Models for over reported income

Abstract

Over reported income is commonly expressed as Z = X/Y, where X denotes the true income and Y a multiplicative error taking values in (0, 1). If Y has the power function distribution then it is well known that X is Pareto distributed if and only if Z is also. Often, the gamma distribution is preferred as a model for income. In this note, we derive the exact distribution of Z as well as the associated percentage points assuming X is gamma distributed.