Housing in Greenwich Village (the broader area including the West Village) has become outrageous in recent decades. A significant reason, but not the only reason, is because the main NYU campus is located in the middle of the small area and doesn't provide nearly enough housing (only ~20%) for the on-campus enrollment.

NYU's on-campus enrollment is also roughly 50% foreign students. I don't have any data about it, but just from experience I've suspected that foreign students are willing and able to pay higher prices for housing.