Power Laws with Different Exponents in Firm-Size Variables

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.