It is inaccurate to say there are far fewer apartments in Manhattan-- many neighborhoods simply became less crowded, as improving mass transit meant that a working-class family sharing a one- or two-room apartment could have a larger space in Brooklyn, Queens, the Bronx, or suburbs beyond, while maintaining access to Manhattan wages. There are also far more single adults and childless couples in the American population (especially in NYC) versus 1910, meaning an apartment that once held a family now often houses only one or two people. The New York Times had an interesting piece on this last year[0]:
"Manhattan was much more crowded in the early 1900s than today, especially in the tenements of the Lower East Side. For example, in 1910, 66 people lived on the four residential floors of 94 Orchard Street. Today, the buildilng houses 15 people."
And, in fact, many apartments have been built in Manhattan since 1910-- much of far northern Manhattan wasn't developed by then, and middle-class apartment houses were in their infancy.
That isn't to say that Manhattan (and its commuter-shed) doesn't need more new housing, and that the 1963 zoning code governing NYC isn't problematic: for example, inner Brooklyn, on top of tons of subway lines still has parking requirements, and many areas are zoned such that any new construction would be less dense than what it would replace. The problem isn't a supply that's been reduced, but rather a supply that isn't increasing along with demand, with zoning laws that often prevent any densification that would lead to increased supply.
The same problem holds in Silicon Valley, except that its zoning holds it to a suburban, rather than an insufficiently urban, form.
[0]http://www.nytimes.com/interactive/2012/03/01/realestate/man...