If demand outpaces supply of any housing, prices will rise and there will be unmet demand. Rather than the city council trying to fight the market, it could encourage developers to make any kind of housing. People will move from different levels of housing and rents will drop correspondingly. Take a look at rent controlled areas and note the scarcity of units.

I think the real problem here is that after all is said and done, it is costing $1M to build a BMR 2+2. There just isn't enough money to subsidize the number of units needed. I really think we need to work on the demand side which at the top level is driven by too many good jobs. Let's not build any more office space (at least for businesses that are not directly serving the local community) until we can get into some kind of balance. Historically (FDR said in 1933) 1/3 of the population is under-housed, so some of the shortfall is a long-overdue and legitimate need to upgrade housing opportunities all the way down the spectrum, but trying to do that by taxing new market rate units (someone has to pay for the BMR units) just drives up housing costs for the market rate buyers and renters which pressures them to bid up the cost of what used to be lower cost housing, hence the plight of the missing middle. We've got to get demand under control!