Review of Industrial Organization (2005) 27:303–328
DOI 10.1007/s11151-005-5053-z
© Springer 2005
Technology and the Size Distribution of Firms:
Evidence from Dutch Manufacturing
ORIETTA MARSILI
Rotterdam School of Management, Erasmus University, Burg. Oudlaan 50, 3000 DR
Rotterdam, The Netherlands, E-mail: [email protected]
Abstract. Empirical studies have shown that the size distribution of firms can be described
as a Pareto distribution. However, these studies have focused on large firms and aggregate
statistics. Little attention has been placed on the role of technology in shaping firm size
distributions. Using a comprehensive dataset of manufacturing firms and the Community
Innovation Survey from the Netherlands, the paper investigates the relationship between
firm size and technology. It shows that technological factors shape the distribution of firm
size, suggesting that the Pareto law is not an invariant property and that technology can
constrain the “self-organising” character of industrial economies.
Key words: firm size, Gibrat’s law, innovation, Pareto distribution.
JEL Classifications: L11, L60, O33.
I. Introduction
The Pareto law is a well-known property of the size distribution of firms.
It says that the frequency of firms in a population above a certain size is
inversely proportional to the firm size. In logarithms, this relationship can
be represented graphically as a straight line.
A number of studies have tested empirically this hypothesis and formulated models able to generate Pareto-like distributions (Steindl, 1965;
Ijiri and Simon, 1977). These studies have been extremely influential in
industrial economics and elsewhere. For example, there has been a renewed
interest in the Pareto distribution across many different disciplines. This
interest focuses on the properties of the Pareto distribution, as a power
law able to describe the organisation of different scientific and social systems (Krugman, 1996; Bak, 1997). If distributed according to a power law,
the structure of the industrial system would depend only on the interaction between its components and not on external factors or the individual
behaviour. In addition, a similar structural form would be observed at
different levels of aggregation, such as countries and sectors. Finally, this
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ORIETTA MARSILI
structure is consistent with the firm size being the cumulative outcome of
purely random factors, according to Gibrat’s “law of proportionate effect”
(Gibrat, 1931).
Running alongside these studies of the Pareto, there has been an
empirical tradition investigating cross-sector differences of industrial structures (Cohen and Levin, 1989). In particular, the approaches of the
“Schumpeterian” tradition have stressed the importance of the innovative
behaviour of firms in shaping performance and market structure (Nelson
and Winter, 1982; Geroski, 1994). Innovation, it is argued, is one of the
main factors behind the process of competition of firms in the market, and
it varies across sectors in response to the nature of the technology specific
to the sector. Differences in technologies help to explain the cross-sectors
differences in market structures.
Yet, as pointed out by Sutton (1998), rarely are these two traditions in
industrial economics integrated. While the former addresses the general properties in the structure of a population of firms, the latter focuses on the
constraints that both technology and demand conditions, specific to each
industry, impose on such structure. Although the random factors underlying
the Pareto law play an important role, Sutton argued that there are “bounds”
to the structure that can be observed across sectors (Sutton, 1998).
Combining the two literatures, it is expected that distributions resembling the Pareto law are more likely to be observed at the aggregate level,
as outcome of aggregation and random effects, and less so at the sectoral
level, as outcome of more stringent technological constraints. In response
to this expectation, this paper examines, first, the goodness of fit of the
Pareto law for Dutch manufacturing firms, observed at the aggregate level
and at the sectoral level. Then, it introduces technological variables of
sectors into the estimation and relates them to the departure from the
Pareto law.
The data draws on two databases collected by the Central Bureau of Statistics Netherlands (CBS). First, the Business Register dataset provides information on the number of employees, here used as a measure of size, of all
the firms registered for tax purposes in the Netherlands. In particular, this
dataset allows the estimation of the Pareto law to be extended to small firms,
down to zero employees (self-employment) and thus to overcome the limitations of earlier studies based on samples of large firms. Second, the second
Community Innovation Survey (CIS-2) provides information on the innovative activities of an extensive sample of firms in the Netherlands.
The results show that for the Dutch case of a small and open economy,
even at the aggregate level of the manufacturing sector, the size distribution
displays a systematic departure from the Pareto law. This departure is more
evident than, for example, was observed for US firms (Axtell, 2001). At
the level of industrial sectors, the departure from the Pareto law is more
TECHNOLOGY AND THE SIZE DISTRIBUTION OF FIRMS
305
remarked and depends on the nature of technology. Specifically, if innovation is incremental, instead of radical, and persistent over time, the presence of firms in the central classes of the distribution is higher than under
the Pareto law. This result gives empirical support to Ijiri and Simon’s
(1974) argument that autocorrelation in firm growth rate may lead to a
greater share of medium-sized firms than the Pareto.
This paper is organised as follows. Section II reviews the empirical and
theoretical literature on the Pareto law. Section III introduces the data and
the variables to estimate the size distribution and to measure technological
conditions of sectors. Section IV discusses the results of the estimation of
the Pareto law and the effects of technology. Section V is the conclusions.
II. Background and Literature Review
Power law behaviour of empirical distributions can be observed in many
diverse fields, such as the distribution of words in a text (Simon, 1955) or
of earthquake magnitudes as one of the many examples in physical sciences (see Bak (1997) for an overview). In social sciences, typical examples are the distributions of income (Pareto, 1987), firm size (Ijiri and
Simon, 1977), cities population (Krugman, 1995), scientific publications
(Katz, 1999) and the returns from innovation (Scherer, 1998; Scherer and
Harhoff, 2000; Scherer et al., 2000). When applied to the size distribution
of firms, this power law behaviour is generally referred to as Pareto law,
and it is expressed in terms of the frequency of units with size greater than
s, that is, in terms of the right-cumulative distribution function (CDF),
F (·). Therefore, it holds that:
F (s) = F r{S ≥ s} = (s/s0 )−α ,
s >0
(1)
where s0 is the minimum size and α is a positive parameter. The Pareto
law is easy to be represented graphically, as its CDF is a straight line on
a double-logarithmic scale.1 The coefficient α, slope of this line, is a measure of market concentration. It is equal, in absolute value, to the relative
frequency of small firms in the distribution (Simon and Bonini, 1958).
The importance of the Pareto law for the size distribution of firms
resides in its direct link with the Gibrat’s law, and the properties of the
dynamic processes of growth and turnover of firms. Testing the Pareto law
can be seen as an alternative test to Gibrat’s law. The Gibrat’s law states
that firm growth can be characterised by a random walk, expressed in
logarithms. In other words, it assumes that firms’ growth rates are purely
random variables (“white noise”) and mutually independent. The Gibrat’s
1
Another method to describe a power law, especially used in physics, is through the
Zipf’s law or “rank-size” rule. This looks at the relationship between the ranks of firms
and their corresponding size, relationship that is linear in logarithms.
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ORIETTA MARSILI
law per se generates a Lognormal distribution with infinite variance in the
limit (Steindl, 1965). However, assuming Gibrat’s law, or weaker versions
of it, and constant rates of entry of new firms, should generate Pareto-like
distributions in steady state (Ijiri and Simon, 1977).2
Empirical studies testing the Gibrat’s law have shown that growth rates,
especially in large firms, have a significant random component (Sutton,
1997). Although the randomness of firm growth leads to size distributions
that resemble the Pareto law, the latter appeared to represent the actual
shape only in a first approximation. In particular, the empirical CDFs in
logarithms tend to be non-linear. They display a concave shape, suggesting that the Pareto law underestimates the frequencies of the medium-sized
classes. For example, Ijiri and Simon (1974) observed similarly concave pattern for the size of the 500 firms of the Fortune list, and Scherer (1998) for
the returns to innovation.
This early empirical evidence on the size distribution refers to large
firms, indicating that the Pareto law may be a property of the upper tail of
the distribution. More recent studies that extend the analysis of the shape
of the size distribution to larger samples or to the entire population of
firms display contrasting results. Stanley et al. (1995) observe that the Lognormal distribution well represents the Compustat population of publicly
traded firms with the exception of the upper tail of the distribution. However, Cabral and Mata (2003) argue that the Lognormal distribution also
fails to provide a good representation of the data at the lower tail of the
distribution. In fact, in a study of the complete population of Portuguese
manufacturing firms, Cabral and Mata observe that the Lognormal underestimates the skewness of the distribution. Finally, Axtell (2001) finds support for the Pareto law using Census data for the entire population of US
firms.
Divergent interpretations have been given to explain the departure from
the Pareto distribution, and the concavity of the CDF curve on a log–log
scale. Ijiri and Simon (l974) argued that it results from the existence of
auto-correlation in firm growth. In addition, merges and acquisitions can
contribute to such a departure as, they suggested, external growth is more
constrained by firm size (and less random) than internal growth (Ijiri and
Simon, 1971). In both interpretations, the authors maintained the assumption of constant returns to scale entailed in Gibrat’s law. In contrast,
Vining (1976) argued that the concavity of the distribution originates in the
existence of decreasing returns to scale.
2
Weaker versions of Gibrat’s law refer to the average growth rates (instead of the individual values) being independent on size, or introduce autocorrelation (Ijiri and Simon,
1977). They generally produce a Yule distribution, within the same class of skewed
distributions identified by Simon (1955).
TECHNOLOGY AND THE SIZE DISTRIBUTION OF FIRMS
307
Because of the departure from the Pareto law, it has been suggested that
more than one Pareto law applies to different size classes. For each size
class, the Pareto law has a different slope (Gopikrishnan et al., 1999), Reed
(2001) showed that a “double” Pareto distribution – that is, a Pareto distribution with different slope at the two tails – can be generated as outcome of growth processes a la Gibrat. This distribution is because of the
aggregation of different cohorts of firms, which are observed at different,
randomly distributed, ages.
Recently, attention has turned to the Pareto distribution as a “general
property” able to represent the structure of different social and natural
phenomena (Buchanan, 1997). As a power law, two properties have been
related to the Pareto distribution (Krugman, 1996). First, a power law is
indicative of a self-organising system. That is, the properties of the overall system “emerge” out of the interaction of the single parts and cannot
be ascribed to the characteristics of its single components. If firm size were
distributed according to Pareto, the distribution would not be affected by
the purposeful action of firms or by external factors and constraints. Second, a power law is indicative of a self-similar system. It displays scaleindependent property, which can be observed at any level of aggregation.
This property is satisfied by the Pareto distribution, when the coefficient α
is lower than two. In this interval, the size distribution does not depend on
the central moments, as the variance and the higher moments are asymptotically infinite.3 A particular case, or Zipf’s law, is represented by the
parameter α being equal to one (Krugman, 1996).
Whether or not, firm size distribution follows a power law is a matter for discussion. In fact, the size distribution may not be invariant over
time. Macro-economic and institutional changes may influence the coefficient of the size distribution of firms in the long run (Henrekson and
Johansson, 1999). Also the size distribution may not be scale independent. If the Pareto distribution holds for the industrial system a whole,
it should be invariant across industrial sectors or countries. The empirical evidence suggests, however, that there are systematic departures from
the Pareto law at the level of industrial sectors (Kwoka, 1982) and that a
scale-dependent distribution, such as the Lognormal, may provide a better
fit in some industries (Quandt, 1966; Silberman, 1967). Furthermore, industry cases have shown that neither the Pareto nor the Lognormal distribution may fit the empirical distributions; for example a “double-humped”
curve was observed for the top pharmaceutical firms (Bottazzi et al., 2001).
Alternatively, by using a non-parametric approach to the estimation of the
size distribution, Machado and Mata (2000) have found that the effects of
3
A self-similar distribution may have finite mean, for 1 < α ≤ 2.
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ORIETTA MARSILI
industry characteristics on firm size vary at the different quartiles of the
size distribution.
The evidence at the sectoral level thus suggests that structural factors may constrain and shape the size distribution of firms. To identify these factors, it is necessary to turn to another literature, which
focuses on the interpretation of cross-sectors differences in market structure. In particular, the approaches following the Schumpeterian tradition
have stressed the importance for market structure of the innovative behaviour of firms and the nature of technology influencing such behaviour
(Nelson and Winter, 1982; Dosi, 1988; Sutton, 1998). When the intensity
of R&D expenditure was used to measure innovative activities, however,
a weak relationship with market structure was found in the empirical literature (Cohen, 1995). Such a relationship appeared to be influenced by
the characteristics of the underlying technologies. These were expressed by
a wide set of variables: technological opportunity and appropriability conditions (Levin et al., 1985), the cumulativeness of innovation (Breschi et
al., 2000), and the relative importance of product and process innovations
(Gort and Klepper, 1982). These conditions distinguish between different
“technological regimes” across industrial sectors (Dosi, 1982; Nelson and
Winter, 1982; Marsili, 2001). In an “entrepreneurial” regime, innovation is
mainly generated by the entrepreneurial activity and creativity of small and
new firms, leading to low market concentration. In a “routinised” regime,
innovation originates in the formal R&D activity of large and established
firms, leading to high market concentration (Breschi et al., 2000). In particular, the empirical studies suggest that market concentration is positively
associated with the contribution of scientific knowledge to technological
opportunity, cumulativeness of innovation and process innovation. In contrast, it is negatively associated with the opportunities for innovation stemming from the vertical chain of production, in particular from suppliers,
and product innovation. For the purpose of this paper, these effects would
be reflected in changes of the slope of the Pareto distribution. No predictions on the higher moments of the distributions, and therefore on the
departure from Pareto, can be made, however, from this literature.
This paper addresses two questions: is the Dutch firm size distribution
a Pareto and, if not why not? The data will be considered as a whole, and
I will return to these questions, and propose answers in the conclusions.
III. The Data
The analysis uses two micro-economic databases collected from the Central
Bureau of Statistics Netherlands (CBS): the Business Register database and
the Second Community Innovation Survey (CIS-2) in the Netherlands.
TECHNOLOGY AND THE SIZE DISTRIBUTION OF FIRMS
309
The Business Register is used for estimating the size distribution, and
the CIS is used for measuring the characteristics of technologies that may
affect the size distribution.
The Business Register database reports employment statistics and sector of activity at 6-digit of the Standard Industrial Classification of all the
manufacturing firms registered in the Netherlands. The population includes
also firms with zero employees, referred to as self-employment.
Table I reports the descriptive statistics of the size distribution of the
population of Dutch firms, including self-employment, over the period
1996–1998. On average around 61,000 firms are observed each year. The
mean firm size is 16.3 employees (18.1 for firms with size greater than
0). The positive value of the skewness coefficient confirms that the size
distribution is skewed to the right, that is, the long tail of the distribution
is in the positive direction (Greene, 2000, p. 64). The high value of the kurtosis coefficient indicates that the size distribution tends to be leptokurtic,
that is, the distribution is more “peaked” and has “fatter tails” than the
normal distribution.
Looking in more detail into the dataset, Table II reports the frequency
of firms in the population by size class from 1996 to 1998. The size
distribution is highly skewed. It is characterised by a large prevalence of
self-employed firms, on average about 45% of the entire population. The
percentage of firms is much lower, to a value of just above 15%, for the
class of firms with one employee. Gradually, it decreases with the increase
of the size class, to the minimum of 0.4%, for the highest size class of firms
with more than 500 employees. This pattern is fairly invariant over time.
Although very small firms represent a large share of the manufacturing
sector, traditionally they have not been included in the estimation of the
Pareto law. The data used in earlier studies were generally left-censored databases. Recently, as more extensive micro-economic databases have become
available, empirical studies have tested the Pareto law on the entire range
of firm size. For the United States, for example, Axtell (2001) used data on
Table I. Descriptive statistics of the size distribution of the
population of manufacturing firms in the Netherlands
Mean
Std. deviation
Skewness
Kurtosis
N
1996
1997
1998
16.7
197.4
133.8
24648.7
60792
16.1
178.5
142.0
27721.6
62198
16.2
178.5
143.0
27966.6
61721
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ORIETTA MARSILI
Table II. Percentage distribution of firms by size class
Size class
1996
1997
1998
0
1
2–4
5–9
10–19
20–49
50–99
100–199
200–499
500 and more
Total
43.9
15.5
13.4
9.2
7.4
5.8
2.4
1.3
0.8
0.4
100
45.7
15.0
13.0
8.7
7.2
5.6
2.4
1.3
0.7
0.4
100
46.3
15.0
12.2
8.5
7.4
5.7
2.4
1.2
0.8
0.4
100
self-employment and the firms with at least one employee. His study extended
previous results based on the distribution of publicly traded firms from the
Compustat database (Stanley et a1., 1995). In this study, I use the comprehensive Business Register database to estimate the Pareto law for all the firms
with employment and self-employment from 1996 to 1998.
To characterise the nature of technology, the CIS-2 dataset is used.
This dataset provides information on the innovative activities of firms
in the Netherlands in 1994 to 1996. The survey was done by Statistics
Netherlands and it includes all of the private sector firms with at least
10 employees. In manufacturing, a total of 3299 responses were obtained
with a response rate of 71 per cent. This sample is representative of
a population of 10,260 firms of which 6069 are innovators. To calculate
indicators of the nature of technologies across different sectors, I used a
classification of sectors between 2- and 4-digit level. This aggregates sectors
at 6-digit level according to Statistics Netherlands’ standard classification
of industries as of 1993. As a result, 62 sectors were defined.
IV. Empirical Results
The first problem to be addressed is whether the size distribution follows
the Pareto law for the aggregate manufacturing and for industrial sectors. The empirical exploration of the Pareto law in Dutch manufacturing
is carried out in two stages. First, I examine its properties in the aggregate manufacturing, using non-parametric and parametric methods. Then,
I examine differences in the shape of the distribution across sectors and
introduce industry-fixed effects in the estimation of the Pareto law.
TECHNOLOGY AND THE SIZE DISTRIBUTION OF FIRMS
311
Figure 1. The size distribution of Dutch manufacturing firms: 1996–1998. Note:
Log–log plot of the right cumulative distribution function of firms with at least one
employee.
1.
FIRM SIZE DISTRIBUTION IN AGGREGATE MANUFACTURING
Preliminary insight into the shape of the distribution is given by plotting the empirical cumulative distribution functions in 1996 to 1998, in
Figure 1. This shows that the empirical distributions largely overlap, and
that they display a concave shape of the distribution throughout the
considered time period.
To assess whether the Pareto law is appropriate to represent the size distribution of firms in the Netherlands, Figure 2 presents the p–p plots of the
theoretical distribution and the empirical distribution in 1998. These graphs
plot the theoretical cumulative distribution function against the empirical
one. Because the Pareto law is considered to fit better the upper tail of the
size distribution, the distributions are plotted for the whole population and
separately for different size classes. For the firms that have size larger than 0,
three size classes are compared: small firms with less than 10 employees (67
per cent of firms); large firms with 500 and more employees (0.7 per cent
of firms); and the extreme upper tail of the distribution as composed of
firms with 1000 and more employees (0.1 per cent). The graphs show that
there is a departure of the empirical distribution from the Pareto law for
the overall population. This departure from Pareto is especially evident for
the class of small firms. In contrast, the Pareto law fits well the data at the
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ORIETTA MARSILI
Figure 2. Pareto p–p plots of the size distribution of firms in 1998.
upper tail of the distribution, although this tail represents only about 0.7
per cent of the population of firms.
Does the Lognormal distribution fit better the overall size distribution
of firms when small firms are included in the population, as in this case?
To address this question, Figure 3 reports the p–p plots comparing the
empirical distribution functions to the Lognormal distribution in the total
population and by size class. The plots suggest that again there is a systematic departure of the empirical distribution of the population of firms
from the theoretical one. By size class, the Lognormal distribution fits well
the data for the class of small firms (below 10 employees), which is a significant proportion of the population (67 per cent). Yet, the upper tail of
the large firms in the population departs considerably from Lognormality.
Alternative methods for fitting the Pareto law to the data and estimating
the exponent of the Pareto law can be found in the literature. One method
uses the log–log linear regression of the cumulative distribution function.
TECHNOLOGY AND THE SIZE DISTRIBUTION OF FIRMS
313
Figure 3. Lognormal p–p plots of the size distribution of firms in 1998.
The other method is based on the Hill estimator (Weron, 2001). I begin by
estimating the Pareto law using the linear regression of the empirical CDF
on a log–log scale, for the entire population of firms with employees. For
this purpose, Equation (1) is transformed into logarithms and time-fixed
effects are introduced to account for changes in the slope, α, of the Pareto
law. The basic model specification is:
log F (x) =
αt dt log(x), x > 0
(2)
t
where x is firm size, F (·) is the right-cumulated distribution function up to
size x and dt is a dummy for time t (t = 1996, 1997, 1998). The constant
term in Equation (2) is set equal to 0, as by definition the right-cumulated
distribution function is equal to one at the minimum size. The equation is
estimated by using pooled ordinary least squares (OLS). This method of
estimation is most commonly used, although, because of non-linearity, the
standard assumptions for OLS regression do not hold. This method should
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ORIETTA MARSILI
be considered heuristic, recognising that too much weight should not be
put on estimated standard errors and p-values, although the method is still
useful for comparison with previous results (Scherer, 1998).
Then, I extend the estimation of the Pareto law to include firms with
zero employees. This requires the following transformation of the basic
model (Axtell, 2001):4
log F (x) =
αt dt log(x + 1), x ≥ 0.
(3)
t
The size classes for all the regressions were defined by using a binning
system of 0.3 on logarithmic scale, with the highest open size class set at
the midpoint of 9.5 In the equations, each size class is identified by the
corresponding midpoint.
Following Scherer (1998) a quadratic term is added to the basic model
to account for the concavity of the distribution. In addition, time-fixed
effects of all coefficients are included, producing the following equation:
log F (x) =
α2t dt [log(x)]2
α1t dt log(x) +
(4)
t
t
where α1 is the coefficient of the Pareto law and α2 measures the departure
of the size distribution from the Pareto law. The distribution is concave if
α2 < 0, and convex if α2 > 0. A similar transformation to Equation (3) was
applied for the population of firms with self-employment.
Table III reports the estimated coefficients of the linear and quadratic
model for the population of firms with at least one employee and that
including also self-employment. For both populations, the results of the
linear model would lead one to conclude that the Pareto law provides a
good representation of the size distribution, with an R-squared of about
0.93 in either case. This value of the goodness-of-fit is, however, lower than
observed in other countries, in particular the US, for which an R-squared
approximately, equal to one was observed (Axtell, 2001). The coefficient of
the Pareto law is equal to 0.90 in 1996 for the firms with employees and
slightly higher for those including self-employment, and it did not significantly vary between 1997 and 1998.
The departure from the Pareto law is even more evident when the quadratic term is added. In Table III, for both populations of firms, the coefficient α2 is statistically significant and negative in 1996, thus displaying
Let X be the size of a firm with employees (x ≥ 1) following a Pareto distribution F (·). The right-cumulated distribution function of the S size of any firm including
self-employment (s ≥ 0), then defined by G(s) = Fr{S ≥ s} = Fr{X − 1 ≥ s} = Fr{X ≥ s + 1} =
F (s + 1) = (s + 1)−α .
5
Alternative binning systems of 0.15, 0.20 and 0.40 have been applied in the aggregate
manufacturing. These have given rise to similar shapes of the size distribution.
4
315
TECHNOLOGY AND THE SIZE DISTRIBUTION OF FIRMS
Table III. Results from pooled OLS regressions of F (x) in aggregate manufacturing (absolute t-statistics in parentheses)
Firms with employees
All firms
Variable
Linear
Quadratic
Linear
Quadratic
Size
− 0.898∗∗∗
(−34.68)
− 0.022
(−0.60)
− 0.027
(−0.73)
− 0.417∗∗∗
(−18.59)
0.032
(1.00)
0.042
(1.33)
− 0.07∗∗∗
(−22.1)
−0.008∗
(−1.75)
−0.01∗∗
(−2.24)
81
0.997
0.07∗∗∗
− 0.914∗∗∗
(−35.94)
− 0.02
(−0.57)
− 0.023
(−0.63)
− 0.443∗∗∗
(−21.24)
0.035
(1.18)
0.051∗
(1.73)
− 0.069∗∗∗
(−23.3)
−0.008∗
(−1.93)
−0.011∗∗
(−2.58)
84
0.997
0.06∗∗∗
Size (d1997 −d1996 )
Size (d1998 −d1996 )
Size2
Size2 (d1997 −d1996 )
Size2 (d1998 −d1996 )
DF
Adjusted R 2
Increase in
adjusted R 2
84
0.930
87
0.935
Notes: The population of all firms includes self-employment. Firm size and the CDF of
size, F (x), are in logarithms. ∗∗∗ significant at 1%; ∗∗ significant at 5%; ∗ significant at 10%.
the existence of a concavity of the distribution. In addition, the concavity tended to increase in 1997 and 1998. This result is in contrast to the
evidence available for the US economy. For US companies, Axtell (2001)
observed a slightly concave distribution. It was associated, however, to the
behaviour at the two extreme size classes only and therefore was interpreted as the statistical outcome of finite size cut-offs at the two extremes
of very small and very large firms (Axtell, 2001). In contrast, the Dutch
data, based on a more disaggregated definition of the size classes, suggest
a more pronounced concavity.
In order to compare more closely the estimates of the Pareto law based
on the Dutch Business Register data in 1996 to 1998, with the estimates
obtained by Axtell (2001) from the US Census data in 1997, I also apply
Axtell’s definition of size classes. As in Axtell, I calculate the size classes with bins of increasing size, in powers of three, and carry out OLS
estimation of the log–log regression of the empirical CDF. It is worth noting that the main difference in the data is that the Dutch data include
only the manufacturing sector while the data in Axtell’s work deal with
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ORIETTA MARSILI
the economy as a whole, including for example services firms. Table IV
reports the results of the estimation for the Dutch data in the same form
as reported in Axtell (2001). The estimated coefficient of the Pareto law, on
this more aggregate definition of size classes, is fairly close to one (ranging
between 0.987 and 1.025), either including or excluding self-employment
in the population. These values are just above Axtell’s estimates of 0.994
and 0.995, for firms with employees and all businesses respectively. However, the R-squared values in Table IV suggest that the goodness of fit of
the Pareto law is lower for the Dutch data (ranging between 0.949 and
0.957) than for the US data (equal to 0.995 and 0.994 in Axtell’s estimation). A goodness of fit comparable to the value reported by Axtell can
be obtained for the Dutch data when adding a squared term to the linear
log–log regression (ranging between 0.993 and 0.998). These findings confirm that the size distribution of Dutch firms shows a greater departure
from the Pareto law, in the form of a concavity of the CDF curve, than
the size distribution of US firms.
Hill Estimator
Because the OLS estimates of the log–log linear regression can be biased,
and tend to overestimate the true slope of the Pareto distribution, alternative methods have been applied. In particular, Hill (1975) has proposed a
maximum likelihood estimator for the tail index α of a class of Generalised
Pareto distributions, with their upper tails that converge to the ordinary
Pareto distribution of exponent α (Weron, 2001). If X(1) , X(2) , . . . , X(N) are
the order statistics of firm size in the sample, that is, X(1) ≥ X(2) ≥ · · · ≥ X(N) ,
then the Hill estimate of α based on the k largest observations is:
−1
k
1
αHill (k) =
(log X(i) − log X(k+1) ) .
k
(5)
i=1
Table IV. Power law exponents of Dutch firms according to Axtell’s (2001) size classes
Year
Type
Estimated coefficient
Adjusted R 2
1996
Firms with employees
All businesses
Firms with employees
All businesses
Firms with employees
All businesses
0.987
1.002
1.013
1.025
1.015
1.025
0.956
0.957
0.949
0.950
0.949
0.949
1997
1998
(0.042)
(0.042)
(0.047)
(0.047)
(0.047)
(0.047)
317
TECHNOLOGY AND THE SIZE DISTRIBUTION OF FIRMS
Table V. Results from pooled OLS regressions of F (x) in industrial sectors (absolute
t-statistics in parentheses)
Sector-fixed effects
Variable
Linear
Quadratic
Linear
Quadratic
Size
−0.658∗∗∗
(−95.44)
−0.006
(−0.64)
−0.014
(−1.48)
−0.508∗∗∗
(−22.03)
−0.011
(−0.33)
0.002
(0.05)
−0.028∗∗∗
(−6.8)
0.001
(0.12)
−0.003
(−0.55)
3647
0.703
[−1.148, −0.280]+
(a)
−0.010∗∗
(−2.01)
−0.013∗∗
(−2.45)
[−1.060, 0.163]+
(b)
−0.005
(−0.65)
−0.007
(−0.89)
[−0.180, 0.043]+
(c)
−0.001
(−0.73)
−0.001
(−0.78)
3525
0.981
Size (d1997 −d1996 )
Size (d1998 −d1996 )
Size2
Size2 (d1997 −d1996 )
Size2 (d1998 −d1996 )
DF
Adjusted R 2
Increase in
adjusted R 2
3650
0.692
0.012∗∗∗
3589
0.913
0.068∗∗∗
Notes: Firm size and the CDF of size, F (x), are in logarithms. ∗∗∗ significant at 1%;
∗∗
significant at 5%; ∗ significant at 10%. (+) The range of the coefficients of the sector
dummies is shown in order to conserve space. (a) Significant at 1% in all sectors. (b) Significant at 1% in 76% of sectors, at 5% in 8% of sectors, at 10% in 3% of sectors. (c) Significant at 1% in 94% of sectors and at 10% in 3% of sectors.
In order to establish whether k converge to a value, which will then
be used to estimate the coefficient α of the Pareto distribution, the values
of αHill (k) are plotted against k and the value of k is selected in correspondence of a region in which the plot levels off (Weron, 2001). Figure 4
reports the Hill estimates for the size distribution for the entire population
of firms in 1998. The plot of the Hill estimates appears to be fairly stable
in the range of k approximately, between 80 and 130, which corresponds
to an upper tail of about 0.1 per cent of firms. For this tail, the estimates
of αHill (k) indicate a Pareto coefficient ranging between 1.5 and 1.7. These
results suggest that the Pareto law fits well the size distribution at the very
extreme upper tail, with finite mean (α > 1) and infinite variance (α < 2).
2.
FIRM SIZE DISTRIBUTION BY SECTOR
In order to assess whether the Pareto law is invariant across industrial
sectors, I present the p–p plots for four industrial sectors, at two-digit level of
318
ORIETTA MARSILI
Figure 4. Hill estimates of α for the size distribution of firms in 1998.
industrial classification in Figure 5. These sectors can be considered typical
of both high technology sectors and more traditional industrial sectors. The
visual inspection of these plots indicates that a variety of patterns can be
observed across sectors. Indeed, depending on the industrial sector, the Pareto law seems to underestimate the presence of small firms, of medium sized
firms and of large firms.
To estimate the departure from Pareto at the sector level, I first estimate
a baseline model by applying Equations (2) and (4) to the sectoral data of
the size distribution, with equal coefficients across sectors. Then, I add to
this component common across sectors, sector-fixed effects, and I compare
them to the baseline model with equal coefficients. The following equation,
in the more general version, is thus defined:
log F (x) =
α1tj dt dsj log(x) +
α2tj dt dsj [log(x)]2
(6)
tj
tj
where x is the firm size, F (·) is the right-hand side CDF, dsj is a dummy
variable for sector j , dt is a dummy variable for time t, α1tj and α2tj are
parameters respectively, for the coefficient of the Pareto law at time t and
sector j , and the deviation from Pareto law.6 Time-fixed effects are allowed
for the cross-sectors means only (that is, α1tj = α1t and α2tj = α2t , for t > 1).
Table V presents the estimates of the pooled OLS regressions for the linear and quadratic specification of the baseline model and the model with
sector-fixed effects. The population is that of firms with employment.
The linear model for the Pareto law is given by setting α1tj = αtj and α2tj = 0 in
Equation (6).
6
TECHNOLOGY AND THE SIZE DISTRIBUTION OF FIRMS
319
Figure 5. Pareto p–p plots for selected industrial sectors.
As shown in Table V, assuming identical coefficients across sectors leads
to a poor fit. With respect to this baseline model, the model with sectorfixed effects produces a remarked improvement, with a goodness-of-fit comparable to that found in the aggregate manufacturing. Specifically, the
increase in the adjusted R-squared is equal to 0.22 in the linear model
and to 0.28 in the quadratic model. Both differences are statistically significant at 0.1 per cent. In particular, adding the quadratic term improves
the R-squared especially when sector-fixed effects are considered. For this
model, the coefficients of the time-dummies are not statistically significant
(column [4]). This suggests that cross-sectors differences in the slope and
shape of the distribution are significant and, on average, persistent over the
considered time period.
320
ORIETTA MARSILI
Looking more in detail into the sector-specific coefficients of the quadratic term, the sign of the coefficient differs across sectors. Most often,
the coefficients are statistically significant and of negative sign (56 sectors
out of the 62). Within this group, machinery industries have the lowest
coefficients and most evident concavity. However, in four industries (telecommunication equipment, computers, motor vehicles and glass products),
the coefficient is statistically significant and of positive sign. Finally, in two
industries (photographic equipment and publishing), the coefficient is not
statistically significant.
3.
THE ROLE OF TECHNOLOGY
This section investigates whether the departure from the Pareto law can be
explained on the ground of differences in the nature of technology.
With this aim, the sector-specific fixed effects of Equation (6) are distinguished into a component common to all sectors and a component depending on the industry technological variables. Therefore, in Equation (6) it is
set αktj = αkt + δ k S j , (k = 1, 2) where S j is the vector of technological variables of sector j , assumed invariant over the considered period, and δ k a
vector of parameters. This produces the following equation:
log F (x) =
α1t dt log(x) +
t
α2t dt [log(x)]2
t
+δ1 Sj log(x) + δ2 S j [log(x)]2 .
(7)
Technological Variables
Four categories of variables are included in the vector S j of technological variables: (i) the level of technological opportunity, (ii) the cumulativeness of innovation, (iii) the sources of technological opportunity and (iv)
the relative importance of product and process innovation. The variables
are constructed at the level of industrial sector, using CIS-2 data.
Direct measures of innovative activity are derived through a combination of input and output indicators. These are (a) the intensity of R&D
expenditure as the ratio of the total R&D expenditure in the period
1994–1996 on the total sales in 1996, (b) the percentage of turnover on the
total sales in 1996 attributed to innovative products, distinguished in the
three categories of products new for the firm, products new for the market and improved products and (c) the percentage of innovators in 1994 to
1996 that carry out R&D activities on a permanent basis, as opposed to
occasional or not at all. These five variables are then summarised by applying a principal component analysis. This approach allowed me to extract
TECHNOLOGY AND THE SIZE DISTRIBUTION OF FIRMS
321
two main factors.7 The first factor is positively correlated with all the measures of innovation intensity and can be interpreted as the general level
of technological opportunity, or “potential” for innovation (Technological
opportunity). The second factor is positively correlated with the turnover
due to improved products and the share of innovators with permanent
R&D activities; it can be interpreted as an indicator of the incremental and
cumulative nature of innovation (Cumulativeness).
In general, measures of the sources of technological opportunity are
derived from innovation surveys by looking at the sources of information
that are relevant for industrial innovation (Cohen et al., 1987). In particular, the CIS-2 listed l2 sources, which are rated by the firm on a fourpoint Likert scale, ranging from 0 “not-used” to 3 “very important”. The
importance of each source for innovation, at the level of industrial sector, is thus measured as the percentage of innovators that rated the source
as “important” or “very important”. The information generated from this
step was summarised, via a principal component analysis, into four main
factors. The first factor is positively correlated with the contribution of
public research from Universities and other research institutes, and the contribution of codified sources of publicly available knowledge (patent disclosure and computer based information). This factor is interpreted as a
measure of the relevance of scientific knowledge for innovation (Science).
The second factor is positively correlated with the contribution of information from suppliers and from publicly available sources of “professional”
knowledge, such as conferences and journals8 and fairs and exhibitions.
It is regarded as indicative of supplier-dominated industries (Suppliers),
following Pavitt (1984) interpretation. The third factor reflects the contribution of customers, in combination with the use of in-house sources
(Users). The last, fourth factor contrasts the contribution of competing
firms within the industry to the contribution of innovation centres, which
act as “bridging” institutions between the industry and the public system.
I label it as the industry factor (Industry).
The relative importance of product versus process innovations is
measured by the ratio between the number of firms with at least one
product innovation and with at least one process innovation (PDT/PCS).
Because responses to the innovation survey may differ between small and
large firms, I controlled for the relative size of innovative firms in a sector. This was measured as the ratio between the average sales of innovative firms and the average sales of the population in the sector at 1996
(Innovator size). This variable reflects the existence of scale economies in
7
The results of the principal component analysis are not reported here for reason of
space and they are available upon request.
8
This category includes both academic and professional journals.
322
ORIETTA MARSILI
the innovation process. Finally, to account for the relationship between
market concentration and the mobility of firm market shares (Caves, 1998)
an index of persistence of firms size (Persistence) is built on the basis of
the Business Register dataset. This index measures the percentage of all the
continuing firms that remain within the same size class between 1997 and
1998.
Note that the aim of estimating the model is not to fully identifying
the determinants of the size distribution of firms. I do not wish to exclude
the possibility that other variables could be important, for example, market
demand.
Table VI presents the results of the OLS estimates of Equation (7) for
the linear and quadratic specification. The variables Innovator size and
Persistence are added as control variables.
Table VI shows that the technological variables have a significant effect
on the slope of the Pareto law. I begin by examining the results for
the linear estimation of the model. These results show that adding the
technology variables increases significantly the R-squared with respect to
the baseline model of identical sectoral coefficients. Specifically, the level of
technological opportunity, the cumulativeness of innovation and the contribution of knowledge from the science system, industry and users have a
positive effect on the Pareto coefficient (which decreases in absolute value),
leading to higher market concentration. In contrast, the contribution of
knowledge from suppliers and the prevalence of product innovation on
process innovation have a negative effect on the Pareto coefficients, leading
to lower market concentration. This findings mirror the results of Breschi
et al. (2000).
Adding a quadratic term to the model leads to a further statistically
significant increase of the R-squared. Although this increase is modest, it
modifies the overall pattern of relationships with the technological variables. In particular, the level of technological opportunity does not have
a statistically significant effect on the slope and concavity (or convexity)
of the distribution. This confirms the weak results found for this variable
in other studies (Levin et al., 1985). Interestingly, the cumulativeness of
innovation has a significant negative effect on the quadratic term. Cumulativeness of learning thus increases the concavity of the distribution with
respect to the Pareto law. This result is consistent with Ijiri and Simon
(1974) argument that autocorrelation in firm growth would result in a
concave distribution of firm size. One possible interpretation could be that
the cumulativeness of innovation is a source of the autocorrelation in the
firm growth processes.
With regard to the sources of knowledge, the contribution of science
has a significant (positive) effect on the shape of the distribution. Science-based sectors show decreasing concavity of the distribution, that is, a
323
TECHNOLOGY AND THE SIZE DISTRIBUTION OF FIRMS
Table VI. Results from pooled OLS regressions of F (x) on technology variables (absolute
t-statistics in parentheses)
Variable
Size
Technological
opportunity
Cumulativeness
Science
Suppliers
Users
Industry
PDT/PCS
Innovator size
Persistence
Size2
Technological
opportunity2
Cumulativeness2
Science2
Suppliers2
Users2
Industry2
PDT/PCS2
Innovator size2
Persistence2
Linear
−0.447∗∗∗
(−31.01)
0.019∗∗∗
(4.27)
0.044∗∗∗
(12.19)
0.073∗∗∗
(17.69)
−0.057∗∗∗
(−17.36)
0.027∗∗∗
(6.91)
0.055∗∗∗
(14.97)
−0.169∗∗∗
(−16.53)
−0.468∗∗∗
(−21.86)
0.020∗∗∗
(4.45)
0.045∗∗∗
(12.07)
0.073∗∗∗
(17.65)
−0.058∗∗∗
(−17.39)
0.028∗∗∗
(7.04)
0.054∗∗∗
(14.77)
−0.171∗∗∗
(−16.58)
0.018
(1.34)
–
Quadratic
−0.431∗∗∗
(−31.76)
0.016
(1.31)
0.141∗∗∗
(12.24)
0.026∗∗
(2.02)
−0.071∗∗∗
(−6.46)
−0.002
(−0.18)
0.063∗∗∗
(5.47)
−0.015
(−0.96)
−0.497∗∗∗
(−28.64)
0.014
(1.16)
0.139∗∗∗
(12.03)
0.026∗∗
(2.05)
−0.074∗∗∗
(−6.72)
−0.005
(−0.46)
0.064∗∗∗
(5.62)
–
–
–
−0.001
(−0.5)
−0.020∗∗∗
(−9.42)
0.011∗∗∗
(4.81)
0.004∗
(1.76)
0.006∗∗∗
(2.91)
0.000
(0.23)
−0.032∗∗∗
(−13.58)
–
–
0.00
(−0.13)
−0.020∗∗∗
(−9.24)
0.011∗∗∗
(4.65)
0.004∗∗
(2.19)
0.007∗∗∗
(3.44)
0.00
(0.04)
−0.038∗∗∗
(−21.45)
−0.003
(−1.11)
0.022∗∗∗
(4.37)
324
ORIETTA MARSILI
Table VI. Continued
Variable
DF
Adjusted R 2
Increase in
adjusted R 2
Linear
3643
0.796
0.105∗∗∗a
Quadratic
3642
0.796
3634
0.821
0.025∗∗∗
3633
0.823
Notes: Firm size and the CDF of size, F (x), are in logarithms. ∗∗∗ significant at 1%;
∗∗
significant at 5%; ∗ significant at 10%. − The variable was excluded from the estimation
because of collinearity. a Difference calculated with respect to the baseline linear model of
colum [1] in Table V.
relatively lower presence of medium-sized firms. In contrast, the contribution of knowledge from suppliers has a significant negative effect on the
slope of the distribution and a positive (although slightly significant) effect
on the curve shape. In other words, supplier-dominated sectors (Pavitt,
1984) are characterised by low market concentration and a slightly less
concave distribution. Sources within the industry tend to increase the level
of market concentration. The contribution of users leads to a less concave
distribution, but it has no statistically significant effect on the slope of the
distribution. Finally, the nature of innovation has a statistically significant
(negative) effect on the coefficient of the quadratic term. Industries where
product innovations are dominant over process innovations display a more
pronounced concavity of the distribution.
The control variable for the existence of scale advantages in innovation
does not have a statistically significant effect in either the linear or quadratic versions of the model. In contrast, a significant effect is observed
for the measure of persistence within the distribution. In particular, the
degree of persistence within the distribution has a significant and positive
effect on the quadratic term of the model. Therefore, increasing mobility
of firms across size classes leads to increasing concavity of the distribution.
Thus, distributions with more mobility are characterised by higher presence of firms in the central size classes than under the Pareto law, and less
skewness towards small firms.
V. Conclusions
This paper has had two tasks, the first preparatory for the second. First,
it has provided an empirical investigation of the properties of the size distribution. Using data on the Dutch manufacturing firms, these properties
have been explored at the level of the aggregate manufacturing and across
industrial sectors. Second, the paper has attempted to establish whether
TECHNOLOGY AND THE SIZE DISTRIBUTION OF FIRMS
325
systematic departures from the Pareto law emerged at the different levels
of analysis and whether such departures could be related to the nature of
technology. In order to explore the two questions, the employment data
from the Business Register of the population of manufacturing firms in the
Netherlands in 1996 to 1998 have been linked with the data from the CIS
for the Dutch manufacturing sector in 1994 to 1996.
Overall, I found that the Pareto law fits the size distribution of firms
only as a first approximation. At the aggregate level, the size distribution
displays a certain concavity. This result is consistent with previous studies
(Ijiri and Simon, 1974; Scherer, 1998). The results based on a finer tabulation of size classes suggest that the concavity of the distribution is not a
pure statistical outcome of the extreme classes definition (Axtell, 2001).
The Dutch manufacturing system appears to depart more systematically
from the structure of a self-organising system, than was observed for example in the US economy. This result suggests that the Dutch system might
be more sensitive to external factors than to its own internal processes,
compared to a large economy as the US. In the Netherlands, the size distribution is characterised by lower skewness towards small firms and higher
presence of medium sized firms, than expected under a power law. Here
power law behaviour is observed only at the extreme upper tail of the
distribution, corresponding to less than one per cent of the population.
The analysis at the sectoral-level showed that the departures from the
Pareto law were more pronounced than at the aggregate level. A variety
of distributional forms appears to emerge across industrial sectors. In other
words, sectoral characteristics shape the distribution of firm size.
These sectoral characteristics are often linked to features of technology
present in these sectors (Nelson and Winter, 1982; Dosi et al., 1995). Three
characteristics appear here to be associated with concave distributions: the
cumulative nature of innovation, the dominance of product on process
innovation, and the mobility of firms in their relative position. A possible
interpretation would be that the Schumpeterian process of dynamic competition that originates in the continuous introduction of new products varieties enables small firms to growth, to reach the middle range of the size
distribution. In contrast, more radical innovations, generated by scientific
developments, are necessary to reach the upper tail of the size distribution
and maintain such a position over time. This result provides a different
perspective, based on the nature of technology, on the interpretation of the
departures from Pareto earlier suggested by Ijiri and Simon (1974). Their
suggestions were based on the autocorrelation of firm growth rates. I want
to suggest this autocorrelation might originate in the persistence of innovation due to the cumulative learning processes of firms (Mazzucato and
Geroski, 2002).
326
ORIETTA MARSILI
Future research is required on the relationship between technology and
the size distribution of firms. It would also be interesting to find out
whether the properties of the Pareto law are invariant across countries,
especially given their different investment profile in technology. Indeed, a
tentative comparison with results obtained in a similar study suggests that
the Dutch manufacturing departs visibly from a self-organising system that
characterised the US firms. The study was focused on departures from the
Pareto law as measured through a polynomial fit of the empirical distribution. A further question arising from my study concerns the class of
theoretical distributions and the underlying processes of growth that may
account for the observed departures. My results hinted that a variety of
distributional forms might characterise different sectors. Further research
should attempt to explore broader classes of theoretical distributions to
represent more fully the variety of industrial patterns.
Acknowledgements
I wish to thank John Kwoka, George van Leeuwen, Roy Thurik and
two anonymous referees for their helpful comments and suggestions. The
empirical part of this research has been carried out at the Centre for
Research of Economic Microdata at Statistics Netherlands. The views
expressed in this paper are those of the author and do not necessarily
reflect the policies of Statistics Netherlands.
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