Tuesday, July 21, 2015

Why the Belief Function Matters

A debate on the monetary transmission mechanism was recently reignited on the blogs with a post by Noah Smith, posts from Nick Rowe and Brad De-Long and a response to Noah from John Cochrane. This was all triggered by a set of slides prepared by Michael Woodford and Maria Garcia Schmidt for a Riksbank Conference in June 2015. A good starting point is the summary here back in 2014 by John Cochrane. 

The question: If the Fed raises the interest rate will it cause more or less inflation? The answer is complex and the topics that must be dealt with in formulating that answer are at the heart of monetary economics. 

In my own work, I emphasize two central points

1. Monetary rational expectations models always have multiple equilibria.

2. The right way to deal with this is by explicitly modeling how people form beliefs using a concept that I call the belief function.

Flash back to 1968. Rational expectations was not part of our vocabulary but economists still needed to model the passage of time. The standard approach was the temporary equilibrium model that John Hicks developed in Value and Capital.

In the temporary equilibrium model, time proceeds in a sequence of weeks. Each week, people meet in a market. They bring goods to market to trade. They also bring money and bonds. The crucial point of temporary equilibrium theory is that the future price is different from our belief of the future price. To complete a model of this kind, we must add an equation to explain how beliefs are determined. I call this equation, the belief function.

Now jump forward to 1972. Robert Lucas wrote an influential paper that changed the way we think about monetary economics. Lucas was concerned that a variable called expectations was floating around in our models and that this variable had the potential to influence outcomes through its impact on the beliefs of people. That was messy and it was inconsistent with Lucas’ intuition that prices and quantities should be pinned down by fundamentals: preferences, endowments and technology. He suggested that we model the future price as a random variable with a probability distribution.

How do the people  deal with this possibility? They must form subjective beliefs about what will happen. Instead of forming a point belief about the future price, they form a complete subjective probability distribution. Now comes the coup de grace. Lucas argued that, if people live in a stationary world where the same events are repeated again and again, that rational people will come to learn the true distribution.

Lucas argued that, although we may not know the future exactly: we do know the exact probability distribution of future events.
Following the work of John Muth, he called this idea, rational expectations.

Rational expectations is a powerful idea. If expectations are rational, then we do not need to know how people form their beliefs. The belief function that was so important in temporary equilibrium theory can be relegated to the dustbin of history. We don’t care how people form beliefs because whatever mechanism they use, that mechanism must be right on average. Who can argue with that?

That is a clever argument. But it suffers from a fatal flaw. General equilibrium models of money do not have a unique equilibrium. They have many. This problem was first identified by the English economist Frank Hahn, and despite the best attempts of the rational expectations school to ignore the problem: it reappears with a alarming regularity. Rational expectations economists who deny an independent role for beliefs are playing a game of whack a mole.

More recently, the multiplicity problem arose in a paper by Jess Benhabib, Stephanie Schmitt-Grohé and Martín Uribe (BSU). In new-Keynesian models with rational expectations, the central bank sets the interest rate in response to inflation using a response function that is called a Taylor Rule.  If the Fed chooses a rule that is aggressive in response to inflation, the New-Keynesians thought that the equilibrium would be unique. BSU showed that they are wrong.

This is not an esoteric point. It is at the core of the question that I pose at the beginning of this post: 
If the Fed raises the interest rate will it cause more or less inflation?  And it is a point that policy makers are well aware of as this piece by Fed President Jim Bullard makes clear.

What is the solution? It is one thing to recognize that the world is random, and quite another to assume that we have perfect knowledge. If we place our agents in models where many different things can happen, we must model the process by which they form beliefs. I made this argument in my 1993 book, the Macroeconomics of Self-Fulfilling Prophecies   where I referred to the mechanism that selects an equilibrium in a rational expectations model as a belief function. It is time to embrace the idea that the belief function matters. 


  1. Thanks Roger!

    By the way, did the old temporary equilibrium guys talk much about "self-fullfilling forecasts"? Maybe they did with bank runs, but I can't think of other examples.

    I've just had a weird thought: is there any relation between what you call a "belief function" and a "social construction of reality"? I've been using that phrase to talk about the "language" spoken by central banks, and how their actions are interpreted. If the central bank raises the nominal interest rate, what does that *mean*?

  2. Roger and Nick,

    Nick asks whether there’s “any relation between what [Roger calls] a ‘belief function’" and what Nick calls the "social construction of reality”? Perhaps only Roger can answer this question, but it raises some related questions perhaps you both might be willing to answer.

    The phrase, “social construction of reality,” has been used by sociologists (P. Berger and T. Luckmann wrote a book with this title) and philosophers (John Searle wrote a book entitled “The Construction of Social Reality”). Roughly speaking, the upshot of this view is that reality isn’t given to us in the form of brute facts, but rather is constructed from the concepts and other resources available to us. One implication of this view is that some of these “social constructions” will be common to every (sane) person, and some won’t be.

    Nick invokes this idea in his discussion of how the actions of central banks “are interpreted.” From the vantage point outlined above, there will most likely be multiple interpretations of central bank actions, and many of these interpretations will be informed by different economic theories. In other words, these central bank actions and their likely consequences will be interpreted differently insofar as market participants are influenced by competing economic theories.

    So, my questions are: 1) Roger, does your “belief function” admit divergent beliefs? And 2) Nick, does the effect of monetary policy in your scheme of thought depend on the distribution of alternative “interpretations of central bank actions”?

  3. Roger, do you see any tie-in in your post here with these two most recent posts by David Glasner: [1], [2] (actually, why didn't that second one make your list?).

    What about Scott Sumner's post here? Right there in the title he says looking for alternatives to Ratex is a waste of time.

    1. Actually, this is the sentence fragment of yours that makes me think that there's some overlap with Glasner's posts: "...we must model the process by which they form beliefs."


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