I disagree that it increases costs more than it increases inflation adjusted wages. Do you have some sort of evidence to support that?
It always works like that. That's how this system works. Guaranteed the people paying minimum wages, usually large companies, are filtering their money through other materials and then the wages end up in the hands of the worker, unless they are storing excessive inventory (which most big companies are smart enough to avoid) in which case they might be borrowing to just pay wages. In the last case, the workers benefit slightly more than they do in the first case, but in the long haul, they don't really benefit at all.
The proof, is in the pudding though. Look at the wealth disparity gap and how it has steadily increased for decades now. Policies like minimum wage, I don't want to say they are the main contributors of this, but anything that causes the creation of more money out of nothing definitely contributes. Mostly it's the Federal Reserve, and you can directly correlate the GINI coefficient with the expansion of credit and the money supply every time. This isn't to say there aren't other factors, but the Fed main driving factor. Just pull up some historical graphs, you will see.
If minimum wage were an effective policy the disparity between the rich and poor would not be so ever expanding and people wouldn't have an increasingly difficult time paying the bills. Which has been happening for a few decades now (again, there is information available proving this, although I somehow doubt you would argue this specific point).
I'm sort of rambling now as I often do, my apologies if I've gone off track.
