A Unified Approach to Measuring Poverty and Inequality
considers the entire income distribution; thus, it is also a crude way of understanding inequality. For income distribution x, let the upper partial mean for percentile p be denoted by WUPM(x; p) and the lower partial mean for percentile p' be denoted by WLPM(x; p'). A partial mean ratio is also commonly reported as a ratio of both partial means ranging from one to ∞. However, as with the quantile ratio, we formulate the partial mean ratio in such a way that it ranges from zero to one. The p/p' partial mean ratio is represented by the following formula: IPMR (x; p / p′ ) =
WUPM (x; p) − WLPM (x; p′ ) W (x; p′ ) = 1 − LPM . WUPM (x; p) WUPM (x; p)
(2.17)
The higher the partial mean ratio, the higher the level of inequality across two percentiles of a society’s population. A partial mean ratio is zero when both upper and lower partial mean incomes are equal. A quantile ratio reaches its maximum value of one when the lower partial mean income WLPM(x; p') is zero and the upper partial mean income is positive. Note that if all people in the society have equal incomes, then any partial mean ratio is zero. However, a partial mean ratio of zero does not necessarily imply that incomes are equally distributed across all people in the society. The most well-known partial mean ratio was devised by Simon Kuznets and is known as the Kuznets ratio. It is based on two income standards: the mean of the poorest 20 percent of the population and the mean of the richest 40 percent of the population. Using our formulation, the Kuznets ratio equivalent inequality measure of distribution x is denoted by IPMR(x; 20/40). How should the number IPMR(x; 20/40) = 0.8 be interpreted? Again, there are several ways to interpret this measure: • The difference in mean income between the richest 20 percent of the population and the poorest 40 percent of the population is 80 percent of the mean income of the richest 20 percent of the population. • The mean income of the poorest 40 percent of the population is (1 − 0.8) = 0.2 or 20 percent or one-fifth of the mean income of the richest 20 percent of the population. • The mean income of the richest 20 percent of the population is 1/(1 − 0.8) = 5 times larger than the mean income of the poorest 40 percent of the population.
90