Kinetic and mean field description of Gibrat's law

TL;DR

This paper develops a kinetic model for firm growth based on Gibrat's law, revealing diverse mean field limits and showing that large-time behavior leads to a lognormal distribution with exponential variance growth.

Contribution

It introduces a novel kinetic framework for Gibrat's law and analyzes its mean field limits, connecting kinetic and diffusion models for firm size evolution.

Findings

Large-time behavior is described by a lognormal distribution.

The variance of the distribution grows exponentially over time.

An explicit formula for the Fourier transform of the lognormal distribution is provided.

Abstract

We introduce and analyze a linear kinetic model that describes the evolution of the probability density of the number of firms in a society, in which the microscopic rate of change obeys to the so-called law of proportional effect proposed by Gibrat. Despite its apparent simplicity, the possible mean field limits of the kinetic model are varied. In some cases, the asymptotic limit can be described by a first-order partial differential equation. In other cases, the mean field equation is a linear diffusion with a non constant diffusion coefficient that models also the geometric Brownian motion and can be studied analytically. In this case, it is shown that the large-time behavior of the solution is represented, for a large class of initial data, by a lognormal distribution with constant mean value and variance increasing exponentially in time at a precise rate. The relationship between the kinetic and the diffusion models allow to introduce an easy-to- implement expression for computing the Fourier transform of the lognormal distribution.