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

7. Matching on the Marriage Market: Theory 311
with base r
σ
σ and height r (r − 1+x0 ) − σ
r (r − 1). Such equality of measures
must hold throughout the assignment profile.
The slope of each matching function is related to the local scarcity of
men relative to women. Men are locally scarce if there are more women
than men at the assigned incomes (φ (y) ,y)=(x, ψ (x)). Or, equivalently,
if an increase in the husband’s income is associated with a smaller increase
in the income of the matched wife. That is,
dx
dy = φ0 (y) =r g(y)
> 1,
f(φ (y))
(7.22)
dy
dx = ψ0 (x) = 1 f(x)
< 1.
r g(ψ (x))
Men are locally abundant if these inequalities are reversed.
7.2.4 Multidimensional matching
The previous discussion explicitly refers to a one-dimensional framework.
Assortative matching is harder to define when several dimensions (or several
traits) are involved; moreover, conditions like supermodularity or singlecrossing
do not have an obvious extension to a multidimensional setting.
Still, they can be generalized; again, the reader is referred to Chiappori,
McCann and Neishem (2010) or Ekeland (2010) for recent presentations.
The main insights can briefly be described as follows. Assume that X and
Y are finite dimensional. Then:
1. The Spence-Mirrlees condition generalizes as follows: if ∂ x h (x0,y)
denotes the superdifferential of h in x at (x0,y), then for almost all
x0, ∂ x h (x0,y1) is disjoint from ∂ x s(x0,y2) for all y1 6= y2 in Y .This
is the ‘twisted buyer’ condition in Chiappori, McCann and Neishem
2010.
2. If the ‘twisted buyer’ condition is satisfied, then the optimal match
is unique; in addition, it is pure, in the sense that the support of the
optimal measure Φ is born by the graph of some function y = φ (x);
that is, for any x there exists exactly one y such that x is matched
with y with probability one.
3. There exists a relaxation of the ‘twisted buyer’ condition (called the
‘semi-twist’) that guarantees uniqueness but not purity.
The notion of ‘superdifferential’ generalizes the standard idea of a linear
tangent subspace to non differentiable functions. If h is differentiable, as
is the case is most economic applications, then ∂ x h (x0,y) is simply the
linear tangent (in x) subspace to h at (x0,y), and the condition states that
for almost all x0, there exists a one to one correspondence between y and