A Stochastic Model for Pareto's Law and the Log-Normal Distribution under the Detailed Balance and Extended-Gibrat's Law

摘要

We verify that Takayasu-Sato-Takayasu (TST) model satisfies not only Pareto's law but also the detailed balance under Gibrat's law, by using numerical simulation. We employ a tent-shaped function as multiplicative noise. We also numerically confirm that the reflection law is equivalent to the equation which gives the Pareto index μ in TST model. We extend the model modifying the stochastic coefficient under a Non-Gibrat's law, and also numerically observe the detailed balance. The obtained pdf is power-law in the large scale region, and is the log-normal distribution in the middle scale one. We also study the dependence of Pareto index on the average of the additive noise.