A Characterization of the Pareto Distribution

Abstract

A positive random variable $X$ whose mean exists, has a Pareto distribution if, and only if, $E(X \mid X > x) = h + gx$ for $g > 1$. This characterization was motivated by the fact that the Pareto distribution has been widely used to model income. Now, suppose that individuals under-report their income for income tax purposes. If one assumes that for a given income, the average under-reporting is a constant fraction of the amount by which the income exceeds the tax exempt level, then the average under-reporting error for a given reported income is a linear function of the reported income if, and only if, incomes follow a Pareto distribution.