Except minimum wage doesn't do that. In most scenarios, due to the persistent unemployment level, there's a race to the bottom on wages which creates poverty wages for the "low skilled" labour, creating an underclass, with all inherent problems of that. Firms don't have to compete for workers by increasing wages because there's always unemployed workers looking for any job to avoid homelessness. This is how you get the working poor population in the US. The minimum wage puts a floor to this process. When the floor is above the poverty line, it eliminates the working poor population.

Besides, the minimum wage doesn't stop firms from offering more to workers if they felt they didn't get enough applications or skill.