Final Report Summary - COMPLEMENTS (Identifying the sign and strength of complements in production) Executive Summary:We have provided a method to solve an old problem in economics: Are better workers employed by more productive firms? Theory tells us that if worker ability and the job productivity are complements in production, in a competitive market the more skilled workers will be employed in the more productive jobs. We like to believe that top lawyers tend to work for the best law firms, and the best doctors are employed by the most technologically advanced hospitals. But how big are those complementarities? Recent research using fixed effects estimates indicates that they are small and insignificant, or even negative, in which case workers and firms are substitutes instead of complements. Even if we believe that the better lawyers work in the better firms, the evidence suggests that they would be equally productive in a worse firm. We establish that the fixed effect estimates are bound to give non significant effects. In addition we develop a method derived from the theory that allows for way to estimate those complements.The key insight comes from the fact that when there are frictions in the search process, wages are not only a function of the productivity, but also of the opportunity cost from matching with a potentially more suitable firm/worker. This renders the wage non-monotone in the firm type, and as a result, the fixed effects estimates are biased.We have managed to establish following results. First, it is virtually impossible to identify whether Assortative Matching between worker and firm types is Positive or Negative. This stems from the fact that wages are non-monotonic and that they reach a maximum around the optimal allocation without frictions. Second, we have shown that we can nonetheless identify the strength of sorting, i.e. the magnitude of the complements. Ultimately, the efficiency of the allocation only depends on the strength and not on the sign. We have proposed a procedure to measure the strength of sorting in the presence of search frictions. The procedure takes into account that the observed wages reflect a combination of search costs and the benefits from complementarities. Third, we have been able to model the firm in greater detail. One of the reasons why estimating complements is a challenge methodologically is because the data used to do so is exclusively wage data. One can readily test for complementarities if there are data on profitability of jobs as well. The problem is that we have plenty of profit data at the firm level, but not at the job level. Even the CEO in a firm will not be able to accurately assign which fraction of the firm profits is due to which job. This is inherently impossible given that production is a team endeavor. While some variation in observed matches may be provide a solution to attribute the share of each job, this is a daunting task given that we also need to use the matched wage data. We propose an alternative by formulating a theory of production that allows for the attribution of the profits to jobs from firm level profits and wages.