Not regulating the price was tried for 30 years, and led to many families unable to afford housing, or spending 80%+ of post-tax income on housing.
That's not self-evident to me. Why not?
The most relevant law I can think of is that if you live outside the city limits you might not be able to send your kids to a school in Berlin.
Now going up to 200 floors is very expensive. However on a per unit area it is only a few times what the cheapest building style* is. So it does require more expensive rent, but not that much more expensive, and as rent gets expensive you expect apartment sizes to fall to balance some of it out.
* cheapest that can meet reasonable building codes. I'm ignoring the uninsulated, fire-prone shacks you see in poor countries which are cheaper by far.
[0] IIRC “if you come home to find your wife has changed her hair and redecorated, you went into the wrong apartment”, according to the DDR museum. I’m sure it lost something in the translation.
[0] DDR? I’m not actually sure who made the decision, now I think about it