It may be way too soon to say anything sensible about what Trump thinks about the Fed or will do about it, but as the first person to have publicly called for appointing Janet Yellen as Chair (back in 2009), I figure I am more situated to stick my neck out to say something, especially when it looks like what is coming is a big contradictory mess.
For the moment the Fed seems to be laying low, having made almost no change in policy or projections as reported by the diligent Tim Duywww.economistsview.typepad.com/economistsview/2017/02/fed-watch-fomc-employment-report-warsh.html [trying to link to economistsview.typepad.com/economistsview/2017/02/fed-watch-fomc-employment-report-warsh.html , but not succeeding]. They remain open to maybe tightening after March if the employment report improves notably, but otherwise seem to be on a "steady as you goes" path for the moment, doing almost nothing. This on top of a letter from Congressman McHenry demanding they stop cooperating with any international banking entities until Trump makes appropriate appointments. And Tim adds a comment on a recent column by former Fed gov Kevin Warsh, who indulged in criticizing the Fed by demanding that it follow policies it is already following. In this latest post Duy suggests that perhaps Warsh is running for Fed Chair, which means one has to appear to criticize Fed policy, even if one is not really. Which raises the question of what Trump will do.
Let us start with something that hardly anybody has noticed, but is just taken for granted: that Trump almost certainly will not reappoint Janet Yellen as Chair. This just seems to follow from his general attitude that all incumbents are no good, and especially anybody appointed by Obama, except for FBI Director Comey. Why she is no good is not immediately obvious, and in fact several times over the last two years he praised her "low interest rate policies" noting that as an old real estate developer he has always been a fan of such policies. However, in June of this past year when the Fed did not raise the fed funds target rate, he denounced her personally and the Fed more generally for not doing so, charging that their failure to do so was part of a plot to goose the economy in an effort to help elect Hillary Clinton. Given that rhetoric it seems unlikely that he would reappoint her, although it could still come to pass that if when her term comes up next year markets seem to like her as well as GOP commentators, he could change his mind.
Let me note how this expected non-reappointment is a violation of existing norms regarding the Fed, that it mostly been above obvious partisan politics. The sign of this is that the last three Fed Chairs were each reappointed by an incoming president of a different party when that president came into office. Thus, Republican Ronald Reagan reappointed Dem Fed Chair Paul Volcker. Dem Bill Clinton reappointed GOP Alan Greenspan, and Dem Barack Obama reappointed GOP Ben Bernanke. But it is all but taken for granted that there is no way Trump will appoint Dem Yellen, and it seems that few observers are batting an eyelash over this, although as near as I can tell she has done a pretty good job and deserves reappointment.
Which brings us to the hard fact that as of now Trump's attitude towards what kind of person he might appoint to the Fed as either just an ordinary governor (and there are two current vacancies) or as Chair to replace Yellen is highly unclear. On the one hand he has made noises about supporting hard money gold bugs, who presumably would push for higher interest rates soon. OTOH, we have already seen that "as a real estate developer" he has long liked low interest rates, and we now have a situation where he has been bashing various foreign countries for supposedly holding their currencies at overly low values relative to the dollar, with the dollar having risen since his election thanks to rising interest rates that happened immediately on his election in the markets, not pushed by the Fed, due to the expectation that he will be increasing the budget deficit. All this simply points to that he does not remotely have a coherent view yet of what his broader economic policy is or will be, much less what his preferred monetary policy. If he increases budget deficits while appointing monetary hawks to the Fed, he may well end up with a Reaganesque mid-80s boom of the dollar to be associated with an unpleasant further hollowing out of our manufacturing sector that he has supposedly been elected to help out.
It is unclear if he is cognizant of the contradictions inherent in his current set of supposed economic policy positions, much less what he will do if and or when he becomes aware that he is facing such contradictions.