you CAN. But that takes a while. There's always something under construction in Califnornia and a good portion of that is trying to build more apartments and housing. The time to make those don't keep up with the influx of people emmigrating to LA or otherwise leaving the parents' homes to seek their own lifestyle.
It's honestly a more feasible solution to try and convince certain companies to build in a city that is otherwise "vacant". Make it a place where people will work around, have the existing land and homes appreciate in value. That's why Amazon and Apple had such aggressive bid wars from cities to try and get them to make new campuses in their respective city areas.
We have a shortage of dense housing because it is illegal to build dense housing in the places it should exist, and where it is legal, community review processes quickly shut it down
It's a a long and expensive process and I wouldn't be surprised if some deals fell through something because the minimum expenses exceeded what some potential clients expected.
In which dense cities in Canada (or the USA) did this work, resulting in a desirable dense city with cheap housing?
You could argue that you can just keep building up and up until you cross the threshold of demand affecting the price, but without a market incentive to do that, it's wishful thinking at best. "Well the government could fund it" you could say, and indeed it can. But speaking of economic research, everytime the government has done such a thing, it causes widespread economic depression of the local area.
Housing does not suffer from induced demand like road capacity does. This is easy to disprove by reading those studies. Sure, building nicer better neighborhoods does make them more desirable but it doesn't follow that they have to be more expensive, as long as you build enough homes for people to live in. Taken to its conclusion the induced demand argument against density is an argument against improving quality of life in general, and pretty nihilistic.
[1] https://ideas.repec.org/p/fer/wpaper/146.html
[2] https://www.dropbox.com/s/oplls6utgf7z6ih/Pennington_JMP.pdf...
[3] https://docs.wixstatic.com/ugd/7fc2bf_ee1737c3c9d4468881bf14...
Tokyo, with its reasonable zoning, seems to have figured out how to build densely without being absurdly expensive. The key here is that the "market" is indeed allowed to figure it out, and not as restricted by NIMBYism.
The best thing the government can do right now is to get out of the way in regards to building more units.
They issue more permits for new housing units in a year than the entire state of California does, even though California has more than double the population. (or for an East Coast comparison - they issue around 9x the housing permits that NYC does, even though the population is <50% larger).
There have been many articles written on this topic by many publications that you can find, but here's one citation: https://www.ft.com/content/023562e2-54a6-11e6-befd-2fc0c26b3...
Tokyo metro area is 14,034 km2.
Meanwhile LA is 14% parking lots or so I’ve heard.