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Facebook employees working in London are paid, say, half as much as the ones in SV.

If we follow your reasoning, London employees are half as productive as the SV ones.

If I knew my colleague in the same position as me was making twice as much as I do just by virtue of physical location, I'd be pretty miffed. I probably won't work at much more than half capacity, if even that much.

I'd assume they would keep total cost to Facebook roughly similar, which depending on the state you move to it might result in less take home pay for the employee?

How so, the same argument about making dollars go further applies to someone already hired.