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

104 3. Preferences and decision making
particular commodity is either purely public or purely private, although
many of our results would extend to more general settings.
We introduce some notation that will be used throughout the chapter.
There are N public goods, and the market purchase of public good j is
denoted Qj; theN-vector of public goods is given by Q. Similarly, private
goods are denoted qi with the n-vector q. Each private good bought is
divided between the two partners so that a receives q a i
of good i and b
receives qb i = qi − qa i . Hence the vectors a and b receive are qa and qb respectively, with qa +qb = q. Anallocation is a N +2n-vector ¡ Q, qa , qb¢ .
The associated market prices are given by the N-vector P and the n-vector
p for public and private goods respectively.
We assume that each married person has her or his own preferences over
the allocation of family resources. Denote a’s utility function by U a ¡ Q, qa , qb¢ and b’s by U b ¡ Q, qa , qb¢ . This general formulation allows that a is concerned
directly with b’s consumption and also that b’s consumption of private
goods impacts on a’s preferences between her own private goods and
the public goods. Any kind of externality is allowed. The presence of the
partner’s private consumption does not mean necessarily that members are
altruistic to each other; for instance, it could simply represent the partner’s
smoking that bothers the other member by reducing their utility. Throughout
the book, we assume, unless stated otherwise, preference orderings are
continuous and convex and can be represented by utility functions U s (.),
s = a, b, that are continuously differentiable and strictly concave.
In the subsequent chapters in the first half of this book we shall be
discussing the resolution of conflicts that arise between partners if U a (.)
and U b (.) represent different preferences. It is important to acknowledge,
however, that if marriage is sometimes a battleground, it is can also be a
playground. In the context of the family, love or affection might be operating
and conflicts are thereby considerably attenuated. We return to this
below.
Although quite reasonable, the form just described is too general to be
used in most contexts - if only because it is difficult to incorporate such
preferences into a model in which agents live alone for some part of their
life-cycle. Consequently the literature generally takes a special case which
is known as caring. 2 For this we first assume agents a and b have felicity
functions ua (Q, qa ) and ub ¡ Q, qb¢ respectively. The most general form has
U a ¡ Q, q a , q b¢ = W a ¡ u a (Q, q a ) ,u b ¡ Q, q b¢¢
U b ¡ Q, q a , q b¢ = W b ¡ u a (Q, q a ) ,u b ¡ Q, q b¢¢ , (3.1)
where both W a (., .) and W b (., .) are monotone increasing functions. The
2 Sometimes the term altruistic is used for preferences taking this form. Pollak (2006)
has suggested the term deferential since each person defers to the judgment of the other
regarding their consumption.