> In practice, new housing increases the denominator (number of students) more the than numerator (tax dollars). This is because the existing tax base includes non-residential land that only provides tax dollars without increase school costs. So any new housing is always a net loss.

The assumption here is that you would have only new residential housing. But zoning that allows for higher density construction also allows for new businesses that pay taxes and have no children.

And you would also have to be making assumptions about the ratio of taxes paid to number of kids. But the people who move into small cheap apartments and condos are most often college students and retirees, who don't have kids yet (and wouldn't continue to live there if they did) or their kids are already grown. From which you get new tax dollars but no new students. Whereas the people who can afford a 3- or 4-bedroom unit have money and pay an above-average amount of tax, so they might have kids but they're paying their own way.

This is also not just a fraction with a numerator and a denominator. Many of a government's expenses are fixed costs, e.g. most single-lane suburban roads are nowhere near capacity and would not need to be expanded or even experience congestion if the amount of traffic on them doubled. Each new resident you get paying taxes increases the tax base over which you can amortize those fixed costs.

Moreover, the ratio is irrelevant. If someone is paying $21,000 in taxes and then their presence requires the government to spend $20,000 more in services, you'd have a net gain of $1000 from their presence regardless of what the ratio is for some other people or existing residents.

> The flip side of that is that people who don’t have those houses want to take that away without paying the market price for that land.

They are paying the market price for that land. In order to build a multi-story building, you first have to buy a piece of land to put it on.

What they're doing is making more efficient use of that land, by creating e.g. 10 housing units on a lot currently being used for 1. That reduces the market price of housing units by supply and demand, but the higher price was only ever a result of artificial scarcity.

> Historicaly, being close to high paying jobs was a major driver.

So a shorter commute makes up for a higher price. But if you prohibit housing construction, this trade off only gets worse. The job is in the city, you could have lived in a 6-unit building 20 minutes away, but if those are prohibited then you need 6 times as much land for the same number of residents and now they either have to pay more even for the 20 minute commute or take the far away housing with a 60+ minute commute.

Whereas if you allow the 6-story buildings, anyone who wants the 20-minute commute can live there and anyone who wants to live in a single-family home can live five blocks away with nearly the same commute but for a lower price than they'd have had to pay when those higher density buildings weren't there satisfying housing demand.

> Though I’ll be honest to not understand why anyone would pay a premium to live like a sardine.

Density and small units are not inherently associated. You can put 4000 sq ft units in a multi-story building and still have ten times the density as a single-family home of the same size as one unit.

> The air quality alone is enough reason to move out of the city.

Tell that to the suburban ignoramuses who think it's a good idea to burn wood:

https://www.familiesforcleanair.org/wp-content/uploads/2013/...

That graphic isn't even to scale. The pellet stove looks like it's only a little over twice as bad as an oil furnace when it's actually more than a hundred times worse, and the wood stoves and fireplaces people have are an order of magnitude worse than that.