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

6. Uncertainty and Dynamics in the Collective model 259
The intuition for this property is easy to grasp. Assume there exists two
states s and s 0 such that the equality does not hold - say:
u 0a (ρ s) /dρ
u 0b (ys − ρ s) /dρ <
Then there exists some k such that
πsu 0a (ρ s) /dρ
u0a (ρ s 0) /dρ
u0b (ys0 − ρs0) /dρ
0) /dρ
πs 0u0a (ρ s 0) /dρ 0
dW b = πsu 0b (ys − ρs) ε − πs0u0b (ys0 − ρs0) kε > 0
and both parties gain from that trade, contradicting the fact that the initial
allocation was Pareto efficient.
The sharing rule ρ is thus a solution of equation (6.21), which can be
rewritten as:
u 0a (ρ) =μu 0b (ys − ρ) (6.22)
where ρ = ρ ¡ ya s ,yb ¢
s . Since the equation depends on the weight μ, there
exists a continuum of efficient risk sharing rules, indexed by the parameter
μ; the larger this parameter, the more favorable the rule is to member b.
As an illustration, assume that agents have Constant Absolute Risk Aversion
(CARA) preferences with respective absolute risk aversions equal to
α and β for a and b respectively:
u a (x) =− exp (−αx) ,u b (x) =− exp (−βx)
Then the previous equation becomes:
α exp £ −αρ ¡ y a s ,y b¢¤ £ ¡ a
s = μβ exp −β ys + y b s − ρ ¡ y a s ,y b¢¢¤ s
which gives
ρ ¡ y a s ,y b¢ β ¡ a
s = ys + y
α + β
b¢ 1
s −
α + β log
µ
μβ
α
We see that CARA preferences lead to a linear sharing rule, with slope
β/(α + β); the intercept depends on the Pareto weight μ.