Non-Gibrat's law in the middle scale region

TL;DR

This paper uses numerical simulations to explore the TST model's adherence to detailed balance under Gibrat's law, extends it to Non-Gibrat's law, and confirms the resulting distributions match theoretical predictions.

Contribution

It extends the TST model to include Non-Gibrat's law and confirms the resulting distribution transitions from power-law to log-normal in different scale regions.

Findings

TST model satisfies detailed balance under Gibrat's law.

Reflection law confirms Pareto index in the model.

Distribution transitions from power-law to log-normal.

Abstract

By using numerical simulation, we confirm that Takayasu--Sato--Takayasu (TST) model which leads Pareto's law satisfies the detailed balance under Gibrat's 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 $\mu$ in TST model. Moreover, we extend the model modifying the stochastic coefficient under a Non-Gibrat's law. In this model, the detailed balance is also numerically observed. The resultant pdf is power-law in the large scale Gibrat's law region, and is the log-normal distribution in the middle scale Non-Gibrat's one. These are accurately confirmed in the numerical simulation.