Pierre André Chiappori (Columbia) "Family Economics" - Cemmap

252 6. Uncertainty and Dynamics in the Collective model
that would improve both agents’ welfare in expected utility terms -which
is exactly the meaning of programs (6.8) and (6.9).
Finally, note that the intertemporal version of the problem obtains simply
by replacing the state of the world index s by a time index t and the
probability πs of state s with a discount factor - say, δ t .
6.2.2 Constraints on commitment
Limits to commitment can generally be translated into additional constraints
in the previous programs. To take a simple example, assume that
in each state of the world, one member - say b - has some alternative option
that he cannot commit not to use. Technically, in each state s, there
is some lower bound ū b s for b’s utility; here, ū b s is simply the utility that
b would derive from his fallback option. This constraint obviously reduces
the couple’s ability to share risk. Indeed, it may well be the case that, in
some states, efficient risk sharing would require b’swelfaretogobelowthis
limit. However, a contract involving such a low utility level in some states
is not implementable, because it would require from b more commitment
than what is actually available.
The technical translation of these ideas is straightforward. Introducing
the new constraint into program (6.8) gives:
max
Qs,q a s ,qb s
X
s
πsu a ¡ Qs, q a s, q b¢ s
subject to P 0 sQs+p 0 ¡ a
s qs + q b¢ ¡ a
s ≤ ys + y b s
X
πsu b ¡ Qs, q a s, q b¢ b
s ≥ ū
s
¢ for all s, (6.10)
and u b ¡ Qs, q a s, q b¢ b
s ≥ ūs for all s (6.11)
Let μs denote the Lagrange multiplier of constraint (Cs); the program
can be rewritten as:
X
πsu a ¡ Qs, q a s, q b¢ X
s + (μπs + μs) u b ¡ Qs, q a s, q b¢ s
max
Qs,q a s ,qb s
or equivalently:
max
Qs,q a s ,qb s
s
subject to P 0 sQs+p 0 ¡ a
s qs + q b¢ ¡ a
s ≤ ys + y b¢ s for all s
X
s
πs
s
∙
u a ¡ Qs, q a s, q b µ
¢
s + μ + μ
s
u
πs
b ¡ Qs, q a s, q b¢ s
¸
¡ a
qs + q b¢ ¡ a
s ≤ ys + y b¢ s for all s
(6.12)
subject to P 0 sQs+p 0 s
Here, ex post efficiency still obtains: in each state s, the household maximizes
the weighted sum ua ³
+ μ + μ ´
s u πs
b . However, the weight is no longer