Chapter 1: Introduction
income of the two, and b denote the larger income, it is natural to measure inequality by the relative distance between a and b, such as I = (b − a)/b, or some other increasing function of the ratio b/a. Indeed, scale invariance and the weak transfer principle essentially require this form for the measure. Suppose that instead of evaluating the inequality between two people, we want to measure the inequality between two equal-sized groups. A natural way of proceeding is to represent each group’s income distribution using an income standard. This yields a pair of representative incomes—one for each group—that can then be compared. Where a denotes the smaller of these two incomes and b the larger, it is natural to measure inequality between the two groups as I = (b − a)/b, or some other increasing function of the ratio b/a. For example, if the distributions are the earnings of men and women and the income standard is the mean, then b/a would be the ratio of the average income for men to the average income for women—a common indicator of inequality between the two groups. As will be discussed below, this “between-group” approach is useful in decompositions of inequality by population subgroup and also in the measurement of inequality of opportunities. The general idea that inequality depends on two income standards is also relevant when evaluating the overall inequality in a population’s distribution of income. But instead of applying the same income standard to two distributions, we now apply two income standards to the same distribution. One of the income standards (the upper standard) places greater weight on higher incomes, and the second (the lower standard) emphasizes lower incomes; so for any given income distribution, the lower-income standard’s value is never larger than the upper-income standard’s value. This is true, for example, when the lower standard is the geometric mean and the upper is the arithmetic mean or, alternatively, when the lower is the 10th percentile income and the upper is the 90th percentile income. Inequality is then seen as the extent to which the two income standards are spread apart: where a denotes the lower-income standard and b the upperincome standard, overall inequality is I = (b − a)/b, or some other increasing function of the ratio b/a. Common Examples Virtually all inequality measures in common use are based on twin income standards. This is easily seen in the case of the 90/10 ratio, and generalizes to any quantile ratio b/a, where a corresponds to the income at a percentile p of
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