Many argue that minimum wages can prevent efficiency losses from monopsony power. We assess this argument in a general equilibrium model of oligopsonistic labor markets with heterogeneous workers and firms. We decompose welfare gains into an efficiency component that captures reductions in monopsony power and a redistributive component that captures the way minimum wages shift resources across people. The minimum wage that maximizes the efficiency component of welfare lies below $8.00 and yields gains worth less than 0.2% of lifetime consumption. When we add back in Utilitarian redistributive motives, the optimal minimum wage is $11 and redistribution accounts for 102.5% of the resulting welfare gains, implying offsetting efficiency losses of −2.5%. The reason a minimum wage struggles to deliver efficiency gains is that with realistic firm productivity dispersion, a minimum wage that eliminates monopsony power at one firm causes severe rationing at another. These results hold under an EITC and progressive labor income taxes calibrated to the U.S. economy.