Computes the Palma index (also known as S90/S40 ratio)
S90S40(x, weight)the value of index
income vector
vector of weights
Anna Saczewska-Piotrowska, Alicja Wolny-Dominiak
The Palma index is the ratio between the income share of the top \(10\%\) and the bottom \(40\%\). The weighted Palma index (with weights \(w_1,w_2,...,w_n\)) is given by: $$S90/S40 = \frac{\sum_{i=1}^n{x_iw_i}\boldsymbol{1}_{x_i>q_{w(0.9)}}}{\sum_{i=1}^n{x_iw_i}\boldsymbol{1}_{x_i \leq q_{w(0.4)}}},$$ where \(x_i\) is an income of individual \(i\), \(n\) is the number of individuals, \(q_{w(0.9)},q_{w(0.4)}\) are the 0.9 and 0.4 quantiles, respectively, \(\boldsymbol{1}_{(\cdot)}\) denotes the indicator function, which is equal to 1 when its argument is true and 0 otherwise.
Cobham A., Sumner A.(2013). Is it all about the tails? The Palma measure of income inequality. Working Paper No. 343, Center for Global Development