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

258 6. Uncertainty and Dynamics in the Collective model
for some constants c i ,i= a, b. Finally,forγ 6= 0and γ 6= 1,wehave:
u i (x) =
for some constants c i and γ i ,i= a, b.
¡ c i + γ i x ¢ 1−1/γ i
1 − 1/γ i
• moreover, the ‘shape’ coefficients γ must be equal:
γ a = γ b
The intuition of this result is that in the ISHARA case, the sharing
rule that solves (6.19) is an affine function of realized income. Note that
ISHARA is not simply a property of each utility independently: the second
requirement imposes a compatibility restriction between them. That said,
CARA utilities always belong to the ISHARA class, even if their coefficients
of absolute risk aversion are different (that’s because they correspond to
γ a = γ b =0). On the other hand, constant relative risk aversion (CRRA)
utilities, which correspond to c a = c b =0,areISHARAifandonlyifthe
coefficient of relative risk aversion, equal to the shape parameter γ i in that
case, is identical for all members (it was equal to one for both spouses in
our log example).
6.3.2 Efficient risk sharing in a one-commodity world
Characterizing efficient risk sharing
We now characterize ex ante efficient allocations. We start with the case
in which prices do not vary; as seen above, we can then model efficient risk
sharing in a one commodity context. A sharing rule ρ shares risk efficiently
if it solves a program of the form:
X £ a
max πs u
ρ
¡ ρ ¡ y a s ,y b¢¢ b
s + μu ¡ y a s + y b s − ρ ¡ y a s ,y b¢¢¤ s
s
for some Pareto weight μ. Thefirst order condition gives:
u 0a ¡ ρ ¡ y a s ,y b¢¢ 0b
s = μ.u ¡ y a s + y b s − ρ ¡ y a s ,y b¢¢ s
or equivalently:
u0a (ρs) u0b = μ for each s (6.21)
(ys − ρs) where ys = ya s + yb s and ρs = ρ ¡ ya s ,yb ¢
s .
This relationship has a striking property; namely, since μ is constant, the
left hand side does not depend on the state of the world. This is a standard
characterization of efficient risk sharing: the ratio of marginal utilities of
income of the agents remains constant across states of the world.