Comparing the Size of the Middle Class Using the Alienation Component of Polarization

This paper shows how to compare the size of the middle class in income distributions using a polarization index that do not account for identification. We derive a class of polarization indices where the antagonism function is constant in identification. The comparison of distributions using an index from this class motivates the introduction of an alienation dominance surface, which is a function of an alienation threshold. We first prove that a distribution has a large alienation component in polarization compared to another if the former always has a larger dominance surface than the latter regardless of the value of the alienation threshold. Then, we show that the distribution with large dominance surface is more concentrated in the tails and has a smaller middle class than the other distribution. We implement statistical inference and test dominance between pairs of distributions using the asymptotic theory and Intersection Union tests. Our methodology is illustrated in comparing the declining of the middle class across pairwise distributions of twenty-two countries from the Luxembourg Income Study data base.