So, like guests who wouldn't leave, Yellen and Fischer can hang around on the board until their governorships expire, even while they are no longer chair and vice chairman. This would limit Trump's ability to remake the board because, by law, there can be only seven governors.
There are currently two open slots, as the Senate refused to confirm two nominees by President Barack Obama. Trump will get to fill those seats and he might consider one of them as a chair.
But the question is whether he gets another two appointees, giving him four, or a clear majority, of the seven? It's up to Yellen and Fischer.
Vince Reinhart, chief economist at Standish Mellon Asset Management and a former senior staffer at the Fed, thinks the two will follow tradition and step down.
"You don't want to go down in history with an asterisk next to your name,'' Reinhart said. He added that Trump is also likely to appoint a vice chair for supervision of banks (again, this comes from among the seven board members), and this could precipitate the resignation of Fed Governor Daniel Tarullo, who has been playing this role de facto. Reinhart thinks Tarullo won't stick around while his work creating the rules on Dodd-Frank is potentially amended or dismantled.
So, the law provides for the possibility that Trump's influence on the Fed could be muted. But tradition suggests it's more likely to be profound, giving him as many as five appointees.
And yet profound toward what ends is unclear. Trump has said he likes low rates and most presidents tend to support them. But he accused the Fed during the campaign of keeping rates low for political reasons (to help Obama) and said that markets were in a bubble because of Fed policy. The possibility that Trump could enact big tax cuts and government spending programs that run up large deficits could put him in conflict with the Fed if inflation takes off. They could need to raise rates more than they now forecast.
Yet the Fed's low-interest rate policies have come under withering criticism from Republicans. GOP favorites to head the Fed have focused on what are termed "hard-money" economists, such as Stanford Economist John Taylor. He has advocated for higher rates and a rule-based system for setting monetary policy.