two univariate marginals are ordered in the usual sense.

Unfortunately, apart from the case of independent marginals, neither the Lorenz “

Taguchi nor the Lorenz “ Arnold approach has a simple economic interpretation.

Koshevoy and Mosler (1996) took a different route. Their multivariate Lorenz

“curve” is a manifold in (d þ 1) space. They started from a special view of the univariate

Lorenz curve: De¬ning a dual Lorenz curve via L(u) ¼ 1 À L(1 À u), 0 u 1, they

introduced the Lorenz zonoid as the convex region bordered by L and L. Clearly, the area

between these two curves is equal to the classical Gini coef¬cient.

The Koshevoy “Mosler multivariate Lorenz curve is now given by a

generalization of this idea to (d þ 1) space. Let Ld be the set of d-variate random

Ð

variables that have ¬nite and positive expectation vectors, E(Xj ) ¼ Rd xj dF(x) . 0,

þ

j ¼ 1, . . . , d, and set

x ¼ (˜ 1 , . . . , xd )` ,

˜ ˜ ˜ j ¼ 1, . . . , d:

xj ¼ xj =E(Xj ),

x

˜

Thus, X is the normalization of X with expectation 1d ¼ (1, . . . , 1)` . If we set

"° #

°

6(h) ¼ h(x) dF(x), h(x)˜ dF(x) ,

x

Rd Rd

þ þ

for every (measurable) h : Rd ! [0, 1], the d þ 1 dimensional Lorenz zonoid

þ

LZ(F) is de¬ned as

LZ(F) ¼ {z [ Rdþ1 j z ¼ (z0 , z1 , . . . , zd )` ¼ 6(h)}: (2:47)

The Lorenz zonoid is therefore a convex compact subset of the unit (hyper) cube

dþ1

in Rþ ”it may be shown to be strictly convex if F is an absolutely continuous

distribution”containing the origin as well as the point 1dþ1 ¼ (1, . . . , 1)` in Rdþ1 .

Now that we have de¬ned a generalization of the area between the Lorenz and

inverse Lorenz functions as a convex set in d þ 1 space, it remains to de¬ne a

generalization of the curve itself. The solution is as follows. Consider the set

( )

°

Z(F) ¼ y [ Rd y ¼ h(x)˜ dF(x), h : Rd ! [0, 1] measurable : (2:48)

x

þ þ

d

Rþ

Z(F) is called the F zonoid. Note that if (z0 , z1 , . . . , zd )` [ LZ(F), then

(z1 , . . . , zd )` [ Z(F). Z(F) is contained in the unit cube of Rd and consists of all

þ

˜

total portion vectors x held by subpopulations. In particular, for d ¼ 1 the F zonoid

Z(F) is equal to the unit interval. Now for a given (z1 , . . . , zd )` [ Z(F), we have

(z0 , z1 , . . . , zd )` [ LZ(F) if and only if z0 is an element of the closed interval

between the smallest and largest percentage of the population by which the portion

vector (z1 , . . . , zd )` is held.

41

2.1 SOME CONCEPTS FROM ECONOMICS

The function lF Z(F) ! Rþ de¬ned by

lF (y) ¼ max{t [ Rþ j (t, y) [ LZ(F)} (2:49)

is now called the d-dimensional inverse Lorenz function and its graph is the

Koshevoy “Mosler Lorenz surface of F. Because LZ(F) and Z(F) are convex sets, lF

is a concave function that is continuous on Z(F) > Rd , with lF (1d ) ¼ 1. If F has no

þ

mass at the origin, then lF (0) ¼ 0. In this sense, it represents a natural generalization

of the univariate Lorenz curve. It may also be shown that the Lorenz surface

determines the distribution F uniquely up to a vector of scaling factors. [It should be

noted that the Lorenz surface is not necessarily a surface in the usual sense but a

manifold that may have any dimension between 1 and d. Its dimension equals the

dimension of Z(F).]

In terms of a distribution of commodities, lF (y) is equal to the maximum

percentage of the population whose total portion amounts to y. The Lorenz zonoid

has the following economic interpretation: To every unit of a population the vector X

˜

of endowments in d commodities is assigned. This unit then holds the vector X of

portions of the mean endowment. A given function h may now be considered a

selection of a subpopulation. Of all those units that have endowment vector x or

Ð

˜

portion vector x, the percentage h(x) is selected. Thus, h(x) dF(x) is the size of the

Ð

subpopulation selected by h, and h(x)˜ dF(x) amounts to the total portion vector

x

held by this subpopulation.

The Koshevoy“ Mosler multivariate Lorenz order is now de¬ned as the set

inclusion ordering of Lorenz zonoids, that is,

F !LZ G :( LZ(F) $ LZ(G): (2:50)

)

It has the appealing property that it implies the classical Lorenz ordering of all

univariate marginal distributions.

In view of the geometric motivation of the classical Gini coef¬cient (2.21), the

question of what a multivariate Gini index might look like arises. Koshevoy and

Mosler (1997) discussed a multivariate Gini index de¬ned as the volume of their

Lorenz zonoid LZ(F), speci¬cally

1

G :¼ vol[LZ(F)] ¼ E(jdet QF j), (2:51)

Qd

(d þ 1)! j¼1 E(Xj )

where QF is the (d þ 1) ‚ (d þ 1) matrix with rows (1, Xi ), i ¼ 1, . . . , d þ 1, and

X1 , X2 , . . . , Xdþ1 are i.i.d. with the c.d.f. F.

It follows that this multivariate Gini coef¬cient may be equal to zero without all

commodities being equally distributed. In fact, it will be equal to zero if at least one

of the commodities is equally distributed or if two commodities have the same

42 GENERAL PRINCIPLES

distribution. (It will also be equal to zero if there are fewer income receiving units

than commodities, that is, if n , d.)

2.1.6 Zenga Curves and Associated Inequality Measures

Fairly recently, an alternative to the Lorenz curve has received some attention in the

Italian literature. Like the Lorenz curve, the Zenga curve (Zenga, 1984) is de¬ned

via the ¬rst-moment distribution, hence, we require E(X ) , 1. The Zenga curve is

now de¬ned in terms of the quantiles of the size distribution and the corresponding

¬rst-moment distribution: For

F(1) (u) À F À1 (u)

À1

F À1 (u)

, 0 , u , 1,

Z(u) ¼ ¼1À (2:52)

À1 À1

F(1) (u) F(1) (u)

the set