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The growth of business firms: Theoretical framework and empirical evidence

Contributed by H. Eugene Stanley, November 2, 2005
Abstract
We introduce a model of proportional growth to explain the distribution P_{g}(g) of businessfirm growth rates. The model predicts that P_{g}(g) is exponential in the central part and depicts an asymptotic powerlaw behavior in the tails with an exponent ζ = 3. Because of data limitations, previous studies in this field have been focusing exclusively on the Laplace shape of the body of the distribution. In this article, we test the model at different levels of aggregation in the economy, from products to firms to countries, and we find that the predictions of the model agree with empirical growth distributions and sizevariance relationships.
Gibrat (1, 2), building on the work of the astronomers Kapteyn and Uven (3), assumed the expected value of the growth rate of a business firm's size to be proportional to the current size of the firm, which is called the law of proportionate effect (4, 5). Several models of proportional growth have been subsequently introduced in economics to explain the growth of business firms (6–8). Simon and coworkers (9–12) extended Gibrat's model by introducing an entry process according to which the number of firms rise over time. In the framework of Simon and coworkers, the market consists of a sequence of many independent “opportunities” that arise over time, each of size unity. Models in this tradition have been challenged by many researchers (13–17) who found that the firmgrowth distribution is not Gaussian but displays a tent shape.
Here we introduce a general framework that provides a unifying explanation for the growth of business firms based on the number and size distribution of their elementary constituent components (18–25). Specifically, we present a model of proportional growth in both the number of units and their size, and we draw some general implications on the mechanisms that sustain businessfirm growth (7, 11, 21, 26–28). According to the model, the probability density function (PDF) of growth rates is Laplace in the center (13) with powerlaw tails (29, 30) decaying as P_{g} (g) ∼ g ^{–ζ}, where ζ = 3.
Also, because of data limitations, previous studies in this field focus on the Laplace shape of the body of the distribution, which, however, is an unconditional object (31). Using a database on the size and growth of firms and products, we characterize the shape of the whole growthrate distribution.
We test our model by analyzing different levels of aggregation of economic systems, from the “micro” level of products to the “macro” level of industrial sectors and national economies. We find that the model accurately predicts the shape of the PDF of growth rate at all levels of aggregation studied.
The Theoretical Framework
We model business firms as classes consisting of a random number of units. According to this view, a firm is represented as the aggregation of its constituent units such as divisions (22), businesses (20), or products (21). Accordingly, on a different level of coarse graining, a class can represent a national economy composed of economic units such as firms. In this article, we study the logarithm of the 1year growth rate of classes g ≡ log[S(t + 1)/S(t)], where S(t) and S(t + 1) are the sizes of classes in the year t and t + 1 measured in monetary values [gross domestic product (GDP) for countries, sales for firms and products]. Our model is illustrated in Fig. 1. Two key sets of assumptions in the model are that the number of units in a class grows in proportion to the existing number of units (Assumptions 1–4) and the size of each unit fluctuates in proportion to its size (Assumptions 5 and 6).
Assumption 1. Each class α consists of K _{α}(t) number of units. At time t = 0 (time step measured by year, generally), there are N(0) classes consisting of n(0) total number of units. The initial average number of units in a class is thus n(0)/N(0).
Assumption 2. At each time step, a new unit is created. Thus, the number of units at time t is n(t) = n(0) + t.
Assumption 3. With birth probability b, this new unit is assigned to a new class so that the average number of classes at time t is N(t) = N(0) + bt.
Assumption 4. With probability 1 – b, a new unit is assigned to an existing class α with probability P _{α} = (1 – b) K _{α}(t)/n(t), so K _{α}(t + 1) = K _{α}(t) + 1.
For simplicity, we do not consider the decrease of the number of units in a class. In reality, elementary units enter and exit. Because we are considering the case of a growing economy, it is legitimate to assume that the entry rate is higher than the exit rate. On the average, the net entry rate of units can be simplified as a positive constant. In the model, the net entry rate of units is fixed at 1. Thus, at large t, it gives results equivalent to the ones that would have been obtained when considering a value for the exit rate of units.
Our goal is to find P(K), the probability distribution of the number of units in the classes at large t. This model in two limiting cases, (i) b = 0, K _{α} = 1 (α = 1, 2,... N(0)) and (ii) b ≠ 0, N(0) = 1, n(0) = 1, has exact analytical solutions P(K) = N(0)/t{t/[t + N(0)]} ^{K} [1 + O(1/t)] (32, 33) and lim _{t} _{→∞} P(K) = (1 + b′)Γ(K)Γ(2 + b′)/Γ (K + 2 + b′), where b′ = b/(1 – b) (34), respectively.
In the general case, the exact analytical solution is not known, and we obtain a numerical solution by computer simulations and compare it with the approximate meanfield solution (see, e.g., chapter 6 of ref. 35 and Appendix A).
Our results are consistent with the exactly solvable limiting cases as well as with the empirical data on the number of products in the pharmaceutical firms and can be summarized as follows. In the limit of large t, the distribution of K in the old classes that existed at t = 0 converges to an exponential distribution (36), where λ = 1 – 1/K(t) and K(t) is the average number of units in the old classes at time t, K(t) = {[n(0) + t]/n(0)}^{1–} ^{b} ·n(0) ^{b} /N(0). The distribution of units in the new classes created at t > 0 converges to a power law with an exponential cutoff, where φ = 2 + b/(1 – b) and f(K) decays for K → ∞ faster than P _{old}(K). The distribution of units in all classes is given by The meanfield approximation for P _{new}(K) is given by where K′ = K{n(0)/[n(0) + t]}^{1–} ^{b} .
Assumption 5. At time t, each class α has K _{α}(t) units of size ξ _{i} (t), i = 1,2,... K _{α}(t), where K _{α} and ξ _{i} > 0 are independent random variables taken from the distributions P(K _{α}) and P _{ξ}(ξ _{i} ), respectively. P(K _{α}) is defined by Eq. 3 and P _{ξ}(ξ _{i} ) is a given distribution with finite mean μ_{ξ} and standard deviation σ_{ξ} . We also assume that lnξ _{i} has finite mean m _{ξ} = 〈lnξ _{i} 〉 and variance . The size of a class is defined as .
Assumption 6. At time t + 1, the size of each unit is decreased or increased by a random factor η _{i} (t) > 0 so that where η _{i} (t) > 0, the growth rate of unit i, is an independent random variable taken from a distribution P _{η}(η _{i} ), which has a finite mean μ_{η} and standard deviation σ_{η} . We also assume that lnη _{i} has finite mean m _{η} ≡ 〈ln η _{i} 〉, and variance .
The growth rate of each class is defined as
Here we neglect the influx of the new units, so K _{α} = K _{α}(t + 1) = K _{α}(t). The resulting distribution of the growth rates of all classes is determined by where P(K) is the distribution of the number of units in the classes, computed in the previous stage of the model, and P_{g} (gK) is the conditional distribution of growth rates of classes with a given number of units determined by the distribution P _{ξ}(ξ) and P _{η}(η).
The analytical solution of this model can be obtained only for certain limiting cases, but a numerical solution can be computed easily for any set of assumptions. We investigate the model numerically and analytically (see Appendix B) and find:

The conditional distribution of the logarithmic growth rates P_{g} (gK) for the firms consisting of a fixed number of units converges to a Gaussian distribution for K → ∞, where V is a function of parameters of the distribution P _{ξ}(ξ) and P _{η}(η), and ḡ is the logarithm of mean growth rate of a unit, ḡ = ln μ_{η}. Thus, the width of this distribution decreases as . This result is consistent with the observation that large firms with many production units fluctuate less than small firms (7, 18, 22, 37).

For g ≫ V _{η}, the distribution P_{g} (g) coincides with the distribution of the logarithms of the growth rates of the units: In the case of powerlaw distribution P(K) ∼ K ^{–φ}, which dramatically increases for K → 1, the distribution P_{g} (g) is dominated by the growth rates of classes consisting of a single unit K = 1; thus, the distribution P_{g} (g) practically coincides with P _{η}(ln η _{i} ) for all g. Indeed, our empirical observations confirm this result.

If the distribution P(K) ∼ K ^{–φ}, φ > 2 for K → ∞, as happens in the presence of the influx of new units b ≠ 0, P_{g} (g) = C _{1} – C _{2}g^{2φ–3}, for g → 0, which in the limiting case b → 0, φ → 2 gives the cusp P_{g} (g) ∼ C _{1} – C _{2}g (C _{1} and C _{2} are positive constants), similar to the behavior of the Laplace distribution P _{L}(g) ∼ exp(–gC _{2}) for g → 0.

If the distribution P(K) weakly depends on K for K → 1, the distribution of P_{g} (g) can be approximated by a power law of g: P_{g} (g) ∼ g^{–3} in wide range , where K(t) is the average number of units in a class. This case is realized for b = 0, t → ∞ when the distribution of P(K) is dominated by the exponential distribution and K(t) → ∞ as defined by Eq. 1. In this particular case, P_{g} (g) for can be approximated by

In the case in which the distribution P(K) is not dominated by oneunit classes but for K → ∞ behaves as a power law, which is the result of the meanfield solution for our model when t → ∞, the resulting distribution P_{g} (g) has three regimes: P_{g} (g) ∼ C _{1} – C _{2}g^{2φ–3} for small g, P_{g} (g) ∼ g^{–3} for intermediate g, and P_{g} (g) ∼ P(ln η) for g → ∞. The approximate solution of P_{g} (g) is obtained by using Eq. 8 for P_{g} (gK) for finite K, meanfield solution Eq. 4 in the limit t → ∞ for P(K), and replacing summation by integration in Eq. 7: For b ≠ 0 the integral above cannot be expressed in elementary functions. In the b → 0 case, Eq. 11 yields the main result which combines the Laplace cusp for g → 0 and the powerlaw decay g^{–3} for g → ∞. Note that because of replacement of summation by integration in Eq. 7, the approximation Eq. 12 holds only for .
In Fig. 2a we compare the distributions given by Eq. 10, the meanfield approximation Eq. 11 for b = 0.1, and Eq. 12 for b 3 0. We find that all three distributions have very similar tentshape behavior in the central part. In Fig. 2b we also compare the distribution Eq. 12 with its asymptotic behaviors for g → 0 (Laplace cusp) and g → ∞ (power law) and find the crossover region between these two regimes.
The Empirical Evidence
To test our model, we analyze different levels of aggregation of economic systems, from the micro level of products to the macro level of industrial sectors and national economies.
First, we analyze a database, the pharmaceutical industry database (PHID), that records sales figures of the 189,303 products commercialized by 7,184 pharmaceutical firms in 21 countries from 1994 to 2004, covering the whole size distribution for products and firms and monitoring the flows of entry and exit at both levels (kindly provided by the epris program). Then, we study the growth rates of all U.S. publicly traded firms from 1973 to 2004 in all industries, based on Security Exchange Commission filings (Compustat). Finally, at the macro level, we study the growth rates of the GDP of 195 countries from 1960 to 2004 (World Bank).
Fig. 3 shows that the growth distributions of countries, firms, and products are well fitted by the distribution in Eq. 12 with different values of V_{g} . Indeed, growth distributions at any level of aggregation depict marked departures from a Gaussian shape. Moreover, even if the P_{g} (g) of GDP can be approximated by a Laplace distribution, the P_{g} (g) of firms and products are clearly more leptokurtic than Laplace. Based on our model, the growth distribution is Laplace in the body, with powerlaw tails. In fact, Fig. 4 shows that the central body part of the growthrate distributions at any level of aggregation is well approximated by a doubleexponential fit. Fig. 5 reveals that the asymptotic behaviors of g at any level of aggregation can be well fitted by power law with an exponent ζ = 3.
Our analysis in The Theoretical Framework predicts that the powerlaw regime of P_{g} (g) may vary depending on the behavior of P(K) for K → 1 and the distribution of the growth rates of units. In the case of the PHID, for which P(1) ≫ P(2) ≫ P(3)..., the growthrate distribution of firms must be almost the same as the growthrate distribution of products (as we stated in The Theoretical Framework). Hence, the powerlaw wings of P_{g} (g) for firms originate at the level of products. Because the PHID does not contain information on the subunits of products, we cannot test our prediction directly, but we can hypothesize that the distribution of the product subunits (number of customers or shipping ways) is less dominated by small K but has a sufficiently wide powerlaw regime because of the influx of new products. These rather plausible assumptions are sufficient to explain the shape of the distribution of the product growth rates, which is well described by Eq. 12.
The PHID allows us to test the empirical conditional distribution P_{g} (gK) and the dependence of its variance σ^{2} on K, where K is the number of products. We find that σ ∼ K ^{–0.28}, which is significantly smaller than behavior. This result does not imply correlations among product growth rates on the firm level (21) but can be explained by the fact that for skewed distributions of product sizes P _{ξ}(ξ) characterized by large V _{ξ}, the convergence of P_{g} (gK) to its Gaussian limit Eq. 8 is slow and the growth rates of the firms are determined by the growth of the few large products. Using the empirical values for the PHID μ_{ξ} = 3.44, V _{ξ} = 5.13, μ_{η} = 0.016, V _{η} = 0.36 and assuming lognormality of the distributions P _{ξ}(ξ) and P _{η}(η), we find that the behavior of σ can be well approximated by a power law σ ∼ K ^{–0.20} for K < 10^{3}. For this set of parameters, the convergence of P_{g} (gK) to a Gaussian distribution takes place only for K > 10^{5}. This result is consistent with the observations of the powerlaw relationship between firm size and growthrate variance reported earlier (13, 18, 19, 38).
Discussion
Business firms grow in scale and scope. The scope of a firm is given by the number of its products. The scale of a firm is given by the size of its products. A firm such as Microsoft gets few big products, whereas Amazon sells a huge variety of goods, each of small size in terms of sales. In this article we argue that both mechanisms of growth are proportional. The number of products that a firm can launch successfully is proportional to the number of products that it has already commercialized. Once a product has been launched, its success depends on the number of customers who buy it and the price they are willing to pay. To a large extent, if products are different enough, the success of a product is independent from other products commercialized by the same company. Hence, the sales of products can be modeled as independent stochastic processes. Moreover, sometimes, new products are commercialized by new companies. As a result, small companies with few products can experience sudden jerks of growth resulting from the successful launch of a new product.
In this article, we find that the empirical distribution of firm growth rates exhibits a central part that is distributed according to a Laplace distribution and powerlaw wings P_{g} (g) ∼ g ^{–ζ}, where ζ = 3. If the distribution of number of units K is dominated by singleunit classes, the tails of firm growth distribution are primarily due to smaller firms composed of one or few products. The Laplace center of the distribution is shaped by big multiproduct firms. We find that the shape of the distribution of firm growth is almost the same in the presence of a small entry rate and with zero entry. We also find that the predictions of the model are accurate in the case of product growth rates, which implies that products can be considered as composed of elementary sale units, which evolve according to a random multiplicative process (6). Although there are several plausible explanations for the Laplace body of the distribution, which can be considered as an unconditional object (18, 31), the powerlaw decay of the tails has not been observed previously. We introduce a simple and general model that accounts for both the central part and the tails of the distribution. The shape of the business growthrate distribution is due to the proportional growth of both the number and the size of the constituent units in the class. This result holds in the case of an open economy (with entry of new firms) as well as in the case of a closed economy (with no entry of new firms).
Acknowledgments
We thank S. Havlin, J. Nagler, and F. Wang for helpful discussions and suggestions. We thank the National Science Foundation and Merck Foundation (epris program) for financial support.
Footnotes

↵ † To whom correspondence may be addressed. Email: dffu{at}bu.edu or hes{at}bu.edu.

Author contributions: F.P., S.V.B., and H.E.S. designed research; D.F., F.P., S.V.B., M.R., K.M., K.Y., and H.E.S. performed research; S.V.B. contributed new reagents/analytic tools; D.F., S.V.B., M.R., K.M., and K.Y. analyzed data; and D.F., F.P., S.V.B., M.R., and K.M. wrote the paper.

Conflict of interest statement: No conflicts declared.

This paper was submitted directly (Track II) to the PNAS office.

Abbreviations: PDF, probability density function; PHID, pharmaceutical industry database; GDP, gross domestic product.
 Copyright © 2005, The National Academy of Sciences
References

↵
Gibrat, R. (1930) Bull. Stat. Gén. 19 , 469. pmid:10771136

↵
Gibrat, R. (1931) Les InégalitésÉconomiques (Librairie du Recueil Sirey, Paris).

↵
Kapteyn, J. & Uven, M. J. (1916) Skew Frequency Curves in Biology and Statistics (Hoitsema Brothers, Groningen).

↵
Zipf, G. (1949) Human Behavior and the Principle of Least Effort (Addison–Wesley, Cambridge, MA).

↵
Gabaix, X. (1999) Q. J. Econ. 114 , 739–767.

↵
Steindl, J. (1965) Random Processes and the Growth of Firms: A Study of the Pareto Law (Griffin, London).

↵
Sutton, J. (1997) J. Econ. Lit. 35 , 40–59.
 ↵

↵
Simon, H. A. (1955) Biometrika 42 , 425–440.

Simon, H. A. & Bonini, C. P. (1958) Am. Econ. Rev. 48 , 607–617.

↵
Ijiri, Y. & Simon, H. A. (1975) Proc. Natl. Acad. Sci. USA 72 , 1654–1657. pmid:16578724

↵
Ijiri, Y. & Simon, H. A. (1977) Skew Distributions and the Sizes of Business Firms (North–Holland, Amsterdam).
 ↵

Plerou, V., Amaral, L. A. N., Gopikrishnan, P., Meyer, M. & Stanley, H. E. (1999) Nature 433 , 433–437.
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵

↵
Kalecki, M. R. (1945) Econometrica 13 , 161–170.

Mansfield, D. E. (1962) Am. Econ. Rev. 52 , 1024–1051.
 ↵
 ↵

↵
Reed, W. J. & Hughes, B. D. (2002) Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 66 , 067103.

↵
Kotz, S., Kozubowski, T. J. & Podgórski, K. (2001) The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance (Birkhauser, Boston).

↵
Johnson, N. L. & Kotz, S. (1977) Urn Models and Their Applications (Wiley, New York).
 ↵
 ↵

↵
Stanley, H. E. (1971) Introduction to Phase Transitions and Critical Phenomena (Oxford Univ. Press, Oxford).

↵
Cox, D. R. & Miller, H. D. (1968) The Theory of Stochastic Processes (Chapman & Hall, London).
 ↵
 ↵
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