Refinements of Muirhead's Lemma and income inequality

It is well known that if income distribution y is majorized by income distribution x, then y can be reached from x by a succession of well-defined transfers from the rich to the poor. It is of interest to characterize special cases of such systems of transfers that are “simple” either in terms of number of transfers or of total amount transferred. Two such characterizations can be found in the economic literature. This paper provides a number of extensions and generalizations.