How does imperfect competition in the labor market affect unemployment policies?

We consider a continuum of workers ranked according to their abilities to acquire education and two firms with different technologies that imperfectly compete in wages to attract these workers. Once employed, each worker bears an education cost proportional to his/her initial ability, this cost being higher in the high-technology firm. At the Nash equilibrium, we show that the unemployed workers are those with the lowest initial abilities. We then study different policies that subsidy either the education cost or wages and compare them. We found that the first best allocation can only be implemented by selective policies. We then analyze second best non-selective policies that do not discriminate between workers and firms and show that, in terms of welfare, subsidizing education costs or wages is strictly equivalent.