This paper analyzes the introduction of the German minimum wage in 2015 in a structural model geared to quantitatively assess its long-run economic effects. We first employ a simple neoclassic model where wages equal their marginal product, then extend this model to two sector economy, and finally introduce search and matching frictions. Even though all model variants remain highly stylized, they yield quantitative insights on the importance of different mechanisms and channels through which minimum wages affect outcomes in the long run. In this framework, the minimum wage has a strong negative effect on employment. When sectors are differently affected by the minimum wage, sectoral relative price changes play an important quantitative role. Other labor market policies and institutions are important for the transmission of minimum wage policy on labor market market outcomes.