收入分配的规模依赖及其影响
Size-Dependency of Income Distributions and Its Implications
摘要：
本文着重讨论了收入分配的规模依赖性，即收入分配曲线与一国人口的关系。利用推广的Lotka-Volterra模型对美国1996-2007年的经验收入数据进行拟合，发现一个重要的参数$λ$可以与当年美国的规模（人口）的$β$次方成正比。我们指出，收入分配的规模依赖性是一个非常重要的性质，但在以往的研究中很少涉及，它具有两个重要的含义：（1）异速生长模式，即不同年份人口与GDP之间的幂律关系，它可以从规模依赖的收入分配中得到数学推导，并得到经验数据的支持;（2）临界现象中概率密度函数的反常标度性的联系，因为收入分布的重标度形式对于许多相关随机变量之和的极限分布具有完全相同的数学表达式。
英文摘要：
This paper highlights the size-dependency of income distributions, i.e. the income distribution curves versus the population of a country systematically. By using the generalized Lotka-Volterra model to fit the empirical income data in the United States during 1996-2007, we found an important parameter $λ$ can scale with a $β$ power of the size (population) of U.S. in that year. We pointed out that the size-dependency of the income distributions, which is a very important property but seldom addressed by 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, which 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 the exactly same mathematical expression for the limit distribution of the sum of many correlated random variables asymptotically.
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