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

264 6. Uncertainty and Dynamics in the Collective model
sharing it is the distribution of income, instead of its realization, that (may)
affect individual powers.
In practice, however, this raises a difficult econometric issue. Testing for
efficient risk sharing requires checking whether observed behavior satisfies
the mutuality principle, that is pooling of income realization. However, by
the previous argument, this requires being able to control for distributions,
hence to distinguish between ex post realizations and ex ante distributions.
On cross-sectional data, this is impossible.
It follows that cross-sectional tests of efficient risk sharing are plagued
with misspecification problems. For instance, some (naive) tests of efficient
risk sharing that can be found in the literature rely on a simple idea: since
individual consumption should not respond to idiosyncratic income shocks
(but only to aggregate ones), one may, on cross sectional data, regress
individual consumption (or more specifically marginal utility of individual
consumption) on (i) indicators of aggregate shocks (for example, aggregate
income or consumption), and (ii) individual incomes. According to this
logic, a statistically significant impact of individual income on individual
consumption, controlling for aggregate shocks, should indicate inefficient
risk sharing.
Unfortunately, the previous argument suggests that in the presence of
heterogeneous income processes, a test of this type is just incorrect. To get
an intuitive grasp of the problem, assume that two agents a and b share risk
efficiently. However, the ex ante distributions of their respective incomes
are very different. a’s income is almost constant; on the contrary, b may be
hit by a strong, negative income shock. In practice, one may expect that
this asymmetry will be reflectedintherespectiveParetoweights;sinceb
desperately needs insurance against the negative shock, he will be willing
to accept a lower weight, resulting in lower expected consumption than a,
as a compensation for the coverage provided by a.
Consider, now, a large economy consisting of many independent clones of
a and b; assume for simplicity that, by the law of large numbers, aggregate
resources do not vary. By the mutuality principle, efficient risk sharing implies
that individual consumptions should be constant as well; and since a
agents have more weight, their consumption will always be larger than that
of b agents. Assume now than an econometrician analyzes a cross section
of this economy. The econometrician will observe two features. One is that
some agents (the ‘unlucky’ b’s) have a very low income, while others (the
lucky b’s and all the a’s) have a high one. Secondly, the low income agents
also exhibit, on average, lower consumption levels than the others (since
they consume as much as the lucky b’s but less than all the a’s). Technically,
any cross sectional regression will find a positive and significant
correlation between individual incomes and consumptions, which seems to
reject efficient risk sharing - despite the fact that the mutuality principle
is in fact perfectly satisfied, and risk sharing is actually fully efficient. The
key remark, here, is that the rejection is spurious and due to a misspecifi-