Multidimensional Sorting under Random Search

We analyze sorting in a standard market environment characterized by search frictions and random search, but where both workers and jobs have multi-dimensional characteristics. We first offer a definition of multi-dimensional positive (and negative) assortative matching in this frictional environment. According to this notion, matching is positive assortative if a more skilled worker in a certain dimension is matched to a distribution of jobs that first-order stochastically dominates that of a less skilled worker. We then provide conditions on the primitives of this economy (technology and distributions) under which positive sorting obtains in equilibrium. We show that in several environments of interest, the main restriction on the primitives is a single-crossing condition of the technology, although in general further restrictions on type distributions are needed. Guided by our theoretical framework, we conduct simulation exercises to quantify the errors in assessing sorting, mismatch and policy by wrongly assuming that heterogeneity is one-dimensional when it is really multi-dimensional.