But we know that u Cf u C f u C f u C f because children in Cf or C fare either

But we know that u cf u c f u c f u c f because

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But we know that u ¯ C f + u ¯ C f u ¯ C f + u ¯ C f , because children in ¯ C f or ¯ C f are either unaffected by the deviation or have their link severed. Therefore this implies that u C f + u C f > u C f + u C f and hence i C f and j C f π ij > i C f u i + j C f u j . But because the matching ( μ, u ) is individual-stable, we have a contradiction. This first result on the relationship between individual stability and familial stability enables us to derive interesting properties of family-stable matchings. From the liter- ature on matching, we know that individual-stable matchings always exist and that they always maximize the sum of total marital surplus 13 . This implies that Theorem 12 See Becker (1973). 13 Shapley and Shubik (1971), Becker (1973), Browning et al. (2014). 11
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1 suffices to prove the existence of family-stable matchings . Moreover, thanks to the equivalence result of the transferable utility framework, we know that there always exists a set of shares of marital surplus that satisfy familial stability for assignments that maximize aggregate surplus . So our model predicts that if parents allowed their children to choose their own partners, the ensuing matching would be stable for fam- ilies. This would argue for promoting individual choice of spouse instead of parental matchmaking in societies where arranged marriage is still prevalent, especially since individual choice should always lead to efficient social outcomes. By contrast, we next show that parental matchmaking may lead to inefficient match- ings. Proposition 1 A matching can be family-stable and inefficient. Consider the two-men-two-women case and the family partition F 1 illustrated in Figure 2. Family f 1 is composed of two sons i 1 and i 2 , while families f 2 and f 3 are composed of one daughter each, respectively j 1 and j 2 . Note that with this family partition, assignments ( μ 1 ) i 1 - j 1 , i 2 - j 2 , represented by dashed lines, and ( μ 2 ) i 1 - j 2 , i 2 - j 1 , represented by thick lines, are feasible. Let us assume that matching μ 1 is inefficient, while matching μ 2 is efficient. f 2 f 3 f 1 i 2 j 2 j 1 i 1 Figure 2: Families versus individuals In this configuration, if individuals chose their spouse, the outcome would be efficient matching μ 2 , as theory predicts. However, when families decide who their children will marry, they may end up stuck with inefficient matching μ 1 . The intuition for this is that even if both son i 1 and daughter j 2 as individuals have an incentive to sever 12
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their respective links so as to marry, family f 1 would prevent a marriage between its son i 1 and j 2 if the loss generated thereby in terms of utility for its second son i 2 is too large. In this case, inefficient matching μ 1 is family-stable but not stable for individuals. If there were no families, individuals would be able to sever their links and remarry in order to reach the efficient assignment. But families forbid such deviations. Inefficient matchings emerge when potential deviations for a family are such that some of its members end up single or worse off, and the benefits its other members obtain from the new matching are not sufficient compensation. This happens
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