Extraordinary Observations' Rob Pitingolo has an interesting essay examining the dynamics of American housing markets. Why are there shortages even if supply grows? It's the demand side of the equation.
Pitingolo goes into more detail in his post, and provides cool graphics, too!
I [argued] that we can't really talk about the "housing market" because housing isn't homogeneous. People are willing to pay more for high-quality housing than they are for low-quality housing. So if you change the type and quality of housing in a neighborhood, you alter the underlying structure of the market itself, and the price at which people are willing to pay to live there, all else equal.
But let's set that aside for a moment and assume, for argument's sake, that housing is a commodity. From a theoretical perspective, we can draw the housing market as a series of simple charts, similar to what you might remember from an intro to microeconomics course.
In a neighborhood housing market, the demand curve is downward sloping. There's nothing notable about this; and in the short-run, the supply curve is a vertical line. This is consistent with the reality that in most neighborhoods you can't start construction whenever you want to. In fact, it's often a huge hassle to construct new units, for a variety of reasons, NIMBYism being one of them. So in the short-run, the supply is fixed.
The theory continues that in order to meet the high or pent-up demand, as well as make housing more affordable, we ought to increase the supply of housing. [...] If all goes well, the resulting equilibrium will be a lower price and higher quantity of housing.
But it doesn't end there, because we haven't accounted for the demand curve, which very well may shift. This could be because new amenities have popped up in the neighborhood; but also because the neighborhood is getting safer, cleaner and generally becoming a more desirable place to live. It could even be for a completely exogenous reason. Whatever the case, let's imagine that the end result is a rightward shift in the demand curve.
If the shift in the demand curve is great enough, it can completely overwhelm the shift in the supply curve, leading to an equilibrium with higher prices.
Now, I'm not using this as an argument to say we shouldn't work to increase density in cities or metro areas. But I do want to show that the simplistic "increase supply to make housing affordable" argument doesn't always hold, for theoretical reasons, especially at the neighborhood-level, and in urban areas that are already relatively dense and desirable.
Pitingolo goes into more detail in his post, and provides cool graphics, too!