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Lorenz map, inequality ordering and curves based on multidimensional rearrangements

We propose a multivariate extension of the Lorenz curve based on multivariate rearrangements of optimal transport theory. We define a vector Lorenz map as the integral of the vector quantile map associated to a multivariate resource allocation. Each component of the Lorenz map is the cumulative share of each resource, as in the traditional univariate case. The pointwise ordering of such Lorenz maps defines a new multivariate majorization order. We define a multi-attribute Gini index and complete ordering based on the Lorenz map. We formulate income egalitarianism and show that the class of egalitarian allocations is maximal with respect to our inequality ordering over a large class of allocations. We propose the level sets of an Inverse Lorenz Function as a practical tool to visualize and compare inequality in two dimensions, and apply it to income-wealth inequality in the United States between 1989 and 2019.


Introduction
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. This visualization property 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. 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 et al. [2011] and Arnold and Sarabia [2018].
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 et al. [2011]. As pointed out by Arnold [2008], 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, Atkinson and Bourguignon [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 Decancq and Lugo [2012], Aaberge and Brandolini [2014], Andreoli and Zoli [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 (1972aTaguchi ( , 1972b, Arnold [1983], and Koshevoy andMosler (1996, 1999). They are reviewed in Marshall et al. [2011] and Sarabia and Jorda [2014] and discussed in more details in the next subsection. We contribute to this literature with a vector version of the Gastwirth [1971] formulation of the Lorenz map, based on a multivariate reordering of the data, with optimal transport rearrangement theory. With this reordering, we emulate the features of the Lorenz curve that most contributed to its success: we derive a tool to compute the share of each resource held by the subset of the population below any given rank, and we provide tools to visualize and compare inequality in multivariate distributions.
The solution of quadratic optimal transport problems, see Villani (2003Villani ( , 2009), can be seen as rearrangements of random vectors. For a given random vector X ∈ R d , interpreted as the resource allocation in the population, under mild regularity conditions, there is a uniform random vector U ∈ [0, 1] d and a gradient ∇ψ X of a convex function such that X = ∇ψ X (U ). The vector U can be interpreted as a vector of multivariate ranks of individuals in the population, and the Lorenz map at r ∈ [0, 1] d can be defined as the cumulative vector share of resources held by individuals with vector ranks below r, componentwise. Equivalently, we define the Lorenz map as the integral of the vector quantile of Chernozhukov et al. [2017]. Correspondingly, we define a Lorenz ordering from the pointwise dominance of Lorenz maps. The Lorenz map and Lorenz ordering defined here reduce to the traditional Lorenz map and ordering in the univariate case. Our multivariate Lorenz map also shares two important properties with the traditional Lorenz map. First, it characterizes the distribution of the random vector it is constructed from. Second, we define a multiattribute notion of egalitarian allocation and we show that such egalitarian allocations have maximal Lorenz maps in a large class of allocations.
Since the multivariate Lorenz map is an integrated quantile map, the Lorenz ordering based on pointwise dominance of Lorenz maps is a multivariate majorization ordering, based on optimal transport reorderings. We show that the Lorenz ordering is able to rank two allocations, when one is a monotone mean preserving spread of the other. The notion of monotone mean preserving spread is a multivariate extension of the univariate monotone mean preserving spread of Quiggin [1992], also called Bickel-Lehmann increases in dispersion (Bickel and Lehmann [1976]). It captures increases in inequality in the marginals, as well as increases in correlation. We show that our Lorenz ordering can rank two allocations, where one is the result of a correlation increasing transfer of the other, that existing multivariate Lorenz orderings are unable to rank (see section 2.3). We also construct a multivariate Gini index based on the distance to the Lorenz map of an egalitarian allocation where all individuals have identical endowments. This Gini index preserves the Lorenz ordering and also reduces to the traditional Gini in the univariate case.
For our visualization device, we define an Inverse Lorenz Function at a given vector of resource shares as the share 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 function. In two dimensions, the αlevel sets of this cumulative distribution function, which we call α-Lorenz curves, are non crossing downward sloping curves that shift to the south-west when inequality increases, as defined by the Lorenz ordering. Because this compelling visual tool for the comparison of multi-attribute inequality is one of the major objectives we pursue with this work, we report all our results in the bivariate case, and propose an illustration to the analysis of income-wealth inequality in the United States between 1989 and 2019.
Closely related work. The original definition of the Lorenz curve is implicit. Denote F Y the cumulative distribution function of random variable Y . The Lorenz 1{·} denotes the indicator function. In the multivariate case, denote F X the cumulative distribution of random vector X and let F i be the cumulative distribution of the i-th marginal. Arnold [1983] extends the implicit formulation to multiple univariate ranks F 1 (x 1 ) = p 1 and F 2 (x 2 ) = p 2 , and defines a scalar-valued Lorenz map L X (p 1 , p 2 ). Koshevoy and Mosler [1996] keep F X (x) = p for random vector X, but extend univariate shares to vector shares. The Lorenz zonoid of Koshevoy and Mosler [1996] relates any proportion of the population to its share of each of the resources. The Lorenz zonoid is difficult to compute in general. We discuss the relation between our Lorenz map, the Lorenz surface of Taguchi (1972aTaguchi ( , 1972b and the Lorenz zonoid of Koshevoy and Mosler [1996] in section 1 after equation (1.3). Koshevoy and Mosler [1999] propose an Inverse Lorenz Function that can be interpreted as the maximum share of the population that holds a given vector share of resources, but it is the solution of a complex optimization problem. Grothe et al. [2021] define an Inverse Lorenz Function as the probability that for each i, a share z i of resource i is collectively held by individuals whose rank in the distribution of resource i is below F i (X i ).
They characterize this Inverse Lorenz Function with a coupling argument. Multivariate Lorenz dominance is defined and discussed in Koshevoy [1995], Koshevoy and Mosler [2007], Banerjee [2010]. See Arnold and Sarabia [2018] for a comprehensive account. Gajdos and Weymark [2005] discuss multivariate versions of the Gini index based exclusively on functionals of individually reordered marginals, as opposed to the multivariate reorderings proposed here. Koshevoy and Mosler [1997], Banerjee Grothe et al. [2021], Mosler [2013], Sarabia and Jorda [2020] also propose multivariate Gini indices based on multivariate Lorenz curve proposals. Finally, optimal transport rearrangements are used to define multivariate comonotonicity and quantiles in Ekeland et al. [2012],  and Puccetti and Scarsini [2010]. More recently, subsequent to our work, Hallin and Mordant [2022] also adopt a transportation approach to the definition of multi-attribute Lorenz curves. They adopt a center-outward approach, which is better suited to define notions of middle class.
Notation. All random elements are defined on the same complete probability space.
Let U [0, 1] d be the uniform distribution on [0, 1] d . The indicator function 1{x ∈ A} takes value 1 if x ∈ A and zeo otherwise. We call R d + the set {z ∈ R d : z ≥ 0}, where ≥ denotes the componentwise inequality. We denote X = (X 1 , . . . , X d ) a random vector on R d . Let P X stand for the distribution of the random vector X.
Following Villani [2003], we let g # ν denote the image measure (or push-forward ) of a measure ν by a measurable map g : R d → R d . Explicitly, for any Borel set A, g # ν(A) := ν(g −1 (A)). The generalized inverse of a function f is denoted f −1 . The symbol ∇ denotes the gradient, and D the Jacobian. The convex conjugate of a convex lower semicontinuous function ψ is denoted ψ * . We use the standard convention of calling weak monotonicity non-decreasing or non-increasing, as the case may be.

Vector Lorenz Maps and Curves
We propose a method to compare, measure and visualize inequality of allocations of two resources. A resource allocation is modeled as a random vector X = (X 1 , X 2 ) with support X ⊆ R 2 + . The distribution P X of X is assumed to have unit mean, to avoid issues of normalization or change of units of measurement. Most of the definitions and properties we present here are valid in higher dimensions. However, the crucial visualization aspect is lost, so we don't pursue a greater level of generality.
1.1. Lorenz map. The Gastwirth [1971] formulation of the Lorenz curve for a scalar allocation relies on the quantile function. Let Y be a scalar allocation with mean 1 and cumulative distribution function F Y . Then V := F Y (Y ) ∼ U [0, 1] is the rank of individuals in the distribution. The value L Y (q) at rank q of the Lorenz curve associated with allocation Y is the cumulative share of individuals with rank V ≤ q.

Hence
To analyze inequality in allocation X, we can first look at inequality in each marginal allocation X 1 and X 2 , using the Lorenz curves L 1 := L X 1 and L 2 := L X 2 .
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 co-monotonic allocation (the wealthier individuals have higher incomes) is more unequal than the admittedly unrealistic countermonotonic allocation (the wealthier individuals have lower income). In section 2.3, we discuss an example of such a comparison. In this example, an allocation where the first half of the population holds all the resources is compared with an allocation where the first half of the population holds the totality of the first resource, while the second half of the population holds the totality of the second resource.
To take dependence into account, we propose to emulate the Gastwirth [1971] formulation by measuring cumulative shares of both resources for all individuals, whose vector rank U is below r = (r 1 , r 2 ). The vector rank we adopt is a uniform random vector on [0, 1] 2 . It is obtained from the vector allocation X via a unique transformation analogous to the probability integral transform V = F Y (Y ) in the scalar case described above. Its associated vector quantile, inverse of the vector rank, is guaranteed when X has independent marginals to simply be the vector of univariate quantile functions. This multivariate quantile function can then be integrated over [0, r 1 ] × [0, r 2 ] to obtain a vector analogue of the Gastwirth [1971] formulation of the Lorenz curve.
We adopt the vector quantile and rank notions proposed in Chernozhukov et al.
[2017], based on optimal transport theory. The vector quantile of random vector X is defined, to emulate the traditional quantile function, as the unique cyclically monotone function that pushes the uniform distribution on [0, 1] 2 into the distribution of random vector X. Existence and uniqueness follow from Theorem 1 in McCann [1995], which extends the Brenier [1991] polar factorization theorem (see also Rachev and Rüschendorf [1990]) beyond finite second moments. We give a brief review of the theory of vector ranks and quantiles, and the underlying notions in optimal transport theory, in appendix A.1. Here we only give the elements required to define our vector Lorenz map.
Definition 1 (Vector quantiles). Let λ be the uniform distribution on [0, 1] d , and let P X be the distribution of a random vector X on R d . There exists a convex The function ∇ψ X exists and is unique, λ-almost everywhere. We define the vector quantile of X as the function ∇ψ X , and we call the function ψ X a potential of X, following the optimal transport literature.
Existence and uniqueness in definition 1 follow from proposition 3 in appendix A.1.
In case d = 1, gradients of convex functions are nondecreasing functions, hence the vector quantile of definition 1 reduces to the classical quantile function.
The proportion of the resources held by individuals ranked below r is the cumulative integral of the vector quantile, as in the scalar case.
As stated in proposition 3 in appendix A.1, when X has absolutely continuous distribution P X , its quantile function is P X -almost everywhere invertible with inverse ∇ψ * X , where ψ * X is the convex conjugate of ψ X (see the introduction for notation and definitions from convex analysis). In that case, the transformation U = ∇ψ * X (X) is the vector analogue of the probability integral transform V = F Y (Y ) discussed above. The random vector U = ∇ψ * X (X) ∼ U [0, 1] 2 is the vector rank of the individual with endowment X, in the terminology of Chernozhukov et al. [2017]. In this case, the Lorenz map of definition 2 can be rewritten: This clarifies the interpretation of L X (r) as the cumulative endowment X of all individuals with vector rank ∇ψ * X (X) below r.
Relation with other multivariate Lorenz concepts. Expression (1.3) also clarifies the relation with the Lorenz surface of Taguchi (1972aTaguchi ( ,1972b and the Lorenz zonoid of Koshevoy and Mosler [1996]. With a bivariate allocation, its Lorenz zonoid (Koshevoy and Mosler [1996]) relates any function φ : R 2 → [0, 1] to the vector Hence, to a proportion of the population E[φ(X)], it associates their share of each of the resources. The Lorenz surface of Taguchi (1972aTaguchi ( ,1972b) is a subset of the Lorenz zonoid, obtained with the choice of functions φ x (·) := 1{· ≤ x}, all x ∈ R 2 .
Both Taguchi (1972aTaguchi ( ,1972b and Koshevoy and Mosler [1996] relate univariate population proportions to share proportions. 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 difference, under the conditions for equation (1.3), we can make a direct connection. If, in (1.4), we use the class of functions φ r (·) := 1{∇ψ * X (·) ≤ r}, for each rank r ∈ [0, 1] d , where ∇ψ * is the vector rank defined in section 1, we obtain the surface {(r 1 r 2 , L X (r 1 , r 2 )); r = (r 1 , r 2 ) ∈ [0, 1] 2 }, which is therefore a subset of the Lorenz zonoid. This subset is different from the Taguchi (1972aTaguchi ( ,1972b Lorenz Examples. We now explore special endowments and compute the corresponding Lorenz maps. First, we consider the case, where all individuals are endowed with the same quantity of resources. Example 1 (Identical allocations). If X = (1, 1) almost surely, then ∇ψ(r) = (1, 1) for all r ∈ [0, 1] 2 , and L X (r) = (r 1 r 2 , r 1 r 2 ). The image of L X is the diagonal in [0, 1] 2 .
Second, we check that our definition is compatible with scalar definitions when both resources are independently distributed.
Example 2 (Independent Resources). Let the components X 1 and X 2 of X be independent with univariate Lorenz curves L 1 and L 2 respectively. Then, the Lorenz map is L X (r 1 , r 2 ) = (r 2 L 1 (r 1 ), r 1 L 2 (r 2 )). This expression of the Lorenz map has the following interpretation. Consider the first component r 2 L 1 (r 1 ). The share of resource 1 held by people with multivariate rank in [0, r 1 ] × [0, 1] is the marginal share, equal to the univariate Lorenz function. Since the two resources are independent, this share is uniformly distributed along the other dimension, so that people with ranks in [0, r 1 ] × [0, r 2 ] command a share r 2 L 1 (r 1 ). The other component is interpreted analogously. When r 1 = 1, the Lorenz map takes values L X (1, r 2 ) = (r 2 , L 2 (r 2 )).
That is, the image of {(1, r 2 ) : 0 ≤ r 2 ≤ 1} under L X is the Lorenz curve L 2 of the second resource X 2 (and symmetrically when r 2 = 1).
Third, we derive the Lorenz map in the case of endowments X = (X 1 , X 2 ) with the same components, i.e., X 1 = X 2 almost surely.
Example 3 (Comonotonic Resources). Let the components X 1 and X 2 of the endowment X be almost surely equal. Then, X 1 and X 2 have identical distributions. The optimal transport map from the uniform distribution on [0, 1] 2 to the distribution of X = (X 1 , X 2 ) is then (u 1 , u 2 ) → (ψ (u 1 + u 2 ), ψ (u 1 + u 2 )), where z → ψ (z) is the optimal map from σ to the distribution of X 1 , where σ : which has density on [0, 2] given by 1 − |1 − z|. Each component of the Lorenz curve is then given by In case X 1 and X 2 are uniformly distributed on [0, 2], the optimal transport map ψ for z < 1 is given by ψ (z) = z 2 , so that ψ(z) = z 3 /3. We then have, when r 1 + r 2 ≤ 1, and when r 1 + r 2 > 1. The image of this Lorenz map is once again the diagonal in [0, 1] 2 .

Inverse Lorenz Function and α-Lorenz
Curves. In the scalar case discussed above, inverting the Lorenz curve L Y defined in (1.1) yields the inverse Lorenz curve The scalar inverse Lorenz curve at y is therefore shown in (1.5) to be equal to the maximum proportion of the population with cumulative share of the resource equal to y. In the vector case, the right-hand side of (1.5) becomes where z = (z 1 , z 2 ) ∈ [0, 1] 2 , inequality ≤ is understood component-wise, and the probability is taken with respect to the uniform random vector U on [0, 1] 2 . The expression above is no longer the inverse of the Lorenz map L X , but it can still be interpreted as the share of the population with cumulative shares of both resources equal to z = (z 1 , z 2 ). This motivates the following definition.
The α-Lorenz curves are the objects involved in our proposed multivariate inequality visualization technique. We will see how to compare the inequality of different endowments based on the shape and relative positions of their respective α-Lorenz curves. First, we revisit the three special cases of the previous section.
Example 1 continued [Identical allocations] The ILF l X (z) of the endowment X = (1, 1) is 0 when z 1 z 2 = 0 and By definition, the α-Lorenz curves C α X are right angles defined by min{z 1 , z 2 } = α.
where l 2 is the univariate inverse Lorenz curve of X 2 .   In the univariate case, the Lorenz curve of the identical endowment Y = 1 almost surely, is L Y (q) = q, which is sometimes called the egalitarian line. The Lorenz curve of any other endowment Y ≥ 0 is below the egalitarian line, i.e., L Y (q) ≤ q, for all q ∈ [0, 1]. For d = 2 the identical allocation of example 1 is a direct extension of the univariate notion of egalitarian. We show here that the Lorenz map and Inverse Lorenz Function of the identical allocation provide similar bounds in the multi-attribute case. For this, we require endowments with components that display a form of positive association defined in assumption 1. We will argue in section 3 that defining egalitarianism solely by identical allocations is too restrictive in the case of multiple resources, and we will show that a much larger class of allocations have Lorenz maps dominated by an egalitarian allocation from definition 9, which includes the identical allocation.
are monotonically increasing as functions of u 2 and u 1 , respectively, where the vec- This assumption imposes a type of positive dependence between X 1 and X 2 through their ranks. More precisely, assumption 1 imposes a form of positive regression dependence, as in Lehmann [1966], between one resource and the other's rank. A sufficient condition for assumption 1 is supermodularity of the potential function ψ X of endowment X, as shown in lemma 1 below. We also show in lemma 1, that supermodularity of the potential function ψ X also implies positive quadrant dependence of the two components X 1 and X 2 of X i.e., P( Lehmann [1966]. Lemma 1 (Supermodular potential). Suppose X has a supermodular potential func- Then, assumption 1 holds, and X 1 and X 2 are positive quadrant dependent.
For endowments satisfying assumption 1, we show that Lorenz map and Inverse Lorenz Function of the identical allocation serve as upper and lower bounds, respectively.
Property 3. The Lorenz map of any endowment X satisfying assumption 1 is component-wise dominated by the Lorenz map of the identical allocation in example 1. Moreover, the Inverse Lorenz Function of endowment X is bounded below by the ILF of the identical allocation.
Without assumption 1, some allocations may have a Lorenz map that is componentwise larger than the Lorenz map of the identical allocation for some ranks. To illustrate the point, consider the potential ψ X (u) = (u 1 − u 2 ) 2 /2 + u 1 + u 2 . It corresponds to an endowment X, whose distribution is supported on the line X 1 + X 2 = 2. Calculating the Lorenz map, we obtain Notice, in particular, that L 1 (r) > r 1 r 2 in the region where r 1 > r 2 . If the implicit relative price of resource 2 is 1, endowment X is an egalitarian endowment, since all individuals have equal budgets. However, this endowment does not satisfy assumption 1 and its Lorenz map is not dominated by (r 1 r 2 , r 1 r 2 ) as we have shown. This apparent departure from properties of the scalar Lorenz curve is due to the fact that an endowment with X 1 + X 2 = 2 a.s. can also be considered egalitarian, as we discuss in section 3.
Visually, inequality can be assessed by the departure of α-Lorenz curves from those of the identical allocation. This visual comparison is facilitated by the fact that they are shaped like indifference curves.
Property 4. (i) The α-Lorenz curves C α X are the level curves of a bivariate cdf, hence they are downward sloping, non decreasing in α and they do not cross. In addition,

Inequality orderings
2.1. Lorenz ordering. Consider two allocations X and X, with respective Lorenz maps L X and L X . If L X (r) ≥ L X (r) for some vector rank r, the same proportion of the population with vector ranks below r commands a larger share of both resources in allocation X than in allocation X. If this is true for any vector rank r in [0, 1] 2 , then, we say that allocation X is more unequal than allocation X.
Definition 4. An allocation X is said to be more unequal in the Lorenz order than an allocation X if L X (r) ≥ L X (r) for all r ∈ [0, 1] 2 . We denote this as X L X.
As a partial ordering based on cumulative sums of vector quantiles, the relation X L X is a multivariate extension of the concept of majorization of Hardy et al. [1934]. It is different from existing multivariate notions of majorization reviewed in Marshall et al. [2011] and Arnold and Sarabia [2018], in that it relies on a multivariate reordering of the random vector allocation. The relation X L X is equivalent to stochastic dominance of the random vector L over L X (U ) (see Section 3.8 of Müller and Stoyan [2002]). As in the scalar case, it is a partial ordering. Under assumption 1, the maximal element of (X , L ) is the identical allocation by property 3.
We can also define an increasing inequality order based on the Inverse Lorenz Functions. Consider two allocations X and X, with respective Inverse Lorenz Functions l X and l X . If l X (z) ≤ l X (z) for some vector of shares z, a larger proportion of the population commands the same share of resources in allocation X than in allocation X. If this is true for any vector z of resource shares in [0, 1] 2 , then, we say that allocation X is more unequal than allocation X.
Definition 5. An allocation X is said to be more unequal in the weak Lorenz order than an allocation X if l X (z) ≥ l X (z) for all z ∈ [0, 1] 2 . We denote this as X l X.
The relation X l X is equivalent to lower orthant dominance of the random vec- Müller and Stoyan [2002]). In the scalar case, the orderings of definitions 4 and 5 both coincide with the traditional Lorenz ordering. In higher dimensions, however, the equivalence doesn't hold. Nonetheless, as the name indicates, the weak Lorenz inequality order of definition 5 is weaker than the Lorenz order of definition 4, as we show in the following proposition.
Proposition 1. An allocation X is more unequal in the weak Lorenz order than an allocation X, i.e., X l X (definition 5) if X is more unequal in the Lorenz order, i.e., X L X (definition 4).
This inequality dominance of definition 5 can be visualized on [0, 1] 2 through the relative positions of α-Lorenz curves. Indeed, the α-Lorenz curves are, by definition, curves with equation l X (z) = α. In other words, Suppose X is less unequal in the Lorenz order than X. For any z ∈ C α X , by definition of the Lorenz order, l X (z) ≤ l X (z) = α. So z ∈ C α X with α ≤ α. This can be visualized as a shift to the north-east of the α-Lorenz curves of the less unequal allocation X relative to the α-Lorenz curves of the less unequal allocation X.  tributed marginals and Plackett copulas, for α = 0.9. The coefficient σ of the lognormal distributions takes value σ = 1 or 1.5, whereas the dependence parameter κ of the Plackett copula takes value κ = 2 (lower dependence) or 10 (higher dependence). In case of marginals with different σ, the asymmetry is reflected in the α-Lorenz curves.
Moreover, other things equal, inequality increases with σ, which measures inequality in the marginals, and with κ, which measures dependence of the copula.

Increasing marginal inequality and increasing correlation. A desirable
feature of the Lorenz inequality ordering of definition 4 is its ability to rank two allocations X and X, when the latter is obtained from the former through a transfer that increases inequality of the marginals or that increases the degree of dependence between the marginals. We formalize this feature with a specific type of multivariate transfer called monotone mean preserving spread, using the terminology of Quiggin [1992] in univariate risk analysis. Such transfers involve ultramodular functions, which were introduced in Marinacci and Montrucchio [2005] and applied to multidimensional inequality and risk analysis in Müller and Scarsini [2012], and which we now define.
Therefore, if the gradient ∇ψ of an ultramodular function ψ, with E [∇ψ] = 0, is added to an allocation with multivariate gradient ∇ψ X , it corresponds to a transfer with the following property: the amount of the transfer of each resource is non decreasing (component-wise) in the multivariate rank of the individual. In other words, individuals richer in resource j receive more of both resources, for j = 1, 2. We call such an unequivocally inequality increasing transfer monotone mean preserving spread.
Definition 7 (Monotone Mean Preserving Spread, MMPS). A distribution with multivariate quantile ∇ψ X is said to be more dispersed than a distribution with multivariate quantile ∇ψ X if there is an ultramodular function ψ such that for all u ∈ [0, 1] 2 , ∇ψ X (u) − ∇ψ X (u) = ∇ψ(u). (2.1) In the univariate case, equation (2.1) states that the difference between the quantile function of X and that of X is a non decreasing function. Such a difference is a monotone mean preserving spread (Quiggin [1992]), also called Bickel-Lehmann increase in dispersion (Bickel and Lehmann [1976]). A related extension in the theory of multivariate risks was proposed in Charpentier et al. [2016].
As in the univariate case, the relation " X is an MMPS of X" is a transitive relation, hence defines a partial order of increasing inequality. The following lemma states that a monotone mean preserving spread is also an increase in inequality according to our Lorenz order in definition 4.
Lemma 2 (Monotonicity in MMPS). If X is an MMPS of X, then X L X.
The multivariate Lorenz order of definition 4 therefore ranks an allocation as more unequal if a marginal is more unequal, as defined by univariate monotone mean preserving spreads. It also ranks an allocation as more unequal if the marginal resource allocations are more correlated. The former point is immediate from lemma 2 and definition 7. The latter point requires more explanation. Lemma 1 shows that allocations with a supermodular potential are positive quadrant dependent. When an allocation X is an MMPS of X, the potential ψ X of X is more supermodular than the potential ψ X of X, in the sense that for all (u 1 , u 2 ) ∈ R 2 . Relation (2.2) is the basis of a partial order of increasing dependence.

Detection of correlation increasing transfers. Consider two endowments
with identical marginal distributions but radically different joint distributions. Call Allocation X the random vector that equals (2, 0) and (0, 2) with equal probabilities.
Hence, every member of the first half of the population holds an equal endowment of the first resource, while every member of the second half of the population holds an equal endowment of the second resource. Compare it with Allocation X, which is the random vector that equals (2, 2) and (0, 0) with equal probabilities. Hence, every member of the first half of the population holds an equal endowment of the both resources, while the second half of the population holds nothing.

Multivariate Gini Index.
Lorenz ordering is a partial ordering of multivariate distributions. For a complete inequality ordering, we also propose an extension of the classical Gini index to compare inequality in multi-attribute allocations.
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 the Lorenz map r → (r 1 r 2 , r 1 r 2 ). The deviation measure we propose is where, as before, L 1 and L 2 are the first and second components of L X , respectively.
Note that the expression can be extended to allow for weights to reflect relative importance of the two resources, without changing the analysis below. After normalization, which yields the following definition.

Definition 8 (Gini Index). (2.4) defines the Gini index of allocation X.
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 function displays inequality aversion by giving more weight to the outcomes of lower ranked individuals than to those of higher ranked ones. We show in the appendix that the same interpretation is valid for our multivariate Gini, which takes the form In expression (2.5), ∂ψ X (u)/∂u 1 + ∂ψ X (u)/∂u 2 is the sum of the two 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 1 u 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.

Examples 2 and 3 continued We compare Gini indices in the independent case
with the perfect comonotonicity case, where X 1 and X 2 have the same (uniform on [0, 2]) marginal distributions. We verify (analytically for r 1 + r 2 < 1 and numerically using Wolfram for r 1 + r 2 ≥ 1) that ∂ψ X (u)/∂u 1 + ∂ψ X (u)/∂u 2 is smaller in the comonotonic case, than in the independent case. Hence the Gini index (and the measure of inequality) is larger in the comonotonic case.
Example 4 (Countermonotone Resources). If we have X 1 +X 2 = 2 a.s., then ∇ψ 1 (u)+ ∇ψ 2 (u) = 2 for almost all u, and we obtain L 1 (r 1 , r 2 ) + L 2 (r 1 , r 2 ) = 2r 1 r 2 ≥ r 2 L 1 (r 1 ) + r 1 L 2 (r 2 ), so that, in particular, the Gini index in the countermonotone case is the same as in the case of the identical allocation, i.e., equal to 0, and both are smaller than the Gini of the allocation with independent resources. This is consistent with the fact that these allocations X are considered egalitarian according to distribution. This property is referred to as anonymity. Second, the Gini index of definition 8 preserves the Lorenz inequality ordering, in the sense that higher inequality according to L implies a larger value of the Gini index. In other words, X L X implies G(X) ≤ G( X). We refer to this property as monotonicity. Finally, it satisfies a multivariate extension of comonotonic independence proposed in , as a generalization of the scalar requirement in Weymark [1981]. If individuals are ranked identically in endowments X and X, and X is more unequal than X, then, adding to both X and X a third endowment Z cannot reverse the inequality ordering, if Z ranks individuals as X and X do. Details and proofs are given in appendix B.

Egalitarian multi-attribute allocations
The identical allocation with Lorenz map (r 1 r 2 , r 1 r 2 ) is a very special instance of egalitarian allocation. We extend this narrow notion of egalitarian allocation to include income egalitarianism, in the terminology of Kolm [1977]. In the special case where the two resources are transferable with relative price p of the second resource, then an allocation is deemed egalitarian if all agents have the same budget endowment, i.e., if X 1 + pX 2 = 1 + p (where the constant value 1 + p is derived from the normalization EX 1 = EX 2 = 1). In the general case of non (or imperfectly) transferable resources, we call egalitarian the allocations with equalized shadow budgets.
Another way to interpret egalitarianism of such an allocation, beyond shadow budget equality, is through the perfect compensation of inequality in the marginal resource allocations by perfect negative correlation between resource allocations. The vector quantile and Lorenz map of egalitarian allocations can be characterized in the following way.
Proposition 2. Let (U 1 , U 2 ) be a random vector with distribution U [0, 1] 2 . (i) An egalitarian X allocation such that X 1 + pX 2 = 1 + p, admits potential ψ X (u 1 , u 2 ) = and allocation X is equal in distribution to (1 + pv (pU 1 − U 2 ), 1 − v (pU 1 − U 2 )); (ii) The Lorenz map is given by (iii) If, in addition, F −1 denotes the quantile function (generalized inverse) of X 1 , where H p is the cdf of the random variable pU 1 − U 2 ; see appendix D for an explicit expression for H p (z).
We see in proposition 2 that the distribution of the allocation X is entirely de- Assumption 2. For some p > 0, the potential ψ X of allocation X satisfies for all z ∈ [−1, p]: Before stating the main result of this section, which is an extension of property 3, we discuss sufficient conditions for assumption 2 and examples of classes of allocations that satisfy assumption 2. The following lemma provides sets of sufficient conditions based on a suitable choice of p.
Lemma 3 (Sufficient condition for assumption 2). An allocation with potential ψ X satisfies assumption 2 if any of the following conditions hold.
(i) The potential ψ X is supermodular.
(ii) The potential ψ X satisfies: is positive and constant equal to p over [0, 1] 2 and, for all z ∈ [−1, p], the Hessian of ψ X is constant over pu 1 − u 2 = z.
The first sufficient condition in lemma 3, i.e. supermodularity of the potential ψ X , imposes a form of positive dependence between the two resources, which implies assumption 2 (and 1). However, assumption 2 also accommodates allocations that do not satisfy positive dependence. For instance, the mixture of an egalitarian allocation with a positively dependent one satisfies assumption 2.
Sufficient condition (2) in lemma 3 also show that positive dependence is not required for assumption 2. Indeed, a special case of condition (2) in lemma 3 is the case, where ψ X is a quadratic function, hence has a constant Hessian. Indeed, in that case, convexity of ψ X immediately yields (3.1).
Example 6. All allocations with quadratic potential ψ X (u 1 , u 2 ) = a 1 u 1 + a 2 u 2 + a 11 u 2 1 + a 12 u 1 u 2 + a 22 u 2 2 with a 1 , a 11 , a 2 , a 22 , a 12 ∈ R, i.e., allocations of the form X = (a 1 +2a 11 U 1 +a 12 U 2 , a 2 +2a 22 U 2 +a 12 U 1 ), with (U 1 , U 2 ) ∼ U [0, 1] 2 , satisfy assumption 2 Sufficient condition (2) in lemma 3 can also be used to show that allocations where the two marginal resource allocations are independent also satisfy assumption 2. More generally, a large class of allocations defined as deviations from independence satisfy assumption 2 as formalized in the following example.
Under assumption 2, we can complement property 3 and emulate the traditional property of Lorenz curves, which are maximal at perfect equality. Here we show that egalitarian allocations dominate all allocations that satisfy assumption 2, and are themselves undominated thereby forming a class of distributions that are maximal under the Lorenz order.
Property 3 continued [Lorenz map maximal at egalitarian allocations] For any allocation X satisfying assumption 2, there is an egalitarian allocation X such that X L X, i.e., L X (r) ≤ L X (r) for all r ∈ [0, 1] 2 . In addition, if two egalitarian allocations are ranked in the Lorenz order, then they are equal.

Empirical Illustration
In this section, we propose estimators for the Lorenz map, Inverse Lorenz Function, α-Lorenz curves and Gini index and how to calculate them. We then apply these to the analysis of wealth-income inequality in the U.S. with data from the Survey of Consumer Finances (SCF) between the years 1989-2019.

Estimation of the Lorenz map, Inverse Lorenz Function, and Gini.
Suppose we have a sample X 1 , . . . , X n from a probability distribution P X with sampling weights w i such that Σ i w i = 1 and normalized to have sample mean one. Let P n be the empirical distribution relative to the weighted sample {(X 1 , w 1 ), . . . , (X n , w n )} and let λ be the Lebesgue measure over [0, 1] 2 .
Vector Quantiles and Lorenz maps. Let ∇ ψ n be an estimator for the population vector quantile ∇ψ X of Definition 1. The empirical Lorenz map is defined as the plug-in estimator One natural estimator for the vector quantile is the solution to the transport problem between the rank space and the data. By Proposition 3 in Appendix A.1, there exists a convex function ψ X : [0, 1] 2 → R ∪ {+∞} such that ∇ ψ n # λ = P n . The function ∇ ψ n exists and is unique, λ-almost everywhere. Since ∇ ψ n pushes λ forward to P n , it takes a.e. finitely many values in {X i } n i=1 . Hence, ψ n is a piecewise affine convex function, and there exists a unique (up to scalar addition) h ∈ R n such that and h is the solution of a convex optimization problem; see for instance Proposition 6 of Ekeland et al. [2012]. The resulting estimator ∇ ψ n induces a cell decomposition of the domain [0, 1] 2 into n convex polytopes, one for each observation in the sample, and each with area equal to the value of the corresponding sample weight w i . This partitioning is called a power diagram; see Aurenhammer [1987].
Viewing the transport problem as solving for the power diagram of X i with weights h has computational advantages. This is exploited in Aurenhammer et al. [1998] where they connect solving for the optimal weight vector h to an unconstrained convex optimization problem. This allows for efficient (damped) Newton or quasi-Netwon methods as in Mérigot [2011] and Kitagawa et al. [2019] for which the gradient and Hessian of the objective function are easily computed, see Lévy [2015] 1 .
The cells of the resulting power diagram are defined by We can therefore rewrite the estimator of the Lorenz map (4.1) as where X i1 and X i2 are the components of X i and the Lebesgue measure is simply the ordinary area of the convex polytope formed by the intersection of the cell W i and the rectangle [0, Inverse Lorenz Function. The estimator L n can then be used to generate a pseudo The Inverse Lorenz function l X can then be estimated with the empirical distribution of this pseudo-sample: and the α-Lorenz curves are given by the level sets of l(z).
Gini Index. Finally, the Gini index is estimated directly from the plug-in of the empirical Lorenz map: where X i1 and X i2 are the two components of individual i's endowment in the sample and the weights are which sum to one over i = 1, . . . , n.
Calculation of our Gini index can be done using Monte Carlo integration techniques.
One way is to take an average of a sample of empirical Lorenz maps evaluated from uniform random sample of ranks to estimate ( The survey over-samples higher income families who are likely to be wealthy. This is intended to counteract low survey response from high income and wealthy households.
Consequently we estimate the vector quantile function for a weighted sample. Details of the sampling technique and a discussion of specific features and issues with the data set are given in appendix C. We refer to inequality in the marginal distribution of income as income inequality and inequality in the marginal distribution of wealth as wealth inequality. We refer to inequality displayed by α-Lorenz curves as overall inequality. Figure 6 shows univariate Lorenz curves for income and for wealth for the    as well as the corresponding identical allocation curves. A common property for all clusters of curves is that they are disproportionately pulled towards the wealth axis and further away from the egalitarian curves indicating that wealth inequality is more pronounced than income inequality. For each year we see the effect on the curves of positive dependency through the curvature-the more curvature on the curve, the less positive dependence there is between the marginals. For example, for each α-level the year 2016 has the most curvature and this intuition is supported by the Kendall's τ in Figure 7 being the smallest value among these years. with support contained in a convex set X , the following holds: there exists a convex function ψ * : X → R ∪ {+∞} such that ∇ψ * # P = λ. The function ∇ψ * exists, is unique and equal to (∇ψ) −1 , P -almost everywhere.
Proposition 3 is an extension of Brenier [1991] (see also Rachev and Rüschendorf [1990]). It removes the finite variance requirement, which is undesirable in our context. Proposition 3 is the basis for the definition of vector quantiles and ranks in Chernozhukov et al. [2017]. In our context, it is applied with uniform reference measure. 2 In case d = 1, gradients of convex functions are nondecreasing functions, hence vector quantiles and ranks reduce to classical quantile and cumulative distribution functions. As the notation indicates, the function ψ * of proposition 3 is the convex conjugate of ψ. In case of absolutely continuous distributions P on R d with finite variance, the vector rank function solves a quadratic optimal transport problem, i.e., vector rank R minimizes, among all functions T such that T (X) is uniform on [0, 1] d , the quantity E X − T (X) 2 , where X ∼ P .

Appendix B. Properties of the Gini coefficient
The Gini index of definition 8 is in [0, 1] under assumption 1. It satisfies a set desirable properties for an inequality index. The first two are explained in the main text and repeated here: Property G1 (Anonymity). The Gini index of allocation X is a function of the distribution P X of X.
Property G2 (Monotonicity). The Gini index is non decreasing in the Lorenz increasing inequality ordering. In other words, X L X implies G(X) ≤ G( X).
It also satisfies a multivariate extension of comonotonic independence proposed in , as a generalization of the scalar requirement in Weymark [1981]. Two scalar endowments Y and Y are comonotonic if they are both increasing transformations of the same uniform random variable V on [0, 1]. Suppose Y and Y have respective cdfs F and F . Then Y and Y are comonotonic if they have the same Now, in case of random vectors X and X, comonotonicity is defined in the same way by the fact that X and X have the same vector ranks. The following definition is due to  and Ekeland et al. [2012], where it is called µcomonotonicity 3 .
Definition 10 (Vector comonotonicity). Random vectors X 1 , . . . , X J on R 2 are said to be comonotonic if there exists a uniform random vector U on [0, 1] 2 such that X j = ∇ψ j (U ) almost surely, where ∇ψ j is the vector quantile of definition 1 associated with the distribution of X j , for each j ≤ J.
The following property states that mixing two equally comonotonic endowments with the same third endowment, comonotonic with the first two, does not change the inequality ordering.
Property G3 (Comonotonic Independence). If X, X and Z are comonotonic allocations, and G(X) ≤ G( X), then, for all µ ∈ (0, 1), The interpretation of property G3 is very compelling for an inequality index. If individuals are ranked identically in endowments X and X, and X is more unequal than X, then, adding to both X and X a third endowment Z cannot reverse the inequality ordering, if Z ranks individuals as X and X do.
We bring these statements together in the following lemma: Proposition 4. The Gini index of definition 8 satisfies properties G1 to G3.
Proposition 4 is not a characterization of our proposed Gini index, even when we extend the definition to allow for different weights given to the different resources. In other words, there may be other inequality indices that also satisfy properties G1-3.
Property G2 is the preservation of the multivariate majorization order we introduce, based on multivariate rearrangements. Our multivariate Gini index preserves this majorization order, but not the traditional multivariate majorization of Kolm [1977]. An example of multivariate Gini that satisfies properties G1, G3 and preserves the multivariate majorization order of Kolm [1977] is G(X) := E[U ∇ψ X (U )] − 1, proposed in , where the expectation is taken with respect to U ∼ U [0, 1] 2 . However, G is unsuitable as an inequality index in our context as can be seen with the following two observations. First, the following expression shows that G only depends on L 1 (p, 1) + L 2 (1, p), hence, on very specific features of the dependence between the two components X 1 and X 2 of allocation X. (B.1) Second, and more troubling still, for any given fixed marginals for X 1 and X 2 , G is maximized when the two components X 1 and X 2 of endowment X are independent, which runs against the intuition that increased dependence can increase inequality 4 .
where G(X 1 ) and G(X 2 ) denote the classical scalar Gini index of components X 1 and X 2 respectively.
Appendix C. Specific features and issues with the data source We review some known issues with the data set that impact our analysis. See Calculation of the Gini index follows a similar procedure. Accounting for the multiple imputation in the calculation of standard errors is an important issue, but is not revelant to our visualization technique. For more information on multiple imputation and inference with imputed values, see Rubin [1996].
C.4. Computational issues. At times within the data, values can be repeated (e.g. many zeros), which have been found to pose issues calculating the quantile maps required for our Lorenz curves. Typically if this happens a few empty cells will be produced, which implies that the solution is invalid. A work-around for this is to introduce some noise to separate the data points. We added a random number between 0 and 10 for most implicates and added a random number between 0 and 200 to the data in 2007 and 2016. These numbers are too small to affect disparity when income and wealth are well over the tens of thousands in most of the stratum of the population so the visuals will not be influenced by this correction. In this section, we omit the X subscript of ψ X for notational compactness.
is distributed identically to (X 1 , X 2 ). Since X 2 = ∂ψ ∂u 2 (U 1 , U 2 ) is monotonically increasing in U 2 , we have U 2 = ∂ψ ∂u 2 −1 (U 1 , X 2 ). Hence Under the stated assumption, X 1 is increasing in U 1 and X 2 . Since X 1 , X 2 d = (X 1 , X 2 ), we have where F i (·|U j ) denotes the cumulative distribution function of X i conditional on U j .
Proof of property 3. We only need to show for one component of the Lorenz map and the second follows with similar reasoning. We have for the first component (0)). Note that H(0) = 0 as an integral over a degenerate interval, and H(1) = 1, as this is the mean of the normalized X 1 . So H(r 2 ) ≤ r 2 and so L 1 (r) ≤ r 1 r 2 , as desired.
Proof of property 4. (i) Let l X (z 1 , z 2 ) = α. Since l X is a cdf, hence non decreasing in both arguments, the generalized inverse z 2 := l −1 X (z 1 ; α) is non increasing in z 1 non decreasing in α. (ii) See Claim 1 in Brock and Thomson [1966].

D.2. Proofs for section 2.
Proof of proposition 1. X L X is equivalent to first order stochastic dominance of L X (U ) over L X (U ), where U ∼ U [0, 1] d (see Section 6.B page 266 of Shaked and Shanthikumar [2007]). Hence, X L X implies P(L X (U ) ∈ S) ≤ P(L X (U ) ∈ S) for any lower set S, so that X L X implies X l X, given that the sets [0, z] are lower sets.
Proof of lemma 2. Suppose Y is an MMPS of X, so that there is an ultramodular function ψ, such that (2.1) holds, with 1 ∇ψ(u)du = 0 (since E[Y ] = E[X] by assumption). We want to see if L Y ≤ L X , i.e., r ∇ψ(u)du ≤ 0. Consider the first component: by convexity. We then show that which holds by supermodularity of ψ. A similar reasoning applies to the second component and the result follows.
Proof of proposition 4. Property G1 is automatically satisfied since G is defined as a function of the allocation's distribution. Property G2 follows immediately from equation 2.4. We now show that Property G3 is satisfied. Take µ ∈ (0, 1). Let X, X and Z be comonotonic according to definition 10, and assume G(X) ≤ G( X), where the second and last equalities hold by lemma 4. The result follows.
We have L 1 (r 1 )1 = L 1 (r 1 ) = B ∂ψ ∂u 1 (u)du + Note that this inequality holds for any dependence structure between X 1 and X 2 .
Proof of lemma 5. This is seen by applying integration by parts a few times. Start Combining these, It is a similar line of reasoning for the second component.
Proof of (2.5). Using lemma 5, we can rewrite G as which is the desired result.
Convexity of ψ implies convexity of v. The normalization Lemma 6 (Explicit formula for H p (z)). The cumulative distribution function of Z = pU 1 − U 2 with (U 1 , U 2 ) ∼ U [0, 1] 2 is given by the following.
We now show that egalitarian allocations do not dominate each other. Suppose an egalitarian allocation X (v,p) with potential v(pu 1 − u 2 ) + u 1 + u 2 dominates an allocation X ( v, p) with potential v( pu 1 − u 2 ) + u 1 + u 2 . Then Hence, for all (r 1 , r 2 ) ∈ [0, 1] 2 , so that both allocations have the same Lorenz map, hence are equally distributed.