Size Dependency of Income Distribution and Its Implications

We systematically study the size dependency of income distributions,i.e.income distribution versus the population of a country.Using the generalized Lotka--Uolterra model to fit the empirical income data for 1996-2007 in the U.S.A,we find an important parameter A that can scale with aβpower of the size(population)of the U.S.A.in that year.We point out that the size dependency of income distributions,which is a very important property but seldom addressed in previous studies,has two non-trivial implications:(1)the allometric growth pattern,i.e.the power-law relationship between population and GDP in different years,can be mathematically derived from the size-dependent income distributions and also supported by the empirical data;(2)the connection with the anomalous scaling for the probability density function in critical phenomena,since the re-scaled form of the income distributions has asymptotically exactly the same mathematical expression for the limit distribution of the sum of many correlated random variables.