Robust MD welfare comparisons (K. Bosmans, L. Lauwers) & E. Ooghe.
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Transcript of Robust MD welfare comparisons (K. Bosmans, L. Lauwers) & E. Ooghe.
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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)
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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.
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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
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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’?
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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
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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:
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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
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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 )
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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
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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 …
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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
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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
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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)
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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
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