Norm-based measures of inequality: A property-focused evaluation

Norm-based inequality measures are developed within the unequally distributed/relative unequally distributed (UD/RUD) framework by applying the L1 norm and the squared L2 norm to the cumulative distribution and quantile function (CDQF), the quantile function (QF), and their combined representation. All six indices maintain key properties such as scale, replication, and translation invariance, as well as anonymity, and their behavior under Pigou-Dalton transfers and subgroup decomposition is examined analytically. The indices based on the cumulative distribution and quantile function combine the vertical and horizontal gaps, giving them representation decomposability and making them the most informative overall measures of inequality. A Monte Carlo study on six contrasting income distributions corroborates these theoretical advantages, confirming finite, stable values even for mixed-sign and heavy-tailed supports.