So you mean the rents in the cities are rising so exorbitantly because of the influx of 3 million migrants, who all also have their rent paid by the authorities? I would have my doubts about that. Especially whether these are the apartments that are in demand and missing on the market.
Yes, in Germany there are about 40 million tenants, of whom an estimated three-quarters live in small towns and cities. Now, if you add 3 million new tenants to these 30 million tenants (none of the newcomers will be able to buy a condominium or a house in any significant proportion), these are numbers that no market (which was already rather tight before) can absorb without disruption. That's really the case!
If most of these new tenants receive a market rent that the other 30 million tenants find increasingly difficult to afford, there will be a displacement competition in the market. How severe this will be is certainly debatable. But not the "if".
I don't know how it is in Berlin, but here in DD there are still plenty of apartments for refugees. But these are not the ones families want. What is missing are the nicely located new buildings within a 5-minute walk to school, work, and shopping centers. But building those costs more than 8 euros rent.
In Berlin, there are not enough apartments; this can be seen very clearly in the rising rents (without interventions by politics). New buildings are initially irrelevant. If you mean buildings constructed from 2009 onwards, we are talking about roughly 10% of the stock. Of course, we are in a new-build forum here, but the main issue is with the 90% "old stock." These do not need to be built; they are just as they are. And they are also highly sought after if located in Berlin.
The Senate pays a family of five a flat of up to 105 m² (800 euros cold rent for 2018, possibly plus 10%), and a single person up to 50 m² (404 euros cold rent, possibly plus 10%). Families mainly come from Eastern Europe. They thus occupy a 3-room flat for 5 people for 850 euros cold rent and thereby displace the long-established family of 3 or the single person who could afford their 80 m² based on the low rents so far and now complains because of rising rents. And the poverty migrants from the rest of the world are mostly single men who occupy the smaller apartments. Proximity to school and daycare is secondary here; proximity to compatriots is often rated much higher. Anyway, all of these apartments are missing from the market or make it even tighter. And since the rent paid by the authorities automatically increases more or less in line with the rent index, these tenants are "safe" from rising prices, while the family that has to earn their rent on the labor market increasingly gets the short end of the stick.