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

3. Preferences and decision making 115
case we have to consider is the one in which only one person contributes.
If this is person a, thefirst order condition in (3.29) holds for her. Person
b spends all of his income on the private good, so that:
u b Q
u b q
µ
ˆQ,
b Y
≤
p
P
p
(3.32)
with a strict inequality if the agent is not on the margin of contributing
to the public good. In this case a redistribution of income from a to b will
generally change the market demand since b will increase his demand for
the private good and a generally will not change her demands to exactly
offset these revisions. Thus we have market demands:
ˆq = ˆq a b Y
+
p = ξa (P, p, Y a )+
ˆQ = ˆ Q a = Ξ ¡ P, p, Y a ,Y b¢
Y b
p
(3.33)
In both cases, the non-cooperative procedure leads to an inefficient outcome
(except for the cases in which one or the other has all of the income);
this is the standard under-provision for the voluntary contributions public
goods game. To see that for the case of an interior solution, add the two
first order conditions (3.29), yielding
u a Q
u a q
³
ˆQ, a
ˆq ´
+ ubQ ub ³
ˆQ, b
ˆq
q
´
=2 P
p ,
while Samuelson’s (1954) condition for an efficient allocation of public
goodsrequiresthat
u a Q
u a q
³
ˆQ, a
ˆq ´
+ ubQ ub ³
ˆQ, b
ˆq
q
´
= P
. (3.34)
p
That is, the sum of the willingness to pay for the public good of the two
partners, should equal to the opportunity cost of the public good in terms
of the private good. In this regard, there is an under provision of the public
good. 6
We now present an example to illustrate some of the points made here.
Normalize prices to unity, P = p =1, and take preferences represented by
u a = q a Q α and u b = q b Q. The parameter α governs how much a likes the
public good; if α>1 then she values it more than b if they have the same
private consumption. We set Y a = ρ and Y b =(1− ρ) so that household
6 Results on dynamic contributions games suggest that inefficiencies cam be eliminated
if players contribute sequentially and cannot reduce previous contributions; see,
for example, Matthews (2006).