A Unified Approach to Measuring Poverty and Inequality
by the total number of people in the distribution. The mean can also be viewed as the average height (or, in mathematical terms, the integral) of the quantile function. It is the income level that all people would achieve if they were given an equal share of overall resources. Another size indicator, median income, is the income at the midway point of the quantile function, with half the incomes below and half above. Most empirical income distributions are skewed so that the mean (which includes the largest incomes in the averaging process) exceeds the median income (which is unaffected by the values of the largest incomes). Still another measure of size is given by the mean income of the lowest fifth of the population, which focuses exclusively on the lower incomes in a distribution. Each of these indicators is an example of an income standard, which reduces the overall income distribution to a single income level indicating some aspect of the distribution’s size.
What Is an Income Standard? To understand what a measure or index means, explicitly stating a set of properties that it should satisfy is helpful. In the case of an income standard, there are several requirements that go beyond the basic symmetry and population invariance discussed above: • Normalization states that if all incomes happen to be the same, then the income standard must be that commonly held level of income—a natural property indeed. • Linear homogeneity requires that if all incomes are scaled up or down by a common factor, then the income standard must rise or fall by that same factor. • Weak monotonicity requires the income standard to rise, or at least not fall, if any income rises and no other income changes. These basic requirements ensure that the income standard measures the size of the income distribution as a “representative” income level that responds “in the right way” when incomes change (for example, these requirements rule out envy effects). It is easy to see that the size indicators discussed in the previous section—mean, median, and mean of the lowest fifth—conform to these general requirements.
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