The Fed chairman’s job has stayed in Republican hands partly because the party had more credibility on inflation for much of the late 20th century. Bill Clinton twice renominated Mr. Greenspan, both times in presidential-election years. Mr. Obama renominated the current chairman, Ben S. Bernanke, with the financial crisis still simmering.

The same forces that led to these renominations may now be helping Mr. Summers. While the differences between him and Ms. Yellen have at times been exaggerated, he does seem to be somewhat to her right, on monetary policy and regulation. Some aides who were advising Mr. Clinton when he selected Mr. Greenspan are now advising Mr. Obama, and it is not hard to imagine why they may see Mr. Summers as a more natural bridge from a Republican-run Fed.

Some Fed watchers think Mr. Summers may be more worried than Mr. Bernanke about the risks of the Fed’s efforts to lift economic growth. But views on monetary policy do not always track neatly with ideology. Both Mr. Greenspan and especially Mr. Bernanke took positions on interest rates that might have seemed more typically liberal.

The area where politics matters most is probably regulation, Mr. Blinder noted. The next leader of the Fed will have a large role in implementing the Dodd-Frank financial regulation law. The Fed will face multiple decisions about when markets should function without interference and when regulators should try to protect consumers or deflate nascent bubbles.

Almost as intriguing, the next leader will have to decide whether to enter the political fray now and then (as Mr. Greenspan did) or avoid it (as Mr. Bernanke generally has). The most obvious topic for a Democratic Fed leader to emphasize is the sharp growth in income inequality, which many scholars think has destabilized the economy. With many families facing stretched budgets, they have taken on large debts and become less able to spend.

“I think the defining issue of our time is: Does the economic, social and political system work for the middle class?” Mr. Summers has said. Ms. Yellen, a monetary-policy expert by training, has spoken less about the topic but has described it as “disturbing.”

Any modern Fed chairman is likely to use a series of boilerplate phrases about the economy — on the importance of education and fiscal rectitude, among other things. The larger choice is whether to go further, as Mr. Greenspan did, and try to influence the country’s economic debate.