Simultaneous Search with Heterogeneous Firms and Ex Post Competition

We study a search model where workers can send multiple applications to high and low productivity firms. Firms that compete for the same candidate can increase their wage offers as often as they like. We show that there is a unique equilibrium where workers mix between sending both applications to the high and both to the low productivity sector. Efficiency requires however that they apply to both sectors because then the coordination frictions are lowest. For many configurations, the equilibrium outcomes are the same under directed and random search. Allowing for free entry creates a second source of inefficiency