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

262 6. Uncertainty and Dynamics in the Collective model
Consider a two agent household, with two commodities - one labor supply
and an aggregate consumption good. Assume, moreover, that agent b is risk
neutral and only consumes, while agent a consumes, supplies labor and is
risk averse (with respect to income risk). Formally, using Cobb-Douglas
preferences:
U a (c a ,l a )= (laca ) 1−γ
and U
1 − γ
b ¡ c b¢ = c b
with γ>1/2. Finally, the household faces a linear budget constraint; let
wa denote 2’s wages, and y (total) non labor income.
Since agent b is risk neutral, one may expect that she will bear all the
risk. However, in the presence of wage fluctuations, it is not thecasethat
agent a’s consumption, labor supply or even utility will remain constant.
Indeed, ex ante efficiency implies ex post efficiency, which in turn requires
that the labor supply and consumption of a vary with his wage:
l a ρ + waT
= ,c
2wa
a ρ + waT
=
2
where ρ is the sharing rule. The indirect utility of a is therefore:
V a (ρ, wa) = 2γ−1
1 − γ (ρ + waT ) 2−2γ w −(1−γ)
a
while that of b is simply V b (y − ρ) =y − ρ.
Now, let’s see how ex ante efficiency restricts the sharing rule. Assume
there exists S states of the world, and let wa,s,ys and ρs denote wage, non
labor income and the sharing rule in state s. Efficient risk sharing requires
solving the program:
max
ρ
X
leadingtothefirst order condition:
s
£ a
πs V (ρs,wa,s)+μV b (ys − ρs) ¤
∂V a (ρ s,wa,s)
∂ρ s
= μ ∂V b (ys − ρ s)
∂ρ s
In words, efficient risk sharing requires that the ratio of marginal utilities
of income remains constant - a direct generalization of the previous results.
Given the risk neutrality assumption for agent b, this boils down to the
marginal utility of income of agent a remaining constant:
which gives
∂V a
∂ρ =2γ (ρ + waT ) 1−2γ w −(1−γ)
a = K
ρ =2K 0 .w 1−γ
1−2γ
a − waT