Emergence of power laws with different power-law exponents from reversal quasi-symmetry and Gibrat’s law
To explore the emergence of power laws in social and economic phenomena, the authors discuss the mechanism whereby reversal quasi-symmetry and Gibrat’s law lead to power laws with different powerlaw exponents. Reversal quasi-symmetry 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 quasi-symmetry, Gibrat’s laws, and power-law distributions were observed. We note that relations between power-law exponents and the parameter of reversal quasi-symmetry in the same year were first confirmed. Reversal quasi-symmetry not only of two variables but also of three variables was considered. The authors claim the following. There is a plane in 3-dimensional space (log K, log L, log Y ) with respect to which the joint PDF PJ (K, L, Y ) is invariant under the exchange of variables. The plane accurately fits empirical data (K, L, Y ) that follow power-law distributions. This plane is known as the Cobb-Douglas production function, Y = AKαLβ which is frequently hypothesized in economics.
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 cluster-size distribution of the spin follows power laws. Using renormalization group methods realizes these conformations to power law as critical phenomena of phase transitions .