Winner of the New Statesman SPERI Prize in Political Economy 2016

Saturday 10 August 2013

Expectations driven liquidity traps

For macroeconomists

This is my own take on the idea of expectations driven liquidity traps (as opposed to liquidity traps where the natural real interest rate is low and unobtainable). I note some of the literature that has promoted these thoughts at the end, but I am not trying to summarise what these papers actually say, but rather to give my own thinking on how such a trap could arise. The usual health warning on such occasions applies: if you think I have got something wrong, or missed something important from the literature, please let me know.

Consider the diagram below, which represents the simplest possible model. Real interest rates are always constant, which is the 45 degree line. Monetary policy follows the Taylor principle, but nominal rates cannot go below zero, so the bold monetary policy line kinks. There is one ‘locally stable’ equilibrium at the inflation target (let us call that the ‘intended’ equilibrium), and one ‘indeterminate’ equilibrium when we are at the ZLB (which involves negative inflation).

It is often said that the intended equilibrium is ‘globally unstable’. (Michael Woodford in Interest and Prices  - page 123 onwards - talks about global ‘multiplicity of equilibria’.) By this is meant that, in the absence of imposing an endpoint constraint that has to be met, there are infinitely many rational expectations solutions to the model, many of which involve inflation exploding. I trace one: if we start at A, the monetary authority raises nominal interest rates, but for constant real rates that must mean that expected inflation next period is even higher etc etc.

John Cochrane says: “Transversality conditions can rule out real explosions, but not nominal explosions.” As a result, he suggests, we cannot rule out travelling along this unstable path. After all, hyperinflations do occur. I am less worried about this. Hyperinflations occur when monetary policy makes no attempt to stabilise inflation. Here we have a model where everyone understands it does, so it makes sense to impose an endpoint on any dynamic path. 

For example, what happens when interest rates and inflation go up when we are at A. Do agents say to themselves ‘hyperinflation here we come’. Of course not. This is inconsistent with the model, which involves an inflation target. They say instead ‘that was unexpected - we must have got something wrong’. We only travel along the unstable path for as long as agents do not revise their ‘beliefs’ (in this case, expectations about the inflation target and the real interest rate). Once they revise their beliefs, whether it is their belief about the inflation target or the real interest rate, inflation is likely to fall towards the intended steady state. [1]

Note that we cannot just say - suppose we start at A, as if history put us there. History does not put us there: in this forward looking model history is irrelevant. Given the Taylor principle, there are only two reasons we could be at A within the context of this model: agents get the real interest rate wrong, or the inflation target wrong. Once we allow beliefs to be revised, it seems inconceivable that hyperinflations would occur within the context of this model.

In looking at how beliefs change we are applying a simple notion of learning. The fact that learning helps stabilise inflation around the intended steady state should not be surprising, because what we are in effect doing is adding some backward dynamics into the model. A locally stable steady state with forward looking dynamics will tend to flip to a stable steady state with backward dynamics. This property is helpful, because we probably do not know the mixture of backward and forward looking dynamics we have in the real world, so it is good that policies should be robust to this.

A consumer has to eventually get on to their stable saddlepath because it is stupid for them to accumulate infinite wealth and stupid for others to carry on lending them more and more (no Ponzi games). But things in this model are not so very different - all we are saying here is that we are working with a model in which we rule out hyperinflation because that is a stupid thing for central banks to allow. But unlike the consumer case, it is not impossible that central banks could allow it, which is why we sometimes see hyperinflation. [2]

If we start off with inflation below the inflation target, then we can apply a symmetrical argument. Nominal interest rates will fall. This is inconsistent with agents’ beliefs, so if they revise these beliefs it seems likely that inflation will rise rather than carry on falling. But suppose they do not revise their beliefs. In that case we do not shoot off to hyper negative inflation. This path will converge on the ZLB steady state. This steady state is not ‘locally stable’, but ‘indeterminate’.

Indeterminacy means that the model does nothing to tie down the initial point. We could start anywhere below the intended steady state, and a solution of the model would get us to the indeterminate steady state. While this may sound desirable, it is not, because we normally want the model to give us a unique dynamic path. With a forward looking model where history does not matter we need something to give us our starting point. Often indeterminate steady states flip to unstable points if we change from forward looking to backward looking dynamics.

This is where the desirability of the Taylor principle comes from. If we replace the Taylor rule plus the Taylor principle by a constant nominal interest rate that passes through the intended steady state, then the fact that this steady state would be indeterminate is conventionally seen as a very strong argument against constant nominal interest rate policies. The ZLB is just a particular constant interest rate policy.

To put this point another way, recall that in this purely forward looking model history is irrelevant. We cannot say ‘history means we start somewhere, and then we converge to the indeterminate steady state’. Now incorrect beliefs could start us anywhere, but beliefs are not completely independent of the model and subsequent dynamic paths. All along the approach to the ZLB equilibrium, events are contradicting those initial beliefs.

However, it may be as unrealistic to assume beliefs are continually revised as it is to assume they are never revised. Suppose beliefs are not revised for some time, and the initial belief involves an inflation target which is below the actual target. Inflation is below target, which leads to interest rates falling, which if real rates are constant implies still lower inflation next period. If beliefs do not get revised, we do not go to hyper disinflation, but to the ZLB steady state. Suppose agents only revise their beliefs once they get close to the ZLB steady state. What will happen then?

Recall that originally agents thought that the inflation target was a bit below the actual target (1% rather than 2%, say). Inflation has now fallen much further (to -3%, say). Is it possible that they might conclude that they originally overestimated the true inflation target? If they ignored the fact that the ZLB is a constraint, they might decide that current stability implied that the inflation target was -3%. The central bank cannot demonstrate that this is incorrect by lowering nominal rates, because of the ZLB. This is why this situation is very different from the hyperinflation case.

In a model this simple, we have stretched credibility a bit to get us to a point where we stay at the ZLB steady state. Agents ignore all the observations on the path towards that position, each of which was inconsistent with a -3% inflation target. But if you add in additional uncertainty, allowing the real interest rate to temporarily change for example, things get more complicated. Agents could interpret falling nominal rates when inflation was 1% as being due to temporarily lower real interest rates.

So for a time, at least, we could stay at the ZLB steady state because of ‘self-fulfilling’ but mistaken expectations. If we allow real interest rates to change, then at some point real interest rates will rise and agents will recognise this. Instead of nominal rates rising (as they should if the inflation target was -3%), they will stay at zero, which should make agents revise their belief about the inflation target. So the ZLB steady state remains transitory. But we could stay stuck in the ZLB steady state because of mistaken beliefs for some time: for as long as beliefs remain unchanged or no information arrives that makes them change.

Does this story of an expectation driven liquidity trap fit the evidence better than stories based on an unobtainable negative natural real rate? Or is it instead just a cute (‘liberating’) theoretical construct with zero application. I think it is difficult to argue that something like this applies today to countries like the US or UK. Expectations of inflation are still positive, and central bank inflation targets are clearly positive and pretty credible. (The concept of pessimistic beliefs, or animal spirits, might well be more applicable in the context of other models with different unobservable variables.)

However, if we take the idea seriously at all, it does suggest that one-sided inflation targets are dangerous. Central banks that have a target of 2% or less invite speculation that they would settle for zero inflation if that came around, which would make falling into an expectations driven liquidity trap that much easier. Perhaps the major economy where the central bank’s intentions towards inflation have been least clear, and therefore the potential for an expectations driven liquidity trap greatest, has been (until very recently) Japan.

Some literature:

Benhabib, J, and Farmer, R (2000) ‘Indeterminacy and Sunspots in Macroeconomics’ , in John
Taylor and Michael Woodford (eds.): Handbook of Macroeconomics, North Holland.

Benhabib, J, Schmitt-Grohe, S and Uribe, M (2002) ‘Avoiding Liquidity Traps’, Journal
of Political Economy 110(3), pp. 535–563. (pdf)

Cochrane, John, 2011, “Determinacy and Identiļ¬cation with Taylor Rules”, Journal of Political Economy 119(3), pp. 565–615. (pdf)

Farmer, R (2012a) “Confidence, Crashes and Animal Spirits,” Economic Journal, Vol. 122, No. 559, Pages, 155-172

Mertens, K and Ravn, M (2012) ‘Fiscal Policy in an expectations driven liquidity trap’ (pdf)

[1] With asset market bubbles, we can get the rather interesting possibility that we continue to travel along the explosive path, not because expectations of the fundamentals are wrong, but because agents think they can make money along that path but get out before the bubble bursts. However, this does not seem to apply to inflation and monetary policy.

[2] Of course it is not completely impossible that some people are misers or get away with Ponzi schemes, which illustrates the point that the difference in rationale for imposing end point conditions in each case is not that great.


  1. Very interesting. This sounds to me like a good argument for:
    1. Solid and clear messaging from the central bank that below-target inflation is not acceptable.
    2. A higher inflation target.

  2. Interesting, this idea seems to line up with a post Scott Sumner made a month back about the Fed unconsciously targeting a 1% inflation target.

    Here's the link:

  3. A consumer has to eventually get on to their stable saddlepath because it is stupid for them to accumulate infinite wealth

    When I look out at the world, I see an economy dominated by profit-making entities that are organized precisely around an effort to accumulate infinite wealth. But Simon Wren-Lewis thinks that would "stupid," so I guess we don't need to worry about how real economic actors behave.

  4. Though we may disagree on the relative slopes of AS or AD curves in a liquidity trap, on this perhaps we can agree. Given all the complications of understanding what's going on at the zero lower bound and the horrible consequences of hitting it according to some analysis, perhaps an inflation target of 5% would have bee more advisable than the common 2%. I'm sure this would have led to some reasonable long run price and wage indexation schemes that would have eliminated much of the bad consequences of this in terms of things like higher price dispersion (and I trust government revenue agencies and politicians would also find a reasonable way to index tax brackets). And the potential gains in terms of more margin of manoeuver for conventional monetary policy in a deep recession would probably be worth it (the way optimal policy analysis is usually done inside a representative agent complete financial markets environment probably severely underestimates this benefit- but I suspect a more realistic heterogeneous agent model would give you a higher optimal inflation target). Certainly, the economic debates in the last few years would have been far less confusing and possibly more civilised under a 5% inflation target.

  5. Has there been a bout of hyperinflation - defined as inflation rising 50% per month - that has not been caused by a political-military problem? If I think of the Mississippi Bubble, French Revolution, Russian Revolution, Weimar 1923, Hungarian Pengo 1945, and Zimbabwe more recently, I don't really see the central bank (if one existed) as the core of the problem!

    Shiller has it that 2004-6 was the period of slowing down before the US home price bubble burst at its 2006 peak, so there was an indication to get out of around two years from a bubble that began in 1997, if that's the sort of person you are.

  6. First reactions:

    1. I'm afraid you lost me a little on this post, beginning with your third paragraph (immediately under the diagram). When people (or Michael Woodford) say "the intended equilibrium is locally stable but globally unstable", don't they just mean that if we start at A we get back to the intended equilibrium (the arrows go the other way than in your diagram), but if we start at too low an inflation rate we don't go back to the intended equilibrium?

    2. There are two sorts of Phillips Curves: Those with price level inertia but no inflation inertia, where the price level cannot jump but the inflation rate can jump (e.g. Calvo); and those with both price level inertia and inflation inertia, where neither the price level nor the inflation rate can jump (e.g. Taylor-style multiperiod overlapping wage contracts). (The difference is that Calvo assumes firms are allowed to change prices at random with probability 1/n, while Taylor assumes each firm is allowed to change prices every n periods, which sounds like a trivial difference but has big implications, because the second assumption creates inflation inertia while the first only creates price level inertia). Even if we ignore the ZLB, and transversality conditions, and learning, the possibility of unstable paths would seem to depend on the sort of Phillips Curve we assume, and whether the inflation rate can jump more quickly than the central bank can adjust the nominal interest rate according to the Howitt/Taylor principle. (I haven't got my head around this yet.)

  7. This comment has been removed by the author.

  8. I would take issue with the idea of 'natural interest rate', the 45 degree line. That line is an abstraction in a Daisyworld type of model.
    (Daisyworld comes from Lovelock's paper on a world populated by 2 kinds of daisies.) That makes it a second order conclusion of such a model, taken as a foundation principle.

    And yet, the real world is chaotic (in the mathematical sense) presenting insufficient information to allow predictive modeling with accuracy very far into the future. As in weather models, "known unknowns" (what we know we don't know) gather significance until the model breaks.

    So all we can really say is that nominal interest rates cannot go below zero (within the system we have constructed) at least in banking. And yet, we know from the example of Japan that societies can implement across-the-board wage and contract value cuts on a regular basis for a couple of decades and have a stable, functional society that is economically competitive with others. This is a de facto implementation of compounded negative interest - at least for the society at large.

    Don't have time to go further right now, but I will close by saying that a quantity (natural interest rate) which is not determinable by any procedure on real data is perhaps something we should dispense with. It is a concept that may be holding us back from more interesting conclusions, rather like the phlogiston theory of heat, or the idea of the ether (that which pervaded everything) held back physicists until they gave it up in favor of particle physics and relativity.

  9. I agree with Scott Sumner's article: “Why I don’t believe in liquidity traps”.


    Of course liquidity traps can exist in specific and unrealistic circumstances, but basically if money is printed and dished out to the population they’ll spend a significant proportion of it. That’s what the empirical evidence shows. Or have I missed something?


Unfortunately because of spam with embedded links (which then flag up warnings about the whole site on some browsers), I have to personally moderate all comments. As a result, your comment may not appear for some time. In addition, I cannot publish comments with links to websites because it takes too much time to check whether these sites are legitimate.