Fiscally stable income distributions under majority voting, Lorenz curves and bargaining sets

Abstract

We explore two variants of the Bargaining Set in a simple majority game on income distributions in order to understand the apparent stability of tax schedules in democratic societies, despite the fact that the core of such games is empty (no majority Condorcet winner). Those variants are sharper than in the literature (Mas-Colell (1989), Shitovitz (1989), Zhou (1994)), by requiring that counterobjections try to guarantee their initial income levels to all members of the minority who stand to lose in an objection. A first variant defines as usual an income disbribution to be stable if there is no objection against it that is “justified”, i.e. for which there is no counterobjection satisfying the above requirement. A second variant allows objecting majorities to look one more step ahead. An objection is “weakly justified” if, whenever there is a counterobjection, the objecting majority can beat it while guaranteeing their income levels to all of its members. An income distribution is strongly stable if there is no weakly justified objection against it.

These two variants generate sharper solution sets than when applied to large market games as in Mas-Colell (1989), Shitovitz (1989). Stable income distributions can indeed be characterized by their degrees of inequality. An income distribution is stable if and only if its Lorenz curve has no point in common with the graph C of f: [1/2,1] → [0,1], with f(b) = 1 − 1/(2b), for b > 1/2. It is strongly stable if and only if it is the egalitarian one.

Paper prepared for the Third International Conference on Mathematical Analysis in Economic Theory, Research Center for Mathematical Economics, Keio University, Tokyo, December 20–22, 2004, and for the Second International Conference on Economic Theory, Institute of Economic Research, Kyoto University, Kyoto, December 17–18,2004. Financial support of both institutions is gratefully acknowledged. I had useful conversations with Thibault Gajdos, Stéphane Gauthier and Guy Laroque while doing the research work toward this paper. Comments and suggestions from participants to the conferences in Tokyo and Kyoto, as well as to seminars at CREST, GREQAM, Marseille and GREDEG, Sophia Antipolis, are gratefully acknowledged. I am also grateful to the referee, whose comments and suggestions generated significant improvements of the exposition of the paper. The usual caveat applies. I am grateful for the efficient typing of Nadine Guedj.