Full text of Economic Review (Federal Reserve Bank of Cleveland) : 1999 Quarter 2, Vol. 35, No. 2

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Vol. 35, No. 2

ECONOMIC REVIEW
1999 Quarter 2
The Third Industrial Revolution:
Technology, Productivity,
and Income Equality

2

by Jeremy Greenwood

Accounting for Capital Consumption
and Technological Progress

13

by Michael Gort and Peter Rupert

Defining Capital in Growth Models
by Michael Gort, Saqib Jatarey, and Peter Rupert

FEDERAL RESERVE BANK
OF CLEVELAND

19

1
http://clevelandfed.org/research/review/
Economic Review 1999 Q2

ECONOMIC REVIEW
1999 Quarter 2
Vol. 35, No. 2

The Third Industrial Revolution:
Technology, Productivity,
and Income Equality

2

by Jeremy Greenwood
The author examines periods of rapid technological change for coincidences of widening inequality and slowing productivity growth. He contends that while in the short run the introduction of technologies offers
profits to investors and premiums for skilled workers, in the long run the
rising tide of technological change lifts everybody’s boat.

Accounting for Capital Consumption
and Technological Progress

13

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by Michael Gort and Peter Rupert
Methods currently used to calculate capital consumption, the capital stock,
and the sources of economic growth do not adequately measure the underlying growth in inputs due to technological advance. This lack affects tax
policy as well as programs targeting potential areas of economic growth.
The authors present a model designed to surmount the deficiencies of current calculation methods.

Defining Capital in Growth Models

19

by Michael Gort, Saqib Jafarey, and Peter Rupert
The authors analyze the measurement of the capital stock when technological advance is embodied in capital. The source of the problem is that capital is not homogeneous across vintages. Which measure of the capital
stock to use is dictated by the question being addressed.

Editors: Michele Lachman
Lisa McKenna
Deborah Zorska
Design: Michael Galka
Typography: Liz Hanna

Opinions stated in Economic
Review are those of the authors
and not necessarily those of the
Federal Reserve Bank of Cleveland or of the Board of Governors
of the Federal Reserve System.

Material may be reprinted if the
source is credited. Please send
copies of reprinted material to
the editors.

ISSN 0013-0281

2

The Third Industrial
Revolution: Technology,
Productivity, and
Income Inequality
by Jeremy Greenwood

Introduction
Did 1974 mark the beginning of a third industrial revolution—an era of rapid technological
progress associated with the development of
information technologies? Did the quickened
pace of technological advance lead to greater
income inequality? Is a productivity slowdown
related to these phenomena?
The story told here connects the rate of technological progress with the level of income
inequality and productivity growth. Imagine that
a leap in the state of technology occurs and is
embodied in new machines, such as those used
in information technologies (IT). Now suppose
that adopting these technologies involves a significant cost in terms of learning, and that
skilled workers have an advantage at learning.
Then the advance in technology will be associated with an increased demand for the skills
needed to implement them. Hence, the wages
of skilled labor relative to unskilled labor (the
skill premium) will rise, and income inequality
will increase. In their early phases, new technologies may not operate efficiently due to
inexperience. The initial incarnation of ideas
into equipment may be far from ideal. Productivity growth may seem to stall as the economy

Jeremy Greenwood is a professor
of economics at the University of
Rochester. Thanks go to Marvin
Kosters for his helpful comments.
This article is based on “1974,”
a paper written in collaboration
with Mehmet Yorukoglu, who
played a vital role in pursuing
this line of research. This article
was also published as a monograph, under the title used here,
by the American Enterprise Institute (1997).

makes the (unmeasured) investment in knowledge needed if the new technologies are to
approach their full potential. The coincidence of
rapid technological progress, widening wage
inequality, and a slowdown in productivity
growth has precedents in economic history.

I. The Information Age
Figure 1 shows the price of a piece of new producer equipment relative to the price of a unit
of nondurable consumer goods or services over
the period following World War II. The marked
drop in the relative price of producer equipment is a reflection of the high rate of technological progress in the producer-durables sector. Specifically, technological progress enables
ever-larger quantities of investment goods to be
produced with a given amount of labor and
capital, a process that drives down the prices of
such goods. This type of advance is dubbed
investment-specific technological progress
because it affects the investment-goods sector
of the economy.
The price of equipment fell faster after 1974
than before, as the slope of the trend line
shows. If the decline in new equipment prices

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F I G U R E

1

Price of New Equipment

F I G U R E
Price of New Computers

SOURCE: Greenwood, Hercowitz, and Krusell (1997).

F I G U R E

2

SOURCE: Yorukoglu (1998).

3

IT Investment and the
Productivity Slowdown

F I G U R E

4

Measures of
Wage Inequality

SOURCES: National Income and Product Accounts; and Citibase.

can be taken as a measure of improved efficiency in production, then the pace of technological advance jumped up around 1974. Some
economists estimate that 60 percent of postwar
U.S. growth may derive from the introduction
of new, more efficient equipment.1 The rapid
advance in technology since 1974 is undoubtedly linked to IT development. The price of a
new computer has plummeted over the postwar period at an average rate of about 19 percent annually (see figure 2).2 Hence, a new
computer costing $5,000 in 1987 would have
been priced at $2 million in 1955. Figure 3 illustrates the phenomenal rise of IT investment as
a fraction of total equipment investment (less
than 7 percent in 1954 compared with 50 percent now).
Growth in labor productivity stalled with the
rise in IT investment, as figure 3 also shows.
Labor productivity, which gauges the amount
of gross domestic product (GDP) created per

SOURCE: Juhn, Murphy, and Pierce (1993).

hour of work, is often taken as a measure of
how efficient labor is in the economy. The
more GDP each worker can produce, the better
off the economy is. Before 1974, labor productivity grew at about 2 percent annually; after
that year, at a paltry 0.8 percent. This change is
often termed the “productivity slowdown.” Isn’t
it paradoxical that at a time of massive technological advance resulting from the introduction
of information technologies and the like, the
advance in a worker’s productivity should stall?
By most accounts, wage inequality began to
increase around 1974 (figure 4 shows some
postwar measures of income inequality).3 The
■ 1 Greenwood, Hercowitz, and Krusell (1998) break down U.S. postwar growth into its sources in terms of investment-specific and other forms
of technological progress.
■ 2 Yorukoglu (1998).
■ 3 The data are from Juhn, Murphy, and Pierce (1993, table 1.B).

4

percentage gap between the average wage
earned by the upper quartile (above the
seventy-fifth percentile) and the average wage
earned by the lower quartile (the twenty-fifth
percentile and below) remained roughly constant between 1959 and 1970. From 1970 to
1988, this gap increased 22 percentage points.
That is, the 53 percent gap in wage income that
existed between the two groups in 1970 had
widened to 75 percent by 1988. The other
measures behaved similarly.

II. The Industrial
Revolution
The Industrial Revolution, which began in 1760,
epitomizes investment-specific technological
progress. This period witnessed the birth of several technological miracles.4 Crompton’s mule
revolutionized the spinning of cotton. Next,
Watt’s energy-efficient engine brought steam
power to manufacturing. The main cost of a
steam engine was operating it: It was a hungry
beast. A Watt steam engine cost £500–800.5
Operating a steam engine, though, was enormously expensive. Each consumed £3,000 of
coal per annum.6 By comparison, 500 horses,
which apparently could produce the same
amount of work, cost only £900 in feed. Thus,
the pursuit of an energy-efficient steam engine
was on. The older Newcomen steam engine of
1769 needed 30 pounds of coal per horsepower
hour, while a Watt engine of 1776 required 7.5
pounds. By 1850 or so, this number had been
reduced to 2.5. So the cost of steam power fell
dramatically over the course of the Industrial
Revolution. When the spinning mule was harnessed to steam power, the mechanization of
manufacturing was inexorable. By 1841, the real
price of spun cotton had fallen by two-thirds.
In 1784, Cort introduced his puddling and
rolling technique for making wrought iron, a
product vital for the industrialization of Britain.
Between 1788 and 1815, wrought iron production increased 500 percent. Its price fell from
£22 to £14 per ton from 1801 to 1815, even
though the general price level rose 50 percent
between 1770 and 1815.
Last, the foundation of the modern machinetool industry was laid when Wilkinson designed
a gun-barreling machine that could make cylinders for Watt’s steam engines, and Maudley
introduced the heavy-duty lathe.
Skill undoubtedly played an important role
in technological innovation and adoption. The
Industrial Revolution produced a handful of
miracles, but many historians also view it as an

age of continuous and gradual smaller innovations—an age of learning. Implementing and
operating brilliant inventions and effecting the
subsequent innovations is often demanding,
skill-intensive work. It took three months, for
instance, for someone brought up in a mill to
learn how to operate either a hand mule or a
self-acting mule.7 Learning to maintain the former required three years, while the latter demanded seven. A worker continued to acquire
knowledge concerning improvements in the
machinery throughout his lifetime. It seems reasonable to conjecture that the demand for skill
rose in the Industrial Revolution, since “for the
economy as a whole to switch from manual
techniques to a mechanized production required hundreds of inventors, thousands of
innovating entrepreneurs, and tens of thousands
of mechanics, technicians, and dexterous rank
and file workers.”8 Mokyr (1994) emphatically
rejects the notion that Britain’s more advanced
science accounted for the development of the
Industrial Revolution. He maintains that ideas
flowed from the Continent to Britain, and then
working technologies flowed back from Britain
to the Continent. He quotes an engineer of the
day, who observed that “the prevailing talent of
English and Scottish people was to apply new
ideas to use and to bring such applications to
perfection, but they do not imagine as much as
foreigners.” Mokyr concludes that “Britain’s
technological strength during the industrial revolution depended above all on the abundance
and quality of its skilled mechanics and practical
technicians who could turn great insights into
productive applications.”9 In fact, income inequality rose throughout the Industrial Revolution (see figure 5).10
The diffusion of new technologies is often
slow because the initial incarnations of the
underlying ideas are inefficient. Getting new
technologies to operate at their full potential
■ 4 This is chronicled in Mokyr (1994).
■ 5 McPherson (1994, p. 16).
■ 6 The classic source is Landes (1969, pp. 99–103), who quotes a
writer’s 1778 comment that “the vast consumption of fuel in these engines
is an immense drawback on the profits of our mines, for every fire-engine
of magnitude consumes £3,000 of coals per annum. This heavy tax
amounts almost to a prohibition.”
■ 7 As reported by von Tunzelmann (1994).
■ 8 Mokyr (1994, p. 29).
■ 9 Ibid., p. 39
■ 10 This is documented in Lindert and Williamson (1983, table 3).

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F I G U R E

5

The Industrial Revolution

Revolution (see figure 5).13 Before the coming
of the new era, productivity was growing at
0.4 percent a year; once it began, productivity
growth fell to an annual rate of 0.2 percent.
Was this decline, which lasted 40 years, connected to the teething pains of adopting new
technologies? As the revolution spread, productivity growth picked up; 70 years in, it was
growing at a much more robust 0.5 percent.
Thus, the fruits of the Industrial Revolution
took time to ripen.

III. The American
Antebellum Period
The Industrial Revolution spread to the United

SOURCES: Harley (1993) and Lindert and Williamson (1983).

F I G U R E

6

The American Antebellum Period

States around 1840, a time of tremendous
investment-specific technological progress. The
nation industrialized at a rapid clip. Figure 6
shows the dramatic decline in the price of new
equipment relative to all goods.14 Presumably,
this decrease reflects the improved efficiency of
new-equipment production, which allowed
more of it to be produced for less. One would
expect this decline in the price of new equipment to encourage more investment. For the
period 1774–1815, the real stock of equipment
per capita grew at roughly 0.7 percent annually.
Between 1815 and 1860, however, the average
annual growth rate was a very robust 2.8 percent, which jumped to a whopping 4.5 percent
from 1860 to 1900. Two examples illustrate the
extraordinary pace of industrialization. In 1830,
there were just 30 miles of railroad tracks in the
United States. By 1840, there were 2,808 miles,
while in 1860 the number was 30,000.15 Likewise, the aggregate capacity of U.S. steam engines more than quadrupled between 1840 and
1860 (from 760,000 to 3,470,000 horsepower).
That capacity rose another one-and-a-half times
(to 5,590,000 horsepower) by 1870. The antebellum period saw a dramatic surge in the skill

SOURCES: Abramovitz and David (1973, table 2); Gallman (1992); and
Williamson and Lindert (1980, appendix D).

■ 11 Again, as related by von Tunzelmann (1994).

may take considerable time, so their productivity may be low at first. Cort’s famous puddling
and rolling process had a long incubation
period and was commercially unsuccessful at
first.11 Royalties had to be slashed to encourage
adoption. Apparently, “both entrepreneurs and
workers had to go through a learning period,
making many mistakes that often resulted in
low outputs of uneven quality.12 Productivity
growth fell in the initial stages of the Industrial

■ 12 C.K. Hyde, Technological Change and the British Iron Industry,
as cited by von Tunzelmann (1994, p. 277).
■ 13 Calculated by Harley (1993, table 3.5).
■ 14 This series is based on calculations using data presented in
Gallman (1992).
■ 15 In 1840, roughly 30 percent of pig-iron production was devoted
to manufacturing tracks, and the railway used 30 percent of the country’s
steam-power capacity (McPherson [1994, chap. 3]).

6

premium (see figure 6).16 Not surprisingly,
skilled workers such as engineers, machinists,
boilermakers, carpenters, and joiners, saw their
wages rise relative to those of common laborers. Last, labor productivity growth slowed in
the 1840s just as the American Industrial Revolution was gaining momentum; the annual
growth rate of labor productivity is plotted in
figure 6.17

IV. The Hypothesis
The adoption of new technologies involves a
significant cost in terms of learning; skill facilitates this learning process. That is, skill is
important for adapting to change. There is considerable evidence of learning effects. For
example, using a 1973–86 data set consisting of
2,000 firms from 41 industries, Bahk and Gort
(1993) find that a plant’s productivity increases
15 percent over its first 14 years because of
learning effects.
There is also evidence that skill plays an
important role in facilitating the adoption of
new technologies. Farmers with high education
levels adopt agricultural innovations earlier
than farmers with low levels. Findings reported
in Bartel and Lichtenberg (1987) support the
joint hypothesis that 1) educated workers have
a comparative advantage in implementing new
technologies because they assimilate new ideas
more readily; and 2) the demand for educated
relative to less-educated workers declines as
experience with a technology is gained. For
each year equipment ages, skilled labor’s share
of the wage bill drops 0.78 percentage point.
This suggests that less skilled labor is needed as
production experience with equipment is
gained through time. Using a cross-country data
set, Flug and Hercowitz (1996) find that a rise
in equipment investment leads to an increase in
the skill premium and to higher relative employment for skilled labor. In particular, an
increase of 1 percentage point in the ratio of
equipment investment to output produces an
increase of 1.90 percentage points in the ratio
of skilled to unskilled employees. The inference drawn is that when investment in equipment is high, so is the demand for skilled labor,
which eases the adoption process.
The hypothesis to be developed here is different from the capital–skill complementarity
hypothesis.18 The latter states that skilled labor
is more complementary with capital in production than is unskilled labor or (more or less
equivalently) that capital substitutes better for
unskilled than for skilled labor. The recent rise

in the skill premium is consistent with capital–
skill complementarity and an increase in the
rate of investment-specific technological
advance.19 The idea in the current paper, however, is that successful adoption of a new technology requires skilled labor. Moreover, as a
technology becomes established, the production process substitutes away from expensive
skilled labor toward less costly unskilled labor.
Therefore, times of heightened technological
progress should see an increase in the demand
for skilled labor, which has a comparative advantage in speeding up and easing the process
of technological adoption. Such times should
therefore be associated with a rise in the skill
premium. If this notion is correct, the skill premium should decline once the recent burst of
investment-specific technological progress subsides as IT matures.20
How large are the costs of technological
adoption? Calculations suggest that the costs
of adopting new technologies exceed invention costs by a factor of 20 to 1, and that adoption costs may amount to 10 percent of GDP.21
Surely, the costs of technological adoption must
be large. How else to explain the long diffusion
lags for new technologies as well as the continual investment in older technologies at the
household, firm, and national levels? And, surely, a large part of these adoption costs must be
in acquiring or developing the skills needed to
implement the new technologies.

V. The Learning Curve
As an example of the importance of learning
effects, consider the Lawrence Number 2 Mill,
an antebellum cotton mill studied by David
(1975). The plant, built in 1834 in Lowell, Massachusetts, kept detailed inventories of its equip■ 16 These data are reported in Williamson and Lindert (1980,
appendix D).
■ 17 The numbers are from Abramovitz and David (1973, table 2).
■ 18 The hypothesis was originally advanced by Griliches (1969).
A more recent formulation can be found in Krusell et al. (1996).
■ 19 Krusell et al. (1996) make this case.
■ 20 By contrast, this is not an implication of the capital–skill complementarity hypothesis. Suppose that skilled labor is more complementary with equipment than is unskilled labor. Then, other things being equal,
the skill premium should rise as long as the stock of equipment increases.
That is, there should be a secular or long-run rise in the skill premium. For
more detail, see Krusell et al. (1996).
■ 21 See Jovanovic (1997).

7

F I G U R E

7

Lawrence Number 2
Cotton Textile Mill

SOURCE: David (1975).

F I G U R E

8

Learning Curve for
Information Technologies

SOURCE: Yorukoglu (1998).

ment showing that no new machinery was
added between 1836 and 1856. Thus, it seems
reasonable to infer that any productivity
increases that occurred in these years were due
purely to learning effects. In fact, the plant’s
output per hour of work during this period
grew 2.3 percent annually. Figure 7 shows the
plant’s learning curve. The four observations

pertain to years when the plant was known to
be operating at full capacity.
Learning curves from angioplasty surgery,
flight-control simulation, munitions manufacturing, and steel finishing are documented in
Jovanovic and Nyarko (1995); the literature
abounds with additional examples. Yorukogolu
(1998) has used data from 297 firms during the
1987–91 period to study the learning curve for
information technologies. His results, plotted
in figure 8, show strong learning effects. The
service flow (analogous to horsepower for a
steam engine) captured from new computers
increases dramatically over time, growing at
approximately 28 percent (compounded) annually. Two words of caution are offered here.
First, as the error bands show, the range of estimates is quite wide because the data set permits studying only a small number of firms for
a short period of time.22 Second, a firm uses
more than computers to produce output. If
computers account for 5 percent of output,
then this translates into an output growth rate
due to learning alone of about 1.4 percent
(.28 3 .05 3 100 percent) a year.
Often, learning about a technology occurs
through use by the final purchaser. For some
products, such as software, important operating
characteristics are revealed only after intensive
use. The manufacturer may then adjust the
product in response to feedback from purchasers, a process that may take many iterations (Rosenberg [1982]). The aircraft industry
provides an excellent case of such learning by
using: As confidence about the operating characteristics of the DC-8 airplane grew through
experience, the manufacturer increased the
engines’ thrust while reducing fuel consumption, and modified the wings to decrease drag.
These modifications eventually allowed the airplane to be “stretched,” which more than doubled its capacity from 123 seats to 251. The
result was a dramatic improvement in operating
costs, notably a 50 percent saving in fuel costs
per seat-mile. For complicated products, where
reliability is a major concern, maintenance
experience proves invaluable. In the case of
aircraft, maintenance may account for 30 percent of the operating costs associated with
labor and materials (this excludes the revenue
lost during downtime). The costs of servicing
new types of jet engines fall dramatically after
their introduction. After a decade of operation,
maintenance costs typically have dropped to
30 percent of their initial level.
■ 22 The error bands show the 95 percent confidence intervals.

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F I G U R E

9

Diffusion of Diesel Locomotives

SOURCE: U.S. Department of Commerce, Bureau of the Census,
Historical Statistics of the United States, 1790–1970.

VI. The Diffusion Curve
The adoption of new technologies is notoriously slow. The initial incarnations of new
ideas are often expensive and bug-infested.
The impact of investment-specific technological
progress on income and productivity is likely to
be regulated by two interrelated factors: the
speed of learning and the speed of diffusion.
The more costly it is for economic agents to
learn about a new technology, the slower the
technology’s diffusion. But the faster a new
technology diffuses through an economy, the
easier it may be to learn about it. Thus, there is
a feedback loop between the cost of adoption
and the extent of adoption. If a new technology represents a radical or discrete departure
from past technologies, society’s knowledge
about it may be quite limited at first. As the
technology’s use becomes widespread, society’s stock of experience with it increases, and
the technology’s productivity rises.
New technologies are expensive when they
are first produced, but prices drop as the manufacturer gains production experience. This
encourages adoption, which in turn fuels further price declines when learning and scale
effects lower the costs of production. Waves of
imitators enter the industry, making pricing
more competitive. The odds of imitating a new
invention depend on the number of firms that
have already succeeded in adopting the new
invention. The number of firms increases with

time, making imitation easier. Firms also rush in
to produce complementary products, such as
software or communication devices for computers. The original product may then have to
be modified to incorporate them better. Bringing these complementary products on line may
take a lot of time and resources. The availability
of such products encourages further adoption,
and so on. An invention may take a long time
to bear fruit.
There is considerable evidence that the diffusion of innovations is slow. In a classic
study, Gort and Klepper (1982) examined 46
product innovations, beginning with phonograph records in 1887 and ending with lasers
in 1960. They traced diffusion by examining
the number of firms producing the new product over time. Only two or three firms on average were producing each new product for the
first 14 years after its commercial development;
then the number of firms increased sharply (by
an average of six firms per year over the next
10 years). Prices fell rapidly after a product’s
inception (down 13 percent annually for the
first 24 years). Using a 21-product subset of the
Gort and Klepper data, Jovanovic and Lach
(1996) find that the output of a new product
took approximately 15 years to rise from the
10 percent to the 90 percent diffusion level.
They also cite evidence from a study of 265
innovations that it took 41 years on average to
move from 10 percent to 90 percent diffusion.
Finally, it took the steam locomotive 54 years
to move from the 10 percent to the 90 percent
diffusion level in the U.S. and the diesel (a
lesser innovation) 12 years. The diffusion curve
for diesels (figure 9) shows that it took about
25 years after their introduction in 1925 for
diesels to account for half of the locomotives
in use.

VII. The Computer
and the Dynamo
The metamorphosis of a novel idea into a productive technology can take a long time.23
Because a technology’s development is
uncharted at its infancy, a lot of time and
resources can go into exploring the various
paths that may be taken. Electricity and computers are two interesting examples of this
uncertain process. Ironically, one of the Industrial Revolution’s least productive inventions
formed the foundation of the current Information Age. Sometime between 1823 and 1832,
■ 23 The section title is borrowed from David (1991).

9

F I G U R E

1 0

The Electrification of America

SOURCE: David (1991, tables 2 and 3).

Charles Babbage created his “Difference
Engine,” a mechanical computer. The insight
for this device came partly from a binary-coded
loom, invented in 1801 by Jean-Marie Jacquard,
which used punch cards to control fabric patterns. But as recently as 50 years ago, the coming of the Information Age was still not obvious. Just after World War II, Popular Mechanics
(March 1949) wrote, “Where a calculator on the
ENIAC is equipped with 18,000 vacuum tubes
and weighs 30 tons, computers in the future
may only have 1,000 vacuum tubes and weigh
only 1½ tons.”

The Electrification
of America
The electrification of America, masterfully
chronicled and analyzed by David (1991), illustrates the delays in successfully exploiting new
technologies. The era of electricity dawned
around 1900, in the midst of the Second Industrial Revolution, which typically is considered
to have started in the 1860s and ended in the
1930s. It saw the birth of the modern chemical
industry and the internal combustion engine, in
addition to electricity. Electricity was obviously
useful as a source of lighting in homes and
businesses, but it had to supplant water and
steam as a source of power in manufacturing.24
This process was complicated by the large
stocks of equipment and structures, already in
place, that were geared to these power sources.

Thus, in the early stages, electricity tended to
be overlaid onto existing systems. In particular,
the mechanics of steam- and waterpower
favored having a single power unit drive a
group of machines, and early electric motors
retained the group-drive system of belts and
shafting that had been used by steam- and
waterpower. Hence, early electric motors were
also used to drive a group of machines. The
benefits of electricity derived from the savings
in power requirements and the greater control
over machine speed. Not surprisingly, electric
power tended to be used mostly by industries
that were expanding rapidly, since new plants
could be designed to accommodate this power
source better.
By around 1910, it was apparent that
machines could be driven with individual electric motors. This realization had a great impact
on workplace productivity. The apparatus used
in the group-drive system could be abandoned,
so factory construction no longer had to allow
for the heavy shafting and belt housing required
for power transmission. The labor demands of
maintaining that system were also eliminated.
Furthermore, the production process became
more flexible for several reasons: It was no
longer necessary to shut down the entire power
system for maintenance or parts replacement.
Because each machine could be controlled
more accurately, the quantity and quality of output increased. Machines could now be located
and moved about more freely to accommodate
the production process. Last, the workplace
became considerably safer. Figure 10 shows the
diffusion of electric motors in manufacturing.25
Horsepower from electric motors, as a fraction
of the total mechanical drive in manufacturing
establishments, followed a typical S-shaped diffusion pattern. Labor productivity growth in
manufacturing slowed down at the time of electricity’s introduction.26
In 1890, an astute observer might have
understood the importance of electricity for
lighting homes and powering factories. He
would not, however, have been able to predict
how it would transform lives through the other
inventions it would spawn: radio, television,
and computers.

■ 24 While only 3 percent of households used electric lighting in
1899, almost 70 percent did by 1929 (David [1991, table 3]).
■ 25 The data source is David (1991, table 3).
■ 26 Ibid., table 2.

10

The Computerization
of America
As with electrification, the harvest of the IT
revolution has not been immediate. When the
era of computers began in the 1950s, they were
used primarily in academic and industrial research to perform calculations that were impractical or impossible to do manually (Jonscher
[1994]). Number-crunching costs declined rapidly over this period. Between 1950 and 1980,
the cost per MIP (million instructions per second) fell 27–50 percent annually, spurring the
use of computers as calculating devices. In a
feedback loop, widespread adoption led to further price reductions as computer manufacturers
rode up their learning curves. In the 1960s,
computers became file-keeping devices used by
businesses to sort, store, process, and retrieve
large volumes of data, thus saving on the labor
involved in information-processing activities.
The cost of storage probably fell at an annual
rate of 25–30 percent from 1960 to 1985. More
recently, computers have evolved into communication devices, beginning with the advent of
remote accessing and networking in the 1970s.
This allowed a partial liberation of the computer
from the “clean room,” but that umbilical cord
was not completely cut until the 1980s, with the
introduction of the personal computer and the
spread of networking.
IT is likely to streamline corporate structures
significantly by economizing on the number of
workers employed in information collection and
processing. The goal of any firm is simple: Maximize profits. To achieve it, the firm’s organizational structure must be capable of detecting
profit opportunities, directing actions to harvest
them, and monitoring and evaluating returns on
its activities. These activities largely involve handling and processing information. By 1980,
there were 1.13 times as many information
workers as production workers, compared to
just 0.22 in 1900. IT can do much of this information collection and processing more efficiently than workers can, eliminating the need
for battalions of clerks, pools of secretaries,
scores of purchasing and sales agents, and layers of supervisors and administrators. Through
IT, headquarters, design centers, plants, and
purchasing and sales offices can be linked
directly to one another. Over time, such major
changes in business structure will inevitably
raise labor productivity as it becomes possible
to create more output with less labor. Studies
such as Brynjolfsson and Hitt (1993) indicate
that this is happening already.

How realistic is the hypothesis just presented?
To judge this, Greenwood and Yorukoglu (1997)
have developed an economic model of the
Information Age, which they simulate on a computer. The model incorporates two ingredients.
First, firms face a learning curve when they
adopt a new technology. Second, firms can
travel up this curve faster by hiring skilled labor.
With the dawning of the Information Age, the
growth rate of labor productivity slumps in the
model economy, and income inequality widens.
The effects of the Information Age gradually
work their way through the system over time. In
the model, productivity growth does not surpass
its old level for about 20 years, and the level of
productivity does not cross its old trend line—
the path it would have traveled had it continued
at its former growth rate—for 40 years. Unskilled wages fall in the initial stages of the
Information Age. Twenty years elapse before
this loss in unskilled wages is recovered, and
about 50 go by before unskilled wages cross
their old growth path. Interestingly, during the
early stages of the Information Age, the stock
market booms as it capitalizes the higher rates
of return offered by new investment opportunities. For many in the economy, waiting for the
benefits of technological miracles will be like
watching the grass grow—but it will grow.

VIII. Conclusion
Plunging prices for new technologies, a surge
in wage inequality, and a slowdown in the advance of labor productivity—could these herald
the dawn of another industrial revolution? Just
as the steam engine shook eighteenth-century
England, and electricity rattled nineteenthcentury America, are information technologies
now rocking the twentieth-century economy?
The story told here is simple. Technological
innovation is embodied in the form of new
producer durables or services, whose prices
decline rapidly in periods of high innovation.
Adopting new technologies is costly. Setting up
and operating new technologies often involves
acquiring and processing new information.
Because skill facilitates this adoption process,
times of rapid technological advance should be
associated with a rise in the return to skill. At
the dawn of an industrial revolution, the longrun advance in labor productivity pauses temporarily as economic agents undertake the
(unmeasured) investment in information
required to get new technologies operating
closer to their full potential.

11

How will this affect people’s lives? In the
long run, everybody will gain. Technological
progress, which implies that a unit of labor can
eventually produce more output, makes a unit
of labor more valuable. Given time, this translates into higher wages and standards of living
for all. Clearly, everybody today is better off
because of Britain’s Industrial Revolution, but
this was not true in 1760. So what about the
short run? Skilled workers will fare better than
unskilled ones, but this disparity will shrink
over time for two reasons. First, as information
technologies mature, the level of skill needed
to work them will decline. Firms will substitute
away from expensive skilled labor toward more
economical unskilled labor. As this happens,
the skill premium will decline. Second, young
workers will tend to migrate away from lowpaying unskilled jobs toward high-paying
skilled ones. This tendency will increase the
supply of skilled labor and reduce the amount
of unskilled labor, easing pressure on the skill
premium. Moreover, the wealthy will do better
than the poor in the short run because the
introduction of new technologies leads to exciting profit opportunities for those with the
wherewithal to invest in them. These profit
opportunities will shrink over time as the pool
of unexploited ideas dries up. On average, the
old have more capital to invest than the young.
Thus, young, unskilled agents will fare worst in
the short run. But in the long run, the rising
tide of technological advance will lift everybody’s boat.

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12

Harley, C. Knick. “Reassessing the Industrial
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–––––––– , and Saul Lach. “ Product Innovation and the Business Cycle,” International
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1997), pp. 3–22.
–––––––– , and Yaw Nyarko. “A Bayesian
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Brooks Pierce. “Wage Inequality and the
Rise in Returns to Skill,” Journal of Political
Economy, vol. 101, no. 3 ( June 1993),
pp. 410–42.
Krusell, Per, Lee E. Ohanian, José-Victor
Ríos-Rull, and Giovanni L. Violante.
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Williamson. “Reinterpreting Britain’s Social
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pp. 94–109.

McPherson, Natalie. Machines and Growth:
The Implications for Growth Theory of the
History of the Industrial Revolution. Westport,
Conn.: Greenwood Press, 1994.
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1830,” in Roderick Floud and Donald
McCloskey, eds., The Economic History of
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Nathan Rosenberg, ed., Inside the Black
Box: Technology and Economics. New York:
Cambridge University Press, 1982.
von Tunzelman, Nick. “Technology in the
Early Nineteenth Century,” in Roderick
Floud and Donald McCloskey, eds., The Economic History of Britain since 1700, 2d ed.
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Yorukoglu, Mehmet. “The Information Technology Productivity Paradox,” Review of Economic Dynamics, vol. 1, no 2 (April 1998),
pp. 551–92.

13

Accounting for Capital
Consumption and
Technological Progress
by Michael Gort and Peter Rupert

Introduction
Oscar Wilde defined a cynic as a person “who
knows the price of everything and the value of
nothing,” and a sentimentalist as one “who sees
an absurd value in everything but doesn’t know
the market price of any single thing.” Most
economists would probably object to the first
definition, for to know the price of something
is to know what value society (that is, the market) places on the last unit. And while few people regard the Internal Revenue Service as sentimental, it has, at least implicitly, adopted the
practice of placing values on capital goods,
usually without knowing their prices.
Computing the value of the stock of capital,
especially in the face of technological advance,
is a large task, complicated by the fact that
assets may lose value over time because of
physical wear and tear as well as obsolescence. When calculating income, owners of
capital are allowed to deduct from earnings the
amount of capital that is consumed by the production process (depreciation), termed capital
consumption in the National Income and Product Accounts (NIPA). Deriving a measure of the
aggregate capital stock entails adding up assets
that have very different lives, hence very different depreciation patterns.

Michael Gort is a professor of
economics at the State University
of New York at Buffalo, and Peter
Rupert is an economic advisor at
the Federal Reserve Bank of
Cleveland.

Difficult as it may be, obtaining fairly precise estimates of the capital stock is important.
One area where reliable estimates are necessary is that of growth accounting. As its name
suggests, its goal is to determine the underlying sources of economic growth in order to
account for the growth in output. How do we
create more and more output over time? At a
very simple level, the inputs that produce the
output might be increasing, or technological
advance in the economy might give us more
from the same expenditure on inputs.
At a slightly deeper level, suppose that the
only two inputs are physical capital (computers, trucks, and so forth) and labor. Output is
derived from these two inputs through some
production process. Now suppose that output
is observed to be growing over time. If there
are no measurement problems, it is possible to
determine what underlies growth in the economy. Observed growth, for example, might be
attributed to growth in the labor force, more
computers, or both. Simple enough.
To complicate things a bit more, suppose
that output is observed to be growing faster
than the measured growth in inputs. Now
what? It is possible that there is an input not

14

included in the simple, two-factor (capital and
labor) model. For example, there might be a
change in how labor and capital are combined,
as when new business practices enable better
communication. Obviously, this could be difficult to measure with any accuracy. Such unmeasured influences go into a catchall component called total factor productivity.
But there is another explanation for the gap
between input growth and output growth. Suppose inputs are not measured correctly due, for
example, to technological growth in one or both
of them. Imagine that given some labor input,
the capital stock (say computers) is increased.
The complication would arise if the new computer were twice as fast as the old one and,
therefore, able to produce much more. If that
feature were not taken into account, the new
computer would be added as if it were an old
one, and growth in the capital stock would be
mismeasured. Hence, too little of the economy’s
growth would be attributed to capital’s contribution and too much to total factor productivity.
Such technological growth in capital is known
as capital-embodied technological change.
The same could be true of labor, giving rise
to labor-embodied technological advance. Understanding where growth comes from has important implications for policy making. With
accurate measurement, policies can be designed
to devote resources to the most productive uses.
For example, it would be possible to assess the
contribution to growth of spending an additional $1 billion on education programs, thus
increasing the level of human capital. Or to
gauge the impact of spending that sum to promote research and development in the computer industry.
To obtain an accurate measure of capital, it is
important to know not only how productive a
new vintage of capital is, but also how quickly
the old capital loses value. Obviously, the faster
an asset is used up in the production process,
the higher the investment rate needed to keep
the stock of capital constant. But assets may
also become obsolete (that is, used up) in a different sense. A computer loses value over time
because newer models are so much better per
dollar spent, not because its keyboard doesn’t
work properly or its hard drive is leaking oil.
The amount of capital consumption the Internal
Revenue Service allows will have a substantial
impact on the rate of investment in the economy. In fact, the depreciation allowance has
been used to increase investment in specific
industries. Pollution control facilities, rehabilitation of low-income housing, the railroad rolling
stock, and coal-mining safety equipment are

instances of such specific targeting. In addition,
depreciation has been used as a countercyclical
policy instrument. For example, when the economy began to overheat in 1966, the investment
tax credit implemented in 1962 to spur investment was suspended, as were accelerated depreciation methods for real property. By the
end of 1967, the economy had begun to weaken and those policies were reinstated.1

I. Aggregating
and Measuring
a Heterogeneous
Capital Stock
A two-step procedure is conventionally used to
measure capital consumption, whether for
depreciation of individual firms or for aggregate
estimates tied into the NIPA. First, the asset’s
useful economic life is estimated (based mainly
on estimates of the Internal Revenue Service).
Second, the asset’s original cost is allocated over
the estimated useful life to measure each year’s
capital consumption (depreciation). To compute
the aggregate stock of capital in the NIPA, each
year’s investment is deflated by a price index,
and depreciation for it is computed separately.
By aggregating current capital consumption
charges from all past investments, each year’s
estimate of aggregate capital consumption in
real terms is obtained. And by aggregating the
net deflated investments from previous years
(net of all current and past capital consumption
charges) a so-called “perpetual inventory” capital stock is derived for each year in the NIPA.
This approach has several problems. First,
the estimates of useful life are of undetermined
reliability. Second, methods of allocating original cost to derive capital consumption, the most
common being “straight line” and “declining
balance,” are quite arbitrary.2 Third, with only a
few exceptions, price indexes used for deflation
do not take account of changes in the quality of
capital over time. Thus, the resulting investment
streams, when aggregated over time, are not
expressed in homogeneous efficiency units.
Fourth, depreciation or capital consumption
lumps together obsolescence and physical decay, making it impossible to identify the separate
effects of technological change—as opposed to
wear and tear on the net stock of capital.
■ 1 See Brazell, Dworin, and Walsh (1989) for a more in-depth
discussion.
■ 2 The past several years have seen efforts to obtain better estimates
of both useful lives and depreciation patterns. See Survey of Current Business (1998).

15

Gort, Greenwood, and Rupert (1999) seek to
surmount all these problems in their estimates
of obsolescence and physical decay for structures. Focusing on office buildings and using
data provided by the Building Owners and
Managers Association, they estimate 1) the rate
of obsolescence over the life of a building; 2)
the rate of physical decay as a building ages;
and 3) the implications of these estimates for
economywide growth in capital and for the
contribution to economic growth of the underlying measured inputs: equipment, structures,
and labor. The authors also derive the contribution of disembodied technological progress
(total factor productivity).3
Contrary to the common assumptions that
technological progress is limited to equipment
and that a building’s life span is largely defined
by its rate of physical decay, the authors find a
substantial rate of technological advance. Such
advance explains a significant fraction of economywide capital growth and changes the share
attributed to total factor productivity.
These estimates are made possible by data
based on market prices. Specifically, a relation is
established between a building’s age on the one
hand and both the total rental revenue and the
gross operating profit generated from rentals on
the other. The authors estimate the net effects of
a building’s age (or vintage) on (a) the rental
revenue per square foot and (b) the gross operating profit per square foot. After allowing for
the effects of several other variables such as the
building’s location, variable (a) gives the effect
of vintage on the decline in the gross flow of
productive services as the building ages, and
(b) gives the effect of vintage on the decline in
income that the building generates.
The key idea is that a new building should
rent for more because it embodies more advanced technology. Here, rent’s rate of decline
measures the technological advance of structures in the economy. In addition, it will be
more profitable for a newer building to employ
equipment and labor that uses a more recent
technology.
Decoupling obsolescence from physical
wear and tear is a formidable task because economic depreciation is defined as the rate at
which an asset loses value over time.4 Both
obsolescence and physical wear and tear contribute to the decline in asset value; moreover,
different types of assets will exhibit different
patterns of decay attributable to those underlying components. For example, the useful service life of the computer used to type this article is quite short (about three years). Evidently,
nearly all of computers’ age-related decline in

value results from technological advance. Each
year, computers become much faster, have more
memory and storage, and so on, but virtually
no loss due to physical wear and tear. In other
words, the three-year-old computer produces
almost exactly the same amount of output as
when it was brand new, but it has lost value because it is vastly inferior to a new model. Automobiles differ from computers in that while
there certainly are technological improvements
(such as ABS brakes, air bags, and so on), physical wear and tear play a much larger role. Many
of a car’s internal parts must be replaced or
repaired long before it loses all of its value.
Gort, Greenwood, and Rupert (1999) infer
that the decline in revenue results from technological change, that is, obsolescence. This conclusion is based on the fact that building owners
must maintain, both by rental contract and by
local ordinances, the safe and effective use of
the building through appropriate repair and
maintenance outlays. Office buildings cannot be
used if they have water leaks, have nonfunctioning heating and plumbing systems, unsafe
elevators, loose bricks, and so on. Repair and
maintenance costs therefore must cover this
physical decay, at least insofar as it affects the
safe, effective use of office space. These expenditures can be viewed as investments to cover
and inhibit physical depreciation. And, as
shown below, repair and maintenance costs rise
systematically as a building ages. Over time,
they cut into a building’s revenue and therefore
influence its useful service life. It should be
stressed that the implied definition of obsolescence is a very broad one, which captures all
sources of decline associated with economic
progress, including architectural changes that
allow better use of space, light, and so on.
Engineering advances enable the occupants
of a building to work in greater comfort. For
example, anti-sway devices, located in the tops
of skyscrapers, limit the extent of the buildings’
movement. “Sky lobbies” permit an elevator
car to move into an alcove when admitting or
discharging passengers, allowing the next car
to pass. Advances in other areas, such as the
introduction of computers, can also lead to a
form of obsolescence, since the need for routing new fiber-optic cables to set up networks

■ 3 Total factor productivity can be thought of as a factor that scales
up the value of all inputs to equal the output. For example, if inputs of all
factors of production equal $5 and produced output that is sold at $10,
then total factor productivity would equal 2.
■ 4 In statistical or econometric terms, the problem is one of identification. See Hall (1968) or Hulten and Wykoff (1981).

16

requires that a building’s interior be amenable
to such changes.
Gort, Greenwood, and Rupert (1999) incorporate existing data into a theoretically based
economic model that uses these data to impose
discipline on the behavior of the model itself.
These and other building-specific data were obtained from analyses performed by the Building
Owners and Managers Association International,
which has been collecting data on individual
office buildings across the United States and
Canada for over 70 years. The collected data include information on size, expenditures for repair and maintenance, region, occupancy rates,
and, most importantly for this exercise, rent.5
Two important facts emerge from the data.
First, rent per square foot declines with the age
of the building.6 Second, repair and maintenance costs increase.7
The results from regression analyses show
that after adjusting for inflation, rent per square
foot declines about 1.5 percent annually, and
repair and maintenance costs rise about 2 percent annually.
Because rents are declining with age while
maintenance costs are increasing, a building
will eventually cease to be profitable and will
be razed to make room for a newer, more productive one. That is, it will be replaced by a
structure with the latest advances in technology, such as faster elevators, better heating,
ventilation, air conditioning, and safety equipment, adjustable interior space, and so on.
With the estimates and restrictions placed on
it, the model shows that the growth rate of technology in office buildings has been about 1 percent annually. That, in conjunction with the fact
that technological progress in equipment (by
one estimate) has been about 3.2 percent annually,8 allows U.S. output growth from capital
accumulation to be broken down into its underlying components.9
Specifically, structures are found to account
for approximately 15 percent of economic
growth, and equipment for approximately 37
percent. The remaining 48 percent is attributed
to labor inputs and total factor productivity; that
is, it cannot be attributed to any specific factor.
The model also allows an exact measurement of the capital stock. Note that in the presence of technological change, aggregating
across different vintages becomes a daunting
task, because one must know how much better
each successive generation of capital is. Further, the embodiment of technological change
in capital means that changes in each generation must be converted into a common unit to

make aggregation across different vintages possible. However, results from the model of the
pace of technological growth make it possible
to determine the exact number of efficiency
units of capital. For example, the NIPA show
that the growth rate of nonresidential structures
per person-hour between 1959 and 1996 has
been 0.75 percent annually.
Results from the Gort, Greenwood, and
Rupert model suggest that this growth rate is
2.4 percent annually, a substantial difference.
Likewise, the NIPA estimate of the growth rate
in the stock of equipment is 2.5 percent annually, while the model puts it closer to 4.4 percent annually. This suggests that the NIPA substantially underestimates the size of the capital
stock, once one takes into account technological advances embedded in new capital are
taken into account.

II. Conclusion
Current methods used to calculate capital consumption, the stock of capital, and the sources
of growth in the economy do not adequately
measure the underlying growth in inputs due to
technological advance. This has implications for
tax policy as well as the design of programs targeting specific areas that can lead to higher
growth in the economy.

■ 5 Since the data are proprietary in nature, the Association provided
them without exact building identifiers. The data used in Gort, Greenwood,
and Rupert (1999) were based on the years 1988–96.
■ 6 This result is based on a regression in which the dependent variable is the log of real rent per square foot and the independent variables
are age, region of the country, calendar year, and a constant term.
■ 7 A similar regression was used to determine the exact rate of
increase in repair and maintenance costs with age.
■ 8 Taken from Greenwood, Hercowitz, and Krusell (1997) and based
on prices from Gordon (1990).
■ 9 This is based on the assumption that other types of nonresidential structures have seen the same rate of technological progress.

17

Appendix
This technical appendix provides the underlying mathematical framework of the model,
although it leaves out many details, such as the
parameters used in the calibration. The reader
is referred to Gort, Greenwood, and Rupert
(1999) for those missing details.
Production is undertaken at a fixed number
of locations, distributed uniformly on the unit
interval, and requires the use of three inputs:
equipment, structures, and labor. Each location
is associated with a stock of structures of a certain age or vintage. Equipment and labor can
be hired each period on a spot market. Let production at a location using structures of vintage
j be given by
(A1)

o ( j ) = zke ( j )αek s ( j )αsl ( j )β,

where z is the economywide level of total factor
productivity, and ke ( j ), k s ( j ), and l ( j ) are the
inputs of equipment, structures, and labor.
Denote the number of locations using structures
of vintage j by n( j ), and let the maximum age
of structures be T. Then ∫0T n( j )dj = 1. Aggregate output is thus
(A2)

y = ∫0T n( j )zke ( j )αek s ( j )αsl ( j )βdj.

where re is the economywide rental price for
equipment and w is the wage rate. The manager’s date-0 problem can be written as the following value function:

(A5) V [ks, 0(0)] =
max
ks, T (0),T

5E

T

0

[πt (t) – µ (t)ks, 0(0)/v0 ]e –ι tdt

6

+ e –ι t [V (ks, 0(0)) – ks,T (0)/vT ] ,
where ι represents the time-invariant interest
rate, and the initial maintenance cost is a fraction µ (0) of the building’s purchase price. As
the building ages, these costs grow exogenously at rate γµ + γy , where γy is the economy’s growth rate.
At each point in time, the equipment manager has ke units of equipment that he can rent
out at re . He must decide how much to invest,
ιe , in new equipment. This investment can be
financed at the fixed interest rate ι . The optimal
control problem governing the accumulation of
equipment is summarized by the current-value
Hamiltonian:
H = re ke – ie + λ [ie q – δe ke ].

Output can be used for four purposes: consumption, c, investment in new equipment, ie,
investment in new structures, is , and investment
in repair and maintenance on old structures, im.
Hence,

Let a consumer’s lifetime utility function be
given by

(A3)

Now, the consumer is free to lend in terms of
bonds, a, earning the return ι . In addition to
the interest he realizes on his lending activity,
w, and the profits from his locations (net of any
repair and maintenance costs and investment in
structures). The law of motion governing his
asset accumulation reads

c + ie + is + im = y.

Imagine constructing a new building at some
location. Suppose that a unit of forgone consumption can purchase v new units of structures. Then, building ks (0) units of new structures would cost ks (0)/v units of consumption.
Let v grow at the fixed rate γv ; this denotes
structure-specific technological progress.10
Structures remain standing until they are replaced. Expenditures on repair and maintenance keep buildings in their original condition.
Those costs grow over time, µ ( j ) = e (γµ + γy )j.
The static profit-maximizing decision at a
location using structures of vintage j is represented by
(A4)

π ( j ) = max
ke ( j),l,( j)

∫0

∞

lncte –ptdt.

w + ι a + ∫0 n(j)[π (j)
T

da/dt =

– µ (j)ks(j)e γv j /v]dj – n(0)ks (0)/v – c.

The balanced growth path can be uncovered
using a guess-and-verify procedure.
Now, consider the economy’s cross-section
of buildings at a point in time. It is easy to calculate that the percentage change in rents as a

zke ( j )αek s ( j )αsl ( j )β

5 – r k ( j ) – wl( j ), 6,
e e

■ 10 The focus of the analysis is on balanced growth paths. As a
result, some variables, such as aggregate output, will grow over time at
constant rates; others, such as the interest rate, will be constant.

18

function of age (the rent gradient δs ) should be
given by

αs
(A6) δs = 1 – α – β γs ,
e
since the stock of structures declines at rate γs
as a function of age, while factor prices remain
constant. This formula gives a measure of obsolescence in buildings. In the absence of depreciation, a new building rents for more than an
old one only because it offers more efficiency
units of structures.
The model can then be calibrated using such
information as the rate of decline in rents for
buildings, the average annual growth rate of
output, and so on, to obtain the underlying
sources’ contribution to growth.

References
Brazell, David W., Lowell Dworin, and
Michael Walsh. A History of Federal Tax
Depreciation Policy. Washington, D.C.:
U.S. Department of the Treasury, Office of
Tax Analysis, 1989.
Cooley, Thomas F., and Edward C. Prescott.
“Economic Growth and Business Cycles,” in
Thomas F. Cooley, ed., Frontiers of Business
Cycle Research. Princeton, N.J.: Princeton
University Press, 1995, pp. 1–38.
Gordon, Robert J. The Measurement of
Durable Goods Prices. Chicago: University of
Chicago Press, 1990.
Gort Michael, Jeremy Greenwood, and
Peter Rupert. “Measuring the Rate of Technological Progress in Structures,” Review of
Economic Dynamics, vol. 2, no. 1 (January
1999), pp. 207–30.
Greenwood, Jeremy, Zvi Hercowitz, and Per
Krusel. “The Macroeconomic Implications of
Investment-Specific Technological Change,”
American Economic Review, vol. 87, no. 3
(June 1997), pp. 342–62.
Hall, Robert E. “Technical Change and Capital
from the Point of View of the Dual,” Review
of Economic Studies, vol. 35 (January 1968),
pp. 35–46.

Hotelling, Harold. “A General Mathematical
Theory of Depreciation,” Journal of the
American Statistical Association, vol. 20
(1925), pp. 340–53.
Hulten, Charles R., and Frank C. Wykoff.
“The Estimation of Economic Depreciation
Using Vintage Asset Prices; An Application
of the Box–Cox Power Transformation,”
Journal of Econometrics, vol. 35, no. 3 (April
1981), pp. 367–96.
Taubman, Paul, and R. H. Rasche. “Economic and Tax Depreciation of Office Buildings,” National Tax Journal, vol. 22, no. 3
(September 1969), pp. 334–46.

19

Defining Capital
in Growth Models
by Michael Gort, Saqib Jafarey, and Peter Rupert

Introduction
The definition or implicit measure of capital
has, in some respects, been left ambiguous in
the literature on economic growth. Most of the
theoretical work on economic growth has
assumed the existence of an aggregate capital
stock that grows at the same rate as output in a
balanced-growth equilibrium. This assumes
away the aggregation problem by treating different units of capital as homogeneous.
However, empirical researchers on the subject of growth accounting, such as Greenwood,
Hercowitz, and Krusell (1992), often must deal
with the possibility that technological advance
is at least partly embodied in new units of capital. In this case, a theoretically consistent measure of the aggregate capital stock requires the
use of efficiency units that give greater weight
to newer vintages. The question then arises as
to the numeraire used to define an efficiency
unit of capital.
In the context of a model where both embodied and disembodied technological progress take place at constant rates, this article
explores two alternative measures of capital—
one using the newly produced capital of each
year as the numeraire for that year (the con-

Michael Gort is a professor of
economics at the State University
of New York at Buffalo; Saqib
Jafarey is a lecturer in economics
at the University of Essex; and
Peter Rupert is an economist at
the Federal Reserve Bank of
Cleveland.

ventional measure of net capital, that is, the
perpetual-inventory method), and the other
using base-year capital as the numeraire. These
two measures are equivalent for any given
year, in that they assign equal relative weights
to different vintages of capital. However, there
are significant differences between the two
measures in the context of time series on capital and, hence, in the resulting measures of the
growth in capital.
While the base-year measure uses a consistent numeraire (that is, the initial year’s capital
stock) from one year to the next, the currentvintage measure changes the numeraire with
each year’s vintage. A meaningful comparison
of capital stocks across years requires a consistent numeraire; a time series of capital stocks in
which the numeraire changes each year would
be useless for many empirical problems.
In growth accounting, for example, the
Solow residual may be estimated by subtracting
the growth rate of capital per worker from the
growth rate of output per worker, provided that
the former is calculated from a time series in
which the capital stock has been aggregated in
terms of the base-year numeraire. As we will
show, the use of a capital stock series, aggregated in units of each year’s vintage, yields an

20

overestimate of the Solow residual, requiring a
correction using an independent estimate of embodied technological advance. Although the use
of a current-year numeraire accounts for the heterogeneity of the capital stock within each year,
it may yield the same upward bias in the estimate of the Solow residual as that which arises
when all capital is treated as homogeneous.
The differences between the two aggregates
are important conceptually, but they also have
broad empirical implications. The magnitude of
the difference depends, of course, on the role of
technological growth “embodied” in the capital
inputs of successive vintages. Gort, Greenwood,
and Rupert (1999) show that technological
growth in the stock of structures has proceeded
at the rate of 1 percent per year. Using a frequently cited estimate of technological growth
in equipment of 3 percent per year for the
postwar period, they conclude that the stock
of structures and equipment has grown at 2.2
percent and 4.4 percent, respectively. In contrast, growth using the conventional perpetualinventory measures of the Commerce Department’s National Income and Product Accounts
was 0.75 percent for structures and 2.5 percent
for equipment.
Other methods have led to even larger estimates of technological growth for capital goods.
Bahk and Gort (1993), for example, estimate the
quality change of capital for a large number of
manufacturing industries. They conclude that a
change in vintage of one year can be expected
to lead to a change in output of 2.5 percent to
3.5 percent. Assuming that capital has a weight
of roughly one-third in the production function,
this implies a rate of quality change in capital of
7.5 percent to 10.5 percent per year. Indeed,
when the effect of vintage is taken into account,
the measure of residual disembodied technological change goes to zero. While the experience
of manufacturing industries may not be characteristic of the economy as a whole, these results
show how massive discrepancies can arise from
using alternative concepts and their corresponding measures of capital.
The behavior of the capital/output ratio in a
balanced-growth equilibrium also depends on
the numeraire chosen for aggregating capital.
When technological advance is at least partly
embodied, the capital/output ratio becomes
constant if, and only if, capital is aggregated in
units of the current year’s vintage; for an aggregate based on the initial year’s vintage, the
ratio grows toward infinity. The long-run behavior of the interest rate varies accordingly—
the return on capital aggregated in current
units becomes constant, while the return on

capital aggregated in base-year units asymptotically approaches zero.
Empirical estimation of dynamic models often
begins by testing the statistical properties of the
relevant time series to determine whether their
behavior is consistent with that assumed in the
model. Since balanced growth implies that capital and output grow at the same rate, an empirical researcher testing the propositions of a
balanced-growth model may expect the capital/
output ratio to remain stationary over time. Such
an expectation would be valid for the capital
stock time series aggregated on a current-year
numeraire, but not for the base-year aggregate.1

I. A Model
of Embodied
Technological
Progress
Assuming constant returns to scale for capital
and labor, the per capita production function
can be written as
(1) yt = At f (kt ),
where At grows at the rate of disembodied
technological progress, µ .
Net additions to per capita stocks of capital
may be measured in efficiency units,
•

(2) kt = φt it – δkt ,
where it is per capita investment at time t; δ is
the rate of capital depreciation; and φt is a
measure of the quality of new capital goods,
translating one physical unit of investment at
time t into equivalent units of base-year capital.
Additionally, we assume that φ 0 = 1 and φt
grows at the constant rate γ.
Current investment is related to current output through
(3) yt = it + ct ,
where ct is per capita consumption at time t.
Equation (3) assumes that investment and consumption goods are perfect substitutes and that
quality improvements in new capital goods

■ 1 In practice, econometric tests of growth theory often evade this
point by running cross-country regressions. This is done, however, partly
because of data availability and partly because of expediency in avoiding
modeling problems that would arise from time-series tests. It does not
change the fact that time-series econometric models are the proper vehicles for testing the implications of growth theory (for a critique of crosscountry regressions, see Levine and Renelt [1992]).

21

entail no expenditure of current output. This is
consistent with the approach taken by Solow
(1960), Fisher (1965), and Greenwood, Hercowitz, and Krusell (1992).2
An alternative measure of capital is given by
k
^
kt = φ t .

(4)

t

^

Both kt and kt measure capital in efficiency
units; however, the choice of the numeraire is
different. To provide a clearer interpretation of
these two measures, it may be helpful to decompose each measure into its underlying investment flow in a discrete-time analog:
kt = (1 + γ )tit + (1 + γ )t –1(1 – δ )it –1
+ (1 + γ )t –2 (1 – δ )it –2 + ... + (1 – δ )ti 0
and
^

kt = it +

(1 – δ )
(1 – δ )2
i +
i
(1 + γ ) t –1 (1 + γ )2 t –2

+ ... +

(1 – δ )t
i .
(1 + γ )t 0

Thus, while kt aggregates capital by augmenting more recent vintages to reflect their
greater efficiency relative to the base-year vin^
tage, kt depreciates past vintages in order to
correct for their relative obsolescence. While kt
is consistent across years in its choice of numer^
aire, kt changes numeraire with each new vintage and reflects the perpetual-inventory method
of aggregating each year’s capital stock in efficiency units of that year’s vintage.
^
The difference between kt and kt is potentially large and has drastic implications not only
for tests of cointegration among a set of variables including output and capital, but also for
estimating models where the capital stock is a
key variable. For example, making the conservative assumption that obsolescence represents
only half of conventionally measured depreciation, Gort and Wall (1998) estimate that kt for
the aggregate U.S. economy (excluding government and agriculture) grew at an annual rate of
6.06 percent during the 1947–89 period. In
^
contrast, the equivalent measure of growth of kt
was only 3.77 percent per year.
^
Per capita output in terms of kt is given by
^

(5) yt = At f (φt kt ),
■ 2 An alternative approach, taken by Hulten (1992), is to adjust each
year’s output to reflect the increased quality of investment goods. Thus,
yt = φt it + ct . This approach, however, leads to netting out the effects of
embodied technological progress along the balanced-growth path.

^

and the rate of change of kt is given by
•

^

^

(6) kt = it – (δ + γ ) kt .
The measurement of returns to capital, then,
^
depends on whether kt or kt is used. Let rt and
^
rt denote the marginal product of an extra
physical unit of base-period and new capital,
respectively. Thus,
(7a) rt = At f ′(kt )
^

(7b) r^t = Atφf ′(φt kt ).
Multiplying (7a) by φt , it follows that
(8) r^t = φt rt .
Note that r^t represents the marginal product
of an extra unit of new capital at time t. To the
extent that (gross) additions to the aggregate
capital stock can take place only through newly
produced units of capital, r^t (and not rt ) represents the relevant return to physical capital. In
the presence of risk-free financial assets, the
interest rate would equal r^t in equilibrium.

II. Analysis
of Growth Paths
The analysis of efficient growth paths usually
begins with the Euler equation describing the
growth rate of per capita consumption. With embodied technological advance, this is given by
(9) σ

c•t
= r^t – ( ρ + δ + γ ),
ct

where σ and ρ are preference parameters
denoting the elasticity of intertemporal substitution and the rate of time preference, respectively. Solving equation (9) for r^t and using
equation (8) gives
(10)

r^t = φt rt = σ

c•t
+ ρ + δ + γ.
ct

In a balanced-growth equilibrium, per capita
consumption grows at a constant rate; there^
fore, the return to kt becomes constant, while
the return to kt decreases at the rate γ .
The rate at which per capita quantities grow
along a balanced-growth path can be solved
using a “sources of growth” equation, obtained
by totally differentiating the production function
with respect to time. Given the two measures of

22

capital, the sources of growth in the relevant
equation can be expressed as
•

•

•

3

•

•
(11a) yt = At + θt k t = µ + θt k t
•
yt
At
kt
kt
and
•

^

•

4

•

^

•
(11b) y t = At + θt k t + φt = µ + θt γ + θt k t ,
^
^
yt
At
kt
φt
kt

where θt ≡ rt kt /yt is capital’s share in output at
any time t.
•
In calculating the Solow residual, AA, atten^
tion must be paid to the distinction between kt
^
and kt . Mistakenly using data for kt in equation (11a), as if it represented measures of kt ,
would lead to an upward bias in the estimated
contribution of disembodied technological progress. However, even if the distinction between
^
the two measures of capital is kept• in mind, kt
cannot be used by itself •to get at AA, without independent estimates of φφ as in equation (11b).
In a balanced-growth equilibrium, the capital/output ratio remains constant. With differing
growth rates between the two measures of capital, however, the capital/output ratio cannot
stay constant for both measures. Solving equation (11a) under the conjecture that kt grows at
the same rate as yt yields the common growth
rate ( µ / 1 – φt ). This is unacceptable for two reasons: First, the rate of embodied technological
progress, γ, does not appear in the solution for
the final growth rate. Second, since rt declines
over time, so will θt , and the long-run growth
rate itself will be a declining function of time.
Is a balanced growth path with constant
^
long-run growth consistent with kt growing at
the same rate as yt ? If so, the output share of
capital, θt , would also remain constant because
r^t is constant in the balanced-growth equilib^ ^
rium, and θt may be written as rt kt /yt . With
this construction, the long-run rate of balanced
growth may be derived using equation (11b).
^
Denoting the common growth rate of kt and yt
by g, equation (11b) implies
•

^

•

(12) yt
yt

=

kt
^

kt

=g =

µ + θγ ,
–
1–θ

–
where θ denotes the value of θt in long-run
equilibrium. Thus, in a balanced-growth equilib^
rium, yt and kt both grow at the common rate
g, while the capital stock measured in baseperiod units grows at the higher rate of (g + γ ).

Indeed, since the two measures of capital
grow at different rates whenever embodied
progress occurs, they cannot be expected to be
cointegrated. Thus, in studying cointegration of
the capital stock and other economic variables,
the hypotheses to be tested should be constructed in light of the difference between them.
In testing the propositions of balanced
^
growth, theory suggests the use of kt . This also happens to be the conventional measure of
perpetual-inventory net stocks on the assumption, of course, that the depreciation rate used
to construct such stocks correctly captures both
^
obsolescence and physical decay. However, kt
requires
an independently derived measure of
•
φ in order to calculate disembodied technologi•
φ
cal progress, AA .

III. Conclusion
When technological progress is embodied in
capital, measuring the capital stock is problematic because capital is not homogeneous across
vintages, especially in the context of time series.
The choice of the measure of the capital stock is
dictated by the question being addressed. It has
been shown that the relevant measure differs
for balanced growth and growth decomposition
models. In particular, when aggregating heterogeneous capital across vintages, a numeraire
must be consistent with the underlying theory.
A further complication is that little progress has
been made in obtaining independent estimates
of embodied technological progress—a necessary step in the empirical implementation of a
broad range of growth models.

23

References
Bahk, Byong-Hyong, and Michael Gort.
“Decomposing Learning by Doing in New
Plants,” Journal of Political Economy,
vol. 101, no. 4 (August 1993), pp. 561–83.
Fisher, Franklin. “Embodied Technical
Change and the Existence of an Aggregate
Capital Stock,” Review of Economic Studies,
vol. 32, no. 92 (October 1965), pp. 263–88.
Gort, Michael, Jeremy Greenwood, and
Peter Rupert. “Measuring the Rate of Technological Progress in Structures,” Review of
Economic Dynamics, vol. 2, no. 1 (January
1999), pp. 207–30.
———, and Richard A. Wall. “Obsolescence,
Input Augmentation, and Growth Accounting,” European Economic Review, vol. 42,
no. 9 (November 1998), pp. 1653–65.
Greenwood, Jeremy, Zvi Hercowitz, and
Gregory W. Huffman. “Investment, Capacity Utilization and the Real Business Cycle,”
American Economic Review, vol. 78, no. 3
(June 1988), pp. 402–17.
———, ———, and Per Krusell. “The LongRun Implications of Investment-Specific
Technological Change,” American Economic
Review, vol. 87, no. 3 (June 1997),
pp. 342–62.
Hulten, Charles R. “Growth Accounting when
Technical Change is Embodied in Capital,”
American Economic Review, vol. 82, no. 4
(September 1992), pp. 964–80.
Levine, Ross, and David Renelt. “A Sensitivity
Analysis of Cross-Country Growth Regressions,” American Economic Review, vol. 82,
no. 4 (September 1992), pp. 942–63.
Solow, Robert M. “Investment and Technical
Progress,” in Kenneth J. Arrow, Samuel
Karlin, and Patrick Suppes, eds., Mathematical Methods in the Social Sciences. Stanford,
Calif.: Stanford University Press, 1960,
pp. 89–104.