The log-normal distribution from Non-Gibrat's law in the middle scale region of profits

Employing profits data of Japanese firms in 2003--2005, we kinematically exhibit the static log-normal distribution in the middle scale region. In the derivation, a Non-Gibrat's law under the detailed balance is adopted together with following two approximations. Firstly, the probability density function of profits growth rate is described as a tent-shaped exponential function. Secondly, the value of the origin of the growth rate distribution divided into bins is constant. The derivation is confirmed in the database consistently. This static procedure is applied to a quasi-static system. We dynamically describe a quasi-static log-normal distribution in the middle scale region. In the derivation, a Non-Gibrat's law under the detailed quasi-balance is adopted together with two approximations confirmed in the static system. The resultant distribution is power-law with varying Pareto index in the large scale region and the quasi-static log-normal distribution in the middle scale region. In the distribution, not only the change of Pareto index but also the change of the variance of the log-normal distribution depends on the parameter of the detailed quasi-balance. As a result, Pareto index and the variance of the log-normal distribution are related to each other.