One of the more sophisticated arguments for minimum wage stems from the Efficiency Wages Hypothesis (EWH). The EWH asserts that firms will sometimes pay higher-than-market wages for their workers. These wages reduce turnover and increase productivity, making the wages more viable for the firms. However, it is important to note that with EWH, there is still unemployment in the industry: higher-than-equilibrium wages reduce quantity demanded and increase quantity supplied from the equilibrium point, creating a surplus of labor (unemployment).
Minimum wage activists will cite the EWH for reasons for the minimum wage, claiming the reduced turnover and increased productivity is a positive for the firms. That much is true. But how does the EWH increase productivity and reduce turnover? Workers may be feeling better with a higher wage, so they’ll naturally work harder. That’s possible. But the real reason is the cost of losing the job is now higher. With persistent unemployment in the industry, the threat of firing forces workers to work harder in order to keep their jobs (thus increasing productivity). Turnover is reduced not out of some sense of loyalty to the firm now paying higher wages but because there are fewer jobs available and they are being competed for by more workers!
In short, an Efficiency Wage (especially if legally mandated like the minimum wage) gives employers more power over workers; it reduces worker bargaining power and reduces worker ability to leave if conditions are unfavorable to them.
The minimum wage is a very cruel policy. The minimum wage as an efficiency wage is even more so.