We examine wage competition in a model where identical workers choose the number of jobs to apply for and identical firms simultaneously post a wage. The Nash equilibrium of this game exhibits the following properties: (i) an equilibrium where workers apply for just one job exhibits unemployment and absence of wage dispersion; (ii) an equilibrium where workers apply for two or for more (but not for all) jobs always exhibits wage dispersion and, typically, unemployment; (iii) the equilibrium wage distribution with a higher vacancy-to-unemployment ratio first-order stochastically dominates the wage distribution with a lower level of labor market tightness; (iv) the average wage is non-monotonic in the number of applications; (v) the equilibrium number of applications is non-monotonic in the vacancy-to-unemployment ratio; (vi) a minimum wage increase can be welfare improving because it compresses the wage distribution and reduces the congestion effects caused by the socially excessive number of applications; and (vii) the only way to obtain efficiency is to impose a mandatory wage that eliminates wage dispersion altogether.