A Unified Approach to Measuring Poverty and Inequality
Example 2.11: Consider the six-person income vector x = ($80, $100, $800, $1,000, $50,000, $70,000), which is divided into two subgroups x' = ($80, $100, $800) and x" = ($1,000, $50,000, $70,000). Suppose the poverty line, z = $1,100, is the same across both subgroups. Note that N' = 3, N" = 3, and N = 6; thus, N'/N = N"/N = 3/6 = 0.5 is the population share of each group. The headcount ratio of x is PH(x;z) = 4/6 = 2/3; the headcount ratio of x' is PH(x';z) = 3/3 = 1; and the headcount ratio of x" is PH(x";z) = 1/3. Thus, the overall headcount ratio may be obtained from the subgroup headcount ratios. The population-weighted average headcount ratio of the subgroups is 0.5P(x';z) + 0.5P(x";z) = 0.5 × 1 + 0.5 × 1/3 = 2/3.
PG (x; z) = WA (g * ) =
1 N z − x *n . ∑ N n =1 z
(2.37)
The poverty gap measure may also be understood and interpreted by using two income standards. The higher income standard is the poverty line z itself, obtained by taking an arithmetic mean of the nonpoverty censored distribution x¯*z . The lower income standard is obtained by applying the arithmetic mean to the censored income distribution x*. Thus, the poverty gap measure can be expressed as WA (xz* ) − WA (x * ) z − WA (x * ) 1 N z − x *n PG (x; z) = = = ∑ . WA (xz* ) z N n =1 z
(2.38)
There is a third way to interpret the poverty gap measure, which is as a product of the headcount ratio and the average normalized income shortfall among the poor. The average normalized income shortfall among the poor 1 q is PIG (x; z) = ∑ n =1(z − x n )/z. The poverty gap measure can be expressed as q PG (x; z) =
N−q q 1 q z − xn ×0+ × ∑ = PH × PIG (x; z). N N q n =1 z
(2.39)
The poverty gap measure lies between zero and one. Zero is obtained when there are no poor in the society. A value of one is obtained when
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