On the decomposition of the extensions of the Gini index that are based on the ‘metallic’ sequences of number theory

Abstract The present paper extends the work of Subramanian (Econ Bull 41(4):2309–2319, 2021), as well as that of Creedy and Subramanian (Exploring a new class of inequality measures and associated value judgements: Gini and Fibonacci-type sequences, 2022), by showing how this new extension of the Gini index may be decomposed by income sources, income classes and population subgroups. It also gives an empirical illustration applying this extension of the Gini index to the analysis of the inequality in expenditure on social protection among European countries in 2018. In the decomposition by benefit function it appears that the housing function consistently exhibits the smallest contribution, due to its low expenditure weight, while the old-age function has the lowest inequality but the highest contribution, due to its substantial expenditure weight. The family/children function's contribution increases when a higher weight is given to lower expenditures, highlighting a pronounced inequality at lower expenditure levels. Conversely, the old-age function shows higher inequality among countries with higher expenditures. In the breakdown by welfare systems, we observe greater inequality between systems than within them.