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

3. Preferences and decision making 113
the simple situation in which preferences are egotistic, and all commodities
are privately consumed. The noncooperative solution boils down to the
following two programs:
max
qa {ua (q a ) subject to p 0 q a = Y a }
max
qb © b
u ¡ q b¢ subject to p 0 q b = Y bª
(3.24)
In words, the noncooperative solution simply implies in that case that each
agent chooses his/her level of consumption independently of the other; that
is, they live side by side, but without any economic interaction. 5 Then the
consumption of individual s is simply this individual’s demands at prices
p and income Y s . Denote the demand functions for s by ξ s (p,Ys ³
).Note
that the allocation ξ a (p,Ya ) , ξ b ¡ p,Yb¢´ is Pareto efficient: clearly, the
utility of, say, a can only be increased by an income transfer from b, which
would reduce b’s welfare. Generally the associated household demands
ξ ¡ p,Y a ,Y b¢ = ξ a (p,Y a )+ξ b ¡ p,Y b¢
(3.25)
will not satisfy income pooling or the Slutsky conditions. The special case
in which income pooling and the Slutsky conditions will hold is if the classic
aggregation conditions hold. That is, if the two agents have linear Engel
curves with each partner having the same slope for any good:
ξ a i = φ a i (p)+ϕ i (p) Y a
ξ b i = φ b i (p)+ϕ i (p) Y b
(3.26)
so that the household demand for good i is given by:
¡ a b
ξi p,Y ,Y ¢ = φ a i (p)+φ b i (p)+ϕi (p) ¡ Y a + Y b¢
= φ a i (p)+φ b i (p)+ϕi (p) Y (3.27)
In this very special case income pooling holds in the sense that the household
demands do not depend on the distribution of income. The distribution
of the goods within the household will, however, depend on the distribution
of income.
3.4.2 One public good.
Whenever a direct interaction between members is introduced - either because
of public consumption, or because one member’s consumption has
an external effect on the other member’s well being - inefficiencies are very
5 Of course, this does not preclude the existence of non-economic interactions - love,
sex, conversation or others.