matchings that do not maximize the sum of total marital surplus may be family

Matchings that do not maximize the sum of total

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matchings that do not maximize the sum of total marital surplus, may be family-stable. This is in sharp contrast with individual-stable matchings, as the central result of the transferable utility framework is that individual stability implies aggregate surplus maximization. Second, even efficient matchings may be stable for families but not for individuals. In this case, the difference lies in the shares of surplus, not in the assignment itself: there are some shares of surplus that support efficient matchings as family-stable, but not as individual-stable. Finally, we find that family partitioning has a direct impact on the characterization of family-stable matchings. In particular, for the family partition such that each family is composed of one son and one daughter, familial stability implies individual stability. Our first result is that individual stability implies familial stability. This result may seem counter-intuitive, as we usually place arranged marriages and love marriages in opposition. The intuition for this result is that, as we are in a transferable utility framework, the utility generated by a marriage between a son and a daughter from different families can be transferred entirely and without friction to their respective families. It is as if the benefits the two individuals experience from a love marriage could be perfectly shared with their respective parents. Indeed, when children individ- ually maximize their own utility on the marriage market, these utility maximizations directly benefit the family as a whole. However, in a non-transferable utility frame- 11 See Shapley and Shubik (1971), Becker (1973), Browning et al. (2014). 10
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work, the utility of the parents could be misaligned with the utilities of their children. For instance, parents could care only about the wealth or the education of their chil- dren’s partners, while grown-up children could care about shared interests or affinity, as documented in urban China in Huang et al. (2012). In this case, individual stabil- ity would not imply familial stability. In such a framework, we could observe sharp differences in terms of outcomes on the marriage market depending on whether the decision-maker is the family or the individual. Our results should help determine which assumptions on utility will be most relevant to different arranged marriages settings. We now state this result formally in Theorem 1. Theorem 1 An individual-stable matching is always family-stable. Before proceeding to the proof, we introduce the notations u C f and ¯ C f . Let u C f be the sum of the utilities of the members in C f , and ¯ C f be such that C f ¯ C f = f . Proof. Consider a matching ( μ, u ) which is stable for individuals. A matching ( μ, u ) is individual-stable if u i + u j > π ij if i and j are not married, and u i + u j = π ij if i and j are married 12 . Assume this matching is not family-stable. Therefore ( f, f ) , ( C f , C f ) of the same size, ( μ , u ) , which satisfy conditions 1, 2 and 3 of Definition 1. Indeed we have that u f + u f > u f + u f u C f + u ¯ C f + u C f + u ¯ C f > u C f + u ¯ C f + u C f + u ¯ C f .
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