The Size of Fiscal Shocks And Their Corresponding Multipliers by xumiaomaio


									 The Size of Fiscal Shocks And Their Corresponding
Multipliers: Fiscal Policy Analysis With a Second Order
                                          Aaron Butz
                                    Department of Economics
                                     University of Virginia

                                          October 25, 2011

           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

    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
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
        See Gali, Lopez-Salido and Valles (2007) and Leeper, Plante and Traum (2010).
        See Ramey and Shapiro (1998), Ramey (2011), Burnside, Eichenbaum and Fisher

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
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
    Work by Zubairy (2010) shows that one can also get a positive consumption response to government
spending if agents have deep habit preferences

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
                          max             E0     β t dt log(cjt − bcjt−1 ) − ϕt ψ
                cjt ,xjt ,ujt ,bjt ,ajt+1

Where the general preference shock and the labor supply shock follow these AR(1) processes:
                                             ln(dt ) = ρd ln(dt−1 ) + υt
                                            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
                                          + (1 −   τtl )
                                                      wt (h)ljt (h)dh +            + ajt +
                                                                            Πt               pt
                                               = cjt + xjt + bjt + qjt+1,t ajt+1 dωj,t+1,t +       (1)
    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

level PPt . The arrow-debreu securities represent assets that cover every state of the world.
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:                                                         (       )
                     Qt : kjt = (1 − δ(ut ))kjt−1 + (1 − µt S          )xjt
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:
                                   log(µt ) = ρµ log(µt−1 ) + υt
    I leave the possibility of complimentarities between leisure and consumption, as in Baxter and King
(1993), for future work.

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)

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.
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
function for the packer for labor from union h is then:
                                             (        )−η
                                               wt (h)
                                    lt (h) =              Ld

Using the zero profit condition one can derive wt which is the aggregate wage:
                                        (∫     1                  ) 1−η

                                wt =               wt (h)1−η dh                         (3)

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:
                                   wt+1 (h) = wt (h) t
   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
            +((1 − ω)λR + ωλN )
                       t+τ        jt+τ
                                                     (1 − τt+τ ) wt (s)lt+τ (s)ds

                                             Πt + s

s.t.                                    ( τ           )−η
                                         ∏ Πχw wt (h)
                           lt+τ (h) =                                     Ld
                                               Πt + s wt+τ

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)
                              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 − ω)λR + ωλN Ld
                                                          t       t   t
                                     ( 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
                                           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                    ∗
                       wt = θw wt−11−η
                                                   + (1 − θw )(wt )1−η
Where wt solves the labor union’s optimization problem for being able to set wages in period

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
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

Intermediate goods producers will have Cobb-Douglass production functions:

                              yt (i) = At (kt−1 (i))α (lt (i))1−α − ϕ

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:
                                 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
                             τ =0
                                           λt   s=1

s.t.                                    (                     )−ϵ
                              yit+τ =           Πχ
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

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)
                                                              (          )1−ϵ
                                                                  Πχ            Π∗ 2
                              gt   =   λt Π∗ yt
                                                  + βθp Et          t            t
                                                                                   g                 (12)
                                                                  Πt+1          Πt t+1
       Where Π∗ represents the ratio
                                              .   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
                              1=   θp p1−ϵ
                                       t−1                      + (1 − θp )(Π∗ )1−ϵ
                                                                             t                       (14)

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:
                        log(τtl ) = ρτ l log(τt−1 ) + (1 − ρτ l )(τss + ϕτl Bt ) + υt
                                              l                    l        ˆ       τ

                       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
government solvency.
   The government has a budget constraint that equates revenues and net borrowing to
expenditures in this manner:
                              gt + Bt−1                            k
                                                = τtl wt lt + τtk rt ut kt−1 + Bt                    (15)
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.
     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.

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
                                                        t          =            cjt dj
                                                                       ∫ ω
                                                       cN =
                                                        t                       cjt dj
                                              ∫    1
                                   Ct =                cjt dj = (1 − ω)cR + ωcN
                                                                        t     t
                                                       ∫   1
                                         Xt =                      xjt dj = (1 − ω)xR
                                                       ∫       1
                                         Bt =                      bjt dj = (1 − ω)bR
                                                       ∫       1
                                         Kt =                      kjt dj = (1 − ω)kt
        To close the model I must specify a resource constraint as follows:
                                                   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
    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.

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-
     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
    The nominal interest rate is not included because I would have to drop another fiscal variable to avoid

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-
      See Fernandez-Villaverde, Rubio-Ramirez and Santos (2006)

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
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
variable, Π∗ , directly affects inflation through equation 14. This effect is very strong when
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
     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.
     The second derivative of marginal utility with respect to the terms ct and ct−1 is −2dt (ct − bct−1 )−3

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

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

                         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

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
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
                                             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
    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.

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

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

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
    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

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

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
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)
                  + 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
     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.

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
    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

                                                                                                                                      −3        Inv.                              −3        Con.                              −3        GDP
                                                                                                                          x 10                                        x 10                                        x 10                                                  Debt
                                                                                                                      1                                           4                                           6                                         0.04

                                                                                                               0.5                                                                                            4                                         0.03
                                                                                                                      0                                                                                       2                                         0.02

                                                                                         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.
                                                                                                                                                                          2                                           2
                                                                                                                     −0.5                                                                                                                                 2
                                                                                                                                                                          0                                           0

                                                                                                       Small Shock

                                                                                                                          −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

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                                                                                          4
                                                                                                                                                          0                             1
                                                                                                                       0                                                                                          2

                                                                                                     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
                                                                                                                     0                                                                   2

pointwise probability intervals.
                                                                                                              −1                                                                         0                               0.02

                                                                                              Large Shock

                                                                                                              −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

                                                                                                                                                               2                                 2                         4
                                                                                                                                                               0                                 0                         2

                                                                                                       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

1.35                                              g1(λhigh,π*)






       0.65         0.7        0.75         0.8                 0.85

       Figure 5: Union First Order Condition.


                                                                                                              2nd order IRF
                                                                                                              1st order IRF
                 Marginal Utility


                                                    Interest Rate

                  of Ricardian



                              x 10

                                                           x 10












                              x 10

                                                           x 10

                                                                                                     x 10






                                                    Government Transfer






                              x 10

                                                                                                     x 10






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








   2001     2002    2003    2004    2005     2006    2007     2008   2009

       Figure 7: The state varying government spending multiplier.
                     Government Transfer Impact Multiplier







  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
500                              500

  0                                0                                       0
       0   0.01     0.02                2    4            6          8          0   0.02     0.04
             σd                                  σµ           x 10                  σe
1500                             1500                                    2000

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
600                                      600                   1000

400                                      400
200                                      200

  0                                        0                       0
  0.09     0.1         0.11                    0   θp0.1   0.2     0.55          0.6   0.65
600                                      600                                θw
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


  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.























                                                               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

      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

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

Table 14: Standard Deviation Estimates For Selected Variables.

            Shock Output      Investment    Debt
             All  0.0117        0.0432     0.1012
             υ    0.0012        0.0015     0.0163
             υ τl 0.0003        0.0002     0.0123
             υ τk 0.0002        0.0008     0.0192
             υ    0.0004        0.0007     0.0176
             υa   0.0013        0.0039     0.0159
             υ    0.0042        0.0072     0.0107
             υ    0.0114        0.0443     0.0603
             υϕ   0.0008        0.0032     0.0037
              m   0.0018        0.0059     0.0231

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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
    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

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.


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