1. Multidimensional Social Welfare Dominance with 4 th Order Derivatives of Utility Christophe Muller Aix-Marseille School of Economics February 2014 conf

2. Dominance and Robustness Poverty, Inequality, Social Welfare Robust judgments: accepted by people with different norms Unidimensional settings: Karamata (192?), Kolm (1969) and Atkinson (1970) Poverty Orderings, Generalized Lorenz Test Degree of aversion to inequality

3. Multi-Dimensional Setting Less obvious how to obtain powerful rules Atkinson and Bourguignon 82, 87; Koshevoy 95, 98; Moyes 99 Bazen and Moyes 03, Gravel and Moyes 12, Muller and Trannoy 11, 12 etc Signs of 4 th order derivatives generally not used because believed to be hard to interpret How to gain discriminatory power?

4. The contribution Introducing Welfare Shock Sharing Providing normative interpretations to sign conditions for 4 th degree derivatives of utility Characterisation of a new asymmetric condition: U 1112 < 0 New Necessary and Sufficient condition for SD results for several classes of utilities Poverty Gap translations Inverse SD extensions More to come…

5. Signs of derivatives of utility Two attributes Signs of derivatives of utility as normative conditions ∆ W = W F - W F* = ∫ ∫ U(x, y) d ∆ F(x, y) Continuous distributions U defined and ‘enough’ differentiable over x in ]0, a 1 ] and y in ]0, a 2 ]; or any intervals Benchmark: U 1, U 2 ≥ 0; U 12,U 11, U 22 ≤ 0 U 111, U 112, U 122, U 222 ≥ 0 Not always necessary to assume all of the above U 1111, U 1112, U 1122, U 1222, U 2222 ≤ 0

6. Welfare Shock Sharing Extending welfare notions by defining ‘social shocks’ and stating solidarity Let be 2 individuals with same bivariate endowments: (x, y) and (x, y). Then, consider the effect of some welfare shocks on this society Welfare shocks may be: losses of some attributes, risks affecting some attributes,… SWFs additives in individual utility functions

7. Let be any endowments (x,y) ∈ R ₊ ². Let c and d > 0. Let ε be a centered numerical random variable and δ be a centered numerical random variable independent of ε (i) A social planner is said to be welfare correlation averse if x-c > 0 and y-d > 0 implies that the social planner prefers the state {(x-c,y);(x,y-d)} to the state {(x,y);(x-c,y-d)} That is: `sharing fixed losses affecting different attributes is good'

8. (ii) A social planner is said to be welfare prudent in x if x+ε > 0 and x-c > 0 implies that the planner prefers the state {(x-c,y);(x+ε,y)} to the state {(x-c+ε,y);(x,y)} `sharing a fixed loss and a centred risk affecting the same attribute is good' (iii) A social planner is said to be welfare cross- prudent in x if y+δ > 0 and x-c > 0 implies that the planner prefers the state {(x,y+δ);(x-c,y)} to the state {(x,y);(x-c,y+δ)} `sharing a fixed loss and a centred risk affecting different attributes is good'

9. (iv) A social planner is said to be welfare temperate in x if x+ε > 0, x+δ > 0 and x+δ+ε > 0 implies that the planner prefers the state {(x+δ,y);(x+ε,y)} to the state {(x,y);(x+δ+ε,y)} `sharing centred risks affecting the same attribute is good' (v) A social planner is said to be welfare cross- temperate if x+ε > 0 and y+δ > 0 implies that the planner prefers the state {(x+ε,y);(x,y+δ)} to the state {(x,y);(x+ε,y+δ)} `sharing centred risks affecting different attributes is good'

10. (vi) A social planner is said to be welfare- premium correlation averse in x if x+ε > 0, x-c+ε > 0 and y-d > 0 implies that the planner prefers the state {(x-c,y);(x,y-d);(x+ε,y);(x+ε-c,y-d)} to {(x,y);(x-c,y-d);(x+ε-c,y);(x+ε,y-d)} `Sharing fixed losses affecting different attributes is good, while less important under background risk in the first attribute'

11. Equivalences (a) Welfare correlation aversion is equivalent to U ₁₂ ≤ 0 (b) Inequality aversion is equivalent to U ₁₁ ≤ 0 (c) Welfare prudence in x is equivalent to U ₁₁₁ ≥ 0 (d) Welfare temperance in x is equivalent to U ₁₁₁₁ ≤ 0 (e) Welfare cross-prudence in x is equivalent to U ₁₂₂ ≥ 0 (f) Welfare cross-temperance is equivalent to U ₁₁₂₂ ≤ 0 (g) Welfare premium correlation aversion in x is equivalent to U ₁₁₁₂ ≤ 0

12. Proof Let c be a fixed loss and ε be a centred risk Jensen’s gap for a function w: v(x,y) = w(x,y;c) - Ew(x+ε,y;c), where w(x,y;c) = U(x,y) - U(x-c,y) utility loss due to income fall v ₂ (x,y) = w ₂ (x,y;c) - Ew ₂ (x+ε,y;c) ≤ 0 iff w ₁₁₂ ≤ 0, that is: U ₁₁₁₂ ≤ 0

13. v ₂ (x,y) = w ₂ (x,y;c) - Ew ₂ (x+ε,y;c) ≤ 0, all c iff w(x,y;c) - Ew(x+ε,y;c) - w(x,y-d;c) + Ew(x+ε,y-d;c) ≤ 0, for all c and d Then, U(x,y) - U(x-c,y) - EU(x+ε,y) + EU(x-c+ε,y) - U(x,y-d) + U(x-c,y-d) + EU(x+ε,y-d) - EU(x-c+ε,y-d) ≤ 0 Therefore, for a 4-person society: U(x-c,y) + U(x,y-d) + EU(x+ε,y) + EU(x-c+ε,y-d) ≥ U(x,y) + U(x-c,y-d) + EU(x-c+ε,y) + EU(x+ε,y-d) Interpretation by decomposing in two groups

14. {(x-c,y); (x,y-d);(x+ε,y); (x+ε-c,y-d)} preferred to {(x,y); (x-c,y-d);(x+ε-c,y); (x+ε,y-d)} Utility premium p(x,y) = U(x,y) - EU(x+ε,y) p(x-c,y) + p(x,y-d) is preferred to p(x,y) + p(x-c,y-d) ‘Welfare-premium correlation aversion’ Also useful for new notions of risk apportionment

15. (s 1,s 2 )-increasing concave function and s-directionally concave functions ‘(s 1,s 2 )-icv: (s 1,s 2 )-increasing concave’: (-1) k ₁ + k2 +1 [∂ k ₁ +k2 /∂ k ₁ x ∂ k2 y] g ≥ 0 For k i = 0,..., s i ; i = 1, 2; s i non-negative integers and 1 ≤ k ₁ +k 2 ‘s-idircv: s-increasing directionally concave (-1) k ₁ +k2+1 [∂ k ₁ +k2 /∂ k ₁ x ∂ k2 y] g ≥ 0 for k ₁ and k 2 non-negative integers and 1 ≤ k ₁ +k 2 ≤ s, s is a non negative integer ≥ 2

16. R s = {(r ₁, r ₂ ) ∈ N²| 1 ≤ r ₁ +r ₂ = s} Let s be an integer greater of equal to n Let U q be the set of generators of a set of utility functions q. Then, U s-idircv = ⋂ {(r ₁,r ₂ ) ∈ Rs} U (r ₁,r ₂ )-icv

17. H x (x) = ∫ ₀ x F x (s)ds L x (x) = ∫ ₀ x ∫ ₀ t F x (s)dsdt M x (x) = ∫ ₀ x ∫ ₀ u ∫ ₀ t F x (s)dsdtdu H(x,y) = ∫ ₀ x ∫ ₀ y F(s,t)dsdt H x (x; y) = ∫ ₀ x F(s,y)ds L x (x; y) = ∫ ₀ x ∫ ₀ s F(u,y)duds M x (x; y) = ∫ ₀ x ∫ ₀ s ∫ ₀ u F(t,y)dtduds Idem by substituting the role of x and y

18. SD Results For any distributions : ∆F = F – F * All usual signs for first and second derivatives of utility (A&B82):1 st +2 nd +U 112,U 122 ≥ 0, U 1122 ≤ 0 F SD F * is equivalent to: (1) For all x, ∆H x (x) ≤ 0 (2) For all y, ∆H y (y) ≤ 0 (3) For all x, y, ∆H x (x, y) ≤ 0

19. (3,1)-icv: U ₁,U ₂ ≥ 0; U ₁₁,U ₁₂ ≤ 0; U 112,U 111 ≥ 0; U 1112 ≤ 0 ( a) △ L x (x; y) ≤ 0, for all x, y (b) △ H x (a 1 ; y) ≤ 0, for all y (c) △ F y (y) ≤ 0, for all y Idem for (1,3)-icv

20. 4-icv: U ₁ ≥ 0 ; U ₁₁ ≤ 0; U 111 ≥ 0; U 1111 ≤ 0 Unidimensional: results already known Now a good reason to assume U 1111 ≤ 0 because sharing risk is good for social welfare ( a) △ M x (x) ≤ 0, for all x (b) △ L x (a 1 ) ≤ 0 (c) △ H x (a 1 ) ≤ 0 Idem with y

21. 4-idircv: U ₁,U ₂ ≥ 0; U 11,U 12,U 22 ≤ 0; U 111,U 112,U 122,U 222 ≥ 0; U 1111,U 1222,U 1122, U 1112, U 2222 ≤ 0 Has a class of generators that is the intersection of the classes of generators of (s 1, s 2 )-icv functions sets with (s 1, s 2 ) in {(2,2),(3,1),(1,3),(4,0),(0,4)} So far, the generators of this class were not known

22. Change in variable in the complex plan z = x + i y = ρ e iθ Modulus ρ = |z| = sqrt (x 2 + y 2 ) θ = Arg z in [0, π/2] since x, y > 0 Theorem: 4-idircv in (x,y) is equivalent to 4-icv in ρ

23. With a 1 = a 2 = + infinity ( a) △ M ρ (ρ) ≤ 0, for all ρ (b) △ L ρ (π/2) ≤ 0 (c) △ H ρ (π/2) ≤ 0

24. Conclusion A new normative approach: Welfare Shock Sharing Normative interpretations of the signs of 4 th degree derivatives of utilities A new characterization for U 1112 < 0 Necessary and Sufficient SD results for several classes of functions To come: Poverty gaps and Lorenz To come: More dimensions and higher degree To come: Generalised polar stochastic dominance More on risk analysis