I'm not sure that's correct. I think the reason that you see a lot at that height is because building codes change quite a bit as you go past a certain height and the costs go up as well.
http://www.structuremag.org/?p=10934
The main height limit is local politics around density, not building codes.
3-4 is also where you need to start thinking about adding an elevator. That adds to the cost and reduces the area available for units.
And so? Is it affordable live in Paris? No, it is not. I just looked up rents there, and a 500sf studio will run you around $1300-$1500USD a month. Given average salaries in the area, this does not compare favorably to SF. (As a matter of fact, the city of Paris has just enacted strict rent-control laws, because rents were seen as unaffordable.)
Also, be more specific: what part of SF are you talking about? If you're talking about the 1/3rd of SF that could be characterized as "city", the density is already quite high. Increasing the density of already dense areas won't meaningfully impact rents. But meanwhile, average density numbers completely ignore that the western 2/3 of SF looks like suburbs:
https://www.google.co.jp/maps/@37.7501516,-122.4626812,3a,75...
That's in the city of San Francisco. I dropped a pin randomly in the west side of the city. You can see Sutro Tower!
When we talk about density in SF, these are the areas that matter. But even if you could build higher density in these areas, you'd still have to buy the land (which would suddenly be much more expensive), build transit, infrastructure, and so on. There is no inexpensive solution to this problem.
Yeah, lots of discussion treats SF as if it was the "downtown" urban core of the Bay Area, but in fact only a part of SF is that way, and a lot of SF, if still urban, not the kind of core that it is often compared to.
I'm actually not opposed to higher density or building up. I'm just pointing out that it isn't a panacea, and certainly won't work if you focus on the wrong places.
Cost of land goes up with density, but that's the point. You build more and spread that cost among more units. Up to a point, it is economical to build higher in that situation (then for skyscrapers you start to see cost per unit going up again). Look at all the 3 or 4 stories apartment buildings sprouting in the city right now. Most of them could be double the height without increasing construction costs per unit.
If your artificially reduce the number of units that can be built on a plot of land, you're forcing to spread the price of land among fewer units and you increase their cost.