This happens when families are composed of several children of the same gender

This happens when families are composed of several

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members obtain from the new matching are not sufficient compensation. This happens when families are composed of several children of the same gender 14 and are facing smaller families or families with few children of the complementary gender. In these cases, the bigger families could oppose potential deviations, as they would be more likely to involve one of their children ending up single. In this configuration, stable matchings differ in the assignment itself, depending on whether the decision-maker is the family or the individual. We may actually observe matchings that are not predicted by the classical theory on matching, but which can be explained if we take families into account. For instance, if we assume that each son is characterized by a single characteristic x , that each daughter is characterized by a single characteristic y and that there is complementarity (substitution) in traits, i.e. that the marital surplus is a supermodular (submodular) function of the attributes of the two partners, the classical matching model predicts positive (negative) assortative mating. By contrast, with these same assumptions, matchings with no positive (negative) assortative mating can be family-stable. Relying on our example, we now prove the existence of such inefficient matchings, showing that there exists a set of shares of surplus that support the inefficient matching μ 1 as family-stable. By assumption, μ 1 is the inefficient matching, and μ 2 is the efficient one, which means that π 12 + π 21 > π 11 + π 22 . We consider possible deviations 14 This issue is also addressed by Vogl (2013): “For instance, siblings of the same gender participate in the same marriage market, sharing a pool of potential spouses. In some ways, they are like any other participants on the same side of the market, but their membership in the same family introduces special constraints on their marriages." (p.1018). 13
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of pairs of families from the inefficient assignment. We first note that families f 2 and f 3 cannot deviate together, both being composed of one single daughter. The only two possible family deviations from the inefficient assignment are (1) the deviation involving f 1 and f 2 , in which case they would form i 2 - j 1 ; and (2) the deviation involving f 1 and f 3 , in which case they would form i 1 - j 2 . Let us consider the first family deviation: f 1 and f 2 could decide to sever their existing links to marry i 2 and j 1 . In particular, family f 1 would sever its link with f 3 to marry its son i 2 to j 1 from family f 2 instead of j 2 from family f 3 . This threat generates an upper bound on the share u j 2 that f 3 can expect from f 1 in the marriage i 2 - j 2 . f 1 and f 2 would have an incentive to deviate if u f 1 + u f 2 < u f 1 + u f 2 u i 1 + u i 2 + u j 1 < u i 2 + u j 1 . By definition, u i 2 = π 22 - u j 2 , therefore, the highest share that f 3 could expect from f 1 in the marriage i 2 - j 2 is u j 2 such that f 1 and f 2 are indifferent between the inefficient assignment and deviation, formally u j 2 such that u i 1 + u i 2 + u j 1 = u i 2 + u j 1 . We replace u i 2 by its expression in terms of u j 2 and find u j 2 π 11 + π 22 - π 21 15 . We follow the
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  • Spring '10
  • JAMES
  • J2, family dimension

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