Robust Welfare Comparisons and Ethical Foundations in Multidimensional Setting

1. 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 αj’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 UN ; |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

9. Representation • Representation (R):There exist C1maps 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 )

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 εin B, with εT > 0 & εN = 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 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

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 allstrictly 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 allstrictly increasing and strictly concave φ for all strictly increasing ψ for all a in RL, with al1≥al2if 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