But that isn't right now, I do not believe. There are corners (particularly SOMA) where cost of living far outstrips even the glitziest corners of Manhattan, but those are few and far between. In general, living in SF is still cheaper than living in Manhattan or the "near" neighborhoods of Brooklyn and Queens (i.e., not a 45-minute train ride from everything).
And therein lies the problem with these cost of living calculators - they have no awareness of equivalencies and idiosyncracies of these places. Why are we rolling Staten Island into this calculation when comparing SI to anywhere in SF is plain lunacy?
Similarly, the lifestyle between the Outer Richmond and North Beach are very different - as are rent prices. But yet these websites essentially add them up and divide by two, as if that number represents something meaningful.
If you want to evaluate cost of living differences between two cities, indices are probably the worst way to do it.
They exist, but they are few and far between. If you offered me a $1850 studio in Chelsea my first thought would be "what's under the floorboards". If you told me it's a $1850 1BR I'd check your temperature. The only places you'd reliably find studios < $2K would be Chinatown and the LES in general - and there are many reasons why those places are inexpensive.
I'm not sure if Manhattan is as cheap as you remember...
Hell, even Brooklyn is ridiculous now. Williamsburg is more expensive than most of Manhattan, Park Slope slightly cheaper but not by much. DUMBO, get outta here, DUMBO is giving TriBeCa a run for its money. $1850 for a studio is unrealistic in all of the above places, though if you're looking in the winter and have luck on your side, possibly doable in not-Park-Slope-but-lets-pretend areas.