4.3 Bourguignon (1999) inequality class (svybmi)

Bourguignon (1999Bourguignon, François. 1999. “Comment to ’Multidimensioned Approaches to Welfare Analysis’ by Maasoumi, E.” In Handbook of Income Inequality Measurement, edited by Jacques Silber, 477–84. London: Kluwer Academic.) proposes a multidimensional inequality index that possesses interesting properties related to the correlation among the welfare dimensions measured. The estimator used in convey comes from the formula presented in Lugo (2007Lugo, María Ana. 2007. “Comparing Multidimensional Indices of Inequality: Methods and Application.” In Inequality and Poverty, 213–36. doi:10.1016/S1049-2585(06)14010-7.) and is defined as: \[ B_{I} = 1 - \frac{1}{ \widehat{N} } \frac{ \sum_{ i \in S } w_i \bigg[ \sum_{ j \in d } w_j x_{ij} \bigg]^{ \alpha / \beta } }{ \bigg[ \sum_{ j \in d } w_j \mu_{ij} \bigg]^{ \alpha / \beta } }, \] where \(\alpha \geqslant 0\) is an inequality-aversion parameter and \(\beta \leqslant 1\) is a parameter defining the degree of substitution among dimensions.

This measure is strong scale-invariant when \(\beta = 0\), although Bourguignon (1999Bourguignon, François. 1999. “Comment to ’Multidimensioned Approaches to Welfare Analysis’ by Maasoumi, E.” In Handbook of Income Inequality Measurement, edited by Jacques Silber, 477–84. London: Kluwer Academic.) demonstrates that strong scale-dependent measures might be interesting in the context of multidimensional inequality. Also, it can be shown that stronger correlation among dimensions leads to less inequality if \(\beta > \alpha\).

For additional usage examples of svybmi, type ?convey::svybmi in the R console.