Advances in Mathematical Physics

Volume 2016, Article ID 9303480, 6 pages

http://dx.doi.org/10.1155/2016/9303480

## Firm Growth Function and Extended-Gibrat’s Property

^{1}Kanazawa Gakuin University, 10 Sue, Kanazawa, Ishikawa 920-1392, Japan^{2}National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan^{3}Department of Informatics, The Graduate University for Advanced Studies, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan^{4}PRESTO, Japan Science and Technology Agency, 7 Gobancho, Chiyodaku, Tokyo 102-0076, Japan^{5}Graduate School of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan^{6}The Canon Institute for Global Studies, 5-1 Marunouchi 1-chome, Chiyoda-ku, Tokyo 100-6511, Japan

Received 21 September 2015; Accepted 6 March 2016

Academic Editor: Doojin Ryu

Copyright © 2016 Atushi Ishikawa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We analytically show that the logarithmic average sales of firms first follow power-law growth and subsequently follow exponential growth, if the growth-rate distributions of the sales obey the extended-Gibrat’s property and Gibrat’s law. Here, the extended-Gibrat’s property and Gibrat’s law are statistically observed in short-term data, which denote the dependence of the growth-rate distributions on the initial values. In the derivation, we analytically show that the parameter of the extended-Gibrat’s property is identical to the power-law growth exponent and that it also decides the parameter of the exponential growth. By employing around one million bits of exhaustive sales data of Japanese firms in the ORBIS database, we confirmed our analytic results.

#### 1. Introduction

As in natural science, statistical laws are also frequently observed in social science. In econophysics, the statistical laws observed in the behavior of people and firms and their universality have been thoroughly investigated [1, 2]. The statistical laws observed in various social quantities at a point in time are especially well known. One of the main arguments is power-law distribution [3–23]. The power-law distributions of firm-size variables in calendar year (e.g., sales, assets, and number of employees) over size threshold have been observed in a number of years and countries as follows:Here, is the probability density function (PDF) of . Exponent is called Pareto’s index [3]. At the same time, mid-sized variables under size threshold have also frequently been described by log-normal distribution.

Short-term statistical laws have also been investigated, which were observed in firm-size variables at two successive points in time : (quasi-)inverse symmetry and (non-)Gibrat’s law. Inverse symmetry denotes that the system is static and symmetric under the time reversal exchange of variables [24, 25]. Quasi-inverse symmetry means that the system is quasi-static and can be expressed as symmetric under the exchange of variables , where and are parameters [26, 27]. Gibrat’s law suggests that conditional PDF of the growth rate defined by is independent of initial value [24, 25]. This law is observed in the large-scale range over size threshold . Non-Gibrat’s law, which reflects the dependence of growth-rate distribution on initial value , is observed in the mid-scale range under [28–31].

The statistical laws, which are observed in firm-size variables at a point in time, are related to the statistical laws observed in them at two successive points in time . Fujiwara et al. showed that the power law (1) observed in large-sized variables is derived from Gibrat’s law [32, 33] under inverse symmetry [24, 25]. The log-normal distribution observed in mid-sized variables is inferred from non-Gibrat’s law under inverse symmetry [28–31]. Quasi-statistically varying power-law and log-normal distributions are derived from Gibrat’s and non-Gibrat’s laws under quasi-inverse symmetry, respectively [26, 27].

At the same time, interesting studies, related to long-term statistical laws, have been reported in empirical data analyses of the network of firms. Miura et al. showed that the number of business connections of firms exponentially increases as they age [34]. Mizuno et al. reported that the sales of firms depend on the number of business connections [35]. Such observations suggest that the sales of firms exponentially grow with age. In this context, the emergence of an early power-law growth is natural for firms that started small. Small firms must grow rapidly to reach the trajectory of exponential growth. This result is probably related to Luttmer’s report that the number of employees grows rapidly in the beginning but soon loses momentum [36]. This can be interpreted as power-law growth. Besides firm-size variables, Petersena et al. observed power-law growth in the progress of researchers defined by the number of publications of scientific papers [37].

In this study, we propose a law of the newly formulated firm growth derived from short-term statistical laws. If there are laws in firm growth [38], they are long-term statistical laws that are different from statistical laws in the short-term. However, these laws might be related. We verify this conjecture in the rest of our paper, which is organized as follows. The law of firm growth must be related to non-Gibrat’s and Gibrat’s laws that describe the change of the growth-rate distributions that are dependent on the initial values. In the next section, we review Gibrat’s law and introduce an extended-Gibrat’s property instead of non-Gibrat’s law. We also analytically derive firm growth that first obeys the power-law function and subsequently follows an exponential function. After that, using our database, we demonstrate that firm growth is analytically derived and observed in the empirical data. The last section concludes this paper.

#### 2. Results

##### 2.1. Gibrat’s Law and Extended-Gibrat’s Property

Gibrat’s law is represented using conditional PDF as follows:Here, does not depend on initial value . This Gibrat’s law is observed in a range over size threshold . On the other hand, under threshold , does depend on . For the case of sales [31], the positive growth-rate distributions gradually decrease as increases, and the negative growth-rate distributions hardly change. Using conditional PDF of logarithmic growth rate which is related to by , this property can be expressed aswhere is a constant. In this paper, we call (3) and (4) the extended-Gibrat’s property, which is different from the previously examined non-Gibrat’s law [30, 31]. Later, we confirm the extended-Gibrat’s property with empirical data.

##### 2.2. Analytical Derivation of Firm Growth

Next, we analytically derive firm growth from the extended-Gibrat’s property ((3) and (4)) and from Gibrat’s law (2). To estimate firm growth, it is convenient to identify a starting point in time for every firm. We consider a firm’s incorporation as its starting point and consider its age from its year of foundation. For simplicity, we examine the average values instead of the distribution of variables and denote the logarithmic average value of the firm-size variable at firm age as . The foundation year is .

We define the growth rate from to asThe dependence in growth rate on is significant in the extended-Gibrat’s property. We hypothesize that Gibrat’s law and the extended-Gibrat’s property, which are observed in two successive calendar years , are also valid in two successive firm ages . Therefore, in the following discussions, calendar year in can be replaced by firm age .

In extended-Gibrat’s property range , using (3) and (4), the average value of is expressed aswhere integrations and are assumed to converge and are denoted as and , respectively. In (6), the dependence of on is important. When the second term is negligible compared with the first term in (6), is approximated byHere, . Equation (7) can be rewritten as follows:By combining (5) and (8), we obtain the following recurrence formula:When is sufficiently larger than 1, (9) has a solution:We used Maclaurin expansion as .

In Gibrat’s law range , the average value of is a constant: . Therefore, the recurrence formula takes the following form:where is a constant. Equation (11) has a solution:In (12), is a constant and is given by

Consequently, using the extended-Gibrat’s property and Gibrat’s law, we show that the growth of the logarithmic average of sales first follows a power-law function when under and next follows an exponential function above . In this discussion, the parameter of the extended-Gibrat’s property is identified by the power-law exponent, and also decides the parameter of exponential growth .

##### 2.3. Data Analysis

We employ the ORBIS database, provided by Bureau van Dijk [39], which contains around 150 million pieces of firm-size data from all over the world. In the database, we analyze around one million pieces of Japanese firm-size data for 2007 and 2008. Since the number of active firms in Japan is estimated to be around one million [40], this database is exhaustive. We denote the sales data of Japanese firms from 2007 and 2008 as with .

First, we confirm Gibrat’s law (2) and the extended-Gibrat’s property ((3) and (4)). Figure 1 depicts the conditional PDFs of logarithmical sales growth in logarithmically equal-sized bins in thousands of US dollars. The values are adjusted using the Consumer Price Index. In Figure 1(b), where , the growth-rate distributions hardly change as increases. This corresponds to Gibrat’s law (2). On the other hand, in Figure 1(a) where , the positive growth-rate distributions gradually decrease as increases, and the negative growth-rate distributions hardly change. The later property corresponds to (4). Figure 2 shows that, for , hardly changes as increases when . This figure also verifies (3).