This paper studies screening and signaling roles of the offer wage and investigates its impact on matching efficiency. It develops a matching model of a large job market populated by observationally indistinguishable, heterogeneous firms and workers. Heterogeneity is introduced in the simplest way, by assuming two basic types of firms and workers, where one type has an advantage over the other: firms prefer good quality workers and workers prefer firms with better attributes, wages being equal. However, good quality workers are much more productive in firms with better attributes, hence efficiency requires firms and workers of the same type be matched. In the model, firms offer wages and workers make application decisions. This simple model generates a rich class of predictions in the form of perfect Bayesian equilibria, relating wage offers and matching efficiency to the distribution of unobservable characteristics: If the proportion of "good" firms to "bad" workers is large, perfect matching occurs through wage offers that do both signaling and screening. In another equilibrium, wages signal firm types but do only partial screening if the good worker population is sufficiently large. Both firm types offer the same wage in equilibrium if the market is predominantly populated by good workers and good firms. Other equilibria exhibit Gresham's Law in the job market: pessimistic workers and firms of the good type withdraw and take their outside options. The screening/signaling motive for wage offers thus has the potential of explaining a variety of wage patterns.