But, again, those policy combinations do not require rent control. The question would then be which creates more affordable housing: rent control plus those policy combinations or those policy combinations without rent control?
You're missing the temporal aspect here. The question isn't which housing policy produces the most affordable housing. A supply shortage's one and only solution is investment in supply (with the government helping that along or not). But development can only go so fast and so rent controls prevent the renters from being bled dry in the meantime. Even in situations where rent isn't intolerably high, rent control laws are a safeguard that can prevent a sudden migration boom from adversely affecting lower class rents while development catches up.
Edit: I have an obligation here I think to elaborate on that final sentence because clearly rent control laws could prolong that catch-up period... but this dovetails quite nicely with Kara's point and your question about why the state has to be involved in the building. Since investors have no idea where to build (because the profit rate is arbitrarily equalized between municipalities), the state has to step in to receive complaints about shortages or analyze vacancy rates. It could then either subsidize building costs in impacted cities (the market solution) or it can simply build its own housing (the public housing solution). Thus even in the market case the state has a central role to play.
All this is to say that rent-control does reduce the the quantity of housing available in a market (while benefiting those who already have units and don't need to move) but other policies can be put in place to counteract that--policies that have been unpopular in North America for a while, at least those involving the construction of public housing. If the state enacts rent control, according to you, it needs to be involved in building units.
In Ontario, modern rent control was introduced in 1975, and it has correlated with a decrease in the construction of purpose-built rental units. 83% of purpose-built rental units were built before 1980 and only 7% since 2000. This is worse in Toronto: less than 4% of rentals were built since 2000. I doubt rent-control is solely or primarily to blame (zoning laws, popularity of condos, land speculation, etc.), but the government has not done its part to mitigate the harm.
Since about 1995, rent control laws have been changed three different times. During that time, only about 20,000 units of public housing have been constructed, increasing the supply by less than 10% while the population has grown by more than 30%. Only about 5% of low-income Ontarians on wait lists (most don't bother to get on the lists) are housed in these units in a given year. It's easy and inexpensive to change rent control laws, but it's difficult and costly to build units.
