Harvard could afford to charge nothing for any of its students. However, by having a form of price discrimination where they gouge the wealthy, they then get to subsidize their low-income students, so that they don't have to miss out on as many of the non-economic opportunities that their classmates engage in, and still provide all the amenities (nice gyms, libraries, clubs and activities) that the upper-class students expect.

The issue is, after the most endowed schools ($/student), there is a steep drop where schools suddenly aren't able to provide the same opportunities for their low-income students, but still compete for students capable of paying in full. This leads to escalating costs, as schools construct nicer gyms, study areas and dorms, pricing out the mid-to-low income students.