PRESENTATION ON THE TOPIC ON INEQUALITY COMPARISONS PRESENTED BY MAG.SYED ZAFAR SAEED

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1 PRESENTATION ON THE TOPIC ON INEQUALITY COMPARISONS PRESENTED BY MAG.SYED ZAFAR SAEED Throughout this paper, I shall talk in terms of income distributions among families. There are two points of contention. One is the issue of cardinality vs. ordinality. Practitioners of the cardinal approach compare distributions by means of summary measures such as a Gini coefficient, variance of logarithms, and the like. Gini coefficient is commonly used as a measure of inequality of income or wealth. The variance of logarithms is a widely used inequality measure which is well known to disagree with the Lorenz criterion. Für purposes of ranking the relative inequality of two distributions, the cardinality of the usual measures is not only a source of controversy, but it is also redundant. Accordingly, some researchers prefer an ordinal approach, adopting Lorenz domination as their criterion. The difficulty with the Lorenz criterion is its incompleteness, affording rankings of only some pairs of distributions but not others. Current practice in choosing between a cardinal or an ordinal approach is now roughly as follows: Check for Lorenz domination in the hope of making a clear comparision; if Lorenz domination fails, calculate one or more cardinal measures. In Section I, Gary Fields and John Fei (1978) base their analysis on three axioms an ordering for income distributions or an index of inequality should satisfy: scale irrelevance, symmetry, and desirability of rank-preserving equalization. Then, in Section 2, I shall use these axioms to investigate and strength previous results by Rothschild and Stiglitz and others regarding the consistency of alternative orderings in terms of Lorenz domination. The principal result of this paper is that the three axioms are sufficient to justify the Lorenz criterion for comparing the relative inequality of two distributions. Like the Lorenz criterion, the axiomatic system so constructed is incomplete.the Lorenz-curve is used in economics and ecology to describe inequality in wealth or size. Section-3. Those that do are the Gini coefficient, coefficient of variation, Atkinson index and Theil index. Hopefully, future researchers will add to our axioms so as to narrow down this incompleteness. 1.THREE AXIOMS FOR INEQUALITY COMPARISIONS I now introduce three properties which I shall propose as axioms for inequality comparisons. Not only do these seem reasonable to me but in addition they have been used by previous writers on inequality. It would seem reasonable and desirable for the measure of inequality to be independent of the level of income. Hence, I assume. A1. AXIOM OF SCALE IRRELEVANCE: X = a Y (a 0 ) implies X = Y. This axiom allows us to normalize all distributions X in Ω + according to the fraction of income received by each family:

2 [ X = ( X 1 X 2. X n )] = [ = ( 1 2 n )] where i = X i / (X 1 + X 2 + +X n ) for i = 1, 2,,n. The totality of all such normalized patterns, Ω c, is the subset of points = ( 1, 2... n ) of Ω + satisfying the conditions where Ω + is the non-negative orthant of the space of all possible income distributions. As Atkinson argues, it might however also be justified that the inequality aversion rises with the income level. This property would be captured by using the variance of a distribution as an index for its inequality. A2 AXIOM OF SYMMETRY: If (i 1, i 2,.,i n ) is any permutation of ( 1, 2,.,n ) then ( X 1 X 2 X n )= (Xi 1 Xi 2..Xi n ). A2 is sometimes referred to as the axiom of anonymity in the literature (see Sen 10). Ω o will be referred to as the monotonic rank-preserving set. The next axiom which I shall introduce is: A3. AXIOM OF RANK-PRESERVING EQUALIZATION: In Ω o, if X =E(Y), then X >Y. Formally, this rank preserving equalization is denoted by X = E (Y ). The self evident justification for this axiom is simply that it is reasonable to regard as more equal a distribution which can be derived from another by a richer person giving a part of his income to a poorer person.. Let X and Y be two patterns of income distribution in Ω +. I shall say that X is obtained from Y by a rank preserving equalization. Fields and Fei (1978) try to characterize transfers which result in a more equal income distribution in the economy. Fields and Fei refer to these transfers as rank preserving equalizations, although they are also known as Pigon Dalton transfers in the literature. 2. LORENZ DOMINATION

3 A frequently used ordinal measure for inequality (see for example Kakwani (1977) is given by Lorenz dominance: The Lorenz curve gives the proportion of total income in the economy the poorest x% of the population (for 0 < x < 1 ) hold (see Atkinson (1970) for a continuous formulation). If the Lorenz curve for some income distribution X lies above the Lorenz curve for some other income distribution Y, the distribution X is unambiguously more equal than Y and X is said to Lorenz dominate Y. If the Lorenz curves for two income distributions intersect, it is not possible to determine which distribution is more equal. In this case it might for example be that an income distribution X is less equal for low income households than Y (meaning that for low x the Lorenz curve for Y lies above that for X ), whereas X is more equal for high income households ( meaning that for high x the Lorenz curve for X lies above that for Y). Hence, Lorenz dominance can be characterized as follows:- X Lorenz-dominates Y ( in notation, Lx L Y ) when X 1 +X 2 + +Xi Y 1 + Y 2 + +Y i for j =1, 2,, n - 1 X 1 + X 2 + +Xi >Y 1 +Y 2 + +Y i for some j < n. X Y* if and only if Lx L Y. The necessary condition of the theorem (i.e., X Y* implies that the Lorenz curve of X dominates that of Y) is a well-known result. The sufficient condition of the theorem states that whenever the Lorenz curve of X dominates that of Y (i.e., Lx LY), X can be obtained from Y by a sequence of equalizations which are rank-preserving. Fields and Fei (1978) show their three axioms induce an ordering which is equivalent to the ordering of income distributions induced by Lorenz dominance. 3. CARDINAL INDEX APPROACH TO INEQUALITY COMPARISIONS. The traditional approach for comparing the inequality of two distributions is to compute a cardinal index of inequality I with domains Ω +. I = f(x) =f(x 1 X 2 X n ), X i = 0 (measures inequality and therefore more equal distributio has a lower index). Examples are the Gini coefficient, coefficient of variation, range, and others which we shall consider below. Inequality comparisons are made according to the following definition: DEFINITION:- Pre-Ordering Induced by an Index: A real- valued index of inequality

4 I =f(x) induces a complete pre-ordering R as follows: for all X, Y Ω +, XRY when f(x) f (Y). The following inequality indices satisfy our three axioms i.e Axiom of Scale Irrelevance, Axiom of Symmetry and Axiom of Rank-Preserving Equalization. 2 / n and (A) Coefficient of Variation: C = / where = i (X i - X) = i Xi/n (B) Gini Coefficient: G = [X 1 + 2X 2 + nx n ]. n n 2 X (C) Atkinson Index: (D) Theil Index: T = X i log nx i. i X 2 X 1 1/ 1 X n, A X X X n 4. CONCLUSION Beginning by assuming three axioms, we showed that the axiomatic (a self-evident truth) system so constructed is sufficient to justify the Lorenz criterion for inequality comparisons.. However, like the Lorenz criterion, the axiomatic system is incomplete. Past researchers have achieved completeness by the use of cardinal inequality measures. We showed that many but no means all of the commonly used indices satisfy our three axioms. The ones which do satisfy the axioms agree on the ranking of distributions whose Lorenz curves do not intersect. We have seen that our three axioms are incomplete insofar as they cannot determine the ordinal ranking uniquely. A practicable and desirable direction for future research is to investigate what further axioms could be introduced to complete the axiomatic system or at least to reduce further the zones of doubtfulness.

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