A Unified Approach to Measuring Poverty and Inequality
and is the normalized shortfall of the generalized mean of the observed censored income distribution x* from the generalized mean of the nonpoverty censored income distribution x¯*. Again, the generalized mean satisfies the normalization property of income standards; thus, the generalized mean of the nonpoverty censored income distribution is the poverty line itself. The CHUC index for a ≤ 1 can be expressed as PCHUC (x; z) =
WGM (xz*;a ) − WGM (x *;a ) z − WGM (x *;a ) = . WGM (xz*;a ) z
(2.52)
The CHUC index lies between zero and one. The minimum value of zero is obtained when there are no poor people in a society. However, the maximum value of the CHUC index cannot be larger than one. When everyone in a society is poor, having equal income, this measure is equal to the average normalized income shortfall. It satisfies all invariance and dominance properties. However, not all measures in this class are additively decomposable. For a = 1, the CHUC index is the poverty gap measure, and for a = 0, the CHUC index is a monotonic transformation of the Watts index. Advantages and Disadvantages of Each Measure We have shown that the two basic measures—the headcount ratio and the poverty gap measure—do not satisfy transfer-related properties and so are not sensitive to inequality across the poor. Besides not being sensitive to inequality, the headcount ratio does not satisfy monotonicity, which, if it is used as a target for public policy, may cause inefficiency in public spending. All of the subsequent advanced poverty measures, in contrast, are sensitive to inequality across the poor. The SST index and the mean gap measure are both equal to the poverty gap measure when everyone in a society is poor and no inequality exists among them. These two measures become larger than the poverty gap measure when the income gap remains the same, but inequality among the poor increases. Each advanced measure, however, has its own pros and cons. Let us begin with the SST measure. We know from our previous discussion that this measure is not subgroup consistent, which means that it may lead to inconsistent outcomes when group-level analysis is of interest. This measure is also not transfer sensitive, which means that if a similar amount of transfer
126