Robust MD welfare comparisons

(K. Bosmans, L. Lauwers) & E. Ooghe

2

Overview

UD setting Axioms & result Intersection = GLD

From UD to MD setting: Anonymity Two problems

Notation Axioms General result

Result1 + Kolm’s budget dominance & K&M’s inverse GLD Result2 + Bourguignon (89)

3

UD setting

Axioms to compare distributions:

Representation (R): Anonymity (A) : names of individuals do not matter Monotonicity (M) : more is better Priority (P): if you have an (indivisible) amount of the single

attribute, then it is better to give it to the ‘poorer’ out of two individuals

Result : with U strictly increasing and strictly

concave.

Iiii

Iiii yUxUYX

, Iii

Iii yUxUYX

nni xxxXnI ,,,, and ,,2,1 1

4

Robustness in the UD setting

X Y for all orderings which satisfy R, A, M, P

for all U strictly increasing & strictly concave

X Y

Ethical background for MD dominance criteria?

(or … ‘lost paradise’?)

GL

, Iii

Iii yUxU

5

From UD to MD setting

Anonymity only credible, if all relevant characteristics are included … MD analysis!

Recall Priority in UD setting:

“if you have an (indivisible) amount of the single attribute, then it is better to give it to the ‘poorer’ out of 2 individuals”

Two problems for P in MD setting: Should P apply to all attributes? How do we define ‘being poorer’?

nmni xxxXmJnI

,,,, and ,,2,1,,,2,1 1

6

Should P apply to all attributes?

Is P an acceptable principle for all attributes? e.g., 2 attributes: income & (an ordinal index of) needs?

(Our) solution: given a cut between ‘transferable’ and ‘non-transferable’ attributes, axiom P only applies to the ‘transferable’ ones

Remark: whether an attribute is ‘transferable’ or not is not a physical characteristic of the attribute, but depends on whether the attribute should be included in the

definition of the P-axiom, thus, …, a ‘normative’ choice

7

How do we define ‘being poorer’?

In contrast with UD-setting, ‘poorer’ in terms of income and ‘poorer’ in terms of well-being do not necessarily coincide anymore

(Our) solution: Given R & A, we use U to define ‘being poorer’

Remark: Problematic for many MD welfare functions; e.g.:

attributes = apples & bananas (with αj’s=1 & ρ = 2),

individuals = 1 & 2 with bundles (4,7) & (6,4), respectively, but:

Ii Jj

ijIi

i jxxU

1

1

1

.416

1

74

1-0.071429-0.070238

46

1

714

1

8

Notation

Set of individuals I ; |I| > 1

Set of attributes J = T U N ; |T| > 0

A bundle x = (xT,xN), element of B =

A distribution X = (x1,x2,...), element of D = B|I|

A ranking (‘better-than’ relation) on D

NT RR

9

Representation

Representation (R): There exist C1 maps Ui : B → R ,

s.t. for all X, Y in D, we have

note: has to be complete, transitive, continuous & separable differentiability can be dropped, as well as continuity over

non-transferables (but NESH, in case |N| > 0) for all i in I, for all there exists a s.t. Ui(xT , xN ) > Ui(0 , yN )

Iiii

Iiii yUxUYX

TTx R

NNN yx R,

10

Anonymity & Monotonicity

Anonymity (A): for all X, Y in D, if X and Y are equal up to a permutation (over individuals), then X ~ Y

Monotonicity (M): for all X, Y in D, if X > Y, then X Y note:

interpretation of M for non-transferables M for non-transferables can be dropped

11

Priority

Recall problems 1 & 2

Priority (P): for each X in D, for each ε in B, with εT > 0 & εN = 0

for all k,l in I, with

we have

note: can be defined without assuming R & A …

,,,, ,,,, lklk xxxx

lk xUxU

12

Main result

A ranking on D satisfies R, A, M, P iff there exist a vector pT >> 0 (for attributes in T)

a str. increasing C1-map ψ: → R (for attributes in N) a str. increasing and str. concave C1-map φ: R → R , a → φ(a)

such that, for each X and Y in D, we have

Ii

iN

iTTIi

iN

iTT yypxxpYX

NR

13

Discussion

Possibility or impossibility result? Related results:

Sen’s weak equity principle Ebert & Shorrock’s conflict Fleurbaey & Trannoy’s impossibility of a Paretian egalitarian

… “fundamental difficulty to work in two separate spaces”

Might be less an objection for dominance-type results This result can be used as an ethical foundation for two, rather

different MD dominance criteria: Kolm’s (1977) budget dominance criterion Bourguignon’s (1989) dominance criterion

Ii

iN

iTTIi

iN

iTT yypxxpYX

14

MD Dominance with |N| = 0

X Y for all orderings which satisfy R, A, M, P

for all strictly increasing and strictly concave φfor all vectors p>>0

for all vectors p>>0

(Koshevoy & Mosler’s (1999) inverse GL-criterion)

Ii

i

Ii

i ypxp

, iGL

i ypxp

zyw

nw

n zxw

nw

ni

Ii

i

Ii

i

,w

i

Ii

i

Ii

i

,w II

1111maxmax

1010

15

MD dominance with |T| = |N| = 1

X Y for all orderings which satisfy R, A, M, P

for all strictly increasing and strictly concave φ

for all strictly increasing ψ

for all a in RL, with al1 ≥ al2 if l1 ≤ l2 , with

L = L(X,Y) the set of needs values occuring in X or Y

FX(.|l) the needs-conditional income distribution in X

Ii

iN

iTIi

iN

iT yyxx

, ||,,

YXLl

a

z YYXLl

a

z X

ll

lzFlzF