A Unified Approach to Measuring Poverty and Inequality
the distribution and b is the income at a higher percentile q of the distribution. The quantile incomes are relatively insensitive income standards, and hence they yield inequality measures that are somewhat crude and that disagree with the weak transfer property that is traditionally regarded as a basic property of inequality measures. Nonetheless, they succeed at conveying tangible information about the distribution—namely, the extent to which two quantile incomes differ from one another—and can be informative, if crude, measures of inequality. The Kuznets ratio has as its twin income standards the mean of those from 40 percent downward and the mean of those from 80 percent upward, respectively. This can be generalized to any ratio of two standards of this form by varying the cutoffs. The resulting measure, which we call the partial mean ratio, is given by b/a, where a is the lower partial mean at p and b is the upper partial mean at q. The case where p = 10 percent and q = 90 percent is often called the decile ratio. Another related measure is the income share of the top 1 percent, which is a multiple of the partial mean ratio with p = 100 percent and q = 99 percent. Although each partial mean ratio satisfies four basic properties of an inequality measure, the component income standards are still rather crude and focus on only a limited range of incomes. Those falling outside the range are ignored entirely, while the income distribution within the range is also not considered. The resulting measure is thus insensitive to certain transfers. As before, though, the twin standards and their ratio convey tangible and easily understood information about the income distribution. The Gini coefficient is defined as the expected (absolute) difference between two randomly drawn incomes divided by twice the mean. Calculating the Gini coefficient is therefore straightforward: 1. Create an N × N matrix having a cell for every possible pair of incomes, and place the absolute value of their difference in the cell. 2. Add all the entries and divide by the number of entries (N2) to obtain the expected value of the absolute difference between two randomly drawn incomes. 3. Divide by two times the mean income of the distribution to obtain the Gini coefficient. It is a natural indicator of how “spread out” incomes are from one another. The Gini coefficient has as its twin income standards the mean and the Sen mean and can be written as I = (b − a)/b, where b is the mean and a is
16