Competitive advantage doesn't really mean much. Let's take some states with low minimum wages for example. In Alabama, minimum wage is the federal minimum wage (they don't have a state minimum wage). Most fast food workers are going to be making around that. Forcing an increase of nearly 100% is going to be a huge ($7.25 currently) hit on businesses. What kind of effect is this going to have on the economies? Grocery stores, restaurants, and most places staffed by entry level positions will have to raise their prices.
The feds are going to raise minimum wage to $15/hr today. It ain't happening. It's a false narrative to think that will actually happen, as even labor union leaders admit. It's an open salvo in a negotiation. Most people think, say, $11/hr is more reasonable.
My issue isn't even the people who are getting the increase. It is more the people just above that (my GF, colledge educated, started off just above this new minimum wage...) who are going to slide into poverty. Their wages are not likely to increase faster than the cost of living will, thus sliding them closer to minimum wage workers than the opposite direction.
I don't really see a refutation of my point here, so I assume you think the wage increase more than makes up for businesses passing on that labor cost in the form of higher prices, yes?
In any case, there are always winners and losers in every law. But I don't see how someone making just above the minimum wage loses. These generally aren't static groups of people. They gain experience and skills from a (few) year(s) working at just above minimum wage, and then move on to higher paying jobs. I'm not sure what makes you think that won't keep up with the cost of living, as costs of living generally isn't at all impacted by minimum wage increases.