Using the zeta function to explain 'downside' and 'upside' inequality aversion

This paper presents a single-parameter generalization of the Gini coefficient of inequality. The generalization yields a unique sequence of measures parametrized by the integer k which runs from minus infinity to plus infinity, and is based on the zeta function (defined on the set of integers). Using suitably normalized income weights, one can generate a family of welfare functions and associated inequality measures. For k belonging to {…,-3,-2,-1}, one has a family of decreasingly ‘upside inequality aversion' measures; when k is zero, one has the familiar ‘transfer-neutral' Gini coefficient; and for k belonging to {1,2,3,…}, one has a family of increasingly ‘downside inequality aversion' measures. As k tends to minus infinity, the underlying social welfare function mimics a utilitarian rule, and as k tends to plus infinity, the Rawlsian rule. When k is 1, the corresponding inequality measure turns out to be the Bonferroni coefficient.