The State-Dependent Effects of Tax Shocks Eric Sims University of Notre Dame & NBER Jonathan Wolff Miami University August 11, 2017 Abstract This paper studies the state-dependent effects of shocks to distortionary tax rates in a dynamic stochastic general equilibrium (DSGE) model augmented with a number of real and nominal frictions. The tax output multiplier is defined as the change in output for a one dollar change in tax revenue caused by a shock to distortionary tax rates on consumption, labor income, or capital income. We find that magnitudes of each tax multiplier vary considerably across states of the business cycle. Tax cuts are most stimulative for output in states where output is high and are comparatively less stimulative in periods of recession. To evaluate the normative desirability of tax cuts as a tool to combat recessions, we also consider the statedependence of what we define as the tax welfare multiplier, which measures the change in a measure of aggregate welfare conditional on a tax cut scaled by the response of tax revenue. Welfare multipliers for each tax are found to be strongly negatively correlated with simulated output. We consider the robustness of these baseline results to several alternative modeling specifications. These include a rich array of alternative tax policies, anticipation, rule-of-thumb households, and a passive monetary policy regime. JEL Codes: E30, E60, E62 Keywords: fiscal policy, tax policy, business cycle, welfare We are particularly grateful to Tim Fuerst, Robert Lester, Michael Pries, Morten Ravn, Nam Vu, participants at the Fall 2014 Midwest Macro Conferences, seminar participants at the University of Notre Dame, Miami University, Bowling Green State University, and several anonymous referees for several comments which have substantially improved the paper. The usual disclaimers apply.
1 Introduction Recent events have sparked a renewed interest in the macroeconomic effects of fiscal policy. This revival has been fueled by the confluence of sluggish labor markets, large public debts, and inadequately accommodative monetary policy in many developed countries following the Great Recession. This paper focuses on the macroeconomic effects of cuts to distortionary tax rates. Namely, we seek to provide answers to the following questions. How stimulative are tax cuts? How much do these effects vary over the business cycle? From a normative perspective, is it desirable to cut taxes during periods in which output is low? The framework in which we address these questions is a medium-scale dynamic stochastic general equilibrium (DSGE) model similar to Christiano, Eichenbaum and Evans (2005); Schmitt- Grohé and Uribe (2005); Smets and Wouters (2007); and Justiniano, Primiceri and Tambalotti (2010, 2011). The model features price and wage rigidity as well as several real frictions, including habit formation in consumption, variable capital utilization, and investment adjustment costs. Monetary policy is governed by a Taylor rule. A government consumes some output, and finances this expenditure with a mix of debt, lump sum taxes, and distortionary taxes on consumption, labor, and capital. We fit the model to U.S. data by estimating a subset of the model parameters via Bayesian maximum likelihood and use conventional calibration methods for those parameters which remain. We solve the model via a higher order perturbation. We define the tax output multiplier to be equal to the change in output for a one dollar change in total tax revenue following an exogenous shock to a distortionary tax rate. We focus on multipliers at two horizons: on impact (the period of the change in the tax rate) and the maximum response (the maximum change in output following a change in a tax rate). These definitions follow Barro and Redlick (2011) and Mertens and Ravn (2012, 2014a). Because we solve the model via a higher order perturbation, the multiplier for each tax rate may differ across states. Our principal quantitative exercise involves simulating states from the model and computing multipliers starting from each point in the simulated state space. We then study the properties of the distributions of multipliers across states. We find that the average values of the consumption, labor, and capital tax multipliers are 0.58, 0.94, and 1.47, respectively. 1 That is, a one dollar decline in tax revenue from a cut to the capital income tax rate stimulates output by approximately one dollar and fifty cents on average. For each of the three kinds of tax rates, we find that there is significant variation in the magnitudes of the multipliers across states. The capital tax multiplier ranges from a low of 1.25 to a maximum value of 1.80. The range for the labor tax multiplier is 0.75 to 1.17. The consumption tax multiplier varies least across states, with a range of 0.54 to 0.62. For each kind of tax rate, the output multipliers are modestly countercyclical (i.e. negatively correlated with simulated output). This countercyclicality 1 Here and throughout the remainder of the paper, multipliers are presented as positive numbers so as to streamline presentation and facilitate comparison with the spending multiplier literature. As defined, tax multipliers in our model are always negative, since any tax change that stimulates output results in less tax revenue (i.e. we are always to the left of the peak of the Laffer Curve ). In addition, when we refer to the output multiplier throughout the Introduction, we are referring to the maximum output multiplier as defined in the paragraph above. 1
seems suggestive that there is more bang for the buck when cutting taxes during a recession. However, we find that the countercyclicality of the output multipliers is in actuality driven by a procyclical response of tax revenue to tax shocks. When we compute output multipliers scaling by the steady state response of tax revenue to a tax shock, we find that the output multipliers are strongly procyclical. That is, tax cuts are most stimulative when output is relatively high, rather than low. Do these co-movements suggest that countercyclical tax cuts are undesirable? To address this question, we adapt terminology from Sims and Wolff (2017) and construct what we call tax welfare multipliers. The welfare multiplier is defined as the consumption equivalent change in welfare (the present discounted value of flow utility) after a shock to a distortionary tax rate divided by the response of tax revenue. Not surprisingly given the distortionary nature of taxes in our model, we find that welfare multipliers are large and positive on average. Relative to the output multipliers, we find significantly more state-dependence in the welfare multiplier for each tax rate. Furthermore, we find that the welfare multipliers for each type of tax are strongly countercyclical. Correlations with simulated output are between -0.82 and -0.89. 2 From a normative perspective, these results suggest that tax cuts in our model are in fact most desirable during periods where output is low. Intuition for our results follows from the time-varying nature of distortions in our model. The interaction between imperfect competition in goods and labor markets with nominal stickiness gives rise to time-varying price and wage markups. The combined effects of these markups can be summarized by the labor wedge (Chari, Kehoe and McGrattan 2007), which is highly countercyclical both in the data and in our model. Put differently, the overall level of distortion in the economy is countercyclical. Cutting taxes, which eases a policy-imposed distortion, is naturally most valuable in periods where other distortions are highest. This accounts for the strong countercyclicality of the welfare multipliers. In contrast, when distortions are highest, output responds least to tax stimulus, which provides intuition for why we find that the output effects of tax shocks are procylical. The mild countercyclicality of the traditionally defined output multiplier is driven by the procyclical response of tax revenue. Tax revenue responds least to a tax cut in periods where the tax base is low, which correlates with periods when output is low. There is an extensive literature on the economic effects of tax shocks. Early contributions include Friedman (1948), Ando and Brown (1963), Hall (1971), Barro (1979), Braun (1994), and McGrattan (1994). More recent contributions include Blanchard and Perotti (2002), Romer and Romer (2010), and Mertens and Ravn (2011, 2012, 2014a). Reduced form empirical approaches yield wide ranges of tax cut multipliers. For example, Blanchard and Perotti (2002) find tax multipliers near one while Romer and Romer (2010) estimate maximum tax cut multipliers around three. 3 Our analysis based on a fully-specified DSGE model is closest to Mertens and Ravn (2011), 2 Unlike the output multipliers, the cyclicalities are qualitatively the same whether one scales the welfare response by the tax revenue response in a given state or by the tax revenue response evaluated in the steady state. When scaling by the tax revenue response in steady state, the correlations with output are somewhat less negative (ranging from -0.50 to -0.75), owing to the procyclical response of tax revenue. 3 A drawback of the purely empirical approach taken by these authors is that it does not distinguish between different kinds of tax rates when thinking about the effects of a tax cut. 2
Chahrour, Schmitt-Grohé and Uribe (2012) and Leeper, Walker and Yang (2013). We extend the DSGE-based literature in examining the state-dependence of tax multipliers as well as looking at the normative implications of tax rate changes. Much of the DSGE-based literature on the economic effects of distortionary tax rates has focused on the roles of policy anticipation, financing method, the presence of credit constrained households, and the interaction with the stance of monetary policy. Steigerwald and Stuart (1997), Yang (2005), House and Shapiro (2008), Perotti (2012), Mertens and Ravn (2012), and Leeper, Walker and Yang (2013) study the implications of anticipation lags for the transmission of tax shocks and generally find that anticipation in tax processes can have a significant impact on the size of multipliers. In an extension of our baseline model, we consider the presence of anticipation lags of 2-6 quarters between when a tax change is announced and when it takes effect. We find that anticipation serves to increase the magnitude of multipliers, but has little impact on the volatility and comovement of output and welfare multipliers over the state space. In addition to anticipation, several studies have noted the importance of financing method on the effectiveness of fiscal policy shocks. Christ (1968), Yang (2005), Mountford and Uhlig (2009), and Leeper, Plante and Traum (2010) note that, in the presence of forward looking agents, the combination of debt, lump sum, and distortionary taxes used to finance present policy changes has a significant impact on the effectiveness of policy. Following Leeper, Plante and Traum (2010), we consider several alternative estimations of the model which consider the role of distortionary taxes in financing government debt. We find that the taxes employed to stabilize debt do play a role in determining the magnitude, volatility, and comovement of tax multipliers with simulated output. However, we generally find that our basic results are robust to different financing regimes, almost always finding that welfare multipliers are strongly countercyclical. Recent work by Agarwal, Liu and Souleles (2007), Galí, López-Salido and Vallés (2007), McKay and Reis (2016), and others suggests that the presence of credit constrained consumers might impact the magnitude of fiscal multipliers. We therefore consider an extension of the baseline model which incorporates a fist-to-mouth consumer population in the spirit of Campbell and Mankiw (1990). We find the magnitude for each type of tax multiplier to be significantly impacted by the presence of this population while the state-dependence and cyclicalities of the multipliers remain relatively unchanged. Increasing the rule-of-thumb consumer population from 5 to 45 percent of the population results in an 18 percent increase in the average consumption tax output multiplier, a 13 percent increase in the average labor tax output multiplier, and a 6 percent increase in the average capital tax output multiplier. Average welfare multipliers for the consumption and labor tax rates are increasing in the rule-of-thumb population, while the average welfare multiplier for the capital tax is decreasing in this population. For all three types of taxes, the output effects of tax shocks are more positively correlated with simulated output, and the welfare multipliers more countercyclical, the larger is the rule-of-thumb population share. Much of the surge in interest concerning the effectiveness of fiscal policy centers on the recent period of passive monetary policy. In several studies dating back to Krugman (1998), Eggertsson and Woodford (2003), and more recently Christiano, Eichenbaum and Rebelo (2011), passive 3
monetary policy regimes have been shown to dramatically increase the effectiveness of fiscal policy shocks. However, this work has focused almost entirely on government spending and has largely ignored the impact of tax cuts in a passive policy regime. Noteworthy exceptions include Mertens and Ravn (2014b) and Boneva, Braun and Waki (2016). We consider a modified version of our baseline model in which monetary policy is passive in response to a tax shock for a known, exogenously determined horizon. We find that tax multipliers monotonically decrease in the length of the passive policy regime; the same is also true of our tax welfare multipliers. In addition, we find that increasing the length of the passive policy regime reduces the standard deviations of multipliers across states. If monetary policy is passive for a longer period of time, the output effects of tax shocks become more procyclical while the welfare multipliers become less negatively correlated with simulated output. Our paper is also related to a small but growing empirical literature studying state-dependence. Generally employing either sign restrictions or a narrative approach, this literature finds significant state-dependence as well as smaller effects of tax stimulus when the economy is characterized by slack. Candelon and Lieb (2013) is an early example employing sign restrictions in a regime switching framework and finding procyclical tax stimulus effectiveness. Arin, Koray and Spagnolo (2015) use a narrative approach in a regime switching model and also find smaller tax multipliers in times of slow growth. Hussain and Malik (2016) study whether increases and decreases in tax rates have differential effects. Our work expands upon the fiscal multiplier literature in several ways. First, we provide the first analysis (of which we are aware) of tax multiplier state-dependence in a fully specified DSGE model. We confirm the relatively large state-dependence documented by several empirical studies and also demonstrate that, when using the steady state tax revenue response, output multipliers are smaller in times of slack (in line with the recent empirical literature). Second, we provide a normative analysis which strongly supports the use of countercyclical tax stimulus. Third, our fully-specified DSGE approach allows us to isolate the state-dependence of specific tax rates, rather than bundling all tax cuts into one multiplier. Finally, we consider the impact of several realistic modeling alternatives which have been shown to impact the effectiveness of tax policy and provide intuition for how these realistic assumptions might impact state-dependence. The remainder of the paper proceeds as follows. Section 2 describes the medium-scale DSGE model. Section 3 discusses the parameterization of the model. In Section 4, we conduct our main simulation exercises to study the magnitude, state-dependence, and co-movement of tax multipliers. Section 5 introduces the extensions of our basic framework. The final section concludes. 2 Medium-Scale DSGE Model This section presents a medium-scale dynamic stochastic general equilibrium (DSGE) model in the spirit of Christiano, Eichenbaum and Evans (2005), Schmitt-Grohé and Uribe (2005), Smets and Wouters (2007), and Justiniano, Primiceri and Tambalotti (2010, 2011). The model features a representative household, a continuum of intermediate goods producers, and a single final good 4
producer. In addition, we incorporate a government with a rich array of financing options including distortionary consumption, labor, and capital taxes, lump sum taxes, and non-state contingent bonds. Among the real frictions present in the model are monopolistic competition, investment adjustment costs, habit formation, variable capital utilization, and the aforementioned distortionary taxes. The model also contains nominal frictions in the form of price and wage stickiness as well as price and wage indexation. Below, we describe the optimization problem of each agent, and conclude the section with a full definition of an equilibrium in this model. 2.1 Firms A single, perfectly competitive final good firm bundles the output of each of the j (0, 1) intermediate goods firms into a single product for consumption and investment by the household. The technology used in transforming these intermediate goods into a final good is given by the following CES aggregator: ( 1 Y t = 0 ) ɛp ɛp 1 ɛp 1 Y t (j) ɛp dj The output of this final good firm is denoted by Y t while the output of intermediate goods producer j is denote by Y t (j). The elasticity of substitution between intermediates is measured by ɛ p > 1 and the prices of each intermediate good j, P t (j), are taken as given by the final good producer. The final good firm s profit maximization problem results in the following demand schedule for each intermediate goods firm j: by: ( ) Pt (j) ɛp Y t (j) = Y t j (2) P t Using (1) and (2), as well as the firm s zero profit condition, the aggregate price index is given (1) ( 1 P t = 0 ) 1 P t (j) 1 ɛp 1 ɛp dj (3) Intermediate goods firms produce output using labor, N d,t (j), and capital services, Kt (j), according to the production function: Y t (j) = A t Kt (j) α N d,t (j) 1 α (4) The exogenous variable A t is a neutral productivity shock common to all intermediate goods firms. Capital services (the product of physical capital and utilization) are rented on a periodby-period basis from the household at the real rental rate rt k. Labor employed by firm j, N d,t (j), is paid a real wage w t. Cost minimization by intermediate goods firm j results in the following optimality conditions: 5
mc t = w1 α t (rt k ) α (1 α) α 1 α α (5) A t K t (j) N d,t (j) = α w t 1 α rt k j (6) Real marginal cost is defined as mc t and is given by (5). All intermediate firms face common factor prices. This, coupled with the assumption that all firms face a common productivity shock, implies that intermediate goods firms will choose capital services and labor in the same ratio. Each period, a fraction (1 θ p ) of randomly chosen firms have the opportunity to update their price, where θ p [0, 1). The opportunity to update price is independent of pricing history. Non-updating firms have the opportunity to index their price to lagged inflation with indexation parameter ζ p [0, 1]. Prices are set to maximize the present discounted value of real profit returned to the household, where discounting is via the household s stochastic discount factor as well as the likelihood of the chosen price remaining in place multiple periods. Given a common real marginal cost, all updating firms select a common reset price which we denote by P # t. To stationarize the model, we define inflation as π t = P t /P t 1 1 and reset price inflation as π # t P # t /P t 1 1. Employing these new variables, the optimal reset price for each firm can be written recursively as: 1 + π # t = ɛ p ɛ p 1 (1 + π t) X 1,t X 2,t u p,t (7) X 1,t = mc t µ t Y t + θ p β(1 + π t ) ζpɛp E t (1 + π t+1 ) ɛp X 1,t+1 (8) X 2,t = µ t Y t + θ p β(1 + π t ) ζp(1 ɛp) E t (1 + π t+1 ) ɛp 1 X 2,t+1 (9) The variable µ t is the household s marginal utility of income. Equations (5)-(9) characterize the optimal behavior of the production side of the economy. The exogenous variable u p,t is a reduced-form price markup shock as in Smets and Wouters (2007). While we do not model its microfoundations, it could be motivated as a time-varying elasticity of substitution (see, e.g., Justiniano, Primiceri and Tambalotti 2010). 2.2 Household We follow Schmitt-Grohé and Uribe (2005) in populating the economy with a single representative household. The household supplies labor to a continuum of labor markets of measure one, indexed by h (0, 1). The demand for labor in each market is given by: ( ) wt (h) ɛw N t (h) = N d,t, h (10) w t The real wage charged in market h is given by w t (h), N d,t is aggregate labor demand from intermediate goods firms, and ɛ w > 1 is the elasticity of substitution among labor in different markets. Wage stickiness is introduced à la Calvo (1983) each period, the household can adjust the wage in a randomly chosen fraction θ w of labor markets, where θ w [0, 1). Nominal wages in non-updated 6
markets can be indexed to lagged inflation at rate ζ w [0, 1]. Total labor supplied by the household is N t, which must satisfy N t = 1 0 N t(h)dh. Combining this condition with (10), we get: 1 ( ) wt (h) ɛw N t = N d,t dh (11) 0 The term inside the integral in (11) is a measure of wage dispersion, to be discussed below. Household welfare is defined as the present discounted value of flow utility from consumption, C t, and leisure, L t = 1 N t : w t V 0 = E 0 β t ν t U (C t bc t 1, 1 N t ) (12) t=0 The period utility function is increasing and concave in each argument and allows for non-separability between consumption and leisure. The parameter 0 b < 1 measures the degree of internal habit formation in consumption and 0 < β < 1 is a discount factor. The exogenous variable ν t is an exogenous preference shock. Physical capital, K t, accumulates according to: ( )) It K t+1 = Z t (1 S I t + (1 δ)k t (13) I t 1 Investment in new physical capital is denoted by I t and 0 < δ < 1 is the depreciation rate. As in Christiano, Eichenbaum and Evans (2005), S( ) measures an investment adjustment cost and satisfies S(1) = S (1) = 0, and S (1) = κ 0. The exogenous variable Z t is a shock to the marginal efficiency of investment as in Justiniano, Primiceri and Tambalotti (2010). The flow budget constraint faced by the representative household is: (1 τ n t ) 1 0 (1 + τ c t ) C t + I t + Γ(u t )K t + B t P t w t (h)n t (h)dh + (1 τ k t )r k t u t K t + (1 + i t 1 ) B t 1 P t + Π t T t (14) The nominal price of goods is denoted by P t. Distortionary tax rates on consumption, labor income, and capital income are denoted by τ c t, τ n t, and τ k t, respectively. The stock of nominal bonds with which the household enters the period is denoted by B t 1. The nominal interest rate on bonds taken into period t+1 is i t. The household pays a lump sum tax to the government, T t. Distributed profit from firms is given by Π t. Utilization of physical capital is given by u t. Utilization incurs a resource cost measured in units of physical capital given by the function Γ( ). It has the following properties: Γ(1) = 0, Γ (1) = ψ 0 > 0 and Γ (1) = ψ 1 0. The following conditions characterize optimal behavior by the household: (1 + τ c t )µ t = ν t U C (C t bc t 1, 1 N t ) βbe t ν t+1 U C (C t+1 bc t, 1 N t+1 ) (15) µ t = βe t µ t+1 (1 + i t )(1 + π t+1 ) 1 (16) 7
(1 τ k t )r k t = Γ (u t ) (17) ( ) ( ) ] It 1 = q t Z t [1 S S It It µ t+1 + βe t q t+1 Z t+1 S I t 1 I t 1 I t 1 µ t ( It+1 I t ) ( ) 2 It+1 (18) I t q t = βe t µ t+1 µ t [ (1 τ k t+1)r k t+1u t+1 Γ(u t+1 ) + (1 δ)q t+1 ] (19) F 1,t w # t = ɛ w u w,t (20) ɛ w 1 F 2,t F 1,t = ν t U L (C t bc t 1, 1 N t )w ɛw t N d,t + θ w βe t (1 + π t ) ɛwζw (1 + π t+1 ) ɛw F 1,t+1 (21) F 2,t = µ t (1 τ n t )w ɛw t N d,t + θ w βe t (1 + π t ) ζw(1 ɛw) (1 + π t+1 ) ɛw 1 F 2,t+1 (22) In these conditions µ t is the Lagrange multiplier on the flow budget constraint; q t is the ratio of the multiplier on the accumulation equation and the flow budget constraint. The optimal real reset wage, w # t, can be written recursively and is the same across all markets. If wages are flexible (i.e. θ w = 0), then optimality conditions related to the labor market reduce to setting the real wage equal to a markup over the marginal rate of substitution between consumption and leisure. Similarly to the price-setting problem of intermediate goods producers, u w,t is an exogenous reduced form wage markup shock. It could alternatively be interpreted as a time-varying intratemporal preference shock (Chari, Kehoe and McGrattan 2009). 2.3 Government Fiscal policy in our model is governed by a system of spending, tax, and budget rules. The fiscal authority has the opportunity to raise revenue via distortionary and lump sum taxes. Any discrepancy between revenue and cost can be settled by the issuance of one period non-state contingent bonds. These bonds are denoted by B g t. The real flow budget constraint for the government is given by: G t + i t 1 B g t 1 P t = τ c t C t + τ n t w t N t + τ k t r k t K t + T t + Bg t Bg t 1 P t (23) We assume that government spending obeys an exogenous AR(1) process in the log, where G is the non-stochastic mean of government spending: ln G t = (1 ρ g ) ln G + ρ g ln G t 1 + s g ε g,t (24) The shock ε g,t is drawn from a standard normal distribution and s g is the standard deviation of 8
the shock. We do not explicitly model the usefulness of government spending. As is standard, we could assume that the household receives flow utility from government purchases as a way to model the desirability of public expenditure. As long as utility from government spending is additively separable from utility over consumption and leisure, the nature of this utility flow is irrelevant for equilibrium dynamics. Given an exogenous time path for government spending, a long run debt target, B g, and an exogenous stock of initial debt, B g t 1, taxes must react to debt sufficiently so as to support a nonexplosive equilibrium. We assume that the tax instruments obey stationary AR(1) processes which feature a built-in reaction to deviations of existing debt from the long run target. These processes are: ( ) T t = (1 ρ T )τ T + ρ T τt 1 T + (1 ρ T ) γt b (B g t 1 Bg ) + γ y T (ln Y t ln Y t 1 ) + s T ε T,t (25) ( ) τt c = (1 ρ c )τ c + ρ c τt 1 c + (1 ρ c ) γc(b b g t 1 Bg ) + γc y (ln Y t ln Y t 1 ) + s c ε c,t (26) ( ) τt n = (1 ρ n )τ n + ρ n τt 1 n + (1 ρ n ) γn(b b g t 1 Bg ) + γn(ln y Y t ln Y t 1 ) + s n ε n,t (27) ( ) τt k = (1 ρ k )τ k + ρ k τt 1 k + (1 ρ k ) γk b (Bg t 1 Bg ) + γ y k (ln Y t ln Y t 1 ) + s k ε k,t (28) Each of these processes features a non-stochastic steady state value of the tax, a persistence parameter, and a reaction coefficient to deviations of debt from target. The coefficients on the deviation of debt from its long run target are given by γt b, γb c, γn, b and γk b. We require the value of these coefficients to be such that the equilibrium features a non-explosive path of government debt. We also include an automatic stabilizer mechanism wherein the tax rates react to output growth. The automatic stabilizer mechanism is governed by the parameters γ y T, γy c, γ y n, and γ y k. We also consider exogenous shocks to the distortionary tax rates, the effects of which are the principal source of inquiry in the paper, as well as the lump sum tax (though we do not study the effects of lump sum tax shocks). These shocks are drawn from standard normal distributions with standard deviations of s c, s n, s k, and s T. Monetary policy is governed by a Taylor interest rate feedback rule which responds to deviations of inflation from target as well as to output growth: [ ] i t = (1 ρ i )i + ρ i i t 1 + (1 ρ i ) φ π (π t π) + φ y (ln Y t ln Y t 1 ) + s i ε i,t (29) The monetary policy rule is subject to an exogenous shock, ε i,t, which is drawn from a standard normal distribution with standard deviation s i. We restrict the parameters of the policy rule to the region consistent with a determinate rational expectations equilibrium. 9
2.4 Exogenous Processes and Market-Clearing In addition to the processes for the distortionary tax rates, monetary policy rule, and government spending process, the model features five other exogenous processes. These are the neutral productivity variable, A t, the marginal efficiency of investment, Z t, the intertemporal flow utility shock, ν t, and the price and wage markup shocks, u p,t and u w,t. Each of these follow mean zero AR(1) processes in the log with shocks drawn from standard normal distributions. These distributions have time invariant standard deviations of s a, s z, s ν, s up, and s uw, respectively. ln A t = ρ a ln A t 1 + s a ε a,t (30) ln Z t = ρ z ln Z t 1 + s z ε z,t (31) ln ν t = ρ ν ln ν t 1 + s ν ε ν,t (32) ln u p,t = ρ up ln u p,t 1 + s up ε p,t (33) ln u w,t = ρ uw ln u w,t 1 + s uw ε w,t (34) Integrating across demand functions for intermediate goods, making use of the fact that all firms hire capital services and labor in the same ratio, and imposing market-clearing for labor yields the following aggregate production function: Y t = A t K t α N 1 α d,t v p t (35) The term v p t as: is a measure of price dispersion arising from staggered price-setting. It can be expressed ] v p t = (1 + π t) [(1 ɛp θ p )(1 + π # t ) ɛp + θ p (1 + π t 1 ) ɛpζp v p t 1 (36) Setting aggregate labor supply from the household to demand from firms yields: ( wt(h) N t = N d,t v w t (37) The variable vt w = ) 1 ɛw 0 w t dh is a measure of wage dispersion and drives a wedge between aggregate labor demand and labor supply. Similarly to price dispersion, it can be written recursively as: ( w # ) ɛw ( ) ɛw ( vt w wt 1 (1 + πt 1 ) ζw = (1 θ w ) + θ w (1 + π t ) t w t Aggregate inflation evolves according to: w t ) ɛw v w t 1 (38) (1 + π t ) 1 ɛp = (1 θ p )(1 + π # t )1 ɛp + θ p (1 + π t 1 ) ζp(1 ɛp) (39) 10
Similarly, the aggregate real wage obeys: ( wt 1 ɛw = (1 θ w ) w # t ) 1 ɛw + θw w 1 ɛw t 1 (1 + π t 1) ζw(1 ɛw) (1 + π t ) ɛw 1 (40) Imposing that the household holds the government s debt at all times and that the flow budget constraints for the household and government both hold with equality yields the aggregate resource constraint: Y t = C t + I t + G t + Γ(u t )K t (41) Finally, we include a recursive representation of the household s value function as an equilibrium condition of the model, which allows us to examine how welfare responds to shocks to tax rates: V t = ν t U(C t bc t 1, 1 N d,t v w t ) + βe t V t+1 (42) 3 Functional Forms, Calibration, and Estimation In this section, we discuss the functional form assumptions as well as the methodology we use to parameterize the model. 3.1 Functional Forms We assume that period utility from consumption and leisure takes the following form: U(C t bc t 1, 1 N t ) = ((C t bc t 1 ) γ (1 N t ) 1 γ ) 1 σ 1, σ > 0, 0 < γ < 1 (43) 1 σ This functional form is consistent with balanced growth while also allowing for non-separability in consumption and leisure. For the special case in which σ = 1, the utility function assumes the log-log form of γ ln C t + (1 γ) ln(1 N t ) in which the marginal utilities of consumption and leisure are independent of one another. forms: The capital utilization and investment adjustment cost functions, respectively, take the following 3.2 Parameterization Γ(u t ) = ( ψ 0 (u t 1) + ψ ) 1 2 (u t 1) 2 (44) ( ) It S = κ ( ) 2 It 1 (45) I t 1 2 I t 1 In total, the model contains fifty-four parameters. A little more than one-third of these parameters are related to the assumed tax processes. In our baseline parameterization, we calibrate a number of parameters and estimate the remaining parameters via Bayesian maximum likelihood. The 11
remainder of this section describes the methods used to derive values for each parameter as well as a brief discussion of the sensitivity of the model to some key parameters of interest. 3.2.1 Calibration The calibrated parameters are { α, β, π, δ, ɛ p, ɛ w, ψ 0, ψ 1, G, B g, τ c, τ n, τ k, T } as well as the debt and automatic stabilizer parameters governing our tax processes. We set α = 1/3, a conventional value. The discount factor is set to β = 0.99 and we assume zero trend inflation, π = 0. Together, these parameters imply a steady state risk free interest rate of approximately four percent annualized. The price and wage elasticity parameters ɛ p and ɛ w are both set to 11, implying steady state price and wage markups of approximately ten percent. These are broadly consistent with the empirical evidence. 4 We set steady state government spending, G, such that the steady state government spending share of output is 20 percent. Steady state government debt, B g, is chosen such that the steady state debt-gdp ratio is 50 percent. The depreciation rate on physical capital is set to δ = 0.025, implying annual depreciation of approximately 10 percent. The value of ψ 0 is pinned down via the normalization of steady state utilization to unity. This requires that ψ 0 = 1 β (1 δ). Estimation of models such as the one in this paper typically drive ψ 1 to a very small number; following Christiano, Eichenbaum and Evans (2005), we set ψ 1 = 0.01, implying that the cost of capital utilization is close to linear. To calibrate the steady state values of τ c, τ n, and τ k, we construct historical tax rate series using data from the national income and product accounts (NIPA). This approach follows Leeper, Plante and Traum (2010). 5 As our model is very similar to theirs, the constructed series have a relatively clean mapping to our model. Our sample covers the period 1984q1-2008q4. This results in steady state values of τ c = 0.0169, τ n = 0.2104, and τ k = 0.1975. These values are similar to House and Shapiro (2006), Leeper and Yang (2008), Uhlig (2010), and Leeper, Plante and Traum (2010), though small differences result from different sample periods. The steady state value of lump sum taxes, T, is then chosen to ensure that the government s flow budget constraint holds in steady state, given our assumption of a steady state debt-gdp ratio of 50 percent and a steady state government spending share of output of 20 percent. As a baseline, we assume that the distortionary tax rates do not react to debt and that no taxes respond to output. That is, we set γ b c = γ b n = γ b k = 0 and γy T = γy c = γ y n = γ y k = 0. We set γ T b sufficiently high so that government debt is non-explosive. 6 While perhaps unrealistic, these assumptions are meant to facilitate comparison with the existing literature. In particular, this specification gives rise to a clean interpretation of the thought experiment of changing a distortionary tax rate if tax rates reacted to debt deviations from target, changes in one tax rate would endogenously induce changes in other tax rates. 4 See, for instance, Basu and Fernald (1997). 5 We direct the reader to the Appendix accompanying Leeper, Plante and Traum (2010) for detailed instructions to construct these series. 6 In our baseline exercise, we set γb T = 0.05. Since the exact timing of lump sum taxes is irrelevant given that distortionary tax rates do not react to debt, our baseline results would be identical with higher values of γb T, or if we assumed that lump sum taxes adjusted to balance the government s budget period-by-period. 12
3.2.2 Bayesian Maximum Likelihood The remaining parameters of our model are estimated via Bayesian maximum likelihood. 7 parameters include {b, θ w, θ p, φ y, φ π, ρ i, κ, ζ w, These ζ p, σ, γ}, as well as the parameters governing the persistence and volatility of the exogenous processes for A t, Z t, ν t, u w, u p, G t, and the four tax processes. Our estimation strategy employs U.S. data covering the period 1984q1 through 2008q4. The beginning date is chosen because of the structural break in aggregate output volatility in the mid- 1980s, while the end date of the sample is chosen so as to exclude the zero lower bound period. We use eleven observable aggregate series in the estimation, corresponding to the number of shocks in the model to be estimated. We follow Leeper, Plante and Traum (2010) in the choice of observables. These series include the growth rates of consumption, investment, labor, government spending, and government debt as well as the levels of inflation, the nominal interest rate, and growth in tax revenue from lump sum, consumption, labor, and capital tax rates. Where applicable, series are from the BEA s national income and product accounts. Consumption is defined as the sum of personal consumption expenditures on nondurable goods and services. Investment is the sum of personal consumption expenditure on durable goods and gross private fixed investment. Hours worked is constructed as the product of average weekly hours in the non-farm business sector with total civilian employment aged sixteen and over. The nominal interest rate is the three month Treasury Bill rate. Inflation is the growth rate of the price index for personal consumption expenditures. Nominal series are converted to real by deflating by this price index, and where relevant series are converted to per-capita terms by dividing by the civilian non-institutional population aged sixteen and over. For the construction of the tax and debt series, we direct the reader to Leeper, Plante and Traum (2010). Table 1 displays the results of our estimation. The estimated parameters are largely in-line with existing parameter estimates in the literature. 8 The estimated price rigidity parameter is θ p = 0.72 and the estimated Calvo parameter for wages is θ w = 0.51. These imply mean durations between price and wage changes of about 3.6 and 2 quarters, respectively. We find modest amounts of price and wage indexation; both parameters are approximately 0.50. The estimated habit formation parameter is b = 0.75, which is quite standard. Our estimated values for the parameters governing curvature in preferences are γ = 0.24 and σ = 2.40. These are similar to the assumed values in Christiano, Eichenbaum and Rebelo (2011). Our baseline estimate of the investment adjustment cost parameter is κ = 4.11, also a relatively standard value in the literature. The estimated Taylor rule features a smoothing component ρ i = 0.75, a strong reaction to inflation (φ π = 1.63), and a modest reaction to output growth (φ y = 0.13). Remaining persistence parameters and standard deviations are found in the second section of Table 1. 7 To estimate the model, we employ Bayesian methods using a first order approximation of the model. While estimating the non-linear version of the model is desirable, estimating a non-linear model with the number of state variables specified above is computationally challenging. Parameters estimated using the linear approximation of the model are then used to solve the model via higher order perturbation. 8 We henceforth engage in a minor abuse of terminology and consider the mean of the posterior distribution of parameters as the estimated parameter values. 13
Overall, the model solved with the mean of the estimated parameters fits the data well. The estimated volatility of output growth is about 0.5 percent (close to its value in the data), consumption growth is about 60 percent as volatile as output, and investment growth is about 3 times more volatile than output. The growth rates of output, consumption, and investment are all significantly autocorrelated, as in the data. Productivity and marginal efficiency of investment shocks account for approximately 30 percent of the unconditional variance of output growth. Likewise, price markup shocks count for approximately 35 percent of output s variance. The next most important sources of output volatility are preference shocks, monetary policy shocks, and government spending shocks, which explain nearly 30 percent of the simulated volatility. Wage markup shocks and the different tax shocks account for the remaining 9 percent. 4 Baseline Results In this section, we simulate the model outlined and parameterized in previous sections to quantify the effects of tax cuts on output and welfare over the state space. We begin by briefly outlining the solution and simulation methodology. We then provide a definition of our tax output and welfare multipliers before concluding the section with a discussion of the results. 4.1 Solution Methodology and Multiplier Definitions We solve the model using the parameters outlined above via third order perturbation. 9 Solving the model via a perturbation of order higher than one is necessary to examine state-dependence. To construct tax multipliers, we first generate impulse response functions to each tax shock. The impulse response function of the vector of endogenous variables, X t, to a shock to tax rate j, is defined as follows: IRF(h) = {E t X t+h E t 1 X t+h ε j,t = s j, S t 1 } (46) The impulse response function at forecast horizon h is the difference between forecasts of the endogenous variables at time t (the period of the shock) and t 1, conditional on the realization of a shock of value s j in period t. In a higher order perturbation, the impulse response function in principle depends upon the initial realization of the state, S t 1, in which a shock hits. It may also depend on the size and sign of the shock, though we do not focus on these elements at this time. Given our non-linear solution methodology, these impulse responses are computed via simulation. First, we start with an initial realization of the state, S t 1 (e.g. the non-stochastic steady state). Then we draw shocks from standard normal distributions and simulate data out to horizon H, where we take H = 20. This process is repeated N = 150 times. Averaging across the N different simulations out to horizon H yields E t 1 X t+h, for h = 0,..., H. We then repeat this process, but subtract s j from the realization of the j th shock in the first period of each simulation. 10 9 Our results are quite similar if we instead use a second order perturbation. 10 Since we are studying the effects of tax cuts, we consider negative shocks to tax rates. 14
Averaging across the N simulations with the shock in the first period yields E t X t+h ε j,t = s j. The difference between these two constructs is the impulse response function. Computing these impulse response functions for different initial values of the state, S t 1, is the means by which we examine state-dependence. The states themselves (other than the non-stochastic steady state) are generated via simulation. Our definition for the tax output multiplier follows Barro and Redlick (2011) and Mertens and Ravn (2012, 2014a). We define the output multiplier for a shock to a distortionary tax rate as the ratio of the change in output to a change in tax revenue following a tax shock. This definition gives the extra (real) output generated from a change in a tax rate for every extra (real) dollar of tax revenue. We define output multipliers for h = 0, 1,..., H forecast horizons where H = 20. Formally, the output multiplier to shock j at forecast horizon h is defined as: YM j (h) = dy t+h dtr t ε j,t = s j, S t 1 for j = c, n, or k (47) The presentation of our results focuses on two multiplier horizons: the impact multiplier, which sets h = 0, and the max multiplier, which is defined as the ratio of the maximum output response to the impact revenue response. 11 As it is based on the impulse response function, the multiplier explicitly depends upon the state in which a shock occurs. Both the numerator, dy t+h, and the denominator, dtr t, can be sources of state-dependence. 4.2 Baseline Simulation Results For our benchmark exercise, we simulate 1,100 periods starting from the non-stochastic steady state and discard the first 100 periods as a burn-in. From each remaining 1,000 simulated states, we compute impulse responses to the three negative distortionary tax shocks. Each shock is considered individually and represents a cut to a single tax rate. In our initial simulation of states, we set the standard deviations of the tax rate shocks to zero; this ensures that any state-dependence of the tax multipliers arises for reasons other than tax rates being abnormally high or low. Table 2 presents some multiplier summary statistics from these simulations. For each of the three types of distortionary tax shocks, we present statistics on impact and maximum output multipliers. In our model, these multipliers are both negative i.e. decreases in tax rates stimulate output, but result in lower tax revenue on impact. For ease of exposition, we scale each multiplier by negative one so that they appear as positive numbers. We present statistics on the mean, minimum, and maximum values of each type of multiplier for each type of tax across the remaining 1,000 simulated periods. We also present the standard deviations of each multiplier over the 1,000 different states to provide a measure of volatility for each multiplier. Finally, we show the correlation of each type of multiplier with the simulated level of log output. In terms of average values, the consumption tax multiplier is 0.58, the labor tax multiplier is 11 The maximum output response to any of the three tax shocks typically occurs at horizons between h = 5 and h = 10. The maximum tax revenue response is generally on impact. 15
0.94, and the capital tax multiplier is 1.47. 12 To take the capital tax as an example, this value means that a change in the tax rate which generates a one dollar decline in total tax revenue generates a maximum output response of approximately one-and-a-half dollars. These magnitudes are comparable to recent theoretical studies with a common debt financing exercise by Leeper and Yang (2008) and Uhlig (2010), but considerably lower than recent empirical studies by Mertens and Ravn (2012, 2014a), who find multipliers of up to 2 on impact and up to 3 after six quarters. 13 For all three types of taxes, the average impact multiplier is smaller than the average max multiplier, suggesting that the peak effect of tax changes on output occurs after several periods. To visualize this point, Figure 1 presents impulse responses of output from each of the 1,000 simulated states for each of the tax shocks. The impulse responses are scaled by the impact response of tax revenue giving them a multiplier interpretation at each horizon. The 1,000 unique impulse responses are presented in gray while an average response at each horizon is presented in black. We find that tax shocks generally have their largest effect after approximately 5-7 quarters. 14 Each tax multiplier exhibits a high degree of state-dependence over our 1,000 period simulation. This is visually apparent upon inspection of Figure 1, but can also be seen by examining the standard deviation of each multiplier across states in Table 2. The rank order of multiplier volatilities across types of tax is the same as the ranking of average magnitudes. The standard deviation of the maximum capital tax multiplier is 0.09, with a min-max range of close to 0.55. The standard deviation of the labor tax multiplier is 0.07 with a min-max range of roughly 0.40. The consumption tax multiplier is least volatile, with a standard deviation of 0.01 and a min-max range of roughly 0.08. Figure 2 plots time series of multipliers across the 1,000 simulated states. Gray shaded regions demarcate periods in which simulated output is in its lowest 20 th percentile; one can think of such episodes as being recessions. From the figure, it appears as though these multipliers are countercyclical, spiking upwards during periods identified as recessions. One might be concerned that some of the state-dependence in the multipliers documented in Table 2 is driven not by different output responses to tax rate changes across states but rather by different tax revenue responses. For example, it is likely that tax revenue will respond less to a change in a tax rate in states when the tax base is relatively small. 15 For that reason, Table 3 constructs output multipliers using the tax revenue response from the non-stochastic steady 12 In a previous version of this paper, our baseline exercises were performed using a common tax persistence parameter to facilitate comparison across all taxes. The persistence of the tax process plays a key role in determining the magnitude of tax multipliers. Given that the estimated persistence parameters in each tax process are not identical, it is difficult to compare the average multipliers across different tax rates. The rank ordering of average multipliers, however, is the same as above when we fix the persistence parameters to be equal across tax rates. 13 As previously noted, many empirical studies of tax multipliers often group revenue from all taxes together. Comparisons against other specified-dsge models are therefore cleaner. 14 This pattern is common across tax studies. See, for example, Mountford and Uhlig (2009), Leeper, Plante and Traum (2010), or Mertens and Ravn (2014a). 15 To see this clearly, suppose that T R t = τ tt B t, where T R t is tax revenue, τ t is a tax rate, and T B t is the tax base. Totally differentiating about a point holding T B t fixed, one gets dt R t = dτ tt B. This will be smaller when T B is small. Of course, the tax base also reacts to tax shocks, so it is not completely clear that tax revenue will react less to a tax shock in periods where the tax base is small. Nevertheless, in our quantitative simulations we find that the total revenue response has a correlation with simulated output of approximately 0.95 for labor and capital tax cuts, and 0.83 for consumption tax cuts. 16