A Unified Approach to Measuring Poverty and Inequality
that is, a' = (1 + g– a)a, and income standard b changes to b' over time with growth rate g– b, that is, b' = (1 + g– b)b. The inequality measure then changes from I = 1 – a/b to I' = 1 – a'/b'. To have a fall in inequality, we require I' < I or 1 – a'/b' < 1 – a/b, which occurs when g– a > g– b. Therefore, for a reduction in inequality, the smaller income standard a needs to grow faster than the larger income standard b. Consider the example of the Gini coefficient, which is constructed from two income standards. The larger income standard is the arithmetic mean WA, and the smaller income standard is the Sen mean WS. Let us denote the growth rate of the mean income by g– and the growth rate of the Sen mean by g–S. The Gini coefficient will register a fall in inequality when the growth rate of the Sen mean is larger than the growth rate of the arithmetic mean, that is, g–S > g–. Similarly, inequality over time, in terms of the Gini coefficient, increases when g–S < g–. What about the Atkinson’s measures and the generalized entropy measures? Measures in these classes, including Theil’s second measure, are based on the arithmetic mean and on any income standard from the class of general means. For a < 1, the arithmetic mean is the larger income standard, and the other general mean–based income standard is the smaller income standard. In this case, if the growth rate of the smaller income standard of order a is denoted by g–GM(a), then inequality decreases when g–GM(a) > g–. If inequality is evaluated by Theil’s second index, then inequality falls when the growth of geometric mean g–GM(0) is larger than that of the arithmetic mean, that is, g–GM(0) > g–. For a > 1 in the generalized entropy measure, the arithmetic mean is the smaller income standard, and the other general mean–based income standard is the larger one. Inequality falls, according to these indices, when the growth rate of the arithmetic mean g– is higher. Is there any way to tell if all inequality measures in the Atkinson family and the generalized entropy family have fallen? Yes, it is possible to do so just by looking at the general mean growth curve, as described in figure 2.12. A generalized mean growth curve is the loci of the growth rates of all income standards in the class of general means. Comparing distributions x and y for the general mean growth curve gGM(x,y) in figure 2.12 shows that all inequality measures in Atkinson’s class and the generalized entropy class agree that the inequality has fallen because the growth rates of the lower income standards are higher than g¯. The growth rates of the larger income standards are lower than g¯. However, for the general mean growth curve gGM(x',y') in the same figure, all inequality measures in Atkinson’s class and
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