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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 anefficiencycomponent that captures reductions in monopsony power and aredistributivecomponent 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.