## Nick Rowe: It’s the End of NK Models as We Know Them, and I Feel Fine

[**UPDATE:** See Nick Rowe’s comments in the post for further clarification, plus I tweaked a few things in the post itself in response to his comments.–RPM]

In a previous post, I explained that the academically respectable, formal New Keynesian model of the economy was collapsing before our very eyes. I mentioned that Nick Rowe was fully aware of–indeed was a chief contributor to–this collapse, and yet he didn’t think it a very big deal. Now I realize that the two go hand in hand: Precisely because Nick has the temperament of Confucius, he can afford to be completely frank about the state of mainstream macro theory. It’s not good.

In this recent post, Nick walks through some of the deep problems with various models, and in doing he gives his take on John Cochrane’s recent critique of liquidity trap models. Here’s Nick:

What John Cochrane is saying is that inflation is indeterminate in the Neo-Wicksellian/New Keynesian model, even if you just assume that the output gap asymptotes to zero.

…

John Cochrane proposes an alternative solution to the same set of New Keynesian equations…John Cochrane is not (as I read him) saying his solution is the right one. He is saying it is no less right than the standard solution.

Both those solutions are consistent with the IS curve and Phillips curve of the New Keynesian model. Both are mathematically correct. Neither of these solutions is “pathological” in the sense of causing inflation to go to plus or minus infinity. Both solutions would be equally stable or unstable in the sense of staying or not staying on that path if the central bank threatened to respond if they strayed from the equilibrium path. And those are just two solutions from an infinite number of solutions. And there is nothing in the model itself that tells us which of those solutions is the “right” one. There are multiple equilibria.

Now there are some models with multiple equilibria where the modeller knew there were multiple equilibria right from the start, and proudly called the readers’ attention to the fact that the model had multiple equilibria, because he thought it was an important and desirable feature of the model that reflected something about the world. The Diamond-Dybvig model of bank runs (which has two equilibria, one with a bank run and one without) is like that.

But the New Keynesian model is not like that. Nobody said “Hey look! I’ve just built this New Keynesian model which is a really neat model because it has an infinite number of equilibria, and so it can explain why sometimes we get recessions and sometimes we get booms, and those recessions and booms just happen; they aren’t caused by anything at all, except animal spirits, or sunspots! And you can get totally different responses to exactly the same shocks and exactly the same monetary policy responses to those shocks, just because!” The New Keynesian model was never taught that way. It was taught as saying that recessions and booms and inflation and deflation were caused by bad monetary policy which didn’t or couldn’t respond to shocks correctly and quickly enough.

How should we respond to all this?

…and then Nick lists some options.

Let me just shine the spotlight on Nick and say that this is really important, folks. Unfortunately, this is analogous to the Great Debt Debate, when a grumpy economist (Don Boudreaux then, John Cochrane now) challenged something very fundamentally that Krugman et al. have been saying throughout the crisis. Nick (then and now) joined in, with his own analogies and idiosyncratic way of putting it. However, I’m afraid that because Nick is so immersed in the literature, that only professional economists will really get what he’s saying. The problem there is that professional economists are the last people on earth to judge a dispute on whether a big chunk of professional economics has been spinning its wheels for the last few decades.

Some open questions that I would love to resolve if I had tenure:

1) Is Nick right that the standard New Keynesian model predicts that ~~a sudden~~ an expected and large *drop* in future government spending can restore full employment in the present? If so, isn’t that kind of important? If both sides of the fiscal policy debate actually believed their rhetoric, this should solve everything: We ~~cut~~ announce cuts to government spending for next fiscal year and fix the debt problem, and we put everyone back to work right away.

2) Krugman has been claiming with a large degree of confidence that Keynesian models show that the normal laws of economics are turned upside down once we hit the zero lower bound. Cochrane and Nick Rowe are claiming that actually, if you write out the equations of a standard New Keynesian model, there are an infinite of equilibria. Yes, some behave the way Krugman says, but others behave the way Fama says. (Furthermore, Cochrane claims that the Krugman-esque equilibria have some really weird properties, such as things blowing up when you introduce epsilon of price stickiness, whereas the Fama-esque equilibria don’t blow up. But Nick didn’t confirm that aspect of Cohrane’s post.) So, is there a contradiction here? It’s true that Krugman doesn’t usually call himself a New Keynesian, but then again when he took Casey Mulligan out to the woodshed he was happy to lump “old or new, it doesn’t matter” into the same category of Keynesian models against Mulligan’s (alleged) ignorance.

3) Both Cochrane and Krugman (sic) have said that the standard Keynesian models (at least if we are looking at “liquidity trap-esque” equilibria) imply large price *deflation* in the kind of slump we’ve had. Krugman has dealt with that problem by adding what (to me) is analogous to Einstein’s cosmological constant: Krugman sees that prices aren’t falling as fast as the model originally predicted, so he plugs in an assumption that wages can’t fall fast. So, is it better to say–for economists who pride themselves on being good empiricists–that we should rule out the liquidity trap equilibria as being unfaithful to the data?

Unlike with the Great Debt Debate, I don’t think I can get sucked too much into this one, at least not until I clean off my consulting plate and that will take at least a month. So for those of you who can actually understand technical macro papers, do it for the children!

Even if it takes you a long time to get around to going deeper into this topic, please do write more about this. I find these kinds of posts to be the most interesting.

And if it leads to something even half as good as the culminating post of the Great Debt Debate, it will well be worth the wait đź™‚

There is trap we are actually in but it is not the liquidity trap, it is rather the QE-trap. And the only exit Bernanke can find is for him to leave the boat..

Thanks Bob!

Some minor corrections:

1. “The problem there is that professional economists are the last people on earth to judge a dispute on whether a big chunk of professional economics has been spinning its wheels for the last few decades.”

It’s only the last 1.5 decades. Not quite so bad.

2. “Is Nick right that the standard New Keynesian model predicts that a sudden and large drop in government spending can restore full employment?”

I am saying that the standard NK model (with the “standard” solution) predicts that a sudden announced *future* drop in government spending (to be implemented after the ZLB passes) will restore full employment. (And do so just as well as a sudden temporary rise in government spending that is announced will end after the ZLB passes).

Yes I am right. That one is easier than all that burden of the debt debate.

3. “Furthermore, Cochrane claims that the Krugman-esque equilibria have some really weird properties, such as things blowing up when you introduce epsilon of price stickiness, whereas the Fama-esque equilibria donâ€™t blow up. But Nick didnâ€™t confirm that aspect of Cohraneâ€™s post.”

I think you might be misunderstanding John Cochrane there. Or maybe i missed it.

4. “So, is it better to sayâ€“for economists who pride themselves on being good empiricistsâ€“that we should rule out the liquidity trap equilibria as being unfaithful to the data?”

It looks to me like something is empirically wrong with the Phillips Curve, but that is a separate question to whether we can rule out liquidity trap equilibria.

Ah. On my point 3, I think you should have said that John Cochrane said that things blow up when you *don’t* introduce at least an epsilon of price stickiness.

On point 2, here are my two posts:

http://worthwhile.typepad.com/worthwhile_canadian_initi/2013/06/new-keynesian-countercyclical-fiscal-policy-isnt-what-you-probably-think-it-is.html

http://worthwhile.typepad.com/worthwhile_canadian_initi/2013/03/fiscal-policy-with-old-and-new-keynesian-is-curves.html

“I am saying that the standard NK model (with the â€śstandardâ€ť solution) predicts that a sudden announced *future* drop in government spending (to be implemented after the ZLB passes) will restore full employment. (And do so just as well as a sudden temporary rise in government spending that is announced will end after the ZLB passes).”

With the difference that in the first case the economy actually produces stuff the people really or at least rather want than in the second case. And, that in the second case, as indicated by the word â€śtemporaryâ€ť, you’d still have the problem to exit from this ramped up government spending which was not done for the sake of producing whatever government officials decided to produce, but only to provide work (like in the worst case ditch digging). Stopping this program might just get you into the same situation as before, just back to square 1, right?

And I guess the second case in which government spending rises refers to deficit spending. Increasing spending funded by direct taxation wouldnâ€™t do it right?

sklien:

In both cases, people need to believe what the government says: that it will in fact cut spending in future.

It depends on how big the government already is (whether it’s bigger or smaller than long-run optimal) and if the government can buy useful things quickly. In Canada’s case, the (Conservative) government increased spending temporarily, but mostly on stuff it would have bought sooner or later anyway. It just preponed the building of some roads and bridges etc by a few years.

“It just preponed the building of some roads and bridges etc by a few years.”

I have the impression many people believe, that in such a case, everything is completely fine and that money isn’t wasted at all. It is just preponed. This is wrong of course. I am sure you are aware of it but I just want to spell it out here since I rarely see this anyone doing.

For every machine, road, bridge or whatever there is a specific point at which it is most useful or expressed the other way around least wasteful to renew it. To do it later or earlier means you are wasting / you make a loss.

Therefore preponing might not be completely unnecessary like ditch digging, but already the intention of preponing something to just provide work by definition means (ex ante) it is suboptimal and a certain part of it is just like ditch digging.

An extreme example is having made a new road today, and because we know it need to be renewed in 10 years anyway, we might just renew it tomorrow again. This would be just as bad as ditch digging.

Normally, preponing an investment is wasteful, yes.

But if real interest rates are 0% or negative (they were), and if the investment does not depreciate (these ones wouldn’t depreciate much) then it is not wasteful.

For example, it makes sense to store apples when the nominal interest rate minus the expected rate of inflation for apples is negative and exceeds the depreciation rate on stored apples.

If this were the case then businesses would do it as well, then there was no need for the government to do it, wouldnâ€™t they?

skylien: presumably private firms and people were doing it too. And if enough of them had been doing it, real rates wouldn’t have been 0%, so presumably there weren’t enough of them doing it.

Nick,

Right, so obviously there is a problem. You are claiming there is money lying on the streets, which the private market just doesnâ€™t pick up. How can this be?

Only if picking up this money is not riskless. And I would argue that your assumption of negligible depreciation of investments financed during times of negative real interest rates like now is wrong.

Negative real interest rates donâ€™t tell you in what to invest. You cannot assume that future depreciation (which depends on future earning power of your investment) is equal to past depreciation of similar investments.

That real rates are 0 or negative means that something is wrong in the economy, right?

In the standard NK model, Ricardian Equivalence holds, so it doesn’t matter whether increased spending is tax-financed or deficit-financed.

Ok, thanks.

BTW, the problem of infinite number of solutions only applies to a subset of NK models: those I call “Neo-Wicksellian” that don’t include Money in the model. Admittedly, that subset is the ones most NK people have been working with for the last few years. But Paul Krugman’s model does contain M, so is immune to that critique. On the other hand, Paul Krugman doesn’t 100% understand his model. He thinks it doesn’t have a Pigou effect, but it does have a Pigou effect, and given sufficient price flexibility the Pigou effect would bring his model to full employment even at the ZLB.

There isn’t an infinite number of solutions. As you point out yourself, it would require the real rate of interest to equal the nominal rate of interest, but real-number variables are never exactly equal, so in practice the only solutions of interest are the zero solution and the infinite solution.

That’s not as shocking as you might think at first glance; the system is an integrator and any positive input to an open-loop integrator gives positive infinity output, any negative input gives negative infinity output (consider these to be the exponential of the dollar value so exp(+inf) => +inf and exp(-inf) => 0 if it helps explain what I’m getting at here). Thus, the two open-loop steady state solutions for a system with infinite DC gain are exactly what you would expect.

However, the economic system never runs in open loop, does it? Someone always takes the reigns and tries to control it. Control algorithms are another matter entirely.

Tel: “As you point out yourself, it would require the real rate of interest to equal the nominal rate of interest,”

No.

Rewrite that slightly to the more common format:

C(t) = Î± . C(t-1)

Î± = (1+r(t))/(1+n)

http://en.wikipedia.org/wiki/Exponential_growth#Difference_equation

Presuming that “r” and “n” are constant, and also that C > 0 and Î± > 0, we can say that whenever Î± > 1 it will exponentially grow to infinity, and whenever Î± < 1 it will exponentially decay to zero,

and there’s nothing else that can happen. Plot that against time on semi-log paper and you get a straight line, which says that what we really have here is an integrator with constant input (or with input determined by the Central Bank at any rate, and should that be constant, you see a straight line output from the integrator).So those are the only two steady state open loop solutions: infinity and zero.

With constant “r” and arbitrary initial conditions it simply cannot be done. Even with non-constant “r(t)” and also some physical limits on how the system reacts in finite time it still cannot be done

for all tbut you can at least build a feedback controller that will converge on C(t) = 100 after finite time has passed (more than one time step).Building a feedback controller to stabilise a “plant” (see footnote), where that plant behaves like an integrator is a well studied problem, going back 100 years or more. Examples include: a cruise controller on an automobile; a gantry crane in a shipyard; a ball and beam system; or try it at home by balancing a broomstick on your hand.

By the way, there’s something topsy-turvy somewhere because r > n should be decay and r < n should be growth, but I think I've been faithful to the original equation that I cut and pasted, so leave it as an exercise to the reader to track down who blundered here (I'm one bottle down and not likely to get better tonight).

Just to block the future gotcha, I don't personally believe the Central Bank really can drive consumption in such a simple manner, but what I'm doing here is responding to someone else's simplified Keynesian model, just to demonstrate the implications of that particular model, not to give it a stamp of approval.

Footnote:I use the word “plant” in the engineering sense, that may or may not have anything to do with vegetation, but it’s become the standard placeholder in control theory literature, regardless of whether it suits the context.Tel:”… or try it at home by balancing a broomstick on your hand.”

That’s the analogy I was using (except I called it a “pole”). And you can do that, providing you can move your hand more quickly than the changing gusts of wind.

But then I realised: in the NK model this is a broomstick that has no memory of where it is right now. It can jump sideways at any time for no reason at all. C(t) is not determined until people see r(t) and also form an expectation of C(t+1).

BTW, you made an arithmetic mistake in alpha.

After double-checking, I still believe the topsy-turvy comes from your side of the fence. Consider the Keynesian stimulus situation: r = 10% and n = 5%

(1+n)/(1+r(t)) = 1.047619

C(t)/C(t+1) = 1.047619

Given that C(t) represents consumption now, and C(t+1) represents future consumption. Present consumption is 4.8% greater than future consumption, so this “stimulus” is driving exponential decay, which will continue for all time steps until “r” is changed.

I very much doubt you will find a Keynesian who will agree with that conclusion.

Regarding the broom that does not know where it is; I don’t understand how you make the leap from a difference equation to a case where memory of present state is lost in an instant.

It is easier to balance a broomstick on your hand than it is to balance a pencil point-down on your fingertip, because the pencil will fall over too quickly for a human to react. We could argue that this is a reaction time race between the Central Bank and the Market… I agree that if the controller is too slow (compared to what it is trying to control), then it will never work.

Instantaneous step change in consumption sounds implausible, both to me and I suspect also to your typical sticky-prices Keynesian.

Bah! Ignore the first bit from above.

http://worthwhile.typepad.com/worthwhile_canadian_initi/2013/09/new-keynesians-just-assume-full-employment-without-even-realising-it.html

Right, I keep wanting to think “n” is nominal interest rates, but it isn’t defined that way in this situation.

What I was trying to say above is this: Consider the Keynesian stimulus situation:

n = 10% and r = 5%That is to say, the central bank sets “r” at a

lower ratethan individual time preference rate. Then you get the numbers:(1+n)/(1+r(t)) = 1.047619

C(t)/C(t+1) = 1.047619

Then you get exponential decay, as explained above.

Not that I’m wanting to in any way defend Keynes (new or old) but I’ll point out that the “P*” in Nick Rowe’s narrative is not measurable, nor will it ever be. Thus, it is some artefact of the model, and nothing more than that. Maybe for calculation purposes working with infinite values might impose some technical difficulties (although integral calculus has done OK) but you can’t use that to prove the model wrong… you can prove the model to be annoying… but not wrong… Only “P” is a real entity, and in the Keynesian model “P” is constrained by the sticky price to remain non-infinite and non-zero.

Thus, this is more of an imagined problem than a real problem.

I might go one step further in doing the impossible and validate the Keynesian model against the real-world observations… if the only two values “P*” ever took were zero and infinity, it would never the less quite closely match what reserve banks do in practice. That is, gradually drop interest rates until they see inflation winding up, and then increase interest rates until inflation is under control, repeat as it goes with this style of hysteresis controller, never finding the correct value for the interest rates, but regularly crossing from one side to the other. It may be a whack way to organize one’s affairs, but it does fit the observable facts, and that’s the general gist of a model.

Just to squeeze in a bit of Keynesian bashing here (and correct me if I’m missing the obvious) but do they have any model of “sticky” prices? I mean, sure I hear a lot about sticky prices, I hear that wages are more sticky than supermarket prices, and that wages can go up, but they can never go down… but outside the hand waving, is there an actual model for sticky prices?

If no such model exists, then you can’t relate “P*” to “P” other than by just having a got at it, and see what happens. That would seem to me to be a more serious concern.

Tel: “That is, gradually drop interest rates until they see inflation winding up, and then increase interest rates until inflation is under control, repeat as it goes with this style of hysteresis controller, never finding the correct value for the interest rates, but regularly crossing from one side to the other. It may be a whack way to organize oneâ€™s affairs, but it does fit the observable facts, and thatâ€™s the general gist of a model.”

That works with the Old Keynesian model with the Old Keynesian IS curve and Old Keynesian Phillips Curve. It doesn’t work with the New Keynesian model.

Okay, and so are a few really stupid questions based on the amount of information that I read in this post.

Both those solutions are consistent with the IS curve and Phillips curve of the New Keynesian model. Both are mathematically correct. Neither of these solutions is â€śpathologicalâ€ť in the sense of causing inflation to go to plus or minus infinity. Both solutions would be equally stable or unstable in the sense of staying or not staying on that path if the central bank threatened to respond if they strayed from the equilibrium path. And those are just two solutions from an infinite number of solutions. And there is nothing in the model itself that tells us which of those solutions is the â€śrightâ€ť one. There are multiple equilibria.1 ) So basically the NK model fits ALL economic situations, regardless of action taken the equilibrium will be met So long as the actions taken are consistent. However since there are an infinite number of solutions there are an equal number of equlibria? So since there are infinite possibilities there are infinite outcomes, can I not at this point say, “well duh”? I mean this would be the case no matter what we did in the economy any time any place. I suppose if your point is that Krugman is wrong that economics does not turn on its head at zero bound points then I am in agreement with this. But I suppose the real question is, is this really what Krugman is saying?

2) Okay, so what you are suggesting is that Krugman is wrong because the real world did not follow expectations… Well would it with Government interference? I mean honestly the only way for a model to be, well correct, is that there are no additional variables added into the equation, as additional variables enter in would not the equilibrium change?

I suppose the best way for me to talk about economics is in reference to sound waves. Imagine you have a large number of pressures all working at the same time, the various oscillation and amplitudes being tied to a master control, but each ‘channel’ if you will having its own volume control as well. The economy would mix together to create ‘equilibrium’ so long as no additional tones or modulations were added and the amplitudes were not changed.

What I am saying is simply this given any inputs and so long as no one messes with the array, the result will be a sum of all oscillations within the resulting wave form, ergo equilibrium. Now you can raise the ‘level’ of all of this by adding additional power, however this simply AMPLIFIES the curves rather than what one would believe of smoothing them out. Bigger high’s lower low’s. However the system continues to remain in equilibrium centered around the Zero line, or whatever trend line you wish to present.

I am sure my analogy is incorrect but it is what I am picturing as you all are speaking about this subject.

Again I am simply an ignorant layman who studies this stuff rather than argues about it in acedimia, I find the discussion fascinating but often times superfluous as all answers seem to arrive at the same conclusion. Belief and a search for justification of said belief. Not to say this is bad. Again I was with Bob Murphy when he thought that the devaluation of the Dollar would have sparked a larger inflation via CPI then has occurred ( it did inflate the money supply after all ) But I suppose the Euro zone mitigated this a great deal making the USA still look like the best looking horse in the glue factory.

That was supposed to be ‘well be correct’ rather than ‘well correct’ lol

Innocent: “So basically the NK model fits ALL economic situations, regardless of action taken the equilibrium will be met ”

No. That’s wrong. And it’s not what I’m saying.

Ah, okay

Nick:

Thanks for the good comments. I’ll update the post tonight to make sure I’m accurately representing your positions, but I think you are misreading Cochrane. This is his paragraph that I had in mind:

The thin red dashed lines marching toward the vertical axis show what happens as you reduce price stickiness. (I only showed inflation, output does the same thing.) As you reduce price stickiness, it all gets worse — output at any given date falls dramatically.For price stickiness epsilon away from a frictionless market, output falls to zero and inflation to negative infinity.Now maybe I phrased it in a misleading way by saying if you introduce epsilon of price stickiness–as opposed to reducing price stickiness down to epsilon–but what I said is true, right?

Bob: OK. What John cochrane meant is that in the limit, as prices approach perfect flexibility, output approaches zero, and inflation approaches minus infinity.

He didn’t phrase it very well.

Right, Nick, that’s what I thought he meant. đź™‚ And his broader point is that that’s kind of weird, that since it’s price stickiness that drives the “problem”–i.e. you only get recessions in a rational expectations model if there are frictions in markets–you would normally expect things to get better, i.e. recessions less severe in reaction to a shock of given size, as the price stickiness is dialed down. Yet the opposite happens in the Krugman-esque equilibria.

However, Cochrane says, in the other family of equilibria, where micro 101 applies, the recession gets more moderate as you dial down the price stickiness. So on that criterion, you would think the Fama-esque equilibria are more reasonable than the Krugman-esque ones.

Do you (a) understand what I’m saying and then (b) agree that this is what Cochrane is saying?

Bob:

a. I understand what you are saying now. (Took me two tries, but maybe that’s me!)

b. It’s very probably what Cochrane is saying, because it’s right (and it follows from everything else he says). I would have to re-read to be 100% sure it’s what he’s saying.

Bob: “However, Cochrane says, in the other family of equilibria, where micro 101 applies, the recession gets more moderate as you dial down the price stickiness.”

I think that should be: “…the *boom* gets more moderate as you dial down the price stickiness.”

Yep, it’s weird.

It’s funny how nobody is pointing out that equilibrium is never reached in the real world, so of course any calculated equilibria on the basis of ideas, would be one among many, since there are many ideas.

I hope that at some point these theorists recognize the fact that praxeology provides a set of objective constraints from which every idealistic equilibria can be rejected as impossible.

The conclusion of infinite equilibria arises when economic propositions are stripped from their praxeological foundation, and kept hanging in mid air as abstract verbal statements subject to no objective rules.

In this article de Gruawe provides some understanding. He says estimating multipliers depends on the models we use. The models in turn depend on the kind of recession we have. If we have a normal recession, then new keynesian model can work but in an abnormal recession like this one, we need Keynes model.