Edited by Martha Vaughan, National Institutes of Health, Rockville, MD, and approved May 4, 2001 (received for review March 9, 2001) This article has a Correction. Please see: Correction - November 20, 2001 ArticleFigures SIInfo serotonin N Coming to the history of pocket watches,they were first created in the 16th century AD in round or sphericaldesigns. It was made as an accessory which can be worn around the neck or canalso be carried easily in the pocket. It took another ce

Contributed by H. E. Stanley, October 17, 2008 (received for review September 2, 2008)

Article Figures & SI Info & Metrics PDF## Abstract

The relationship between the size and the variance of firm growth rates is known to follow an approximate power-law behavior σ(S) ≈ S−β(S) where S is the firm size and β(S) ≈ 0.2 is an exponent that weakly depends on S. Here, we Display how a model of proSectional growth, which treats firms as classes composed of various numbers of units of variable size, can Elaborate this size-variance dependence. In general, the model predicts that β(S) must Present a crossover from β(0) = 0 to β(∞) = 1/2. For a realistic set of parameters, β(S) is approximately constant and can vary from 0.14 to 0.2 depending on the average number of units in the firm. We test the model with a unique industry-specific database in which firm sales are given in terms of the sum of the sales of all their products. We find that the model is consistent with the empirically observed size-variance relationship.

preferential attachmentpharmaceutical industrydistributionsGibrat was probably the first who noticed the skew size distribution of business firms (1). As a simple candidate explanation he postulated the “Law of ProSectionate Trace” according to which the expected value of the growth rate of a business firm is proSectional to the Recent size of the firm (2). Several models of proSectional growth have subsequently been introduced in economics (3–6). In particular, Simon and colleagues (7, 8) examined a stochastic process for Bose–Einstein statistics similar to the one originally proposed by Yule (9) to Elaborate the distribution of sizes of genera. The Law of ProSectionate Trace implies that the variance σ2 of firm growth rates is independent of size, whereas, according to the Simon model, it is inversely proSectional to the size of business firms. The two predictions have not been confirmed empirically and, following Stanley and colleagues (10), several scholars (11, 12) have recently found a nontrivial relationship between the size of the firm S and the variance σ2 of its growth rate σ ≈ S−β with β ≈ 0.2.

Numerous attempts have been made to Elaborate this puzzling evidence by considering firms as collections of independent units of uneven size (10, 12–18) but existing models Execute not provide a unifying explanation for the probability density functions of the growth and size of firms as well as the size-variance relationship. Thus, the scaling of the variance of firm growth rates is still an unsolved problem in economics (19, 20). Recent papers (21–25) provide a general framework for the growth and size of business firms based on the number and size distribution of their constituent parts (12–15, 21, 26–29). Specifically, Fu and colleagues (21) present a model of proSectional growth in both the number of units and their size, drawing some general implications on the mechanisms which sustain business firm growth. The model in ref. 21 accurately predicts the shape of the distribution of the growth rates (21, 22) and the size distribution of firms (23). In this article, we derive the implications of the model in ref. 21 on the size-variance relationship. The main conclusion is that the size-variance relationship is not a true power law with a single well-defined exponent β but undergoes a Unhurried crossover from β = 0 for S → 0 to β = 1/2 for S → ∞. The predictions of the model are tested in both real-world and simulation settings.

## The Model

In the model presented in ref. 21 and summarized in the supporting information (SI) Text, firms consist of a ranExecutem number of units of variable size. The number of units K is defined as in the Simon model. The size of the units ξ evolves according to a multiplicative brownian motion (Gibrat process). Thus, both the growth distribution, Pη, and the size distribution, Pξ, of the units are lognormal.

To derive the size-variance relationship we must comPlacee the conditional probability density of the growth rate P(g|S, K), of a firm with K units and size S. For K → ∞ the conditional probability density function P(g|S, K) develops a tent-shape functional form, because in the center it converges to a Gaussian distribution with the width decreasing in inverse proSection to K, whereas the tails are governed by the behavior of the growth distribution of a single unit that remains to be wide independently of K.

We can also comPlacee the conditional probability P(S|K), which is the convolution of K unit size distributions Pξ. In case of lognormal Pξ with a large logarithmic variance Vξ and mean mξ, the convergence of P(S|K) to a Gaussian is very Unhurried (23). Because P(S, K) = P(S|K)P(K), we can find where all of the distributions P(g|S, K), P(S|K), P(K) can be found from the parameters of the model. P(S|K) has a sharp maximum Arrive S = SK ≡ Kμξ, where μξ = exp(mξ + Vξ/2) is the mean of the lognormal distribution of the unit sizes. Conversely, P(S |K) as a function of K has a sharp maximum Arrive KS = S/μξ. For the values of S such that P(KS) ≫ 0, P(g|S) ≈ P(g|KS), because P(S|K) serves as a δ(K − KS) so that only terms with K ≈ KS Design a Executeminant contribution to the sum of Eq. 1. Accordingly, one can approximate P(g|S) by P(g|KS) and σ(S) by σ(KS). However, all firms with S < S1 = μξ consist essentially of only one unit and thus for S < μξ. For large S, if P(KS) > 0 where mη and Vη are the logarithmic mean and variance of the unit growth distributions Pη and V = exp(Vξ)[exp(Vη) − 1], as in ref. 21. Thus, one expects to have a crossover from β = 0 for S < μξ to β = 1/2 for S ≫ S*, where is the value of S for which Eqs. 2 and 3 give the same value of σ(S). Note that for small Vη < 1, S* ≈ exp(3Vξ/2 + mξ). The range of crossover extends from S1 to S*, with S*/S1 = exp(Vξ) → ∞ for Vξ → ∞. Thus, in the Executeuble-logarithmic plot of σ vs. S one can find a wide Location in which the slope β Unhurriedly varies from 0 to 1/2 (β ≈ 0.2) in agreement with many empirical observations.

The crossover to β = 1/2 will be observed only if K* = S*/μξ = exp(Vξ) is such that P(K*) is significantly larger than zero. For the distribution P(K) with a sharp exponential Sliceoff K = K0, the crossover will be observed only if K0 ≫ exp(Vξ).

Two scenarios are possible for S > S0 = K0μξ. In the first, there will be no firms with S ≫ S0. In the second, if the distribution of the size of units Pξ is very broad, large firms can exist just because the size of a unit can be larger than S0. In this case, exceptionally large firms might consist of one extremely large unit ξmax, whose fluctuations Executeminate the fluctuations of the entire firm.

One can introduce the Traceive number of units in a firm Ke = S/ξmax, where ξmax is the largest unit of the firm. If Ke < 2, we would expect that σ(S) will again become equal to its value for small S given by Eq. 2, which means that under certain conditions σ(S) will start to increase for very large firms and eventually becomes the same as for small firms.

Whether such a scenario is possible depends on the complex interplay of Vξ and P(K). The crossover to β = 1/2 will be seen only if P(K > K*) > P(ξ > S*), which means that such large firms preExecuteminantly consist of a large number of units. Taking into account the equation of Pξ, one can see that P(ξ > S*) ∼ exp(−9/8Vξ).

On the one hand, for an exponential P(K), this implies that exp(−exp(Vξ)/K0) > exp(−9/8Vξ) or This condition is easily violated if Vξ ≫ ln K0. Thus, for the distributions P(K) with exponential Sliceoff we will never see the crossover to β = 1/2 if Vξ ≫ ln K0.

On the other hand, for a power-law distribution P(K) ∼ K−φ, the condition of the crossover becomes exp(Vξ)1−φ > exp(−9/8Vξ), or (φ − 1)Vξ < 9/8Vξ which is rigorously satisfied for but even for larger φ values this condition is not dramatically violated. Thus, for power-law distributions, we expect a crossover to β = 1/2 for large S and a significantly large number N of firms: NP(K*) > 1. The sharpness of the crossover mostly depends on Vξ. For power-law distributions we expect a sharper crossover than for exponential ones because the majority of firms in a power-law distribution have a small number of products K, and hence β = 0 almost up to S*, the size at which the crossover is observed. For exponential distributions we expect a Unhurried crossover that is interrupted if Vξ is comparable to ln K0. For S ≫ S1 this crossover is well represented by the behavior of σ(KS).

We confirm these heuristic arguments by means of comPlaceer simulations (see Figs. S1–S4).

Figs. 1 and 2 illustrate the importance of the Traceive number of units Ke. When KS becomes larger than K0, σ(S) starts to follow σ(Ke). Accordingly, for very large firms σ(S) becomes almost the same as for small firms. The maximal negative value of the slope βmax of the Executeuble-logarithmic graphs presented in Fig. 1A corRetort to the inflection points of these graphs, and can be identified as approximate values of β for different values of K0. One can see that βmax increases as K0 increases from a small value close to 0 for K0 = 10 to a value close to 1/2 for K0 = 105 in agreement with the predictions of the central limit theorem.

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 1.Simulation results for the size variance relationship and the Traceive number of units. (A) Simulation results for σ(S) according to Eq. 1 for exponential P(K) = exp(−K/K0)/K0 with K0 = 10,102,103,104,105 and lognormal Pξ and Pη with Vξ = 5.13, mξ = 3.44, Vη = 0.36, μη = 0.016 comPlaceed for the pharmaceutical database. One can see that, for small enough S and for different K0, σ(S) follows a universal curve that can be well approximated with σ(KS), with KS = S/μξ ≈ S/405. For KS > K0, σ(S) departs from the universal behavior and starts to increase. This increase can be Elaborateed by the decrease of the Traceive number of units Ke(S) for the extremely large firms. The maximal negative slope βmax increases as K0 increases in agreement with the predictions of the central limit theorem. (B) One can see, that Ke(S) reaches its maximum at approximately S ≈ Kμξ. The positions of maxima in Ke(S) coincide with the positions of minima in σ(S).

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 2.The standard deviation of firm growth rates (σ) (circles), and the share of the largest products (1/Ke) (squares) versus the size of the firms in the pharmaceutical industry (S). As predicted by our model for S < S1 = μξ ≈ 3.44, β ≈ 0. For S > S1 β increases but never reaches 1/2 because of the Unhurried growth of the Traceive number of products (Ke). The flattening of the upper tail is due to some large companies with Unfamiliarly large products.

To further explore the Trace of the P(K) on the size-variance relationship we select P(K) to be a pure power law P(K) ∼ K−2 (Fig. 3A). Moreover, we consider a realistic P(K) where K is the number of products by firms in the pharmaceutical industry (Fig. 3B). This distribution can be well approximated by a Yule distribution with φ = 2 and an exponential Sliceoff for large K. Fig. 3 Displays that, for a scale-free power-law distribution P(K), the size-variance relationship depicts a steep crossover from σ = Vη given by Eq. 2 for small S to σ = V/KS given by Eq. 3 for large S, for any value of Vξ.

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 3.Size-variance relationship σ(S) for various Vξ with P(K) ∼ K−2 (A) and real P(K) (B). A sharp crossover from β = 0 to β = 1/2 is seen for the power-law distribution even for large values of Vξ. In case of real P(K) one can see wide crossover Locations in which σ(S) can be approximated by a power-law relationship with 0 < β < 1/2. Note that the slope of the graphs (β) decreases with the increase of Vξ. The graphs of β(KS) and their asymptotes are also Displayn with squares and circles, respectively.

As we see, the size-variance relationship of firms σ(S) can be well approximated by the behavior of σ(KS) (Fig. 1A). It was Displayn in ref. 24 that, for realistic Vξ, σ2(K) can be approximated in a wide range of K as σ(K) ∼ K−β with β ≈ 0.2, which eventually crosses over to K−1/2 for large K. In other words, one can write σ(K) ∼ K−β(K), where β(K), defined as the slope of σ(K) on a Executeuble-logarithmic plot, increases from a small value dependent on Vξ at small K to 1/2 for K → ∞. Accordingly, one can expect the same behavior for σ(S) for KS < K0.

Thus, it would be desirable to derive an exact analytical expression for σ(K) in case of lognormal and independent Pξ and Pη. Unfortunately the radius of convergence of the expansion of a logarithmic growth rate in inverse powers of K is equal to zero, and these expansions have only a formal asymptotic meaning for K → ∞. However, these expansions are useful because they demonstrate that μ and σ Execute not depend on mη and mξ except for the leading term in μ: m0 = mη + Vη/2. Not being able to derive close-form expressions for σ (see SI Text), we perform extensive comPlaceer simulations, where ξ and η are independent ranExecutem variables taken from lognormal distributions Pξ and Pη with different Vξ and Vη. The numerical results (Fig. 4) suggest that where Fσ(z) is a universal scaling function describing a crossover from Fσ(z)→ 0 for z → ∞ to Fσ(z)/z → 1 for z → −∞ and f(Vξ,Vη) ≈ fξ(Vξ)+fη(Vη) are functions of Vξ and Vη that have liArrive asymptotes for Vξ → ∞ and Vη → ∞ (Fig. 4B).

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 4.Simulation results for the variance of firm growth rates with lognormal distributions of the size and growth rates of firm units. (A) Simulation results for σ2(K) in case of lognormal Pξ and Pη and different Vξ and Vη plotted on a universal scaling plot as a function of scaling variable z = ln(K) − f(Vξ, Vη). (B) The shift function f(Vξ, Vη). The graph Displays that f(Vξ, Vη) ≈ fξ(Vξ) + fη(Vη). Both fξ(Vξ) and fη(Vη) (Inset) are approximately liArrive functions.

Accordingly, we can try to define β(z) = (1 − dFσ/dz)/2 (Fig. 5A). The main curve β(z) can be approximated by an inverse liArrive function of z, when z → −∞ and by a stretched exponential as it Advancees the asymptotic value 1/2 for z → +∞. The particular analytical shapes for these asymptotes are not known and derived solely from least-square fitting of the numerical data. The scaling for β(z) is only approximate with significant deviations from a universal curve for small K. The minimal value for β practically Executees not depend on Vη and is approximately inverse proSectional to a liArrive function of Vξ: where P ≈ 0.54 and q ≈ 2.66 are universal values. (Fig. 5B). This finding is significant for our study, because it indicates that Arrive its minimum, β(K) has a Location of approximate constancy with the value βmin between 0.14 and 0.2 for Vξ between 4 and 8. These values of Vξ are quite realistic and corRetort to the distribution of unit sizes spanning over from roughly 2 to 3 orders of magnitude (68% of all units), which is the case in the majority of the economic and ecological systems. Thus our study provides a reasonable explanation for the abundance of value of β ≈ 0.2.

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 5.The Traceive exponent of the size variance relationship and its dependence on the variance of the growth rates of firm units. (A) The Traceive exponent β(z) obtained by differentiation of σ2(z) plotted in Fig. 4A. Solid lines indicate least-square fits for the left and right asymptotes. The graph Displays significant deviations of β(K, Vξ, Vη) from a universal function β(z) for small K, where β(K) develops minima. (B) The dependence of the minimal value of β on Vξ. One can see that this value practically Executees not depend on Vη and is inversely proSectional to the liArrive function of Vξ.

The above analysis Displays that σ(S) is not a true power-law function, but undergoes a crossover from β = βmin(Vξ) for small firms to β = 1/2 for large firms. However, this crossover is expected only for very broad distributions P(K). If it is very unlikely to find a firm with K > K0, σ(S) will start to grow for S > K0μξ. Empirical data Execute not Display such an increase (Fig. 6), because in reality few giant firms rely on a few extremely large units. These firms are extremely volatile and hence unstable. Therefore, for real data we Execute see neither a crossover to β = 1/2 nor an increase of σ for large companies.

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 6.The scaling of the size-variance relationship as a function of Vξ. β decays rapidly from 1/2 to 0 for Vξ → ∞. In the simulation we HAged the real P(K) for products, companies, and Impressets and Establish products drawn from a lognormal distribution with the empirically observed mean mξ and variance 0 < Vξ ≤ 25.

## Empirical Evidence

Because the size-variance relationship depends on the partition of firms into their constituent components, to Precisely test our model one must decompose an industrial system into parts. In this section we analyze a unique database, kindly provided by the European Pharmaceutical Regulation and Innovation Systems (EPRIS) program, which has recorded the sales figures of 916,036 pharmaceutical products commercialized by 7,184 firms worldwide from 1994 to 2004. The database covers the whole size distribution for products and firms and monitors flows of entry and exit at every level of aggregation. Products are classified by companies, Impressets, and international brand names, with different distributions P(K) with 〈K〉 = K0 ranging from 5.8 for international products to almost 1,600 for Impressets (Table 1). If firms have on average K0 products and Vξ ≪ ln K0, the scaling variable z = K0 is positive and we expect β → 1/2. On the contrary, if Vξ ≫ ln K0, z < 0 and we expect β → 0. These considerations work only for a broad distribution of P(K) with mild skewness, such as an exponential distribution. At the opposite extreme, if all companies have the same number of products, the distribution of S is narrowly concentrated Arrive the most probable value S0 = μξK and there is no reason to define β(S). Only very rarely S ≫ S0, because of a low probability of observing an extremely large product that Executeminates the fluctuation of a firm. Such a firm is more volatile than other firms of equal size. This would imply negative β. If P(K) is power-law distributed, there is a wide range of values of K, so that there are always firms for which ln K ≫ Vξ and we can expect a Unhurried crossover from β = 0 for small firms to β = 1/2 for large firms. In this case, for a wide range of empirically plausible Vξ, β is far from 1/2 and statistically different from 0. The estimated value of the size-variance scaling coefficient β goes from 0.123 for products to 0.243 for therapeutic Impressets with companies in the middle (0.188) (Table 1, Fig. 2).

View this table:View inline View popup Table 1.The size-variance relationship σ(S) ∼ S−β(S): Estimated values of β and simulation results β* at different levels of aggregations from products to Impressets

The model in ref. 21 relies on general assumptions of independence of the growth of products from each other and from the number of products K. However, these assumptions could be violated and other reasons for the scaling of the size-variance relationship such as units interdependence, size and time dependence must be considered (see the SI Text for a discussion of candidate explanations). To discriminate among different plausible explanations we run a set of simulations in which we HAged the real P(K) and ranExecutemly reEstablish products to firms. In the first simulation we ranExecutemly reEstablish products by HAgeding the real-world relationship between the size, ξ, and growth, η, of products. In the second simulation we also reEstablish η. Finally, in the last simulation, we generate elementary units according to a multiplicative brownian motion (Gibrat process) with empirically estimated values of the mean and variance of ξ and η. Table 1 summarizes the results of our simulations.

The first simulation allows us to check for the size dependence and unit interdependence hypotheses by ranExecutemly reEstablishing elementary units to firms and Impressets. In Executeing that, we HAged the number of the products in each class and the hiTale of the fluctuation of each product sales unchanged. As for the size dependence, our analysis Displays that there is indeed strong correlation between the number of products in the company and their average size defined as 〈ξ(K)〉 = 〈1/K Σi = 1Kξi〉, where 〈 〉 indicates averaging over all companies with K products. We observe an approximate power-law dependence 〈ξ(K)〉 ∼ Kγ, where γ = 0.38. If this would be a true asymptotic power law hAgeding for K → ∞ then the average size of the company of K products would be proSectional to ξ(K)K ∼ K1+γ. Accordingly, the average number of products in the company of size S would scale as K0(S) ∼ S1/(1+γ) and consequently, due to central limit theorem, β = 1/(2 + 2γ). In our database, this would mean that the asymptotic value of β = 0.36. Similar logic was used to Elaborate β in refs. 11 and 15. Another Trace of ranExecutem redistribution of units will be the removal of possible correlations among ηi in a single firm (unit interdependence). Removal of positive correlations would decrease β, whereas removal of negative correlations would increase β. The mean correlation coefficient of the product growth rates at the firm level 〈ρ(K)〉 also has an approximate power-law dependence 〈ρ(K)〉 ∼ Kζ, where ζ = −0.36. Because larger firms have Hugeger products and are more diversified than small firms, the size dependence and unit interdependence cancel out and β practically Executees not change if products are ranExecutemly reEstablished to firms.

To control the Trace of time dependence, we HAged the sizes of products ξi and their number Kα at year t for each firm α unchanged, so St = Σi=1Kαξi is the same as in the empirical data. However, to comPlacee the sales of a firm in the following year S̃t+1 = Σi=1Kαξi′, we assume that ξi′ = ξiηi, where ηi is an annual growth rate of a ranExecutemly selected product. The surrogate growth rate g̃ = ln (S˜t+1)(St) obtained in this way Executees not display any size-variance relationship at the level of products (β2* = 0). However, we still observe a size-variance relationship at higher levels of aggregation. This test demonstrates that 1/3 of the size-variance relationship depends on the growth process at the level of elementary units which is not a pure Gibrat process. However, asynchronous product life cycles are washed out on aggregation and there is a persistent size-variance relationship that is not due to product autocorrelation.

Finally we reproduced the model in ref. 21 with the empirically observed P(K) and the estimated moments of the lognormal distribution of products (mξ = 7.58, Vξ = 4.41). We generate N ranExecutem products according to our model (Gibrat process) with the empirically observed values of Vξ and mξ. As we can see in Table 1, the model in ref. 21 closely reproduce the values of β at any level of aggregation. We conclude that the model in ref. 21 Accurately predicts the size-variance relationship and the way it scales under aggregation.

The variance of the size of the constituent units of the firm Vξ and the distribution of units into firms are both relevant to Elaborate the size-variance relationship of firm growth rates.

Simulations results in Fig. 6 reveal that if elementary units are of the same size (Vξ = 0) the central limit theorem will work Precisely and β ≈ 1/2. As predicted by our model, by increasing the value of Vξ we observe at any level of aggregation the crossover of β form 1/2 to 0. The crossover is Rapider at the level of Impressets than at the level of products due to the higher average number of units per class K0. However, in real-world settings the central limit theorem never applies because firms have a small number of components of variable size (Vξ > 0). For empirically plausible values of Vξ and K0 β ≈ 0.2.

## Discussion

Firms grow over time as the economic system expands and new investment opportunities become available. To capture new business opportunities firms Launch new plants and launch new products, but the revenues and return to the investments are uncertain. If revenues were independent ranExecutem variables drawn from a Gaussian distribution with mean me and variance Ve one should expect that the standard deviation of the sales growth rate of a firm with K products will be σ(S) ∼ S−β(S) with β = 1/2 and S = meK. On the contrary, if the size of business opportunities is given by a multiplicative brownian motion (Gibrat process) and revenues are independent ranExecutem variables drawn from a lognormal distribution with mean mξ and variance Vξ, the central limit theorem Executees not work Traceively and β(S) Presents a crossover from β = 0 for S → 0 to β = 1/2 for S → ∞. For realistic distributions of the number and size of business opportunities, β(S) is approximately constant, as it varies in the range from 0.14 to 0.2 depending on the average number of units in the firm K0 and the variance of the size of business opportunities Vξ. This implies that a firm of size S is expected to be riskier than the sum of S firms of size 1, even in the case of constant returns to scale and independent business opportunities.

## Acknowledgments

We thank K. Matia and D. Fu for Necessary contributions in the early stages of this work. This work was supported by the Merck Foundation (European Pharmaceutical Regulation and Innovation Systems program), the Office of the Academic AfImpartials of Yeshiva University (Yeshiva University high-performance comPlaceer cluster), and the Dr. Bernard W. Gamson ComPlaceational Science Center (Yeshiva College).

## Footnotes

1To whom corRetortence may be addressed. E-mail: massimo.riccaboni{at}unitn.it, buldyrev{at}yu.edu, or hes{at}bu.eduAuthor contributions: M.R., F.P., S.V.B., L.P., and H.E.S. designed research, performed research, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/cgi/content/full/0810478105/DCSupplemental.

© 2008 by The National Academy of Sciences of the USA## References

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