Among my family we often joke about "struggle money." It refers to the amount of money that your employer gives you to move somewhere with a higher cost of living. The origin of the word was a translation by my mother in law of a German accounting term, but none of us can remember the original word. What was this word?
Based on your additional information that your mother in law was in Berlin during the years of the so called reunification, i.e. around 1990, the original term for her struggle money may quite possibly have been
This term is in use even in current German legislation, see here for example the Auslandsverwendungsverordnung (Regulation for being sent on duty abroad).
Amusingly in another forum (dict.leo.org) somebody is searching for an appropriate English word for it; a suggestion from the US published there is differential cost allowance.
According to the translation »struggle money« I can imaging it's
though this term is usually used for one time costs.
You seem to look for an official accounting term, otherwise I would also propose the rarely used, but existing term
(origin: a soldier's extra pay)
as it fits your translation very well.
From the word meaning I would translate it to
But this is not related to the place of living.
There was an expression after the re-union of Germany in the 1990ies. People who move to east Germany got additional money that people called "Buschzulage". Busch is here in the meaning of jungle.
The reason for the use of this expression was that salaries in east Germany are lower than in the west part. Although the living costs im the east were lower in the first years there was an additional "Aufwandsentschädigung" for the moving people. There was no obvious reason for it, moving from north to south did not result in additional money. So the joke was "There must be a jungle in the east" and therefore they get a "Buschzulage".
The salaries are still lower but the differences are less. But the living costs are nearly identical, except for some areas around rich cities like Munich and Hamburg.