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Things that got me bloody Chicklets and dings back in the old days of BP: I imagine that some of the problem the school faces is enmeshed with their image. Michigan has long enjoyed a reputation of being one of the premier public universities in the US. Speaking academically, I imagine they see their competition as UC Berkley, Wisconsin, Texas, and North Carolina. They do not take their position lightly or for granted. They have worked long and hard to earn that reputation and to keep it. Historically, they were one of the first public universities to establish an endowment fund (1817 - as in 19th century) with the expressed purpose of allowing them to attract leading scholars in all fields and retain them. They have lobbied the Michigan legislature for support of higher education and the state has long led the rest of the states in support to higher education as a result. Because of their prestige, Michigan is first in line when the hogs are released to feed at the state's funding trough.
In contrast, Ohio's legislature has seldom equaled Michigan's support for higher education and Ohio State's position as the state flagship university has been up and down, especially during Jim Rhodes' governorship. Add to this the fact that Ohio State did not begin to establish an endowment fund until 1985, and while it has grown impressively, it still lags far behind Michigan's.
So what is this leading up to: For the last fifteen years Michigan has had to retire one coach, hire and quickly fire two coaches, and then reach deep into their coffers to hire Harbaugh, setting a record for the amount of compensation for a college head coach. During the same time period, Ohio State refurbished and expanded Ohio Stadium, created world class facilities for most of its non-revenue sports, and paid it off with funds from the athletic department, not the school's general fund. Michigan is in the process of doing the same, but as I understand it, they have not been able to pay off the debt as they go along.
Now they're faced with the prospect of another buy out, another search, another need to match what the highest bidder might offer to a rising star coach. And here's where I think the rub might come in. Harbaugh's salary is one thing, but look at the rest of the football payroll, Don Brown was getting a million, Jay "Crayon Eater" Harbaugh was paid 350 grand to produce receivers who consistently dropped crucial passes. At he same time - and this gets back to the heart of the University of Michigan - the pay scale for one of the premier public schools in the US shows that the highest paid full professors are pulling down $250,000 to actually teach, research, write, and publish. Tell me that doesn't grate. Tell me it isn't a frequent topic of discussion when the academics sit down with the president and the board at the faculty club. Berkeley long ago put the brakes on athletics. Texas lets the oil men have their play pen as long as the money comes from them and not the general fund. Chapel Hill could give a shit about football and if basketball fucks up again with "jocks only" courses there will be hell to pay. Wisconsin manages to get along with lowered standard of expectations. That leaves Michigan in a far different situation than - dare I say, Ohio State - where I'm sure the faculty has its issues with football's place, but where even when COVID wipes out football and basketball revenue, the athletic department has the funds to cover the losses.
In the end, I'm sure Michigan cares far more about retaining its status as a premier public university than in winning a national championship and that means keeping the worker bees in the academic hive happy.