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

286 6. Uncertainty and Dynamics in the Collective model
with respect to t a of
E(u a )=lnq1 +lnq2 + α ln Q +
Z∞
θf(θ)dθ + F ( ˆθ)( ˆθ), (6.59)
with the first order condition
1
+
q1
γ
q2
=[1−δF( ˆ α
θ]
2(1 + ta .
)
(6.60)
Similarly, the choice of t b as a function t a is determined by the maximization
with respect to t b of
E(u b Z∞
)=lnq1 +lnq2 + α ln Q + θf(θ)dθ + F ( ˆθ) ˆθ + F ( ˆθ)[ln β − ln(1 − β],
with the first order condition
ˆθ
ˆθ
(6.61)
wb +
q1
γwb
=[1−δF( q2
ˆθ)+f( ˆ β
θ)ln
1 − β ]
α
2(1 + tb . (6.62)
)
We see that the expected marginal reward from exerting effort is smaller
to the husband (note that for β < 1 β
2 , ln 1−β < 0). The husband takes
into account her lower consumption, and thus higher marginal utility from
consumption, following divorce. He responds by shifting additional time
in the first period into work so that his future wage will be higher. This
defensive investment in market work by the husband causes an inefficient
time allocation. Examining conditions (6.60) and (6.61), we see that the
requirement for cost minimization is not satisfied.
When partners cannot commit on the allocation of time, commitments
made at the time of marriage should adjust. One may assume that the
husband has a higher bargaining power at the time of marriage, because of
his higher wage and thus higher consumption as single. However, it makes
sense for the husband to give up some of his power, which will raise the
"pie" available during marriage that he and his the wife enjoy equally.
Returning now to the case of equal division and efficient allocation of
time, we can provide some further analysis of the investment decision. Using
the efficiency conditions (and constant returns to scale) we have that, in
an interior solution,
Q = √ wb (1 + t b ). (6.63)
We also have that
q1 = 1+w b − t a − w b t b
= 2(1+w b ) − 2 √ wbQ (6.64)