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

118 3. Preferences and decision making
³
where e is an N-vector of ones. Let ˆQ s
1, ... ˆ Qs ´
N for s = a, b be a Nash
equilibrium. 7 We say that person s contributes to good j if ˆ Q s j
> 0. Let
ma (respectively, mb ) be the number of goods to which a (respectively, b)
contributes. Browning at al (2010) show that if all public goods are bought
( ˆ Qs j > 0 for at least one s) theneitherma + mb = N or ma + mb = N +1
(generally). This striking result shows that there is at most one public good
to which both partners contribute. 8
To see the result informally, suppose that both partners simultaneously
contribute to two public goods, i and j. Then both set the marginal rates
of substitution between the two goods to unity (the relative prices) and
hence equalize the mrs’s:
ua i
ua j
= ub i
u b j
(3.36)
Without restrictions on preferences and incomes, this is unlikely to hold.
Moreover, if it does hold, if we make an infinitesimal change in Y a or Y b
the property (3.36) will generally not hold.
If there is some overlap in contributions (ma + mb = N +1)thenwe
have the local income pooling result, just as in the one public good case
when both contribute. The result that each partner has a set of public
goods which are his or her ‘domain’ suggests a gender division of allocation
within the household. Note, however, that the goods that each takes as
their domain is determined endogenously by preferences and the division
of income within the household. As we move from b having all the income
to a having all the income (holding total income constant) the number of
goods that she contributes to will generally rise and the number of goods
to which he contributes will generally fall.
We illustrate with an example with egoistic preferences from Browning
et al (2010) for the case of two public goods, G and H. Let the two partners
have preferences represented by the pair of Cobb-Douglas utility functions
u a (q a ,G,H) = lnq a + 5 8
ln G +
3 9 H
u b (q b ,G,H) = lnq b + 15 1
ln G + ln H
32 2
The relative weights on the two public goods are 45 15
24 and 16 for a and
b respectively; that is, a likes good G relative to good H, morethanb.
Figure 3.3 shows the purchases of public goods against a’s share of income.
When a has a low share of income (region I on the x-axis) she does not
7 We assume enough to ensure the existence of at least one Nash equilibrium. We do
not impose uniqueness.
8 This result is generic in the sense that it is possible to find ‘knife-edge’ configurations
of preferences and incomes for which the two partners contribute to more than one
common public good.