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T echnical C hange, D iffu s io n ,
and P ro d u c tiv ity
Jeffrey R. Campbell

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Working Papers Series
Macroeconomic Issues
Research Department
Federal Reserve Bank of Chicago
December 1993 (WP-93-16)

FEDERAL RESERVE BANK
OF CHICAGO

Technical C h a n g e ,

Diffusion, a n d

Productivity

Jeffrey R. Campbell*
November 1993

Abstract
This paper presents a quantitative theoretical framework which addresses
two empirical regularities. The first is the sizable volume of resource reallo­
cation between plants. The second is the large and procyclical fluctuations in
m easured total factor productivity. It investigates the role of capital realloca­
tion in generating productivity fluctuations by modeling plant entry and exit
as a channel for the diffusion of process innovations. Only new plants have
access to the leading edge production process, but the scrap capital from ex­
iting plants is available for their construction. Therefore, concurrent increases
in entry and exit cause m easured productivity to rise. Because it explicitly
accounts for technological heterogeneity across plants, the model is capable
of structuring a sim ultaneous analysis of the plant level and macroeconomic
data. In a version of the model estim ated with only macroeconom ic d a ta , the
measured Solow residual mimics the univariate time series properties of its em­
pirical counterpart. Consideration of heterogeneous p lan ts’ roles in technology
diffusion provides a more reasonable alternative to the interpretation of the
Solow residual as a neutral, exogenous, productivity shock.

1

Introduction

This paper presents a quantitative theoretical framework which simultaneously ad­
dresses two sets of empirical regularities, one found in plant level data and the other
in the macroeconomy. The first concerns the nature of resource reallocation between
plants. Using data on manufacturing plants, Dunne, Roberts, and Samuelson (1988;
* D ep a rtm en t o f E c o n o m ic s, N o rth w estern U n iversity, E v a n sto n , IL 6 0 2 0 8 . T h e a u th o r g ra tefu lly
ack n ow led ges fin an cial su p p o r t th ro u g h an A lfred P. S loan D o cto ra l D isse r ta tio n F ellow sh ip and
research su p p o rt from th e Federal R eserve B an k o f C h ica g o . M a n o lo B a lm a se d a , K y le B a g w ell, V .
V . C hari, Larry C h r istia n o , M artin E ich en b a u m , B o H onore, E k aterin i K y ria zid o u , P a u l M an n on e,
M onika M erz, R o b ert P orter, M arfa L u isa de la T orre, and sem in a r p a r tic ip a n ts a t th e Federal
R eserve B ank o f C h ica g o and a t N o rth w estern U n iv ersity co n trib u ted very u sefu l c o m m e n ts on th is
research. A ll errors are th e a u th o r ’s resp o n sib ility .




1

1989a; 1989b) observed that the volume of resource reallocation between plants is
enormous. Between 40% to 50% of plants operating during a given census year no
longer exist 5 years later. Exiting plants produce nearly 20% of manufacturing output.
The second empirical regularity is the macroeconomic observation that fluctuations
in the Solow (1957) residual are large and procyclical. The standard error of total
factor productivity growth is 0.85%,and its correlation with output growth is 0.80.1
The identification of total factor productivity movements with exogenous, neutral,
technological change is a common feature of research from the real business cycle
literature. The models of Kydland and Prescott (1982), Hansen (1985), and King,
Plosser, and Rebelo (1988a; 1988b) are prominent examples of work which embodies
this interpretation. As Summers (1986) noted, a problem with this understanding
of total factor productivity movements is the lack of obvious candidate sources for
such large shocks. The difficulty in finding technological advances which uniformly
expand all producers’ production possibilities makes this interpretation of total factor
productivity fluctuations problematic. 2
Many examples of technological innovation come to mind, but few of them are
costless to implement. Furthermore, their diffusion throughout the economy’s pro­
duction sector is rareh' instant or uniform. A reasonable explanation of productivity
fluctuations based on technological change must account for these facts. Restricting
the impact of technical innovations to new capital goods is one way of addressing
these issues. In the vintage capital models of Solow (1960; 1962a; 1962b) and Green­
wood, Hercowitz, and Krusell (1992), the only channel for technology diffusion is
the accumulation of new capital goods. A shortcoming of this interpretation is its
inability to explain the observed productivity fluctuations. Because the flow of new
investment is small relative to the stock of capital, fluctuations in the productivity of
new capital can have only minor immediate effects on the Solow residual.
To improve the vintage capital model’s performance along this dimension, this
work examines a second channel for technology diffusion: capital reallocation. Pre­
sumably, reallocating capital from exiting plants to their replacements increases its
productivity. In a study of plant level data from the U.S. manufacturing sector,
Bartlesman and Dhrymes (1992) argue that the reallocation of economic activity
between production sites is an important source of a g g r e g a t e productivity growth.
Reallocating existing capital goods does not change their technical characteristics, so
this can not be a channel for product diffusion. However, production processes do
vary across plants. Focusing on process diffusion, rather than on product diffusion, al­
lows capital reallocation and vintage capital effects to be simultaneously incorporated
into a technology diffusion model.
The large observed volume of plant entry and exit suggests that this is a major
medium for capital reallocation. To investigate its role in the generation of Solow
'Sections 5 and 6 provide details about these estimates.
2For an alternative identification which relies less on a literal interpretation of “technology” , see
Hansen and Prescott (1993).




2

residual fluctuations, the paper integrates a selection model of entry and exit, such
as those of Jovanovic (1982) and Hopenhayn (1992a; 1992b), into a standard, gen­
eral equilibrium, business cycle framework with aggregate uncertainty. In the model
economy, only newly constructed plants have access to the leading edge production
process. The entering plants implement these new production processes with varying
degrees of success. At any time, a fraction of each plant’s capital is recoverable as
scrap. In a competitive equilibrium, plants with productivity below an endogenously
determined threshold exit. Increasing the return to exiting, the plants’ scrap value,
raises the threshold. A higher threshold increases average plant productivity and the
volume of exit. The construction of new plants can use the scrap capital from exiting
plants. As in the model of Greenwood, Hercowitz, and Huffman(1988), the economy’s
only source of aggregate uncertainty is the productivity of new plants. An exogenous
improvement in the leading edge production process increases the return to capital
reallocation, so the pace of entry and exit accelerates. When the productivity dif­
ference between exiting and entering plants is large, the rise in reallocation causes a
sizable increase in measured total factor productivity.
The union of a selection model of entry and exit with a business cycle model
is capable of addressing issues which neither could alone. In contrast to standard
macroeconomic models which use a representative plant to stand for the economy’s
production sector, the framework explicitly recognizes the observed productivity het­
erogeneity among plants. On the other hand, the inclusion of aggregate uncertainty
in a computable model of entry and exit allows an analysis of the cross sectional
productivity response to process innovations. A surprising result of this study is that
a lower cost of reallocation I'educes the model’s ability to generate large productivity
fluctuations. The productivity difference between entering and exiting plants is lower
when the scrap value of old plants is high, so a given amount of reallocation has a
smaller effect on measured aggregate productivity.
Whether the model adequately reproduces the behavior of the observed Solow
residual is an empirical issue. Many of the model’s parameters can be estimated using
macroeconomic data. Others, those governing the uncertainty of process implemen­
tation and the cost of capital reallocation, are most naturally estimated using plant
level data. For example, the parameters governing the plants’ idiosyncratic uncer­
tainty can be estimated using employment and capital asset data from an unbalanced
panel of plants. Because a plant exits when it falls below a productivity threshold, the
estimation easily fits into the Tobit model employed by Heckman (1976; 1979). Be­
cause this data is not readily available, this paper presents preliminary results based
entirely on macroeconomic data and published work with the Longitudinal Research
Database. The model’s parameters are estimated using a just identified version of
Hansen’s (1982) generalized method of moments estimator. The empirical results
are encouraging. In the estimated model, the volume of capital reallocation jumps
between 30% and 35% following an unexpected 1 % improvement in the productivity
of the leading edge production process. This generates an observed Solow residual




3

which resembles a random walk. Following the reallocation, output, consumption,
and investment all respond much as they would to an exogenous technology shock.
The increase in capital reallocation is transitory, only lasting one period. Thereafter,
entry and exit temporarily decline because there are relatively few low productivity
plants remaining. The decline coincides with the increase in output, so capital reallo­
cation is countercyclical. This agrees with the observation of Davis and Haltiwanger
(1990; 1992) that the reallocation of workers across sites is countercyclical.
Wald tests largely support the null hypothesis that the model adequately repro­
duces the Solow residual’s univariate time series properties. The finding that process
innovation diffusion can cause Solow residual fluctuations like those observed in the
U.S. economy provides a more reasonable alternative to their interpretation as neu­
tral, exogenous, technology shocks. Unfortunately, this paper’s explanation shares
a weakness with the real business cycle interpretation: neither of them provides a
reasonable account of productivity decline.3
Computation of the model’s equilibrium required an extension of standard dy­
namic solution techniques to handle plant level heterogeneity. A set of Euler equa­
tions and a transversality condition characterize the model’s competitive equilibrium.
As in the solution method of King, Plosser, and Rebelo (1987), log-linear expansions
around a non-stochastic, steady state growth path approximate the equations. The
continuous specification for plant level heterogeneity implies that they are non-trivial
functional equations. To compute the solutions of this linear dynamical system, the
functional equations are approximated using quadrature methods. This produces a
dynamical system with a large, but finite, dimensional state space. The analysis of
this approximate system can use standard methods, such as those of Blanchard and
Kahn (1980) and King, Plosser, and Rebelo (1987).
The paper’s next two sections describe the model economy and discuss its com­
petitive equilibrium. Section 4 decomposes changes in the traditionally measured
Solow residual into those reflecting capital quantity and capital quality mismeasurement. Section 5 presents the identification and estimation strategy used to evaluate
the model’s empirical performance. In particular, it shows how plant level data could
be used to estimate parameters governing the capital reallocation process. Section
6 presents the results of the empirical analysis. The paper’s last section contains
concluding remarks.
3Other authors have offered different explanations for the procyclicality of observed productivity.
Summers (1986) and Burnside, Eic.henbaum, and Rebelo (1990) and Sbordone (1993) entertain the
hypothesis that firms’ labor hoarding behavior is the source, of these fluctuations. Baxter and King
(1991), Caballero and Lyons (1992), focus on the presence of aggregate externalities. Hall (1988)
showed that imperfect competition in product markets can cause spurious procyclical productivity
movements.




4

2

The M odel

The model of this section differs from a standard equilibrium macroeconomic frame­
work by explicitly modeling plant level technological heterogeneity. As in Solow
(1960; 1962a; 1962b), and Greenwood, Huffman, and Krusell (1992), newly con­
structed plants have access to the leading edge technology. Unlike these models,
previously produced capital goods may be reallocated to new plants. This provides
a second channel for the diffusion of technological innovations through the economy.
The remainder of the model is standard. There are many identical consumers,
who provide the economy’s labor and own its equity. Capital goods are traded in
complete markets. A single representative firm rents all factors of production. Its
manager chooses the allocation of labor among the plants and makes the capital
reallocation decisions. She makes these choices to maximize the firm’s profits. The
constant returns to scale technology ensures that profits are zero in a competitive
equilibrium.
This section describes the technology available to the model economy’s production
sector, the representative consumer’s preferences and endowments, and the physical
environment within which they operate. Section 3 describes the market structure,
presents the agents’ optimization problems, and defines a competitive equilibrium.

2.1

The Production Sector

A continuum of atomistic plants populates the economy’s production sector. This
sector uses labor and plants as inputs to produce the aggregate good. This good may
then be used either for consumption or for the construction of new plants. Each plant
has a capital stock, .s’, and an idiosyncratic productivity level, 6 ‘. The superscript
i denotes variables which are specific to a single plant. The production technology
available to these plants is
y = (sie6'y-ana.

(1)

The plant’s labor input is n and its output of the aggregate good is y . Restricting the
elasticity of output with respect to labor input, a, to lie strictly between zero and one
guarantees that the plant’s technology is strictly concave. Because this technology
obeys constant returns to scale, the plants’ size distribution (as measured by capital)
does not effect the economy’s production possibilities. The production sets available
to a single plant with one unit of capital and N otherwise identical plants each with
•s’ = 1 / N are the same.
Building a new plant requires investing one unit of the aggregate good. When the
plant is constructed, it has access to the leading edge production process. New plants
implement this process with varying degrees of success. Their original productivity
levels reflect this. If the plant is constructed in period t, its initial value of 9 l is
normally distributed with mean z t and standard error a .




0£~yV(zt,a2)

5

(2)

The productivity of a plant with an average implementation of the leading edge
production process is z t . This is an index of the leading edge technology. It follows a
random walk with a positive drift.
zt
e\

=
~

(iz +

zt_i +

e tz

(3)

N ( 0 , < t z2 )

The exogenous technological progress embodied in z t is the model economy’s only
source of growth and uncertainty. Although new plants are o n a v e r a g e more pro­
ductive than existent plants, nothing distinguishes a new plant with idiosyncratic
productivity level O' from an otherwise identical incumbent.
At the end of every period, a fraction, 6, of each plant’s capital is lost to depreci­
ation. The plants may n o t engage in replacement investment. What remains of the
plant may then be used in one of two ways. It may be retired, in which case .s units
of the aggregate good are recovered as its scrap value. The scrap value is positive but
less than one. Alternatively, the plant may be left intact. In that case, it receives an
idiosyncratic shock to its productivity level before the next period.
Q't+1

=

+ £<+i

et+i

~

Ar(0,<r2)

(4)

That is, the plant’s idiosyncratic productivity level follows a random walk. The
unit root in the plant productivity process implies that the level of the leading edge
production technology during its construction, 2 t, will have a permanent effect on its
productivity. In this sense, the model includes vintage capital effects like those in
the models of Solow (1960; 1962a; 1962b) and Greenwood, Hercowitz, and Krusell
(1992). The random walk’s innovation is normally distributed with zero mean and
standard error a . The standard error is the same as that governing the distribution
of the plant’s initial value of 6'. This is not essential and is adopted only for the sake
of parsimony.

2.2

Consumers

There are many identical infinitely lived consumers who value two goods, consumption
and leisure. The population of consumers grows exogenously at the constant rate
fip . Each consumer has a time endowment of one unit each period, which she must
allocate between leisure and labor. The utility function
—n<) g'ves
her preferences over state contingent sequences of these two goods. Her discount
factor is (3, which lies strictly between zero and one. Her momentary utility function,
u(c(, 1 - n(), is
u ( c t , 1 - n t ) = ln(ct) + v(l - n t )
(5)
To ensure the concavity of u(-, •), u(-) is continuous, twice differentiable, and weakly
concave. If v(l —n<) = k ln(l —n * )5 then her preferences take the familiar log-log




6

form. If u(l —nt) = « ( 1 —n(), then her preferences correspond to those derived by
Hansen (1985) in an indivisible labor model. These two preference specifications will
appear below in the model’s computational analysis.
In the model economy, firms and consumers trade labor, the aggregate good, and
capital goods in complete markets. Given the specification of preferences and tech­
nology, the next section describes the market structure governing trade, presents the
agents’ optimization problems, defines an equilibrium for the economy, and discusses
computational issues.

3

Competitive

Equilibrium

In a competitive equilibrium, the economy’s agents trade the aggregate good, labor,
and capital goods of all productivity types in complete markets. This section be­
gins by outlining the model economy’s market structure. Although several different
market structures can support a competitive equilibrium, one is chosen to simplify
the analysis. Presentations of firms’ and consumers’ maximization problems follow
this. It finishes by defining a competitive equilibrium and outlining the computational
method used.

3.1

Market Structure

Three types of agents populate the model economy, production firms, construction
firms, and consumers. They trade capital goods, labor, and the aggregate consump­
tion good in competitive markets. As in the models of Lucas (1978) and Prescott
and Mehra (1980), the equilibrium prices for capital assets are of particular interest.
They are the basis for capital reallocation decisions. Therefore, the market structure
in the model economy is that from Prescott and Mehra (1980).
At the beginning of each period, each consumer owns an identical portfolio of
the economy’s productive assets. They sell these assets and their labor services to
the production firms. Production firms only exist for one period. They produce the
aggregate good with the technology described by equation ( 1 ). After production, the
firms decide which of the surviving plants to keep intact and which to salvage for their
scrap value. Then the firms sell their stock of the aggregate good, consisting of what
they produced and recovered as scrap, to the construction firms and the consumers.
The aggregate good is the numeraire, and its spot price always equals one. It is
perishable, so the consumer must consume her purchases within the current period.
The construction firms also exist for only one period. They purchase the aggregate
good from the production sector and use it to create new plants. At the end of the
period, firms in both the production sector and the construction sector liquidate,
selling their stock of plants to the consumers. To maintain wealth homogeneity, a
fraction of each consumer’s portfolio is taxed and granted to young consumers. In
the next period, nothing differentiates a young from an old consumer.




(

This market structure, as opposed to one in which consumers rented capital goods
to firms every period, naturally prices the economy’s capital assets. 4 The technology
available to firms in both sectors obeys constant returns to scale. Therefore, firms earn
zero profits in equilibrium. As in a standard macroeconomic model, each sector acts
as if it has a single, representative, price taking firm. This division of the production
sector into two representative firms is arbitrary but analytically useful.

3.2

The Production Problem

Given the prices of all plants and the wage rate, the representative production man­
ager hires labor and trades plants to maximize its profits. The size distribution of the
plants does not change the firm’s production possibilities, so without loss of general­
ity, we may consider the case where all of its plants have one unit of capital. Let w t
denote the wage rate in period t and let <?°(0 ) and q ] { 9 ) respectively designate prices
of a plant with one unit of capital and productivity level $ at the beginning and end
of this period. If asset prices equal discounted expected dividend streams, then asset
prices will be increasing in 0. A plant’s scrap value is invariant to its value of 6 , so
the representative firm will chose to scrap only those plants below a threshold, 0 t .
Those plants with productivity levels above the threshold will remain in production.
This threshold scrap rule is similar to those found in Hopenhayn (1992a) and Jovan ovic. (1982). After incorporating the plant retirement decision, the representative
production firm’s profit maximization problem is
max

y + .s (l—S )

f

k(9)d9

J - oo

fc(0),fc(0), n(fl),u

+ jst<ii w m M subject to:

y

»

m

S-oo k ( 9 ) e e^ ~ a h i ( 9 ) a d 9

=

STooWMWO

=

(i-*)j^(*=*)fc(*)<0

(6)

The objective function’s first two terms are the firm’s total output and the value
of its scrap capital. The third term is the value of its stock of capital at the end
of the period. The number of plants of type. 9 which the firm may sell at the end
of the period is k ( 0 ) . The final terms are the cost of its beginning of period capital
purchases and its wage bill. The first constraint on the firm’s problem says that
A dditional markets in state contingent claims on capital assets and the aggregate good could
be added at the expense of considerable extra notation, but equilibrium prices for physical assets
would not change. Because the consumers have identical preferences and endowments, the net trade
in contingent claims must be zero.




8

summing the output of all plants yields the firms total output. The amount of labor
the firm assigns to a single plant with productivity 9 is n ( 0 ) . The second constraint
restricts the total labor allocated among the plants to equal that purchased by the
representative firm. The last constraint gives the stock of plants the firm may sell at
the end of the period as a function of its purchases and scrap decisions. It embodies
the capital depreciation rate S, and the law of motion for 0. At the end of the period,
each of its plants has 1 —S units of capital. These will have the same value to the
firm as 1 —8 plants of the same productivity level, each with one unit of capital. The
function <f>(z) denotes the standard normal p.d.f.
The envelope theorem allows this problem to broken into two steps. First consider
the problem of maximizing the firm’s output given its capital and labor inputs. This
is the labor allocation problem.
/ OO
k ( 0 ) e e^ ~ ° ‘h i ( 9 ) a d 9

-oo

subject to:

n

^

=

k(0 )n (0 )d 0

This problem has a simple and familiar solution. Define the firm’s
to be

effective capital

stock

f e ek (0 )d 0 .
(8 )
J—OO
The effective capital stock is the sum of the number of plants of each type, weighted
by their productivity level. With this notation in place, the solution to the labor
allocation problem is
k =

(9 )

y = k '-ana

A Cobb-Douglas productionfunction in labor ande f f e c t i v e capital represents the
firm’s production possibilities.Using the same functionalform, Solow (1960)derived
this result in a model of vintage capital.
Substituting the solution to the labor allocation problem into the profit maxi­
mization problem yields
rct

=

max

n ,0t,A:(0),fc(0)

+ feT
subject to:

k

m

P ' an" + s(l —8 )

c im m d e ~

/“

[

k(0)d&

J -oo

q°t { e ) m d e

=

al k ( 6 ) d 0

=

(i - < ) j t !«¥)*■•(«)<»

-

wn

(10)

The first order conditions for this problem are




( k V - Q

w

=

Or ( —1

(11)

\n
9

s

r

&

(12)

* )- * £ * ) *

<S(») = (!-«)(£) ”«' + !{»<«.)(!-«)*
+i{» > «,}(i -«)

r

J —oo

i}0)-<K—
(7

(7

(W)
)<ie

The indicator function, 1 {-}, equals one if the condition it contains is true, and
zero otherwise. Equation ( 1 1 ) is a standard labor demand condition equalizing the
marginal product of labor and the wage rate. Equation ( 1 2 ) defines the firm’s optimal
choice of 6 t , the capital reallocation threshold. It says that the scrap value of the
marginal plant must equal the expected end of period asset price if it remains in
production. Equation (13) is an asset pricing equation. It constrains an asset’s
beginning of period price to equal the dividends it returns plus its value at the end
of the period. If the asset is scrapped, this value equals that of the scrap capital.
Otherwise, it equals the expected end of period asset price.

3.3

The Construction Problem

A large number of construction firms use a one to one technology for converting the
aggregate good into new plants. Firms are risk neutral and maximize their expected
profits. In a competitive equilibrium, the firms must earn zero profits. The condition
which guarantees this is
i = r &'(*)-*(—
J—oo
(7
y

(7

W
J

(i4)

The left hand side of equation (14) is the cost of constructing one plant. The right
hand side is the expected price of the completed plant. In this sense, marginal q
always equals one.

3.4

The Consumer’s Problem

Each consumer maximizes her expected utility by choosing state contingent sequences
of consumption, {ct}£2 0 , labor, {«(}£2 0i an<l asset holdings {fc((0 )}£2 j taking wages,




10

a s s e t p r ic e s , a n d h e r in it ia l a s s e t h o ld in g s a s g iv e n .
max

* ! > > ( « ) + « ( 1 _ n 0)
t=o

subject to:

0

=

ct +

ff°o o q l ( 6 ) k t + 1 ( 6 ) d 0

^

- u > t n t - f ? eoq W ) k t (e)d e

given

k0(6)

To maintain wealth homogeneity, a fraction of each consumer’s portfolio is taxed at
the end of every period and granted to the young consumers. The presence of e/ip in
the consumer’s budget constraint reflects this inter-generational transfer assumption.
The mathematical expectations operator with respect to the information set at time
t is E t . In addition to the static budget constraint in (15), the necessary conditions
for a solution to the consumer’s problem are
-« " « !(» )
Ct

=

E ,p —

—w t
ct

=

i / ( l - n t)

c (+1

, f +1(«)

(16)
(17)

Equation (16) is familiar from the asset pricing literature. It says that the expected
product of the intertemporal marginal rate of substitution with the returns to holding
an plant with productivity 6 equals one. It relates an asset’s price at the end of the
current period to its price at the beginning of the subsequent period. Equation (17)
is a standard static labor supply equation. A set of decisions for consumption, labor
supply, and asset holdings will be a solution for this problem if they satisfy equation
(16), equation (17), and the transversality condition,
1 f°°
lim E qP 1— /
q?(0 )k t(6)dd = 0.

*—►00

3.5

Ct J —oo

(18)

Market Clearing, Equilibrium, and Computation

In a com petitive equilibrium the firms’ and consumers’ problems are connected through
the imposition of market clearing conditions.
D e fin itio n 1 A c o m p e t i t i v e e q u i l i b r i u m f o r th e e c o n o m y i s a s t a t e c o n t i n g e n t s e ­
q u e n c e o f p e r c a p i t a c o n s u m p t i o n , {C'e}£f.0, la b o r , { N t } % l 0 , g r o s s i n v e s t m e n t , { I t } f l 0 ,
a s s e t h o l d i n g s , { /v ^ # ) } ^ ,, w a g e r a t e s ,
a n d a s s e t p r i c e s , {<7?( 0 )}t^o and {<?*(0 )}£lo>
such th a t
1.




{Art}£f.0> a n d {A't(0 )}£ 2.j i s a s o l u t i o n t o t h e c o n s u m e r ’s p r o b l e m g i v e n
{<7?(0)}£ o> a n d { q } { 0 ) } Z o -

11

2. I<t(0), e>lpK t+ \{ 6 ) — ^<j>
a t tim e t g iven w t)

qf(0),

l u a n ^ N t so lv e the re p re se n ta tiv e f i r m ’s p ro b lem
a n d q }{6 )-

3. The p ro d u c tio n a n d c o n stru c tio n f ir m s earn ze ro p ro fits.
Jh

The sequ en ces o f p e r c a p ita l a s s e t h oldin gs a n d g ro ss in v e s tm e n t s a tis fy the
c a p ita l a ccu m u la tio n r e str ic tio n .

e -W « ) = ( ■ - * ) £ > ( ^ )

K M * +l* ( ^ )

*

Because the first and second welfare theorems apply to the model economy, its
com petitive equilibrium quantities will correspond to those a social planner would
choose to maximize the utility of a representative consumer.
max

+

}(=0i
fe.WlSo.WWS,

0

subject to:

e--K w

=

t=0

c t+ It- Y t-St

Y, =

N f U Z . e ’K . W M ) ' - "

s, =

( i - s ) s & K lm d t

(0)

=

(1 - l )

+7*

i t

K M

(19)

*

( ¥ ) I>

The solution to this problem is a set of decision rules expressing the social planner’s
choice variables as functions of the current state, A'*(0), and the exogenous shock,
z t . The strategy for com puting the model econom y’s com petitive equilibrium is to
approximate the solution of problem (19).
The solution of similar social planning problems is common in the real business
cycle literature. W hat distinguishes this problem from those previously studied is the
nature of the choice variable, A't(0). Because it is a function rather than a scalar, the
standard solution m ethods are not im m ediately applicable. This hurdle is overcome
by using quadrature approximations of the relevant functional equations. This ap­
proximation reduces a functional dynamical system to a standard vector dynamical
system with a large, but finite, state space. Applying standard m ethods for solving
linear dynamical system s then produces the desired solution.
Eliminating all sources of non-stationarity is the first step in solving a problem
like (19). First note that the center of the distribution K ( 6 ) will continually shift




12

to the right as z t grows. Recasting the problem in terms of K ( 6 —z t ) removes this
source of non-stationarity. The aggregate production function can be re-written as
Yt

=

( e ^ r z‘N t ) a ( f ° ° e e~ Zt K t ( $ -

z t)d O y ~ a

(20)

The Cobb-Douglas functional form for the production function guarantees that tech­
nical change can be written in l a b o r augmenting form, even though it is embodied in
the plants. This technical progress is a source of growth for the economy. Scaling
all of the social planner’s choice variables but hours worked by e ~ z ~Zt, yields a social
planning problem for an equivalent economy which is stationary. This transforma­
tion is familiar from the work of King, Plosser, and Rebelo (1988a; 1988b). Following
Christiano (1988), this is called the star economy. The star economy is locally stable
around its non-stochastic, steady state.
To find an approximate solution to this social planning problem, replace its first
order necessary conditions with log-linear approximations around its steady state.
Because the capital stock is a function rather than a scalar, these approximate first
order conditions are f u n c t i o n a l equations. Quadrature approximations, the evaluation
of which only requires the function’s values at a finite number of points, replace
the functional equations.5 This approximation produces a finite dimensional linear
dynamical system. Although its dimension is much greater than that of a standard
problem, its solutions can be found by applying standard linear algebraic techniques.
The approximate system of equations possesses a continuum of solutions. The
unique one which also satisfies the social planning problem’s transversality condition
is an approximate solution to problem (19). Rescaling the solution by e ~ z ~ Zt yields
the desired approximate solution to the original problem.
The log-linear nature of the approximation method yields decision rules of the
form
Inf/w+.W)

=

S( K /'■<(«),-'<)

ln(A',) =

h lM h -J B j.z,)

<2>)

The functionals g ( - , •) and h ( - , •) are linear in their arguments. Composing g ( - , •) with
itself produces the moving average representation of \ n ( I \ t ( 6 ) ) in terms of z t . The
moving average representations of all the other variables can then be computed by
exploiting the log-linearity of the policy functions. With these in hand, computing
the correlations and standard errors for stationarity inducing transformations of the
endogenous variables is straightforward. A technical appendix (in progress) describes
the computational strategy used here in greater detail.
5See Press, Teukolsky, Vetterling, and Flannery (1992) for an explanation of quadrature approx­
imation of integrals.




13

4

Solow

Residual M e a s u r e m e n t

An econometrician measuring total factor productivity with data on output, Y t , la­
bor input, N t , and ef fective, capital input, /?(, would find no evidence of technology
shocks. However, the measure of capital input used in most traditional Solow residual
accounting exercises is at best a poor approximation of effective capital input. The
capital stock data found in the national income and product accounts suffers from two
shortcomings. First, it does not capture the increase in average plant productivity
resulting from capital reallocation and new investment. This results in mismeasurement of average plant q u a l i t y . Second, variation in the capital reallocation rate will
cause the effective capital depreciation rate to change. The capital input data are con­
structed by assuming a constant rate of depreciation. This will cause mismeasurement
of capital q u a n t i t y . Together, these two forms of effective capital mismeasurement will
cause the measured Solow residual to vary, even absent a technology shock of the sort
found in real business cycle models.
Traditional Solow residual accounting uses data on output, labor input, and a
measure of capital input, K t , to measure total factor productivity as
a:t = ln(Ft) - a ln(Art) -

(1

- o) ln( K t ).

(2 2 )

Under the assumption of competitive labor and output markets, as in the model
economy, the elasticity of output with respect to labor input, a is correctly identified
with labor’s share of output.
The most common source of I \ t is the national income and product accounts. To
construct it, the Department of Commerce applies a perpetual inventory method to
the flow of newly constructed equipment. In the model economy, newly constructed
equipment, N I t , equals gross investment, I t , less the scrap capital from exiting plants,
St.
K t+i = ( l - S ) K t + N I t

(23)

The national income accountant assumes that a fraction of the capital stock, S, is
destroyed every period. Newly constructed equipment replenishes this and adds to
next period’s capital stock.
To decompose the Solow residual into its two sources of measurement error, con­
sider using the true stock of plants, K * =
K t ( 6 ) d 0 . Movements of the Solow
residual constructed with this measure of capital will only reflect shifts of the pro­
ductivity distribution of plants, not the mismeasurement of their number. We can
write, the measured Solow residual as
= (1 - «)(ln(/?i) - In{ K f i ) + (1 - a)(ln(A7) - In( I < t ) ) .

(24)

The i d e a l Solow residual is first term on the right hand side of (24). Its only source
of variation is quality mismeasurement. It is proportional to the log of the average
plant productivity. Exogenous growth of the leading edge technology causes this to




14

grow. Furthermore, changes in the diffusion of the leading edge technology will cause
analogous movement in x t . The remainder is error from quantity mismeasurement.
Changes in the capital reallocation rate will cause both sources of measurement
error to fluctuate. If the response of capital reallocation to shocks in z t is large enough,
their combined effect will produce large fluctuations in the measured Solow residual.
The next two sections assess whether the model adequately reproduces the Solow
residual’s observed behavior. Section 5 presents the identification and estimation
methods used to assign values to the model’s parameters and measure moments in the
data of particular interest. Section 6 reports the results of the empirical analysis. In
the estimated model, the measured Solow residual mimics the random walk behavior
of its empirical counterpart.

5

Identification a n d

Estimation

This section and the next address two quantitative questions. First, can the mea­
surement error associated with technology diffusion adequately explain the observed
Solow residual’s behavior? Second, if this view of technology shocks is correct, can
they account for a substantial fraction of output variance? The econometric strategy
for answering the first question is to jointly estimate the model’s parameters and the
Solow residual’s univariate autocorrelations using a just identified version of Hansen’s
(1982) generalized method of moments estimator. The hypothesis that the model re­
produces the Solow residual’s autocorrelations can be cast as a non-linear restriction
on the estimated parameters and moments. Because they are jointly estimated, Wald
tests of this restriction are then easily constructed. A measure of how well the model
replicates observed output fluctuations is the ratio of output’s standard error in the
model economy to that in the U.S. economy. As in Eichenbaum (1991), this ratio
is introduced as a parameter into the estimation. This allows quantification of the
uncertainty about the model’s performance along this dimension.
The estimation uses six quarterly time series. The first is the non-institutional
civilian population in the United States over age 16. The remainder all cover the
non-farm, non-government sector of the U.S. economy. They include labor’s share
of income, per capita real output (in 1987$), per capita real measured capital input,
and two measures of per capita hours worked, household hours and establishment
hours. The sample period begins in the first quarter of 1965 and ends in the last
quarter of 1987. Data extending through the first quarter of 1993 is available, but is
not yet entered into the computer. The data appendix contains more details about
data sources and construction.
Throughout this exercise, three of the model’s parameters will be held fixed. The
consumers’ subjective discount rate, /?, is set equal to 1.03-1/4. This implies a 3%
annual risk free interest rate on the steady state growth path of an analogous econ­
omy with no aggregate uncertainty. The capital depreciation rate, 6 , is set equal
to 0.025. Finally, the fraction of a consumer’s time endowment spent at work in




15

non-stochastic steady state is set equal to 1/3. The last two parameters are set to
maintain comparability with the real business cycle literature.
The model’s remaining parameters can be divided into two groups. The first are
familiar from the real business cycle literature. They include the elasticity of output
with respect to labor input, a , the mean and standard error of technological progress,
p. and cr2, and the rate of population growth, pp. These can be identified using
only restrictions involving macroeconomic observables. The remainder describe the
productivity distribution of the production sector and the cost of capital reallocation.
With plant level data, their estimation would be relatively straightforward. However,
without access to a data set such as the Longitudinal Research Database. (LRD)6,
estimation requires an alternative strategy. The preliminary empirical investigation
uses only macroeconomic data and the results of published empirical work with the
LRD.
Identification of the first set of model parameters and second moments of the U.S.
economy is the topic of the subsection 5.1. That which follows details how the second
set could be estimated with plant level data and explains the procedure actually used
in its absence. Subsections 5.3 and 5.4 set forth the joint estimation and testing
procedures respectively.

5.1

Macroeconomic Identification

In addition to the standard error of output growth a y , the population second moments
of interest are the standard error of Solow residual growth, crx, the contemporaneous
correlation between these two variables, p xy, and Solow residual growth’s first four
autocorrelations, p \ ... p 4 . With accurate data on per-capita output, per-capita hours
worked, and the measured per-capita capital input, these can be easily estimated using
the orthogonality restrictions which define them. Data on output and capital input
are readily available. There exist two measures of hours worked. The first is from the
Bureau of Labor Statistics’ survey of establishment payrolls. The second is derived
from the Current Population Survey. These are referred to as establishment hours
and household hours respectively. Substituting these two series into equation (2 2 )
yields the establishment and household Solow residuals. Use e x t and h x t to denote
them.
Following Prescott (1986) and Christiano and Eichenbaum (1992), the estimation
allows for the possibility that establishment and household hours are error ridden
measures of true labor input. Because the primary data sources are different, it is
reasonable to suppose that the two measurement errors are independent. In this
case, the cross moments computed with e x t and h x t consistently estimate the vari­
ance and autocorrelations of the true, unobserved Solow residual. In the presence of
measurement error in hours worked, the following orthogonality conditions identify
6 For a description of this data set, covering the population of manufacturing plants, see McGuckin
and Pascoe (1988).




16

t h e p o p u la t io n s e c o n d m o m e n t s o f in t e r e s t f r o m t h e U .S . e c o n o m y .
E [ A y t - f i y]

= 0

E [ A e x t — flex]

= 0

E [ A h x t — fikx ]

= 0

E [ ( A y t - fiy)2

- oj]

= 0

E [ ( A e x t - f i ex) ( A h x t -

f i hx

) - ct'
2]

=

£[(At/< - fly)(AeXt- flex) ~ PxyCTxCTy]
E [(A h x t

- f i h x ) ( A e x t - i - f iex) -

E [(A hxt

-

f i hx) { A e x t - 4 - f i ex)

-

pl^l]

0(25)

=

0

=

0

=

0

The mean growth rates of output, the establishment Solow residual, and the household
Solow residual are /zy, /xex, and fijlx respectively. Although the model implies that the
three growth rates are the same, they are left unconstrained in the estimation.
Simple first moment restrictions provide identification for three of the model’s
parameters, a, //r, and \ i p . Under the assumption of competitive labor and output
markets, the elasticity of output with respect to labor input equals labor’s share of
income. This allows the identification of a with the restriction
E [a -

w tN t

Yt

(26)

] = o.

The model economy’s average population growth rate is
with its empirical analog.
E[Apt- pp]= 0

fiv .

This is easily identified
(27)

As noted above, the economy’s only source of per capita growth is the exogenous
advancement of the leading edge technology. The growth rate of output will equal
that of technological change when expressed in l a b o r augmenting form. This implies
that fiz can be identified with the condition
fiy - -— -fiz = 0.
a

(28)

The log-linear approximation of the social planning problem’s Euler equations
and their numerical solution are invariant to the value of a . . Because the endogenous




17

variable’s correlations are scale invariant, they can be calculated using their moving
average representations without taking a stand on the size of the unobserved tech­
nology shocks. This is not true when considering the standard error of output. In
the real business cycle literature, the technology shock’s standard error is chosen to
match the Solow residual’s. A similar procedure is available to estimate the standard
error of this model’s unobserved technology shocks. This is chosen so that the stan­
dard error of the traditionally measured Solow residual in the model economy is the
same as that in the U.S. data.
<7 x

-

V £az

= 0

(29)

The relative volatility of the measured Solow residual’s growth rate and that of the
true technology shock in the model economy is V™. This is a function of the model’s
parameters, but this notation is suppressed for convenience. To calculate the mea­
sured Solow residual from the model economy, the construction of the NIPA capital
stock data must be simulated. This was done using a log-linear approximation of (23)
along the non-stochastic, steady state growth path. The assumed depreciation rate,
<5, was chosen so that the capital measure is correct along this path.7
With the estimate of <j z in hand, the definition of A can provide its identification.
*yA -

5.2

V™ os

= 0

(30)

M icroeconom ic Identification

The scrap value of old firms, s, and the standard error of the idiosyncratic productiv­
ity shock, cr, do not have obvious empirical counterparts in the macroeconomic data.
However, with a plant level data set, they can be estimated using the model’s implica­
tions for the productivity and employment distribution across plants. The following
discussion presumes that the Longitudinal Research Database (LRD) is available for
the estimation.
Consider cr, the standard error of the plant level productivity innovation. If two
adjoining quarterly observations on plant level value added, capital, and labor input
were available, then cr could be easily estimated with the sample standard error of
productivity growth across those plants which survived the first period. This strategy
has two shortcomings. First, such data is not available in the LRD. Second, the
assumption that 6\ follows a random walk implies that the unconditional distribution
of a plant’s growth rate is the same as that conditional upon its survival. It is not
hard to modify the model so that this condition is violated but its macroeconomic
implications are the same. To estimate cr using a cross section of plant level data,
both of these problems must be overcome.
‘In the estimated models, this involves only a small adjustment to 6, the true rate of capital
depreciation.




18

To deal with the first problem, considerr the representative firm’s labor allocation
problem. Its solution implies that employment at a plant with productivity 0\ and
capital stock s \ equals
N f = e e‘s \ N t / K t
(31)
The model implies a one to one correspondence between a plant’s labor input, (ad­
justed for its size) and its productivity level. This allows inference about the process
governing 6\ using only data on N \ and s \ . Now consider a plant in production in
period t that survives until period t + 1. The growth rate of its employment to capital
ratio over this period will be
n’<+1 -

n\ = e \+l

+ ln(JVl+1//? t+1) - ln(JV,//?t)

(32)

The logarithm of the plant’s labor to capital ratio is n \ . The first term reflects
the contribution of the idiosyncratic shock to employment growth at the plant. The
remainder is the growth rate of the aggregate labor to effective capital ratio over this
period. The LRD contains annual observations on a plant’s beginning and end of
period capital stock as well as quarterly observations on hours worked. This allows
the construction of two adjoining quarterly observations, from the last quarter of one
year and the first quarter of the next, of n \ . Equation (32) can substitute for equation
(4) to provide an estimate of a .
This alternative estimator is implementable, but it also suffers a sample selection
problem. As Heckman (1976; 1979) showed, if the distribution of a plant’s growth
rate conditional upon its survival is different from the unconditional distribution,
then the regression equation (32) is misspecified. The usual estimates of k and a are
inconsistent. If a plant’s productivity level follows a random walk, this will not be a
problem. To change this without effecting the model’s macroeconomic implications,
add a temporary, idiosyncratic productivity disturbance. That is, let 6\ be plant i ’s
productivity level at time t . It is the sum of two components, a permanent part and
a transitory part.
*i = * ; + 4
(3»)
The permanent component obeys the same stochastic process as it does above. The
temporary component is i . i . d . across plants and across time.
(34)
The rule for labor allocation across plants remains the same. Plant z’s labor input is
a function of 0\ and s \ .
N i ( 6 ) = e §i‘s \ N t / K t
(35)
This distribution of i]'t across plants is time invariant. Because rft is also independent
from 6 \, the representative firm’s aggregate production function is unchanged up to a
multiplicative constant. Furthermore, the asset pricing equations, (13) and (16), the




19

zero profit condition on entry, (14), and the exit threshold, (12), remain unchanged.
That is, only the distribution of the permanent component has aggregate implications.
The existence of rj\ breaks the simple relationship between a plant’s productivity
growth and the growth of its labor to capital ratio. Modifying equation (32) to reflect
this yields
tt’t+i —nt = £I+i +

Vt+i

—Vt + hi(-N<+i/K t+ 1 ) —In ( N t / K

t)

(36)

The variance of the right hand side of (36) is a 2 + 2a'2, so using the cross sectional
standard error of adjusted labor growth to estimate a is inappropriate.
Adding temporary idiosyncratic productivity shocks also weakens the tight con­
nection between a plant’s observed productivity level and its exit behavior. It is still
the case that a plant will exit if 6\ < 6 t . However, the exit condition changes when
written in terms of observables.
n \ ~ Vt

<

+ ln(ATt//? t)

(37)

This induces a sample selection problem. The error term in inequality (37) is cor­
related with a plant’s observed adjusted labor growth rate. This implies that the
distribution of the growth rate conditional on survival is not the same as the uncon­
ditional distribution. Therefore, the sample variance of adjusted labor growth is not
even a consistent estimator of cr2 -f 2 a 2.
In the presence of temporary idiosyncratic, productivity shocks, the use of ordinary
least squares to estimate equation (32) is inappropriate. However, the application of
maximum likelihood methods yields consistent and asymptotically efficient estimates
of a and a , r In the taxonomy of Amemiya (1985), equation (36) and inequality (37)
form the basis of a type 2 Tobit model.
To use the type 2 Tobit framework, rewrite equation (36) and inequality (37) in
terms of unobserved dependent variables, the plant’s adjusted labor growth rate and
the difference between its permanent productivity level and the exit threshold, and
observed regressors, the plant’s current labor to capital ratio and a constant.
=

Ki

+ e,i

y?2 =

k2

+

Vi i

x i2

y *i

yn

=

0

+ £«'2

(38)

»fy*2>o
otherwise

The plant’s (possibly) unobserved growth rate is y* j. The difference between 6 \ and
8 t is y*2. The exit threshold in period t and growth rate of the aggregate capital to
labor ratio from t to t + 1 are subsumed in K\ and « 2 - The plant’s observed labor to
capital ratio in the first period is X i 2 - The error terms are idiosyncratic to the plant,




20

and so are

i.i.d.

across observations.
Vti
y*
x ,12

= n!+i -

n\

—

-

=

n*

£*i = et+i +
12

ac2

—

0 t-

Vt+1

Vt

-vi

=

\ n ( N t+ \ / h t + i ) i - H N t / K t )

=

- 0 t ~ H N t/ K t )

covariance matrix of the disturbance vector,
E [eiii'}

=

erf

cr12

crv2

cr'j

—

ii

=

v 2 + 2<x>

(39)
\

Divide the sample of plants in production during period t into two groups, those
which exit before period t + 1 and those which remain in production. The first N o
observations are in the former group, and there are Ar total observations. Using
equation (10.7.6) in Amemiya (1985), the model’s likelihood function can be written
l

=

n & l i - * ( £ ( < * + **))]

x

n ^ A.0+1<I>

+ i i2) +

- «i)}

(40)

The function $(•) is the c.d.f. of a standard normal distribution. After imposing the
restrictions on <Ti, ct2, and <r12 from (39), the function may be maximized over aci,
ac2, ct, and a 1} to produce consistent and asymptotically efficient estimates of these
parameters.8
In the absence of the necessary plant level data, this estimation strategy is in­
feasible. The preliminary empirical work presented in section 6 used a value of a
derived from previous work with the LRD. Hopenhayn and Rogerson (1993) report
the results of an ordinary regression analysis on an equation like (32). There are
8Note that in this case, crin the standard error of the selection equation, is identified. This is
because the coefficient on the plant’s adjusted employment in this equation is known a priori to
equal one.




21

three differences between the equation they estimate and (32). First, instead of using
the ratio of hours to the capital stock, they regress employment on lagged employ­
ment and a constant. Second, the span of time is five years instead of one quarter.
Third, it does not impose a random walk on the process for 9 . Although it is clearly
inappropriate for the present purposes, this is the best available estimate of a . The
implied estimate of a from this work is 0.0364.
The cost of reallocating capital to a plant with the leading edge technology is 1 —s.
Raising this cost will, all else equal, lower the threshold below which a manager retires
a plant for its scrap capital. The change in the scrap decision will imply changes in
both the rate of capital reallocation and the cross sectional productivity distribution.
This suggests that exit rates, productivity levels, or other functions of the cross
sectional productivity distribution could be used to identify s.
To build intuition for how changing s effects the productivity distribution across
plants, figure 1 displays the probability distribution of 9 —f i z t and the exit threshold,
9 t —f i s t along the steady state growth path for three different values of s. For these
calculations, a was set to equal labor’s average share of income in the non-farm private
economy, f i: was set so that the average rate of growth in the model equaled that of
output in the data, f i p was set to the average population growth rate, and a = 0.0364.
The next subsection contains details the data used to produce these estimates.
Lowering .s causes plants to delay exit. This reduces the volume of capital realloca­
tion, and increases the productivity advantage that a newly constructed plant enjoys
over incumbents. Table 1 reports statistics calculated from these distributions. The
capital reallocation rate, the fraction of plants exiting in any period, doubles when s is
increased from 0.8 to 0.9. The difference between the productivity of the leading edge
process and that of the production sector’s average plant declines comparably. Two
forces influence this difference: a selection effect and a vintage capital effect. The. exit
of less productive plants will decrease the difference, and the exogenous improvement
of new plants will increase it. Clearly, the latter effect dominates. The average plant’s
productivity level is that of the leading edge technology one to three years earlier.
Decreasing .s depresses the selection effect, and so increases this difference.
In the absence of data on plant entry and exit, indirect evidence must be employed.
The model incorporates two channels of technology diffusion, capital reallocation and
new plant construction. The volume of new investment is small relative to that of
the total stock of plants. Therefore, if the accumulation of new plant’s is an impor­
tant channel of technology diffusion relative to capital reallocation, then productive
innovations will only gradually diffuse throughout the production sector. Because s
determines the cost of using capital reallocation, it is reasonable to suppose that it
would effect the relative importance of the two technology diffusion channels. Even
in the absence of rapid technology diffusion, output is responsive to changes in z t .
An innovation in the leading edge production process will change the relative price
of investment (in efficiency units), and consumption. The intertemporal income and
substitution effects associated with this price change will cause output to fluctuate




22

even if technology diffusion is gradual.
This intuition suggests that it is possible to identify s using a second moment from
the macroeconomic data. Consider one, the contemporaneous correlation of output
growth and Solow residual growth. Figure 2 plots this moment versus s over the range
[0.75,0.95]. When s lies below this range, the volume of exit is ridiculously small.9
If s is close to one, the impact of selection on average plant productivity overcomes
the vintage capital effect, so new entrants are less productive than incumbents. In
this case, increases in investment actually decrease average productivity. 10 Clearly,
over much of the parameter space, the correlation is decreasing in .s. To understand
the intuition for this, consider the response of technology diffusion to an innovation
in the leading edge technology. As noted above, increasing s raises the productivity
threshold below which plants exit. This increases the volume of capital reallocation,
so a given percentage change in its value will have a greater impact on average plant
productivity. On the other hand, the average productivity of exiting plants is higher,
so any given amount of reallocation will increase average productivity less. Over most
of the parameter space, the second effect dominates the first. The impact of a shock
to z t on average plant productivity is decreasing in s . Only when s is close to one
does decreasing the cost of diffusion increase the volatility of effective capital input.
Although it is preferable to work with plant level data, this variation in the corre­
lation of output growth and Solow residual growth allows the identification of .s with
the moment restriction
P * V - P : y = 0.
(41)
The first term is the correlation of output growth and Solow residual growth estimated
from the data. The second term is the analogous statistic implied by the model.

5.3

Estim ation

Collect the orthogonality conditions, (25), (26), (27), (28), (29), (30), and (41) into
a vector equation.
£[<7(7 o, A't)] = 0
(42)
The vector 7 is a parameter vector containing the model parameters and population
moments of interest. The true value of 7 is 7 0 . The operator E[ - ] denotes uncondi­
tional expectation. The vector X t contains the data required to evaluate the function
9\Vhen s = 0.75, the exit rate along the non-stochastic, steady state growth path is 0.007.
10When s = 0.95, there is no productivity difference between the average incumbent and entering
plants.




23

O'
Pv
Pz
0z

A yt
A kt
Aen t

A
5

<rx

x t=

Pxy
p\

Ayt_4
Afct_4
Aeti(_4
Ahnt

(43)

A pt

.4

Px
fly

l$t

p tx
phx

The growth rates of per-capita establishment and household hours worked are Aent
and A hnt respectively. Labor’s share of output is lst, and the population growth rate
is A p t . The dimension of </(•, •) equals that of 7 , so the parameters are just identified.
In this case, Hansen’s (1982) GMM estimator satisfies the sample analog of (42).
(44)
1 t- 0

The function g ( •, •) is continuous and differentiable in all of its arguments. Under
the hypothesis that the model is the true data generating process, X t is a stationary,
normally distributed process. If it can be shown that the condition (42) is only
satisfied at 70, then the consistency and asymptotic normality results of Hansen (1982)
can be applied to give the distribution of 7 . Conditional on a value of s, there exists a
unique solution to the remaining orthogonality conditions in (42). Figure 2 suggests
that if the parameter space is restricted to exclude large and small values of s, then
a unique value of s satisfies equation (41).11
With this caveat, y / T (7 —70) is asymptotically normally distributed.
V

T (

7

-

70 )

~

Af(0, E , )

(45)
e7

=

t - ust 11

The matrix T =
an(i § is the spectral density matrix of <7(70, X t ) eval­
uated at frequency zero. To compute the variance-covariance matrix of 7 , T is es­
timated with its sample analog and S with the procedure suggested by Newey and
West (1987).
11 With the model specifications estim ated, there was always a unique, solution to the sample
analog of (41).




24

5.4

Testing

In a standard model of vintage capital, like that of Greenwood, Hercowitz, and Krusell
(1992), the only channel for technology diffusion is the accumulation of new capital.
Because the flow of investment is small relative to the stock of capital, this implies
that technical innovations only gradually diffuse through the economy. In this case,
total factor productivity growth is positively autocorrelated. The observed Solow
residual’s growth rate is uncorrelated with its lagged values. Simple vintage capital
models fail to replicate this observation. The model of this paper adds a second
diffusion channel, capital reallocation. Unless this channel is relatively important, the
model will also fail to predict that the Solow residual follows a random walk. This
suggests that a reasonable strategy for evaluating the ability of capital reallocation
to account for observed total factor productivity fluctuations is to test whether the
model can reproduce the Solow residual’s autocorrelation structure.
The null hypothesis that the model reproduces the Solow residual’s autocorrelation
structure can be written as
~ P f
pI - p T

= 0
= 0

P i ~ P llj

= 0.

p\

H0 :

(46)

The j ’tli autocorrelation of the Solow residual implied by the model is p"u . The
analogous population moment is p3x . This null hypothesis can be expressed as j
continuous and differentiable non-linear restrictions on the elements of 7 .
Ho

: f?(7 o) = 0

(47)

As Ogaki (1992) shows, applying the delta method and (45) yields the asymptotic
distribution of \ / T R { 7 ) under HoV T R (7 )

~

A'(0,E«)
(48)

Eh =

I ^ l i 1

The matrix T/j is the gradient of R (-) with respect to 7 . Under H o , the standard
distributional result about quadratic forms of normally distributed vectors implies
that the Wald test statistic is asymptotically y 2 distributed with j degrees of freedom.
T R ( j Y H ; 'R ( j ) ~ x ‘

(49)

The next section reports the results of applying these Wald tests to the estimated
model when j = 1 ,..., 4.




6

E m p iric a l R e s u lts

This section contains the results of applying the empirical methodology detailed in
the previous section to the U.S. data. The estimation procedure was applied to two
different model specifications. In the first, the consumers’ momentary preferences are
log-linear in leisure. In the second, they are linear in leisure. These are referred to as
log and Hansen preference specifications respectively. In both of the estimated mod­
els, capital reallocation is an important channel for technology diffusion. The volume
of entry and exit increases following a positive shock to z t . This produces a sizable,
permanent increase in the measured Solow residual. Thereafter, the innovation grad­
ually diffuses through the production sector through the accumulation of new capital
goods. The model’s Solow residual adequately mimics the dynamic behavior of its
empirical counterpart. The Wald tests provide very little evidence against the null
hypothesis that the autocorrelations of Solow residual growth are the same in the
model and U.S. economies.

6.1

P roductivity in the U .S. Econom y

The macroeconomic phenomenon which motivates this work is the large, procyclical
movement in the Solow residual. Table 2 provides estimates of the standard errors
of total factor productivity and output growth, the contemporaneous correlation be­
tween these two variables, and the first four autocorrelations of Solow residual growth.
It also provides the estimates’ standard errors. Quarterly fluctuations in Solow resid­
ual growth are nearly as large as those of output growth. The standard errors of both
are close to 1%, and the estimated correlation between the two is 0.81. Individually,
the autocorrelations of Solow residual growth are all insignificantly different from
zero. This is the foundation of the observation that the Solow residual essentially
follows a random walk.

6.2

The E stim ated M odels

Tables 3 and 4 report the estimates of the model’s parameters for both preference
specifications. By construction, the parameter estimates for a, jip , and fxz are the same
across the specifications. Table 3 reports their estimates standard errors. Labor’s
average share of output is 2/3. Annual population growth averages 1.6%. The average
annual rate of per capita output growth over this period is 1.5%. Equation (28)
implies that the productivity of the leading edge production process grows at double
this rate.
The parameter estimates for s, cr-, and A differ across specifications. Table 4
reports their estimates and standard errors. Both specifications indicate that the cost
of reallocating a unit of capital between plants is about 0.11 units of the aggregate
good. This implies that the productivity distribution along the non-stochastic, steady




26

state growth path will be nearly identical in the two specifications. Figure 3 graphs
this distribution when s = 0.89. Table 5 reports its population statistics. The
distribution is slightly skewed and centered to the left of the leading edge productivity
level. The average plant’s productivity is that obtainable with the leading edge
production process one year earlier. However, the marginal exiting plant is 24%
less productive than the average entrant. About 2% of all plants exit each quarter.
In the model economy, a positive shock to the leading edge production process
lowers the price of effective investment in terms of consumption goods and leisure.
As Barro and King (1984) and Greenwood, Hercowitz, and Huffman (1988) have
noted, this will have intertemporal income and substitution effects for a represen­
tative consumer. The substitution effect raises investment, raises labor effort, and
lowers current consumption. The income effect from the price change works against
this. The upper panels of figures 4 and 5 plot the responses of output, consumption,
net investment, hours worked, and average labor productivity to a unit impulse in
z t . In the estimated models, the substitution effect dominates. With the log prefer­
ence specification, consumption slightly declines, hours worked increase 0.25%, output
increases 0.16%, and investment increases 1%. With the Hansen preference specifi­
cation, labor supply is much more elastic.. Accordingly, the signs of the variables’
responses are the same, but their magnitudes are greater.
An improvement in the leading edge production process also influences the rep­
resentative firm’s plant retirement decisions. By increasing new plants’ productivity,
process innovations make capital reallocation more attractive. This is its primary
effect. The lower left panels of figures 4 and 5 plot the responses of entry and exit to
a one percent shock to z t . In both models, a one percent innovation to the leading
edge technology causes a 30% to 35% temporary increase in the volume of capital
reallocation, .S'<. The following period, very few low productivity firms are left, so
reallocation temporarily declines. The surge in capital reallocation moves capital
from the lowest productivity plants to the highest. This has noticeable effects upon
the measured Solow residual. The lower right hand panel of the figures plots the
response of the ideal Solow residual and that measured using the NIPA capital stock
to a shock to z t . The increase in average plant productivity causes a 0.45% jump in
the ideal Solow residual. However, the increase in the depreciation rate significantly
dampens this effect. The measured Solow residual increases only 0.15% following the
reallocation. Thereafter, it slowly climbs to a permanently higher level.
In the period following the capital reallocation, the resulting productivity increase
causes macroeconomic aggregates to react much as they would in an RBC model
to a standard technology shock. Output appears to follow a random walk. There
is a temporary increase in investment, and consumption slowly grows to its new
permanent level. Hours worked rise temporarily and then fall back to their steady
state. As would be expected, hours react more severely when the preferences are
linear in leisure.
The timing of reallocation and output growth produces an interesting result. Al­




27

though the productivity increase from capital reallocation causes output growth, exit
is co u n tercyclical. After the surge in exit, few low productivity plants remain. This
causes both entry and exit to decline one period after a shock to z t . At the same
time, the increase in average plant productivity causes output to expand. In the
estimated model with the log preference specification, the correlation between out­
put growth and exit growth is —0.40. With the Hansen preference specification,
it is —0.43. In studies of job reallocation across manufacturing plants, Davis and
Haltiwanger (1990; 1992) observed that the reallocation of jobs across manufactur­
ing plants is countercyclical. The selection model with stochastic production process
improvement naturally reproduces the finding of countercyclical resource reallocation.
The measured Solow residuals from the model economies do not follow a random
walk, but they are not dissimilar from one either. Table 6 presents the first four
autocorrelations of Solow residual growth in the estimated models. In both cases,
they are all very close to zero. Wald tests of the null hypothesis, (46), for j = 1 ,... ,4
confirm this judgment. Table 7 presents these test statistics. With the log preference
specification, the test statistics never exceed conventional significance levels. With
Hansen preferences, the test statistic exceeds the 10% significance level when j = 4.
In the other cases, the tests fail to reject the null hypothesis.
The response of the measured Solow residual to an innovation in z t is weak relative
to that of output. The estimates of A and cr~ in table 4 reflect this. In order to match
the Solow residual’s standard error, the underlying technology shock must fluctuate
6% per quarter with the log preference specification. With the Hansen preferences,
this estimate is only slightly lower. Although these seem large, they are not ridicu­
lously so. In a standard RBC model with a Gobb-Douglas production function, the
standard error of capital embodied technological change must equal 0.255% to match
crx . The large point estimates of A reflect the estimates of cr. The standard error of
output growth in the model with log preferences is 1.5 times its value in the U.S.
economy. The analogous statistic for the model with Hansen preferences is 2.3. It
would be ridiculous to say that the model accounts for 230% of the observed fluctu­
ations in output. The statistic 1/ A has a more sensible interpretation. This is the
fraction of Solow residual fluctuations which the model reproduces when crz is set to
match the observed volatility of output. With this identifying assumption for cr., the
log and Hansen specifications account for 66% and 43% of Solow residual volatility
respectively. Neither of these identifying assumptions for cr, is satisfactory. An esti­
mate of cr, based on the volatility of capital reallocation in the U.S. economy, would
provide a sounder judgement of whether capital reallocation and technology diffusion
are responsible for significant fractions of output and productivity fluctuations.

6.3

P roductivity D ynam ics and R eallocation C osts

In light of the seemingly precise estimates of the model’s parameters in tables 3 and
4, it seems odd that the estimates of A have such large standard errors. The source




28

of this imprecision is the sensitivity of the observed Solow residual to changes in 5 .
Figures 6, 7, and 8 illustrate this by graphing the Solow residual’s response to process
innovations for three different values of s . In figure 6, 5 = 0.8944, its estimate with
the log preference specification. The standard error of this estimate, 0.016 is added
and subtracted from this baseline value in figures 7 and 8. In the estimated model,
the Solow residual increases 0.13% after a 1% innovation in z t . After the reallocation,
output has increased 0.4%. Lowering s increases the response of the Solow residual
without greatly changing the path of output. When s = 0.878, the Solow residual
increases by 0.16% in the period after a technology shock. On the other hand, when
.s = 0.91, this is dampened to 0.09%.
Increasing plants’ scrap value has two effects on the response of productivity
to increases in capital reallocation. First, if s is higher, then plants with higher
productivity are retired. Second, a greater volume of plants will be retired. The first
effect implies that a given amount of capital reallocation will have le ss of an impact on
average plant productivity. However, the greater volume associated with the second
effect may overcome this. In the estimated model the first effect dominates the second.
The upper right panels of figures 6, 7, and 8 confirm this. Increasing .s dampens the
response of effective capital input, A'*, to innovations in z t . The greater volume of
capital reallocation can not counter the smaller average productivity improvement,
but it does cause the volume of capital lost during reallocation to increase. This is
in spite of the decline in the average capital lost at each retiring plant. These two
effects combine to dampen the measured Solow residual’s fluctuations and leave the
ideal Solow residual unchanged.
Examining larger changes in .s makes the influence of capital reallocation costs on
productivity dynamics more apparent. Figure 9 is analogous to figure 4, but with s =
0.925. After an improvement in the leading edge production process, the measured
Solow residual slowly rises. In no way could this reproduce the movements of observed
total factor productivity. Figures 10 and 11 sharply contrast with this. They graph
the model’s responses when .s = 0.85 and s = 0.75 respectively. In the first case, the
model’s behavior is qualitatively the same, but the influence of capital reallocation
on the Solow residual is much stronger. Measured total factor productivity increases
0.2% following a 1% innovation in z t . When s is further lowered to 0.75, the standard
macroeconomic aggregates react to technology shocks in a different fashion. Capital
reallocation causes the measured Solow residual to jump almost instantly to its new,
long run level. The income effect associated with this productivity increase causes
hours worked, investment, and output to f a ll in the period of a process innovation. In
the following period, they rise as they do in the estimated model. Because the Solow
residual is much more responsive to process innovations, the model requires smaller
shocks to replicate a x . In this case, A = 1.25.




29

7

C o n c lu d in g R e m a rk s

Although the empirical results are tentative, they are also promising. In principle, the
endogenous diffusion of process innovations through capital reallocation is capable of
reproducing the observed Solow residual’s behavior. Exogenous advancement in the
leading edge technology causes the volume of capital reallocation to drastically rise.
The shift of resources from the least productive plants to those with access to the
leading edge technology causes measured total factor productivity to suddenly and
permanently rise. Tests of the hypothesis that this reproduces the Solow residual’s
random walk behavior can not reject it.
The interpretation this work gives the Solow residual stands in contrast to that
of the Real Business Cycle literature. Authors such as Kydland and Px-escott (1982),
Hansen (1985), and King, Plosser, and Rebelo (1988a; 1988b) identify it with an
exogenous, neutral technology shock effecting the economy’s entire pioduction sector.
The explanation for the Solow residual fluctuations based on technology diffusion
has two advantages over this simple model of technological improvement. First, the
technology shocks only effect a subset of plants, those under construction. In this
sense, the model relies less on exogenous changes in the aggregate production set
to generate fluctuations. Second, the model’s microeconomic implications allow the
application of plant level data to test the theory.
One such implication concerns the cyclical behavior of resource reallocation. Be­
cause it eliminates low productivity plants, the increase in reallocation causes a re­
duction in exit the following period, when output growth increases. This produces a
negative correlation between the growth rate of exit and that of output. Although
there are no jobs in the model, this provides an interesting perspective of the obser­
vation by Davis and Haltiwanger (1990; 1992) that inter-plant job reallocation in the
manufacturing sector is countercyclical.
The ability to address microeconomic empirical issues in the context of macroe­
conomic. fluctuations is one of the model’s strengths which the preliminary empirical
analysis did not fully exploit. Using plant level data in estimation would produce
more reliable estimates of the model’s microeconomic parameters and test a selection
model of entry and exit. This will be the focus of future empirical work. Producing
an accurate assessment of the model’s ability to reproduce, the observed total factor
productivity fluctuations is difficult without this analysis.
8

D a ta A p p e n d ix

The empirical work used six quarterly time, series: The United States’ population, real
GDP, The NIPA measure of the capital stock, hours worked from the establishment
survey, hours worked from the household survey, and labor’s share of output.




30

• The population measure is the civilian non-institutional population over age 16.
It is produced by the Bureau of Labor Statistics.
• Real GDP is measured as seasonally adjusted total GDP in 1987$ less the output
of farms and government.
• The capital stock, as reported in the NIPA, is the fixed, non-residential private
capital for all industries measured at constant cost less the analagous measure
for farms.
• Establishment hours were constructed by multiplying the number of employees
of private, non-farm establishments by the average weekly hours worked produc­
tion or non-supervisory employees at these same establishments. This data is
compiled from a survey of employer payrolls by the Bureau of Labor Statistics.
The resulting time series was seasonally adjusted using the EZ-X11 computer
program.
• Household hours were constructed in the same manner as establishment hours.
The data were compiled from the current population survey by the Bureau of
Labor Statistics.
R e fe re n c e s
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34

Table 1: Statistics from Model Cross Sectional Productivity Distribution

= 0.80 s = 0.85 s = 0.90
-0.032
-0.057
Mean Productivity -0.078
0.106
0.120
Standard Error
0.135
0.022
0.015
Reallocation Rate
0.010
-0.307
-0.230
Exit Threshold
-0.380
The mean productivity is measured relative to z t.
The exit threshold is measured relative to z t.
The reallocation rate equals the fraction of plants exiting each quarter.




s

Table 2: Estimated Second Moments
Estimate Standard Error
0.0008
0.0085
crx
0.0113
0.0011
ay
0.041
0.805
pxy
0.058
0.110
pi
0.155
0.089
pI
0.095
0.073
pI
0.002
0.083
Pi
standard error of total factor productivity growth.
standard error of output growth.
j ’th autocorrelation of total factor productivity growth.
Moment

crx
(jy
pi

=
=
=

Table 3: Common Parameter Estimates
Parameter
cv

a
pz
/.ip

=
=
=

Estimate Standard Error
0.66
0.004
0.0067
0.0026
Pz
0.0041
0.0004
Up
elasticity of output with respect to labor input.
mean quarterly growth rate of leading edge technology.
mean quarterly population growth rate.

Table 4: Specification Parameter Estimates

Log Preferences
Parameter Estimate Standard Error
0.8944
0.016
s
0.009
0.061
<?z
A
0.19
1.54
Hansen Preferences
Parameter Estimate Standard Error
0.8877
0.049
5
G z
0.056
0.009
A
2.27
0.26
s

=
az =
A =

scrap value of exiting plants.
standard error of leading edge technology improvement.
ratio of output growth standard error in model and U.S. economies.

Table 5: Statistics from Cross Sectional Productivity Distribution: Estimated Model
= 0.8944
Mean Productivity
-0.034
Standard Error
0.108
Reallocation Rate
0.021
Exit Threshold
-0.239
The mean productivity is measured relative to z t .
The exit threshold is measured relative to z t .
The reallocation rate equals the fraction of plants exiting each quarter.




s

36




Table 6: Population Autocorrelations from Estimated Models

Log Preferences
Estimate Standard Error
0.032
-0.050
0.012
0.033
0.012
0.054
0.012
0.061

Moment
Pi
pI
pI

pi

Moment
Pi
pi
pI
pi
fPx

=

y

Hansen Preferences
Estimate Standard Error
0.064
-0.045
0.032
0.006
0.007
0.050
0.008
0.055

th autocorrelation of total factor productivity growth.

Table 7: Wald Tests of Solow Residual Behavior
Log Preferences
Test Statistic Degrees of Freedom
0.813
i
PAx
2.64
2
PA x ') P ax
3.23
3
PAx'*' *") PAx
8.20*
4
PAx'*' *"> PAx
★ Surpasses the 10% critical value.
Moments

Moments
P ax
PAxi P A
1

x

3

PAx v *•> PAx
1
4
PAx')' ' "> PAx

Hansen Preferences
Test Statistic Degrees
0.558
2.57
2.88
6.93

37

of Freedom
i
2
3
4

° -0 .5

-0 .4

- 0 .J

-0 .2

-0.1

0 -

F igure 1: Cross Sectional Productivity Distributions




fj.z * i

-0 .0

0.1

0.2

0.3

o

cn

o
CD

d
o

x co
<*- o '
G
O
CO

LO
o
o
rO
o
^ ________________L_
° 0.75
0.80

s

F igu re 2: Identification of 5




.

■

I

° -0 .4 -0 .3

11

■

-0 .2

1

1

-0.1

1

1

-0 .0

G -

*

1 --

1 ............. 1—

0.1
fxz*t

F igure 3: Cross Sectional Productivity Distribution: Estimated Model




0.2

1

1

- — 1.

0.3

0.4

0.5

i\

•.ww. Output
.... Consumption
Net Investment
Percent

Hours
Looor Productivity

'0

4

a

12

16

20

24

28

32

36

40

>0

4

8

12

16

20

24

28

32

36

Meosured SolowResiduol
Ideol SolowResiduol

Percent

Scrop
•• Cross Investment
{■
’/

8

12

16

20

24

28

32

36

40

Quorters

F igure 4: Responses in Estimated Model: Log Preferences




40

Output
Consumption
Net Investment

Hours
lobor Productivity

12

IS

20

Quarters

to
Scrop
.v. Ooss Investment
a.

*0

4

8

12

16

20

24

28

32

36

40

Quarters

F igure 5: Responses in Estimated Model: Hansen Peferences




24

28

0.<
0.2

NIPACopltol Stock
.. •• Totol Capital Stock
.w Effective Copltol Input

- 1 .0

- 0 .6

-0 .2

Percent

Output
Consumption
Net Investment

0

4

6

12

16

20

24

26

32

36

40

0.5

Quarters
Meosured SolowResidual
Ideol SolowResidual

Percent
12

16

20

24

Quorters

F igu re 6: Productivity Response: s=0.895




2ft

32

36

2

Scrop
Cross Investment

0.3

0.4

4^O
C

0

4

6

12

16

20

Quarters

24

26

32

36

40

;A \

s 'V

Percent

*

Output
.... Consumption
Net Investment

* .y

^,Vy,Vf
........ ••••

-..

•

0

4

0

12

••;;;*7/*

G.

NIPACapitol Stock
•• •• Total Capital Stock
• Effective Capital Input
18

20

24

26

32

38

40

• 0

4

6

12

18

Quarters

:r

20

24

26

32

36

40

Quarters
MeosuredSolowResidual
Ideal SolowResidual

Percent

Scrap
Cross Investment
\i/

II

0

4

6

12

16

20

24

Quarters

F igure 7: Productivity Response: s=0.910




28

32

36

40

”

0

4

8

12

16

20

Quarters

24

26

32

36

40

F igure 8: Productivity Response: s=0.878




•••• Hours

Output
Consumption
Net Investment

• 0

4

6

12

16

20

24

26

32

36

O.J
0.1
- 0 .2

- 0 .1

- 0 .0

Percent

***Lobor Productivity

40

0

4

6

12

16

24

26

32

36

40

0.5

T

Measured SolowResidual
Ideol SolowRestduol

Percent

>

Z

Scrop
C^oss Investment

0.3

0 .4

C3

20

Ouorters

Ouorters

..V*
y.
12

16

20

24

Ouorters

F igure 9: Productivity Response: s=0.9‘25




28

32

36

40

0

4

6

12

16

20

Ouorters

24

28

32

36

40

o
r
°

t.

>v
\
> 'v
•
Percent

i

••• Consumption
Net Investment

S-,

f

'

o
■

,
* •*■*•.A. .

.••... ........................
.. •“

16

a.

o

•

12

52

od
20

Quorters

24

28

32

36

40

o

S?

•
•n
*c. o
»
u
• <N
d t'\..

Percent

Scrap
Cross Investment

5

'0

”

•• ... ..'. v. ... ... ... ... ................ ... ..
-------- ----------------- ■'

1

o
1?

16

20

24

Quarters

F igu re 10: Productivity Response: s=0.8-5




... Ideal SolowResidual

o

28

32

36

40

12

16

20

Quorters

24

2ft

32

36

40

\\
? V

Percent

Output
Consumption
Net Investment

12

16

20

24

26

32

Quarters

Scrap
Cross Investment
Percent

^ o

16

20

24

Quarters

F igu re 11: Productivity Response: s=0.75




26

32

36