A Generalized Gibrat's Law*

Abstract

Many economic and non‐economic variables such as income, wealth, firm size, or city size often distribute Pareto in the upper tail. It is well established that Gibrat's law can explain this phenomenon, but Gibrat's law often does not hold. This note characterizes a class of processes, one that includes Gibrat's law as a special case, that can explain Pareto distributions. Of particular importance is a parsimonious generalization of Gibrat's law that allows size to affect the variance of the growth process but not its mean. This note also shows that under plausible conditions Zipf's law is equivalent to Gibrat's law.

Zipf's law for cities in the regions and the country

“…7 We find that city growth rates are indeed independent of initial city size. The normalized mean growth rate of zero and the normalized variance of one fall inside the 99% pointwise confidence bands throughout the entire range of city sizes.…”

On the distribution of city sizes

“…Among the most prominent probabilistic models are the ones by Champernowne [5], Simon [21], and more recently, Gabaix [11], and Córdoba [6]. The fundamental insight obtained by these authors is that Gibrat's law, or proportional growth, can lead to Pareto distributions.…”

“…This paper studies necessary conditions to explain a Pareto distribution of city sizes within a class of standard urban models. For this purpose, we build on the statistical results of Gabaix [11] and Córdoba [6] who find that, under some mild restrictions, a Pareto distribution of city sizes, and Zipf's distribution in particular, can only result from Gibrat's law, a law that requires the growth process (of cities) to be independent of size. Our goal is to translate these statistical results into meaningful economic restrictions about preferences, technologies, and stochastic driving forces of standard economic models of cities.…”

New Evidence on Gibrat’s Law for Cities

“…These theoretical models include those of Gabaix (1999), Duranton (2006Duranton ( , 2007 and Córdoba (2008).…”