## Amse-aixmarseille.fr

On Multidimensional inequality with variable household
DEC, Univ. of Pescara & DEPS, Univ of Siena
Ernesto Savaglio (DEC, Univ. of Pescara & DEPS,
We provide an analogue to the classic HLP theorem consisting inproving that a version of the result due to Schur and Ostrowski onthe class of majorization order-preserving functions also holds in ourmultidimensional setting.

We address the issue of assessing multidimensional inequality byintroducing a new ordering that compare (discrete) multidimensionaldistributions representing households that di¤er in severalcharacteristics besides income and that could have di¤erent size andweights.

Ernesto Savaglio (DEC, Univ. of Pescara & DEPS,
We address the issue of assessing multidimensional inequality byintroducing a new ordering that compare (discrete) multidimensionaldistributions representing households that di¤er in severalcharacteristics besides income and that could have di¤erent size andweights.

We provide an analogue to the classic HLP theorem consisting inproving that a version of the result due to Schur and Ostrowski onthe class of majorization order-preserving functions also holds in ourmultidimensional setting.

Ernesto Savaglio (DEC, Univ. of Pescara & DEPS,
For any x, y 2 @, the following two conditions are equivalent:
holds for any f :

**R **!

**R **that is convex and symmetric, that is

invariant under any permutation of the coordinates.

Ernesto Savaglio (DEC, Univ. of Pescara & DEPS,
The second approach studies the properties of multidimensionalevaluative inequality statistics following a two-steps procedure(Gajdos and Weymark (2005), Kolm (1977), Maasoumi (1986, 1999),Tsui (1995, 1999)).

The third approach applies some tools of convex analysis andintroduces some new attractive analytical criteria to comparemultivariate distributions (Joe and Verducci (1993), Koshevoy (1995,1998), Koshevoy and Mosler (1996), Savaglio (2006, 2011)).

From one-to-multi dimensional inequality measurement
Criteria borrowed from the literature on stochastic dominance whichare consistent with some very speci…c classes of social evaluationfunctionals. (Atkinson and Bourguignon (1982, 1987), Kolm (1977),List (1999), Mosler (1991)).

Ernesto Savaglio (DEC, Univ. of Pescara & DEPS,
The third approach applies some tools of convex analysis andintroduces some new attractive analytical criteria to comparemultivariate distributions (Joe and Verducci (1993), Koshevoy (1995,1998), Koshevoy and Mosler (1996), Savaglio (2006, 2011)).

From one-to-multi dimensional inequality measurement
Criteria borrowed from the literature on stochastic dominance whichare consistent with some very speci…c classes of social evaluationfunctionals. (Atkinson and Bourguignon (1982, 1987), Kolm (1977),List (1999), Mosler (1991)).

The second approach studies the properties of multidimensionalevaluative inequality statistics following a two-steps procedure(Gajdos and Weymark (2005), Kolm (1977), Maasoumi (1986, 1999),Tsui (1995, 1999)).

Ernesto Savaglio (DEC, Univ. of Pescara & DEPS,
From one-to-multi dimensional inequality measurement
Criteria borrowed from the literature on stochastic dominance whichare consistent with some very speci…c classes of social evaluationfunctionals. (Atkinson and Bourguignon (1982, 1987), Kolm (1977),List (1999), Mosler (1991)).

The second approach studies the properties of multidimensionalevaluative inequality statistics following a two-steps procedure(Gajdos and Weymark (2005), Kolm (1977), Maasoumi (1986, 1999),Tsui (1995, 1999)).

The third approach applies some tools of convex analysis andintroduces some new attractive analytical criteria to comparemultivariate distributions (Joe and Verducci (1993), Koshevoy (1995,1998), Koshevoy and Mosler (1996), Savaglio (2006, 2011)).

Ernesto Savaglio (DEC, Univ. of Pescara & DEPS,
rs (JME, 2011): convex analysis, convex geometry, linear operator
h (??, 2016): convex analysis; matrix theory;
vp (ET, 2006): convex analysis, matrix theory.

Ernesto Savaglio (DEC, Univ. of Pescara & DEPS,
h (??, 2016): convex analysis; matrix theory;
vp (ET, 2006): convex analysis, matrix theory.

rs (JME, 2011): convex analysis, convex geometry, linear operator
Ernesto Savaglio (DEC, Univ. of Pescara & DEPS,
vp (ET, 2006): convex analysis, matrix theory.

rs (JME, 2011): convex analysis, convex geometry, linear operator
h (??, 2016): convex analysis; matrix theory;
Ernesto Savaglio (DEC, Univ. of Pescara & DEPS,
In sample, weights are needed in order to correct for possible biases inthe data;
In grouped data, the available information typically consists of thefrequencies (and income means) for every group;
Theoretical works on equivalence scales has stressed the importance ofweights for distributional analysis in the case of heterogeneoushouseholds.

What about di¤erent weights of individuals?
Ernesto Savaglio (DEC, Univ. of Pescara & DEPS,
In grouped data, the available information typically consists of thefrequencies (and income means) for every group;
Theoretical works on equivalence scales has stressed the importance ofweights for distributional analysis in the case of heterogeneoushouseholds.

What about di¤erent weights of individuals?
In sample, weights are needed in order to correct for possible biases inthe data;
Ernesto Savaglio (DEC, Univ. of Pescara & DEPS,
Theoretical works on equivalence scales has stressed the importance ofweights for distributional analysis in the case of heterogeneoushouseholds.

What about di¤erent weights of individuals?
In sample, weights are needed in order to correct for possible biases inthe data;
In grouped data, the available information typically consists of thefrequencies (and income means) for every group;
Ernesto Savaglio (DEC, Univ. of Pescara & DEPS,
What about di¤erent weights of individuals?
In sample, weights are needed in order to correct for possible biases inthe data;
In grouped data, the available information typically consists of thefrequencies (and income means) for every group;
Theoretical works on equivalence scales has stressed the importance ofweights for distributional analysis in the case of heterogeneoushouseholds.

Ernesto Savaglio (DEC, Univ. of Pescara & DEPS,
Let K 2

**N**. For each n 2

**N **we de…ne:

(x1, ., xn) : xi 2

**R**k , 1

Given X = (x1, ., xn) 2 Γkn, we denote by

**M**m,n the set of all

row-stochastic m

Ernesto Savaglio (DEC, Univ. of Pescara & DEPS,
Let (X , a) 2 Λkm and (Y , b) 2 Λkn. Then, we say that (X , a) ish-majorized by (Y , b), denoted to (X , a)
exists a matrix R 2

**M**m,n such that

Ernesto Savaglio (DEC, Univ. of Pescara & DEPS,
population of three and two individuals respectively endowed with thesame two goods with di¤erent a- uence and two weights’systemsrepresenting for example two equivalence scales of two di¤erent States:
a = (0.2, 0.3, 0.5) and b = (0.4, 0.6) .

Since there exists a row-stochastic matrix:
that Y has a greater level of disparity than X .

Ernesto Savaglio (DEC, Univ. of Pescara & DEPS,
Let (X , a) 2 Λkm and (Y , b) 2 Λkn. The following conditions areequivalent:
(ii ) For any continuous convex function

*φ *:

**R**k !

**R**
∑ ai

*φ *(xi) ∑ bi

*φ *(yj).

Ernesto Savaglio (DEC, Univ. of Pescara & DEPS,
h is a preorder on Λk , and if m = n and ai = bj = 1 ,
n, then Theorem 1 and Birkho¤-Von Neumann’s Theorem
(see Marshall and Olkin (1979)) imply that the h-majorization inducesthe partial ordering analyzed among others in Marshall and Olkin(1979) chapter 15 de…nition A.2 and commonly referred to asstandard multivariate majorization. Hence:
Ernesto Savaglio (DEC, Univ. of Pescara & DEPS,
If X , Y 2 Γkn then the following conditions are equivalent:(i ) The inequality ∑ni=1

*ϕ *(xi )
∑ni=1

*ϕ *(yi ) holds for any convex function

*ϕ *:

**R**k !

**R**;

(ii ) There exists some doubly stochastic matrix P such that X = PY .

for k = 1, Corollary amounts to the well-known (vector) majorizationpreorder as studied by Hardy, Littlewood and Polya (1934).

Ernesto Savaglio (DEC, Univ. of Pescara & DEPS,

Source: http://www.amse-aixmarseille.fr/sites/default/files/mars.pdf

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