Interestingly we also find that a matching can be efficient and family stable

Interestingly we also find that a matching can be

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deviate together is bounded, inefficient outcomes are still likely to arise. Interestingly, we also find that a matching can be efficient and family-stable but not individual-stable. Proposition 2 A matching can be efficient and family-stable but not stable for indi- viduals. This means that the assignment itself might be the same for families and individuals, while the shares of surplus that support it as stable differ. We find that the set of shares of surplus that support efficient assignments as family-stable includes the set of shares of surplus that support them as individual-stable. The intuition here is that when we consider families instead of individuals, constraints are less binding, and therefore families may accept a wider range of sharings-out of surplus than individuals. Consider again the two-men-two-women case presented previously. We now study the efficient matching. For individuals, we follow Browning et al. (2014), who characterize the shares of surplus that support the efficient matching μ 2 as individual-stable 17 . The authors show that all pairs ( u j 1 , u j 2 ) satisfying the inequalities π 12 - π 11 u j 2 - u j 1 π 22 - π 21 with π 21 u j 1 0 and π 12 u j 2 0 yield imputations u j 1 , u j 2 , u i 1 = π 12 - u j 2 , and u i 2 = π 21 - u j 1 , which support μ 2 as stable for individuals. Indeed we observe that when the decision-maker is the individual, the share of surplus that woman j 2 can expect to obtain is bounded and depends on the share of surplus that woman j 1 obtains 18 . For families, we first consider the family partition F 1 , already described. We characterize formally the set of the shares of surplus that support the efficient assignment μ 2 as family-stable with F 1 , and compare it with the set of surplus that supports it as individual-stable. Note that for this purpose, it is important to 17 See Example 1 in Section 8.1 of Browning et al. (2014). 18 Browning et al. (ibid.) explain p.318 “Woman j 2 , who is matched with man i 1 , cannot receive in that marriage more than π 12 - π 11 + u j 1 because then her husband would gain from replacing her by woman j 1 . She would not accept less than u j 1 + π 22 - π 21 because then she can replace her husband with man i 2 , offering to replace his present wife.” Notations are adapted. 16
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choose a family partition for which assignments μ 1 and μ 2 are both possible 19 . This is so that we can isolate the impact of family decision-making on the set of the shares of surplus which support the stable matching, from the impact of siblings of different sex, who cannot marry. We follow the same reasoning as before and consider possible deviations by pairs of families from the efficient assignment. We find that all pairs ( u j 1 , u j 2 ) satisfying inequalities u j 1 π 12 + π 21 - π 22 and u j 2 π 12 + π 21 - π 11 with π 21 u j 1 0 and π 12 u j 2 0 yield imputations u j 1 , u j 2 , u i 1 = π 12 - u j 2 and u i 2 = π 21 - u j 1 that support the efficient assignment as family-stable. It is worth noting that, unlike when the marriage decision is taken by individuals, there is no lower bound on u j 1 and u j 2 other than 0, and u j 1 and u j 2 are independent of each other. The reason for this is that family partition F 1
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  • Spring '10
  • JAMES
  • J2, family dimension

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