That is what I'm trying to understand, since a lot of people here imply that "increase in supply will lower prices".
In the model, the price of a unit will change as you adjust either supply or demand. If there is lots of demand and little supply, the price is high. If there is lots of supply, the demand cannot force the price higher. Assuming the seller wants to sell, they have to compete with other sellers. The more units for sell, the less negotiating power they have because someone more desperate can sell cheaper.
My point is: prices will have to increases for supply to increase as well. Saying that increasing supply will keep prices low makes no sense if right now prices are too low for private interests to build more housing.
Supply won't keep prices "low" and certainly not lower than they are now. How would that work? How would they make profit?