A Unified Approach to Measuring Poverty and Inequality
Subgroup Consistency In certain empirical applications, there is a natural concern for certain identifiable subgroups of the population as well as for the overall population. For example, one might be interested in the achievements of the various states or subregions of a country to understand the spatial dimensions of growth. When population subgroups are tracked alongside the overall population value, there is a risk that the income standard could indicate contradictory or confusing trends. A natural consistency property for an income standard might be that if subgroup population sizes are fixed but incomes vary, then when the income standard rises in one subgroup and does not fall in the rest, the overall population income standard must rise. This property is known as subgroup consistency; and using a measure that satisfies it avoids inconsistencies arising from this sort of multilevel analyses. In fact, several income standards discussed above do not survive this test and, hence, may need to be avoided when undertaking regional evaluations or other forms of subgroup analyses. The mean of the lowest 20 percent is subject to this critique because a given policy could succeed in raising the mean of the lowest 20 percent in every region of a given country; yet the mean of the lowest 20 percent in the overall population could fall. The same is true of the Sen mean or the median. In contrast, every general mean satisfies the consistency requirement. In fact, it can be shown that the general means are the only income standards that are subgroup consistent while satisfying some additional basic properties. Moreover, each of the general means has a simple formula that links regional levels of the income standard to the overall level. If one were to go further and specify an additive aggregation formula across subgroup standards—a requirement that might be called additive decomposability—the only general mean that would survive is the mean itself. The overall mean is just the population-weighted sum of subgroup means.
Dominance and Unanimity One motivation for examining several income standards at the same time is robustness: Do conclusions about the direction of change in the distribution size using one income standard (say, the mean) hold for others (say, the nearby generalized means)? A second reason might be focus or an identified
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