Company Size Distribution for Developing Countries ARTICLE in PHYSICA A: STATISTICAL MECHANICS AND ITS APPLICATIONS · JANUARY 2006 Impact Factor: 1.73 · DOI: 10.1016/j.physa.2005.04.027

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ARTICLE IN PRESS

Physica A 359 (2006) 607–618 www.elsevier.com/locate/physa

Company size distribution for developing countries R. Herna´ndez-Pe´rez,1, F. Angulo-Brown, Dionisio Tun1 Departamento de Fı´sica, Escuela Superior de Fı´sica y Matema´ticas, Instituto Polite´cnico Nacional, Edif. 9 U.P. Zacatenco, Me´xico D.F. 07738, Me´xico Received 8 February 2005 Available online 14 June 2005

Abstract We analyze company size distribution for developing countries using the framework proposed by Ramsden and Kiss-Haypa´l [Physica A 277 (2000) 220]. Although this distribution does not ﬁt developing countries data as good as it does to developed ones, the parameters of the distribution (y and r) for developing countries are remarkably different to those for developed countries. This result supports the hypothesis that parameter y plays a role analogous to the temperature of the economy, which could be related to the level of economic development, as reported previously by Saslow [Am. J. Phys. 67 (1999) 1239]. Also, this supports the hypothesis that r is related to the competitive exclusion in economics, as r tending to zero implies the competition free limit case where company size distribution is predicted to be a power-law, as reported by Takayasu and Okuyama [Fractals 6 (1998) 67]. Finally, we report the goodness of ﬁt for two functions: a ﬁnite-size scaling and a log–normal. We found that these functions ﬁt the data better in some cases. However, this is not in itself sufﬁcient evidence that those functions are an appropriate representation of the phenomenon. r 2005 Elsevier B.V. All rights reserved. Keywords: Econophysics; Company size; Zipf law; Power-law

Corresponding author. 1

E-mail address: [email protected] (R. Herna´ndez-Pe´rez). Present address: Sate´lites Mexicanos (SATMEX). Centro de Control Satelital Iztapalapa, Me´xico.

0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.04.027

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1. Introduction Company size distribution has been approached in different ways [1], since the pioneering work by Zipf, who established that assets of US corporations followed approximately a power-law: sr H1=r, where sr is the size of the company ranked in the position r in a list ordered by asset size, beginning with the largest. Power-laws appear widely in physics, biology, earth and planetary sciences, economics and ﬁnance, computer science, demography and the social sciences [2–5]. Amaral et al. analyzed data of all publicly traded US manufacturing ﬁrms within the years 1974–1993, using several proxies for company size such as sales, number of employees, assets, etc. [6]. They found that the distribution of ﬁrm size remains stable for the 20-year timespan study, when taking into account the ﬂuctuation in size which in turn is one of the two new universal scaling models in economics [7]. Additionally, using records of quarterly sales of a large database of pharmaceutical products in the European Union and North America, Matia et al. [8] studied the statistical properties of the internal structure of a ﬁrm and its growth and conﬁrmed the features predicted in Ref. [2]. Also for US companies, Axtell found that Zipf distribution holds for data from several years when size is deﬁned as the number of employees or the amount of receipts [9]. For Japanese companies, Okuyama et al. reported power-law distribution over more than three decades in income scale [10]. Additionally, D’Hulst and Rodgers obtained a log–normal distribution with powerlaw tails for the size of ﬁrms, taking the number of customers as a proxy of business size [11]. By analyzing international company data, Takayasu and Okuyama found that company size distributions are not universal but clearly depend on country [13]. They introduced a stochastic model of company sizes which can reproduce country’s statistical features. In particular, they showed that their model predicts a power-law size distribution in the totally free competition limit. Ramsden and Kiss-Haypa´l analyzed the size distribution of companies in different countries, mainly focused in developed economies, using the annual net revenue as proxy for size [12]. They found that the so-called Simple Canonical Law (SCL), a sort of modiﬁed Zipf law, ﬁtted the data well. However, neither of the parameters of the distribution appeared to be well correlated with traditional economic indicators. Recently, Gaffeo et al. analyzed the average size distribution of a pool of ﬁrms from G7-countries over a 13-year period and for different proxies for ﬁrm size [14]. They found that the empirical distributions are all consistent with a power-law but the resulting distribution generally is not of the Zipf type. The works mentioned above have two things in common. First, the fact that the power-law distribution of ﬁrm size seems to hold for multiple years and for various deﬁnitions of size (for instance: number of employees, revenue or assets). Second, these works are strongly focused on developed countries. However, to test the robustness of these results it is necessary to widen the sample of data used, that is applying the analysis not only to US companies or any other developed countries but also to developing ones. Are the distributions obtained in previous analysis applicable to developing economies? If this is the case, what is the economical meaning of the parameters introduced in these distributions?

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In the present work, we extend the analysis of Ramsden and Kiss-Haypa´l (R&K) to developing countries and then, we compare the obtained results to those from developed economies. This comparison allows us to discuss and support the economical meaning of the parameters that were proposed by R&K. Additionally, we ﬁt the data to two other functions: a ﬁnite-size scaling and a log-normal. Even though these functions ﬁt the data better for some countries than SCL does, this is not in itself sufﬁcient evidence that those functions are an appropriate representation of the phenomenon.

2. Firms data As we consider that the results of the analysis should hold for companies distributions no matter what the year and where the data are from, we looked for all the available companies ranking data for developing countries issued on-line by recognized local business magazines. Available data resulted to be within the time period from 1999 to 2001.

3. Results and discussion As mentioned above, R&K approached company size distributions using the socalled SCL sr ¼ Pðr þ rÞ1=y ,

(1)

where sr is the annual net revenue of the company with rank r, r and y are the parameters of the distribution and P is a normalizing coefﬁcient P 1=y (P1 ¼ N ðr þ rÞ ). As they showed, this function seems to ﬁt properly the r¼1 set of data they used [12]. For the parameters y and r, R&K found signiﬁcant differences between countries. Also, these parameters did not appear to correlate well with traditional economic indicators. Even though, the authors proposed that the role of y could be analogous to that the temperature plays in protein transitions from one conformational substate to another; therefore, countries with low y might ﬁnd it harder to make transitions from one economic activity substate to another and also are relatively frozen in their present state [12]. On the other hand, they proposed that r could be correlated to the tendency of an economy to concentrate the economical activity in few high yield sectors. Moreover, they proposed a link between r and the competitive exclusion in the economy: as r tends to zero the system becomes non-degenerate, i.e., there are enough niches for all entrepreneurs or, in other words, competition is tolerated and the distribution is power-law. This is consistent with the result predicted by Takayasu–Okuyama’s model [13], which states that power-law size distribution is obtained in the totally free competition limit.

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In Table 1, we summarize the ﬁtted SCL parameters for the developing countries we considered, while Fig. 1 shows some representative distributions. Comparing this table to the results from R&K, it can be seen that the residuals of the ﬁtted SCL to developing countries data are larger than that obtained by R&K, even though as R&K did [12], to ﬁt the data we also used a nonlinear least-squares LevenbergMarquardt algorithm. Although the SCL distribution for developing countries does not ﬁt as good as it does for developed countries, we found that the value of the ﬁtted parameters y and r for developing countries are remarkably different to those reported by R&K for developed countries. Fig. 2 allows the comparison between the values of y we obtained for developing countries and those obtained by R&K for developed ones. It can be seen that only Venezuela and Mozambique have values of y that are greater than the lowest reported by R&K (South Africa and Netherlands) [12], while the remaining developing countries have values of y lower than the developed ones. Therefore, according to the hypothesis that y could be analogous to the role of temperature, one could say that developing countries are colder than developed ones, i.e., the thermodynamic temperature of an economy might be associated with the level of economic development. This assumption is consistent with the analogy between economic systems and thermodynamics made by Saslow [16]. The result that we obtained also supports the hypothesis of R&K that countries with low y might ﬁnd it harder to change their present state, which is a feature of most developing countries

Table 1 Parameters ﬁtting to SCL for data from developing countries Country

Number of ﬁrms, N

Source

y

Argentina Bolivia Brazil Chile Colombia Ecuador Mexico Mozambique Peru Venezuela

501 100 150 100 300 51 500 100 1963 219

a b c d e f g h i j

0.801 0.701 0.689 0.706 0.818 0.771 0.694 0.673 0.755 0.610

r

22.956 3.250 3.093 3.352 26.238 8.675 3.740 2.755 14.534 0.759

rss

0.052 0.208 0.114 0.274 0.142 0.319 0.309 0.103 0.733 1.224

Notes: a Colombian news website La Nota (http://www.lanota.com) with data from Mercado magazine (data for 1999). b Nueva Economı´a magazine (http://www.nuevaeconomia.com.bo/guia.php?guia=11) (data for 1999). c http://www.iel-ideies.com.br/menu_esq/inf_empresarial/inf_revista/inf_revista_150_ 2001_rank_geral.htm (data for 2001). d La Nota with data from The Commercial Burse of Santiago de Chile (data for 1999). e La Nota, with data from Colombian government agencies: Confeca´maras, National General Accounting and Societies and Companies Division (data for 1999). f La Nota (data for 1999). g Expansio´n , mexican business magazine (data for 2001). h Magazine Revista 100 Maiores Empresas de Moc- ambique (by KMPG) (http://www.editando.pt) (data for 2001). i National Institute for Statistics and Census, Peruvian government (http//www.inei.gob.pe) (data for 1999). j Dinero, venezuelan business magazine (http//www.dinero.com.ve) (data for 2000).

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Fig. 1. Illustrative log–log plots of company size vs. rank for: (a)Argentina, (b) Chile, (c) Mexico and (d) Mozambique. Dashed lines are the ﬁtted SCLs.

where introducing new policies is a difﬁcult process. For instance, in 1982, Mexico faced a deep economic crisis, and despite Mexican administration implementing both a prudent macroeconomic policy and deep structure reforms, the recovery from the crisis was a difﬁcult and slow process [17]. Also, Mexico has been struggling since a long time to achieve additional economic structural reforms, such as the tax policy as well as the energy policy, which will provide the capital necessary to increase Mexican economy’s productivity and competitiveness [18]. Moreover, the failed efforts of the developing countries to formalize the ownership of the assets accumulated by the poorest social sectors are another example of the difﬁculties developing countries face to make their economies growth [19]. For the case of the ﬁtted r for developing countries, Fig. 3 shows the comparison between both developed and developing countries. As can be seen, developing countries have a higher value of r than the developed ones, which could imply, according to the hypothesis proposed by R&K [12], that in developing countries the competition between companies is less tolerated than in developed countries. The reason for this difference, following the logic of R&K, is that in the developing

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1.4

Developed series Developing series

1.2

θ

1

0.8

0.6

0.4

0

5

10

15

20

Country Fig. 2. Results from R&K (developed countries) and from the present work (developing countries) where both series are in ascending order for y.

countries there are few niches to ﬁll because consumer behavior has not yet generated many niches. In many countries large companies themselves, through marketing and global trade, generate previously nonexistent new tastes and desires. The developed countries consume all different product variations, while the developing ones usually get a narrow variety of goods. Moreover, as r affects the upper ranks (larger companies) the most, competition for the upper ranks is less tolerated for greater values of r. Larger companies compete more for resources. Even if they work in different industries, there is a ﬁnite personal or company income they can share through selling their products, and apart from food and water, customers need to decide whether they buy a car or renovate the house from the same amount of money. This is why larger companies compete with each other even in different industries. For the developing countries that we analyzed, there are few very large companies (oil companies, mostly) with noticeable high incomes that standout in the distribution; but, for the developed countries the distribution is smoother. As mentioned above, SCL does not seem to ﬁt the developing countries data as good as it does for developed countries. Therefore, we ﬁt the data to other two

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30

Developed series 25

Developing series

ρ

20

15

10

5

0

0

5

10 Country

15

20

Fig. 3. Results from R&K (developed countries) and from the present work (developing countries) where both series are in ascending order for r.

different distributions. The ﬁrst one is shown in Eq. (2), named Finite Size-Scaling (FSS). This function is a rearranged form of the one used by Blok and Bergersen to ﬁt the cumulative distribution of avalanches in the Game of Life while analyzing the effect that boundary conditions have on it [15]: sr ¼ ra er=b .

(2)

As can be seen, this function has two components: a power-law and an exponential. The exponent of the power law in Ref. [15], (1 b), was renamed as a, while the time-constant in the exponential (tc ) was redeﬁned as b. The parameter tc is called the critical lifetime, and represents a typical timescale up to which the power law is a valid description of the dynamics [15]. For small values of r, bigger companies, the power-law component dominates up to a certain rank, depending on the value of b, according to Ref. [15]. This suggests that the size distribution for the bigger companies seems to follow a power-law, which has been reported to be the case for the average size distribution of a pool of ﬁrms of G7 countries [14].

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Table 2 Parameters ﬁtting to FSS and log–normal functions for data from developing countries FSS

Log– normal

Country

Number of ﬁrms, N

a

b

rss

s

m

rss

Argentina Bolivia Brazil Chile Colombia Ecuador Mexico Mozambique Peru Venezuela

501 100 150 100 300 51 500 100 1963 219

0.576 0.687 0.696 0.445 0.819 1.432 0.810 0.975 0.615 0.425

400.000 52.538 89.366 36.765 5775.100 31.460 200.000 112.505 666.667 34.483

0.085 0.116 0.212 0.089 0.106 0.351 0.081 0.074 0.575 0.658

1.7094 1.4951 1.8116 1.3423 1.6709 1.7030 1.5313 1.9376 1.5429 0.9405

4.7752 2.5792 2.5737 2.6434 4.5889 3.0902 3.2938 2.2264 4.8749 2.8099

0.1817 0.1616 0.1368 0.1778 0.2897 0.3621 0.1108 0.1120 0.6566 0.8749

100

log (relative size)

log (relative size)

100

10-1

10-2

10-3 0 10

101

(a)

102

10-1

10-2

10-3 100

103

10-1

log (relative size)

log (relative size)

102

103

102

103

log (rank) 100

100

(c)

101

(b)

log (rank)

10-2

10-1

10-2

10-3

10-4 0 10

101

102 log (rank)

103

10-3

(d)

100

101 log (rank)

Fig. 4. Illustrative log–log plots of company size vs. rank for: (a) Argentina, (b) Chile, (c) Mexico and (d) Mozambique. Solid lines are the ﬁtted FSS.

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Table 2 shows the parameters ﬁtting of the FSS. Comparing to Table 1, it can be seen that for most of the countries considered, the residuals for the FSS ﬁt are lower than those for the SCL. Fig. 4 shows some representative FSS distributions. Excluding Peru, the correlation coefﬁcients between a and b with H (the Shannon P entropy [20] for the ranking H ¼ N p log p i , where pi is the proportion of output i i of the companies of the ith industry sector) are 0.31 and 0.58, respectively. This might suggest that the FSS parameters depend on the evenness of the economical activity, according to R&K. To test this hypothesis, we ran the data ﬁt on companies of only one industry sector (H ¼ 0). We found that for the sectors and countries considered, the FSS ﬁts the data better than SCL does. Therefore, suggesting that the distribution for one business category is similar to the distribution for the whole categories, which was reported to be the case for some developed countries [13]. When considering only one business category we will have that H ¼ 0, therefore, the correlation coefﬁcient does not longer make sense. Thus, the FSS parameters may depend on other features of the competitive dynamics between the ﬁrms, and not only in the evenness of the economical activity. Among other factors, the role a certain company plays in this dynamics is determined by its size because the bigger companies have a greater variety of advantageous attributes (scale economies, wider access to ﬁnancial sources, bargain power, etc.) than the smaller ones. Therefore, 100

log (relative size)

10-1

10-2

10-3

10-4 100

101 log (rank)

Fig. 5. Illustrative FSS (solid line) for only one sector: Venezuelan ﬁnancial companies.

102

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these attributes may explain the difference between the shape of the distribution of size for both big and small companies which is well captured by the FSS function (see Fig. 5). On the other hand, as the curvature present in the data points in Fig. 1 suggests they may be better represented by a log–normal distribution, we ﬁt the data of companies income to this kind of distribution, namely [21]: 1 1 2 p ﬃﬃﬃﬃﬃ ﬃ sr ¼ exp 2 ðlogðrÞ mÞ , 2s rs 2p

(3)

where s and m are the parameters of the distribution. Studies on companies growth rate have reported to ﬁt log–normal distributions [2,8,11]. Table 2 shows the results of this ﬁtting, while Fig. 6 depicts representative distributions. It can be seen that for some countries, the residuals are lower than those from the FSS ﬁt. This suggests that the ﬁrms may be undergoing a Gibrat kind of growth, although Gibrat’s model has found to be approximately correct [6]. However, data for several years for each country are required to analyze the growth rate of companies in developing countries and determine whether the hypothesis of Gibrat growth is supported.

Fig. 6. Illustrative log–log plots of company size vs. rank for: (a) Argentina, (b) Chile, (c) Mexico and (d) Mozambique. Dashed lines are the ﬁtted log–normal functions.

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Even though the ﬁt results for FSS and log– normal distributions are in some cases better than the obtained from SCL, this is not in itself sufﬁcient evidence that those functions are an appropriate representation of the phenomenon. Further analysis could help determine whether these distributions could represent appropriately the company size distribution as well as provide a better understanding of the economic meaning behind the parameters of these distributions.

4. Conclusions Although the SCL distribution for developing countries does not ﬁt as good as it does for developed ones, we found that the values of ﬁtted parameters y and r for developing countries are remarkably different to those reported by R&K for developed countries. Most of the developing countries considered in the present work have values of y lower than the developed ones. This supports the hypothesis of R&K that y could be analogous to the role of temperature. This assumption is consistent with the analogy between economic systems and thermodynamics made by Saslow [16]. This result also supports the hypothesis of R&K that countries with low y should ﬁnd it harder to change their present state, which is a feature of most developing countries where introducing new policies is a large and complex process. For instance, in 1982, Mexico faced a deep economic crisis, and despite Mexican administration implemented both a prudent macroeconomic policy and deep structure reforms, the recovery from the crisis was a difﬁcult and slow process [17]. Also, Mexico has been struggling since a long time to achieve additional economic structural reforms, such as the tax policy as well as the energy policy, which will provide the capital necessary to increase Mexican economy’s productivity and competitiveness [18]. Moreover, the failed efforts of the developing countries to formalize the ownership of the assets accumulated by the poorest social sectors are another example of the difﬁculties developing countries face to growth their economies [19]. For the case of parameter r, developing countries have a higher value than the developed ones, which could imply that developing countries have a more widespread economic activity, according to the hypothesis proposed by R&K [12]. The reason for the difference between r values for developed and developing countries, following the logic of R&K, is that in the developing countries there are few niches to ﬁll because consumer behavior has not yet generated many niches. Moreover, as r affects the upper ranks (larger companies) the most, competition for the upper ranks is less tolerated for greater values of r. Larger companies compete more for resources. Even if they work in different industries, there is a ﬁnite personal or company income they can share through selling their products, and apart from food and water, customers need to decide whether they buy a car or renovate the house from the same amount of money. This is why larger companies compete with each other even in different industries. Finally, we report the goodness of ﬁt for two additional functions: a ﬁnite-size scaling and a log–normal. Even though the ﬁt results for FSS and log– normal are in some cases better than the obtained from SCL, this is not in itself sufﬁcient evidence

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that those functions are an appropriate representation of the phenomenon. Further analysis could help determine whether these distributions could represent appropriately the company size distribution as well as provide a better understanding of the economic meaning behind the parameters of these distributions. Acknowledgements We thank Dr. L. Guzma´n-Vargas for comments on the manuscript and for his assistance to obtain some references. We also acknowledge the reviewers’ suggestions and criticisms, which helped to improve this manuscript. References [1] R.N. Mantegna, H.E. Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance, Cambridge University Press, Cambridge, 2000. [2] L.A.N. Amaral, S.V. Buldyrev, S. Havlin, M.A. Salinger, H.E. Stanley, Power law scaling for a system of interacting units with complex internal structure, Phys. Rev. Lett. 80 (1998) 1385. [3] M.E.J. Newman, Power laws, Pareto distributions and Zipf’s law, (cond-mat/0412004). [4] L. Guzma´n-Vargas, F. Angulo-Brown, Simple model of the aging effect in heart interbeat time series, Phys. Rev. E 67 (2003) 052901. [5] L. Guzma´n-Vargas, A. Mun˜oz-Diosdado, F. Angulo-Brown, Inﬂuence of the loss of time-constants repertoire in pathologic heartbeat dynamics, Physica A 348 (2005) 304. [6] L.A.N. Amaral, S.V. Buldyrev, S. Havlin, H. Leschhorn, P. Maass, M.A. Salinger, H.E. Stanley, M.H.R. Stanley, Scaling behavior in economics: I. Empirical results for company growth, J. Phys. I France 7 (1997) 621. [7] H.E. Stanley, L.A.N. Amaral, X. Gabaix, P. Gopikrishnan, V. Plerou, Similarities and differences between physics and economics, Physica A 299 (2001) 1. [8] K. Matia, D. Fu, S.V. Buldyrev, F. Pammolli, M. Riccaboni, H.E. Stanley, Statistical properties of the internal structure of a ﬁrm, Europhys. Lett. 67 (3) (2004) 498. [9] R.L. Axtell, Zipf distribution of US ﬁrm sizes, Science 293 (2001) 1818. [10] K. Okuyama, M. Takayasu, H. Takayasu, Zipf’s law in income distribution of companies, Physica A 269 (1999) 125. [11] R. D’Hulst, G.J. Rodgers, Business size distributions, Physica A 299 (2001) 328. [12] J.J. Ramsden, Gy. Kiss-Haypa´l, Company size distribution in different countries, Physica A 277 (2000) 220. [13] H. Takayasu, K. Okuyama, Country dependence on company size distributions and a numerical model based on competition and cooperation, Fractals 6 (1998) 67. [14] E. Gaffeo, M. Gallegati, A. Palestrini, On the size distribution of ﬁrms: additional evidence from the G7 countries, Physica A 324 (2003) 117. [15] H.J. Blok, B. Bergersen, Effect of boundary conditions on scaling in the Game of Life, Phys. Rev. E 55 (1997) 6249. [16] W.M. Saslow, An economy analogy to thermodynamics, Am. J. Phys. 67 (1999) 1239. [17] Nora Lustig, Me´xico: Hacia la reconstruccio´n de una economı´ a, Colmex & FCE, Mexico, 2002, second ed., p. 24 (in Spanish). [18] Fourth Government Report - President of Mexico, Presidencia de la Repu´blica, Mexico, 2004, p. 97 (in Spanish) (http://cuarto.informe.presidencia.gob.mx/). [19] Hernando de Soto, El misterio del capital, Diana, Mexico, 2002, ﬁrst ed., pp. 196–197 (in Spanish). [20] C.E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J. 27 (1948) 379. [21] E. Limpert, W.A. Stahel, M. Abbt, Log–normal distributions across the sciences: keys and clues, BioScience 51 (5) (2001) 341.