By combining features of rationing models and hedonic models in a novel way, this paper develops a structural model of categorical labor force behavior to help explain several puzzles in data on unemployment and discouraged workers. It traces the links among minimum wages or other rigidities, hiring and firing decisions by firms, and labor force participation decisions by individuals of differing skill levels. A key comparative static result is that a rise in an effective minimum wage increases the labor force participation of more skilled marginal workers but reduces the participation of less skilled marginal workers.