### 6.4.4 Application: Poverty Measures

For a poverty index, we need a poverty line that may be an exogenously given constant ζ or may depend on the income distribution ζ(F). An important class of poverty indices can then be described in the following way:

$P\left(F\right):={\displaystyle \int p\left(y,\zeta \left(F\right)\right)}\mathrm{d}F\left(y\right)$

(6.78)

where p is a poverty evaluation function that is nonincreasing in y and takes the value zero for y ≥ ζ(F). Once again we need the IF, which is given by

$\mathrm{IF}\left(z;P,F\right)=p\left(z,\zeta \left(F\right)\right)-P\left(F\right)+{\displaystyle \int {p}_{\zeta}\left(y,\zeta \right)}\mathrm{d}F\left(y\right)\mathrm{IF}\left(z;\zeta ,F\right)$

(6.79)

where p_{ζ} is the differential of p with respect to its second argument (Cowell and Victoria-Feser, ...