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

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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.

Introduction

In recent times, inequality in the distribution of income and wealth has become a major issue and has spurred a renewed interest in both the political and the academic debate (Stiglitz 2012; Piketty 2014; Atkinson 2015; Stiglitz 2015). It is now a well-known fact that existing economic disparities, both between countries and individuals within a single country, tend to persist—if not worsen—with harmful effects on economic growth and increased risk of turning into a crisis (e.g. Cingano 2014; Ostry et al. 2014; Dabla-Norris et al. 2015; Berg and Ostry 2017; see also Rose 2018). These outcomes are in sharp contrast to the predictions of most economic theories, according to which there should be a tendency to convergence in average incomes between richer and poorer countries as well as a more equitable distribution of resources among the citizens of a country once full industrial development has been reached (Kuznets 1955).

Concern with inequality has a long-standing tradition in economics, dating back to the early work of Pareto (1895, 1896, 1897a, b), who first observed that roughly 80% of total income/wealth is owned by the top 20% of the population. Later, the American economist Lorenz (1905) introduced the Lorenz curve, one of the most widely adopted tools for measuring the extent of income and/or wealth inequality. This curve shows how much the actual distribution of income or wealth varies from an equal distribution. If there is complete equality—people receive exactly the same amount of income or wealth—the Lorenz curve coincides with the diagonal of a unit square, whereas worsening distribution (i.e. more inequality) moves the curve away from the diagonal line (and vice-versa).

Economists have also been resorting to various inequality measures for summarizing the degree of inequality in income and wealth distributions with a single number. Among them is the Gini coefficient, which has long been accepted as the workhorse measure of inequality. Named after the Italian statistician Gini, who first introduced it in 1914, this time-honored inequality metric is still widely used by scholars involved in the analysis of income and wealth distribution, as well as used in numerous technical reports from research-oriented organizations. Simplicity and ease of interpretation—thanks to its intuitive graphical relation to the Lorenz curve—as well as various extensions suggested later by several scholars, have certainly contributed to the popularity of the Gini index in economics and outside of it (e.g. Giorgi 1990, 1993, 2005, 2011; Giorgi and Gigliarano 2017; Giorgi and Gubbiotti 2017).

Success and longevity of the Gini index have not, however, been without criticism. For instance, some of the arguments made over the years to abandon it are (Atkinson 1970): (i) the fact that the Gini index does not embed any functional linkage between inequality and social welfare; (ii) its habit of assigning more weight to transfer of income near the modal value of the distribution rather than at the tails; and (iii) its lack of additive decomposition by groups (within and between). However, while these criticisms were made in order to improve the Gini coefficient’s performance in the analysis of income and wealth inequality (Giorgi 1990, 1993, 2005, 2011; Giorgi and Gubbiotti 2017), the criticism will explore questions about the ability of the Gini index to be used to measure inequality at all. In particular, we will assert that the Gini only tells us about the degree of the concentration of transferable quantities,Footnote 1 it does not capture other key aspects of inequality, such as the degree of heterogeneity and asymmetry embodied in income and/or wealth distribution (Gallegati et al. 2016). On the contrary, studying the asymmetry of the Lorenz curve through an adequate measure allows us to account for all the aforementioned features.

In 1914 Corrado Gini wrote: «The ratio that we are proposing in this note as the appropriate measure of concentration, can also be obtained by improving a graphical method already introduced by some authors, as Lorenz, Chatelain, Séailles, in order to evaluate inequality in the distribution of wealth. [...] The less unequal is the wealth distribution, the less accentuated is the concentration curve, that tends to a straight line (egalitarian line) in the case of equi-distribution » (Gini 1914, translation, p. 23).

While referring to his famous concentration index, Gini traces the logic for its interpretation. In a non-axiomatic way, Gini’s logic moves from (the null hypothesis of) distributional inequality, that can be detected through the Lorenz curve, to conclude with the statement that “the more the distribution is unequal, the more it is concentrated”, where “unequal” means that the distribution presents an asymmetric shape revealing that few individuals hold almost all of the total amount of the transferable quantity (e.g. the “rich”) while the largest share of individuals (e.g. the “poor”) accumulates an amount that does not balance with that of the former. It is worth noting that Gini proposes his index as an appropriated measure of concentration that can be useful in measuring inequality, not the contrary. By contrast, a popular interpretation instead evaluates the degree of inequality, and can be summed up like this: “the more the distribution is concentrated, the more it is unequal”. That is precisely the contrary of the Gini’s logic. Such an alternative logic is intuitive enough but it does not come without drawbacks. It leads to a misinterpretation of the Gini index and introduces the severe methodological error of considering the Gini concentration index as the measure of inequality while, by Gini’s own admission, it appropriately measures one aspect of distributional inequality, concentration.

This logical and methodological error is faulty conditioned by the fact that one often bears in mind inequality essentially as a matter of income or wealth distributions, and that the rich get richer due to income and high saving rates while the poor get poorer because of modest labor income and low saving rates. Also, this causes one to implicitly assume, without proof (i.e. axiomaticallyFootnote 2), that all the distributions that generate Gini index estimates are associated, under comparison, to Lorenz curves that exhibit concordant positive asymmetry profiles in favor of the rich side of the distribution. Finally, this often happens regardless of possible reciprocal intersections of the Lorenz curves, and nullifies the possibility of considering more unequal the distribution with higher concentration.

If the asymmetry-concordance assumption were confirmed by the facts, then one might indulge in the luxury of trusting this other way of interpreting the data, but this kind of test is not always implemented and, in any case, drawing conclusions about inequality from concentration is an incorrect practice when Lorenz curves intersect, as almost always happens. As Gini told us, «This graphical approach presented two drawbacks, promptly acknowledged by Lorenz and by King: (a) it does not provide a precise measurement of concentration; (b) it does not allow to assess, not even in some circumstances, when or where concentration is stronger. In fact, if two curves cross each other (Fig. 2), it is not always possible to say if one denotes a stronger concentration than the other. This drawback, that can be deemed not relevant for the comparison of phenomena of the same nature (e.g. the concentration of incomes for two different years or countries), is particularly serious for comparing phenomena of different nature, whose distribution’s shape differs » (Gini 1914, translation, p. 24).Footnote 3

Unfortunately, and more often than not, one refers only to statistical estimates of the Gini index without examining the underlying Lorenz curve. As a consequence, implicitly (and perhaps unconsciously) one assumes that all the Lorenz curves of income distributions exhibit the same concordant positive asymmetry profiles and, even more heroically, that they, too, do not intersect. Based on this axiom, one then feels confident in measuring the intensity of inequality across income distributions by means of the Gini concentration index. Consistently with Gini’s logic, we explain that, at least in terms of statistical significance, if only one were sure that two Lorenz curves exhibit concordant asymmetry profiles (either positive or negative) then the more concentrated one may also be the more unequal, provided that the measure of inequality involved is not biased by the intersecting profiles of the curves. Said differently, if we have two income distributions with Lorenz curves that are similarly asymmetric (e.g. revealing commensurable long right-tails), then their shapes are almost similar with an asymmetry measure of the same sign (i.e. concordant). Provided that these Lorenz curves do not intersect, the one with higher concentration is also the most unequal, but if they intersect then concentration alone is not enough to state what is the curve associated to the most unequally distributed one: to this end, we need to jointly consider concentration and asymmetry in a single inequality measure that is able to account for both intensity and direction without being affected by the intersection of the Lorenz curves. In general, the axiomatic perspective of the alternative logic leads to the wrong use of the Gini index because it cannot control for the cases of intersecting Lorenz curves, nor can it account for the shape of asymmetry.

To overcome these limitations, a non-axiomatic statistical point of view is needed: of course, income inequality is the most relevant but, statistically speaking, it is only a propaedeutic case. Therefore, an appropriate measure of distributional inequality must be general and suitable for the analysis of every transferable-quantity Lorenz curve. It must be able to account for discordant and concordant profiles of asymmetry of the Lorenz curves to consider the direction of inequality, and, to be sound in measuring its intensity for comparison purposes, it must not be biased by the fact that the Lorenz curves may intersect. Beyond the standard practice of using the Gini index, we prove that the Zanardi (1964, 1965) index of asymmetry of the Lorenz curveFootnote 4 is an appropriate measure of inequality as long as, among other desirable properties, it fulfils all the above-mentioned requirements.

The paper makes two relevant contributions to the existing literature, again calling into question the criticality of the Gini index as a measure of inequality. First, it proves both in analytical and empirical terms that studies focusing solely on the Gini index might not consider crucial information that in turn could affect the validity of researchers’ conclusions and cause. Second, draws attention to the good properties of the Zanardi index, which are still little known among researchers. With the exception of the contributions of Tarsitano (1988) and Gallegati et al. (2016), to the best of the authors’ knowledge there are no other articles in English concerning the Zanardi index. In addition, because the Zanardi index was published in Italian, its scope had, and still has, a very limited scholarly reach.

The paper is organized as follows. Section 2 provides an operational definition of inequality, discusses the asymmetry of the Lorenz curve, and sheds light on what we consider are the main drawbacks of the Gini index. Section 3 introduces the Zanardi (1964, 1965) index, which characterizes an important aspect of the shape of the Lorenz curve, namely asymmetry. Using data from the Luxembourg Income Study Database, Sect. 4 provides empirical estimates of both the Gini and the Zanardi indexes to show that the former is not enough to adequately measure inequality. Section 5 is the conclusion.

An operational definition of inequality and the limits of the Gini index

Researchers measure inequality using income distribution. We consider the income variable as any disposable monetary economic resource earned by a receiver and used in transactions,Footnote 5 and the inequality as the degree of imbalance in the income distribution.Footnote 6 Therefore, this paper is exclusively concerned with the distributional inequality of disposable income\(\mathcal {W}\) among receivers in an economy.

Following Gallegati et al. (2016), the operational definition of distributional inequality is:

Definition 1

A not negative and transferable quantity \(\mathcal {W}\) for which:

  1. (a)

    a unique point on the Lorenz curve exists that conventionally separates between the poorly and richly endowed classes of interacting (e.g. transacting) individuals (“complexity: heterogeneity and interaction”);

  2. (b)

    the observations on individuals in the two conventionally separated groups are ranked in descending order and accumulate increasing values of \(\mathcal {W}\) (“concentration”);

  3. (c)

    the Lorenz curve is asymmetric (“asymmetry”),

is said to be “unequally distributed”.

According to Definition 1, an unequally distributed quantity performs distributional inequality, that can be explained in terms of complexity notions. In what follows, we will develop this notion with formal details. Furthermore, the previous definition puts emphasis on three main aspects that are related to each other to stress that distributive inequality is a composite notion; these are: (a) complexity, in terms of heterogeneous partition of endowments and interactive behaviour by transfers among individuals; (b) concentration of distributed endowments; (c) asymmetry of the concentration curve related to the underlying distribution.

As discussed in Sect. 1, inequality is usually measured by means of the Gini index \(\mathcal {G}\), which compares the observed distribution of income in a society with an ideal case in which everyone earns exactly the same amount of income. In a graphical way, this information can be analyzed through the well-know Lorenz curve, which is a graphical representation of the cumulative income distribution. As reported in Fig. 1, the Lorenz curve shows for the bottom \({p}\%\) of households the percentage of the total income (\({q}\%\)) they hold.Footnote 7

If \(p=q\) the Lorenz curve is a straight line, and no inequality emerges in the data. This implies that, for example, the first \(30\%\) of the households hold \(30\%\) of the total income. Thus, any departure from this 45\(^{\circ }\)-line, which expresses perfect equality, represents the degree of inequality. It follows that if everybody shares the same level of income, the cumulative percentage of total income (q) held by any proportion of the population would also be equal to p, and therefore the Lorenz curve would be \(q=\mathcal {L}(p)\), which formalizes the concept that the income shares and the population shares are identical.

Fig. 1
figure1

The \(\mathcal {L}\)-curve and its characteristic points

As expressed above, the useful content of the Lorenz curve is its distance, \(p-\mathcal {L}(p)\), from the 45\(^{\circ }\) line of perfect income equality. Given that the Lorenz curve is defined on the unit-square, it implies that the surface of the lower triangle in Fig. 1 is equal to 1 / 2. To obtain a coefficient with values bounded between 0 and 1, we simply take twice the integral of \(p-\mathcal {L}(p)\), i.e.:

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

which is exactly the usual Gini coefficient.

The Gini index measures the concentration of \(\mathcal {W}\) as the area enclosed by the arc \(\widetilde{OE}\) and the equi-concentration segment \(\overline{OE}\): the more \(\mathcal {G}\rightarrow 1^-\), the more \(\mathcal {W}\) is concentrated, whereas the more \(\mathcal {G}\rightarrow 0^+\), the more \(\mathcal {W}\) is diffused. In practice, \(\mathcal {G}\) is commonly used to measure inequality by considering only one of the aspects of distributional inequality, that is, the divergence from the equi-concentration neglecting both the complexity (i.e. heterogeneity and interaction) and the asymmetry (i.e. the shape) of the underlying distribution.

At this point, it should be clear that concentration and inequality are related concepts, but they could explain exactly the same aspect just in a specific case, i.e. the case of symmetric distribution.

To better define and distinguish these two concepts, and to better understand the limit of the concentration index as a measure of inequality, we must introduce the asymmetry of the Lorenz curve, which could be easily understood by defining its discriminant and critical points.

As represented in Fig. 1, there exists only one discriminant point \(D(p_d,q_d)\) on any Lorenz curve, which intersects the segment \(q=1-p\) connecting the two points of coordinates (0, 1) and (1, 0):Footnote 8 Therefore, \(\mathcal {L}(p_d)=q_d=1-p_d\). Also, only one critical point \(C(p_c,q_c)\) exists on a Lorenz curve,Footnote 9 located at the maximum distance from the equi-concentration segment \(\overline{OE}\) of equation \(q=p\): Therefore, \(\mathcal {L}'(p_c)=1\) such that \(\mathcal {L}(p_c)=q_c\).Footnote 10 The point D discriminates between the poor and the rich groups of the distribution,Footnote 11 while the critical point C identifies the peak about the expected value of \(\mathcal {W}\).

The limits of \(\mathcal {G}\) as an effective inequality measure emerge when two different distributions characterized by the same concentration show differences in term of asymmetric shapes. In fact, it is possible that two Lorenz curves with different asymmetry profiles could perform the same \(\mathcal {G}\) index simply because they intersect each other at the same discriminant point \(D\left( p_{d},q_{d}\right) \), see Fig. 2. In this scenario, one cannot grasp the inequality arising from the disparity of concentration between poor and rich: indeed, according to the “asymmetry” criterion of Definition 1, inequality not only has an intensity, due to heterogeneity and concentration, but also a direction, measured by the asymmetry of the Lorenz curve that evaluates the difference between the groups’ concentrations: a positive asymmetry concerns the case where the rich group is more concentrated than the poor one; in the opposite case, asymmetry turns out being negative.

The asymmetry of the Lorenz curve

Figure 2 illustrates the case in which two Lorenz curves, i.e. two distributions, might have the same concentration while exhibiting opposite asymmetric shapes. In real-world economies, income distributions are unequal in the sense of Definition 1, which implies that a few receivers share a lot of income while a lot of receivers share a few of it.Footnote 12

Fig. 2
figure2

Two Lorenz curves with the same Gini index and opposite asymmetry with their characteristic points

To understand the imbalance, one can resort to the general asymmetry index as reported by Damgaard and Weiner (2000):

$$\begin{aligned} \mathcal {S}=\mathcal {F}(\mu )+\mathcal {L}(p_\mu )=p_\mu +q_\mu >0, \end{aligned}$$
(2)

where \(\mu \) is the expected value of \(\mathcal {W}\), \(p_\mu \) is the cumulative share of receivers below the expected value, and \(q_\mu \) is the cumulative share of income below the expected value. Since the distribution \(\mathcal {F}\) of the quantity \(\mathcal {W}\) on the (wp)-plane is equivalent to \(\mathcal {L}\) on the (pq)-plane, \(\mathcal {S}\) relates a distribution to its Lorenz curve, given that the axis of symmetry is \(\mathcal {F}(w)+\mathcal {L}(p_w)=1\) or, equivalently, \(q_w=1-p_w\). Geometrically, for a symmetric Lorenz curve the discriminant and the critical points coincide at the intersection of \(\mathcal {L}(p_w)=q_w\), with \(q_w=1-\mathcal {F}(p_w)\), such that the arcs \(\widetilde{OD}\) and \(\widetilde{DE}\) are congruent by reflection with respect to \(q=1-p\).

It is worth noting that in the case of perfect symmetry \(\mathcal {S}=1\) the discriminant and critical points coincide. This specific case does not imply equi-concentration; rather, it shows that all that matters in measuring inequality is the concentration index, because no distributional inequality in the poor and the rich side emerge—put differently, there is some degree of proportional heterogeneity in the two sides of the distribution. In all the other cases, more reasonable in real-word economies, the discriminant and the critical points do not coincide. Precisely, when \(\mathcal {S}>1\) the critical point C is above the axes of symmetry, hence above the discriminant point D (see Fig. 1 and the solid line in Fig. 2); conversely, in the case of \(\mathcal {S}<1\) the critical point C is below the axis of symmetry, that is below the discriminant point D (see the dashed line in Fig. 2).

Therefore, it becomes clear that asymmetry is more relevant to detect inequality than concentration, because it allows for distinguishing cases like those in Fig. 2, while concentration alone is suitable only in the case of perfect symmetry, which is a very peculiar case.

According to the discriminant point definition, \(D(p_d,q_d)\in \overline{M_0M_1}\) distinguishes the poor group, for which \(p\in [0;0.5]\) and \(q\in [0;0.5]\), from the rich one, for which \(p\in [0.5;1]\) and \(q\in [0.5;1]\). As discussed above, it always exists and is unique, hence it is the key point to introduce the minimal heterogeneity, which is the case when \(p_d\in [0.5;1]\) and \(q_d\in [0;0.5]\). The discriminant point is not only relevant for heterogeneity, but for asymmetry and concentration as well. Indeed, the ratios:

$$\begin{aligned} 0\le \left( R_d^p=\frac{q_d}{p_d}\right) \le \left( R_d^r=\frac{1-q_d}{1-p_d}\right) \le 1, \end{aligned}$$
(3)

which measure the average per-capita income in the poor and rich groups respectively, depends precisely on D. Hence, on average:

$$\begin{aligned} O_d=\frac{R_d^r}{R_d^p}\ge 1, \end{aligned}$$
(4)

evaluates the proportion of a rich per-capita income compared to a poor one.

Consider now Figs. 1 and 2, and remember that, by definition, \(p_d\ge 1-p_d \) and \(q_d\le 1-q_d\) such that \(p_d\in [0.5,1]\) and \(q_d\in [0,0.5]\). Let us then describe the following limit cases:

  1. Case (a)

    If \(p_d\approx 1-p_d\), then \(q_d\approx 1-q_d\): the shares of poor and rich receivers are almost balancing, and therefore the cumulative shares of income are almost equal; as a consequence, \(R_d^r\approx R_d^p=1\) and \(O_d\approx 1\), so that \(D\approx M_0\), which implies both absolute no-concentration and perfect symmetry, i.e. \(\mathcal {G}\approx 0\) and \(\mathcal {S}\approx 1\) together with \(D\approx C\). This specific case reveals the degenerate case \(\mathcal {W}\sim \overline{W}\), where all the receivers earn the same (average) income. More formally:

    $$\begin{aligned} \begin{aligned} \{p_d\rightarrow 0.5^+\}&\Leftrightarrow \{q_d\rightarrow 0.5^-\}\Rightarrow \\ \{R_d^r\approx R_d^p\Leftrightarrow O_d\approx 1\}&\Leftrightarrow \{\mathcal {G}\approx 0\wedge \mathcal {S}\approx 1\}\Rightarrow \mathcal {W}\sim \overline{W}. \end{aligned} \end{aligned}$$
    (5)

    This might be termed as the case of absolute equality.

  2. Case (b)

    The more \(p_d> 1-p_d\) the more \(q_d< 1-q_d\), hence \(R^p_d\rightarrow 0^+\wedge R^r_d\rightarrow +\infty \) such that \(O_d\rightarrow +\infty \), that is \(D\rightarrow M_1\), which implies both absolute concentration and extreme positive asymmetry: \(\mathcal {G}\approx 1\) and \(\mathcal {S}\gg 1\). In this second degenerate case \(\mathcal {W}\sim W\) hence there are no receivers but one individual that earns the whole (aggregate) income:

    $$\begin{aligned} \begin{aligned} \{p_d\rightarrow 1^-\}&\Leftrightarrow \{q_d\rightarrow 0^+\}\Rightarrow \\ \{R_d^r\rightarrow +\infty \wedge R_d^p\rightarrow 0^+&\Leftrightarrow O_d\rightarrow +\infty \}\Leftrightarrow \{\mathcal {G}\approx 1\wedge \mathcal {S}\gg 1\}\Rightarrow \mathcal {W}\sim W. \end{aligned} \end{aligned}$$
    (6)

    This might be termed as the case of absolute inequality.

  3. Case (c)

    Among the previous limited cases there is no finite set of configurations. Some may perform positive or negative asymmetric profiles consistently with different values of \(\mathcal {G}\) (say regular cases), but in others the same \(\mathcal {G}\) is consistent with opposite profiles of asymmetry (say ambiguous cases). The ambiguous cases are characterized by different Lorenz curves that share the same discriminant point D, with the same \(\mathcal {G}\) index but opposite values of \(\mathcal {S}\), which reveal opposite asymmetry profiles (see Fig. 2).

For opposite reasons, both the absolute equality and absolute inequality are degenerate cases. Case (a) reveals that all the receivers are one receiver, so we can consider this as a case of homogeneity, not to be confused with a uniform distribution configuration. Case (b) is a different kind of degenerate case: since only one receiver earns income, then this individual is the only one allowed to trade, unless we do not accept unrealistic levels of debt, and this individual can only transfer income by lending, purchasing something or donating portions of income to others; until a second individual earns some income, there would not be a meaningful distribution beyond an all or nothing configuration. Hence, this second case can also be considered a sort of homogeneity. Therefore, it is interesting to notice that both Case (a) and Case (b) are consistent with a so-called representative receiver framework, for which no distribution is meaningful since this receiver is either alone or equivalent to all the others. One, then, is enough to represent them all. Moreover, Case (c) noticeably posits that the discriminant point D is not enough to disambiguate the ambiguous cases, as well as \(\mathcal {G}\) is not enough to measure inequality, for which it takes considering the critical point C, too. Stated otherwise, the Gini index of concentration can be used as a meaningful measure of inequality only if we are sure about the negative/positive asymmetry of the Lorenz curve but, when comparing two distributions, we also have to be sure that the two Lorenz curves do not intersect. If they do, then \(\mathcal {G}\) is still not enough to appropriately measure inequality.

According to its definition, the critical point \(C(p_c,q_c)\) is a point on the \(\mathcal {L}\)-curve at the maximum distance from the equi-concentration segment \(\overline{OE}\). Both the regular and ambiguous sub-cases discussed in Case (c) define not finite sets of configurations. Moreover, it is not difficult to see that the intrinsic property of ambiguous cases is that the critical points of two intersecting Lorenz curves are equidistant from the \(q=1-p\) line and both belong to the same tangent line.

If \(p_c>p_d\), then C is above D and the \(\mathcal {L}\)-curve exhibits imbalance toward the higher income classes: this implies \(\mathcal {S}>1\), meaning that the rich group is more concentrated than the poor one. Hence, \(\mathcal {S}>1\)means income unfavorably distributes over the poor receivers, i.e the poor are more gradually poor while the rich are more increasingly rich and share the largest part of total income. This is represented in Fig. 1 and by the solid line in Fig. 2.

If \(p_c<p_d\), then C is below D and the \(\mathcal {L}\)-curve exhibits imbalance toward the lower income classes: this means \(\mathcal {S}<1\) and more concentration of the poor group than the rich one. Hence, \(\mathcal {S}<1\)means income favorably distributes over the poor receivers, i.e. the income unfavorably distributes over the rich side in the sense that the rich are more gradually rich while the poor are more decreasingly poor and share the smallest part of total income among a few. This is represented by the dashed line in Fig. 2.

Therefore, by means of the discriminant and critical points of the Lorenz curve we can cover all the notions described in the Definition 1 and, most relevantly, the notion of asymmetry. Still missing is a measure of distributional inequality that we will discuss in the next section.

The Zanardi index

This section introduces the Zanardi index to measure the distributional inequality of the quantity \(\mathcal {W}\) according to Definition 1. The Zanardi index is defined as follows:Footnote 13

$$\begin{aligned} \mathcal {Z}_d=\mathcal {Z}\left( K_d, \mathcal {G}^p,\mathcal {G}^r,\mathcal {G}\right) =2K_d\frac{\mathcal {G}^r-\mathcal {G}^p}{\mathcal {G}}\in [-1,+1]. \end{aligned}$$
(7)

The index varies between \(-\,1\) and \(+\,1\),Footnote 14 is centered at 0 and embeds the following information: (i) heterogeneity introduced by means of the discriminant point as \(K_d=(p_dq_d)/2\); (ii) concentration introduced with the Gini index \(\mathcal {G}\); (iii) asymmetry introduced via the parameter \(\delta \) which denotes the disparity of concentration between the rich and the poor, i.e. \(\delta =\mathcal {G}^r-\mathcal {G}^p\). The Zanardi index is thus able to account for all of the three concepts outlined in Definition 1.

Index (7) also satisfies certain desirable properties any inequality index should respect:Footnote 15

  1. 1.

    \(\mathcal {Z}_{d}\) is independent of any characteristic of individuals other than their income;

  2. 2.

    \(\mathcal {Z}_{d}\) does not change when all incomes change proportionally;Footnote 16

  3. 3.

    \(\mathcal {Z}_{d}\) is invariant to replications of the original population;

  4. 4.

    \(\mathcal {Z}_{d}\) changes when income transfers occur among individuals in the income distribution.Footnote 17

Bounded between \(-\,1\) and 1, the Zanardi index is also able to capture all three cases discussed in Sect. 2.1. Following is an interpretation of this index.

\(\mathcal {Z}_d=0\) happens only if \(\delta =0\), that is if the two groups are equivalently within concentrated (\(\mathcal {G}^r=\mathcal {G}^p\)). In this case there is no imbalance in the distribution because the Lorenz curve is perfectly symmetric (\(\mathcal {S}=1\)) and the two characteristic points coincide (\(D\equiv C\)).Footnote 18 This is consistent with the previous case Case (a) but it does not ensure absolute inequality because \(\mathcal {G}\) may be significantly different from 0. Therefore, if \(\mathcal {S}=1\Leftrightarrow \mathcal {Z}_d=0\) are statistically significant to hypothesis testing then all that matters is \(\mathcal {G}\). Hence, if two distributions have perfectly symmetrical Lorenz curves, then the one with higher concentration is more unequal, because it is more concentrated. Said another way, if the Zanardi index \(\mathcal {Z}_d\) is not significantly different from 0, equivalently to the fact that the asymmetry index \(\mathcal {S}\) of the Lorenz curve is not significantly different from 1, this is the only peculiar case for which the Gini index is an appropriate measure of distributional inequality. Indeed, in this case, neither heterogeneity (i.e. \(D(p_d,q_d)\equiv C(p_c,q_c)\)) nor asymmetry (\(\mathcal {S}=1\Leftrightarrow \mathcal {Z}_d=0\)) can be considered while concentration rules the scenario: the more the distribution is concentrated (\(\mathcal {G}>0\)) the more it is unequal.

Consider the case \(\mathcal {Z}_d\in (0;1]\). This happens when \(\delta >0\), hence the rich group is more within-concentrated than the poor one (\(\mathcal {G}^r>\mathcal {G}^p\)): in this case we have a positive imbalance in the distribution and some degree of inequality because the Lorenz curve is not symmetric (\(\mathcal {S}>1\)). The characteristic points do not coincide, i.e. C is above D and the peak of the expected value belongs to the rich side of the distribution. This kind of imbalance is therefore consistent with a quantity \(\mathcal {W}\) that favorably distributes among the rich and unfavorably among the poor, as is the case represented by the solid line in Fig. 2. It means that the many poor receivers share a small portion of the total income and they are less heterogeneous than the rich receivers. Therefore, in terms of (4), the more \(\mathcal {Z}_d\rightarrow 1^-\), the more \(O_d\rightarrow +\infty \), the more it will approach \(\mathcal {G}\approx 1\) with \(\mathcal {S}\gg 1\) as described in the previous Case (b) of absolute inequality.

We now consider the opposite case \(\mathcal {Z}_d\in [-1,0)\). This happens when \(\delta <0\), hence the poor group is more within-concentrated than the rich one (\(\mathcal {G}^r<\mathcal {G}^p\)). In this case we have a negative imbalance in the distribution and some degree of inequality because the Lorenz curve is still not symmetric (\(\mathcal {S}<1\)) and the characteristic points do not coincide: unlike before, D is now above C and the peak of the expected value belongs to the poor side of the distribution. In this case, the imbalance in the distribution of \(\mathcal {W}\) is in favor of the poor, and income unfavorably distributes among the rich as in the case represented by the dashed line in Fig. 2. It means that the many poor receivers share a lot of total income and they are more heterogeneous than the rich. Therefore, the more \(\mathcal {Z}_d\rightarrow -1^+\), the more \(O_d\approx 0\), the more \(\mathcal {G}\approx 0\) with \(\mathcal {S}\ll 1\).

Finally, being normalized upon \(\mathcal {G}\), \(\mathcal {Z}_d\) it is a suitable index to measure inequality not only in the regular sub-cases of Case (c) but also when two Lorenz curves intersect. Therefore, the Zanardi index is also able to disambiguate the ambiguous sub-cases of the previous Case (c) because, unlike the Gini index, it considers both the intensity and the direction of inequality, as discussed above.

Empirical estimates of inequality: Gini versus Zanardi

In this section we provide empirical estimates of the Gini and Zanardi indexes, considering a large set of economies during the years 1980–2014. Precisely, the source of income distribution data is the Luxembourg Income Study Database (LIS), where micro-data on household income are collected and organized on the basis of five-year waves up to 2000 (waves I to V) and on a three-year basis since 2004 (waves VI to IX). The high coverage both at the temporal and the geographical level allows us to cover 49 countries belonging to different continents: Europe, North America, Latin America, Africa, Asia, and Australia.

One of the main advantage of using the LIS is that it provides cross-national-survey data harmonization by combining the single surveys conducted at the country level. This procedure guarantees that income aggregates are uniformly defined across countries, so that data can be compared more reliably than data obtained from each country (Gottschalk and Smeeding 2000).

For the purposes of this paper, we focus on the income category defined in the LIS dataset as “disposable household income”, i.e. the sum of monetary and non-monetary income from several sources less the amount of income taxes and social contributions paid, which is usually the preferred measure for income distribution analysis (Canberra Group 2011).

Data on income are expressed in national currency units and “equivalized” by the square root of household size to account for the different composition and needs of the households.

Moreover, all households where data on disposable income are missing or exactly equal to zero are excluded from the analysis, and person-level adjusted weights (i.e. the product of household weights and the number of household members) are used to re-calibrate income indicators for the total population.

Finally, data have been bottom-coded at 1% of equivalized mean income and top-coded at 10 times the median of non-equivalized income.

Main results

The first element we consider is the relative heterogeneity of poor and rich income classes separated by the discriminant point \(D\left( p_{d},q_{d}\right) \). As shown in columns 2 to 4 of Table 1, the mean fraction of relatively poor income earners (\(p_{d}\)) amounts to 61.29% and accrues a share of total income averaging 38.71%, whereas the richest 38.71% (\(1-p_{d}\)) holds 61.29% (\(1-q_{d})\) of total income.

Table 1 Summary statistics for the Zanardi asymmetry index and related components

The ratio \(O_{d}\), measuring the divide between the rich and poor in terms of per-capita income, clearly indicates that income of the rich is on average 2.6 times that of the poor, thus confirming that the two-class partition of income distribution operated by the discriminant point has heterogeneous relative size of both the income and population shares (“heterogeneity” criterion of Definition 1).

In columns 5 and 6 of Table 1 we report the Gini indexes for the poor (\(\mathcal {G}^{p}\)) and the rich (\(\mathcal {G}^{r}\)), and the Gini coefficient for the overall population is found in column 7. On average, the concentration of income among the poor (\(\mathcal {G}^{p}=0.3987\)) is higher than the concentration among the rich (\(G^{r}=0.3887\)), and both are greater than the overall concentration (\(\mathcal {G}=0.3180\)). This is an obvious consequence of the fact that the poor receive a smaller proportion of total income than their population share, while the relatively smaller fraction of rich people gets the largest proportion (“concentration” criterion of Definition 1).

The Gini concentration index is clearly inadequate as a description of income inequality. In fact, it is possible that two Lorenz curves have the same \(\mathcal {G}\) but different asymmetry profiles, so as to intersect each other at \(D\left( p_{d},q_{d}\right) \). In this scenario, one cannot grasp the inequality arising from disparity of concentration between poor and rich: indeed, according to the “asymmetry” criterion of Definition 1, inequality has not only an intensity due to heterogeneity and concentration, but also a direction, measured by the asymmetry of the Lorenz curve, the sign of which is provided by the class divide \(\delta =\mathcal {G}^{r}-\mathcal {G}^{p}\). Clearly, the more \(\delta \rightarrow 0\), the more similar will be the concentration of the two groups. However, this does not imply that the distribution of income is not concentrated: in fact, as Fig. 3a shows, when \(\delta =0\) the Gini index \(\mathcal {G}\) displays different—in some cases very large—values of overall concentration.

Fig. 3
figure3

a Overall concentration (\(\mathcal {\delta ,G}\)), b odds ratio (\(O_d,\mathcal {G}\)) and c class divide (\(\delta ,O_d\)) estimated on the LIS database

Furthermore, in Fig. 3b \(\mathcal {G}\) increases with \(O_{d}\), meaning that overall concentration is higher (lower) where per-capita disparities are wide (small).Footnote 19 But this is not enough to characterize inequality: specific values of \(\delta \) are consistent with different levels of \(O_{d}\) as portrayed in Fig. 3c, hence the overall measures \(O_{d}\) and \(\mathcal {G}\) do not capture the inner differences in income distribution that allow an evaluation of both the intensity and direction of inequality.

As we argued in the previous section, the Zanardi index of asymmetry captures the most significant aspects of a Lorenz curve and thus is a valuable tool for measuring inequality according to Definition 1. Summary statistics that describe the distribution of the Zanardi index as estimated using the LIS income data are given in the last column of Table 1. Note that although \(\mathcal {Z}_{d}\) can vary between \(-\,1\) and 1, the minimum and maximum values for \(\left| \mathcal {Z}_{d}\right| \) are only slightly greater than 0.1. This required us to test the null hypothesis of symmetry (\(H_{0}: \mathcal {Z}_{d}=0\)) against the general alternative of asymmetry (\(H_{a}: \mathcal {Z}_{d}\ne 0\)) at the 5% level of significance.Footnote 20 For each LIS wave, Fig. 4 summarizes the frequency of data sets for which we were not able to reject the null hypothesis of symmetry and the number of cases for which the alternative of asymmetry can not be rejected.

Fig. 4
figure4

Number of LIS data sets with symmetric or asymmetric Lorenz curve: see the LIS List of Datasets at http://www.lisdatacenter.org/documentation/list-of-datasets/

Overall, it emerges that 222 out of 284 estimates of the Zanardi index are significantly different from zero, i.e. more than 78% of the samples analyzed have a Lorenz curve that is negatively asymmetric (because \(\mathcal {Z}_{d}<0\)) or positively asymmetric (because \(\mathcal {Z}_{d}>0\)).

The cases for which we find evidence of significant asymmetry in the Lorenz curve are scattered against the corresponding Gini estimates in Fig. 5.

Fig. 5
figure5

Overall Gini index against a the Zanardi asymmetry index, b the Gini index for the poor, and c the Gini index for the rich; estimates on the LIS database

The lack of correlation between \(\mathcal {G}\) and the Zanardi index of asymmetry is noteworthy and confirmed by a Pearson correlation coefficient value of 0.057.Footnote 21 In particular, Fig. 5a shows that cases of negative or positive asymmetry of the Lorenz curve can coexist with the same level of overall concentration. Conversely, panels (b) and (c) of the same figure display a positive correlation between \(\mathcal {G}\) and the concentration of income among the poor and the rich that is evidently appreciable—the Pearson correlation values are quite similar and equal to approximately 0.952.Footnote 22 This means that, by sampling only rich observations or just poor observations in the different samples, the \(\mathcal {G}\) index would not be able to capture any significant difference compared to the case of mixed and heterogeneous samples—i.e. \(\mathcal {G}\) is almost indistinguishable from \(\mathcal {G}^{p}\) and \(\mathcal {G}^{r}\). That is, for any given value of \(\mathcal {G}\), either if the sampling distributions were drawn only from the rich or only from the poor side of the income earners population, this would make no difference to the Gini coefficient. On the contrary, even in these extreme cases the Zanardi index would capture the intrinsic asymmetry profile of the sampling Lorenz curves, thus describing the intrinsic inequality of the sampling distributions wherever they were drawn from.

A look at the time-series evolution of inequality

As discussed in the previous section, the empirical analysis clearly suggests that the distribution of income is significantly asymmetric. Here we analyse the evolution of inequality over time for some of the countries under study.

Figure 6 shows the temporal evolution of the Gini and Zanardi indexes for those economies with more years/waves of consecutive data. For the sake of exposition, these economies are clustered into the following four groups: Anglo-Saxon countries (Canada, the United Kingdom and the United States); continental European countries (France and Germany); Southern European countries (Italy and Spain); Northern European countries (Denmark, Norway and Sweden).

Fig. 6
figure6figure6

a Temporal evolution of the Gini (bullets-left scale) and Zanardi indexes (squares-right scale), Anglo-Saxon countries; estimates on the LIS database. b Temporal evolution of the Gini (bullets-left scale) and Zanardi indexes (squares-right scale), continental European countries; estimates on the LIS database. c Temporal evolution of the Gini (bullets-left scale) and Zanardi indexes (squares-right scale), Southern European countries; estimates on the LIS database. d Temporal evolution of the Gini (bullets-left scale) and Zanardi indexes (squares-right scale), Northern European countries; estimates on the LIS database

There is, evidently, a misleading understanding of the inequality concept that would not be apparent if we were only focused on the overall Gini index. Indeed, by exploring the inequality status of the different groups of economies as measured by their Gini and Zanardi indexes, some striking differences emerge. Looking at Anglo-Saxon countries we found an increasing trend of the Gini concentration ratio in the United States, as one would expect. At the beginning of the period, the Zanardi index shows a strong negative asymmetry that tends to decrease during the years, with values of the index still negative but closer to zero in 2014. To correctly interpret the trend, one should bear in mind that moving from a strong negative asymmetry during the 1970s toward the case of a “perfectly symmetric” Lorenz curve in 2014 does not imply a less unequal distribution of income: as we have explained so far, in cases of symmetry, the direction of inequality is null but does not imply absence of intensity. In fact, in agreement with the well-known anecdote claiming that the top 3% of the U.S. population owns almost 95% of total income, the lower negative values achieved by the Zanardi index over the period point out that the position of the poor has worsened over the last decades. The higher negative asymmetry (or asymmetry to the left) of the Lorenz curve in the first period is the expression of a distribution in which a majority of relatively poor U.S. citizens shares a given quota of total income, while a few rich U.S. citizens possess the rest. In terms of Fig. 2, less negative values of the Zanardi index over the years implies a shift from a situation described by the dashed red line to a situation represented by the solid blue line. The mass below the poor side tends to decrease, moving in favour of the rich side, and mimicking in this way a situation in which the distribution is characterized by a large fraction of relatively poor U.S. citizens and a small fraction of rich U.S. citizens who share the largest part of total income. A similar tendency, but with lower values of concentration, is found for Canada.

The worst situation seems to be faced by the United Kingdom. The Gini index increases from the beginning up to the 2000s, with a slightly decreasing tendency over the last years, whereas a very different path is drawn by the Zanardi index. According to the latter, starting from higher levels of inequality in the 1970s, the United Kingdom faced a period of decreasing inequality, which reached a minimum value at the end of the 1970s. Thereafter, income inequality grew sharply during the 1980s and up to the onset of the global financial crisis, with a lower increasing trend since then and until 2014. Such analysis suggests that the gap between the very rich and everyone else continued growing after the early 1990s.

Conversely, continental European countries like France and Germany show a relatively stable level of concentration—the Gini index does not fluctuate so much, with values close to 0.25–0.30—while the series of the Zanardi index exhibit a positive imbalance of distribution. In this case, the total income of the two economies is unequally distributed between the poor and the rich, with a higher proportion of poor citizens sharing a small part of the total income and a restricted part of rich who share the biggest portion of it.

The case of Southern European countries (Italy and Spain) reveals another interesting pattern. The Gini index remains quite stable during the years, with values close to 0.30–0.35. On the other hand, the Zanardi index shows a fluctuating trend over the years, with a decreasing path starting in the 2000s. According to the interpretation thereof given in Sect. 2.1, this recent tendency reflects a relative imbalance of the distribution that favoured the poor over the rich, in the sense that the rich became more gradually rich while the poor were more decreasingly poor and shared the smallest part of total income among a few. However, this does not exclude that other intra-distributional movements might have happened within the group of the “poor” with direct implications for welfare analysis, like for instance a growing percentage of people below a pre-specified poverty line.

Finally, even in countries that used to enjoy relatively low levels of income inequality (like the Nordic countries) there has been over the years a substantial increase in the gap between rich and poor. Countries like Denmark and Sweden have recently recorded less negative values of the Zanardi index, which has been increasing over time since the 1990s, from strongly negative values of the index to values closer to zero, revealing a disadvantage for the poor side of the distribution.

To summarize, our findings suggest the existence of a distinct Anglo-Saxon pattern—where the increase in overall concentration is accompanied by a decreasing negative asymmetry of the Lorenz curve that increases concentration in favour of the rich and disadvantages the poor—as opposed to continental Europe—where the level of concentration and the positive asymmetry of the Lorenz curve hardly changed during the past 30 years or so, favouring the rich over the poor.Footnote 23 The Northern European countries tend be somewhere inbetween the European pattern and the Anglo-Saxon pattern, with stable (or slightly increasing) levels of concentration and less negative values of the Zanardi index over the years. Finally, the results obtained for Southern European countries (Italy and Spain) shows that overall income concentration has not changed significantly from the second half of the 1990s—thus evolving closer to the trends observed in continental Europe—while the Zanardi index has displayed a W-shaped dynamics by alternating periods with \(\mathcal {Z}_{d}<0\) and periods with \(\mathcal {Z}_{d}>0\), i.e. periods more in favor of the poor and others that favour the rich.

Concluding remarks

The overemphasis on the Gini coefficient as the measure of income inequality follows from an axiomatic logic of interpretation. Contrary to the logic traced by Gini, stating that “the more it is unequal, the more it is concentrated”, the widespread alternative logic misleadingly claims that “the more it is concentrated, the more it is unequal”. Beyond any sort of axiomatic approach, this paper has proven that the Gini is only able to grasp concentration, but it does not capture other key aspects of inequality, such as the degree of heterogeneity and asymmetry embodied in the income distribution assumed as a propaedeutic representative distribution of any other transferable quantity. For a more unambiguous idea of income inequality, fundamental complementary information on income distribution can be determined by means of the Zanardi asymmetry index of the Lorenz curve. The Zanardi index is particularly suited for measuring inequality since it captures the most significant aspects of the Lorenz curve as the dual representation of a distribution, and thus it provides a useful and sound measure of inequality beyond the commonly employed and misleadingly interpreted Gini index.

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