VIEWS: 4 PAGES: 42 POSTED ON: 2/25/2012
The Size of Fiscal Shocks And Their Corresponding Multipliers: Fiscal Policy Analysis With a Second Order Approximation∗ Aaron Butz Department of Economics University of Virginia October 25, 2011 Abstract I use a New Keynesian model with non-Ricardian agents to analyze the role of ﬁscal policy. To estimate the structural parameters of this model I use a second order approximation to the policy functions with a bayesian likelihood approach. From the estimated model I calculate present value multipliers. Because of the second order approximation I show that these multipliers will vary with the size of the shock being used to generate the impulse. I discuss the implications of this ﬁnding for policymak- ers and modelers by interpreting a counterfactual scenario with the estimated model solutions. ∗ I am very grateful for all the guidance and support received from Chris Otrok and Eric Young. I would also like to thank Valentina Michelangeli and Marika Santoro for all their help in building the model. Lastly, I am very appreciative of the help I received programming various routines from Katherine Holcomb and UVACSE. 1 Introduction In the current global crisis ﬁscal policy has been used with the intent of creating jobs and restoring output. In 2009 The United States Congress passed the American Recovery and Reinvestment Act which set aside $787 billion dollars to be issued as tax cuts and gov- ernment expenditures. This amount would total approximately 5% of GDP. Nearly a year later Congress passed the Statutory Pay-As-You-Go Act which would require that any new spending or tax cuts be ”budget neutral”. As the crisis progressed in the United States policymakers eventually found themselves at a standoﬀ over the United States debt-ceiling in 2011. Two themes seem to have emerged from the debt-ceiling crisis; active ﬁscal policy and ﬁscal restraint. The debate continues today over the role and eﬀectiveness of ﬁscal pol- icy when constrained by high levels of debt. I analyze the eﬀectiveness of ﬁscal policy in a general equilibrium model under a simple endogenized form of ﬁscal restraint. As the literature shows there is no current agreement on the size of ﬁscal multipliers. Work using structural vector autoregressions to estimate ﬁscal multipliers has yielded dif- ferent estimates. Fatas and Mihov (2001) ﬁnd the government consumption multiplier to be signiﬁcantly larger than unity on impact. Blanchard and Perotti (2002) ﬁnd this multiplier to be slightly less than unity while Mountford and Uhlig ﬁnd that it is closer to .65. First order approximations to medium scale DSGE models generate the same spectrum of results1 . While the multiplier literature is large few papers have considered the eﬀectiveness of ﬁscal policy under diﬀerent states of the economy. Christiano, Eichenbaum and Rebelo (2010) consider the size of the multiplier when the nominal interest rate is unable to respond. Using a general equilibrium model they ﬁnd that the multiplier increases as the nominal interest rate reaches zero. In the VAR literature there are several papers that look to identify government defense spending shocks and therefore distinguish between war time and peace time ﬁscal policy2 . As far as I can determine there is nothing in the literature which examines the eﬀect of debt and other state variables in the economy on the multiplier. To understand the eﬀects of ﬁscal policy in a general equilibrium framework I construct a dynamic stochastic general equilibrium (DSGE) model. As is typical of DSGE models my model uses micro-level theory to construct and explain aggregate variable movements. Many DSGE models exist for interpreting ﬁscal policy. Work by Leeper, Plante and Traum (2010), Zubairy (2010), Traum and Yang (2010), Gali et al. (2007), Forni, Monteforte and Sessa, and Colciago (2010) comprise a small set of medium to large scale DSGE models capable of interpreting ﬁscal policy innovations. I will consider the eﬀects of four possible innovations or shocks to ﬁscal policy; government consumption, labor tax, capital tax and transfers. I will approximate the solution to my model with a second order taylor expansion around the steady state. This is facilitated by work done by Schmitt-Grohe and Uribe. I will estimate my structural model with a bayesian likelihood approach. I will use data for all the ﬁscal processes in my model. These series will include government consumption, labor taxes, capital taxes, transfers and government debt. I will also use data for several other real variables in the economy. Using a bayesian likelihood approach I will be able to consider the posterior distribution of parameters. This will be much more informative than 1 See Gali, Lopez-Salido and Valles (2007) and Leeper, Plante and Traum (2010). 2 See Ramey and Shapiro (1998), Ramey (2011), Burnside, Eichenbaum and Fisher 2 a point estimate. The concept of the multiplier is derived strictly from estimating a linear solution to a model. By deﬁnition multipliers are a one-size-ﬁts-all approximation to ﬁscal innovations. Because my approximated solution contains squared terms I will be able to consider the eﬀect of the magnitude of ﬁscal policy innovations on model variables. Ultimately this will lead to a diﬀerent approach to reporting standard ﬁscal multipliers than is found in the SVAR or linearized DSGE model literature. I will consider the eﬀects of ﬁscal policy under large and small ﬁscal innovations. I will show that many of the dynamics of ﬁscal policy are dependent on the size of the ﬁscal innovations. I will also show with a second order approximation that impact multipliers will change with the states of the economy. For example, the interaction terms in the second order approximation will allow me to consider the ﬁrst order eﬀect of government debt on the impact multiplier for government spending. The paper reports present value multipliers for diﬀerent size ﬁscal innovations. I explain the diﬀerence between multipliers of diﬀerent size innovations through analysis of impulses and estimated variances of the model variables. I also discuss the state varying nature of the ﬁscal multipliers under a second order approximation and estimate the size of these multipliers over time. Finally, I perform a counterfactual exercise in which I remove the ﬁscal innovations. 2 A Model With Rule of Thumb Consumers My model has the features of a typical New Keynesian model combined with ﬁscal variables. The model is based on work from Smets and Wouters (2007), Christiano, Eichenbaum and Evans (2005), Fernandez-Villaverde and Rubio-Ramirez (2008), Traum and Yang (2010), and Leeper, Plante and Traum (2009). To create a model with sticky wages and rule of thumb consumers, or credit-constrained consumers, I follow the work of Colciago (2011) and Traum and Yang (2010). As Gali, Lopez-Salido and Valles (2007) show, the presence of rule- of-thumb consumers is crucial to creating the positive aggregate consumption response of an increase to government spending 3 . Like most New Keynesian models my model features a nominal rigidities in the price of the consumption good and wages ` la Calvo (1983) and real rigidities in investment and capital utilization through adjustment costs. There are nine exogenous shocks in the economy. There are two preference shocks, an investment adjustment cost shock, a total factor productivity shock, and ﬁve shocks in the processes describing the government and monetary authority. 2.1 Ricardian Households I assume a continuum of households in the economy indexed by j ∈ (0, 1). Households in the interval (ω, 1) are able to access credit markets and own the capital stock. All households in this model will derive utility from consumption and leisure which they discount geometrically over time. I assume preferences for consumption and leisure, the disutility of labor, are 3 Work by Zubairy (2010) shows that one can also get a positive consumption response to government spending if agents have deep habit preferences 3 additively separable.4 Agents receive no utility from government spending. I assume a log-habit form for households’ preferences over consumption. Work done by Constantinides (1990), Boldrin, Christiano and Fisher (2001) shows the importance of these preferences in generating persistence in output. Agents will maximize utility over an inﬁnite horizon by choosing paths for consumption, investment, capital utilization, and savings in government bonds or Arrow-Debreu securities. Labor will be demand driven. Wages will be set by unions that maximize household utility. The optimization problem for household j ∈ (ω, 1) is then: { } ∑∞ (LS )1+γ jt max E0 β t dt log(cjt − bcjt−1 ) − ϕt ψ cjt ,xjt ,ujt ,bjt ,ajt+1 t=0 1+γ Where the general preference shock and the labor supply shock follow these AR(1) processes: d ln(dt ) = ρd ln(dt−1 ) + υt ϕ ln(ϕt ) = ρϕ ln(ϕt−1 ) + υt The household faces a ﬂow budget constraint (1) where revenue from taxed wages, taxed capital rents, and the government transfer must equal consumption, investment and net savings. Let τtl and τtk be tax processes that I will deﬁne later. The household can save through government bonds, bjt , money balances, mjt , or arrow-debreu securities, ajt+1,t . ∫ bjt−1 rt−1 mjt−1 λt : zt + (1 − τtk )ujt rjt kjt k + (1 − τtl ) wt (h)ljt (h)dh + + ajt + Πt pt ∫ mjt = cjt + xjt + bjt + qjt+1,t ajt+1 dωj,t+1,t + (1) pt The Ricardian household’s savings of one unit of consumption at time t-1 can be converted into rΠt units of consumption at time t. Where Πt represents the change in the price t−1 level PPt . The arrow-debreu securities represent assets that cover every state of the world. t−1 Generalizing for the sake of notation, I represent the probability of any state with dωj,t+1,t . Investment from the household will be converted to capital according to the following rule: ( ) xjt Qt : kjt = (1 − δ(ut ))kjt−1 + (1 − µt S )xjt xjt−1 Where S is a function such that S(1) = 0, S ′ (1) = 0 and S ′′ (1) > 0. I also assume δ(1) = 0 and δ ′′ (1) > 1. To insure that utilization is one in the steady state I must set δ ′ (1) equal to the steady state real return on capital. The investment cost shock will evolve according to the following rule: µ log(µt ) = ρµ log(µt−1 ) + υt 4 I leave the possibility of complimentarities between leisure and consumption, as in Baxter and King (1993), for future work. 4 2.2 Non-Ricardian Households Non-Ricardian households are unable to access borrowing markets or capital markets. They simply consume their income for any given period. ∫ cjt = (1 − τt ) wt (h)ljt (h)dh + zt ∀j ∈ (0, ω) l (2) 2.3 Unions There exists a continuum of unions which control a continuum of labor inputs, lt (h) for h ∈ (0, 1). Households are uniformly distributed over these unions. The real wage for labor in union h at time t will be wt (h). Unions are monopolistic suppliers of these inputs via their control over the wages. All unions face a zero proﬁt labor packer who must produce a composite labor good, Ld . The packer produces the composite labor good with a C.E.S. t production function on all union labor inputs. The labor packer will choose union labor inputs to minimize costs, given union wages and a necessary product of Ld . The demand t function for the packer for labor from union h is then: ( )−η wt (h) lt (h) = Ld t wt Using the zero proﬁt condition one can derive wt which is the aggregate wage: (∫ 1 ) 1−η 1 wt = wt (h)1−η dh (3) 0 Assume that there is probability θw that a union will not be able to adjust their wage in a given period. When they are unable to adjust their wage it will be partially indexed to changes in the price level using the following rule: w Πχ wt+1 (h) = wt (h) t Πt+1 Unions will then set wages to maximize an objective of aggregate utility over all their households. The maximization problem for union h ∈ (0, 1) is then: [ (∫ )1+γ ∑ ∞ lt+τ (s)ds max Et (βθw )τ −dt+τ ϕt+τ ψ wt (h) 1+γ τ =0 ∫ ] ∏ Πχw τ +((1 − ω)λR + ωλN ) t+τ jt+τ t+s−1 (1 − τt+τ ) wt (s)lt+τ (s)ds l s=1 Πt + s s.t. ( τ )−η ∏ Πχw wt (h) t+s−1 lt+τ (h) = Ld t+τ s=1 Πt + s wt+τ 5 Because all households supply the same amount of labor to each union all Ricardians will have the same level of consumption and all non-Ricardians will have the same level of consumption. Therefore marginal utility of consumption for the Ricardian households will be: λR = λjt ∀j ∈ (ω, 1) t (4) Marginal utility of consumption for non-Ricardian households will be: dt dt+1 bβ λN = − ∀j ∈ (0, ω) (5) t cjt − hcjt−1 cjt+1 − hcjt The ﬁrst order condition for the union problem is a pair of inﬁnite sums which must be expressed recursively with helper variables. The ﬁrst sum represents marginal utility from consumption due to a change in the real wage. ( )−η wt (h) [ ] ft =(η − 1) 1 (1 − ω)λR + ωλN Ld t t t wt ( χw )1−η ( )η (6) Πt wt+1 (h) 1 + βθw Et ft+1 Πt+1 wt (h) The second equation represents marginal utility from leisure due to a change in the real wage. ( )−η−1 2 s γ wt −1 ft =ηψdt ϕt (Lt ) wt L d t wt (h) ( χw )−η ( )η+1 (7) Πt wt+1 (h) 2 + βθw Et ft+1 Πt+1 wt (h) ft1 = ft2 (8) Since all the unions face the same labor demand from the packer the objective function will be the same across unions. I will choose the symmetric equilibrium such that all unions that are able to set wages in a given period will set the same wage. Therefore if union i and union j are able to set wages in period t then: wt (i) = wt (j) Combining this with equations 4 and 3 one can see that risk in wage setting becomes idiosyncratic and the aggregate real wage will evolve according to: ( χw )1−η πt−1 ∗ 1−η wt = θw wt−11−η + (1 − θw )(wt )1−η πt ∗ Where wt solves the labor union’s optimization problem for being able to set wages in period t. 6 2.4 Firms I will assume that there exists a continuum of intermediate goods and their respective ﬁrms i ∈ (0, 1). Like the labor market there also exists a zero proﬁt ﬁnal good packer with a C.E.S. production function over all intermediate goods. If the packer chooses the lowest cost d combination of intermediate goods given prices, pt (i), to produce an amount of output, yt , then her demand function must be: ( )−ϵ pt (i) d yt (i) = yt pt Intermediate goods producers will have Cobb-Douglass production functions: yt (i) = At (kt−1 (i))α (lt (i))1−α − ϕ d d Where kt−1 (i) is the capital rented by the ﬁrm and lt (i) is the amount of packed labor rented by the ﬁrm. Where the aggregate productivity shock follows the process: A log(At ) = ρA log(At−1 ) + υt The parameter ϕ represents ﬁxed costs and is typically set so that economic proﬁts are zero in steady state. Intermediate Good Producers face a two stage problem. First, given rental rates and a level of output they must decide the lowest cost combination of capital and labor. Equating marginal product to marginal cost ratios gives us the equation: α wt d kt−1 (i) = l (i) (9) 1 − α rt t Combining this equation with the production function which has constant returns to scale one can then derive the marginal cost of output: ( )1−α 1−α α α 1 wt rt mct (i) = (10) 1−α α At Intermdiate goods ﬁrms, like labor unions, are given probability θp that they will not be able to set their wages in a given period. These ﬁrms will then maximize proﬁts through the choice of price for their goods subject to the demand of the labor packer. Firms will discount future proﬁts using, λt , from the households. Their problem can be represented as: ( τ ) ∑∞ ∏ τ λt+τ χt+s−1 pit max Et (βθp ) Π − mct+τ pit τ =0 λt s=1 pt+τ s.t. ( )−ϵ ∏ τ pit yit+τ = Πχ t+s−1 d yt+τ s=1 pt+τ Since all ﬁrms face the same demand from the ﬁnal good packer I assume a symmetric equilibrium where all ﬁrms able to set price at time t choose p∗ . Like the union problem, the t 7 ﬁrst order condition to the producer problem equates two inﬁnite sums. These sums can be redeﬁned using the following helper variables: ( χ )−ϵ 1 d Πt 1 gt = λt mct yt + βθp Et gt+1 (11) Πt+1 ( )1−ϵ Πχ Π∗ 2 2 gt = λt Π∗ yt t d + βθp Et t t g (12) Πt+1 Πt t+1 p∗ Where Π∗ represents the ratio t t pt . Now I can express the ﬁrst order condition as: 1 2 gt = gt (13) Like wages, it can be shown that the aggregate intermediate good price evolves according to: ( )1−η χp Πt−1 1= θp p1−ϵ t−1 + (1 − θp )(Π∗ )1−ϵ t (14) Πt 2.5 Government Tax and Expenditure Processes Like Leeper et al. (2009) I assume that labor income taxes, capital income taxes, government spending and transfers follow these linear processes: l log(τtl ) = ρτ l log(τt−1 ) + (1 − ρτ l )(τss + ϕτl Bt ) + υt l l ˆ τ τk log(τtk ) = ρτ k log(τt−1 ) + (1 − ρτ k )(τss + ϕτk Bt ) + υt k k ˆ log(gt ) = ρg log(gt−1 ) + (1 − ρg )(gss + ϕg Bt ) + υt ˆ g log(zt ) = ρz log(zt−1 ) + (1 − ρz )(zss + ϕz Bt ) + υt ˆ z ˆ ¯ Where Bt = log(Bt−1 /B). The parameters ϕz , ϕg , ϕτk , ϕτl represent stabilizers which insure 5 government solvency. The government has a budget constraint that equates revenues and net borrowing to expenditures in this manner: rt−1 gt + Bt−1 k = τtl wt lt + τtk rt ut kt−1 + Bt (15) Πt In steady state government consumption, gss , is calibrated to be close to 7 percent of output. The transfer is calibrated to be close to 11 percent of output. These percentages are derived from the averages in my data. Finally, the debt to output ratio is set to 68 percent. This is done by setting the labor income tax to 21 percent and capital income tax to 23 percent. 5 It can be shown from the ﬁrst order taylor expansion of the government budget constrait that without any stabilizers the path of debt would be unstable and the ﬁrst order approximation of the model would not satisfy the Blanchard Kahn conditions. 8 Lastly, the central monetary authority sets the interest rate according to the following rule: ( )γr ( )γπ (1−γr ) ( )γy (1−γr ) rt rt−1 Πt yt = mt (16) rss rss Πss yss I will restrict the parameter γπ to values larger than one in order to guarantee that the taylor principle is satisﬁed. 2.6 Aggregation As I stated before, Ricardian households are subject to idiosyncratic risk through the wage setting process. Since I assume complete markets the invidual Ricardian household variables, cjt , kjt , xjt and bjt will be identical across Ricardian households. Aggregating these variables gives: ∫ 1 cR t = cjt dj ω ∫ ω cN = t cjt dj 0 ∫ 1 Ct = cjt dj = (1 − ω)cR + ωcN t t 0 ∫ 1 Xt = xjt dj = (1 − ω)xR t ω ∫ 1 Bt = bjt dj = (1 − ω)bR t ω ∫ 1 Kt = kjt dj = (1 − ω)kt R ω To close the model I must specify a resource constraint as follows: d yt = Ct + gt + Xt (17) 3 Estimation The model is estimated with US quarterly data from 1985Q1 to 2008Q1 using Bayesian methods. The length of data was limited partially by the processing power required to estimate likelihood with the particle ﬁlter. Like Traum and Yang (2010) the time period of my data was chosen mainly because it is believed that monetary policy subscribed to a taylor rule (Taylor (1993)) during this period and ﬁscal policy was mostly passive.6 I use nine observables; labor income tax revenue, capital tax income revenue, consump- tion, investment, government consumption, debt, wages, the government transfer payment 6 An analysis of active ﬁscal policy would not be suitable for a DSGE model of the business cycle since detrended ﬁscal variables can ﬂuctuate up to 20 percent or more, i.e. the transfer of 2008Q2 or the nominal interest rate in 2008Q3. 9 and inﬂation. A detailed description of the data can be found in the appendix.7 The data is log transformed and then detrended using the Hodrick-Prescott ﬁlter. While one should attempt to endogenously explain the non-stationary components of the data via a balanced growth path in the model, as in Chang, Doh and Schorefheide (2007), the implications of a balanced growth path are not always consistent with the data. For instance, in my model government consumption should share the same trend as output under a balanced growth path. However, our data does not seem to corroborate this path as the share of government consumption in output will range from nine percent to ﬁve percent. Assuming diﬀerent trends for each observable may raise concern, however, Canova (2009) explains that there is still no consensus on how to detrend macro variables. I construct second order approximations to the logs of the model variables following Schmitt Grohe (2004) and estimate the likelihood of the structural parameters of the model using the particle ﬁlter as outlined by Fernandez-Villaverde (2010) and An and Schorfheide (2007). Priors are speciﬁed in Table 1. If a set of parameters does not have a determinate solution to the model the ﬁlter will return zero likelihood. To maximize this likelihood func- tion I then use the optimization routine GLOBAL from Csendes. I use a global optimization routine because as Fernandez-Villaverde and Rubio-Ramirez (2010) show, bayesian likeli- hood functions can contain many local maximums. To check identiﬁcation at this mode I will compute a forward diﬀerence hessian and check its condition number to make sure it is below machine precision. Finally, to draw from the posterior distribution I use Metropolis- Hastings. The particle ﬁlter must be initialized with an initial distribution, or swarm, of state variables. To construct this initial swarm I simulate the model with the given structural shock variances. From this simulation I estimate the variances of the state variables for use in the initial distribution. In order to evaluate the likelihood of each particle the particle ﬁlter assumes normally distributed forecast errors. These errors represent the likelihood of a set of observables given a set of states. With a second order approximation the control variables of the model are quadratic functions of the states and shocks. Unlike the Kalman ﬁlter, which uses a linear transformation of the i.i.d. normal structural shocks to compute the unconditional variances of the normally distributed control variables, I cannot assume the control variables are distributed normally and therefore must assume normal distributions for these forecast errors. I assume the variance of each forecast error is simply a fraction of the variance of the observable data. As in Fernandez Villaverde and Rubio Ramirez (2004) I draw a large number of particles (80000) to avoid resampling problems. This is necessary not only because of the large number of states and exogenous variables in my model but because of the small forecast errors for several observables. With the particle ﬁlter structural shock recovery is very simple. The shocks in each particle will be assigned a weight equal to the likelihood of the particle and then averaged for the entire observation. The particle ﬁlter oﬀers another advantage over the Kalman ﬁlter in that it is more robust to starting values. The Kalman ﬁlter and its non-linear variant, the Unscented Kalman Filter, unconditionally assume the initial unobserved state to be the steady state; unless otherwise speciﬁed by the econometrician. Since the particle ﬁlter assumes an initial distribution of the unobserved 7 The nominal interest rate is not included because I would have to drop another ﬁscal variable to avoid singularity. 10 initial state it is more robust to the choice of starting period of the data. Ultimately, because I will show the existence of strong second order eﬀects in my model, neglecting these eﬀects in estimation could produce ﬁrst order estimation errors in the Kalman ﬁlter likelihood8 . To assure myself that the estimation procedure is valid I simulate data given a set of structural parameters and then attempt to recover those same parameters with the ﬁlter. Using 8 processors with 32 gigabytes of memory on the University of Virginia Computa- tional Science and Engineering cluster, evaluation of the ﬁlter takes a little under a minute. Computational time is high because the model has 19 state variables. While only 10 of these variables are nonlinear in their laws of motion the bulk of the computational time is multiplying the matrices for the state transitions for 80,000 particles. These computational constraints limit the number of draws I am able to make with Metropolis Hastings. It will also prevent thorough convergence testing i.e. Brooks and Gelman (1998). 3.1 Priors The choice of which parameters to estimate brings one naturally to the question of iden- tiﬁcation. Iskrev (2008) shows many Calvo parameters in the Smets-Wouters model are unidentiﬁed due to the structure of the model. For my model I will ﬁx the wage and price markup parameters, η and ϵ, to 9. This implies a markup of around 11 percent which is in line with many estimates. As Chari, Kehoe and McGrattan (2008) show, a wage markup shock can be observationally equivalent to the labor supply shock in my model. I assume constant markups and choose to include the labor supply shock because it has a much more obvious structural interpretation. I choose to ﬁx several other parameters with values that are generally accepted in the literature. I set α, the income share of capital in the Cobb-Douglass production function to the standard .36. I set β, the discount factor, to .99. This implies an annual real interest rate of around 4 percent. Depreciation is set to .015 in the steady state. I set the interest rate smoothing parameter, γr , to .9 and steady state inﬂation or target inﬂation, πss , to average quarterly inﬂation in my data .006558. Finally, I set, ψ, the scalar on the marginal rate of substitution to 9. This scalar mostly determines the steady state of labor hours which I will set close to .3. I set inverse Frisch elasticity, to 2.15 so that Frisch elasticity is close to .47. Following Leeper et al. I center the stabilizers on ﬁscal processes, ϕg ,ϕτl ,ϕτk and ϕz , around .15 and give their prior distributions a relatively small variance. This reﬂects my belief that ﬁscal response to debt is relatively small. I also set the prior mean of the output gap response in the taylor rule to .15. Schmitt Grohe (2004a) shows that in optimal taylor rules this should be even closer to 0. Since I do not include hours in the observables the parameter θw should be diﬃcult to identify. As in Smets Wouters (2007) I set a strong prior on θw since my likelihood function achieves a maximum when this parameter is above .95 (wages reset every 5 years on average). The rest of the prior distributions are set to be diﬀuse over their relevant regions. Given these diﬀuse priors it is easy to choose parameters such that government con- sumption or transfer payments can crowd in or crowd out private consumption or private investment. I can also choose paramters such that the impact government spending multi- 8 See Fernandez-Villaverde, Rubio-Ramirez and Santos (2006) 11 plier is above or less than one.9 Therefore I have no restrictions, a priori, as to the occurence of any of these eﬀects or the size of the multipliers. 3.2 The importance of second order terms I choose to use a second order approximation to the model because certain policy rules are not linear over their relevant variables. In particular, I observe large second order eﬀects in marginal utility of non-Ricardian consumption and the marginal utility of Ricardian con- sumption which lead to large second order eﬀects in Ricardian consumption and investment. As one can see from equation 5 that a high habit will lead to high marginal utility of consumption at the steady state. Campbell and Cochrane (1999) explain that this marginal utility function becomes highly nonlinear as steady state consumption goes to zero or b goes to one so a ﬁrst order approximation may not adequately capture movements in welfare. As evidenced by the large second derivative with respect to the cross terms ct and ct−1 a second order approximation to marginal utility will capture the intertemporal interaction of consumption on marginal utility10 . This second derivative is negative. Therefore a ﬁrst order approximation fails to capture this intertemporal interaction of consumption and overstates marginal utility when a consumption series positively deviates from it’s steady state. Figure 6 shows the diﬀerence in impulse response functions of ﬁrst order and second order decision rules to a ten percent increase in the government transfer. I construct these impulses using a mode of my likelihood function obtained from GLOBAL. A ten percent increase in the transfer might be considered large; however my detrended data shows the transfer moving as much as 15 percent from trend. The impulse for investment shows crowding out at the ﬁrst order. This result is consistent with Gali et al. (2007) and Traum and Yang (2010). However, the second order impulse clearly shows crowding in. The reasons for this disparity in ﬁrst and second order approximations can be traced back to the initial calculation of marginal utility. While overstating marginal utility of Ricardian consumption has an obvious eﬀect on Ricardian household intertemporal decisions, like savings and investment, via the lagrange multiplier λt , it also has a large eﬀect on price setting. The diﬀerence in Ricardian marginal utility approximated at the ﬁrst and second order will aﬀect the choice of Π∗ in t the intermediate good producer problem. Figure 5 uses the intermediate good producer ﬁrst order conditions, 11 and 12, to show that if I assume the producer knows it will be able to reset it’s price tomorrow, and output will be held constant, an increase in the marginal utility of Ricardian consumption today will lead to an increase in the producer’s price, Π∗ , t today. Since the intersection of the left and right hand side of 13 determines p∗ increasing t the marginal utility of the Ricardian, which the ﬁrm uses to discount the future, shifts both sides of the equation in a manner that will increase the producer’s price, p∗ , today. The t variable, Π∗ , directly aﬀects inﬂation through equation 14. This eﬀect is very strong when t the parameter, θp is low. In Figure 6 one can see that the ﬁrst order approximation shows higher inﬂation than does the second order. Because increasing marginal utility today means there is an increase in 9 The parameter ω determines the proportion of non-Ricardians in the economy and therefore largely determines the real wage response to government spending. This response of real wages will determine whether the multiplier is less than or greater than one. 10 The second derivative of marginal utility with respect to the terms ct and ct−1 is −2dt (ct − bct−1 )−3 12 the value of consumption today it should be obvious that the real interest rate must increase when marginal utility today increases. This ﬁrst order increase in the real interest rate means that the ﬁrst order approximation must overstate the nominal interest rate since inﬂation is also higher at the ﬁrst order than it is at the second order. The high nominal interest rate crowds out private investment at the ﬁrst order while the second order approximation shows less movement in marginal utility and inﬂation such that private investment actually becomes crowded in. This analysis shows that the second order eﬀects of marginal utility have signiﬁcant eﬀects on real variables in my model of the economy given certain parameter speciﬁcations. 3.3 Posteriors The means and other features of my posterior distributions are listed in Table 1. The posterior draws for my parameters can be found in Figures 9 and 10. Like Leeper, Plante and Traum (2010) I ﬁnd the structural shock with the largest standard deviation to be the investment cost shock. As Schorfheide and An point out posteriors constructed with the particle ﬁlter are typically tighter and more well deﬁned than those constructed from the Kalman ﬁlter. My ﬁfth and ninety-ﬁfth percentiles are relatively close for several parameters (i.e. θw ). Estimating my model with the particle ﬁlter also results in much smaller estimates for the standard deviations of the shocks than with the Kalman ﬁlter. A partial explanation for this result is that the simulation performed with the second order decision rules at the beginning of the Particle Filter to estimate the unconditional variances of the state variables will likely send variables oﬀ to inﬁnity when standard deviations grow large. If a simulation fails the particle ﬁlter assigns zero likelihood to the parameter vector which generates the failed simulation. Also, since my shocks have been log transformed, the small standard deviations have a much clearer structural interpretation. For instance, the model estimates that percent change in the government transfer has a standard deviation of 1.5% between quarters (assuming debt is at the steady state). Because of the nonlinear decision rules I am unable to construct a variance decomposition for my model. I am, however, able to estimate the variance of the model variables attributable to each structural shock by simulating with each structural shock alone. The estimated standard deviations can be found in Table 14. One should be aware that these variances will not sum to the variance of the full simulation and they cannot be decomposed further without a careful consideration of the distribution created by the second order decision rules. I am also able to recover the shocks using the prcodeure discussed earlier. Figure 11 shows the smoothed exogenous shock processes. Figure 12 shows other smoothed variables of the model. For my model the structural shocks with the largest standard deviations are the investment cost shock and the labor preference (supply) shock. From Table 14 it is clear that output is driven mainly by the investment cost shock, followed by non-sytematic monetary policy and then the general preference shock. This ﬁnding is consistent with the New Keynesian model results of Justiniano et al. (2010) who ﬁnd that the main driver of business cycles is a shock to the marginal eﬃciency of investment. For investment I ﬁnd a surprisingly low variance due to the capital tax shock. This is because changes in the marginal tax rate of capital will change the return on capital in the ﬁrst order condition for utilization. This change in the capital tax rate can be immediately oﬀset by a change 13 Table 1: Priors and Posteriors. Parameter Prior Mean Prior St. Dev. Dist. Post. 5% Post. Mean Post. 95% Structural Parameters b 0.7 0.3 B 0.7986 0.8512 0.9096 κ 33.9 10 B 2.9146 8.7856 14.4524 θw 0.9 0.01 B 0.8909 0.9048 0.9178 θp 0.5 0.3 B 0.7645 0.8085 0.8436 χw 0.5 0.3 B 0.5688 0.6131 0.6496 χp 0.5 0.3 B 0.6541 0.7145 0.759 γu 0.15 0.05 B 0.0409 0.1298 0.2026 ω 0.15 0.05 B 0.0925 0.1465 0.2029 Taylor Rule Parameters γy 0.15 0.05 B 0.056 0.0986 0.1639 γπ 1.75 0.125 N 1.5965 1.8102 2.0538 γr 0.8 0.1 0.7136 0.745 0.7721 Shock Process Parameters ρa 0.5 0.3 B 0.4038 0.4498 0.5005 ρd 0.5 0.3 B 0.2933 0.3316 0.3838 ρµ 0.5 0.3 B 0.3909 0.457 0.5306 ρϕ 0.5 0.3 B 0.1028 0.1298 0.162 Fiscal Policy Parameters ρτk 0.5 0.3 B 0.7206 0.7758 0.82 ρτl 0.5 0.3 B 0.4711 0.531 0.5838 ρg 0.5 0.3 B 0.6197 0.6637 0.7205 ρz 0.5 0.3 B 0.4214 0.4665 0.5508 ϕg 0.15 0.05 B 0.0401 0.0829 0.119 ϕz 0.15 0.05 B 0.0492 0.083 0.1158 ϕτl 0.15 0.05 B 0.1234 0.1564 0.1929 ϕτk 0.15 0.05 B 0.0563 0.0862 0.1175 Shock Standard Deviations σg 0.01 ∞ IG 0.01 0.0136 0.0178 στk 0.01 ∞ IG 0.011 0.0163 0.0221 στl 0.01 ∞ IG 0.0124 0.0196 0.0278 σz 0.01 ∞ IG 0.0109 0.0145 0.0186 σa 0.01 ∞ IG 0.0063 0.0087 0.0115 σd 0.01 ∞ IG 0.0125 0.0179 0.0254 σµ 0.01 ∞ IG 0.0474 0.0823 0.1209 σϕ 0.01 ∞ IG 0.0248 0.1084 0.2402 σe 0.01 ∞ IG 0.0024 0.0028 0.0032 14 in utilization which in turn changes period depreciation. Because utilization costs are low variable utilization will equalize returns to capital across changes in the capital tax rate and investment will not move much with the capital tax. The exogenous ﬁscal shock with the largest standard deviation is the capital tax process. This ﬁnding is consistent with Traum and Yang and Leeper et al. even though they use a much longer series of data. The structural parameter means from my posteriors all take on values that are accepted in the literature. The habit parameter, b, has a a mean of approximately .85. The size of this parameter suggests evidence of excess smoothing of consumption in the data. While this value might be high it is not unprecedented11 . The parameter κ is the second derivative of my investment cost function S( xxt ). My estimate of about 8.8 is close to what is found in t−1 the literature. Wage stickiness is also slightly high (.9048). This value means that unions are able to reset wages on average about every two and a half years. This is a typical length for ”long term” union contracts. The value of wage indexation is close to .6131 which is close to what Smets and Wouters (2007) ﬁnd for U.S. data using Calvo pricing in a Ricardian representative agent model. The estimated fraction of Non-Ricardians in my model, .1465, is somewhat close to Traum and Yang who ﬁnd a value of .18. The parameter, γu represents the second derivative of the depreciation function δ(u). The relatively small estimate of this parameter indicates a low cost to increasing utilization. Finally, my Taylor rule parameters corroborate the Taylor principle with a high interest rate response to inﬂation, 1.8102. 4 Results 4.1 Present Value Multipliers Like Leeper et al.(2010), Blanchard and Perotti(2002), and Mountford and Uhlig (2011) I calculate the present value multipliers for my posterior distribution. These multipliers are calculated by summing the changes in output discounted by the interest rate. ∑n t+j j=0 β ∆Yt+j M = E t ∑n t+j ∆G (18) j=0 β t+j Under a linear approximation to the decision rules the size of the shock will not change this present value measure of the multiplier. This is a result of the decision rules being homogenous of degree one in the state variables. If I assume that hx is the ﬁrst derivative of the state transition rule, x is any vector of states and σ a scalar then the following equation demonstrates this property: σEt xt+j = σ(hx )j xt = (hx )j σxt (19) Under a second order approximation the present value multipliers are no longer invariant to the size of the shock. For my analysis I will consider two shock values to the ﬁscal variables; small and large. For the small shock value I will use .01; or, equivalently, a one percent 11 Justiniano et al. (2010) have a ninety ﬁfth percentile of .84 for the habit parameter. Mine is .903. Altig et al. (2011) estimate habit to be .7 which is my prior mean. Fernandez-Villaverde and Rubio-Ramirez (2010) estimate the habit parameter to be .97. 15 change. The small shock is less than the standard deviation of all ﬁscal shocks. For the large shock value I will use .1; or a ten percent change. The large shock can be equal to eight standard deviations of some ﬁscal variables. This should not be unusual as the ﬁscal shock distribution estimated by my model seems to demonstrate a higher level of kurtosis than a normal distribution would exhibit. Looking at the H.P. ﬁltered log transformed observables I notice that the transfer drops by 15% in the ﬁrst quarter of 1991 and by 8% in the third quarter of 2006. Although I do not use data from after the ﬁrst quarter of 2008 the transfer for the fourth quarter exceeded ten percent in growth. Labor and capital taxes also fall well below ten percent of trend with the Bush tax cuts. While a ten percent shock may be far outside the model standard deviations for ﬁscal variables it is certainly not an extrapolation. One way in which I will show the importance of the second order approximation will be in the diﬀerent multipliers between large and small shocks. Trivially, one can show that if the second order terms have little eﬀect on the decision rules for ﬁscal variables large and small multipliers would be identical. The Bayesian estimates of the ﬁscal multipliers are reported in tables 2 through 13. The multipliers for small and large shocks do not appear very diﬀerent for a shock to government consumption. As is typical for a New Keynesian model with non-Ricardian agents I ﬁnd impact multipliers for government consumption to be greater than one. However, unlike Gali et al. and others, I do not ﬁnd the impact multipliers to be as high as 1.5. Moreover, if I change the horizon for the output multiplier calculation the value drops well below unity. Like, Gali et al., I ﬁnd private consumption crowded in by government consumption at impact. Ultimately, however, the result of increased government consumption is the crowding out of private consumption in the long run. This crowding out occurs slightly sooner with the large shock. I also ﬁnd that private investment is crowded out by government consumption. The only reason the ouput multiplier is above one at impact is the crowding in of private consumption at impact. Why does crowding in of private consumption occur at impact? The answer is because of the eﬀect of increased demand on non-Ricardians. As labor increases so does the income and therefore the consumption of the non-Ricardian. After impact, as wages and investment begin to adjust more, the result of increased government consumption is less discounted additional output than what the government pays for to increase its consumption. The output multiplier for a negative shock to the capital tax is positive in the short run but negative in the long run. Again, the multipliers for large and small shocks do not appear to be very diﬀerent. This negative long run response to expansionary ﬁscal policy can be explained by the response of automatic stabilizers. The drop in tax revenue immediately increases debt. One period after this increase in debt automatic stablilizers in the ﬁscal processes will decrease government consumption and the transfer by close to one percent in the case of the large shock. Both these decreases will lead to an eventual fall in output. The transfer drop will lead to a decrease in non-Ricardian consumption and thus aggregate consumption. The increase in investment is not large enough to counteract this drop in output in the long run. In future work one way to allow for more ﬂexibility in ﬁscal processes and still preserve the condition for ﬁscal solvency would be to make stabilizers respond to debt from earlier states or include a time-to-adjust feature. This could also represent the lag in policy making. The output multiplier for an increase to the government transfer provides my most in- teresting result. The impact mutipliers for a government transfer are small but similar for 16 the small and the large shock. However, as I increase the horizon of the calculation, after twenty periods the small shock multiplier mean and median are negative and the large shock multplier mean and median are positive. This is the result of a larger disparity in the invest- ment multiplier. At all horizons the small shock 95th percentile of the investment multiplier clearly shows crowding out while the 95th percentile of the large shock investment multiplier shows crowding in. Log linear models with non-Ricardian agents will predict crowding out from a government transfer (i.e. Traum and Yang). However, as explained earlier, second order terms will approximate marginal utility better. As a result of lower marginal util- ity on impact for the non-Ricardian and the Ricardian agents wage setting for unions and intertemporal discounting will be diﬀerent between ﬁrst and second order approximations. This result is not in direct contrast with Ricardian equivalence since the Ricardian agent ultimately lowers investment. However, because ﬁscal processes are endogenous and not all agents can borrow I cannot expect typical Ricardian behavior from the aggregate vari- ables. The increased transfer means increased debt and ultimately higher taxes with lower government consumption. Anticipation of these ﬁscal movements along with strong habit preferences can drive the Ricardian to actually increase investment, albeit slightly, after a government transfer shock. Lastly, the multipliers for a decrease in the labor tax show similar results to that of a government transfer. One should recall that both these ﬁscal shocks have a large and direct eﬀect on non-Ricardian consumption and the highly nonlinear marginal utility of non- Ricardian consumption. The output multipliers for a labor tax decrease are also small but very similar on impact for the small and large shocks. Again, as the horizon for the multiplier calculation increases large disparities appear between the large and small shock multipliers. The mean of the large shock multiplier is larger than the mean and median of the small shock multiplier. I attribute this diﬀerence again to diﬀerences in the investment multiplier for a large and a small shock. In the instances of a labor tax decrease and a transfer increase ﬁscal variables will respond countercyclically to the transfer increase; but non-Ricardian consumption will respond procyclically. Because these eﬀects can oﬀset eachother initially output will not fall and Ricardian income will rise. This rise in income will be relatively large and, depending on the measure of marginal utility, will lead to diﬀerent decisions for consumption and investment. This cursory explanation does not even include the eﬀect of non-Ricardian marginal utility on the wage setting process. It should be clear that second order eﬀects can be very important in my model. The diﬀerence in the transfer and labor tax multipliers for large and small shocks reinforces this point. In the following sections I will explain how this diﬀerence in multipliers should be relevant to modelers and policymakers. 4.2 Impulses Figures 1 through 4 show the Bayesian impulse response functions for the four large and small ﬁscal shocks. In ﬁgure one can clearly see the crowding in of investment that occurs with a large shock to the government transfer shock and the crowding out that occurs with a small shock. While this contrast only occurs in the 95th percentile probability band it means that we cannot rule out crowding in with a large shock. All other dynamics appear relatively similar between shock sizes. In the capital tax and government consumption impulses it is hard to spot any diﬀerences except in scale between the large and small shocks. This 17 might suggest that there is little advantage to modeling government consumption shocks and capital tax shocks at the second order in New Keynesian models. Finally, the labor tax shock shows slight diﬀerence between large and small shock impulses for investment. The impulses for labor tax and transfer shocks show diﬀerent movements in model variables can be depending on the size of the shocks. This certainly suggests an important role for second order approximations in ﬁscal policy analysis. 4.3 A Counterfactual Simulation In May of 2003 the United States Congress passed the Jobs and Growth Tax Relief Rec- onciliation Act of 2003 (JGTRR). The act immediately lowered income taxes for nearly all households and accelerated the tax code changes that were to be phased in by 2006 due to the Economic Growth and Tax Relief Reconciliation Act (EGTRR) of 2001. While most of the changes insituted by JGTRR and EGTRR were to marginal income tax rates, capital gains taxes were also lowered by a signiﬁcant amount. As their names imply the aims of both these policies were to stimulate short run growth through consumer spending and long run growth through private investment. For my counterfactual simulations I will consider the state variables of the economy at the end of the ﬁrst quarter of 2003 recovered from a weighted average of the particles from the ﬁlter. From these states I will solve for the structural shocks that reconcile the model variables of output, consumption, government consumption, labor tax revenue, capital tax revenue, government debt and inﬂation with their corresponding observable values in the data. The structural shocks recovered from the model clearly show an immediate shock to the marginal tax rates τ k and τ l . The recovered shocks to these variables show that they dropped 15% and 8%, respectively, from their steady state values in the second quarter of 200312 . To consider the factors of recovery in the U.S. from the 2001 recession I will run two counterfactual scenarios. I will consider the path from the state variables in 2003 generated by the set of recovered structural shocks excluding the shocks to labor tax rates, capital tax rates, government consumption and the government transfer. I will consider this a counterfactual in which non-systematic ﬁscal policy is inactive. One should recall that ﬁscal policy will not remain fully inactive during this scenario due to the stabilizer response to debt in the ﬁscal policy rules. I will also consider the path from the state variables generated by the set of recovered structural shocks excluding the monetary shock, mt . This counterfactual scenario should resemble the path of the economy without non-systematic monetary policy. Figure 13 shows the paths of the counterfactual scenarios. The observable path of output shows that the U.S. economy was still recovering from the 2001 recession in 2003. The ﬁtted model coincides nearly exactly with observable output over this time. The counterfactual path that excludes non-systematic ﬁscal policy shows that output would have continued to recover without non-systematic ﬁscal policy. However, out still would have been higher under active ﬁscal policy. The counterfactual path that excludes non-systematic monetary policy falls well below the observable path of output and the counterfactual path which excludes non-systematic ﬁscal policy. Despite the large tax cuts which my estimation procedure recovers from the data as large structural shocks to 12 A 10% drop from the steady state value of the tax rate of labor would be approximately a two percentage point drop in the actual rate 18 the tax rate processes, my model demonstrates that ﬁscal policy was not as crucial in the recovery as monetary policy. The counterfactuals for consumption conﬁrm that monetary policy was crucial to the recovery. It even appears that upon impact ﬁscal policy crowded out private consumption. The counterfactuals for debt show that without non-systematic ﬁscal policy national debt would have dropped nearly 5% by the second quarter of 2005. Without non-systematic monetary policy the interest rate increases due to ﬁscal policy would have inﬂated the debt by nearly 10% in the same time. Finally, the counterfactual paths for in- vestment show that monetary policy was able to elevate investment slightly in the recovery process. The counterfactual for investment that excludes non-systematic ﬁscal policy nearly coincides with the simulated path of investment; therefore ﬁscal policy did not crowd out private investment. Recall that I do not use investment as an observable to recover the shocks13 . All of these results are consistent with the multipliers discussed earlier. My model shows that tax cuts do not crowd out private investment. Tax cuts and perhaps ﬁscal policy altogether seem to have a much smaller eﬀect on real aggregate variables than monetary policy. This result is conﬁrmed by Traum and Yang. 4.4 State Varying Impact Multipliers Because of the second order solution I can examine the ﬁrst order eﬀects of the states on the impact multiplier. This is because we derive the impact multiplier from the ﬁrst order approximation terms. For example, the ﬁrst order coeﬃcient of government spending on output is the simple impact multiplier. ¯ yt = y + A1 (bt−1 − ¯ + A2 (kt−1 − k)... + Am υt + ... ¯ b) g (20) Am is the impact multiplier of government spending under a ﬁrst order approximation. However, when we include the second order terms one can see that the interaction terms of g υt with other states represent the ﬁrst order eﬀects of states on the multiplier. ¯ yt =¯ + A1 (bt−1 − ¯ + A2 (kt−1 − k)... y b) ¯ ¯ + B1 (bt−1 − ¯ 2 + B2 (kt−1 − k)(bt−1 − ¯ + B3 (kt−1 − k)2 + ... b) b) (21) ¯ + Bn (bt−1 − ¯ t + Bn+1 (kt−1 − k)υt + ... + Am υt + ... b)υ g g g ImpactM ultiplier I will neglect the square term of the government spending shock and calculate the value of these multipliers for the states I recovered for the counterfactual exercises. I will extend the length of the recovered states and observables to the last quarter of 2008. Figure 7 shows the impact government consumption multiplier over time and ﬁgure 8 shows the transfer multiplier over time. The government consumption impact multiplier will move between 1.04 and 1.19 beween 2001 and 2009. It appears again that government consumption has small second order eﬀects. The transfer multiplier, however, nearly doubles in the last quarter of 2008. This could be due to large second order eﬀects; or the large monetary, preference and investment cost shocks which I recovered for that period. If this eﬀect is due to second 13 To include investment in the shock recovery procedure I would have to exclude output, consumption or government consumption. All four variables will enter into the resource constraint. 19 order terms it is consistent with the ﬁndings of Christiano, Eichenbaum and Rebelo who show that government multipliers are larger at the zero bound. The orginal intent of this paper was to analyze the second order interaction terms of debt with ﬁscal innovations. All of these interaction terms with debt are small; but negative. This would suggest the level of debt has little eﬀect on ﬁscal policy. However, the second order coeﬃcient on the interest rate is large and merits further exploration in a model capable of handling the zero bound and other ﬁnancial frictions. The varying multipliers in both these graphs suggest that ﬁscal policy will be slightly more or less eﬀective under diﬀerent states of the world. 5 Conclusion My paper has studied the eﬀects of ﬁscal policy using a New Keynesian model. I have estimated the deep parameters of the model using ﬁve data series which detail U.S. ﬁscal policy and four data series that describe the real aggregate variables of the economy. I have shown the importance of using a second order approximation in estimating and calculating the eﬀects of large ﬁscal policy shocks. In calculating ﬁscal multipliers with a second order approximation I have shown that the size of the shock will change the dynamics of ﬁscal policy. In the case of the government transfer a large shock can potentially crowd in private investment while a small shock will lead to crowding out. Ultimately, this means that the output multiplier for a transfer will vary at long horizons depending on the size of the shock. I have also shown that similar dynamics can exist for a large and small labor tax shock. Both these results come from the direct eﬀect of the shocks on non-Ricardian consumption. I ﬁnd that the government consumption multiplier is greater than unity on impact. However, at longer horizons this multiplier quickly falls below one. I also ﬁnd that government consumption strongly crowds out investment. While government consumpton crowds in private consumption on impact at longer horizons the ultimate eﬀect is crowding in. For all shocks but government consumption I ﬁnd multipliers substantially less than unity. And, at horizons of three years or longer all ﬁscal multipliers are less than unity. In addition, my counterfactual exercise reveals that ﬁscal policy was not as big a factor in the U.S. economic recovery in late 2003. With and without the large tax shocks of 2003 counterfactual investment and consumption paths remained nearly the same. Monetary policy, however, was largely responsible for elevating investment and consumption and driving the 2003 recovery. Finally, I show how multipliers vary with the states of the world. Although these multipier do not vary signiﬁcantly in my model until the 2008 crisis; I show how further analysis can be conducted with a larger model capable of handling the structural shocks of the ﬁnancial crisis. 20 −3 Inv. −3 Con. −3 GDP x 10 x 10 x 10 Debt 1 4 6 0.04 0.5 4 0.03 2 0 2 0.02 0 Large Shock −0.5 0 0.01 −1 −2 −2 0 0 10 20 30 0 10 20 30 0 10 20 30 0 20 40 −4 Inv. −4 Con. −4 GDP −3 Debt x 10 x 10 x 10 x 10 0 4 4 4 lines represent the 90% pointwise probability intervals. 3 2 2 −0.5 2 0 0 Small Shock 1 −1 −2 −2 0 0 20 40 0 20 40 0 20 40 0 20 40 Figure 1: Bayesian IRFs for an increase in the government transfer payment. Red dotted −3 Inv. −4 Con. −4 GDP x 10 x 10 x 10 Debt 4 5 4 0.06 2 2 0.04 0 90% pointwise probability intervals. 0 0 0.02 Large Shock −2 −5 −2 0 0 20 40 0 20 40 0 20 40 0 20 40 −4 Inv. −5 −5 −3 x 10 x 10 Con. x 10 GDP x 10 Debt 4 5 3 6 2 2 4 0 1 0 2 0 Small Shock −2 −5 −1 0 0 20 40 0 20 40 0 20 40 0 20 40 Figure 2: Bayesian IRFs for a decrease in the capital tax. Red dotted lines represent the −3 Inv. −3 Con. −3 GDP x 10 x 10 x 10 Debt 1 3 4 0.06 2 0 2 pointwise probability intervals. 0.04 1 −1 0 0.02 Large Shock 0 −2 −1 −2 0 0 20 40 0 20 40 0 20 40 0 20 40 −5 −4 −4 −3 x 10 Inv. x 10 Con. x 10 GDP x 10 Debt 5 4 4 6 0 2 2 4 −5 0 0 2 −10 Small Shock −15 −2 −2 0 0 20 40 0 20 40 0 20 40 0 20 40 Figure 3: Bayesian IRFs for a decrease in the labor tax. Red dotted lines represent the 90% −3 Inv. −3 Con. −3 GDP x 10 x 10 x 10 Debt 0 10 2 0.04 −2 0.03 5 1 −4 0.02 0 0 Small Shock −6 0.01 −8 −5 −1 0 0 20 40 0 20 40 0 20 40 0 20 40 −3 −3 −3 Debt x 10 Inv. x 10 Con. x 10 GDP represent the 90% pointwise probability intervals. 0 10 2 0.04 −2 0.03 5 1 −4 0.02 0 0 −6 0.01 Large Shock −8 −5 −1 0 0 20 40 0 20 40 0 20 40 0 20 40 Figure 4: Bayesian IRFs for an increase in government consumption. Red dotted lines 1.45 g1(λlow,π*) t 1.4 g2(λlow,π*) t 1.35 g1(λhigh,π*) t g2(λhigh,π*) t 1.3 1.25 1.2 1.15 1.1 1.05 1 0.65 0.7 0.75 0.8 0.85 π* t Figure 5: Union First Order Condition. 40 30 2nd order IRF 1st order IRF 30 Marginal Utility Consumption Interest Rate 20 of Ricardian 20 10 10 −3 −4 x 10 x 10 0 0 0 −1 −2 2 0 −2 30 30 30 Consumption Consumption 20 20 20 Investment Aggregate Ricardian 10 10 10 −3 −3 −3 x 10 x 10 x 10 0 0 0 5 0 −5 1 0 −1 1 0.5 0 30 30 30 Government Transfer 20 20 20 Inflation Output 10 10 10 −3 −4 x 10 x 10 0 0 0 2 0 −2 0.5 −0.5 0 5 0 −5 Figure 6: First and Second Order Impulses For a Ten Percent Increase to The Government Transfer.The structural parameters for these IRFs mostly lie within the 95 percent conﬁdence intervals constructed in the Estimation section. The key parameter values for generating these second order diﬀerences were b = .87, γu = .28, and θw = .91 Government Consumption Impact Multiplier 1.2 1.18 1.16 1.14 1.12 1.1 1.08 1.06 1.04 2001 2002 2003 2004 2005 2006 2007 2008 2009 Figure 7: The state varying government spending multiplier. Government Transfer Impact Multiplier 0.5 0.45 0.4 0.35 0.3 0.25 0.2 2001 2002 2003 2004 2005 2006 2007 2008 2009 Figure 8: The state varying government transfer multiplier. σg στ στ l k 1500 1500 1500 1000 1000 1000 500 500 500 0 0 0 0 0.01 0.02 0.01 0.02 0 0.02 0.04 σz σa 1500 1500 σφ 1000 1000 1000 500 500 500 0 0 0 0 0.01 0.02 2 4 6 8 0 0.02 0.04 −3 σd σµ x 10 σe 1500 1500 2000 1000 1000 1000 500 500 0 0 0 0 0.1 0.2 0 0.05 0.1 0 0.005 0.01 Figure 9: Posteriors for Standard Deviations of Exogenous Shocks. φg φτ b k 600 600 600 400 400 400 200 200 200 0 0 0 0.8 φτ 0.85 0.9 0 φz0.1 0.2 0.1 0.12 ρg 0.14 l 600 600 1000 400 400 500 200 200 0 0 0 0.09 0.1 0.11 0 θp0.1 0.2 0.55 0.6 0.65 κ 600 600 θw 500 400 400 200 200 0 0 0 0 50 0.75 0.8 0.85 0.85 0.9 0.95 χw γpi ω 1000 500 500 500 0 0 0 0.55 0.6 0.65 1.6 1.8 2 0 0.2 0.4 Figure 10: Posteriors for Selected Parameters. Government Consumption Labor Tax Capital Tax 0.05 0.02 0.05 0 0 0 −0.05 −0.02 −0.05 1980 2000 2020 1980 2000 2020 1980 2000 2020 Transfer Productivity Demand 0.05 0.01 0.02 0 0 0 −0.05 −0.01 −0.02 1980 2000 2020 1980 2000 2020 1980 2000 2020 Investment Cost −3 Philo −3 e x 10 x 10 0.2 5 5 0 0 0 −0.2 −5 −5 1980 2000 2020 1980 2000 2020 1980 2000 2020 Figure 11: Smoothed Shocks At The Posterior Mean. Debt −3 Interest Rate Capital x 10 0.2 5 0.05 0 0 0 −0.2 −5 −0.05 1980 2000 2020 1980 2000 2020 1980 2000 2020 −3 v −3 vw Wages x 10 x 10 1 5 0.01 0 0 0 −1 −5 −0.01 1980 2000 2020 1980 2000 2020 1980 2000 2020 Ricardian Consumption −3 Inflation Investment x 10 0.02 5 0.1 0 0 0 −0.02 −5 −0.1 1980 2000 2020 1980 2000 2020 1980 2000 2020 Figure 12: Smoothed States At The Posterior Mean. 2005.5 2005.5 2005.5 2005.5 2005 2005 2005 2005 2004.5 2004.5 2004.5 2004.5 2004 2004 2004 2004 2003.5 2003.5 2003.5 2003.5 0 0 0.2 −0.2 0 0.1 −0.1 0 0.02 −0.02 0.02 −0.02 Debt Inv. GDP Cons. Figure 13: 2003 Counterfactual Simulation. Solid lines represent observables. Dotted lines represent the ﬁtted paths of the variables. Dashed lines represent the counterfactual sim- ulation with no non-systematic monetary policy. Dashed and dotted lines represent the counterfactual scenario with no non-systematic ﬁscal policy. The investment observable was not used to construct the ﬁtted paths. Table 2: Output multipliers for an increase in the government transfer. Small Shock Large Shock N 95% 50% Mean 5% 95% 50% Mean 5% 1 0.252 0.164 0.172 0.104 0.265 0.172 0.181 0.112 4 0.220 0.136 0.141 0.078 0.251 0.152 0.163 0.091 8 0.167 0.085 0.088 0.015 0.256 0.131 0.142 0.049 12 0.131 0.042 0.042 -0.044 0.279 0.114 0.123 0.002 20 0.088 -0.030 -0.033 -0.159 0.272 0.056 0.070 -0.100 30 0.050 -0.095 -0.105 -0.277 0.238 -0.002 0.005 -0.216 Table 3: Investment multipliers for an increase in the government transfer. Small Shock Large Shock N 95% 50% Mean 5% 95% 50% Mean 5% 1 -0.002 -0.002 -0.003 -0.003 0.003 -0.001 -0.001 0 4 -0.012 -0.011 -0.016 -0.013 0.026 -0.007 -0.005 0.003 8 -0.027 -0.03 -0.039 -0.026 0.077 -0.019 -0.011 0.018 12 -0.043 -0.051 -0.063 -0.032 0.111 -0.035 -0.02 0.031 20 -0.087 -0.092 -0.108 -0.047 0.098 -0.073 -0.057 0.028 30 -0.14 -0.126 -0.147 -0.064 0.037 -0.11 -0.101 0.004 Table 4: Consumption multipliers for an increase in the government transfer. Small Shock Large Shock N 95% 50% Mean 5% 95% 50% Mean 5% 1 0.254 0.166 0.175 0.107 0.266 0.173 0.183 0.112 4 0.250 0.166 0.173 0.108 0.268 0.176 0.185 0.118 8 0.256 0.175 0.181 0.113 0.301 0.198 0.208 0.130 12 0.277 0.192 0.197 0.118 0.343 0.230 0.238 0.146 20 0.317 0.221 0.220 0.114 0.400 0.273 0.277 0.159 30 0.340 0.233 0.221 0.079 0.429 0.295 0.290 0.134 Table 5: Output multipliers for a decrease in the labor tax. Small Shock Large Shock N 95% 50% Mean 5% 95% 50% Mean 5% 1 0.265 0.170 0.179 0.107 0.254 0.163 0.172 0.104 4 0.238 0.153 0.159 0.093 0.237 0.152 0.161 0.096 8 0.202 0.127 0.131 0.073 0.223 0.138 0.147 0.083 12 0.183 0.105 0.111 0.051 0.224 0.127 0.138 0.067 20 0.163 0.066 0.070 -0.026 0.220 0.104 0.107 0.000 30 0.145 0.019 0.018 -0.124 0.198 0.062 0.060 -0.090 34 Table 6: Investment multipliers for a decrease in the labor tax. Small Shock Large Shock N 95% 50% Mean 5% 95% 50% Mean 5% 1 -0.003 -0.001 -0.003 -0.004 0.002 -0.001 -0.002 -0.002 4 -0.008 -0.005 -0.01 -0.011 0.009 -0.004 -0.005 -0.004 8 -0.01 -0.012 -0.02 -0.015 0.028 -0.008 -0.008 0.001 12 -0.012 -0.021 -0.029 -0.012 0.043 -0.015 -0.012 0.011 20 -0.036 -0.047 -0.058 -0.013 0.031 -0.038 -0.037 0.015 30 -0.081 -0.075 -0.093 -0.026 -0.016 -0.066 -0.072 -0.001 Table 7: Consumption multipliers for a decrease in the labor tax. Small Shock Large Shock N 95% 50% Mean 5% 95% 50% Mean 5% 1 0.268 0.172 0.182 0.110 0.255 0.164 0.173 0.105 4 0.261 0.173 0.182 0.112 0.255 0.169 0.178 0.111 8 0.274 0.192 0.197 0.125 0.276 0.191 0.199 0.128 12 0.303 0.220 0.225 0.146 0.313 0.222 0.231 0.151 20 0.358 0.267 0.269 0.173 0.375 0.271 0.280 0.179 30 0.390 0.286 0.287 0.167 0.410 0.300 0.301 0.175 Table 8: Output multipliers for an increase in government consumption. Small Shock Large Shock N 95% 50% Mean 5% 95% 50% Mean 5% 1 1.131 1.074 1.069 1.000 1.179 1.120 1.115 1.043 4 1.028 0.963 0.949 0.833 1.063 0.995 0.981 0.861 8 0.895 0.816 0.789 0.621 0.926 0.842 0.816 0.644 12 0.804 0.706 0.677 0.472 0.835 0.731 0.704 0.494 20 0.704 0.578 0.547 0.292 0.736 0.604 0.573 0.309 30 0.626 0.481 0.432 0.112 0.655 0.501 0.455 0.126 Table 9: Investment multipliers for an increase in government consumption. Small Shock Large Shock N 95% 50% Mean 5% 95% 50% Mean 5% 1 -0.034 -0.016 -0.028 -0.026 -0.034 -0.017 -0.029 -0.026 4 -0.108 -0.055 -0.092 -0.084 -0.107 -0.056 -0.093 -0.085 8 -0.211 -0.116 -0.182 -0.164 -0.207 -0.118 -0.184 -0.164 12 -0.288 -0.167 -0.249 -0.218 -0.284 -0.17 -0.251 -0.218 20 -0.378 -0.23 -0.321 -0.268 -0.375 -0.235 -0.326 -0.27 30 -0.44 -0.265 -0.361 -0.293 -0.441 -0.271 -0.367 -0.296 35 Table 10: Consumption multipliers for an increase in government consumption. Small Shock Large Shock N 95% 50% Mean 5% 95% 50% Mean 5% 1 0.145 0.088 0.093 0.049 0.152 0.093 0.098 0.052 4 0.090 0.037 0.038 -0.011 0.095 0.040 0.041 -0.010 8 0.021 -0.031 -0.032 -0.091 0.026 -0.028 -0.030 -0.091 12 -0.021 -0.073 -0.077 -0.148 -0.015 -0.071 -0.075 -0.149 20 -0.057 -0.127 -0.135 -0.240 -0.053 -0.125 -0.133 -0.242 30 -0.103 -0.197 -0.210 -0.378 -0.099 -0.196 -0.210 -0.384 Table 11: Output multipliers for a decrease in the capital tax. Small Shock Large Shock N 95% 50% Mean 5% 95% 50% Mean 5% 1 0.034 0.011 0.014 0.003 0.033 0.011 0.014 0.003 4 0.075 0.027 0.034 0.009 0.074 0.027 0.034 0.009 8 0.095 0.026 0.034 -0.007 0.094 0.027 0.035 -0.005 12 0.087 0.004 0.013 -0.042 0.088 0.007 0.015 -0.038 20 0.034 -0.059 -0.052 -0.125 0.037 -0.054 -0.047 -0.118 30 -0.034 -0.120 -0.122 -0.209 -0.030 -0.113 -0.116 -0.200 Table 12: Investment multipliers for a decrease in the capital tax. Small Shock Large Shock N 95% 50% Mean 5% 95% 50% Mean 5% 1 0.020 0.012 0.016 0.008 0.019 0.012 0.015 0.008 4 0.046 0.033 0.036 0.019 0.045 0.032 0.036 0.019 8 0.070 0.057 0.055 0.030 0.068 0.055 0.054 0.029 12 0.078 0.070 0.059 0.033 0.077 0.068 0.058 0.032 20 0.074 0.064 0.042 0.019 0.071 0.062 0.041 0.019 30 0.064 0.021 0.010 -0.005 0.061 0.020 0.009 -0.005 Table 13: Consumption multipliers for a decrease in the capital tax. Small Shock Large Shock N 95% 50% Mean 5% 95% 50% Mean 5% 1 0.002 -0.001 -0.002 -0.006 0.002 -0.001 -0.001 -0.005 4 0.017 0.008 0.008 0.001 0.017 0.008 0.008 0.001 8 0.036 0.016 0.017 0.001 0.035 0.016 0.017 0.001 12 0.053 0.022 0.024 -0.001 0.052 0.022 0.024 -0.001 20 0.086 0.034 0.037 -0.006 0.084 0.034 0.036 -0.005 30 0.115 0.043 0.047 -0.016 0.112 0.043 0.046 -0.014 36 Table 14: Standard Deviation Estimates For Selected Variables. Shock Output Investment Debt All 0.0117 0.0432 0.1012 g υ 0.0012 0.0015 0.0163 υ τl 0.0003 0.0002 0.0123 υ τk 0.0002 0.0008 0.0192 z υ 0.0004 0.0007 0.0176 υa 0.0013 0.0039 0.0159 d υ 0.0042 0.0072 0.0107 µ υ 0.0114 0.0443 0.0603 υϕ 0.0008 0.0032 0.0037 m 0.0018 0.0059 0.0231 37 References [1] David Altig, Lawrence J. Christiano, Martin Eichenbaum, and Jesper Lind´. Firm- e speciﬁc capital, nominal rigidities and the business cycle. Review of Economic Dynamics, 14(2):225 – 247, 2011. [2] Sungbae An and Frank Schorfheide. Bayesian analysis of dsge models. Econometric Reviews, 26(2-4):113–172, 2007. [3] Marianne Baxter and Robert G. King. Fiscal policy in general equilibrium. The Amer- ican Economic Review, 83(3):pp. 315–334, 1993. [4] Olivier Blanchard and Roberto Perotti. An empirical characterization of the dynamic eﬀects of changes in government spending and taxes on output. The Quarterly Journal of Economics, 117(4):1329–1368, 2002. [5] Michele Boldrin, Lawrence J. Christiano, and Jonas D. M. Fisher. Habit persistence, asset returns, and the business cycle. The American Economic Review, 91(1):pp. 149– 166, 2001. [6] Stephen P. Brooks and Andrew Gelman. General methods for monitoring convergence of iterative simulations. Journal of Computational and Graphical Statistics, 7(4):pp. 434–455, 1998. [7] Guillermo A. Calvo. Staggered prices in a utility-maximizing framework. Journal of Monetary Economics, 12(3):383 – 398, 1983. [8] YONGSUNG CHANG, TAEYOUNG DOH, and FRANK SCHORFHEIDE. Non- stationary hours in a dsge model. Journal of Money, Credit and Banking, 39(6):1357– 1373, 2007. [9] V.V. Chari, Patrick J. Kehoe, and Ellen R. McGrattan. New keynesian models: Not yet useful for policy analysis. Working Paper 14313, National Bureau of Economic Research, September 2008. [10] Lawrence Christiano, Martin Eichenbaum, and Sergio Rebelo. When is the govern- ment spending multiplier large? Working Paper 15394, National Bureau of Economic Research, October 2009. [11] Lawrence J. Christiano, Martin Eichenbaum, and Charles L. Evans. Nominal rigidities and the dynamic eﬀects of a shock to monetary policy. Journal of Political Economy, 113(1):pp. 1–45, 2005. [12] Andrea Colciago. Rule of thumb consumers meet sticky wages. Journal of Money, Credit and Banking, 43:325–353, 03 2011. [13] George M. Constantinides. Habit formation: A resolution of the equity premium puzzle. Journal of Political Economy, 98(3):pp. 519–543, 1990. 38 [14] Antonio Fatas and Ilian Mihov. The eﬀects of ﬁscal policy on consumption and em- ployment: Theory and evidence. CEPR Discussion Papers 2760, C.E.P.R. Discussion Papers, 2001. u a [15] Jes´s Fern´ndez-Villaverde. The econometrics of dsge models. SERIEs: Journal of the Spanish Economic Association, 1:3–49, 2010. 10.1007/s13209-009-0014-7. u a [16] Jes´s Fern´ndez-Villaverde and Juan F. Rubio-Ram´ ırez. Estimating dynamic equilib- rium economies: linear versus nonlinear likelihood. Journal of Applied Econometrics, 20(7):891–910, 2005. u a [17] Jes´s Fern´ndez-Villaverde, Juan F. Rubio-Ram´ ırez, and Manuel S. Santos. Conver- gence properties of the likelihood of computed dynamic models. Econometrica, 74(1):93– 119, 2006. [18] Lorenzo Forni, Libero Monteforte, and Luca Sessa. The general equilibrium eﬀects of ﬁscal policy: Estimates for the euro area. Journal of Public Economics, 93(3-4):559 – 585, 2009. [19] Nikolay Iskrev. Evaluating the information matrix in linearized dsge models. Economics Letters, 99(3):607 – 610, 2008. [20] John Bailey Jones. Has ﬁscal policy helped stabilize the postwar u.s. economy? Journal of Monetary Economics, 49(4):709 – 746, 2002. [21] Alejandro Justiniano, Giorgio E. Primiceri, and Andrea Tambalotti. Investment shocks and business cycles. Journal of Monetary Economics, 57(2):132 – 145, 2010. [22] Andrew Mountford and Harald Uhlig. What are the eﬀects of ﬁscal policy shocks? Working Paper 14551, National Bureau of Economic Research, December 2008. [23] Valerie A. Ramey. Identifying government spending shocks: It’s all in the timing*. The Quarterly Journal of Economics, 126(1):1–50, 2011. e ın [24] Stephanie Schmitt-Groh´ and Mart´ Uribe. Optimal ﬁscal and monetary policy under imperfect competition. Journal of Macroeconomics, 26(2):183 – 209, 2004. Monetary Policy. e ın [25] Stephanie Schmitt-Groh´ and Mart´ Uribe. Solving dynamic general equilibrium mod- els using a second-order approximation to the policy function. Journal of Economic Dynamics and Control, 28(4):755 – 775, 2004. [26] Frank Smets and Rafael Wouters. Shocks and frictions in us business cycles: A bayesian dsge approach. The American Economic Review, 97(3), 2007. [27] John B. Taylor. Discretion versus policy rules in practice. Carnegie-Rochester Confer- ence Series on Public Policy, 39:195 – 214, 1993. 39 [28] Nora Traum and Shu-Chun Yang. When does government debt crowd out investment? Caepr Working Papers 2010-006, Center for Applied Economics and Policy Research, Economics Department, Indiana University Bloomington, May 2010. [29] Sarah Zubairy. Deep habits, nominal rigidities and interest rate rules. Mpra paper, University Library of Munich, Germany, 2010. 40 A Data My data is mostly from the National Income and Product Accounts Tables from the National Bureau of Economic Research. To create real observables from nominal series I divide by the GDP deﬂator for personal consumption expenditures (Table 1.1.4 line 2). I will also use the deﬂator series for my observable inﬂation. Most of my observable transforms follow Traum and Yang (2010). Consumption and Investment. Consumption and investment are taken from Table 1.1.5 lines 2 and 7 respectively. While I do not use investment as an observable it is used to construct GDP. Capital and Labor Taxes. Following Jones (2002) I compute the average personal income tax from the equation: F IT + SIT τp = (22) W + P RI/2 + CI where F IT is federal income taxes and SIT is state and local income taxes (Table 3.2 line 3). W is wages and salaries (Table 1.12 line 3). P RI is proprietor income (Table 1.12 line 9). CI denotes capital income which is deﬁned as half of proprietor’s income, rental income (Table 1.12 line 12), corporate proﬁts (Table 1.12 line 13) and net interest (Table 1.12 line 18). Average labor income tax is then: τ p (W + P RI/2 + CSI) τl = (23) EC + P RI/2 where CSI denotes total contributions to government social insurance (Table 3.2 line 11) and EC denotes compensation of employees (Table 1.12 line 2). Multiplying this tax rate by employee compensation and half of propietor income gives me my observable labor tax revenue. Finally, the average capital income tax rate is calculated as τ p CI + CT τk = CI + P T (24) W + P RI/2 + CSI where CT is taxes on corporate income (Table 3.2 line 7) and P T is property taxes (Table 3.3 line 8). Multiplying this tax rate by corporate proﬁts and property taxes gives me my observable for capital tax revenues. Government Consumption. Government consumption is current government expen- ditures (Table 3.2 line 20) and government net purchases of non-produced assets (Table 3.2 line 44), minus government consumption of ﬁxed capital (Table 3.2 line 45). Transfers. The transfer payment is deﬁned as net current transfers, net capital transfers, and subsidies (Table 3.2 line 32), minus the tax residual. Net current transfers are current transfer payments (Table 3.2 line 22) minus current transfer receipts (Table 3.2 line 16) minus . Net capital transfers are capital transfer payments (Table 3.2 line 43) minus capital transfer receipts (Table 3.2 line 39). The tax residual is deﬁned as current tax receipts (Table 3.2 line 2), contributions for government social insurance (Table 3.2 line 11), income receipts 41 on assets (Table 3.2 line 12), and current surplus of government enterprises (Table 3.2 line 19) minus total tax revenue (consumption, labor, and capital tax revenues). Hours Worked. Hours worked is simply the product of the index for nonfarm busi- ness, all persons, average weekly hours duration (U.S. Department of Labor) and civilian employment for sixteen years and over (CE16OV from Bureau of Labor Statistics). Wages. Wages are derived from the Major Sector Productivity and Costs index (PRS85006103 from Bureau of Labor Statistics). Output. Output will be the sum of the observables I have mentioned for consumption, investment and government consumption. I avoid singularity from the resource constraint, 17, by excluding investment from the set of observables. Debt. I construct the debt series using an initial observation from the Dallas Federal Reserve Bank series. I calculate debt given the previous period’s debt by adding interest payments (Table 3.2 line 29), government consumption, transfers and subtracting growth in the monetary base(AMBSL from the St. Louis Federal Reserve Bank) and capital and labor tax revenues. All variables, except inﬂation, will be divided by a population index (derived from LNS10000000 from the Bureau of Labor Statistics) and then log transformed. Finally, all variables will be detrended with the Hodrick-Prescott ﬁlter. 42