A Unified Approach to Measuring Poverty and Inequality
Recall that the general mean of a distribution converges toward the maximum or largest element in a vector or distribution. The largest element in the gap vector g* belongs to the poorest person in the society. We have already discussed the properties that the headcount ratio, the poverty gap measure, and the squared gap measure satisfy. Thus, we know what properties the FGT family of indices satisfies when a = 0, 1, and 2. The additional property that the measures in this family satisfy is transfer sensitivity when a > 2, which implies that if a similar amount of transfer takes place between two poorer poor people and two richer poor people, then this measure is able to distinguish between these two situations. An aspect that is not so intuitive in this family of measures is interpretation of the inequality aversion parameter. A larger value of a implies greater aversion to inequality among the poor. However, when there is no inequality in the society, should the poverty measure alter because of a change in Îą? For example, suppose that in a society of 100 people, everyone is poor and all people have an equal income of $500. If the poverty line is z = $1,000, then the normalized income gap of each person is one-half in this society. Given that there is no inequality in the society, it should not matter how averse the society is to inequality because there is no inequality. However, the FGT family of measures may not remain the same for all Îą. For the simple example considered above, PFGT(x;z,1) = PG(x;z) = 1/2 and PFGT(x;z,2) = PSG(x;z) = 1/4. However, this problem can be easily solved by calculating a monotonic transformation of the original FGT family of measures as P'FGT(x;z,a) = [PFGT(x;z,a)]1/a = WGM(g*; a) for a > 0.
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Note that this formula is not valid for the headcount ratio when a = 0. For the example above, P'FGT(x;z,a) = 1/2 for all a > 0 because the general mean satisfies the normalization property of income standards. Mean Gap Measure The mean gap measure of poverty can be obtained by taking the Euclidean mean (WE) of the normalized income shortfalls. This is a monotonic transformation of the squared gap measure. More specifically, the mean gap measure is the square root of the squared gap measure. The mean gap measure can be expressed as
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