Putty-Clay and Investment: A Business Cycle
Analysis
Simon Gilchrist
Boston University
John C. Williams
Board of Governors of the Federal Reserve System
This paper develops a general equilibrium model with putty-clay technology, investment irreversibility, and variable capacity utilization. Low
short-run capital-labor substitutability induces the putty-clay effect of
a tight link between changes in capacity and movements in employment and output. Permanent shocks to technology or factor prices
generate a hump-shaped response of hours, persistence in output
growth, and positive comovement in the forecastable components of
output and hours. Capacity constraints result in asymmetric responses
to large shocks with recessions deeper than expansions. Estimation of
a two-sector model supports a significant role for putty-clay capital in
explaining business cycle and medium-run dynamics.
I.
Introduction
In this paper we develop a dynamic stochastic general equilibrium model
based on the putty-clay technology introduced by Johansen (1959). The
putty-clay model possesses a number of attractive features typically absent from business cycle models, including irreversible investment, varWe are indebted to John Shin, Steven Sumner, and Joanna Wares for exemplary research
assistance. We wish to thank Flint Brayton, Jeff Campbell, Thomas Cooley, Russell Cooper,
Sam Kortum, John Leahy, Scott Schuh, Dan Sichel, John Willis, and two anonymous
referees for helpful comments. We also thank the National Science Foundation for financial support (SES-9876548). The opinions expressed here are not necessarily shared
by the Board of Governors of the Federal Reserve System or its staff.
[Journal of Political Economy, 2000, vol. 108, no. 5]
q 2000 by The University of Chicago. All rights reserved. 0022-3808/2000/10805-0006$02.50
928
putty-clay and investment
929
1
iable capacity utilization, and endogenous machine replacement. We
investigate the implications of putty-clay technology for macroeconomic
dynamics at business cycle frequencies. A key finding is that low shortrun capital-labor substitutability native to the putty-clay framework induces the putty-clay effect of a tight link between changes in capacity and
movements in employment and output. In the short run, an expansion
of employment quickly confronts sharp increases in marginal costs, owing to capacity constraints. Once new capacity is in place, a sustained
boom in employment and output can occur, as firms fully employ new
machines while maintaining high rates of utilization on preexisting
capacity.
Putty-clay technology has important implications for business cycle
dynamics. In particular, permanent shocks to technology or factor prices
generate a prolonged hump-shaped response of hours, persistence in
output growth, and positive comovement between the forecastable components of output and hours. These features of the business cycle, documented by Cogley and Nason (1995) and Rotemberg and Woodford
(1996), are absent in standard business cycle models, where the response
of hours peaks on the impact of the shock and the dynamic response
of output closely follows that of the shock.2 In addition, capacity constraints inherent to the putty-clay framework imply asymmetric responses
to large shocks, with recessions steeper and deeper than expansions,
consistent with the empirical evidence documented by Neftçi (1984)
and Sichel (1993).
The empirical relevance of putty-clay technology is confirmed by minimum distance estimation of a two-sector model that formally nests both
the putty-clay model and Solow’s (1960) vintage capital model within a
common econometric framework. We find that the distance between
model and data moments is minimized for an estimated putty-clay share
of total output that is on the order of 50–70 percent. The data overwhelmingly reject the restriction of no role for putty-clay capital.
In addition to its empirical relevance, the putty-clay model provides
an intrinsically appealing description of investment, capacity utilization,
and machine replacement. In this framework, capital goods embody the
level of technology and the choice of capital intensity made at the time
of their creation. Ex ante, the choice of capital intensity—the amount
of capital to be used in conjunction with one unit of labor—is based
on a standard Cobb-Douglas production function. Ex post, the production function has the Leontief form with a zero-one utilization decision
1
Putty-clay models have a long history in both the growth (Johansen 1959; Solow 1962;
Phelps 1963; Sheshinski 1967; Cass and Stiglitz 1969) and investment literatures (Bischoff
1971; Ando et al. 1974).
2
Kydland and Prescott (1982) provide a classic example of a real business cycle (RBC)
model with putty-putty technology.
930
journal of political economy
based on the output per hour of a given piece of capital relative to the
prevailing wage rate. Over time, as the economy grows and real wages
rise, older vintages of capital become too costly to operate given their
labor requirements, and they are mothballed or scrapped.3
Our model incorporates what we view as the essential features of the
putty-clay framework, including variable capacity utilization and investment irreversibility.4 Although the assumption of ex post Leontief technology may at first seem unrealistically stark, the resulting aggregate
production function embeds, depending on the model’s parameterization, both the relatively flat short-run marginal cost curve usually associated with a putty-putty production function and a reverse L-shaped
marginal cost curve traditionally associated with the putty-clay framework. Moreover, the model’s microfoundations are largely consistent
with empirical evidence on the importance of plant shutdowns as a shortrun adjustment margin (Bresnahan and Ramey 1994) and the lumpiness
of investment at the plant level (Cooper, Haltiwanger, and Power 1999).
II.
The Model
In this section we describe the model and derive the equilibrium conditions. As noted above, each capital good possesses two defining qualities: its level of embodied technology and its capital intensity. In addition to aggregate technological change, we allow for the existence of
idiosyncratic uncertainty regarding the productivity of investment projects. As in Campbell (1998), the introduction of heterogeneity within
vintages smooths the aggregate allocation and simplifies computation
of the equilibrium. More important, such idiosyncratic uncertainty implies the existence of a well-defined aggregate production function despite the Leontief nature of the microeconomic utilization choice.
Once in place, capital goods are irreversible; that is, they cannot be
converted into consumption goods or capital goods with different embodied characteristics, and they have zero scrap value. Firms can choose,
however, whether or not to operate a given unit of capital depending
on the profitability of doing so in the current economic environment.
We assume that there are no costs of taking machines or workers onand off-line. As such, the utilization choice is purely atemporal. The
3
The effects of technological lock-in motivate the machine replacement problem first
addressed by Johansen (1959) and Calvo (1976) and more recently formalized in a dynamic
programming environment by Cooper and Haltiwanger (1993) and Cooley, Greenwood,
and Yorukoglu (1997).
4
In Atkeson and Kehoe (1999), the energy intensity of production is the putty-clay
factor, but their dynamic analysis focuses on the case in which capital is always fully utilized.
See also Struckmeyer (1986). Cooley, Hansen, and Prescott (1995) model variable capital
utilization but do not impose the irreversibility of factor proportions central to the puttyclay effect described in this paper.
putty-clay and investment
931
optimal utilization choice for each unit of capital is determined by the
difference between the (labor) productivity of the capital and the cost
of utilizing the capital, which in the absence of other costs equals the
wage rate. If the productivity of a unit of capital exceeds the wage rate,
the capital is used in production; otherwise, it is not. In equilibrium,
the wage rate, capacity utilization rate, and levels of employment, production, consumption, and investment are jointly determined by the
dynamic optimizing behavior of households and firms. To characterize
the equilibrium allocation, we first discuss the production technology
and then describe utilization and investment decisions.
A.
The Production Technology
Each period a set of new investment “projects” becomes available. Constant returns to scale implies an indeterminacy of scale at the level of
projects, so without loss of generality, we normalize all projects to employ
one unit of labor at full capacity. We refer to these projects as “machines.” The productive efficiency of a machine initiated at time t is
affected by three terms: the amount of capital invested per machine, kt,
the economywide level of vintage technology, vt, and an idiosyncratic
shock to productivity, mt. We assume that capital goods require one
period for initial installation and fail at the exogenously given rate d.5
The economywide level of vintage technology, vt, follows a stochastic
process with mean gross growth rate (1 1 g)12a. The idiosyncratic shock
mt 1 0 is drawn from a continuous distribution with unit mean and probability density function (pdf) denoted by fm(7). The realized level of
machine productivity is assumed to be permanent and thus embodied
in the machine. For the sake of notational clarity, in the following discussion we abstract from disembodied aggregate technological
change—that is, economywide shifts in total factor productivity—of the
type typical in the RBC literature. Such a process is incorporated at the
end of this section.
Final-goods output produced at time t by a machine built in period
t 2 j with embodied technology mt2j vt2j and capital k t2j is
a
a
a
Y(m
t
t2j vt2j k t2j) p 1{L t(mt2j vt2j k t2j) p 1}mt2j vt2j k t2j ,
(1)
5
Although our model allows for endogenous depreciation owing to machine obsolescence, the long-run rate of economic growth is too low to fully explain capital depreciation
purely through this mechanism. Introducing exogenous machine failure thus captures
wear and tear, which constitutes a major portion of measured rates of depreciation. From
a qualitative point of view, setting d p 0 has no consequence as long as economic growth
is positive and some form of machine retirement occurs along the balanced growth path.
Quantitatively, rapid obsolescence, either endogenous or exogenous, reduces the economic consequences of irreversible capital-labor ratios embedded in the putty-clay
framework.
932
journal of political economy
a
t2j
where L t(mt2j vt2j k ) is labor employed at the machine, and the zeroa
one indicator function 1{L t(mt2j vt2j k t2j
) p 1} reflects the Leontief nature
of machine production with unit labor capacity.6 Let
a
X t { mtvk
t t
(2)
denote the efficiency level (labor productivity) of a particular machine.
To characterize the production possibilities of this economy, we note
that, once produced, machines are distinguished only by their efficiency
level X.7 Let Ht(X) denote the quantity of machines of efficiency level
X that are available for production at time t. Integrating over X yields
expressions for aggregate output,
E
`
Yt p
1{L t(X ) p 1}XH(X
)dX,
t
(3)
0
and aggregate labor,
E
`
Lt p
L t(X )H(X
)dX,
t
(4)
0
where L t(X ) p 1 if machines of efficiency X are utilized.
The quantity of machines with efficiency X available for production
at time t 1 1 equals the quantity of machines that survive from last period
plus the quantity of new machines with efficiency X that are put into
place at time t. Owing to the idiosyncratic variation in project outcomes,
the realized efficiency of a new machine, Xt, is randomly distributed
a
around X t p vk
t t , the mean efficiency of new machines. The quantity
of machines of a given efficiency X evolves according to
Ht11(X ) p (1 2 d)H(X
) 1 fx(X; X t)Q t,
t
(5)
where
fx(X; X t) p (X t)21fm
()
X
Xt
is the pdf of Xt, and Q t is the aggregate quantity of new machines that
are put in place at time t.
In the absence of government spending or other uses of output,
aggregate consumption, Ct, satisfies
6
In equilibrium, machines are used at full capacity or not at all; hence, we could
equivalently express eq. (1) using the more traditional Leontief notation:
a
Yt(mt2jvt2jkt2j
) p mt2jvt2jkat2j min [Lt(mt2jvt2jkat2j), 1].
7
Because all existing machines have a constant probability of exogenous failure d, regardless of machine age, we may eliminate the time subscript associated with the realized
efficiency of a given machine.
putty-clay and investment
933
C t p Yt 2 It,
(6)
where It p k tQ t denotes total investment expenditures. Along with labor
supply, which is assumed to grow at a constant rate n, equations (3)–(6)
define the production possibilities for this economy.
B.
The Utilization and Investment Decision
The only variable cost to operating a machine is the cost associated with
employing labor, the wage rate Wt. Idle machines incur no loss in current
resources.8 Given the Leontief structure of production, these assumptions imply a cutoff value for the minimum efficiency level of machines
used in production: those with productivity X ≥ Wt are run at capacity,
whereas those less productive are left idle.
Conditional on the current wage rate, equation (3) implies that aggregate output may be expressed as
Yt p
E
XH(X
)dX,
t
(7)
X 1Wt
and equation (4) implies that aggregate labor may be similarly expressed
as
Lt p
E
H(X
)dX.
t
(8)
X 1Wt
Define the time t discount rate for time t 1 j income by R˜ t,t1j . Before
making investment decisions, project managers observe the economywide level of vintage technology vt. The idiosyncratic shock to individual
machines mt is revealed only after kt is chosen, however. These timing
assumptions imply that capital intensity is chosen to maximize the present discounted value of profits to the machine
max
k t,{L t1j (X t)}`
jp1
{
E t 2k t 1
O
`
jp1
}
R˜ t,t1j(1 2 d) j(X t 2 Wt1j)1{L t1j(X t) p 1} ,
(9)
where expectations are taken over the time t idiosyncratic shock and
future values of wages and interest rates.
t
Expected net income in period t 1 j from a vintage t machine, pt1j
,
conditional on Wt1j, is given by
8
The model can be extended to allow for a fixed labor cost per unit of capital. Under
such a specification, it is optimal to permanently scrap inefficient machines. Because such
machines nearly always lie idle, such a modification would not alter the putty-clay effect
or the main results of this paper.
934
t
pt1j
p (1 2 d) j
journal of political economy
E
(X 2 Wt1j)fx(X; X t)dX.
X 1Wt1j
Substituting this expression for net income into equation (9) eliminates
the future choices of labor from the investment problem. The remaining
choice variable is kt. An increase in kt increases expected net income by
raising X t, the mean value of machine efficiency. Taking the derivative
of the profit function with respect to kt yields the first-order condition
for an interior solution for kt:
{O
`
Et
­pt1j
R˜ t,t1j
­k t
jp1
t
}
p 0.
(10)
New machines are put into place until the value of a new machine
(the present discounted value of net income) is equal to the cost of a
machine kt:
{O
`
Et
t
R˜ t,t1jpt1j
jp1
}
p k t.
(11)
This is the free-entry or zero-profit condition. This condition must hold
as long as there is gross investment in period t; if the cost of a machine
exceeds the value of a machine for all admissible values of kt, no investment is undertaken.
C.
Preferences
To close the model, we specify the economic relationships that determine labor supply and savings decisions and define the rational expectations equilibrium. We assume that the economy is made up of representative households whose preferences are given by
O
`
Et
Nt1s b sU(C t1s, L t1s, Nt1s),
(12)
sp0
where
U(C, L, N ) p
1
12g
[ ( )]
C
12g N
12
L
z
N
is utility per period, b P (0, 1), g 1 0, z 1 0, and Nt p N 0(1 1 n)t is the
labor endowment of the representative household. Households optimize
over these preferences subject to the standard intertemporal budget
constraint. We assume that claims on the profit streams of individual
machines are traded; in equilibrium, households own a diversified portfolio of all such claims, implying that the discount factor, R̃t,t1j , used by
putty-clay and investment
935
firms to discount future profits, is consistent with the household’s intertemporal decisions
R̃t,t1j p b j
Uc,t1j
Uc,t
,
(13)
where Uc,t1s denotes the marginal utility of consumption. The first-order
condition with respect to leisure and work is given by
Uc,tWt 1 UL,t p 0,
(14)
where UL,t denotes the marginal utility associated with an incremental
increase in work (decrease in leisure).
The rational expectations equilibrium is defined to be the set of
sequences of prices and quantities such that each household and firm
solves its respective maximization problem as described above, taking
prices as given, and all markets clear. The derivation of the deterministic
steady state and its properties are found in Gilchrist and Williams (1998).
In conducting the dynamic analysis, we focus on deviations from the
balanced growth path. The solution methodology for the dynamic analysis is described in the Appendix.
D.
The Lognormal Distribution
To parameterize the model, it is necessary to choose a distributional
form for mt, the idiosyncratic shock to machine productivity. A natural
choice is to assume that mt is lognormally distributed:
log mt ∼ N(2 2 j 2, j 2 ),
1
where the mean correction term 2 12 j 2 implies E(mt) p 1. The lognormal
distribution for mt implies that Xt is also lognormally distributed, with
E(log X tFX t) p log X t 2 2 j 2.
1
Hence, the distribution of machines expressed in equation (5) evolves
according to
Ht11(X ) p (1 2 d)H(X
)
t
(1j (log X 2 log X 1
1 (jX )21f
t
1
2
)
j 2 ) Q t,
(15)
where f(7) denotes the pdf of the standard normal.9
9
As emphasized in Gilchrist and Williams (1998), the lognormal distribution facilitates
the analysis of aggregate quantities while preserving the putty-clay characteristics of the
microeconomic structure.
936
journal of political economy
Given the lognormal distribution, the expressions for aggregate output and aggregate labor developed in subsection C may be considerably
simplified by developing an explicit notion of capacity utilization. Capacity utilization for vintage s at time t is the ratio of actual output
produced from the capital of that vintage to the level of output that
could be produced at full capacity,
∫X 1W Xfx(X; X s)dX
t
∫ Xfx(X; X s)dX
`
0
p 1 2 F(z ts 2 j),
(16)
where F(7) denotes the cumulative distribution function of the standard
normal, and
1
1
z ts { (log Wt 2 log X s 1 2 j 2 )
j
(17)
measures the productivity difference between the efficiency of the marginal machine in use at time t and the mean efficiency of a vintage s
machine. The expression on the right-hand side of equation (16) follows
from the formula for the expectation of a truncated lognormal random
variable.10 Conditional on the current wage, the mean output of a vintage s machine at time t equals [1 2 F(z ts 2 j)]X s. An increase in the
current wage reduces capacity utilization and hence output from any
given vintage.
To illustrate these ideas, figure 1 shows the distribution of machine
productivity for the model calibrated to parameters specified below. This
figure also shows the distribution of machine productivity for the most
recent vintage (right scale). Its position on the horizontal axis reflects
both the current level of embodied technology and the capital intensity
of new machines. Owing to trend growth in technology and relatively
long-lived capital, the average productivity of the most recent vintage
is substantially higher than the average productivity of existing machines. Note that trend growth in investment—due to population growth
and technological change—causes the aggregate distribution to be
skewed.
The wage is shown by the vertical line in figure 1. Capacity utiliza10
Capacity utilization may be expressed as
[
]
E(XsFXs 1 Wt)
Pr (Xs 1 WtFWt).
E(Xs)
We then apply the following formula: If log (m) ∼ N(z, j2), then
E(mFm 1 x) p
1 2 F(g 2 j)
E(m),
1 2 F(g)
where g p [log (x) 2 z]/j (Johnson, Kotz, and Balakrishnan 1994, vol. 1).
putty-clay and investment
937
Fig. 1.—Distribution of machine productivity
tion—the ratio of actual output to output under full utilization—is obtained by computing the integral of XH(X) over the shaded region and
dividing by the integral of XH(X) computed over the entire curve. Obsolescence through embodied technological change implies that, on
average, old vintages have lower utilization rates than new vintages.
When equations (7), (15), and (16) are combined, aggregate output
may be expressed as a function of current capacity utilization rates and
past investment decisions:
O
`
Yt p
[1 2 F(z tt2j 2 j)](1 2 d) jQ t2jX t2j .
(18)
jp1
Given the current wage Wt, the share of vintage s machines in use at
time t is
s
Pr (X s 1 WF
t Wt) p 1 2 F(z t ).
(19)
When equations (8), (15), and (19) are combined, aggregate labor may
be expressed as a function of current labor requirements and past investment decisions:
O
`
Lt p
[1 2 F(z tt2j)](1 2 d) jQ t2j .
(20)
jp1
To derive the optimality conditions for investment, we may express
938
journal of political economy
expected net income in period t from a vintage s machine, pts, conditional on Wt, in terms of expected utilization rates:
pts p (1 2 d)s{[1 2 F(z ts 2 j)]X s 2 [1 2 F(z ts)]Wt}.
The choice of kt has a direct effect on profitability through its effect on
the expected value of output X t. It also has a potential indirect effect
through its influence on utilization rates. For any given realization of
mt, a higher choice of kt raises the probability that a machine will be
utilized in the future. This increase in utilization raises both expected
future output and expected future wage payments. Because the marginal
machine earns zero quasi rents, this indirect effect has no marginal
effect on profitability, however:11
­pts
[
]
1
1
p (1 2 d)s f(z ts 2 j)X s 2 f(z ts)Wt p 0.
­z
j
j
s
t
Accordingly, the first-order condition for kt may be expressed as
{O
`
k t p aE t
}
t
R˜ t,t1j(1 2 d) j[1 2 F(z t1j
2 j)]X t .
jp1
(21)
The zero-profit condition may now be expressed as
{O
`
kt p Et
t
R˜ t,t1j(1 2 d) j[1 2 F(z t1j
2 j)]X t
jp1
O
}
M
2
t
R˜ t,t1j(1 2 d) j[1 2 F(z t1j
)]Wt1j .
jp1
(22)
The first term on the right-hand side of equation (22) equals the expected present discounted value of output adjusted for the probability
that the machine’s idiosyncratic productivity draw is too low to profitably
operate the machine in period t 1 j. The second term likewise equals
the expected present value of the wage bill, adjusted for the probability
of such a shutdown. Equations (21) and (22) jointly imply that, in
11
The condition that the marginal machine earns zero quasi rents applies for a general
distribution function. For the lognormal case, one may use the formula for the pdf of a
random normal variable to show that ­pst /­zst p 0 follows directly from eq. (17). Rearranging this expression implies that MPLst p Wt, where
MPLst {
f(zst 2 j)
Xs
f(zst )
is the marginal product of labor for vintage s machines, i.e., the increment to average
machine output [1 2 F(zst 2 j)]Xs divided by the increment to average labor input 1 2
F(zst ) obtained through a reduction in zst .
putty-clay and investment
939
equilibrium, the expected present value of the wage bill equals 1 2 a
times the expected present value of revenues.
In simulations reported below, we compare the effects of shocks to
disembodied technological change to the effects of shocks to embodied
technological change. To allow for disembodied technological change,
let At denote the current level of economywide total factor productivity,
which is assumed to be known at the time current production and
investment decisions are made. With disembodied technological
change, machine output for a vintage s machine at time t satisfies
Y(X
t
s) p 1{L t(X s) p 1}A t X s. A machine with efficiency level X is operated
at capacity if A t X ≥ Wt. The term z ts in equation (17) is redefined as
1
1
z ts { [log (Wt) 2 log (A tX s) 1 2 j 2 ].
j
Equation (18) may then be expressed as
O
`
Yt p
[1 2 F(z tt2j 2 j)](1 2 d) jQ t2j A tX t2j ,
jp1
and other model expressions are changed accordingly.
E.
The Solow Vintage Model
When the restriction that ex post capital-labor ratios are fixed is relaxed,
the model described above collapses to the vintage capital model initially
introduced by Solow (1960). For the Solow vintage model, define the
capital aggregator Kt:
O
`
Kt {
1/a
vt2j
(1 2 d) jIt2j
jp1
1/a
p (1 2 d)K t21 1 vt21
It21 ,
(23)
where It denotes gross aggregate capital investment. In the Solow vintage
model, embodied technological change simply enters the model
through the capital accumulation equation in the form of a reduction
in the relative cost of new capital goods.
Aggregate production in period t is given by
Yt p exp
(
1 2 a j2
A t K taL12a
,
t
a 2
)
(24)
where Lt denotes aggregate labor input, At represents disembodied technological change, and exp {[(1 2 a)/a](j 2/2)} is a scale correction that
results from aggregating across machines with differing levels of idiosyncratic productivity. In this economy, a mean-preserving spread to
940
journal of political economy
idiosyncratic productivity causes an increase in disembodied productivity
at the aggregate level. Equations (6), (23), and (24) describe the production technology of the Solow vintage model.
III.
Model Dynamics
In this section we describe the model’s dynamic response to persistent
shocks to factor prices; in Section IV we turn our attention to permanent
shocks to technology. We focus primarily on three fundamental features
of the business cycle: the comovement between output and hours, the
persistence of output growth, and the asymmetric response to positive
versus negative shocks. As documented below, the putty-clay model possesses a powerful internal propagation mechanism that yields substantial
improvements over the Solow model in all three dimensions.
A.
The Short-Run Marginal Cost Curve
Insight into the dynamic responses of the putty-clay model is provided
by the short-run marginal cost (SRMC) curve, which shows the relationship between the level of output and the marginal cost of production
(or, equivalently, the inverse of the marginal product of labor). In deriving this relationship, we are holding the wage and the stock of capital
constant. Figure 2 shows the SRMC curve, in log deviations from the
deterministic steady state, for the Solow vintage model and three versions of the putty-clay model with different degrees of idiosyncratic
uncertainty (measured by j). The slope of the marginal cost curve equals
the percentage increase in labor input necessary to obtain a 1 percent
increase in aggregate output. For the Solow vintage model with the
Cobb-Douglas production function, the SRMC curve is linear in logs.
The SRMC curve of the putty-clay model, however, is distinctly nonlinear. In the short run, an expansion of output occurs through increased utilization of marginal machines. At low levels of output, there
are many idle machines with relatively high efficiency, and output can
be expanded at relatively low cost. At high levels of output, the efficiency
level of the marginal machine drops sharply, and an expansion of output
causes a rapid increase in marginal cost.
The degree of curvature displayed by the SRMC curve depends on
the degree of idiosyncratic uncertainty j. For very low values of j, the
putty-clay SRMC curve becomes vertical for levels of output a few percent
above steady state. As j increases, the SRMC curve of the putty-clay
model becomes less sharply curved and approximates that of the Solow
vintage model as j becomes large. In this way, the model developed in
this paper embeds both the reverse L-shaped marginal cost curve tra-
putty-clay and investment
941
Fig. 2.—Short-run marginal cost curve
ditionally associated with putty-clay technology and the log-linear marginal cost curve of the Cobb-Douglas production function.
B.
Calibration
The models are calibrated using parameter values taken from Kydland
and Prescott (1991) and Christiano and Eichenbaum (1992), except for
the trend growth rates, which are U.S. averages over 1954–96. We assume
that a period is one quarter of one year. In annual basis terms, the
calibrated parameters are b p 0.956, g p 1, z p 3, g p 0.018, n p
0.015, d p 0.084, and a p 0.36. The results reported in this paper are
not sensitive to reasonable variations in these parameters. In our calibration of the model, the only parameter for which we do not have a
strong prior is the variance of the idiosyncratic component of a machine’s productivity, j2. As discussed above, for large j, the SRMC curve
is very close to Cobb-Douglas. By lowering j, we increase the curvature
of the SRMC curve and increase the degree to which the model displays
dynamics unique to the putty-clay structure. To make clear distinctions
942
journal of political economy
between the Solow vintage model and the putty-clay model, we set
j p 0.15.12
C.
Capital Cost Shocks
We start by characterizing the effect of a temporary but persistent shock
to the cost of producing capital goods relative to consumption goods.
This is identical to an increase in technology embodied in capital goods;
henceforth, we describe it as such.13 We assume that embodied technology follows the process
(1 2 rv L) ln vt p (1 2 rv L)t(1 2 a) ln (1 1 g) 1 u t,
where ut is an independently and identically distributed innovation, L
is the lag operator, and t(1 2 a) ln (1 1 g) represents trend productivity
growth. We set the autocorrelation coefficient of the shock process at
rv p 0.95, implying a half-life for the shock of just under 14 quarters.
Figure 3 shows the impulse responses of output and labor hours for
both the putty-clay model (upper panel) and the Solow vintage model
(lower panel) to a one-percentage-point positive shock to embodied
technology.
The difference in output dynamics between the two models is striking.
For the putty-clay model, output rises very little initially, steadily increases for a period of four years, and gradually returns to steady state.
For the Solow vintage model, the peak response occurs at the onset of
the shock, after which output declines monotonically: output dynamics
simply mirror shock dynamics with no evidence of any cyclical pattern.
In the putty-clay model, output exhibits a clear hump-shaped response
that creates a slowly unfolding and long-lasting business cycle.
The dynamic response of hours also exhibits a strong cyclical pattern
in the putty-clay model. Labor hours expand at the onset of the shock
as the economy seeks to increase its investment rate. The initial expansion in hours is muted, however, owing to the sharply increasing slope
of the SRMC curve embedded in the putty-clay model. As more capital
is brought on-line, the marginal cost curve shifts out and labor expands
further. As a result, the peak labor response occurs two years after the
onset of the shock.
12
Lowering j to 0.1 does not alter model results in any substantial manner; lowering j
much further causes numerical problems for the dynamic solution methods. On the other
hand, raising j to 0.5 for almost all essential purposes replicates the Solow vintage model
dynamics, whereas intermediate values (j p 0.2–0.25) provide results that are a combination of the Solow and putty-clay models with low j.
13
It is, not, however, the same as a distortionary shock to the cost of capital goods from
a change in taxes or a shock that might result from a monetary disturbance in a model
with nominal rigidities. Such shocks affect the equilibrium allocation but not the feasible
allocation for the economy.
Fig. 3.—Persistent cost of capital shock
944
journal of political economy
The slow but sustained rise in hours above steady-state levels occurs
because the benefits to building new capital goods are much greater if
existing, efficient capital is not scrapped in the process. With ex post
Leontief technology, labor cannot be reallocated across machines to
equate marginal products. To benefit from new machines without losing
the productive services of existing capital, the economy must hire new
workers to operate these machines. Eventually, as the productive value
of these machines falls, labor returns to steady state. We refer to this
dynamic linkage between labor and machines as the putty-clay effect.
In this exercise, the capital cost shock has important implications for
the dynamic link between investment and productivity—both in the
aggregate and at the machine level. Aggregate movements in total factor
productivity, conventionally measured as the Solow residual, are displayed in figure 3 for both the putty-clay and the Solow vintage models.
In both models, movements in the Solow residual reflect the rapid
expansion in investment that occurs as the economy installs new capital
goods that embody the latest technology. For the putty-clay model, this
rapid expansion in investment is shown in the upper panel of figure 4.
Figure 4 also decomposes aggregate investment into the quantity of
new machines, Q, and the capital intensity of each new machine, k. At
the onset of the shock, Q rises and k falls as a large number of low–capital
intensity machines are produced. This rapid expansion of inexpensive
(in terms of forgone consumption) machines facilitates the increase in
labor input. This reliance on low efficiency capital does not persist,
however. As the real interest rate falls and the real wage rises, firms
substitute into high–capital intensity capital goods.
The capital intensity of new machines remains at elevated levels for
a number of years as households store the benefits of the temporary
shock through increased saving, implying capital deepening. As shown
in the lower panel of figure 4, rising capital intensities dramatically offset
the exponential rate of decay in technology and consequently provide
a sustained increase in the efficiency levels of new machines for a number of years after the shock has occurred. Because machines are longlived, this sustained increase in efficiency levels of new machines translates into a sustained increase in aggregate labor productivity over a
long horizon.
D.
Nonlinear Dynamics
The nonlinearities embedded in the SRMC curve also imply an asymmetric response to positive versus negative shocks. For small shocks,
these asymmetries are negligible, and the response to positive and negative shocks is roughly the same. For larger shocks, however, these asym-
Fig. 4.—Persistent cost of capital shock: putty-clay model
946
journal of political economy
metries can be substantial: the response to negative shocks is more rapid
and larger than that to positive shocks.
One plausible source of persistent movements in factor costs sizable
enough to generate measurably asymmetric responses is a change in
marginal tax rates associated with a tax reform. To display these asymmetries, we consider the differential effect of a 10 percent increase
versus decrease in the marginal cost of labor. We embed the labor cost
shock by inserting a wedge between the wage paid by firms and the
wage received by workers. Formally, firms pay (1 1 ht)Wt for each unit
of labor employed, where ht is a payroll tax that is rebated lump-sum
to the household. We assume
ln (1 1 ht) p rh ln (1 1 ht21) 1 et,
where et is an independently and identically distributed innovation. For
the simulations reported here, we set rh p 0.98. Figure 5 reports the
outcome of these simulations. For purposes of comparison, the response
of an increase in labor cost is displayed with the opposite sign.
An increase in labor cost causes an immediate shutdown of machines
as the economy moves down the relatively flat portion of the SRMC
curve. This shutdown produces a sharp contraction in output and hours
in both the initial and subsequent periods. In contrast, a decrease in
labor cost has only a limited effect on either output or hours until new
capital is put in place. Evidence of nonlinearity diminishes after four
to six years as capacity eventually adjusts. As a result, in the putty-clay
model, large contractionary shocks cause steep immediate declines in
output and hours, whereas large expansionary shocks generate a humpshaped dynamic response even more pronounced than in the case of
small expansionary shocks. Of particular interest here is the fact that
the asymmetries are the most pronounced for labor, a result supported
by Neftçi’s (1984) nonlinear time-series analysis.
IV.
Permanent Technology Shocks
An important unresolved issue in macroeconomics is the extent to which
permanent innovations in technology can explain output fluctuations
at the business cycle frequency. While past research has claimed varying
degrees of success, more recent work has tempered enthusiasm for business cycle theories based on permanent technology shocks. Cogley and
Nason (1995) and Rotemberg and Woodford (1996) demonstrate that
the standard Solow model with permanent disembodied productivity
shocks is unable to match the persistence and comovement properties
of key aggregate variables. Christiano and Eichenbaum (1992) show
that such models predict excessive contemporaneous correlation between output growth and labor productivity growth. In this section, we
putty-clay and investment
947
Fig. 5.—Asymmetries in response to labor cost shocks. Impulse responses to a 10 percent
increase in the cost of labor are displayed with the reverse sign.
extend this literature in two directions by analyzing the effects of permanent embodied, as well as disembodied, technology shocks and allowing for putty-clay technology. Greenwood, Hercowitz, and Krusell
(1997) argue that the evidence supports embodied technology as the
primary source of technological change, making the analysis of such
shocks of particular interest. In what follows, we focus on three features
of the business cycle that the Solow model has difficulty replicating:
persistence in output growth, positive comovement between forecastable
components of output and hours, and the correlation between the
growth of output and productivity. As shown below, the putty-clay model
948
journal of political economy
TABLE 1
Output Persistence with Permanent Shocks
Solow Vintage Model
Moments
r(Dyt, Dyt21)
j(Dyˆt,4)/j(Dyt,4)
j(Dyˆt,8)/j(Dyt,8)
j(Dyˆt,16)/j(Dyt,16)
Putty-Clay Model
Data
Disembodied
(1)
Embodied
(2)
Disembodied
(3)
Embodied
(4)
Estimate
(5)
Standard
Error
(6)
.01
.06
.08
.09
.07
.25
.30
.31
.04
.14
.17
.18
.80
.83
.77
.67
.31
.64
.71
.65
.08
.06
.07
.06
Note.—In the columns labeled “disembodied” (“embodied”), the only source of disturbances is permanent disembodied (embodied) technology shocks. The variable y denotes the log of output, and Dyˆt,j { Et(yt1j 2 yt), where expectations are based on date t information; Dyt,j { yt1j 2 yt. The terms j(x) and r(x, y) denote the asymptotic values of the
standard deviation of x and the correlation of x and y, respectively. Data sample pertains to 1960–97.
generates dynamic responses to permanent technology shocks that accord well with these key properties of the data.
Table 1 reports two types of statistics summarizing the persistence
properties of the two model economies in response to permanent technology shocks. In this table and table 2 below, j(x) and r(x, y) denote
the asymptotic values of the standard deviation of x and the correlation
of x and y, respectively. For purposes of comparison, the point estimates
and standard errors for U.S. data from 1960 to 1997 are shown in
columns 5 and 6. (See the Appendix for a description of the data and
empirical methodology.)
The putty-clay model delivers significantly more persistence in output
growth than the Solow model. This finding holds for the two alternative
measures of persistence that we consider: the autocorrelation of output
growth and the ratio of standard deviations of forecastable output
growth to total output movements, j(Dyˆt,k)/j(Dyt,k). Here, forecastable
refers to the conditional expectation Dyˆt,k p E t{yt1j 2 yt}. Not surprisingly,
embodied technology shocks generate much more persistence in output
growth than disembodied shocks do. This is especially true for the puttyclay model, which, when driven solely by permanent embodied technology shocks, actually overpredicts the persistence in output growth
observed in U.S. postwar data.
The intuition behind these results is provided by the impulse responses to permanent technology shocks, plotted in figure 6. In both
the putty-clay and Solow vintage models, the long-run effect of a 1 2
a percent increase in technology is to raise output, consumption, and
investment by 1 percent while leaving the long-run level of labor unchanged. In the case of disembodied shocks (shown in the panels on
the right), total factor productivity increases immediately, causing an
immediate expansion of output. For both models, this initial increase
in output in response to a disembodied shock to technology represents
a large fraction of the permanent increase; nearly all of the output
putty-clay and investment
949
Fig. 6.—Permanent productivity shocks
movement is unforecastable, and output growth displays very little
persistence.
If technology shocks are embodied in capital (shown in the panels
on the left), productivity does not increase until the economy invests
in new capital goods. In the Solow vintage model, a positive shock to
embodied technology still causes a large immediate expansion in output,
owing to a surge in hours in response to the high rate of return to
950
journal of political economy
TABLE 2
Permanent Technology Shocks
Solow Vintage Model
Moments
Unconditional:
j(Dc)/j(Dy)
j(Dh)/j(Dy)
j(Di)/j(Dy)
j(Dp)/j(Dy)
r(Dct, Dyt)
r(Dht, Dyt)
r(Dit, Dyt)
r(Dpt, Dyt)
r(Dct, Dct21)
r(Dht, Dht21)
r(Dit, Dit21)
r(Dpt, Dpt21)
Forecastable :
r(Dcˆt,4, Dyˆt,4)
r(Dhˆ t,4, Dyˆt,4)
r(Diˆt,4, Dyˆt,4)
r(Dpˆ t,4, Dyˆt,4)
b(Dcˆt,4, Dyˆt,4)
b(Dhˆ t,4, Dyˆt,4)
b(Diˆt,4, Dyˆt,4)
b(Dpˆ t,4, Dyˆt,4)
Putty-Clay Model
Data
Disembodied
(1)
Embodied
(2)
Disembodied
(3)
Embodied
(4)
Estimate
(5)
Standard
Error
(6)
.55
.36
2.15
.65
.99
.98
.99
.99
.10
2.03
2.02
.06
1.13
1.57
6.03
.69
2.83
.95
.97
2.71
.05
2.03
2.02
.18
.70
.09
1.76
.91
.99
.96
.99
1.00
.10
.20
2.01
.03
2.87
.91
8.14
.77
2.12
.68
.53
.50
.06
.20
2.02
.73
.36
.66
1.82
.68
.52
.74
.60
.76
.34
.60
.50
.60
.02
.04
.11
.04
.06
.04
.05
.03
.08
.06
.07
.06
1.00
21.00
21.00
1.00
3.18
21.68
24.44
2.68
1.00
21.00
21.00
1.00
3.18
21.68
24.44
2.68
1.00
.44
2.98
.95
1.61
.14
2.49
.86
1.00
.44
2.98
.95
1.61
.14
2.49
.86
.95
.86
.76
.76
.22
.59
1.70
.41
.07
.06
.07
.10
.07
.09
.24
.09
Note.—In the columns labeled “disembodied” (“embodied”), the only source of disturbances is permanent disembodied (embodied) technology shocks. The variables c, y, h, i, and p denote the log of consumption, output, hours,
investment, and output per hour, respectively. The term Dyˆt,j { Et(yt1j 2 yt), where expectations are based on date t
information; Dyt,j { yt1j 2 yt. The terms j(x), r(x, y), and b(x, y) denote the asymptotic values of the standard deviation
of x, the correlation of x and y, and the regression coefficient of x on y, respectively. The data sample pertains to
1960–97.
investment. Because of the rapid expansion in production, the initial
output response represents a large fraction of the permanent response,
and output movements are mostly unforecastable. In the putty-clay
model, however, a positive shock to embodied technology causes only
a small initial expansion. Output and hours rise over time as new capital
is brought on-line, with labor reaching its peak response a number of
years after the shock occurs. As a result, most of the output dynamics
are forecastable, and output growth displays a high degree of persistence
in response to embodied shocks to technology.
In addition to the limited degree of internal propagation, Rotemberg
and Woodford (1996) criticize the high negative correlations between
forecastable movements in output and hours implied by the Solow vintage model in response to permanent technology shocks. This point is
formalized in table 2, which reports a variety of statistics summarizing
the dynamic properties of the two models. For the Solow vintage model,
the negative correlation between forecastable movements in output and
putty-clay and investment
951
hours follows from the fact that the peak response in hours occurs at
the onset of the shock, whereas output continues to grow as capital
accumulation proceeds. Thus hours are falling and output is rising. The
putty-clay model’s slow expansion of output and hours reverses this
correlation and provides a closer match with the data on this dimension.
Finally, we consider the model’s predictions regarding the contemporaneous correlation between output and productivity growth. If hours
were held constant, both models would predict a perfectly positive correlation between growth in output and productivity. As can be seen from
table 2, with disembodied shocks, this correlation is nearly unity as the
effect of the movements in hours on productivity is dwarfed by the effects
of the shock itself.
In the case of embodied technology shocks, the model’s predictions
differ greatly. The Solow vintage model predicts a highly negative correlation between growth in output and productivity. This negative correlation results from the immediate expansion of hours, which produces
a large decline in productivity at the same time as the largest increase
in output. In contrast, the putty-clay model predicts a positive correlation
between output and productivity growth. This difference lies in the
muted expansion of hours, which does less to offset the positive comovement in productivity and output directly resulting from the shock.
The putty-clay model thus provides an explanation for why the correlation between growth in output and productivity may be positive but
less than unity, even in the absence of shocks to demand.
Although embodied shocks help explain a number of empirical regularities, some results reported in table 2 are not consistent with the
business cycle. In particular, embodied shocks imply negative comovement between consumption growth and output growth and excess volatility of consumption and investment relative to the data.14 This excess
volatility occurs because factor cost movements create investment patterns that overwhelm the usual desire to smooth consumption.
V.
Estimation
The results in Section IV highlight the putty-clay model’s ability to explain key business cycle facts such as persistence in output growth and
positive forecastable comovement between output and hours. In this
section we provide a more formal evaluation of the empirical relevance
14
The response of consumption to an innovation to embodied technology depends
crucially on the specification of preferences. Using a putty-putty model, Greenwood, Hercowitz, and Huffman (1988) show that if there is no intertemporal substitution effect on
labor effort, consumption can immediately rise in response to a positive shock to embodied
technology.
952
journal of political economy
of a putty-clay production process in matching key moments of U.S.
aggregate data.
To perform this evaluation, we construct a two-sector model that nests
both the putty-clay model and the Solow vintage model within the same
econometric framework. We assume that sector 1 output is produced
using the putty-clay production process described above, whereas sector
2 output is produced using the Solow vintage technology. Letting At
denote the level of disembodied technology, we assume that final-goods
output is a Cobb-Douglas function of sectoral output: Yt p A tY1tlY2t12l.
We then estimate l, the share of output obtained from the putty-clay
sector. If our estimate of l is close to unity, the data effectively put a
large weight on putty-clay production in order to match the vector of
moments that we consider. If our estimate of l is close to zero, the data
suggest little if any role for putty-clay in explaining the moments we
choose to match.
To address recent criticisms regarding standard RBC-style momentmatching exercises, our methodology relies on both unconditional moments traditionally emphasized in the RBC literature and forecastable
moments emphasized by Rotemberg and Woodford (1996).15 The unconditional moments include the standard deviations of the growth rates
of investment and hours relative to the standard deviation of output,
the correlations of investment and hours with output, and the first-order
autocorrelations of these variables. The forecastable moments include
correlations and regression coefficients among forecastable changes in
output, investment, and hours over a four-quarter horizon. These moments also include the ratio of the standard deviation of the forecastable
change in output relative to the standard deviation of the total change
in output at this horizon.
To compute moment estimates for the historical data, we specify a
vector autoregression (VAR) process for (yt, h t, i t), the logs of output,
hours, and investment in the data. As shown in the Appendix, the statistical properties and hence all relevant moments of this VAR are summarized by an unknown parameter vector y. We estimate y using standard time-series techniques and then use the resulting parameter
ˆ along with the variance of
estimate to compute a set of moments g(y),
these moments Vg.
Our estimation strategy is to choose l, the share of output accounted
for by the putty-clay sector, along with other relevant parameters, to
minimize the distance between model moments and data moments. For
15
Christiano and Eichenbaum (1992) provide a generalized method of moments procedure for estimating and evaluating business cycle models based on unconditional moments. A contribution of this paper is to provide a generalized method of moments
procedure that allows for both types of moments within a unified econometric framework.
putty-clay and investment
953
a given vector w of model parameters, we use our model solution to
compute gM(w), the model’s analogue of g(y). By minimizing
ˆ 0 V 21[g (w) 2 g(y)]
ˆ
L(w) p [g M(w) 2 g(y)]
g
M
with respect to w, we obtain the minimum distance estimator ŵ. Given
ˆ provides a x2 test for equality
T observations of historical data, T # L(w)
ˆ
ˆ
between g M(w) and g(y).
For our first moment-matching exercise, we consider two sources of
fluctuations: disembodied technological change and embodied technological change. We assume that shocks follow an AR1 process and
then freely estimate [rA, rv] , the autocorrelations of the shock processes,
and jv /(jA 1 jv), the relative importance of embodied shocks, where jx
is the standard deviation of the innovation to the AR1 process for random variable x. To generalize our results beyond technology shocks, we
also consider a specification that includes the labor cost shock, ht, introduced above. Under this specification, we also estimate the autocorrelation and the relative importance of labor cost shocks.
Table 3 reports the results of this exercise. Column 1 reports the
results from a benchmark putty-putty model whose parameters are imposed rather than estimated. For this benchmark model, we assume that
all shocks are disembodied with rA p 0.95. Column 2 shows the results
based on estimation using the seven unconditional moments reported
in the middle of the table. Columns 3 and 4 show results based on the
full set of unconditional and forecastable moments, with column 4 also
allowing for labor cost shocks. This table also reports the test statistic
T # L(w) for the various specifications. For comparison purposes, we
report both unconditional and forecastable moments for all specifications. Although they are not used in the estimation, we also report a
comparable set of consumption moments.
Estimation results based on unconditional moments imply a large
share for the putty-clay production technology—on the order of 50
percent.16 The estimate of jv /(jA 1 jv) is about 0.2, suggesting a substantial role for embodied technology shocks in explaining aggregate
16
The putty-clay share is estimated with a standard error of 0.02, although this number
should be treated with some caution given that we reject a test of equality between model
and data moments. The basic finding of a substantial role for putty-clay technology is
robust, however, to a variety of changes in specification, including changes in preferences,
the introduction of convex adjustment costs to investment, and the inclusion of other
disturbances such as changes in government spending. This finding is also robust to
reasonable changes in the horizon used to compute forecastable moments and to a procedure that matches moments computed from consumption data rather than investment
data. As a final check of robustness, we also considered an alternative to estimating
jv/(jA 1 jv) by fixing this ratio at 0.6, Greenwood, Hercowitz, and Krusell’s (1997) estimate
of the share of postwar technological change embodied in capital. Under this specification,
setting rA p rv p 1 and jL p 0, we estimate a putty-clay share of 0.69.
954
journal of political economy
TABLE 3
Estimation of the Two-Sector Model
Model Parameters and
Moments
Putty-clay share
jv/(jv 1 jA)
jh/(jv 1 jA 1 jh)
rv
rA
rh
Unconditional moments:
j(Dh)/j(Dy)
j(Di)/j(Dy)
r(Dht, Dyt)
r(Dit, Dyt)
r(Dyt, Dyt21)
r(Dht, Dht21)
r(Dit, Dit21)
Forecastable moments:
r(Dhˆ t,4, Dyˆt,4)
r(Diˆt,4, Dyˆt,4)
b(Dhˆ t,4, Dyˆt,4)
b(Diˆt,4, Dyˆt,4)
j(Dyˆt,4)/j(Dyt,4)
T # L(w) :
Unconditional moments
All moments
Consumption moments:
j(Dc)/j(Dy)
r(Dct, Dyt)
r(Dct, Dct21)
r(Dcˆt,4, Dyˆt,4)
b(Dcˆt,4, Dyˆt,4)
PuttyPutty
Benchmark
(1)
Two-Sector
Model
Estimate
(5)
Standard
Error
(6)
(3)
(4)
.47
.23
.68
.01
.64
.00
.64
.95
1.00
1.00
.98
.95
.57
2.85
.99
1.00
2.02
2.04
2.03
.39
2.49
.69
.87
.08
.21
.07
.30
2.60
.96
.99
.06
.24
.04
.67
2.67
.75
.99
.09
.25
.08
.65
1.80
.74
.61
.32
.61
.50
.04
.11
.04
.05
.08
.06
.07
.89
.94
.80
3.60
.27
2.78
2.89
2.49
21.84
.17
.90
.84
.43
2.99
.25
.82
.80
.53
2.80
.26
.89
.86
.64
1.83
.62
.04
.05
.10
.27
.07
708
1,148
361
786
444
.28
.94
.19
2.08
2.04
.73
.72
.09
.99
2.15
.40
.91
.18
.23
.19
.35
.93
.28
.30
.27
.36
.52
.34
.95
.22
.02
.06
.08
.07
.07
0
0
(2)
Data
.94
.99
Note.—See the notes from table 1. The term jx is the standard deviation of the innovation to the AR1 process for
random variable x. The parameters of the putty-putty benchmark are imposed. The estimate of rh is constrained at the
imposed upper bound of .99.
dynamics. Relative to the putty-putty benchmark, allowing for embodied
shocks and a nonzero weight on putty-clay technology provides a substantial gain in terms of fit, reducing T # L(w) by nearly 50 percent.
The estimated values of rA and rv reach their upper bound of unity,
emphasizing the importance of highly persistent shocks when matching
unconditional moments.
Estimation results based on the full set of moments also imply a large
putty-clay share—on the order of two-thirds—regardless of the shock
processes. Again, introducing putty-clay technology provides a substan-
putty-clay and investment
955
tial gain in fit, which is further improved through the introduction of
labor cost shocks. For either specification, the estimated model also
does reasonably well at matching consumption moments.
Estimation results based on the full set of moments also imply that
factor-cost shocks provide an important source of fluctuations. Although
the estimate of jv /(jA 1 jv) drops to zero and the persistence of disembodied shocks falls to 0.94, our estimates imply that highly persistent
movements in ht substantially improve the model’s ability to match empirical moments. Computing variance decompositions, we find that labor cost shocks account for 15 percent of output fluctuations at the
one-year horizon and 47 percent at the five-year horizon. Similar to
Braun’s (1994) findings in the context of a putty-putty model, these
results imply that disembodied shocks to technology do well at explaining output movements at high frequencies, whereas persistent shocks
to factor costs do better at lower frequencies.
VI.
Conclusion
By combining investment irreversibilities, capacity constraints, and variable capacity utilization, the putty-clay model developed in this paper
provides a rich framework for analyzing a number of issues regarding
investment, labor, capacity utilization, and productivity. In this paper
we highlight some implications for employment, output, and investment
at business cycle and medium-run frequencies. Compared to standard
models based on putty-putty technology, the putty-clay model displays
a substantial degree of persistence and propagation for both output and
hours in response to shocks to factor costs and technology. And, unlike
standard models, the putty-clay model generates forecastable comovements between labor, output, and consumption consistent with the data.
Finally, owing to the existence of a nonlinear marginal cost curve, the
putty-clay model generates interesting asymmetries with recessions
steeper and deeper than expansions.
Beyond its descriptive appeal, the putty-clay production process is also
found to be empirically relevant for explaining business cycle and medium-run dynamics. Estimates obtained from a two-sector model that
minimize the distance between moments generated by the model and
those obtained from the data place a sizable weight on putty-clay production—on the order of one-half to two-thirds. This finding is robust
to the specification of the shock processes and the choice of moments
used in the estimation. In addition to supporting a major role for puttyclay technology, our results suggest that factor price shocks may be key
to explaining fluctuations, especially at the medium-run frequencies of
two to eight years.
956
journal of political economy
Appendix
A.
Solution Methods
The state space of the model consists of the shocks plus the mean machine
efficiency and total investment for each past vintage, X t2j , Q t2j , j p 1, … , `.
We approximate this state space in one of two ways. For the simulation illustrating
the asymmetric response to the labor cost shock shown in figure 5, an extended
path algorithm based on Fair and Taylor (1983) is used in which the life span
of machines is truncated at 120 quarters (30 years). This method is computationally very expensive. The second method, used for all other simulations and
the estimation exercises, uses a log linearization of the equilibrium conditions,
evaluated at the deterministic steady state. We approximate the weighted infinite
sum of leads and lags of variables by simple time-series processes. For example,
for some variable u and coefficient sequence {a j }1`, we approximate the sum
`
jp1 a ju t2j by the process Vt { u t21 /B(L), where
O
B(L) p b 0 2 b 1 L 2 b 2 L2 2 … 2 bp Lp
and p is finite. For the analysis in this paper we used p p 1 and chose values
corresponding to b0 and b1 that minimize the weighted squared deviations between the approximate time-series representation and the true lag or lead structure. The resulting approximated linear model is solved at trivial computational
cost using the Anderson and Moore (1985) implementation of the BlanchardKahn (1980) method. For simulations of moderately sized shocks, the two methods yield nearly identical answers.
B.
Econometric Methodology
We begin by specifying a stochastic process for the log of output, hours, and
investment in the data that depends on a parameter vector n, which is to be
estimated. Our approach follows Rotemberg and Woodford’s (1996) method of
specifying a tightly parameterized low-order VAR system to characterize the data.
Our data consist of chain-weighted real private nonfarm output, total private
hours, and chain-weighted real nonresidential business fixed investment (equipment and structures) for the period 1960–97. Relative to gross domestic product,
the output and hours series exclude government and farm output as well as the
imputed output obtained from owner-occupied housing. The output and hours
series are thus defined in a mutually consistent manner. With the exception of
investment in structures and machinery by the agricultural sector and the households and institutions sector, the investment series is also consistent with the
output and hours series.
After first removing a linear time trend from the hours series, we assume that
the first difference of the log of output, the log of the investment-output ratio,
and the log of hours, (Dyt, i t 2 yt, h t), can be represented using a two-lag stationary VAR representation. Defining
u t0 p [Dyt, i t 2 yt, h t, Dyt21 , i t21 2 yt21 , h t21], E(u tu t0) p Su ,
0
et0 p [ety, eti, eth], E(etet0 ) p Se, E(etet1s
) p 0 for s ( 0,
we express the stochastic process for ut in companion form as
putty-clay and investment
957
u t p Au t21 1 vt, A p
[IP0] , v p [0e ] , E(v v ) p S ,
0
t t
t
t
v
where P is a matrix of VAR coefficients. The parameter vector y and its associated
variance is then
yp
vec(P)
S ^S
, V p[
[vech(S
)]
0
21
u
e
w
e
]
0
,
S 22
where S22 is a function of Se and the duplication matrix Dn satisfying
Dnvech(Se) p vec(Se) (Hamilton 1994, pp. 301–2).17
Given this specification for the stochastic process for output, hours, and investment, we compute second moments of the first differences of these variables
as well as second moments of the forecastable components of the kth differences,
where the forecast is made conditional on time t 2 k information. To obtain
expressions for these second moments as functions of the underlying VAR parameters, we first express the kth differences Dyt,k , Di t,k , and Dh t,k as functions
of data known at time t and shocks that occur between t and t 1 k. For x p
y, i, h we have
O
k
Dx t,k { x t1k 2 x t p b kxu t 1
d jxvt1j ,
jp1
where
O
O
k
b ky p e 10
Ai, b ki p b ky 1 e 20 (Ak 2 I ), b kh p e 30 (Ak 2 I ),
ip1
k2j
d jy p e 10
As, d ji p d jy 1 e 20 Ak2j, d jh p e 30 Ak2j.
sp0
When we take expectations as of time t, the forecastable components of the kth
difference of x are
Dxˆ t,k { E t{x t1k 2 x t} p b kxu t.
The covariance between the kth differences of x and y is
O
k
jxy,k { E(Dx t,k Dyt,k) p (b kx)Su(b ky)0 1
(d jx)Sv(d jy)0
jp1
whereas the covariance between the forecastable components of the kth differences is
ˆ t,k Dyˆt,k) p (b kx)Su(b ky)0.
jxy,k
ˆˆ { E(Dx
Using these expressions, we define two sets of moments. The first set of
moments,
[
]
0
jDi jDh jDi,Dy jDh,Dy
g1 p
,
,
,
,r ,r ,r ,
jDy jDy jDi jDy jDhjDy Dy Di Dh
17
The “vec” operator transforms an n # n matrix into an n2 # 1 vector by stacking the
columns. The “vech” operator transforms an n # n matrix into an [n(n 1 1)/2] # 1 vector
by vertically stacking those elements on or below the principal diagonal.
958
journal of political economy
includes the standard deviations of Dit and Dht relative to the standard deviation
of Dyt; the unconditional correlations of Dit, Dht with Dyt; and the autocorrelations
of Dyt, Dit, and Dht, respectively. The second set of moments,
g2 p
[
0
jˆiy,k
jhy,k
jˆiy,k
jhy,k
jy,k
ˆˆ
ˆˆ
ˆ
ˆ
ˆ
, 2 ,
,
,
,
2
jy,k
jy,k
jˆi,kjy,k
jh,k
jy,k
ˆ jy,k
ˆ
ˆ
ˆ
ˆ
]
includes regression coefficients and correlations between the predictable components of Dit,k and Dht,k and the predictable component of Dyt,k. The second
set of moments also includes the ratio of the variance of the predictable component of Dyt,k relative to the total variance of Dyt,k. Recognizing that g1 and g2
are functions of A, Su, Sv, and hence the VAR parameter vector y, we obtain
our moment vector of interest: g(y) p [g 10 g 20 ]0. An estimate of the variance of
this moment vector is
Vg p
­g ­g
V .
­y0 y ­y
We estimate the model parameter vector
[
w p a, rA, rv , rh ,
]
jv
jh
,
jv 1 jA jv 1 jA 1 jh
by minimizing the distance between g(y) and gM(w), where gM is the model’s
analogue of g(y). To obtain gM as a function of w, we rely on the fact that our
model solution is linear and may be expressed in the first-order companion
form
X t p A X t21 1 BUt,
0
where E(UtUt ) p I and Xt is a vector of model variables, with yt p e y0 X t, ct p
ec0 X t, and h t p e h0 X t. The moment vector gM may then be computed as a function
of A and B. Because the matrices A and B are (nonlinear) functions of the
underlying model parameters w, the moment vector gM is also a function of w.
Standard errors for w may then be obtained from
Vw p
­g M 21 ­g M
Vg
.
­w 0
­w
(
)
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