Income inequality measurement: a fresh look at two old issues

Abstract

The literature on income inequality measurement is split into (1) the ethical approach, from which the Atkinson–Kolm–Sen and Kolm–Pollak classes of indices are derived, and (2) the axiomatic approach, which mainly leads to the generalised entropies. This paper shows how to rationalise, under utilitarianism, the generalised entropies as ethical indices. In this framework a generalised entropy index is consistent with the principle of transfers if and only if the underlying utility function is increasing. This unconventional interpretation explains the strange behaviour of the generalised entropies for some values of the inequality aversion parameter, as identified by Shorrocks (Econometrica 48:613–625, 1980). In that case, the underlying utility function is convex. Then, it provides a solution to escape the so-called Hansson–Sen paradox (Hansson in Foundational problems in the special sciences. Reidel Publishing Company, Dordrecht, 1977; Sen in Personal income distribution. North-Holland, Amsterdam, pp 81–94, 1978) that affects the standard ethical indices and which corresponds to a counterintuitive increase in inequality as a result of a concave transformation of the utility function. A normalised version of the generalised entropies behaves appropriately after such a transformation.

Appendices

A. Inequality indices with unequal mean distributions

When no restriction is placed on the mean income, indices are obtained from Definitions 1 and 2 by normalising the income distributions. We denote by $\hat{\varvec{x}} {{\mathrm{=}}}(\hat{x}_1,\hat{x}_2,\ldots ,\hat{x}_n)$ the reduced distribution of $\varvec{x}\in \mathscr {D}^{n}$ where $\hat{x}_i {{\mathrm{=}}}x_i / \mu (\varvec{x})$, and by $\tilde{\varvec{x}} {{\mathrm{=}}}(\tilde{x}_1,\tilde{x}_2,\ldots ,\tilde{x}_n)$ the centered distribution where $\tilde{x}_i {{\mathrm{=}}}x_i - \mu (\varvec{x})$. By using Definition 1, the so-called Atkinson–Kolm–Sen class of relative ethical indices (Kolm 1969; Atkinson 1970; Sen 1973) is defined by $I_{r}(\varvec{x}) = I_{U_r}(\hat{\varvec{x}})$, where $r< 2$ and:

$$\begin{aligned} U_{r}(x) = \left\{ \begin{array}{lll} \frac{1}{r-1}\,x^{r-1}, &{} \text {if}\quad r\ne 1, \\ \ln x + 1, &{} \text {if}\quad r= 1. \end{array} \right. \end{aligned}$$

(A.1)

It follows that $I_{r}(\varvec{x}) = 1 - \Xi _{U_r}(\varvec{x})/\mu (\varvec{x})$. Equivalently, the Kolm–Pollak class of absolute ethical indices (Kolm 1969; Pollak 1971) is defined by $I_a(\varvec{x}) = I_{U_a}(\tilde{\varvec{x}})$, where $a> 0$ and:

$$\begin{aligned} U_{a}(x) {{\mathrm{=}}}\left\{ \begin{array}{lll} -\frac{1}{a}\,e^{-a\,x}, &{} \text {if}\quad a\ne 0, \\ x, &{} \text {if}\quad a= 0. \end{array} \right. \end{aligned}$$

(A.2)

Hence $I_a(\varvec{x}) = \mu (\varvec{x}) - \Xi _{U_a}(\varvec{x})$. A relative (resp. absolute) ethical index measures the relative (resp. absolute) income loss resulting from inequality. We also point out that, by definition, the utility functions $U_{r}$ and $U_{a}$ are strictly increasing and concave.

By applying Definition 2, the class of the relative generalised entropies (Bourguignon 1979; Cowell 1980; Shorrocks 1980, 1984) can be written as $G_r(\varvec{x}) {{\mathrm{=}}}G_{\phi _r}(\hat{\varvec{x}})$ with $r\in \mathbb {R}$ and:

$$\begin{aligned} \phi _{r}(x) {{\mathrm{=}}}\left\{ \begin{array}{lll} \frac{1}{r(r-1)}\,x^{r}, &{} \text {if}\quad r\ne 0,1, \\ x\,\ln x, &{} \text {if}\quad r= 1, \\ -\,\ln x, &{} \text {if}\quad r= 0. \end{array} \right. \end{aligned}$$

(A.3)

Finally the class of the absolute generalised entropies (Bosmans and Cowell 2010), can be written as $G_a(\varvec{x}) {{\mathrm{=}}}G_{\phi _a}(\tilde{\varvec{x}})$ with $a\in \mathbb {R}$ and:

$$\begin{aligned} \phi _{a}(x) {{\mathrm{=}}}\left\{ \begin{array}{lll} \frac{1}{a^2}\,e^{-ax}, &{} \text {if}\quad a\ne 0, \\ \frac{1}{2} x^2, &{} \text {if}\quad a= 0. \end{array} \right. \end{aligned}$$

(A.4)

Whatever the values of $r$ and $a$, $\phi _{r}$ and $\phi _{a}$ are always strictly convex.

The Atkinson–Kolm–Sen (resp. Kolm–Pollak) class is ordinally equivalent to a part of the relative (resp. absolute) generalised entropies class. Indeed $I_{r} = 1 - \left[ (r-1)(r-2) G_{r-1} + 1 \right] ^{\nicefrac {1}{(r-1)}}$ if $r\ne 1$ and $I_{1} = 1 - \exp (-G_{0})$, so that $I_{r}$ is an increasing monotonic transformation of $G_{r-1}$ as soon as $r< 2$. Equivalently $I_{a} = \ln (a^2 G_{a}+1)/{a}$, so that $I_{a}$ is an increasing monotonic transformation of $G_{a}$ when $a> 0$. This relationship is well-established in the literature. In this paper we provide a new relationship, more transparent, through an original interpretation of the generalised entropies.

B. Proofs

Proof of Proposition 1

By definition, $I_{U}(\varvec{x}) = \mu - U^{-1}\left( \frac{1}{n} \sum _{i=1}^n U(x_i) \right)$. Hence we can write $I_{U}(\varvec{x}) = \frac{1}{n} \sum _{i=1}^{n}\left[ U^{-1} \left( U(x_i) \right) - U^{-1} \left( \mu (U)\right) \right]$. Recalling that $\sum _{i=1}^{n}\left[ U(x_i) - \mu (U) \right] = 0$, one obtains the desired result. $\square$

Proof of Proposition 2

We know that $G_\phi (\varvec{x}) = D_\phi (\varvec{x}|\!| \mu \varvec{1}_n)$ for any $\varvec{x}\in \mathscr {D}^{n}(\mu )$. The proof is based on the Legendre duality which characterises Bregman divergences (see Boissonnat et al. 2010). By definition, $\phi \,: \mathscr {D}\rightarrow \mathbb {R}$. The conjugate function of $\phi$, denoted by $\phi ^*$, is defined as follows:

$$\begin{aligned} \phi ^*(t) = \sup _{x \in \mathscr {D}} \left( t x - \phi (x) \right) . \end{aligned}$$

(B.1)

Recalling that $\phi$ is continuously differentiable and strictly convex, it follows that we also have $\phi ^*(t) = t\,{(\phi ')}^{-1}(t) - \phi \left( {(\phi ')}^{-1}(t) \right)$. Moreover, it can be shown that ${(\phi ')}^{-1} = {(\phi ^*)}'$ and ${(\phi ^*)}^{-1} = \phi '$. By letting $F=D_{\phi ^*}( \phi '(\mu ),\dots ,$ $\phi '(\mu ) |\!| \phi '(x_1),\!\dots ,\!\phi '(x_n) )$ we first notice that, by definition:

$$\begin{aligned} F = \frac{1}{n} \sum _{i=1}^{n}\left[ {(\phi ^*)}(\phi '(\mu )) - {(\phi ^*)}(\phi '(x_i)) - (\phi '(\mu )-\phi '(x_i)) {(\phi ^*)}' (\phi '(x_i)) \right] . \end{aligned}$$

(B.2)

Moreover ${(\phi ^*)}(\phi '(x)) = \phi '(x) x - \phi (x)$, and $(\phi '(\mu )-\phi '(x)) {(\phi ^*)}' (\phi '(x)) = (\phi '(\mu )-\phi '(x)) x$ by applying the property ${(\phi ^*)}^{-1} = \phi '$. It follows that:

$$\begin{aligned} F = \frac{1}{n} \sum _{i=1}^{n}\left[ \phi '(\mu ) \mu - \phi (\mu ) - \phi '(x_i) x_i + \phi (x_i) - (\phi '(\mu )-\phi '(x_i)) x_i \right] . \end{aligned}$$

(B.3)

After simplification, one obtains $F = \frac{1}{n} \sum _{i=1}^{n}\left[ \phi (x_i) - \phi (\mu ) \right] = D_\phi (\varvec{x}|\!| \mu \varvec{1}_n)$. If now we define U as $U = \phi '$, one immediately observes that $\phi ^*= \varphi$ with $\varphi$ as stated in the proposition. Hence we also have from the definition of F that $F = D_\varphi (U(\mu ) \varvec{1}_n|\!| \varvec{U})$, which completes the proof. $\square$

Proof of Proposition 3

Let $\varvec{x}\in \mathscr {D}^{n}(\mu )$. We know that, by definition, $G_\phi (\varvec{x}) = D_\phi (\varvec{x}|\!| \mu \varvec{1}_n) = \frac{1}{n} \sum _{i=1}^{n}\left[ \phi (x_i) - \phi (\mu ) - (x_i - \mu ) \phi ^{\prime }(\mu ) \right]$. By denoting $U {{\mathrm{=}}}\phi ^{\prime }$ and using the fundamental theorem of calculus, one has $\phi (x_i) - \phi (\mu ) = -\int _{x_i}^{\mu } U(x) dx$ and $- (x_i - \mu ) \phi ^{\prime }(\mu ) = \int _{x_i}^{\mu } U(\mu ) dx$, from which the result is deduced. The function $\phi$ is strictly convex, hence U is strictly increasing. $\square$

Proof of Lemma 1

Notice first that:

$$\begin{aligned} \frac{d}{dt}V \left( U^{-1}(t) \right) = \frac{V' \left( U^{-1}(t) \right) }{U' \left( U^{-1}(t) \right) }. \end{aligned}$$

(B.4)

If $V \left( U^{-1}(t) \right)$ is a strictly concave function of t then, for all $s,t \in \mathbb {R}$ with $s < t$, we have:

$$\begin{aligned} \frac{V' \left( U^{-1}(s) \right) }{U' \left( U^{-1}(s) \right) } > \frac{V' \left( U^{-1}(t) \right) }{U' \left( U^{-1}(t) \right) }. \end{aligned}$$

(B.5)

Let $s = U(x)$. After substitution in (B.5), and assuming that $(0 <)\, V'(x) \le U'(x)$, it follows that:

$$\begin{aligned} \frac{V' \left( U^{-1}(t) \right) }{U' \left( U^{-1}(t) \right) } < 1,\quad \forall t > U(x). \end{aligned}$$

(B.6)

By letting $t = U(z)$ with $z > x$, one obtains $V'(z) < U'(z)$ for all $z>x$. If moreover there exists $y \in \mathscr {D}$ such that $x<y$ and $V(y)=U(y)$, one immediately obtains the desired result. $\square$

Proof of Proposition 4

We first demonstrate that (a) implies (b). Let $V {{\mathrm{=}}}\psi '$ and $U {{\mathrm{=}}}\phi '$, and choose any distribution $\varvec{x}\in \mathscr {D}^{n}(\mu )$. One first remarks that:

$$\begin{aligned} I_\phi = G_{\phi _{\star }},\ \text {where}\ \phi _{*}(x) = \frac{1}{U'(\underline{x})}\phi (x) + \beta \,x + \gamma ,\ \text {for any}\ \beta , \gamma \in \mathbb {R}. \end{aligned}$$

(B.7)

It follows that ${\phi _{*}}''(\underline{x}) = {U_{*}}'(\underline{x})=1$. By choosing $\beta = c - U(\mu )/U'(\underline{x})$ and $\gamma =0$ it follows that, in addition, $U_{*}(\mu )=c$. Equivalently we have:

$$\begin{aligned} I_\psi = G_{\psi _{\star }},\ \text {where}\ \psi _{*}(x) = \frac{1}{V'(\underline{x})}\psi (x) + \delta \,x + \eta ,\ \text {for any}\ \delta , \eta \in \mathbb {R}. \end{aligned}$$

(B.8)

Hence ${\psi _{*}}''(\underline{x}) = {V_{*}}'(\underline{x})=1$ and, by choosing $\delta = c - V(\mu )/V'(\underline{x})$ with $\eta =0$, we have $V_{*}(\mu )=c$. By using Proposition 3, we know that:

$$\begin{aligned} G_{\phi _{\star }}(\varvec{x}) = \sum _{i\,:\,x_i \le \mu } {\int _{x_i}^{\mu } \left[ {U_{*}}(\mu ) - {U_{*}}(x) \right] dx} + \sum _{i\,:\,x_i > \mu } {\int _{\mu }^{x_i} \left[ {U_{*}}(x) - {U_{*}}(\mu ) \right] dx}, \end{aligned}$$

(B.9)

and:

$$\begin{aligned} G_{\psi _{\star }}(\varvec{x}) = \sum _{i\,:\,x_i \le \mu } {\int _{x_i}^{\mu } \left[ {V_{*}}(\mu ) - {V_{*}}(x) \right] dx} + \sum _{i\,:\,x_i > \mu } {\int _{\mu }^{x_i} \left[ {V_{*}}(x) - {V_{*}}(\mu ) \right] dx}. \end{aligned}$$

(B.10)

Because $U_{*}(\mu ) = V_{*}(\mu )=c$, by substracting (B.10) from (B.9), one obtains:

$$\begin{aligned} G_{\phi _{\star }}(\varvec{x}) - G_{\psi _{\star }}(\varvec{x})&= \sum _{i\,:\,x_i \le \mu } {\int _{x_i}^{\mu } \left[ {V_{*}}(x) - {U_{*}}(x) \right] dx} \nonumber \\&\quad + \sum _{i\,:\,x_i > \mu } {\int _{\mu }^{x_i} \left[ {U_{*}}(x) - {V_{*}}(x) \right] dx}, \end{aligned}$$

(B.11)

From Lemma 1, recalling that ${U_{*}}'(\underline{x})= {V_{*}}'(\underline{x})$ and $U_{*}(\mu ) = V_{*}(\mu )$, we deduce that ${V_{*}}(x) > {U_{*}}(x)$ for all $x < \mu$ and ${V_{*}}(x) < {U_{*}}(x)$ for all $x > \mu$. From (B.11) we have $G_{\phi _{\star }}(\varvec{x}) - G_{\psi _{\star }}(\varvec{x}) > 0$, or equivalently $I_\psi (\varvec{x}) < I_\phi (\varvec{x})$. Hence statement (a) implies statement (b). To prove that the converse implication is false, consider two increasing functions U and V such that ${U}'(\underline{x})= {V}'(\underline{x})$, $U(\mu ) = V(\mu )$, ${V}(x) > {U}(x)$ for all $x \in [\underline{x},\mu )$, ${V}(x) < {U}(x)$ for all $x \in (\mu ,\overline{x}]$, and such that U is concave. It follows that $I_\psi (\varvec{x}) < I_\phi (\varvec{x})$ for any $\varvec{x}\in \mathscr {D}$, where $V {{\mathrm{=}}}\psi '$ and $U {{\mathrm{=}}}\phi '$ (see Fig. 1). Since V is not necessarily concave it cannot be, in all cases, a concave transformation of U (which is already concave). $\square$