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Robust MD welfare comparisons (K. Bosmans, L. Lauwers) & E. Ooghe

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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)

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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.

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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’?)

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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’?

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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

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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 α j ’s=1 & ρ = 2), individuals = 1 & 2 with bundles (4,7) & (6,4), respectively, but:

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8 Notation Set of individuals I ; |I| > 1 Set of attributes J = T U N ; |T| > 0 A bundle x = (x T,x N ), element of B = A distribution X = (x 1,x 2,...), element of D = B |I| A ranking (‘better-than’ relation) on D

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9 Representation Representation (R) : There exist C 1 maps U i : 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. U i (x T, x N ) > U i (0, y N )

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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

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11 Priority Recall problems 1 & 2 Priority (P): for each X in D, for each ε in B, with ε T > 0 & ε N = 0 for all k,l in I, with we have note: can be defined without assuming R & A …

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12 Main result A ranking on D satisfies R, A, M, P iff there exist a vector p T >> 0 (for attributes in T) a str. increasing C 1 -map ψ: → R (for attributes in N) a str. increasing and str. concave C 1 -map φ: R → R, a → φ(a) such that, for each X and Y in D, we have

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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

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14 MD Dominance with |N| = 0 X Y for all orderings which satisfy R, A, M, P for all strictly increasing and strictly concave φ for all vectors p >> 0 for all vectors p>>0 (Koshevoy & Mosler’s (1999) inverse GL-criterion)

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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 R L, with a l 1 ≥ a l 2 if l 1 ≤ l 2, with L = L(X,Y) the set of needs values occuring in X or Y F X (.|l) the needs-conditional income distribution in X

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