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

260 6. Uncertainty and Dynamics in the Collective model
Similarly, if both spouses exhibit Constant Relative Risk Aversion (CRRA)
with identical relative risk aversion γ, then:
u a (x) =u b (x) = x1−γ
1 − γ
and the equation is:
£ ¡ a
ρ ys ,y b¢¤−γ £ a
s = μ ys + y b s − ρ ¡ y a s ,yb ¢¤−γ
s
which gives
where
ρ ¡ y a s ,y b¢ ¡ a
s = k ys + y b¢ s
k =
μ− 1
γ
1+μ − 1 γ
(6.23)
(6.24)
Therefore, with identical CRRA preferences, each spouse consumes a fixed
fraction of total consumption, the fraction depending on the Pareto weight
μ. Note that, in both examples, ρ only depends on the sum ys = y a s + y b s,
and
0 ≤ ρ 0 (ys) ≤ 1
Properties of efficient sharing rules
While the previous forms are obviously specific totheCARAandCRRA
cases, the two properties just mentioned are actually general.
Proposition 6.1 For any efficient risk sharing agreement, the sharing rule
ρ is a function of aggregate income only:
ρ ¡ y a s ,y b¢ ¡ a
s =¯ρ ys + y b¢ s =¯ρ (ys)
Moreover,
0 ≤ ¯ρ 0 ≤ 1
Proof. Note, first, that the right hand side of equation (6.22) is increasing
in ρ, while the left hand side is decreasing; therefore the solution in ρ must
be unique. Now, take two pairs ¡ ya s ,yb ¢ ¡
a
s and ¯y s , ¯y b ¢
a
s such that ys + yb s =
¯y a s +¯y b s. Equation (6.22) is the same for both pairs, therefore its solution
must be the same, which proves the first statement. Finally, differentiating
(6.22) with respect to ys gives:
and finally:
u 00a (¯ρ)
u 0a (¯ρ) ¯ρ0 = u00b (ys − ¯ρ)
u 0b (ys − ¯ρ) (1 − ¯ρ0 ) (6.25)
¯ρ 0 (ys) =
− u00b (ys−¯ρ)
u0 (ys−¯ρ)
− u00a (¯ρ)
u0a (¯ρ) − u00b (ys−¯ρ)
u0b (ys−¯ρ)
(6.26)