The arms race issue is real, though, and students are less price-sensitive than you might otherwise expect because so many of them are paying for all of it with debt, and the dollar figure looks so obscene everywhere that I suspect lots of students just choose not to think about it. One school's dining hall being fancier is immediately appreciable on a college tour. One school's ridiculous high number that you won't have to pay for years vs. a slightly different ridiculous number at a different one feels less concrete, even though that difference is what paid for the fancier dining hall. Given that, there are big incentives for schools to go ahead and invest in the fancy things and win the students.
I think it is real, but I don't think expenditures drive it all that much. What does drive it is the perceived value through pricing. If the going rate for public schools is $5k/yr, the private colleges put their nameplate rate at $25-30k/yr. Significantly more, but that higher price sends a quality signal, and the students they want don't actually pay it, so they aren't negatively impacted by the high cost.
When the state school price increases to $10k/yr or even $15k/yr, the private schools can move up their nameplate price to $40k/yr or $50k/yr and it doesn't seem ridiculous anymore, and still retains the signaling value.
There has been some increase in student services costs based on the expenditures numbers I posted (about 4%/yr real increase), but that's not enough to explain the 50-100% increases in nameplate tuition at both public and private schools. That is driven by de-investment in post-secondary education by state legislatures, which directly results in tuition increases by public schools to balance the books, and then the similar increases by private schools to maintain their relative cost.
