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

302 7. Matching on the Marriage Market: Theory
(aij) that maximizes the linear objective (7.6) (or (7.9)) subject to the linear
constraints (7.7) and (7.8) (resp. (7.10) and (7.11)). We can therefore
use the standard tools of linear programming - specifically, duality theory.
Associated with the maximization of aggregate surplus which determines
the assignment is a dual cost minimization problem that determines the set
of possible divisions of the surplus. Specifically, one can define a dual variable
ui for each constraint (7.10) and a dual variable vj for each constraint
(7.11); the dual program is then:
subject to
min(
ui,vj
MX
i=1
ui +
NX
vj) (7.12)
j=1
ui + vj ≥ zij, i∈ {1, ..., M}, j∈ {1, ..., N} (7.13)
ui ≥ 0, vj≥ 0.
The optimal values of ui and vj can be interpreted as shadow prices of
the constraints in the original maximization problem (the primal). Thus,
ui+vj = zij if a marriage is formed and ui+vj ≥ zij otherwise. 2 This result
is referred in the literature as the complementarity slackness condition, see
for instance Gale (1978). It has a very simple interpretation. Any man i is
a resource that be can allocated to any woman, but only one woman, in
society. Similarly woman j is a resource that can be allocated to any man
in society, but only one man. The shadow price of each constraint in (7.10)
describes the social cost of moving a particular man (woman) away from
the pool of singles, where he (she) is a potential match for others. The sum
of these costs ui + vj is the social cost of removing man i and woman j
from the pool while zij is the social gain. Thus if ui + vj >zij, thecosts
exceed the gains and the particular marriage would not form. However, if
a marriage is formed then ui + vj = zij and each person’s share in the
resulting surplus equals their opportunity cost in alternative matches.
The crucial implication of all this is that the shadow price ui is simply
the share of the surplus that Mr.i will receive at the stable matching (and
similarly for vj); consequently, conditions (7.13) are nothing else than the
stability conditions, stating that if i and j are not matched at the stable
matching, then it must be the case that the surplus they would generate
if matched together (that is zij) isnotsufficient to increase both utilities
above their current level!
These results have a nice interpretation in terms of decentralization of
the stable matching. Indeed, a stable assignment can be supported (implemented)
by a reservation utility vector, whereby male i enters the market
2 Conversely, aij can be seen as the dual variable for constraint (7.13). In particular,
if aij > 0, then the constraint must be binding, implying that ui + vj = zij.