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The size variance relationship of business firm growth rates

Contributed by H. E. Stanley, October 17, 2008 (received for review September 2, 2008)
Abstract
The relationship between the size and the variance of firm growth rates is known to follow an approximate powerlaw 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 show how a model of proportional growth, which treats firms as classes composed of various numbers of units of variable size, can explain this sizevariance dependence. In general, the model predicts that β(S) must exhibit 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 industryspecific 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 sizevariance relationship.
Gibrat was probably the first who noticed the skew size distribution of business firms (1). As a simple candidate explanation he postulated the “Law of Proportionate Effect” according to which the expected value of the growth rate of a business firm is proportional to the current size of the firm (2). Several models of proportional 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 explain the distribution of sizes of genera. The Law of Proportionate Effect implies that the variance σ^{2} of firm growth rates is independent of size, whereas, according to the Simon model, it is inversely proportional 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 σ^{2} of its growth rate σ ≈ S^{−β} with β ≈ 0.2.
Numerous attempts have been made to explain this puzzling evidence by considering firms as collections of independent units of uneven size (10, 12–18) but existing models do not provide a unifying explanation for the probability density functions of the growth and size of firms as well as the sizevariance 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 proportional 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 sizevariance relationship. The main conclusion is that the sizevariance relationship is not a true power law with a single welldefined exponent β but undergoes a slow crossover from β = 0 for S → 0 to β = 1/2 for S → ∞. The predictions of the model are tested in both realworld and simulation settings.
The Model
In the model presented in ref. 21 and summarized in the supporting information (SI) Text, firms consist of a random 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, P_{η}, and the size distribution, P_{ξ}, of the units are lognormal.
To derive the sizevariance relationship we must compute the conditional probability density of the growth rate P(gS, K), of a firm with K units and size S. For K → ∞ the conditional probability density function P(gS, K) develops a tentshape functional form, because in the center it converges to a Gaussian distribution with the width decreasing in inverse proportion to
We can also compute the conditional probability P(SK), which is the convolution of K unit size distributions P_{ξ}. In case of lognormal P_{ξ} with a large logarithmic variance V_{ξ} and mean m_{ξ}, the convergence of P(SK) to a Gaussian is very slow (23). Because P(S, K) = P(SK)P(K), we can find where all of the distributions P(gS, K), P(SK), P(K) can be found from the parameters of the model. P(SK) has a sharp maximum near S = S_{K} ≡ K_{μξ}, where μ_{ξ} = exp(m_{ξ +} V_{ξ}/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 near K_{S} = S/μ_{ξ}. For the values of S such that P(K_{S}) ≫ 0, P(gS) ≈ P(gK_{S}), because P(SK) serves as a δ(K − K_{S}) so that only terms with K ≈ K_{S} make a dominant contribution to the sum of Eq. 1. Accordingly, one can approximate P(gS) by P(gK_{S}) and σ(S) by σ(K_{S}). However, all firms with S < S_{1} = μ_{ξ} consist essentially of only one unit and thus for S < μ_{ξ}. For large S, if P(K_{S}) > 0 where m_{η} and V_{η} are the logarithmic mean and variance of the unit growth distributions P_{η} and V = exp(V_{ξ})[exp(V_{η}) − 1], as in ref. 21. Thus, one expects to have a crossover from β = 0 for S < μ_{ξ} 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 V_{η} < 1, S* ≈ exp(3V_{ξ}/2 + m_{ξ}). The range of crossover extends from S_{1} to S*, with S*/S_{1} = exp(V_{ξ}) → ∞ for V_{ξ} → ∞. Thus, in the doublelogarithmic plot of σ vs. S one can find a wide region in which the slope β slowly 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*/μ_{ξ} = exp(V_{ξ}) is such that P(K*) is significantly larger than zero. For the distribution P(K) with a sharp exponential cutoff K = K_{0}, the crossover will be observed only if K_{0} ≫ exp(V_{ξ}).
Two scenarios are possible for S > S_{0} = K_{0}μ_{ξ}. In the first, there will be no firms with S ≫ S_{0}. In the second, if the distribution of the size of units P_{ξ} is very broad, large firms can exist just because the size of a unit can be larger than S_{0}. In this case, exceptionally large firms might consist of one extremely large unit ξ_{max}, whose fluctuations dominate the fluctuations of the entire firm.
One can introduce the effective number of units in a firm K_{e} = S/ξ_{max}, where ξ_{max} is the largest unit of the firm. If K_{e} < 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 V_{ξ} 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 predominantly consist of a large number of units. Taking into account the equation of P_{ξ}, one can see that P(ξ > S*) ∼ exp(−9/8V_{ξ}).
On the one hand, for an exponential P(K), this implies that exp(−exp(V_{ξ})/K_{0}) > exp(−9/8V_{ξ}) or This condition is easily violated if V_{ξ} ≫ ln K_{0}. Thus, for the distributions P(K) with exponential cutoff we will never see the crossover to β = 1/2 if V_{ξ} ≫ ln K_{0}.
On the other hand, for a powerlaw distribution P(K) ∼ K^{−φ}, the condition of the crossover becomes exp(V_{ξ})^{1−φ} > exp(−9/8V_{ξ}), or (φ − 1)V_{ξ} < 9/8V_{ξ} which is rigorously satisfied for but even for larger φ values this condition is not dramatically violated. Thus, for powerlaw 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 V_{ξ}. For powerlaw distributions we expect a sharper crossover than for exponential ones because the majority of firms in a powerlaw 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 slow crossover that is interrupted if V_{ξ} is comparable to ln K_{0}. For S ≫ S_{1} this crossover is well represented by the behavior of σ(K_{S}).
We confirm these heuristic arguments by means of computer simulations (see Figs. S1–S4).
Figs. 1 and 2 illustrate the importance of the effective number of units K_{e}. When K_{S} becomes larger than K_{0}, σ(S) starts to follow σ(K_{e}). Accordingly, for very large firms σ(S) becomes almost the same as for small firms. The maximal negative value of the slope β_{max} of the doublelogarithmic graphs presented in Fig. 1A correspond to the inflection points of these graphs, and can be identified as approximate values of β for different values of K_{0}. One can see that β_{max} increases as K_{0} increases from a small value close to 0 for K_{0} = 10 to a value close to 1/2 for K_{0} = 10^{5} in agreement with the predictions of the central limit theorem.
To further explore the effect of the P(K) on the sizevariance 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 cutoff for large K. Fig. 3 shows that, for a scalefree powerlaw distribution P(K), the sizevariance relationship depicts a steep crossover from σ =
As we see, the sizevariance relationship of firms σ(S) can be well approximated by the behavior of σ(K_{S}) (Fig. 1A). It was shown in ref. 24 that, for realistic V_{ξ}, σ^{2}(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 doublelogarithmic plot, increases from a small value dependent on V_{ξ} at small K to 1/2 for K → ∞. Accordingly, one can expect the same behavior for σ(S) for K_{S} < K_{0}.
Thus, it would be desirable to derive an exact analytical expression for σ(K) in case of lognormal and independent P_{ξ} and P_{η}. 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 do not depend on m_{η} and m_{ξ} except for the leading term in μ: m_{0} = m_{η} + V_{η}/2. Not being able to derive closeform expressions for σ (see SI Text), we perform extensive computer simulations, where ξ and η are independent random variables taken from lognormal distributions P_{ξ} and P_{η} with different V_{ξ} and V_{η}. The numerical results (Fig. 4) suggest that where F_{σ}(z) is a universal scaling function describing a crossover from F_{σ}(z)→ 0 for z → ∞ to F_{σ}(z)/z → 1 for z → −∞ and f(V_{ξ},V_{η}) ≈ f_{ξ}(V_{ξ})+f_{η}(V_{η}) are functions of V_{ξ} and V_{η} that have linear asymptotes for V_{ξ} → ∞ and V_{η} → ∞ (Fig. 4B).
Accordingly, we can try to define β(z) = (1 − dF_{σ}/dz)/2 (Fig. 5A). The main curve β(z) can be approximated by an inverse linear function of z, when z → −∞ and by a stretched exponential as it approaches the asymptotic value 1/2 for z → +∞. The particular analytical shapes for these asymptotes are not known and derived solely from leastsquare 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 does not depend on V_{η} and is approximately inverse proportional to a linear function of V_{ξ}: where P ≈ 0.54 and q ≈ 2.66 are universal values. (Fig. 5B). This finding is significant for our study, because it indicates that near its minimum, β(K) has a region of approximate constancy with the value β_{min} between 0.14 and 0.2 for V_{ξ} between 4 and 8. These values of V_{ξ} are quite realistic and correspond 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.
The above analysis shows that σ(S) is not a true powerlaw function, but undergoes a crossover from β = β_{min}(V_{ξ}) 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 > K_{0}, σ(S) will start to grow for S > K_{0}μ_{ξ}. Empirical data do not show 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 do see neither a crossover to β = 1/2 nor an increase of σ for large companies.
Empirical Evidence
Because the sizevariance relationship depends on the partition of firms into their constituent components, to properly 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, markets, and international brand names, with different distributions P(K) with 〈K〉 = K_{0} ranging from 5.8 for international products to almost 1,600 for markets (Table 1). If firms have on average K_{0} products and V_{ξ} ≪ ln K_{0}, the scaling variable z = K_{0} is positive and we expect β → 1/2. On the contrary, if V_{ξ} ≫ ln K_{0}, 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 near the most probable value S_{0} = μ_{ξ}K and there is no reason to define β(S). Only very rarely S ≫ S_{0}, because of a low probability of observing an extremely large product that dominates 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 powerlaw distributed, there is a wide range of values of K, so that there are always firms for which ln K ≫ V_{ξ} and we can expect a slow crossover from β = 0 for small firms to β = 1/2 for large firms. In this case, for a wide range of empirically plausible V_{ξ}, β is far from 1/2 and statistically different from 0. The estimated value of the sizevariance scaling coefficient β goes from 0.123 for products to 0.243 for therapeutic markets with companies in the middle (0.188) (Table 1, Fig. 2).
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 sizevariance 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 keep the real P(K) and randomly reassign products to firms. In the first simulation we randomly reassign products by keeping the realworld relationship between the size, ξ, and growth, η, of products. In the second simulation we also reassign 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 randomly reassigning elementary units to firms and markets. In doing that, we keep the number of the products in each class and the history of the fluctuation of each product sales unchanged. As for the size dependence, our analysis shows that there is indeed strong correlation between the number of products in the company and their average size defined as 〈ξ(K)〉 = 〈1/K Σ_{i = 1}^{K}ξ_{i}〉, where 〈 〉 indicates averaging over all companies with K products. We observe an approximate powerlaw dependence 〈ξ(K)〉 ∼ K^{γ}, where γ = 0.38. If this would be a true asymptotic power law holding for K → ∞ then the average size of the company of K products would be proportional to ξ(K)K ∼ K^{1+γ}. Accordingly, the average number of products in the company of size S would scale as K_{0}(S) ∼ S^{1/(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 explain β in refs. 11 and 15. Another effect of random redistribution of units will be the removal of possible correlations among η_{i} 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 powerlaw dependence 〈ρ(K)〉 ∼ K^{ζ}, where ζ = −0.36. Because larger firms have bigger products and are more diversified than small firms, the size dependence and unit interdependence cancel out and β practically does not change if products are randomly reassigned to firms.
To control the effect of time dependence, we keep the sizes of products ξ_{i} and their number K_{α} at year t for each firm α unchanged, so S_{t} = Σ_{i=1}^{Kα}ξ_{i} is the same as in the empirical data. However, to compute the sales of a firm in the following year S̃_{t+1} = Σ_{i=1}^{Kα}ξ_{i}′, we assume that ξ_{i}′ = ξ_{i}η_{i}, where η_{i} is an annual growth rate of a randomly selected product. The surrogate growth rate g̃ = ln
Finally we reproduced the model in ref. 21 with the empirically observed P(K) and the estimated moments of the lognormal distribution of products (m_{ξ} = 7.58, V_{ξ} = 4.41). We generate N random products according to our model (Gibrat process) with the empirically observed values of V_{ξ} and m_{ξ}. 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 correctly predicts the sizevariance relationship and the way it scales under aggregation.
The variance of the size of the constituent units of the firm V_{ξ} and the distribution of units into firms are both relevant to explain the sizevariance relationship of firm growth rates.
Simulations results in Fig. 6 reveal that if elementary units are of the same size (V_{ξ} = 0) the central limit theorem will work properly and β ≈ 1/2. As predicted by our model, by increasing the value of V_{ξ} we observe at any level of aggregation the crossover of β form 1/2 to 0. The crossover is faster at the level of markets than at the level of products due to the higher average number of units per class K_{0}. However, in realworld settings the central limit theorem never applies because firms have a small number of components of variable size (V_{ξ} > 0). For empirically plausible values of V_{ξ} and K_{0} 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 open new plants and launch new products, but the revenues and return to the investments are uncertain. If revenues were independent random variables drawn from a Gaussian distribution with mean m_{e} and variance V_{e} 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 = m_{e}K. On the contrary, if the size of business opportunities is given by a multiplicative brownian motion (Gibrat process) and revenues are independent random variables drawn from a lognormal distribution with mean m_{ξ} and variance V_{ξ}, the central limit theorem does not work effectively and β(S) exhibits 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 K_{0} and the variance of the size of business opportunities V_{ξ}. 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 important 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 Affairs of Yeshiva University (Yeshiva University highperformance computer cluster), and the Dr. Bernard W. Gamson Computational Science Center (Yeshiva College).
Footnotes
 ^{1}To whom correspondence may be addressed. Email: massimo.riccaboni{at}unitn.it, buldyrev{at}yu.edu, or hes{at}bu.edu

Author 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
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