In the short run when (nominal) prices are not so flexible, there will be a trade-off between (nominal) minimum wages and unemployment rate. The mechanics is simple. If businesses cannot change the price they charge their customers, they will optimize their cost. Since a person’s real wage theoretically equals the person’s marginal product of labor (or in English, productivity), businesses will try to maintain workers whom are reasonably productive with respect to the wage paid. In reality, this could mean either tighter selection of workers or even firing of unproductive workers. More often than not, it would likely only lead to tighter worker selection criteria. Regardless, with labor population growth, it would lead to lower hiring compared to pre-minimum wage and then immediate creating greater unemployment among the labor force, with all else being constant.
In the long run when prices finally adapt, the relationship between minimum wage and unemployment can be rendered impotent. Prices adapting means erosion of minimum wages by inflation. The more prices adapt, the less productivity is required given the equivalent fixed minimum wage level. This thus opens up more space for less productive workers to have a shot at employment in sector which the minimum wage law covers and in turn, applies a downward pressure on the unemployment rate.
Unless, of course, if the minimum wage level is updated in line with some measure of inflation. In that case, the negative relationship within minimum wage and unemployment rate will be sustained.
There is one important point that I wish to highlight if it is not so apparent already. While the negative relationship between minimum wage and unemployment will weaken over time in the face of inflation and non-update of the law, the effect of unemployment is real due to sticky prices in the short run.
With a real minimum wage, the effect is permanent.