Two new definitions of decreasing inequality

It is standard to measure inequality using the Lorenz quasi-ordering, which can be connected with progressive transfers of Pigou-Dalton (Hardy et al.). Yet, it has been often claimed that the principle of progressive transfers fails to capture inequality in some situations. In particular, polarization is an important feature of inequality that is not modelized by all such transfers: polarization conveys the idea that groups can appear in a distribution, and each individual of a group can identify to the persons of the same group, and compare to the persons of other groups. In this work, we define two new quasi-orderings on income distribution of equal mean which aim to explicitely incorporate the concept of polarization, the second one moreover to take into account possible social tensions. More precisely, we use the notions of "absolute satisfaction" and "absolute deprivation". It starts with the following idea, already modeled in [Chateauneuf and Moyes]: the social status of an individual (approximated by its income in this paper) can create deprivation or satisfaction, depending on her social position. Besides, there are evidences that each individual tends to compare her situation to other people situations. In this paper, we define a poor individual as one belonging to the fifty percent poorest and a rich individual as one belonging to the fifty percent richest. Our first quasi-ordering assumes each poor individual compares herself with poorer individuals while each rich individual compares herself with richer ones. Our second quasi-ordering assumes each poor individual compares herself with richer ones while rich individuals compare herselves with poorer ones. We give necessary and sufficient conditions for the rank-dependent utility social welfare function of Yaari to represent these quasi-orderings. Different aspects and interpretations of these quasi-orderings are discussed. Finally, we look for transfers generating our quasi-orderings. ∗PSE-CES, Université de Paris 1. †PSE-CES, Université de Paris 1. ‡Prism Sorbonne, Université de Paris 1.