Download W(h)ither the Stock of Public Capital?

DISCUSSION PAPERS IN ECONOMICS
Working Paper No. 99-03
W(h)ither the Stock of Public Capital?
Kenneth R. Beauchemin
Department of Economics, University of Colorado at Boulder
Boulder, Colorado
January 1999
Center for Economic Analysis
Department of Economics
University of Colorado at Boulder
Boulder, Colorado 80309
© 1999 Kenneth R. Beauchemin
Kenneth R. Beaucheminl
Revised: January 1999
1Department
of Econonucs, University of Colorado at Boulder, Boulder, CO 80309.
Phone: (303) 492-2651. Fa-x: (303) 492-8960. E-~Iail: [email protected].
I am grateful to Martin Boileau, Todd Clark, Dean Corbae, JoAnne Feeney,Beth Ingram,
Narayana Kocherlakota. GeneSavin, Kei-~Iu Yi and Chuck Whiteman for helpful comments
and suggestions. An earlier ,-ersionof this paper was presentedat the 1994annual meeting of
the Society for Economic Dynamics and Control and the 1996 Monetary Policy Roundtable
meeting hosted by the Federal ReserveBank of Kansas City. I thank the participants at
these gatherings for useful feedback.
Abstract
This paper maintains that the commonly used measure of the aggregate stock of public capital is conceptually divergent from the "true"' public capital input in private
production technologies. Consequently, the published measure has the potential to
misrepresent the historic growth profile of the public capital input and the pro duct ivity of government purchases in general. As a test of this hypothesis: the identifying
assumption of a standard stochastic growth model is combined with observations on
output, consumption, labor hOUl-S
and government purchases to deduce public capital
paths that are mutually consistent with observed flows and economic theory. It is
found that the inferred series grows more rapidl~' than the published series during
the post-1973 period thereby pro\iding evidence that government under-investment
is not an important source of declining U.S. producti,.ity
Key words: public capital; productivity; gro,\-th
JEL classification: E62, 040.
growth.
1
Introduction
In recent years, the stock of publicly-owned capital -as measured by the national
income accounts -has declined relative to the private capital stock. Since 1970. the
ratio of nonmilitary public capital to private capital (including consumer durables),
declined from a peak value of .284 to approximately .237 in 1993. Including military
capital, the ratio attains a peak of .365 in 1963 and declines to .257 in 1993. (see
figure 1). Similar observations formed the basis for an extensive research program
statisti<;:ally relating the diminishing rate of public capital formation to the slowdown in productivity
growth that began in the early- to mid-1970s.1,2 This paper,
while accepting, a priori, the notion that some portion of government purchases are
ultimately useful in private production activities, questions the validity of using the
measured stock of public capital to draw inferences regarding the productivity growth
slowdown and as a guide to public policy.
Although measurementdifficulties exist for both privately and publicly-owned capital, important differencesmake assessingthe public capital stock inherently more
problematic.
The primary difference is conceptual.
How does one decide on the
proper fraction of current government purchases to be capitalized as an input to private production technologies? Although it is hard to deny the complementary nature
of private capital and "core infrastructure" items such as highways and streets, \vater
systems and mass transit facilities, the link between private production and pub-
lic capital structures such as hospitals, courthouses.civic centers, and conser,-ation
1Aschauer (1989a, 1989b) are early references to what became the 'public
literature.
capital hypothesis'
This work is surveyed by Gramlich (1994). This article also contains a good description
of the measured stock of public capital and its components.
2United States capital stock data are published by the Bureau of Economic Analysis in Fixed
Reproducible Tangible Wealth in the United States. The series are updated annually in the August
Survey of Current Business.
1
facilities is less clear. These ambiguities imply that the measured stock of public
capital is potentially "too large."
Alternatively,
some components of government
..consumption" expenditures conceivably expand future production possibilities and,
as such, are part of a broader and unmeasured concept of public capital. For exampIe, the public education expense of a teacher's salary is not capitalized even though
it augments the stock of human capital and productive capacity suggesting that the
measured stock of public capital is potentially "too small." These arguments indicate
a potential rift between an economic theorist's conception of the public capital input
and the vie" reflected by the official measures.:!
This paper adopts the extreme vie\\" that the productivity-enhancing
stock of pub-
lic capital is not observed. To study the interaction between public investment and
productivity.
an alternative measurement method is proposed that combines observ-
able time series with the identifying assumptions implied by a standard stochastic
growth model to deducea path for the public capital input. In this sense, the study
follows in the tradition of growth accounting due to Solow (1957)
In its simplest
form, growth accounting combines time-series data on inputs to production with a
neoclassical. constant returns to scale production function to infer the history of to-
tal factor productivity. This deducedseries is then interpreted as the contribution
of technologic~l progress to economic growth.
For the purposes of this study, this
simple gro\\"th accounting is inadequate because public capital is not only taken to
be productive, but also unobserved. Clearly, a richer apparatus is required to simul-
taneouslyidentify the growth paths of both total factor productivity and productive
3Gramlich (1994) points out two additional problems associated with the measure of public
capital in relation to private capital. First, the servicesof public capital are not sold in markets
making it difficult to value public assetsaccording to the future stream of benefits that they produce.
Second, once purchased,public capital items are rarely re-sold. As a consequence,economic rates
of depreciation are almost never directly observedso it is difficult to ascertain service lives.
2
The plan of the paper is as follows. Section 2 sets out a modified version of a
standard stochastic grov.rth model that includes a go'\-ernment that purchases consumption and investment. goods The model \\ith indivisible labor is also discussed
Section 3 describes the implememation of the procedure and Section 4 presents the
results and their implications for aggregate labor productivity.
2
The Model
Section 5 concludes.
Environment
This section introduces a standard stochastic groVv"thmodel modified to include a
government that purchases goods from the private sector that can be used for either
investment or consumption pUl'pO~es.The first-order conditions of this model provide
the restrictions necessary to identif\- the gro\\'th path of public capital. The measurement procedure \vill be executed under both the standard labor market institution
with a finely divisible labor suppl~' and the indivisible labor setup of Hansen (1985)
and Rogerson (1988).4
2.1 The Standard Model
Consider an economy inhabited by an infinitely-lived
representative agent and a
government. The agent owns a constant returns to scale production technology
Yt = At
that transforms the labor and capital inputs into a single output good, Yt, where
nt is the fraction of period-t spent \vorking, /t'ptrepresents the beginning-of-period-t
stock of private capital and kgt ~imilarly denotes the stock of public capital.
The
4Braun and McGratten (1993) ana Baxter and King (1993) also introduce public inputs to
standard dynamic general equilibrium modelsto assessthe quantitative impacts of assorted policy
experiments.
4
:omplements
exogenous shock to total factor productivity is given by .-it. The parameters a and
b are each contained in (0,1); the former gives the share of output earned by capital
(broadly defined), and the latter attaches relative weights to the two types of capital.
The parameter p E [-1,00)
governs the substitutability
bet\\-een private and public
capital. This specification allows the model to nest a continuum of economies which
are distinguished by the technological relationship between the two capital types.
If p
-1, public and private capital are perfect substitutes:
they become perfect
in the limit as p becomes infinite.
Output in this environment is divided between private consumption, private investment, and government purchases:
Yt = Ct+ it + gt
The one-period ahead value of private capital is the sum of the current undepreciated
quantity and the current flow of private investment
kp,t+l
kpt(1
fJp) + it
(2)
where 8p is the rate of depreciation on private capital.5
During period t. the government spends 9t which is financed contemporaneously
with lump-sum ta.--::es
levied on households. In a fashion similar to the private capital
accUllrulation technology, government purchases are apportioned between consurnption and investment uses so that
k9,t+l = kgt(1 -8g) + O"t9t
(3)
5In creating the correspondencebetweenthe measurementsthat are available for the U.S. economy and the model variables, investment (i) is defined as the sum of businessfixed investment and
householdinvestment. Household investmentcombinesthe purchase of consumer durables and new
homes. Since the national income accountsdo not value the flow of senices arising from household
durables, it is imputed by assuming that the flow is proportional to the deterministic steady state
interest rate plus the rate of depreciation of privately-owned capital (tp). The flow is then added to
the published consumption-of-servicesseries. Seethe appendLxfor details.
5
Lhe
where O"tis the fraction of period-t government expenditures devoted to building
the productive public capital stock and 6g is the depreciation rate of public capital.
As seen belo,v. neither type of government expenditure affects the utility of private
agents.6
representative agent maximizes expected lifetime utility given by
(4)
where /3 is the subjective discount factor of the agent and ()t is the period-t realization
of an exogenous shock to preferences. Utility in anyone period depends positively on
consumption and leisure; the time endowment in each period is normalized to one.
Changes in {}t represent changes in the agent's willingness to substitute market activities (measured by consumption) for non-market activities (measured by leisure). An
increase in (}t, for example, is strictly interpreted as a willingness of the representative
agent to work longer hours which subsequently drives down the marginal product of
labor. A sustained increase in the gro\vth rate of (}t, therefore, results in a period of
slower labor productivity gro\vth.
In eachperiod t, it is assumedthat the agentknows the history of the state given
by
Ot
{As,
(}s, 0" s, kps,
kgB, 9s
S .$: t}
In this environment, the agent faces two external risks to the rate of return on private
capital. First. the rate of return is affected directly by the total factor productivity
(technology) shock At. Positive movements in At raise the marginal products of all
inputs uniformly. Second,the rate of return is affectedby movementsin the amount
of productive public capital available (kgt). A public investment boom raises the
marginal productivity of private capital and labor while driving down the marginal
UThis restriction could be rela.xed without altering the analysis provided that government pur-:hases
enter the momentary utilit}, function in an additively separable fashion,
6
1
product of public capital. Both shocks influence the optimal behavior of the agent in
the same way: positi,-e shocks encourage work and discourage present consumption
in favor of private capital accumulation- Negative shocks naturally have the opposite
influence.
Since there are no distortions or externalities produced by this economy, the allocation that solves the centralized planning problem goes through as the competitive
equilibrium. 7 This problem is written as follows:
00
max Eo L {3t[()t log Ct+ log (1 -nt)]
t=O
Sot
Ct
-O"t) gt + kp,t+l + kg,t-r-l = (1 -lip)
kpt
1 -8g)
kgt
Yt
given (1), (2), kpO> O. kgo > 0 and {gt}~o.
The solution to this problem must satisfy equation (1) and the following first-order
conditions:
Ct/ (}t
1-
~ = /3Et~
Ct
Ct+l
{ 1 -Dp
= (1 -a) ~
nt
(5)
nt
(~ ) [1 + (~ ) </>-;:1
]
-1
+ Q
kp,t+l
Ct+ kp,t+l = Yt + (1
1 -b
{jp) kpt -gt
(6)
(7)
where
~
c/>t+l -k
.
p,t+l
Equation (5) is the intratemporal efficiency condition equating the marginal rate
of substitution between consumption and leisure to the marginal product of labor.
7The assumption of constant returns to scale coupled \vith the agent's role as a competitive
profit-maximizing firm implies a portion of private output, equal in value to the stock of public
capital, multiplied by its marginal product, is not distributed as income to the private factors of
production. In the specifiedenvironment, the integrity of the competitive equilibrium is maintained
by implicitly assuming that the governmentextracts a lump sum payment from the firm equal to
this amount to maintain the zero-profit condition.
7
ences.
Equation (6) is the intertemporal efficiency condition equating expected discounted
marginal utility of consumption acrosstime periods. and equation (7) is the resource
constraint.
2.2 The Model with Indivisible
Labor
It is now apparent from the intertemporal efficiency condition (6) that the time
path of the unobserved preference shock, ()t, identified by the intratemporal efficiency
condition (5), plays a role in determining the intertemporal allocation of resources.
Consequently, the measurement of public capital is not independent of the institutional nature of the labor market" The popular alternative to the labor market structure embodied above, is the indi,isible labor mod~l of Hansen (1985) and Rogerson
(1988) in which volatility in hour5 \\orked occurs through employment variation rather
than the continuous adjustment of work effort. Since it is likely that \"ariation in ac-
tual aggregatelabor hours is generatedby a combination of hours and employment
adjustment, the measurement procedures are performed under both institutions.
The model with indivisible labor is obtained with a: simple substitution of preferIf n. is the constant fraction of the time endowment that is ~'orked with some
probability 7rt > 0, then per-capita hours worked is nt = 1itn. The pre,.ious preference
specification (4) is then replaced b~'
Eof y lOtlog Ct '+
t=O
resulting in the intratemporal efficiency condition
Ct/Ot
(1 -0:) ~
ii/ log (1 -ii)
nt
All other specifications and necessary conditions are unchanged from the standard
economy.
8
where'Y is a p x 1 vector of parameters. Unlike preference and technology parameters.
economic theory and independent evidence provides no guidance in selecting these
values. They must, however, be chosen to satisfy the first-order conditions (5) (7).
Equivalently, the elements of 1 are assigned so that a sample analog of the
orthogonality condition implied by the stochastic Euler equation (6) is satisfied:
T-l
L (Xt+l -1) Zt = 0,
Zt E Slt
(9)
t=l
where
Xt+l -
1 -8p
b )
1+ ( ~
°t-'-l
Yt+l
+ Q; -,
kp,t+l
-1
-p
J
and Zt is a vector of instruments contained in the agent's information set Slt and T is
the numberof data observations. It is assumedthat there are as many instruments as
there are parameters in the government's policy function so that (9) represents apxp
system of equations with,
as the unique solution. In other words, the procedure for
assigning ,-alues to the parameters of f is the exactly-identified version of Hansen's
(1982) generalized method-of-moments estimator. The parameter vector is computed
using annual data from the period 1956-93 (T = 38).8
The government policy rule (8) is further restricted so that f is a linear combination
of elements in the agent's information set describing the current state:
f (Ot;,)
= Wt 'Y
(10)
where Wt is a (1 x p) vector of variables observed in period t and f is a (p x 1) vector
of parameters that satisfies the sample orthogonality condition (9). To facilitate computation, only the variables in Ot that do not need to be inferred from the procedure
itself appear in the vector Wt With this restriction in mind~ it is assumed that
Wt
1~~(}
,
,
Yt
, t
Yt
1!A description of the data is contained in the appendix.
10
so that p = 4 with the corresponding parameter vector")' = (")'°"')'1"')'2"')'3)' Given
that the instruments need only be observed by the agent in period t, the vector of
instruments is assumed to be identical to the vector of explanatory state variables:
Zt
Wt
It should be understood that there is no model of government behavior intended
the function (10) is merely descriptive of the government:s past behavior.
Viewed from this perspective, the parameter values comprising I do not have a strict
economic interpretation
Calibration
To complete the task of identifying the shocks, it is necessaryto calibrate the model
economy. The most straightforward settings are those for a and /3. In calibrating
capital's share of income, it must be recognized that the flow of income deriving from
the stock of public capital is not included in the official measure of U.S. national
income. Furthermore. the value of the stock must be known prior to computing its
income flow
Given that this is not possible. a value of a
36 is chosen following
the work of Kydland and Prescott (1982), Hansen (1985) and Prescott (1986) in
which income from public capital is not imputed.
Labor's share is the remaining
1 -Q = .64.9 The subjective discount factor is chosen to be /3
1.03-1 implying a
deterministic steady-state real interest rate of 3.0 percent.10
In setting the remaining technology parameters, band p, it is useful to view the
measurement procedure as a composition of level effects and growth effects. Once the
initial stock of public capital is determined, a complete sequence of period-to-period
growth rates are sufficient to characterize the imputed time path of public capital.
9The results are not significantly impacted by the Q = .40 setting suggested by Cooley and
Prescott (1995) in accounting for income generated by (published measures of) public capital.
!UAgain, reasonable adjustments of this parallleter leave the general results and conclusions intact.
11
5
Conclusion
This paper began ,vith the assumption that publicly provided capital augments
private production possibilities and also plays a quantitati\"ely significant role in economic growth. It also maintains that the existing measures of public capital do not
accurately reflect the economic theorist's view of the true public capital input, and
therefore, may not be represemati\e of the true input's historic time profile. It is
shown that a time-series for public capital that is mutually consistent with a standard stochastic gro"--th model and the data on flows of aggregate output, consump-
tion. governmentpurchasesand labor hours, exhibits a markedly different historical
pattern than its measured counterpart. In particular, the \iew that public underinvestment is partially responsible for the slowdown in labor productivity
growth is
not supported
It is important
to bear in mind that the results depend on the identifying as-
sumption of a neoclassical gro"lh model.
Given the widespread application and
understanding of this framework. the results serve as a benchmark from which other
investigations can be launched. The intent of the research is not to replace traditional
accounting methods. Different models will generate different numbers comprising the
inferred history of the public capital input; the issue is whether these competing
modelswill changethe generalshapeof the time profile or the basic conclusionscontained herein. It is hoped that the results of this study are sufficiently compelling to
generate future research along these lines and to inform accounting-based economic
measurementtechniques.
17
18
Appendix
This appendix contains the sources of the raw data and definitions of the variables
used in the study. Data used are annual, real aggregate data of the United States
for the period 1956 to 1993 unless otherwise indicated. In cases where the raw flow
data are published at a monthly or quarterly frequency: the values are averagedto
an annual frequency.
The data on output, consumption,and governmentpurchasesare constructed from
the following Citibase variables (Source: National Income and Product Accounts,
Table 1.2)
la. GC.!~:-Q(personal consumption expenditures, nondurable goods)
2a. GC SQ (personal consumption expenditures, services)
3a. GC DQ (personal consumption expenditures, durable goods)
4a. G P I Q (gross private domestic investment.)
5a. GGEQ (government purchases of goods and services)
Series la -5a are converted into per capita quantities using the total labor force
(Citibase variable LHG).
6a. c
The following aggregations are used:
GCNQ + GCSQ + csd
7a. i = GPIQ + GCDQ
8a. 9 = GGEQ
9a. y = c + i + 9
where c3d is the flow of services generated by the per capita stock of consumer
durables. The service flow (csd) is computed using the expression
csdt
(r + 8p) kht
wherer = ~i3 is the deterministicsteadystate (real) interest rate and is the kht
beginning of period-t real net stock of consumer durables as reported by the Bureau
of Economic Analysis (Katz and Herman 1997) converted to a per capita quantity.
The labor hours series used in the study is constructed from the following Citibase
variables:
lOa. LHOURS:
lla.
LHC:
average weekly hours (constructed from household survey data)
civilian labor force.
Per capita hours are expressed as a proportion of the time endowment, which is set
to 80 hours per week (16 hours per day times 5 days per week):
12a. 'n = (LHOU RBI LHC) 180.
19
REFERENCES
1] Aschauer. D. A
Public Expenditure Productive'
Journal of Monetary
Economics, 23, 177-200,
21
D. A. (1989b)
Public Capital Crowd Out Private Capital? :" Journal
of Monetary Economics, 24. 171-188.
[3] Baxter, ~I. and King R. G. (1993), "Fiscal Policy in General Equilibrium,"
American
Economic Review, 83. 315-334.
Braun, R. A. and McGratten, E. R. (1993), "The ~Iacroeconomics of War and Peace,
NBER ~Iacroeconomics Annual, l\IIT Press: Cambridge, MA.
Burnside, C., Eichenbaunl. ~1 and Rebelo, S. (1993), "Labor Hoarding and the Business Cycle,"
Journal of Political
Economy, 100. 245-273.
Cooley, T. F. and Prescott. E. C. 1995), "Economic Growth and Business Cycles
T. F. Cooley (ed
in
Frontiers of Business Cycle Research, Princeton, NJ: Prince-
ton University Press
[7] Gramlich, E. M. (1994)
Investment
A Review Essay,1 Journal of
Economic Literature. 32, 1176-1196.
[8] Hansen, G. D. (1985),
Labor and the Business Cycle,
Journal of Mon-
etary Economics, 16. 309-327.
[9] Hansen, L. P. (1982), ;'Large Sample Properties of Generalized Method of Moments
Estimators," Econometrica. 50, 1029-1054.
[10] Ingram. B. F., Kocherlakota, ~.~ and Savin~ N. E.
'Is
(1989a)
Aschauer.
'Does
[4]
[5]
[6]
'Infrastructure
'Indivisible
'Explaining
1994)
Business
Cycles: A Multiple Shock_-1..pproach.':
Journal of Monetary Economics,34, 415-428.
20
llId
'Using
199.
,urement:
'Time
Ingram.
B. F,
Kocherlakota, N
Savin.
:\"
E
Theory for Mea-
An .:-\nalysis of the Cyclical Beha\"ior of Home Production."" Journal
of Monetary
Economics, 40. -4035-456.
[121 Prescott, E. C. (1986), "Theory .-\head of Business-Cycle ~Ieasurement," CarnegieRochester Conference on Public Policy, 24, 11-44.
[13] Katz, A. J. and Herman, S. W. (1997), "Improved Estimates of Fixed Reproducible
Tangible Wealth, 1929-95,' SurI.'eyof Cu~nt
K v"dland.
F. E. and Prescott. , E. C (1982)
c
Business, 77 (May), 69-92.
to Build and Aggregate Fluctua-
tions," Econometrica, 50, 13-,1.5-1370.
15] Rogerson, R. (1988), "Indivisible Labor, Lotteries and Equilibrium, " Journal
of Mon-
etary Economics, 21, 3-16.
[16] Solow, R. (1957), "Technical Change and the Aggregate Production Function," Review of Economic Studies, 39. 312-330.
21
Table I
Parameter Settings
a
(0.20,0.25)
Q
{j
.B
0.36 1.03-1
p
0.035 (.5,1,2..5)
n
0.466
Note: Bold face indicates the benchmarkparameters (divisible labor
economy used for presentationpurposesonly The ..shift length" n
setting is only required in the indivisible labor economy
22
Table IIa
(Divisible Labor)
*The measurementexperiment with rho equal to unity is a useful benchmark
for comparisonin that the simulated public to private capital ratio of 0.311 is
identical to the one reported by the Bureau of Economic Analysis.
23
-0.609
-0.23
-0.48
Table lIb
(Indivisible Labor)
Policy rule parameters:
'Yo (constant)
7.494
3.747
'Yl (kpt/Yt)
2.915
1.458
0.729
0.292
12 (gt/Yt)
13.365
6.682
3.341
1.336
13 (()t)
-6.093
-:3.047
-1.874
0.749
-1.523
Average annual gro\vth of kg/kp (0/(,)
1956
-93
-1.75
-0.88
-0.44
-0.17
1970 -93
4.97
2.45
1.22
0.49
1974 -93
7.55
3.71
1.84
0.73
0.97
0.93
Average annual growth of A (Cl(
1956 -93
1.18
1970 -93
-0.97
1974 -93
-1.46
Mean of kg/ kp:
1.04
-0.09
-0.70
0.096
0.299
24
-0.32
0.542
-0.09
0.781
FIG. 1. Public-pri\'ate
capital ratios, 1956-93. Source: Fixed Reproducible Tangible
Wealth in the United States, U.S. Department of Commerce.
25
0
~
I!J
~
0
0
~
0
I!J
00;
0
0
00;
0
ro
(l)
.c
I!J
I'0
CJ
r.,
Ln
t'!
-,
0
'--'
'
,---~--,
,---,-
0
CD
'-
...
c:i
Ln
Ln
d
0
1I1
01955
1960
1965
,975
380
1985
Year
FIG. 5. RealizaTionsof the preference shock (}t
29
!990
1995