I have a cousin who lives in LA. He rents a studio for something like $1100 a month, in a decent part of the city. Apparently it's rent-controlled, which is something I've never heard of as far as Los Angeles is concerned.

To be honest, I heard of all this through his mother. I kind of suspect he's lying to her and is actually a drug dealer and his place costs a lot more than he's letting on.

is it not possible to move to a place with houses which are lower in price?

In Australia it's not really possible because everyone did that and now houses are high in price literally everywhere but surely in America they would have more states and places that have lower house prices?