Post-Match Investment and Dynamic Sorting between Capital and Labor

By integrating frictional sorting into the neoclassical framework with post-match investment by firms, this paper analyzes the social optimum and the equilibrium. A main theoretical result is that post-match investment restores positive assortative matching (PAM) to the social optimum in the long run, but increases the likelihood of negative assortative matching (NAM) at the beginning of a match. The stronger is the complementarity between capital and the skill in production, the more likely is NAM to occur at the beginning of a match. I calibrate the model to show that sorting is NAM initially but becomes PAM on average over time. Workers with a higher skill have higher labor productivity and wages throughout the match. They also have steeper time profiles of these variables. There is sizable within-skill inequality in labor productivity and wages. Moreover, an increase in the matching efficiency reduces the labor share and increases the skill premium.