Using an intertemporal model of saving and capital accumulation with two types of agents we demonstrate that it is impossible for any binding minimum wage to increase the after‐tax incomes of workers if the production function is Cobb–Douglas with constant returns to scale, or if there are no differences in ability among workers. Moreover, it is impossible to increase the incomes of employed workers through minimum wage legislation, even under decreasing returns to scale and heterogeneity of ability among workers, unless the support provided to unemployed workers is far below what they would earn in the absence of minimum wages.