Refinements of Muirhead's Lemma and Income Inequality

It is well known that if income distribution y is majorized by income distribution x (i.e. y Lorenz-dominates x at constant sum, the welfare standard of y being more equally distributed than x) then y can be reached from x by a succession of transfers "from the rich to the poor" that do not go beyond permuting their incomes. A number of results have been published that characterize special cases of such system of transfers that are "simple" either in terms of number of transfers or of total amount transferred. We show that some of those results are flawed and we provide a number of new ones.