A Unified Approach to Measuring Poverty and Inequality
IGini (x) =
N N 1 ∑ ∑ x n − x n′ . 2N 2 × WA (x) n =1 n’=1
(2.22)
Note that equation (2.22) may be broken into two components: WA(x) N 2 (the mean of the distribution) and (∑N n=1∑ n'=1|xn – xn'|)/2N (the average difference between pairs of incomes). The second component is divided by its number of elements, 2N2. There are 2N2 elements because each element in x is compared with another element in x including itself twice. This original Gini coefficient formula can be simplified further. The second component of the Gini coefficient can be written as 1 N N 1 N N x x W x − = ( ) − ∑ ∑ n n′ A ∑ ∑ min {x n , x n′}= WA (x) − Ws (x), (2.23) N 2 n = 1 n ′= 1 2 N 2 n =1 n ′ =1 where WS(x) is the Sen mean of distribution x. Therefore, the Gini coefficient may be simply formulated by using the arithmetic mean and the Sen mean. Like any measure in Atkinson’s class, the Gini coefficient can be expressed as IGini (x) =
WA (x) − WS (x) W (x) = 1− S . WA (x) WA (x)
(2.24)
Thus, the Gini coefficient is the difference between the arithmetic mean and the Sen mean divided by the arithmetic mean. The coefficient lies between zero and one, and inequality increases as the index moves from zero to one. The minimum inequality level, zero, is obtained when income is equally distributed across everyone in the society. Like Atkinson’s measures, if IGini(x) = 0, then, by implication, income in the society is equally distributed. Again, this is because any inequality measure in this family is constructed by considering the entire distribution (see example 2.7). What is the interpretation of IGini(x) = 0.292? First, IGini(x) is based on two income standards: the arithmetic mean of x and the Sen mean of x. The arithmetic mean represents the level of welfare obtained when the overall income is distributed equally across all people in the society. This is an ideal situation when there is no inequality in the society. The Sen mean, in contrast, is an ede income, which, if received by everyone in the society, would yield the same welfare level as in x. So IGini(x) = 0.292 implies that the loss of welfare because of inequality in distribution x is 29.2 percent of
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