A Unified Approach to Measuring Poverty and Inequality
decision requiring political discourse to obtain a solution. Regardless of the method for choosing r, the resulting tools allow a useful decomposition of poverty into an absolute poverty group (those below the absolute poverty line) and a hybrid or relative group (those above the absolute but below the hybrid poverty line). This is analogous to the above decomposition into the ultra-poor and the poor above the ultra-poverty line and likewise could be helpful in guiding differential policy responses for the two groups. Atkinson and Bourguignon (2000) combine absolute and relative poverty lines in a different way. When mean income is low enough that za > zr, they suggest that the absolute poverty line would be appropriate and hence the income elasticity of the poverty line is zero in this region. However, when incomes are high enough for zr > za, the relative poverty line should apply, yielding a unitary income elasticity of the poverty line. The hybrid poverty line of a country is then the maximum of the absolute poverty line and the relative poverty line, or z = max{za,zr}. Atkinson and Bourguignon use data on poverty lines and mean incomes to calibrate the absolute and relative lines. Ravallion and Chen (2011) argue that an income elasticity of one is implausible and posit a weak relativity axiom that requires poverty to fall if all incomes rise by the same proportion. They then provide the alternative hybrid poverty line formula z = max{za,f+zr}, where f > 0 is interpreted as the fixed cost of social inclusion. They set the three parameters of their proposed formula with the aid of data. Although the line of Atkinson and Bourguignon (2000) does not satisfy the weak relativity axiom for the standard scale invariant poverty measures, the lines of Foster (1998) and Ravallion and Chen (2011) do.
Categorical and Ordinal Variables The previous analysis applies to any cardinal welfare indicator, where cardinality requires that values convey more information than just more or less. Nonmonetary examples of cardinal variables might include calories, years of schooling, or hectares of land. Many other variables are more appropriately interpreted as ordinal, because their values are only indicators of order. Others might be categorical and have no values or underlying ordering at all. Examples of ordinal variables include self-reported health and subjective well-being. Categorical variables include sanitation facilities or the
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