How? All it does is set the rate of rent increase to e.g. 4% a year, it does nothing about the zoning of the lot. For example, people always cite something like someone in greenwich village getting a deal on an apartment, and that is somehow constraining development, while ignoring that the neighborhood of greenwich village is built to over 90% of its zoned capacity, and you cant add very many more units there even if you could get away with renting them for a million dollars a month and had permission to level whatever building was on the lot previously. Plenty of cities actually saw significant downsizing to zoned capcity since the 1960s, when things were certainly more affordable by most accounts. Sometimes by factors of 4 or more fold reductions in zoned capacity when you account for population size. The city of LA itself for example is built to over 90% of its zoned capacity. 2.
1. Some people will naturally want to move out of their rent-controlled apartment, for example to be closer to a new job or to get more space for a newborn. Those people are disincentivized to move because rent control means they are currently paying below-market rent.
Incidentally, that also increases traffic congestion because nobody wants to move across town to be closer to work. So you commute to where I live and I commute to where you live, instead of us both living within walking distance of work.
2. Rent control normally does not include an adjustment for inflation. And, when it does, the rule is usually "rent can increase by x% of rent or y% of CPI, whichever is lower ". That means that over decades, rent falls in real-dollar terms.
Since potential landlords can do math (they have that technology) they can see that the ROI of an apartment building (measured in decades) is lower in places with rent control. That disincentivizes them to buying or building apartments.
Lastly, it seems like you're really trying to hang your hat on zoned capacity. The thing is that zoning can change. If the ROI looks right, landlords will lobby the city for a zoning variance to increase capacity. If it doesn't, they won't.
1. http://furmancenter.org/files/sotc/SOC2009_State_of_New_York... 2. https://la.curbed.com/2015/4/8/9972362/everything-wrong-with...