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

314 7. Matching on the Marriage Market: Theory
where
Differentiating:
which implies that
F (y, x) = ∂H
∂y (x, y, v (y)) + v0 (y) ∂H
(x, y, v (y)) . (7.26)
∂v
∂F
∂y
∂F
∂y
+ ∂F
∂x φ0 (y) =0 ∀y,
≤ 0 if and only if ∂F
∂x φ0 (y) ≥ 0.
The second order conditions can hence be written as:
µ 2 ∂ H
∂x∂y (φ (y) ,y,v(y)) + v0 (y) ∂2
H
(φ (y) ,y,v(y)) φ
∂x∂v 0 (y) ≥ 0 ∀y.
(7.27)
Here, assortative matching is equivalent to φ 0 (y) ≥ 0; this holds if
∂2H ∂x∂y (φ (y) ,y,v(y)) + v0 (y) ∂2H (φ (y) ,y,v(y)) ≥ 0 ∀y. (7.28)
∂x∂v
Since v 0 (y) ≥ 0, asufficient (although obviously not necessary) condition
is that
∂2H ∂x∂y (φ (y) ,y,v(y)) ≥ 0 and ∂2H (φ (y) ,y,v(y)) ≥ 0. (7.29)
∂x∂v
One can readily see how this generalizes the transferable utility case. Indeed,
TU implies that H (x, y, v (y)) = h (x, y) − v (y). Then∂2H ∂x∂v =0and
the condition boils down to the standard requirement that ∂2H ∂x∂y = ∂2h ∂x∂y ≥
0. General utilities introduces the additional requirement that the cross
derivative ∂2H ∂x∂v should also be positive (or at least ‘not too negative’).
Geometrically, take some point on the Pareto frontier, corresponding to
some female utility v, and increase x - which, by assumption, expands the
Pareto set, hence shifts the frontier to the North East (see Figure 7.2). The
condition then means that at the point corresponding to the same value
v on the new frontier, the slope is less steep than at the initial point. For
instance, a homothetic expansion of the Pareto set will typically satisfy this
requirement.
The intuition is that whether matching is assortative depends not only on
the way total surplus changes with individual traits (namely, the usual idea
that the marginal contribution of the husband’s income increases with the
wife’s income, a property that is captured by the condition ∂2H ∂x∂y ≥ 0), but
also on how the ‘compensation technology’ works at various income levels.