Minimum unit prices (MUPs) have been proposed on the grounds that they can reduce alcohol consumption of the heaviest drinkers, without significantly burdening moderate drinkers. This paper examines the case for MUPs in an optimal tax framework. Such a policy can improve welfare when two conditions are both satisfied. First, beverage quality and quantity should be substitutes. Second, there should be more distortion to consumption of cheaper alcohol than to more expensive varieties. The consequences of a MUP for the optimal corrective tax are explored with a calibrated numerical example. This example illustrates how the optimal tax rate might be higher when used in isolation, than when a MUP is also being used.