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

6. Uncertainty and Dynamics in the Collective model 255
The efficiency assumption can now take two forms. Ex post efficiency
requires that, in each state s of the world, the allocation of consumption is
efficient in the usual, static sense: no alternative allocation could improve
both utilities at the same cost. That is, the vector cs = ¡ ca s,cb ¢
s solves:
max u a (c a s) (6.15)
under the constraints:
X
i
u b ¡ c b¢ b
s ≥ ūs ¡ a
pi,s ci,s + c b ¢ a
i,s = ys + y b s = ys
As before, we may denote by μs the Lagrange multiplier of the first constraint;
then the program is equivalent to:
max u a (c a s)+μsu b ¡ c b¢ s
under the resource constraint. The key remark is that, in this program, the
Pareto weight μs of member b may depend on s. Expostefficiency requires
static efficiency in each state, but imposes no restrictions on behavior across
states.
Ex ante efficiency requires, in addition, that the allocation of resources
across states is efficient, in the sense that no state-contingent exchange can
improve both agents’ expected utilities. Note that, now, welfare is computed
ex ante, in expected utility terms. Formally, the vector c =(c1, ..., cS) is
efficient if it solves a program of the type:
max X
πsu a (c a s) (6.16)
s
under the constraints:
X
πsu
s
b ¡ c b¢ s ≥
b
ū (6.17)
X ¡ a
pi,s ci,s + c b ¢
i,s =
a
ys + y b s = ys, s=1,...,S (6.18)
i
Equivalently, if μ denotes the Lagrange multiplier of the first constraint,
the program is equivalent to:
max X
πsu a (c a s)+μ X
πsu b ¡ c b¢ X £ a a
s = πs u (cs)+μu b ¡ c b¢¤ s
s
s
under the resource constraint (6.18).
One can readily see that any solution to this program also solves (6.15) for
μ s = μ. Butex ante efficiency generates an additional constraint - namely,
the Pareto weight μ should be the same across states. A consequence of this
requirement is precisely that risk is shared efficiently between agents.
s