Ranking income distributions using the geometric mean and a related general measure.

• Author: Moore, Robert E.

1. Introduction

   The recent application of stochastic dominance ranking rules to the evaluation of income distributions has stimulated new interest in welfare orderings. While the power of stochastic dominance rules is quite remarkable [19; 6], there are still important cases where incomplete orderings emerge. Bishop, Formby, and Thistle, (hereafter referred to as BFT) [6] use rank dominance (first degree stochastic dominance) and generalized Lorenz dominance (second degree stochastic dominance) to order the income distributions of two international data sets. While the generalized Lorenz (rank) dominance orders approximately 82-84% (75-78%) of the pairwise comparisons of the two data sets, some of the most interesting cases are among those left unordered. For example, Japan's income distribution dominates 16 others, but is unordered compared to 9 distributions.

   This paper proposes a ranking index that provides a complete ordering of income distributions. This ordering is consistent with the partial ordering from generalized Lorenz dominance. The ranking index includes a subjective parameter ([Epsilon]) that allows the observer to set the desired degree of equity preference relative to efficiency preference and meets the minimum requirements of Schur-concavity and the weak Pareto principle for all admissible values of [Epsilon], while satisfying strict Schur-concavity and the strong Pareto principle for all interior values of [Epsilon]. It gives a measure that is decomposable into the arithmetic mean and a measure of dispersion. Furthermore this generalized index encompasses as special cases, the arithmetic mean ([Epsilon] = 0), the Rawlsian criteria ([Epsilon] [approaches] [infinity]), and the geometric mean ([Epsilon] = 1).

   Rank dominance [18; 20; 21] is based on the strong Pareto principle and is shown by Saposnik [20; 21] to be equivalent to first degree stochastic dominance (FDSD). BFT [6] note that a possible objection to rank dominance is that it does not take account of the degree of income inequality. Generalized Lorenz dominance [19; 11; 12] is based on the strong Pareto principle and the principle of transfers and is shown by Thistle [24] to be equivalent to second degree stochastic dominance (SDSD). While FDSD may be objected to for not taking account of the degree of income inequality, SDSD may be objected to for very weak equity preference. To illustrate this, consider a lexicographic ranking obtained by using the arithmetic mean and an inequality measure such as the Gini coefficient only as a tie-breaker when the means of two distributions are the same. This lexicographic ranking incorporates equity preference via the inequality measure in essentially a second order manner. Significantly, SDSD produces a partial ordering that does not conflict with the complete lexicographic ordering. Apparently, then, equity preference is an order of magnitude lower than efficiency preference in SDSD rankings as well. BFT [6, 1409] come to largely the same conclusion and report that: ". . . much of the power of generalized Lorenz dominance to order distributions derives from efficiency preference rather than equity preference."

   The incomplete ordering provided by both FDSD and SDSD leaves some ambiguity in ranking distributions. Some economists have argued that this ambiguity is appropriate and that the subjective weighting should be left to the policy analysts and decision makers [17]. However, with a rigorously specified criterion that provides a complete ordering, the effect of the subjective weights on the final rankings can be determined. Hence, there remains a role for a ranking index with a subjective parameter that allows the observer to determine the desired degree of equity preference relative to efficiency preference and which can provide complete orderings of income distributions.

2. Measuring Welfare

   There are broadly acceptable requirements that may serve as minimum desirable properties for a welfare index that can be summarized in two basic axioms for a welfare function, W, and an income distribution, y.

   AXIOM 1. Weak Pareto...