A Unified Approach to Measuring Poverty and Inequality
population. The vertical axis denotes the corresponding value of a quantile function in terms of income. Of course, no one in the society can have any income above Qx (100). Half of the population has an income less than the median bM, so Qx (50) = bM. The shaded area underneath the quantile function is the mean x¯ of the distribution x. Having introduced these basic concepts, we discuss income standards in the next section.
Income Standards An income standard gauges the size of a distribution by summarizing the entire distribution in a single income level. Some income standards can be viewed as stylized measures of a society’s overall level of well-being. Others focus more narrowly on one part of the distribution or have no general welfare interpretation. We begin this section by introducing common properties that an income standard should satisfy. We denote any income standard by W and use subscripts to indicate specific measures or indices. Desirable Properties An income standard can satisfy several basic properties. We refer to the first two properties—symmetry and population invariance—as invariance properties because they describe changes in the distribution that leave the income standard unaltered. The second pair of properties—weak monotonicity and the weak transfer principle—are called dominance properties because they require the income standard to rise (or not fall) when the income distribution changes in a particular way. Finally, normalization and linear homogeneity are calibration properties that ensure the income standard is measured by income. The additional property of subgroup consistency is not a part of the basic properties, but it is desirable when evaluating income standards of subpopulations. Symmetry requires that switching two people’s incomes leaves the income standard evaluation unchanged. In other words, a person should not be given priority on the basis of his or her identity when calculating a society’s income standard. Thus, symmetry is also known as anonymity. In technical terms, symmetry requires the income standard of distribution x to be equal to the income standard of distribution x', if x' is obtained from x by a permutation of incomes.
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