An analogy of the size distribution of business firms with Bose–Einstein statistics

Physica A 389 (2010) 3837–3843

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An analogy of the size distribution of business firms with Bose–Einstein statistics R. Hernández-Pérez ∗ SATMEX, Av. de las Telecomunicaciones S/N CONTEL Edif. SGA-II. México, D.F. 09310, Mexico

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Article history: Received 30 December 2009 Received in revised form 20 April 2010 Available online 1 June 2010 Keywords: Econophysics Firms’ size distribution Bose–Einstein statistics

abstract We approach the size distribution of business firms by proposing an analogy of the firms’ ranking with a boson gas, identifying the annual revenue of the firms with energy. We found that Bose–Einstein statistics fits very well to the empirical cumulative distribution function for the firms’ ranking for different countries. The fitted values for the temperature-like parameter are compared between countries and with an index of economic development, and we found that our results support the hypothesis that the temperature of the economy can be associated with the level of economic development of a country. Moreover, for most of the countries the value obtained for the fugacity-like parameter is close to 1, suggesting that the analogy could correspond to a photon gas in which the number of particles is not conserved; this is indeed the case for real-world firms’ dynamics, where new firms arrive in the economy and other firms disappear, either by merging with others or through bankruptcy. © 2010 Elsevier B.V. All rights reserved.

1. Introduction One of the complex systems in economics studied by physicists and mathematicians is that constituted by business firms in different economies [1,2], and the study of firm size distribution has received much attention, primarily for developed economies [3–7], although other works have extended the analysis to emergent economies [8,9] and global rankings [10]. One approach to the study of firm size distribution is based on the analysis of firm rankings, focused on the statistical properties of firm rankings, where the cumulative distribution function (CDF) for sizes is often obtained, the main result being that the CDF is very close to a power law (or to a modified power law [6,8]), and that this distribution holds for different proxies for size, diverse time spans and different countries [3–6,8–11]. Some of the previous works have proposed a mapping of the firms set to a physical system—for instance, in Refs. [6,8] where it is proposed that the exponent of the modified power law could be identified with the temperature of the economy, which in turn could be related to the level of economic development [12] or with the average amount of money per economic agent [13]. Moreover, for the study of money and wealth distribution in individuals, there are relevant studies on wealth, money and income distributions related to the Boltzmann–Gibbs [13–16] and the κ -generalized [17] distributions. In the present work, we propose an analogy of the firms’ ranking to a boson gas, analyzing firm rankings for different countries, by identifying the energy levels with the firms’ annual revenue and fitting the CDF corresponding to the Bose–Einstein statistics to the empirical CDF obtained from the firms’ ranking. Additionally, we compare the values of the parameters of the distribution (the Boltzmann parameter β = 1/kT and the fugacity z = eβµ , where µ is the chemical potential) between different countries, to assess whether there is a correlation between the temperature-like parameter for the country and its economic development, as measured by the Human Development Index.

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2. Description of the Bose–Einstein statistics As is known, the distribution of occupation numbers ni in the Bose–Einstein (BE) statistics is given by gi ni = −1 β , z e i −1

(1)

where gi is the number of single-particle states with average energy i , β = 1/kT is the Boltzmann parameter, T is the temperature, z = eβµ is the fugacity and µ is the chemical potential. Notice that in this approach, the single-particle spectrum is split into packets with gi energy levels, which is similar to obtaining histograms for the distribution of particles in the energy spectrum. Let us recall that to obtain Eq. (1) in the microcanonical ensemble, the following assumptions are made: (i) there is no limit on the maximum number of indistinguishable particles that can occupy a one-particle energy state, which leads to a bosonic statistical weight:

Ω=

Y  g i + ni − 1  ni

i

;

(ii) the system is in equilibrium with a thermal reservoir at temperature T ; (iii) both the internal energy and the total number of particles are conserved: E=

X

N =

X

ni  i ,

i

ni .

i

Finally, the entropy, given by Boltzmann’s formula S = k ln Ω , is maximized under the constraints (iii) to find the most probable distribution in the equilibrium state (Eq. (1)) [18]. 3. Business firms as a boson-like gas We start our analogy by identifying the energy of a certain firm with its annual revenue. The identification of money with energy has been employed in the study of wealth distributions [13–15]. Nevertheless, it is important to define an energy scale that allows us to compare results for firms in different countries. As revenue data are usually expressed in local currency and they are for different time periods, it would be helpful to express them in a reference currency (for instance, US dollars for a predefined year), considering the corresponding deflator to account for inflation. To accomplish this task, it is necessary to get the exchange rate and the deflator data for the countries under consideration. Then, we propose that energy is mapped to the firm revenue, defining i as the energy for the ith level corresponding to revenue si , which is already expressed in US dollars. In this way, the lowest value considered for the annual revenue of a firm is zero, corresponding to the ground state. Below, we describe the identification of the assumptions for the Bose–Einstein distribution ((i) through (iii)) with the business firms set, for the analogy with a boson-like gas. A. No exclusion for the energy levels. We observe that there is no a priori constraint that prevents one or more firms having the same revenue; therefore, there is no limit on the number of firms that can occupy a certain revenue range (energy level). B. The firms system is in thermal equilibrium with a reservoir. We assume that the firms’ ranking is closed and is in thermal equilibrium with a thermal reservoir. In our analogy, the system is the firms’ ranking, which is closed since it contains a fixed number of firms; and, the thermal reservoir is the economy of the country—as well as other countries, since firms can be global. The assumption of equilibrium is an idealization that is common in the economic literature, even though the real economies might never be in equilibrium [15]. C. Conservation of energy and of number of particles. Since we are considering the firm rankings to be closed, and that the energy is a function of the annual revenue, the total internal energy would be a function of the total revenue of the firms in the ranking; therefore this total revenue is conserved. Additionally, the total number of firms in the ranking is conserved. As can be seen, assumptions (A) to (C) are equivalent to assumptions (i) to (iii) from the previous section; therefore, we propose the analogy that firms follow a bosonic-like statistics (Eq. (1)). Moreover, since we are considering that the energy scale starts from zero, the fugacity must have values 0 ≤ z ≤ 1; thus the chemical potential will be restricted to µ ≤ 0 [19]. Moreover, we assume that, like the bosons, the firms are indistinguishable. Business firms can be considered indistinguishable in the sense that we could imagine swapping the incomes of two firms, and nothing in the distribution of firms could tell us a priori that these two firms have swapped incomes. Also, firms can be seen as indistinguishable since all of them have the same purpose: to make profit out of production; thus the business process is the same for all of them. However, from a different perspective, we may think that firms are distinguishable because each one has its own identity and its own market niche.

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Fig. 1. Business firms’ rankings for some selected countries. Note that US firms have a significantly higher income compared to firms in other countries.

4. Data for firms Most of the data for developing countries under consideration are the same as those used in a previous work [8]. Ranking data were issued on-line by recognized local business magazines and they were within the time period from 1999 to 2001. We also consider data for Mexico for the years 2002 and 20031 and for South Africa for the years 2005 and 2007.2 Additionally, data for US firms were obtained from the Fortune 500 ranking for the year 2003. Moreover, we include in the analysis the Fortune 500 Global3 ranking to use as a benchmark. Since the data are for a narrow period of time, we approximate the US deflator as equal to 1, and we converted to US dollars the firms’ revenues data for the different countries using the exchange rate at the end of the year under consideration.4 Fig. 1 shows the business firms’ rankings for some selected countries. 5. Fit to the Bose–Einstein distribution For analyzing the goodness of fit of the firms’ ranking to the BE distribution, we could first start by obtaining histograms for the data which provide the occupation numbers and then perform the fitting to the BE distribution (Eq. (1)), where gi is related to the histogram bin width. This approach is the simplest one since it considers the single-particle spectrum divided into a certain number of bins, with each bin having a certain average energy. Notice that in this approach, each bin has a different number of single-particle states, and that every country has its own number of bins since the numbers of firms are not the same across the pool of countries. However, the choice of bin width is normally arbitrary; thus this choice represents a trade-off between the number of bins analyzed, i.e. the resolution of the frequency distribution, and the accuracy with which each value of the distribution is estimated (fewer observations per bin provide a poorer density estimate). Therefore, to avoid the issue of selecting the proper bin width and the problem of empty bins, we perform the fit of the theoretical CDF over the empirical CDF, which is straightforward to construct for the firm rankings data. 5.1. Theoretical CDF for the boson-like gas To obtain the theoretical CDF for the boson gas, we start with the probability density function for the BE statistics: a p() = −1 β , z e −1 where a is a normalization constant.

1 Data source: Las 500 de Expansión. Expansión, Mexican business magazine (http://www.cnnexpansion.com/). 2 Data source: South Africa’s top companies (http://www.topcompanies.co.za/). 3 Data source: Fortune Global 500 (2005) (http://money.cnn.com/magazines/fortune/global500/). 4 Data source: US Federal Reserve (http://www.federalreserve.gov/releases/g5a/).

(2)

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Table 1 Estimates for the Bose–Einstein distribution function. Country

N

β

z

m

M

R2

KS

Argentina Bolivia Brazil Chile Colombia Ecuador Mexico (2001) Mexico (2002) Mexico (2003) Mozambique Peru South Africa (2005) South Africa (2007) USA Venezuela Fortune Global

501 100 150 100 300 51 500 500 500 100 1963 319 209 500 219 500

4.35 29.13 33.99 1.41 14.63 7.21 1.34 0.98 0.64 66.76 108.03 0.40 0.27 0.08 4.86 0.05

1.0000 0.9129 1.0000 0.9786 1.0000 0.6923 1.0000 1.0000 1.0000 1.0000 0.5108 0.9988 1.0000 1.0000 0.9906 1.0000

0.0600 0.0017 0.0062 0.0303 0.0308 0.0857 0.0271 0.0278 0.0354 0.0022 2.85 × 10−7 2.96 × 10−5 0.0333 2.9483 1.81 × 10−5 12.4310

6.6021 0.4427 0.8790 4.1642 3.6551 6.3303 45.4851 51.3069 58.0136 0.4717 1.2077 23.3259 29.2449 246.5250 2.5209 287.9890

0.9638 0.9855 0.9597 0.9953 0.9890 0.9900 0.9841 0.9833 0.9939 0.9800 0.9857 0.9974 0.9936 0.9913 0.9953 0.9939

Rejected Not rejected Rejected Not rejected Not rejected Not rejected Rejected Rejected Not rejected Not rejected Rejected Not rejected Not rejected Not rejected Not rejected Not rejected

Since the energies (revenues) of the business firms in the rankings do not contain either the ground state or an infinite energy state, the CDF is obtained as follows: f () =



a

m

z −1 eβξ

Z

−1

dξ

where m is the minimum energy value in the ranking. After carrying out the integration, and rearranging terms, the CDF is given by f () =

a



 log

β

eβ − z



eβm − z



− β( − m ) .

From the normalization condition,

Z

M m

a z −1 eβξ − 1

dξ = 1,

where M is the maximum energy value in the rank, we obtain the constant a:



a = β log



eβM − z eβm − z



− β(M − m )

−1

.

Therefore, the CDF is given by f () =

log log





eβ −z eβm −z

eβM −z eβm −z





− β( − m )

− β(M − m )

.

(3)

5.2. Parameters of the distribution Fig. 2 shows for selected countries the empirical CDF and the fitted CDF in a semilogarithmic plot to ease the visualization. As can be seen, the fit is remarkably good. Moreover, the firms from different countries are allocated in distinct regions of the energy spectrum; that is considered to be the same for all the countries. For instance, it is observed that the firms from the United States occupy the levels with higher energies. Moreover, Table 1 shows the results for the fitted parameters β , z, minimum m and maximum M values of energy (expressed in US billion dollars) for each country, along with the number of firms in the rank (N) and the residual errors of the fit (R2 ). As can be seen, most of the values of z are very close to 1, with many of them numerically equal to 1 (recall that for photons z = 1). In order to assess the hypothesis that the temperature could be mapped to the level of economic development, we sort countries in descending order with respect to the temperature, calculated as the inverse value of β . Fig. 3 shows the values of the temperature-like parameter for each country in a semilogarithmic fashion to ease the comparison. Moreover, we include the Human Development Index (HDI) for each country for the year corresponding to each ranking5 (see Table 2). The

5 Data source: Human Development Reports by the United Nations Development Programme (http://hdr.undp.org/).

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a

b

c

d

e

f

Fig. 2. Empirical () and theoretical (–) distribution functions for: (a) Argentina, (b) Chile, (c) Colombia, (d) Mexico, (e) South Africa, and (f) United States.

Table 2 Values for the temperature parameter and the Human Development Index. Country

T

HDI

Argentina Bolivia Brazil Chile Colombia Ecuador Mexico (2001) Mexico (2002) Mexico (2003) Mozambique Peru South Africa (2005) South Africa (2007) USA Venezuela Fortune Global

0.230 0.034 0.029 0.710 0.068 0.139 0.744 1.023 1.572 0.015 0.009 2.471 3.745 11.924 0.206 20.396

0.842 0.648 0.777 0.825 0.765 0.726 0.800 0.802 0.814 0.356 0.743 0.674 0.683 0.944 0.770 1.000

HDI is a summary measure of development used to rank and compare countries; and it is a more comprehensive measure of economic development, compared to the per capita income, since it is a composite index measuring average achievement in three basic dimensions of human development: a long and healthy life, access to knowledge and a decent standard of living [20]. In addition, the inset in Fig. 3 shows a scatter semilogarithmic plot of HDI versus T . The correlation coefficient between these parameters is 0.6079, indicating a non-negligible correlation between the value of the temperature-like parameter and the level of economic development.

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Fig. 3. Comparison of the values for the temperature-like parameter for each country obtained from the BE fit. Notice that for the US firms the corresponding temperature is higher than for the developing countries. Inset: Scatter semilogarithmic plot of HDI versus T , with the correlation coefficient equal to 0.6079.

5.3. Goodness of fit In order to assess whether the fitted distribution correctly models the firms’ ranking data, we perform the well-known Kolmogorov–Smirnov (KS) test. The KS test is based on two empirical distribution function statistics, namely D+ and D− , that are the largest vertical differences when the empirical distribution function Fn (x) is greater or smaller than the cumulative distribution function F (x), respectively. Formally, D+ = supx {Fn (x) − F (x)} and D− = supx {F (x) − Fn (x)} [21]. The KS statistic is D = sup |Fn (x) − F (x)| = max(D+ , D− ).

(4)

x

√

In the KS test, the null hypothesis that the sample comes from F (x) is rejected at level α for nD ≥ Kα , where Kα is obtained from Pr(K ≤ Kα ) = 1 − α , according to the KS distribution [21]. Table 1 summarizes the results of the goodness of fit tests for each of the countries considered, in which we can see that for most cases, the null hypothesis that the sample comes from the boson distribution has not been rejected. 6. Discussion Our attempt to find an analogy between the distribution of business firms and a statistical physics distribution is consistent with the spirit of econophysics, whose aim is to use concepts and mathematical methods from statistical physics in the study of statistical properties of complex economic systems, allowing analogies with some concepts from statistical physics. In the present case, our analogy indicates that for developing countries the firms tend to cluster towards the lowest energy levels, resembling the Bose–Einstein condensation that occurs in boson systems for low temperatures. In the case of business firms, following the analogy of the temperature with the level of economic development, we see that the colder economies contain firms that have an overall lower revenue than others from developed countries. We found that there is a non-negligible correlation between the temperature-like parameter estimated from the BE distribution and the level of economic development as measured by the HDI, which supports the hypothesis that the temperature of an economy could be identified with the level of economic development [6,8,12] or the average amount of money per agent [12–15,22,16]. Moreover, given that the fitted fugacity is very close to 1 for most of the countries, we propose that the firms might actually be compared to photons where the number of particles is not conserved. This is indeed the case for business firms dynamics, where all the time new firms arrive in the economy as well as other firms disappear, either by merging or through bankruptcy. 7. Conclusions We propose an analogy of the firms’ size distribution with the Bose–Einstein statistics, based on the occupation rules for occupying the single-particle spectrum for bosons, considering that there is no a priori constraint as regards several firms occupying the same revenue level. The parameters of the Bose–Einstein distribution, namely the Boltzmann parameter β = 1/kT and the fugacity z = eβµ , were obtained by fitting the theoretical CDF over the empirical one for each country. After comparing the corresponding value of the temperature-like parameter 1/β for each country with the corresponding

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value of the Human Development Index, we find that the most developed countries tend to have a higher temperature, which supports previous studies suggesting that, in analogy with thermodynamic systems, the temperature of an economy maps to its level of development [6,8,12] or the average amount of money per economic agent [13]. Moreover, the values obtained for the fugacity-like parameter for most of the countries were very close to 1, which suggests that the BE statistics in these cases is close to that for photons (with z = 1), for which the number of particles is not conserved since photons constantly create and annihilate. This turns out to be the case for business firms’ dynamics, where all the time new firms are arriving in the economy and others disappearing from it. The firms’ distribution is well fitted by the Bose–Einstein statistics for the ranking data that we have collected, mostly for developing countries. However, we were not able to obtain data for other developed countries; thus we consider it worth extending this boson gas analogy to business firms from other countries, and assessing the validity of the hypothesis that the temperature of an economy can be identified with the level of economical development. Also, it would be interesting to follow this approach in the study of individual income and wealth distributions. Acknowledgements The author thanks F. Angulo-Brown and D. Tun for helpful advice and fruitful discussions. Also, thanks are due to the anonymous referees whose suggestions led to improvement of the manuscript. References [1] H.E. Stanley, L.A.N. Amaral, S.V. Buldyrev, P. Gopikrishnan, V. Plerou, M.A. Salinger, Self-organized complexity in economics and finance, Proc. Natl. Acad. Sci. 99 (2002) 2561. [2] R.N. Mantegna, H.E. Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance, Cambridge University Press, UK, 2000. [3] R.L. Axtell, Zipf distribution of US firm sizes, Science 293 (2001) 1818–1820. [4] K. Okuyama, M. Takayasu, H. Takayasu, Zipf’s law in income distribution of companies, Physica A 269 (1999) 125–131. [5] E. Gaffeo, M. Gallegati, A. Palestrini, On the size distribution of firms: additional evidence from the G7 countries, Physica A 324 (2003) 117–123. [6] J.J. Ramsden, G. Kiss-Haypál, Company size distribution for different countries, Physica A 277 (2000) 220–227. [7] P. Cirillo, An analysis of the size distribution of Italian firms by age, Physica A 389 (2010) 459–466. [8] R. Hernández-Pérez, F. Angulo-Brown, D. Tun, Company size distribution for developing countries, Physica A 359 (2006) 607–618. [9] J. Zhang, Q. Chen, Y. Wang, Zipf distribution in top Chinese firms and an economic explanation, Physica A 388 (2009) 2020–2024. [10] Q. Chen, L. Chen, K. Liu, Firm Size Distribution in Fortune Global 500, in: J. Zhou (Ed.), Complex Sciences: First International Conference, Complex 2009, Springer, Germany, 2009, pp. 1774–1782. [11] R. D’Hulst, G.J. Rodgers, Business size distributions, Physica A 299 (2001) 328–333. [12] W.M. Saslow, An economic analogy to thermodynamics, Am. J. Phys. 67 (1999) 1239–1247. [13] A. Dragulescu, V.M. Yakovenko, Statistical mechanics of money, Eur. Phys. J. B 17 (2000) 723–729. [14] A. Dragulescu, V.M. Yakovenko, Evidence for the exponential distribution of income in the USA, Eur. Phys. J. B 20 (2001) 585–589. [15] V.M. Yakovenko, J.B. Rosser Jr., Colloquium: statistical mechanics of money, wealth and income, Rev. Modern Phys. 81 (2009) 1703–1725. [16] A. Banerjee, V.M. Yakovenko, Universal patterns of inequality, New J. Phys. (2010) (in press), arXiv:0912.4898. [17] F. Clementi, T.D. Matteo, M. Gallegati, G. Kaniadakis, The κ -generalized distribution: a new descriptive model for the size distribution of incomes, Physica A 387 (2008) 3201–3208. [18] K. Huang, Statistical Mechanics, John Wiley & Sons Inc., New York, 1963. [19] L.E. Reichl, A Modern Course in Statistical Physics, Wiley-VCH Verlag GmbH & Co., Weinheim, 2009. [20] United Nations Development Programme, Human Development Report 2009, Palgrave Macmillan, New York, 2009. [21] R.B. D’Agostino, M.A. Stephens (Eds.), Goodness-of-fit Techniques, Marcel Dekker, New York, 1986. [22] V.M. Yakovenko, Statistical mechanics approach to econophysics, in: R.A. Meyers (Ed.), Encyclopedia of Complexity and Systems Science, Springer, 2009, arXiv:0709.3662v4.