental Rate Percent Percent Percentage points 4 12 0.1 3 8 0 2 4 -0.1 1 0 0 -0.2 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 Solid line: Model with habit-persistence; Dotted line: Benchmark investment adjustment cost model. B.: Model with a Hybrid New-Keynesian Phillips Curve Capital Investment Rental Rate Percent Percent Percentage points 4 12 0.1 3 8 0 2 4 -0.1 1 0 0 -0.2 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 Solid line: Model with hybrid new-Keynesian Phillips curve; Dotted line: Benchmark investment adjustment cost model. C.: Two-sector Model with Limited Cross-sectoral Factor Mobility Capital Investment Rental Rate Percent Percent Percentage points 4 12 0.1 3 8 0 2 4 -0.1 1 0 0 -0.2 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 Solid line: Two-sector model; Dotted line: Benchmark investment adjustment cost model.

FIGURE 7: Capital Tax Rate Cut, Alternative Adjustment-Cost Specifications A.: Capital Adjustment Costs Capital Investment and Consumption Rental Rate Percent Percent Percentage points 4 12 0.4 3 8 0.2 2 4 0.0 1 0 -0.2 I C 0 -4 -0.4 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 B.: Factor-Ratio Adjustment Costs Capital Investment and Consumption Rental Rate Percent Percent Percentage points 4 12 3 3 8 2 2 4 1 1 0 0 I C 0 -4 -1 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 C.: Investment Adjustment Costs Capital Investment and Consumption Rental Rate Percent Percent Percentage points 4 12 0.10 3 8 0.05 2 4 0.00 1 0 -0.05 I C 0 -4 -0.10 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20

FIGURE 8: Comparison of Two Temporary Equal-Revenue Investment Incentive Policies Capital Investment Percent Percent 3 10 8 2 6 4 1 2 0 0 0 4 8 12 16 20 24 28 32 36 40 0 4 8 12 16 20 24 28 32 36 40 Tax Revenue Output Per period flow, percent Percent 4 2.0 0 1.5 -4 1.0 -8 0.5 -12 0.0 0 4 8 12 16 20 24 28 32 36 40 0 4 8 12 16 20 24 28 32 36 40 Solid line: 30 percent partial expensing allowance Dotted line: 19.5 percentage point cut in the capital tax rate