Here are two possible routes Trump could take:
- He'll leave things essentially as they are, as the Fed has suggested it plans to raise interest rates gradually in the coming years in response to improving economic conditions.
- He'll reshape the Federal Reserve by diluting its independence and its power over interest rates.
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The case for the status quo
Some experts expect things at the Fed to carry on in the trajectory they have been heading, meaning higher interest rates to come, as is favored by conservative economists, but there will be no great shake-up.
"President Trump is going to be different from candidate Trump, and President Trump needs the economy to keep humming along," said Greg McBride, CFA, Bankrate's chief financial analyst.
That means, in McBride's estimation, a rapid acceleration of interest rate hikes is unlikely.
"From the standpoint of monetary policy and the economy, higher interest rates, sharply higher interest rates, would not be beneficial to ongoing economic growth," he said.
In December, the Federal Reserve projected it could increase short-term interest rates as many as three times in 2017. That more hawkish approach -- which in the short term could impact how much interest consumers pay on credit card balances and how much savers earn on certificates of deposit -- may be what Trump wants, said Steven Gattuso, CFA, an assistant professor of economics at Canisius College in Buffalo, New York.
"There's the potential for rates to be rising certainly faster than they are with the current Fed," Gattuso, a financial planner at Courier Capital, said.
But, he said, an increasingly aggressive Fed coupled with any stimulus spending Trump proposes -- he's called for both broad tax cuts and a large infrastructure package -- "could heat up" inflation quickly.
"If the rates rise faster than the economy can absorb, that certainly could tip the economy into recession," Gattuso said.
Inflationary pressure is already mounting, though, said John Chang, the head of research at Marcus & Millichap, a Calabasas, California real estate brokerage firm.
Indeed, the Federal Reserve this month bolstered language in its statement following its latest policy meeting suggesting they would be looking closely for evidence inflation is picking up.
"There are a lot of unknowns, more unknowns than any previous transition," Chang said. "The only thing I can really say is that Trump's a business person, and I think most of the business people are at least hopeful that the decisions that he makes will be good for the economy."
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Will Trump radically change the Fed?
Late last year, some reports suggested Congressional Republicans, emboldened by a Republican president, would seek to restrict the Fed's independence and move to make rate adjustments more automated.
It's a suggestion President Ronald Reagan considered in 1980, but later abandoned after meeting then-Fed Chair Paul Volcker, whom he later re-appointed to the top leadership post.
Trump has not said what his plans for the Fed are, but Gary Cohn, head of the White House National Economic Council, has been critical of the Fed's role in regulating banks.
Meanwhile, Yellen has not said if she expected to encounter an attempt to change the Fed's structure.
"I'm not going to offer the incoming president advice about how to conduct himself in policy," she said. "I'm a strong believer in the independence of the Fed. We have been given the independence by Congress to make decisions about monetary policy ... and that is what I intend to stay focused on."
Trump may prove more hawkish than people expect -- and that could force borrowers to pay more in interest charges -- but could he restructure the Fed by his words alone? That's one lingering unanswered question.
"I do think that tweeting and off-the-cuff comments aside, the Fed is still going to be subject to the same Monday morning quarterbacking from Congress," McBride said. "But as long as they have the ability to operate without any political constraints, I think that's the key point."