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clevelandfed.org/research/workpaper/1995/wp9509.pdf
Working P a ~ e 9509
r
OPTIMAL FISCAL POLICY, PUBLIC CAPITAL, AND THE
PRODUCTIVITY SLOWDOWN
by Steven P. Cassou and Kevin J. Lansing
Steven P. Cassou is a professor of economics at the State
University of New York at Stony Brook, and Kevin J.
Lansing is an economist at the Federal Reserve Bank of
Cleveland. For helpful comments, the authors thank David
Altig, Costas Azariadis, Terry Fitzgerald, Timothy Fuerst,
Gary Hansen, Gregory Huffman, Per Krusell, Finn Kydland,
B. Ravikumar, Peter Rupert, Randall Wright, and seminar
participants at the Federal Reserve Banks of Cleveland and
New York, and the 1995 Midwest Macroeconomics
Conference.
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 authors and are not necessarily those of the Federal
Reserve Bank of Cleveland or of the Board of Governors of
the Federal Reserve System.
September 1995
clevelandfed.org/research/workpaper/1995/wp9509.pdf
ABSTRACT
This paper develops a quantitative theoretical model for the optimal provision of public capital.
We show that the ratio of public to private capital in the U.S. economy from 1925 to 1992 evolves
in a manner that is generally consistent with an optimal transition path derived from the model.
The model is also used to quantify the conditions under which an increase in the stock of public
capital is desirable and to investigate the effects of hypothetical nonoptimal fiscal policies on
productivity growth.
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clevelandfed.org/research/workpaper/1995/wp9509.pdf
1. Introduction
In recent years, the link between public capital and private sector production has
been a subject of considerable debate among policymakers and researchers. Although
the idea that public capital may represent an important productive input is not new
(for example, see Arrow and Kurz [1970]), work by Aschauer (1989, 1993) and Munnell
(1990) stimulated renewed interest in this area because their empirical results suggested
that large gains could be had by expanding public investment. These researchers also
claimed that the observed decline in the rate of public capital accumulation during
the 1970s and 1980s contributed significantly t o the slowdown in the growth rate of
U.S. labor productivity over the same period. Subsequent studies have added t o the
debate by attempting t o confirin (or refute) the productive effects of public capital
using increasingly sophisticated empirical methods.' Up t o this point, however, little
attention has been given t o addressing these issues from a theoretical perspective.
In this paper, we develop a quantitative theoretical model for the optimal provision of
public capital. We show t11a.t the ratio of public t o private capital in the U.S. economy
from 1925 t o 1992 evolves in a manner that is generally consistent with a n optimal
transition path derived from a simple endogenous growth framework. Moreover, we are
able t o quantify the conditions under which a n increase in the stock of public capital is
justified in terms of maximizing the utility of a representative household. We find that
even when the output elasticity of public capital is as high as 0.10, a n increase in public
capital from current levels is not called for. Finally, we show that a nonoptimal public
investment policy of the type that might be interpreted as reflecting U.S. experience
'For instance, Aaron (1990), Tatom (1991), and Holtz-Eakin (1992) have s l ~ o w nt h a t empirical
methods which incorporate omitted variables, adjustments for nonstationarities, or more disaggregated
d a t a find t h a t the output elasticity of public capital is not statistically different from zero. In contrast,
Lynde and Richmond (1992), Finn (1993), and Ai and Cassou (1995) show that empirical techniques
which properly handle reverse causality concerns continue to support large contributions to output from
public capital.
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over the last 30 years can account for only a small portion of the productivity slowdown
that began in the early 1970s. In contrast, we show that the trend of increasing tax
rates in the U.S. economy offers a better explanation for the productivity slowdown in
the context of our model.
To perform our analysis, we embed a version of the empirical public capital model
used by Aschauer (1989), Munnell (1990), and others in an equilibrium framework
with an optimizing government.2 The optimizing framework is similar to one used by
Glomm and Ravikumar (1994). Our model differs from theirs in three fundamental
ways. First, both private and public capital stocks are long-lived. Second, labor supply
is endogenous, and third, the relevant stock of public capital for production is the per
capita (or per firm) quantity. Our motivation for each of these features is as follows:
By modeling capital as long-lasting, we are able t o capture the lengthy transitional
dynamics of an economy moving toward its balanced growth path. With endogenous
labor supply, the model can be used t o investigate changes in the growth rate of labor
productivity arising from changes in the capital stocks. Finally, by specifying public
capital as a per capita quantity, we link our model to previous empirical specifications in
the literature which typically do not include any explicit congestion
effect^.^ The model
is used t o explore the optimal transitional dynamics for an economy moving toward a
balanced growth path and t o quantify the effects of some hypothetical nonoptimal fiscal
policies on productivity g r ~ w t h . ~
'Early work by Kydland and Prescott (1977), Barro (1979), and Lucas and Stokey (1983) laid
the groundwork for evaluating government optimization problems. Most of the recent work has been
on applications to the Ramsey optimal-tax problem (e.g., Lucas [1990], Zhu [1992], Jones, Manuelli
and Rossi [1993], Chari, Christian0 and Icehoe [1994], and Cassou [1995]). Recently, Barro (1990) and
Glomm and Ravikumar (1994) have investigated the spending side of the government budget constraint.
3Specifying public capital as a per capita quantity incorporates an implicit congestion effect associated with the size of the population. This differs from the explicit congestion effect in Glomm and
Ravikumar (1994), where congestion is linked t o the size of the private capital stock.
4Some recent research that also investigates transitional dynamics in neoclassical models includes
King and Rebelo (1993) and Mulligan and Sala-i-Martin (1992).
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The remainder of this paper is organized as follows. Section 2 presents the model.
Section 3 describes the transitional dynamics of optimal fiscal policy. Section 4 describes
how we obtain parameter values to carry out our quantitative exercises. Section 5
presents the quantitative exercises, and section 6 concludes.
2. The Model
The model economy consists of a private sector that operates in competitive markets
and a benevolent government that solves a dynamic version of the Ramsey (1927) optimal tax problem. The private sector is typical of macroeconomic models with agents
behaving optimally, taking government policy as given. In formulating its policy, the
government takes into account the rational responses of the private sector. Our description of the economy proceeds in two steps and reflects this Stackelberg game hierarchy.
2.1. The Private Sector
The private sector consists of alarge but fixed number of households. Each household
is the owner of a single firm that produces output yt at time t according to the technology
where 0 < Ao, 0 < 8; for i = 1,2,3, and 81 f 82
f
B3 = 1.5 With this technology,
there are three factors of production: the per capita stock of private capital kt, the per
capita labor supply lt, and the per capita stock of public capital
The firm chooses
kt and lt, but takes kg,t as exogenously supplied by the government. Defining kg,t as a
per capita quantity ensures that there are no scale effects associated with the number of
firms. Output is also affected by ht, which is an index of knowledge outside the firm's
5Empirical research by Aschauer (1989), Munnell (1990), Ai and Cassou (1995), and others finds
support for a technology specification with 81 82 8 3 = 1.
+ +
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control that augments the productive capacity of labor. Following Romer (1986), it is
assumed that knowledge grows proportionally to, and as a by-product of, accumulated
-
private investment and research activities, such that ht = kt, where
Kt
is the average
capital stock across firms. With this specification, the assumption that firms view ht as
outside their control requires that there be a sufficiently large number of firms so that
no single firm has an impact on Et. Furthermore, because all firms are identical, zt = kt
in equilibrium. Thus, the condition
is imposed after firms choose their optimal labor and capital input
level^.^
It is assumed that firms operate in competitive markets and maximize profits
where wt denotes the real wage and
Since dl
rt
denotes the real rental rate on private capital.
+ d2+ d3 = 1, the firm earns an economic profit equal to public capital's share of
output. Our assumptions about firm ownership iinply that all housel~oldsreceive equal
amounts of t o t d profits.7 We assume that these profits are distributed t o households as
dividends and taxed as ordinary income. T h e market clearing prices for private capital
-
'The technology specification, the equilibrium condition ht = kt = k t , and the condition 91 +&+& =
1, imply constant returns to scale in the two reproducible factors kt and
Consequently, the model
exhibits endogenous growth. I<ocherlakota and Yi (1995) find evidence in U.S. d a t a supporting growth
models that emphasize public capital. See Barro and Sala-i-Martin (1992) for a review of modeling
structures that exhibit endogenous growth.
'It is possible to allow for different numbers of households and firms. However, as long as ownership
of firms is uniform across households, each household will realize exactly the same level of profits as
here.
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and labor inputs and the resulting firm profits are given by
Tt
=
B'y",
kt
w t = -@2yt
It
and
The infinitely-lived representative household chooses {ct, lt, it, kt+l : t
> 0)
t o max-
imize
subject t o
kt+l = ~ ~ k i - ~ i , 6 ko
, given,
where 0
< ,B < 1, 0 5 B, 1 < y , 0 < A1, and 0 < S 5 1. In this specification, ct denotes
private consumption a t time t, it is private investment, and rt is the income tax rate.
The household operates in competitive markets and takes government tax policy rt,
knowledge accumulation ht, and dividends rt as being determined outside of its control.
Three features of the household's problem warrant comment. First, the average
capital stock affects the marginal utility of leisure via the knowledge accumulation term.
This specification, which can be motivated by household production theory, ensures that
the supply of labor, It, remains stationary along the balanced growth path.' Second, the
parameter y controls the elasticity of household lamborsupply. As y becomes very large,
the level of labor supplied approaches one, and the model reduces t o one with a fixed
labor supply. Third, the law of motion for private capital given by (4) implies a nonlinear
relationship between current investment and next period's capital. When S = 1 and
'See Greenwood, Rogerson, and Wright (1994).
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A1 = 1, capital depreciates fully after one period, as in Glomm and Ravikumar (1994).
When 0
< S < 1, capital is long lasting. This nonlinear specification has been used by
Hercowitz and Sampson (1991) and can be viewed as reflecting adjustment costs as in
Lucas and Prescott (1971).
Using standard techniques, it can be shown that t h e household's decision rules are
given by
ct = (1 - ao)(l - rt)yt,
(5)
2.2. The Public Sector
The government chooses an optimal program of taxes and expenditures t o maximize
the discounted utility of the household. In addition t o public investment, government
expenditures include purchases of other goods and services, gt, which do not contribute
t o production or household utility. We model nonproductive expenditures as a constant
fraction
4
> 0 of total output, such that gt = 4yt, but
assume that the policymaker
views gt as exogenous. This specification is a simple way of ensuring that gt continues
t o represent a significant fraction of output in this growing economy.10
To finance expenditures, the government imposes a t a x on income a t t h e rate rt such
that
ig,t
+ gt
=
TtYt
(8)
' ~ appendix
n
showing the derivation of these decision rules and other analytical results in the paper
can be obtained from the authors upon request.
''Alternatively, we could introduce gt as an additively separable argument in the household utility
is constant in equilibrium.
function (3). In this case, we obtain the same result-that the ratio
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is the government budget constraint a t time t, where iglt represents public investment.
Public investment contributes to future public capital stocks according to the following
law of motion, which is analogous to (4):
1-6.6
kg,t+i = Aik,,, z,,,,
kg,o given.
The government's problem can be formalized as choosing {rt,ig,t,kg,t+1,ct, It, it, kt+l :
t 2 0), so as to maximize (3) subject to (4), ( 5 ) , ( G ) , (7), (8), and (9). Because the
model is analytically tractable, standard optimization procedures yield the following
optimal policy rules:
Zg,t
where a1 =
&.
= alyt,
(10)
Notice that the tax rate is constant over time and that it can
be decomposed into two parts, one for public iilvestmeilt igYtand one for nonproductive
expenditures gt.
3. Transitional Dynamics of Optimal Fiscal Policy
The model's tractable nature allows us to obtain closed-form expressions describing
the optimal transition path for an economy with initial conditions that lie off the balanced growth path. To characterize the trailsitional dynamics, we begin by computing
the optimal ratio of public t o private capital, R*, when the economy is in balanced
growth. The intuition for the transitional changes is straightforward. If the current
value of Rt =
% is less than the balanced growth ratio R*, then optimal policy would
call for an increase in Rt over time until R* is reached. On the other hand, if Rt is
greater than R*, then a decline in Rt over time would be consistent with optimal fiscal
policy.
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To derive an expression for R* as a function of the model's parameters, we combine
the household and government decisioil rules with the laws of motion for the two capital
stocks (4) and (9). Because there are two state variables, kt and kg,t, the decision rules
must be solved jointly t o obtain the equations that govern the optimal transition path
leading t o R* .
Substituting the optimal decision rules (6), (7), and ( l l ) , and the production equations (1) and (2), into (4) yields
Equation (12) is the equilibrium law of inotioil governing the evolution of private capital
when there is optimal behavior on the part of households, firms, and the government.
Similarly, ( l o ) , ( l l ) , (7), ( I ) , and (2) call be substituted into (9) t o yield the equilibrium
law of motion for public capital:
Dividing (13) by (12) gives
This equation implies that along the balanced growth path, that is, when
"*
= .,(1_ab1-4)
yields R* =
9
=
9,
which is constant. Making use of the expressions for ao, a l , and (11)
where T is the constant tax rate. By combining (12) and R*, the
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following expression for the per capita growth rate can be obtained:
l1
4. Calibration of the Model
In general, parameters are assigned values based on empirically observed features of
the U.S. economy. However, for some parameters, such as the output elasticity of public
capital 83, there is no general consensus regarding the appropriate value. Since 83 is
important for determining R*, we a.ttempt to remain objective by exploring a range
of values.12 We also explore a range of values for the parameter 6, which appears in
the laws of motion for the two 'capital stocks. In this case, the range is motivated by
the lack of empirical attention given to the nonlinear specification for the relationship
between current investment and next period's capital stock.
We choose baseline parameter values as follows: A discount factor of /3 = 0.962
implies that the real return on private assets along the balanced growth path is equal
to 4 percent. Following Greenwood, Hercowitz, and Huffman (1988), we set y = 1.60,
which implies that the intertemporal elasticity of substitution in labor supply l / ( y - 1)
is equal to 1.7. Although the share of output used to compensate workers has been
relatively constant over time, estimates of O2 are influenced by the way in which certain
types of income are apportioned between labor and capital. For example, proprietor's
income, indirect business taxes, and imputed services from consumer durables may affect
+
"To derive this result, we make use of the expression O1 + O2
03 = 1. Consequently, this is a
necessary condition for balanced growth in the model.
he range of direct empirical estimates for o3 a t 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 03 that are not statistically different from zero.
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estimated values of 82. The output elasticity of labor, O2 = 0.60, is chosen based on
empirical work by Christian0 (1988), Ai and Cassou (1995), and others and is close t o
the value of 0.58 used by King, Plosser, and Rebelo (1988). The value
6 = 0.17 implies
a ratio of nonproductive governme~ltspending t o output of 0.17, consistent with the
postwar U.S. average.
Most macroeconomic research employs a linear law of motion for capital accumulation. It is well known, however, that this specification does not yield closed-form
decision rules except in the special case of 100 percent depreciation. For this reason,
we employ the nonlinear form given in (4) and (9). Even in these nonlinear forms,
however, 6 controls the depreciation rate of existing capital.13 Using this specification,
Hercowitz and Sampson (1991) report a point estimate of 6 = 0.34, with a standard
deviation of 0.26, using annual d a t a on U.S. private capital from 1954 t o 1987. Given
the imprecise nature of the estimate, we explore a wide range of values for 6. With 83
set a t its baseline value (described below), we find that 6 = 0.10 provides a reasonable
fit of the U.S. time series of Rt =
% from 1925 t o 1992. This is the period for which
d a t a on public and private capital stocks are available.14 We also investigate values up
t o 6 = 1.0, which coincides with t h e value implicitly used by Glomm and Ravikumar
(1994).
We examine values for O3 in t h e range 0
5 O3 5 0.20. For each O3 in this range,
we define O1 = 1 - O2 - 83 t o maintain the necessary condition for balanced growth
in t h e model. Two combinations of O1 and O3 are of particular interest. The first is
.O1 = 0.277 and 83 = 0.123, which, together with 6 = 0.10, yield an optimal transition
path that is consistent with the U.S. time series of Rt =
2 over most of the sample
131n the nonlinear law of motion, 6 is most properly interpreted as the elasticity of the next period
capital stock with respect to current investment.
''The capital series are in 1987 dollars and were obtained from Fixed Reproducible Tangible Wealth
i n the United States, U.S. Department of Commerce (1993). The series for kg,t includes nonmilitary
government-owned equipment and structures. The series for kt includes privately owned equipment and
structures. The "capital input" measure of of the net stock was used for all capital data.
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period. This combination of parameters implies R* = 0.60, which is slightly higher
than the maximum value observed in postwar U.S. data. The second combination we
examine is O1 = 0.30 a.nd O3 = 0.10, which implies R* = 0.44. This ratio coincides with
the observed value a t the end of our sample in 1992. This case is important because it
shows that current levels of public capital in the U.S. economy can be consistent with
optimal fiscal policy, even when 133 is as large as 0.10.'~
T h e remaining parameters, Ao, Al, and B, affect the scaling of the model and were
calibrated using O1 = 0.271, O3 = 0.123, and 6 = 0.10. The value of B = 3.76 implies
that household labor supply 11 is approximately equal to 0.3 along the balanced-growth
path. Given a time endowment normalized t o one, this meails that households spend
approximately one-third of their discretionary time in market work. The constants
A. = 4.36 and A1 = 1.16 imply tha.t the ratio of private investment t o output is 0.15
and the steady-state growth rate of labor productivity is 2.77%. This growth rate
coincides with the U.S. average from 1947 t o 1969. Our decision to calibrate the growth
rate to this 23-year subsample of U.S. data is motivated by our interest in examining
the degree to which nonoptiinal fiscal policies can account for a productivity slowdown
of the magnitude observed in the U.S. economy during the early 1970s.
5. Policy Evaluation
In this section, we examine how well our model can account for the evolution of the
stock of public capital relative t o private capital in the U.S. economy over the last 70
years. Figure 1 shows the U.S. time series of Rt =
over the period 1925 to 1992. The
series, which is plotted as a dashed line in the figure, grew a t a rapid pace throughout
the 1930s before experiencing a temporary acceleration during World War 11. After the
I5If consumer durables are included in k t , then the ratio Rt =
0.31 and 83=0.09 imply R* = 0.37 in the calibration.
% in 1992 is 0.37. In this case, 81 =
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war, the ratio declined for a few years and then settled into a long, slow growth period
that peaked in the mid-1960s. Over the last 30 years, the ratio has displayed a generally
declining trend.
For comparison, figure 1 also plots the optimal transition paths implied by our
model for three different parameter settings. In general, the model predicts a rapid
initial growth in the ratio of public to private capital, followed by a leveling off as the
economy converges to the balanced growth ratio R*. Although the U.S. data do not
display this monotonicity, the model's optimal transition path with
83
= 0.123 and
S = .10 is generally consistent with the data up until about the mid-1960s, particularly
if one views the war years as being influenced by a temporary shock. When
83
= 0.10
and S = .lo, the optimal transition path lies below the U.S. data for most of the sample
period.
Figure 1 also shows the optimal transition path when
83
= 0.123 and S = 1.0.
Looking to the far right of the figure, we see that S has a quantitatively small impact on
the balanced growth ratio R * . ' ~Although S ha.s little effect on R*, it strongly influences
the length of time needed for the transition. As one would expect, higher levels of 6
lead to more rapid transitions. When S = 1.0, the transition occurs in a single jump
after the initid period. This illustrates a limitation of the Glomm and Ravikumar
(1994) model for analyzing transitional dynamics. In the policy analysis that follows,
we restrict our attention to the case of 6 = 0.10, since this yields a reasonable transition
path in comparison to U.S. data.
5.1. O p t i m a l Policy a n d t h e R e c e n t Decline in P u b l i c Capital
In recent years, many policymakers and researchers have voiced concern that the
decline in the ratio of public to private capital over the last 30 years is evidence that
161t can be shown that
> 0.
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the United States has been underinvesting in public capital.17 However, figure 1 shows
that this conclusion does not necessarily follow. In particular, a declining ratio of
public to private capital can be consistent with optimal fiscal policy, even when public
capital contributes in a significant way t o private output. When 93 = 0.10, the optimal
transition path in figure 1 lies below the U.S. time series of Rt =
% over the postwar
period. Thus, a decline in the U.S. ratio over this period might be interpreted as bringing
the economy closer to the optimal balanced growth ratio R*.
To explore the robustness of this result, consider figure 2, which shows the effect
of varying 93 on R*. AS public ca.pita.1 becomes more productive (93 increases), the
optimal ratio R* along the balanced-growth path increases rapidly. Figure 2 shows that
when 0
< 93 5 .lo, then R* 5
'
0.44. Note that 0.44 is the ra.tio observed a t the end
of the sample in 1992. Thus, when 0
< 93 < .lo, the model implies that an increase in
the ratio of public t o private capital from current levels is not called for. However, if
93
> 0.10, then figure 2 shows that R* > 0.44. In this case, the model implies that the
ratio of public to private capital should be increased.
It is important to note that our analysis does not resolve the debate over whether the
U.S. economy is underinvested in public capital because the optimal ratio R* depends
crucially on the size of g3, which is the subject of much uncertainty. However, our
model identifies some middle ground tlzat neither side of the public-capital debate has
formally recognized. Proponents of expanding public investment tend t o make their case
using empirical evidence that shows O3
> 0. This result, together with the observation
that the ratio of public to private capital has been declining over time, is often cited
as evidence of nonoptimal fiscal policy. In contrast, opponents of expanding public
investment tend to make their case by testing Ho : 83 = 0. Our analysis slzows that this
condition is much stronger than is needed to establish that the data do not call for a n
17see, for example, Economic Report of the President, 1994, p.43.
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increase in public investment. Even with 83 as high as 0.10, our model suggests that the
decline in the U.S. ratio of public to private capital over the last 30 years is no cause for
concern. This argument is made even stronger by the fact that empirical estimates of
83 tend t o be very imprecise. For example, Finn (1993) estimates the output elasticity
of public highway capital t o be 0.16. However, the 95 percent coilfidence interval on
this estimate ranges from a low of 0.001 t o a high of 0.32. Thus, even for relatively
large point estimates of 83, the data do not necessarily imply that the "true" value of
83 would call for an increase in public investment.
5.2. P u b l i c C a p i t a l a n d t h e P r o d u c t i v i t y Slowdown
The debate on the productive effects of public capital is often linked t o discussions
regarding the slowdown in the growth rate of U.S. labor productivity that began in
the early 1970s. Some researchers argue that underinvestment in public capital is a t
~ this section, we take up this issue by
least partially responsible for the s l o ~ d o w n . 'In
examining how some hypothetical nonoptimal fiscal policies can affect the growth rate
of labor productivity within the context of our model.
For our first experiment, we investigate the consequences of a nonoptimal public
investment policy. In the previous section, we pointed out that the observed decline in
the U.S. ratio of public t o private capital can be reconciled with optimal fiscal policy
when d3 5 0.10. However, if the optimal transition path from 1925 t o 1992 is more
appropriately described by the case with 83 = 0.123 in figure 1, then the recent decline
in the U.S. ratio would not be optimal. For this experiment, we adopt the latter view
and set 83 = 0.123 (and 81 = .277), which implies R* = .GO. Next, as an input to
the model, we construct an exogenous series for public investment, iglt,such that the
resulting time path for Rt =
% coincides with the path observed in the U.S. economy
"see, for example, Aschauer (1993) and Munnell (1990).
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from 1947 t o 1992. The constructed series for Rt is shown as a crossed line in figure
3a, while the solid line shows the optimal transitioil path computed earlier in figure 1.
Since the constructed Rt lies below the optillla1 tra.nsition pa.th leading to R* = 0.60,
and tends t o move further awa.y over time, we interpret this experiment as capturing
the type of nonoptimal public investment policy that is often cited as a possible cause
of the U.S. productivity slowdown.
For this experiment, the t a x rate is held constant a t the optimal level implied by
( l l ) , and nonproductive government expenditures, g t , are determined as a residual such
that the government's budget constraint (8) is satisfied each period.lg In this way,
we isolate the effect of a declining public capital ratio on productivity, holding other
important policy variables, such as tax rates, constant. Finally, we assume that the
private sector reacts optima.lly t o government policy, according to the decision rules ( 5 ) ,
(61, and (7).
Figure 3b displays the results of this experiment. The crossed line shows the growth
trend of labor productivity in the model, given the nonoptimal public investment policy.
T h e solid line shows labor productivity when public investment policy is optimal, that
is, when Rt follows the optimal transition path leading t o R*. The dashed line shows
U.S. labor productivity from 1947 to 1992. 111 comparison t o the optimad policy case,
the nonoptimal policy produces a mild productivity slowdown beginning around 1970.
Notice, however, that this slowdowil is much less pronounced than the one observed for
the U.S. economy. This experiment shows that a nonoptimal public investment policy
of the type that might be interpreted as reflecting U.S. experience over the last 30 years
can account for only a small portion of the productivity slowdown. This suggests that
other forces may have contributed to the slowdown. One alternative, which can be
IgThe optimal tax rate r * is computed from (11) using O3 = 0.123, 6 = 0.10, and 4 = 0.17. Nonproductive expenditures are then given by gt = r ' y t Since gt is determined as a residual for this
experiment, the ratio
is no longer constant.
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investigated using the same methodology, represents ailother type of nonoptimal fiscal
policy, namely, increasiilg tax rates.
For the secoitd experiment, we introduce an exogenous series of tax rates, rt, that
coincides with an average tax rate series for the U.S.
Because this series is
not constant, but displays an increasing trend over time, we interpret it as nonoptimal.
To isolate the effect of this nonoptimal tax policy, we construct an exogenous series
for public investment, igtt,such that the resulting series for Rt =
% generated by the
model follows the optimal transition path leading to R* = 0.60. As before, the private
sector reacts optimally and the level of nonproductive expenditures gt is determined as
a residual such that the government budget constraint is satisfied each period.
The results of the second experiment are displayed i n figures 4a and 4b. Figure 4b
shows that a policy of nonoptimal tax rates call also generate a productivity slowdown.
The slowdown is much more severe than in the first experiment, however. The key
difference between the two exercises is that in the first experiment, the government
misallocates resources between ig,t and gt, while tax revenue as a fraction of total output
remains constant. In the second experiment, the sha.re of total resources claimed by the
government increases over time.
Figure 4b shows that labor productivity in the model displays an abrupt change in
trend around 1970 that is strikingly similar to the trend shift in U.S. labor productivity
that occurred a t about the same time. The cause of this trend shift in the model can
be traced to the period of sharply increasing average tax rates in the late 1960s and
early 1970s (see figure 4a). This experiment shows that the existence of a productivity
slowdown need not imply that public investment policy is nonoptimal.
2 0 ~ computed
e
the average tax rate series for the U.S. economy by dividing total federal, state, and
local government receipts for each year (Citibase series GGFR+GGSR+GGFSIN+GGSSIN) by GDP.
This approach yields an average tax rate that is roughly consistent with the model's use of a production
tax to finance all government expenditures. The resulting tax rate series displays an upward trend
which is very similar to that observed for the average marginal tax rate on labor income estimated by
Barro and Sahasakul (1986).
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As a final experiment, we introduce both types of nonoptimal fiscal policy into
the model. T h e results of this exercise are summarized in figures 5a and 5b. As one
might expect, the productivity slowdown in the model now becomes even more severe.
This occurs because an increasing fraction of total resources are now being devoted t o
nonproductive public expenditures gt. Interestingly, the simulated productivity trend
from the model provides a very close match t o the U.S. productivity trend. Table 1
provides a quantitative comparison of the productivity effects in each of the three policy
experiments.
To summarize, our experiments show that a ilonoptiinal public investment policy
does not, by itself, provide a coilvinciilg explailation for the U.S. productivity slowdown.
However, it may have been a contributiilg factor, together with the trend toward increasing tax rates. Finally, we note that ma.ny other explana.tioi~shave been put forth
t o help explain t h e U.S. productivity slowdown. Some of the a1terna.tive hypotheses
include: (1) a return t o "normal" productivity growth from the unsustainably high
growth rates experienced after the Great Depression and World War 11; (2) changes in
demographic factors that have tended t o reduce the quality of the labor force; (3) a falloff in the rate of research and development spending; (4) increased costs of complying
with governmeilt regulations (such as mandated pollution control expenditures); and
(5) increases in energy costs due to oil price
6. Conclusion
This paper showed that optimal transitional dynamics in a simple endogenous growth
model can account for much of the behavior of the stock of public capital in the U.S.
economy over the last 70 years. Moreover, we showed that the observed decline in
"See Munnell (1990), Tatom (1991), Aschauer (1993), and the references cited therein for a more
detailed discussion of these alternative hypotheses.
clevelandfed.org/research/workpaper/1995/wp9509.pdf
the U.S. ratio of public t o private capital since the mid-1960s might be interpreted
as a movement toward the optimal balanced growth ratio, even for output elasticities
as high as 0.10. Finally, we found that a nonoptimal public investment policy of the
type consistent with U.S. d a t a does not have much impact on the growth rate of labor
productivity in our model, suggesting that other explanations for the U.S. productivity
slowdown should be considered.
clevelandfed.org/research/workpaper/1995/wp9509.pdf
References
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Munnell, ed., Is There a Shortfall in Public Ca.pital Investment?, Proceedings of
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forthcoming in Applied Economics.
Arrow, Kenneth J. and Mordecai I<urz, 1970, Public investment, the rate of return,
and optimal fiscal policy (Baltimore: Johns Hopkins Press).
Aschauer, David A., 1989, Is public expenditure productive?, Journal of Monetary
Economics 23, 177-200.
, 1993, Public capital and economic growth, Jerome Levy Economics
Institute,
Public Policy Brief No. 4.
Barro, Robert J., 1979, On the determination of public debt, Jourilal of Political
Economy 87, 940-971.
, 1990, Government spending in a simple model of endogenous growth, Journal
of Political Ecoilomy 98, S103-125.
Barro, Robert J. and Chaipat Sahasakul, 1986, Average marginal tax rates from social
security and the individual income tax, Journal of Busiiless 59, 555-566.
Barro, Robert J. and Xavier Sala-i-Martin, 1992, Public finance in models of economic
growth, Review of Economic Studies 89, 645-661.
Cassou, Steven P., 1995, Optimal tax rules in a dynamic stochastic economy with
capital, Journal of Ecoilomic Dyilainics and Control 19, 1165-1197.
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Chari, V.V., Lawrence J. Christiano, and Patrick J. Icehoe, 1994, Optimal fiscal policy
in a business cycle model, Journal of Political Economy 102, 617-652.
Christiano, Lawrence J., 1988, Why does inventory investment fluctuate so much?,
Journal of Monetary Econonlics 21, 247-280.
Finn, Mary, 1993, Is all government capital productive?, Federal Reserve Bank of
Richmond, Economic Quarterly 79, 53-80.
Glomm, Gerhard and B. Ravikumar, 1994, Public investment in infrastructure in a
simple growth model, Journal of Economic Dynamics and Control 18, 1173-1187.
Greenwood, Jeremy, Zvi Hercowitz, and Gregory W. Huffman, 1988, Investment, capacity utilization, and the real business cycle, American Economic Review 78, 402-417.
Greenwood, Jeremy, Richard Rogerson, and Randall Wright, 1994, Household production in real business cycle theory, in T. F. Cooley, ed., Frontiers of Business Cycle
Research (Princeton, NJ: Princeton University Press).
Hercowitz, Zvi, and Michael Sampson, 1991, Output growth, the real wage, and employment fluctuations, American Economic Review 81, 1215-1237.
Holtz-Eakin, Douglas, 1992, Public-sector capital and the productivity puzzle, NBER
working paper 4122.
Jones, Larry E., Rodolfo Manuelli, and Peter E. Rossi, 1993, Optimal taxation in
models of endogenous growth, Journal of Political Economy 101, 485-517.
Jorgenson, Dale and Martin A. Sullivan, 1981, Inflation and corporate capital recovery,
in C.R. Hulten, ed., Depreciation, inflation, and the taxation of income from capital
(Washington, DC: Urban Institute Press).
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King, Robert G., Charles I. Plosser, and Sergio T. Rebelo, 1988, Production, growth
and business cycles I. The basic neoclassical model, Journal of Monetary Economics
21, 195-232.
King, Robert G. and Sergio T . Rebelo, 1993, Transitional dynamics and economic
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Economics and Statistics, forthcoming.
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inconsistency of optimal plans, Jourilal of Political Economy85, 473-491.
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Econometrica 39, 659-681.
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in an economy without capital, Journal of Monetary Economics 12, 55-94.
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Mulligan, Casey B. and Xavier Sala-i-Martin, 1992, Transitional dynamics in two-sector
models of endogenous growth, NBER working paper 3986.
Munnell, Alicia H., 1990, Why has productivity growth declined?: Productivity and
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Romer, Paul, 1986, Increasing returns and long-run growth, Journal of Political Economy 94, 1002-1037.
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Table 1: Annual Productivity Growth Rates
/ 1947-1969
U.S. Data
Experiment
# 1:
Suboptimal Public Investment
Experiment #2:
Suboptimal Tax Policy
Experiment #3:
Joint Suboptimal Policy
Source: Authors' calculatioils,
1970-1992
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FIG 1: RATIO O F PUBLIC TO PRIVATE CAPITAL
S0
z
/
-
I
-
0 3 = 0 . 1 2 3 . 6 = 1.0
\
0
\
\
/ - - -
,
/
/
- - .'-
0, = 0.123. 6 = 0.10
\
U.S. rotio k,Jk,
\
\
'\
-
- _ /
O3 = 0.100, 6 = 0.10
-
0
20
v
-
0
0
9
%25
1930
1935
1940
1945
1950
1955
1960
Yeor
SOURCE: Authors' calculations.
1965
1970
1975
1980
1985
1990
1995
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Figure 3a: EXPERIMENT: A DECLINING PUBLIC CAPITAL RATIO
647
Observed Declining Ratio
1952
1957
1962
1967
1972
1977
1982
Year
Figure 3b: EFFECT ON LABOR PRODUCTIVITY
-
U.S. Productivity GNP/LHOURS. 1947- 1992
Model Productivity w/ Optimal Ratio
Model Productivity w/ Declining Ratio
Year
SOURCE: Authors' calculations.
1987
1992
1997
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Figure 4 0 : EXPERIMENT: AN INCREASING TAX RATE
0
-
0
9
Optimal Tax Rate from Model
Observed Increasing Tax Rate
20
0
2
-
0
t-
t_, o
J0
rr
X
0
I-
*0
0
20
0
2
n
0
..
N
0
0
N
-
0
2
h47
1952
1957
1962
1967
1972
1977
1982
1987
1992
1997
1987
1992
1997
Year
Figure 4b: EFFECT ON LABOR PRODUCTIVITY
-
U.S. Productivity GNP/LHOURS. 1947- 1992
Model Productivity w/ Optimal Tax Rate
Model Productivity w/ Increasing Tax Rate
=!
9947
1952
1957
1962
1967
1972
Year
SOURCE:
A u t h o r s ' calculations.
1977
1982
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Figure 50: EXPERIMENT: A DECLINING RATIO AND AN INCREASING TAX RATE
-
h47
Observed Declining Ratio
Optimal Tax Rate from Model
Observed Increasing Tax Rate
1952
1957
1962
1967
1972
1977
1982
1987
1992
1997
1987
1992
1997
Year
Figure 5b: EFFECT ON LABOR PRODUCTIVIN
-
U.S. Productivity GNP/LHOURS. 1947-1992
Model Productivity w/ Optimal Policy
Model Productivity w/ Observed Policy
o!
7947
1952
1957
1962
1967
1972
Year
SOURCE: Authors' calculations.
1977
1982
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Appendix
Firm Optimization
The firm's optimization problem is straightforward. They choose lt and kt to maximize
This implies
and
nt =
Yt
- 02%
-
Q1yt = (1 - 01 - Q2)yt.
Household Optimization
Write the problem as choosillg {ct, It, it, kt+l : t 2 0) to maximize
subject to
ct
+
it = (1 - rt)(wtlt
+ ~ t k t+ nt),
Using the results from the firm's optiinization problem and the production function, the
Lagrangian for this problem can be written as
L(.) =
5~'
{log (ct - ~htl:)
t=O
+
~t
[(1
- rt)(wtlt
+ r t k + nt)
- ct -
t+l
eY]).
t
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The first-order conditions are
w.1 = ( 1
dX t
-
rt)(wtlt
+ rtkt + n t ) -
Substituting (14) into (15) and using rt =
9and wt
ct
=
-
it = 0.
(17)
yields
Using ( 1 ) and (2) and solving ( 1 8 ) for lt yields
To find the other decisioll rules we use the method of undetermined coefficients. We
guess the functional forms
where a0 and bo are collstailts to be determined. Substituting these into (16) and solving
for a0 gives
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We also need to verify that our guess was correct by verifying that bo is, in fact, a
constant. To do this, we use (14) and (20) t o obtain
Substituting this expression into (17) and using (18), we can solve for bo:
Substituting the expression for bo into (21) and combining with (18) gives
We can interpret a0 as the marginal propensity to save out of after-tax income.
Government Optimization
The government views gt as exogenous and does not include it as part of its optimization
decision. The government problem is t o choose { T ~iglt,
, kg,t+t,ct,lt,it, kt+l : t
maximize
subject to
> 0)
to
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It will be useful to reduce the number of constraints by eliminating some of the variables.
We begin by writing (6) in two forms:
Substitute (22) into (7) t o eliminate r t , yielding
Next, substitute this expressioil for lt, together with (4),into (1) t o obtain
where
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The next step employs (8), (23), (4), (9), and (24) to obtain the following version of the
government budget constraint:
where gt is viewed as exogenous by the policymaker. An expression for the argument of
the household utility function can be obtained using (5), (20), (22), and (4). The result
is
We now can write the Lagrangian for the government problem as
The first-order conditions are
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To find the decision rules, we again use the method of undetermined coefficients. We
guess the functional forms
where
a1
and bl are constants to be determined. Substituting these expressions into
( 2 7 ) and solving for
a1
gives
a1
=
P 603
1 -P(1-6).
To find the optimal decision rule for rt , we substitute the expression for
a1
into (8) and
make use of gt = 4yt to obtain
rt
=
a1
+ 4.
We verify that (29) and ( 3 0 ) are correct by sllowing that bl is in fact a constant. To do
this, we substitute the optimal tax rate into (6) to obtain
Substituting this expression, together with ( 2 9 ) and (30), into ( 2 6 ) and solving for b1
yields
!h
b1 =
7
+
Peo6 - ( 1 - a1
-
4) [ I -
P ( 1 - 611
1 - P(1 - 6 )
Since this is constant, our guess is confirmed.
Derivation of t h e Equilibrium Laws of M o t i o n for P r i v a t e a n d P u b l i c C a p i t a l
The laws of motion ( 1 2 ) and ( 1 3 ) are relatively straightforward to derive. The
only tricky part is to first obtain an alternate expression for the production function.
clevelandfed.org/research/workpaper/1995/wp9509.pdf
Substituting (7) and (2) into (1) gives
Substituting (G), ( l l ) , and (31) into (4) and rearranging gives
Similarly, substituting (29) and (31) into (9) and rearranging gives
Making use of (12) and R* yields the following expression for the per capita growth
rate:
@FedFRASER