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

306 7. Matching on the Marriage Market: Theory
different implications); we will discuss such tests later on.
Finally, the impact of traits on the value of being single does not affect
these considerations, because the welfare of each person as single depends
only on his own traits. Therefore, in the aggregate, the output that individuals
obtain as singles is independent of the assignment. Although the
value of being single does matter to the question who marries, it does not
affect who marries whom, in equilibrium.
Examples
In many models, the surplus function takes a specific form. Namely, the two
traits x and y can often be interpreted as the spouses’ respective incomes.
Following the collective approach described in the previous Chapters, we
may assume that a couple consisting of a husband with income x and a wife
with income y will make Pareto efficient decisions; then it behaves as if it
was maximizing a weighted sum of individual utilities, subject to a budget
constraint. The important remark is that the constraint only depends on
the sum of individual incomes. Then the Pareto frontier - or in our specific
case the value of the surplus function h (x, y) which defines it - only depends
on the sum (x + y); 5 that is:
h (x, y) = ¯ h (x + y)
The various properties described above take a particular form in this context.
For instance, the second cross derivative hxy is here equal to the
second derivative ¯ h00 . It follows that we have assortative matching if ¯ h is
convex, and negative assortative matching if ¯ h is concave. The interpretation
is as above: a convex ¯ h means that an additional dollar in income is
more profitable for wealthier people - meaning that wealthier husbands are
willing to bid more aggressively for a rich wife than their poorer competitors.
Conversely, if ¯ h is concave then the marginal dollar has more value
for poorer husbands, who will outbid the richer ones.
In models of this type, the TU assumption actually tends to generate
convex output functions, hence assortative matching. To see why, consider
a simple model of transferable utility in the presence of a public good.
Preferences take the form
ui = cig(q)+fi(q), (7.17)
where c and q denote private and public consumption, respectively. The
Pareto frontier is then
ua + ub = h(Y )=max[(Y
− q)g(q)+f(q)],
q
5 Of course, while the Pareto set only depends on total income, the location of the
point ultimately chosen on the Pareto frontier depends on individual incomes - or more
specifically on the location of each spouse’s income within the corresponding income
distribution. These issues will be analyzed in the next Chapter.