Woodford’S Reflexive Equilibrium

 For macroeconomists

Karl Whelan lately tweeted: “Read Cochrane as well as Woodford on neo-Fisherism today. Cochrane - clear as well as sentiment provoking. Woodford - unclear as well as rambling.” I grip almost the clarity of John Cochrane’s writing, both inward absolute terms as well as relative to Michael Woodford. But on this occasion I think Woodford has a to a greater extent than realistic approach. So hither is my examine to explicate the number that both are addressing, as well as Woodford’s version of learning. The ii papers Karl is referring to tin plough over notice live on constitute here as well as here.

The ‘problem’ that both address is that inward the touchstone New Keynesian model a fixed involvement charge per unit of measurement policy involves an interplanetary space number of rational expectations equilibrium paths. Another way of proverb the same affair is that the initial restrict inward prices is non tied down, but if y'all direct to select a starting quest the subsequent path would save rational expectations. This multiple equilibrium number typically agency that macroeconomists would consider this monetary policy government every bit problematic, but Cochrane says that in that place is no logical ground to spend upward these paths, as well as Woodford agrees. However Woodford argues that this policy is problematic, because if y'all direct some detail way of selecting a detail equilibrium (and Cochrane does suggest one), it volition non live on learnable inward the feel Woodford describes. (The sentiment that indeterminate rational expectations solutions are non learnable is non new, every bit I banker's complaint below.)

What is Woodford’s reflexive approach to learning? For me the most intuitive way to depict it is that it is really similar to Fair as well as Taylor’s method of finding the solution to a dynamic economical model involving rational expectations, although it may live on that this only reflects my background. (Woodford’s give-and-take of how his sentiment relates to the literature, which opens amongst this analogy, is really readable as well as tin plough over notice live on constitute inward department 2.4.) The method starts past times assuming some arbitrary values for expectations variables inward the model, as well as solves it. This gives a solution to the model conditional on those arbitrary expectations. Now accept that solution, as well as recompute using these solution values every bit expectations. Iterate until the solution hardly changes, as well as accept that solution every bit the rational expectations equilibrium. The logic is that if some laid of expectations (almost) reproduce themselves inward this way, they are (almost) model consistent.

Woodford’s reflexive learning is really similar, although he would impose some arbitrary, as well as small, cutting off for the number of iterations (=n). This has diverse interpretations, but the 1 I similar is that each catamenia a proportion of the population fully recomputes their expectations assuming rationality (or iterates a large number of times), piece others stick to their previous expectations. Another interpretation (which could likewise receive got diversity) is to appeal to ‘level k thinking’, which has been observed inward experiments. The reflexive learning sentiment is based on locomote past times Evans as well as Ramey, as well as is closely related to the E-stability concept developed past times Evans as well as Honkapohja: Woodford explains why he prefers his approach. Evans as well as Honkapohja receive got likewise applied their learning technique to this really issue, amongst similar results: run across George Evans here for example.

Woodford shows, both analytically as well as amongst numerical examples, how the reflexive equilibrium converges to the rational expectations equilibrium every bit the number of iterations n increases if monetary policy is described past times a Taylor dominion that obeys the Taylor principle, but does non for a fixed nominal involvement charge per unit of measurement policy. To quote:

“It is truthful that nether the supposition of a permanent interest-rate peg, the alone forward-stable PFE are ones that converge asymptotically to an inflation charge per unit of measurement determined past times the Fisher equation as well as the interest-rate target (and thus, lower past times 1 pct quest for every 1 percent reduction inward the involvement rate). But for most possible initial conjectures (as starting points for the procedure of belief revision proposed above), none of these perfect foresight equilibria correspond, fifty-fifty approximately, to reflective equilibria — fifty-fifty to reflective equilibria for some really high story of reflection n.”

There is much to a greater extent than inward the paper, but on the number of reflective equilibrium a natural conjecture (mine non Woodford) is whether all indeterminate solution paths neglect to live on a reflexive equilibrium. In other words is this a rationale for ignoring indeterminate solutions, or perchance to a greater extent than appropriately, designing policy to avoid them? Using the analogy amongst the Fair-Taylor algorithm, it may depend on the human relationship betwixt iterative stability as well as dynamic stability. When in that place was much to a greater extent than role of iterative methods for model solution I think in that place was a literature on this (and it may withal live on alive), as well as I look to recall both similarities but likewise differences, but beyond that I receive got no idea.

I am non qualified to address the extent to which Woodford’s sentiment of a reflexive equilibrium adds to the learning literature, but it is straightaway outset to aspect every bit if the number that a fixed involvement charge per unit of measurement policy is non stable nether learning is robust. As James Bullard says inward a recent presentation (HT ‘acorn’ inward comments), this may live on “a variety of “victory” for the learning literature”. 

Postscript (31/12) See this banker's complaint from Evans as well as McGough (in a Mark Thoma post) which I think is consistent amongst what I tell here.