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.         

The wheels on the bus

I have an image in my mind. Its a bus running downhill, and its brakes have failed. There are four men in the front cab. The two men in the middle are both trying to control the steering wheel to keep the bus on the road. The man to their right has control of the accelerator, and is pushing on the gas hoping this will crash the bus to the right. The fourth man to their left controls nothing, but as his pleas to stop pressing the accelerator fall on deaf ears, he begins to wonder whether it would be better for the passengers to grab the wheel and crash the bus to the left. The three other drivers do not agree on very much, except that it is all the fault of the guy on the left, and now appear to be thinking about throwing him off. As the bus hurtles downhill swerving from side to side, its passengers are battered, some injured, and a few are jumping off.

I do not need to explain the symbolism. I tried to change the image to explain why the man on the right refuses to stop pressing on the accelerator of growing primary surpluses, but gave up because the real reason is that he wants to crash the bus anyway. (The argument that the Eurozone’s rules do not allow debt write-offs is just nonsense.) Otherwise I think the image works well. The two men in the centre represent Tsipras and maybe Hollande. Hollande is saying that if only you would let me have the wheel (‘structural reform’) all would be well, but in truth the main reason the passengers are being injured (unemployment and welfare cuts) or are jumping (migration) is the speed of the bus.

The central question is whether the men in the middle are delusional. By keeping the Greek economy on the road that is the Eurozone are they only going to prolong the agony with the same inevitable crash which is Grexit?

There is only one reason for optimism that I can see, although it assumes yet further reductions in Greek living standards. The hill the bus is travelling along will begin to flatten out and the road might even start to rise as Greece becomes more competitive in terms of price. I outlined here why that has not yet boosted the Greek economy to the extent it has in Ireland, but if unemployment remains at or above 25% Greece should get even more competitive. Instability and unwise Troika interventions may delay the process, but eventually the tourists will come. The Eurozone does contain a natural correction mechanism: it is just slow and painful.

If this does eventually lead to sustained growth in Greece, it does not excuse what has gone before: recoveries do not justify recessions, and government profligacy does not have to imply a 25% fall in GDP! However this correction mechanism is not bound to succeed, if it is countered by another dynamic, which is one that has been and continues to be imposed by the Troika. That dynamic is austerity chasing primary surpluses when that austerity makes the economy shrink. Macromodels would probably tell us which dynamic will win out, but they will not factor in a deterioration in the financial position of banks (already not good as Frances Coppola points out) as the economy stagnates, and the deteriorating social and political situation that austerity brings.

So the eventual outcome still depends on the decisions of the Troika. It always has of course. The truth that their apologists find so uncomfortable is that the Troika has been in charge of the economy since 2010, and therefore is responsible for the mess we are now in. The idea that all would be well if only Greece had undertaken every item of structural reform they specified (and a lot was done) is just silly. Now it appears as if it is all the fault of the former Greek finance minister, because he dressed funny, or kept wanting to talk about economics, or did some contingency planning - it is so absurd you couldn’t make it up.

One ray of hope offered by Anatole Kaletsky is that now “ritual humiliation” has been achieved, the Troika will be more forgiving. I wish he was right, but this argument fails to account for the German finance minister who clearly believes that exit is the best option. He wants the bus to crash for the sake of the other cars on the road. An optimistic view would be that the shock [1] of what was done to Greece a few weeks ago will bring others to their senses, and Schäuble’s influence on the Eurogroup (and strangely the IMF) will decrease. I fear the larger truth is that the non-German bloc in the Eurozone does not have an alternative economic vision to offer (although it clearly exists), and will never face Germany down.

[1] Link added 31/07

After Oxi, what next?

A lot of the commentary on Greece fails to see why the Greek No vote changes anything. This view tends to see the stance of the Eurozone group as simply expressing their own voters’ preferences which will not be changed by what happened yesterday. Here is an alternative reading.

It starts from a simple observation. The Troika will get far less of its money back (if any!) if Greece is forced out of the Eurozone. (I say forced out because Greece does not want to leave, so Greek exit is first and foremost an ECB decision: if you think otherwise read Karl Whelan and Matthew Klein and Paul De Grauwe. [1]) That is why creditors are generally weak in negotiations of this kind. Things are different in this case only because the creditors include the ECB, and Greece wants to stay in the Eurozone. The Troika has played this for all it is worth. They were relying (you could say gambling) on the Greek people, one way or another, deciding that they would agree to the Troika’s demands because they feared Greek exit more.

So far this strategy has failed. First they pushed Tsipras further than he could possibly go, hoping perhaps that Syriza would collapse in recriminations. Tsipras’s response was a unifying referendum. They then gambled that Greece would say no, and they lost that too. Tsipras continues to offer the Troika the chance to be more reasonable. He followed the referendum not with triumphalism but by removing his finance minister. This was both a signal - I really want a deal, even though it will in all probability inflict further (unnecessary) pain on Greece - and a lifeline, because the Troika can now say that an important obstacle to a deal has been removed. (An obstacle, because Varoufakis was too open - something politicians and much of the press hate - and too honest about the other side’s lack of economics.)

Now the Troika seem to face a simple choice. Agree a deal and get a little more heat from your political opponents at home for ‘giving in’, or force Greek exit with the risk that you will get a lot more heat when Greece defaults and people realise you have lost all their money. If they are really just interested in getting as much of their money back as possible, it would seem crazy to throw away their best card by forcing Greece out of the Eurozone.

Of course rationality may not prevail, or interests may be rather different. The IMF may continue to be an unhelpful nuisance. (If you think my criticism of their role was harsh, read this from Peter Doyle.) Some within the Troika will be happy to go for Greek exit because they think nationalist sentiment can overcome any kickback from the subsequent Greek default. Others may fear a deal may encourage anti-austerity sentiment in their own indebted countries.

Unfortunately there is a third possibility, which is probably the worst possible outcome. To prevent any loss of face, the Troika may continue to gamble, waiting for days or even weeks, and watch ECB pressure, together with reluctance by Tsipras to introduce a new currency, gradually bring chaos to the Greek economy. Only then will it negotiate, allowing any deal to be portrayed as the result of desperation by the Greek government. In which case, recent European politics will have reached a new all time low.    

[1] Postscript: Martin Sandbu provides a very clear account.

The Eurozone’s cover-up over Greece

Whenever I write about Greece, a large proportion of comments (maybe not a majority) could be summarised as follows: how can you side with Greece when its economy is so inefficient and its governments so inept and after everything we have done for them. I have no illusions about the inefficiencies and corruption endemic within the Greek economy. Nor do I want to become an apologist for any Greek government.

What does seem to me very misguided is the idea that European policymakers have already been generous towards Greece. The general belief is that had they not stepped in austerity in Greece would have been far worse. This seems simply wrong. If European policymakers have been generous to anyone, it is the Greek government’s original creditors, which include the banks of various European and other countries.

Suppose that Eurozone policy makers had instead stood back, and let things take their course when the markets became seriously concerned about Greece at the beginning of 2010. That would have triggered immediate default, and a request from the Greek government for IMF assistance. (In reality at the end of 2009 the Euro area authorities indicated that financial assistance from the Fund was not “appropriate or welcome”: IMF 2013 para 8) In these circumstances, given the IMF’s limited resources, there would have been a total default on all Greek government debt.

If that had happened, the IMF’s admittedly large assistance programme (initially some E30 billion, but increased by another E12 billion in later years), would have gone to cover the primary deficits incurred as Greece tried to achieve primary balance. That E42 billion is very close to the sum of actual primary deficits in Greece from 2010 (which includes the cost of recapitalising Greek banks).

What that means is that the involvement of European governments has not helped Greece at all. With only IMF support, Greece would have suffered the same degree of austerity that has actually occurred. The additional money provided by the European authorities has been used to pay off Greece’s creditors, first through delaying default in 2010 and 2011, and then by only allowing partial default in 2012. (I’m not sure the two groups see the division that way, but if some of the IMF money was intended to pay off Greece’s creditors, you have to ask why the IMF should be doing that.)

It is pretty clear why the European authorities were so generous to Greece’s creditors. They were worried about contagion. (For more on this, see Karl Whelan here.) The IMF agreed to this programme with only partial default, even though their staff were unable to vouch that the remaining Greek public debt was sustainable with high probability (IMF 2013, para 14).

The key point is that the European authorities and the IMF were wrong. Contagion happened anyway, and was only brought to an end when the ECB agreed to implement OMT (i.e. to become a sovereign lender of last resort).This was a major error by policymakers - they ‘wasted’ huge amounts of money trying to stop something that happened anyway. If Eurozone governments had needlessly spent money on that scale elsewhere, their electorates would have questioned their competence.

This has not happened, because it has been so easy to cover-up this mistake. Politicians and the media repeat endlessly that the money has gone to bail out Greece, not Greece’s creditors. If the money is not coming back, it becomes the fault of Greek governments, or the Greek people. That various Greek governments, at least until recently, agreed to participate in this deception is lamentable, although they might respond that they were given little choice in the matter. (Some of a more cynical disposition might have wondered how many of the creditors were rich Greeks.)

The deception has now developed its own momentum. What should in essence be a cooperative venture to get Greece back on its feet as soon as possible has become a confrontation saga. If the story is that all this money has gone to Greece and they still need more, harsh conditions including further austerity must be imposed to justify further 'generosity'. Among the Troika, hard liners can play to the gallery by appearing tough, perhaps believing that in the end they will be overruled by more sensible voices. The problem with this saga is similar to the problem with imposing further austerity - you harm the economy you are supposed to be helping. (Some see a more sinister explanation for what is currently going on, which is an attempt at regime change in Greece.)

That this is happening is perhaps not too surprising: politicians act like politicians often act. The really sad thing is that playing to the gallery seems to work: politicians using the nationalist card can deflect criticism that should be directed at them for their earlier mistakes. It happens all the time of course: see Putin and the Ukraine, or Scotland and the 2015 UK election. I wonder whether there will ever come a time when this cover-up strategy fails. Futile though it might be, I just ask those who might see this as an ungrateful nation always demanding more to realise they are being played.

Why Sen is right about what is being done to Greece

At first sight the negotiations between Greece and the Troika seem to be simply a battle about resources: how much of the pie that is Greek national income their creditors should receive. There have been many similar types of battle over the years - what makes this one unusual is that the creditors have a unique weapon on their side. With primary surplus approximately achieved, Greece’s bargaining position would normally be extremely strong. The Eurozone creditors would be desperate to salvage what they could from their foolish decision to effectively buy some privately owned Greek government debt. The only reason the Troika is able to call the shots is that it can threaten to eject Greece from the Eurozone. [1]

Part of the deliberate mystification that goes on here is to present Eurozone exit as if it somehow automatically follows if there is a Greek default. But of course Greece has already defaulted, and it remains in the Eurozone. Greece wants to remain in the Eurozone. What will stop them if they do default will be a run on their banks, and a refusal of the lender of last resort for their banks - the ECB - to act in that capacity. Again this will be presented by the ECB as inevitable given the ECB’s own rules. But as Karl Whelan points out, the ECB in reality has considerable discretion, and it has been using that discretion in its role as part of the Troika.

Still, even if the sides are a little unequal in their power, is it still just a battle over resources? One side advocates left wing/populist/humanitarian policies and the other side policies that are more of the consensus/neoliberal/tough variety. [2] One side has suffered a massive fall in GDP partly as a result of previously agreements, while the other is negotiating over what is to them peanuts. So there is plenty of opportunity to pick sides based on preferences. Both sides would almost certainly be better off with an agreement, so it also makes sense for people to appeal for flexibility, which is why I signed this letter.

But to pick sides, or to call for flexibility from both, misses the main point. Yes, this is a battle over resources, but it is a battle where one side is using its power to pursue a policy that is very short-sighted, involving incredible hubris, and which is ultimately self defeating. The Troika are not acting in the long term interests of those they represent. This is I believe what Amartya Sen was saying when he compared these negotiations to the Versailles agreement. 

The most obvious example of this are the Troika’s continuing demands for positive primary surpluses and the austerity measures required to achieve them. This is just stupid. It continues to waste large amounts of valuable resources, as well as inflicting real suffering. It certainly is not in the interests of Greece, but it is almost certainly not in the interests of the creditors either. If you shrink the pie, you are less likely to get the amount of pie you have a claim to. (Martin Sandbu goes through the maths here.) It is disgraceful that key parts of the IMF plays along (or worse) with this. In the past I have described how heterogeneous the IMF is, but taking absolutely no notice of what your own research department says about austerity is crazy. Ambrose Evans-Pritchard talks about the Fund's own credibility and long-term survival being at stake. Keynes, who helped found the organisation, would be turning in his grave.

What about all the reforms that the Troika is insisting on? It is tempting to look at each in turn, and apply our economics expertise to come to an evaluation. If a reform demanded by the Troika might tend to increase the long run pie, then perhaps they are right to insist on it. This neglects one small issue: democracy. The Greek people have elected a government with a mandate. The Troika appear to be acting as if this is neither here nor there, and that is deeply worrying, at least to those of us who are very weary of yet further economic and political integration. (For those who have that integration as their ultimate goal, Greece can be seen as an annoying obstacle that should be thrown aside. [3])  

The hubris of the Troika is incredible. They have convinced themselves that they must override the democratic wishes of the Greek people because the Troika have the wisdom about what is good for the Greek economy. This is the same body that with its superior wisdom prevented full default, and imposed ridiculously strong austerity on Greece and crashed the economy as a direct result. To cover up these errors they play to stories in the media about the lazy and privileged Greek people, stories that largely disintegrate when confronted with evidence. Now they shrug their shoulders and say I have to keep on the same disastrous path because my electorate gives me no choice!

Amartya Sen is right. This is our Versailles treaty moment. It could be so very different. No doubt some of Syriza’s mandate may be unwise, but their own economists may recognise that and welcome the excuse to shelve them. The Troika and Syriza’s negotiating team could be cooperating in that endeavour, but I’m pretty sure that is not what is happening. On austerity the priority of the Troika should be to eliminate the Greek output gap, which means in the short term less rather than more austerity. This would not just be in the interest of the Greek people but also the interests of the rest of the Eurozone.

[1] This is why I think these negotiations are less like bargaining over a mutually beneficial exchange (rolling over lending in exchange for reforms), and more like a discussion with a mugger over which of your personal possessions you hand over.

[2] Or add any other description you prefer - my point is that they differ and people have strong views about them both in principle and practice, and therefore pick sides accordingly.

[3, postscript] Karl Whelan shows here that Giavazzi’s piece is as grounded in facts as some other FT op-eds.