You're missing that in most high priced housing markets, you can't just build an apartment building anywhere you can build a single family home.
edit: and it's also obvious that density can be affected by zoning laws, but I was assuming those are held constant (because a, they kind of are constant in that they don't change much, and b, the present phenomenon of rents rising sharply since 2005 is, I think, not generally believed to be caused primarily by a change in zoning regulations)
You recognize that there are occasions when a developer has the option to build more density or less density on a specific piece of land, yes? I was using an exaggerated example (1 vs 16) to illustrate the more general concept of less-vs-more. I am arguing that, other things being equal, higher wealth inequality will lead to developers choosing to build less density, because an increase in wealth inequality implies that the rich have more to spend than the poor (relative to whatever previous period you're comparing to that had less wealth inequality).