1. Unequal Longevities and Compensation Marc Fleurbaey (CNRS, CERSES, U Paris 5) Marie-Louise Leroux (CORE, UC Louvain) Gregory Ponthiere (ENS, PSE) Social Choice and Welfare Meeting Moscow, 24 July 2010.

2. Introduction (1) Facts Large longevity differentials (even within cohorts). Distribution of age at death: 1900 Swedish female cohort Sources: Human Mortality Database

3. Introduction (2) Which allocation of resources under unequal longevities? Existing literature relies on classical utilitarianism: Bommier Leroux Lozachmeur (2009, 2010) Leroux Pestieau Ponthiere (2010) Problem: utilitarianism implies transfers from short-lived to long-lived agents…against the intuition of compensation! Which compensation of the short-lived? Some problems raised by the compensation of short-lived: -Short-lived persons can hardly be identified ex ante. -It is impossible to compensate short-lived persons ex post.

4. Introduction (3) Our contributions This paper is devoted to the construction of a measure of social welfare that is adequate for allocating resources among agents who turn out to have unequal longevities. We show from some plausible ethical axioms that an adequate social objective is the Maximin on the Constant Consumption Profile Equivalent on the Reference Lifetime (CCPERL). For any agent i and any lifetime consumption profile of some length, the CCPERL is the constant consumption profile of a length of reference ℓ* that makes the agent i indifferent with his lifetime consumption profile. We propose to compare allocations by focusing on the minimum of such homogenized consumptions.

5. Introduction (4) Comparing allocations Allocation A Allocation B C i j j Age Age

6. Introduction (5) The CCPERL Allocation A Allocation B C ~ i C ~ i ~ j ~ j ℓ* Age ℓ* Age Under Maximin on CCPERL allocation B is preferred to A.

7. Introduction (6) Our contributions We also compute the optimal allocation of resources in various contexts where the social planner ignores individual longevities before agents effectively die. We consider different degrees of observability of individual preferences and life expectancies (FB and SB). A key result is that the social planner can improve the lot of short-lived agents by inducing everyone to save less. Outline 1.The framework 2.Ethical axioms 3.Two characterizations of social preferences 4.Optimal allocation in a 2-period model with heterogeneity

8. The framework (1) N = set of individuals, with cardinality |N|. T = the maximum lifespan (T ℕ). x i is a lifetime consumption profile, i.e. a vector of dimension T or less. X = U ℓ=1 T ℝ + ℓ is the set of lifetime consumption profiles x i. The longevity of an individual i with consumption profile x i is a function λ: X → ℕ such that λ(x i ) is the dimension of the lifetime consumption profile, i.e. the length of existence for i. An allocation defines a consumption profile for all individuals in the population N: x N := (x i ) iN X |N|. Each individual i has well defined preference ordering R i on X (i.e. a reflexive, transitive and complete binary relation). I i denotes the indifference and P i the strict preference.

9. The framework (2)  is the set of preference orderings on X satisfying two properties: -For any lives x i and y i of equal lengths, preference orderings R i on x i and y i are assumed to be continuous, convex and weakly monotonic (i.e. x i ≥ y i => x i R i y i and x i >> y i => x i P i y i ). -For all x i X, there exists (c,…,c) ℝ + T such that x i I i (c,..,c), i.e. no lifetime consumption profile is worse or better than all lifetime consumption profiles with full longevity. This excludes lexicographic preferences wrt longevity. A preference profile for N is a list of preference orderings of the members of N, denoted R N := (R i ) iN  |N|. A social ordering function ≿ associates every preference profile R N with an ordering ≿ RN defined on X |N|.

10. Ethical axioms (1) Axiom 1: Weak Pareto (WP) For all preference profiles R N  |N|, all allocations x N, y N X |N|, if x i P i y i for all i N, then x N ≻ RN y N. Axiom 2: Hansson Independence (HI) For all preference profiles R N, R N ’  |N|, and for all allocations x N, y N X |N|, if for all i N, I(x i, R i ) = I(x i, R i ’) and I(y i, R i ) = I(y i, R i ’), then x N ≿ RN y N if and only if x N ≿ RN’ y N. where I(x i, R i ) is the indifference set at x i for R i defined such that I(x i, R i ) := {y i X | y i I i x i }. ~ The social preferences over two allocations depend only on the individual indifference curves at these allocations.

11. Ethical axioms (2) Axiom 3: Pigou-Dalton for Equal Preferences and Equal Lifetimes (PDEPEL) For all preference profiles R N  |N|, all allocations x N, y N X |N|, and all i, j N, if R i = R j and if λ(x i ) = λ(y i ) = λ(x j ) = λ(y j ) = ℓ, and if there exists δ ℝ ++ ℓ such that y i >> x i = y i – δ >> x j = y j + δ >> y j and x k = y k for all k ≠ i, j, then x N ≿ RN y N ~ For agents identical on everything (longevities, preferences) except consumptions, a transfer from a high-consumption agent to a low-consumption agent is a social improvement.

12. Ethical axioms (3) Axiom 4: Pigou-Dalton for Constant Consumption and Reference Lifetime (PDCCRL) For all preference profiles R N  |N|, all allocations x N, y N X |N|, and all i, j N, such that λ(x i ) = λ(y i ) = λ(x j ) = λ(y j ) = ℓ*, and x i and x j are constant consumption profiles, if there exists δ ℝ ++ ℓ* such that y i >> x i = y i – δ >> x j = y j + δ >> y j and x k = y k for all k ≠ i, j, then x N ≿ RN y N ~ If two agents have a longevity of reference ℓ*, a transfer that lowers the constant consumption profile of the rich and raises the profile of the poor is a social improvement.

13. Characterization of social preferences (1) Definition For any i N, R i  and x i X, the CCPERL of x i is the constant consumption profile x i such that λ(x i ) = ℓ* and x i I i x i Theorem Assume that the social ordering function ≿ satisfies WP, HI, PDEPEL and PDCCRL on  |N|. Then ≿ is such that for all R N  |N|, all x N, y N X |N|, min(x i ) > min(y i ) => x N ≻ RN y N i N i N where x i is the CCPERL of agent i under allocation x N. ~ Under axioms WP, HI, PDEPEL and PDCCRL, the social ordering satisfies the Maximin property on the CCPERL.

14. Characterization of social preferences (2) Alternative characterization of social preferences Take longevity as a continuous variable; a consumption profile is now a function x i (t) defined over the interval [0, T]. Axiom 5: Inequality Reduction around Reference Lifetime (IRRL) For all preference profiles R N  |N|, all allocations x N, y N X |N|, and all i, j N, such that λ(x i ) = ℓ i, λ(y i ) = ℓ i ’, λ(x j ) = ℓ j, λ(y j ) = ℓ j ’, and some c ℝ ++ is the same constant per-period level of consumption for x i, y i, x j, y j, if ℓ j, ℓ j ’ ≤ ℓ* ≤ ℓ i, ℓ i ’ and ℓ j - ℓ j ’ = ℓ i ’ - ℓ i > 0 and x k = y k for all k ≠ i, j, then x N ≿ RN y N

15. Characterization of social preferences (3) -Remark: the plausibility of IRRL depends on the monotonicity of preferences wrt longevity. => our alternative theorem will focus on preference profiles with monotonicity of preferences wrt longevity. Replacing the two Pigou-Dalton axioms by IRRL yields the following alternative characterization of the social ordering. Theorem Assume that the social ordering function ≿ satisfies WP, HI and IRRL on  * |N|. Then ≿ is such that for all R N  * |N|, all x N, y N X |N|, min(x i ) > min(y i ) => x N ≻ RN y N i N i N

16. Optimal allocation in a 2-period model (1) Assumptions Minimum longevity = 1 period. Maximum longevity T = 2 periods. Reference longevity ℓ* = 2 periods. Total endowment: W. Utility of death normalized to 0. Intercept of temporal utility function non negative (u(0) ≥ 0). Time-additive lifetime welfare + Expected utility hypothesis: Ex ante (expected) lifetime welfare: u(c ij ) + π j β i u(d ij ) Ex post lifetime welfare: Short: u(c ij ) ; Long: u(c ij ) + β i u(d ij )

17. Optimal allocation in a 2-period model (2) Assumptions (continued) Two sources of heterogeneity: -Time preferences: 0 < β 1 < β 2 < 1 -Survival probabilities: 0 < π 1 < π 2 < 1 No individual savings technology: Consumption bundles (c ij, d ij ) for agents with time preferences parameter β i and survival probability π j must be consumed as such. Solving strategy: 4 groups ex ante (low / high patience and life expectancy). The solution requires to compute 8 CCPERL (i.e. as each group will include ex post short-lived and long-lived agents).

18. Optimal allocation in a 2-period model (3) Maximin on CCPERL (FB: perfect observability of β i, π j ) Max Min x i = (c ijℓ, c ijℓ ) s.t. Σ i,j c ij + π j d ij = W Solution: c 21 = c 22 > c 11 = c 12 > d 21 = d 22 = d 11 = d 12 = 0 -d ij = 0 to compensate the short-lived as much as possible; -more consumption for patient agents, who need more compensation for a short life than impatient agents. Maximin on CCPERL (SB: imperfect observability of β i, π j ) Max Min x i = (c ijℓ, c ijℓ ) s.t. Σ i,j c ij + π j d ij = W s.t. IC constraints Solution: c 11 = c 12 = c 21 = c 22 > d 21 = d 22 = d 11 = d 12 = 0

19. Optimal allocation in a 2-period model (4) Extensions and generalizations The utility of zero consumption: u(0) < 0 FB: Maximin CCPERL gives d ij = d* such that u(d*) = 0 to old agents, and c ij > c 1j. SB: Maximin CCPERL gives d ij = d*; and equal c ij to all agents. Reference longevity ℓ* = 1 (still with u(0) > 0) FB: Maximin CCPERL equalizes all c ij and gives d ij = 0. SB: Maximin CCPERL equalizes all c ij and gives d ij = 0 (= FB). Savings technology for all (still with u(0) > 0) FB: Maximin CCPERL differentiates endowments W ij according to patience (+) and survival probability (+). SB: Maximin CCPERL equalizes all W ij.

20. Concluding remarks Can one compensate short-lived agents? Our answer: YES WE CAN! Our solution: to apply the Maximin on CCPERL. That social objective follows from intuitive ethical axioms. The optimum involves differentiated compensation. The optimum involves decreasing consumption profiles... … At odds with the observed profiles (inverted U shaped)… Main limitation: exogenous survival.