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.
Introduction (1)
Facts
• Large longevity differentials (even within cohorts).
• Distribution of age at death: 1900 Swedish female cohort
0,1
0,09
0,08
probability
0,07
0,06
0,05
0,04
0,03
0,02
0,01
0
0
10
20
30
40
50
60
70
80
age
probability of age x at death
Sources: Human Mortality Database
90
100
110
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.
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.
Introduction (4)
Comparing allocations
Allocation A
C
Allocation B
C
i
i
j
j
Age
Age
Introduction (5)
The CCPERL
Allocation A
~i
C
Allocation B
C
~i
~j
~j
ℓ*
Age
ℓ*
Age
Under Maximin on CCPERL allocation B is preferred to A.
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.
2.
3.
4.
The framework
Ethical axioms
Two characterizations of social preferences
Optimal allocation in a 2-period model with heterogeneity
The framework (1)
• N = set of individuals, with cardinality |N|.
• T = the maximum lifespan (T ϵ ℕ).
• xi is a lifetime consumption profile, i.e. a vector of dimension
T or less.
• X = Uℓ=1T ℝ+ℓ is the set of lifetime consumption profiles xi.
• The longevity of an individual i with consumption profile xi is
a function λ: X → ℕ such 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.
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 xi ϵ X, there exists (c,…,c) ϵ ℝ+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|.
Ethical axioms (1)
• Axiom 1: Weak Pareto (WP)
For all preference profiles RN ϵ |N|, all allocations xN, yN ϵ
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, yN ϵ 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) := {yi ϵ X | yi Ii xi}.
~ The social preferences over two allocations depend only on
the individual indifference curves at these allocations.
Ethical axioms (2)
• Axiom 3: Pigou-Dalton for Equal Preferences and Equal
Lifetimes (PDEPEL)
For all preference profiles RN ϵ |N|, all allocations xN, yN ϵ
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 δ ϵ ℝ++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.
Ethical axioms (3)
• Axiom 4: Pigou-Dalton for Constant Consumption and
Reference Lifetime (PDCCRL)
For all preference profiles RN ϵ |N|, all allocations xN, yN ϵ
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 δ ϵ ℝ++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.
Characterization of social preferences (1)
• Definition
For any i ϵ N, Ri ϵ  and xi ϵ X, the CCPERL of xi is the
constant consumption profile xi such that λ(xi) = ℓ* and xi Ii xi
• Theorem
Assume that the social ordering function ≿ satisfies WP, HI,
PDEPEL and PDCCRL on |N| . Then ≿ is such that for all
RN ϵ |N|, all xN, yN ϵ X|N|,
min(xi) > min(yi) => xN ≻RN yN
iϵN
iϵN
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.
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, yN ϵ
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 ϵ ℝ++ 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
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 RN ϵ *|N|, all
xN, yN ϵ X|N|,
min(xi) > min(yi) => xN ≻RN yN
iϵN
iϵN
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) + πj βi u(dij)
Ex post lifetime welfare: Short: u(c ) ; Long: u(c ) + β u(d )
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).
Optimal allocation in a 2-period model (3)
• Maximin on CCPERL (FB: perfect observability of βi , πj)
Max Min xi = (cijℓ, cijℓ)
s.t. Σ i,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. Σ i,j cij + πjdij = W
s.t. IC constraints
• Solution: c11 = c12 = c21 = c22 > d21 = d22 = d11 = d12 = 0
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 cij and 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 Wij according
to patience (+) and survival probability (+).
SB: Maximin CCPERL equalizes all W .
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.