We discuss a mechanism through which inversion symmetry (i.e., invariance of a joint probability density function under the exchange of variables) and Gibrat’s law generate power-law distributions with different tail exponents. Using a dataset of firm size variables, that is, tangible fixed...

In this study, the authors examine exhaustive business data on Japanese firms, which cover nearly all companies in the mid- and large-scale ranges in terms of firm size, to reach several key findings on profits/sales distribution and business growth trends. Here, profits denote net profits....

By analyzing a huge amount of point-of-sale data collected from Japanese supermarkets, we find power law relationships between price and sales numbers. The estimated values of the exponents of these power laws depend on the category of products; however, they are independent of the stores,...

In this study, we investigate interfirm networks by employing a unique data set containing information on more than 800,000 Japanese firms, about half of all corporate firms currently operating in Japan. First, we find that the number of relationships, measured by the indegree, has a fat-tail...

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 Gibrat's law, we derive a relation between the change of Pareto index and a symmetry in the detailed...

In order to understand the characteristics common to distributions which have both fractal and non-fractal scale regions in a unified framework, we introduce the concept of a typical scale. We employ a model of 2d gravity modified by the R2 term as a tool to understand such distributions through...

We study distributions which have both fractal and non-fractal scale regions by introducing a typical scale into a scale invariant system. As one of models in which distributions follow power law in the large-scale region and deviate further from the power law in the smaller-scale region, we...

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...

Employing profits data of Japanese companies in 2002 and 2003, we confirm that Pareto's law and the Pareto index are derived from the law of detailed balance and Gibrat's law. The last two laws are observed beyond the region where Pareto's law holds. By classifying companies into job categories,...

Employing profits data of Japanese firms in 2003 and 2004, we report the proof that a Non-Gibrat's 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...