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

370 8. Sharing the gains from marriage
contexts, it is clearly too restrictive in other situations.
In this section, we explore the more general framework introduced in
Chapter 7, in which utility is not (linearly) transferable. That is, although
compensations between spouses are still possible, they need not take place
at a constant ‘exchange rate’: there is no commodity the marginal utility of
which is always identical for the spouses. In particular, the matching model
is no longer equivalent to a linear optimization problem. The upside is that,
now, any change affecting the wife’s and husband competitive positions
(for example, a change in income distributions) will potentially affect all
consumptions, including on public goods - which allows for a much richer
set of conclusions. The downside is that the derivation of individual shares
from the equilibrium or stability conditions is more complex. It remains
feasible,however.Wefirst present the general approach to the problem,
then we concentrate on a specific andtractableexample.
8.3.1 Recovering individual utilities: the general strategy
We use the same framework as in Chapter 7. Male income is denoted by
x and female income is denoted by y; the Pareto frontier for a couple has
the general form
u = H(x, y, v) (8.42)
with H(0, 0,v)=0for all v. Ifamanwithincomexremains single, his
utility is given by H(x, 0, 0) andifawomanofincomeyremains single
her utility is the solution to the equation H(0,y,v)=0.Bydefinition,
H(x, y, v) is decreasing in v; weassumethatitisincreasinginx and y,
that is that a higher income, be it male’s or female’s, tends to expand
the Pareto frontier. Also, we still consider a continuum of men, whose
incomes x are distributed on [0, 1] according to some distribution F ,anda
continuum of women, whose incomes y are distributed on [0, 1] according
to some distribution G; letrdenote the measure of women. Finally, we
assume that an equilibrium matching exists and that it is assortative - that
is, that the conditions derived in Chapter 7 are satisfied; let ψ (x) (resp.
φ (y)) denote the spouse of Mr. x (of Mrs. y).
As previously, the basic remark is that stability requires:
u (x) =maxH(x,
y, v (y))
y
where the maximum is actually reached for y = ψ (x). First order conditions
imply that
∂H
∂y (φ (y) ,y,v(y)) + v0 (y) ∂H
(φ (y) ,y,v(y)) = 0.
∂v
or:
v 0 ∂H
∂y
(y) =−
∂H
∂v
(φ (y) ,y,v(y))
(φ (y) ,y,v(y)).
(8.43)