No, your standard of living absolutely won’t increase. I would actually say almost everyone living in that kind of city is (perhaps unknowingly?) choosing to sacrifice in that column. Depending on what you value, however, your quality of life may increase dramatically. And any efforts in frugality will likely yield a greater absolute return than they would in a less costly locale.

Wages don't rise exactly like that. There's a threshold under which it doesn't make much sense at all. If you're making under 100K - and certainly under 50k - I wouldn't be surprised if you could be much more financially comfortable in, say, Dallas. Chop your rent budget in half, lower your tax burden, you'll probably get a bigger chunk of monthly spending money.