Atushi Ishikawa

Pareto index induced from the scale of companies

Employing profits data of Japanese companies in 2002 and 2003, we confirm that Paretos law and the Pareto index are derived from the law of detailed balance and Gibrats law. The last two laws are observed beyond the region where Paretos law holds. By classifying companies into job categories, we find that companies in a small-scale job category have more possibilities of growing than those in a large-scale job category. This kinematically explains that the Pareto index for the companies in the small-scale job class is larger than that for the companies in the large-scale job class.

Annual change of Pareto index dynamically deduced from the law of detailed quasi-balance

Employing data on the assessed value of land in 1983–2005 Japan, we investigate the dynamical behavior in the high scale region of non-equilibrium systems. From the detailed quasi-balance and Gibrats law, we derive a relation between the change of Pareto index and a symmetry in the detailed quasi-balance. The relation is confirmed in the empirical data nicely.

Derivation of the distribution from extended Gibrat's law

Employing profits data of Japanese companies in 2002 and 2003, we identify the non-Gibrats law which holds in the middle profits region. From the law of detailed balance in all regions, Gibrats law in the high region and the non-Gibrats law in the middle region, we kinematically derive the profits distribution function in the high and middle range uniformly. The distribution function accurately fits with empirical data without any fitting parameter.

Pareto law and Pareto index in the income distribution of Japanese companies

In order to study the phenomenon in detail that income distribution follows Pareto law, we analyze the database of high income companies in Japan. We find a quantitative relation between the average capital of the companies and the Pareto index. The larger the average capital becomes, the smaller the Pareto index becomes. From this relation, we can possibly explain that the Pareto index of company income distribution hardly changes, while the Pareto index of personal income distribution changes sharply, from a viewpoint of capital (or means). We also find a quantitative relation between the lower bound of capital and the typical scale at which Pareto law breaks. The larger the lower bound of capital becomes, the larger the typical scale becomes. From this result, the reason there is a (no) typical scale at which Pareto law breaks in the income distribution can be understood through (no) constraint, such as the lower bound of capital or means of companies, in the financial system.

The uniqueness of firm size distribution function from tent-shaped growth rate distribution

Employing profits data of Japanese firms in 2003 and 2004, we report the proof that a Non-Gibrats law in the middle scale region of profits is unique under the law of detailed balance. This uniquely leads to the probability distribution function (pdf) of profits. In the proof, two approximations are employed. The pdf of growth rate is described as tent-shaped exponential functions and the value of the origin of the growth rate distribution is constant. These approximations are confirmed in the database. The resultant profits pdf fits with the empirical data consistently. This guarantees the validity of the approximations.

Quasi-Statically Varying Power-Law and Log-Normal Distributions in the Large and the Middle Scale Regions of Japanese Land Prices

Employing data on the assessed value of land in 1974–2008 Japan, we exhibit a quasistatically varying distributions not only in the large scale region but also in the middle scale one. In the derivation, an extended Gibrat’s law under the detailed quasi-balance is adopted together with two approximations. The resultant distribution is quasi-static power-law with the varying exponent in the large scale region and the quasi-static log-normal distribution with the varying standard deviation in the middle scale one. In the distributions, not only the change of the exponent but also the change of the standard deviation depends on the parameter of the detailed quasi-balance. These results are consistently confirmed by the empirical data.

A new method for measuring tail exponents of firm size distributions

We propose a new method for estimating the power-law exponent of a firm size variable, such as annual sales. Our focus is on how to empirically identify a range in which a firm size variable follows a power-law distribution. As is well known, a firm size variable follows a power-law distribution only beyond some threshold. On the other hand, in almost all empirical exercises, the right end part of a distribution deviates from a power-law due to finite size effect. We modify the method proposed by Malevergne et al. (2011) so that we can identify both of the lower and the upper thresholds and then estimate the power-law exponent using observations only in the range defined by the two thresholds. We apply this new method to various firm size variables, including annual sales, the number of workers, and tangible fixed assets for firms in more than thirty countries.

Non-Gibrat's Law in the Middle Scale Region

By using numerical simulation, we confirm that Takayasu--Sato--Takayasu (TST) model which leads Paretos law satisfies the detailed balance under Gibrats law. In the simulation, we take an exponential tent-shaped function as the growth rate distribution. We also numerically confirm the reflection law equivalent to the equation which gives the Pareto index

Shape of growth-rate distribution determines the type of Non-Gibrat’s Property

\mu

Fractal structure with a typical scale

in TST model. Moreover, we extend the model modifying the stochastic coefficient under a Non-Gibrats law. In this model, the detailed balance is also numerically observed. The resultant pdf is power-law in the large scale Gibrats law region, and is the log-normal distribution in the middle scale Non-Gibrats one. These are accurately confirmed in the numerical simulation.