I mean if you look at it from the other way… they agree to pay $X for the work but give you an increase of $Y to support a higher cost of living in certain area

I mean if you look at it from the other way… they agree to pay $X for the work but give you an increase of $Y to support a higher cost of living in certain areas to be near the office. This is being spun as a decrease instead of seeing that people actually have had an increase because of CoL. which no longer applies if you move. If you agreed to $X and then move to say the Bay Area would you expect them to keep your salary the same?