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

266 6. Uncertainty and Dynamics in the Collective model
general population), theory 13 suggests that the matching should actually
be negative assortative (that is, more risk averse agents should be matched
with less risk averse ones) - so that heterogeneity should be, if anything,
larger within risk sharing groups than in the general population. 14
Finally, can we test for efficient risk sharing without this assumption?
The answer is yes; such a test is developed for instance in Chiappori,
Townsend and Yamada (2008) and in Chiappori, Samphantharak, Schulhofer-
Wohl and Townsend (2010). However, it requires long panels - since one
must be able to disentangle the respective impacts of income distributions
and realizations.
6.4 Intertemporal Behavior
6.4.1 The unitary approach: Euler equations at the household
level
We now extend the model to take into account the dynamics of the relationships
under consideration. Throughout this section, we assume that
preferences are time separable and of the expected utility type. The first
contributions extending the collective model to an intertemporal setting
are due to Mazzocco (2004, 2007); our presentation follows his approach.
Throughout this section, the household consists of two egoistic agents who
live for T periods. In each period t ∈ {1, ..., T }, lety i t denote the income of
member i.
We start with the case of a unique commodity which is privately consumed;
c i t denotes member i’s consumption at date t and pt is the corresponding
price. The household can save by using a risk-free asset; let
st denotes the net level of (aggregate) savings at date t, andRt its gross
return. Note that, in general, y i t, st and c i t are random variables
We start with the standard representation of household dynamics, based
on a unitary framework. Assume, therefore, that there exists a utility function
u ¡ c a ,c b¢ representing the household’s preferences. The program de-
13 See, for instance Chiappori and Reny (2007).
14 An alternative test relies on the assumption that agents have CARA preferences.
Then, as seen above, the sharing rule is an affine function, in which only the intercept
depends on Pareto weights (the slope is determined by respective risk tolerances). It
follows that variations in levels of individual consumptions are proportional to variations
in total income, the coefficient being independent of Pareto weights. The very nice
feature of this solution, adopted for instance by Townsend (1994), is that it is compatible
with any level of heterogeneity in risk aversion. Its main drawback is that the
CARA assumption is largely counterfactual; empirical evidence suggests that absolute
risk aversion decreases with wealth.