Vol. 3, 2009-11 | April 7, 2009 | http://www.economics-ejournal.org/economics/journalarticles/2009-11 Power-Law and Log-Normal Distributions in Temporal Changes of Firm-Size Variables Atushi Ishikawa Kanazawa Gakuin University, Kanazawa, Japan Abstract In this paper the author shows that signed temporal changes of firm size variables follow the power-law for large changes; while, for middle changes a log-normal distribution is found. In the analyses, the author employed three databases: high- income data, high-sales data and positive-profits data of Japanese firms. It is particularly worth noting that the growth rate distributions in temporal changes of the firm size data have no wide tail, unlike the distributions observed in assets and sales of firms, the number of employees and personal income data. An Extended-Gibrat’s Law was also found in the growth rate distributions of temporal changes of firm size variables, which induces both the power-law and the log-normal distributions in the temporal changes of firm size under the Detailed Balance. Special issue “Reconstructing Macroeconomics” JEL: D30, D31, D39 Keywords: Econophysics; temporal change of firm size; Pareto’s law; log-normal distribution; (non-)Gibrat’s law; detailed balance Correspondence: Atushi Ishikawa; Kanazawa University, Kanazawa 920-1392, Japan, e-mail address: [email protected] The author is grateful to the Yukawa Institute for Theoretical Physics at Kyoto University, where this work was initiated during the YITP-W-07-16 on “Econophysics III – Physical Approach to Social and Economic Phenomena” and especially to Dr. S. Fujimoto for a lot of useful discussions and comments about K–S test. Thanks are also due to Professor H. Kasama for careful reading of the manuscript. This work was supported in part by a Grant–in–Aid for Scientific Research (C) (No. 20510147) from the Ministry of Education, Culture, Sports, Science and Technology, Japan. © Author(s) 2009. Licensed under a Creative Commons License - Attribution-NonCommercial 2.0 Germany 2 Economics: The Open-Access, Open-Assessment E-Journal 1 Introduction Power-law distributions are frequently observed in economic data such as as- sets, the number of employees, personal income, sales, profits and income of firms (denoted by x). The Power Law is known as Pareto’s Law (Pareto 1897) and the probability density function (pdf) is represented as μ 1 PPL(x)=Cx− − for x>xth , (1) where C is a normalization and the power μ is called the Pareto index. In general, the power-law is valid only for large-sized companies (Badger 1980; Montroll and Shlesinger 1983). The size threshold of the region of large-sized firms is denoted by xth. It has been hypothesized that the pdf follows the log-normal distribution for middle-sized firms below the size threshold xth: 1 ln2 (x/x¯) PLN(x)= exp x<xth . (2) x√2πσ2 − 2σ2 ∙ ¸ Here,x ¯ is a mean value and σ2 is a variance. Therefore, we concluded that further research of the power-law distribution was warranted because a small number of firms dominate a large percent of the overall sales and profits. Furthermore, since the majority of firms are middle-sized, we also decided to look at the log-normal distributions. It has been shown that these distributions can be explained by laws observed in a massive amount of digitized economic data. Fujiwara et al. (2003, 2004) point out that Pareto’s Law can be derived from the Law of Detailed Balance and from Gibrat’s Law (Gibrat 1931). Along these lines, Ishikawa (2006a, 2007a) shows that the log-normal distribution can also be deduced using the Law of Detailed Balance and Non-Gibrat’s Law. The Detailed Balance is time-reversal symmetry as observed in the equilibrium system. On the one hand, Gibrat’s Law states that the conditional pdf of the growth rate is independent of the initial value. On the other hand, Non- Gibrat’s Law describes the dependence of the initial value. Gibrat’s Law is observed only in companies in the large scale region, and Non-Gibrat’s Law in companies in the middle scale region. It is interesting to note that there are two types of growth rate distributions. The shape of the growth rate distribution of profits or income of firms (Fig. 11)is different from that of assets and sales of firms, the number of employees or personal income (Fig. 2). This difference is observed not only for companies in the large scale region but also for companies in the middle scale region. The point is that the difference might be related to the difference between two Non-Gibrat’s Laws in companies in the middle scale region. On the one hand in Fig. 1, the probability of positive growth decreases and the probability of negative growth increases as the bin size of the classification of x increases in the middle scale region (Ishikawa 2006a, 2007a). On the other hand in Fig. 2, the probability of positive and negative growth decreases as the bin size of the classification of x increases (Aoyama 2004, Aoyama et al. 2004). This size dependence for companies in the middle scale region 1All figures appear in the Appendix. www.economics-ejournal.org Economics: The Open-Access, Open-Assessment E-Journal 3 is significant due to the preponderance of companies and individuals in this size range. In this study, we propose that the shape of the growth rate distribution is de- termined by the type of economic variable. In concrete terms, the variables are calculated by any subtraction or not. In this study we will prove our hypothesis by employing data from sales, profits and income. 2 Firm Size Distributions In this section, we will review the derivation of Pareto’s Law and the log-normal distribution from the Detailed Balance and (Non-)Gibrat’s Law, and confirm the laws by employing data for sales, profits and income of Japanese firms. In Japan, firms having an annual income of more than 40 million yen have been officially designated as “high-income firms”; there are approximately 70 thousand of these firms. The exhaustive database was published by Diamond Inc. Top 500 thousand sales data of Japanese firms are available on the database “CD Eyes 50” published by TOKYO SHOKO RESEARCH, LTD. This database is thought to be approximately exhaustive. The database includes both positive and negative profits data. The number of firms with positive profits is approximately 300 thousand and that with negative profits is approximately 40 thousand. In this study we will exclude firms with negative profits, the number of which is significantly less than that with positive profits. This is due to the fact that the negative profits data gathered from high-sales data are exclusive. Despite the fact that we do not have a complete picture of the positive profits data for companies in the middle scale region; nonetheless, we decided to employ this data to investigate the consistency of the laws with the data. In this study, we will examine the following three databases: high-income data (Database I), high-sales data (Database II) and positive-profits data (Database III). 2.1 Pareto’s Law from the Detailed Balance and Gibrat’s Law Let firm sizes at the two successive points in time be denoted by x1 and x2.The growth rate R is defined as the ratio R = x2/x1. The Detailed Balance and Gibrat’s Law (Gibrat 1932) are represented as follows: Detailed Balance • The joint pdf P12(x1,x2) is symmetric under the exchange x1 x2: ↔ P12(x1,x2)=P12(x2,x1) . (3) Gibrat’s Law • The conditional pdf of the growth rate Q(R x1) is independent of the initial | value x1: Q(R x1)=Q(R) , (4) | www.economics-ejournal.org 4 Economics: The Open-Access, Open-Assessment E-Journal where the conditional pdf Q(R x1)isdefined as | P1R(x1,R) Q(R x1)= (5) | P (x1) by using the pdf P (x1)andthejointpdfP1R(x1,R). These laws are confirmed in Databases I — III. In order to compare analyses in the next section, we will investigate data for 2003 (x0), 2004 (x1) and 2005 (x2). In the scatter plot of each database, the Detailed Balance (3) is approximately confirmed by applying the one-dimensional Kolmogorov—Smirnov (K—S) test (see 2 Appendix).Figs.3—5showtime-reversalsymmetryundertheexchangex1 x2. Gibrat’s Law (4) is also confirmedforeachdatabase.Figs.6—8showthatthe↔ conditional pdf of the growth rate is approximately independent of the initial value, if the initial value is larger than some threshold xth.Herethepdfforr =log10 R defined by q(r x1) is related to that of R by | log q(r x1)=log Q(R x1)+r +log (ln 10) . (6) 10 | 10 | 10 Note that large negative growth is not available if there is a lower bound of the data. This is notably observed in Figs. 3 and 6 for high-income Database I. This is also observed in Figs. 4 and 7 for high-sales Database II; however, the lower bound is probably obscure.3 The Detailed Balance and Gibrat’s Law are confirmed by employing personal income data in Japan (Fujiwara et al. 2003), and assets and sales data in France and the number of employees in UK (Fujiwara et al. 2004). In the literature (Fujiwara et al. 2003, 2004), Pareto’s Law is analytically derived from the Detailed Balance and Gibrat’s Law. By using the relation P12(x1,x2)dx1dx2 = P1R(x1,R)dx1dR under the exchange of variables from (x1,x2)to(x1,R), these two joint pdfs are related to each other P1R(x1,R)=x1P12(x1,x2) .