I believe you might be thinking of month-to-month, which is not a traditional lease. On all fixed term leases over a month rent cannot be changed during the term of the lease.
If you have a lot of money and you want to build/buy a rental property, would you do it in a) a growing city with no rent control where your discounted future cash flows are unbounded or b) a city where there is a law placing a ceiling on your returns.
All else being equal, you are going to choose the city without rent control to build your new housing units in (or invest in something other than real estate altogether), thus illustrating how rent control disincentives new investment in housing, and specifically low-income rental housing.
> All else being equal
Okay but all else is not equal, and in the real world, it is never equal. These thought experiments do nothing to improve conditions in the real world, since they are based on flawed models of how people behave.
By that logic, nothing affects anything else and we should all give up trying to think or plan.
I think it is wise to produce better models with new information - not to ignore everything or to knowingly use false models. People do not - and should not - act according to those 'rational' ideas of maximum profit.
> Rent control deters the creation of new housing.
I keep seeing this repeated but I don't understand. Why does rent control deter creation of new housing?
> This reduce in supply raises the price
But you've glossed over why supply is reduced. I don't see any reason that supply would be reduced with basic moderate rent control like this one.
Edit: I have tripped some wire (?) and am not allowed to post any more. In response to the comment below,
> https://www.google.com/url?sa=t&source=web&rct=j&url=https:/...
That seems to be Washington, not California. According to http://www.hrc-la.org/doc.asp?id=36 it seems to read that some leases allow rent increases mid-lease.