Winner of the New Statesman SPERI Prize in Political Economy 2016

Tuesday 22 December 2015

Woodford’s reflexive equilibrium

 For macroeconomists

Karl Whelan recently tweeted: “Read Cochrane and Woodford on neo-Fisherism today. Cochrane - clear and thought provoking. Woodford - unclear and rambling.” I agree about the clarity of John Cochrane’s writing, both in absolute terms and relative to Michael Woodford. But on this occasion I think Woodford has a more realistic approach. So here is my attempt to explain the issue that both are addressing, and Woodford’s version of learning. The two papers Karl is referring to can be found here and here.

The ‘problem’ that both address is that in the standard New Keynesian model a fixed interest rate policy involves an infinite number of rational expectations equilibrium paths. Another way of saying the same thing is that the initial jump in prices is not tied down, but if you choose to select a starting point the subsequent path would preserve rational expectations. This multiple equilibrium result typically means that macroeconomists would regard this monetary policy regime as problematic, but Cochrane says that there is no logical reason to reject these paths, and Woodford agrees. However Woodford argues that this policy is problematic, because if you choose some particular way of selecting a particular equilibrium (and Cochrane does suggest one), it will not be learnable in the sense Woodford describes. (The idea that indeterminate rational expectations solutions are not learnable is not new, as I note below.)

What is Woodford’s reflexive approach to learning? For me the most intuitive way to describe it is that it is very similar to Fair and Taylor’s method of finding the solution to a dynamic economic model involving rational expectations, although it may be that this just reflects my background. (Woodford’s discussion of how his idea relates to the literature, which opens with this analogy, is very readable and can be found in section 2.4.) The method starts by assuming some arbitrary values for expectations variables in the model, and solves it. This gives a solution to the model conditional on those arbitrary expectations. Now take that solution, and recompute using these solution values as expectations. Iterate until the solution hardly changes, and take that solution as the rational expectations equilibrium. The logic is that if some set of expectations (almost) reproduce themselves in this way, they are (almost) model consistent.

Woodford’s reflexive learning is very similar, although he would impose some arbitrary, and small, cut off for the number of iterations (=n). This has various interpretations, but the one I like is that each period a proportion of the population fully recomputes their expectations assuming rationality (or iterates a large number of times), while others stick to their previous expectations. Another interpretation (which could also have diversity) is to appeal to ‘level k thinking’, which has been observed in experiments. The reflexive learning idea is based on work by Evans and Ramey, and is closely related to the E-stability concept developed by Evans and Honkapohja: Woodford explains why he prefers his approach. Evans and Honkapohja have also applied their learning technique to this very issue, with similar results: see George Evans here for example.

Woodford shows, both analytically and with numerical examples, how the reflexive equilibrium converges to the rational expectations equilibrium as the number of iterations n increases if monetary policy is described by a Taylor rule that obeys the Taylor principle, but does not for a fixed nominal interest rate policy. To quote:
“It is true that under the assumption of a permanent interest-rate peg, the only forward-stable PFE are ones that converge asymptotically to an inflation rate determined by the Fisher equation and the interest-rate target (and thus, lower by one percentage point for every one percent reduction in the interest rate). But for most possible initial conjectures (as starting points for the process of belief revision proposed above), none of these perfect foresight equilibria correspond, even approximately, to reflective equilibria — even to reflective equilibria for some very high degree of reflection n.”

There is much more in the paper, but on the issue of reflective equilibrium a natural conjecture (mine not Woodford) is whether all indeterminate solution paths fail to be a reflexive equilibrium. In other words is this a rationale for ignoring indeterminate solutions, or perhaps more appropriately, designing policy to avoid them? Using the analogy with the Fair-Taylor algorithm, it may depend on the relationship between iterative stability and dynamic stability. When there was much more use of iterative methods for model solution I think there was a literature on this (and it may still be alive), and I seem to remember both similarities but also differences, but beyond that I have no idea.

I am not qualified to address the extent to which Woodford’s idea of a reflexive equilibrium adds to the learning literature, but it is now beginning to look as if the result that a fixed interest rate policy is not stable under learning is robust. As James Bullard says in a recent presentation (HT ‘acorn’ in comments), this may be “a sort of “victory” for the learning literature”. 

Postscript (31/12) See this note from Evans and McGough (in a Mark Thoma post) which I think is consistent with what I say here.         


  1. Good post.

    I'm being a bit pedantic, but [should it be] "Another way of saying the same thing is that the initial jump in prices [inflation??] is not tied down, but if you choose to select a starting point the subsequent path would preserve rational expectations."

    Because the inflation rate can jump with new information in Calvo Phillips Curves, but the price level cannot jump.

    I think that's right.

  2. Do you think this conclusion depends on exactly which variables are chosen as being subject to the reflexive learning process?

    When I've used Fair Taylor type methods to progressively eliminate expectation errors in dynamic models, I find it generally quite difficult to get convergence when the price level is taken as the "learned" variable. I find a much more effective method is to use expected values for a future real balance and compare this with the extrapolated nominal balance to deduce an expected price level. The expected value of the real balance is then the variable that is adjusted for subsequent runs.

    Doing it this way, I have no problem getting convergence under fixed nominal interest rate policy scenarios.

  3. But central banks are not supposed to have "interest rate" policies. Interest rates are sometimes the instruments to achieve certain objectives -- ngdp growth target, price level growth target, even a disastrous inflation rate ceiling policy, employment growth. And sometimes they are just the results of other instruments purchasing/selling LT securities or foreign exchange. I think this falls under the "never reason from a price change" interdict.

  4. Woodford says

    “It is true that under the assumption of a permanent interest-rate peg, the only forward-stable PFE are ones that converge asymptotically to an inflation rate determined by the Fisher equation and the interest-rate target (and thus, lower by one percentage point for every one percent reduction in the interest rate)."

    In fact, Woodford neither explains why this equilibrium is "rational", nor by virtue of which mechanism it would be attained in an economy with rational agents. And, in my view, it is less relevant that he considers this equilibrium as unlikely to be attained in practice than the fact that he qualifies it as a rational (and even perfect foresight) equilibrium. Indeed, qualifying it as rational and perfect foresight has encouraged John Cochrane to deem the Neo-Fisherian battle as at least half won...

    So, starting from the Fisher's equation - which establishes how capital's owners protect their capital against inflation - Neo-Fisherites use the equation to argue that central banks pegging a low interest rate lead agents to expect low inflation: this is indeed how Neo-Fisherites explain low-flation today and the failure of central banks to fight it through low interest rate policies. They thus conclude that central banks should be advised to raise their interest rate peg, if they want to achieve higher inflation...

    There seems to be some kind of "immaculate inflation" mechanism at work in Neo-Fisherian models, whereby rational people respond to the central bank's commitment to a high interest rate policy stance by raising prices and by expecting rising prices.

    What lies inside that mechanism? What would be the dynamics required for the economy to converge to Woodford's (Neo-Fisherian) PFE? How do Neo-Fisherites explain their economics?

  5. This comment has been removed by the author.

  6. Simon: Equilibrium selection results based on learning should be viewed somewhat sceptically. It is precisely when agents fail to learn an equilibrium, that they would most readily detect the misspecification in their learning method. So a failure of learnability is a stronger argument against the learning method than it is an argument against an equilibrium.
    In a somewhat unpolished working paper, I show that with a moderately more sophisticated learning method (maximum likelihood, rather than least squares), agents are able to learn any equilibrium, including sunspots and indeterminate ones. The paper may be downloaded from here:

  7. Economics between Angelology and Nonentitylogy
    Comment on ‘Woodford’s reflexive equilibrium’

    In the Middle Ages savants were heavily occupied with questions like had Adam a navel? or how many angels can dance on the head of a pin? Methodologically economics is roughly at the same stage.

    The question of how to pick a path among an infinite number of rational expectations equilibrium paths is a fine specimen of Angelology*. Has anyone on this blog noticed that there is no such thing as an equilibrium in the economy? Since 1990 equilibrium is officially dead and the historians of economic thought ‘have finally hammered down the nails in this coffin’ (Blaug, 2001, p. 160).

    The logical conclusion is to entirely reconstruct economics without the concept of equilibrium. In methodology this is called a paradigm shift “There is another alternative: to formulate a completely new research program and conceptual approach. As we have seen, this is often spoken of, but there is still no indication of what it might mean.” (Ingrao et al., 1990, p. 362)

    Obviously, there is until this very day ‘no indication of what it might mean’ to do economics without the nonentity equilibrium and the green cheese behavioral assumptions of constrained optimization and rational expectations.

    The loudspeakers of the profession simply cannot get their head around the fact that they are irrecoverably lost in a scientific parallel universe. As Krugman recently confirmed on his blog “most of what I and many others do is sorta-kinda neoclassical because it takes the maximization-and-equilibrium world as a starting point ...”.

    Recently, though, there have been signs of progress because Krugman talks more of Trump than of maximization-and-equilibrium. It would significantly improve the intellectual level of theoretical economics if all reflexive equilibrists could follow this trend and focus more on Trump or other important and interesting manifestations of what Hegel called the Weltgeist**.

    Egmont Kakarot-Handtke

    Blaug, M. (2001). No History of Ideas, Please, We’re Economists. Journal of Economic Perspectives, 15(1): 145–164.
    Ingrao, B., and Israel, G. (1990). The Invisible Hand. Economic Equilibrium in the History of Science. Cambridge, MA, London: MIT Press.

    * See Wikipedia
    ** See Wikipedia


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