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–Volterra model to fit the empirical income data for 1996–2007 in the U.S.A., we find an important parameter λ that can scale with a β power 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.