Chapter 2: Income Standards, Inequality, and Poverty
remains unchanged when compared to the income standard, even after all incomes are scaled up or down by the same factor. This similarity supports the idea that the relative inequality level remains unchanged.7
Scale Invariance: If distribution x' is obtained from distribution x' such that x' = cx, where c > 0, then I(x') = I(x). The fourth property, normalization, requires that if incomes are the same across all people in a society, then no inequality exists within the society and the inequality measure should be zero. Normalization is a natural property. For example, if the income vector of a three-person society is ($20k, $20k, $20k), then the inequality measure should be zero. Even if everyone’s income increases 10-fold and the new income vector is ($200k, $200k, $200k), the inequality measure should still be zero.
Normalization: For the income distribution x = (b, b ,..., b), I(x) = 0. The fifth property is the transfer principle, which requires that a regressive transfer between two people in a society should increase inequality and a progressive transfer between two people should reduce inequality. Regressive and progressive transfers were defined earlier for income standards.
Transfer Principle: If distribution x' is obtained from distribution x by a regressive transfer, then I(x') > I(x). If distribution x" is obtained from distribution x by a progressive transfer, then I(x") < I(x). In inequality measurement, there is also a weaker version of the transfer principle, which requires that a regressive transfer between two people in a society not decrease inequality and that a progressive transfer between two people not increase inequality. Thus, the weaker principle allows the possibility that the level of inequality may remain unaltered because of progressive or regressive transfers.
Weak Transfer Principle: If distribution x' is obtained from distribution x by a regressive transfer, then I(x') â&#x2030;Ľ I(x). If distribution x" is obtained from distribution x by a progressive transfer, then I(x") â&#x2030;¤ I(x).
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