Level-adjusted S-Gini index and its complementary index as a pair of sensitivity-adjustable inequality measures

The one-parameter family of S-Gini indices is the most representative of generalized Gini indices. This paper proposes a variant of the S-Gini index, called the level-adjusted S-Gini index (abbreviated as the aS-Gini index), together with its complementary one-parameter index, called the complementary level-adjusted S-Gini index (caS-Gini index). The relation of the new indices to the original index corresponds to that of the generalized entropy (GE) index to the Atkinson index, in a sense. The complementary index is introduced to overcome an issue arising from the failure of the aS-Gini index to satisfy some properties exhibited by the GE index. The combination of the aS-Gini and caS-Gini indices enables us to measure the extent of inequality in size distributions containing small portions of negative values, such as net wealth distributions, by different levels of sensitivity to higher values than to lower values. The caS-Gini index, as well as the S-Gini and aS-Gini indices, is also a generalization of the standard Gini index because the index is geometrically expressed as the area of a figure enclosed by a transformed egalitarian curve and a transformed Lorenz curve with a constant multiplier. For a specific parameter value, its expression coincides with the well-known