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Working Paper Series
The Effects of Fiscal Policy in a
Neoclassical Growth Model
WP 97-08
Michael Dotsey
Federal Reserve Bank of Richmond
Ching Sheng Mao
National Taiwan University
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Working Paper 97-8
THE EFFECTS OF FISCAL POLICY IN A NEOCLASSICAL GROWTH
MODEL
Michael Dotsey"
Federal Reserve Bank of Richmond
and
Ching Sheng Mao*
National Taiwan University
June 1997
*We would like to thank Mary
especially Alan Stockman for
are those of the authors and
Reserve Bank of Richmond nor
Finn, Marvin Goodfriend, Robert King, and
many helpful comments. The views expressed here
do not necessarily represent those of the Federal
the Federal Reserve System.
1.
INTRODUCTION
This paper studies the effects of fiscal policies--depicted as
stochastic changes in government spending and distortionary tax rates--when
the government cannot use lump sum taxes to achieve intertemporal budget
balance. This framework contrasts the more standard analysis in which
spending and taxes follow exogenous Markov process and where lump sum taxation
is used to balance the government's budget. Although we also model tax rates
and spending as following Markov processes, the transition probabilities of
these processes depend on the ratio of government debt to gnp.
The ratio of
debt to gnp, will have consequences for the future choices of government
spending and distortionary taxation and hence will affect real economic
activity. The paper, therefore, is able to contribute to current public
discussions over the economic effects of debt and deficits and to the effects
of policies that attempt to reduce the deficit through cuts in government
expenditures or increases in distortionary taxation.
Our depiction of fiscal policy gives bite to the restriction
imposed by intertemporal budget balance since debt can not be viewed as a
residual of policy that is dealt with via lump sum means.
The results
generated in our model can differ substantially from those in standard
stochastic models.
For example, the effects due to changes in the tax rate on
capital depend on both the debt to gnp ratio and the persistence of the tax
process.
Even for processes that are fairly persistent, increases in the tax
rate on capital can lead to increases in investment and this counterintuitive
result is more likely to happen at very high or very low levels of the debt to
gnp ratio.
behavior.
Thus the debt to gnp ratio has interesting qualitative effects on
2
Also, the economic effects of changes in government debt depend on
the way that intertemporal budget balance is attained.
If budget balance is
primarily due to future changes in the tax rate on capital then debt crowds
out investment. But unlike a standard Keynesian model higher debt ratios are
associated with lower real interest rates.
If on the other hand budget
balance results from changes in the path of tax rates on labor, then
investment is actually crowded in.
It is only when government spending varies
and taxes are held fixed that crowding out and higher interest rates are
associated with higher ratios of debt.
Our model of fiscal policy implies that the debt to gnp ratio is
mean reverting, which is consistent with evidence in Kremers (1989),
(1990),
and Bohn (1991b).
King
The model, despite its simplicity, also generates
debt behavior that is reasonably consistent with U.S. data.
The paper also represents an extension and alternative method for
analyzing the effects of fiscal policy from the perfect foresight models of
Judd (1985,
1987)
and Baxter and King (1993).
We essentially take the central
messages of Bizer and Judd (1989) and Judd (1985)
seriously by both
investigating a model that explicitly incorporates uncertainty and that also
includes an elastic labor supply. The modeling strategy, as mentioned, allows
us to incorporate the behavior of public debt in a meaningful way, which
represents an extension of the literature on stochastic fiscal policy.
paper is thus most closely related to Dotsey (1994))
The
but the model analyzed
below is much richer than the one studied in that paper.
The inclusion of
elastic labor supply adds important behavioral elements to the model and
allows us to more realistically investigate the effects that debt has on
economic activity.
3
The paper proceeds as follows.
In section 2 we present the basic
model and in section 3 we describe the effects of stochastic taxation.
Section 4 investigates stochastic government spending while section 5 analyzes
the welfare implications of using capital taxation versus taxing labor.
A
notable feature of our model is that it is optimal to significantly tax
capital.
Section 6 compares the fiscal policy generated by our methodology
with actual fiscal policy, and section 7 concludes the paper.
2.
THE MODEL
The basic model is a standard neoclassical growth model into which
we introduce distortionary taxation and government spending. These variables
are modeled as Markov processes. To maintain intertemporal government budget
balance the transition probabilities are functions of the debt to gnp ratio.
The stochastic process characterizing fiscal policy is endogenous and the
government debt is mean reverting as in (Dotsey (1994)).
Kremers (1989)
nor King (1990)
debt, and Bohn (1991b)
Bohn (1991a)
Empirically, neither
can reject mean reversion in U.S. government
finds evidence that debt levels are mean reverting.
also shows that historically U.S. deficits have been eliminated
both by reductions in spending and increases in tax rates. Our model is
consistent with these observations. Because all but the stochastic part of
the model is standard, we give only a brief description of the model.
Firms
Firms maximize profits, d,, which are remitted to households, by
producing output via a constant return to scale technology that employs both
capital, k, and labor, n.
Both factors are rented from individuals. Capital
4
is always supplied inelastically while we consider both inelastic and elastic
labor supply.
PF:
Formally,
max
d, = f(k,,n,) - rtkt - w,n,
(k,,n,)
where r is the rental rate on capital and w is the real wage.
The first order
conditions equate each factor's marginal product with its rental rate.
Individuals
Individuals maximize lifetime utility which depends on both
consumption and leisure. They are endowed with one unit of time each period
and an initial stock of capital.
Individuals make their labor-leisure,
consumption, and investment-saving decisions taking as given wage rates and
rental rates.
They also purchase one period government debt at a price pt.
Each bond pays one unit of consumption in the succeeding period.
Consumers
observe the current state of fiscal policy summarized by beginning of period
per capita government debt, B,, current tax rates on capital and labor income,
rk and T", and the current level of government spending. They also know
current aggregate economic magnitudes such as output, the capital stock,
employment, investment, and end of period debt B,,,. Formally, the
individual's problem, PI, is written
PI:
max
U = E,,%8' uk,J-n,)
{c,,n,,b,+,k,+,)
subject to
c, + it + P$,+, 5 (I-T:)w,n,t (1-7:&k,
+ b, + TR, + T$k,
kt+, = (l-6)k, t it
where TR is aggregate per capita transfers, @k,
is a depreciation allowance,
and lower case variables indicate values at the individual level.
Maximization yields the following first order conditions
(la)
uz(ct, l-n,) = u,(c,, l-n,)(l-7#+
(lb)
u,(cJ-n,)
UC)
PtUJCt
= BE,{W-r:+,b-,+, + 7:+,6 + U-a)l~,(c,,J-n,,)~
l-n,) = BEtul(ct+,,l-n,,,)
where uj refers to the partial derivative with respect to the jth argument.
Fiscal Policy
The government spends resources and finances its spending through
taxes and debt.
(2)
Debt evolves according to
ptBt+,= G, + B, - &K,
- 7ptRt + TR,
where capital letters refer to per capita aggregate quantities. G is
government spending, B is the stock of one-period bonds outstanding, and TR is
the level of transfers. Tax rates on capital and labor income,
7k
and
7”,
and
the ratio of government spending to gnp, g, depend on the debt to gnp ratio,
6
ii.' Government budget balance is achieved through changes in distortionary
taxation and government spending. Specifically, we model the elements of
fiscal policy as a two-state Markov process with transition probabilities
given by
(3a)
prob
(7,+,
=
(3b)
prob
(7t+l
= Th 1 7,
7a
1
7,
=
7J
= min {max[(l-yb,)"', 01, 1)
= rh) = max {min[&"',
(4a) prob (g,,,= g1 1 g, = gp) = max (min[&"",
(4b) prob (it+,= gh 1 &
11, 0)
11, 0)
= 9,) = min (max[(I-&)"?
01, I)
where the subscripts e, h refer to low and high values respectively. These
transition probabilities imply that the debt to gnp ratio is bounded and only
rarely lies outside the interval [0, l/r]. As b approaches a value of l/y,
taxes will be high and spending will be low with probability one.
As long as
a combination of high taxes and low spending reduces debt, the debt to gnp
ratio will be driven down.
Similarly as b approaches zero the economy will be
in a low-tax, high-government-spending state and the debt will rise.
Thus,
'We focus on the ratio of government spending to gnp rather than the
level of spending because the ratio is stationary making it easy to extend our
analysis to economies with steady state growth. One could easily add growth
to our model by including technical progress in labor productivity. In that
case one could interpret our model as represening deviations from trend as in
King, Plosser, and Rebel0 (1988).
7
there is some tendency for debt to revert toward its mean.2
In what follows
we will call this policy a managed debt policy.
The parameters /Land q control the persistence of the tax and
spending processes. As these parameters increase the probabilities of
remaining in a given tax or spending state increase for any value of the debt
to gnp ratio.
Equilibrium
Equilibrium is described by a set of functions representing
quantities and prices that solve the firms and consumers maximization
problems, do not let either consumers or the government borrow more than can
be repaid, and obey the following aggregate equilibrium conditions.
(5)
C, + I, + G, = f(K,,
(6)
b, = B,
(7)
k, = K,
(8)
nt = N,
$1
2The debt to gnp ratio can temporarily move outside [0, l/r] because next
period's taxes and spending depend on this period's debt to gnp ratio. For
Given this
example, the current state could be 7, = 7a, gt = ghh,ht = (l/-y)-&.
state it is possible that next period's taxes and spending will not change.
Thus tomorrow's debt/gnp could exceed l/r and the debt/gnp two periods hence
could be larger still. However, since &+, > l/r implies 7,+a=
7h
and
=& the debt to gnp ratio will start to decline. Since a combination of
7a, g can only increase 6 by so much, 6 is bounded above. Similarly, b is
bounted below. Further our process for fiscal policy rules out any Ponzi
gt+2
games.
That is lim E,[P,B,,/~ (l/P,)] = 0 for equilibrium paths in this
T-
model.
s=t
8
We solve for equilibrium by first using equation (5) to eliminate
consumption.
Equation (la) together with the relationship w, = f2[K,, N,], and
equations (7) and (8) are then used to solve for labor nt = n(k,, bt,
gt9 k,,,)= n(s,, k,,,)where the state s, = (k,, b,,
7:’
7:,
719
7:,
g,). We then
substitute for labor in equation (lb) to yield an equation determining capital
accumulation,
(9)
u,[fW,,
n(s,, k,,,)) + U-W,
- gt - kt+,$ l-n(s,,
= BE,[(1-r:+,)f,(~t,,n(st+,,~t~2))
x u,[f(k,+,, nb,,,,
+ 7:+
kt+2)) + WW,+,
k,+,)l
+ (I-VI
- gt+l - kt+29 l-n(st+19 kt+2)1.
Equation (9) is a nonlinear second order stochastic difference equation.
Given n(s, k') where the "'
indicates next period's value of a variable, we
solve for the function, k' = h(s) which is the fixed point of (9). This
equilibrium policy function for k' then yields the equilibrium policy function
for labor n, because n was a function of arbitrary k'.
At each step of the
iteration we use equations (lc) and (2) to determine b' based on the current
state s and the policy functions n and h.
The algorithm is similar to the
discrete state space method described in Baxter (1991)
and Dotsey and Mao
(1992).
3.
STOCHASTIC
TAXES
We .canhighlight the effects of distortionary taxation by
comparing an equilibrium generated by a policy with managed debt with the
standard case in which taxes follow an exogenous Markov process.
Our
-9
comparisons are based on an examination of policy functions, impulse response
functions, and impact effects. To understand the effects of fiscal policy, we
proceed sequentially by first taking the simplest case--a stochastic tax rate
on capital and a fixed tax on labor with inelastic labor supply--and then
proceed to the more general cases.
The experiments in this section are dynamic stochastic analogs to
comparative static analysis. Our fundamental concern is understanding the
workings of a fairly intricate fiscal policy process. We use post-Korean War
U.S. data as a rough guide for calibrating the models.
We fix the ratio of
government spending to gnp at .18, which is the ratio reported in Christian0
and Eichenbaum (1991).
We also fix the level of transfers at 5% of gnp.
In
our experiments the debt to gnp ratio essentially lies between 0 and l/2.
Until recently, measured government debt/gnp has remained within this range.
Picking a limited range also helps conserve on grid points.
Our remaining parameter values are within the realm of most real
business cycle models.
Labor's share of gnp is chosen to be .68, utility is
logarithmic and separable in consumption and leisure, the discount factor is
.97, and the depreciation rate on capital is .06. We parameterize the utility
function so that individuals spend 20% of their time working.3
(a) Fixed Labor SUDD~Y with the Variable Tax Rates on Income from Caoital
In this example we allow the tax rate on capital to vary and use a
persistance parameter of j4=3. With this parameter, tax rates are unlikely to
3The parameterization lies within the ranges of a number of RBC models,
in particular Kydland and Prescott (1982), Hansen (1985),
Greenwood, Hercowitz
and Krusell (1992)) King, Plosser and Rebel0 (1988), Rebel0 and Stokey (1993),
and Finn (1995).
10
change for most of the values for the debt/gnp ratio.4 The tax rate on
capital takes on the value of either .20 or .50. The mean of the tax rate is
.38 with a standard deviation .168 and an AR1 coefficient of .57. This
parameterization is roughly consistent with one of the series reported in
Auerbach and Hines (1988)
which has a mean of .40, a standard deviation of
.141, and an AR1 coefficient of .82. We choose somewhat lower than actual
persistence to illustrate an interesting result, that it can be optimal for
agents to invest more when taxes are high even when tax rates on capital are
persistent.
The policy functions for capital and consumption, and the
equilibrium function for the real after-tax rate of interest are displayed in
Figure 1.
The policy functions are
middle of capital's ergodic set.
drawn for a capital value chosen from the
As shown, the capital stock in the high tax
state (dotted line) lies above the capital stock in the low tax state.
This
result implies that investment is higher when taxes are high even though a
high tax rate today generally implies a high tax rate next period.
This
result is the same as the one in Dotsey (1994) for an economy using a linear
technology and occurs for the same reason. A high tax rate today lowers the
debt to gnp ratio implying that the future path of taxes will be lower and
that investment is profitable. This response is only optimal if tax rates are
not too persistent.
If we set g=4 implying an AR1 coefficient on taxes of
.70, agents will invest less when taxes are high.
Therefore, for a tax
4For example, the probabilities of taxes remaining in the low-tax state
for debt/gnp ratios of (-.lO,
-.063,
-.026,
,011, .047, .084, ,121, .158,
.195, .232, .268, .305, .342, ..379,,416, .453, ,489, .526,
.60) are
f:;“,
1.0, 1.0, .99, .96, .93, .901 .87, .83, .79, .75, .71, .%i, .59, .52,
.24, 0, 0, 0). It IS not until the debt/gnp ratio reaches I49 that next
period's tax rate is more likely to be high than low.
11
process displaying persistence that conforms more closely to the data
investment will fall when the tax rate rises.
Further, investment declines
with debt because higher debt levels imply higher future taxes.
The above result stands in sharp contrast to the standard tax
literature5, where labor supply is typically fixed and taxes follow a Markov
process. As long as tax rates are positively correlated the standard case
implies that high taxes today result in higher future tax rates and less
current investment.
The policy function for consumption is a mirror image of the
policy function for capital. With inelastic labor supply investing more
implies consuming less.
The equilibrium function for interest rates is also
shown in Figure 1 and its shape is related to the policy function for
consumption.
Interest rates are lower in the high tax state due to the upward
slope of the consumption policy function. When taxes are high today, debt and
consumption will fall next period, while if taxes are low, debt and
consumption will rise.
This implies that for any given debt level interest
rates in the high tax state lie below those in the low tax state, a result
that is contrary to that presented in the perfect foresight model of Judd
(1987). The interest rate equilibrium functions are also downward sloping
attaining their lowest value when debt is high.
In the high tax-high debt
state there is little probability that a low tax rate will occur tomorrow,
hence the expected consumption decline is relatively large implying a low real
interest rate.
In the low tax state there is a reasonably high probability
'For example see Coleman (1991) or Dotsey (1990).
In a nonstochastic
environment see Judd (1987),
Abel (1982), Abel and Blanchard (1983),
Becker
(1985)) Brock and Turnovsky (1981)) Danthine and Donaldson (1985)) and Hall
(1981).
12
that high taxes will occur tomorrow, implying a relatively small expected
increase in consumption and hence a lower rea1 interest rate.
S imilarly rates
are higher when the debt is low.
The extent to which debt is non-neutral in our model can be
illustrated by the elasticity of the various policy functions with respect to
debt around the steady state debt to gnp ratio (see Table 1) and by the
correlations between debt and other endogous variables (see Table 2).
An
increase in debt crowds out investment and slightly increases consumption.
The non-neutrality in this model differs from a standard Keynesian model
because real rates in this model are negatively related to the level of debt.
These features also appear in the correlation coefficients which show a
negative correlation between debt and investment as well as a negative
correlation between debt and the real interest rate.
(b) Variable Labor SUDD~Y with Variable Tax Rates on Income from Canital
For these experiments we keep the same parameter values but allow
labor to vary, which creates another degree of freedom in the model.6 With
labor fixed, changes in investment must be offset one for one with changes in
consumption. With variable labor that need not be the case since output can
adjust contemporaneously. Variable labor allows consumption to be much
smoother and at the same time allows investors to take advantage of low
persistent marginal tax rates.
The policy functions for capital, labor, consumption, and the
equilibrium function for the real after-tax interest rate are depicted in
Figure 2.
The policy functions for capital and consumption differ from those
'Varying labor represents a significant extension over Dotsey (1994).
13
in the fixed labor case.
With varying labor, agents now invest more, work
more, and consume less in the low tax state over much of the debt space.
Persistence of the tax processes also plays a role in the shape of
the policy functions. Reducing the persistence of the tax series by setting
~=2, which implies p=.46 yields the same qualitative results as the fixed
labor case.
Crossovers in the policy functions occur because the expected
duration of remaining in any particular state depends on the value of the debt
to gnp ratio.
For example, if debt were high and taxes were low, agents would
expect taxes to rise and stay high for a greater number of periods than if
taxes were currently high.
Hence they invest less in the low tax state. As
in the previous example, the policy functions for consumption imply that the
real interest rate will be higher in the low tax state and negatively related
to debt.
Evaluating the elasticities of the various policy functions with
respect to debt and the correlation coefficients leads to the conclusion that
only half of the standard Keynesian story occurs. Higher debt crowds out
investment but reduces the interest rate.
(c) Variable Labor with a Varvinq Labor Tax and Fixed Tax on Capital Income
We next examine the effects of varying the tax on labor income
rather than the tax on capital. Here we allow labor tax rates to vary between
.23 and .31. With ~=7, these rates have a mean of .28, a standard deviation
of .039, and an AR1 coefficient of .79. Using post-Korean War data our tax
process matches the one constructed by Barro and Sahasakul (1986), which has a
mean of .278, a standard deviation of .039, and an AR1 coefficient for their
detrended series of .78.
14
Intratemporal substitution effects in the labor-leisure decision
dominate the results.
Individuals substitute labor effort into low tax
states, driving up the marginal productivity of capital and hence increasing
investment demand.
Greater labor effort results in more output and more is
invested. As debt rises, the probability of high taxes next period increases
thus inducing individuals to take even greater advantage of the current low
tax rate.
In the low tax state, high debt means that future taxes are more
likely to be high so the incentive to work is greater than when debt is low.
Thus the policy function for labor effort is upward sloping (see Figure 3).
Because the policy function for both labor and capital are now
upward sloping (a non-Keynesian result) the policy function for consumption is
downward sloping even though there is more output available at high levels of
debt.
Agents, however, consume and invest more in the low tax state due to
increased labor effort and greater output. As in the previous case interest
rates are higher when taxes are low.
This is because capital and, therefore,
next period's consumption increase when taxes are low.
That is, shifts in the
consumption policy function dominate movements along the function.
The variable tax on labor income creates crowding in rather than
crowding out, just the opposite of the standard Keynesian story.
The policy
function for investment has a positive elasticity and positive correlation
with respect to debt while the real interest rate is negatively correlated
with debt.
The managed debt case also yields somewhat greater impact effects
than the standard exogenous Markov case because of the stronger intertemporal
substitution effects on labor effort (see Table 3).
With debt management,
lower current taxes imply a higher future path of taxes making agents work
15
even harder today.
The greater impact on effort feeds over into output and
investment.
(d) Taxina Both Labor and Caoital
In this example both labor and capital are taxed.
The capital tax
rates take values of .18 and .53 with a persistence parameter of fi=9. This
degree of persistence implies an empirically relevant value for the AR1
coefficient of .80. Taxes on labor again vary between .23 and .31 with a
persistence parameter of fi=9. The AR1 coefficient on labor taxes is,
therefore, .79. The results are a hybrid of the results in the last two
sections. The large divergence in policy functions (Figure 4) between high
and low tax states reflects the responsiveness of labor to a tax on wage
income. The negative slope of the capital and labor policy functions as well
as the positive slope .of the consumption policy function reflect the influence
of the tax on capital.
Because this case is hybrid of the previous two
experiments, the elasticity of investment with respect to debt is greatly
diminished from the case when only
7k
varies. Thus when both factors of
production are taxed there is much less crowding out than in the case where
only income from capital is taxed. The interest rate, however, varies
indirectly with government debt and thus only half of the traditional
Keynesian story holds.
4.
GOVERNMENT SPENDING
This section examines the effects of government spending. To
highlight the differences from standard models, we first keep tax rates
constant throughout and allow lump sum taxes to balance the budget when
spending follows an exogenous two state Markov process. When there are no
16
lump sum taxes government spending must adjust so that the debt to gnp ratio
is bounded. We allow government spending relative to gnp to vary between .14
and .22.
Its mean is .174 in the following experiments and its standard
deviation is .04.. The parameter q is varied between 6 and 1 implying AR1
coefficients of .74 and .ll.
This allows us to explore the effects that
persistence has on economic activity. Thus our spending process with ~=6
matches the key features of the government spending series reported by
Christian0 and Eichenbaum (1992).
constant rate of 26%.
The government taxes production at the
After isolating the effects of government spending, we
allow tax rates and spending to vary simultaneously.
(a) Persistent Government Sending
We assume that government spending is useless. The economic
response to changes in government spending, therefore,-mainly arise through
wealth and crowding out effects. The policy functions in Figure 5, show that
agents work harder and consume less when spending is high.
Although high
government spending causes high output through increased labor effort, output
rises by less than government spending. Hence next period's capital stock
falls.
As debt rises the expected future path of government spending
falls.
The policy function for labor is, therefore, downward sloping with
respect to debt while the consumption policy function is upward sloping. As
labor hours decrease, output and the capital stock fall.
Hence debt crowds
out investment. High government spending raises interest rates motivating
agents to work harder and consume less. As the debt rises, implying less
future government spending, labor effort, capital, and consumption growth
17
decline.
Thus the equilibrium function for interest rates is downward sloping
with respect to debt.
Even though the equilibrium function for the interest rate is
negatively related to debt, the correlation between interest rates and debt is
positive. The intuition can be seen by examining the economy's response to a
high government spending shock, which is displayed in Figure 6.
Debt rises
when spending is above its average value causing spending to eventually fall
below its steady state expected value. This mild oscillatory behavior in
spending sets up oscillatory behavior in the other variables. As spending
falls and debt rises, labor effort declines. However, declining government
spending allows agents to increase consumption and investment even though
output mimics the behavior of labor. The real rate is generally above its
steady state value as a result of consumption growth, so the correlations
between debt and investment and debt and interest rates resemble the
predictions of standard Keynesian models.
Investment is below average when
the debt is relatively high while interest rates are above average.
With the exception of labor (and as a result output), the behavior
of the other endogenous variables is not strikingly different from what occurs
when spending follows an exogenous Markov process. The impact effects in
Table 4 show that labor responds with more vigor to an increase in government
spending when spending follows a Markov process.
In the debt management case
higher spending raises the level of debt implying that future spending must be
lower than it otherwise would have been. The wealth effects are, therefore,
smaller than when spending is exogenous.7
7We calculated the present value of government spending to be about 10%
'lessfor the managed debt policy in this example.
18
(b) The Effects of Lowerinq Persistence
When the persistence in government spending is greatly reduced by
setting q=l implying an AR1 coefficient on spending of .lO, the results for
the exogenous Markov process and the managed debt process are very similar
(see Figure 7).
Changes in government spending are transitory and have
smaller wealth effects.
Thus the impact effects of a rise in spending are
much smaller (see Table 4 and Figure 7).
These results are consistent with
those in Aiyagari, Christiano, and Eichenbaum (1991)
and Baxter and King
(1993). Also, because government spending changes states so frequently the
debt doesn't fluctuate very much and the path of shocks generated by each
process are almost identical. As a result all endogenous variables behave in
a like manner.
(c)
The Effects of Verv Hiqh Persistence
In this experiment we examine McGrattan's (1992)
suggestion that
very high persistence in government spending can lead to increased investment
in the high spending state.
To generate high persistence we set q=lOO which
corresponds to an AR1 coefficient of .92. We find that with log utility and
hence a relative risk aversion parameter of o=l it is possible for investment
to be higher when spending is high, but only over a narrow range of the debt
space.
With an exogenous Markov process for spending, investment is higher
when spending is high, but this result is sensitive to the degree of relative
risk aversion.
With increased risk aversion (0=2) investment is lower when
spending is high in both the managed debt and exogenous Markov process cases.
The reason for the disparity in results is that with debt
mangement the wealth effects of high or low government spending are almost
identical near the boundaries of the debt space.
If, for example, debt levels
19
are very high the probability that next period's government spending will be
low and stay low is high no matter what the current state. Therefore, labor
effort and consumption do not differ by very much across spending states and
the major difference across the two states is in investment. In particular,
investment is lower in the high spending state. An analogous argument
indicates that investment is lower in the high spending state when debt is
very low.
It is only in the middle of the debt space that the wealth effects
of high spending can cause enough of an increase in labor effort and decline
in consumption that investment is higher. The large increase in labor effort
also increases the marginal product of capital reinforcing the wealth effects
on consumption and investment. When government spending follows an exogenous
Markov process the persistence of the process is independent of debt levels.
Therefore, wealth effects and the accompanying substitution effects are either
strong enough to encourage investment when spending is high or they are not.
An increased persistence in government spending and the
accompanying higher investment in the high spending state results in greater
consumption variability as well.
With CRRA utility, an increase in relative
risk aversion implies a reduction in the elasticity of intertemporal
substitution of consumption. With agents less willing to substitute
intertemporally, investment becomes less variable, and therefore it is less
likely that investment will rise in response to high government spending.
(d) Taxes and Soendino Both Vary
In this case we now add persistent taxes and compare how
simultaneously varying taxes and spending affects behavior. These comparisons
are done by examining the impulse response functions in figures 8 and 9 which
are responses to a high spending-low tax shock and a high spending-high tax
20
shock.
The impulse responses are generated by averaging over 2000
realizations of 50 periods each.
The combination of low taxes and high spending is more
expansionary than just lowering taxes or increasing spending. The tax induced
substitution effects augment the wealth effects of government spending
implying that labor effort increases by a large amount. This increases output
by enough so that the impact effect on both consumption and investment is
positive.
When the initial impulse to taxes is high, (Figure 9) the impact
effect of fiscal policy is reversed. With an increase in the tax rate
substitution effects outweigh wealth effects and labor effort falls.
The fall
in labor effort results in lower output, consumption, investment, and a drop
in the real rate of interest. Thus the expansionary effect on output of
government spending programs can be totally overturned if they are financed
out of current tax revenue. This latter result is consistent with the
analysis in Baxter and King (1993).
5.
WELFARE COMPARISONS BETWEEN CAPITAL AND LABOR TAXATION
The model also allows us to check the relative efficiency of using
capital taxation versus labor taxation.
In particular, we analyze if it is
more costly to vary the tax rate on labor, or capital, or both.
The tax
processes evaluated are similar to those in section 3d, and hence represent
processes that are representative of actual U.S. tax rates.
The experiment,
therefore, answers the question of which tax rate should be the primary
21
instrument for maintaining budget balance conditional on the mean of the other
tax rate being set at its optimal value.'
To perform this experiment, we fix g at .18 and the transfer to
gnp ratio at .05. We then compare the discounted utility of the
representative individual when
7k
and
7”
are set at their optimal values with
the discounted utility that arises when only
and when both
7k
and
7”
vary.
7k
varies, when only
7”
varies,
In the cases where tax rates vary around their
optimal values we parameterize the processes so that the standard deviations
and AR1 coefficients are approximately equal to what one observes in the
actual data.9
The derivation of the optimal tax rates follow the methodology in
Zhu (1990). For the case of no transfers, the social planners first-order
condition for efficient capital accumulation is:
(11)
u’(cJ = BE,~~(1-~,+,)f,(k,+,, n,,,) + (Wlu’
(ct+,)>,
while for the representative agent it is:
(12)
u’(c,)
Setting 7: =
=
j?E,{[(1-~~+,)f,&+,,n,,,)+ (1-U + &luf(ct+l(~o
stf,W,P,)
f,W,JJ
4
will result in the path for capital under a
competitive equilibrium being identical to that chosen by the planner.
Further setting
7:
=gt will result in an equivalence between the marginal
'A full depiction of optimal taxation under uncertainty can be found in
Zhu (1990).
'Recall for 7k: u = .14 and p = .82, and for 7”: CJ= .039 and p = .79.
The values for 7k are (.34, .68) and for 7” are (.24, .34).
22
conditions that determine labor-leisure choices for the representative agent
and the planner. The solution is first best and is analogous to the solution
presented in Jones, Manuelli, and Rossi (1991).
When transfers are also involved and the inital capital tax is
constrained to its mean, then these transfers will be optimally financed by
taxing labor"
Thus the optimal,tax on labor income is
7:
=gt t tr/(l-a),
where tr is the percent of output transfered by the government.
Given our
parameterization the optimal tax rate on labor should be .254 and that on
capital should be .392. The latter value is quite high and substanially
differs from the steady state value of zero found in models that treat the.
level of government spending as exogenous. Also, these values are very close
to their actual means of .28 and .40.
When
7,
= ,254 and 71(= .392 the discounted utility of the
representative agent is -18.09.
Allowing
7,
to vary around its optimal value
so that its standard deviation is .038 and its first-order autoregressive
parameter is .76 yields a utility value of -18.18.
If instead one lets the
tax on capital fluctuate around its optimal value with a standard deviation of
.135 and first-order autoregressive parameter of .75 the agent's discounted
utility is -18.13. Allowing both tax rates to vary yields a discounted
utility of -18.19.
Thus variation in one tax or the other around its optimal
value has very little effect on welfare.
It may, therefore, be a matter of
indifference which tax is used for budget balancing purposes so long as its
mean is set correctly.
"Note, if utility was not separable in consumption and leisure, then
financing transfers solely through the tax on labor would no longer be optima7
(see Zhu (1990)).
23
6.
EMPIRICAL RELEVANCE OF THE MANAGED DEBT POLICY
In this section we investigate the empirical relevance of the
managed debt policy by examining if this policy can account for the behavior
of debt, and if it is consistent with the behavior of tax rates and government
spending. Because some of our empirical work will use frequency domain
techniques we prefer a fairly long date set. We, therefore, test if our model
is consistent with the actual post-1916
data set given in Bohn (1991a).”
Because we are concerned with a more detailed investigation of our
methodology's ability to replicate actual data we require some essential
modifications. First we extend the range of the admissable debt to gnp ratio
to [-.l,
1.11
so that it is in accord with actual experience. Second, because
the managed debt policy as described by (3a-4b) produces excessive oscillitory
behavior, we allow taxes and government spending to follow exogenous markov
processes on the interior of the debt space, but respond to debt when near the
boundary. We also use three states for the tax rate and we set
7k
= 7”
since
Bohn's data only includes average tax rates.
The model generates tax rates that have a mean of .16, a standard
deviation of .04, and an AR1 coefficient of .87, while it generates government
spending that has a mean of .17, a standard deviation of .12, and an AR1
coefficient of .78. The comparable statistics for the data are .14, .04, and
"We use his data because it doesn't net out any components of government
spending. If we are to have any chance of matching the series on debt we must
either use inclusive measures or model the different components of spending
separately. We start in 1916 because that is the inception of income taxes,
does not appear to be
and the data over the entire sample, 1800-1988,
generated by the simple model in this paper (i.e. the mean of government
spending and tax revenue vary greatly over the last two centuries). To match
the data we would need more than one fiscal policy regime. As it is the model
is forced to confront two major wars in order to get enough data points for
the spectra to have any meaning. What we would like is 100 years of postKorean war data.
24
.89 for tax rates and .16, .08, and .80 for government spending. Thus we
start out by rep1icating some essential features of the two fiscal po7icy
processes.
(a) The Behavior of Debt
To compare the behavior of debt generated by our modified fiscal
policy process with the behavior of actual government debt, we examine both
the spectrum of actual and model generated government debt as well as the
coherence between the two sets of data.12 In generating model data on debt we
use the same tax rate and government spending series as Bohn (1991a)
and then
derive model behavior by linearly interpolating between the theoretical policy
functions. Thus the two data sets are comparable.
The results of this exercise are displayed in figure 10.
The
spectrum for the model has less power at low frequencies than the actual data,
but the general shape of the spectra are fairly comformable (i.e. both spectra
peak at low frequencies). The coherence between the model and the data is
generally fairly high.
The lowest value of the coherence occurs at a
periodicity of 20 years, which roughly corresponds to intervals between major
wars.
Thus our model of debt does not accurately reflect war time behavior.
At business cycle frequencies, however, the coherence exceeds .90 which is
much higher than that displayed by real business cycle models for many
relevant economic magnitudes (see Watson (1993)).
We, therefore, find the
overall results of this exercise encouraging.
"The spectra were estimated using linearly detrended data. Since the
model data do not display any trend the mode7 data is in deviation from mean
form.
25
(b) The Behavior of Tax Rates and Government Soendinq
In this experiment we work with filtered actual data.13 For
actual data, the behavior of taxes and government spending are depicted by the
following two regressions (t-statistics are in parenthesis).
(13a)
7,
=
.03 t
LW
(13b)
.81 7,.,
(7.39)
gt = -.12 t
.89 gt-,
(.30) (13.5)
t .59 7,,2
(4.26)
-
t
005 b,.,
(1.75)
.42 b,-,
(8.42)
[R'=.50]
[R'=.74]
The regression results from filtered model data are obtained from a sample of
1000 observations. These results are depicted by
=
.004 t
.75 7t-1
(.09) (25.5)
t
.016 b,_,
(2.09)
(14a)
7,
(14b)
g,= - .02 t
.78 gt-, c3 AH) gtm2 - .36 b,-,
(.08) (25.9)
.
(10.86)
[R2=.56]
[R2=.74]
Comparing the two sets of regressions, one notices some important
similarities. Debt affects fiscal policy in the data similarly to the way it
affects policy in the model.
Also, the coefficients on the first-order lags
are approximately the same across the two sets of regressions. One important
difference, however, is the number of significant lags in the data versus the
model.
Taxes in the data appear to be generated by an AR2, while model data
is depicted as an ARl.
The opposite appears to be true for government
spending. Since government spending in the model is generated by a firstorder Markov process the significant coefficient on the second lag must be due
to the filter. Although the model and the data do not match exactly, the
13The data are filtered using Harvey's and Jaeger's (1993)
procedure.
26
results of this analysis are encouraging and indicate that the theoretical
methodology of this paper can be used to capture empirically relevant
behavior.
(c) Forecastinq Tax Rates and Government SDendinq
As a final experiment, we analyze the theoretical models one step
ahead forecasting ability.
1916-1988
We do this by taking actual data over the period
(from Bohn's data) and using our probability model to derive
expected values for each succeeding period's taxes and government spending.
The results of this exercise are depicted in Figure 11.
Regarding government spending the model does quite well, the RMSE
is .07 and the only serious forecasting errors occur during World War II,
although spending during the depression is also somewhat overpredicted. This
latter result occurs because .08 is the lowest expected value of the ratio of
government spending to gnp produced by our calibrated statistical model when
the debt-gnp ratio is the interior of [-.l,
1.11.
For similar reasons the
expected value of the tax rate is overestimated in the early portion of the
sample and is largely responsible for the RMSE of .035. Overall, the
forecasting performance, especially over the post World War II period, leads
us to conclude that our methodology is flexible enough to capture important
features of the data.
6.
CONCLUSION
This paper has examined an alternative methodology for studying
the effects of fiscal policy.
Our model of fiscal policy takes the
consequences of intertemporal budget balance seriously and at the same time
allows for uncertainty in the fiscal policy process. The combination of these
27
two elements is able to generate behavior that is, in some instances,
strikingly different from standard results. Namely debt is non-neutral, the
expansionary effects of government spending are dampened, and the taxation of
capital can have surprising and counterintuitive results. The model generates
cases where debt crowds in investment and the behavior of the real interest
rate differs from behavior portrayed in standard Keynesian models.
The model
is also consistent with empirical evidence on U.S. fiscal policy as well as
with the behavior of U.S. government debt.
We feel, therefore, that our
methodology represents a promising alternative for investigating the effects
of fiscal policy in a dynamic stochastic general equilibrium framework.
28
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31
TABLE 1
Elasticity of Policy Functions Around Steady State Debt/GNP Ratios’
c
.048
.047
.036
.035
-.014
-.012
.015
.014
.020
.022
n
.ooo
.ooo
-.063
-.061
.024
.023
-.026
-.025
-.036
-.039
i
-.178
-.174
-.310
-.309
.105
.I38
-.llO
-.160
-.171
-.242
Y
.ooo
.ooo
.0004
.0004
.ooo
.ooo
.OOOl
.OOOl
.ooo
.ooo
r
-.131
-.I71
-.213
-.233
-.OlO
-.007
-.lOl
-.146
-.046
-.044
Note:
Case 1: labor fixed, rk varies ( plc = 3), f fixed, g fixed.
Case 2: labor varies, rk varies ( pk = 3), y fixed, g fixed.
Case 3 : labor varies, ~~fixed, r” varies ( pn = 7),g fixed.
Case 4: labor varies, rk varies ( pk = 9), 7” varies ( pn = 9), g fixed.
Case 5: labor varies, rk fixed,s"fixed,gvaries
(q= 6).
32
TABLE 2
Correlation
Coeflicients with Respect to the Debt to GNP ratio
(3000 observations)
I
I
-.97
-.20
.56
na
-.57
7k varies, n varies ( pk = 3)
-.93
-.31
-.Ol
-.89
-.92
7”varies (p” = 7)
-. 10
-.81
.40
-.20
.04
-.52
-.57
.53
-.37
-.20
-.42
.96
-.79
-.61
-.82
-.33
.99
-.99
-.44
-.74
-.67
-.61
.12
-.32
-.33
-.75
-.54
.33
-.33
-.28
~&varies, n fixed (pi
7’ and f
gvaries
~aq(~”
= 3)
=9,
pn =9)
(q=6)
g varies (7 = 1)
gvaries(v=6),
g
rkandrnvaxy(pk
varies (q = 1 S), 7k
=3,pn=3)
and:” vary ( pk = 9, ,y = 9)
33
TABLE 3
Impact Effects for a Deciine in Taxes
(measured as minus the ratio of the percent deviation
from steady state values to the percent deviation
in the decline in taxes)
Managed Debt
Y
n
c
i
L
I!!
7k varies,
II fixed ( pk = 3)
0
0
.OOl
-.003
.339
0
r’varies,
fz varies ( pk = 3)
.003
.004
-.002
.024
.3 16
-.OOl
.380
.561
.070
1.494
.532
.210
.194
.Oll
.567
.688
.059
f varies ( p” = 7)
7’
and
7”
vaqQk
=9, /,” =9)
.131
I
I
Markov
izass
rk varies, n fixed ( p,, = .57)
0
0
-.017
.060
.301
0
fk varies, 11varies ( pp = .59)
.018
.026
-.015
.135
.327
-.008
.331
.486
.091
1.182
.433
,212
,139
.205
-.004
.651
.523
.047
7”
varies ( p,” = .79)
rk and
7n
vary (p,,
= .80, p,. = .79)
34
TABLE 4
Impact Effects for a Rise in Government
Spending/GNP
Manaeed Debt
r
n
c
f
L
211
7k and gn
fixed, q = 6
.040
.058
-.032
-.674
.103
-.018
7’ and 7”
fixed, 77= 1
.015
.022
-.013
-.859
.064
-.007
.255
.378
.014
.096
.904
.107
,268
.395
.016
.165
1.318
.113
.078
.113
-.065
-.428
.171
-.036
.032
.047
-.026
-.770
.092
-.015
.364
.538
-.052
.778
1.193 *
.083
.403
.596
-.077
1.032
1.584
.073
r’andr”vary
(pk =3,$=3),
q=6
rk and
7n vary
( pk =9, p* =9), q = I5
rk and
7”
fixed, pg = .74
r’andffixed,
rkandz%xy
pg= .ll
(p* = .62, p,“= .59),
p-=.70
zk and 7” vary
pn=.75
( p,,
=
.77,
p,” =
.75),
Policy
Functions
(TV varies:
pK=3,
Labor
fixed)
Labor
Capital
I,
I I,
0.04
Z-o.10
1,
0.18
I I L I,,
0.32
0.46
, 1,
0.60
,
,
,
,
,
d
iii
cwd
d/
’
0.04
’
’
’
0.18
0.32
Debt/GNP
I1
18
0.46
”
0.60
Real Rate
Nl
’
1
0.32
Debt/GNP
Consumption
’
I1
0.18
Debt/GNP
r-.
%- ’
+O.lO
I,
0.46
”
\\” ”
i\
\\
\\
\\
-‘\ \\
”
”
’ ““I
I
I
0.60
Debt/GNP
Figure 2
Policy
Functions
(TV varies:
pK=3,
Labor
varies)
Labor
Capital
3,
,
z-0.10
,
)
0.04
,
,
,
1,.
I
0.32
0.18
I
I,.
0.46
1,
0.60
Debt/GNP
Debt/GNP
Consumption
Real Rate
d
i
d-O.10
Debt/GNP
“““““’
0.04
0.18
0.32
Debt/GNP
0.46
0.60
Policy
Functions
(7” varies:
p”=7,
Labor
varies)
Labor
Capital
I I1 1 I I I I I I II 11 ”
i2
l-Jde I t- III
z-_ I
d
O-Jzd
;:
L-o.10
0.04
0.32
0.18
0.46
0.60
$0.10
I
I
I
I
I
I
I
I
I
_e-I
.,,1,,,,,,,11111
0.04
,,,,,,,,,,,,,,,,
0.04
0.16
0.32
Debt/GNP
0.32
0.18
0.46
0.60
Real Rate
Consumption
2
I.
Debt/GNP
Debt/GNP
; -0.10
-_--
0.46
0.60
b”“““”
d-O.10
0.04
0.18
0.32
Debt/GNP
0.46
0.60
Figure 4
Policy
Functions
(g fixed,
7K varies:
pK=9,
T” varies:
labor
Capital
Ki-,,,,,,a.111111’*
’ ’
d-O.10
0.60
0.46
0.32
0.16
0.04
tir
’
d-O.10
~-0.10
1
0.04
0.04
-----8
d-O.10
-
’
’
’
0.04
’
’
’
0.18
’
’
’
0.32
Debt/GNP
a
11
11
0.18
11
0.32
“1
0.46
0.60
Real Rate
Consumption
’
I1
Debt/GNP
Debt/GNP
:-
pn=9)
I
0.46
a
I
*
0.60
iF””
d-0.10
1
0.04
0.10
0.32
Debt/GNP
1
0.46
I
*
*
0.60
Policy
Functions
(g varies:
1
*
s,
0.04
I
I1
0.18
I
II
0.32
?
and 7” fixed)
Labor
Capital
T-o.10
7=6,
1.
0.46
Ifi
’
0.60
Debt/GNP
Debt/GNP
Real Rate
Consumption
:d
cos*
‘6 -0.10
0.04
0.18
0.32
Debt/GNP
0.46
0.60
Debt/GNP
Figure 6
Impulse
Responses
(g varies:
Capital
Spending/GNP
r)=6,
7K and
Tax
2
20
30
40
50
ouptut
Public
0.
2.
20
30
period
period
period
\
.
‘--__-,
10
Investment
Consumption
Hours
20
period
period
$ : ?f,,
T: .a’
Tax
3
10
period
Labor
Labor
4
r)
d
3
d
2
- 0
7” fixed)
Real
debt
. c.------_
Rate
--________------’
9.
0.
N’
&
$0
Deriod
10
20
30
period
40
50
period
40
50
Jmpulse
Responses
(g varies:
Capital
Spending/GNP
~=l,
7K and T” fixed)
Labor
Tax
3
*)
Ii
s
d
;;;
I\ ,..---- -I.-.-.----------.--.-------.:-,-
z
20
10
20
30
z
40
!
zo
10
20
30
40
50
period
period
Labor
Tax
Jnvestment
Consumption
Hours
$L.--
l-y
:
o
:
______________-_-_-_________________
I
$
:,,”
20
10
20
30
40
20
,
period
Ouptut
Public
L
2:
t.
0,
-b ,;:
*. .:
0.:
n
db
2
,-/-
----------
_---.___----
----a--__-
I’
:-t’
lb
20
30
period
40
5’0
10
20
30
period
period
Real
debt
Rate
-.-*._
r-l.
C’
-----____e_-__
___.._--____-------r’
d
5;.
9.
0.
z,0
+
I
10
20
30
period
40
period
40
I
Figure 8
Impulse
Responses
(g varies:
q= 6, -rK varies:
Capital
Spending/GNP
pK=3,
T” varies:
$‘=3)
Labor
Tax
Tax
L-e
8
’ :’
:\ ’’
‘.,”
*--.-
’8..I’
J
10
20
30
40
10
20
30
40
50
period
period
period
Lobor
f
Investment
Consumption
Hours
l
‘“r
. ,r---,--.---~
-..
3.\:
*
20
period
10
20
Public
Ouptut
30
40
period
period
Real
debt
Rate
I’:
[
,
8
,.
/
0
:
:
10
\
\
, s.’
##
20
--..-
30
period
*.---
----.--
40
go
period
10
20
30
period
40
1
Impulse
Responses
(4
varies:
Spending/GNP
10
20
30
40
77=6,
7K varies:
Capital
Tax
20
30
40
:
20c
\----
10
20
Public
40
50
period
Real
debt
Rote
-~‘-‘.\___..----.-._-:
F/l. . . - 1
0” 1:
20
30
period
ouptut
”
Investment
.-(
period
.-.\
Tax
period
Consumption
Hours
I
gy : \
10
$‘=3)
Lobor
period
“!
0.
z
2 0
T” varies:
:
period
Labor
pK=3,
10
30
20
period
40
50
$0
I
10
20
30
period
40
50
period
Figure 10
Frequency
(Actual: 1916Doma in Comparison
Spectrum: Debt/GNP
C
1988
-
\_I
0.2
0.3
vs. Theoretica
0.4
0.5
cycles per year
Coherence:
Debt/GNP
0
d 0.0
0.1
0.2
cycles
0.3
per
yeor
0.4
0.5
Figure
One-Step-Ahead
II
Forecost:
Spending/GNP
0
In
d
,
’
I
3
I
I
%:d
i
4
1
!
I
1
I
I 1
I 1
I1
\I
--m-w-
,
0
d
7
I
1920
1930
l
I
1940
19so
1970
1960
1990
1990
Year
One-Step-Ahead
Forecast:
Tax
Rate
L
.
1960
@FedFRASER