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

380 8. Sharing the gains from marriage
In principle, it is possible to calculate these coefficients directly by solving
the market equilibrium (that is the linear programming problem) associated
with a stable assignment. More interestingly, one can use data on
actual marriage patterns and the observed attributes of participants in a
"marriage market" to estimate the gains from marriage of these individuals
(relative to remaining single). 14 Basically, the preferences for different
types of spouses are "revealed" from the choice probabilities of individuals.
Taking the simplest case without covariates, we see
Pr (i ∈ I is matched with j ∈ J)
ln
Pr (i ∈ I is single)
Pr (j ∈ J is matched with i ∈ I)
ln
Pr (j ∈ J is single)
= UIJ − UI0
= UIJ − UJ0. (8.59)
Estimating separate multinomial logit for men and women, one can estimate
the utilities for each gender in a marriage of each type. Summing the
estimated utilities one can recover, for each matching of types (I,J), the
systematic output of the marriage ξ IJ (which, under the normalization that
being single yields zero utility, equals the total surplus ZIJ). The estimated
matrix ZIJ can then be analyzed in terms of the assortative matching that
it implies. Of particular interest is whether or not this matrix is supermodular
(implying positive assortative mating) or not. As noted by Choo
and Siow (2006) and Siow (2009), in the absence of covariates the super
modularity of ZIJ is equivalent to the supermodularity of
ln (μ(I,J))2
σ(I)σ(J)
where μ(I,J) is the total number of type (I,J) marriages and σ(I) and
σ(J) are the number of single men and single women, respectively. Such
supermodularity requires that for all I 0 >I and J 0 >J
ln μ(I0 ,J0 )μ(I,J)
μ(I,J0 )μ(I0 > 0.
,J)
Siow (2009) uses census data on married couples in the US, where the
husband and wife are 32-36 and 31-35 respectively. In each couple, the wife
and the husband can belong to one of five possible schooling classes (less
than high school, high school, some college, college and college plus). He
compares the marriage patterns in the years 1970 and 2000 and finds that
in each of the two years strict supermodularity fails to hold as in some
of the off diagonal cells, the log odds ratio is negative. Looking at the
14 Because the probabilities in (8.58) are unaffected by a common proportionality factor,
some normalization is required. A common practice is to set the utility from being
single to zero for all individuals.