Net tuition and fees for students at private 4-year colleges is about $14,600 in 2018-2019. That results in a total of $58k/student in revenue, not $150k. https://trends.collegeboard.org/college-pricing/figures-tabl...
As public institutions, UC campuses see even less: https://trends.collegeboard.org/college-pricing/figures-tabl...
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.
The college board has good data on the cost side: https://trends.collegeboard.org/college-pricing
And the department of education has good data on the expenditures side:
Private: https://nces.ed.gov/programs/digest/d17/tables/dt17_334.30.a...
Public: https://nces.ed.gov/programs/digest/d17/tables/dt17_334.10.a...
[0] https://www.topuniversities.com/student-info/student-finance...
You can see UCLA's financials here: https://ucla.app.box.com/v/acct-pdf-AFR-16-17
Tuition and fees netted $833MM in 2017. The cost of the undergraduate schools alone was well over a billion dollars.
[0] https://www.latimes.com/local/education/la-me-uc-spending-20...