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

134 3. Preferences and decision making
The main result that Samuelson provides is that if income is redistributed
so as to maximize a given social welfare function, then the family aggregate
consumptions will satisfy the Slutsky conditions. That is a family will act
in the same manner as a single person.
Becker (1991) criticizes Samuelson for not explaining how a social welfare
function arises. In the context of moral judgements, each person can have
a private utility that is definedonoutcomesaffecting them directly, and a
social utility function that reflects preferences on the outcomes for all family
members. So it is unclear how partners agree on a single common social
welfare function. One mechanism suggested by Becker is that one person,
the ‘head’, has most of the family resources and is sufficiently altruistic
that they will transfer resources to the other member. If the dependents’
consumption is a normal good for the head, all family members will align
their actions with the head’s preferences, as any improvement in the income
under the command of the head raises their utilities. It is then the case
that the family as a group acts as if a single objective is being maximized.
This is the Rotten Kid Theorem mechanism outlined in subsection 3.4.3.
In that noncooperative voluntary contributions model, one of the partners
may effectively be a dictator if they control most (but not necessarily all)
of household resources. In that case a unitary model obtains locally if one
partner is wealthier and they are the sole contributor to the public good.
Another important case is when the preferences display transferable utility
(TU); see subsection 3.1. Indeed, under TU, program (3.40) becomes
max μ ¡ f a ¡ q a −1, Q ¢ + G (Q) q a¢ ¡ b
1 + f ¡ q b −1, Q ¢ + G (Q) q b¢ 1 (3.50)
(where qs −1 denotes the quantity of s’s private goods, except the first one)
under the budget constraint. The first surprising feature of the TU assumption
is that if the optimum has qa 1 and qb 1 both positive, then μ is
necessarily equal to unity. To see this, set the price of the first good to
unity and substitute for q b 1 using the budget constraint:
max μ ¡ f a ¡ q a −1, Q ¢ + G (Q) q a¢ 1
+f b ¡ q b −1, Q ¢ + G (Q) ¡ x − P 0 Q − p 0 −1
Taking the derivative with respect to q a 1
we see that:
¡ a
q−1 + q b ¢ ¢ a
−1 − q1 (3.51)
μG (Q) − G (Q) =0 (3.52)
which implies μ =1so that the UPF is a line with a constant slope of −1.
Thus the Pareto weight cannot depend on prices, income or any distribution
factors. Therefore the partners will always agree to act in a manner which
shifts the frontier out as far as possible by the choice of ¡ Q, q a −1, qb ¢
−1 .In
fact they will agree to maximize the sum of their individual utilities given
by:
f a ¡ q a −1, Q ¢ + f b ¡ q b −1, Q ¢ + G (Q) ¡ x − P 0 Q + p 0 −1
¡ a
q−1 + q b ¢¢
−1 (3.53)