A Unified Approach to Measuring Poverty and Inequality
decomposability property makes the first Theil measure useful in understanding within-group and between-group inequalities. The second Theil measure can be decomposed as IT2(x) = w'IT2(x') + w"IT2(x") + IT2(x¯', x¯"),
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where the weights are w' = (N'/N) and w" = (N"/N). Inequality and Welfare The Gini coefficient and the inequality measures in Atkinson’s family share a social welfare interpretation. As we have already discussed, they can be expressed as I = (x¯ − a)/x¯, where x¯ is the mean income of the distribution x and a is an income standard that can be viewed as a welfare function (satisfying the weak transfer principle). Note that the distribution in which everyone has the mean income has the highest level of welfare among all distributions with the same total income, and the distribution’s measured welfare level is just the mean itself. This finding results from the normalization property of income standards. Thus, the mean WA(x) = x¯ is the maximum value that the welfare function can take over all income distributions of the same total income. When incomes are all equal, a = WA(x) and inequality is zero. When the actual welfare level a falls below the maximum welfare level WA(x), the percentage welfare loss I = (WA(x) − a)/WA(x) is used as a measure of inequality. This is the welfare interpretation of both the Gini coefficient and the Atkinson’s class of measures. The simple structure of these measures allows us to express the welfare function in terms of the mean income and the inequality measure. A quick rearrangement leads to a = WA(x)(1 – I), which can be reinterpreted as saying that the welfare function a can be viewed as an inequality-adjusted mean. If there is no inequality in the distribution, then (1 – I) = 1 and a = WA(x). If the inequality level is I > 0, then the welfare level is obtained by discounting the mean income by (1 – I) < 0. For example, if we take I to be the Gini coefficient, IGini(x), then the Sen mean (or Sen welfare function) can be obtained by multiplying the mean by [1 – IGini(x)], that is, WS(x) = WA(x)[1 – IGini(x)]. Similarly, if we take I to be the Atkinson’s measure with parameter a = 0, IA(x; 0), then the welfare function is the geometric mean, and the geometric mean can be obtained by multiplying the mean by [1 – IA(x; 0)], that is, WG(x) = WA(x)[1 – IA(x; 0)].
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