Monotonic matching in search equilibrium

It is well-known from Becker's [Becker, G., 1973. A theory of marriage, Part I. Journal of Political Economy 81, 813-846] analysis of the neoclassical marriage market that matching is positive assortative if agent-types are complements in household production. Shimer and Smith [Shimer, R., Smith, L., 2000. Assortative matching and search. Econometrica 68, 343-369] have shown that this result does not generally extend to markets with search frictions, and they provide additional conditions on the production function that guarantee assortative matching in these settings. Here, the relationship between the Shimer-Smith restrictions and equilibrium matching pattern is considered, and alternative proofs of their results on assortative matching are provided.