Statistical Properties in Firms’ Large-scale Data

In this chapter, I present my views on the future direction of economics from the history of the development of physics. In physics, statistical rules found in macroscopic systems guide the theoretical framework of microscopic systems. In both physics and economics, macroscopic systems are aggregates of microscopic components. Therefore, the methodology of physics should be an important guide to the development of economics. From this perspective, this chapter describes the structure of this book.

The distribution of the growth rate does not depend on the initial value in a large-scale range of firm-size variables, such as operating revenues, assets, current profits, and the number of employees. This is called Gibrat’s law. On the other hand, in the mid-scale range of firm-size variables, the growth-rate distribution changes regularly based on the initial value. This idea is referred to as the non-Gibrat’s property in this book. In this chapter we show that when the system is in equilibrium, a log-normal distribution of firm-size variables is derived from the non-Gibrat’s property in the mid-scale range. When the growth-rate distribution is linear on the log–log axis, it is analytically proven that the form of the system’s non-Gibrat’s property at equilibrium is uniquely determined. We then confirm that these properties can be observed with high accuracy in empirical data using the positive current profit data for Japanese firms in 1998 and 1999.

In this chapter, we show that when a system changes quasi-statically, the power-law and log-normal distributions, which change quasi-statically, are derived from Gibrat’s law in the large-scale range and from the non-Gibrat’s property in the mid-scale range. Then using Japan’s publicly announced land price data from 1974 to 2020, analytical discussions can be confirmed accurately.

Similar to current profits discussed in Chap. 2, operating revenues and total assets also follow a power-law distribution in the large-scale range and a log-normal distribution in the mid-scale range in a certain year. Also, observing such short-term changes as two consecutive years, there is a time-reversal symmetry in the joint probability density function of such firm-size variables over “a certain year” and “next year.” Furthermore, Gibrat’s law also holds, which states that the conditional growth-rate distribution of firm-size variables does not depend on the initial values in a large-scale range. However, unlike the positive current profits discussed in Chap. 2, the distribution of such growth rates as operating revenues and total assets is not linear on the logarithmic axis; it has a downward curvature. Even in this case, the initial dependence of the conditional growth-rate distribution in the mid-scale range is regular. Here we extend our discussion in Chap. 2 to present a non-Gibrat’s property that describes its regularity. We show that log-normal distribution is derived from time-reversal symmetry and the extended non-Gibrat’s property and conclude that the results are consistent with the empirical data.

With the Orbis database provided by Bureau van Dijk, we analyzed the dependence of firms’ operating revenues, total assets, and number of employees (firm-size variables) on firm age in Japan and France from 2010 to 2013. As a result, we confirmed that the geometric mean value of the firm-size variables obeys a power-law growth for its first 10 years and subsequently follows exponential growth. Using numerical simulations, these long-term properties of firm-size growth were derived from short-term growth law and properties that were observed in two successive years. First, early power-law growth under a size threshold comes from the extended non-Gibrat’s property. Second, subsequent exponential growth over the threshold is derived from Gibrat’s law.

It is as important to consider how a firm will cease its activities as how it will continue those same activities. The short-term inactive rate of firms is observed in the long term as their age distribution. In this chapter, we identify the age dependence of the inactive rate of firms and link it to the long-term property of firm-age distribution. Specifically, we investigated the inactive rates of firm activities by comparing their 2014 and 2015 statuses in Germany, Spain, France, the United Kingdom, Italy, Japan, Korea, and the Netherlands. We found that in Japan, the inactive rate of firm activities does not depend on a firm’s age. In other countries, however, the inactive rate of young firms, which was higher than that of more established firms, gradually fell and eventually became constant as firms aged. This inactive rate leads to the following conclusion. In Japan, firm-age distribution decreases exponentially; in other countries, it exponentially decreases asymptotically, but young firms shift slightly upward. Using empirical data, we compared the inactive rate of firms with parameters measured from firm-age distribution and confirmed our analytical discussion.

In this chapter, we investigate the dependence of the inactive rate of firms on the following main financial variables: total revenue, net income, total assets, and net assets. We used worldwide information on German, Spanish, French, British, Italian, Japanese, Korean, and Dutch firms recorded in the 2015 and 2016 editions of the comprehensive Orbis database of listed and unlisted firms. We confirmed that the inactive rate of firms is constant regardless of the size of the financial variables in the large-scale data range. In the mid-scale data range, the inactive rate of firms increases under a power law as the financial variables decrease. The boundary between the large- and mid-scale data ranges corresponds to the boundary between the power-law and log-normal distributions of the financial data.

In Chap. 8, we discuss the relationship between the power-law distributions observed in the large-scale range of different types of firm-size variables at the same time. We focused on operating revenues, tangible fixed assets, and the number of employees as firm-size variables and measured their power indices by country and year. We confirmed that the power indices of the three types of firm-size variables barely changed over time and that they differ depending on the type of variable. In Chap. 3, using time-reversal symmetry, we connected the power-law distributions in which the exponent changes due to the quasi-static time evolution of a system composed of single variables. After applying this argument to a system consisting of two sets of variables, we conclude that the ratio of the two power indices and the slope of the symmetry axis coincide by simultaneously considering the quasi-inversion symmetry from one firm-size variable to another. Our analytical conclusions are based on empirical data from firms in Japan, Spain, France, the United Kingdom, and Italy because they have sufficient data and their axis of symmetry can be easily observed. We confirmed that our analytical result is accurately established within the error range.

We directly observed a Cobb–Douglas symmetric plane using the index of surface openness, which is used in geography, and successfully identified it. Based on this observation, we measured the capital shares (capital elasticity) and labor shares (labor elasticity) and compared them with the results of multiple regression analysis used in economics. We confirmed consistent agreement in seven countries: Japan, Germany, France, Spain, Italy, the United Kingdom, and the Netherlands. Thus, we show that the Cobb–Douglas production function can be clearly captured in empirical data as a geometric entity with a quasi-inverse symmetry of variables. Based on the above discussion, we theoretically clarified why the Cobb–Douglas production function is better fit to empirical data in economics, because it uniquely derives the fact that their variables follow a power-law distribution.