A couple of weeks ago, Yale University filed a federal tax return showing that Richard C. Levin received an $8.5 million payout when he retired in 2013 after two decades as the university’s president. On Monday, the Chronicle of Higher Education released its new rankings of presidential salaries at public universities.

From these two data sources one can learn that Levin made more money in 2013 (the lump sum plus other compensation of $1.15 million) than the 10 highest-paid public university presidents and chancellors combined did in fiscal 2014.

So does that mean Levin was overpaid or public university leaders are underpaid? I think it’s mostly the latter — at least if you think markets are better at setting pay than politicians are.

Here are the top five from the Chronicle of Higher Education’s public university pay list:

• Rodney A. Erickson, Pennsylvania State University, $1.49 million

• R. Bowen Loftin, Texas A&M University, $1.13 million

• Joseph A. Alutto, Ohio State University, $996,169

• Elson S. Floyd, Washington State University, $877,250

• Paula Allen-Meares, University of Illinois at Chicago, $872,458

And here are the top five from its most-recent private university pay list, which covers 2012:

• Shirley Ann Jackson, Rensselaer Polytechnic University, $7.14 million

• John L. Lahey, Quinnipiac University, $3.76 million

• Lee C. Bollinger, Columbia University, $3.39 million

• Amy Gutmann, University of Pennsylvania, $2.47 million

• Charles R. Middleton, Roosevelt University, $1.76 million

Yale’s Levin came in 10th in the 2012 ranking, at $1.38 million. The giant check he got in 2013 was a retention bonus, agreed upon in 2004, to motivate him to stick around for a second decade in office. Shirley Ann Jackson’s big 2012 included a similar $5.89 million retention incentive — her pay in the previous four years had fluctuated between $1.1 million and $1.69 million. The other four in the top five also saw their pay boosted by anomalously large one-time payouts.

One-time payouts are still money, though, and even without them it’s clear that private university presidents, 36 of whom made $1 million or more in 2012, get paid a lot more than their public university counterparts. This isn’t because their jobs are harder. Public universities tend to be bigger — the public institutions in the top-five list above have more than 200,000 students among them; for the five private universities it’s about 77,000. They also tend to have less financial maneuvering room — of the 15 universities with the biggest endowments, according to Bloomberg data, only four (University of Texas, Texas A&M, University of Michigan and University of California) are public. And their administrators have to deal with more complex governance issues, as they are answerable in the end not just to boards but to elected officials.

That pressure from elected officials is the biggest factor in keeping public university salaries down. Most public university presidents aren’t the highest paid people on campus - - that honor usually goes to the football or basketball coach or somebody at the medical school. But they tend to make a lot more money than state lawmakers, and make for much better political targets than, say, Nick Saban. Gregory Fenves, the new president of the University of Texas at Austin, recently asked for a starting salary of $750,000 instead of the $1 million he was offered because the higher salary “would affect the ability of the president to work with the Texas Legislature.”

So, clearly, private university salaries represent something closer to the outcome of a market for higher-education leadership talent. And it is a market, albeit one with lots of frictions and quirks. A complaint often made of corporate directors is that they’re beholden to chief executive officers to hold on to their cushy gigs, and thus overpay them. Board members at private universities, though, tend to be donors to the universities, not recipients of university cash.

So why do private university boards make such big payouts? The private university presidents with the highest pay tend to fall into two camps:

• Empire builders at lesser-known universities.

• Long-serving presidents of rich, prestigious universities.

RPI’s Jackson and Quinnipiac’s Lahey are in the latter category. Jackson raised $1.4 billion during her first decade in office (she took over in 1999) and brought new renown to a venerable but oft-overlooked technical school; Lahey has since the 1980s built Quinnipiac from a little-known commuter college into a university that is beginning to build a national profile. It’s no surprise that their institutions’ boards are willing to pay a premium to keep them around.

At more established universities such as Columbia, the University of Pennsylvania and Yale, it seems like a combination of boards populated by very wealthy people who don’t think $1 million is very much and high-profile leaders who probably could get pretty good jobs elsewhere — or go back to being professors with lots of lucrative extracurricular opportunities — rather than begging for money 10 hours a day, which as best I can tell is the main thing university presidents do.

Yale’s Levin, an economics professor at the university before he was named president in 1993, was seen as a very successful president. The university was obviously already great when he took over, but it was less financially secure than some of its peers, located in a decaying city, and beset by conflicts with staff, faculty and students. The two presidents who preceded Levin, A. Bartlett Giamatti and Benno C. Schmidt Jr., had both left after relatively short tenures and moved on to prominent and lucrative jobs — Giamatti as commissioner of Major League Baseball and Schmidt as CEO of the for-profit school operator Edison Schools.

Levin stayed for two decades during which Yale’s endowment burgeoned, New Haven made a comeback and everybody got along most of the time. This wasn’t all Levin’s doing, of course (David Swensen, Yale’s legendary endowment manager, took office eight years before Levin did), but he was an adept, no-drama manager who made big admirers out of the members of Yale’s governing board.

As one of them, former Procter & Gamble CEO John Pepper, told the Wall Street Journal: “He could have been in investment banking, he could have been in venture capital, he could have run a corporation. Obviously, if he’d gone into other fields, the compensation would be orders of magnitude greater.”

Such comparisons can be misleading — I would bet Levin never dreamed of becoming an investment banker, for example. Still, after leaving Yale’s presidency he did take a job as CEO of a venture-capital-backed startup, online-education provider Coursera. It seems likely that he got paid so much at Yale mainly because he really was valuable.

Now, there’s a whole other set of questions about whether paying the head of a non-profit organization that kind of money really is smart. There are potential costs in terms of public relations and intra-university cohesion. There may even be a risk of goading lawmakers into revisiting private universities’ tax-exempt status. And Levin, who had spent his entire career at Yale before becoming president, might have stuck around even without such a big retention bonus. Still, private university paychecks such as his are surely a better measure of the economic value of university presidents than what the public university presidents get.