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Or there is not enough housing driving prices up (which then makes everything more expensive)

Rental prices in Germany are heavily controlled.

There's your problem. You can't regulate your way out of supply shortage by capping prices below their market value. Venezuela tired that, ask them how it's going.

You either build more housing to math the demand, or you restrict the demand from the number of people coming to move in.

I said this without judgement as an answer to this comment:

> If there are at least 50 applicants, arguably, perhaps the price is too low...

Yes, prices are too low. Yes, you cannot increase prices. Status quo.

Generally I agree to your point - if demand is high, prices will be high. People should move somewhere else.

Quite opposite!

We need to bring more migrants, so there are 500 applicants. That will bring price down for sure!