Assessing Inequality in the Presence of Growth: an Expository Essay

Abstract

In a great deal of empirical work on distributional analysis, the only sorts of inequality measures employed are ‘relative’ ones. The present paper argues the case for a more plural approach to inequality measurement, in which both ‘absolute’ and ‘intermediate’ conceptualizations of inequality are admitted. In particular, it is suggested that there is a strong case for the employment of intermediate measures of inequality in assessing over time changes in inequality and, through that route, the inclusiveness or otherwise of the process of growth in per capita income. The purpose of the paper is two fold: exposition and persuasion.

Appendix

On the Parabolic Shape of the Krtscha Iso-Inequality Path

Let x = (x ₁, x ₂) be an unequal, increasingly ordered two-person distribution, with x ₁ + x ₂ ≡ X, and let y = (y ₁, y ₂), with y ₁ + y ₂ ≡ Y be a distribution derived from x through an increase of ∆ in total income from X to Y, so that Y − X = Δ > 0. Imagine now that ∆ is comprised of some K finite and equal increments, so that the rise in total income from X to Y is seen to be composed of a sequence of transitions from one distribution to another. These transitions are supposed to preserve inequality, and are of the following nature: from x to x ¹ ≡ (x ¹₁ , x ¹₂ ) (with ${x}_1^1+{x}_2^1\equiv {X}_1=X+\frac{\varDelta }{K}$); from x ¹ to x ² ≡ (x ²₁ , x ²₂ ) (with ${x}_1^2+{x}_2^2\equiv {X}_2={X}_1+\frac{\varDelta }{\operatorname{K}}$);… and from x ^K − 1 to x ^K ≡ (x ^K₁ , x ^K₂ ), with ${x}_1^{\mathrm{K}}+{x}_2^{\mathrm{K}}\equiv Y={X}_{\operatorname{K}-1}+\frac{\varDelta }{\operatorname{K}}$. Note further that each income vector is required to be derived from the preceding income vector by distributing one-half the incremental total income $\frac{\varDelta }{\operatorname{K}}$ in the proportions as they obtain in the preceding vector, and the other one-half equally between the two persons. The vectors x, x ¹, x ², …, x ^K − 1, x ^K are points on the Krtscha iso-inequality curve. Given how x ¹ is derived from x, we can write:

$$ {x}_1^1={x}_1+\left(\frac{1}{2}\right)\left(\frac{x_1}{X}\right)\left(\frac{\varDelta }{K}\right)+\left(\frac{1}{2}\right)\left(\frac{\varDelta }{2K}\right)=\left(\frac{1}{4 KX}\right)\left(4 KX{x}_1+2\varDelta {x}_1+\varDelta X\right). $$

It follows then that the difference between the income-share of the poorer person in distribution x ¹ and his income-share in distribution x is given by

$$ {\delta}_1\equiv \frac{x_1^1}{X_1}-\frac{x_1}{X}=\left(\frac{1}{4 KX\left( KX+\varDelta \right)}\right)\left(4K\mathrm{X}{x}_1+2\varDelta {x}_1+\varDelta X\right)-\left(\frac{x_1}{X}\right) $$

(A1)

It can be verified that the right hand side of Eq. (A2) simplifies to the following expression:

$$ {\delta}_1=\left(\frac{\varDelta }{4X\left(K\mathrm{X}+\varDelta \right)}\right)\left({x}_2-{x}_1\right)>0\ \mathrm{since},\mathrm{by}\ \mathrm{definition}\kern1em {x}_2>{x}_1. $$

(A2)

Defining δ ₂,…,δ _K analogously to δ ₁, we can prove, exactly along the lines just demonstrated, that δ _k  > 0 for all k = 2, …, K. That is to say, in order to preserve the level of inequality, the income-share of the poorer person has to keep increasing with every successive increment of income. This is the same thing as saying that the Krtscha iso-inequality locus will, in the limit, veer toward the ‘equal division’ outcome as income increases. Hence the parabolic shape of the locus, as featured in Fig. 3 in the text.