Currently, that's not a helpful way of thinking about, for example, the members of the FOMC, and why they might disagree with each other. Fundamentally, these people don't agree on the theories they are using to guide their decisions. Some people might argue that preferences determine the choice of theories (that seems to be the crux of what Krugman has to say), but I don't think so. Here's a better guide to how people think:
New Keynesians: These are people who have absorbed modern macroeconomics, and have bought into the framework developed by Mike Woodford and others. That framework was an outgrowth of the neoclassical growth model (Cass-Koopmans), Brock and Mirman, and Kydland-Prescott, incorporating monetary factors, and with a role for monetary policy. This is the theory that is dominant on the committee. It's what's principally driving the forward guidance aspect of current monetary policy, i.e. promises about future policy that are supposed to have beneficial effects today. The hard-core New Keynesians are Williams (San Francisco), Evans (Chicago), and Kocherlakota (Minneapolis). Bullard (St. Louis) has some New Keynesian sympathies, as does Plosser (Philadelphia), I think.
Old Keynesians: This is some version of IS-LM, AD-AS, Phillips curve, much like what you would find in some undergraduate textbooks (not in my favorite one, as Mankiw would say). This is pretty loosey-goosey, and can incorporate almost anything. Fortunately for the Old Keynesians, the New Keynesians (for just this reason) have taken pains to describe what they do in the language of Old Keynesians - you can often find words like "IS curve" and "Phillips curve" in there. The rationale for quantitative easing (QE) - a cornerstone of current policy - is pretty much Old Keynesian. To the extent anyone justifies it, it's done using pre-1970 theory. The "transmission" mechanism that Fed officials describe for QE - purchase assets, interest rate falls, investment goes up - sounds like something straight out of a principles of economics textbook. The most prominent Old Keynesian on the FOMC is Janet Yellen.
Old Monetarists: This is literally a dying breed. Hard-core monetarism is represented best by the views of Milton Friedman: (i) monetary factors are very important, but attempts by the central bank to intervene to influence real activity in a good way are likely to go haywire; (ii) there exists a stable money demand function; (iii) central banks should be targeting money growth so as to control inflation. Views (ii) and (iii) died in most, if not all, central banks in the 1980s. But Plosser has some old monetarist views.
There are some outliers in there. George (Kansas City), Fisher (Dallas), and Lockhart (Atlanta) are hard to place. Rosengren is a Keynesian with some background in banking, and Lacker doesn't fit into a neat macro pigeonhole. The others I don't know much about.
Increasingly important, and I think surprisingly overlooked in light of the financial crisis, are the backgrounds of monetary policymakers in general economics. What do they know about banking theory, incentives, and information problems? What do they know about banking and monetary history and how the financial systems in different countries work? In the next financial crisis, we want a person running our central bank who understands what too-big-to-fail is about, the nature of long-run moral hazard, and how moral hazard can work against the central bank even during a crisis.
As well, in the context of run-of-the-mill macro monetary policy, different views about the mechanism by which monetary policy works can be irrelevant. Everyone appears to accept that there is some short-run nonneutrality of money at work. Everyone accepts that it is important that the central bank control inflation. The key differences are in views about the persistence of monetary nonneutrality, and how to spot inefficiencies that the central bank is capable of correcting. Some people look at the state of the world now and think that the difference between real GDP today and what it would have been if it had continued to grow at 3% per year since the end of World War II is all inefficiency. You can find other people who think that where real GDP is today is about the best we can do. There are a lot of other people who aren't sure one way or the other.
So, how would either Larry Summers, or Janet Yellen, fit in as leader of this group? Brad DeLong tells us that other people think that Summers is a "right-wing hyena," but he thinks Summers is a good guy. DeLong fits nicely in the Old Keynesian camp, so maybe Summers is an Old Keynesian. But clearly Paul Krugman does not like Summers. He seems to like Yellen better. Krugman is certainly an Old Keynesian too, and maybe DeLong is just sticking up for his friend and coauthor Summers, so maybe Summers is not so Old Keynesian after all.
Janet Yellen is firmly Old Keynesian. Her training happened before the revolution in macroeconomics took place, and it's quite clear that she thinks about the world in a conventional IS-LM, Phillips curve, demand-management style.
If we were to choose a Fed chair based on academic records, then this would be no contest. Summers, in spite of being occupied as a policy wonk for a considerable period of time, still has a REPEC ranking of 25. That's a #25 ranking in the world among economists, based on publications in academic journals, quality of publications, citations to those publications, etc. Yellen's ranking is 805. Summers was tenured at Harvard at age 28, while Yellen never made it past the assistant professor rank there. Of course we know that central banking requires some different skills from what it takes to publish papers. Some of our colleagues should definitely not be let out in public.
A good Fed Chair should be collegial. That means making FOMC members feel comfortable in expressing their views, so that diverse information can be forged into some kind of coherent synthesis, as part of the policy process. Bernanke has been very good at exploiting the strengths of the unusually-decentralized Federal Reserve System. The regional Presidents have their own staffs, their own views, and Bernanke by all reports is willing to listen, and has used ideas from the regional Feds as part of the monetary policy program. As well, the regional Presidents seem free to speak their minds in public, and they often do. I think all of that is healthy.
Some clues to how Summers thinks are in the transcript and video for this forum on "new economics." The worst it gets is when Summers says this:
When I was in the government, I got a lot of papers in the mail. To the first approximation, I attempted to read all the ones that used the words ‘leverage,’ ‘liquidity,’ ‘deflation’ or ‘depression.’ And I attempted to read none of the ones that used the words ‘neoclassical,’ ‘choice theoretic,’ ‘real business cycle,’ or ‘optimizing model of.’ (laughter) There were more in the second category than there were in the first. But there were a reasonable number in the first, and they told you a lot.
There is a lot in Badgett [sic] that is about the crisis we just went through. There’s more in Minksy and perhaps more still in Kindleberger.2 There are enormous amounts that are essentially distracting, confusing, and problem denying in the stuff that is the substance of the first year courses in most PhD programs.
So I think economics knows a fair amount. I think economics has forgotten a fair amount that’s relevant. And it has been distracted by an enormous amount.
You can find other things in that forum, and in what Summers has said in other contexts, that put him in a more favorable light, but I think that quote is telling. It's interesting to read Bagehot, or Minsky, or Kindleberger, but we're going to find a lot more serious ammunition to bring to bear on thinking about financial problems in work on money, banking, information economics, mechanism design, contracts, and incentive problems, that people have been working on in the last 30 years or so. Summers has revealed himself here to be shockingly closed-minded - he wants to ignore a large segment of research that the mainstream of the economics profession takes very seriously. A leader of central bankers needs to dig into all the alternatives and understand them, even if he or she doesn't agree with everything. Summers seems to impress people as always wanting to demonstrate that he is the smartest person in the room. He is quick to come to a conclusion, and can be an intellectual bully. Not exactly Mr. Collegial.
If you have never seen Janet Yellen in action, here's a speech she gave at the Haas Business School at Berkeley. She's clear, articulate, and thoughtful, I think. Yellen has a reputation in the Fed system as being a good listener - she's certainly collegial. There is no reason to expect that she would not follow Bernanke's example in encouraging independent thinking in the hinterland - places like St. Louis. Yellen may be overly inclined to see everything going on in the U.S. economy as a demand management problem, but maybe that's not the worst flaw we could have in a central banker.
Summers has the additional problem of being too closely allied with various power structures. He is part of the Cambrige, MA clique. He has worked for hedge funds and too-big-to-fail banks. Not good for central bank independence.
I think I'm with Krugman on this one. Though there are potentially better choices around, Janet Yellen would be fine, and I think Larry Summers is just too risky a bet, though he might surprise us.