There's yet another staffing change looming for the already new-look Trump administration: the chair of the Federal Reserve.

The current chair, Janet Yellen, will finish out her four-year term early next year. At that point, President Trump will have to nominate a replacement, to be confirmed by the Senate. And the question of just who that replacement will be is already roiling the White House.

The chair of the Federal Reserve leads the central bank's voting officials, wielding enormous influence over monetary decisions and the Fed's regulatory duties. Politico reports that Gary Cohn — currently leading Trump's Council of Economic Advisers — is the frontrunner for the job. But he isn't an uncomplicated pick, with his history as a registered Democrat and support of free trade.

Still, the bench for other possible replacements is extremely short. "I think there's a 30 percent chance [Trump] renominates Yellen," one anonymous bank executive told Politico, "not because they would ever want to nominate her, but because they'd run out of people and time." On the other hand, some Senate GOP staff and senior administration officials are responding to the idea of renominating Yellen by effectively saying, "No way, no chance, never, ever gonna happen," Politico reports.

In many ways, Trump's dilemma is an outgrowth of the strange and self-contradictory ideological space he's tried to carve out for himself: ostensibly populist, yet friendly to the desires of the GOP donor class. As another major bank official told Politico, Trump is basically looking for a Fed chair who wants low interest rates and who also supports financial deregulation. And that's just not easy to find.

Let's start with interest rates. The basic tradeoff is that higher interest rates tamp down inflation, but also discourage job creation and wage growth; low interest rates do the opposite. The financial industry prefers high interest rates and low inflation, as those conditions make it a lot easier for banks big and small to make hefty profits.

For other wealthy Americans and business sectors, the situation is a bit more complicated: Many of them can actually benefit from lower interest rates, because credit and capital are cheaper. But by encouraging tight job markets and higher wages, low interest rates also increase the cost of labor — which most companies would prefer to avoid.

The people who really benefit from low interest rates, then, and who are harmed the most by high rates, are ordinary working Americans — low-wage and minority workers most of all. As the economy heats up and labor becomes scarce, workers can demand bigger pay hikes and better working conditions from their employers, with people at the bottom of the economy seeing the fastest gains. Conversely, high interest rates control inflation by throwing the most vulnerable Americans out of work.

Trump sits in a strange sweet spot here: He comes from real estate, which certainly benefits from cheap labor, but not to the same extent as, say, the service sector. And Trump's own past real estate deals relied on huge (and often disastrous) amounts of borrowing. Moreover, sitting presidents generally become more popular when voters are seeing widespread economic gains.

So it's not surprising that Trump, as president, has characterized himself as a "low interest rates person," despite criticizing low rates as a candidate.

But financial deregulation is a different story.

The Federal Reserve has a lot of regulatory responsibilities overseeing the big banks and major financial firms, particularly since the passage of Dodd-Frank. For obvious reasons, the financial industry really doesn't like those regulations: They make it harder to take big risky bets with huge potential payoffs; they chew up time, labor, and paperwork; and they increase the cost of doing business.

But it's not just big banks — leaders across all industries pretty much hate regulation as much as the financial industry does.

Trump himself has never wavered from a staunch deregulatory message. During the campaign, he repeatedly promised to slash "job-killing regulations." Now that he's in office, his Treasury Department has already issued a report calling for a whole suite of financial regulations to be rolled back, mostly through agency decisions and executive orders that don't require congressional cooperation. That report has also been endorsed by Randal Quarles, Trump's nominee for the Fed's top regulatory position.

Once again, the people who do need tough regulations are ordinary Americans. As consumers, borrowers, and homeowners, they need to be protected from fraud and abuse by the financial sector. They also have an obvious stake in preventing the financial industry from taking on big risky bets and complicated schemes, because such shenanigans have a proven track record of wrecking the economy.

There's a pretty clear class divide here. If you're primarily concerned with the interests of workers, you're going to favor lower interest rates and more regulation. If you're primarily concerned with what benefits the wealthy, big businesses, and the financial industry itself (as many Republicans do), you're going to favor higher interest rates and less regulation.

All of which puts Trump in a tough spot when it comes to choosing a Fed chief.

Yellen is widely seen as a monetary "dove," largely responsible for the Fed's slow pace on interest-rate hikes. But she might also throw up roadblocks to any deregulatory push. Meanwhile, Cohn would certainly support Quarles' bid for a regulatory rollback, but could well give in to the financial industry's conventional wisdom and support higher interest rates as well if he becomes Fed chair. He's a Goldman Sachs alumnus, after all.

Cohn's background could cause consternation among more populist or Tea Party-type Republicans in the Senate, but he's still more likely to get past the Senate than Yellen. And Trump's own track record of sticking to his guns when his supposed pro-worker populism butts up against GOP orthodoxy is not impressive in the slightest. So the next Fed boss may well be Gary Cohn.

Finding a Fed chair who supports both of Trump's postures — low interest rates and lighter regulations — is sort of like finding someone who supports both single-payer health care and major tax cuts for millionaires. Such people may well exist. But they're pretty rare.