Peichl et. al (2008) defined an affluence index. Weighted index (with weights \(w_1,w_2,...,w_n\)) is given by:
$$R^{CHA}_{\beta}(\boldsymbol{x},\boldsymbol{w},\rho_w) = \frac{\sum_{i=1}^n(1-(\frac{\rho_w}{x_i})^\beta)\boldsymbol{1}_{x_i > \rho_w}w_i}{\sum_{i=1}^n{w_i}}, \beta > 0,$$
where \(x_i\) is an income of individual \(i\), \(n\) is the number of individuals, \(\rho_w\) is the richness line,
\(\boldsymbol{1}_{(\cdot)}\) denotes the indicator function, which is equal to 1 when its argument is true and 0 otherwise.
Index satisfies transfer axiom \(T1\) (concave): a richness index should increase when a rank-preserving progressive transfer between two rich individuals takes place.