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 a βpower 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.
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