Identifying Sorting -- in theory

Assortative matching between workers and firms provides evidence of the complementarities or substitutes in production. The presence of complementarities is important for policies that aim to achieve the optimal allocation of resources, e.g. unemployment insurance. We argue that using wage data alone, it is virtually impossible to identify whether assortative matching is positive or negative. Even though we cannot identify the sign of the sorting, we can identify the strength, i.e. the magnitude of the crosspartial and the associated welfare loss. We first show that the wage for a given worker is non-monotonic in the type of his employer. This is due to the fact that in a sorting model, wages reflect the opportunity cost of mismatch. We analytically show that this non-monotonicity prevents standard firm fixed effects to correlate with the true type of the firm. We then propose an alternative procedure that measures the strength of sorting in the presence of search frictions. Knowing the strength of sorting facilitates the measurement of the output loss due to mismatch.