Rising demand and inelastic supply leads to rising prices. This is a good thing. The price premium the wealthy are willing to pay allows schools to massively subsidize their brain drain of the brightest students. If you get into stanford and your parents make less than 60k, you don't pay tuition at all. The alternative is rationing which leaves just as many students out in the cold but doesnt allow the school to capture extra rents.

So the other issue is inelastic supply. This is partially down to the regulatory environment, but that isn't the whole picture. I haven't done enough research to understand why there aren't more schools starting up. I suspect it's an accreditation barrier to entry issue.