Technology and the Size Distribution of Firms: Evidence from Dutch

DOI 10.1007/s11151-005-5053-zReview of Industrial Organization (2005) 27:303–328 © Springer 2005

Technology and the Size Distribution of Firms:Evidence from Dutch Manufacturing

ORIETTA MARSILIRotterdam School of Management, Erasmus University, Burg. Oudlaan 50, 3000 DRRotterdam, The Netherlands, E-mail: [email protected]

Abstract. Empirical studies have shown that the size distribution of firms can be describedas a Pareto distribution. However, these studies have focused on large firms and aggregatestatistics. Little attention has been placed on the role of technology in shaping firm sizedistributions. Using a comprehensive dataset of manufacturing firms and the CommunityInnovation Survey from the Netherlands, the paper investigates the relationship betweenfirm size and technology. It shows that technological factors shape the distribution of firmsize, suggesting that the Pareto law is not an invariant property and that technology canconstrain 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 isinversely proportional to the firm size. In logarithms, this relationship canbe represented graphically as a straight line.

A number of studies have tested empirically this hypothesis and for-mulated models able to generate Pareto-like distributions (Steindl, 1965;Ijiri and Simon, 1977). These studies have been extremely influential inindustrial economics and elsewhere. For example, there has been a renewedinterest in the Pareto distribution across many different disciplines. Thisinterest focuses on the properties of the Pareto distribution, as a powerlaw able to describe the organisation of different scientific and social sys-tems (Krugman, 1996; Bak, 1997). If distributed according to a power law,the structure of the industrial system would depend only on the interac-tion between its components and not on external factors or the individualbehaviour. In addition, a similar structural form would be observed atdifferent levels of aggregation, such as countries and sectors. Finally, this

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structure is consistent with the firm size being the cumulative outcome ofpurely random factors, according to Gibrat’s “law of proportionate effect”(Gibrat, 1931).

Running alongside these studies of the Pareto, there has been anempirical tradition investigating cross-sector differences of industrial struc-tures (Cohen and Levin, 1989). In particular, the approaches of the“Schumpeterian” tradition have stressed the importance of the innovativebehaviour of firms in shaping performance and market structure (Nelsonand Winter, 1982; Geroski, 1994). Innovation, it is argued, is one of themain factors behind the process of competition of firms in the market, andit varies across sectors in response to the nature of the technology specificto the sector. Differences in technologies help to explain the cross-sectorsdifferences in market structures.

Yet, as pointed out by Sutton (1998), rarely are these two traditions inindustrial economics integrated. While the former addresses the general prop-erties in the structure of a population of firms, the latter focuses on theconstraints that both technology and demand conditions, specific to eachindustry, impose on such structure. Although the random factors underlyingthe 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 resem-bling 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 sectorallevel, as outcome of more stringent technological constraints. In responseto this expectation, this paper examines, first, the goodness of fit of thePareto law for Dutch manufacturing firms, observed at the aggregate leveland at the sectoral level. Then, it introduces technological variables ofsectors into the estimation and relates them to the departure from thePareto law.

The data draws on two databases collected by the Central Bureau of Sta-tistics Netherlands (CBS). First, the Business Register dataset provides infor-mation on the number of employees, here used as a measure of size, of allthe firms registered for tax purposes in the Netherlands. In particular, thisdataset allows the estimation of the Pareto law to be extended to small firms,down to zero employees (self-employment) and thus to overcome the limita-tions of earlier studies based on samples of large firms. Second, the secondCommunity Innovation Survey (CIS-2) provides information on the innova-tive 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 distributiondisplays a systematic departure from the Pareto law. This departure is moreevident than, for example, was observed for US firms (Axtell, 2001). Atthe 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 innova-tion is incremental, instead of radical, and persistent over time, the pres-ence of firms in the central classes of the distribution is higher than underthe Pareto law. This result gives empirical support to Ijiri and Simon’s(1974) argument that autocorrelation in firm growth rate may lead to agreater share of medium-sized firms than the Pareto.

This paper is organised as follows. Section II reviews the empirical andtheoretical literature on the Pareto law. Section III introduces the data andthe variables to estimate the size distribution and to measure technologicalconditions of sectors. Section IV discusses the results of the estimation ofthe 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 manydiverse fields, such as the distribution of words in a text (Simon, 1955) orof earthquake magnitudes as one of the many examples in physical sci-ences (see Bak (1997) for an overview). In social sciences, typical exam-ples are the distributions of income (Pareto, 1987), firm size (Ijiri andSimon, 1977), cities population (Krugman, 1995), scientific publications(Katz, 1999) and the returns from innovation (Scherer, 1998; Scherer andHarhoff, 2000; Scherer et al., 2000). When applied to the size distributionof 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 thans, that is, in terms of the right-cumulative distribution function (CDF),F(·). Therefore, it holds that:

F(s)=Fr{S ≥ s}= (s/s0)−α, s >0 (1)

where s0 is the minimum size and α is a positive parameter. The Paretolaw is easy to be represented graphically, as its CDF is a straight line ona double-logarithmic scale.1 The coefficient α, slope of this line, is a mea-sure of market concentration. It is equal, in absolute value, to the relativefrequency of small firms in the distribution (Simon and Bonini, 1958).

The importance of the Pareto law for the size distribution of firmsresides in its direct link with the Gibrat’s law, and the properties of thedynamic processes of growth and turnover of firms. Testing the Pareto lawcan be seen as an alternative test to Gibrat’s law. The Gibrat’s law statesthat firm growth can be characterised by a random walk, expressed inlogarithms. In other words, it assumes that firms’ growth rates are purelyrandom variables (“white noise”) and mutually independent. The Gibrat’s

1 Another method to describe a power law, especially used in physics, is through theZipf’s law or “rank-size” rule. This looks at the relationship between the ranks of firmsand their corresponding size, relationship that is linear in logarithms.

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law per se generates a Lognormal distribution with infinite variance in thelimit (Steindl, 1965). However, assuming Gibrat’s law, or weaker versionsof it, and constant rates of entry of new firms, should generate Pareto-likedistributions 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 distributionsthat resemble the Pareto law, the latter appeared to represent the actualshape only in a first approximation. In particular, the empirical CDFs inlogarithms tend to be non-linear. They display a concave shape, suggest-ing that the Pareto law underestimates the frequencies of the medium-sizedclasses. For example, Ijiri and Simon (1974) observed similarly concave pat-tern for the size of the 500 firms of the Fortune list, and Scherer (1998) forthe returns to innovation.

This early empirical evidence on the size distribution refers to largefirms, indicating that the Pareto law may be a property of the upper tail ofthe distribution. More recent studies that extend the analysis of the shapeof the size distribution to larger samples or to the entire population offirms display contrasting results. Stanley et al. (1995) observe that the Log-normal distribution well represents the Compustat population of publiclytraded firms with the exception of the upper tail of the distribution. How-ever, Cabral and Mata (2003) argue that the Lognormal distribution alsofails to provide a good representation of the data at the lower tail of thedistribution. In fact, in a study of the complete population of Portuguesemanufacturing firms, Cabral and Mata observe that the Lognormal under-estimates the skewness of the distribution. Finally, Axtell (2001) finds sup-port for the Pareto law using Census data for the entire population of USfirms.

Divergent interpretations have been given to explain the departure fromthe Pareto distribution, and the concavity of the CDF curve on a log–logscale. Ijiri and Simon (l974) argued that it results from the existence ofauto-correlation in firm growth. In addition, merges and acquisitions cancontribute to such a departure as, they suggested, external growth is moreconstrained by firm size (and less random) than internal growth (Ijiri andSimon, 1971). In both interpretations, the authors maintained the assump-tion of constant returns to scale entailed in Gibrat’s law. In contrast,Vining (1976) argued that the concavity of the distribution originates in theexistence of decreasing returns to scale.

2 Weaker versions of Gibrat’s law refer to the average growth rates (instead of the indi-vidual values) being independent on size, or introduce autocorrelation (Ijiri and Simon,1977). They generally produce a Yule distribution, within the same class of skeweddistributions identified by Simon (1955).


Because of the departure from the Pareto law, it has been suggested thatmore than one Pareto law applies to different size classes. For each sizeclass, the Pareto law has a different slope (Gopikrishnan et al., 1999), Reed(2001) showed that a “double” Pareto distribution – that is, a Pareto dis-tribution with different slope at the two tails – can be generated as out-come of growth processes a la Gibrat. This distribution is because of theaggregation of different cohorts of firms, which are observed at different,randomly distributed, ages.

Recently, attention has turned to the Pareto distribution as a “generalproperty” able to represent the structure of different social and naturalphenomena (Buchanan, 1997). As a power law, two properties have beenrelated to the Pareto distribution (Krugman, 1996). First, a power law isindicative of a self-organising system. That is, the properties of the over-all system “emerge” out of the interaction of the single parts and cannotbe ascribed to the characteristics of its single components. If firm size weredistributed according to Pareto, the distribution would not be affected bythe purposeful action of firms or by external factors and constraints. Sec-ond, a power law is indicative of a self-similar system. It displays scale-independent 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 onthe central moments, as the variance and the higher moments are asymp-totically infinite.3 A particular case, or Zipf’s law, is represented by theparameter α being equal to one (Krugman, 1996).

Whether or not, firm size distribution follows a power law is a mat-ter for discussion. In fact, the size distribution may not be invariant overtime. Macro-economic and institutional changes may influence the coeffi-cient of the size distribution of firms in the long run (Henrekson andJohansson, 1999). Also the size distribution may not be scale indepen-dent. If the Pareto distribution holds for the industrial system a whole,it should be invariant across industrial sectors or countries. The empiri-cal evidence suggests, however, that there are systematic departures fromthe Pareto law at the level of industrial sectors (Kwoka, 1982) and that ascale-dependent distribution, such as the Lognormal, may provide a betterfit in some industries (Quandt, 1966; Silberman, 1967). Furthermore, indus-try cases have shown that neither the Pareto nor the Lognormal distribu-tion 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 thesize 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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industry characteristics on firm size vary at the different quartiles of thesize distribution.

The evidence at the sectoral level thus suggests that structural fac-tors may constrain and shape the size distribution of firms. To iden-tify these factors, it is necessary to turn to another literature, whichfocuses on the interpretation of cross-sectors differences in market struc-ture. In particular, the approaches following the Schumpeterian traditionhave stressed the importance for market structure of the innovative behav-iour of firms and the nature of technology influencing such behaviour(Nelson and Winter, 1982; Dosi, 1988; Sutton, 1998). When the intensityof R&D expenditure was used to measure innovative activities, however,a weak relationship with market structure was found in the empirical lit-erature (Cohen, 1995). Such a relationship appeared to be influenced bythe characteristics of the underlying technologies. These were expressed bya wide set of variables: technological opportunity and appropriability con-ditions (Levin et al., 1985), the cumulativeness of innovation (Breschi etal., 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 andWinter, 1982; Marsili, 2001). In an “entrepreneurial” regime, innovation ismainly generated by the entrepreneurial activity and creativity of small andnew firms, leading to low market concentration. In a “routinised” regime,innovation originates in the formal R&D activity of large and establishedfirms, leading to high market concentration (Breschi et al., 2000). In par-ticular, the empirical studies suggest that market concentration is positivelyassociated with the contribution of scientific knowledge to technologicalopportunity, cumulativeness of innovation and process innovation. In con-trast, it is negatively associated with the opportunities for innovation stem-ming from the vertical chain of production, in particular from suppliers,and product innovation. For the purpose of this paper, these effects wouldbe reflected in changes of the slope of the Pareto distribution. No pre-dictions on the higher moments of the distributions, and therefore on thedeparture from Pareto, can be made, however, from this literature.

This paper addresses two questions: is the Dutch firm size distributiona Pareto and, if not why not? The data will be considered as a whole, andI will return to these questions, and propose answers in the conclusions.

III. The Data

The analysis uses two micro-economic databases collected from the CentralBureau of Statistics Netherlands (CBS): the Business Register database andthe Second Community Innovation Survey (CIS-2) in the Netherlands.


The Business Register is used for estimating the size distribution, andthe CIS is used for measuring the characteristics of technologies that mayaffect the size distribution.

The Business Register database reports employment statistics and sec-tor of activity at 6-digit of the Standard Industrial Classification of all themanufacturing firms registered in the Netherlands. The population includesalso firms with zero employees, referred to as self-employment.

Table I reports the descriptive statistics of the size distribution of thepopulation of Dutch firms, including self-employment, over the period1996–1998. On average around 61,000 firms are observed each year. Themean firm size is 16.3 employees (18.1 for firms with size greater than0). The positive value of the skewness coefficient confirms that the sizedistribution is skewed to the right, that is, the long tail of the distributionis in the positive direction (Greene, 2000, p. 64). The high value of the kur-tosis coefficient indicates that the size distribution tends to be leptokurtic,that is, the distribution is more “peaked” and has “fatter tails” than thenormal distribution.

Looking in more detail into the dataset, Table II reports the frequencyof firms in the population by size class from 1996 to 1998. The sizedistribution is highly skewed. It is characterised by a large prevalence ofself-employed firms, on average about 45% of the entire population. Thepercentage of firms is much lower, to a value of just above 15%, for theclass of firms with one employee. Gradually, it decreases with the increaseof the size class, to the minimum of 0.4%, for the highest size class of firmswith more than 500 employees. This pattern is fairly invariant over time.

Although very small firms represent a large share of the manufacturingsector, traditionally they have not been included in the estimation of thePareto law. The data used in earlier studies were generally left-censored da-tabases. Recently, as more extensive micro-economic databases have becomeavailable, empirical studies have tested the Pareto law on the entire rangeof firm size. For the United States, for example, Axtell (2001) used data on

Table I. Descriptive statistics of the size distribution of thepopulation of manufacturing firms in the Netherlands

1996 1997 1998

Mean 16.7 16.1 16.2Std. deviation 197.4 178.5 178.5Skewness 133.8 142.0 143.0Kurtosis 24648.7 27721.6 27966.6N 60792 62198 61721

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Table II. Percentage distribution of firms by size class

Size class 1996 1997 1998

0 43.9 45.7 46.31 15.5 15.0 15.02–4 13.4 13.0 12.25–9 9.2 8.7 8.510–19 7.4 7.2 7.420–49 5.8 5.6 5.750–99 2.4 2.4 2.4100–199 1.3 1.3 1.2200–499 0.8 0.7 0.8500 and more 0.4 0.4 0.4Total 100 100 100

self-employment and the firms with at least one employee. His study extendedprevious results based on the distribution of publicly traded firms from theCompustat database (Stanley et a1., 1995). In this study, I use the compre-hensive Business Register database to estimate the Pareto law for all the firmswith 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 firmsin the Netherlands in 1994 to 1996. The survey was done by StatisticsNetherlands and it includes all of the private sector firms with at least10 employees. In manufacturing, a total of 3299 responses were obtainedwith a response rate of 71 per cent. This sample is representative ofa population of 10,260 firms of which 6069 are innovators. To calculateindicators of the nature of technologies across different sectors, I used aclassification of sectors between 2- and 4-digit level. This aggregates sectorsat 6-digit level according to Statistics Netherlands’ standard classificationof 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 followsthe Pareto law for the aggregate manufacturing and for industrial sec-tors. The empirical exploration of the Pareto law in Dutch manufacturingis carried out in two stages. First, I examine its properties in the aggre-gate manufacturing, using non-parametric and parametric methods. Then,I examine differences in the shape of the distribution across sectors andintroduce industry-fixed effects in the estimation of the Pareto law.


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 oneemployee.

1. FIRM SIZE DISTRIBUTION IN AGGREGATE MANUFACTURING

Preliminary insight into the shape of the distribution is given by plot-ting the empirical cumulative distribution functions in 1996 to 1998, inFigure 1. This shows that the empirical distributions largely overlap, andthat they display a concave shape of the distribution throughout theconsidered time period.

To assess whether the Pareto law is appropriate to represent the size dis-tribution of firms in the Netherlands, Figure 2 presents the p–p plots of thetheoretical distribution and the empirical distribution in 1998. These graphsplot the theoretical cumulative distribution function against the empiricalone. Because the Pareto law is considered to fit better the upper tail of thesize distribution, the distributions are plotted for the whole population andseparately 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 (67per cent of firms); large firms with 500 and more employees (0.7 per centof firms); and the extreme upper tail of the distribution as composed offirms with 1000 and more employees (0.1 per cent). The graphs show thatthere is a departure of the empirical distribution from the Pareto law forthe overall population. This departure from Pareto is especially evident forthe class of small firms. In contrast, the Pareto law fits well the data at the

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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 0.7per cent of the population of firms.

Does the Lognormal distribution fit better the overall size distributionof 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 theempirical distribution functions to the Lognormal distribution in the totalpopulation and by size class. The plots suggest that again there is a sys-tematic departure of the empirical distribution of the population of firmsfrom the theoretical one. By size class, the Lognormal distribution fits wellthe data for the class of small firms (below 10 employees), which is a sig-nificant proportion of the population (67 per cent). Yet, the upper tail ofthe large firms in the population departs considerably from Lognormality.

Alternative methods for fitting the Pareto law to the data and estimatingthe exponent of the Pareto law can be found in the literature. One methoduses the log–log linear regression of the cumulative distribution function.


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 byestimating the Pareto law using the linear regression of the empirical CDFon a log–log scale, for the entire population of firms with employees. Forthis purpose, Equation (1) is transformed into logarithms and time-fixedeffects are introduced to account for changes in the slope, α, of the Paretolaw. The basic model specification is:

log F(x)=∑

t

αtdt log(x), x >0 (2)

where x is firm size, F(·) is the right-cumulated distribution function up tosize x and dt is a dummy for time t (t = 1996,1997,1998). The constantterm in Equation (2) is set equal to 0, as by definition the right-cumulateddistribution function is equal to one at the minimum size. The equation isestimated by using pooled ordinary least squares (OLS). This method ofestimation is most commonly used, although, because of non-linearity, thestandard assumptions for OLS regression do not hold. This method should

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be considered heuristic, recognising that too much weight should not beput on estimated standard errors and p-values, although the method is stilluseful for comparison with previous results (Scherer, 1998).

Then, I extend the estimation of the Pareto law to include firms withzero employees. This requires the following transformation of the basicmodel (Axtell, 2001):4

log F(x)=∑

t

αtdt log(x +1), x ≥0. (3)

The size classes for all the regressions were defined by using a binningsystem of 0.3 on logarithmic scale, with the highest open size class set atthe midpoint of 9.5 In the equations, each size class is identified by thecorresponding midpoint.

Following Scherer (1998) a quadratic term is added to the basic modelto account for the concavity of the distribution. In addition, time-fixedeffects of all coefficients are included, producing the following equation:

log F(x)=∑

t

α1t dt log(x)+∑

t

α2t dt [log(x)]2 (4)

where α1 is the coefficient of the Pareto law and α2 measures the departureof the size distribution from the Pareto law. The distribution is concave ifα2 <0, and convex if α2 >0. A similar transformation to Equation (3) wasapplied for the population of firms with self-employment.

Table III reports the estimated coefficients of the linear and quadraticmodel for the population of firms with at least one employee and thatincluding also self-employment. For both populations, the results of thelinear model would lead one to conclude that the Pareto law provides agood representation of the size distribution, with an R-squared of about0.93 in either case. This value of the goodness-of-fit is, however, lower thanobserved in other countries, in particular the US, for which an R-squaredapproximately, equal to one was observed (Axtell, 2001). The coefficient ofthe Pareto law is equal to 0.90 in 1996 for the firms with employees andslightly higher for those including self-employment, and it did not signifi-cantly vary between 1997 and 1998.

The departure from the Pareto law is even more evident when the qua-dratic term is added. In Table III, for both populations of firms, the coeffi-cient α2 is statistically significant and negative in 1996, thus displaying

4 Let X be the size of a firm with employees (x ≥ 1) following a Pareto distribu-tion F(·). The right-cumulated distribution function of the S size of any firm includingself-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 aggregatemanufacturing. These have given rise to similar shapes of the size distribution.


Table III. Results from pooled OLS regressions of F(x) in aggregate manufacturing (abso-lute t-statistics in parentheses)

Firms with employees All firms

Variable Linear Quadratic Linear Quadratic

Size −0.898∗∗∗ −0.417∗∗∗ −0.914∗∗∗ −0.443∗∗∗

(−34.68) (−18.59) (−35.94) (−21.24)Size (d1997−d1996) −0.022 0.032 −0.02 0.035

(−0.60) (1.00) (−0.57) (1.18)Size (d1998−d1996) −0.027 0.042 −0.023 0.051∗

(−0.73) (1.33) (−0.63) (1.73)Size2 −0.07∗∗∗ −0.069∗∗∗

(−22.1) (−23.3)Size2 (d1997−d1996) −0.008∗ −0.008∗

(−1.75) (−1.93)Size2 (d1998−d1996) −0.01∗∗ −0.011∗∗

(−2.24) (−2.58)DF 84 81 87 84Adjusted R2 0.930 0.997 0.935 0.997Increase in 0.07∗∗∗ 0.06∗∗∗

adjusted R2

Notes: The population of all firms includes self-employment. Firm size and the CDF ofsize, 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 concav-ity tended to increase in 1997 and 1998. This result is in contrast to theevidence available for the US economy. For US companies, Axtell (2001)observed a slightly concave distribution. It was associated, however, to thebehaviour at the two extreme size classes only and therefore was inter-preted as the statistical outcome of finite size cut-offs at the two extremesof very small and very large firms (Axtell, 2001). In contrast, the Dutchdata, based on a more disaggregated definition of the size classes, suggesta more pronounced concavity.

In order to compare more closely the estimates of the Pareto law basedon the Dutch Business Register data in 1996 to 1998, with the estimatesobtained by Axtell (2001) from the US Census data in 1997, I also applyAxtell’s definition of size classes. As in Axtell, I calculate the size clas-ses with bins of increasing size, in powers of three, and carry out OLSestimation of the log–log regression of the empirical CDF. It is worth not-ing that the main difference in the data is that the Dutch data includeonly the manufacturing sector while the data in Axtell’s work deal with

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the economy as a whole, including for example services firms. Table IVreports the results of the estimation for the Dutch data in the same formas reported in Axtell (2001). The estimated coefficient of the Pareto law, onthis more aggregate definition of size classes, is fairly close to one (rangingbetween 0.987 and 1.025), either including or excluding self-employmentin the population. These values are just above Axtell’s estimates of 0.994and 0.995, for firms with employees and all businesses respectively. How-ever, the R-squared values in Table IV suggest that the goodness of fit ofthe Pareto law is lower for the Dutch data (ranging between 0.949 and0.957) than for the US data (equal to 0.995 and 0.994 in Axtell’s estima-tion). A goodness of fit comparable to the value reported by Axtell canbe obtained for the Dutch data when adding a squared term to the linearlog–log regression (ranging between 0.993 and 0.998). These findings con-firm that the size distribution of Dutch firms shows a greater departurefrom the Pareto law, in the form of a concavity of the CDF curve, thanthe 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, alterna-tive methods have been applied. In particular, Hill (1975) has proposed amaximum likelihood estimator for the tail index α of a class of GeneralisedPareto distributions, with their upper tails that converge to the ordinaryPareto distribution of exponent α (Weron, 2001). If X(1),X(2), . . . ,X(N) arethe 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:

αHill(k)=[

1k

k∑

i=1

(log X(i) − log X(k+1))

]−1

. (5)

Table IV. Power law exponents of Dutch firms according to Axtell’s (2001) size classes

Year Type Estimated coefficient Adjusted R2

1996 Firms with employees 0.987 (0.042) 0.956All businesses 1.002 (0.042) 0.957




Table V. Results from pooled OLS regressions of F(x) in industrial sectors (absolutet-statistics in parentheses)

Sector-fixed effects

Variable Linear Quadratic Linear Quadratic

Size −0.658∗∗∗ −0.508∗∗∗ [−1.148,−0.280]+ [−1.060,0.163]+

(−95.44) (−22.03) (a) (b)Size (d1997−d1996) −0.006 −0.011 −0.010∗∗ −0.005

(−0.64) (−0.33) (−2.01) (−0.65)Size (d1998−d1996) −0.014 0.002 −0.013∗∗ −0.007

(−1.48) (0.05) (−2.45) (−0.89)Size2 −0.028∗∗∗ [−0.180,0.043]+

(−6.8) (c)Size2 (d1997−d1996) 0.001 −0.001

(0.12) (−0.73)Size2 (d1998−d1996) −0.003 −0.001

(−0.55) (−0.78)DF 3650 3647 3589 3525Adjusted R2 0.692 0.703 0.913 0.981Increase in

adjusted R2 0.012∗∗∗ 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 sectordummies is shown in order to conserve space. (a) Significant at 1% in all sectors. (b) Sig-nificant at 1% in 76% of sectors, at 5% in 8% of sectors, at 10% in 3% of sectors. (c) Sig-nificant 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 thenbe used to estimate the coefficient α of the Pareto distribution, the valuesof αHill(k) are plotted against k and the value of k is selected in corre-spondence of a region in which the plot levels off (Weron, 2001). Figure 4reports the Hill estimates for the size distribution for the entire populationof firms in 1998. The plot of the Hill estimates appears to be fairly stablein the range of k approximately, between 80 and 130, which correspondsto an upper tail of about 0.1 per cent of firms. For this tail, the estimatesof αHill(k) indicate a Pareto coefficient ranging between 1.5 and 1.7. Theseresults suggest that the Pareto law fits well the size distribution at the veryextreme 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 industrialsectors, 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 typicalof both high technology sectors and more traditional industrial sectors. Thevisual inspection of these plots indicates that a variety of patterns can beobserved across sectors. Indeed, depending on the industrial sector, the Pa-reto law seems to underestimate the presence of small firms, of medium sizedfirms and of large firms.

To estimate the departure from Pareto at the sector level, I first estimatea baseline model by applying Equations (2) and (4) to the sectoral data ofthe size distribution, with equal coefficients across sectors. Then, I add tothis component common across sectors, sector-fixed effects, and I comparethem to the baseline model with equal coefficients. The following equation,in the more general version, is thus defined:

log F(x)=∑

tj

α1tj dtdsj log(x)+∑

tj

α2tj dtdsj [log(x)]2 (6)

where x is the firm size, F(·) is the right-hand side CDF, dsj is a dummyvariable for sector j , dt is a dummy variable for time t , α1tj and α2tj areparameters respectively, for the coefficient of the Pareto law at time t andsector j , and the deviation from Pareto law.6 Time-fixed effects are allowedfor 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 lin-ear and quadratic specification of the baseline model and the model withsector-fixed effects. The population is that of firms with employment.

6 The linear model for the Pareto law is given by setting α1tj = αtj and α2tj = 0 inEquation (6).


Figure 5. Pareto p–p plots for selected industrial sectors.

As shown in Table V, assuming identical coefficients across sectors leadsto a poor fit. With respect to this baseline model, the model with sector-fixed effects produces a remarked improvement, with a goodness-of-fit com-parable to that found in the aggregate manufacturing. Specifically, theincrease in the adjusted R-squared is equal to 0.22 in the linear modeland to 0.28 in the quadratic model. Both differences are statistically sig-nificant at 0.1 per cent. In particular, adding the quadratic term improvesthe R-squared especially when sector-fixed effects are considered. For thismodel, the coefficients of the time-dummies are not statistically significant(column [4]). This suggests that cross-sectors differences in the slope andshape of the distribution are significant and, on average, persistent over theconsidered time period.

320 ORIETTA MARSILI

Looking more in detail into the sector-specific coefficients of the qua-dratic term, the sign of the coefficient differs across sectors. Most often,the coefficients are statistically significant and of negative sign (56 sectorsout of the 62). Within this group, machinery industries have the lowestcoefficients and most evident concavity. However, in four industries (tele-communication equipment, computers, motor vehicles and glass products),the coefficient is statistically significant and of positive sign. Finally, in twoindustries (photographic equipment and publishing), the coefficient is notstatistically significant.

3. THE ROLE OF TECHNOLOGY

This section investigates whether the departure from the Pareto law can beexplained on the ground of differences in the nature of technology.

With this aim, the sector-specific fixed effects of Equation (6) are distin-guished into a component common to all sectors and a component depend-ing on the industry technological variables. Therefore, in Equation (6) it isset αktj =αkt + δkSj , (k =1,2) where Sj is the vector of technological vari-ables of sector j , assumed invariant over the considered period, and δk avector of parameters. This produces the following equation:

log F(x)=∑

t

α1t dt log(x)+∑

t

α2t dt [log(x)]2

+δ1Sj log(x)+ δ2Sj [log(x)]2. (7)

Technological Variables

Four categories of variables are included in the vector Sj of technologi-cal variables: (i) the level of technological opportunity, (ii) the cumulative-ness of innovation, (iii) the sources of technological opportunity and (iv)the relative importance of product and process innovation. The variablesare constructed at the level of industrial sector, using CIS-2 data.

Direct measures of innovative activity are derived through a combina-tion of input and output indicators. These are (a) the intensity of R&Dexpenditure as the ratio of the total R&D expenditure in the period1994–1996 on the total sales in 1996, (b) the percentage of turnover on thetotal sales in 1996 attributed to innovative products, distinguished in thethree categories of products new for the firm, products new for the mar-ket and improved products and (c) the percentage of innovators in 1994 to1996 that carry out R&D activities on a permanent basis, as opposed tooccasional or not at all. These five variables are then summarised by apply-ing a principal component analysis. This approach allowed me to extract


two main factors.7 The first factor is positively correlated with all the mea-sures of innovation intensity and can be interpreted as the general levelof technological opportunity, or “potential” for innovation (Technologicalopportunity). The second factor is positively correlated with the turnoverdue to improved products and the share of innovators with permanentR&D activities; it can be interpreted as an indicator of the incremental andcumulative nature of innovation (Cumulativeness).

In general, measures of the sources of technological opportunity arederived from innovation surveys by looking at the sources of informationthat are relevant for industrial innovation (Cohen et al., 1987). In partic-ular, the CIS-2 listed l2 sources, which are rated by the firm on a four-point Likert scale, ranging from 0 “not-used” to 3 “very important”. Theimportance of each source for innovation, at the level of industrial sec-tor, is thus measured as the percentage of innovators that rated the sourceas “important” or “very important”. The information generated from thisstep was summarised, via a principal component analysis, into four mainfactors. The first factor is positively correlated with the contribution ofpublic research from Universities and other research institutes, and the con-tribution of codified sources of publicly available knowledge (patent dis-closure and computer based information). This factor is interpreted as ameasure of the relevance of scientific knowledge for innovation (Science).The second factor is positively correlated with the contribution of informa-tion 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 con-tribution of customers, in combination with the use of in-house sources(Users). The last, fourth factor contrasts the contribution of competingfirms within the industry to the contribution of innovation centres, whichact 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 ismeasured by the ratio between the number of firms with at least oneproduct innovation and with at least one process innovation (PDT/PCS).Because responses to the innovation survey may differ between small andlarge firms, I controlled for the relative size of innovative firms in a sec-tor. This was measured as the ratio between the average sales of innova-tive 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 ofspace and they are available upon request.

8 This category includes both academic and professional journals.

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the innovation process. Finally, to account for the relationship betweenmarket concentration and the mobility of firm market shares (Caves, 1998)an index of persistence of firms size (Persistence) is built on the basis ofthe Business Register dataset. This index measures the percentage of all thecontinuing firms that remain within the same size class between 1997 and1998.

Note that the aim of estimating the model is not to fully identifyingthe determinants of the size distribution of firms. I do not wish to excludethe possibility that other variables could be important, for example, marketdemand.

Table VI presents the results of the OLS estimates of Equation (7) forthe linear and quadratic specification. The variables Innovator size andPersistence are added as control variables.

Table VI shows that the technological variables have a significant effecton the slope of the Pareto law. I begin by examining the results forthe linear estimation of the model. These results show that adding thetechnology variables increases significantly the R-squared with respect tothe baseline model of identical sectoral coefficients. Specifically, the level oftechnological opportunity, the cumulativeness of innovation and the con-tribution of knowledge from the science system, industry and users have apositive effect on the Pareto coefficient (which decreases in absolute value),leading to higher market concentration. In contrast, the contribution ofknowledge from suppliers and the prevalence of product innovation onprocess innovation have a negative effect on the Pareto coefficients, leadingto lower market concentration. This findings mirror the results of Breschiet al. (2000).

Adding a quadratic term to the model leads to a further statisticallysignificant increase of the R-squared. Although this increase is modest, itmodifies the overall pattern of relationships with the technological vari-ables. In particular, the level of technological opportunity does not havea statistically significant effect on the slope and concavity (or convexity)of the distribution. This confirms the weak results found for this variablein other studies (Levin et al., 1985). Interestingly, the cumulativeness ofinnovation has a significant negative effect on the quadratic term. Cumu-lativeness of learning thus increases the concavity of the distribution withrespect to the Pareto law. This result is consistent with Ijiri and Simon(1974) argument that autocorrelation in firm growth would result in aconcave distribution of firm size. One possible interpretation could be thatthe cumulativeness of innovation is a source of the autocorrelation in thefirm growth processes.

With regard to the sources of knowledge, the contribution of sciencehas a significant (positive) effect on the shape of the distribution. Sci-ence-based sectors show decreasing concavity of the distribution, that is, a


Table VI. Results from pooled OLS regressions of F(x) on technology variables (absolutet-statistics in parentheses)

Variable Linear Quadratic

Size −0.447∗∗∗ −0.468∗∗∗ −0.431∗∗∗ −0.497∗∗∗

(−31.01) (−21.86) (−31.76) (−28.64)Technological 0.019∗∗∗ 0.020∗∗∗ 0.016 0.014opportunity (4.27) (4.45) (1.31) (1.16)Cumulativeness 0.044∗∗∗ 0.045∗∗∗ 0.141∗∗∗ 0.139∗∗∗

(12.19) (12.07) (12.24) (12.03)Science 0.073∗∗∗ 0.073∗∗∗ 0.026∗∗ 0.026∗∗

(17.69) (17.65) (2.02) (2.05)Suppliers −0.057∗∗∗ −0.058∗∗∗ −0.071∗∗∗ −0.074∗∗∗

(−17.36) (−17.39) (−6.46) (−6.72)Users 0.027∗∗∗ 0.028∗∗∗ −0.002 −0.005

(6.91) (7.04) (−0.18) (−0.46)Industry 0.055∗∗∗ 0.054∗∗∗ 0.063∗∗∗ 0.064∗∗∗

(14.97) (14.77) (5.47) (5.62)PDT/PCS −0.169∗∗∗ −0.171∗∗∗ −0.015 –

(−16.53) (−16.58) (−0.96)Innovator size 0.018 –

(1.34)Persistence – –Size2 – –Technological −0.001 0.00opportunity2 (−0.5) (−0.13)Cumulativeness2 −0.020∗∗∗ −0.020∗∗∗

(−9.42) (−9.24)Science2 0.011∗∗∗ 0.011∗∗∗

(4.81) (4.65)Suppliers2 0.004∗ 0.004∗∗

(1.76) (2.19)Users2 0.006∗∗∗ 0.007∗∗∗

(2.91) (3.44)Industry2 0.000 0.00

(0.23) (0.04)PDT/PCS2 −0.032∗∗∗ −0.038∗∗∗

(−13.58) (−21.45)Innovator size2 −0.003

(−1.11)Persistence2 0.022∗∗∗

(4.37)

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Table VI. Continued

Variable Linear Quadratic

DF 3643 3642 3634 3633Adjusted R2 0.796 0.796 0.821 0.823Increase in 0.105∗∗∗a 0.025∗∗∗

adjusted R2

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 estimationbecause of collinearity. aDifference calculated with respect to the baseline linear model ofcolum [1] in Table V.

relatively lower presence of medium-sized firms. In contrast, the contribu-tion of knowledge from suppliers has a significant negative effect on theslope of the distribution and a positive (although slightly significant) effecton the curve shape. In other words, supplier-dominated sectors (Pavitt,1984) are characterised by low market concentration and a slightly lessconcave distribution. Sources within the industry tend to increase the levelof market concentration. The contribution of users leads to a less concavedistribution, but it has no statistically significant effect on the slope of thedistribution. Finally, the nature of innovation has a statistically significant(negative) effect on the coefficient of the quadratic term. Industries whereproduct innovations are dominant over process innovations display a morepronounced concavity of the distribution.

The control variable for the existence of scale advantages in innovationdoes not have a statistically significant effect in either the linear or qua-dratic versions of the model. In contrast, a significant effect is observedfor the measure of persistence within the distribution. In particular, thedegree of persistence within the distribution has a significant and positiveeffect on the quadratic term of the model. Therefore, increasing mobilityof firms across size classes leads to increasing concavity of the distribution.Thus, distributions with more mobility are characterised by higher pres-ence of firms in the central size classes than under the Pareto law, and lessskewness 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 dis-tribution. Using data on the Dutch manufacturing firms, these propertieshave been explored at the level of the aggregate manufacturing and acrossindustrial sectors. Second, the paper has attempted to establish whether


systematic departures from the Pareto law emerged at the different levelsof analysis and whether such departures could be related to the nature oftechnology. In order to explore the two questions, the employment datafrom the Business Register of the population of manufacturing firms in theNetherlands in 1996 to 1998 have been linked with the data from the CISfor the Dutch manufacturing sector in 1994 to 1996.

Overall, I found that the Pareto law fits the size distribution of firmsonly as a first approximation. At the aggregate level, the size distributiondisplays a certain concavity. This result is consistent with previous studies(Ijiri and Simon, 1974; Scherer, 1998). The results based on a finer tabu-lation of size classes suggest that the concavity of the distribution is not apure statistical outcome of the extreme classes definition (Axtell, 2001).

The Dutch manufacturing system appears to depart more systematicallyfrom the structure of a self-organising system, than was observed for exam-ple in the US economy. This result suggests that the Dutch system mightbe more sensitive to external factors than to its own internal processes,compared to a large economy as the US. In the Netherlands, the size dis-tribution is characterised by lower skewness towards small firms and higherpresence of medium sized firms, than expected under a power law. Herepower law behaviour is observed only at the extreme upper tail of thedistribution, corresponding to less than one per cent of the population.

The analysis at the sectoral-level showed that the departures from thePareto law were more pronounced than at the aggregate level. A varietyof distributional forms appears to emerge across industrial sectors. In otherwords, sectoral characteristics shape the distribution of firm size.

These sectoral characteristics are often linked to features of technologypresent in these sectors (Nelson and Winter, 1982; Dosi et al., 1995). Threecharacteristics appear here to be associated with concave distributions: thecumulative nature of innovation, the dominance of product on processinnovation, and the mobility of firms in their relative position. A possibleinterpretation would be that the Schumpeterian process of dynamic compe-tition that originates in the continuous introduction of new products vari-eties enables small firms to growth, to reach the middle range of the sizedistribution. In contrast, more radical innovations, generated by scientificdevelopments, are necessary to reach the upper tail of the size distributionand maintain such a position over time. This result provides a differentperspective, based on the nature of technology, on the interpretation of thedepartures from Pareto earlier suggested by Ijiri and Simon (1974). Theirsuggestions were based on the autocorrelation of firm growth rates. I wantto suggest this autocorrelation might originate in the persistence of inno-vation due to the cumulative learning processes of firms (Mazzucato andGeroski, 2002).

326 ORIETTA MARSILI

Future research is required on the relationship between technology andthe size distribution of firms. It would also be interesting to find outwhether the properties of the Pareto law are invariant across countries,especially given their different investment profile in technology. Indeed, atentative comparison with results obtained in a similar study suggests thatthe Dutch manufacturing departs visibly from a self-organising system thatcharacterised the US firms. The study was focused on departures from thePareto law as measured through a polynomial fit of the empirical distri-bution. A further question arising from my study concerns the class oftheoretical distributions and the underlying processes of growth that mayaccount for the observed departures. My results hinted that a variety ofdistributional forms might characterise different sectors. Further researchshould attempt to explore broader classes of theoretical distributions torepresent more fully the variety of industrial patterns.

Acknowledgements

I wish to thank John Kwoka, George van Leeuwen, Roy Thurik andtwo anonymous referees for their helpful comments and suggestions. Theempirical part of this research has been carried out at the Centre forResearch of Economic Microdata at Statistics Netherlands. The viewsexpressed in this paper are those of the author and do not necessarilyreflect the policies of Statistics Netherlands.

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Technology and the Size Distribution of Firms: Evidence from Dutch