Haughton and Khandker
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be nowhere lower for year 1 at all points up to the maximum poverty line, and at least somewhere higher. What happens above zmax is not relevant, for the study of poverty at least. In the example in figure 5.3, there is second-order dominance in the relevant range, and we may consider that poverty has indeed fallen. Intuitively, using the poverty gap index as the measure of poverty is equivalent to saying that the sum of the poverty gaps (that is, the poverty deficit) is smaller in year 2 than in year 1, no matter what poverty line is used, provided it is below zmax. Example: To illustrate the two dominance tests, consider an initial state in which four people have consumption in amounts (100, 110, 140, 150) in year 1 and these change to (110, 112, 128, 150) in year 2. Has poverty changed between year 1 and year 2? To help answer this question, consider the numbers in table 5.6. If the poverty line is 100, a quarter of the population is poor in year 1 and none are poor in year 2. It would appear that poverty has fallen. But if the poverty line is set at 130, the poverty rate was 0.5 in year 1 and actually rose to 0.75 in year 2. In other words, whether poverty (as measured by the headcount rate) has risen or fallen turns out to be sensitive to the choice of poverty line. Formally, the poverty incidence curves cross, so we do not have first-order stochastic dominance. Now consider the poverty deficit curve. If the poverty line is 120, the value of the poverty deficit curve is 1.25, obtained by summing the values of the poverty incidence curves (that is, 0.25 + 0.50 + 0.50 in this case). The poverty deficit was unambiguously higher (or at least not lower) in year 1 than in year 2 (see table 5.6, final three columns), no matter what poverty line was chosen. In this case there is secondorder stochastic dominance, and we have a moderately robust finding that poverty has fallen. This makes some intuitive sense: between year 1 and year 2, average Table 5.6 Comparison of Poverty Incidence and Poverty Deficit Curves Using Different Poverty Lines Poverty incidence curve, F(z)a Poverty line (z) 100 110 120 130 140 150
Year 1 0.25 0.50 0.50 0.50 0.75 1.00
> = = < = =
Poverty deficit curve, D(z )b
Year 2
Year 1
0 0.50 0.50 0.75 0.75 1.00
0.25 0.75 1.25 1.75 2.50 3.50
Year 2 > > > = = =
0 0.50 1.00 1.75 2.50 3.50
Source: Example designed by the authors. Note: Assumes a society of four people with consumption of (100, 110, 140, 150) in year 1 and (110, 112, 128, 150) in year 2.
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a. Poverty incidence curves cross. b. Poverty deficit curves do not cross.