A Unified Approach to Measuring Poverty and Inequality
urban areas, states, provinces, and other geographic regions; across ethnic and religious groups; across genders; or across age groups. One may want to evaluate the source of inequality, such as whether overall income inequality is due to unequal income distribution within sex or unequal income distribution across sex. The eighth property is additive decomposability, which requires overall inequality to be expressed as a sum of within-group inequality and betweengroup inequality. Within-group inequality is a weighted sum of subgroup inequalities. Between-group inequality is the inequality level obtained when every person within each subgroup receives the subgroup’s mean income. Kanbur (2006) discussed the policy significance of this type of inequality decomposition. It is often found that the contribution of the between-group term is much lower than the within-group term, and, thus, policy priority is directed toward ameliorating within-group rather than between-group inequality. These types of policy conclusions should be carefully drawn, because the lower between-group term may receive much larger social weight than its within-group counterpart. Also, the between-group term’s share of overall inequality may increase as the number of groups increases. How to incorporate these issues into inequality measurement requires further research, and solving these issues is beyond the scope of this book. However, if the policy interest is in understanding how the between-group inequality as a share of total inequality has changed over time for a fixed number of groups, then the decomposability property is very useful. To formally outline the additive decomposability property, we will use two groups to simplify the interpretation, but the definition can be extended to any number of groups. Suppose the income vector x with population size N is divided into two subgroup vectors: x' with population size N' and x" with population size N" such that N' + N" = N. Let us denote the means of these three vectors by x¯, x¯', and x¯". The additive decomposability property can be stated as follows:
Additive Decomposability: If income distribution x is divided into two subgroup distributions x' and x", then I(x) = W'I(x') + W"I(x") + I(x¯',x¯"), where W' and W" are weights. The between-group contribution is I(x¯', x¯")/I(x) and the within-group contribution is [W' I(x') + W" I(x")]/I(x), as seen in example 2.5.
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