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

114 3. Preferences and decision making
likely to appear. To bring out the essential features of the analysis, let
us assume that there is only one public good and one private good and
that each person has egotistic preferences; see Chen and Wooley (2001)
and Browning, Chiappori and Lechene (2010). Given that we have a public
good and individual incomes a natural, noncooperative process to consider
is a voluntary contributions game in which each person contributes to the
purchase of the public good and then uses any money remaining to buy the
private good for themselves. That is, the two agents have the problems:
© a
max u ¡ Q a + Q b ,q a¢ subject to PQ a + pq a = Y aª
Q a ,q a
max
Q b ,q b
© u b ¡ Q a + Q b ,q b¢ subject to PQ b + pq b = Y bª
(3.28)
where Q s denotes agent s’s contribution to the public good. Assuming that
both goods are normal, this interaction has exactly one Nash equilibrium,
which can take one of two forms. In the first form, both agents contribute
to the public good. Since this is an interior solution for both we have:
u a Q
ua q
ub Q
ub q
³ ˆQ, ˆq a ´
³ ˆQ, ˆq b ´
If we sum the budget constraints we have:
= P
p
= P
p
P ˆ Q + p ¡ ˆq a +ˆq b¢ = Y a + Y b
(3.29)
(3.30)
³ ´
Thus we have three equations in three unknowns ˆQ, a b ˆq , ˆq with a solution:
ˆq a = ξ a ¡ P, p, Y a + Y b¢
ˆq b = ξ b ¡ P, p, Y a + Y b¢
ˆQ = Ξ ¡ P, p, Y a + Y b¢
(3.31)
We conclude that the household’s market demand for both the public good,
ˆQ, and the private good, ˆq =ˆq a +ˆq b depends only on total household income
Y a + Y b and not on how it is distributed. In other words, we have income
pooling even though we have a non-unitary model. This is an example of
the remarkable neutrality result due to Warr (1983) (see also Bergstrom,
Blume and Varian (1986) and Bernheim (1986)). This shows that while
income pooling is a necessary condition for the unitary model, it is not
sufficient.
It is important to note that income pooling here is a local property and
holds only as long as both partners contribute to the public good. The other