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.