Adv. Math. Econ. 11, 77-93 (2008)

A Distance and a Binary Relation Related to Income Comparisons

Hidetoshi Komiya

Faculty of Business and Commerce, Keio University, Hiyoshi, Kohoku-ku, Yokohama 223-8521, Japan

Received: October 31, 2007 Revised: December 3, 2007

JEL classification: I32

Mathematical Subject Classification (2000): 15A51, 60E15, 91A05

Abstract. We define a distance and a binary relation among income distributions which is closely related to Lorenz dominance. An income distribution is represented by a vector (x_{1}, x_{2}, …, x_{n}) when the society under consideration consists of n individuals or households. The component x_{i} denotes the income of the i-th individual and the sum ∑_{i=1}^{n} x_{i} is the total wealth of the society. The distance is defined on the n-dimensional Euclidean space R^{n} mathematically, and it gives indices of difference between two income distributions with not only the same total wealth but also the different total wealths. Thus, the distance might give a criterion for income distributions taking account of equity and efficiency.

Key words: Lorenz dominance, distance, binary relation, minimax theorem