Download Scale effects in Schumpeterian models of economic growth

J Evol Econ (1999) 9: 157±185
Scale e€ects in Schumpeterian models
of economic growth
Elias Dinopoulos1 , Peter Thompson2
1
2
Department of Economics, University of Florida, Gainesville, FL 32611, USA
Department of Economics, University of Houston, Houston, TX 77204±5882, USA
Abstract. Early models of Schumpeterian growth incorporate scale e€ects
predicting that large economies grow faster than small economies, and that
population growth causes accelerating per capita income growth. An absence of clear empirical evidence for these scale e€ects has led some researchers to question the foundations underlying the Schumpeterian
approach to growth. This paper reviews empirical evidence on the relationship between scale and growth, and recent attempts to construct
Schumpeterian growth models without scale e€ects.
Key words: Economic growth ± R&D ± Scale e€ects
JEL-classi®cation: O2; O3
1 Introduction
The idea that scale matters for economic growth is one of the oldest ideas
in economics. William Petty (1682), observing the reconstruction of London after the Great Fire, seems to have been the ®rst to identify a possible
link:
As for the arts of delight and ornament, they are best promoted by the
greatest number of emulators. And it is more likely that one ingenious
curious man may rather be found amongst 4 millions than among 400
persons.
* We would like to thank Steven Klepper for suggesting the topic of this paper, and two
referees for some extremely useful suggestions.
Correspondence to: E. Dinopoulos
158
E. Dinopoulos, P. Thompson
Petty's insight survived through centuries of economic thought1, and
disappeared from the study of economic growth only after the development
of the neoclassical model, in which scale plays no role. During the last
decade, however, the question of scale has received renewed attention from
growth theorists. This attention is a direct result of the formalization of
Schumpeterian models of R&D-driven growth2, which have made the link
between scale and growth precise and explicit.
Consider the following simple version of a Schumpeterian model. Output is governed by a Ricardian production function
Y …t† ˆ A…t†LY …t†
…1†
in which Y(t) is output, LY(t) is labor devoted to the production of ®nal
goods, and A(t) denotes the state of knowledge at time t. Knowledge advances according to
_
A…t†
ˆ cLA …t†
A…t†
…2†
where LA(t) = L(t))LY(t) is labor devoted to knowledge creation, and L is
the size of the work force3. Dividing both sides of equations (1) and (2) by
population, and noting that the fractions of labor allocated to manufacturing, LY(t)/L(t), and to R&D, LA(t)/L(t), must be constant in any steady
state, one obtains
_
_
y…t†
A…t†
LA …t†
ˆ
ˆc
L…t† ;
…3†
y…t† A…t†
L…t†
where y(t)=Y(t)/L(t) is output per capita.Equations (2) and (3) together
generate testable propositions.
Equation (3), relating resource endowments to growth, predicts that a
country such as China should have an enormous growth advantage over,
1
The idea can be found in the opening chapter to Wealth of Nations [``Men are much more
likely to discover easier and readier methods of attaining any object when the whole
attention of their minds is directed towards that single object than when it is dissipated
among a great variety of things.'' (Smith, 1776, Book I, ch. 1)], and survived in much the
same form to the beginning of the twentieth century [``If we could separate the individuals
whose knowledge, correlated and combined, is expressed in the ocean steamship or great
modern building, it is doubtful if their separate-knowledge would suce for more than the
construction and tools of the savage.'' (George, 1898, Book I, ch. VI)]. Schumpeter's
(1942) views on the importance of the size of the resource base for inventive activity are
well known.
2
Romer (1983, 1986) initiated this new approach to economic growth, but clear expositions of Schumpererian models can also be found in Aghion and Howitt (1992, 1998),
Grossman and Helpman (1991), Romer (1990), and Segerstrom, Anant and Dinopoulos
(1990). These papers constitute what we shall refer to as the early Schumpeterian models.
Readers not familiar with the microeconomic foundations of Schumpeterian growth
models can ®nd a concise overview in Dinopoulos (1994)
3
Romer (1990), along with many other authors, assumes that human capital (or skilled
labor) drives the knowledge creation process. The prediction of scale e€ects survives this
relabeling.
Scale e€ects in Schumpeterian models of economic growth
159
say, Hong Kong, and that population growth induces rising income growth.
Equation (2), relating input use to growth, predicts that increases in R&D
e€ort should be accompanied by increases in the rate of growth.
These predictions are robust to model details and to equilibrium choice.
An equation similar to (3) is the outcome of planning problems in all early
Schumpeterian models and, even though market imperfections cause
competitive equilibria to diverge from optimal plans, fully-speci®ed competitive equilibria also preserve the prediction that growth rates increase
with economy size4. Nonetheless, in the decade following Romer's ®rst
contributions, theorists have been curiously ambivalent about the importance that should be attached to the e€ects of scale. Romer (1986, 1994,
1996) views scale e€ects as a primary motivation for his theory. Grossman
and Helpman (1991), and Rivera-Batiz and Romer (1991) have treated scale
e€ects as an important outcome of their models, using them to predict a
link between international integration and growth. Other theorists have
simply ignored scale e€ects. Indeed, to Lucas (1993) they were simply ``a
nuisance implication that we want to dispose of.''
Two in¯uential papers by Jones (1995a, 1995b) have recently forced
economists to take the question of scale e€ects seriously. Jones' analysis
goes to the heart of what we call the ``scale e€ects problem''. First, he
pointed out that scale e€ects are central to the logic underlying existing
Schumpeterian models ± that individuals discover new products and processes ± and they cannot simply be ignored as nuisance parameters. Second,
he could ®nd no empirical support for the scale e€ects predicted so forcefully by these models. Third, he argued that a plausible solution to the scale
e€ects problem leads to a model that fundamentally alters the long-run
properties of Schumpeterian models. In particular, Jones' solution removes
the property that long-run growth rates are endogenous. In a short period
of time, Jones has stimulated several responses. Some of these have followed Jones and constructed models in which the long-run growth rate is
exogenous; others have developed alternative models in which there are no
scale e€ects but long-run growth remains endogenous.
The development of the new class of models that exhibit long-run
growth without scale e€ects requires a more precise terminology. We will
use the term ``Schumpeterian growth'' to refer to a particular type of economic growth that is generated through the introduction of new goods or
processes, as opposed to physical or human capital accumulation. We will
use the term ``endogenous'' [``exogenous''] to distinguish between Schumpeterian growth models in which long-run growth can [cannot] be a€ected
by permanent policy changes5. For instance, the Romer (1990) model is an
4
Moreover, similar scale e€ects arise in models of learning by doing. See Krugman (1987),
Lucas (1993), Parente (1994), Stokey (1988), and Young (1991, 1993).
5
In our view, the term ``Schumpeterian growth'' is preferable to the term ``R&D-based
growth'' because there are models of exogenous long-run growth where the introduction
of new products results from deterministic ®xed costs in an environment without knowledge spillovers. Since R&D investment is highly uncertain and is usually associated with
knowledge spillovers, the term R&D-based growth is somewhat narrower in scope when
applied to this type of models.
160
E. Dinopoulos, P. Thompson
endogenous Schumpeterian growth model with scale e€ects, whereas Jones'
(1995b) model could be classi®ed as an exogenous Schumpeterian growth
model without scale e€ects.
In this paper we assess the current state of a€airs on the scale e€ects
problem. Our analysis focuses on three related questions. First, what can
one infer from the empirical evidence on scale e€ects? Second, can
Schumpeterian growth models be constructed that remove the scale e€ects
property? Third, can empirical observations distinguish between competing
theoretical solutions to the scale e€ects problem?
2 The evidence
It is worth pointing out immediately that if scale e€ects are present in the
data they are not at all obvious. Figure 1, which plots average annual
per capita income growth rates for 1960±88 against initial population for
a broad cross-section of countries, reveals no relationship between size
and growth. But one should perhaps not expect the evidence to be so
obvious.
As we have noted, Schumpeterian models have three important observable consequences for long-run economic growth. First, population
growth should cause per capita income growth to accelerate. As population
has grown signi®cantly everywhere during the twentieth century, time series
of income per capita should exhibit accelerating growth. Second, larger
economies are predicted to grow faster, a prediction that should be evident
in international cross-sectional regressions. Third, changes in the level of
inputs used in knowledge creation should be accompanied by changes in the
rate of growth. This section brie¯y reviews the evidence that has been
brought to bear on each of these three predictions. An assessment of the
conclusions that can be drawn is given at the end of the section.
Fig. 1. Scale and Per Capita Income Growth, 1960±1988
Scale e€ects in Schumpeterian models of economic growth
161
Table 1. Romer's and Jones' tests for changes in per capita GDP growth
I
Romera
United Kingdom
France
Denmark
United States
Germany
Sweden
Italy
Australia
Norway
Japan
Canada
II
Jonesb
Date of ®rst
observation
p
Change in mean growth rate,
1900±29 vs. 1950±87
(annual percentage rate)
1700
1700
1818
1800
1850
1861
1861
1861
1865
1870
1870
.63*
.69**
.70**
.68**
.67*
.58
.76**
.64
.81**
.67**
.64
1.64*
1.47*
0.77
0.10c
2.24*
1.19
2.17**
1.83**
1.28
3.99**
0.62
Asterisks denote signi®cance at the (*) ten percent and (**) ®ve percent levels.
a
p is the probability that, for any two growth rates, the later one is the larger. Asterisks
denote the probability of observing a value of p at least as large as the observed value
under the null hypothesis that the true probability is 0.5. Romer's last decade is 1970±79.
b
Jones reports t-tests of the di€erence in mean growth rates for the two periods.
c
First period is 1880±1929.
Sources: Romer (1986, Table 3) and Jones (1995a, Table 2).
2.1 Time-series evidence on income growth
Romer's evidence for accelerating growth is replicated in column I of
Table 1. Comparing average growth rates across decades, Romer calculated
^, that for any two growth rates, the later one is
the sample probability, p
larger. Rank correlation tests were used to test the null hypothesis that the
true probability, p, is equal to 0.5. Of eleven countries tested, all point
estimates of p exceeded 0.5, and the null was rejected (at ten percent signi®cance) in eight of them6. Similar evidence is provided by Jones (1995a),
who compares the mean growth rates for 1900±1929 with post-war growth
over the period 1950±1987. The mean growth rate was higher in the later
period for all ®fteen countries examined, although t-tests revealed a signi®cant di€erence in only six cases7. While there are problems with these
data8 , few economists would argue against the claim that long-run average
6
Romer (1986) also claims additional support for accelerating growth from Reynold's
(1983) overview of growth in developing countries: ``growth rates appear to be increasing
not only as a function of calendar time but also as a function of the level of development.''
7
We report in column II of Table 1 Jones' results only for the eleven countries analyzed by
Romer.
8
The data are from Maddison (1982). De Long (1988) has shown that selection bias (of ex
post industrialized countries) in the data set generates inferences that do not necessarily
apply to broader samples.
162
E. Dinopoulos, P. Thompson
growth rates have risen over the course of the last few centuries. After all,
growth rates during the industrial revolution would today correspond to
periods of recession.9
Jones (1995a) has also suggested that one can test his model against
earlier Schumpeterian growth models with scale e€ects, by testing whether
growth rates have persistent components. He tested for unit roots in the
1900±87 growth rates of 14 OECD countries. In every case, tests of the null
hypothesis that growth rates have a unit root are strongly rejected10. Jones
points out that endogenous Schumpeterian growth models with scale e€ects
are consistent with this evidence only if the permanent changes in the underlying variables believed to drive long-run growth have just happened to
o€set each other. It would, he claims, be little short of miraculous if this had
occurred.
Interestingly enough, Kocherlakota and Yi (1997) have suggested that
the miracle described by Jones may well have happened. They construct and
test a simple model in which long-run growth may depend (positively) on
public capital and (negatively) on distortionary taxation. Using long timeseries data for the United States and the United Kingdom, they conclude
that public capital and taxation both have permanent e€ects on growth, but
that their e€ects have almost exactly o€set each other. Moreover, they point
out that one should expect their e€ects to be o€setting: As public capital
investment rises and stimulates growth, the government's budget constraint
requires that it raises taxes, with o€setting growth e€ects, to ®nance its
expenditures. Therefore, absent direct tests on the persistence of shocks to
policy, the stationarity of growth rates is at best suggestive of the absence of
scale e€ects, and does not necessarily imply that long-run growth is exogenous.
2.2 International cross-sectional evidence on economy size and growth
The simple scatterplot of growth and initial population in Figure 1 revealed
no relationship between income growth and population. In cross-sectional
9
Although his data are far less reliable, Kremer (1993) provides evidence that growth rates
have been rising since prehistoric times. He combines a scale-dependent production
function with a neo-Malthusian population growth function, and argues that accelerating
population growth over the last one million years or so is indicative of scale e€ects.
10
Jones' results are consistent with a large body of evidence on unit roots in per capita
GDP levels. Numerous studies since Nelson and Plosser's (1982) pioneering work have
concluded that the logarithm of per capita GDP is integrated of order one. If the logarithm of per capita income is integrated of order one, then it necessarily follows that per
capita income growth is integrated of order zero, as Jones showed. The case for stationarity in growth rates has been further strengthened by recent evidence (Ben-David and
Papell, 1995; Lumsdaine and Papell, 1997) that, once one allows for trend breaks, income
per capita levels are themselves stationary between breaks. The trend breaks are, of
course, permanent components. But they are most often associated with the Great Depression, World War II, and the ®rst oil-price shock, events that have typically been
outside the purview of growth models.
Scale e€ects in Schumpeterian models of economic growth
163
regressions, Backus, Kehoe and Kehoe (1992) similarly found no evidence
of scale e€ects: A regression of per capita GDP growth on the log of initial
GDP yielded an insigni®cant slope coecient, and an R2 of just 0.02, half
the value one would expect to obtain when the dependent variable is a
vector of random numbers11. Controlling for additional determinants of
growth did not reverse this result.
However, Backus et al. did ®nd evidence of scale e€ects when attention
is restricted to the manufacturing sector. A simple regression of growth in
manufacturing output per worker on the log of initial manufacturing output for 67 countries yielded a signi®cant and positive coecient, and initial
manufacturing scale explained one third of the sample variation in growth
rates. Their results imply that each doubling of the size of the manufacturing sector is on average associated with a 0.6 percent increase in the
subsequent growth rate. The scale coecient remained positive through a
variety of model speci®cations, and in most cases remained signi®cant.
Moreover, the result was robust to variations in sample coverage that exclude particularly high or low incomes per capita.
2.3 The scale of inputs and economic growth
We turn now to the scale of input use. A useful starting place is Jones'
(1995b) evidence on R&D e€ort in the United States. Jones presents evidence similar to that given in Figure 2. Between 1950 and 1988, the number
of scientists and engineers engaged in R&D grew by a factor of ®ve. Over
the same period, productivity growth was constant, or perhaps even declining. A similar story can be told for most OECD countries. Moreover,
the story does not change when one restricts attention to productivity
growth in the manufacturing sector.
Backus et al. (1992) investigated the e€ects of input scale in a crosssectional setting. They conducted cross-country regressions on equations of
the form
A_
LA
ˆ b0 ‡ b1 LA ‡ b2
;
…4†
A
L
which allows for discriminating tests between the scale and intensity of
inputs used in knowledge creation. Their regressions used the growth rates
of GDP per capita and manufacturing per worker, while LA was variously
measured by the number of students, the number of scientists, and R&D
expenditure. Figure 3 summarizes the relevant point estimates obtained
from these speci®cations. The ellipses centered on each pair of point estimates indicate the 95 percent joint con®dence interval. The conclusions that
11
In principle, we prefer some measure of population as a measure of scale. Initial GDP
confounds scale e€ects and variations in initial per capita income that are related to
transition dynamics in the neoclassical model caused by initial variations in the capitallabor ratio. As Mankiw, Romer and Weil (1992), among many others, have shown, initial
per capita GDP may be negatively related to subsequent growth rates, leading to a
downward bias in the scale coecient.
164
E. Dinopoulos, P. Thompson
Fig. 2. R&D E€ort and GDP Growth in the US, 1950±1990
emerge seem clear. There is no support for scale e€ects in GDP, while
intensity e€ects matter only when inputs are measured by the number of
students. In contrast, scale e€ects in the manufacturing sector are evident
for all three input measures.
Fig. 3. Scale and intensity E€ects on Growth, 1970±1985
Scale e€ects in Schumpeterian models of economic growth
165
2.4 Firm growth and scale e€ects
The problem of scale e€ects arises also in models of industry evolution.
After all, the Schumpeterian growth models are little more than industry
models to which resource constraints and a large dose of symmetry have
been appended. The relationship between ®rm growth and size has been
investigated extensively by industrial organization researchers in connection
to Gibrat's law, which implies that the growth rate of ®rms is independent
of their size (i.e., there are no ®rm-level scale e€ects)12. The results reported
in this literature have been mixed. Several studies have rejected Gibrat's law
by reporting a negative statistical correlation between size and ®rm growth
(Evans, 1987a,b; Hall, 1987). These studies support the conjecture that
Gibrat's law holds for large ®rms. For instance, Hall (1987, p583) states
that ``Gibrat's Law is weakly rejected for the smaller ®rms in my sample
and accepted for the larger ®rms'', and Evans (1987a, p567) states that
``®rm growth decreases at a diminishing rate with ®rm size even after
controlling for the exit of slow-growing ®rms from the sample. Gibrat's
Law therefore fails although the severity of the failure decreases with ®rm
size''. Despite their mixed results, however, these studies clearly point in the
direction of no scale e€ects at the level of the ®rm.
In contrast, several recent studies have investigated the determinants
and e€ects of industry shake outs, and have suggested that scale e€ects may
underlie factors that generate shake outs.13 This literature has postulated
scale e€ects in the form of ``learning'' about a technological breakthrough
(Jovanovic and MacDonald, 1994) or R&D scale economies associated
with process innovations within the context of multiproduct ®rms (Klepper,
1996). The relatively short duration of shake outs points out to the possibility of transitional scale e€ects that might be important in understanding
historical episodes. However, it would not be risky to conclude that, if
anything, ®rm-level evidence does not support the assumption of long-run
growth scale e€ects.
2.5 What can we conclude from the empirical evidence?
The empirical evidence on scale e€ects is, to say the least, unclear. First,
growth rates have accelerated over the course of a century or more, but they
have failed to accelerate in the face of increasing R&D e€ort during the last
forty years. Second, there is no evidence for scale e€ects across countries in
GDP, but there is in manufacturing output. Third, scale e€ects do not
appear in ®rm-level data, yet they may lie behind the phenomenon of industry shake outs. Not surprisingly, the authors of these various studies
12
Sutton (1997) provides an excellent recent survey of the literature and evidence on
Gibrat's hypothesis.
13
Shake outs are associated with a relatively short period of time of about 20±25 years,
when the number of producers tends ®rst to rise and then to fall dramatically. Klepper and
Simons (1997) have documented the existence of sharp shakeouts for the U.S. automobiles, tires, televisions and penicillin industries.
166
E. Dinopoulos, P. Thompson
have reached con¯icting conclusions. Romer (1986) claims that increasing
growth rates constitute prima facie evidence for scale e€ects and, in Romer
(1996), he argues that
scale e€ects should no longer be treated the way a growth accountant such
as Dennison did, as a kind of afterthought that had something to do with
plant size. They should be treated the way Adam Smith did, as one of the
fundamental aspects of our economic world.
Jones (1995a,b), in contrast, dismisses the evidence on accelerating growth,
and points instead to the increases in R&D e€ort:
one might worry about the relevant unit of observation (the world vs. a
single country) or the lags associated with R&D, but it should be clear . . .
that these concerns cannot overturn the rejection of scale e€ects. The assumption embedded in the R&D equation that the growth rate of the
economy is proportional to the level of resources devoted to R&D is obviously false.
Finally, Backus et al. take something of a middle ground:
[The] theories do better in general when confronting the data for manufacturing than they do for aggregate output. At least in the case of R&D
expenditures this should come as no surprise: Most R&D expenditures that
are sector speci®c go to manufacturing.
On the face of it, Jones' time-series evidence against scale e€ects is compelling. However, we think that Jones has overstated to some degree the
import of the evidence. First, changes in reporting practices and requirements have increased the fraction of R&D that is now formally documented14. Second, a simple enumeration of scientists and engineers
overstates the real increase in R&D intensity15. Third, there is evidence
suggesting that diculties in measuring quality growth and service-sector
productivity have become more pronounced over time, and consequently
that growth rates are understated to a greater extent now than they were,
say, thirty years ago16. Despite the importance of these issues, we none14
For example, Hall (1993) has noted that the introduction of tax credits for incremental
R&D investment in the United States may have provided new incentives to report R&D
expenditures.
15
As R&D uses multiple inputs, the number of R&D scientists and engineers is not a
sucient statistic for R&D e€ort. Between 1970 and 1988, private sector expenditure on
R&D in the United States rose from 1.6 percent to 1.8 percent of GDP, a much more
moderate increase. Aghion and Howitt (1998, ch. 12) point out that total expenditure on
R&D in the United States has been between 2.2 percent and 2.9 percent of GDP in every
year since 1957, with no obvious tendency to rise over time.
16
Following an exhaustive study of measurement errors in many service sectors, Baily and
Gordon (1988) conclude that as much as twenty percent of the post-1973 productivity
slowdown can be attributed to unobserved productivity growth. Thompson and Waldo
(1997) have estimated the consumer's Euler equation obtained from a structural model of
process and product innovation in the presence of measurement error, concluding that
®fteen percent of the slowdown can be attributed to unobserved quality growth. For
related evidence see Brooke (1992), Griliches (1994), and Hausman (1997).
Scale e€ects in Schumpeterian models of economic growth
167
theless doubt that attempts to deal with them would overturn Jones' empirical puzzle.
Accelerating growth rates provide only indirect evidence of scale e€ects,
and the attention we should pay to them depends on whether there exist
alternative and more plausible explanations17. There are several competing
explanations. First, much of the increase in post-war growth rates can be
attributed to standard neoclassical transition dynamics18. Five of the six
signi®cant increases in growth rates that Jones found were for countries
whose stocks of physical capital had su€ered signi®cant damage during
World War II, and whose rapid post-war growth rates may re¯ect only a
return to steady state. Second, the knowledge production function has itself
also undergone technical change. It seems self-evident that if technological
change a€ects the ®nal-good production process it can also a€ect the
knowledge creation process. Accelerating growth may consequently be
driven by innovations that increase c in equation (2), thereby raising the
productivity of resources allocated to knowledge creation. Third, changes in
the institutional environment, incidental to the formal model, may have
also played a critical role in promoting more rapid technological change.
The manufacturing scale e€ects identi®ed by Backus et al. are dramatic.
Should we believe them? Yet again, we think there are good reasons to be
cautious about these results. First, scale may be no more than a proxy for
omitted variables. Omitted factors that limit growth may be the same as, or
at least correlated with, omitted factors that limit the size of the manufacturing sector or the scale of inputs into knowledge creation. Backus et al.
do report some supplementary regressions with additional explanatory
variables drawn from prior empirical literature. The point estimates of the
scale coecient are reduced, although not signi®cantly so19.
Second, if growth rates are autoregressive, it is easy to generate spurious
scale e€ects from cross-sectional regressions. We conducted a regression of
1970±1985 growth rates of per capita GDP against their counterparts for
the period 1960±70. A sample of 119 countries yielded a slope coecient of
0.27, indicating that countries which grew faster than average during 1970±
85 (the period covered by Backus et al.) were also likely to have grown
faster than average during 1960±70. It immediately follows that, even if
growth rates in the ®rst period were orthogonal to initial size (i.e. there are
no scale e€ects), growth rates in the second period will be positively correlated with 1970 scale.
While we think the empirical question of scale e€ects requires further
work, we have made some tentative conclusions. In our view, the evidence
17
Kremer (1993), for example, acknowledges that his population data are also consistent
with numerous models lacking scale e€ects.
18
Incorporating physical capital into Schumpeterian models is straighforward, and has
the e€ect of combining the transitional dynamics of the Solow model with the endogenous
growth properties of the R&D-based growth model.
19
The supplementary regressions are reported for the economy size regressions, but not
for the input regressions. The set of supplementary regressors used is also rather small,
although Levine and Renelt (1992) and Sala-i-Martin (1997) report that at least 60
variables have been found to be signi®cant in one growth regression or another.
168
E. Dinopoulos, P. Thompson
does not provide much support for long-run aggregate scale e€ects. We
have no doubt that economies can be small enough to sti¯e innovation20,
but it does not seem plausible that scale is an important determinant of
growth in modern economies. As we have argued, evidence to the contrary
must be viewed with considerable caution. In short, we are more or less in
accord with Romer's (1996) recent position that scale e€ects may be important for understanding episodes in economic history, but we also think
that one should not rely on them to explain di€erences in growth rates
among modern nations.
3 Removing scale e€ects
Despite considerable ambiguity in the evidence, theorists have begun to
construct Schumpeterian growth models that exclude scale e€ects. This
section reviews the theoretical progress that has been made on the scale
e€ects problem. No attempt is made to do justice to the rich detail contained in the models reviewed below. We continue to let equation (1) describe production of the ®nal good, and we restrict attention to steady-state
_
growth paths in which A…t†=A,
LY(t)/L(t) and LA(t)/L(t) are constant. Along
the steady-state path, per capita income grows at the same rate as technology, and we can focus our attention on the latter variable.
3.1 Technology in Schumpeterian growth models
Schumpeterian models provide substance to the technology parameter,
A(t), in one of two ways. In the ®rst (e.g. Grossman and Helpman, 1991,
ch. 3; Romer, 1990), A(t) is related to the number, v(t), of horizontally
di€erentiated varieties of goods that have been developed by time t. In these
variety-expansion models the level of technology can be written as
A…t† ˆ v…t†1=…rÿ1†
…5†
where r is the elasticity of substitution between varieties. Variety matters in
these models either because total factor productivity increases when ®rms
use a greater variety of intermediate goods, or because consumers have a
taste for variety in consumption. In either case, the number of varieties is
assumed to depend on e€ective R&D investment, I(t), according to
v_ …t† ˆ I…t†v…t† :
…6†
The second class of models (e.g. Segerstrom, Anant and Dinopoulos, 1990;
Grossman and Helpman, 1991, ch. 4; Aghion and Howitt, 1992) focuses on
quality improvements, and interpret technology as
A…t† ˆ kq…t†
…7†
where k > 1 is equal to one plus the quality increment of a good relative
to its immediate predecessor in an industry, and q(t) is the number of
20
The isolated communities of feudal Europe provide an obvious example.
Scale e€ects in Schumpeterian models of economic growth
169
innovations that have occurred since time zero21. The arrival of innovations
in an industry is random and governed by a homogeneous Poisson process
whose intensity (i.e. the expected number of innovations per unit of time)
equals the e€ective R&D investment, I(t). Given a large number of identical
independent industries, the law of large numbers implies that aggregate
growth is deterministic and satis®es
_ ˆ I…t† :
q…t†
…8†
For both classes of models, one obtains
_
A…t†
ˆ cI…t† ;
A…t†
…9†
where c ˆ 1=…r ÿ 1† in the variety-expansion class of models, and c ˆ ln k
in the quality-improvement models.
Both classes of models incorporate important knowledge spillovers that
ensure that growth persists in the long run. In the variety-expansion setting,
the proportional increase in income secured by an additional variety falls as
the number of existing varieties rises [see eq. (5)]; to o€set this, the cost of
developing a new variety, in terms of e€ective R&D e€ort, must fall as the
number of existing varieties increases [eq. (6)]. In quality-improvement
models each innovation secures a constant proportional increase in the level
of quality [eq. (7)], so that the expected cost of an innovation can be
constant [eq. (8)].
The scale e€ects property in both classes of models arises from the
functional form that relates e€ective R&D to economy resources. A generalized form of this relationship can be written as
I…t† ˆ
LA …t†
X …t†
…10†
where LA(t) is the amount of labor devoted to R&D, and X(t) captures the
degree of R&D diculty in the sense that higher values of X require more
labor to achieve the same level of I22. Combining (9) and (10), we have:
_
A…t†
LA …t† L…t†
ˆc
;
…11†
A…t†
L…t† X …t†
and, as the fraction of the labor force devoted to R&D, LA(t)/L(t), must be
constant in the steady state, the removal of scale e€ects requires that the
steady-state growth rate of X(t) be exactly the same as the population
growth rate. In a cross-sectional dimension, scale e€ects are absent only
when X is proportional to L. The challenge for theorists has been to construct plausible models that satisfy this stringent steady-state condition. We
review two distinct approaches that have o€ered possible solutions.
21
Productivity growth in a single-good economy can be modeled in an analogous fashion.
In this case, k measures one plus the proportional increase in total factor productivity
obtained from an innovation.
22
In multi-factor models LA can be replaced with any constant returns to scale production
function that generates R&G services without a€ecting the essence of the scale of e€ects
problem.
170
E. Dinopoulos, P. Thompson
3.2 Exogenous Schumpeterian growth models
Jones (1995b), and Segerstrom (1998) have removed scale e€ects by assuming that R&D becomes more dicult over time. In Scherer's (1965)
terminology technological opportunities are assumed to diminish systematically over time. Jones solves a variety expansion model identical to
Romer (1990) except for an assumption that X(t)=A(t)1)/. Segerstrom
analyzes a quality improvement model that incorporates the assumption
that X_ …t†=X …t† ˆ …1 ÿ /†I…t†. Except for inconsequential di€erences in X(0)
and A(0), the two assumptions are equivalent and yield a knowledge production function of the form
_
A…t†
LA …t†
ˆc
;
…12†
A…t†
A…t†…1ÿ/†
which states that the proportional advances in the level of technology
(whether measured by the number of varieties or the level of quality) obtained from a given amount of R&D e€ort declines monotonically as
technology advances.
Jones and Segerstrom justify this assumption with an identical assertion:
R&D becomes more dicult over time because ``the most obvious ideas are
discovered ®rst.'' But, in an innovative paper, Kortum (1997) has provided
some interesting theoretical microfoundations for the assumption of increasing R&D diculty, without requiring that more obvious ideas be
discovered ®rst. Kortum models research as a search process that consists
of drawing random eciency levels from a probability distribution that
represents technological opportunities. Given a current productivity level
A(t), the probability that a draw is more productive than the current state of
the art is given by S…t†h …1 ÿR F …A…t†††, where h > 0. F(A(t)) is the stationary
t
search distribution, S…t† ˆ ÿ1 LA …s†ds is the research stock, and S(t)h is the
spillover function. Kortum's central result is that, irrespective of the value
of h and the particular distribution F, the productivity of R&D declines
over time23. In the special case that F is a Pareto distribution, Kortum
derives a simple expression for productivity growth
_
LA …t†
A…t†
ˆ j…1 ‡ h† R t
;
…13†
A…t†
ÿ1 LA …s† ds
where j is the parameter of the Pareto distribution denoting the mean
magnitude of quality improvements (i.e., the size of innovations).
Equations (12) and (13) are equivalent and each can be derived by differentiating A(t)=S(t)b with respect to time. Equation (12) corresponds to
b ˆ 1=…1 ÿ /†, while equation (13) is obtained by setting b ˆ j…1 ‡ h†. Thus
Kortum's model provides a novel interpretation to the diminishing-opportunities parameter / in the exogenous Schumpeterian growth models
23
While this result at ®rst seems quite remarkable, the fact that it does not depend on the
value of h should not be taken to imply that knowledge spillovers are unrestricted. As
Kortum readily acknowledges, the assumed form of the search distribution imposes a limit
on the potency of spillovers even for large values of h.
Scale e€ects in Schumpeterian models of economic growth
171
developed by Jones and Segerstrom. Because …1 ÿ /† ˆ 1=‰j…1 ‡ h†Š, diminishing opportunities are stronger if R&D spillovers are weak (i.e., h is
small) or if the stationary search Pareto distribution has a low mean (i.e., j
is small).
_
Let g…t† ˆ A…t†=A…t†
denote the growth rate of technology, and let gL
denote the constant growth rate of population. Di€erentiating (12), using
the fact that LA(t) grows at the rate gL in steady state, yields:
_
g…t†
ˆ gL ‡ …/ ÿ 1†g…t† :
g…t†
…14†
If g(t) is greater than [less than] gL =…1 ÿ /†, the growth rate is falling [rising]. Thus, the steady-state path is stable, satisfying g ˆ gL =…1 ÿ /), and the
level of technology is related to R&D e€ort by
1
c…1 ÿ /†LA …t† 1ÿ/
:
…15†
A…t† ˆ
gL
That is, the growth rate does not depend on any measure of scale. Increases
in the steady-state level of R&D raise technology and income per capita at
any point in time, but they do not raise the growth rate. Consequently,
policies that increase R&D have level e€ects but not growth e€ects.
3.3 Endogenous Schumpeterian growth models without scale e€ects
A second approach to the scale e€ects problem, suggested by Young
(1998) and subsequently analyzed independently by Aghion and Howitt
(1998, ch. 12), Dinopoulos and Thompson (1998), Howitt (1999), Peretto
(1998), and Peretto and Smulders (1998), introduces the concept of localized intertemporal R&D spillovers to generate long-run endogenous
Schumpeterian growth without scale e€ects. These studies remove the
scale e€ects property by essentially the same mechanism as the one employed by exogenous Schumpeterian growth models, namely by assuming
that aggregate R&D is becoming more dicult over time as it is spread
over more ®rms. However, their novel contribution is that they append
other growth channels that maintain the endogeneity of long-run
Schumpeterian growth.
We will illustrate the concept of localized intertemporal knowledge with
a taste structure that, following Dinopoulos and Thompson (1998), incorporates horizontal and vertical product di€erentiation. We will assume
knowledge spillovers in the vertical (quality) dimension but none in the
horizontal (variety) dimension. To this taste structure, we will append the
simplest variety creation process introduced in Aghion and Howitt (1998),
namely that variety creation is the result of costless imitation.
Consider a closed economy consisting of a continuum of in®nitely-lived
identical households. The size of each dynastic household grows at an exogenous constant rate, gL>0, which equals the rate of population growth.
The representative household maximizes the following intertemporal utility
function:
172
E. Dinopoulos, P. Thompson
Z1
Uˆ
e…gL ÿq†t ln D…t† dt :
…16†
0
where q > 0 is the constant subjective discount rate and D(t) is a subutility
de®ned by
2
3r=…rÿ1†
Zv…t†
6
7
:
…17†
D…t† ˆ 4 a…i; t†1=r d…i; t†…rÿ1†=r di5
0
D(t) is the quality-augmented Dixit-Stiglitz consumption index; v(t) denotes
the number of varieties that have been developed at time t; a(i,t) and d(i,t)
are the level of quality and per capita consumption of variety i at time t, and
r > 1 is both the constant elasticity of substitution between varieties and
the elasticity of demand for any single variety v(i)
Let p(i,t) denote the price of good i, and let E(t) denote per capita
consumption expenditure. At each instant of time, the
R v…t†consumer maximizes
D(t) subject to the budget constraint E…t† ˆ 0 p…i; t† d…i; t†di. This
standard maximization problem yields the instantaneous per capita demand
d…i; t† ˆ
a…i; t† p…i; t†ÿr E…t†
:
v…t†
R
1ÿr
a…i; t†p…i; t†
…18†
0
The instantaneous aggregate demand for each variety is c(i,t)=d(i,t)L(t),
where L(t) is the level of population at time t. Symmetry of tastes across
varieties implies that ®rms charge an identical and constant price,
p…i; t† ˆ r=…r ÿ 1†, for every product, and (18) can be written as
r ÿ 1 a…i; t† E…t†
;
…19†
d…i; t† ˆ
r
A…t† v…t†
R
v…t†
where A…t† ˆ 0 a…i; t† di =v…t† is the economy's average quality level at
time t. Per capita consumption of good i depends on its quality level relative
to the economy's average and on expenditure per variety. Since all ®rms
charge the same price, varieties with higher quality levels command higher
market shares.
The evolution of per capita expenditure, E(t), is governed by the familiar
di€erential equation
_
E…t†
ˆ r…t† ÿ q ;
E…t†
…20†
where r(t) is the instantaneous market interest rate.
Assume that labor is the only factor of production, and that it can be
allocated between two activities: Manufacturing of ®nal products, where
one unit of labor produces one unit of output, and quality-enhancing R&D.
The level of quality associated with variety i at time t is given by
a…i; t† ˆ kq…i;t† , where k > 1 denotes the quality improvement between two
Scale e€ects in Schumpeterian models of economic growth
173
adjacent innovations within variety i, and q(i,t) is the number of quality
innovations that variety i has experienced since time zero. The arrival of
innovations is governed by a Poisson process with intensity I(i,t). Assuming
symmetry and applying the law of large numbers, it can be shown that the
evolution of average quality A…t† ˆ kq…t† is given by
_
A…t†
ˆ I…t† ln k ;
A…t†
…21†
where I(t) is the R&D investment per variety. It is immediately obvious
from this formulation that there are knowledge spillovers related to quality
improvements, that any change in I(t) alters long-run growth, and that the
removal of scale e€ects requires a steady-state equilibrium with constant
I(t).
Following Aghion and Howitt (1998, Chapter 12) we assume that the
creation of new varieties is the result of costless imitation and, for simplicity, we suppose that the quality level associated with a variety js created
at time s equals the average quality at time t [i.e., a…js ; t† ˆ A…s†]. Each
individual has the same exogenous probability of imitating and the arrival
of imitation is governed by an exogenously given intensity h. Therefore, the
aggregate rate of variety accumulation is simply v_ …t† ˆ hL…t†, where
_ ˆ gL L…t†. The ratio k(t)=L(t)/v(t) is governed by the equation
L…t†
_k…t† ˆ gL k…t† ÿ hk…t†2 , and so in steady state the number of varieties is
proportional to the level of population:
v…t† ˆ
h
L…t† :
gL
…22†
In the long run, the number of varieties grows at the rate of population
growth, gL, and, in the absence of international technological linkages,
larger economies will produce a larger number of varieties.
The model is closed with the full employment of labor condition, which
can be written as
Zv…t†
c…i; t†
L…t† ˆ v…t†I ‡
0
I …r ÿ 1†E
;
…23†
ˆ L…t† ‡
k
r
where
R v…t† v(t)I is the amount of labor devoted to quality-enhancing R&D and
c…i; t†di is the amount of labor devoted to manufacturing.
0
The model exhibits a steady-state equilibrium in which the intensity of
quality-enhancing R&D per variety and per capita consumption expenditure are constant over time. The scale of the economy, measured by L(t),
does not a€ect the full employment condition in the long-run, and equation
(18) implies that r…t† ˆ q Substituting (19) into (17) and performing the
integration yields
D…t† ˆ
…r ÿ 1†
EA…t†1=…rÿ1† v…t†1=…rÿ1† ;
r
…24†
174
E. Dinopoulos, P. Thompson
di€erentiation of which provides an expression for the endogenous rate of
Schumpeterian growth:
_
_
D…t†
1 A…t†
1 v_ …t†
ˆ
‡
D…t† …r ÿ 1† A…t† …r ÿ 1† v…t†
1
‰…ln k†1A ‡ gL Š :
ˆ
…r ÿ 1†
…25†
Because the number of varieties is proportional to the scale of the economy,
(24) implies that the level of instantaneous utility depends positively on
scale. However, the rate of growth of per capita instantaneous utility is
unrelated to scale, as (25) demonstrates. The long-run growth rate of the
economy depends positively on the rate of population growth as in models
of exogenous Schumpeterian growth [compare equations (25) and (14)], but
it also depends positively on the magnitude of quality innovations, and on
the intensity of quality-enhancing R&D. Any policy that a€ects the magnitude or intensity of innovations has long-run growth e€ects. In the absence of quality growth (i.e., when k ˆ 1), long-run growth becomes
exogenous and depends only on the rate of population growth. In general,
however, the absence of population growth does not reduce the long-run
growth rate to zero.
The role of variety-creation in removing the scale e€ects property is also
apparent from equations (22) and (23). If the evolution of varieties is
characterized by intertemporal knowledge spillovers [i.e. v_ …t†=v…t† ˆ hL…t†],
then the term gL in (25) should be replaced by hL…t†, and the long-run rate
of growth depends on scale and eventually becomes unbounded as L(t)
increases exponentially over time. In addition, if the long-run number of
varieties v(t) remains constant, the full employment condition (23) implies
that I and long-run growth increase exponentially over time. Therefore the
endogenous Schumpeterian growth models without scale e€ects o€er a
generalization both of earlier Schumpeterian growth models with scale effects and of exogenous Schumpeterian growth models without scale e€ects.
One way to interpret the channel through which localized knowledge
spillovers remove scale e€ects is to rewrite the aggregate (economy wide)
R&D-based knowledge production function, (21), as
_
A…t†
…ln k†v…t†I k…ln k†LA …t†
LA …t†
ˆ …ln k†I ˆ
ˆ
ˆc
;
A…t†
v…t†
L…t†
X …t†
…26†
where c ˆ …ln k† is a positive constant, and X(t)=v(t)=L(t)/k can be interpreted as the degree of aggregate R&D diculty. In other words (26) is
identical to (11), where the degree of R&D diculty at time t, equal to the
number of varieties, grows at the rate gL 24.
Finally, it is worth mentioning that, in the presence of localized intertemporal knowledge spillovers the aggregate knowledge production function can be written as a function of the share of labor devoted to R&D,
24
This speci®cation of the aggregate knowledge production function has been used in
several recent studies (see Dinopoulos and Segerstrom 1998; Sener 1997; among others)
under the name of the permanent e€ects on growth (PEG) model.
Scale e€ects in Schumpeterian models of economic growth
175
LA(t)/L(t), [see (26)] contrary to theoretical objections raised by Jones
(1995b) regarding the microfoundations for this speci®cation25.
Costless imitation might be the simplest assumption that generates a
linear equation relating the number of varieties to the level of population,
but it is not the only one. Young (1998) and Dinopoulos and Thompson
(1998) assume that the establishment of a new variety entails a ®xed cost,
and derive an equation that is equivalent to (22). Peretto (1998) and Peretto
and Smulders (1998) employ a similar assumption in a model of growth
through variety accumulation where, instead of quality improvements,
®rms produce a ¯ow of varieties and the establishment of a new (multiproduct) ®rm requires a ®xed cost. Free entry in the process of variety
accumulation results in an equation similar to (22). Howitt (1999) obtains
the same result by assuming that the variety-creation function depends on
labor and ®nal output through a constant returns to scale production
function, and that there is free entry in varieties.
4 On the empirics of Schumpeterian growth models without scale e€ects
All the models reviewed in Section 3 are consistent with the post-war time
series evidence presented by Jones (1995a,b) and the cross-country evidence
reported by Backus et al. (1992): Resources devoted to R&D increase exponentially, whereas the rate of long-run growth is constant over time.
Initially, several studies (e.g., Jones, 1995a,b; Feenstra et al., 1997; Eaton
and Kortum, 1999a) focused on testing the di€erences between endogenous
Schumpeterian growth models with scale e€ects and exogenous Schumpeterian growth models without scale e€ects. These studies lend empirical
support to the latter class of models and have been interpreted, somewhat
prematurely, as providing evidence for the exogeneity of long-run Schumpeterian growth.
The recent development of endogenous Schumpeterian growth models
without scale e€ects has now provided a more demanding alternative
against which to test exogenous Schumpeterian growth models. While much
remains to be done in this area (indeed, this section will read much like a list
of ideas for future research), we summarize here some observations on the
empirics underlying di€erences between the models.
4.1 Patent statistics
One testable proposition that di€erentiates models incorporating variety
creation from those that are based on only quality improvements is the
25
Jones (1995b) states: ``First, equation (26) is inconsistent with the microfoundations of
the R&D models developed by Romer/Grossman-Helpman/Aghion-Howitt. These microfoundations imply that new ideas are discovered by individuals so that the number of
innovations is inherently tied to the number of persons engaged in R&D. A speci®cation
devoid of scale such as (26) has the counterfactual implication that an economy with only
one unit of labor can produce as many innovations (or at least can generate equivalent
TPF growth) as an economy with 1 million units of labor''.
176
E. Dinopoulos, P. Thompson
long-run prediction on the ¯ow of new products. The former class of
models (e.g., Jones, 1995b; Dinopoulos and Thompson, 1998; and Young,
1998) predict that the ¯ow of innovations increases at the rate of population
growth, and therefore that the ¯ow of innovations per R&D worker is
constant over time. In contrast, in models of exogenous Schumpeterian
growth that abstract from variety accumulation (e.g., Kortum, 1997; and
Segerstrom, 1998), the ¯ow of innovations is constant and the ¯ow of
innovations per researcher declines over time. Assuming that a constant
fraction of all innovations are patented, patent statistics can be used, in
principle, as an approximate measure of the ¯ow of innovations to distinguish between these two classes of models26.
Consider ®rst the standard time-series evidence on patents. Since the
turn of the century, patents granted annually by the U.S. patent oce to
U.S. residents have ¯uctuated between 25,000 and 60,000, with no consistent trend (see series B and C in Fig. 4). This evidence clearly lends prima
facie support to the models of Kortum (1997) and Segerstrom (1998).
However, the evidence is muddied by several features of the data. First, the
aggregate data confound two interesting phenomena: The constant rate of
patent awards comprises the sum of a strong positive trend in patents
granted to U.S. corporations (which rose from less than 5,000 at the beginning of the century to almost 40,000 by 1970; see series A in Fig. 4) and a
countervailing decline in patents granted to individuals (which fell from
30,000 early in the century to less than 14,000 by 1970; see series B in Fig.
4). As the models we have reviewed are clearly intended to represent ®rm
behavior, there is some doubt about which time series we should consider.
What did it mean to be an individual inventor in, say, 1920? Is the shift
from individual to corporate patenting a re¯ection of changes in who is
inventing, or merely changes in who appropriates the rents from invention?
And are there di€erences in the propensity of individuals and corporations
to patent inventions? We do not know the answers to these questions, yet
obtaining them is clearly a precursor to carrying out reliable tests between
competing models.
A second complication in the data is that the behaviors of patent grants
and patent applications have been somewhat di€erent. In particular, applications for U.S. patents by U.S. residents have doubled since 1985,
reaching over 120,000 in 1995 (Kortum and Lerner, 1997). The surge in
patents since 1985 has occurred in a period of constant and even declining
R&D intensity which suggests a rising patents per researcher ratio. One
possible explanation is that the recent surge in patenting re¯ects institutional changes, possibly the establishment of the Court of Appeals of the
Federal Circuit in 1982. However, Kortum and Lerner (1997), simulating a
version of Kortum's (1997) model, cast doubt on the institutional explanation and conclude instead that the surge in patent applications re¯ects an
26
However, it must be emphasized that patent statistics should be used with caution for
inferences regarding the empirical performance of growth models that do not explicitly
model the microfoundations of endogenous patenting behavior of ®rms, and that abstract
from other mechanisms that provide protection to intellectual property rights. (see Griliches, 1990).
Scale e€ects in Schumpeterian models of economic growth
177
Fig. 4. U.S. Patent Grants and Applications, 1990±1993
increase in U.S. innovation27. The surge in applications has not yet been
matched by an increase in patent grants, however, and it remains to be seen
whether the incremental applications actually do constitute inventions.
A third complication is that patent data do not readily capture changes
in the value of patents. Schankerman and Pakes (1986) have attempted to
estimate the value of patents in three European countries from patent renewal rates, concluding that patent values had risen over time to o€set the
decline in patents per researcher28. Thompson (1996) produces some contradictory evidence. Using stock market data from U.S. publicly traded
corporations, he concluded that the stock market valuation of an innovation declined between 1973 and 1991, and that this decline was due in part
to a rise in technological opportunity. Finally, Kortum (1997) points to a
rapid increase in the fraction of domestic innovations which are also patented internationally. This trend to broader patent protection, he contends,
27
Note, however, that Kortum's model predicts not only a temporary surge in patenting
caused by a permanent increase in research productivity, but also a temporary increase in
R&D intensity. The latter prediction is not consistent with the U.S. evidence as Kortum
and Lerner (1997) acknowledge.
28
We are somewhat skeptical of the reliability of evidence from patent renewal data.
Patent renewal costs are extremely low and, at least for corporations, it may be cheaper to
renew a patent than to calculate the bene®ts of renewal. Currently, the basic ®ling fee in
the U.S. is $790 ($395 for small entities). Patent attorney oces advertising on the internet
are o€ering patent ®ling services for as low as $1,600, including all application fees.
178
E. Dinopoulos, P. Thompson
is consistent with the claim that patents have become more valuable over
time. It is fair to conclude that the time series evidence on the ¯ow of
innovations is mixed and does not yet provide decisive support to any class
of Schumpeterian growth models without scale e€ects.
4.2 Scale e€ects in income levels
All the models reviewed in section 3 predict scale e€ects in income levels.
For instance, in models with variety accumulation the level of instantaneous utility is an increasing function of the number of varieties, and the
level of varieties is proportional to the level of population [see eqs. (15) and
(23)]. Thus, these models predict that larger economies have higher income
per capita in the long run. The models do not predict that (abstracting from
international linkages) China will grow more rapidly that Hong Kong, but
they do predict that it will be enormously wealthier.
However, one needs to distinguish between models where varieties take
the form of intermediate goods (e.g., Aghion and Howitt, 1998; Howitt,
1999; Jones, 1995b; Peretto, 1996; Young, 1998) and models where varieties
are ®nal consumption goods (e.g., Dinopoulos and Thompson, 1998). The
former predict that measured GDP per capita exhibits scale e€ects, since the
accumulation of varieties is captured by increases in total factor productivity, which are re¯ected in ®nal output. The latter predict that the level of
instantaneous utility exhibits scale e€ects and that these level e€ects refer to
di€erences in the cost of living. However, it is now well-known that GDP is
not a measure of the cost of living29. In fact, GDP de¯ators do not even
attempt to capture the welfare gains from increased variety of ®nal consumption goods. In practice, a small number of products within a narrowly
de®ned category (e.g., 30 inch color televisions) are sampled to obtain an
average price, and the consumer price index is not enhanced by increased
number of varieties within each category. That is, GDP statistics treat all
goods within a product category as perfect substitutes. As Hausman (1997)
has shown, failure to account for new ready-to-eat cereal brands has resulted
in an overestimation of the consumer price index for cereals by about 20 to
25 percent. The prediction of scale level (as opposed to growth) e€ects is an
issue that awaits further empirical investigation regarding the relevance of
both endogenous and exogenous long-run Schumpeterian growth models.
4.3 Closed-economy models in an open world
It might also be useful to remind the reader that all these growth models
have abstracted from the role international market linkages and the international di€usion of knowledge issues30. These linkages create diculties in
29
Armknecht et al. (1997) have an excellent discussion on the limitations of the U.S. CPI
in accounting for new varieties.
30
Notable exceptions are the theoretical models of Aghion and Howitt (1998, ch. 12),
Dinopoulos and Segerstrom (1998), Eaton and Kortum (1999a, b), and Howitt (1997),
who develop multi-country models of Schumpeterian growth without scale e€ects.
Scale e€ects in Schumpeterian models of economic growth
179
interpreting and using international cross-sectional data to test some of the
implications of growth models without scale e€ects. Consider, for example,
Kortum's (1997) model and assume that there is perfect international diffusion of knowledge which renders the P
location of research irrelevant. Then
the world ¯ow of research is LA …t† ˆ niˆ1 LiA (t), where LiA …t† is research
employment in country i. Equation (13) then determines the world growth
rate, which will be common for all countries. Although all countries share
the same level of productivity, the inventive output of each country (and
hence the ¯ow of innovations per unit of time) becomes proportional to its
research e€ort, but there are no scale e€ects on growth because all countries
grow at the same rate. Eaton and Kortum (1998b) analyze the more realistic
case of imperfect international technology di€usion which allows for differences in productivity levels across countries, but equal growth still prevails in steady state31.
Notwithstanding their possible sensitivity to open-economy extensions,
the models do o€er a novel insight concerning the determinants of crosscountry income per capita di€erences through level scale e€ects. For example, consider the thought experiment of opening trade between two
identical economies with the structure described in Section 3.3. There will
be intra-industry trade in varieties which will generate income per capita
level e€ects, but which will not have long-run growth e€ects32. Since intraindustry trade is more prevalent among advanced countries with similar
factor endowments, the level e€ects of trade on income per capita should
be greater among advanced countries. This prediction has profound implications for the performance of cross-country income per capita regressions and indicates that multisectoral and multicountry models that
incorporate both trade and technology transfer might enhance our understanding of cross-country productivity di€erences and of the wealth of
nations33. Schumpeterian growth models without scale e€ects, but incorporating variety expansion, have provided a new link between the static
trade literature on patterns of trade and gains from trade, and the new
growth literature.
Incorporating open economy features into the models might also
moderate the strong degree of level scale e€ects implied by the closed
economy models of growth without scale e€ects. This is certainly desirable,
as the scale e€ects in a closed economy may be staggering. Consider, for
example, the U.S. and France, two countries with similar R&D intensities.
As the U.S. employs four times as many researchers as does France, the
31
The same result is also obtained in Dinopoulos and Segerstrom (1999).
Indeed, the income per capita level e€ects are identical to the ones generated by static
models of intra-industry trade with monopolistic competition and manufacturing scale
economies (e.g., Krugman, 1979).
33
For example, it may explain why Mankiw, Romer and Weil (1992) found that the
income per capita predictions of the closed-economy neoclassical model do not perform
well on a cross-section restricted to OECD countries. However, Dinopoulos and
Thompson (1999) and Klenow and RodrõÂ guez-Clare (1997) have provided evidence that
the neoclassical model does not perform well on a cross-section of any sample of countries.
32
180
E. Dinopoulos, P. Thompson
steady-state equation (15) implies that the U.S. should be at least four times
richer. For / ˆ 0:5, the U.S. should be 16 times richer than France, while
for / ˆ 0:9 it should be one million times richer! The extent to which multicountry models will moderate this implication of the closed-economy
models awaits further research.
4.4 Does variety creation require R&D?
Finally, the variety expansion models of Section 3 raise an interesting
measurement issue associated with resources devoted to variety creation.
On the one hand, the process of variety accumulation does not entail any
uncertainty and, what is more important, there are no knowledge spillovers.
These features suggest that resources devoted to variety development might
not be associated with R&D investment, but rather should be interpreted as
®xed capital start-up costs. This view is reinforced by the obvious kinship
between growth models with variety accumulation and no scale e€ects, and
the literature on variety-based static monopolistic competition introduced
by Dixit and Stiglitz (1977), in which the number of varieties depends on
®xed manufacturing costs. On the other hand, unlike the static models
where, by necessity, the creation of varieties occurs instantly (as opposed to
sequentially) and manufacturing scale economies limit the number of
available products, this class of models assumes that the ®xed costs are paid
before any manufacturing is taken place. This assumption is standard is
R&D-based models of economic growth.
The treatment of variety creation in Section 3.3 therefore raises the
following question: Does variety creation require R&D, or physical capital
investment? The answer matters, of course, for how one should test the
models with R&D and investment data. But it also matters for a rather
di€erent reason. In Young's (1998) model, in which vertical and horizontal
innovations occur simultaneously as a result of the same R&D investment,
proportional R&D subsidies do not a€ect the steady-state intensity of R&D
per variety. Put another way, the standard policy tool to raise R&D e€ort
has no e€ect, even though increases in R&D e€ort do have permanent
e€ects on growth. Dinopoulos and Thompson (1998) have argued that this
awkward result turns on the question of whether or not the costs of variety
creation are, in fact, R&D. This issue might be resolved in the future, but
these considerations provide support for our use of the term ``Schumpeterian'' instead of ``R&D-based'' growth to characterize this strand of
literature.
5 Conclusions
This paper addressed three questions. First, what evidence is there for the
scale e€ects predicted by early Schumpeterian growth models? Second,
how can one modify the early models to remove the scale e€ects? Third,
what empirical evidence might be brought to bear on the modi®ed
models?
Scale e€ects in Schumpeterian models of economic growth
181
Our answer to the ®rst question is brief: The evidence for the existence of
scale e€ects is weak, although they might have been present in history.
There is no clear evidence to suggest that larger economies grow faster.
We described two main theoretical approaches that have generated
Schumpeterian growth models without scale e€ects. In the ®rst, aggregate
R&D e€ort aimed at proportional technology increments is assumed to
become more dicult over time, perhaps because the obvious ideas are
discovered ®rst. In the second, ®rm-level R&D aimed at proportional technology increments is not becoming more dicult over time, but economywide R&D is, because it is di€used over a greater number of varieties in
larger economies. Both approaches generate steady states in which scale has
level, but not growth, e€ects. Exogenous Schumpeterian growth models
predict that long-run per capita income growth is proportional to the rate of
population growth. Endogenous Schumpeterian growth models remove the
scale e€ects property in much the same way as exogenous Schumpeterian
growth models, but they maintain the endogeneity of long-run growth by
introducing the notion of localized intertemporal knowledge spillovers. The
latter models generate an expression for long-run growth that consists of two
additive terms: A term that is proportional to the rate of population growth
and a term that can be a€ected by a variety of permanent policy changes.
The identi®cation of the scale e€ects property of earlier Schumpeterian
growth models, and the response to Jones' criticism of them, have profound
implications for the theory of economic growth. At the theoretical level, the
existence of endogenous Schumpeterian growth models without scale e€ects
serves as a strong reminder that the removal of scale e€ects does not necessarily imply the removal of policy endogeneity of long-run Schumpeterian
growth. However, it must be emphasized that the removal of scale e€ects
reduces considerably the set of policies that a€ect long-run Schumpeterian
growth. In addition, the construction of Schumpeterian growth models
without scale e€ects provides the ®rst step towards a generalization of the
neoclassical growth model. The generalization, we anticipate, will eventually result in a uni®ed theory of economic growth in which endogenous
technology levels and growth rates are appended to the standard neoclassical model of economic growth.
We are not fully satis®ed with the empirical testing of the Schumpeterian
growth models summarized in this paper. Theoretical solutions to the scale
e€ects problem have been naturally couched in terms of closed economy
models, yet the empirical evidence that can be brought to bear on the
models is muddied by measurement problems and by the potentially important e€ects of trade and international technology di€usion. Jones
(1995b) has suggested that because of international technology ¯ows one
should not even attempt to test the models against modern international
cross-sectional data. We do not share this view, although we do acknowledge that one must be very cautious when interpreting results. There are of
course numerous empirical studies on technology di€usion34, and there is a
34
Griliches (1992) and Nadiri (1993) review the empirical literature. For subsequent research see Coe and Helpman (1995), Houser (1996), Keller (1997), Lichtenberg (1996) and
Rogers (1996).
182
E. Dinopoulos, P. Thompson
sizable theoretical literature on the relationship between trade, foreign direct investment, international technology di€usion and growth35. But the
theoretical link between scale e€ects and international linkages has yet to be
made explicit, although a few recent studies have started investigating this
issue. This is, we think, likely to be a very fruitful and challenging avenue
for further research on the scale e€ects problem.
References
Aghion P, Howitt P (1992) A model of growth through creative destruction. Econometrica 60: 323±351
Aghion P, Howitt P (1998) Endogenous growth theory. MIT Press, Cambridge, MA
Armknecht, PA, Lane WF, Stewart KJ, (1997), New Products and the U.S. Consumer
Price Index. In: Bresnahan TF, Gordon RJ (eds), The economics of new goods.
University of Chicago Press, Chicago
Backus DK, Kehoe PJ, Kehoe TJ (1992) In search of scale e€ects in trade and growth.
Journal of Economic Theory 58: 377±409
Baily MN, Gordon RJ (1988) The productivity slowdown, measurement issues, and the
explosion of computer power. Brookings Papers on Economic Activity 1988(2): 347±420
Barro R, Sala-i-Martin X (1995) Economic growth. McGraw-Hill, New York
Ben-David D, Papell DH (1995) The great wars, the great crash, and steady state growth:
Some new evidence about an old stylized fact. Journal of Monetary Economics 36:
453±475
Brooke GM (1992) The economics of information technology: Explaining the productivity
paradox. Review of Economic Dynamics 1
Bureau of the Census (1976) Historical statistics of the United States. Colonial times to
1970. US Department of Commerce, Washington, DC
Coe DT, Helpman E (1995) International R&D spillovers. European Economic Review
39: 859±887
De Long JB (1988) Productivity, convergence, and welfare: Comment. American Economic Review 78: 1138±1154
Dinopoulos E (1994) Schumpeterian growth: An overview. Osaka City University Economic Review 29: 1±21
Dinopoulos E, Segerstrom P (1999) A Schumpeterian model of protection and relative
wages. American Economic Review (forthcoming)
Dinopoulos E, Thompson P (1999) R&D-based long-run growth in a cross-section of
countries. Journal of International Economics (forthcoming)
Dinopoulos E, Thompson P (1998) Schumpeterian growth without scale e€ects. Journal
of Economic Growth 3: 313±335
Dixit A, Stiglitz JE (1977) Monopolistic competition and optimum product diversity.
American Economic Review 67: 297±308
Eaton J, Kortum S (1999a) Engines of growth: Domestic and foreign sources of innovation. Japan and the World Economy (forthcoming)
Eaton J, Kortum S (1999b) International technology di€usion: Theory and measurement.
International Economic Review (forthcoming)
Evans DS (1987a) The relationship between ®rm growth, size, and age: Estimates for 100
manufacturing industries. Journal of Industrial Economics 35: 567±581
35
See for example, Barro and Sala-i-Martin (1995), and Grossman and Helpman (1991).
Dinopoulos and Thompson (1999) use international cross-sectional data to assess the
performance of a closed-economy endogenous growth model without scale e€ects based
on Romer's (1990) model. They conclude that incorporating international technology
di€usion would signi®cantly improve the model's empirical performance.
Scale e€ects in Schumpeterian models of economic growth
183
Evans DS (1987b) Tests of alternative theories of ®rm growth. Journal of Political
Economy 95: 657±674
Feenstra R, Madani D, Yang T, Liang C (1997) Testing endogenous growth in South
Korea and Taiwan. NBER Working Paper 6028
George H (1898) The science of political economy. Doubleday, New York
Griliches Z (1990) Patent statistics as economic indicators: A survey. Journal of Economic
Literature 28: 1661±1707
Griliches Z (1992) The search for R&D spillovers. Scandinavian Journal of Economics
(Suppl) 29±47
Griliches Z (1994) Productivity, R&D, and the data constraint. American Economic
Review 84: 1±23
Grossman GM, Helpman E (1991) Innovation and growth in the global economy. MIT
Press, Cambridge, MA
Hall BH (1987) The relationship between ®rm size and ®rm growth in the U.S. manufacturing sector. Journal of Industrial Economics 35: 583±606
Hall BH (1993) R&D tax policy during the eighties: Success of failure? Tax Policy and the
Economy, 7: 1±36
Hausman JA (1997) The valuation of new goods under perfect and imperfect competition.
In: Bresnahan T, Gordon R, (eds) The economics of new products. University of
Chicago Press, Chicago
Houser C (1995) Estimating international R&D knowledge spillovers. Eastern Economic
Journal 22: 401±412
Howitt P (1997) Endogenous growth and cross-country income di€erences. The Ohio
State University (mimeo)
Howitt P (1999) Steady endogenous growth with population and R&D inputs growing.
Journal of Political Economy (forthcoming)
Jones CI (1995a) Time series tests of endogenous growth models. Quarterly Journal of
Economics 110: 495±525
Jones CI (1995b) R&D-based models of economic growth. Journal of Political Economy
103: 759±784
Jovanovic B, MacDonald G (1994) The life cycle of a competitive industry. Journal of
Political Economy 102: 322±347
Keller W (1997) Are international R&D spillovers trade-related? Analyzing spillovers
among randomly matched trading partners. NBER working paper no. 6065
Klenow P, RodrõÂ guez-Clare A (1997) The neoclassical revival in growth economics: Has it
gone too far? In: Bernanke B, Rotemberg J (eds) NBER macroeconomics annual 1997.
MIT Press, Cambridge MA, pp 73±102
Klepper S (1996) Entry, exit, growth and innovation over the product cycle. American
Economic Review 86: 562±583
Klepper S, Simons KL (1997) Technological extinctions of industrial ®rms: An inquiry
into their nature and causes. Industrial and Corporate Change 6: 379±460
Kocherlakota NR, Yi K-M (1997) Is there endogenous long-run growth? Evidence from
the United States and the United Kingdom. Journal of Money, Credit, and Banking
29: 235±260
Kortum S (1997) Research, patenting, and technological change. Econometrica 65: 1389±
1419
Kortum S, Lerner J (1997) Stronger protection or technological revolution: What is behind the recent surge in patenting? NBER Working Paper 6204
Kremer M (1993) Population growth and technological change: One million B.C. to 1990.
Quarterly Journal of Economics 106: 450±457
Krugman PR (1979) Increasing returns, monopolistic competition, and international
trade. Journal of International Economics 9: 469±479
Krugman PR (1987) The narrow moving band, the Dutch disease, and the economic
consequences of Mrs. Thatcher. Journal of Development Economics 27: 41±55
184
E. Dinopoulos, P. Thompson
Levine R, Renelt D (1992) A sensitivity analysis of cross-country growth regressions.
American Economic Review 82: 942±963
Lichtenberg F (1996) International R&D spillovers: A re-examination. NBER working
paper no. 5668
Lucas RE (1993) Making a miracle. Econometrica 61: 251±272
Lumsdaine RL, Papell DH (1997) Multiple trend breaks and the unit root hypothesis.
Review of Economics and Statistics 79: 212±218
Maddison A (1982) Phases of capitalist development. Oxford University Press, Oxford
Mankiw NG, Romer D, Weil DN (1992) A contribution to the empirics of economic
growth. Quarterly Journal of Economics 107: 407±437
Nadiri MI (1993) Innovations and technological spillovers. NBER working paper no 4423
Nelson C, Plosser C (1982) Trends and random walks in macroeconomics. Journal of
Monetary Economics 10: 139±162
Parente SL (1994) Technology adoption, learning-by-doing, and economic growth.
Journal of Economic Theory 63: 346±369
Peretto P (1998) Technological change and population growth. Journal of Economic
Growth 3: 238±311
Peretto P, Smulders S (1998) Specialization, knowledge dilution, and scale e€ects in an IObased growth model. Mimeo, Center for Economic Research, Tilberg University
Petty W (1682) Another essay on political arithmetick, concerning the growth of the city
of London. London
Rivera-Batiz L, Romer PM (1991) Economic integration and endogenous growth.
Quarterly Journal of Economics 106: 531±555
Reynolds LG (1983) The spread of economic growth to the third world: 1850±1980.
Journal of Economic Literature 21: 941±980
Rogers M (1995) International knowledge spillovers: A cross-country study. In: Dowrick
S (ed) Economic approaches to innovation. Edward Elgar, Aldershot
Romer PM (1983) Dynamic competitive equilibria with externalities, increasing returns,
and unbounded growth. PhD dissertation, University of Chicago
Romer PM (1986) Increasing returns and long-run growth. Journal of Political Economy
94: 1002±1037
Romer PM (1990) Endogenous technological change. Journal of Political Economy 98:
S71±102
Romer PM (1994) The origins of endogenous growth. Journal of Economic Perspectives 8:
3±22
Romer PM (1996) Why, indeed in America? Theory, history, and the origins of modern
economic growth. NBER working paper no 5443
Sala-i-Martin X (1997) I just ran two million regressions. American Economic Review
Papers and Proceedings 87: 178±183
Schankerman M, Pakes A (1986) Estimates of the value of patent rights in European
countries during the post-1950 period. Economic Journal 96: 1052±1076
Scherer FM (1965) Firm size, market structure, opportunity and the output of patented
inventions. American Economic Review 55: 1097±1125
Schumpeter JA (1942) Capitalism, socialism and democracy. Harper and Row, New York
Segerstrom PS (1998) Endogenous growth without scale e€ects. American Economic
Review 88: 1290±1310
Segerstrom PS, Anant TCA, Dinopoulos E (1990) A Schumpeterian model of the product
life cycle. American Economic Review 80: 1077±1092
Sener F (1997) Knowledge-based trade, wage inequality and unemployment. University of
Florida (mimeo)
Smith A (1776) An inquiry into the nature and causes of the wealth of nations, Various
publishers
Stokey NL (1988) Learning by doing and the introduction of new goods. Journal of
Political Economy 96: 701±717
Scale e€ects in Schumpeterian models of economic growth
185
Summers R, Heston A (1988) A new set of international comparisons of real product and
price level estimates for 130 countries, 1950±85. Review of Income and Wealth 34: 1±26
Sutton J (1997) Gibrat's legacy. Journal of Economic Literature 35: 40±59
Thompson P (1996) Technological opportunity and the growth of knowledge. Journal of
Evolutionary Economics 6: 77±97
Thompson P, Waldo D (1997) Process versus product innovation: Do consumption data
contain any information? University of Houston (mimeo)
Young A (1991) Learning by doing and the dynamic e€ects of international trade.
Quarterly Journal of Economics 106: 369±405
Young A (1993) Invention and bounded learning by doing. Journal of Political Economy
101: 443±472
Young A (1998) Growth without scale e€ects. Journal of Political Economy 106: 41±
63