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 cross-partial 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.

Using administrative panel data on the entire Danish population we document a new set of facts characterizing occupational mobility. For most occupations, mobility is U-shaped and directional: not only low but also high wage earners within an occupation have a particularly large probability of leaving their occupation, and the low (high) earners tend to switch to new occupations with lower (higher) average wages. Exceptions to this pattern of two-sided selection are occupations with steeply rising (declining) productivity, where mainly the lower (higher) paid workers within this occupation tend to leave. The facts conflict with several existing theories that are used to account for endogeneity in occupational choice, but it is shown analytically that the patterns are explained consistently within a theory of vertical sorting under absolute advantage that includes learning about workers' abilities.