Lower and upper tail concern and the rank dependent social evaluation functions

Some properties that strengthen the Pigou-Dalton transfer principle have been introduced in the literature, such as the diminishing transfer principle, and its positional version, the principle of positional transfer sensitivity. These principles state that social evaluation function should be more sensitive to transfers that take place lower down in the distribution and reflect a lower tail concern. Another principle which operates in the opposite direction is also introduced, first degree upside positional transfer sensitivity which imposes more sensitivity to transfers further up the scale; and therefore reflects an upper tail concern. In this paper a new principle is proposed, up-down positional transfer sensitivity, which combines the lower tail concern when transfers occur at the low tail of the distribution, and the upper tail concern when they happen at the up tail. Some rationales behind the new property and its implications for the class of rank dependent social evaluation functions are analyzed. It is shown that the Gini function satisfies this property, and a class of social evaluation functions consistent with it is proposed.