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

A Correction to this article was published on 17 October 2019

This article has been updated

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.

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Change history

  • 17 October 2019

    In the original publication of the article, caption of Figure 3 on the third page of Sect. 4.1 was incorrectly published as.

Notes

  1. 1.

    A quantity is transferable if (i) single units may transfer with each other portions of their endowments and (ii) the total value is the algebraic aggregation of units’ endowments.

  2. 2.

    Landini et al. (2018) argue that assuming a hypothesis without proof of validity at the test with facts is the same of assuming it as an axiom, that does not need of proof by definition.

  3. 3.

    Two sampling income distributions can be drawn from different populations, or in different times from the same population, and these are precisely consistent with the second type of phenomena reported by Gini. An exception would be that of a balanced panel, which is not always the case and, nonetheless, it undergoes the first type of phenomena considered by Gini.

  4. 4.

    See also Tarsitano (1987, 1988) and Gallegati et al. (2016).

  5. 5.

    In this sense, income is any real or financial earning flow after taxes and debt commitments that can be transferred to other receivers. The notion of transferability implies that it is possible to reallocate some portion of the endowment W from observation i to observation j such that the endowment \(W_i\) decreases by the same amount, which increases \(W_j\). Therefore, a quantity \(\mathcal {W}\) is transferable if its aggregate or total value W can be partitioned in different shares for each observation. Finally, we do not consider the stock of wealth explicitly, but the same reasoning apply with obvious adjustments.

  6. 6.

    Also, from here onward, terms like society and economy are considered as synonymous and we always refer to them as to complex systems characterized by heterogeneity (i.e. the receivers earn different endowments of income) and interaction (i.e. the receivers exchange or transfer portions of their income by transactions). Nevertheless, our proposal can be extended to other quantities than income. In general, it is appropriated for the analysis of distributional inequality of any transferable quantity.

  7. 7.

    The percentage of households is reported on the x-axis, while the percentage of income on the y-axis.

  8. 8.

    This result of existence and uniqueness can be proved by considering that the Lorenz curve is, by definition, convex and strictly monotonic increasing.

  9. 9.

    Strict increasing monotonicity and convexity of the Lorenz curve ensure this result.

  10. 10.

    The relevance of such a characteristic point is discussed in the following Sect. 2.1.

  11. 11.

    Clearly, this distinction between “poor” and “rich” does not follow conventions commonly used in the income distribution literature. In fact, as is well known, evaluation of poverty begins with an identification step in which the people considered as poor are those with income levels falling below a pre-specified threshold, called the poverty line. As for the rich, the definition closest to the existing literature would specify, usually arbitrarily, a percentage of the total income (like the top 20%, 10%, 5% or even 1%) and identify the population found above this threshold as the rich. Also, the definition could take an arbitrary number of persons—as in the UK Sunday Times list of richest people—or have a minimum cut-off value in order for a person to qualify as rich—as in the US Forbes 400 list. Other alternatives could use the deviation from the mean (median) income or a multiple of this quantity as a parameter—defining the rich as those whose incomes are beyond a determined amount of standard deviation in relation to the mean (median) of the distribution, or those who have more than x times the mean (median) income—or define the rich as those found on the part of the income distribution curve whose shape is similar to the classical Pareto (1895, 1896, 1897a, b) model, which is usually considered as a good approximation of the upper tail of the income distribution.

  12. 12.

    In essence, this is the meaning of the well known “80/20” Pareto law. For an analytic development see Clementi and Gallegati (2016) and Clementi et al. (2016); mathematical justifications are developed in Landini (2016).

  13. 13.

    See Zanardi (1964, 1965), Tarsitano (1987, 1988) and Gallegati et al. (2016). A derivation of the index in terms of Analytic Geometry and transformations on the real plane is proposed in Appendix A.

  14. 14.

    To meet the usual normalization between 0 and 1, it is sufficient to compute the monotonic transformation \(z_d=(\mathcal {Z}_d-1)/2\). In this case, the perfect symmetry or absence of inequality are at 0.5.

  15. 15.

    When an inequality index fulfills some desirable properties, measurement of inequality is said to be “axiomatic”. The axiomatic approach to inequality measurement, however, is purely descriptive and does not embody any value judgments about the nature of inequality (bad or good) and its degree of desirability for a society. Social welfare judgments, by contrast, are made explicit when a “normative approach” to inequality measurement is taken on. For instance, a generalization of the standard Gini index by Donaldson and Weymark (1980, 1983) and Yitzhaki (1983) makes it dependent on the specified degree of inequality aversion—see also Yitzhaki and Schechtman (2005) for a review:

    $$\begin{aligned} \mathcal {G}\left( \nu \right) =\frac{-\nu \;\text {cov}\left\{ y,\left[ 1-F\left( y\right) \right] ^{\nu -1}\right\} }{\mu } \end{aligned}$$

    where \(\text {cov}\left( \cdot ,\cdot \right) \) is the covariance between income levels y and the cumulative distribution of the same income, \(F\left( y\right) \), \(\mu \) is average income, and \(\nu \) is the degree of inequality aversion—note that with \(\nu =2\) the above formula coincides with the standard Gini index. Assigning different values to the inequality aversion parameter \(\nu \) changes the value of the Gini index, as it weights differently incomes in different parts of the income distribution. Particularly, increasing \(\nu \) leads to higher values of \(\mathcal {G}\left( \nu \right) \), because more significance is attached to the incomes of the poorer individuals, and thus \(\mathcal {G}\left( \nu \right) \) is more sensitive to transfers at the lower end of the distribution; if \(\nu \rightarrow \infty \), only the situation of the poorest individual in society matters, so for very high values of \(\nu \) only transfers to the very lowest income group are valued. All of the standard Gini’s useful properties are inherited by \(\mathcal {G}\left( \nu \right) \), which, when embodied into the Zanardi index (7), allows the latter to bring out a clear relation between inequality and social welfare judgments. In particular, rising \(\nu \) leads to smaller values of \(\mathcal {Z}_{d}\), whereas the latter increases for lower values of \(\nu \)—detailed results for this analysis are not shown but are available on request. The intuition for the rise (fall) in \(\mathcal {Z}_{d}\) when \(\nu \) decreases (increases) follows the same lines as in footnote 17.

  16. 16.

    A change of location, instead, do alter \(\mathcal {Z}_{d}\). In fact, the Gini index of the linear transformation \(x=a+by\) is \(\mathcal {G}^{x}=\left[ \mu ^{y}/\left( a+\mu ^{y}\right) \right] \mathcal {G}^{y}\), and, therefore, since poor and rich have a different mean income, it follows that \(\mathcal {Z}^{x}_{d}\ne \mathcal {Z}^{y}_{d}\) (see e.g. Tarsitano 1988).

  17. 17.

    This is the well-known “Pigou–Dalton principle of transfers”. In particular, \(\mathcal {Z}_{d}\) rises with a progressive transfer, i.e. an income transfer from richer to poorer individuals, whereas the index falls with a regressive transfer, i.e. an income transfer from poorer to richer individuals. Notice that the Pigou–Dalton axiom usually requires the inequality measure to fall with a progressive transfer and to rise with a regressive one. However, the interpretation of the change in \(\mathcal {Z}_{d}\) when an income transfer occurs is quite intuitive: in the case of a progressive transfer, for instance, the Lorenz curve moves closer to the 45\(^{\circ }\)-line much more for the poor than for the rich; such a shift in the Lorenz curve necessarily reduces \(\mathcal {G}\) and rises the disparity of concentration \(\delta =\mathcal {G}^r-\mathcal {G}^p\), leading to an increase of \(\mathcal {Z}_{d}\). When there are progressive transfers entirely on the poor side of the distribution, \(\mathcal {Z}_{d}\) increases to a much larger extent: in this case, the portion of the Lorenz curve on the poor side of the distribution moves closer to the 45\(^{\circ }\)-line, while the rich side is unaffected by the transfer; such a shift in the Lorenz curve slightly reduces \(\mathcal {G}\) but it raises \(\delta \) much more, leading to a greater increase of the Zanardi index. Clearly, the opposite holds in the case of a regressive transfer and when emphasis is laid on the rich side of the distribution.

  18. 18.

    Note that \(\mathcal {Z}_{d}=0\) does not imply that income is equally distributed, but only that the Lorenz curve is perfectly symmetric and that income is (un-)equally concentrated within the two sub-distributions of the poorest \(p_{w}\%\) and the richest \(\left( 1-p_{w}\right) \%\). Only in this specific case everything is ruled by the Gini index alone.

  19. 19.

    This represents the Gini’s logic about “the more it is unequal the more it is concentrated” discussed in Sect. 1.

  20. 20.

    Since the distribution of the Zanardi index under the null hypothesis is unknown, a Monte Carlo simulation strategy was used to approximate the null distribution of \(\mathcal {Z}_{d}\). In a nutshell, for each LIS data set our procedure was as follows:

    1. 1.

      First, we calculated the observed value of the Zanardi index, \(\mathcal {Z}^{\text {obs}}_{d}\).

    2. 2.

      Second, we calculated the Zanardi index \(\mathcal {Z}^{j}_{d}\), \(j=1,\ldots ,M\), for \(M=999\) samples simulated independently from a log-normal distribution with unknown parameters replaced by their empirical sample estimates. The log-normal is the best known example of a distribution with a Lorenz curve that is symmetric about the alternate diagonal of the unit square (e.g. Kleiber and Kotz 2003, ch. 4). The distribution was fitted to the empirical sample observations via a simple method of moments, i.e. using the (weighted) mean of the logarithm of data points as an estimate of the location parameter and the log-normal Gini inversion \(\hat{\sigma }=\sqrt{2}\varPhi ^{-1}\left( \frac{1+\mathcal {G}}{2}\right) \) for the shape parameter, where \(\varPhi ^{-1}\left( \cdot \right) \) denotes the inverse normal probability distribution function and \(\mathcal {G}\) is the empirical estimate of the Gini index. Hence, under the null hypothesis simulated data entail \(\mathcal {Z}_{d}=0\) while exhibiting the same overall concentration as the real data.

    3. 3.

      A Monte Carlo estimate of the p-value for the two-tailed test was obtained as (e.g. MacKinnon 2009, p. 186):

      $$\begin{aligned} p\text {-value}=2\min \left[ \frac{1}{M}\sum \limits ^{M}_{i=1}I\left( \mathcal {Z}^{j}_{d}<\mathcal {Z}^{\text {obs}}_{d}\right) ,\frac{1}{M}\sum \limits ^{M}_{i=1}I\left( \mathcal {Z}^{j}_{d}\ge \mathcal {Z}^{\text {obs}}_{d}\right) \right] \end{aligned}$$

      where \(I\left( \cdot \right) \) denotes the indicator function, which is equal to 1 when its argument is true and 0 otherwise.

    4. 4.

      The null hypothesis was rejected in favor of the alternative whenever \(p\,\text {value}<0.05\).

  21. 21.

    The corresponding p-value is 0.399 and leads us to accept the null hypothesis of no correlation between the two indexes. The same result obtains if one uses Kendall \(\tau \) and Spearman \(\rho \) as more general and robust measures of dependence, obtaining \(\tau =0.011\) (\(p\text {-value}=0.804\)) and \(\rho =0.012\) (\(p\,\text {value}=0.856\)).

  22. 22.

    Statistical significance is also very high, with a p-value very close to 0 in both cases. The Spearman and Kendall correlation coefficients are \(\rho ^{p}=0.941\) and \(\tau ^{p}=0.791\) for the poor class, and \(\rho ^{r}=0.912\) and \(\tau ^{r}=0.756\) for the rich. For both the tests computed p-values are almost 0.

  23. 23.

    A similar contrast between Anglo-Saxon countries and continental European countries has received a lot of attention in the recent top income literature, resulting in a number of new insights about income inequality. While top income shares have remained fairly stable in continental European countries over the past three decades, they have increased enormously in the U.S. and other Anglo-Saxon countries. As discussed in Atkinson and Piketty (2007), the rise of top income shares in Anglo-Saxon countries, and in particular in the U.S., is due to the replacement of capital owners (the “rentiers”) by top executives (the “working rich”) at the top of the income hierarchy over the course of the twentieth century. This contrasts with the continental European pattern, where capital incomes are still predominant at the top of the distribution (albeit at lower levels).

  24. 24.

    For the ease of readability, with some abuse of notation \(\mathcal {L}\) either represents the functional and its graph.

  25. 25.

    The reason for which this point is named discriminant can be better understood in the following section.

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Acknowledgements

The authors thank Bruce C. N. Greenwald for discussion. Views and opinions expressed in this article are those of the authors and do not necessarily reflect those of their institutions. The research did not received funding sources. The authors declare that they have no conflict of interest.

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Appendix A: The specification of the Zanardi index

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.

A.1 The Gini transformation

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
figure7

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

A.1.2 Notes on the Lorenz curve

Consider the top-left panel of Fig. 7. The Lorenz curveFootnote 24\(q=\mathcal {L}(p)\) 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 discriminantFootnote 25 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.

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(pq) 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.

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 \(\mathbb {U}''\).

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 \(\mathcal {L}\).

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 \(\mathbb {U}'''\subset Op'''q'''\).

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
figure8

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.

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.

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Clementi, F., Gallegati, M., Gianmoena, L. et al. Mis-measurement of inequality: a critical reflection and new insights. J Econ Interact Coord 14, 891–921 (2019). https://doi.org/10.1007/s11403-019-00257-2

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Keywords

  • Income inequality
  • Gini index
  • Concentration
  • Lorenz curve asymmetry
  • Zanardi index

JEL Classification

  • C18
  • D31
  • D63