Chapter 2: Income Standards, Inequality, and Poverty
• The number can be interpreted as the lowest income of the richest 10 percent of the population, being 10 times (1/(1 – 0.9)) larger than the highest income of the poorest 10 percent of the population. Similarly, IQR(x; 90/50) = 0.75 implies that the lowest income of the richest 10 percent of the population is 1/(1 − 0.75) = 4 times larger than the highest income of the poorest 50 percent of the population. Quantile ratios may be classified into three categories: upper end quantile ratio, lower end quantile ratio, and mixed quantile ratio. The first two categories capture inequality within any one side of the median, and the third category captures inequality in one side of the median versus that of the other side of the median. For example, IQR(x; 90/50) is an upper end quantile ratio, and IQR(x; 50/10) is a lower end quantile ratio, whereas IQR(x; 90/10) is a mixed quantile ratio. What properties does a quantile ratio satisfy? A quantile ratio, as defined earlier, satisfies symmetry, normalization, population invariance, and scale invariance. Thus, a quantile ratio satisfies all four invariance properties. What about the dominance properties? It turns out that a quantile ratio satisfies none of the dominance properties. The following example shows that a quantile ratio does not satisfy the weak transfer principle. Suppose the highest income of the poorest 10 percent of the population is $100 and the lowest income of the richest 10 percent of the population is $2,000. Then IQR(x; 90/10) = ($2,000 − $100)/$2,000 = 0.95. Now, suppose that a regressive transfer takes place between the poorest person in the society and the richest person among the poorest 10 percent of the population such that the highest income in that group increases to $120. Then the post-transfer quantile ratio is IQR(x; 90/10) = ($2,000 − $120)/$2,000 = 0.94. Therefore, the quantile ratio shows a decrease in inequality even when a regressive transfer has taken place. If a quantile ratio does not satisfy the weak transfer principle, then it cannot satisfy its stronger version—the transfer principle, or transfer sensitivity. The quantile ratios are not additively decomposable and also do not satisfy subgroup consistency. Partial Mean Ratio A partial mean ratio is an inequality measure comparing an upper partial mean and a lower partial mean. Like quantile ratios, no partial mean ratio
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