Pigou-Dalton consistent multidimensional inequality measures:

some characterizations

C. Lasso de la Vega, A.de Sarachu, and A. Urrutia

University of the Basque Country

A consistent multidimensional generalization of the Pigou-Dalton transfer principle: an Analysis (2007) The B.E. Journal of Theoretical Economics.

PIGOU-DALTON CONSISTENT MULTIDIMENSIONAL INEQUALITY MEASURES

THE OUTLINE:1. Motivation2. Some basic notions about:

The Pigou-Dalton transfer principle in the one-dimensional framework

The Uniform Majorization

3. The proposal of a proper generalization of the Pigou-Dalton principle :The Pigou-Dalton bundle principle

4. Characterizations of classes of aggregative multidimensional inequality measures consistent with the Pigou Dalton bundle principle

5. Some concluding remarks

1. MOTIVATION

How might multidimensional distributions be compared in order to say that one distribution is more equal than another ?

Multidimensional Context:

Uniform majorization, UM

One dimensional Context:

Pigou-Dalton transfer principle

1 2

1 12 20

2 4 8

3 18 6

A A

person

person

person

X : 1 2

1 10 20

2 6 8

3 18 6

A A

person

person

person

Y :

According to the Pigou-Dalton transfer principle inequality is bound to diminish.

These two distributions can not be compared with UM.

Where

1

1i i j

j i j

y x x

y x x

0,1

xj xi

yj

Xj+

yi

Xi-+

1

1i i

j j

y x

y x

Y B X

2. BASIC NOTIONS about the Pigou-Dalton principle in the unidimensional setting

2i jx x

Bistochastic matrix:

• Non negatives entries

• each row sums to 1

• each column sums to 1

Single attribute:

individual i’s income

1

...

...i

n

x

x

x

2. BASIC NOTIONS about the Pigou-Dalton transfer principle in the unidimensional setting

Pigou-Dalton transfer principle:

if Y can be obtained from X by a finite sequence of Pigou-

Dalton progressive transfers

or equivalently

If Y =BX, where B is a bistochastic matrix I Y I X

I Y I X

X =

1

...

...i

n

x

x

x

2. BASIC NOTIONS ABOUT THE UNIFORM MAJORIZATION

11 1 1

1

1

. .

... ...

. .

j k

i ij ik

n nj nk

x x x

x x xX

x x x Bundle of attributes of the individual i

Individual i’s quantity of attribute j

incomeincome health education

Uniform majorization:

if Y= BX where B is a bistochastic matrix

(Kolm, (1977); Marshall and Olkin, (1979))

( ) ( )I Y I X

•

Attrib 1

Att

rib 2

2 3

1

•

•

•

12 20

4 8

18 6

X =

10 20

6 8

18 6

Y =

10 17

6 11

18 6

'Y =BX

Attrib 1

Att

rib 2

2 3

1

•

•

• Attrib 1

Att

rib 2

2 3

1

•

•

• •

•

•

2. BASIC NOTIONS ABOUT THE UNIFORM MAJORIZATION

Some difficulties arise with UM:

1) The reasons for transferring all attributes in the same proportions are not clear.

2) It is not necessarily the case that each attribute can be considered as transferable, for

instance for educational attainment or health status.

3) This criterion warrants transfers of different directions for different attributes, and is not

limited to cases when one individual is richer than another, being not obvious that

these transfers are inequality reducing.

Attribute 1

Att

ribut

e 2

23

1

•

•

• •

•

Attribute 1

Att

ribut

e 2

23

1

•

•

• •

• •

2. BASIC NOTIONS ABOUT THE UNIFORM MAJORIZATION

Who is the richer person?

If a person has more of all attributes than another, we can unambiguously

consider this person richer than the second.

The Pigou-Dalton Bundle principle, PD, requires that

whenever an individual i is richer than another j, a transfer of at least part of one

attribute from individual i to j that preserves the order, decreases inequality.

This principle gets over the previous difficulties:

1) It is not necessary to transfer all attributes in the same proportions.

2) the attributes which are considered as transferable can be selected.

3) It takes into consideration who is the richer and who is the poorer.

Fleurbaey and Trannoy , Soc Choice Welfare (2003)

3. THE PIGOU-DALTON BUNDLE PRINCIPLE

Following Tsui (1999):

MULTIDIMENSIONAL GENERALIZATION OF THE GENERALIZED ENTROPY FAMILY

4. OUR CHARACTERIZATION RESULTS

4. OUR CHARACTERIZATION RESULTS

•There exist measures in our family that don’t belong to the family derived by Tsui, that is,

measures fullfilling PD and not UM, and vice versa

•There exist measures in the family derived by Tsui that don’t belong to our family, that is,

measures fullfilling UM and not PD

•There exist measures belonging to both families, that is, measures fullfilling PD and UM.

1 2

1 2

/

1 i1 2 i21 i n

/

1 1 2 2

r ri1 i21 i n

r r1 2

r x r x11 0

n r rB X

x x11 0

n

1 0 1where and

Bourguignon (1999)

√

1 2 0when r r

5. FUTURE RESEARCH

1 2

1 2

r ri1 i21 i n

r

/

1 i1 2 i21 i n

/

1 1

r1

2 2

2

r x r x11 0

n rB X

x x11 0

n

r

1 0 1where and

Bourguignon (1999)

A related area of research:

The characterization of all multidimensional inequality measures that are

Pigou-Dalton Bundle consistent.

(without imposing the aggregative principle).

This paper :

• provides a greater understanding of the Pigou-Dalton transfer principle in

the multidimensional framework

• represents a step forward in the derivation of multidimensional inequality

measures by imposing some convenient properties.