The Macro-dynamics of Sorting between Workers and Firms

We develop a stochastic model of random search on the job, with ex-ante heterogeneous workers and firms and aggregate productivity shocks in which firms make state contingent offers and counter offers to workers. The model delivers rich dynamics in which the distributions of unemployed workers, posted vacancies, and worker-firm matches evolve stochastically over time. We prove that the surplus function, which defines the value of a match and fully characterizes the mobility decision of workers, does not depend on the distributions of unemployed workers, posted vacancies or worker-firm matches. This result means that solving the model with aggregate productivity shocks becomes no more difficult than solving the stationary version. A calibrated version of the model does a good job in generating the fluctuations and co-movements with productivity of unemployment, vacancies, job-finding and job-losing rates.