We propose a generalization of the decomposition by population subgroups of the α-Gini index, the so-called multi-level subgroup decomposition. We demonstrate that all components obtained from the decomposition, can integrate in their functional form a parameter of inequality sensitivity being either related to overall inequalities (α) or to between-group non-overlappings (β). Weprove that a decision maker who behaves in accordance with the α-Gini indices is sensitive to the inequalities between the groups when this sensitivity may differ for each partition of groups. For that purpose, we check the axiomatic foundations of this new family of measures providing also a characterization for the α-Gini indices where α > 2. An illustration of the multi-level α-Gini decomposition is performed for the analysis of the income disparities in France in 2005.
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