And I'll assert that it just doesn't matter for this discussion.
Look, as someone said elsewhere, this is an impossible problem. You have too many factors to figure out. Have a family? Want to own a house instead of rent? Do you need a car? What lifestyle do you maintain?
The point here is that we're trying to assess what it would cost to maintain a similar standard of living in two different cities. Food. Transportation costs. Housing. Rents. Healthcare. Forget 52" TVs and trips to Paris for the time being.
"I have a wife and two kids. I want to own a 3 bedroom house with a yard. I need a car, and I drive 15,000 miles a year. We eat <x> number of meals at home each week, and <y> at restaurants. We do <z> things as a family including movies. This takes up 80% of our net income."
Okay, now what would that cost in Minneapolis, and what would that cost in San Francisco? That's all we're trying to ask.
On the other hand, consider the more realistic scenario of renting in NY vs. building house equity elsewhere. Given that house equity is the prevalent savings scenario in the US, the average New Yorker has a higher cost of living and a lower savings rate.
One has to distinguish cost of living for a non-saver from cost-of-living for a saver.