- Email: [email protected]
An analysis of Chinese firm size distribution and growth rate
Physica A 535 (2019) 122344
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
An analysis of Chinese firm size distribution and growth rate Yijiang Zou School of economics, Anyang Normal University, Anyang, 455000, China
highlights • • • •
The firm size distribution in China was studied by employing the empirical data. The Laplace form distribution of firms’ growth rate was estimated by Subbotin family function. The standard deviation distribution of growth rate were studied with assets and employees of these firms. To study the correlation between firms’ growth rate and their ages, the standard deviation of growth rate on the ages of firms was analyzed.
article
info
Article history: Received 20 September 2018 Received in revised form 27 February 2019 Available online 9 August 2019 Keywords: Firm size Growth rates Power law Laplace distribution Chinese firms
a b s t r a c t We analyzed a database comprising more than 100 thousands firms over the period 1998–2008. It was found that both asset and employee could be well fitted with power-law distribution, even firms were classified in different industries. Meanwhile, the growth rate distribution has a Laplace form, which is estimated by Subbotin family function, and it is also robust when firms are in categories. By analyzing annual logarithmic growth rates, we can see that the standard deviation has a power-law shape on both asset and employee, as researchers have found in western countries. In addition, there exists a power-law dependence of the standard deviation on the age of firms, which means that the growth of firms also rely on the years that they have opened. © 2019 Published by Elsevier B.V.
1. Introduction The distribution of firm size has been deeply studied in statistical and economic literature, since the pioneering work of Pareto [1]. The power-law behavior in the upper tail of firm size distribution has been verified for various periods and economies, e.g. for U.S. firms by Axtell [2], Podobnik et al. [3] and Simon and Bonini [4], and for Italian firms by Cirillo [5] and Cirillo and Husler [6]. However, most of empirical evidences on firm analysis are from developed countries. So it is necessary to test the ubiquity and robustness of these findings by widening the scope of this research to developing countries. Gaffeo [7] have analyzed the average size distribution of a pool of the G7 group’s firms over the period 1987– 2000. Chen [8] studied the top 500 Chinese firms and found that their ranks completely obey the Zipf distribution with exponents of 1. Xie [9] investigated the growth dynamics of the primary industry and the population of 2079 counties in mainland China. They find that the annual growth rates are distributed according to Student’s t distribution with the tail exponent less than 2. In this paper, we analyzed the distributional properties of Chinese firms during the period 1998–2008 with abundant data obtained from Chinese Industrial Enterprises Database (CIED). The size distribution was studied by two size measures, namely the total assets and employees. It was found that both the asset and employee distribution were power-law distributed. However, they have different Pareto index (the slope of power-law scaling). E-mail address: [email protected]. https://doi.org/10.1016/j.physa.2019.122344 0378-4371/© 2019 Published by Elsevier B.V.
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Y. Zou / Physica A 535 (2019) 122344
Table 1 The descriptive statistics of assets of Chinese industrial firms for the years from 1998–2008. The unit of asset is 1000 yuan. Year
Obs.
Mean
Median
Sta.dev
Skewness
Kurtosis
Min
Max
1998 2000 2002 2003 2004 2005 2006 2007 2008
160806 158803 179379 194708 274406 270902 299772 335251 372305
67671.25 79475.81 82806.71 86748.84 78844.15 90445.86 97309.97 105364.4 99397.88
12158 13178 12898 13188 11828 13645 16409 15703 15000
598146.9 724051.3 768138.3 804635 1147967 1305670 1403115 1533228 1436910
59.61446 54.62236 52.43374 64.34964 219.5449 211.1767 192.2964 181.975 137.5098
5646.262 4414.765 3906.593 6831.871 76579.84 71044.82 61976.85 57455 34066.6
10 1 1 1 1 1 12 22 1
90322267 85791220 85808770 130525517 435000000 182000000 513000000 564000000 466445550
Table 2 The fitted parameters of asset and employ distribution in the years from 1998–2008. The blank data of employ are absence in the years from 1998–2003.
α
Year
xmin
P-value
asset 1998 2000 2002 2003 2004 2005 2006 2007 2008
α
xmin
P-value
2.57 2.55 2.54 2.53 2.50
2041 2018 2336 2019 1748
0.63 0.94 0.67 0.61 0.72
employee 2.20 2.10 2.05 2.05 2.07 2.06 2.04 2.05 2.05
462394 212102 194144 192000 225292 321308 252442 404500 314429
0.93 0.59 0.31 0.23 0.30 0.18 0.18 0.53 0.39
As known, statistics on the growth rates of business firms have been studied for a long time. Since the original paper on this subject began with the work of Stanley et al. [10] who showed empirically that growth rate of firms was not Gibrat like. One of the stylized facts, the Laplace distribution of growth rates are robust to how to define firm size [11–13]. Alfarano [14] et al.showed that the profit rate distribution of large publicly traded US companies was well described by a Laplace distribution. Fu [15] et al. proposed a model of proportional growth to explain the distribution of business firm growth rates, and they found that the predictions of the model agreed with empirical growth distribution and size-variance relationships. Looking at Chinese industrial firms, we observed that the growth rates distribution were consistently Laplace distributed, and the results were tested by the Subbotin fimaly function. Furthermore, another interesting issue of firm dynamics is the relationship between the size of a company and its growth rate [16–20]. Based on the other stylized fact of Stanley [10] et al. the standard deviation σ of the growth rates scales as a power-law of the initial firm size S. Then several models predict that standard deviation of growth rates amongst all firms with sales scales as a power law σ (g /S) S −β , where g is growth rate of firm size. Although it is known that Gibrat’s assumptions [21] are rejected empirically and theoretically, there are still some empirical findings have not addressed this issue [22,23]. Given that the size distribution of firms depends on the economic structure of each country, and considering China have the most rapid changes in the economic structure in past decades. Therefore, we think it is necessary to verify these results with Chinese firms. This paper is organized as follows: In Section 2, the details of empirical data were introduced and firm size distribution was studied. The growth rate distribution was analyzed in Section 3. In Section 4, the size-dependent standard deviation for growth rates was analyzed from size and age of firms. Conclusions were given in Section 5. 2. Data set and firm size distribution Data were obtained from the Chinese Industrial Enterprises Database conducted by the National Bureau of Statistics (NBS) of China for the years from 1998–2008. It contains various firm size measures, like employees, turnover, profits and assets. In this paper, we take employees and assets as size measures to study firms’ size distribution and their growth rates. The descriptive statistics of assets of Chinese industrial firms was presented in Table 1, and the probability density function (PDF) of asset were showed in Figs. 1(a) and 1(b), and the case of employee were showed in Figs. 2(a) and 2(b). One would find that both asset and employ distribution could be well fitted as power-law distribution. The fitted results of the probability distribution of asset and employ in all the years were showed in Table 2. It was observed that the Pareto indexes of asset distribution varied from 2.04 to 2.20. However, the Pareto index of employ distribution changed with an average value 2.5. This means that the analysis result may be different when we choose different firm size measures. Furthermore, in order to study the influence of different industries, the firms are in categories. The fitting result shows that the distribution of firms in four different industries has power-law form with different Pareto indexes, as shown in Fig. 3.
Y. Zou / Physica A 535 (2019) 122344
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Fig. 1. (a) The probability distribution of asset in 2004, which can be fitted as a power-law distribution with Pareto index 2.07. (b) The asset distribution in 2007, and the Pareto index is 2.05.
Fig. 2. (a) The probability distribution of employ in 2004, which can be fitted as a power-law distribution with Pareto index 2.57. (b) The employee distribution in 2007, and the Pareto index is 2.53.
3. The distribution of growth rates In order to measure growth, the central variable we look at is the growth rate, defined as
( gi (t) ≡ log10
Si (t + ∆t)
)
Si (t)
= log10 Si (t + ∆t) − log10 Si (t),
(1)
where Si (t) and Si (t + ∆t) are the measures of firm size, e.g. assets and employees. Especially, the asset is in units of 103 Chinese Yuan. In this paper, we only study the growth rate in two consecutive years, which means that ∆t = 1 year hear. So we can write a rate in general as r1 ≡ log10
S1 S0
,
(2)
where S1 and S0 are size of firms in two consecutive years. The growth rates distribution of asset were showed in Figs. 4(a) and 4(b), and employee in Figs. 5(a) and 5(b). Remarkably, both asset and employee curves displayed a ‘tent-shaped’ form, like firms in other western countries. Furthermore, in order to verify that the growth rates distribution is not Gaussian, the well-known Subbotin family function is introduced [24], which represents a generalization of several particular cases, such as the Laplace and the Gaussian. In this way, we can provide a fitting for the growth rates distribution, and compare to different possible distributions. The functional form of the Subbotin family is f (x, a, b) =
1 1 b
2ab Γ 1 +
(
1 b
)e
− 1b | x−µ a |
b
,
(3)
where µ is the mean, b and a are two different shape parameters and Γ is the standard Gamma. If b → 1 the Subbotin distribution becomes a Laplace, and a Gaussian for b → 2. Then we estimated these three Subbotin parameters on our data by using maximum likelihood, as shown in Table 3. We can see that parameter b is closer to 1 than to 2, which
4
Y. Zou / Physica A 535 (2019) 122344
Fig. 3. The probability distribution of asset in four different industries in 2004.
Fig. 4. (a) The growth rate distribution of asset in 2004–2005. (b) The growth rate distribution of asset in 2007–2008.
means that the growth rate distribution is more similar to a Laplace one than Gaussian. Furthermore, parameter a seems to increase with years from 2002–2008 in both asset and employee cases, but parameter b is not very different in most of the cases. Meanwhile, the growth rates distribution of asset in categories are also been analyzed, as shown in Fig. 6. The fitting result of the Subbotin family function shows that the growth rate distribution are Laplace distributed, even firms are in categories. 4. Standard deviation of the growth rate Next, we studied the relationship between standard deviation and the growth rate. In order to investigate if and how the standard deviation of the logarithmic growth rates σ0 correlates with the initial size of firms S0 , we divided the initial
Y. Zou / Physica A 535 (2019) 122344
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Fig. 5. (a) The growth rate distribution of employee in 2004–2005. (b) The growth rate distribution of employee in 2007–2008. Table 3 The fitted three Subbotin parameters of asset and employ growth rates distribution, in the consecutive years from 2002–2008. The b and a are two different shape parameters of the Subbotin family, and µ is the mean. Year
b
a
µ
asset 2002–2003 2003–2004 2004–2005 2005–2006 2006–2007 2007–2008
b
a
µ
0.59 0.67 0.69 0.66 0.67 0.62
0.29 0.34 0.36 0.33 0.34 0.35
0.03 0.05 0.05 0.05 0.05 0.02
employee 0.64 0.71 0.72 0.71 0.72 0.65
0.21 0.23 0.25 0.23 0.23 0.24
0.03 0.05 0.07 0.06 0.07 0.03
Table 4 The fitted parameters by least-square fits in the year from 2002–2008. The β is the slope, and the Adj.R2 is the goodness estimation of fit.
β
Year 2002–2003 2003–2004 2004–2005 2005–2006 2006–2007 2007–2008
β
Adj.R2
asset
Adj.R2
employee
−0.16 −0.16 −0.12 −0.17 −0.12 −0.14
(0.02) (0.03) (0.02) (0.03) (0.03) (0.02)
0.83 0.74 0.85 0.78 0.54 0.83
−0.10 −0.12 −0.11 −0.10
(0.03) (0.03) (0.03) (0.02)
0.64 0.59 0.60 0.73
size S0 into different bins, with sizes increasing by powers of 4: 4n−0.5 < S0 < 4n+0.5 , n = 1, 2, 3.... One would find that, for both firm size measures asset and employee, there exist this scaling relation, which is −β
σ (s0 ) ∝ S0 .
(4)
The case of asset were showed in Figs. 7(a) and 7(b), and employee in Figs. 8(a) and 8(b). It was found that the slopes of the scaling relation varied in different years, but they were very close. Meanwhile, the slopes of the case of asset were a bit larger than case of employee, in both two consecutive years. Then we analyzed the standard deviation of both asset and employee growth rate in all consecutive years, as shown in Table 4. It was observed that the slopes in asset case were a bit larger than that in employee case, which showed that it was different when we chose different size measures to describe firm dynamics. Furthermore, comparing to the influence of firm size, we also investigated whether their growth rates changed with age. Here, firm age is simply given by the difference between the year of observation and that date [5]. From young to old, firms were divided into seven groups. Figs. 9(a) and 9(b) showed the standard deviation of asset growth rate by age, and the employee cases were showed in Figs. 10(a) and 10(b). We can see that there exist power-law scaling in both asset and employee by age. However, most of the slopes of case of asset are a bit larger than that of employee. Comparing to the standard deviation of growth rate by size, the slopes are smaller than that of growth rate by age, for both size measures asset and employee. The fitted parameters of standard deviation of growth rate by age are showed in Table 5.
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Y. Zou / Physica A 535 (2019) 122344
Fig. 6. The growth rate distribution of asset in four different industries in 2004–2005.
Fig. 7. (a) Standard deviation σ (s0 ) of asset growth rate of the year 2004–2005. The solid lines are least-square fits to the data with slope β = 0.12 ± 0.02. (b) The parallel analysis of the year 2007–2008. The solid lines are least-square fits to the data with slope β = 0.14 ± 0.02.
5. Conclusion In this paper, we analyzed Chinese firm dynamics by a database comprising more than 100 thousands firms in the period 1998–2008. Firstly, we studied the size distribution of Chinese firms, it was found that both size measures asset and employ could be well fitted with power-law distribution. The slopes of power-law scaling change a little in these years. However, the employee distribution have bigger scaling slopes than that of asset in all of these years. In addition, when firms are in categories, the power-law distribution of firm size have different exponents. Then we analyzed the growth rate distribution, we found that both asset and employee growth rate had Laplace form, not Gaussian distributed.
Y. Zou / Physica A 535 (2019) 122344
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Fig. 8. (a) Standard deviation σ (s0 ) of employee growth rate of the year 2004–2005. The solid lines are least-square fits to the data with slope β = 0.10 ± 0.02. (b) The parallel analysis of the year 2007–2008. The solid lines are least-square fits to the data with slope β = 0.10 ± 0.02.
Fig. 9. (a) Standard deviation σ (s0 ) of asset growth rate by age of the year 2004–2005. The solid lines are least-square fits to the data with slope β = 0.14 ± 0.02. (b) The parallel analysis of the year 2007–2008. The solid lines are least-square fits to the data with slope β = 0.28 ± 0.05.
Fig. 10. (a) Standard deviation σ (s0 ) of employee growth rate by age of the year 2004–2005. The solid lines are least-square fits to the data with slope β = 0.12 ± 0.02. (b) The parallel analysis of the year 2007–2008. The solid lines are least-square fits to the data with slope β = 0.14 ± 0.01.
In order to verify this finding, we estimated our result with Subbotin family function. The fitting results showed that all the parameters b were close to 1, not to 2, even firms are classified in four different industry. Furthermore, by analyzing annual logarithmic growth rates, it is found that the standard deviation also has a power-law shape, as researchers showed in several western countries. Finally, we also observed that the growth rate of firms had dependence on firms’ age. There
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Y. Zou / Physica A 535 (2019) 122344 Table 5 The fitted parameters by least-square fits in the year from 2002–2008. The β is the slope, and the Adj.R2 is the goodness estimation of fit.
β
Year 2002–2003 2003–2004 2004–2005 2005–2006 2006–2007 2007–2008
β
Adj.R2
asset
Adj.R2
employee
−0.19 −0.10 −0.14 −0.24 −0.21 −0.28
(0.05) (0.02) (0.02) (0.03) (0.06) (0.05)
0.67 0.76 0.90 0.92 0.66 0.83
−0.12 −0.09 −0.13 −0.14
(0.02) (0.02) (0.01) (0.01)
0.87 0.84 0.97 0.97
exists a power-law dependence of the standard deviation on the age of firms, which means the growth of firms also rely on the years that they have opened. Acknowledgment This work was supported by Ph.D. Programs Foundation of Anyang Normal University, China under Grant No. 162184118001. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
V. Pareto, Le Cours D’Economie, Macmillan, London, 1897. R. Axtell, Science 293 (2001) 1818. B. Podobnik, D. Horvatic, A. Petersen, B. Uroevi, H. Stanley, Proc. Natl. Acad. Sci. 107 (2010). H. Simon, C. Bonini, Amer. Econ. Rev. 48 (1958) 607. P. Cirillo, Physica A 389 (2010) 459. P. Cirillo, J. Husler, Physica A 388 (2009) 1546. E. Gaffeo, M. Gallegati, A. Palestrini, Physica A 324 (2003) 117. Q. Chen, J. Zhang, Y. Wang, Wireless communications, networking and mobile computing, in: Wicom 08, International Conference on IEEE, 2008, pp. 1. W. Xie, G. Gu, W. Zhou, Physica A 389 (2010) 3876. M.H.R. Stanley, et al., Nature 379 (1996) 804. L. Amaral, S. Buldyrev, S. Havlin, H. Leschhorn, P. Maass, M. Salinger, H.E. Stanley, J. de Phys. I 7 (1997) 621. G. Bottazzi, G. Dosi, M. Lippi, F. Pammolli, M. Riccaboni, Int. J. Ind. Organ. 19 (2001) 1161. G. Fabritiis, F. Pammolli, M. Riccaboni, Physica A 324 (2003) 38. S. Alfarano, M. Milakovic, A. Irle, J. Kauschke, J. Econom. Dynam. Control 36 (2012) 136. D. Fu, F. Pammolli, S. Buldyrev, M. Riccaboni, K. Matia, K. Yamasaki, H.E. Stanley, Proc. Natl. Acad. Sci. 102 (2005) 18801. M. Kelecki, Econometrica 13 (1945) 161. B. Jovanovic, Econometrica 50 (1982) 649. G. Bottazzi, A. Secchi, Econ. Lett. 80 (2003) 415. A. Palestrini, Econ. Lett. 94 (2007) 367. H.M. Gupta, et al., Physica A 375 (2007) 643. R. Gibrat, Les Inegalites Economiques, Sirey, Paris, 1933. G. Bottazzi, E. Cefis, G. Dosi, Ind. Corporate Change 11 (2002) 705. R. Axtell, R. Perline, D. Teitelbaum, et al., Kiel Institute for the World Economy. U.S. Small Business Administration, Office of Advocacy, 2006. M.T. Subbotin, Mat. Sb. 31 (1923) 296.
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
An analysis of Chinese firm size distribution and growth rate Yijiang Zou School of economics, Anyang Normal University, Anyang, 455000, China
highlights • • • •
The firm size distribution in China was studied by employing the empirical data. The Laplace form distribution of firms’ growth rate was estimated by Subbotin family function. The standard deviation distribution of growth rate were studied with assets and employees of these firms. To study the correlation between firms’ growth rate and their ages, the standard deviation of growth rate on the ages of firms was analyzed.
article
info
Article history: Received 20 September 2018 Received in revised form 27 February 2019 Available online 9 August 2019 Keywords: Firm size Growth rates Power law Laplace distribution Chinese firms
a b s t r a c t We analyzed a database comprising more than 100 thousands firms over the period 1998–2008. It was found that both asset and employee could be well fitted with power-law distribution, even firms were classified in different industries. Meanwhile, the growth rate distribution has a Laplace form, which is estimated by Subbotin family function, and it is also robust when firms are in categories. By analyzing annual logarithmic growth rates, we can see that the standard deviation has a power-law shape on both asset and employee, as researchers have found in western countries. In addition, there exists a power-law dependence of the standard deviation on the age of firms, which means that the growth of firms also rely on the years that they have opened. © 2019 Published by Elsevier B.V.
1. Introduction The distribution of firm size has been deeply studied in statistical and economic literature, since the pioneering work of Pareto [1]. The power-law behavior in the upper tail of firm size distribution has been verified for various periods and economies, e.g. for U.S. firms by Axtell [2], Podobnik et al. [3] and Simon and Bonini [4], and for Italian firms by Cirillo [5] and Cirillo and Husler [6]. However, most of empirical evidences on firm analysis are from developed countries. So it is necessary to test the ubiquity and robustness of these findings by widening the scope of this research to developing countries. Gaffeo [7] have analyzed the average size distribution of a pool of the G7 group’s firms over the period 1987– 2000. Chen [8] studied the top 500 Chinese firms and found that their ranks completely obey the Zipf distribution with exponents of 1. Xie [9] investigated the growth dynamics of the primary industry and the population of 2079 counties in mainland China. They find that the annual growth rates are distributed according to Student’s t distribution with the tail exponent less than 2. In this paper, we analyzed the distributional properties of Chinese firms during the period 1998–2008 with abundant data obtained from Chinese Industrial Enterprises Database (CIED). The size distribution was studied by two size measures, namely the total assets and employees. It was found that both the asset and employee distribution were power-law distributed. However, they have different Pareto index (the slope of power-law scaling). E-mail address: [email protected]. https://doi.org/10.1016/j.physa.2019.122344 0378-4371/© 2019 Published by Elsevier B.V.
2
Y. Zou / Physica A 535 (2019) 122344
Table 1 The descriptive statistics of assets of Chinese industrial firms for the years from 1998–2008. The unit of asset is 1000 yuan. Year
Obs.
Mean
Median
Sta.dev
Skewness
Kurtosis
Min
Max
1998 2000 2002 2003 2004 2005 2006 2007 2008
160806 158803 179379 194708 274406 270902 299772 335251 372305
67671.25 79475.81 82806.71 86748.84 78844.15 90445.86 97309.97 105364.4 99397.88
12158 13178 12898 13188 11828 13645 16409 15703 15000
598146.9 724051.3 768138.3 804635 1147967 1305670 1403115 1533228 1436910
59.61446 54.62236 52.43374 64.34964 219.5449 211.1767 192.2964 181.975 137.5098
5646.262 4414.765 3906.593 6831.871 76579.84 71044.82 61976.85 57455 34066.6
10 1 1 1 1 1 12 22 1
90322267 85791220 85808770 130525517 435000000 182000000 513000000 564000000 466445550
Table 2 The fitted parameters of asset and employ distribution in the years from 1998–2008. The blank data of employ are absence in the years from 1998–2003.
α
Year
xmin
P-value
asset 1998 2000 2002 2003 2004 2005 2006 2007 2008
α
xmin
P-value
2.57 2.55 2.54 2.53 2.50
2041 2018 2336 2019 1748
0.63 0.94 0.67 0.61 0.72
employee 2.20 2.10 2.05 2.05 2.07 2.06 2.04 2.05 2.05
462394 212102 194144 192000 225292 321308 252442 404500 314429
0.93 0.59 0.31 0.23 0.30 0.18 0.18 0.53 0.39
As known, statistics on the growth rates of business firms have been studied for a long time. Since the original paper on this subject began with the work of Stanley et al. [10] who showed empirically that growth rate of firms was not Gibrat like. One of the stylized facts, the Laplace distribution of growth rates are robust to how to define firm size [11–13]. Alfarano [14] et al.showed that the profit rate distribution of large publicly traded US companies was well described by a Laplace distribution. Fu [15] et al. proposed a model of proportional growth to explain the distribution of business firm growth rates, and they found that the predictions of the model agreed with empirical growth distribution and size-variance relationships. Looking at Chinese industrial firms, we observed that the growth rates distribution were consistently Laplace distributed, and the results were tested by the Subbotin fimaly function. Furthermore, another interesting issue of firm dynamics is the relationship between the size of a company and its growth rate [16–20]. Based on the other stylized fact of Stanley [10] et al. the standard deviation σ of the growth rates scales as a power-law of the initial firm size S. Then several models predict that standard deviation of growth rates amongst all firms with sales scales as a power law σ (g /S) S −β , where g is growth rate of firm size. Although it is known that Gibrat’s assumptions [21] are rejected empirically and theoretically, there are still some empirical findings have not addressed this issue [22,23]. Given that the size distribution of firms depends on the economic structure of each country, and considering China have the most rapid changes in the economic structure in past decades. Therefore, we think it is necessary to verify these results with Chinese firms. This paper is organized as follows: In Section 2, the details of empirical data were introduced and firm size distribution was studied. The growth rate distribution was analyzed in Section 3. In Section 4, the size-dependent standard deviation for growth rates was analyzed from size and age of firms. Conclusions were given in Section 5. 2. Data set and firm size distribution Data were obtained from the Chinese Industrial Enterprises Database conducted by the National Bureau of Statistics (NBS) of China for the years from 1998–2008. It contains various firm size measures, like employees, turnover, profits and assets. In this paper, we take employees and assets as size measures to study firms’ size distribution and their growth rates. The descriptive statistics of assets of Chinese industrial firms was presented in Table 1, and the probability density function (PDF) of asset were showed in Figs. 1(a) and 1(b), and the case of employee were showed in Figs. 2(a) and 2(b). One would find that both asset and employ distribution could be well fitted as power-law distribution. The fitted results of the probability distribution of asset and employ in all the years were showed in Table 2. It was observed that the Pareto indexes of asset distribution varied from 2.04 to 2.20. However, the Pareto index of employ distribution changed with an average value 2.5. This means that the analysis result may be different when we choose different firm size measures. Furthermore, in order to study the influence of different industries, the firms are in categories. The fitting result shows that the distribution of firms in four different industries has power-law form with different Pareto indexes, as shown in Fig. 3.
Y. Zou / Physica A 535 (2019) 122344
3
Fig. 1. (a) The probability distribution of asset in 2004, which can be fitted as a power-law distribution with Pareto index 2.07. (b) The asset distribution in 2007, and the Pareto index is 2.05.
Fig. 2. (a) The probability distribution of employ in 2004, which can be fitted as a power-law distribution with Pareto index 2.57. (b) The employee distribution in 2007, and the Pareto index is 2.53.
3. The distribution of growth rates In order to measure growth, the central variable we look at is the growth rate, defined as
( gi (t) ≡ log10
Si (t + ∆t)
)
Si (t)
= log10 Si (t + ∆t) − log10 Si (t),
(1)
where Si (t) and Si (t + ∆t) are the measures of firm size, e.g. assets and employees. Especially, the asset is in units of 103 Chinese Yuan. In this paper, we only study the growth rate in two consecutive years, which means that ∆t = 1 year hear. So we can write a rate in general as r1 ≡ log10
S1 S0
,
(2)
where S1 and S0 are size of firms in two consecutive years. The growth rates distribution of asset were showed in Figs. 4(a) and 4(b), and employee in Figs. 5(a) and 5(b). Remarkably, both asset and employee curves displayed a ‘tent-shaped’ form, like firms in other western countries. Furthermore, in order to verify that the growth rates distribution is not Gaussian, the well-known Subbotin family function is introduced [24], which represents a generalization of several particular cases, such as the Laplace and the Gaussian. In this way, we can provide a fitting for the growth rates distribution, and compare to different possible distributions. The functional form of the Subbotin family is f (x, a, b) =
1 1 b
2ab Γ 1 +
(
1 b
)e
− 1b | x−µ a |
b
,
(3)
where µ is the mean, b and a are two different shape parameters and Γ is the standard Gamma. If b → 1 the Subbotin distribution becomes a Laplace, and a Gaussian for b → 2. Then we estimated these three Subbotin parameters on our data by using maximum likelihood, as shown in Table 3. We can see that parameter b is closer to 1 than to 2, which
4
Y. Zou / Physica A 535 (2019) 122344
Fig. 3. The probability distribution of asset in four different industries in 2004.
Fig. 4. (a) The growth rate distribution of asset in 2004–2005. (b) The growth rate distribution of asset in 2007–2008.
means that the growth rate distribution is more similar to a Laplace one than Gaussian. Furthermore, parameter a seems to increase with years from 2002–2008 in both asset and employee cases, but parameter b is not very different in most of the cases. Meanwhile, the growth rates distribution of asset in categories are also been analyzed, as shown in Fig. 6. The fitting result of the Subbotin family function shows that the growth rate distribution are Laplace distributed, even firms are in categories. 4. Standard deviation of the growth rate Next, we studied the relationship between standard deviation and the growth rate. In order to investigate if and how the standard deviation of the logarithmic growth rates σ0 correlates with the initial size of firms S0 , we divided the initial
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Fig. 5. (a) The growth rate distribution of employee in 2004–2005. (b) The growth rate distribution of employee in 2007–2008. Table 3 The fitted three Subbotin parameters of asset and employ growth rates distribution, in the consecutive years from 2002–2008. The b and a are two different shape parameters of the Subbotin family, and µ is the mean. Year
b
a
µ
asset 2002–2003 2003–2004 2004–2005 2005–2006 2006–2007 2007–2008
b
a
µ
0.59 0.67 0.69 0.66 0.67 0.62
0.29 0.34 0.36 0.33 0.34 0.35
0.03 0.05 0.05 0.05 0.05 0.02
employee 0.64 0.71 0.72 0.71 0.72 0.65
0.21 0.23 0.25 0.23 0.23 0.24
0.03 0.05 0.07 0.06 0.07 0.03
Table 4 The fitted parameters by least-square fits in the year from 2002–2008. The β is the slope, and the Adj.R2 is the goodness estimation of fit.
β
Year 2002–2003 2003–2004 2004–2005 2005–2006 2006–2007 2007–2008
β
Adj.R2
asset
Adj.R2
employee
−0.16 −0.16 −0.12 −0.17 −0.12 −0.14
(0.02) (0.03) (0.02) (0.03) (0.03) (0.02)
0.83 0.74 0.85 0.78 0.54 0.83
−0.10 −0.12 −0.11 −0.10
(0.03) (0.03) (0.03) (0.02)
0.64 0.59 0.60 0.73
size S0 into different bins, with sizes increasing by powers of 4: 4n−0.5 < S0 < 4n+0.5 , n = 1, 2, 3.... One would find that, for both firm size measures asset and employee, there exist this scaling relation, which is −β
σ (s0 ) ∝ S0 .
(4)
The case of asset were showed in Figs. 7(a) and 7(b), and employee in Figs. 8(a) and 8(b). It was found that the slopes of the scaling relation varied in different years, but they were very close. Meanwhile, the slopes of the case of asset were a bit larger than case of employee, in both two consecutive years. Then we analyzed the standard deviation of both asset and employee growth rate in all consecutive years, as shown in Table 4. It was observed that the slopes in asset case were a bit larger than that in employee case, which showed that it was different when we chose different size measures to describe firm dynamics. Furthermore, comparing to the influence of firm size, we also investigated whether their growth rates changed with age. Here, firm age is simply given by the difference between the year of observation and that date [5]. From young to old, firms were divided into seven groups. Figs. 9(a) and 9(b) showed the standard deviation of asset growth rate by age, and the employee cases were showed in Figs. 10(a) and 10(b). We can see that there exist power-law scaling in both asset and employee by age. However, most of the slopes of case of asset are a bit larger than that of employee. Comparing to the standard deviation of growth rate by size, the slopes are smaller than that of growth rate by age, for both size measures asset and employee. The fitted parameters of standard deviation of growth rate by age are showed in Table 5.
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Fig. 6. The growth rate distribution of asset in four different industries in 2004–2005.
Fig. 7. (a) Standard deviation σ (s0 ) of asset growth rate of the year 2004–2005. The solid lines are least-square fits to the data with slope β = 0.12 ± 0.02. (b) The parallel analysis of the year 2007–2008. The solid lines are least-square fits to the data with slope β = 0.14 ± 0.02.
5. Conclusion In this paper, we analyzed Chinese firm dynamics by a database comprising more than 100 thousands firms in the period 1998–2008. Firstly, we studied the size distribution of Chinese firms, it was found that both size measures asset and employ could be well fitted with power-law distribution. The slopes of power-law scaling change a little in these years. However, the employee distribution have bigger scaling slopes than that of asset in all of these years. In addition, when firms are in categories, the power-law distribution of firm size have different exponents. Then we analyzed the growth rate distribution, we found that both asset and employee growth rate had Laplace form, not Gaussian distributed.
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Fig. 8. (a) Standard deviation σ (s0 ) of employee growth rate of the year 2004–2005. The solid lines are least-square fits to the data with slope β = 0.10 ± 0.02. (b) The parallel analysis of the year 2007–2008. The solid lines are least-square fits to the data with slope β = 0.10 ± 0.02.
Fig. 9. (a) Standard deviation σ (s0 ) of asset growth rate by age of the year 2004–2005. The solid lines are least-square fits to the data with slope β = 0.14 ± 0.02. (b) The parallel analysis of the year 2007–2008. The solid lines are least-square fits to the data with slope β = 0.28 ± 0.05.
Fig. 10. (a) Standard deviation σ (s0 ) of employee growth rate by age of the year 2004–2005. The solid lines are least-square fits to the data with slope β = 0.12 ± 0.02. (b) The parallel analysis of the year 2007–2008. The solid lines are least-square fits to the data with slope β = 0.14 ± 0.01.
In order to verify this finding, we estimated our result with Subbotin family function. The fitting results showed that all the parameters b were close to 1, not to 2, even firms are classified in four different industry. Furthermore, by analyzing annual logarithmic growth rates, it is found that the standard deviation also has a power-law shape, as researchers showed in several western countries. Finally, we also observed that the growth rate of firms had dependence on firms’ age. There
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Y. Zou / Physica A 535 (2019) 122344 Table 5 The fitted parameters by least-square fits in the year from 2002–2008. The β is the slope, and the Adj.R2 is the goodness estimation of fit.
β
Year 2002–2003 2003–2004 2004–2005 2005–2006 2006–2007 2007–2008
β
Adj.R2
asset
Adj.R2
employee
−0.19 −0.10 −0.14 −0.24 −0.21 −0.28
(0.05) (0.02) (0.02) (0.03) (0.06) (0.05)
0.67 0.76 0.90 0.92 0.66 0.83
−0.12 −0.09 −0.13 −0.14
(0.02) (0.02) (0.01) (0.01)
0.87 0.84 0.97 0.97
exists a power-law dependence of the standard deviation on the age of firms, which means the growth of firms also rely on the years that they have opened. Acknowledgment This work was supported by Ph.D. Programs Foundation of Anyang Normal University, China under Grant No. 162184118001. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
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