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

8. Sharing the gains from marriage 335
FIGURE 8.1. Imputations when z12 + z21 >z22 + z11
z21,z12 >z11) and symmetry holds, z12 = z21, implying that man 2 is also
more productive than man 1 in all marriages. The main feature here is
that the difference v2 − v1 is bounded between the marginal contributions
of replacing woman 1 by woman 2 as spouses of man 1 and man 2. Woman
2 who is matched with man 1 cannot receive in that marriage more than
z12 − z11 + v1, because then her husband would gain from replacing her by
woman 1. She would not accept less than v1 + z22 − z21, because then she
can replace her husband by man 2 offering him to replace his present wife.
The assumption that z12 − z11 >z22 − z21 implies that man 1 can afford
this demand of woman 2, and will therefore "win" her. In this fashion,
the marriage market "prices" the different attributes of the two women.
Symmetric analysis applies if we would replace (v1,v2) with (u1,u2).
Similarly, if z12 + z21 ≤ z11 + z22, implying that the stable match is
‘diagonal’, then all pairs (v1, v2) satisfying the inequalities
z22 − z21 ≥ v2 − v1 ≥ z12 − z11, (8.2)
z11 ≥ v1 ≥ 0,
z22 ≥ v2 ≥ 0,
yield imputations v1,v2,u1 = z11−v1,u2 = z22−v2 that support the stable
assignment along the diagonal. The shaded area in Figure 8.2 describes
all the pairs that satisfy the constraints required for stability expressed