We analyze a game in which firms with private information compete for workers by making a single salary offer. Once salaries are chosen, firms make offers to workers, who care only about salary. Firms and workers are matched according to the Gale-Shapley deferred acceptance algorithm that dominates the theory of two-sided matching. For a two-firm, two-worker model, we prove existence of a Bayesian Nash equilibrium in which each firm type chooses a salary according to a continuous distribution with interval support in the salary space. We find a `separation' of types in equilibrium, in the sense that between two types with a common most preferred worker, one type always makes higher offers than the other type. The type that makes the higher offers depends on the relative marginal values attached to the workers by the different firm types. Moreover, more `popular' workers attract higher average equilibrium salaries.

We also consider an extension of the model to larger markets by replicating the two-firm, two-worker case, in order to examine the effects of market size on competition and equilibrium salaries. In the limit, there is no aggregate uncertainty about the realization of firm types. We characterize the equilibria for this limit case in which there are a continuum of firms and a continuum of workers divided into two equal-sized worker classes. Finally, we conjecture the existence and convergence of the sequence of equilibria in finite replicated markets to the corresponding continuum equilibrium as the number of replications approaches infinity.