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Working Paper 9507
OPTIMAL FISCAL POLICY WHEN PUBLIC CAPITAL
IS PRODUCTIVE: A BUSINESS CYCLE PERSPECTIVE
by Kevin J. Lansing
Kevin J. Lansing is an economist at the Federal Reserve
Bank of Cleveland. For helpful comments and suggestions,
the author thanks Costas Azariadis, Steve Cassou, Eric
Engen, Terry Fitzgerald, Tim Fuerst, Jang-Ting Guo,
Gary Hansen, Axel Leijonhufvud, Peter Rupert, Michael
Wallerstein, seminar participants at UCLA and UC Riverside,
and an anonymous referee. An earlier version of this paper was
published as FRB Cleveland Working Paper 9406, April 1994.
Working papers of the Federal Reserve Bank of Cleveland
are preliminary materials circulated to stimulate discussion
and critical comment. The views stated herein are those of
the author and not necessarily those of the Federal Reserve
Bank of Cleveland or of the Board of Governors of the
Federal Reserve System.
August 1995
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ABSTRACT
This paper develops a dynamic general-equilibrium model with productive public capital
to help account for differences in the business cycle characteristics of public- versus
private-sector expenditures in postwar U.S. data. A specification that allows for multiple
stochastic shocks (to technology and depreciation rates) can reproduce a number of
features describing the cyclical behavior of U.S. public investment and public
consumption as well as other fiscal variables, such as average marginal tax rates and the
government debt-to-output ratio. The model also delivers reasonable predictions for the
behavior of private-sector aggregates. It is less successful, however, in capturing the
large variability of public consumption expenditures in U.S. data, and it overpredicts the
variability of the capital tax relative to the labor tax.
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1. Introduction
In postwar U.S. data, the cyclical behavior of public investment and public consumption is very
different from the behavior of their private-sector counterparts. In comparison, public investment is much
less variable than private investment, while the reverse holds true for consumption, i.e., public
consumption is about twice as variable as private consumption. Moreover, the correlation between public
expenditures and output is very weak. This contrasts sharply with the highly procyclical nature of
private-sector expenditures. In this paper, I develop a dynamic general-equilibrium model with
productive public capital to help account for these types of differences, as well as other features
describing the cyclical behavior of U.S. fiscal policy. The approach represents an effort to carefully
model the behavior of an entity--namely, the government--that directly accounts for about one-quarter
of U.S. GNP and whose policies greatly impact the behavior of agents that generate the remaining threefourths. This approach can be viewed as a natural extension of recent efforts to integrate fiscal policy
into equilibrium models of the business cycle.
This paper builds on the recent work of Chari, Christiano, and Kehoe (1994a,b), who develop
a competitive real business cycle (RBC) model in which a government policymaker chooses an optimal
sequence of distortionary taxes in a dynamic version of the Ramsey (1927) optimal tax problem. The
model developed here differs from theirs in three important respects. First, public capital is introduced
as an input to the economy's production technology. This feature motivates the public investment
process and is consistent with an expanding body of theoretical and empirical research which suggests
that public capital may play an important role in the dynamics of growth and output.' Second, due to
the specification of constant returns to scale in all inputs, competitive firms realize positive economic
profits in equilibrium. This implies that the optimal steady-state tax on capital is positive, consistent with
U.S. observations. Furthermore, I show that when profits are taxed at the same rate as capital rental
income, the stochastic version of the model pins down a unique decentralization that is consistent with
the optimal allocations. In a model without profits, the optimal steady-state tax on capital is zero, and
h he ore tical models with productive public expenditures include Arrow and Kurz (1970), Barro (1990), Barro and Sala-iMartin (1992), Jones, Manuelli, and Rossi (1993a), Baxter and King (1993), and Glomm and Ravikumar (1994). Some recent
empirical applications include Finn (1993) and Kocherlakota and Yi (1995). Aschauer (1993) provides a review of the empirical
evidence regarding the productive effects of public capital.
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there exist many decentralizations that can support the optimal allocation^.^ Third, public consumption
expenditures are endogenized by introducing a separable term in the household utility function. In the
Chari, Christiano, and Kehoe model, government spending follows an exogenous stochastic process.
I compare simulation results from four versions of the model that differ according to the tax
regime (distortionary versus lump-sum taxes) and the process governing stochastic shocks (single versus
multiple shocks). The cyclical behavior of the allocations turns out to be similar under the two tax
regimes, regardless of the shock specification. However, the distortionary tax model encompasses a
larger set of endogenous fiscal variables. The additional variables are the distortionary tax rates on labor
and capital income and the government debt-to-output ratio.
The model with distortionary taxes and multiple shocks (to technology and depreciation rates)
can reproduce the following stylized facts describing the behavior of U.S. fiscal policy: Public
consumption and public investment are much less procyclical than their private-sector counterparts;
private investment is more variable than public investment; the capital tax is more variable than the labor
tax; the correlation between tax rates and output is relatively weak; and the government debt ratio is
about twice as variable as output. The model also delivers reasonable predictions for the cyclical
behavior of private-sector aggregates.
The model's principal shortcomings are its inability to capture the high variability of public
consumption expenditures observed in the data and its overprediction of the variability of the capital tax
relative to the labor tax. Other deficiencies are that the debt-to-output ratio is too countercyclical, and
the correlation between public expenditures and output is still higher than that observed in the data. I
find that the introduction of additional features, such as shocks to the demand for public goods and
higher profit levels, can help remedy some of these shortcomings. I also experiment with an alternative
tax structure in which labor and capital incomes are taxed at the same rate. However, I find that this
specification does not have much impact on the cyclical behavior of the optimal allocations.
Some related research includes Rojas (1993), who examines the optimal behavior of public
%he zero tax result is discussed by Arrow and Kurz (1970), pp. 195-203, and has been further elaborated on by Judd (1985)
and Chamley (1986). The non-uniqueness result is shown by Zhu (1992) and Chari, Christiano, and Kehoe (1994a).
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investment in an RBC model with a capital tax and a continuously balanced budget. An alternative
modeling approach treats fiscal policy as a series of exogenous shocks. For example, Christian0 and
Eichenbaum (1992) include stochastic government spending in the household utility function to shift the
labor s~lpplycurve and thereby help explain the low observed correlation between postwar U.S. labor
hours and real wages (as measured by average labor productivity). Braun (1994), McGrattan (1994), and
Dotsey and Mao (1994) show that similar results can be obtained by introducing stochastic distortionary
taxes to shift the labor supply curve. Ambler and Paquet (1994a) develop an RBC model in which some
of the government's fiscal variables (such as public investment and nonmilitary public consumption) are
endogenous, while others (such as the income tax and military spending) are exogenous.
The remainder of this paper is organized as follows: Sections 2 and 3 describe the model and
define a competitive equilibrium under each tax regime. The computation procedure and the choice of
parameter values is discussed in section 4. Section 5 examines the quantitative implications of the model
and presents the simulation results. Section 6 concludes.
2. The Model
The model economy consists of households, firms, and the government. All goods are produced
using a privately owned technology that exhibits constant returns to scale in the three productive inputs:
labor, private capital, and public capital. This specification implies that private firms earn an economic
profit equal to the difference between the value of output and payments made to the private inputs. The
existence of profits yields a positive optimal tax rate on capital in the steady state for the model with
distortionary taxes3 As owners of the firms, households receive net profits in the form of dividends.
Various options regarding the taxation of these dividends are considered.
3~ones,Manuelli, and Rossi (1993b) show that when profits derive from productive public goods, the absence of a separate
profits tax yields a positive optimal tax rate on capital in the steady state. However, if profits derive from the monopoly power
of firms, Guo and Lansing (1995a) show that the optimal steady-state tax on capital can be positive, negative, or zero. In a
perfectly competitive environment with no profits, the optimal steady-state tax on capital is zero.
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2.1 The Household's Problem
There is a large number of identical, infinitely lived households, each of which maximizes a
stream of discounted utilities over sequences of consumption and leisure:
max E o x p t (lnc, - A h ,
+
Blng,)
OcP c1,
A, B>O.
I 4
In this utility function, P is the household discount factor and c, represents private consumption
goods. The symbol E, is the expectation operator conditional on information available at time t. Each
household is endowed with one unit of time each period and works h, hours during period t. The fact
that utility is linear in hours worked draws on the formulation of indivisible labor described by Rogerson
(1988) and Hansen (1985). This means that all fluctuations in total labor hours are due to changes in
the number of workers employed, as opposed to variations in hours per ~ o r k e rHousehold
.~
preferences
also include a term representing the utility provided by public consumption goods g, . For simplicity, I
interpret g, as representing a per capita quantity of public goods that is free from congestion effects or
specific user charges. The separability in c, and g, implies that public consumption does not affect the
marginal utility of private consumption, a specification supported by parameter estimates in McGrattan,
Rogerson, and Wright (1993). Households view g, as outside their control.
The representative household faces the following within-period budget constraint:
where x, is private investment, kt is private capital, and b,,, represents one-period, real government bonds
carried into period t+l by the household. Households derive income by supplying labor and capital
services to firms at rental rates w, and r, . Two additional sources of household income are the firm's
net profits ft, (which are distributed to households as dividends) and the interest earned on government
bonds r,, b, .
?n postwar U.S. data, about two-thirds of the variance in total hours is due to changes in the number of workers. See
Kydland and Prescott (1990).
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I impose a restriction that prevents the government from taxing away profits. In particular, I
assume that the tax authority does not distinguish between profits and other types of capital income. As
a result, the tax on capital income also functions as a tax on profits, but one with an endogenous upper
bound. This scenario is reflected in equation (2), where net profits
bond interest r,, b, are all taxed at the same rate z,,
. Labor
fit,
capital rental income r, kt, and
income is taxed at the rate z,
. The
term z,,S,k, represents the depreciation allowance built into the U.S. tax code, where 6, is the
depreciation rate. The depreciation rate for tax purposes is assumed to coincide with the rate of
economic depreciation. Finally, T, represents a lump-sum tax.
The following equation describes the law of motion for the private capital stock:
kt+, = (1-S,)k,
+
x,.
In equations (2) and (3), the depreciation rate can vary over time according to an exogenous stochastic
process that will be specified shortly. The depreciation rate for public capital (6,,) is also allowed to
vary. Variable depreciation rates can be thought of as capturing the impact of external forces (for
example, energy prices) on the utilization level or on the degree of obsolescence of the existing capital
stock.5 Households view tax rates, wages, interest rates, dividends, and depreciation as determined
outside their control.
'several recent examples where shocks of this sort have been successfully used in business cycle applications are
Greenwood, Hercowitz, and Huffman (1988), Hercowitz and Sampson (1991), and Ambler and Paquet (1994b).
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The household first-order conditions with respect to the indicated variables and the associated
transversality conditions (TVC) are
TVC:
limEoPth,k,+l=O,
limEoP'h,b,+,=O,
t
t
+-
+-
(4e)
where h, is the Lagrange multiplier associated with the budget constraint (2). The transversality
conditions ensure that (2) can be transformed into an infinite-horizon, present-value budget constraint.
2.2 The Firm's Problem
Output (y, ) is produced by identical private f m s that seek to maximize after-tax profits, subject
to a technology that exhibits constant returns to scale in the three productive inputs, h,, kt, and k,,, where
k,, is the per capita stock of public capital. As with g,, I assume that public capital is free from
congestion effects and specific user charge^.^ The production technology is subjected to serially
correlated exogenous shocks z, that are revealed to agents at the beginning of period t. These shocks
generate equilibrium business cycle fluctuations in the model. Since the focus here is on the (detrended)
business cycle movements of variables, the model abstracts from exogenous technical progress. The
firm's decision problem can be summarized as
?Specifyingk,, as a per capita quantity ensures that there are no scale effects associated with the number of firms (see Barro
and Sala-i-Martin. [1992]). This setup can be viewed as incorporating an implicit congestion effect associated with the number
of firms (which is equal to the number of households here). For a model with an explicit congestion effect that is linked to the
size of the private capital stock, see Glomm and Ravikumar (1994).
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subject to
8
8
8,
Y , = exp(z,) k, ' h, ' k c ,
+ e 2 + e 3= 1 .
ocei<l,
(6)
The above expressions allow for the possibility that the firm's profits may be taxed directly. The
government's use of the tax rate z, for this purpose implies that the tax authority does not distinguish
between households and firms when assessing taxes on capital income. As before, this ensures that
profits will not be completely taxed away. When y= 2, profits are initially taxed at the firm level and
then taxed again as dividends at the household level. When y = 1, dividends are taxed only at household
level. When y= 0, the effective tax rate on profits is zero. In reduced form, the y= 0 case captures the
possibility that the tax authority cannot observe pure profits. The first-order conditions from (5) are
Each household is the owner of a fm and receives the firm's after-tax profits in the form of
dividends. The expression for dividends is
The stochastic process governing the evolution of the technology shock and the capital
depreciation rates is summarized below:
E
-
,
z,, 6,,
&GO
In (9), @ is a 3x3 transition matrix, @, is a 3x1 matrix of constants, and E,,
given.
(9)
is a 3x1 matrix of
serially uncorrelated residuals. The residuals are drawn from a normal distribution with mean 0 and
covariance matrix Ze. When the off-diagonal elements of @ are non-zero, innovations in technology can
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have a direct impact on the capital depreciation rates, and vice versa.
2.3 The Government's Problem
The government chooses an optimal program of taxes, borrowing, and public expenditures to
maximize the discounted utility of the household. Two versions of the government's problem are
considered, depending on the type of tax instrument available to the policymaker. The first-best solution
is obtained with lump-sum taxes. In this case, zht=z, = O for all t. The more realistic second-best
solution is obtained when all taxes are distortionary, namely, when T , = O for all t. To avoid time
inconsistency problems, I assume that the government can commit to a sequence of policies announced
at t = 0. Following the approach of Chari, Christiano, and Kehoe (1994a), I further assume that zm and
r, are specified exogenously such that tax revenue collected at t=O cannot finance all future
expenditures. If the initial levy on private-sector assets is sufficiently large, then the government chooses
z,, = zkt = 0 for some t > f This case is not very interesting because after period
i, the distortionary tax
model looks identical to the model with lump-sum taxes. In per capita terms, the government's budget
constraint and the law of motion for public capital are as follows:
Government expenditures on the left-hand side of (10) include public consumption g,, public
investment x,,
and an adjustment cost for public investment equal to x,, cp(x,,/ k,,). Following Abel
and Blanchard (1983), adjustment costs are formulated as a premium cp( - ) paid for each unit of
investment goods relative to consumption goods. The properties of cp( - ) are cp(0) = 0, cp'( .) > 0, and
cp"( .) > 0. In the computations, a simple quadratic version of cp( .) is employed: cp( - )= ?ha(~,,/k,,)~,
where a > 0. The presence of a small adjustment cost ensures that public investment always remains
positive in the stochastic simulations. This cost can be viewed as reflecting differences in the way that
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investment decisions are made in the public versus the private sector. For example, the decision to
undertake an infrastructure project may require a public debate or a voter referendum to settle issues
about financing or environmental impact.
The summation of the household budget constraint (2) and the government budget constraint (10)
yields the following per capita resource constraint for the economy:
Yt
=
9,
X,
+(P(~G,'~GI)]'
(I2)
Because the resource constraint and the government budget constraint are not independent equations,
equation (12) can be used in place of (10) in formulating the government's problem.
As a condition for equilibrium, government policy must take into account the rational responses
of the private sector, as summarized by (2), (3), (4), (7), and (8). For the distortionary tax case, these
equations can be conveniently summarized by the following "implementability constraint":
Equation (13) is obtained by substituting the first-order conditions of the household and the firm
into the present-value household budget constraint, with T,= 0 for all t.7 When y = 0, the capital tax for
t2
1 does not appear in the implementability constraint. Since z, and r, are specified exogenously, the
government's problem in this case amounts to choosing a set of allocations h, (= 11c, ), h, , g, , kt+,, and
kG,, for all t to maximize household utility (1) subject to (12) and (13). Given the optimal allocations,
the appropriate set of prices r, and w,, and policy variables z,, , z,, , and r,, that decentralize them, can
be computed using the profit-maximization conditions (7), the household first-order conditions (4), and
the household budget constraint (2). For example, the optimal allocations uniquely determine w, through
equation (7b). Given h, and w, , equation (4b) uniquely determines the government's optimal choice for
7 ~ o r specifically,
e
equation (13) is obtained as follows: Multiply both sides of the household budget constraint (2) by A,,
take expectations at t = 0, substitute in (4a)-(4d) and (8), iterate the resulting expression forward and sum over time, and finally,
apply the transversality conditions (4e).
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z,, . The government has much more flexibility, however, in choosing the optimal capital tax and the
optimal interest rate on government debt. The expectation operators in (4c) and (4d) imply that the aftertax returns on capital and bonds (weighted by marginal utility) must be the same "on average." In
response to a series of shocks, the government can satisfy this ex ante arbitrage condition and implement
the optimal allocations using many different decentralizations involving z,, and r,, . Consequently, when
y = 0 (or when 1 - 8, - 8, = O), the model does not uniquely pin down the time-series behavior of these
policy variables. (See Zhu [I9921 and Chari, Christiano, and Kehoe [1994a] for a more complete
description.)
When y> 0, the capital tax for t 2 1 appears in (13). In this case, (4c) must be imposed as a
separate constraint on the government's problem, and z,, for t 2 1 is an additional decision variable. In
contrast to the earlier indeterminate case, the model now implies a unique decision rule for z,, that is
consistent with the optimal allocations. Given this decision rule, equation (2) and the optimal allocations
uniquely determine r,, . Intuitively, the restriction that profits and dividends must be taxed at the same
rate as capital rental income pins down a unique decentralization to support the allocations.'
Under lump-sum taxes, the optimal allocations can be obtained by maximizing (1) subject only
to the resource constraint (12). In this case, government expenditures can be financed by any arbitrary
sequence of lump-sum taxes and government debt, provided that the government budget constraint is
satisfied in present-value terms (see Sargent [1987], chapter 3).
' ~ o h n(1994) provides a related example where a restriction on the menu of available policy instruments pins down a unique
decentralization. In his model, capital incomes derived from different technologies must all be taxed at the same (state-contingent)
tax rate.
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The general version of the government's problem is
subject to
with zo, 60 , 6G0,ko , kGo, b, , ,z , and r, given. The Lagrange multiplier A associated with (13) is
determined endogenously at t = 0 and is constant over time. To impose (4c) as a constraint, I apply the
law of iterated expectations and group terms such that the return function at time t involves only
variables dated t or earlier. This is done to facilitate a recursive solution algorithm. Notice that the firstorder condition with respect to the Lagrange multiplier & recovers (4c). The lagged multiplier &, is
treated as an additional state variable for t 2 1 in the recursive version of (14).' When y= 0, the firstorder condition with respect to z, implies & = 0 for all t. This reflects the fact that (4c) does not need
to be imposed as a separate constraint when y = 0.
With distortionary taxes, the solution to (14) for t 2 1 can be characterized by a set of stationary
g~ncludingp,-, in the state vector at time t is the mechanism by which the commitment assumption is maintained when
choosing z, each period in the recursive version of (14). This solution method was developed by Kydland and Prescott (1980)
in a deterministic setting. Rojas (1993) shows how the method can be applied in a stochastic environment.
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decision rules A, ( s, ,A ), h, ( s, ,A ), g, ( s, ,A ), kt+,( s, ,A ), kc,,( s, ,A ), zkl( s, ,A ), and ~ l(,s, ,A ), where
S,
={z,, 6,, 6Gt,k t , kc, , CI,,).Given these decision rules, a stationary decision rule for the government
bond allocation b,, ( s t ,A ) can be computed as the solution to the following recursive equation:
( k t + l+ b t + l =
)
b
-A
ht+l -',+I
(1-zkt+l)y(1-el-e2)~t+l
+',+I
( k t + 2+ b t + 2 ) ] '
(I5)
Equation (15) is the household budget constraint at t+l (with TI+,= 0), after taking expectations
and substituting in the first-order conditions of the private sector. At t = 0 , the government chooses A,,
h, , go, k, , kc, , and ~b . The t = 0 allocations, together with the decision rules for t 21, determine A for
a given set of initial conditions.
With lump-sum taxes, A = CI, = 0 for all t, and (14) collapses to a standard social planning
problem. The solution to the planner's problem for t 2 0 can be characterized by a set of stationary
decision rules A, ( s t ) , h, ( s t), g,(s,), kt+,( s t ) ,and kc,,( st ), where s,={z,, 6,, 6,, , k t , kc, 1.
3. Defining an Equilibrium
This section defines a competitive equilibrium under each tax regime. In what follows, I assume
that y > 0 so that the solution to (14) pins down a unique decentralization under distortionary taxes. In
this case, a competitive (Ramsey) equilibrium is defined as
(i) A set of stationary decision rules h, ( st ,A ), h, ( st ,A ), g, ( st ,A ), kt+,( st ,A ), kc,,( st ,A ),
zkr( s t , A), and u( s t , A ) that satisfy (14) for t 21, where st ={z,, 6,, 6,, , k t , kc,, U-,
).
(ii) A stationary decision rule for the bond allocation b,, ( s t ,A ) that satisfies (15) for t 21
(iii) A set of allocations A,, ho , g o , k, , and kc, , and Lagrange multipliers ~b and A, that satisfy
(14) and (2) given the decision rules for t 21 and the initial conditions z, , 6,, 6,, , ko , kc,, b, ,
z, and r, .
(iv) A sequence of factor prices r, and w, for t 2 0 defined by (7a) and (7b).
(v) A sequence of labor tax rates z,, for t 2 0 defined by (4b).
(vi) A sequence of bond interest rates r,,, for t 2 1 defined by (2).
Under lump-sum taxes, a competitive equilibrium is defined as
( i ) A set of stationary decision rules h ( st ), h, ( st ), g, ( st ), kt+,( st ), and kc,,( st ) that satisfy (14)
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with A = p,= 0 for t20, where s,={z,, ti,, tiGt,kt, kc, ), given the initial conditions z,, ti0, ti,,,
k, , and kc,
(ii) A sequence of factor prices r, and w, for t 2 0 defined by (7a) and (7b).
(iii) An arbitrary sequence of lump-sum taxes T, and government debt b, that satisfies:
lim
bt+l
=
0
(iv) A sequence of bond interest rates r,, for t 2 0 defined by (2).
4. Computation Procedure and Calibration
The stationary decision rules governing the optimal allocations in the model are obtained by
solving a linear-quadratic approximation of (14). The algorithm makes use of the fact that (14) is
recursive for t 2 1 under distortionary taxes and for t 2 0 under lump-sum taxes. The resulting decision
rules, which are log-linear functions of the state variables, are valid in the neighborhood of the
deterministic steady state.'' Under distortionary taxes, the stationary decision rules depend on the
Lagrange multiplier A, which is computed as follows. First, given an initial guess for A, I compute the
deterministic steady state from the first-order conditions of (14) with respect to A,, h, , g, , k,, , k,,
, p, ,
and 2,. The steady-state version of (15) is then used to compute the steady-state level of government
debt b I repeat this procedure, adjusting A until a desired level of steady-state debt is obtained. Next,
I use the first-order conditions of (14) at t = 0,together with the log-linear decision rules for t 2 1, the
household budget constraint (2) evaluated at t = 0 and t = 1, and the initial conditions z,, ti0, ti,,, k, ,
'O~hecomputation procedure is based on the algorithm described by Hansen and Prescott (1994). Chari, Christiano, and
Kehoe (1994b) assess the accuracy of the log-linear approximation method in an optimal policy problem similar to the one
presented here. They conclude that the approximation yields very accurate results for the allocations, but less accurate results for
the tax rates, in comparison to a minimum-weighted residual method.
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k,,, ,z,
and r,, to compute an initial level of debt b, that is consistent with A and b. The economy
is assumed to be in stationary equilibrium. All simulations represent small fluctuations around the
deterministic steady state.
Parameters are assigned values based on empirically observed features of postwar U.S. data. The
sample period begins in 1954 to avoid the influence of the Korean War. The time period in the model
is taken to be one year, which is consistent with both the time frame of most government fiscal decisions
and the frequency of available data on average marginal tax rates. The discount factor
P (= 0.962)
implies a real rate of interest of 4 percent. The parameter A in the household utility function is chosen
such that the fraction of time spent working is equal to 0.3 in the steady state. This coincides with timeuse studies, such as Juster and Stafford (1991), which indicate that households spend approximately onethird of their discretionary time in market work. The value of B is chosen to yield a steady-state ratio
Fly equal to 0.17,
the average value for the U.S. economy from 1954 to 1992. In computing this
average, public consumption was estimated by subtracting total public investment (including military
investment) from an annualized series for government purchases of goods and services (GGEQ from
Citibase). This was done to reduce double counting, since the GGEQ series does not distinguish between
government consumption and investment goods."
The exponents in the Cobb-Douglas production function are chosen on the basis of two criteria.
First, the selected values of 8, (= 0.32) and 8, (= 0.60) are in the range of the estimated shares of GNP
received by private capital and labor in the U.S. economy (see Christian0 [1988]). Second, given the
steady-state depreciation rate 6, (described below), the output elasticity of public capital 8, (= 0.08) is
chosen to yield a steady-state ratio of public investment to output equal to 0.03. This coincides with the
average ratio of nonmilitary public investment to GNP in the U.S. economy from 1954 to 1992."
With constant returns to scale in all inputs, the value of 8, (= 1- 8, - 8,) also determines the
he specific parameter values used in the computations are A = 2.480, B = 0.267 for the model with distortionary taxes,
and A = 3.330, B = 0.285 for the model with lump-sum taxes.
'%he range of direct empirical estimates for 8, at the aggregate national level is quite large. Aschauer (1989) and Munnell
(1990) estimate values of 0.39 and 0.34, respectively. Finn (1993) estimates a value of 0.16 for highway public capital. Aaron
(1990) and Tatom (1991) argue that removing the effects of trends and taking account of possible missing explanatory variables
(such as oil price shocks) can yield point estimates for 8, that are not statistically different from zero.
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steady-state level of firm profits. Since profits do not affect household decisions at the margin, the
government would like to tax profits as much as possible to obtain non-distortionary revenue. Choosing
T,,
> 0 accomplishes this objective in varying degrees, depending on the value of y. As y increases, the
capital tax collects a larger fraction of revenue from profits. This motivates the government to choose
a higher capital tax and lower labor tax for a given level of profits. Since dividends are subject to double
taxation under the U.S. tax code, I choose y = 2 as the baseline, which implies that the optimal steadystate tax on capital in the model is % = 0.29. This value coincides with the average effective corporate
tax rate in the United States from 1954 to 1980, as estimated by Jorgenson and Sullivan (1981). The
steady-state tax on labor (Yh ) turns out to be 0.22. This is close to the average marginal tax rate on labor
income estimated by Barro and Sahasakul (1986) from 1954 to 1983. When y = 0 , the steady-state tax
rates are 7, = 0.10 and ?, = 0.33.13
The steady-state private capital depreciation rate
8 (= 0.067) is estimated by
a least squares
regression of x, - (kt+,- k t ) on kt. An analogous regression yields the steady-state public capital
depreciation rate 6, (= 0.022).14The steady-state value of the technology shock is set equal to zero. The
transition matrix cP governing the evolution of the shocks is estimated by a vector autoregression using
detrended annual data on z, , St, and S,, from 1954 to 1992.15The constant vector <Po is chosen to agree
13when profits are zero ( 8, + 8, = l), the optimal steady-state tax on capital is zero. If a separate profits tax were available,
the government would tax profits at 100 percent and other capital income at 0 percent in the steady state (see footnote 2).
1 4 ~ a t sources
a
are as follows. The capital and investment series are in 1987 dollars from Fixed Reproducible Tangible
Wealth in the United States, U.S. Department of Commerce (1993). The series for k,, and x,, include nonmilitary governmentowned equipment, structures, and residential components. The series for k, and x, include business equipment and structures,
consumer durables, and residential components. The "capital input" measure of the net stock was used for all capital data.
Annualized series for the following variables were constructed using the indicated quarterly series from Citibase: y, = GNPQ,
c, = GCNQ + GCSQ (nondurables and services), h, = LHOURS (household survey), y,/ h, = GNPQJLHOURS, g, = GGEQ - x,, military investment. The series for b,ly, is federal debt held by the public as a fraction of GNP, where the debt series is from
Federal Debt and Interest Costs, U.S. Congressional Budget Office (1993), table A-2. All variables were normalized by the total
population series PAN from Citibase to obtain per capita quantities analogous to those in the model. Average marginal tax rates
on labor income (z,,) are from Barro and Sahasakul(1986) for 1954-83, McGrattan, Rogerson, and Wright (1993) for 1954-87,
and Mendoza, Razin, and Tesar (1994) for 1965-88. Average marginal tax rates on capital income (z,) are from Jorgenson and
Sullivan (1981, table 11) for 1954-80, McGrattan, Rogerson, and Wright (1993) for 1954-87, and Mendoza, Razin, and Tesar
(1994) for 1965-88.
he technology shock was measured as z, = In y, - 0.321n k, - 0.601n h, - 0.081n k,, . The depreciation rates were measured
as 6, = [x, - (k,,, - k, )] 1k, and 6, = [x,, - (k,,,, - k,, )] 1k,, . Trends were removed by regressing each series on a constant and a
time trend and subtracting the trend. The specification with 6, and 6, in levels yielded a much better fit of the data than did
a logarithmic specification.
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with the steady-state values of the shock variables. The covariance matrix CE is estimated from the data
and imposed in the simulations. The stationary stochastic process used in the simulations is
C E = S'-S, where S
=
I
O
0.00531
6.224e-5
The following matrix shows the correlation coefficients among the residuals:
An interesting aspect of pE is that innovations in the cyclical component of the technology
variable display a weak negative correlation with innovations in the cyclical components of the
depreciation rates. Also, the correlation between innovations in the two depreciation rates is weak
( =0.05).16 The model is also simulated with technology shocks alone. This provides a benchmark for
comparison with the standard real business cycle specification. In this case, the depreciation rates are
constant over time and the evolution of z, is described by
The parameter a ( = 40) is chosen such that cp( .) =0.01 in steady state. This implies that
adjustment costs are equal to 1 percent of public investment, or 0.03 percent of output, in steady state.
Costs of such small magnitude seem reasonable and are sufficient to ensure that x,, always remains
1
%is specification differs from that employed by Ambler and Paquet (1994b). They use Euler equations from their model
together with quarterly investment data to construct an artificial time series of capital stocks and corresponding (quarterly)
depreciation rates. In the resulting specification, depreciation shocks are strongly correlated with one another, but have no
persistence and are not correlated with technology shocks. Here, annual depreciation rates are computed from actual capital stock
data.
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positive during the simulations. Under distortionary taxes, the value of A (= 0.3 13) implies a steady-state
ratio b/yequal to 0.37. This is the average level of U.S. federal debt held by the public as a fraction
of GNP from 1954 to 1992. The initial conditions z,, 6,,
k, , k,, , and ,z
are set to their
corresponding steady-state levels, and r, = r, - 6, . The initial debt ratio b, / y o that is consistent with
these values is 0.362.
Table 1 summarizes steady-state values for the model versus the corresponding U.S. averages
from 1954 to 1992. For the data, three values are shown for the average marginal tax rates
< and ?, .
These correspond to the sample means from different studies (see footnote 14 for sources).
5. Quantitative Implications of the Model
5.1 Optimal Decision Rules
Table 2 shows the optimal decision rules for selected variables in the case of distortionary taxes
and multiple shocks.17 The optimal behavior of public versus private expenditures over the business
cycle can be inferred by examining the coefficients on the shock variables in the decision rules. For x,,
and g,, the coefficients on z, are both positive, implying procyclical behavior. The private-sector
counterparts, x, and c, , are also procyclical, as evidenced by their positive coefficients on z, . Notice that
public and private investment exhibit a sign difference regarding the coefficient on 6,. This feature of
the model, together with the fact that innovations in the depreciation rates tend to be negatively
correlated with innovations in z,, tends to reduce the contemporaneous correlation between public and
private expenditures relative to the single-shock version of the model. This is a desirable outcome, given
the weak correlation between public and private expenditures in postwar U.S. data.
The decision rules also show that the capital tax responds much more strongly to exogenous
shocks than does the labor tax. This reflects the government's use of the capital tax as a tool for
absorbing shocks to its budget (which are caused by changes in the size of the tax base over the business
cycle). An increase in z, expands the tax base, allowing government spending requirements to be met
1 7 ~ othe
r chosen parameter values, it can be shown that the linearized dynamic system implied by (14) possesses a unique
set of stable, stationary decision rules for t 2 1. (See Guo and Lansing [1995b] for more details.) Where possible, the nonlinear
versions of the equilibrium conditions are used to compute the period-by-period values of endogenous variables during the
simulations.
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with a lower value of z,,. Due to the depreciation allowance, an increase in 6, tends to reduce the tax
base, which calls for an increase in z,
. Absorbing
shocks mainly by changes in z,
as opposed to
changes in z,, is efficient because household assets (capital and government bonds) cannot be quickly
adjusted in response to a change in the capital tax. In contrast, the household can instantaneously adjust
labor supply in response to a change in the labor tax.18 Given the behavior of z,
the optimal response
of government borrowing to shocks is relatively small. With constant depreciation rates, the procyclical
nature of the tax base, especially the level of profits, implies strong countercyclical behavior for z, and
z,
. With variable depreciation rates, however, the correlation between output and tax rates is reduced.
Again, this improves comparison with the data.
5.2 Simulation Results
The model's quantitative predictions for the cyclical behavior of fiscal policy and private-sector
aggregates are shown in tables 3 and 4. Figures 1-8 provide a visual comparison for some selected
variables.
The cyclical behavior of the allocations turns out to be similar under the two tax regimes,
regardless of the shock specification. However, the distortionary tax model includes a larger set of
endogenous fiscal variables. The additional variables are the distortionary tax rates on labor and capital
income and the government debt-to-output ratio.
Under both tax regimes, the single-shock specification underpredicts the variability of output and,
consequently, most of the other variables in table 3. Actually, this is a desirable feature, since evidence
suggests that the contribution of technology shocks to fluctuations in U.S. output is less than 100 percent
(see Aiyagari [1994]). Both tax regimes capture the fact that public investment is only about half as
variable as private investment in postwar U.S. data. Moreover, in the model with distortionary taxes, the
capital tax is more variable than the labor tax, a feature which also tends to characterize U.S. tax rates.
The single-shock specification suffers from a number of serious shortcomings. First, public
he optirnality of using a state-contingent capital tax to absorb budget shocks has been shown previously by Judd (1989),
Zhu (1992), and Chari, Christiano, and Kehoe (1994a).
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expenditures in the model (g, and x,,) are strongly procyclical in contrast to the relatively weak
correlation with output observed in the data. The predicted correlations between the components of
public expenditures and output are both close to 1.O. In the data, however, the correlation between public
consumption and output is only 0.05, while the correlation between public investment and output is 0.38.
This is further reflected in table 4, where the predicted correlations between public expenditures and
their private-sector counterparts are much higher than those in the data. Second, the standard deviation
of g, in the model is too low, both in absolute terms and relative to the standard deviation of c, . Third,
under distortionary taxes, the predicted standard deviation of the labor tax is much lower than any of
the values based on U.S. tax rates. Fourth, the model predicts strong countercyclical behavior for tax
rates and the government debt ratio, in contrast to the weak correlations observed in the data. Finally,
the correlation between hours worked and average labor productivity is close to 1.0 in the model versus
a near-zero correlation in the data. This last observation is a well-known deficiency of single-shock RBC
models (see Christian0 and Eichenbaum [1992]).
In general, the addition of multiple shocks improves the model's performance in a number of
ways. The standard deviation of output, hours, and the government debt ratio all increase to values that
are very close to those in the data. Public expenditures become less procyclical, and as a result, the
contemporaneous correlations between public and private expenditures decline. These correlations are
still stronger than those observed in the data, however, especially in the case of g, (see table 4 and
figures 1-4). With multiple shocks, the correlations between the tax rates and output become much
weaker, and the correlation between hours worked and average labor productivity substantially declines.
The last observation confirms earlier results obtained by Hercowitz and Sampson (1991) and Ambler
and Paquet (1994b) using RBC models with a shock to the capital accumulation equation.
The multiple-shock specification does not help much to increase the standard deviations of g,
and z,, , and the government debt ratio remains strongly countercyclical. The correlation between the
debt ratio and output is -0.85 in the model versus -0.18 in the data. A notable drawback to the multipleshock specification is that the standard deviations of x, and T, are now much higher than the
corresponding U.S. values.
To summarize, the model with multiple shocks and distortionary taxes is able to capture the
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following stylized facts describing the behavior of U.S. fiscal policy: Public expenditures (g, and x,,)
are much less procyclical than private expenditures (c, and x, ); private investment is more variable than
public investment; the capital tax is more variable than the labor tax; the correlation between tax rates
and output is relatively weak; and the government debt ratio is about twice as variable as output. The
model's principal shortcomings are its inability to capture the high variability of g, relative to c,, and
its overprediction of the variability of T,, relative to
T,,,
. Some other deficiencies are that b,ly,
is too
strongly countercyclical and that g, and x,, remain more strongly correlated with output than in the data.
Figure 3 reveals that the high variability of public consumption relative to private consumption
in U.S. data is strongly influenced by the sizable expansion of government spending that occurred in the
late 1960s, coinciding with the Vietnam war. To capture this sort of phenomenon, I experimented with
a modified version of the model that included an independent stochastic shock applied to the parameter
B in the household utility function. This shock, which can be viewed as representing changes in the
demand for public goods (perhaps due to wars, riots, or disasters), improved the model's performance
in two ways: first, by increasing the variability of g, relative to c,, and second, by reducing the
contemporaneous correlation between g, and y,. The countercyclical nature of b,ly, was also reduced,
but only to a small degree.lg
Figures 5-8 display the U.S. and model tax rates before detrending. For quantitative comparisons
(tables 3 and 4), detrending is necessary because U.S. labor tax estimates all display a distinct upward
trend, while two out of the three capital tax estimates display a downward trend. These trends have no
counterpart in the model. The fact that the model overpredicts the variability of the capital tax relative
to the labor tax is consistent with the results of Chari, Christiano, and Kehoe (1994a). In their model,
with log utility and technology shocks alone, the standard deviation of the capital tax is 17.67 percent
(for the decentralization with uncontingent debt) versus 0.08 percent for the labor tax. In comparison,
table 3 indicates that the standard deviation of T,, in the single-shock specification is 4.39 percent versus
Ig~mblerand Paquet (1994b) adopt a different approach by introducing an exogenous stochastic component of government
expenditures in their model. These expenditures are calibrated to U.S. data on military spending. The income tax in their model
is also taken to be exogenous (and constant over time). In contrast, I have chosen to endogenize all of-the government's fiscal
variables.
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0.38 percent for z,, . The fact that z, is less variable here is due primarily to the existence of
profits.20
I experimented with varying both the size of firm profits (by changing 8, relative to 8, ) and
the level of dividend taxation (by changing y). As either 8, or y increased, the percentage standard
deviation of z, declined. The intuition for this result is straightforward. Recall that dividends (equal to
after-tax profits) do not distort household decisions because profits are determined outside households'
control. Higher profits or a higher y implies a larger and less elastic tax base for the capital tax.
Consequently, smaller changes in the tax rate can produce the same revenue effect when responding to
shocks. Although higher levels of 8, reduce the standard deviation of z,, the resulting steady-state ratio
?,/?
exceeds the U.S. average, which is undesirable from a calibration standpoint. Similarly, values
of y greater than two become difficult to justify in comparison to the U.S. tax structure.
The model's tax structure imposes a sharp distinction between the taxation of labor and capital
incomes. In reality, however, the personal income tax encompasses income from many different sources,
such as wage earnings and investment income. To investigate the effects of an alternative tax structure,
I introduced the following additional constraint in (14) to require that labor and capital incomes be taxed
at the same rate: (1 - z,,) = A / (A, w,), where w,= 8, y, / h, . Table 5 shows the results of this experiment.
The standard deviation of the optimal flat-rate income tax z,turns out to be very low (0.80 percent with
multiple shocks). Given that labor's share of income is 0.60 in the model, it is not surprising that z,
behaves similarly to z,, under the original tax structure. In steady state, = 0.243, which exceeds
z, = 0.221. Ceteris paribus, a higher labor tax exerts more of a dampening effect on hours worked (and
output) in response to shocks. This effect, and the fact that the tax base for z, is higher than for z,,,,
helps compensate for the absence of the capital tax as a separate shock absorber. This is evidenced by
the standard deviations of the allocations in table 5, which are mostly very close to those in table 3.
20~hereare a number of other differences with the Chari, Christiano, and Kehoe model that are likely to affect the
quantitative comparisons. First, their solution technique involves a minimum-weighted residual method applied to a discrete state
space. Second, simulated tax rates in their model are not subjected to the Hodrick-Prescott filter before computing the statistics.
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6. Concluding Remarks
This paper develops a model that combines elements from the theory of optimal public finance
with an equilibrium view of aggregate fluctuations. Interestingly, the model predicts that if the
government responds systematically to external forces and takes into account the rational responses of
the private sector, then fiscal variables can fluctuate substantially over the business cycle. Moreover,
these policy variations might even be in the best interests of households. More specifically, this paper
demonstrates that an equilibrium business cycle model with optimal fiscal policy can reproduce a number
of features describing the cyclical behavior of public expenditures, tax rates, and the government debt-tooutput ratio in the postwar U.S. economy. Although the model's quantitative predictions do not coincide
with some key aspects of the data, the results provide encouragement for an approach that seeks to
explicitly model government behavior in a general equilibrium framework. The consideration of
additional features, such as voting behavior, monetary policy, or institutions that affect the commitment
mechanism (or lack thereof), are likely to improve our understanding of the links between government
policy and the business cycle.
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Table 1: Model Steady States versus U.S. Averages
Model
Variable
Distortionary Taxes
Lump-Sum Taxes
U.S. Economya
y/F
0.17
0.20
0.22
a ~ othe
r U.S. economy, investment, consumption, and capital averages are for the period 1954 to 1992. Tax
rate averages correspond to the sources and periods listed in footnote 14.
Table 2: Optimal Decision Rules (Distortionary Taxes with Multiple Shocks)
Variable
X,
=
Constant
zt
6,
6ct
In (kt )
In (kc, )
0.071
0.654
0.773
0.626
-0.162
0.047
Source: Author's calculations.
In (
~ ~ - 1 )
-0.020
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Table 3: Business Cycle Statistics for Models and U.S. Economy
Standard Deviation (percent)
Model with
Multiple Shocksa
Model with
Technology Shocks Onlya
Variable
Distortionary
Taxes
Lump-Sum
Taxes
Distortionary
Taxes
Lump-Sum
Taxes
U.S. Econ m
1954-92'by
Yt
1.61
1.66
1.32
1.27
1.60
CI
0.49
0.43
0.56
0.48
0.82
Contem~oraneousCorrelation with Out~ut
Model with
Multi~leShocks
Variable
Distortionary
Taxes
Lump-Sum
Taxes
Model with
Technoloev Shocks Onlv
Distortionary
Taxes
Lump-Sum
Taxes
U.S. Economy
1954-92
aModel statistics are means over 250 simulations, each 39 periods long, after dropping the first 50 periods.
Before computing the statistics, all series were logged (except for the tax rates) and detrended using the Hodrick-Prescott
filter (see Prescott [1986]). The smoothing parameter for the filter was set equal to 10, which is appropriate for annual
data (see Baxter and King [1995]).
b ~ o the
r U.S. economy, data sources are described in footnote 14. Data on average marginal tax rates do not
extend over the full sample. All series were logged andlor detrended as in the model.
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Table 4: Com~arisonof Selected Statistics
Model with
Multiple Shocksa
Model with
Technology Shocks Onlya
Distortionary
Taxes
Lump-Sum
Taxes
Distortionary
Taxes
Lump-Sum
Taxes
U.S. Econgmy
Statistic
O x1
9
~
0.27
0.40
0.61
0.63
0.47
1947-92
0.92
0.87
0.26
COrr (xG9x)
0.5 1
0.71
...........................................................................................................................................................................................................................................
og/oc
1.15
1.26
0.93
1.06
1.98
0.94
0.94
0.10
con (g, c)
0.78
0.8 1
............................................................................................................................................................................................................................................
Oh 10~fi
3.43
3.45
2.02
1.69
2.47
corr (h, ylh)
0.58
0.29
0.89
0.91
0.04
............................................................................................................................................................................................................................................
-11.44
-3.82, 1.22, 2.37
0,10Th
39.9
-0.89
-0.29, 0.47, 0.89
COrr (%,7,)
0.67
Table 5: Model with Flat-Rate Income Tax
Multiple Shocks
Technology Shocks Only
Variable
Standard
Deviation (%)
Correlation with
Output
Standard
Deviation (%)
Correlation with
Output
Yr
1.61
1.OO
1.31
1.OO
aModel statistics are means over 250 simulations, each 39 periods long, after dropping the first 50 periods.
Before computing the statistics, all series were logged (except for the tax rates) and detrended using the Hodrick-Prescott
filter (see Prescott [1986]). The smoothing parameter for the filter was set equal to 10, which is appropriate for annual
data (see Baxter and King [1995]).
b ~ o the
r U.S. economy, data sources are described in footnote 14. Data on average marginal tax rates do not
extend over the full sample. All series were logged andlor detrended as in the model.
clevelandfed.org/research/workpaper/1995/wp9507.pdf
FIG 1: U.S. INVESTMENT 1 9 5 4 - 9 2
FIG 2: MODEL INVESTMENT
D i s t o r t i n g Taxes, Multiple S h o c k s
U.S. Privote lnvestment
U.S. Public lnvestment
Year
Year
FIG 3: U.S. CONSUMPTION 1 9 5 4 - 9 2
FIG 4: MODEL CONSUMPTION
8
D i s t o r t i n g Taxes, Multiple S h o c k s
1
i
;\
I
\
1
\
U.S. Privots Conaumption
U.S. Public Conaumption
I
t
-7
ct
g,
- Privots Consumption
- Public Consumption
I
9
Year
Sources: Author's calculations and Citibase.
Year
clevelandfed.org/research/workpaper/1995/wp9507.pdf
FIG 5:
U.S. LABOR TAX
FIG 6:
-Eatlmoio from Borra & Sohonokul (1986)
- - - Estlmoio from UcCrottan et 01. (1993)
Estimoie from Uendoro et 01.
MODEL LABOR TAX
-T ~ -L Multiple Shocks
- - - The - Technolo~yShocks Only
(1994)
0.05
0.00
5
0
4
8
12
16
Year
FIG 7:
0.65
0.60
0.55
,"
0
-
0.50
0.45
G
0.40
Y
.-a
0
0.35
0
c 0.30
0
w
Y
0.25
:0.20
X
2
0.15
-
0.00
-
U.S. CAPITAL TAX
-
_-
\
/I \,,--'\
\
\
-
0
FIG 8:
24
28
32
36
40
MODEL CAPITAL TAX
0.65
0.60
"
,
\ - - / \
-
--,
'
-
0.55
0.50
0.45
-
-
Y
-
0.100.05
\,------\
20
Year
-
---
a
t-'
Estimate from UcGrotton et 01. (1993)
Estlmote lrom Uendozo et 01. (1994)
Estlmaie from Jorgenaon & Sulllvon (1981)
1956
1960
,
I
I
t
1964
1968
1972
1976
Year
Sources: As noted, and author's calculations.
0.10
0.05
I
1980
1984
1988
1992
0.00
-
- - - TU
TN -
M ~ n l p l eshock.
Technology Shock Only
I
0
4
8
12
8
I
I
16
20
24
Year
28
I
*
32
36
40
@FedFRASER