If DOD is a an example

You do get paid more in DC.
I came from Europe with a very high COLA
The higher BAH in DC More than makes up the difference
Military get BAH in the states, but civilian employees of the government do not earn housing allowances in the United States. Federal civilian employees earn housing allowances and cost-of-living adjustments when stationed overseas, so they nearly always earn more overseas, though, like the military, its a benefits package, not just base salary. I've never heard anyone at State or DoD complain about getting by pay cut by moving overseas. It's usually the opposite.