A Unified Approach to Measuring Poverty and Inequality
vice versa). Further, the poverty gap’s ordering implies (but is not implied by) the FGT index’s ordering. Because the poverty deficit curve is found by taking the area under the poverty incidence curve, a higher poverty incidence curve leads to a higher poverty deficit curve. The same is true for the poverty deficit and poverty severity curves. The poverty orderings of the Watts and CHUC indices can also be easily constructed and lead to another nested set starting with second-order dominance for the poverty gap measure. The poverty ordering for the Watts index, for example, is simply generalized Lorenz (or second-order stochastic) dominance applied to the distributions of log incomes. Each CHUC poverty ordering likewise applies generalized Lorenz dominance to distributions of transformed incomes (see Foster and Jin 1998). Placing an upper limit z* on the range of poverty lines is equivalent to comparing poverty curves (or the poverty incidence, deficit, or severity curves) over this limited range or to using censored distributions associated with z*. For example, the limited range poverty ordering for the poverty gap is equivalent to comparing the generalized Lorenz curves of the censored distributions or to comparing censored welfare levels across all utilitarian welfare functions with identical and increasing utility functions that have diminishing marginal utility. In the above example, we varied the poverty line while holding the poverty measure fixed. We can also vary the poverty measure for a given poverty line to examine robustness to the choice of measure. For example, using a five-dimensional vector, one can depict the poverty levels of the FGT measures for a = 0, 1, and 2; the Watts index; and the SST index. Vector dominance would then be interpreted as a variable measure poverty ordering that ranks distributions when all five measures unanimously agree. An analogous approach using poverty curves can be employed when using poverty measures indexed by a parameter. Consider a poverty curve that depicts the CHUC indices (z − ma (x*))/z for a ≤ 1 and the FGT indices ma (g*))/z for a ≥ 1. We are using the income standard version of each measure (rather than the decomposable version) because of its nice interpretation as a normalized average gap. The poverty measure at a = 1 is the usual poverty gap measure. As a rises, the FGT values progressively rise because the measures with higher a use a general mean that focuses on the higher gaps in the gap vector g*. The extent to which poverty rises as a > 1 rises depends on the generalized entropy inequality in g* for a. To the left, the CHUC values
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