Unequal Longevities and Compensation

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 iindifferent 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 C i 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 • The framework • Ethical axioms • Two characterizations of social preferences • 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 ϵℕ). • xi is a lifetime consumption profile, i.e. a vector of dimension T or less. • X = Uℓ=1Tℝ+ℓ is the set of lifetime consumption profiles xi. • The longevity of an individual i with consumption profile xi is a function λ: X → ℕsuch that λ(xi) 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: xN := (xi)iϵN ϵ X|N| . • Each individual i has well defined preference ordering Ri on X (i.e. a reflexive, transitive and complete binary relation). Ii denotes the indifference and Pi the strict preference.

9. The framework (2) •  is the set of preference orderings on X satisfying two properties: • For any lives xi and yi of equal lengths, preference orderings Ri on xi and yi are assumed to be continuous, convex and weakly monotonic (i.e. xi ≥ yi => xiRiyi and xi >> yi => xiPiyi). • For all xiϵ X, there exists (c,…,c) ϵℝ+T such that xi Ii (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 RN := (Ri)iϵN ϵ|N|. • A social ordering function ≿ associates every preference profile RN with an ordering ≿RN defined on X|N|.

10. Ethical axioms (1) • Axiom 1: Weak Pareto (WP) For all preference profiles RN ϵ|N|, all allocations xN, yNϵ X|N|, if xi Pi yi for all i ϵ N, then xN≻RN yN. • Axiom 2: Hansson Independence (HI) For all preference profiles RN, RN’ ϵ|N|, and for all allocations xN, yNϵ X|N|, if for all i ϵ N, I(xi, Ri) = I(xi, Ri’) and I(yi, Ri) = I(yi, Ri’), then xN≿RN yN if and only if xN≿RN’ yN. where I(xi, Ri) is the indifference set at xi for Ri defined such that I(xi, Ri) := {yiϵ X | yi Ii xi}. ~ 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 RN ϵ|N|, all allocations xN, yNϵ X|N|, and all i, j ϵ N, if Ri = Rj and if λ(xi) = λ(yi) = λ(xj) = λ(yj) = ℓ, and if there exists δϵℝ++ℓ such that yi >> xi = yi – δ >> xj = yj + δ >> yj and xk = yk for all k ≠ i, j, then xN≿RN yN ~ 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 RN ϵ|N|, all allocations xN, yNϵ X|N|, and all i, j ϵ N, such that λ(xi) = λ(yi) = λ(xj) = λ(yj) = ℓ*, and xi and xj are constant consumption profiles, if there exists δϵℝ++ℓ* such that yi >> xi = yi – δ >> xj = yj + δ >> yj and xk = yk for all k ≠ i, j, then xN≿RN yN ~ 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 ϵ N, Riϵ and xiϵ X, the CCPERL of xi is the constant consumption profile xi such that λ(xi) = ℓ* and xi Iixi • Theorem Assume that the social ordering function ≿ satisfies WP, HI, PDEPEL and PDCCRL on |N| . Then ≿ is such that for all RNϵ|N|, all xN, yNϵ X|N|, min(xi) > min(yi) => xN≻RNyN i ϵ N i ϵ N where xi is the CCPERL of agent i under allocation xN. ~ 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 xi(t) defined over the interval [0, T]. • Axiom 5: Inequality Reduction around Reference Lifetime (IRRL) For all preference profiles RN ϵ|N|, all allocations xN, yNϵ X|N|, and all i, j ϵ N, such that λ(xi) = ℓi, λ(yi) = ℓi’, λ(xj) = ℓj, λ(yj) = ℓj’, and some c ϵℝ++ is the same constant per-period level of consumption for xi, yi, xj, yj, if ℓj, ℓj’ ≤ ℓ* ≤ ℓi, ℓi’ and ℓj - ℓj’ = ℓi’ - ℓi > 0 and xk = yk for all k ≠ i, j, then xN≿RN yN

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 RNϵ*|N|, all xN, yNϵ X|N|, min(xi) > min(yi) => xN≻RNyN i ϵ N i ϵ 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(cij) + πjβi u(dij) Ex post lifetime welfare: Short: u(cij) ; Long: u(cij) + βi u(dij)

17. Optimal allocation in a 2-period model (2) • Assumptions (continued) • Two sources of heterogeneity: • Time preferences: 0 < β1 < β2 < 1 • Survival probabilities: 0 < π1 < π2 < 1 • No individual savings technology: Consumption bundles (cij, dij) for agents with time preferences parameter βi and survival probability πj 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 βi , πj) Max Min xi = (cijℓ, cijℓ) s.t. Σi,j cij + πjdij = W • Solution: c21 = c22 > c11 = c12 > d21 = d22 = d11 = d12 = 0 • dij = 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 βi , πj ) Max Min xi = (cijℓ, cijℓ) s.t. Σi,j cij + πjdij = W s.t. IC constraints • Solution: c11 = c12 = c21 = c22 > d21 = d22 = d11 = d12 = 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 dij = d* such that u(d*) = 0 to old agents, and cij >< d* with c2j > c1j. SB: Maximin CCPERL gives dij = d*;and equal cij to all agents. • Reference longevity ℓ* = 1 (still with u(0) > 0) FB: Maximin CCPERL equalizes all cijand gives dij = 0. SB: Maximin CCPERL equalizes all cij and gives dij = 0 (= FB). • Savings technology for all (still with u(0) > 0) FB: Maximin CCPERL differentiates endowments Wijaccording to patience (+) and survival probability (+). SB: Maximin CCPERL equalizes all Wij.

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 differentiatedcompensation. • The optimum involves decreasing consumption profiles... … At odds with the observed profiles (inverted U shaped)… • Main limitation: exogenous survival.