4.2 Bourguignon-Chakravarty (2003) multidimensional poverty class

A class of poverty measures is proposed in Bourguignon and Chakravarty (2003Bourguignon, François, and Satya R. Chakravarty. 2003. “The Measurement of Multidimensional Poverty.” The Journal of Economic Inequality 1 (1). Springer US. doi:10.1023/a:1023913831342.), using a cross-dimensional function that assigns values to each set of dimensionally normalized poverty gaps. It can be defined as: \[ BCh = \sum_{i \in n} \bigg[ \bigg( \sum_{j \in d} w_{j} x_{ij} \bigg)^{\frac{1}{\theta}} \bigg]^\alpha \text{, } \theta > 0 \text{, } \alpha > 0 \] where \(x_{ij}\) being the normalized poverty gap of dimension \(j\) for observation \(i\), \(w_j\) is the weight of dimension \(j\), \(\theta\) and \(\alpha\) are parameters of the function.

The parameter \(\theta\) is the elasticity of subsitution between the normalized gaps. In another words, \(\theta\) defines the order of the weighted generalized mean across achievement dimensions. For instance, when \(\theta = 1\), the cross-dimensional aggregation becomes the weighted average of all dimensions. As \(\theta\) increases, the importance of the individual’s most deprived dimension increases. As Vega, Urrutia, and Diez (2009Vega, Maria Casilda Lasso de la, Ana Urrutia, and Henar Diez. 2009. “The Bourguignon and Chakravarty multidimensional poverty family: A characterization.” Working Papers 109. ECINEQ, Society for the Study of Economic Inequality. https://ideas.repec.org/p/inq/inqwps/ecineq2009-109.html.) points out, it also weights the inequality among deprivations. In its turn, \(\alpha\) works as society’s poverty-aversion measure parameter. In another words, as \(\alpha\) increases, more weight is given to the most deprived individuals. Similar to \(\theta\), when \(\alpha = 1\), \(BCh\) is the average of the weighted deprivation scores.