"The city needs people who make less than X to live here (teachers, social workers etc). We shouldn't have a homogenous city of software engineers and bankers."
This is mistaken. The cost of living is routinely taken into account when people are deciding what wages they will demand. If literally no teachers can afford to live in your city, and you wish your children to be educated, you'll have to pay a lot in order to import a teacher who can then afford to live there. That drives up wages to equilibrium.
Furthermore, if literally no teachers can afford to live in your city, and you wish your children to be educated, you might very well decide to move to somewhere with a lower cost of living. This drives down demand to equilibrium.
The only argument in favor of rent control is that it reduces short-term volatility of rent. This only makes sense if moving entails huge negative externalities. You do see people making this argument (they talk a lot about long-term community cohesion).
This is mistaken. The cost of living is routinely taken into account when people are deciding what wages they will demand. If literally no teachers can afford to live in your city, and you wish your children to be educated, you'll have to pay a lot in order to import a teacher who can then afford to live there. That drives up wages to equilibrium.
Furthermore, if literally no teachers can afford to live in your city, and you wish your children to be educated, you might very well decide to move to somewhere with a lower cost of living. This drives down demand to equilibrium.
The only argument in favor of rent control is that it reduces short-term volatility of rent. This only makes sense if moving entails huge negative externalities. You do see people making this argument (they talk a lot about long-term community cohesion).