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Federal Reserve Bank of Chicago

Why Is Productivity Procyclical? Why
Do We Care?

By: Susanto Basu and John Fernald

WP 2000-11

Why Is Productivity Procyclical? Why Do We Care?

Susanto Basu
University of Michigan and NBER
sbasu@umich.edu
John Fernald
Federal Reserve Bank of Chicago
jfernald@frbchi.org

Abstract: Productivity rises in booms and falls in recessions. There are four main explanations for this
procyclical productivity: (i) procyclical technology shocks, (ii) widespread imperfect competition and increasing
returns, (iii) variable utilization of inputs over the cycle, and (iv) resource reallocations. Recent macroeconomic
literature views this stylized fact of procyclical productivity as an essential feature of business cycles because
each explanation has important implications for macroeconomic modeling. In this paper, we discuss empirical
methods for assessing the importance of these four explanations. We provide microfoundations for our
preferred approach of estimating an explicitly first-order approximation to the production function, using a
theoretically motivated proxy for utilization. When we implement this approach, we find that variable
utilization and resource reallocations are particularly important in explaining procyclical productivity. We also
argue that the reallocation effects that we identify are not “biases”—they reflect changes in an economy’s ability
to produce goods and services for final consumption from given primary inputs of capital and labor. Thus, from
a normative viewpoint, reallocations are significant for welfare; from a positive viewpoint, they constitute
potentially important amplification and propagation mechanisms for macroeconomic modeling.

Prepared for the Conference on Research in Income and Wealth, March 1998. We are grateful for comments
from Zvi Griliches, John Haltiwanger, Michael Horvath, Charles Hulten, Dale Jorgenson, Catherine Morrison,
Plutarchos Sakellaris, and participants in the conference. Basu thanks the National Science Foundation and the
Alfred P. Sloan Foundation for financial support. The views in this paper are solely the responsibility of the
authors and should not be interpreted as reflecting the views of the Federal Reserve Bank of Chicago or of any
other person associated with the Federal Reserve System.

Productivity is procyclical. That is, whether measured as labor productivity or total factor productivity,
productivity rises in booms and falls in recessions. Recent macroeconomic literature views this stylized fact
of procyclical productivity as an essential feature of business cycles, largely because of the realization that
each explanation for this stylized fact has important implications for the workings of macroeconomic models.
In this paper, we seek to identify the empirical importance of the four main explanations for this stylized fact,
and discuss the implications of our results for the appropriateness of different macroeconomic models.
Until recently, economists generally regarded the long-run average rate of productivity growth as
important for growth and welfare; procyclical productivity, by contrast, seemed irrelevant for understanding
business cycles. Economists presumed that high-frequency fluctuations in productivity reflected cyclical
mismeasurement—e.g., labor and capital worked harder and longer in booms—but these cyclical variations
in utilization were not themselves important for understanding cycles.
In the past decade-and-a-half, productivity fluctuations have taken center stage in modeling output
fluctuations, and are now viewed as an essential part of the cycle. Figure 1 charts the Solow residual for the
aggregate U.S. economy. The figure also shows growth in output and growth in “inputs,” defined as a
weighted average of labor and capital growth (we discuss data sources and definitions in Section IV). Mean
productivity growth is positive, so that over time society's ability to produce goods or services that satisfy
final demand is rising faster than its inputs. In addition, productivity growth is quite volatile. The volatility
is not random, but is significantly procyclical: The correlation with output growth is about 0.8.
Macroeconomists have become interested in the cyclical behavior of productivity because of the
realization that procyclicality is closely related to the impulses or propagation mechanisms underlying
business cycles. Even the cyclical mismeasurement that was formerly dismissed as unimportant turns out to
be a potentially important propagation mechanism.
There are four main explanations for high-frequency fluctuations in productivity. First, procyclical
productivity may reflect procyclical technology. After all, under standard conditions, total factor
productivity measures technology. If there are high-frequency fluctuations in technology, it is not surprising
that there are high-frequency fluctuations in output as well. Second, widespread imperfect competition and
increasing returns may lead productivity to rise whenever inputs rise. (Increasing returns could be internal to

2
a firm, or could reflect externalities from the activity of other firms.) Figure 1 shows the key stylized fact of
business cycles, the comovement of inputs and output. With increasing returns, the fluctuations in inputs
then cause endogenous, procyclical fluctuations in productivity. Third, as already mentioned, utilization of
inputs may vary over the cycle. Fourth, reallocation of resources across uses with different marginal
products may contribute to procyclicality. For example, if different industries have different degrees of
market power, then inputs will generally have different marginal products in different uses. Then aggregate
productivity growth is cyclical if sectors with higher markups have input growth that is more cyclical.
Alternatively, if inputs are relatively immobile or quasi-fixed, then marginal products may temporarily differ
across uses; as these resources eventually shift, productivity rises.1
Why do economists now care about the relative importance of these four explanations? In large part,
changes in the methodology of macroeconomics have raised the fact of procyclical productivity to the
forefront. Macroeconomists of all persuasions now use dynamic general equilibrium (DGE) models, and it
turns out that each of the four explanations has important implications for the workings of these models.
Hence, answering why productivity is procyclical sheds light on the relative merits of different models of the
business cycle.
First, if high-frequency fluctuations in productivity reflect high-frequency fluctuations in technology,
then comovement of output and input (i.e., business cycles) are a natural byproduct. The DGE approach to
business-cycle modeling began with so-called real-business-cycle models, which explore the extent to which
the frictionless one-sector Ramsey-Cass-Koopmans growth model can explain business-cycle correlations.
Real-business-cycle models use Solow’s productivity residual—interpreted as aggregate technology
shocks—as the dominant impulse driving the cycle (e.g., Cooley and Prescott, 1995). Other impulses may
affect output in these models, but technology shocks must dominate if the model is to match the key stylized
fact of business cycles: the positive comovement of output and labor input.2

1

For examples of these four explanations, see, respectively, Cooley and Prescott (1995); Hall (1988, 1990); Basu
(1996), Bils and Cho (1994), and Gordon (1993); and Basu and Fernald (1997a, 1997b).
2 Barro and King (1984) provide an early discussion of this issue. Dynamic general equilibrium models without
technology shocks can match this stylized fact with countercyclical markups of price over marginal cost, arising
from sticky prices (as in Kimball, 1995) or from game-theoretic interactions between firms (as in Rotemberg and
Woodford, 1992). Models with an extreme form of increasing returns — increasing marginal product of labor —
can also produce a positive comovement between output and labor input; see, e.g., Farmer and Guo (1994).

3
Second, recent papers show that increasing returns and imperfect competition can modify and magnify
the effects of various shocks in an otherwise-standard DGE model. In response to government demand
shocks, for example, models with countercyclical markups can explain a rise in real wages while models
with increasing returns can explain a rise in measured productivity. Perhaps most strikingly, if increasing
returns are large enough, they can lead to multiple equilibria, in which sunspots or purely nominal shocks
drive business cycles.3 Furthermore, if firms are not all perfectly competitive, then it is not appropriate to use
the Solow residual as a measure of technology shocks, since the Solow residual becomes endogenous.
Taking the Solow residual to be exogenous thereby mixes impulses and propagation mechanisms.
Third, variable utilization of resources turns out to improve the propagation of shocks in DGE models.
If firms can vary the intensity of factor use, then the effective supply of capital and labor becomes more
elastic. Small shocks (to technology or demand) can then lead to large fluctuations.4 If the model has sticky
nominal prices, these elastic factor supplies greatly increase the persistence of the real effects of nominal
shocks.
Fourth, reallocations of inputs, without any change in technology, may cause aggregate productivity to
be procyclical. For example, in the sectoral shifts literature (e.g., Ramey and Shapiro (1998) and Phelan and
Trejos (1996)), demand shocks cause differences in the marginal product of immobile factors across firms.
Output fluctuations then reflect shifts of resources among uses with different marginal products. Basu and
Fernald (1997a) provide a simple stylized example in which aggregation over constant-returns firms with
different levels of productivity lead to the existence of multiple equilibria. Weder (1997) calibrates a DGE
model where durables manufacturing firms have increasing returns while all other producers have constant
returns (calibrated from results in Basu and Fernald, 1997a), and shows that reallocations make multiple
equilibria possible in that model. Of course, reallocations can also help propagate sector-specific technology
shocks, as in Lilien (1982).
In this paper, we seek to identify the importance of the four explanations. Our approach builds on
the seminal contributions of Solow (1957) and Hall (1990). We allow for imperfect competition and non-

3

See, for example, Rotemberg and Woodford (1992), Farmer and Guo (1994), and Beaudry and Devereux (1994).
Rotemberg and Woodford (1995) survey dynamic general equilibrium models with imperfect competition.
4 See, for example, Burnside and Eichenbaum (1996); Dotsey, King, and Wolman (1997); and Wen (1998).

4
constant returns to scale, as well as variations in the workweek of capital or the effort of labor that are
unobservable to the econometrician. In the Solow-Hall tradition, we take the production-function residual
as a measure of sectoral technology shocks. We then aggregate over sectors, since our ultimate focus is
on explaining movements in aggregate productivity.
Our empirical work relies primarily on the tools developed by Basu and Fernald (1997b) and Basu and
Kimball (1997). Both papers allow for increasing returns to scale and markups of price over marginal cost.
Basu and Fernald stress the role of sectoral heterogeneity. They argue that for economically plausible
reasons—e.g., differences across industries in the degree of market power—the marginal product of an input
may differ across uses. Then aggregate productivity growth depends in part on which sectors change inputs.
Basu and Kimball stress the role of variable capital and labor utilization. Solow’s original (1957)
article presumed that variations in capacity utilization were a major reason for the procyclicality of measured
productivity, a presumption widely held thereafter. (See, for example, Gordon, 1993; Abbott, Griliches, and
Hausman, 1998). In essence, the problem is cyclical measurement error of input quantities: true inputs are
more cyclical than measured inputs, so that measured productivity is spuriously procyclical.
Basu and Kimball use the basic insight that a cost-minimizing firm operates on all margins
simultaneously—whether observed or unobserved—ensuring that the marginal benefit of any input equals its
marginal cost. As a result, increases in observed inputs can proxy for unobserved changes in utilization. For
example, when labor is particularly valuable, firms will work existing employees both longer (increasing
observed hours per worker) and harder (increasing unobserved effort).
Our work on these issues follows the Solow-Hall tradition, which makes minimal assumptions and
focuses on the properties of the resulting residual. An alternative literature, surveyed by Nadiri and Prucha
(1999), addresses issues of technology change by attempting to estimate an extensively parameterized
production (or cost) function. That approach can address a wider range of issues because it imposes much
more structure on the problem. If the problem is correctly specified and all necessary data are available, the
more parametric approach offers clear theoretical advantages, since one can then estimate second-order
properties of the production function such as elasticities of substitution. However, that approach is likely to
suffer considerable practical disadvantages, such as the increased likelihood of misspecification and the

5
necessity of factor prices being allocative period-by-period. We are skeptical that observed factor prices are
always allocative—with implicit contracts, for example, observed wages need only be right on average,
rather than needing to be allocative period-by-period. Hence, we argue in favor of an explicitly first-order
approach when possible, since results are likely to be more robust.
Note that this production-function literature sometimes claims to solve the capacity utilization problem
(e.g., Morrison, 1992a,b). Suppose, for example, that the capital stock cannot be changed instantaneously
but that it is nevertheless used all the time. It can still be used more “intensively” by combining the fixed
capital services with more labor and materials. As a result, the shadow value of this quasi-fixed capital stock
may vary from its long-run equilibrium factor cost. Capital’s output elasticity may also vary over time,
reflecting variations in this shadow value. More generally, with quasi-fixity, quantities of inputs (and output)
may differ from long-run equilibrium values; similarly, on the dual (cost) side, short-run variable cost may
differ from long run variable cost at a given output level. If we equate “full capacity” with the firm’s optimal
long-run equilibrium point, then “capacity utilization” (defined by how far actual output or variable cost is
from their appropriately-defined long-run equilibrium values) may vary over time.
For estimating the technology residual, the relevant feature of this literature is that it attempts to
estimate the shadow value of quasi-fixed inputs in order to calculate the time-variation in the output
elasticities. The production-function literature thus tries to control for time-varying output elasticities, while
the capacity-utilization literature tries to measure factor quantities (i.e., the service flow) correctly.5
The enormous confusion between these two logically distinct ideas stems purely from the semantic
confusion caused by both literatures claiming to address “the capacity utilization” problem. In particular,
adherents of the time-varying elasticity approach often claim that their optimization-based methods obviate
the need to use proxies that control for unobserved variations in capital workweek or labor effort, with the
implication that the use of proxies is somehow ad hoc. This implication is wrong. The production-function
literature assumes that the quantities of all inputs are correctly measured; the capacity-utilization literature
devotes itself to correcting for measurement error in the inputs. The two address separate issues.

5

For our purposes, this is the only difference that matters. The structural literature also estimates a variety of other
quantities, such as elasticities of substitution, that may be of independent interest.

6
The easiest way to see the difference between the two concepts is to suppose that output is produced via
a Cobb-Douglas production function. Then the output elasticities are constant, so for our purposes—e.g.,
estimating returns to scale—the time-varying elasticity literature has nothing to add. But there is still a
capacity utilization problem—for example, if workers work harder in a boom than in recessions, but this fact
is not captured in the statistics on labor input. Formally, therefore, the problem of time-varying elasticities is
a second-order issue—it relates to deviations from the first-order Cobb-Douglas approximation—but
capacity utilization is a first-order issue, since it concerns the accurate measurement of the inputs.
After presenting empirical results, we discuss implications for macroeconomics. We discussed some of
these implications above, in motivating our interest in procyclical productivity, so we highlight other issues.
Normative productivity analysis emphasizes the welfare interpretation of the productivity residual. But what
if productivity and technology differ because of distortions such as imperfect competition? Recent
macroeconomic literature often seems to assume that we measure productivity in order to measure
technology; any differences reflect “mismeasurement.” Although variable input utilization is clearly a form
of mismeasurement, we argue that other distortions (such as imperfect competition) are not: Productivity
has clear welfare implications, even in a world with distortions. A modified Solow residual—which reduces
to Solow’s measure if there are no economic profits—approximates to first order the welfare change of a
representative consumer. Intuitively, growth in aggregate output measures the growth in society’s ability to
consume. To measure welfare change, we must then subtract the opportunity cost of the inputs used to
produce this output growth. Input prices measure that cost, even when they do not measure marginal
products. Hence, if productivity and technology differ, then productivity most closely indexes welfare.
Section I provides a relatively informal overview of the issues. We highlight key issues facing the
empirical literature and make recommendations. That section will allow the reader to proceed to the data and
empirical sections IV and V. However, some of our choices in Section I require more formal treatment to
justify fully; Section II provides this treatment. Section III discusses aggregation from the firm to the
economy-wide level. Section IV discusses data, and Section V presents results. Section VI discusses several
macroeconomic implications of our results. Section VII concludes.

7
I. Methods of Estimating Technical Change: Overview
Why do macroeconomists care about fluctuations in productivity? First, and perhaps foremost,
productivity yields information about the aggregate production of goods and services in the economy.
Second, productivity analysis may provide information about firm behavior—e.g., the markup and its
cyclicality, the prevalence of increasing returns to scale, and the factors determining the level of utilization.
There are several possible approaches to empirical productivity analysis. Each has advantages and
disadvantages. In this section, we assess these alternatives, and make recommendations about key decisions
facing an empirical researcher. We summarize the microeconomic foundations of our preferred approach,
which is in the spirit of Solow (1957) and Hall (1990). This discussion may satisfy the interests and needs of
most readers, who can then proceed to the data and results. However, a full justification of our approach
requires a somewhat more technical discussion. We present that technical discussion in Sections II and III.
Our ultimate goal is to understand the aggregate economy. At an aggregate level, the appropriate
measure of output is national expenditure on goods and services, i.e., GDP—the sum of consumption,
investment, government purchases, and net exports. GDP measures the quantity of goods available to
consume today or invest for tomorrow. GDP is intrinsically a value-added measure, since the national
accounts identity assures us that when we aggregate firm-level value added (defined in nominal terms as total
revenue minus the cost of intermediate inputs of materials and energy; in real terms, gross output with
intermediate inputs netted out in some way, as discussed in Section III), aggregate value added equals
national expenditure. The economy’s resources for producing goods and services are capital and labor.
Nevertheless, despite our interest in macroeconomic aggregates, we should begin at the level where
goods are actually produced: at the level of a firm or even a plant. The appropriate measure of output for a
firm is gross output—shoes, books, computers, haircuts, and so forth. Firms combine inputs of capital, labor,
and intermediate goods (materials and energy) to produce this gross output.
Our goal, then, is to explore how firm-level production of gross output translates into production of
aggregate final expenditure. Macroeconomists often assume an aggregate production function that relates
aggregate final expenditure (value added) to inputs of capital and labor. For many purposes, what matters is
whether such a function provides at least a first-order approximation to the economy’s production

8
possibilities—even if an explicit aggregate function does not exist (as it rarely does). For example, in
calibrating a dynamic general equilibrium model, one may care about how much aggregate output increases
if the equilibrium quantity of labor increases, and a first-order approximation should give the right
magnitude.6 We argue that in a world without frictions or imperfections, aggregation is generally very clean
and straightforward. However, with frictions and imperfections such as imperfect competition or costs of
reallocating inputs, the assumption that an aggregate production function exists (even as a first-order
approximation) generally fails—but the failures are economically interesting. These failures also help
explain procyclical productivity.

A. The Basic Setup
We write the firm’s production function for gross output, Yi , in the following general form:

Yi = F i ( K% i , L%i , M i , Ti )

(1)

Firms use capital services K% i , labor services L%i , and intermediate inputs of materials and energy M i . We
write capital and labor services with tildes to remind ourselves that these are the true inputs of services,
which may not be observed by the econometrician. Ti indexes technology, which we define to include any
inputs that affect firm-level production but are not compensated by the firm. For example, Ti comprises both
standard exogenous technological progress and any Marshallian externalities that may exist; we take
technology as unobservable. (For simplicity, we omit time subscripts.)
The services of labor and capital depend on both raw quantities (hours worked and the capital stock),
and the intensity with which they are used. We define labor services as the product of the number of
employees, N i , hours worked per employee H i , and the effort of each worker, Ei . We define capital
services as the product of the capital stock, K i , and the utilization of the capital stock, Z i . (For example,

6 However, DGE models usually also need to relate changes in factor prices (e.g., wages) to changes in the
quantities of inputs. For this purpose, the first-order approximation is not sufficient: one also needs to know the
elasticities of substitution between inputs in production, which is a second-order property.

9

K i might represent a particular machine, whereas Z i represents the machine’s “workweek”—how many
hours it is operated each period). Hence, input services are:

L%i = Ei H i N i ,
K% i = Z i K i .

(2)

We will generally assume that the capital stock and the number of employees are quasi-fixed, so firms cannot
change their levels costlessly.
Let the firm's production function F i be (locally) homogeneous of arbitrary degree γ i in total inputs.
Constant returns then corresponds to the case where γ i equals one. Formally, we can write returns to scale
in two useful, and equivalent, forms. First, returns to scale equal the sum of output elasticities:

γi =

F1i K% i
Yi

+

F2i L%i
Yi

+

F3i M i
,
Yi

(3)

where FJi denotes the derivative of the production function with respect to the Jth element (i.e., the marginal
product of input J). Second, assuming firms minimize cost, we can denote the firm’s cost function by

Ci (Yi ) . (In general, the cost function also depends on the prices of the variable inputs and the quantities of
any quasi-fixed inputs, although we omit those terms for simplicity here.) The local degree of returns to
scale equals the inverse of the elasticity of cost with respect to output:7

γ i ( Yi ) =

Ci (Yi ) Y i ACi
Ci ( Yi )
=
=
,
Yi Ci′ ( Yi )
Ci′ (Yi )
MCi

(4)

where ACi equals average cost, and MCi equals marginal cost. Increasing returns, in particular, may reflect
overhead costs or decreasing marginal cost; both imply that average cost exceeds marginal cost. If
increasing returns take the form of overhead costs, then γι(Yi) is not a constant structural parameter, but
depends on the level of output the firm produces. As production increases, returns to scale fall as the firm
moves down its average cost curve.
As equation (4) shows, there is no necessary relationship between the degree of returns to scale and the
slope of the marginal cost curve. Indeed, increasing returns is compatible with increasing marginal costs, as

10
in the standard Chamberlinian model of imperfect competition. One can calibrate the slope of the marginal
cost curve from the degree of returns to scale only by assuming there are no fixed costs. An important point
is that the slope of the marginal cost curve determines the slopes of the factor demand functions, which in
turn are critical for determining the results of DGE models (for example, whether the model allows sunspot
fluctuations). Several studies have used estimates of the degree of returns to scale to calibrate the slope of
marginal cost; for a discussion of this practice, see Schmitt-Grohe (1997).
Firms may also charge a price Pi that is a markup, µ i, over marginal cost. That is, µi ≡ Pi MC i .
Returns to scale γ i is a technical property of the production function, whereas the markup µi is essentially a
behavioral parameter, depending on the firm’s pricing decision. However, from (4), the two are inextricably
linked:

γi =

Ci (Yi )
Pi Ci (Yi )
=
= µi (1 − sπ i ) ,
Yi Ci′ (Yi ) Ci′ (Yi ) PY
i i

(5)

where sπ i is the share of pure economic profit in gross revenue. As long as pure economic profits are small
(and in our data, we estimate the average profit rate to be at most 3 percent8), equation (5) shows that µi
approximately equals γ i . Large markups, for example, require large increasing returns. This is just what
one would expect if free entry drives profits to zero in equilibrium—e.g., in Chamberlinian monopolistic
competition. Thus, although increasing returns and markups are not equivalent from a welfare perspective,
they are forced to equal one another if competition eliminates profits. As a result, as we show in the next
subsection, one can write the resulting wedge between output elasticities and factor shares in terms of either
parameter.Given low estimated profits, equation (5) shows that strongly diminishing returns (γi much less
than one) imply that firms consistently price output below marginal cost (µi less than one). Since price
consistently below marginal cost makes no economic sense, we conclude that average firm-level returns to
scale must either be constant or increasing. Increasing returns also requires that firms charge a markup, as
long as firms do not make losses.

7
8

See Varian (1984, p. 68) for a proof.
Rotemberg and Woodford (1995) also provide a variety of evidence suggesting that profit rates are close to zero.

11

B. The Solow-Hall Approach
Suppose we want to estimate how rapidly technology is changing. Solow’s (1957) seminal contribution
involves differentiating the production function, and using the firm’s first-order conditions for cost
minimization. If there are constant returns to scale and perfect competition, then the first-order conditions
(discussed below) imply that output elasticities are observed in the data by revenue shares. Hall (1988, 1990)
extends Solow’s contribution to the case of increasing returns and imperfect competition. Under these
conditions, output elasticities are not observed, since neither returns to scale nor markups are observed.
However, Hall derives a simple regression equation, which he then estimates. In this section, we extend
Hall’s approach by using gross-output data and taking account of variable factor utilization.
We begin by taking logs of both sides of (1) and then differentiating with respect to time:

F1i K i % F2i Li % F3i M i
dyi =
dki +
dli +
dmi + dti .
Yi
Yi
Yi

(6)

Small letters denote growth rates (so dy, for example, equals (1 Y )Y& ), and we have normalized the output
elasticity with respect to technology equal to one for simplicity.
As Solow (1957) and Hall (1990) show, cost minimization puts additional structure on equation (6),
allowing us to relate the unknown output elasticities to observed factor prices. In particular, suppose firms
charge a price Pi that is a markup, µi , over marginal cost. (The advantage of the cost minimization
framework is that it is unnecessary to specify the potentially very complicated, dynamic profit maximization
problem that gives rise to this price.) Perfect competition implies µi equals one. Suppose that firms take
the price of all J inputs, PJi , as given by competitive markets.

The first-order conditions for cost-

minimization then imply that:
i
PF
i J = µ i PJi .

(7)

In other words, firms set the value of a factor’s marginal product equal to a markup over the factor’s input
price. Equivalently, rearranging the equation by dividing through by µI, this condition says that firms equate
each factor’s marginal revenue product ( Pi µ i ) FJi to the factor’s price.

12
The price of capital, PKi , must be defined as the rental price (or shadow rental price) of capital. In
particular, if the firm makes pure economic profits, these are generally paid to capital, since the owners of the
firm typically also own its capital. These profits should not be incorporated into the rental price. Equation
(7) still holds if some factors are quasi-fixed, as long as we define the input price of the quasi-fixed factors as
the appropriate shadow price, or implicit rental rate (Berndt and Fuss, 1986). We return to this point in
Section II, where we specify a dynamic cost-minimization problem.
Using equation (7), we can write each output elasticity as the product of the markup multiplied by total
expenditure on each input divided by total revenue. Thus, for example,

F1i Z i K i
P K
= µi Ki i ≡ µi sKi .
Yi
PY
i i

(8)

(The marginal product of capital is F1i Z i , since the services from a machine depend on the rate at which it is
being utilized, i.e., its workweek.) The shares sJi are total cost of each type of input divided by total
revenue. Thus, the input shares sum to less than one if firms make pure profits.
We now substitute these output elasticities into (6) and use the definition of input services from (2):

dyi = µi  sKi dk%i + sLi dl%i + sMi dmi  + dti

= µi  sKi ( dki + dzi ) + sLi ( dni + dhi + dei ) + sMi dmi  + dti
 s dz + s de 
= µi  sKi dki + sLi ( dni + dhi ) + sMi dmi  + µi (1 − sMi )  Ki i Li i  + dti
 (1 − sMi ) 

By defining dx as a share-weighted average of conventional (observed) input growth, and du as a weighted
average of unobserved variation in capital utilization and effort, we obtain our basic estimating equation:

dyi = µi dxi + µi (1 − sMi )dui + dti .

(9)

Using equation (5), we could rewrite equation (9) in terms of returns to scale γi. In that case, the correct
weights to calculate weighted-average inputs are cost shares, which sum to one, rather than revenue shares,
which might not. Hall (1990) used the cost-share approach, but there is no economic difference between
Hall’s approach and ours, and the data requirements are the same. In particular, once we allow for the
possibility of economic profits, we must in any case estimate a required rental cost of capital. Writing the

13
equation in terms of µi turns out to simplify some later derivations, and also facilitates the welfare discussion
in Section VI.
Suppose all firms have constant returns to scale, all markets are perfectly competitive, and all factor
inputs are freely variable and perfectly observed. Then the markup µi equals one, dui is identically zero (or
is observed), and all factor shares are observed as data (since there are no economic profits to worry about).
This case corresponds to Solow’s (1957) assumptions, and we observe everything in equation (9) except
technology change dti, which we can calculate as a residual. However, if Solow’s conditions fail, we can
follow Hall (1990) and treat equation (9) as a regression.
An alternative to the Solow-Hall approach involves estimating many more properties of the production
function (1)—Nadiri and Prucha (1999, in this volume) survey that approach. That approach requires much
more structure on the problem. For example, one must usually postulate a functional form and specify the
firm’s complete maximization problem, including all constraints. If the problem is correctly specified and all
necessary data are available, that approach offers clear theoretical advantages for estimating the second-order
properties of the production function. However, the structural approach may suffer considerable practical
disadvantages, such as the increased likelihood of misspecification and the need for all factor prices to be
allocative period-by-period. In any case, for the first-order issues we focus on here, it should give similar
results. (If it does not, our view is that the Solow-Hall approach is probably more robust).
Regarding (9) as an estimating equation, one immediately faces three issues. First, the econometrician
usually does not observe utilization du directly. In particular, if capital and labor utilization vary, then
growth rates of the observed capital stock and labor hours do not capture the full service flows from those
inputs. In the short run, firms can vary their inputs of capital and labor only by varying utilization. In this
case, the regression suffers from measurement error. Unlike classical measurement error, variations in
utilization du are likely to be (positively) correlated with changes in the measured inputs dx, leading to an
upward bias in estimated elasticities. Below, we draw on recent work in the literature on capacity utilization,
when we attempt to control for variable service flow from inputs.
Second, should one take the output elasticities as constant (appropriate for a Cobb-Douglas production
function or for a first-order log-linear approximation), or time-varying? That is, should one allow the

14
markup and the share-weights in (9) to change over time? If the elasticities are not truly constant over time,
then treating them as constant may introduce bias. However, as we discuss later, attempting to estimate the
time-varying shares may lead to more problems than it solves.
Third, even if the output elasticities are constant and all inputs are observable, one faces the
“transmission problem” noted by Marschak and Andrews (1944): The technical change term, dt, is likely to
be correlated with a firm’s input choices, leading to biased OLS estimates. In principle, one can solve this
problem by instrumenting, but the need to use instruments affects our choice of the appropriate technique for
estimating (6). (One might expect technology improvements to lead to an expansion in inputs, making the
OLS bias positive; Gali (1999) and Basu, Fernald, and Kimball (1998), however, argue that there are
theoretical and empirical reasons to expect a negative bias.)

C. Empirical Implementation
We now discuss the empirical issues noted above, beginning with capacity utilization. In the estimating
equation (9), we need some way to observe utilization growth du or else to control for the measurement error
resulting from unobserved changes in utilization.
Our approach builds on the intuition that firms view all inputs (whether observed by the econometrician
or not) identically. Suppose a firm wants more labor input but cannot instantaneously hire more workers.
Then the firm should equate the marginal cost of obtaining more services from the observed intensive margin
(e.g., working current workers longer hours) and from the unobserved intensive margin (working them
harder each hour). If the costs of increasing hours and effort are convex, firms will choose to use both
margins. Thus changes in an observed input—e.g., hours per worker—provide an index of unobserved
changes in the intensity of work. This suggests a regression of the form:

dyi = µi dxi + ai dhi + dti ,

(10)

where dhi is the growth rate of hours per worker. Earlier work by Abbott, Griliches, and Hausman (1998)
also runs this regression to control for utilization.
In Section II, we construct a dynamic model of variable utilization that provides complete microfoundations for this intuition. That model shows that the regression in (10) appropriately controls for

15
variable effort. In addition, if the cost of varying the workweek of capital takes the form of a shift
premium—e.g., one needs to pay workers more to work at night—then this regression corrects for variations
in utilization of capital as well as effort. (If the cost of varying capital’s workweek is “wear and tear”—i.e.,
capital depreciates in use—then the regression is somewhat more complicated.)
Variable utilization of inputs suggests two additional problems in estimating regressions like (9) or (10)
, both of which make it difficult to observe “true” factor prices at high frequencies. First, firms will vary
utilization only if inputs of capital and labor are quasi-fixed—that is, costly to adjust. Varying utilization
presumably costs the firm something—e.g., a shift premium. Thus, if firms could vary the number of
machines or workers without cost, they would always adjust along these extensive margin rather than
varying utilization. However, if inputs are costly to adjust, then the shadow price of an input to the firm may
not equal its current market price. For example, investment adjustment costs imply that the return to
installed capital may differ from its frictionless rental rate.
Second, varying utilization—especially of labor—is most viable when workers and firms have a longterm relationship. Firm may increase work intensity when demand is high, promising to allow workers a
break in the next downturn. Such a strategy cannot be implemented if most workers are not employed by the
firm when the next downturn comes. The existence of such long-term relationships suggests that wages may
be set to be right “on average,” instead of being the correct spot-market wages in every period. Thus, both
quasi-fixity and implicit labor contracts imply that we may not be able to observe factor prices at high
frequencies.
This inability to observe factor prices period-by-period implies that we probably want to assume
constant, rather than time-varying, elasticities. This is unfortunate, since the first-order equations (8) suggest
that if factor shares vary over time, then output elasticities may vary as well. But if, because of quasi-fixity
and implicit labor contracts, we do not observe the relevant shadow values, then observed shares do not tell
us how the elasticities vary. For estimating the average markup, a first-order approximation may suffice, and
may be relatively unaffected our inability to observe factor prices at high frequency. Of course, if our goal
were to estimate elasticities of substitution between inputs, then we would need to find some way to deal

16
with these problems—we could not use a first-order approximation that simply assumes the elasticities are
one.9
With these considerations in mind, we now assess four methods of estimating the parameters in the
production function. First, one can simply take (6) as an estimating equation and estimate the output
elasticities directly. This approach, which essentially assumes that the production function is Cobb-Douglas,
can be justified as a first-order approximation to a more general production function. But this procedure
requires us to estimate three parameters (or more, if we include proxies for variable utilization) for the output
elasticities using data that are often multicollinear, and also subject to differing degrees of endogeneity and
hence differing OLS biases. The use of instrumental variables is not a complete solution, since most
plausible instruments are relatively weak. The literature on weak instruments suggests that instrumentalvariables methods have difficulty with multiple parameters; see, for example, Shea (1996).
Second, one can impose cost-minimization and estimate equation (9) or (10), while still taking a firstorder approximation. Cost-minimization is a relatively weak condition, and seems likely to hold at least
approximately. Imposing it allows us to move from estimating three parameters to estimating only one (the
markup), thus increasing efficiency. (Of course, assuming a particular model of cost-minimization, if it is
inappropriate, can lead to specification error. For example, if firms are not price takers in factor markets—
e.g., if firms have monopsony power or, in general, face price-quantity schedules for their inputs rather than
single prices—then cost-minimizing conditions and hence the Hall equations (9) or (10) are misspecified.)
In this case, one simply assumes that the shares used in constructing dxi are constant, as is the markup,

µi . As with the first approach, if our interest is in first-order properties such as average markups or returns
to scale, it can in principle give an accurate answer, although the omitted second-order terms may well be
important. (Risk aversion is a second-order phenomenon, but there is an active insurance industry.) A
substantial advantage of the first-order approach is that it does not require that observed factor prices (or
rental rates) be allocative in every period. The average shares are likely to be close to the steady-state shares,
so the approximation will typically be correct even with quasi-fixed inputs or implicit contracts.
9

For example, Bound et al. and others investigate the hypothesis that technological progress is skill-biased, and
hence contributed to the increase in income inequality in the United States in recent decades; Jorgenson (1987)

17
Third, one could continue to estimate (9) or (10), but allow the shares to change period-by-period. If
markups are constant, if factor prices are allocative period-by-period, and if there are no quasi-fixed factors,
then this approach can in principle give a second-order approximation to any production function (Diewert
1976). If these conditions fail, then this approach in essence incorporates some second-order effects, but not
others; it is unclear whether this is preferable to including none of them.
Fourth, one could estimate a flexible, general functional form along with Euler equations for the quasifixed inputs; as Nadiri and Prucha (1999) discuss, this approach provides a complete second-order
approximation to any production technology. If we properly parameterize the firm’s problem, then the
markup need not be constant, and factors can be quasi-fixed—the model provides estimates of the true
shadow values. In principle, this full structural approach provides a complete characterization of the
technology; one can then calculate elasticities of substitution, biases of technological change, and so forth.
For some macroeconomic questions, these parameters are crucial.10
However, this general approach has the disadvantage that one needs to estimate many parameters (a
translog production function for (1), for example, has 25 parameters before imposing restrictions, to say
nothing of the associated Euler equations). Efficient estimation requires various restrictions and identifying
assumptions, such as estimating the first-order conditions (7) along with the production function itself.
Hence, results may be sensitive to misspecification. For example, if wages are determined by an implicit
contract, and hence are not allocative period-by-period, then one would not want an approach that relies
heavily on high-frequency changes in observed factor prices (and hence factor shares) for identification. The
structural approach can be estimated either through the production or cost function—i.e., from the primal or
the dual side. Our concerns apply to both. We discuss these issues at greater length in Section IIC.
Since our primary interest is in measures of technical change, average returns to scale, and the average
markup, in this paper we follow the second approach outlined above, using an explicitly first-order
approximation. (In practice, this gives qualitatively similar results to the third approach, which allows the

argues that part of the productivity slowdown of the 1970s and 1980s is due to energy-biased technical change.
10 For example, a recent strand of business-cycle theory emphasizes the cyclical properties of markups as important
propagation mechanisms for output fluctuations; see, e.g., Rotemberg and Woodford (1992, 1995). Time-series
variation in the markup is a second-order property that can only be estimated with a second-order (or higher)
approximation to the production function.

18
factor shares to vary over time). This second method is essentially the procedure of Hall (1990), generalized
to include materials input and controls for variable factor utilization. Although this approach allows us to
estimate the parameters governing firm-level technology and behavior, our ultimate interest is describing the
evolution of aggregates. Thus, we must take one more step, and aggregate output growth across firms.

D. Aggregation
So far, we have discussed production and estimation in terms of firm-level gross output. Our ultimate
interest is in aggregate value added. In general, no aggregate production function exists that links aggregate
output to aggregate inputs—but the relationship between these aggregates remains of interest. It turns out
that aggregation across firms can introduce a significant new source of procyclical productivity.
In Section III, we derive the following equation for aggregate output (value added) growth dv:

dv = µ V dxV + du + R + dt V .

(11)

µ V is the average “value added” markup across firms, du is an appropriately weighted average of firm-level
utilization rates, R represents various reallocation (or aggregation) effects, and dt V is an appropriately
weighted average of firm-level technology. dxV is an weighted average of the aggregate capital stock and
labor hours. In all cases, the superscript V refers the fact that aggregate output is a value-added measure
rather than a gross-output measure, which requires some minor changes in definitions. (As we discuss in
Section III, for macroeconomic modeling it is the “value-added markup” that is likely to be of interest.)
The major implication of equation (11) is that output growth at the aggregate level is not completely
analogous to output growth at the firm level. The firm-level equation (9) looks similar to equation (11).
Firm-level output growth depends on input growth, the markup, variations in utilization, and technical
change; aggregate output growth depends on aggregate input growth, the average markup, average variation
in utilization, and average technical change. But equation (11) has a qualitatively new term, R.
The reallocation term reflects the effect on output growth of differences across uses in the (social)
values of the marginal products of inputs. Output growth therefore depends on the distribution of input
growth as well as on its mean: if inputs grow rapidly in firms where they have above-average marginal

19
products, output grows rapidly as well. Thus, aggregate productivity growth is not just firm-level
productivity growth writ large. There are qualitatively new effects at the aggregate level, which may be
important both for estimating firm-level parameters and as powerful amplification and propagation
mechanisms in their own right.
For example, suppose that some firms have large markups of price over marginal cost while others have
low markups. Also suppose that all firms pay the same prices for their factors. Then resources such as labor
have a higher marginal product in the firms with the larger markup, as shown by the first-order condition (8).
Intuitively, because these firms have market power, they produce too little output and employ too few
resources; hence, the social value of the marginal product of these inputs is higher. The reallocation term R
captures the fact that aggregate output rises if resources shift from low- to high-markup firms.
To summarize this section, we argue in favor of the following approach to estimating technology
change, controlling for utilization. First, interpret all results as first-order approximations, on the grounds
that the data are probably insufficient to allow reliable estimation of second-order approximations. Second,
estimate equation (10)—a Hall (1990)-style regression with a theoretically justified utilization proxy—at a
disaggregated level. Take the residuals as a measure of disaggregated technology change. Third, use the
aggregation equation (11)—which incorporates the disaggregated estimates of technology change, markups,
and utilization—to identify the importance of the various explanations for procyclical productivity.
The casual reader is now well prepared to proceed to the data and empiral sections of this paper in
Sections IV and V. However, our discussion passed quickly over several technical details. To fully
justify our choice of method, it is useful to address those issues in detail. We do so in Sections II and III.

II. The Meaning and Measurement of Capacity Utilization
Section I raised several issues with empirical implementation of the Solow-Hall approach to estimating
technology, markups, and variations in utilization at a disaggregated level. First, we must decide what prices
to use in calculating weights. With quasi-fixed inputs, the appropriate shadow price is not, in general, the
observed factor price. In addition, even if factors are freely variable, the observed factor prices may not be
allocative: For example, firms and workers may have implicit contracts. Second, we must decide whether to

20
use a first- or second-order approximation to the continuous time equation (10). Third, we must find suitable
proxies for du.
This section provides explicit microfoundations for our preferred approach. We specify a dynamic
cost-minimization problem to provide appropriate shadow values. We then discuss the pros and cons of firstversus second-order approximations. Finally, we use the first-order conditions from the cost minimization
problem to find observable proxies for unobserved effort and capital utilization.
We argue in favor of using an explicitly first-order approximation for the estimating equation, on the
grounds that the data necessary for an appropriate second-order approximation are unavailable. We also
argue for using growth in hours per worker to adjust for variations in effort and capital utilization. We also
spend some time clarifying conceptual confusions over the meaning of capacity utilization. In particular, we
argue that correcting for capacity utilization requires correcting quantities, not merely output elasticities.

A. A Dynamic Cost-Minimization Problem
We now specify a particular dynamic cost-minimization problem. Although the problem is relatively
complicated, specifying it provides insight into several practical issues in attempting to estimate equation
(10) in the previous subsection, and in Section IID provides proxies for unobserved utilization.
We model the firm as facing adjustment costs in both investment and hiring, so that both the amount of
capital (number of machines and buildings), K, and employment (number of workers), N, are quasi-fixed.
We model quasi-fixity for two reasons. First, we want to examine the effect of quasi-fixity per se on
estimates of production-function parameters and firm behavior. Second, quasi-fixity is necessary for a
meaningful model of variable factor utilization. Higher utilization must be more costly to the firm, otherwise
factors would always be fully utilized. If there were no cost to increasing the rate of investment or hiring,
firms would always keep utilization at its minimum level and vary inputs using only the extensive margin,
hiring and firing workers and capital costlessly. Only if it is costly to adjust along the extensive margin is it
sensible to adjust along the intensive margin, and pay the costs of higher utilization.11

11

One does not require internal adjustment costs to model variable factor utilization in an aggregative model (see,
e.g., Burnside and Eichenbaum, 1996), since changes in input demand on the part of the representative firm change
the aggregate real wage and interest rate, so in effect the concavity of the representative consumer’s utility function

21
We assume that the number of hours per week for each worker, H, can vary freely, with no adjustment
cost. In addition, both capital and labor have freely variable utilization rates. For both capital and labor, the
benefit of higher utilization is its multiplication of effective inputs. We assume the major cost of increasing
capital utilization, Z, is that firms may have to pay a shift premium (a higher base wage) to compensate
employees for working at night, or at other undesirable times.12 We take Z to be a continuous variable for
simplicity, although variations in the workday of capital (i.e., the number of shifts) are perhaps the most
plausible reason for variations in utilization. The variable-shifts model has had considerable empirical
success in manufacturing data, where, for a short period of time, one can observe the number of shifts
directly.13 The cost of higher labor utilization, E, is a higher disutility on the part of workers that must be
compensated with a higher wage. We allow for the possibility that this wage is unobserved from period to
period, as might be the case if wage payments are governed by an implicit contract in a long-term
relationship.
Consider the following cost-minimization problem for the representative firm of an industry:
∞

Min

Z ,E ,H ,M ,I , A

s

− r dτ
C (Y ) = ⌠⌡ e ∫0 WNG ( H , E ) V ( Z ) + PM M + WN Ψ ( A N ) + PI K J ( I K )  ds (12)
0

subject to

Y = F ( ZK , EHN , M , T )

(13)

K& = I − δ K

(14)

N& = A

(15)

The production function and inputs are as before. In addition, I is gross investment, and A is hiring net of
separations. WG ( H , E ) V ( Z ) is total compensation per worker (compensation may take the form of an
implicit contract, and hence not be observed period-by-period). W is the base wage; the function G specifies

acts as an adjustment cost that is external to the firm. However, if one wants to model the behavior of firms that
vary utilization in response to idiosyncratic changes in technology or demand—obviously the case in the real
world—then one is forced to posit the existence of internal adjustment costs in order to have a coherent model of
variable factor utilization. (Both of these observations are found in Haavelmo’s (1960) treatment of investment.)
12 Our model can be extended easily to allow utilization to affect the rate at which capital depreciates, as in Basu
and Kimball (1997). We consider the simpler case for ease of exposition. Nadiri and Prucha (1999) show that their
approach of estimating a second-order approximation to the production function can also accommodate variable
depreciation, but do not consider either a shift premium or variable labor effort.

22
how the hourly wage depends on effort, E, and the length of the workday, H; and V(Z) is the shift premium.

WN Ψ ( A N ) is the total cost of changing the number of employees; PI K J (I K ) is the total cost of
investment; PM is the price of materials. δ is the rate of depreciation. We continue to omit time subscripts
for clarity.
Using a perfect-foresight model amounts to making a certainty-equivalence approximation. But even
departures from certainty equivalence should not disturb the key results, which rely only on intra-temporal
optimization conditions rather than intertemporal ones.
We assume that Ψ, and J are convex, and make the appropriate technical assumptions on G in the spirit
of convexity and normality.14 It is also helpful to make some normalizations in relation to the normal or
“steady-state” levels of the variables. Let J (δ ) = δ , J ′ (δ ) = 1, Ψ ( 0 ) = 0. We also assume that the
marginal employment adjustment cost is zero at a constant level of employment: Ψ ′(0 ) = 0.
We use the standard current-value Hamiltonian to solve the representative firm’s problem. Let λ , q,
and θ be the multipliers on constraints (13), (14), and (15) respectively. Numerical subscripts denote
derivatives of the production function F with respect to its first, second and third arguments, and literal
subscripts denote derivatives of the labor cost function G. The six intra-temporal first-order conditions for
cost-minimization are:
Z:

λ K F1( ZK , EHN , M ; T ) = WNG ( H , E ) V ′ ( Z ) .

(16)

H:

λ EN F2 ( ZK , EHN , M ; T ) = WN GH ( H , E ) V ( Z )

(17)

E:

λ HN F2 ( ZK , EHN , M ; T ) = WN GE ( H , E ) V ( Z )

(18)

M:

λ F3( ZK , EHN , M ; T ) = PM

(19)

A:

θ = W Ψ′ ( A N )

(20)

I:

q = PI J ′ ( I K ) .

(21)

The Euler equations for the capital stock and employment are:
13
14

See, e.g., Beaulieu and Mattey (1998) and Shapiro (1996).
The conditions on G are easiest to state in terms of the function Φ defined by ln G(H,E) = Φ(ln H, ln E). Convex

23

q& = [ r + δ ] q − λ Z F1 + PI  J ( I K ) − ( I K ) J ′ ( I K ) 

(22)

θ& = rθ − λ EH F2 + W G ( H , E )V ( Z ) + W  Ψ ( A N ) − ( A N ) Ψ ′ ( A N )  .

(23)

Since λ is the Lagrange multiplier associated with the level of output, one can interpret it as ma.rginal
cost. The firm internally values output at marginal cost, so λ F1 is the marginal value product of effective
capital input, λ F2 is the marginal value product of effective labor input, and λ F3 is the marginal value
product of materials input.15 We defined the markup, µ, as the ratio of output price, P, to marginal cost, so λ
equals:

λ = C ′(Y ) =

P
.
µ

(24)

Note that equation (24) is just a definition, not a theory determining the markup. The markup depends on the
solution of the firm’s more complex profit-maximization problem, which we do not need to specify.
Equations (22) and (23) implicitly define the shadow (rental) prices of labor and capital:

λ Z F1 = [ r + δ ] q − q& + PI  J ( I K ) − ( I K ) J ′ ( I K )  ≡ PK

(25)

λ EH F2 = rθ − θ& + W G ( H , E )V ( Z ) + W  Ψ ( A N ) − ( A N ) Ψ ′ ( A N )  ≡ PL

(26)

As usual, the firm equates the marginal value product of each input to its shadow price. Note that with these
definitions of shadow prices, the atemporal first-order condition (7) holds for all inputs. For some intuition,
note that equation (25) is the standard first-order equation from a q-model of investment. In the absence of
adjustment costs, the value of installed capital q equals the price of investment goods PI , and the “price” of
capital input is then just the standard Hall-Jorgenson cost of capital (r + δ ) PI . With investment adjustment
costs, there is potentially an extra return to owning capital, through capital gains q& (as well as extra terms
reflecting the fact that investing today raises the capital stock, and thus lowers adjustment costs in the future.)
The intuition for labor in equation (26) is similar. Consider the case where labor can be adjusted freely,
so that it is not quasi-fixed. Then adjustment costs ψ are always zero. So is the multiplier θ, since constraint

Φ guarantees a global optimum; assuming Φ11 > Φ12 and Φ22 > Φ12 ensures that optimal H and E move together.
15 For the standard static profit-maximization problem, of course, marginal cost equals marginal revenue, so these
are also the marginal revenue products.

24
(15) does not bind. In this case, as we expect, (26) says that the shadow price of labor input to the firm—the
right side of the equation—just equals the (effort-adjusted) compensation WG ( H , E ) V ( Z ) received by the
worker. Otherwise, the quasi-fixity implies that the shadow price of labor to a firm may differ from the
compensation received by the worker.

B. First-Order Approximations
We now turn to issues of estimation. This subsection discusses how to implement our preferred firstorder approximation; the next subsection compares this approach with second-order approximations.
Equations (6) and (9) hold exactly in continuous time, where the values of the output elasticities adjust
continuously. In discrete time, we can interpret these equations as first-order approximations (in logs) to any
general production function if we assume the elasticities are constant. For a consistent first-order
approximation to equation (9), one should interpret it as representing small deviations from a steady-state
growth path, and evaluate derivatives of the production function at the steady-state values of the variables.
Thus, to calculate the shares in equation (9), one should use steady-state prices and quantities, and hence
treat the shares as constant over time. The markup is then also taken as constant.
For example, in the first order approach, we want the steady-state output elasticity for capital, up to the
unknown scalar µ. Using asterisks to denote steady-state values, we use equations (21), (25), and our
normalizations to compute the steady-state output elasticity of capital:

F1* Z * K *
Y*

=µ

*

PK* K *
P*Y *

≡µ

*

(r

*

+ δ ) PI* K *
P*Y *

.

(27)

Note that the steady-state user cost of capital is the frictionless Hall-Jorgenson (1967) rental price.16 Since
quasi-fixity matters only for the adjustment to the steady state, in the steady state q = PI and q& = 0 .
Operationally, we calculate the Hall-Jorgenson user cost for each period and take the time average of the
resulting shares as an approximation to the steady-state share. We proceed analogously for the other inputs.

16

In practice, one would also include various tax adjustments. We do so in the empirical work, but omit them in
the model to keep the exposition simple.

25
In the final estimating equation for (9), we use logarithmic differences in place of output and input growth
rates, and use steady-state shares for the weights.
Thus, we can construct the index of observable inputs, dx, and take the unknown µ ∗ multiplying it as a
parameter to be estimated. We can use a variety of approaches to control for the unobserved du; we discuss
some of them in subsection IID below. Alternatively, under the heroic assumption that du is always zero or
is uncorrelated with dx, we can estimate equation (9) while simply ignoring du. Hall (1988, 1990) and Basu
and Fernald (1997a) follow this second procedure.
In any case, we have to use instruments that are orthogonal to the technology shock dt, since technology
change is generally contemporaneously correlated with input use (observed or unobserved).17
C. Second-Order Approximations
The first-order approach of constant weights is, of course, equivalent to assuming that for small,
stationary deviations from the steady-state balanced growth path, we can treat the production function as
Cobb-Douglas. Parameters such as the average markup or degree of returns to scale are first-order
properties, so the bias from taking a first-order approximation may not be large.
Nevertheless, the Cobb-Douglas assumption is almost surely not literally true. In principle, using a
second-order approximation allows us to eliminate some of the bias in parameter estimates from using CobbDouglas, and eliminate approximation errors that end up in the residual. It also allows us to estimate secondorder properties of production functions, such as separability and elasticities of substitution. (Nadiri and
Prucha (1999) discuss the benefits of the second-order approach.) Unfortunately, the second order approach
suffers severe practical disadvantages—particularly the increased likelihood of misspecification. Our view is
that these sizeable disadvantages outweigh the potential benefits.
We begin this discussion with a simplest case where one obtains an appropriate second-order
approximation to (9) simply by using weights that change period-by-period. Suppose the production
function takes a more general form than Cobb-Douglas, e.g., translog. With a more general production
function, output elasticities will typically vary over time if relative factor prices are not constant. Also

26
suppose the markup is constant, and that all factor shadow values (i.e., input prices) are observed, as will be
the case if observed prices are allocative and there are no costs of adjusting inputs. With a translog function
(which provides a flexible approximation to any functional form), the discrete-time Tornquist approximation
to the continuous time equation (9) turns out to be exactly correct.18 This approximation requires replacing
growth rates with log differences, and replacing the continuous-time input shares with average shares in
adjacent periods. Thus, for example, the output elasticity for capital is approximated by (1 2 )( sKt + sKt −1 ) ,
and is multiplied by the capital-change term ( ln K t − ln K t −1 ) . In this case, using a Tornquist index of input
use allows a correct second-order approximation.
A major potential shortcoming with using changing shares is that observed factor prices may not be
allocative period-by-period because of implicit contracts or quasi-fixity. Then observed factor shares might
not be proportional to output elasticities period-by-period, and the Tornquist index is misspecified.
A large literature has focused on the problem of quasi-fixity of capital and labor, while largely ignoring
the concern about implicit contracts. We briefly review that approach to estimating second-order
approximations. As discussed in the introduction, the time-varying elasticity literature—e.g., Berndt and
Fuss (1986), Hulten (1986), and the much more parametric literature surveyed in Nadiri and Prucha (1999)—
tries explicitly to deal with quasi-fixity. Quasi-fixity may lead to large variations in input shadow prices,
which in turn may cause the output elasticities to vary over time.19 The shadow prices may differ
substantially from observed prices; in our cost-minimization setup of subsection IIA, equations (25) and (26)
show the relationship between shadow price and observed prices. For example, a firm may find that it has
too many workers, so that the output elasticity with respect to labor is very low. If the firm cannot (or
chooses not to) immediately shed labor, it will be forced to continue paying these workers. Then the

17

Olley and Pakes (1996) propose an insightful alternative to the usual instrumental-variables approach; see
Griliches and Mairesse (1998) for an excellent discussion of the pros and cons.
18 This result follows from Diewert (1976), who shows that the Tornquist index is “superlative”—that is, exact for
functional forms such as the translog that are themselves flexible approximations to general functions—and shows
that superlative indices have desirable index-number properties. For discussions of the concept of “flexibility” and
its limitations, see Lau (1986) and Chambers (1988, ch. 5).
19 As our previous discussion indicates, the output elasticities will generally change whenever input prices (shadow
values) change. Quasi-fixity is only one reason why shadow values change.

27
observed labor share may be higher than the true output elasticity, since the shadow value of labor is less
than the wage.
Hulten (1986) shows that even with quasi-fixity, if there are constant returns and perfect competition,
then one can implement the Tornquist approximation to equation (9) using observed input prices and output
growth as long as there is only a single quasi-fixed input (e.g., capital). Under these conditions, the revenue
shares sum to one. Since the observed input prices give the correct shares for all inputs other than capital,
capital’s share can be taken as a residual. (As we emphasize again below, this approach corrects only for
variations in the shadow values of quasi-fixed inputs, not unobserved changes in the workweek of capital.)
Multiple quasi-fixed inputs cannot be accommodated using the non-parametric approach of equation (9)
, but one can estimate the parameters of a structural cost function. Pindyck and Rotemberg (1983) and
Shapiro (1986), for example, estimate particular parametric forms of the production function along with
Euler equations for the quasi-fixed inputs, such as (22) and (23). As a by-product, this approach provides
estimates of the shadow values of the state variables, and thus one can construct a measure of period-byperiod shadow costs by valuing the input of each quasi-fixed factor at its shadow price.
The problem grows even more challenging when one allows for non-constant returns to scale. Since
one cannot estimate the scale elasticity from the first-order conditions for the cost shares (see, for example,
Berndt (1991), chapter 9), one must estimate both the cost function and the share equations together. But
output is a right-hand-side variable in the cost function; since the error term is interpreted as technical
change, output is clearly endogenous. Pindyck and Rotemberg (1983) deal with this problem by using
lagged variables as instruments, but it is unclear to what extent this procedure alleviates the problem. For
example, technical change—the error term in the cost function—can be (and usually is found to be) serially
correlated. Finally, one can compute the markup from the estimates of returns to scale (in this context, a
time-varying parameter), by using equation (5) and the observed prices and estimated shadow prices to
construct a period-by-period estimate of economic profit.
Morrison (1992a,b) attempts to construct better instruments by estimating industry demand curves
embodying shift variables, and then imputes values of output using the simple, static, monopoly-pricing
formula for the markup. However, there are also major problems with this procedure. First and most

28
importantly, it implies that µ is no longer identified from cost-minimization conditions alone; one has to
subscribe to a particular model of firm behavior and use profit-maximization conditions as well. This change
is a major loss in generality. Second, Morrison’s specific model of firm behavior assumes that firms can
collude perfectly in every period and that prices are completely flexible, and hence is misspecified if the
degree of collusion varies (as stressed, e.g., by Rotemberg and Saloner, 1986, and Green and Porter, 1984),
or if prices are sticky in the short run, leading the actual markup to deviate from the optimal markup (see,
e.g., Ball and Romer, 1990, and Kimball, 1995).
The challenges raised by these issues suggest that misspecification is a serious concern. Even in the
best of cases, where factor prices for variable factors are correctly observed and utilization does not vary, one
needs to correctly specify the problem and constraints, and then estimate a large number of parameters. This
complexity makes misspecification harder to spot.
Of course, this “best of cases” is unlikely to hold, so misspecification is probably much worse. First,
we have strong reasons to think that input prices observed by the econometrician may not be allocative
period-by-period because of implicit contracts. Since many economic relationships are long-term, it may
not be worth recontracting explicitly period-by-period. Then the correct shadow price of a variable input
may differ from its observed price. The shadow prices may satisfy the first-order conditions period-byperiod, while the observed factor payments may satisfy the conditions only on average. With implicit
contracts, the data that are available hinder accurate estimation of a second-order approximation—even if the
full problem and constraints were correctly specified.20
Second, variations in capacity utilization—i.e., in labor effort E and capital’s utilization/workweek Z—
also lead to misspecification. Since these utilization margins are not observed by the econometrician, they
remain as omitted variables in estimating a fully parametric production function. Using the dual approach
does not help. In specifying the cost function, one needs to specify what margins the firm can use to
adjust—and at what price. But it is virtually impossible for the econometrician to correctly observe the

20

Carlton (1983) also stresses that observed materials prices may not be allocative if firms use delivery lags to
clear markets. A similar problem arises from labor composition changes. Solon, Barsky, and Parker (1994) find
that the marginal worker in a boom is of relatively low quality—and hence relatively low paid—so that if labor data
do not adjust for this composition effect, the wage will appear spuriously countercyclical. Cost-function estimation
that uses spuriously cyclical wage data is, of course, misspecified

29
“prices” of varying effort or shifts to a firm. For example, Bils (1987) discusses the importance of the
premium paid for overtime work. The wage function G ( H , E ) in Section IIA incorporates any overtime
premium. However, the fully parameterized dual approach generally prices labor at the average wage rather
than the correct marginal wage.

Hence, the fully parameterized production or cost functions are almost

surely misspecified.
The misspecification arising from omitted capacity utilization is sometimes obscured by the
parameterized second-order literature, in part because that literature sometimes claims to solve the utilization
problem. They are addressing a different issue. That is, there are two distinct concepts of utilization in the
literature: what we call capacity utilization, and the Berndt-Fuss concern that inputs may not always be at
their steady-state levels.
The Berndt-Fuss notion of utilization addresses the possible time-variation in output elasticities caused
by time-varying shadow prices of quasi-fixed inputs. Capacity utilization, on the other hand, refers to the
much earlier idea that there is a particular type of measurement error in the inputs: Certain factors—again,
notably capital and labor—have variable service flow per unit of observed input (the dollar value of
machinery or the number of hours worked). Many commentators have explained the procyclicality of
productivity and the short-run-increasing-returns-to-labor (SRIRL) puzzle by arguing that variations in the
workweek of capital and changes in labor effort increase effective inputs much more than observed inputs.
In terms of equation (9), they argue that du is non-zero, and is positively correlated with dy and dx.
It is easy to see the difference between the two concepts in the case where F is actually known to be a
Cobb-Douglas production function. Since the elasticities are truly constant over time, the first-order
discrete-time implementation of equation (9) is exact. The observed factor shares might vary over time, but
since we know that F is Cobb-Douglas, these variations would reflect the effects of quasi-fixity rather than
changes in output elasticities. Quasi-fixity of inputs thus creates no problems for estimation, since one can
observe and use the steady-state prices and shares. Indeed, the Berndt-Fuss approach would “correct” the
time-varying shares, using constant shares instead. Despite this “correction,” variations in factor utilization
remains a problem. If one cannot somehow control for du, then estimates of the markup or returns to scale
will generally be biased.

30
Formally, therefore, the problems posed by quasi-fixity are second-order. If the first-order
approximation of Cobb-Douglas is actually exact, then quasi-fixity—or, more generally, time-varying
elasticities—cannot matter for the estimation of µ. On the other hand, capacity utilization remains a firstorder problem, since it concerns the measurement of the right-hand-side variables. It continues to pose a
problem even if the Cobb-Douglas approximation holds exactly.
The two concepts are often confused because, as we argued above, quasi-fixity is a necessary condition
for capacity utilization to vary. As a result, the shadow value of an extra worker will be high at exactly the
same time that unobserved effort is high. Because of this relationship, some authors in the productionfunction literature assert that the capacity utilization problem can be solved by allowing elasticities to vary
over time, or by using a dual approach and allowing shadow values to vary from “long run” equilibrium
levels. Many of these authors do not appear to realize that there are two separate problems, and allowing for
quasi-fixity does not solve the problem that the workweek of capital or the effort of workers may vary over
time.21 That is, there is no reason that the shadow value computed from a misspecified variable cost function
will necessarily capture all of the effects of variable utilization.
We conclude this section by returning to the first-order approach. In particular, though estimation
using second-order approximations is likely to be misspecified, the problems that approach addresses—such
as quasi-fixity—are clearly concerns. So given these concerns, how robust is the first-order approach that
ignores them?
It is plausible that the first-order approach is relatively robust. That is, ignoring quasi-fixity probably
does not significantly affect estimates of markups and returns to scale. First, quasi-fixity affects only the
period-by-period computation of input shares, not the growth rate of capital (or any other quasi-fixed input).
Since these shares are constant to a first-order Taylor approximation, any errors caused by failure to track the
movements of the shares is likely to be small. Second, quasi-fixity is likely to be most important for capital
(Shapiro (1986) finds that quasi-fixity is not important for production-worker labor, although it is present for
non-production workers). But mismeasurement of the rental rate of capital affects only capital's share, and
21

The survey by Nadiri and Prucha (1999) does try to address both issues. They focus primarily on the issue of
time-varying elasticities. However, they also allow capital’s workweek to vary, where the cost of increasing capital

31
since the growth rate of capital is almost uncorrelated with the business cycle, errors in measuring capital’s
share are unlikely to cause significant biases in a study of cyclical productivity. Caballero and Lyons (1989)
present simulations indicating that maximum biases from ignoring quasi-fixity of capital are likely to be on
the order of 3 percent of the estimated coefficients. (However, capital utilization is cyclical, and it is also
multiplied by capital’s share.)
In sum, the second-order approach offers great theoretical advantages that, in practice, it cannot deliver.
The failures are understandable, since the problems are difficult. Given these practical difficulties, we prefer
the explicitly first-order approach to estimation. This has costs as well, such as ignoring quasi-fixity for
estimation purposes, and the inability to estimate elasticities of substitution, time variation in the markup,
and other second-order properties. Nevertheless, results on first-order properties are likely to be relatively
robust.

D. Capacity Utilization
Before we can estimate µ from equation (9), we need to settle on a method for dealing with changes in
utilization, du. A priori reasoning—and comparisons between results that control for du and those that do
not— argue that du is most likely positively correlated with dx; thus ignoring it leads to an upward-biased
estimate of µ ∗ . Three general methods have been proposed. First, one can try to observe du directly using,
say, data on shiftwork. When possible this option is clearly the preferred one, but data availability often
precludes its use.22 Second, one can impose a priori restrictions on the production function. Third, one can
derive links between the unobserved du and observable variables using first-order conditions like equations
(16)-(21). Both the second and third approaches imply links between the unobserved du and observable
variables, which can be used to control for changes in utilization.
The approach based on a priori restrictions basically imposes separability assumptions on the
production function. For example, Jorgenson and Griliches (1967) use an idea going back at least to Flux

utilization is “wear and tear”—capital depreciates faster. They do not consider either a shift premium or variable
labor effort (or the consequent problem that observed wages may not be allocative).

32
(1913), and assume that effective capital input is a function of capital services and energy; this idea has
recently been revived by Burnside et al. (1995). (We henceforth refer to it as the “Flux assumption”). To
clarify this idea, it will be useful to separate intermediate goods M into two components: the flow of energy
inputs, W (mnemonic: Wattage); and all other intermediate inputs O (mnemonic: Other). Then the Flux
assumption implies:

(

)

( (

)

)

F K% , EHN , W , O; T = G S K% , W , EHN , O; T .

(28)

Jorgenson and Griliches and Burnside et al. generally also assume that

(

)

S K% , W = min  K% , W  ,

(29)

so that energy input is a perfect index of capital input.23
Using this implication of equation (29) to substitute into equation (9), we have

dy = µ  sK ( dk + dz ) + sL ( dn + dh + de ) + sW dw + so do  + dt
= µ ( sK + sw ) dw + sL ( dn + dh ) + so do  + µ sL de + dt.

(30)

Note that under the maintained hypothesis of equation (28), electricity use proxies for changes in capital
utilization, but does not capture variations in labor effort. (The same is true if one uses direct observations
on utilization, like the number of shifts.) Electricity use is also a much more sensible proxy for the
utilization of heavy machinery than for the services of structures or light machinery like computers, which
are often left on day and night regardless of use. Thus, electricity use is probably a reasonable proxy only
within manufacturing—and even there it does not capture variations in effort.
Basu (1996) attempts to control for both labor effort and capital utilization by using either of two
different separability assumption:

F ( ZK , EHN , W , O; T ) = TG (V ( ZK , EHN ) , S (W , O ) ) ,

(31)

or

22

In the United States, shift-work data are available only for a relatively short time period, and solely for
manufacturing industries. The only data set on worker effort that we know of is the survey of British manufacturing
firms used by Schor (1987).
23 Burnside et. al (1995) also experiment with a CES functional form for H, and allow the ratio of capital to energy
to change depending on their relative prices. The generalization to the CES does not affect their results
significantly, but their measures of the rental price of capital services are questionable.

33

F ( ZK , EHN , W , O; T ) = TG (V ( ZK ,W , EHN ) , S ( O ) ) .

(32)

In both cases, he assumes that G takes the Leontief form:24

G (V , S ) = min [V , S ] .

(33)

The basic intuition behind the separability assumption is the distinction between the materials inputs that are
being used up in production and the inputs that are assembling the materials into final output. In the simplest
case, equation (32), the estimating equation is just

dy = µ do + dt .

(34)

Under the maintained hypothesis of equation (33), variations in materials use capture changes in both capital
and labor utilization.
A problem with separability-based methods of controlling for utilization is that they rely crucially on
the assumption that the production function is homothetic (i.e., relative input demands do not depend on the
level of output).25 In other words, it is important that the function S in the Flux case or the functions V and S
in the Basu case all be homogenous of degree one. Basu (1996) discusses this issue in detail, and argues that
if homotheticity does not hold (or if one does not take additional steps), the estimated values of µ from
equations (30) and (34) are likely to be biased downward. With this caveat, however, the intuitive
approaches presented here offer relatively easy ways to control for changes in utilization.
Bils and Cho (1994), Burnside and Eichenbaum (1996), and Basu and Kimball (1997) argue that one
can also control for variable utilization using the relationships between observed and unobserved variables
implied by first-order conditions like (16)-(21). Our discussion follows Basu and Kimball.
They begin by assuming a generalized Cobb-Douglas production function:

(

F ( ZK , EHN , M ; Z ) = Z Γ ( ZK )

αK

( EHN )

αL

)

M αM ,

(35)

where Γ is a monotonically increasing function. In their case this assumption is not merely a first-order
approximation, because they make use of the second-order properties of equation (35), particularly the fact

24

Basu also experiments with CES specifications for G, but allowing deviations from the Leontief assumption has
little effect on his estimates
25 See Chambers (1988) for a discussion of homotheticity.

34
that the ratios of output elasticities are constant. Although they argue that one can relax the Cobb-Douglas
assumption, we shall maintain it throughout our discussion.
Equations (17) and (18) can be combined into an equation implicitly relating E and H:

H GH ( H , E )
G(H, E)

=

EGE ( H , E )
G(H, E)

.

(36)

The elasticity of labor costs with respect to H and E must be equal, because on the benefit side the elasticities
of effective labor input with respect to H and E are equal. Given the assumptions on G, (36) implies a
unique, upward-sloping E-H expansion path, so that we can write

E = E ( H ),

E ′ ( H ) > 0.

(37)

Equation (37) says that the unobservable intensity of labor utilization E can be expressed as a monotonically
increasing function of the observed number of hours per worker H. This result also holds in growth rates:
thus,

d ln ( EHN ) = dn + dh + de = dn + (1 + ζ ) dh .

(38)

Finding the marginal product of labor from (35) and substituting into the first-order condition for hours
per worker, equation (17), we find:

WNH GH ( H , E ) V (U ) = λγ α LY .

(39)

Substituting the marginal product of capital and equation (39) into (16) yields:

λγ α K

G ( H , E ) V ′(Z )
Y
= λγ α LY
.
Z
HGH ( H , E ) V ( Z )

(40)

Define

g (H ) =

H GH ( H , E ( H ) )
G ( H , E ( H ))

(41)

and

v(Z ) =

ZV ′ ( Z )
V (Z )

.

(42)

v(Z) is thus the ratio of the marginal shift premium to the average shift premium. Rearranging, we get

35

1=

αL v (Z )
.
αK g ( H )

(43)

The labor cost elasticity with respect to hours given by the function g ( H ) is positive and increasing by the
assumptions we have made on G ( H , E ) . The labor cost elasticity with respect to capital utilization given
by the function v(Z) is positive as long as there is a positive shift premium. We also assume that the shift
premium increases rapidly enough with Z to make the elasticity increasing in Z.
First, define

η=

H ∗ g′ ( H ∗ )
g (H∗)

,

(44)

and

ν=

Z ∗ v′ ( Z ∗ )
v(Z∗ )

.

(45)

η indicates the rate at which the elasticity of labor costs with respect to hours increases. ν indicates the
rate at which the elasticity of labor costs with respect to capital utilization increases. Using this notation,
the log-linearization of (45) is simply26

dz =

η
dh .
ν

(46)

Thus, equations (46) and (38) say that the change in hours per worker should be a proxy for changes in
both unobservable labor effort and the unmeasured workweek of capital. The reason that hours per
worker proxies for capital utilization as well as labor effort is that shift premia create a link between
capital hours and labor compensation. The shift premium is most worth paying when the marginal hourly
cost of labor is high relative to its average cost, which is the time when hours per worker are also high.
Putting everything together, we have an estimating equation that controls for variable utilization:

26

This equation is where the Cobb-Douglas assumption matters. Basu and Kimball differentiate (35) assuming that

a L a K is a constant. Their theory allows for the fully-general case where the ratio of the elasticities is a function
of all four input quantities, but they argue that pursuing this approach would demand too much of the data and the
instruments.

36

η 

dy = µ ∗ dx + µ ∗  ζ sL + sK  dh + dt
ν 


(47)

This specification controls for both labor and capital utilization, without making special assumptions about
separability or homotheticity. However, for our simple derivation, the Cobb-Douglas functional form is
important. As we have noted before, our model can be generalized to allow depreciation to depend on
capital utilization. This modification would introduce two new terms into the estimating equation (47), as in
Basu and Kimball (1997).

III. Aggregation over Firms27
Sections I and II emphasize production and estimation in terms of firm-level gross output. Section I
also provided an overview of the relationship between firm-level gross-output measures and aggregate valueadded measures. In this section, we derive and interpret this relationship in greater detail. We aggregate in
two steps. First, we relate firm-level gross output to value-added. Second, we aggregate over firm-level
value added. We then provide an economic interpretation of the various terms in the “aggregation equation.”
We conclude by discussing how aggregation could spuriously generate apparent “external effects” across
firms or industries.
The discussion highlights the potential pitfalls of using value added directly as a production measure.
A long literature from the 1970s (see, for example, Bruno 1978) also argues against the use of value added,
but on very different grounds. That literature emphasized the strong separability assumptions necessary for
the existence of a stable value-added function. By contrast, our argument against value added does not rely
on separability, which is a second-order property of the production function. Instead, we point out that value
added is akin to a “partial Solow residual,” subtracting from gross output growth the growth in intermediates,
weighted by their share in revenue. Firms equate revenue shares to output elasticities only with perfect
competition. Hence, with imperfect competition, some of the productive contribution of intermediate inputs
remains in measured value-added growth—a first-order issue that applies regardless of whether separability
holds or fails.

27

The appendix contains detailed derivations of the equations in this section.

37

A. The Conversion to Value Added
Measures of real value added attempt to subtract from gross output the productive contribution of
intermediate goods. Hence, gross output is shoes, while value added is “shoes lacking leather, made without
power” (Domar 1961, p716), or books without paper or ink.
Despite its unobservable and perhaps unintuitive nature at a firm level, we focus on value added for two
reasons. First, discussing firm-level value added turns out to be a useful intermediate step in moving
between firm-level gross output and aggregate value added. Second, many researchers use data on value
added for empirical work, and the results in this section shed light on the (de)merits of that approach.
Nominal value added, PV V , is defined unambiguously: PV V = PY − PM M . It is less clear how to
decompose nominal value added into price and quantity: We must subtract real intermediate inputs from
gross output in some way, but several methods are possible.
For the national accounting identity to hold, one must deflate nominal value added with the same
method used to deflate nominal final expenditure (Sato 1976). Since the national accounts now use chainlinked indices, we follow suit here. In particular, we use the continuous time analogue to a discrete time
chain-linked index, and define value added at a firm level, dvi , using the standard Divisia definition:28

dvi ≡

dyi − sMi dmi
.
1 − sMi

(48)

After substituting in for output growth from equation (9), we can write this as:

dvi = µ i ( dxiV + dui ) + ( µi − 1)

sMi
dti
,
dmi +
1 − sMi
1 − sMi

(49)

where primary-input growth, dxiV , is defined analogously to aggregate primary input growth

28

See Sato (1976). Until recently, the national accounts used a Laspeyres index for the expenditure side of GDP.
That is, each component is valued using base-year prices, and then components are added. The national accounts
identity then implied that one needed a Laspeyres or double-deflated index for real value added: V DD = Y - M ,
where gross output Y and intermediate inputs M are valued in base-year prices. The chain-linked index gives cleaner
results for productivity analysis, although the basic conclusions are unchanged. See the appendix to Basu and
Fernald (1995).

38

dxiV =

sKi
s
dki + Li dli ≡ sVKi dki + sVLi dli ,
1 − sMi
1 − sMi

(50)

The main implication of equation (49) is that value-added growth is not, in general, simply a function of
primary inputs dxiV . Value added growth is calculated by subtracting from gross output the revenue-shareweighted contribution of intermediate goods. With markups, however, the output elasticity of intermediate
inputs ( µi sMi ) exceeds its revenue share. Hence, some of the contribution of materials and energy is
attributed to value added; that is, value-added growth does not subtract off the full productive contribution of
intermediate inputs. This “extra” productive contribution affects value-added growth.
Of course, value-added growth could still be a function of primary-input growth alone, to the extent that
intermediate inputs move together with primary inputs. To explore this possibility, it is useful to rewrite
equation (49) in the following way (the appendix shows the algebra):

 sMi 
V
V
dvi = µiV dxiV + ( µiV − 1) 
 ( dmi − dyi ) + µi dui + dti ,
1 − sMi 

(51)

where:

µiV ≡ µi

dtiV ≡

1 − sMi
1 − µi sMi

dti
.
1 − µi sMi

(52)

(53)

A useful benchmark case is where intermediate inputs are used in fixed proportions to output; this
assumption is implicit in most dynamic general equilibrium models with imperfect competition (see, e.g.,
Rotemberg and Woodford, 1995). With fixed proportions, equation (52) shows that value-added growth is,
indeed, a function of primary inputs alone. The “value-added markup” µiV , defined by equation (52),
includes the productive contribution of primary inputs as well as the extra contribution of intermediates.
Hence, if µi is greater than one, the value-added markup µiV exceeds the gross-output markup µi .
Nevertheless, equation (51) makes clear that with imperfect competition, if the assumption of fixed

39
proportions fails, then value-added growth depends on the growth rate of the ratio of intermediate inputs to
output.
The value-added markup is plausibly the appropriate concept for calibrating the markup charged by the
representative firm in a one-sector macroeconomic model (see Rotemberg and Woodford 1995). The reason
is that small markups at a plant level may translate into larger efficiency losses for the economy overall,
because of “markups on markups.” That is, firms buy intermediate goods from other firms at a markup, add
value, and again price the resulting good with a markup, generally selling some of it to another firm to use as
an intermediate good. We explore this (and other) intuition in greater detail in the appendix.
That the intensity of intermediate-input use affects value-added growth underlies our argument that the
right approach to estimating even the value-added markup is to use gross-output data. Without making any
auxiliary assumptions about whether materials are used in fixed coefficients, one can estimate (a utilizationcorrected) µ from equation (10), and then transform it into its value-added analogue using equation (52).
(Note that value added remains appropriate as a national accounting concept, since the wedge between the
cost and marginal product of intermediate inputs represents actual goods and services available to society—
it’s just that we cannot in general allocate its production to primary inputs of capital and labor.)
The firm's revenue-weighted value-added productivity residual, dpi , equals dvi − dxiV . Hence,

 sMi 
V
V
dpi = ( µiV − 1) dxiV + ( µiV − 1) 
 ( dmi − dyi ) + µi dui + dti .
1 − sMi 

(54)

A long literature in the 1970s explored whether a “value added function” exists (see Bruno 1978, for
example), and argued that the answer depends on separability properties of the production function. The
equation above shows that with imperfect competition, taking value-added to be a function only of primary
inputs is generally misspecified regardless of whether the production function is separable between value
added and intermediate inputs. Separability is a second-order property of a production function, so its
presence or absence does not affect first-order approximations like equations (49) and (51). However, the
fact that the output elasticity of materials is µi sMi instead of simply sMi is of first-order importance.

40
B. Aggregation
We define aggregate inputs as simple sums of firm-level quantities:

K ≡ ∑ i =1 K i
N

L ≡ ∑ i =1 Li
N

For simplicity, we assume that there is one type of capital and one type of labor. (This can be relaxed
easily.)
In principle, different firms may face different shadow prices for a homogeneous input; this will
generally be the case if some inputs are quasi-fixed. For any input J, let PJi be the shadow price it pays to
rent or hire the input for one period; differences across firms in shadow prices could reflect factor-price
differences or else adjustment costs, as in Section II. We define the aggregate (rental) prices of capital and
labor as implicit deflators—that is, total factor payments divided by aggregate quantities:

PK

∑
≡

PL

∑
≡

N
i =1

PKi K i

K
N
i =1

PLi Li

L

(55)

(56)

We use the standard Divisia definition of aggregate output. This measure weights goods by market
prices and hence avoids substitution bias in the aggregate output and price indices. Divisia aggregates are
defined most naturally in growth rates, and we denote the growth in aggregate output (equivalently,
aggregate value added) by dv. From the national accounting identity, one can define aggregate output in
terms of either production (aggregating value added over firms) or expenditure (aggregating sales for
consumption, investment, government purchases, or export). Welfare (discussed in Section VI) uses the
expenditure side; production (emphasized so far) relates to the value-added side. From the production side,
aggregate output is a Divisia index of firm-level value-added. In growth rates:

dv = ∑ i =1 wi dvi
N

where wi is the firm's share of nominal value added:

(57)

41

wi =

PiV Vi
PV V

We can now substitute in from equation (51) for dvi . Substantial algebraic manipulation (shown in the
appendix) yields our basic aggregation equation:

dv = µ V dxV + du + R + dt V ,

(58)

where:

 PL
dxV =  VL
P V


 PK K 
V
V
 dl +  V  dk ≡ sL dl + sK dk ,

P V

µ V = ∑ i =1 wi µiV ,
N

(59)

du = ∑ i =1 wi µ dui , and
N

V
i

dt V = ∑ i =1 wi dtiV .
N

dxV is growth in aggregate primary inputs, µ V is the average firm value-added markup, du is average firm
utilization growth (weighted by markups), and dt V is average “value added” technology change. R
represents various reallocation effects:

R = Rµ + RM + µ V RK + µ V RL ,

(60)

where:

Rµ = ∑ i =1 wi ( µiV − µ V ) dxiV ,
N

 s 
N
RM = ∑ i =1 wi ( µiV − 1)  Mi  ( dmi − dyi ) ,
1 − sMi 
 P − PK 
N
RK = ∑ i =1 wi sVKi  Ki
 dki , and
 PKi 

(61)

 P − PL 
N
RL = ∑ i =1 wi sVLi  Li
 dli .
 PLi 

Note that aggregate utilization du from equation (59) is the weighted average of firm-level “valueadded” utilization. It is a value-added measure, since it is a form of primary input, albeit one that is
unobserved. Since it is multiplied by value-added returns to scale, it captures the full effect on aggregate

42
output of a change in utilization, incorporating both the contribution of a change in the unobserved quantity
of input, and the contribution of the markup. (As with primary inputs, we could have separated this into a
term reflecting the “average” markup, and a “reallocation of the markup” term, but since our main interest in
utilization is on the total effect, it is simpler to keep them together.)
Aggregate productivity growth is the difference between the growth rates of aggregate output, dv, and
aggregate inputs, dxV :

dp ≡ dv − dxV .

(62)

Note that this is a “modified Solow residual,” since the input weights in dxV need not sum to one. The
shares are factor payments in total nominal value added, and sum to one only if there are no economic
profits. Thus:

dp = ( µ V − 1)dxV + du + R + dt V .

(63)

Equation (63) shows the distinction between aggregate productivity and aggregate technology. If all
firms are perfectly competitive, pay the same price for factors (perhaps reflecting perfect factor mobility),
and do not vary utilization, then all terms other than dt V disappear: Productivity equals technology.
However, with imperfect competition or frictions in product or factor markets, productivity and technology
are not equivalent.29

C. Productivity Interpretation of Reallocation Terms
Aggregate output combines goods using market prices, which in turn measure consumers’ relative
valuations. Suppose we want to know how much consumers (and, therefore, usually society) value having a
marginal input allocated to a particular firm (which we assume produces a single good). That valuation
i
equals the good’s price times the factor’s marginal product: PF
i J . The first-order condition (7) shows that

this valuation equals the firm’s markup times the input price paid by the firm: µi PJi . If markups or input

29

Jorgenson, Gollop and Fraumeni (1987) derive an equation for the case of case of constant returns to scale and
perfect competition, so they omit the terms other than RK and RL . They also allow for heterogeneity in capital and
labor, which we have ignored for simplicity. With heterogeneity, our results generalize easily: For example, if RKk

is the factor-price reallocation term for capital of type k, then RK =

Êk RKk .

43
prices differ across firms, then the marginal “social value” of inputs also differs across firms. Rµ , RK , and

RL reflect shifts of resources among uses with different marginal social values.
Consider the markup-reallocation term, Rµ . By definition, the markup represents the wedge between
the value consumers place on the good—i.e., its price—and its marginal cost. Reallocating resources from
low- to high-markup firms thus shifts resources towards uses where consumers value them more highly. If
the variability of firms’ inputs are correlated with their market power, then imperfect competition affects
aggregate productivity even if the average markup is small. For example, Basu and Fernald (1997a) estimate
that durable-goods industries have larger returns to scale and markups than non-durable-goods industries.
Durable-industries are more cyclical, and employ a larger share of the marginal inputs in a boom. This
marginal reallocation thus contributes to the procyclicality of aggregate productivity.
Now consider the input-reallocation terms RK and RL . Shifting labor from firms where it has a low
shadow value to firms where it has a high shadow value increases aggregate output. Why might shadow
values (or wages) differ across firms? First, labor may not be instantaneously mobile across sectors; sectoral
shifts may lead workers in, say, defense industries to have lower marginal products than they would in
health-care.30 Second, efficiency wage considerations may be more important in some industries than others,
as emphasized by Katz and Summers (1989). Third, unions with monopoly power might choose to charge
different wages to different firms. Whatever the reason, shifting labor to more productive uses increases
aggregate output, even if total input does not change.
Note that the first reason, costly factor mobility, is completely consistent with constant returns and
perfect competition. Differences in marginal products that reflect factor immobility should be temporary—
that is, PK as defined in (25) should not persistently differ from the Hall-Jorgenson cost of capital, and PL as
defined in (26) should not persistently differ from the firm’s compensation payments (see Berndt and Fuss).

30

Whether differences in labor’s marginal product lead to differences in wages depends on whether the adjustment
costs are paid by workers or firms.

44
With costly factor adjustment, these shadow values may differ substantially; hence reallocation effects on
output and productivity may be significant, even in a world with perfect competition and constant returns.31
The materials-reallocation term, RM , reflects the extent to which measured real value added depends
on the intensity of intermediate-input use. Firm-level value added is useful for national accounting,
regardless of technology or market structure. However, with imperfect competition, value-added growth
does not subtract off the full marginal product of intermediate inputs. Growth in primary inputs captures
some of this productive contribution (which is why µiV differs from µi ), but some “wedge” may remain.

RM equals the sum of these wedges. It represents real goods and services, and hence affects aggregate
output and productivity.
Note that RM depends on the size of markups in firms using materials. Consider an economy where
some firms produce intermediate goods using capital and labor, and other firms assemble intermediate goods
into final goods (e.g., Beaudry and Devereux, 1994). The importance of RM depends on the size (and
heterogeneity) of markups in the final-goods industry. This is important because firms may be able to
negotiate multi-part prices with long-term suppliers of their inputs, and thus partially offset the inefficiencies
resulting from imperfect competition in intermediate-goods industries.32 The inefficiency in RM , however,
depends on markups in firms using intermediate goods, not those selling such goods; multi-part pricing for
intermediate goods does not eliminate this inefficiency. (However, the inefficiency is larger in symmetric
models, such as Basu (1995), where all output is also used as materials input.)

31

The “sectoral shifts” literature takes this approach; see, e.g. Phelan and Trejos (1996). Horvath (1995) also
incorporates adjustment costs into a dynamic general equilibrium model, generating effects on aggregate
productivity from input reallocations. Microeconomic productivity literature (e.g. Baily et al., 1992) finds that there
are systematic productivity level differences across firms within narrowly defined industries; to the extent these
productivity differences are not measurement error, they show up either as higher profits from a higher markup, and
hence are reflected in Rm , or higher factor payments, and hence are reflected in RK and RL .
32

We thank Robert Hall for this observation.

45
D. The Definition of Technology Change
Conceptually, aggregate technology change measures the change in aggregate output in response to
firm-level technology shocks, holding primary inputs fixed. Under what conditions does this correspond to
our measure dtV?
With constant returns and perfect competition, Domar (1961) and Hulten (1978) show that our
definition properly measures the outward shift in society’s production possibilities frontier (PPF) when firmlevel technology changes. In Figure 2, this case corresponds to point A, where society allocates resources
optimally.
In this case, all of the µi equal 1, so this “Domar-weighted measure” of aggregate technology equals:

dt V = ∑ i wi

dt i
.
1 − sMi

(64)

Conceptually, Domar weighting converts gross-output technology shocks to a value-added basis by dividing
through by the value-added share, 1 − sM . These shocks are then weighted by the firm’s value-added share.
Even with imperfect competition, the Domar-weighted measure correctly shows how much final output
(the sum of value added over firms) increases, if all of the increase in gross output goes to final sales, with
primary and intermediate inputs remaining unchanged. With perfect competition and perfect factor mobility,
Domar-weighting is correct even if inputs adjust. That is, the consequences for aggregate output and
technology are the same whether the firm sells all of its additional output for final consumption (with a
valuation equal to its price), or instead sells the additional output to other producers, who in turn use the
additional intermediate inputs (with a marginal product equated to the price) to produce goods for final
consumption.
With imperfect competition, however, aggregate technology change is not unambiguously defined. The
economy may produce inside its PPF, at a point like B in Figure 2, or at an allocatively inefficient point on
the PPF, like point C. At these points, the same firm-level technology shocks can affect the distribution of
inputs across firms in different ways, and thus cause different changes in aggregate output. Imperfect
competition or differences in factor payments across firms may lead the same factor to have a different social

46
value for its marginal product in different uses. Hence, changes in the distribution of resources can affect
aggregate output, even if there are no technology shocks and total inputs remain the same.
The definition in equation (59) is essentially a markup-corrected measure. In terms of the gross-output
shocks, it is equivalent to the following:

dt V = ∑ i wi

dt i
,
1 − µi sMi

(65)

This definition correctly measures the increase in aggregate output under standard conditions where an
aggregate production function exists. Suppose there are no factor-market frictions, and that all firms have
the same separable gross-output production function, always use materials in fixed proportions to gross
output, and charge the same markup of price over marginal cost. Under these assumptions, which are
implicit or explicit in most dynamic general-equilibrium models with imperfect competition, there is a
representative producer and an aggregate value-added production function (see, e.g., Rotemberg and
Woodford, 1995). In this case, aggregate technology change corresponds to our definition dt V .
Qualitatively, the Domar-weighted and markup-corrected measures turn out to give very similar
correlations between technology change and various business-cycle indicators. (One empirical note of
caution is that concern the markup-corrected measure is sometimes very sensitive to estimates of the markup.
When we estimate µi for each sector, we hope that our estimates are unbiased. However, the estimate of
value-added technology change in equation (53), dti (1 − µi sMi ) , is convex in µi . Hence, it overweights
sectors where µˆ i exceeds µi, and underweights sectors where µˆ i is less than µi.)

E. Externalities
So far, we have discussed internal increasing returns but not Marshallian externalities. With such
external effects, the economy might display aggregate increasing returns even if all firms produce with
constant returns. Hence, one can model increasing returns without having to model imperfect competition.
As a result, models of growth and business cycles use such externalities extensively (e.g., Romer, 1986;
Baxter and King, 1991; and Benhabib and Farmer, 1994). Externalities are almost surely important for
modeling economic growth, as suggested by the extensive R&D-spillover literature surveyed by Griliches

47
(1992). High-frequency demand spillovers (as discussed in Cooper and John 1988) also seem eminently
sensible. However, apart from Diamond (1982), few have proposed models of short-run technological
spillovers that operate at high frequencies.
Caballero and Lyons (1992) argued, however, that a basic prediction of externality models is a robust
feature of the data: Estimated returns to scale should rise at higher levels of aggregation, as the increasing
returns become internalized. In other words, aggregate productivity should be more procyclical than would
be implied by estimates of industry-level returns to scale. Thus, despite lacking a formal model of short-run
external effects, they concluded that there is strong prima-facie evidence for such externalities.
Caballero and Lyons (1990, 1992) proposed an empirical model to estimate any short-run externalities.
They augmented a firm-level estimating equation like (10) with aggregate inputs as well as firm-level inputs.
In practice, they used industry-level data on output and inputs, and added aggregate output to the estimating
equation to capture any externalities that would otherwise be relegated to the error term, dt. They used NIPA
data on real value added by industry. Thus, their basic estimating equation was essentially:

dvi = µiV dxiV + β dxV + dti ,

(66)

where i indexed a two-digit manufacturing industry, and dxV was growth of aggregate capital and labor in
manufacturing. (Their 1992 paper also included various utilization controls.) They found large, positive,
and statistically-significant values of β in data from the United States and a number of European countries.
Numerous authors have questioned the interpretation and robustness of the Caballero-Lyons’s results.
For example, their results may reflect inappropriate data (Basu and Fernald, 1995; Griliches and Klette,
1996), incorrect econometric method (Burnside, 1996), or inadequate utilization proxies (Sbordone, 1997).
Nevertheless, even if one can dismiss their interpretation, the stylized fact remains that productivity is
more procyclical at higher levels of aggregation. However, equation (63) suggests an alternative
explanation of the Caballero-Lyons stylized fact. Aggregate productivity is more procyclical because the
reallocation terms, R, are procyclical. Thus, we can explain the Caballero-Lyons stylized fact based only on
firm-level heterogeneity, without invoking external effects that have questionable theoretical basis in the
business-cycle context. Indeed, we note that the one well-known economic model of short-run externalities,
Diamond’s (1982) search model, relies on increasing returns to scale in the “matching function” that

48
produces new hires as a function of economy-wide vacancies and unemployment. But efforts to estimate this
matching function directly (e.g., Blanchard and Diamond, 1990) show that it exhibits approximately constant
returns to scale, not large increasing returns. Thus, in the absence of any direct evidence for short-run
externalities, we do not model them explicitly in this paper.33

IV. Data and Method
A. Data
We now construct a measure of “true” aggregate technology change and explore its properties. As
discussed in Section I, we estimate technology change at a disaggregated level, and then aggregate using the
theory in Section III. Our aggregate is the private U.S. economy, and our “firms” are 33 industries; for
manufacturing, these industries correspond roughly to the two-digit SIC level.
Given that each industry includes thousands of firms, it may seem odd to take industries as firms.
Unfortunately, no firm-level data sets span the economy. In principle, we could focus on a subset of the
economy, using the Longitudinal Research Database, say. However, narrowing the focus requires sacrificing
a macroeconomic perspective, as well as panel length and data quality. By focusing on aggregates, our paper
complements existing work that uses small subsets of the economy.34
We use data compiled by Dale Jorgenson and Barbara Fraumeni on industry-level inputs and outputs.
These data comprise gross output and inputs of capital, labor, energy, and materials for a panel of 33 private
industries (including 21 manufacturing industries) that cover the entire U. S. non-farm private economy.
These sectoral accounts seek to provide accounts that are, to the extent possible, consistent with the
economic theory of production. (For a complete description of the dataset, see Jorgenson et al., 1987.)
These data are available from 1947 to 1989; in our empirical work, however, we restrict our sample to 1950
to 1989, since our money shock instrument (described below) is not available for previous years.

33

The search for short-run spillovers remains an active area of ongoing research. See, for example, Bartelsman et
al. (1994), Cooper and Johri (1997), and Paul and Siegel (1999).
34 See, for example, Baily, Hulten, and Campbell (1992), Haltiwanger (1997), Bartelsman and Dhrymes (1998),
and Foster, Krizan, and Haltiwanger (1998). The aggregation theory in Section III implies that our industry data
include various intra-industry reallocation terms, including the analogous terms to Rm , RK , and RL .

49
We weight growth rates (measured as log changes) of capital, labor, and intermediate inputs using the
average shares in revenue over the entire period. To compute capital’s share sK for each industry, we
construct a series for required payments to capital. Following Hall and Jorgenson (1967) and Hall (1990),
we estimate the user cost of capital C. For any type of capital, the required payment is then CPK K , where

PK K is the current-dollar value of the stock of this type of capital. In each sector, we use data on the current
value of the 51 types of capital, plus land and inventories, distinguished by the BEA in constructing the
national product accounts. Hence, for each of these 53 assets, indexed by s, the user cost of capital is

Cs = ( r + δ s )

(1 − ITCs − τ d s ) , s=1 to 53.
(1 − τ )

(67)

r is the required rate of return on capital, and δ s is the depreciation rate for assets of type s. ITCs is the
asset-specific investment tax credit, τ is the corporate tax rate, and d s is the asset-specific present value of
depreciation allowances. We follow Hall (1990) in assuming that the real required return r equals the
dividend yield on the S&P 500. Jorgenson and Yun (1991) provide data on ITCs and d s for each type of
capital good. Given required payments to capital, computing sK is straightforward.
As discussed in Section I, we require instruments uncorrelated with technology change. We use two of
the Hall-Ramey instruments: the growth rate of the price of oil deflated by the GDP deflator and the growth
rate of real government defense spending.35 (We use the contemporaneous value and one lag of each
instrument.) We also use a version of the instrument used by Burnside (1996): quarterly Federal Reserve
“policy shocks” from an identified VAR. We sum the four quarterly policy shocks in year t-1 as instruments
for year t.36

35

We drop the third instrument, the political party of the President, because it appears to have little relevance in
any industry. Burnside (1996) shows that the oil price instrument is generally quite relevant, and defense spending
explains a sizeable fraction of input changes in the durable-goods industries.
36 The qualitative features of the results in Section III appear robust to using different combinations and lags of the
instruments. On a priori grounds, the set we choose seems preferable to alternatives—all of the variables have
strong grounds for being included. In addition, the set we choose has the best overall fit (measured by mean and
median F statistic) of the a priori plausible combinations we considered. Of course, Hall, Rudebusch and Wilcox
(1996) argue that with weak instruments, one does not necessarily want to choose the instruments that happen to fit
best in sample; for example, if the “true” relevance of all the instruments is equal, the ones that by chance fit best in
sample are in fact those with the largest small sample bias. That case is probably not a major concern here, since the
instrument set we choose fits well for all industry groupings; for example, it is the one we would choose based on a

50

B. Estimating Technology Change
To estimate “firm-level” technology change, we estimate a version of equation (10) for each industry.
Although we could estimate these equations separately for each industry (and indeed do so as a check on
results), some parameters—particularly on the utilization proxies—are then estimated rather imprecisely. To
mitigate this problem, we combine industries into four groups, estimating equations that restrict the
utilization parameters to be constant within industry groups.
As discussed in Section II, this estimating equation corresponds to the special case where the cost of
higher capital utilization is a shift premium. In that case, variations in hours per worker fully captures
variations in capital utilization and effort. Thus, for each group we have

dyi = ci + µi dxi + adhi + dti

(68)

The markup µi differs by industries within a group (Burnside (1996) argues for allowing this variation). The
groups are durables manufacturing (11 industries); non-durables manufacturing (10); mining and petroleum
extraction (4); and all others, mainly services and utilities (8). To avoid the “transmission problem” of
correlation between technology shocks and input use, we estimate each system using Three-Stage Least
Squares, using the instruments noted above.
After estimating equation (68), we take the sum of the industry-specific constant ci and residual dtˆt as
our measure of technology change in the gross-output production function. We then insert these industry
estimates in the aggregation equation (63), derived in Section III. Note that this aggregation equation is an
accounting identity. It allows us to decompose aggregate productivity into a technological component plus
various non-technological components, including the effects of markups and various reallocations.
One problem in implementing the decomposition is that we may not, in fact, observe period-by-period
the appropriate “shadow” factor prices PLi and PKi i defined by equations (25) and (26). We deal with this
problem by taking an explicitly first-order approach to the estimating equation in (68), using fixed weights
on capital and labor in constructing dxi. (It is straightforward, though it requires some care, to ensure that the
rule of, say, using the instruments that fit best in durables industries as instruments for non-durables industries, and

51
aggregation equation (63) remains an accounting identity with fixed weights.) This approach is unlikely to
lead to major problems in estimating the markup- and materials-reallocation terms ( Rµ and RM ) in the
accounting identity, since those terms are driven primarily by changes in quantities, rather than changes in
the weights (which, in turn, incorporate the shadow prices).
However, the inability to measures prices is a major problem for measuring the input reallocation terms
( RK and RL ), since those terms depend explicitly on differences in prices across sectors. Jorgenson, Gollop,
and Fraumeni (1987) estimate these terms under the assumption that factor prices are allocative, and that
there are zero profits in all sectors (so that they can “back out” capital’s input PK as a residual). Although we
do not require these assumptions elsewhere, we will show summary statistics from the Jorgenson et al.
estimates—we emphasize that these are meant to be suggestive only. Since the aggregation equation (63) is
an accounting identity, changing our estimate of one component requires a change in other components as
well. We then remove the effect of the input-reallocation terms from the average-markup term (a natural
place to take it from, given equation (A.17) in the appendix). It is worth emphasizing that even if we
mismeasure the input-reallocation terms, this primarily affects our measurement of other non-technological
components of aggregate productivity. In particular, it does not directly affect our estimate of aggregate
technology, which we built up from disaggregated residuals.

V. Results
We now investigate empirically why productivity is procyclical. We seek to identify the importance of
(i) imperfect competition, (ii) reallocations, and (iii) variable utilization. Controlling for these influences
allows us to move from aggregate productivity to aggregate technology. We then explore the cyclical
properties of the “corrected” technology series.
We define aggregate productivity growth as the modified Solow residual defined in equation (62). This
measure differs from the standard Solow residual since the revenue weights need not sum to one; the
difference reflects economic profits or losses. However, we estimate that profits are small (about 3 percent

vice versa.

52
on average, using our estimates of required payments to capital), so the results we report are essentially
unchanged using the standard Solow residual instead.
Table 1 reports summary statistics for three series: (i) the Solow residual; (ii) a series that makes no
utilization corrections, but corrects only for aggregation biases; (iii) a “technology” measure based on (68),
which uses growth in hours-per-worker to correct for utilization. The first measure uses aggregate data
alone. The other two are based on sectoral Solow-Hall-style regression residuals, as described in the
previous section; these residuals are then aggregated using equation (63). Hence, aggregate technology
change is the weighted sum of sectoral technology changes, as described in Sections I and III (see equations
(53) and (59)).
Panel A shows results for the entire non-farm business economy. Our corrected series have about the
same mean as the Solow residual. However, the variance is substantially smaller: The variance of the fullycorrected series is less than one-third that of the Solow residual, so the standard deviation (shown in the
second column) is only about 55 percent as large. The reported minimums show that we do estimate
negative technical change in some periods, but the lower variance of the technology series implies that the
probability of negative estimates is much lower. For example, the Solow residual is negative in 12 out of 40
years; the fully-corrected residual is negative in only 5 out of 40 years.
Panel B gives results within manufacturing alone. Data within manufacturing (especially for output)
are often considered more reliable than data outside manufacturing. In addition, some other papers (such as
Burnside, 1996) focus only on manufacturing, so these results provide a basis for comparison. The results
are qualitatively similar to those for the aggregate economy.
Some simple plots and correlations summarize the comovement in our data: Output and inputs are
strongly positively correlated, and all are positively correlated with the Solow residual.. Figure 1 plots
business-cycle data for the non-farm private economy: output (value-added) growth dv, primary input
growth, dxV , and the Solow residual dp (all series are demeaned). These series comove positively, quite
strongly so in the case of dp and dv. Table 2 shows correlations for these three variables, as well as growth
of total hours worked (dh + dn). Hours correlate more strongly with productivity than do total inputs,

53
reflecting the low correlation of changes in the capital stock with the business cycle. The 95 percent
confidence intervals show that all are significant.
Figure 3 plots our fully-corrected technology series against these three variables. The top panel shows
that technology fluctuates much less than the Solow residual, consistent with intuition that non-technological
factors, such as variable input utilization, increase the volatility of the Solow residual. In addition, some
periods show a phase shift: the Solow residual lags technology change by one to two years. This phase shift
reflects the utilization correction. In our estimates, technology improvements are associated with low levels
of utilization, thereby reducing the Solow residual relative to the technology series. The phase shift, in
particular, appears to reflect primarily movements in hours per worker, which generally increase a year after
a technology improvement. In the model from Section II, increases in hours per worker imply increases in
unobserved effort, which in turn increase the Solow residual.
The middle panel plots aggregate value-added output growth (dv) against technology. There is no clear
contemporaneous comovement between the two series although, again, the series appear to have a phase
shift: Output comoves with technology, lagged one to two years.
Finally, the bottom panel plots the growth rate of primary inputs of capital and labor ( dxV ) and the
same technology series. These two series clearly comove negatively over the entire sample period.
Table 3 shows the correlations between our technology measures and business-cycle variables. Panel A
shows results for the aggregate private economy. With full corrections, the correlation of technology with
output is about zero, and the correlations with inputs are strongly negative: -0.42 for total primary inputs, and
-0.44 for hours alone. Both correlations are statistically significantly negative at the 95 percent level.
The correlations with aggregate technology change differ sharply from those predicted by the usual
RBC model (e.g., Cooley and Prescott, 1995). In particular, in calibrated dynamic general equilibrium
models with flexible prices, technology shocks generally cause a contemporaneous increase in both inputs
and output. These kinds of standard real-business-cycle models explore whether technology shocks lead to
comovement that matches the stylized facts of business cycles. Given that the central stylized fact of
business cycles is the comovement between inputs and output, if technology shocks drive the cycle then
almost any sensible calibration implies that technology improvements increase inputs and output.

54
Basu, Fernald, and Kimball (1998) explore the negative contemporary comovement between
technology and input growth at length. They argue that this negative comovement is consistent with stickyprice models. For example, suppose a firm’s technology improves but the firm cannot change its price. If its
demand curve does not change, then it cannot change its sales—but it can produce that same quantity with
fewer inputs. Over time, of course, the firm (and economy) adjust to the technology change.
In terms of our accounting identities, what explain the movement away from a strong positive
correlation? Figure 4 shows the our estimated utilization series—aggregated from the implicit utilization
series for each industry using the equation for du from (59)—and our estimated reallocation series, both
plotted against the Solow residual. Both utilization and reallocations are procyclical, as shown here by the
positive comovement with the Solow residual. Each contributes about equally to generating the negative
correlation. To see this, we first subtracted the estimated utilization change from the Solow residual; the
correlation with inputs fell to about zero. Similarly, we then subtracted the estimated reallocation terms from
the Solow residual; again, the correlation with inputs fell to about zero.
It is not surprising that utilization is procyclical. After all, utilization is a form of primary input, and
inputs are procyclical. It is less obvious why reallocations are procyclical, since research at a highly
disaggregated level—for example, using firms within a narrowly defined industry—finds that reallocations
tend to be countercyclical.37 Low productivity firms enter and expand disproportionately in booms, and
contract or disappear disproportionately in recessions. Hence, within narrowly defined industries,
reallocations appear to make aggregate productivity less cyclical, not more. (Since we use much more
aggregated data on industries, these intra-industry reallocations will tend to appear as decreasing returns to
scale or—given the close link between returns to scale and markups discussed in equation (5) of Section I—
markups less than one.)
A different process clearly must be at work across aggregated industries. Table 3 presents summary
statistics for the components of the reallocation terms from equation (61). (As discussed in the previous
section, we obtained estimates of the capital- and labor-reallocation terms from Dale Jorgenson and Barbara
Fraumeni.) None of these components has a sizeable mean, but they do have substantial standard deviations.

37

See, for example, Foster, Krizan, and Haltiwanger (1998).

55

Rµ and RM are the most important components of the reallocation term. For the instrumented series, these
terms have standard deviations of 0.72 and 0.51 respectively, compared with a total standard deviation of
1.11 for R. The sum of RK and RL is much less volatile. Rµ , in particular, comoves strongly with inputs,
with a correlation of 0.7 with aggregate dx.
Why is Rµ so important over the cycle? Its variation reflects the fact that high-markup sectors tend to
be more cyclical than average. Hence, in a boom, high markup (and hence, high marginal product) firms
produce a disproportionate share of the marginal output. We can see this in Figure 5, which shows estimates
of the gross-output markup by industry, plotted against the relative cyclicality of the industry.38 (The
cyclicality was estimated by regressing dxiV − dxV on dxV .) High markup industries tend to be durables
manufacturing industries, which also tend to be the most cyclical. In a simulated DGE model, Basu, Fernald,
and Horvath (1996) find that reallocations between durable and non-durable producers provides a significant
propagation mechanism for shocks to the economy.
One might ask whether the reallocation effects we have identified represent the important gap between
(utilization corrected) productivity and technology, or whether the difference between the two is still driven
mostly by the “average markup” effect identified by Hall (1988, 1990). We compared our results to what we
would have found had we used only the “average” correction, i.e. that coming from the presence of the

(µ

V

− 1) dxV term. Using the correlations with dxV as a benchmark, we find that the reallocation effects

are the more important. For the OLS results, productivity corrected for the average effect would have
yielded a correlation with input growth of 0.18, as opposed to 0.23 for the uncorrected series and 0.05 for our
estimated technology series. For the IV results, the correlation would have been 0.12, as opposed to -0.13 for
the estimated series. Thus, our reallocation effects are responsible for at least two-thirds of the correction.
This result should not be surprising since, as we noted, the recent literature finds small average markups.
Our results echo this finding: Without any utilization correction, we find µ V = 1.12 , for the hours-corrected

38

The mean markup in Figure 5 is less than one. As Basu and Fernald (1997a) discuss, the average of sectoral
estimates of gross-output markups is less than one, even though the average of value-added markups tends to be
slightly greater than one. This primarily reflects the difference in weights used to aggregate the two.

56
case we find µ V = 1.05 . The surprising result is that even such small average markups are consistent with
important differences between aggregate productivity and aggregate technology coming from reallocations
across sectors.

VI. Implications for Macroeconomics
A. Productivity as Welfare
Why is aggregate productivity growth interesting? The usual justification is Solow’s (1957) proof that
with constant returns to scale, perfect competition, and no frictions, it measures aggregate technology
change. But in a world with distortions, is the Solow residual merely mismeasured technology?
In this section we summarize the argument of Basu and Fernald (1997b), who suggest that the answer is
No. They show that productivity growth correctly computed from aggregate data (i.e., after eliminating the
mismeasurement caused by changes in utilization) has a natural welfare interpretation, whether or not it also
measures technology change. In particular, the modified aggregate Solow residual ( dp − dxV ) defined in
Section III—which reduces to Solow’s residual if there are no economic profits—measures welfare change
for a representative consumer. This result holds even with imperfect competition in output markets and nonconstant returns to scale in production. Intuitively, growth in aggregate output measures the growth in
society’s ability to consume. To measure welfare change, we must then subtract the opportunity cost of the
inputs used to produce this output growth. Input prices measure that cost, regardless of whether they also
reflect marginal products. For example, increasing the stock of capital requires foregoing consumption, and
the rental price of capital measures the consumer’s opportunity cost of providing new capital, just as the
wage measures the opportunity cost of providing extra labor. Proving our welfare result requires simply that
consumers take prices parametrically. Hence, if productivity and technology differ, then it is productivity
that most closely indexes welfare.
This conclusion is appealing for two reasons. First, it shows that productivity rather than, say, GDP, is
the right measure of economic welfare under fairly general conditions. Second, it shows that even with
distortions, policymakers can compute interesting quantities from aggregate data—we do not always need to
calculate firm-level or aggregate technology change. In the short to medium run, productivity can change for

57
reasons unrelated to technology change. Thus, even with distortions such as imperfect competition, when the
aggregate Solow residual does not in general index technology change, it remains an excellent index of
welfare change. Hence, it remains an appropriate target for policy, as well as a convenient indicator.
In Section III, we showed that aggregate technology change needs to be measured using
disaggregated—ideally, firm-level—data. So why do aggregate data yield a meaningful measure of welfare
change? The welfare properties of the Solow residual follow from the equality of relative market prices to
the consumer’s marginal rates of substitution (MRS) between goods; this includes the equality of the real
wage to the MRS between goods and leisure. These equalities hold even when market prices do not reflect
the economy’s marginal rate of transformation (MRT) between those goods. Equivalently, we need only to
investigate the expenditure side of the National Income Accounts identity; we do not need to know the
production technology of firms or the competitive structure of industries.
There are two qualifications to our argument. First, the ratio of factor prices may not equal the
consumer’s marginal rate of substitution: taxes, for example, create a wedge between the two, since the wage
paid by firms then differs from the wage received by households. The welfare interpretation of the residual
requires factor prices received by households, but this modification is straightforward: all prices should be
those perceived by the household. Second and more seriously, the representative-consumer assumption may
fail. Consumers may have different marginal utilities of wealth or, as in standard efficiency-wage models or
bargaining models, they may face different prices. In this case, one cannot compute aggregate welfare
change from aggregate statistics alone. However, we do not claim that our proposed productivity measure is
a completely general measure of welfare change, merely that it is one under much more general conditions
than the usual Solow residual. It seems a particularly apt measure in the context of recent macroeconomic
models with a representative consumer but with imperfect competition in product markets (e.g., Rotemberg
and Woodford (1992, 1995)), or with multiple sectors and costly factor reallocation (e.g., Ramey and
Shapiro, 1997).
Figure 2 shows the economic intuition underlying our argument. Suppose the economy produces two
goods, both of which are consumed (and possibly also used as intermediate inputs). To keep the graph
simple, assume that the supplies of capital and labor are fixed. The production possibilities frontier (PPF)

58
depicts all feasible ( C1 , C2 ) pairs. An economy without distortions attains the social optimum at point A,
supported by relative prices P1 P2 . Now suppose there are distortions. Then the economy might be at an
allocatively inefficient point on the PPF, like point B, or even within the PPF, like point C. As shown in
Figure 2, these outcomes can be supported by price ratios different from the MRT between C1 and C2 .
Note that in all cases the consumer’s budget line shows the economy’s iso-output line, which
aggregates heterogeneous output using market prices (regardless of whether these prices reflect technological
tradeoffs.).39 Thus, in this example welfare increases only if output increases. (This is a special case of our
general result that welfare increases only if productivity increases, since in this example dk = dl = 0.) Hulten
(1978) shows that under Solow’s conditions — perfect competition and constant returns — aggregate
productivity growth represents both technology improvement and welfare increase. In terms of Figure 2,
Hulten’s result applies to an economy at point A: Output (productivity) can increase only if the PPF shifts out
at point A, i.e. if there is (local) technological improvement.
However, the same is not true at points B and C: Output (productivity) can increase without any change
in technology, as long as distortions lessen. But these productivity improvements raise welfare, since output
and inputs are weighted using prices that reflect the consumer’s MRS between goods. Thus, Basu and
Fernald’s (1997b) finding generalizes Hulten’s (1978) result to the case of imperfect competition and nonconstant returns, and clarifies the essence of his argument linking productivity and welfare.

B. Reallocations as Propagation Mechanisms
It is reasonable for practical macroeconomists to ask when and how they can avoid the perils posed by
the non-existence of an aggregate production function (even to a first-order approximation). It is also
reasonable to ask whether the reallocation effects that lead to aggregation failures can also serve a positive
function, by providing new amplification and propagation mechanisms in macro models. In this section we
take a first pass at these large and difficult questions. Much of our intuition comes from the extended
example of Basu and Fernald (1997a, Section V).

39

For a proof, differentiate the consumer’s budget constraint holding income and prices fixed.

59
When can we ignore heterogeneity in production and act as if a representative firm produces all
output? Doing so means modeling the production side of the economy using an aggregate production
function for GDP. We ask three questions. First, is this procedure ever sensible if the world actually has
significant heterogeneity? Second, what parameters should one use to calibrate the assumed aggregate
production function? In particular, can one use estimates from aggregate data and ignore heterogeneity?
Third, will the model with heterogeneity reproduce some of the interesting macroeconomic properties of the
single-firm model it replaces?
We emphasize that in this subsection, as in the previous one, we abstract from variations in utilization.
Capacity utilization is an important empirical issue, since it implies that certain inputs are unobserved by the
econometrician. From the standpoint of theory, however, the possibility of changing utilization just changes
the calibration of a model by changing elasticities of supply and demand, but is not a qualitatively new
effect. (For example, variable effort implies that the correct labor supply elasticity is larger than the usual
data suggest; variable capital utilization implies that the labor demand curve is flatter than the standard
production function would indicate.) In this section, by contrast, we ask whether reallocations of inputs
constitute a qualitatively new propagation mechanism.
We now discuss whether a representative-firm model can capture the important features of a world
where firms have approximately constant returns to scale, but where there are large reallocation effects.
When it comes to fluctuations in output conditional on changes in aggregate inputs, the answer is often Yes.
The reason is that in many ways reallocations act like increasing returns to scale at a representative firm—in
both cases, a one percent change in aggregate inputs is associated with a greater than one percent change in
output. The major difference is that in the representative-firm economy, the degree of returns to scale is a
structural parameter. In the economy with reallocations, however, the “effective returns to scale” depends on
the nature of the reallocations induced by the driving shocks. Different shocks are likely to induce different
degrees and types of reallocation, leading to variations in the effective returns to scale parameter under
different circumstances—a classic example of the Lucas (1976) critique. However, this variation may
actually prove an advantage under some circumstances. To the extent that the economy responds differently

60
to shocks of different types, variable reallocation effects may help explain why. Basu, Fernald and Horvath
(1996) address this question, among others.
This answer to the first question that we posed at the beginning of this subsection also partially
answers the second, on the issue of calibration. We believe that if any single summary statistic is useful, it is
likely to be the degree of returns to scale (or markups) estimated from aggregate data without composition
corrections. If the Lucas critique problem is not too severe, this parameter will correctly capture the
procyclical behavior of aggregate output and productivity, but a one-sector model calibrated with the average
of the firm-level parameters would be unable to replicate this behavior. But, as the discussion above
indicated, a one-sector model can never be a perfect substitute for a multi-sector model if one wishes to
understand the response to multiple shocks.
The final question—whether the model with heterogeneity reproduces some of the interesting
macroeconomic properties of the single-firm model—is the hardest to answer. Indeed, the example of Basu
and Fernald (1997a) shows that there are no general answers. Here we discuss one of many interesting
issues, the possibility of positive-feedback loops that might magnify the effects of shocks, and in the limit
give rise to multiple equilibria. The work on real rigidities (Ball and Romer, 1990) and “indeterminacy”
(e.g., Farmer and Guo, 1994) has drawn attention to the importance of positive feedback as a propagation
mechanism. Farmer and Guo show that a very strong form of increasing returns—increasing marginal
product of inputs—provides such positive feedback. The intuition is simple. Suppose the labor demand
curve is upward-sloping (due to increasing marginal product of labor) and steeper than the labor supply
curve. Also suppose a shock increases lifetime wealth, causing workers to supply less labor at each real
wage (without affecting labor demand). This leftward shift in the labor supply curve causes equilibrium
labor supply to increase rather than decrease, as standard neoclassical theory predicts. If this effect is
sufficiently strong, the increase in labor and output can be self-justifying. Workers expect higher real wages
now and in the future, which reduces their marginal utility of wealth, which increases the equilibrium real
wage—validating the initial expectation of higher lifetime wealth.40

40

In a dynamic model one also needs an increase in capital accumulation to increase the rate of return to capital.

61
An interesting question, then, is whether reallocation effects can create such positive-feedback loops.
Note that positive feedback depends on changes in marginal factor prices, since these are the prices that are
relevant for economic decisions (saving and labor supply). In fact, increasing returns in the normal sense is
not sufficient to create positive feedback. As we discussed in Section I, returns to scale can come from fixed
costs, and can be quite consistent with diminishing marginal products. Thus, we conjecture that reallocations
coming from differences in markups, Rµ , and the failure of a value-added production function, RM , will
have a positive-feedback effect only if the increasing returns accompanying the markups take the form of
diminishing marginal cost.41
Similar caveats apply to RK , and RL . Recall that these terms capture differences in shadow prices of
the the same input across firms. If the variance in shadow prices comes only from adjustment costs, and
adjustment costs are paid by the firm, then the differences in marginal product will not translate directly to
differences in factor prices. (Of course, adjustment costs usually have general-equilibrium effects on both
prices and quantities.) Again, the reallocations induced by quasi-fixity will typically affect the dynamics of
aggregate output following a shock (as discussed by Ramey and Shapiro, 1997).
However, if the RK , and RL terms come from steady-state differences in factor payments across
sectors, matters might be different. (In the case of labor, for example, these differences might come from
efficiency wages or union wage premia.) If the wage differences are allocative for the firm—that is, if the
firm equates the marginal product of labor to the above–market-clearing wage, as in Katz and Summers
(1989), instead of the wage premium reflecting efficient bargaining—then these reallocations can have
feedback effects. Even in this case, however, positive feedback is not guaranteed, because it is unclear what
form of rationing rule supports long-run differences in wages for identical labor—i.e., how high-paying jobs
are restricted to a subset of workers. Basu and Fernald (1997a, Section V) show that the form of the
rationing rule determines whether long-run wage premia that give rise to a non-zero RL also generate
positive feedback. Thus, the implications of reallocation for macroeconomic models are likely to be quite

41

As discussed in Section I, we believe on theoretical and empirical grounds that free entry eliminates long-run
economic profits, forcing markups to approximately equal the degree of returns to scale.

62
sensitive to institutional assumptions, including institutions that do not directly concern production and firm
behavior.
The general point of this discussion is that reallocation effects are not just a “nuisance term”—they are
potentially important propagation mechanisms for shocks, and in the limit can give rise to multiple equilibria.
This is an important lesson, since recent empirical estimates (including those we provide above) suggest that
average, firm-level returns to scale are approximately constant. We argue that this finding does not
necessarily imply that one should reject macroeconomic parables in which increasing returns at a
representative plant play a central role in explaining economic fluctuations. Ascertaining which paradigm
provides better macroeconomic insights is an important, unresolved question, and the focus of on-going
research.

VII. Conclusions
In this paper, we explore the meaning and measurement of productivity in a world with frictions and
distortions. In such a world, productivity growth might not estimate aggregate technology change. We
provide a general accounting framework that relates growth in aggregate productivity and aggregate
technology. We identify various non-technological terms that reflect not only variations in utilization, but
also changes in the allocation of factors across uses with different marginal products. Marginal products, in
turn, can differ because of frictions or distortions in the economy. These reallocations affect aggregate
output and productivity, without necessarily reflecting technology. Hence, computing aggregate technology
change requires micro data.
The non-technological components should not necessarily be considered “mismeasurement.” Variable
input utilization clearly constitutes mismeasurement, but reallocations do not. In fact, we argue in Section VI
that even with distortions such as imperfect competition, a modified Solow residual appropriately measures
welfare change. Thus, though much of the recent productivity literature emphasizes the use of micro data, in
some circumstances welfare measurement requires only readily-available national income accounts data.42

42 In practice, of course, welfare depends on a broader measure of output than just GDP—e.g., household
production, investment in human capital, and changes in environmental policy. Unfortunately, these items are hard
to measure.

63
Several existing studies provide models of fluctuations in economies that deviate in a variety of ways
from the standard one-sector model of production. In those models, the non-technological sources of
productivity fluctuations are not always clear, nor is the relationship to other models. Our general
framework can aid in understanding and interpreting the fluctuations arising, for example, from sectorspecific technology shocks, vintage capital effects, or imperfect competition with heterogeneity (see,
respectively, Phelan and Trejos (1996), Gilchrist and Williams (1996), and Basu, Fernald, and Horvath
(1996)).
Applying our decomposition to the data raises several practical and methodological issues. We discuss
the pros and cons of estimating first- and second-order approximations to the production function, and
advocate an explicitly first-order approach. We use a model of a dynamically cost-minimizing firm to derive
a proxy for unobserved variations in labor effort and the workweek of capital. We estimate aggregate
technology change by aggregating industry-level shocks that are “purged” of the effects of variable
utilization and imperfect competition.
Variable utilization and cyclical reallocations appear to explain much of the cyclicality of aggregate
productivity. We find that in the short run, technology improvements significantly reduce input use while
appearing to reduce output slightly as well. These results are inconsistent with standard parameterizations of
RBC models, which imply that technology improvements raise input use at all horizons. However, Basu,
Fernald, and Kimball (1998) argue that they are consistent with standard sticky-price models.
Finally, we discuss implications for macroeconomics. We conclude that reallocations are welfarerelevant, and hence not biases. We also conclude that while accounting for reallocation reduces the average
markup and the average degree of returns to scale, an economy with strong reallocation effects can
sometimes display the same behavior as an economy with large increasing returns. Hence, reallocations
constitute a potentially-important propagation mechanism, which can be utilized in multi-sector dynamic
models of business cycles.

64
Appendix: Derivations from Section III
This appendix presents detailed derivations of the equations in Section III. We first derive the
relationship between firm-level gross output and firm-level value-added, and discuss the interpretation of the
value-added equation. We then aggregate firm-level value-added growth to obtain aggregate output growth
as a function of aggregate inputs, technology, utilization, and reallocations of resources.
As discussed in Section III, the Divisia definition of value-added growth is:43

dvi =

 s 
dyi − sMi dmi
= dyi −  Mi  (dmi − dyi ) .
1 − sMi
1 − sMi 

(A.1)

We now want to substitute in for output growth dyi. In Section I, we obtained the following equation
for output growth:

dyi = µi dxi + µi (1 − sMi )dui + dti .
Inserting this equation using the definition of input growth dxi gives:

dyi = µi  sKi dki + sLi ( dni + dhi ) + sMi dmi  + µi (1 − sMi )dui + dti

(A.2)

We can rewrite this equation as:

 sLi

sKi
dyi = µi (1 − sMi ) 
dli +
dki  + µi (1 − sMi )dui + µ i sMi dmi + dti
(1 − sMi ) 
 (1 − sMi )
= µi (1 − sMi )(dxiV + dui ) + µ i sMi dmi + dti ,

(A.3)

where primary-input growth, dxiV , is defined analogously to aggregate primary input growth:

dxiV =

sKi
s
dki + Li dli ≡ sVKi dki + sVLi dli .
1 − sMi
1 − sMi

Note that sVKi and sVLi are shares of capital and labor costs in nominal value added.
Now subtract µi sMi dyi from both sides of (A.3) and divide through by (1 − µi sMi ) . This gives:
Double-deflation is an alternative method, where V Ÿ Y - M , where Y and M are valued at base-year prices. By
differentiating the double-deflated index, we can express the growth in double-deflated value-added in a form
completely parallel to equation (A.1)—the only difference is that the intermediate share is calculated using baseyear prices, rather than current prices. An implication of this difference is that just as substitution bias affects
Laspeyres indices of aggregate expenditure, substitution bias affects double-deflated value added. Hence, as Basu
and Fernald (1995, appendix) show, double-deflated value added suffers all of the biases we identify in this section
for Divisia value-added, plus an additional substitution bias.
43

65

 µi (1 − sMi ) 
 µ i sMi 
dti
 (dxiV + dui ) + 
dyi = 
.
 ( dmi − dyi ) +
1 − µi sMi
 1 − µi sMi 
1 − µi sMi 
We can write this equation as:

 s

dyi = µiV (dxiV + dui ) + µiV  Mi  ( dmi − dyi ) + dtiV ,
 1 − sMi 

(A.4)

where:

µiV ≡ µi

dtiV ≡

1 − sMi
1 − µi sMi

dti
.
1 − µi sMi

(A.5)

(A.6)

Equation (A.4) relates gross output growth to growth rates of primary inputs dxiV , utilization dui , the
intermediates-output ratio dmi − dyi , and technology. Growth in primary inputs and utilization are
multiplied by a “value-added markup” µiV , defined by equation (A.5). We provide an economic
interpretation of µiV below.
We can now substitute for dyi from (A.4) into the definition of value-added growth (A.1):

 sMi 
V
V
dvi = µiV dxiV + ( µiV − 1) 
 ( dmi − dyi ) + µi dui + dti ,
−
1
s

Mi 


(A.7)

The firm's revenue-weighted value-added productivity residual, dpi , equals dvi − dxiV . Hence,

 sMi 
V
dpi = ( µiV − 1) dxiV + ( µiV − 1) 
 ( dmi − dyi ) + µi dui + dti .
−
1
s

Mi 


(A.8)

It is obvious from equations (A.7) and (A.8) that value-added growth is not, in general, simply a
function of primary inputs dxiV . A long literature in the 1970s (e.g., Bruno, 1978) explored whether a “value
added function” exists, and argued that the answer depends on separability properties of the production
function. The equation above shows that with imperfect competition, taking value-added to be a function
only of primary inputs is generally misspecified — regardless of whether the production function is separable

66
between value added and intermediate inputs. Separability is a second-order property of a production
function, so its presence or absence does not affect first-order approximations like equation (A.7). However,
the fact that the output elasticity of materials is µi sMi instead of simply sMi is of first-order importance.
Nevertheless, it will be useful to make that further assumption of separability in order to provide a
simple interpretation of value-added growth. In particular, suppose the production function is separable, as
follows:

(

)

( (

)

)

Yi = F i K% i , Li , M i , TiV = G i V Pi K% i , Li , TiV , H i ( M i ) ,

(A.9)

The firm combines primary inputs to produce “productive value added”, V Pi , which it then combines with
intermediate inputs to produce gross output. (Note that V Pi does not necessarily correspond to the nationalaccounting measure of value-added—that is, the sum of firm-level productive value added need not equal
national expenditure). We can break the cost-minimization problem into two stages: first, minimize the cost
of using primary inputs to produce any level of V Pi ; second, minimize the cost of using productive value
added and intermediate inputs to produce any level of gross output.
In the first stage, the logic from equation (15) implies that the “productive” value-added growth, dv P ,
depends on the revenue-weighted growth in primary inputs dxV , plus technology shocks (without loss of
generality we normalize to one the elasticity of productive value added V Pi with respect to technology):

dviP = µiV dxiV + dtiV .

(A.10)

In the second stage, the firm seeks to minimize the cost of using value added and intermediate inputs to
produce gross output. The cost in the minimation problem is MCiV Vi + PMi M i , where MCiV is the marginal
i
cost of value added to the firm. The first-order condition is then MCiV = PG
i V µ i . Analogously to the

problem in Section I, we can interpret the value-added markup µiV as the ratio of the price of productive
value added to the marginal cost of producing it: µiV = PiV MCiV . Hence,

GV V Pi µi
= V sVi .
G
µi

(A.11)

67
Note that sVi equals PiV Vi PY
i i = ( PY
i − PMi M i ) PY
i , which equals (1 − sMi ) . However, without knowing
more about the shape of the production function (and hence, the slopes of marginal cost of producing V P and
Y), we cannot make any general statements about the magnitude of the value-added markup µiV .
To do so, we make the further substantive assumption that all returns to scale are in V P , arising
perhaps from overhead capital or labor. This requires that G be homogeneous of degree one in V P and H,
and that H be homogeneous of degree one in M. Under these assumptions, the left-hand-side of equation
(A.11) equals (1 − µi sMi ) . Equation (A.11) then reads:

(1 − µi sMi ) =

µi
(1 − sMi ) .
µiV

(A.12)

Rearranging this equation verifies that the value-added markup µiV is the same as we defined in equation
(A.5). Hence, just as µi creates a wedge between an input price and the input’s marginal product in terms of
gross output, µiV appropriately measures the wedge between the input price and the marginal product in
terms of value added.
Note that for macroeconomic modeling, the value-added markup µ V is likely to be the parameter of
interest. Rotemberg and Woodford (1995), for example, make this point in a dynamic general-equilibrium
model with imperfect competition. In their model, there are no factor-market frictions, all firms have the
same separable gross-output production function, always use materials in fixed proportions to gross output,
and charge the same markup of price over marginal cost. Under these assumptions (which are implicit or
explicit in most other dynamic general-equilibrium models with imperfect competition), there is a
representative producer and an aggregate value-added production function. Rotemberg and Woodford show
that the correct aggregate markup corresponds to µ V .
One source of economic intuition for µ V is that under some circumstances, it correctly captures
“economy-wide” distortions, as small markups at the plant level translate into larger efficiency losses for the
economy overall. Suppose, for example, that final output is produced at the end of an infinite number of
stages. At each stage a representative firm with markup µ uses all the output of the previous stage as

68
intermediate input, and also uses primary inputs of capital and labor. Then the percent change in national
income—the output of the last (nth) stage— is

dyn = µ (1 − sM ) dxVn + µ sM dyn −1
= µ (1 − sM ) dxVn + µ sM µ (1 − sM ) dxnV−1 + ( µ sM ) dyn − 2 .
2

We can substitute into this equation for dyn − j , and let j go to infinity. Since each firm is identical, dxiV is
the same for all i as for the aggregate. This gives an infinite sum:
∞

dyn ≡ dv = µ (1 − sM ) dxV ∑ ( µ sM )

j

j =1

= µ (1 − sM ) dxV

1
1 − µ sM

(A.13)

= µ V dxV .
Thus, µ V is plausibly the appropriate concept for calibrating the markup charged by the representative firm
in a one-sector macroeconomic model.
Thus, µ V correctly captures the idea that small deviations from perfect competition “cascade” in going
from gross output to value added, because of the “markup on markup” phenomenon: firms buy intermediate
goods from other firms at a markup, add value, and again price the resulting good with a markup, generally
selling some of it to another firm to use as an intermediate good. Nevertheless, this derivation shows that
there is a limit to how much the effects can cascade or build up.
Even if we want this value-added markup, however, we still in general require data on intermediate
inputs. The reason is that we do not observe V Pi directly, but must infer it from observable gross output and
intermediate inputs.
Returning to equation (A.7), real value-added growth depends on primary input growth, changes in the
materials-to-output ratio, variations in utilization of capital and labor, and technology. The first term shows
that primary inputs are multiplied by the value-added markup. The second term reflects the extent to which
the standard measure of value added differs from “productive” value added V Pi , and hence does not properly
measure the productive contribution of intermediate inputs. Intuitively, the standard measure of value added
subtracts off intermediate input growth using revenue shares, whereas with imperfect competition the

69
productive contribution of these inputs exceeds the revenue share by the markup. The third term shows that
variations in utilization are also multiplied by the value-added markup. The fourth term is the value-addedaugmenting technology shock.
We now aggregate over firms to get aggregate output growth as a function of technology, aggregate
primary inputs growth, and the distribution of inputs. As in Section III, we define aggregate output
growth dv as a share-weighted average of firm-level value-added growth:

dv = ∑ i =1 wi dvi ,
N

where wi = PiV V PV V . Substituting in from equation (A.7) for dvi gives:

 sMi 
N
N
N
N
V
V
dv = ∑ i =1 wi µiV dxiV + ∑ i =1 wi ( µiV − 1) 
 ( dmi − dyi ) + ∑ i =1 wi µi dui + ∑ i =1 wi dti .
−
1
s

Mi 

(A.14)
As in the text, we will define:

du = ∑ i =1 wi µiV dui
N

 s 
N
RM = ∑ i =1 wi ( µiV − 1)  Mi  ( dmi − dyi ) , and
1 − sMi 
dt V = ∑ i =1 wi dtiV .
N

Hence, equation (A.14) becomes:

dv = ∑ i =1 wi µiV dxiV + RM + du + dt V .
N

(A.15)

We now decompose the first term into the effects of the “average” value-added markup µ V and the
distribution of the markup. Rearranging (A.15) gives:

dv = µ V ∑ i =1 wi dxiV + ∑ i =1 wi ( µiV − µ V )dxiV + RM + du + dt V
N

N

= µ V ∑ i =1 wi dxiV + Rµ + RM + du + dt V
N

where

Rµ = ∑ i =1 wi ( µiV − µ V )dxiV .
N

,

(A.16)

70
At this point, we are almost finished. However, we still need to relate the first term on the right-hand side
of (A.16) to aggregate input growth. As in Section III, consider the case where there was only one type of
capital and one type of labor.44 Aggregate labor and capital are arithmetic sums across firms, so that

K = ∑ i =1 K i and L = ∑ i =1 Li . Aggregate primary input growth is the share-weighted growth in
N

N

aggregate capital and labor growth:

dxV = sVK dk + sVL dl .
Using the definitions of sVKi and sVLi , and differentiating the definitions of aggregate K and L, we can
write this as:

Ki
PK
N L
dki + LV ∑ i =1 i dli
K
P V
L
V
V
N P V P K P
N P V P L P
= ∑ i =1 i V i KiV i K dki + ∑ i =1 i V i LiV i L dli .
P V Pi Vi PKi
P V Pi Vi PLi

dxV =

PK K
PV V

∑

N

i =1

Noting the definitions of wi , sVKi , and sVLi , we can write this as:

dxV = ∑ i =1 wi sVKi
N

PK
N
P
dki + ∑ i =1 wi sVLi L dli
PKi
PLi

P −P 
 P − PL 
N
N
N
= ∑ i =1 wi ( sVKi dki + sVLi dli ) − ∑ i =1 wi sVKi  Ki K  dki − ∑ i =1 wi sVLi  Li
 dli
 PKi 
 PLi 
= ∑ i =1 wi dxiV − RK − RL ,
N

(A.17)
where:

44

In general, suppose there are N firms, N L types of labor, and N K types of capital. For each type of capital

K and labor Ll , the aggregate is an arithmetic sum across firms, so that K k = Êi =1 K ik and Ll = Êi =1 Lli .
N

k

N

 P L dl , where P L is total labor compensation to labor of type l.
 P L 

Aggregate capital and labor are then defined as a Divisia index across these types of capital, so that, for example,
aggregate labor growth is dl Ÿ

Êl =1
NL

l
L

l

l

l
L

l

The

L

derivations that follow extend easily to this general case, except that there is a separate input-reallocation term for
each type of labor and capital. Jorgenson, Gollop, and Fraumeni (1987) derive this result explicitly.

71

 P − PK 
N
RK = ∑ i =1 wi sVKi  Ki
 dki ,
 PKi 
 P − PL 
RL = ∑ i =1 w s  Li
 dli .
 PLi 
N

.

V
i Li

By substituting equation (A.17) into equation (A.16), we find:

dv = µ V dxV + du + Rµ + RM + µ V RK + µ V RL + dt V
= µ V dxV + du + R + dt V

.

(A.18)

Since aggregate productivity equals dv − dxV , it immediately follows that:

dp = ( µ V − 1)dxV + du + R + dt .

(A.19)

72
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78
Table 1. Descriptive Statistics for Technology Residuals

A. Private Economy
Mean

Standard
Deviation

Minimum

Maximum

Solow Residual

0.011

0.022

-0.044

0.066

Tech. Residual
(No Util. Correct.)

0.012

0.016

-0.034

0.050

Tech. Residual
(Hours Correct.)

0.013

0.013

-0.013

0.042

B. Manufacturing
Mean

Standard
Deviation

Minimum

Maximum

Solow Residual

0.023

0.035

-0.081

0.080

Tech. Residual
(No Util. Correct.)

0.014

0.030

-0.085

0.072

Tech. Residual
(Hours Correct.)

0.018

0.028

-0.030

0.082

Note: Sample period is 1950-1989. The Solow residual is calculated using aggregate data alone. The two
technology residuals are calculated by aggregating residuals from sectoral regressions of gross-output growth
on input growth, as described in the text. The “hours corrected” residual corrects for variable utilization by
including growth in hours per worker as a separate explanatory variable, in line with the theory developed in
Section II.

79
Table 2. Basic Data Correlations
A. Private Economy
Output
Growth
dv
Output Growth
dv

Input
Growth

dxV

Hours
Growth
dh+dn

Solow
Residual

1

1

dx

0.78
(0.62, 0.88)

Hours Growth
dh+dn

0.80
(0.64, 0.89)

0.91
(0.83, 0.92)

1

Solow
Residual

0.84
(0.72, 0.91)

0.33
(0.02, 0.59)

0.44
(0.15, 0.66)

1

Hours
Growth
dh+dn

Solow
Residual

Input Growth
V

B. Manufacturing
Output
Growth
dv
Output Growth
dv

Input
Growth

dxV

1

1

dx

0.81
(0.66, 0.90)

Hours Growth
dh+dn

0.86
(0.75, 0.92)

0.98
(0.96, 0.99)

1

Solow
Residual

0.84
(0.71, 0.91)

0.36
(0.05, 0.61)

0.46
(0.17, 0.68)

Input Growth
V

1

Note: 95 Percent confidence intervals in parentheses, calculated using Fisher transformation.
Sample period is 1950-1989.

80
Table 3. Correlations of Technology Residuals with Basic Data

A. Private Economy

Solow
Residual

dxV

Hours
Growth
dh+dn

0.46
(0.17, 0.68)

-0.12
(-0.41, 0.21)

-0.06
(-0.37, 0.26)

0.77
(0.63, 0.88)

0.04
(-0.28, 0.35)

-0.42
-0.44
(-0.65, -0.12) (-0.66, -0.14)

0.40
(0.10, 0.64)

Output
Growth
dv

Input
Growth

Tech. Residual
(No Util. Correct.)

Tech. Residual
(Hours Correct.)

B. Manufacturing

dxV

Hours
Growth
dh+dn

Standard
Productivity

0.42
(0.12, 0.65)

-0.14
(-0.44, 0.18)

-0.04
(-0.35, 0.28)

0.79
(0.63, 0.89)

-0.40
(-0.64, 0.10)

-0.64
-0.62
(-0.80, -0.41) (-0.78, -0.38)

-0.05
(-0.36, 0.27)

Output
Growth
dv

Input
Growth

Tech. Residual
(No Util. Correct.)

Tech. Residual
(Hours Correct.)

Note: 95 Percent confidence intervals in parentheses, calculated using Fisher transformation. Sample period
is 1950-1989. Technology residuals are calculated by aggregating residuals from sectoral regressions of
gross-output growth on input growth, as described in the text. The “hours corrected” residual corrects for
variable utilization by including growth in hours per worker as a separate explanatory variable.

81

Table 4: Reallocation by Component, 1959-89
Instrumental Variables Estimates

R

Rµ

RM

µ V RK + µ V RL

Mean

-0.11

-0.17

0.02

-0.03

Standard
Deviation

1.04

0.71

0.47

0.24

Minimum

-2.86

-2.05

-1.09

-0.51

Maximum

1.36

1.08

0.74

0.64

Note: Entries are percentage points per year. R is the sum of the components shown in the columns to the
right. Rµ is reallocations of inputs among industries with different markup estimates. RM is reallocations
of intermediate inputs. RK and RL are reallocations of capital and labor among uses with different factor
prices, as calculated by Jorgenson and Fraumeni. See the text for further details.

82
Figure 1. Aggregate Solow Residual, Input Growth, and Output Growth
0.06
Output

Inputs (K, L)

Growth rate (Demeaned)

0.04

0.02

0

-0.02

-0.04

Solow
Residual
88

86

84

82

80

78

76

74

72

70

68

66

64

62

60

58

56

54

52

50

-0.06

Note: All series are demeaned. Sample period in 1950-89. Data are from Jorgenson, Gollop, and Fraumeni
for the non-farm private business economy. Inputs are a share-weighted average of capital and labor growth.

83
Figure 2

C2

A
B

C

PPF

C1

84
Figure 3. Technology Residual, Solow Residual, Output and Input Growth

0.06

Solow
Residual

Growth rate (demeaned)

Technology
0.04
0.02
0
-0.02
-0.04

Growth rate (demeaned)

0.03

88

86

84

82

80

78

76

74

72

70

68

66

64

62

60

58

56

54

52

50

-0.06

Agg. Inputs

Technology

0.02
0.01
0
-0.01
-0.02
-0.03

84

86

88

84

86

88

82

80

78

76

74

72

70

68

66

64

62

60

58

56

54

52

50

-0.04

Growth rate (demeaned)

0.06

Agg. Output

Technology

0.04
0.02
0
-0.02
-0.04

82

80

78

76

74

72

70

68

66

64

62

60

58

56

54

52

50

-0.06
criw a3.xls--hours corrected tech

Note: The technology series is the hours-adjusted aggregate residual, which measures technology change
(adjusted for variations in utilization) for the non-farm business economy. Aggregate inputs are a shareweighted average of capital and labor growth. All series are demeaned. Entries are log changes. Sample
period is 1950-89.

85
Figure 4: Non-Technological Adjustments to the Solow Residual
0.06
Solow Residual

Growth rate (Demeaned)

0.04

0.02

0

-0.02

Utilization

-0.04

88

86

84

82

80

78

76

74

72

70

68

66

64

62

60

58

56

54

52

50

-0.06

0.04

Growth rate (Demeaned)

Reallocations
0.02

0

-0.02

-0.04

Solow Resid.

88

86

84

82

80

78

76

74

72

70

68

66

64

62

60

58

56

54

52

50

-0.06

Note: Aggregate utilization growth is a weighted average of estimated industry-level utilization growth
(which includes the estimated markup for the industry). Estimated reallocations are the sum of Rµ
(reallocations of inputs among industries with different markup estimates), RM (reallocations of intermediate
inputs), and RK and RL (reallocations of capital and labor among uses with different factor prices).

86
Figure 5: Markups and Cyclicality

Markup (hours-corrected)

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Cyclicality Coefficient
Non-manuf

Durables

Non-Durables

Note: The horizontal axis shows the utilization-corrected markup estimate by industry. The vertical axis
shows the estimated “cyclicality coefficient” for the industry, estimated by regressing industry growth in
primary inputs on aggregate growth in primary inputs.