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

which directly implies that:
½
βptEt
6. Uncertainty and Dynamics in the Collective model 271
∙ Rt+1
pt+1
(ct+1) −γ
¸¾−1/γ >ct
In words, the (marginal utility of) aggregate consumption now follows a
supermartingale.
Regarding now individual consumptions, one can readily check that equations
(6.29) become:
or equivalently:
(1 − μ t) u0a (c a t )
pt
u
μt 0b ¡ cb ¢
t
Et
Et
pt
"
0a u ¡ ca ¢
t+1
u0a (ca t )
"
0b u ¡ cb ¢
t+1
= ¡ ¢ a
1 − μt+1 β Et
= μ t+1β b Et
ptRt+1
pt+1
u0b ¡ cb ¢
t
ptRt+1
pt+1
#
#
"
0b u ¡ cb ¢
t+1
pt+1
= 1
β a
= 1
β b
"
0a u ¡ ca ¢
t+1
pt+1
Rt+1
#
1 − μt 1 − μt+1 μ t
μ t+1
Rt+1
#
(6.37)
(6.38)
In words: under full commitment, the left hand side expressions should
be constant, while they may vary in the general case. A first implication,
therefore, is that whenever individual consumptions are observable, then
the commitment assumption is testable. Moreover, we know that (6.36)
holds for each t. These relations imply that μ t is identifiable from the data.
That is, if Pareto weights vary, it is possible to identify their variations,
which can help characterizing the type of additional constraint that hampers
full commitment.
Finally, individual consumptions are not observed in general, but individual
labor supplies typically are; the same tests can therefore be performed
using labor supplies as indicated above. Again, the reader is referred to
Mazzocco (2007) for precise statements and empirical implementations. In
particular, Mazzocco finds that both the unitary and the collective model
with commitment are rejected, whereas the collective model without commitment
is not. This finding suggests that while static efficiency may be
expected to hold in general, dynamic (ex ante) efficiency may be more
problematic.
6.4.4 Conclusion
The previous results suggest several conclusions. One is that the collective
approach provides a simple generalization of the standard, ‘unitary’