Unequal inequalities. II

This paper analyzes properties of measures of inequality, applied to income inequalities but meaningful for practically any measure of dispersion in economics. We call n the number of persons, i the person's index, x_i person i's income, $\text{x}\text{= Σ(}\text{x}{}_{\mathrm{i}}\text{n}\text{)}$ the average income, x the vector of the x_i's or income distribution, I(x) a real-valued function of x which is the measure (or index) of inequality.Part I (Sects. I–V), which appeared in the last issue of this journal, analyzed several structures or properties, and specific forms, of I. We distinguished several I's: the measures of inequality per person (or “absolute”) I^a, per pound (or “relative”) $\text{I}{}^{\mathrm{r}}\text{=}\text{I}{}^{\mathrm{a}}\text{x}$, and total nI^a. We presented several possible properties of inequality measures, such as: I = 0 if all x_i's are equal (“zero at equality”), I > 0 otherwise (“positivity out of equality”), symmetry of I for x (“impartiality”), $\text{((}\text{?I}\text{?x}{}_{\mathrm{i}}\text{) ? (}\text{?I}\text{?x}{}_{\mathrm{i}}\text{))(x}{}_{\mathrm{i}}\text{? x}{}_{\mathrm{j}}\text{) > 0}$ for x_i ≠ x_j (“rectifiance” of the function I, or “transfers principle,” this being the strict form whereas the weak one is with sign ?), the fact that $\text{(}\text{?(}\text{x}\text{? I}{}^{\mathrm{a}}\text{)}\text{?}{}_{\mathrm{i}}\text{)}\text{(}\text{?(}\text{x}\text{? I}{}^{\mathrm{a}}\text{)}\text{?}{}_{\mathrm{j}}\text{)}$ does not depend upon x_k for k ≠ i,j (“welfare independence,” or, for short, “independence”). Rectifiance plus symmetry is Schur-convexity. Independence plus symmetry plus zero at equality implies that $\text{x}\text{≡}\text{x}\text{? I}{}^{\mathrm{a}}\text{= ?}{}^{\mathrm{?1}}\text{(}\text{1}\text{n}\text{) Σ ?(x}{}_{\mathrm{i}}\text{)]}$ where $\text{x}$ is the “equal equivalent income”; and we will show that, these three properties being satisfied, the following ones are equivalent to each other: positivity out of equality, rectifiance, quasi-convexity, ?'s concavity.Part I largely focused on the study of six related specific measures of inequality, which in particular possess all the above properties: ?, α, and Ξ being positive parameters, they are $\text{I}{}_{\mathrm{c}}{}^{\mathrm{a}}\text{=}\text{x}\text{+ξ ? (}\text{1}\text{n}\text{∑ (x}{}_{\mathrm{i}}\text{+ ξ)}{}^{\mathrm{1?epsi;}}\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>1</mtext><mtext>??</mtext>}}$, $\text{I}{}_{\mathrm{c}}{}^{\mathrm{a}}\text{=}\text{x}\text{+ξ ? ∏ (x}{}_{\mathrm{i}}\text{+ ξ)}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}$, $\text{I}{}_{\mathrm{c}}{}^{\mathrm{r}}\text{=}\text{I}{}_{\mathrm{c}}{}^{\mathrm{a}}\text{x}$, I_r=I_c^r for ξ=O, $\text{I}{}_{\mathrm{r}}{}^{\mathrm{a}}\text{=}\text{x}\text{I}{}_{\mathrm{r}}\text{=I}{}_{\mathrm{c}}{}^{\mathrm{a}}$ for ξ=, $\text{I}{}_{\mathrm{l}}\text{=(}\text{1}\text{α}\text{)}\text{log}\text{(}\text{1}\text{n}\text{) ∑ eα·(x?x}{}_{\mathrm{i}}\text{)]}$ and $\text{I}{}_{\mathrm{l}}{}^{\mathrm{r}}\text{=}\text{I}{}_{\mathrm{l}}\text{x}$. Lower indices c, r, l respectively stand for “centrist,” “rightist,” and “leftist” measures of inequality. I_r and I_l are invariant under respectively equiproportional variation in, or equal addition to, all incomes; measures which have the first of these two properties are said to be “intensive.”We now consider different and more general measures, and other properties. We first reconcile the last two properties by dropping the “indepencence” one (Section VI.). Then, we analyze another mildly equalitarian property, the “principle of diminishing transfers” (Section VII). Section VIII turns to the relations between inequality measures and Lorenz and concentration curves. We then consider the effect on inequality of additions of incomes, and we analyze the properties of “diminishing equality” (Section IX). The effect of unions of populations is the topic of Section X. Finally, the last section (XI) presents the more general relations between the various structural properties of inequality measures.¹