Chicago pays its Professors higher than any other University, once you factor in cost of living

https://www.insidehighered.com/news/2019/04/10/aaup-study-finds-small-gains-faculty-salaries-offset-inflation

Top Salaries for Full Professors at Private Universities, 2018-19 (Average)

1 Columbia University $259,700

2 Stanford University $256,100

3 Princeton University $248,000

4 Harvard University $244,300

5 University of Chicago $241,900

6 Massachusetts Institute of Technology $232,200

7 Yale University $230,900

8 University of Pennsylvania $223,600

9 New York University $218,300

10 Northwestern University $215,200

If you take cost of living into account, Chicago would be #1, since cost of living is 25% higher in Boston, so you would need about $301K to come out even in Boston.
You would need $481K in Manhattan, $301K in Queens, $383K in San Francisco, $246K in New Haven, $224K in Philadelphia

Don’t places like Columbia and Stanford provide at least some of their faculty with a housing benefit? I know that at least at one point profs moving to Stanford were given either a favorable loan - or maybe the university co-owned the place with them and provided most of the funding. Not sure if this applied to all faculty or just tenured.

OP this is a bit of oversimplification. In the top private (and even in top public) there is a whole class of chair professors. They get paid much, much higher than full professor. And the exact amount really depends on each endowed chair.

Let me elaborate. Without using real names, let’s say a T5 school wants to entice a world class researcher from a T50 school to join them. The T5 school may offer the Warren Buffett Distinguished Service Professor position to the world class researcher and the salary may be two to three times more than a base Full Professor pay.

University administration typically does not disclose how much they pay these chair professors. That is between the school and the individual chair professor. And administration obviously does not want to embarrass donors by telling the world how much each chair professorship pays. A lot of donors do not want the details of their donation made public.

But you can be sure there are many of these chair professorships at the ten elite universities you list above. Just look at the Department of Econ at UChicago:

https://economics.uchicago.edu/directories/full/faculty

Every big shot there is a chair professor. So they are paid more than the base full professor rate but we have no idea how much Levitt is paid versus List.

Thus without accounting for salary component of these chair professorships, it is really meaningless to just compare the pay of full professors. You bet really rich schools like Harvard and Princeton will have boatloads of chair professors that get paid sky high salaries.

According to the AAUP survey that this article cites, Duke professors make $214,200. They would have to be paid $280,609 in Chicago to maintain the same living standard.

https://www.insidehighered.com/aaup-compensation-survey

https://www.bankrate.com/calculators/savings/moving-cost-of-living-calculator.aspx

These abstract cost of living numbers are awfully imprecise. How are they comparing “Chicago” or “Philadelphia” – places that cover a huge range of living situations – with San Francisco, Manhattan, or Palo Alto, which no longer have anywhere near the same diversity?

As someone who lives (very nicely) in Philadelphia, has children in Chicago and Brooklyn, and for a while maintained an apartment in Chicago because my spouse’s job was based there, I can certainly attest that there are huge differences in the cost of living place to place. But the difference between Chicago and Philadelphia is a heck of a lot more than the 8% the OP’s post suggests.

Stanford does provide housing subsidies. So, by the way, does NYU, quite extensively. It’s an important recruiting tool for them. Both of those universities own a LOT of real estate around their campuses. Penn and Yale, I believe, use meaningful subsidies to get faculty and staff to live nearby and shore up the immediately surrounding neighborhoods.

UChicago does too. In fact they have a pretty decent blanket program that applies to all employees, not just faculty (and is totally not a useful tool to gentrify woodlawn). Faculty get even nicer ad-hoc deals.

https://civicengagement.uchicago.edu/anchor/uchicago-local/employer-assisted-housing-program/

U of C does have one in house advantage that best many of the schools on the above list: Lab. As far as I know, none of the schools other than U of C on that list can offer enrollment of an elite private K-12 school to its faculty’s children. And faculty pays half only of the Lab sticker prices.