> Has the productivity at the lower margin grown that much? I seriously doubt that.

That's the problem. The minimum wage is a legal instrument, not just an economic process. It is designed for the lower margin to be "overpaid" over what they could get from the market otherwise.

It basically says "everyone working should get at least this much of our collective production". This is what we have decided our society should work like.

Our collective productivity increased much faster than minimal wage, making the minimal wage worker much poorer by comparison.