
Emergence of power laws with different powerlaw exponents from reversal quasisymmetry and Gibrat’s law
Abstract
To explore the emergence of power laws in social and economic phenomena, the authors discuss the mechanism whereby reversal quasisymmetry and Gibrat’s law lead to power laws with different powerlaw exponents. Reversal quasisymmetry is invariance under the exchange of variables in the joint PDF (probability density function). Gibrat’s law means that the conditional PDF of the exchange rate of variables does not depend on the initial value. By employing empirical worldwide data for firm size, from categories such as plant assets K, the number of employees L, and sales Y in the same year, reversal quasisymmetry, Gibrat’s laws, and powerlaw distributions were observed. We note that relations between powerlaw exponents and the parameter of reversal quasisymmetry in the same year were first confirmed. Reversal quasisymmetry not only of two variables but also of three variables was considered. The authors claim the following. There is a plane in 3dimensional space (log K, log L, log Y ) with respect to which the joint PDF P_{J} (K, L, Y ) is invariant under the exchange of variables. The plane accurately fits empirical data (K, L, Y ) that follow powerlaw distributions. This plane is known as the CobbDouglas production function, Y = AK^{α}L^{β} which is frequently hypothesized in economics.
Introduction
In various phase transitions, it has been universally observed that physical quantities near critical points obey power laws. For instance, in magnetic substances, the specific heat, magnetic dipole density, and magnetic susceptibility follow power laws of heat or magnetic flux. We also know that the clustersize distribution of the spin follows power laws. Using renormalization group methods realizes these conformations to power law as critical phenomena of phase transitions [1].