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

340 8. Sharing the gains from marriage
Hence,
zN,M − zN,M−1 ≥ vM − vM−1 ≥ zN−1,M − zN−1,M−1
and we get the upper and lower bounds on vM − vM−1. Now we also know
that woman M − 2 and man N − 2 marry each other. Using the fact that
M − 1 and N − 1 also marry each other we get by the same argument that
zN−1,M−1 − zN−1,M−2 ≥ vM−1 − vM−2
≥ zN−2,M−1 − zN−2,M−2 (8.7)
and so on all the way to the lowest married couple. Because we assume
morewomenthanmen,M>N, woman M − N +1 will marry man 1. For
this particular couple, we have
The boundary condition is therefore
u1 + vM−N+1 = z1,M−N+1
u1 + vM−N ≥ z1,M−N
vM−N = 0
z1,M−N+1 − z1,M−N ≥ vM−N+1 .
We see that along the stable assignment the prices must form an increasing
sequence. This is a consequence of complementarity.
When we set the bounds on couple N,M in (8.7), we referred only to
couple N − 1,M − 1. However, there are M − 1 stability constraints, one
for each woman that man N is not married to:
uN ≥ zN,M−1 − vM−1
uN ≥ zN,M−2 − vM−2...
and also N − 1 stability constraints for woman M regarding for each man
that she is not married to. We now show that the most binding constraint
from all these constraints is the one expressing that man N (woman M)
does not marry woman M − 1 (man N − 1). That is,
zN,M−1−vM−1 ≥ zN,M−2−vM−2 ≥ ··· ≥ zN,M−N+1−vM−N+1 ≥ zN,M−N.
Note, first, that if man N does not want to marry woman M −1, thenhe
doesnotwanttomarrywomanM −2 either; that is, the stability constraint
related to woman M −1 is more binding than that related to woman M −2.
Indeed, we want to show that:
zN,M−1 − vM−1 ≥ zN,M−2 − vM−2. (8.8)