Generalized Pareto Distribution of Firm Sizes: Evidence from China

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Generalized Pareto Distribution of Firm Sizes: Evidence from China

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Yong Tao^*,Ruoxi Liu

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Submitted:

26 April 2024

Posted:

28 April 2024

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Abstract

It has been empirically observed that the upper tail of the firm size distribution follows either the Pareto distribution or the Zipf distribution, with both patterns being explained by Gibrat’s law (Gabaix, 1999). This article analyzed firm revenue data from China, spanning from 2005 to 2013, to examine the whole range of the firm size distribution. Our empirical analysis revealed that firm revenue data over these years is well fitted by a three-parameter generalized Pareto distribution, with the fitted parameters indicating a dichotomy: The size distribution of large-sized firms, namely the upper tail, is asymptotically characterized by a Pareto distribution or a Zipf distribution, whereas the size distribution of smaller and medium-sized firms is approximated by an exponential distribution. This finding suggests that Gibrat’s law should be extended to account for the emergence of a generalized Pareto distribution.

Keywords:

Subject: Business, Economics and Management - Economics

1. Introduction

It has been empirically observed that the upper tail of the firm size distribution follows either the Pareto distribution or the Zipf distribution (Axtell, 2001; Gabaix, 1999, 2009, 2016). Both laws state that the fraction of firms with a size greater than

x

is inversely proportional to

x

itself; that is,

P (X > x) = {(\frac{x}{x_{m}})}^{- a},

(1)

where

a \geq 1

and

x \geq x_{m}

Equation (1) represents the Pareto distribution. In particular, when

a = 1

, it corresponds to the Zipf distribution, which has been proposed to characterize the upper tail of the size distribution of firms and cities (Axtell, 2001; Gabaix, 1999, 2009, 2016; Malevergne et al., 2013). In contrast, the Pareto distribution with

a > 1

is commonly used to describe the upper tail of the distribution of income and wealth (Nirei and Souma, 2007; Gabaix, 2009; Benhabib et al., 2011; Aoki and Nirei, 2017; Jones and Kim, 2018). To account for the origin of the Pareto distribution (1), Gabaix (1999), applying Gibrat’s law, proposed the following random growth model:

d X_{t} = X_{t} (μ d t + σ d Z_{t}),

(2)

where

X_{t}

denotes the size of a firm (or city) at the time

t

μ

and

σ

are two parameters, and

d Z_{t} ~ N (0, d t)

denotes the standard Brownian motion that is independent of the

X_{t}

The stochastic differential equation (2) is a mathematical representation of Gibrat’s law (Gibrat, 1931), which posits that the growth rate a firm’s (or city’s) size,

r = (\frac{d X_{t}}{d t}) / X_{t}

, is independent of its size

X_{t}

. Equation (2) has been widely used to understand the inequality in the upper tail of the distribution (Gabaix, 2009; Benhabib et al., 2011; Aoki and Nirei, 2017; Jones and Kim, 2018). However, the validity of the Pareto distribution (1) in estimating the threshold parameter has been recently questioned (Jenkins, 2017). This challenge may result in a biased estimation of inequality (Charpentier and Flachaire, 2022). To address this problem, the generalized Pareto distribution (GPD) has been shown to provide more reliable results (Jenkins, 2017; Charpentier and Flachaire, 2022). Therefore, there is a need to consider applying the GPD rather than the Pareto distribution.

Empirical observations have suggested that Gibrat’s law (2) does not hold exactly but only asymptotically for large-sized firms (Almus, 2000; Becchetti and Trovato, 2002; Daunfeldt and Elert, 2013). Based on this observation, Tao (2024) proposed an extension of equation (2) by taking smaller-sized firms into account, as follows:

d X_{t} = (1 + η X_{t}) (μ d t + σ d Z_{t}),

(3)

where

η

is a parameter.

Equation (3) asymptotically approaches the functional form of equation (2) as

X_{t} \to \infty

. Tao (2024) has demonstrated that if firm size

x

evolves according to equation (3), the resulting size distribution conforms to the GPD. Since equation (3) extends equation (2) by incorporating smaller-sized firms, the GPD is anticipated to encompass the entire range of the firm size distribution. In this paper, we employ firm revenue data from China, ranging from 2005 to 2013, to examine the whole range of the firm size distribution. For this purpose, our dataset includes a large sample from both large-sized firms and small and medium-sized firms.

2. The Model

Tao (2024) has shown that, if the dynamics of firm size

x

obey the stochastic differential equation (3), the resulting density distribution

f (x, t)

satisfies a Kolmogorov forward equation as follows:

\frac{\partial f (x, t)}{\partial t} = - \frac{\partial [μ (1 + η x) f (x, t)]}{\partial x} + \frac{1}{2} \frac{\partial^{2} [σ^{2} {(1 + η x)}^{2} f (x, t)]}{\partial x^{2}} .

(4)

The solution of equation (4) can be written as (Tao, 2024):

\{\begin{matrix} f (x) = (\frac{η + \frac{1}{θ}}{1 + η x_{0}}) {(\frac{1 + η x}{1 + η x_{0}})}^{- \frac{1}{θ η} - 2} \\ x \geq x_{0} \end{matrix},

(5)

where

θ = - \frac{σ^{2}}{2 μ}

with

μ < 0

Here, we use

F (X \leq x) = \int_{x_{0}}^{x} f (z) d z

to denote the cumulative distribution. Thus, by equation (5), one has (Tao, 2024):

F (X \geq x) = {(\frac{1 + η x}{1 + η x_{0}})}^{- \frac{1}{θ η} - 1},

(6)

where

x \geq x_{0}

Equation (6) represents a generalized Pareto distribution (GPD) with three parameters. To see this, one can rewrite it in the standard form of the GPD:

F (X \geq x) = {[1 + A (\frac{x - x_{0}}{B})]}^{- \frac{1}{A}},

(7)

where

A = \frac{θ η}{1 + θ η}

and

B = \frac{θ (1 + η x_{0})}{1 + θ η}

It is easy to check that, when

η > 0

, the GPD (6) has an asymptotic Pareto tail on the right side of the distribution. In particular, when

η

approaches 0 sufficiently, the GPD (6) can be decomposed into a two-class pattern as follows (Tao, 2024):

F (X \geq x) \approx \{\begin{matrix} e x p (- \frac{x - x_{0}}{θ}) & x_{0} \leq x < x_{m} \approx \frac{1}{η} \\ {(\frac{x}{x_{m}})}^{- \frac{1}{θ η} - 1} & x > x_{m} \approx \frac{1}{η} \end{matrix},

(8)

where

x_{m} = (1 + η x_{0}) / η

3. Data Analysis

We employ firm revenue data from China, which spans from 2005 to 2013, to fit the GPD (6). However, due to the lack of data for the year 2010, this year has been excluded from our analysis. The raw data were collected from the Chinese Industrial Enterprises Database (CIED), which includes a large sample consisting of both state-owned and non-state-owned firms. In existing literature, firm size is typically measured by either the number of employees (Gabaix, 2009) or revenue (Chen et al., 2023). In this paper, we choose to use the firm’s revenue as the index for measuring its size.

To mitigate the potential impact of outliers within the dataset, we have chosen to exclude firm samples with negative or zero revenue, as these may indicate instances of bankruptcy. This implies that we are not considering the birth and death processes of firms. In fact, when these processes are taken into account, the resulting distribution of firm sizes follows a generalized double-Pareto distribution (Tao, 2024). The descriptive statistics of the raw data are presented in Table 1, displaying the number of observations, minimum, maximum, average, and median of each year’s data. To fit the GPD (6), we organized the firm revenue data into cumulative percentages at various revenue quantiles.

Figure 1a shows the fit of the GPD (6) to the data in China from 2005 to 2013. One can observe that the agreement between the GPD (6) and the data is very good. In Table 2, we used nonlinear least squares fitting in MATLAB to estimate three parameters in the GPD (6) and

R^{2}

for each year, with all

R^{2}

values exceeding 0.999. According to Table 2, we find that

η

is approximately in the order of

10^{- 5}

. This implies that

η

sufficiently approaches 0; therefore, by equation (8), the firm size distribution can be decomposed into a two-class pattern, where the size distribution of large-sized firms is asymptotically characterized by the Pareto distribution, while the size distribution of smaller and medium-sized firms is approximated by an exponential distribution. Figure 1b displays the two-class pattern.

The large-sized firms correspond to the upper tail in equation (8), where the Pareto exponent is denoted by

a = 1 + 1 / (θ η)

. We have listed the estimated values of

a

in Table 2, which are in the vicinity of 1 for all years, roughly agreeing with the Zipf distribution. However, smaller and medium-sized firms follow an exponential distribution rather than a Pareto distribution. This suggests that, unlike large-sized firms, smaller and medium-sized firms face fiercer competition and hence generate significantly less economic profit, aligning with the conditions of competitive market theory, where the exponential distribution is indicative of a spontaneous order in the firm size distribution, as described by Tao (2016).

4. Conclusion

Existing research has shown that the upper tail of the firm size distribution follows either the Pareto distribution or the Zipf distribution, with both patterns being explained by Gibrat’s Law in the form of equation (2). Using firm revenue data from China (2005-2013), our empirical analysis suggests that the firm size distribution is adequately represented by a three-parameter generalized Pareto distribution, which is explained by an extension of Gibrat’s Law presented by equation (3). The fitted parameters reveal a dichotomy: The size distribution of large-sized firms, namely the upper tail, is asymptotically characterized by a Pareto distribution or a Zipf distribution, and the size distribution of smaller and medium-sized firms is approximated by an exponential distribution. Our findings suggest extending the version of Gibrat’s law given in equation (2) to the form in equation (3) to accommodate the emergence of a generalized Pareto distribution.

Author Contributions

Yong Tao designed research; Ruoxi Liu organized raw data for the research; Ruoxi Liu analyzed data; Yong Tao wrote the paper.

Data Availability Statement

This study analyzed publicly available datasets from the Chinese Industrial Enterprises Database.

Acknowledgement

This work was supported by the Social Science Planning Project of Chongqing (Grant No. 2019PY40) and the Research project on education and teaching reform in Southwest University (Grant No. 2021JY045)

Author Information

The authors declare no competing interests. Correspondence and requests for materials should be addressed to Yong Tao (taoyingyong@swu.edu.cn) or Ruoxi Liu (liuruoxi64@163.com).

References

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Figure 1. a. Fitting results of generalized Pareto distribution (6). b. Fitting results of the two-class distribution (8).

Table 1. Descriptive statistics of data.

Year	Obs	MIN	MAX	AVG	MEDIAN
2005	269553	10	1.49 $\times$ 108	92094	19270
2006	299704	10	1.88 $\times$ 108	104827	21735
2007	335018	10	1.96 $\times$ 108	119572	25580
2008	368529	10	2.28 $\times$ 108	121711	26143
2009	335535	10	2.1 $\times$ 108	139400	32344
2011	302591	150	2.6 $\times$ 108	271600	77667
2012	323960	10	4.13 $\times$ 108	277935	79831
2013	344831	145	4.77 $\times$ 108	292601	84302

Table 2. Fitting parameters.

Year	$η$	$x_{0}$	$θ$	$a$	$R^{2}$
2005	7.34 $\times$ 10^-5	4.07 $\times$ 10³	1.10 $\times$ 10⁵	1.12	0.999
2006	5.89 $\times$ 10^-5	4.07 $\times$ 10³	1.22 $\times$ 10⁵	1.14	0.999
2007	4.41 $\times$ 10^-5	4.34 $\times$ 10³	1.14 $\times$ 10⁵	1.20	0.999
2008	4.47 $\times$ 10^-5	4.26 $\times$ 10³	1.40 $\times$ 10⁵	1.16	0.999
2009	3.27 $\times$ 10^-5	4.52 $\times$ 10³	1.56 $\times$ 10⁵	1.20	0.999
2011	1.43 $\times$ 10^-5	1.90 $\times$ 10⁴	1.79 $\times$ 10⁵	1.39	0.999
2012	1.47 $\times$ 10^-5	1.84 $\times$ 10⁴	2.17 $\times$ 10⁵	1.31	0.999
2013	1.58 $\times$ 10^-5	1.93 $\times$ 10⁴	2.94 $\times$ 10⁵	1.22	0.999

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Generalized Pareto Distribution of Firm Sizes: Evidence from China

Abstract

1. Introduction

2. The Model

3. Data Analysis

4. Conclusion

Author Contributions

Data Availability Statement

Acknowledgement

Author Information

References

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