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

338 8. Sharing the gains from marriage
Denote the solution for individual utilities (or individual ‘prices’) by
(ûi, ˆvj). From the results on duality (Gale, 1960, chapter 5) we know that
the solution to this problem yields the same value as the solution to the
primal problem. That is
NX
ûi +
i=1 j=1
MX
ˆvj = X
i,j
âijzij,
where â denotes the assignment that solves the primal.
Now compare (8.3) to the dual problem when man N is eliminated:
subject to
min
u,,v
⎛
N−1 X
⎝
i=1
ui +
MX
j=1
ui + vj ≥ zij for i =1, 2...N − 1,j =1, 2...M
vj
⎞
⎠ (8.4)
Denote the solution for prices by (ūi, ¯vj). Again we know that the solution
to this problem yields the same value as the solution to the primal problem.
That is
N−1 X
i=1
ūi +
MX
¯vi = X
j=1
i,j
āijzij.
where ā denotes the assignment that solves the primal associated with (8.4).
Notice that the values (ûi, ˆvj) chosen in the dual problem (8.3) arefeasible
in the dual problem (8.4). It follows that the minimum attained satisfies
or
implying that
N−1 X
i=1
ūi +
MX
¯vj ≤
j=1
N−1 X
i=1
ûi +
MX
ˆvj ,
j=1
X
āijzij ≤ X
âijzij − ûN ,
i,j
i,j
ûN ≤ X
âijzij − X
i,j
i,j
āijzij .
That is, the upper bound on the utility that man N can get is his marginal
contribution to the value of the primal program (that is, the difference
between the maximand with him and without him). Note that to calculate
this upper bound we must know the assignments in both cases, when N is
excluded and N is included. This is easily done if we assume positive or