What actually tends to happen, is that first, a city/county/state/country's economy improves to the point that everyone is making a certain wage, just because of competition for workers. And then, policy-makers notice this, realize that there's an upside (better civic image) and no downside (workers out of jobs) to raising the minimum wage... and so they do.
Thinking of minimum wage as an adjustable variable, rather than an indicator, is trying to use the tail to wag the dog.