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

7. Matching on the Marriage Market: Theory 301
we may describe the stable assignment as a solution to an integer linear
programming problem:
subject to aij ≥ 0 and
max
aij
MX
NX
i=0 j=0
aijζ ij
(7.6)
NX
aij = 1,i=1, 2., .M, (7.7)
j=0
MX
aij = 1,j=1, 2., .N. (7.8)
i=0
MX
NX
A first remark is that since a0j =1− aij and ai0 =1− aij the
program can be rewritten as
max
aij
subject to
MX
where C =
NX
i=1 j=1
i=1
aij
i=1
¡ ¢
ζij − ζi0 − ζ0j + C =max
aij
MX
i=1 j=1
j=1
NX
aijzij + C (7.9)
NX
aij ≤ 1, i=1, 2., .M, (7.10)
j=1
MX
aij ≤ 1,.j =1, 2., .N, (7.11)
i=1
MX NX
ζi0 + ζ0j is the aggregate utility of singles. Therefore,
j=1
the maximization of aggregate marital output over all possible assignments
is equivalent to the maximization of aggregate surplus and, without loss of
generality, we can normalize the individual utilities by setting ζ i0 = ζ 0j =0
for all i and j.
Secondly, one can actually assume that in the problem above, the aij
can be real numbers in the (M − 1)-dimensional simplex (insteadofconstraining
them to be integers). Intuitively, aij can then be interpreted as
the probability that Mr. i marries Mrs. j. Note, however, that given the
linearity of the structure, one solution at least to this generalized problem
is anyway attained with all aij being either zero or one.
The basic remark, at that point, is that the program thus defined is a
standard, linear programming problem; that is, we want to find a vector