Mis-measurement of inequality: a critical reflection and new insights

Abstract

This article documents that the Gini index is an insufficient measure of inequality and, according to the traditional logic of interpretation, that it may lead to incorrect deductions. Since, apart from concentration, it cannot grasp other relevant features of inequality like heterogeneity and asymmetry—which, beyond its intensity, allow for considering the direction of inequality too—we suggest using the less known Zanardi index of asymmetry of the Lorenz curve as an appropriate measure of inequality. Our findings are supported with estimates from the Luxembourg Income Study Database.

Appendix A: The specification of the Zanardi index

For the sake of completeness and to help with the understanding of Sect. 3, while drawing from Gallegati et al. (2016) and references cited therein, to which the interested reader is addressed for further insights, this appendix develops an equivalent specification of the Zanardi index. Section A.1 develops the main steps of the Gini transformation of the Lorenz curve; Sect. A.2 defines the Zanardi index; Sect. A.3 presents some remarks expanding on points raised previously.

1.1 A.1 The Gini transformation

1.1.1 A.1.1 Minimal topology

Let $\mathcal {B}=\{\mathbf {e}_1,\mathbf {e}_2\}$ be the canonical basis spanning the Euclidean vector space $\mathbb {R}^2$: $\mathbf {e}_1=(1\;0)^T$ and $\mathbf {e}_2=(0\;1)^T$. Being the origin of $\mathbb {R}^2$ fixed at the point O(0, 0), the standard Cartesian system Opq of $\mathbb {R}^2$ is oriented according to $\mathcal {B}$ such that the horizontal Op axis defines the abscissa and the vertical axis Oq defines the ordinate, orthogonally intersecting at O.

Let $\mathbf {u_1}=(p_1\;q_1)^T$ and $\mathbf {u}_2=(p_2\;q_2)^T$ be the vectors identifying, respectively, two points $P_1(p_1,q_1)$ and $P_2(p_2,q_2)$ on Opq. The length $|\overline{P_1P_2}|$ of the segment $\overline{P_1P_2}$ is the distance $d(P_1,P_2)$ evaluated by the norm $||\mathbf {u}_1-\mathbf {u}_2||=\sqrt{(p_1-p_2)^2+(q_1-q_2)^2}\ge 0$, where equality holds only if $P_1\equiv P_2$.

As the unit-vectors $\mathbf {e}_1$ and $\mathbf {e}_2$ identify, respectively, points $A_1(1,0)$ and $A_2(0,1)$ while the $\mathbf {u}=\mathbf {e}_1+\mathbf {e}_2$ identifies the point $A_3(1,1)$, together with the origin O(0, 0) these points define the unit-square $\mathbb {U}=[0,1]\times [0,1]\subset \mathbb {R}^2$ represented in the top-left panel of Fig. 7. The subspace $\mathbb {U}\subset \mathbb {R}^2$ is limited and closed (i.e. compact), with interior $\mathring{\mathbb {U}}$ and boundary $\partial \mathbb {U}$, it is completely included in the first (i.e. positive) quadrant of Opq, it is convex and $\forall P_1,P_2\in \mathbb {U}\Rightarrow 0\le |\overline{P_1P_2}|\le \sqrt{2}$: accordingly, $|\overline{OA_1}|=|\overline{OA_2}|=|\overline{A_1A_3}|=|\overline{A_2A_3}|=1$ and $|\overline{OA_3}|=\sqrt{2}$ by consequence of the Pythagora’s theorem.

Let $r_1: q=p$ be the line through O and $A_3$ that includes the diagonal $\overline{OA_3}$ of $\mathbb {U}$. Also, let $r_2:q=1-p$ be the line through $A_1$ and $A_2$ that includes the counter-diagonal $\overline{A_1A_2}$ of $\mathbb {U}$. Then, $r_1\cap r_2=M(1/2,1/2)\in \mathring{\mathbb {U}}$ identifies the center of the unit square.

Fig. 7

The Gini Transformation. [top-left]: Lorenz curve; [top-right]: reflection; [bottom-right]: rotation; [bottom-left]: dilatation

1.1.2 A.1.2 Notes on the Lorenz curve

Consider the top-left panel of Fig. 7. The Lorenz curve²⁴$q=\mathcal {L}(p)$ is completely enclosed in $\mathbb {U}$: points O and $A_3$ are fixed. If $\forall p\in [0,1]\Rightarrow \mathcal {L}\equiv \overline{OA_3}$, then the Lorenz curve belongs to $r_1$ and it represents the case of equi-concentration or maximum diffusion of the transferable quantity $\mathcal {W}$ with distribution $p=\mathcal {F}(w):w\in [w^{\min },w^{\max }]\subset \mathbb {R}_+$. If $\mathcal {L}=\overline{OA_1}\cup \overline{A_1A_3}$, then it represents the case of maximum concentration or minimum diffusion of $\mathcal {W}$. Between these two extrema there is a not-finite numerable set of cases for which $\mathcal {L}$ is enclosed by the rectangle-triangle $O \overset{\triangle }{A_1} A_3$. In all the cases but the maximum concentration one, the Lorenz curve is monotonic and strictly increasing; in all the cases it is convex as its epigraph is a subset of $\mathbb {U}$.

The Lorenz curve admits a unique discriminant²⁵ point $D(p_d,q_d)\in \overline{MA_1}\subset r_2$, given by $\mathcal {L}\cap r_2$, where $p_d\in [1/2,1]$ and $q_d\in [1/2,1]$: the more it slides toward $A_1$ (M), the more the quantity $\mathcal {W}$ is concentrated (diffused). D separates the poor from the rich side of the distribution $\mathcal {F}(w)$ of $\mathcal {W}$ on $\mathcal {L}$.

The Lorenz curve admits a unique critical point $C(p_c,q_c)$, fulfilling the condition $\mathcal {L}'(p)=1$ of parallelism w.r.t. $r_1$ and it is placed on $\mathcal {L}$ at the maximum distance from $r_1$. If $p_c<p_d$, then C is below $D\in r_2$ revealing a leftward-asymmetry or negative distributional imbalance. If $p_c>p_d$ then C is above $D\in r_2$, revealing a rightward-asymmetry or positive distributional imbalance. If $p_c=p_d$, then $q_c=q_d$, hence the Lorenz curve is perfectly symmetric with respect to $r_2$ and $C\equiv D\in r_2$, revealing distributional balance with some degree of concentration that depends on the distance of D from $A_1$ or, equivalently, from M: $p_c=p_d=1/2\Leftrightarrow q_c=q_d=1/2$ is the equi-concentration case; $p_{c}=p_{d}=1\Leftrightarrow q_c=q_d=1$ is the maximum concentration one. Therefore, depending on the relative positions of D and C, the distributional imbalance profile can be classified according to its intensity and direction, beyond the usual conclusions one may draw from the Gini index $\mathcal {G}$: the Zanardi index $\mathcal {Z}_d$ developed in the following section accounts for both intensity and direction of concentration, providing a sounder measure of inequality. The $\mathcal {Z}_d$ is developed on the basis of the Lorenz curve and embeds information from $\mathcal {G}$: to formally specify $\mathcal {Z}_d$ it is worthwhile transforming $\mathcal {L}$ as follows.

1.1.3 A.1.3 Reflection

Let’s consider an isometric (i.e. distance-preserving) transformation $\chi :Opq\rightarrow Op'q'$ that maps the abscissa p into the ordinate $q'$ and the ordinate q into the abscissa $p'$ for any point of $\mathbb {U}\subset Opq$:

$$\begin{aligned} \chi \left\{ \begin{array}{l} p'=q,\\ q'=p. \end{array} \right. \end{aligned}$$

(A.1)

As $\chi$ reflects P(p, q) w.r.t. the line $r_1:q=p$ identifying $P'(p',q')\equiv P'(q,p)$, by applying $\chi$ to any point of the Lorenz curve $\mathcal {L}$ one finds the inverse curve $\mathcal {L}^{-1}$: being $\mathcal {L}$ strictly increasing and convex on Opq, then $\mathcal {L}^{-1}$ is strictly increasing and concave on $Op'q'$. In terms of coordinates, reference points of $\mathbb {U}$ change into points of $\mathbb {U}'$ as $A_1(1,0)\rightarrow A_1'(0,1)$, $A_2(0,1)\rightarrow A_2'(1,0)$, $D(p_d,q_d)\rightarrow D'(q_d,p_d)$ and $C(p_c,q_c)\rightarrow C'(q_c,p_c)$, while O and $A_3$ do not change—but it is convenient considering $A_3\rightarrow A_3'$ even though they are the same point. See the top-right panel of Fig. 7.

1.1.4 A.1.4 Rotation

Let now $\rho _\phi$ be an isometric counter-clockwise rotation of $Op'q'$ by an angle $\phi$ into $Op''q''$:

$$\begin{aligned} \rho _\phi \left\{ \begin{array}{l} p''=p'\cos \phi +q'\sin \phi ,\\ q''=-p'\sin \phi +q'\cos \phi . \end{array} \right. \end{aligned}$$

(A.2)

Notice that both $\chi$ and $\rho _\phi$ are invertible. By setting $\phi =\pi /4$, then $\sin \phi =\cos \phi =\sqrt{2}/2$ and

$$\begin{aligned} \rho _{\pi /4}\circ \chi \left\{ \begin{array}{l} p''=\frac{\sqrt{2}}{2}(p+q),\\ q''=\frac{\sqrt{2}}{2}(p-q), \end{array} \right. \end{aligned}$$

(A.3)

for which the origin remains fixed at O while all the other points change accordingly. Since for $\phi =\pi /2$ it follows that $\sin \phi =\cos \phi =\sqrt{2}/2=\sqrt{2}$, then $\rho _{\pi /4}\circ \chi$ reflects and counter-clockwisley rotates the space as shown in the bottom-right panel of Fig. 7. Therefore, $\rho _{\pi /4}\circ \chi$ maps the original Lorenz curve $q=\mathcal {L}(p)$ on $\mathbb {U}$ into $q''=\mathcal {L}_1(p'')$ on $\mathbb {U}''$.

1.1.5 A.1.5 Stretching

Still on $Op''q''$, consider the homothetic transformation $\varepsilon _\gamma$ of real scale factor $\gamma >0$ mapping $Op''q''$ into $Op'''q'''$ where, clearly, $Op''q''\equiv Op'''q'''$:

$$\begin{aligned} \varepsilon _\gamma \left\{ \begin{array}{l} x=\gamma p'',\\ y=\gamma q''. \end{array} \right. \end{aligned}$$

(A.4)

For $\gamma \in (0,1)$ it realizes a compression of any segment $\overline{(p'',q'')}$ into $\overline{(x,y)}$, $\gamma >1$ realizes a dilatation.

Having set $\phi =\pi /2:\sin \phi =\cos \phi =\sqrt{2}/2=\sqrt{2}$, now set $\gamma =\sqrt{2}>1$ and apply $\varepsilon _{\sqrt{2}}$ to $\rho _{\pi /4}\circ \chi$ as follows:

$$\begin{aligned} \varGamma :=\varepsilon _{\sqrt{2}}\circ \rho _{\pi /4}\circ \chi \left\{ \begin{array}{l} x=p+q,\\ y=p-q, \end{array} \right. \end{aligned}$$

(A.5)

which is called the Gini (direct) transformation that maps $\mathbb {U}\subset Opq$ into $\mathbb {U}'''\subset Op'''q'''$. As shown in the bottom-left panel of Fig. 7, $\varGamma$ reflects, counter-clockwisely rotates and dilates $\mathbb {U}$ into $\mathbb {U}'''$. Therefore, by applying $\varGamma$ to any point of $\mathcal {L}$ on $\mathbb {U}$ one obtains the Gini-transformed Lorenz curve $y=\ell (x)$ on $\mathbb {U}'''$. Notice that $\varGamma$ is invertible:

$$\begin{aligned} \varGamma ^{-1} \left\{ \begin{array}{l} p=\frac{1}{2}(x+y),\\ q=\frac{1}{2}(x-y). \end{array} \right. \end{aligned}$$

(A.6)

Therefore, by applying $\varGamma ^{-1}$ to any point of $\ell$ one finds the corresponding point on $\mathcal {L}$.

1.1.6 A.1.6 Remarks

By means of $\varGamma$ the Gini-transformed Lorenz curve $\ell$ is enclosed by the rectangle-triangle $O \overset{\triangle }{A_2'''} A_3'''$ of hypotenuse $\overline{OA_3'''}$ on $Op'''q'''$, and it is equivalent to the original Lorenz curve $\mathcal {L}$ enclosed by the rectangle-triangle $O\overset{\triangle }{A_1} A_3$ of hypotenuse $\overline{OA_3}$ on Opq. Moreover, the bisector of the rectangle angle at $A_2'''$ is perpendicular to $Op'q'\equiv Opq$ and so is w.r.t. the hypotenuse $\overline{OA_3'''}$ at $M'''$; therefore, $|\overline{A_2'''M''}|=|\overline{A_3A_1}|=1$ while $|\overline{OA_3'''}|=2$. The geometry of $\mathcal {L}$ inside $\mathbb {U}\subset Opq$ is equivalent to that of $\ell$ inside $\mathbb {U}'''\subset Op'''q'''$.

1.2 A.2 The Zanardi index

As noticed above, $\mathcal {L}$ and $\ell$ are geometrically equivalent. For the ease of a clearer understanding consider now Fig. 8. The area inside the segment $\overline{OA_3}$ and the arc $\widetilde{OA_3}$ in the left panel of Fig. 8 is evaluated by the standard Gini index of concentration:

$$\begin{aligned} \mathcal {G}=1-2\int _0^1 \mathcal {L}(p) {\text {d}}p, \end{aligned}$$

(A.7)

while the corresponding area inside the segment $\overline{OA_3'''}$ and the arc $\widetilde{OA_3'''}$ in the right panel is

$$\begin{aligned} \mathcal {A}=\int _0^2 \ell (x) {\text {d}}x. \end{aligned}$$

(A.8)

Fig. 8

The Lorenz curve $\mathcal {L}$ (left) and the Gini-transformed curve $\ell$ (right)

As regards the discriminant point, it can be noticed that $D(p_d,q_d)\in \mathcal {L}\rightarrow _\varGamma D'''(x_d,y_d)\in \ell$ as well as for the critical point it holds that $C(p_c,q_c)\in \mathcal {L}\rightarrow _\varGamma C'''(x_c,y_c)\in \ell$.

As a first feature of the $\varGamma$-transformation, notice that while $p_d\in [1/2,1]$ one always finds $x_d=1$: being $x_d$ fixed and separating the two sides of $\ell$, this is why this point is named discriminant of the poor side $\widetilde{OD}\subset \mathcal {L}$, mapping into $\widetilde{OD'''}\subset \ell$, from the rich side $\widetilde{DA_3}$, mapping into $\widetilde{D'''A_3'''}$; being $\varGamma$ invertible, the vice-versa is always true, and this is why $D(p_d,q_d)$ is named as the discriminant point on $\mathcal {L}$.

By means of the discriminant point $D'''(x_d,y_d)\in \ell$ two areas $\mathcal {A}_0^p$ and $\mathcal {A}_0^r$ can be identified as indicated in Fig. 8, both evaluated as follows:

$$\begin{aligned} \left\{ \begin{array}{l} \mathcal {A}_0^p=\int _{0}^{x_d} \ell (x)dx - \frac{y_d}{2}\\ \; \\ \mathcal {A}_0^r=\int _{x_d}^{2} \ell (x)dx - \frac{y_d}{2} \end{array} \right. \;:\;y_d=\ell (x_d)\;,\;x_d=1. \end{aligned}$$

(A.9)

These are the areas under the curves $\widetilde{OD'''}$ and $\widetilde{D'''A_3'''}$ net of the triangles inside.

The concentration within each of the two sides of the Lorenz curve $\mathcal {L}$ can then be evaluated as:

$$\begin{aligned} \left\{ \begin{array}{l} \mathcal {G}^p=\frac{\mathcal {A}_0^p}{K_d}\\ \; \\ \mathcal {G}^r=\frac{\mathcal {A}_0^r}{K_d} \end{array} \right. \;:\;K_d=\frac{p_dq_d}{2}, \end{aligned}$$

(A.10)

where the normalizing term $K_d$ evaluates the areas of the triangles $O \overset{\triangle }{P} D$ and $D \overset{\triangle }{Q} A_3$.

The Zanardi index is then defined as follows:

$$\begin{aligned} \mathcal {Z}_d=2K_d\frac{\delta }{\mathcal {G}}\in [-1,+1]\;:\;\delta = \mathcal {G}^r-\mathcal {G}^p. \end{aligned}$$

(A.11)

It’s worthwhile to mention a few things. First, $\mathcal {Z}_d$ depends on the discriminant point and on the Gini concentration index. Second, as the concentration divide $\delta$ is normalized by $\mathcal {G}$, then $\mathcal {Z}_d$ returns a pure number evaluating the asymmetry profile of a Lorenz curve, with intensity and direction, that can be safely compared to that of another curve even if they intersect—remarkably, this is not allowed for the Gini measure of inequality. Third, as it considers the concentration divide $\delta$ between the two sides, it provides a direction of asymmetry and not only the intensity of the distributional unbalance. Fourth, together with its interpretation, we proved in Sect. 3 that $\mathcal {Z}_d$ fulfills many desirable theoretical properties that $\mathcal {G}$ cannot fulfill: empirical proofs have been described with statistical details.

1.3 A.3 Remarks

One may wonder why the Gini transformation is needed as $\ell$ and $\mathcal {L}$ are found being equivalent. The fundamental relevance of $\varGamma$ applied to $\mathcal {L}$ is that the discriminant point $D'''(x_d,y_d)$ is all one needs, regardless of the critical point $C'''(x_x,y_c)\in \ell$ or $C(p_c,q_c)\in \mathcal {L}$: remarkably, the critical point corresponds to the peak about the expected value on the distribution $\mathcal {F}(w)$ of the quantity $\mathcal {W}$ and it is well known that the average suffers of outliers; hence, overcoming this limitation is a first improvement.

Moreover, by means of $\varGamma$ not only O and $A_3'''$ are fixed but a third constraint comes naturally: i.e. $x_d=1$ while $y_d$ is free to slide along the perpendicular to $\overline{OA_3'''}$, as well as D may slide along the counter-bisector through M. As a consequence, while discussing the asymmetry profile of $\mathcal {L}$ needs to consider the relative positions of C and D, on $\ell$ one only needs $D'''$, say $y_d$, to evaluate $\mathcal {A}_0^r$ and $\mathcal {A}_0^p$ that give concentrations $\mathcal {G}^r$ and $\mathcal {G}^p$ on the Lorenz curve $\mathcal {L}$ of the distribution $\mathcal {F}$ of the quantity $\mathcal {W}$, that is what one usually deals with in practice. Therefore, it is only by means of $\varGamma$ that one can safely identify $D'''$, and then correctly interpret $D=\varGamma ^{-1}(D''')$: without $\varGamma$, the discriminating role of D would be only conventional; it is by means of the Gini transformation of the Lorenz curve that one may motivate the discriminant role of point D, rigorously and formally.

Also, it goes without saying that once one infers a theoretical model for $\mathcal {L}$ from empirical data, then she may unequivocally find the associated model for $\ell$ and the estimator of $D'''$, as well as those for $\mathcal {A}_0^r$ and $\mathcal {A}_0^p$, to obtain those of $\mathcal {G}^r$ and $\mathcal {G}^p$, unknown to the standard Gini methodology but correctly defined by the Zanardi one: ironically, this is possible only by means of the Gini transformation $\varGamma$. To the best of the authors’ knowledge, these aspects have never been put forth in the literature.

In addition, a further theoretical aspect worth mentioning is that the Zanardi measure of the Lorenz curve asymmetry overcomes the interpretative limitations faced by the Gini measure because it evaluates distributional inequality considering both intensity and direction.

Finally, it is essential to understand that the Zanardi methodology is perfectly consistent with the Gini logic discussed in Sect. 1: i.e. the more a quantity unequally distributes the more it is concentrated, not the contrary, as one wrongly is usually tempted saying. The advantage of considering asymmetry direction beyond intensity of distributional imbalance completes the interpretative framework as one is put in the condition of knowing what part of the distribution is responsible for inequality. All these aspects are mainly relevant in practice for inequality measurement and direct comparison of income distributions, either in the cross-section and time-dependent cases, even though the concentration curves intersect.