Common wisdom holds that the introduction of a non-binding minimum wage is irrelevant for actual wages and employment. Empirical and experimental research, however, has shown that the introduction of a minimum wage can raise even those wages that were already above the new minimum wage. In this paper, we analyze how these findings can be explained by theoretical wage bargaining models between unions and firms. While the Nash bargaining solution is unaffected by minimum wages below initially bargained wages, we show that such minimum wages can drive up wages – and be harmful to employment – when bargaining follows the Kalai-Smorodinsky solution.