A Unified Approach to Measuring Poverty and Inequality
introduced earlier were the Watts index, the SST index, the FGT family of measures for α > 1, and the CHUC family of indices. Each of these distribution-sensitive poverty measures is built on a specific income or gap standard that is closely linked to an inequality measure. For example, the Watts index is closely linked with Theil’s second measure of inequality, the SST index is closely linked with the Gini coefficient, the FGT family of indices for α > 1 is linked with the generalized entropy measures, and the CHUC family of indices is linked with Atkinson’s family of measures. For the Watts index, SST index, and CHUC family of indices, the inequality measure is applied to the censored distribution x*, with greater censored inequality being reflected in a higher level of poverty for a given poverty gap level. The FGT indices for α > 1, however, use generalized entropy measures applied to the gap distribution g*, with greater gap inequality leading to a higher level of poverty for a given poverty gap level. Recall from our earlier discussion in the income standard section that certain income standards can be viewed as welfare functions, and this link provides yet another lens for interpreting poverty measures. The Sen mean used in the SST index and the general means for α ≤ 1 that are behind the CHUC indices can be interpreted as welfare functions. In each poverty measure, the welfare function is applied to the censored distribution to obtain the censored income standard, which is now seen to be a censored welfare function that takes into account poor incomes and only part of nonpoor incomes up to the poverty line. For these measures, poverty and censored welfare are inversely related—every increase in poverty can be seen as a decrease in censored welfare. Dominance and Unanimity A poverty measure assesses the level of poverty within a society by a single number for a given poverty line. Two obvious questions arise: (a) Does a single poverty measure evaluate two distributions in the same way for all poverty lines? and (b) Do all poverty measures evaluate two income distributions in the same way? More specifically, according to the first question, if one distribution has more poverty than another distribution for a particular poverty line, is there any certainty that the former distribution would have more poverty than the latter for any other poverty line? Consider the following example with two four-person income distributions x = ($800, $900, $5,000, $70,000) and x' = ($200, $1,200, $1,600,
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