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

6. Uncertainty and Dynamics in the Collective model 263
where K0 is a constant depending on the respective Pareto weights. In the
end:
l a = K 0 .w −γ
2γ−1
,c a = K 0 1−γ
− 2γ−1
.w
and the indirect utility is of the form:
a
V a = K 00 .w
1−γ
− 2γ−1
a
for some constant K 00 . As expected, a is sheltered from non labor income
risk by his risk sharing agreement with b. However, his consumption, labor
supply and welfare fluctuate with his wage. The intuition is that that agents
respond to price (or wage) variations by adjusting their demand (here labor
supply) behavior in an optimal way. The maximization implicit in this
process, in turn, introduces an element of convexity into the picture. 11
6.3.4 Econometric issues
Distributions versus realizations
We now come back to the simpler, one-commodity framework. As expressed
by Proposition 6.1, efficient risk sharing schemes satisfy the mutuality principle,
which is a form of income pooling: the sharing rules depends only on
total income, not on the agent’s respective contributions y a and y b per se.
This result may sound surprising; after all, income pooling is a standard implication
of the unitary setting which is typically not valid in the collective
framework; moreover, it is regularly rejected empirically.
The answer to this apparent puzzle relies on the crucial distinction between
the (ex post) realization and the (ex ante) distribution of income
shocks.Whenriskissharedefficiently, income realizations are pooled: my
consumption should not suffer from my own bad luck, insofar as it does
not affect aggregate resources. On the other hand, there exists a continuum
of efficient allocations of resources, indexed by some Pareto weights;
different weights correspond to different (contingent) consumptions. The
Pareto weights, in turn, depend on the ex ante situations of the agents;
for instance, if a has a much larger expected income, one can expect that
her Pareto weight will be larger than b’s, resulting in a higher level of consumption.
In other words, the pooling property does not apply to expected
incomes, and in general to any feature (variance, skewness,...) of the probability
distributions of individual income streams. The main intuition of the
collective model is therefore maintained: power (as summarized by Pareto
weights) matters for behavior - the nuance being that under efficient risk
11 Generally, the ability of risk neutral agents to adjust actions after the state is observed
induces a "risk loving" ingredient, whereby higher price variation is preferred,
and which may counterweight the agent’s risk aversion.
a