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

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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.

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’?)

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’?

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

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

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 )

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 ε T > 0 & ε N = 0 for all k,l in I, with we have  note: can be defined without assuming R & A …

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

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

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)

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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