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.         

Defending rational expectations

Whenever I post anything which suggests that the idea of rational expectations was a useful innovation in macroeconomics, Lars Syll writes something to the effect that I am (and therefore most mainstream macroeconomists are) “so wrong, so wrong”. Now why does this bother me? Well, to be honest, it does not bother me very much. As Bob Dylan sang: ‘Yes, I received your letter yesterday (About the time the doorknob broke)’.

But it does bother me a bit. Professor Syll does write very eloquently, and this kind of eloquent prose can appeal to the occasional young economist, who is inclined to believe that only the radical overthrow of orthodoxy will suffice. I meet one or two each year I teach. I remember the feeling: been there, done that. It also appeals to people like Aditya Chakrabortty who are understandably unhappy by the current state of things economic. (See this nice recent post by Diane Coyle on both this particular article but also heterodox critiques more generally.) There is plenty to legitimately criticise about mainstream economics (and its textbooks), so it is a shame Professor Syll wastes his talents on one of its major achievements, which is rational expectations. On this he is, well, so wrong.

This discussion can easily get populated with straw (super)men, so let's be clear about some things. It is not a debate about rational expectations in the abstract, but about a choice between different ways of modelling expectations, none of which will be ideal. This choice has to involve feasible alternatives, by which I mean theories of expectations that can be practically implemented in usable macroeconomic models. In the past, I have attempted to try and start a dialog with heterodox economists on the level of practical macroeconomics, to get beyond the fine words and phrases. It did not seem to work. I tried again in that recent post, asking for practical alternatives to rational expectations. Professor Syll referred me to behavioural economics, or Frydman and Goldberg’s ‘Imperfect Knowledge Economics’. But perhaps I did not make it clear what I meant by practical.

If I really wanted to focus in detail on how expectations were formed and adjusted, I would look to the large mainstream literature on learning, to which Professor Syll does not refer. (Key figures in developing this literature included Tom Sargent, Albert Marcet, George Evans and Seppo Honkapohja: here is a nice interview involving three of them.) Macroeconomic ideas derived from rational expectations models should always be re-examined within realistic learning environments, as in this paper by Benhabib, Evans and Honkapohja for example. No doubt that literature may benefit from additional insights that behavioural economics and others can bring. However it is worth noting that a key organising device for much of the learning literature is the extent to which learning converges towards rational expectations.

However most of the time macroeconomists want to focus on something else, and so we need a simpler framework. In practice that seems to me to involve a binary choice. Either we assume that agents are very naive, and adopt something very simple like adaptive expectations (inflation tomorrow will be based on current and past inflation), or we assume rational expectations. My suspicion is that heterodox economists, when they do practical macroeconomics, adopt the assumption that expectations are naive, if they exist at all (e.g. here). So I want to explain why, most of the time, this is the wrong choice. My argument here is similar but complementary to a recent piece by Mark Thoma on rational expectations.

Suppose we have an equation determining wage or price inflation (a Phillips curve), where inflation expectations appear on the right hand side of the equation. We need some way of determining those expectations. Lots of nice words like ‘non-ergodic’ will not do: we need something simple that can be used to solve the model. To assume, as mainstream macroeconomists once did, that these expectations just depend on past observations about inflation seems to assume that agents are stupid. These agents ignore everything that economists and the media say about inflation: they ignore monetary policy, and whether the economy is in a boom or recession. Now if getting expectations right did not matter too much to these agents, then maybe such naivety would be understandable. But in this case making expectations errors can mean getting real wages or profits wrong, so it matters.

Perhaps you think the alternative is equally unbelievable. Rational expectations, often called model consistent expectations, implies that agents know the model that the modeller has constructed, and use it to generate expectations. This is where the elegant prose comes in - you can make this sound incredible. Of course it will not be literally true, but I think it is a lot nearer the truth than the adaptive expectations alternative. The reason why mainstream economics replaced adaptive expectations with rational expectations in the 1970s was because the new approach was consistent with what economists did elsewhere. Firms may not know the true demand curve for their product and work out the price that maximises profits each period, but that is a better approximation to how they choose prices than a model where they have a fixed mark-up on costs. So it just seemed consistent to also assume that agents used relevant and available information to generate expectations when those expectations mattered. The closest we can get to that, without assuming an elaborate learning model, is to assume rational expectations.

This is an empirical claim. But how else do you make sense of a whole forecasting industry, and the newsworthy character of macro forecasts. Rational expectations at least acknowledges that endeavour, while adaptive expectations pretends it does not exist. And how else do you make sense of the response of Japanese inflation expectations to little more than a policy change: see Carola Binder’s discussion. How can you make sense of all the discussion of forward guidance without the concept of rational expectations?

Most of the references I make to rational expectations in posts are in the context of the history of macroeconomic thought. I suspect the problem some people have is that they associate rational expectations with the New Classical critique of Keynesian economics, and therefore think rational expectations must be anti-Keynesian. This confuses who fought wars with the weapons they used. I see it quite differently. Before rational expectations, mainstream Keynesian theory that incorporated the Phillips curve depended on a rather fragile story of why economic booms (downturns) could occur, which was that workers kept under (over) estimating inflation. New Keynesian theories based on rational expectations are more compelling, and can include the fact that information is both costly and incomplete.

So I just do not get this obsession that some heterodox economists have with rational expectations. I think its fine to criticise mainstream theory (particularly macro theory) for being too wedded to rationality in general: I seem to remember some remarks of my own along those lines. But mainstream macro does take learning, and the problem of costly and limited information, seriously. However for the foreseeable future, rational expectations will remain the starting point for macro analysis, because it is better than the only practical alternative. 

The choice really matters. An economy where agents form their expectations in a naive adaptive way is like an elaborate machine which takes no account of what policymakers are doing. In reality the economy appears more intelligent than this: policy is difficult because people in the economy take actions which anticipate what policymakers might do. This makes designing good policy difficult, but the concept of rational expectations has allowed macroeconomists to tackle this problem. To throw all that away by abandoning rational expectations would not improve macroeconomics, it would impoverish it.