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

378 8. Sharing the gains from marriage
the researcher observes marriage patterns within broad categories, such as
schooling level, race or occupation and observes only some of the individual
attributes that distinguish individuals within these classes. That is, in addition
to their observed class, individuals are characterized by some observed
attributes such as income or age and by some idiosyncratic marriage related
attributes that are observed by the agents in the marriage market but not
by the researcher. We assume that the marital output that is generated by
the match of man i, andwomanj can be written in the form
ζ ij = z I(i)J(j) + α iJ(j) + β jI(i). (8.54)
The first component z I(i)J(j) depends on the class of the two partners, the
second component α iJ(j) depends on man i and the class of woman j and
the third component depends on woman j and the class of man i. This
specification embodies a strong simplifying assumption; the interaction between
two married partners is always via their class identity. In particular
we do not have a term that depends on both i and j. 13 We further assume
that
α iJ(j) = a 0 I(i)J(j) xi + ε iJ(j)
β jI(i) = b 0 I(i)J(j) xj + ε I(i)j (8.55)
where xi and xj are the observed attributes of man i and woman j, respectively,
aIJ and bIJ are vectors of coefficients that represent the marginal
contribution of each male (female) attribute to a marriage between a man
of class I and woman of class J. The error terms ε iJ(j) represent the unobserved
contribution of man i to a marriage with any woman of class J.
Similarly, ε I(i)j represents the contribution of woman j to a marriage with
any man of class I.
A basic property of the matching model with transferable utility that we
discussed in Chapter 7 is the existence of prices, one for each man, vi, and
one for each woman, uj, that support a stable outcome. At these prices,
the matching is individually optimal for both partners in each match. Thus,
equilibrium implies that i is matched with j iff
ui = ξij − vj ≥ ξik − vk for all k, andui≥ξi0, vj = ξij − uj ≥ ξkj − uk for all k, andvj≥ ξj0. (8.56)
Under the special assumptions specified in (8.54) and (8.55), Chiappori,
Salanié and Weiss (2010) prove the following Lemma:
Lemma 8.1 For any stable matching, there exist numbers UIJ and VIJ,I =
1,...,M,J =1, ..., N, with the following property: for any matched couple
13 This simplifying assumption has been introduced in the context of transferable utility
by Choo and Siow (2006). Dagsvik(2002) considers a more general error structure in
the context of non transferable utility (e.g. an exogenous sharing rule).