Level dependence of the adjustment for unbalance and inequality for the Human Development Index

As remarked by Ravallion (J Dev Econ 99:201–209, 2012), the recent switch from arithmetic to geometric mean in the aggregation of the United Nations’ Human Development Index has caused a more severe inequality penalization of the index for less developed countries, with outlying consequences. We clarify and explain this fact and propose an aggregation function, the Trichotomy Mean, that depends on two parameters: one regulates the overall penalization of disequilibria (among or within dimensions) in analogy with Atkinson’s inequality aversion parameter for power means; the other modulates the Level Dependence of the Adjustment, a novel concept describing the behavior—decreasing, increasing, or constant—of penalization of given disequilibria for increasing index level. Unlike the geometric mean (which, incidentally, has decreasing LDA type), the TM remains valid for zero or negative—and does not distort for small positive—values of the input variables, thus permitting less restrictive raw-variable normalizations and to overcome the need for exogenous lower bounds. We compare the three versions of TM with the geometric mean in an empirical analysis on the HDI 2014 data. We finally illustrate the contributions of the TM to the development literature debate.