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Lorenz map, inequality ordering and curves based on multidimensional rearrangements
\address{University of Washington, Penn State, University of Alberta and University of Washington}
The Lorenz curve, first proposed in lorenz:1905, is a compelling visual and simple quantification tool for the analysis of dispersion in univariate distributions. It allows easy visualization of dispersion from the curvature of a convex curve and its distance from the diagonal. The diagonal itself is the Lorenz curve of a degenerate distribution-- an egalitarian allocation where all individuals have the same amount of resource. It also enables quick computations, reading off the curve, as it were, of the share of a resource held by the top or bottom of the allocation distribution for that resource. These features of the Lorenz curve account for much of its enduring appeal among practitioners, policy analysts and policy makers. This appeal is further enhanced by the relation between majorization and the pointwise ordering of Lorenz curves, which provides a way to visualize inequality comparisons between populations and within a given population between time periods. Comprehensive accounts are given in marshall:1979 and arnold:2018:book.
The appealing properties of the Lorenz curve are well captured by the formulation given in gastwirth:1971. In that formulation, the Lorenz curve is the graph of the Lorenz map, and the latter is the cumulative share of individuals below a given rank in the distribution, i.e., the normalized integral of the quantile function. The relation to majorization and the convex order follows immediately, as shown in section C of marshall:1979. As pointed out by arnold:110, this makes the Lorenz ordering an uncontroversial partial inequality ordering of univariate distributions, and most open questions concern the higher dimensional case.
Dispersion in multivariate distributions is not adequately described by the Lorenz curve of each marginal, and a genuinely multidimensional approach is needed. Even for utilitarian welfare inequality, AB:1982 motivate the need for the multidimensional approach initiated by fisher:1956. More generally, the literature on multidimensional inequality of outcomes and its measurement is vast, as evidenced by many recent surveys, see for instance DL:2012, AB:2014, AZ:2020. We only discuss it insofar as it relates to the Lorenz curve.
Multivariate extensions have been proposed for the Lorenz curve, most notably Taguchi:72a (Taguchi:72a, Taguchi:72b), arnold:1983, and koshevoy:1996 (koshevoy:1996, koshevoy:1999)\footnote{More recently, subsequent to our work, HM:2022 also adopt a multivariate rearrangement approach to the definition of multi-attribute Lorenz curves. They adopt a center-outward approach (see HBCM:2021), which is better suited to define notions of middle class.}. They are reviewed in marshall:1979 and SJ:2014 and discussed in more details in section (ref), where we compare them to our proposal. We contribute to this literature with a vector version of the gastwirth:1971 formulation of the Lorenz curve. We provide an implementable criterion to measure and compare inequality in multivariate distributions, which emulates the features of the Lorenz curve that most contributed to its success.
The traditional gastwirth:1971 formulation of the Lorenz curve is an integrated quantile over the lowest ranked individuals. To simplify the argument in the univariate case, model the population as a continuum on $[0,1]$ and suppose the distribution of incomes in the population is continuous. Then the gastwirth:1971 formulation can be thought of involving two stages. Take an income allocation $Y$, which is a random variable on $\mathbb R_+$ with cumulative distribution function $F_Y$. First, reorder individuals in the population so that they are ranked in increasing incomes. Then compute the cumulative share of lowest ranked individuals by integrating $F_Y^{-1}$ from $0$ to $r$ and dividing by the mean. The first step involves the probability integral transform $F_Y(Y)$, which should be thought of in this context as a cardinal to ordinal transformation, since $F_Y(Y)$ is uniform on $[0,1]$, so cardinal information is purged, but $F_Y$ is increasing, so ordinal information is preserved. Our proposal is based on a multivariate version of the cardinal to ordinal transformation involved in the first stage. The latter is the unique map that transforms a $d$ dimensional allocation into a uniform one on $[0,1]^d$, and is cyclically monotone\footnote{Existence and uniqueness are shown in mccann:1995. See section (ref) for details and definitions.} and hence preserves ordinal information. This motivates our definition of the vector Lorenz map as the cumulative integral of the multivariate quantile of CGHH:2017.
The vector Lorenz map we propose, therefore, is the vector of shares of each resource held by individuals below a given rank. The associated Lorenz inequality dominance criterion deems a multivariate allocation more equal if this share of resources is larger for each rank. Hence, our proposal shares the interpretation of the traditional Lorenz curve and Lorenz dominance. It also shares the desirable properties of the Lorenz curve and dominance ordering. Like the Lorenz zonoid of koshevoy:1996 (koshevoy:1996, koshevoy:1999), it characterizes the distribution of an allocation (see section (ref) for a definition and discussion). Unlike the Lorenz zonoid, the vector Lorenz map we propose can be efficiently computed as an unconstrained convex optimization problem and connected to recent developments in computational optimal transport theory\footnote{An account of recent advances is given in peyre:2018.}. Hence, the Lorenz dominance order we propose is an implementable inequality dominance criterion. Using recent advances on the asymptotic properties of multivariate quantiles, surveyed in Hallin:2022, our Lorenz dominance criterion can be the basis for inequality dominance testing that accounts for sampling uncertainty. This contrasts our proposal with the growing literature on multivariate inequality dominance criteria proposed for finite populations. See for instance GM:2012, banerjee:2016, FG:2021 and references within.
Other implementable inequality dominance criteria are proposed in the literature, in koshevoy:1995, koshevoy:1996, koshevoy:2007 and banerjee:2016 and other references surveyed in arnold:2018:book. However, they do not provide an equivalence between the Lorenz dominance criterion and a class of compatible social evaluation functionals. An exception is GM:2012 and FG:2021 who give a comprehensive treatment of the special case of a finite population with a single cardinal transferable attribute combined with an ordinal non transferable one. We characterize the class of social evaluation functionals that are inequality averse in the sense that they are increasing in the Lorenz dominance order. We build on the multivariate extension of the Quiggin:92-Yaari:87 rank dependent decision theory in GH:2012 to show that, as in weymark:1981 for the univariate case, social evaluation functionals are inequality averse if and only if they are rank dependent social evaluation functionals with attribute specific weights decreasing in ranks. We also characterize the class of transfers that increase inequality according to the Lorenz dominance criterion as rank preserving transfers of any attribute from a lower to a higher ranked individual. A special case of such transfers, which we call {\em monotone regressive transfers} weakly increase marginal inequality and dependence between attributes.
To visualize Lorenz dominance, we define an {\em Inverse Lorenz Function} at a given vector of resource shares as the fraction of the population that cumulatively holds those shares. It is characterized by the cumulative distribution function of the image of a uniform random vector by the Lorenz map. Hence, it is a cumulative distribution function by construction, like the univariate inverse Lorenz curve. In two dimensions, the $\alpha$-level sets of this cumulative distribution function, which we call $\alpha$-Lorenz curves, are non crossing downward sloping curves that shift to the south-west when inequality increases, as defined by the Lorenz ordering. For the cases, where allocations are not ranked in the Lorenz inequality dominance ordering, we propose a family of multivariate S-Gini coefficients based on our vector Lorenz map, with the flexibility to entertain different tastes for inequality in different dimensions. Finally, we propose an illustration to the analysis of income-wealth inequality in the United States between 1989 and 2022.
In the first section, we define the Lorenz map, explain its computation, detail its properties and how it compares with alternative proposals. In section (ref), we introduce the Lorenz dominance ordering, its characterization in terms of classes of social evaluation functionals and in terms of transfers compatible with it. Section (ref) illustrates the implementation of our proposed tools, and the final section concludes.
The Lorenz curve was originally proposed in lorenz:1905 to provide a graphical representation of inequality of distribution of a single resource. Let $Y$ be a random variable on $\mathbb R_+$ with cumulative distribution function $F_Y$, which represents the allocation of a resource in a population. The population is modeled as the continuum $[0,1]$.
The Lorenz curve is traditionally defined as the set of points in $[0,1]^2$, parameterized by $y$, with coordinates
where $\mu_Y$ is the expectation of $Y$. See for instance page 149 of arnold:2018:book. gastwirth:1971 points out that the Lorenz curve is given by the graph of the map on $[0,1]$
where $F^{-1}(v):=\inf\{y: v\leq F_Y(y)\}$ is the traditional quantile function. Formulation ((ref)) provides a closed form expression and simple interpretation: For each proportion $q\in[0,1]$, the Lorenz map gives the cumulative share of the resource held by the poorest proportion $q$ of the population. This relies on the well known fact that the quantile function $F_Y^{-1}$ is the only increasing function such that for any uniformly distributed random variable $V$ on $[0,1]$, $F_Y^{-1}(V)$ is distributed identically to $Y$. Hence, integrating $F_Y^{-1}$ from $0$ to $q$ and normalizing produces the cumulative share held by the individuals ranked below $q$.
Conversely, when $F_Y$ admits a density, the probability integral transform $V:=F_Y(Y)$ produces a uniformly distributed random variable $V$ on $[0,1]$, which preserves the ranks of individuals in the population. This holds because $F_Y$ is an increasing map. Hence, the probability integral transform removes cardinal information (by producing a uniformly distributed outcome), while preserving ordinal information (by keeping the rank order of individuals in the population). The probability integral transform $V:=F_Y(Y)$ is the rank associated with allocation $Y$.
If $F_Y$ is not continuous, then $F_Y(Y)$ is no longer uniformly distributed (positive masses of individuals have identical ranks). However, it is still the case that for any uniformly distributed random variable $V$ on $[0,1]$, $F_Y^{-1}(V)=\inf\{y: V\leq F_Y(y)\}$ is distributed identically to $Y$. Hence, the closed form solution for the Lorenz map ((ref)) still holds with the same interpretation: Integrating $F_Y^{-1}$ from $0$ to $q$ and normalizing still produces the cumulative share held by the individuals ranked below $q$.
Consider now an allocation $X:=(X_1,\ldots,X_d)$ of $d$ resources in the population. To analyze inequality in allocation $X$, we can first look at inequality in each marginal allocation $X_1,\ldots,X_d$, using the univariate Lorenz curves $L_1:=L_{X_1},\ldots,L_d:=L_{X_d}$. However, this strategy disregards the effect of dependence. The latter is relevant to inequality, as can be trivially illustrated by the fact that for given wealth and income marginal allocations, the comonotonic allocation (the wealthier individuals have higher income) is more unequal than the admittedly unrealistic counter-monotonic allocation (the wealthier individuals have lower income).
To take dependence into account, we propose to emulate the gastwirth:1971 formulation by measuring cumulative shares of each resource for all individuals, {\em below a certain rank}. Conceptually, this is achieved in two steps. First, we find a transformation that removes cardinal information while preserving individual's ranking in the population, i.e., a cardinal to ordinal transformation. Then we integrate the shares of individuals with lowest rank. The difficulty here, of course, is the absence of a canonical order in $\mathbb R^d$ to define the rank.
As noted in FR:2017, by Borel's Isomorphism Theorem\footnote{See for instance section 13.1 page 487 of Dudley:2002.}, there exist measurable bijective maps $T:[0,1]\rightarrow\mathbb R^d$ such that for any uniformly distributed random variable $V$ on $[0,1]$, $T(V)$ is distributed identically to $X$. However, such maps are unsuitable cardinal to ordinal transformations for two main reasons. First, there is no known explicit construction, hence no way to compute them. Second, even if we could compute such a map, its choice would imply an implicit ad hoc aggregation of the different resources in allocation $X$ in order to arrive at a scalar ranking of individuals in the population.
In order to avoid an implicit ad hoc aggregation of the different resources in $X$, the cardinal to ordinal transformation must be between $\mathbb R^d$ and $[0,1]^d$. Hence, we model the population as a continuum on $[0,1]^d$ and individual ranks are points in $[0,1]^d$. The multivariate quantile transform, and its inverse (the cardinal to ordinal transform, or rank transform), must satisfy the same requirements as in the univariate case: It must map the uniform distribution (no cardinal information) to the distribution of the allocation, and it must be monotonic (so as to preserve ordinal information). The monotonicity of the quantile in the univariate case ensures that the cardinal to ordinal transformation does indeed preserve the rankings of individuals in the population.
To construct an analogue of the gastwirth:1971 Lorenz curve formulation, we therefore need the cardinal to ordinal transformation to satisfy a form of multivariate monotonicity. The classical notion of monotonicity in $\mathbb R^d$, also known as $2$-monotonicity of a map $T:\mathbb R^d\rightarrow\mathbb R^d$, requires
for any pair of vectors $x,x^\prime\in\mathbb R^d$. It can be interpreted as monotonicity on average. For uniqueness of the cardinal to ordinal transformation, we need the stronger version of monotonicity, called {\em cyclical monotonicity}, which characterizes the gradients of convex functions and was introduced by Rockafellar:66. Cyclical monotonicity requires
for any $K$, and any collection of vectors $(x_1,\ldots,x_K)$, setting $x_{K+1}=x_1$. Cyclical monotonicity also characterizes maps $T$ that minimize distortion in the sense that in case the allocation $X$ has continuous distribution with finite variance, $T$ minimizes $\mathbb E\| X-T(X)\|^2$ among all the maps such that $T(X)$ is uniformly distributed on $[0,1]^d$.
The following definition summarizes the properties needed for a cardinal to ordinal transformation as a first step in the Lorenz map construction.
As shown in mccann:1995, there exists a transformation that conforms with definition (ref) and it is unique in the sense that two such transformations are equal almost everywhere. It is proposed as a vector quantile notion in CGHH:2017, and we will refer to it as the vector quantile associated with $X$.
Once we model the population as a continuum on $[0,1]^d$, interpret each point on $[0,1]^d$ as a rank, and define the vector quantile $Q_X$ as a multidimensional rearrangement of the allocation $X$ in rank order, we simply integrate the quantile over the lowest ranks to define a multivariate version of the gastwirth:1971 formulation of the Lorenz curve.
The transformation of $X$ into its normalized version $\tilde X$ prior to integrating the vector quantile is required to remove dependence of the Lorenz map of definition (ref) on units of measurements. Different resources, such as earnings and health, may not be measured with the same units of measurement. The transformation into $\tilde X$ makes the allocation unit free. Hence the Lorenz map satisfies ratio-scale invariance (i.e., invariance to rescaling of the different attributes, or change of units of measurement). Section (ref) discusses an alternative unnormalized version of the definition in the spirit of Shorrocks:83.
When $X$ has absolutely continuous distribution $P_X$, its quantile function is $P_X$-almost everywhere invertible (see for instance theorem 2.1 in CGHH:2017). In that case, the transformation $U=Q_X^{-1}(X)$ is the vector analogue of the probability integral transform $V=F_Y(Y)$ discussed above. The random vector $U=Q_X^{-1}(X)$ is uniformly distributed on $[0,1]^d$, and is the vector rank of the individual with endowment $X$, in the terminology of CGHH:2017. The Lorenz map of definition (ref) can then be rewritten as:
This clarifies the interpretation of $\mathcal{L}_X(r)$ as the cumulative share of all individuals with vector rank below $r$ in the partial order of $\mathbb R^d$.
In the scalar case discussed above, inverting the Lorenz curve $L_Y$ defined in ((ref)) yields the inverse Lorenz curve
where the probability is taken with respect to a uniformly distributed random variable $V$ on $[0,1]$. The scalar inverse Lorenz curve at $y$ is therefore shown in ((ref)) to be equal to the maximum proportion of the population with cumulative share of the resource equal to $y$. In the vector case, the analogue of the right-hand side of ((ref)) can still be used to define an Inverse Lorenz Function.
The expression above is no longer the mathematical inverse of the Lorenz map $\mathcal{L}_X$, but it can still be interpreted as the share of the population with cumulative shares of all resources equal to a predetermined proportion $z=(z_1,\ldots,z_d)$.
We now give a step-by-step method to compute the Lorenz map of a discrete distribution, which may be the allocation in a finite population, or the empirical distribution of a (possibly weighted) sample from an underlying (possibly mixed discrete-continuous) distribution. The full algorithm and a step-by-step guide to implementation in R are given in appendix (ref).
Let $X$ be a random vector in $\mathbb R_+^d$ with discrete distribution. The probability mass function of the distribution of $X$ is given by $\{(x^1,w_1),\ldots,(x^n,w_n)\}$, where $x^1,\ldots,x^n$ are vectors in $\mathbb R_+^d$ and $w_1,\ldots,w_n$ are positive scalar weights summing to $1$.
To illustrate the definition and the computation of the Lorenz map, we now explore examples of specific allocations and compute the corresponding Lorenz maps. First, we illustrate the computation of the Lorenz map for a discrete allocation.
Next, we consider the special case, where all individuals are endowed with the same quantity of resources.
We also check that our definition is compatible with scalar definitions when all resources are independently distributed.
Next, we derive the Lorenz map in the case of allocations $X=(X_1,X_2)$ with the same components, i.e., $X_1=X_2$ almost surely. We stick to $d=2$ for notational simplicity.
Let $X$ be the original allocation. Normalizing $X$ into $\tilde X$ as in definition (ref) by dividing each component by its mean, removes any sensitivity to (changes in) units of measurements. It is a standard approach to achieve ratio-scale invariance. See for instance banerjee:2010 (banerjee:2010, banerjee:2016). However, by construction, it comes with the disadvantage of removing scale effects. In the univariate case, Shorrocks:83 proposes to eschew normalization in order to take scale effects into account in the measurement of inequality.
When units of measurement are not a concern, an alternative definition without the feature described above is defined as
where $Q_X$ is the vector quantile of the original allocation $X$ (not the normalized one). This alternative version also allows the weighting of different resources according to a priori importance to overall inequality. Call $X^\epsilon=(\lambda_1(\epsilon)X_1,\ldots,\lambda_d(\epsilon)X_d)$ the suitably rescaled version of the initial allocation $X$. Define the sequence of weights $(\lambda_1(\epsilon),\ldots,\lambda_d(\epsilon))$ in such a way that $\lambda_{k+1}(\epsilon)/\lambda_k(\epsilon)\rightarrow0$ as $\epsilon\rightarrow0$. In this way, resources are ordered in decreasing importance to inequality, and we can entertain the extreme lexicographic case, where $\epsilon\rightarrow0$.
It follows from CGS:2010 that, when $X$ has an absolutely continuous distribution, as $\epsilon$ tends to $0$, the alternative Lorenz map tends to the map
where $Q_X^{KR}$ is the inverse of the Knothe-Rosenblatt transform $T^{KR}:\mathbb R^d\mapsto[0,1]^d$ of the original allocation $X$ proposed by Rosenblatt:52 and Knothe:57, and defined as follows
The Knothe-Rosenblatt quantile map is the only multivariate quantile map from the uniform on $[0,1]^d$ to $\mathbb R^d$ proposed in the literature other than optimal transport based vector quantiles as in definition (ref). The result above shows that the Knothe-Rosenblatt quantile is not a good alternative to vector quantiles of definition (ref) to base an integrated quantile definition for the Lorenz map, since it relies on an a priori lexicographic ordering of the different resources in the allocation.
In this section, we detail previous proposals for multivariate extensions of the Lorenz curve and list the properties that distinguish our proposal from the former.
Until now, the development of multivariate extensions of the Lorenz curve was hampered by the lack of simple multivariate analogues of ranks and quantiles. Early proposals for bivariate extensions of the Lorenz curve in Taguchi:72a (Taguchi:72a,Taguchi:72b) and arnold:1983 (arnold:1983,arnold:1987) are based on a direct ad-hoc extension of the traditional formula given in ((ref)). Let $(x_1,x_2)\mapsto F(x_1,x_2)$ be the CDF of a bivariate allocation with density $f$ and mean $(\mu_1,\mu_2)$. Taguchi:72a (Taguchi:72a,Taguchi:72b) proposes the bivariate Lorenz surface $L:[0,1]^2\rightarrow[0,1]$ defined implicitly by $(s,t,L(s,t)):=$
In order to treat both dimensions of the allocation symmetrically, arnold:1983 (arnold:1983,arnold:1987) proposes the alternative Lorenz surface parameterized by $(x_1,x_2)$ as the set of points
where $F_1$ and $F_2$ are the marginal CDFs associated with $F$, and $\mu_{12}$ is the expectation of the product $X_1X_2$. A closed form solution, given in SJ:2014, makes the Lorenz surface ((ref)) amenable to parameterization and statistical analysis. However, it does not share the interpretation or any of the properties of the univariate Lorenz curve.
A more successful proposal in that respect, is the Lorenz zonoid of koshevoy:1996. Again, take ((ref)) in the univariate case as the point of departure. It associates a fraction $p$ of the population to the share of the resource collectively held by the poorest fraction $p$ of the population. koshevoy:1996 eschew the need to order the population by associating with a fraction $p$ of the population the share of resources held by any group of individuals making up a fraction $p$ of the population, poor, rich, or mixed. The lower bound is the share held by the poorest individuals (the traditional Lorenz curve), and the upper bound is the share held by the richest individuals (a reverse Lorenz curve). The Lorenz zonoid is defined in koshevoy:1996 as the collection of all such shares for each fraction of the population. It is a convex region in $[0,1]^2$ bounded below by the Lorenz curve and above by the reverse Lorenz curve. More precisely, the Lorenz zonoid is defined as the set of points
where the function $\phi$ ranges over the set $\Phi$ of measurable functions from $\mathbb R_+$ to $[0,1]$. The lower (resp. upper) bound is obtained with the collection of functions $\phi(v):=\mathds 1\{v\leq y\}$ (resp. $\mathds 1\{v> y\}$), $y\in\mathbb R_+$.
Since the definition of the Lorenz zonoid does not rely on ranks or quantiles, the extension to higher dimensions is straightforward. Let $\Phi$ now be the set of measurable functions from $\mathbb R_+^d$ to $[0,1]$ and $X=(X_1,\ldots,X_d)$ be a multivariate allocation with CDF $F$ and mean $(\mu_1,\ldots,\mu_d)$. The Lorenz zonoid of koshevoy:1996 is defined as the set of points $L(X):=$
The Lorenz zonoid is an American football-shaped region in $[0,1]^{d+1}$ with poles at points $(0,\ldots,0)$ and $(1,\ldots,1)$. The Lorenz surface of Taguchi:72a (Taguchi:72a,Taguchi:72b) is a subset of the Lorenz zonoid, obtained when $\Phi$ is restricted to the set of functions $\phi_x(\cdot):=\mathds{1}\{\cdot\leq x\}$, all $x\in\mathbb R^2$. A function $\phi\in\Phi$ defines a point in the zonoid. The interpretation is simple in the case of indicator functions. The latter pick out specific groups of individuals in the population and the corresponding point in the zonoid has first coordinate equal to the fraction of the population involved. The other coordinates are the shares of each of the resources held by this group of individuals.
Definition 4.1 of banerjee:2016 proposes a multivariate inequality ordering in the finite population case. The analogue multivariate Lorenz map in the general case of a (possibly mixed discrete continuous) vector allocation $X$ with normalized version $\tilde X$ can be defined\footnote{We thank Xiaoxia Shi for bringing this to our attention.} for all $r=(r_1,\ldots,r_d)\in[0,1]^d$ by
where, for each $j=1,\ldots,d$, and $Q^{B,j}$ is the quantile function associated with the random variable $\Sigma_{k=1}^d (\tilde X_j+\tilde X_k)/2d$. If the latter were replaced by $\tilde X_j$, the Lorenz map would be the vector of marginal Lorenz curves. Mixing with the average across allocation introduces sensitivity to dependence between the marginal allocations. Note however, that $Q^B:=(Q^{B,j})_{j=1}^d$ is not a valid vector quantile for $\tilde X$, since $Q^B(U)$ is not distributed like $\tilde X$ when $U$ is uniform on $[0,1]^d$. As a result, $L^B(r)$ is not a vector of resource shares, as is the case for the univariate Lorenz curve.
The proposed multivariate extensions of the Lorenz curve in both Taguchi:72a (Taguchi:72a,Taguchi:72b) and koshevoy:1996 relate population proportions to a vector of resource shares. Our proposal differs substantially from these in that it directly relates a specific subset of the population, namely individuals with multivariate rank below $r$ to their share of both resources. Beyond this major conceptual difference, we now investigate properties of our multivariate extension of the Lorenz curve that make it a valuable contribution.
We can use the vector Lorenz map $\mathcal L_X$ of an allocation $X$ introduced in section (ref) as a tool to compare inequality of different allocations. We base an inequality dominance criterion to compare different allocations on the dominance of Lorenz maps. We develop a visualization tool for inequality dominance, and an inequality index for the cases, where the allocations are not Lorenz ordered.
Consider two allocations $X$ and $X^\prime$, with respective Lorenz maps $\mathcal{L}_X$ and $\mathcal{L}_{X^\prime}$. If $\mathcal{L}_X(r)\geq \mathcal{L}_{X^\prime}(r)$ for some vector rank $r$, the same proportion of the population with vector ranks below $r$ commands a larger share of all resources in allocation $X$ than in allocation $X^\prime$. If this is true for any vector rank $r$ in $[0,1]^d$, then, we say that allocation $X^\prime$ is more unequal than allocation $X$.
The Lorenz partial order of definition (ref) is an implementable dominance criterion: The Lorenz maps can be computed and compared. The relation $X\succcurlyeq_{\mathcal{L}} X^\prime$ is equivalent to stochastic dominance of the random vector $\mathcal{L}_X(U)$, with $U\sim U[0,1]^d$, over $\mathcal{L}_{X^\prime}(U)$ (see Section 3.8 of MS:2002). Hence, dominance tests can be derived on the basis of sample analogues of the Lorenz maps to emulate the large literature on inference techniques to compare inequality of distributions of a single attribute. See DD:2000 and references within.
Following the literature on the measurement of inequality, we assess the value of this implementable dominance criterion for inequality comparisons in two ways. First, we analyze the class of social evaluation functionals that are compatible with the Lorenz order, and show that they are rank-dependent social evaluation functionals, with weights decreasing in rank. Second, we identify the class of transfers that increase inequality as defined by this Lorenz criterion.
The first way to gain insight into the relevance of our multivariate Lorenz dominance criterion is to characterize the set of social evaluation functionals that are compatible with it. A social evaluation functional is a map $S$ from an allocation $X$, i.e., a random vector in $\mathbb R_+^d$, to $\mathbb R$, which orders allocations in their social desirability. A social evaluation functional $S$ is compatible with the dominance criterion if $X\succcurlyeq _{\mathcal{L}}X^\prime\Rightarrow S(X)\geq S(X^\prime)$. Compatibility with Lorenz dominance is a form of inequality aversion of the social evaluation functional, since more equal allocations are deemed socially more desirable.
By construction, a social evaluation functional that is compatible with the Lorenz dominance order must satisfy anonymity and ratio-scale invariance. Anonymity, also called law-invariance or symmetry in the literature, refers to the fact that $S(X)=S(X^\prime)$ whenever $X$ and $X^\prime$ are identically distributed. The identity of individuals does not matter in the social evaluation, so that a permutation of individuals in the population leaves $S$ unchanged. Ratio-scale invariance refers to the fact that $S(\lambda^\top X)=S(X)$ for any positive vector $\lambda$. Hence, the social evaluation is not affected by a change in units of measurement.
Next, and more substantively, all social evaluation functionals that are compatible with the Lorenz dominance criterion are rank-dependent social evaluation functionals. Individuals are weighted in the social evaluation according to their rank in the distribution. To define and formalize this statement, start with the case of a single attribute. weymark:1981 shows that social evaluations that satisfy the comonotonic independence property defined below take the form of weighted sums of quantiles.
{\bf Property CI} (Comonotonic Independence). A social evaluation functional $S$ is said to satisfy comonotonic independence if, whenever $X$, $X^\prime$ and $Z$ are comonotonic allocations, and $S(X) \geq S(X^\prime)$, then, for all $\mu\in(0,1)$, $S(\mu X+(1-\mu)Z) \geq S(\mu X^\prime+(1-\mu)Z)$.
In the univariate case, two allocations are called comonotonic if one is a positive increasing function of the other. In other words, individuals are ranked identically in both allocations. Comonotonic independence means that the comparison of two allocations with a common component only depends on the comparison between the two variable components, as long as rankings stay unchanged. As an illustration, when assessing the effect on household income distributions of a policy that only affects women, under perfect assortative matching, one need only look at the change in the distribution of women's income.
The same property of comonotonic independence can be entertained in the multi-attribute case, with the same interpretation. Two allocations are comonotonic if individuals are ranked identically in both allocations. Now, in case of random vectors $X$ and $X^\prime$, comonotonicity is defined in the same way by the fact that $X$ and $X^\prime$ have the same vector ranks. Definition (ref) below follows GH:2012 and EGH:2012, where it is called $\mu$-comonotonicity\footnote{See PS:2010 for a discussion of this and other multivariate comonotonicity concepts.} of $\tilde X^1,\ldots,\tilde X^J$.
With this definition of comonotonicity (which coincides with the usual definition in the single attribute case), comonotonic independence of a social evaluation functional is still defined as property CI. If individuals are ranked identically in allocations $X$ and $X^\prime$, and $X^\prime$ is socially less desirable than $X$, then, adding to both $X$ and $X^\prime$ a third common allocation $Z$ cannot reverse the ordering, if $Z$ ranks individuals as $X$ and $X^\prime$ do.
As in weymark:1981 for the single attribute case, GH:2012 show that comonotonic additive social evaluation functionals are rank dependent, i.e., of the form
for some function $\phi:[0,1]^d\rightarrow\mathbb R_+^d$. To each vector rank $u$, $\phi$ associates the attribute-specific weights of ranked $u$ individual in the social evaluation. We show that social evaluation functionals are only compatible with the Lorenz dominance order if they satisfy comonotonic additivity, hence if they are rank-dependent social evaluation functionals. There remains to determine which functions $\phi$ make social evaluation functional $S_\phi$ compatible with the Lorenz dominance criterion. As we discuss below, they are characterized by inequality aversion.
Inequality aversion of a rank dependent social evaluation functional implies a weighting scheme that gives more weight to lower ranked individuals. In the scalar case analyzed in weymark:1981, an inequality averse rank dependent social evaluation functional is characterized by decreasing weights as ranks increase. We show a similar result in the multivariate case. Social evaluation functionals that are compatible with the Lorenz order of definition (ref) are rank dependent social evaluation functionals with rank-specific weights of the form
where $m_j$ is a non negative measure on $[0,1]^d$, all $j\leq d$.
A special case of weighting scheme satisfying proposition (ref) is the case $m_j=\delta_r$ all $j$, where all individuals below rank $r$ receive weight $1$ and all individuals above rank $r$ receive weight $0$. More generally, individuals can be given different weights for different resource dimensions, but as non negative mixtures of the indicators $\mathds 1\{u\leq r\}$, the weights are always decreasing in ranks.
The second way we evaluate our Lorenz dominance criterion is by identifying transfers of resources between individuals that increase inequality according to this criterion. Since inequality is a cardinal aspect of the distribution, we consider a class of transfers that preserves the multivariate ranks. The transfers we consider are functions $T:[0,1]^d\rightarrow\mathbb R^d$. If the $j$-th component of transfer $T$ is positive (resp. negative), it is added to (subtracted from) the endowment in resource $j$ of individual with rank $u$.
First we show that the transfers that increase inequality according to the Lorenz criterion are the arbitrary combinations of rank preserving transfers of a non negative quantity of one of the resources from an individual with rank $u_1$ to an individual with rank $u_2\geq u_1$.
The inequality in ((ref)) expresses the fact that mass is transferred from lower ranked to higher ranked individuals.
A desirable feature of the Lorenz inequality ordering of definition (ref) is its ability to rank two allocations $X$ and $X^\prime$, when the latter is obtained from the former through a transfer that increases inequality of the marginals or that increases the degree of positive dependence between the marginals. We formalize this feature with a specific type of multivariate transfer we call {\em Monotone Regressive Transfers}. We specialize the discussion to bivariate allocations to avoid wading into concepts of increasing multivariate dependence when $d>2$.
In the univariate case, a monotone regressive transfer reduces to a monotone mean preserving spread (Quiggin:92), also called Bickel-Lehmann increase in dispersion (BL:76). In the multivariate case\footnote{A related extension in the theory of multivariate risks was proposed in CGH:2016.}, monotone regressive transfers weakly increase both marginal inequality and positive dependence. The former happens because each component of the transfer has non negative own derivative, hence is increasing in each component of the rank. The latter happens because the transfer has non negative cross derivative, hence increases the degree of positive dependence between the two resources.
Proposition (ref) shows that the multivariate Lorenz dominance order of definition (ref) therefore ranks an allocation as more unequal if the marginal resource allocations are weakly more unequal and if the marginal resource allocations are weakly more positively dependent. This is in contrast with the Lorenz dominance order based on inclusion of Lorenz zonoids proposed in koshevoy:1996. Indeed, by Proposition 8 in koshevoy:2007, if two allocations have identical marginals, and $X$ dominates $X^\prime$ in the Lorenz dominance order based on Lorenz zonoid inclusion, then $X$ and $X^\prime$ are identically distributed.
The Lorenz dominance ordering of definition (ref) does not satisfy the {\em uniform majorization principle} proposed by kolm:1977. In case of discrete populations, the uniform majorization principle stipulates that inequality should be reduced through multiplication by a doubly stochastic matrix different from a permutation. However, as Dardanoni:93 points out, such transformations can increase correlation and therefore increase inequality in an egregious way. See the discussion in Savaglio:2006. We show that a similar issue arises with the continuous version of the uniform majorization principle. The latter requires an inequality dominance order to be monotonic with respect to the concave order. From theorem 4 of GH:2012, we deduce that a social evaluation functional that satisfies uniform majorization and comonotonic independence must be equal to
up to an affine transformation. We show in appendix (ref) that in the case of bivariate allocation $X=(X_1,X_2)$, $S_{UM}$ is minimized when the two components $X_1$ and $X_2$ of allocation $X$ are independent, which runs against the intuition that increased dependence can increase inequality\footnote{Note that we are considering inequality over outcomes, not welfare inequality. Hence, the point made in AB:1982, that increased correlation may decrease utilitarian welfare inequality when resources are complements, doesn't apply here.}.
Failures of Lorenz dominance can be visualized with the Inverse Lorenz Function. Consider two allocations $X$ and $X^\prime$, with respective Inverse Lorenz Functions $l_X$ and $l_{X^\prime}$. If $l_X(z)\leq l_{X^\prime}(z)$ for some vector of shares $z$, a larger proportion of the population commands the same share of resources in allocation $X^\prime$ than in allocation $X$. Lorenz dominance of allocation $X$ over allocation $X^\prime$ (in the sense of definition (ref)) implies that the relation $l_X(z)\leq l_{X^\prime}(z)$ holds for each resource share vector $z$ (see proposition (ref) in the appendix).
In case of bivariate allocations, the latter can be easily visualized on $[0,1]^2$ through the relative positions of the level sets of the Inverse Lorenz Function, which we call $\alpha$-Lorenz curves, denoted
The $\alpha$-Lorenz curves provide a visualization of Lorenz dominance. We can compare the inequality of different allocations based on the shape and relative positions of their respective $\alpha$-Lorenz curves. Suppose $X$ is less unequal in the Lorenz dominance order than $X^\prime$. Then, by proposition (ref) in appendix (ref), for any $z\in\mathcal C^\alpha_{X^\prime}$, $l_{X}(z)\leq l_{X^\prime}(z)=\alpha$. So $z\in\mathcal C^{\tilde \alpha}_{X}$ with $\tilde \alpha\leq \alpha$. This can be visualized as a shift to the north-east of the $\alpha$-Lorenz curves of the more unequal allocation $X^\prime$ to the $\alpha$-Lorenz curves of the less unequal allocation $X$.
Figure (ref) and (ref) display $\alpha$-Lorenz curves of the multivariate lognormal allocation $X$ defined by
where $\mathcal{N}$ is the normal distribution, $\sigma_1,\sigma_2 > 0$ control the dispersion of the respective marginals and $\rho$ controls the degree of dependence.
In figure (ref), we revisit the three special cases of identical allocations, independent attributes and comonotonic attributes. We also add the counter-monotonic case, where individuals are ranked in opposite order for the two resources, as well as intermediate dependence cases. Figure (ref) shows the $\alpha$-Lorenz curve (with $\alpha=0.9$) of the identical allocation compared to multivariate lognormal allocations with variances $\sigma_1=\sigma_2=1$ and correlation coefficients $\rho=-0.99$ (counter-monotonic), $\rho=-0.6,-0.3$, $\rho=0$ (independent), $\rho=0.3,0.6$, and $\rho=0.99$ (comonotonic).
Visually, inequality can be assessed by the departure of $\alpha$-Lorenz curves from those of the identical allocation. This visual comparison is facilitated by the fact that they are shaped like indifference curves. In addition, correlation information is preserved through the curvature of the $\alpha$-Lorenz curves, which decreases when positive dependence increases.
Figure (ref) compares $\alpha$-Lorenz curves of $6$ different allocations that are multivariate lognormally distributed as in (ref), for $\alpha=0.9$. The parameters $\sigma_1^2$ and $\sigma_2^2$ take values $1$ or $2$, whereas $\rho$ takes values $0.2$ or $0.8$. In case of marginals with different $\sigma$, the asymmetry is reflected in the $\alpha$-Lorenz curves. Moreover, other things equal, inequality increases with $\sigma$, which measures inequality in the marginals, and with $\rho$, which measures correlation.
Finally, figure (ref) shows an example of two multivariate lognormally distributed allocations $X$ and $X^\prime$ such that the marginals of $X$ are more unequal than those of $X^\prime$, but $X^\prime$ is more unequal overall due to positive dependence of its marginals. Specifically, the marginals of $X$ are independent and have the same scale parameter $\sigma^2 = 1.2$, while $X^\prime$ has marginals with correlation parameter $\rho = 0.9$ and scale $\sigma^2 = 1$ each. This shows that the Lorenz inequality dominance ordering does not imply dominance of the marginals, and how it can instead incorporate trade-offs between marginal inequality and positive dependence.
The Lorenz dominance ordering is a partial ordering of multivariate distributions. In many cases, $\alpha$-Lorenz curves may cross. For a complete inequality ordering, we also propose an extension of the classical Gini index to compare inequality in multi-attribute allocations. gajdos:2005 propose a multivariate Gini coefficient based on aggregation across individuals first, then across dimensions, which removes the effect of dependence across attributes. DL:2012 propose to aggregate across dimensions first, then across individuals, in order to keep track of correlation. From the volume of the Lorenz zonoid, a multidimensional Gini coefficient can be derived naturally as in koshevoy:1997. An alternative strategy is followed by arnold:1983, koshevoy:1997, who extend the definition based on the sum of all distances between pairs of individuals.\footnote{Other multivariate Gini indices based on multivariate Lorenz curve proposals include banerjee:2010, grothe:2021, and SJ:2020.}
The univariate Gini index can be interpreted as the average deviation from the egalitarian allocation, the univariate version of our identical allocation. We emulate this interpretation and define a multivariate Gini based on an average deviation from the Lorenz map $r\mapsto(r_1r_2\cdots r_d,\ldots,r_1r_2\cdots r_d)$. The deviation measure we propose is
where $\mathcal{L}_j$ is the $j$-th component of the Lorenz map $\mathcal{L}_X$, with $j=1,\ldots,d$. After normalization, ((ref)) becomes
which yields the following definition.
The traditional Gini index of a univariate allocation can also be characterized as a weighted sum of outcomes, where the weights are increasing linearly in the rank of the individual in the population. Hence, the negative of the Gini, seen as a social evaluation functional displays inequality aversion by giving more weight to the outcomes of lower ranked individuals than to those of higher ranked ones.
The same interpretation is valid for our multivariate Gini. Specifically,
is an inequality averse social evaluation functional of the form ((ref)) with uniform measures on $[0,1]^d$ in ((ref)). So $S(X)$ evaluates the social desirability of an allocation with a weighted sum of individual endowments, where the weights $\Pi_{j=1}^d(1-u_j)$ are component-wise decreasing in the individual's rank $u=(u_1,\ldots,u_d)$.
The Gini index correspondingly takes the form $G(X)=1-(2^d/d)S(X),$ with $S$ as in ((ref)). For instance, in the case of bivariate allocations, ((ref)) takes the form
In expression ((ref)), $Q_1(u)$ and $Q_2(u)$ are the components of $Q_{\tilde X}$, so that $Q_1(u)+Q_2(u)$ is the sum of the two normalized resource allocations of the individual in the population with vector rank $(u_1,u_2)$. Hence, the Gini index is indeed a weighted sum of outcomes, with weights $(u_1+u_2-u_1u_2)$ increasing with the vector ranks $(u_1,u_2)$. It is a genuinely multivariate extension in that the weighting scheme, hence the social evaluation of inequality, depends on multivariate ranks of individuals.
The Gini index of definition (ref) is in $[0,1]$ under assumption (ref). It equals $0$ for the identical allocation. It tends to $1$, when the Lorenz map tends to $0$ (extreme inequality). The Gini index of an allocation with independent components reduces to the average of classical scalar Ginis of both components. Like the classical scalar Gini index, it preserves the Lorenz inequality ordering, in the sense that higher inequality according to $\preccurlyeq_{\mathcal{L}}$ implies a larger value of the Gini index. In other words, $X\succcurlyeq_{\mathcal{L}} X^\prime$ implies $G(X)\leq G(X^\prime)$, so that the negative of the Gini is a compatible social evaluation functional. Hence it inherits the properties of anonymity, scale invariance and comonotonic independence.
The multivariate Gini in expression ((ref)) is the suitably normalized negative of an inequality averse social evaluation functional of the form ((ref)) with uniform measures on $[0,1]^d$ in ((ref)). It can be extended to reflect varying concern for inequality in different attributes. To achieve this, a multivariate Gini coefficient can be defined as $1-cS(X)$, where $c$ is a normalizing constant and $S$ is a social evaluation functional that reflects different degrees of inequality aversion in different attributes.
In the univariate case, to reflect varying degrees of inequality aversion, DW:80 propose a single parameter family of Gini coefficients, called S-Gini, defined by
where $\mathcal L$ is the traditional Lorenz curve, and $\delta$ ranges from $1$, corresponding to indifference to inequality, to the Rawlesian extreme at the limit $\delta\rightarrow\infty$, where only the poorest individual matters\footnote{We were unable to locate a precise statement of this in the literature, so we include it in proposition (ref) in the appendix with a proof for completeness.}.
The S-Gini family of DW:80 can be extended to the assessment of multivariate inequality within our framework. Let $\delta=(\delta_1,\ldots,\delta_d)$ be a $d$-dimensional parameter, where $\delta_j\in[1,\infty)$, $j=1,\ldots,d,$ reflects the concern for inequality in attribute $j$. We define the family of multivariate S-Gini coefficients of inequality of an allocation $X$ with Lorenz map $\mathcal L_X=(\mathcal L_1,\ldots,\mathcal L_d)$ as
where $c_\delta:=2^{d-1}/\sum_{j=1}^d\delta_j^{-1}$ is a normalizing constant, and $S_\delta$ is the social evaluation functional
The normalizing constant $c_\delta$ is chosen such that the multivariate S-Gini $G_\delta$ lies in $[0,1]$ and is zero in case of the identical allocation. There remains to verify that the social evaluation functional $S_\delta$ is indeed of the form ((ref)), and hence compatible with the Lorenz order. Indeed, we have
with $m^{(\delta)}_j(r)=\left(\prod_{l=1,l\neq j}^d r_l\right)[1-(1-r_j)^{\delta_j-1}]$, for each $j=1,\ldots,d$. The multivariate S-Gini thereby incorporates varying degrees of inequality aversion for different attributes. We recover the S-Gini of DW:80 when $d=1$, and the multivariate Gini of section (ref) when $\delta_j=2$, for all $j=1,\ldots,d$, as desired. We also recover Rawelsian limits as $\delta_j$ tends to zero as formalized in proposition (ref) in the appendix.
In this section, we apply our methodology to the analysis of income-wealth inequality in the United States between 1989 and 2022, based on the public version of the triennial Survey of Consumer Finances (SCF). Wealth refers to all assets, financial and otherwise. Details of the sampling technique and a discussion of specific features and issues with the data set are given in appendix (ref). A guide for practical implementation of the computational procedure outlined in section (ref) is given in appendix (ref). We refer to inequality displayed by our measure as overall inequality, while specific marginal inequality is described as wealth or income inequality.
Figure (ref) shows the $\alpha$-Lorenz curves for $\alpha = 0.6,0.8,0.95$ for the years 1989, 2007, 2010, and 2022. There is a general worsening of overall inequality over 3 decades since the curves shift away from the north-east corner. The tight curvature also reflects the positive correlation of income and wealth as in figure (ref). Using figure (ref) as reference, the skew towards the wealth axis indicates inequality from the wealth marginal is dominant at these $\alpha$-levels, as expected.
Figure (ref) shows resource shares of the $25\%$ fraction of the the population below rank $r=(0.5,0.5)$ as well as the $90\%$ fraction\footnote{The fraction of the population is exactly $0.95^2=0.9025$.} of the population below rank $r=(0.95,0.95)$. The shares in both resources of the bottom $90\%$ have been steadily declining and the shares of wealth are lower than the shares of income. Between 2007 and 2010, we see income shares increasing relatively more than the decrease in wealth shares. Wealth and income shares both fell from 2010 to 2022, explaining the shift in the curves from figure (ref). As for the bottom $25\%$, changes over time are minor compared to those of the bottom $90\%$.
Figure (ref) displays the marginal Gini indices for income and for wealth, the multivariate Gini index based on (ref), as well as Kendall's $\tau$ for the dependence between income and wealth over time. The multivariate Gini shows a steady increase in overall inequality. If the resources were independent, the multivariate Gini would be the average of the marginal Ginis. In the present case, the multivariate Gini reveals a positive association between resources since it is higher than the average of the marginal Gini indices. The multivariate Gini shows reduced overall inequality between $2007$ and $2010$. The decreased correlation and income inequality may have been sufficient to offset the rise in wealth inequality.
Inequality analysis across groups can reveal further insights. Figure (ref) shows Gini indices among White and Black respondents as well as among the working age (64 years and below) and retiring age populations (65 years and above). While overall inequality has worsened among White respondents, the inequality among Black respondents has remained steady. When comparing inequality across age groups, they both exhibit a steady increase in overall inequality, however the multivariate Gini among the retiring age group inherits the variability in the income marginal.
In this paper, we propose a new multivariate extension of the Lorenz curve. We propose to emulate the gastwirth:1971 formulation of the Lorenz curve and define a Lorenz map by integrating vector quantiles of CGHH:2017. The value of the Lorenz map is a vector of shares of each resource held by the poorer section of the population, as in the scalar case. Dominance of Lorenz maps defines a multi-attribute inequality dominance partial ordering. This Lorenz ordering is, like its scalar counterpart, an implementable criterion to compare inequality in allocations. It is, also like its scalar counterpart, equivalent to preference by any inequality averse rank dependent social evaluation functional. We propose an Inverse Lorenz Function and its level sets as a multivariate inequality visual comparison tool, and apply it to income and wealth in the United States between $1989$ and $2022$.
Multi-attribute inequality can vary substantially across population groups, as shown in MR:2016 within the information theoretic framework of Maasoumi:86. There is a tension between heterogeneity across covariates and the anonymity axiom, according to which inequality measurement should not depend on individual's identities, but only on the distribution of resource allocations. As kolm:1977 pointed out, this tension is alleviated in part by including more variables in the allocation. This reinforces the motivation for a multidimensional approach to inequality measurement. As for the other potential sources of individual heterogeneity that matters to the social planner, anonymity can be restored by measuring inequality in each subgroup. We illustrate this in figure (ref). Beyond this, the conditional approach of MR:2016 could also be extended to our framework with the use of conditional vector quantiles in CCG:2016.
Finally, we argue that a formal test of multi-attribute inequality dominance can be based on our Lorenz map, in analogy to dominance testing based on the traditional Lorenz curve in DD:2000 and references within. The statistical theory for such a test relies on multivariate stochastic dominance testing and the regularity of optimal transport maps, and is left for future research.