The Comparative Statics of Sorting

In the marriage model, Becker (1973) found that positive (or negative) sorting is efficient with supermodular (or submodular) match payoffs. But characterizing the optimal matching with general production remains unsolved decades later.

Rather than tackle this difficult open problem, we instead ask when match sorting optimally increases. To do this, we first argue that the positive quadrant dependence (PQD) stochastic order on bivariate cdf’s captures an economically meaningful notion of increasing sorting — e.g. a higher correlation of partners.

Our theory turns on synergy: the local cross partial difference or derivative. A natural guess fails: increasing synergies need not raise sorting. But sorting rises if (1) synergy either everywhere increases or proportionately upcrosses through zero, and (2) cross-sectionally, synergy is upcrossing or downcrossing in types.

Our proof develops and exploits new monotone comparative statics methods. It proceeds by induction with finitely many types, and secures the continuum type results by taking limits. Our main results are easy to apply. We illustrate all theorems, applying them to the major post 1990 marriage model papers.

To see the nontriviality of our model, consider this example:

For but one of many example, assume quadratic production: φ(x,y) = αxy+β(xy)^2