Chapter 2: Income Standards, Inequality, and Poverty
Example 2.10: How is the headcount ratio calculated by different methods? Consider the four-person income vector x = ($800, $1,000, $50,000, $70,000). If the poverty line is set at z = $1,100, then two of the four people are poor. Thus, the headcount ratio is PH(x;z) = 2/4 = 0.5 or 50 percent. How can the headcount ratio be calculated using the concept of doubly censored distribution? The doubly censored vector of x is x*z* = (0, 0, $1,100, $1,100) and the nonpoverty censored distribution is x¯*z = ($1,100, $1,100, $1,100, $1,100). Then WA(x¯*z) = 4 × $1,100/4 = $1,100 and WA(x*z*) = 2 × $1,100/4 = $550. Hence, PH(x;z) = ($1,100 − $550)/$1,100 = 0.5. The headcount ratio is the most well-known and most widely used poverty measure because its interpretation is highly intuitive and simple. However, the effectiveness of the headcount ratio depends on which properties the headcount ratio satisfies. It satisfies all invariance properties: symmetry, normalization, population invariance, scale invariance, and focus. However, it does not satisfy any dominance property except subgroup consistency. The headcount ratio is not sensitive to changes in the income level of the poor as long as incomes do not cross the poverty line. This is why the headcount ratio does not satisfy the other dominance properties and monotonicity, which require poverty measures to change as the incomes of the poor change. The headcount ratio satisfies subgroup consistency because the headcount ratio is additively decomposable, as shown by example 2.11. Poverty Gap Measure The second basic poverty measure is the poverty gap measure. Like headcount ratio, it is also widely used. The poverty gap measure (PG) is the average normalized shortfall with respect to the poverty line across the poor. In society X, the normalized income shortfall of a person, say, n, is calculated as (z − x*n)/z, which means that the normalized income shortfall of a nonpoor person is zero. The average normalized income shortfall is the average of all normalized income shortfalls within a society. We denote the normalized gap vector of x by g* = ((z − x*1)/z,…,(z − x*N)/z). Then the poverty gap measure is
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