And anyone with even an ounce of Old Keynesian blood left in his veins, if they understood what the New Keynesians are doing, would be screaming blue murder that we are teaching this New Keynesian model to our students as the main macro model, and that central banks are using this model to set monetary policy.
I have made this point before. And here. But I'm now going to make this point so simply and clearly that any New Keynesian macroeconomist will be able to understand it.
Here is a very simple version of a standard New Keynesian model.
Assume no investment, government spending or taxes, and no exports and imports. There is only consumption. Assume a "haircut" economy of self-employed hairdressers, cutting each other's hair, in which all goods are services, with labour the only input, so counsumption, output, and employment are all the same thing. And so prices and wages are the same thing too.
Assume no exogenous shocks, ever. And no growth either. Nothing exogenous ever changes.
Assume a constant population of very many, very small, identical, infinitely-lived agents, with logarithmic utility of consumption, and a rate of time-preference proper of n.
The individual agent's consumption-Euler equation, with r(t) as the one-period real interest rate, is therefore:
C(t)/C(t+1) = (1+n)/(1+r(t))
Ignore the Zero Lower Bound on nominal interest rates. In fact, just to
make the central bank's job even simpler, ignore nominal interest rates
altogether, and assume the central bank sets a real interest rate r(t).
Suppose the "full employment" (natural rate) equilibrium is (say) 100 haircuts per agent per year consumption, income, and employment. Forever and ever.
The central bank's job is to set r(t) such that C(t)=100, for all t.
Inspecting the consumption-Euler equation, we see that this requires the central bank to set r(t)=n for all t. Assume the central bank does this.
It is obvious that setting r(t)=n for all t only pins down the expected growth rate of consumption from now on. (It pins it down to zero growth.) It does not pin down the level of consumption from now on.
Suppose initially we are at full employment. C(t)=100. Then every agent has a bad case of animal spirits. There's a sunspot. Or someone forgets to sacrifice a goat. So each agent expects every other agent to consume at C(t)=50 from now on. So each agent expects his sales of haircuts to be 50 per period from now on. So each agent expects his income to be 50 per period from now on. So each agent realises that he must cut his consumption to 50 per period from now on too, otherwise he will have to borrow to finance his negative saving and will go deeper and deeper into debt, till he hits his borrowing limit and is forced to cut his consumption below 50 so he can pay at least the interest on his debt.
His optimal response to his changed expectation of other agents' cutting their consumption to 50, if he expects the central bank to continue to set r(t)=n, is to cut his own consumption immediately to 50 and keep it there.
C(t)=50, which means 50% permanent unemployment (strictly, underemployment), is also an equilibrium with r(t)=n. So is any rate of unemployment between 0% and 100%.
What can the central bank do to counter the bad animal spirits?
If it cuts r(t) below n, even temporarily, we know there exists no rational expectations equilibrium in which there is always full employment. All we know is that we must have negative equilibrium growth in consumption for as long as r(t) remains below n. It is not obvious to me how making people expect negative growth in their incomes from now on should cause everyone to expect a higher level of income right now from a higher level of everyone else's consumption right now.
Sacrificing a goat sounds more promising as a method of restoring full employment.
Did every other New Keynesian macroeconomist already know about this, and just swept it under the mathematical rug? Didn't I get the memo?
"Under
the assumption that the effects of nominal rigidities vanish
asymptotically [lim as T goes to infinity of the output gap at time T goes to zero]. In
that case one can solve the [consumption-Euler equation] forward to
yield…"
Bullshit. It's got nothing to do with the effects of nominal rigidities. What he really means is "We need to just assume the economy
always approaches full employment in the limit as time goes to infinity,
otherwise our Phiilips Curve tells us we will eventually get hyperinflation or hyperdeflation, and we can't have our model predicting that, can we?"
That Neo-Wicksellian/New Keynesian nonsense is what the best schools have been teaching their best students for the last decade or so. They have been teaching their students to just assume the economy eventually approaches full employment, even though there is absolutely nothing in the model to say it should.
Remember the Old Keynesian Income-Expenditure/Keynesian Cross diagram? What we have here, if the central bank sets r(t)=n, is a version of that diagram in which APC=MPC=1 for all levels of income, so the AE curve coincides with the 45 degree line. Any level of income between 0 and full employment income is an equilibrium.
New Keynesians simply must put money back into the model.
Worthwhile Canadian Initiative 
notsneaky: I’m not sure about this, but what you are saying sounds to my ears similar to what John Cochrane is saying. (The difference is that JC has the CB responding to inflation, while you have the CB responding to the output gap, but that’s not an important difference in this context).
Actually I’m assuming CB is inflation targeting here too, I just took differences and plugged in the Philips Curve. The actual monetary rule is r(t)=n+api(t), the PC is pi(t)=pi(t-1)+kc(t). So r(t)-r(t-1)=(ak)c(t)=bc(t). So it might very well be the same thing Cochrane is saying, I dunno, I’ll have to go and read his blog. I’ll come back to the MR below.
Simplify: How about this: at the beginning of time, the CB makes the following (“blow up the world”) commitment: “If everyone always chooses C=100, I will always set r=n. But if anyone ever chooses C=/=100, I will choose r=2n, and keep it there forever…”
I think this would work too but am not sure (is that supposed to be r=2n or just a stand in for any “crazy” r?) What’s tripping me up a bit is the exact timing within each period, and trying to combine the assumption of an unexpected shock with perfect foresight afterward. If C=/=0 in period zero but not because of the choice of households but some exogenous shock, does the CB set r=2n? If yes, then I don’t think it works. If no, then I think it does, since all you want is to pin down the correct expectations.
We have c(t+1)=c(t)+(r(t)-n) but – because as you emphasize c can jump – this does NOT apply in the period right after the shock… right? The Euler implies consumption smoothing but unexpected shocks can make you jump from one path to another and only once you’re on the new right path does it hold.
The policy I’m thinking off above involves a monetary rule EXCEPT for the initial period (or given the timing within each period and the lags, the period after the shock) where the CB chooses an r(t) in a “discretionary” fashion to get on the appropriate stable path. It’s MR after that.
So consider a output targeting rule r(t)=n+vc(t). Then r(t)-r(t-1)=v(c(t)-c(t-1)) which means that dr/dt=0 is the same nulcline as dx/dt=0, the vertical line at r=n. In that case if a shock happens the only stable “path” (actually a point) is to just set r=n but that means there’s no adjustment back to c=0. Interest rates only stabilize the economy, they don’t bring it back to full employment. In that case you’re perfectly right. No equilibratin’, self or otherwise. I think this has actually been emphasized in some of the papers and textbook presentations of the NK model (I’m not a NK myself, just playing one here, because I don’t think there’s any genuine NKs involved who are willing to jump in and defend their framework, so I’m just filling in that void)
(Note that reacting to the output gap in a rule based fashion is a different policy than “set whatever r is necessary to get back to full employment”.)
If you have a Taylor style MR, say r(t)=n+bc(t+1)+(1-b)pi(t) (I put c(t+1) in there because of perfect foresight and it reduces the worrying one has to do about the time subscripts) then dr/dt=0 nullcline is a downward sloping line rather than the x-axis or the vertical axis at r=n. But there’s still a stable path.
There is still that crazy implication that if the economy goes into recession (c(0)<0) then what the CB needs to do is first RAISE interest rates, to get on a path where it can cut them later. Like I said it’s weirding me out a bit and I’m very very un-confident that anything I’m saying here is correct.
Subject to that caveat/confession, if I was gonna declare a winner in the NicK vs. NK fight I’d give you most of it. It’s not exactly that the NK models just “assume that the economy comes back to full employment”. It’s rather something like, “the rational expectations consumers know that the CB will pick an initial value of r to get on a stable path which leads back to full employment”, and that it’s MR is “sensible”. The part that seems to get left out of the how the NK models are described is that for all this to work (and it does work with ratex or perfect foresight, I think) the CB has to choose a crazy initial interest rate.
Nick: Arguably the problem stems from the fact that the model which we teach in intro macro as Keynesian is actually more akin to Pigou’s macro model than to Keynes’. https://www.uoguelph.ca/economics/sites/uoguelph.ca.economics/files/2013-06.pdf
Akshay: Thanks!
If agents believe that the effect of the sunspot will be temporary, I think it has no effect at all. Because if each agent thinks his income will rise from 50 back to 100, and that the CB will keep r(t)=n, he will immediately want to spend more than 50, which isn’t a rational expectations equilibrium.
Sjysnyc:
If all else fails, the argument from authority is a good one! (Not that I have much.) When my dentist, doctor, or car mechanic uses it, I put some weight on it.
“So take the fraction of intermediate firms that do adjust prices. They drop their prices substantially and hire lots of labor. They have enough demand at those new lower prices now, and can actually expand their production. So output goes up at those firms that adjusted their prices. And unless you’ve got some strange aggregation function overall demand should be greater than 50.”
But I could equally well counterargue that when they drop their relative prices it simply raises the relative prices of the remaining firms. So that merely causes a re-distribution of production and employment between the two sets of firms. If we had a well-defined vertical AD curve, this would be what must happen. If instead we had a well-defined downward-sloping AD curve, this could not happen. But in this case we do not have a well-defined AD curve at all. It’s like the AD curve is vertical (because the level of P doesn’t matter) but it’s very very thick. Anything can happen.
Put it another way: yes, that fraction of firms cutting prices could result in an increase in aggregate C; that’s an equilibrium. But it’s also an equilibrium if it causes no change in aggregate C, just a redistribution. Sacrificing a goat might work too; or might not work. We have a multiplicity of equilibria.
notsneaky: “(is that supposed to be r=2n or just a stand in for any “crazy” r?) ”
It didn’t have to be 2. Any r(t) greater than n for all t would have the same effect. Because it means C(t) must grow over time along a perfect foresight path.
You lost me a little on what you said immediately after that. I’m still in the model where there are no exogenous shocks (except sunspots). So I’m saying the CB threatens to raise r(t) permanently if agents ever choose C(t) below full employment as a result of seeing a sunspot.
“There is still that crazy implication that if the economy goes into recession (c(0)<0) then what the CB needs to do is first RAISE interest rates, to get on a path where it can cut them later. Like I said it’s weirding me out a bit and I’m very very un-confident that anything I’m saying here is correct.”
Yep. Understood. I’m with you. It’s very similar to writing down the Fisher equation, assuming that money is superneutral so cannot affect the real rate, and saying that if the central banks wants to increase inflation it simply needs to raise the nominal interest rate. It’s similar to having a model with an unstable equilibrium where the comparative statics all have the wrong sign, so an increase in demand causes prices to fall. These are some of the paradoxes you get when you assume the economy always jumps to a perfect foresight equilibrium path. Some economists handle this by assuming some sort of adaptive learning. So that the Rational Expectations equilibrium is treated as the limiting case where learning takes place very quickly. Which I think is a useful approach. Put it this way: what I am (maybe) saying is that the sort of NK model I have here will not converge on full employment under any reasonable sort of learning mechanism where agents learn about their future income from observing their past incomes and interest rates, like atheoretical econometricians.
BSF: that deserves to be read. My off-the cuff reaction though, before reading it: If this NK model I have here simply added a Pigou effect, the problem would be solved. If all agents expect C(t)=50 from now on, they know that prices will fall, and that M(t)/P(t) will rise without limit, so each agent will become infinitely wealthy sometime in the future, but will still be living like an underemployed pauper, which is not individually rational, so each individual will decide to consume more than 50 if he expects his income to be 50, and this cannot be an equilibrium. Since we have a continuum of equilibria (like a ball on a perfectly flat table) all it needs is a tiny Pigou effect to get the economy back to full employment immediately (unless the CB sets r(t) too high).
Nick
Correct me if I’m wrong but I think this reduces (or generalizes?) to saying two things:
1)The Wicksellian cumulative process (asymptotically)goes on infinitely in a world with (asymptotically) vanishing nominal rigidities.
If so, that’s true. Didn’t Wicksell explicitly mention nominal rigidities (‘history’, he called it) as the only reason the cumulative process is bounded?
And every once so often, economies do collapse into Howitt-Wicksell hyperinflations, no? The interesting thing to ponder is why there never is a hyperdeflationary death trap.
2) Think J W Mason has mentioned this but, in a world with otherwise efficient markets, why does one price need to be set ‘exogenously’? Think John Geanakoplos and compatriots had a decent swing at that question with their GEI (General Equilibrium with Incomplete Asset Markets) project, An anthology here : http://www.dklevine.com/archive/refs41115.pdf : in sections 5 and 8 Geanakoplos describes how private agents fail to create the financial numeraire, paving the way for a monetary/price regime.
The problem’s not Wicksell, Nick. The problem’s Arrow-Debreu-Lucas.
And what should/can the NKs immediately place back into the model? Risk.
Ritwik:
I’m still not sure I am understanding you, but I don’t think we are on the same page. let me guess, and try this:
1. Wicksell’s cumulative process was about what happened if the central bank set the wrong rate of interest. Bad stuff happens. And it gets steadily worse over time unless the central bank eventually corrects its mistake. New Keynesians agree. I agree. But New Keynesians say that bad stuff cannot happen if the central bank sets the right rate of interest. I say that that conclusion does not follow from their model. It might follow from a different model (it would follow from a model with an Old Keynesian IS curve, or from Wicksells model), but it doesn’t follow from the NK model. The NK’s are abusing their own model. If they used it properly, they would see that bad stuff might happen even if the CB always sets the right rate of interest.
2. Whoever produces apples must set either the price or quantity (or something) of the apples they produce. That’s true whether it’s a private or government producer of apples. It would also be true if it were a private or government producer of money, instead of apples. There’s a whole separate question of whether a rate of interest is analagous to a price or quantity or something of money. I say it isn’t. And that this makes interest rates a bad instrument for monetary policy. But, if for example we take an Old Keynesian IS curve, and assume the CB sets a rate of interest, the CB cannot set any real rate of interest it wants, without the economy eventually blowing up (or down). It must set exactly the right rate of interest (in the long run). The CB cannot freely choose the real rate of interest.
The problem’s not Wicksell. The problem’s not Arrow-Debreu-Lucas. The problem is New Keynesians taking bits from Wicksell and bits from other models and jamming them all together into an incoherent package that doesn’t make sense.
“And what should/can the NKs immediately place back into the model? Risk.”
No. Money. They’ve implicitly got a monetary exchange economy, without actually having money. They need to put money in properly. Risk makes no difference to my point here.
Nick,
You seem to be indicating that with:
C(t) / C(t+1) = ( 1 + n )/( 1 + r(t) )
Cutting r(t) below n will tend to lower C(t+1) with respect to C(t) unless:
C(t+1) = C(t) * ( 1 + r(t) + dM/dt ) / ( 1 + n )
Where M(t) is the money supply adjusted for liquidity preference. Here, reductions in r(t) that lead to increases in M(t) through a credit channel can increase C(t + 1) with respect to C(t).
Nick,
I don’t think C(t)/2 can be an equilibrium path in the NK model unless the policy rule doesn’t satisfy the Taylor principle. If the CB follows a Taylor rule, asymptotically paths go to +/- infinity or zero output gap.
“Does everybody else in the illuminati know the NK’s are just sweeping the whole thing under the rug?”
In his book, Woodford points out that there is no forward inflation dynamic, i.e. today’s inflation in no way determines tomorrow’s. It is expectations of tomorrow’s inflation that determines today’s. So there is no sense in which a small inflation error can lead to a diverging path. A diverging path is caused by an expectation of asymptotically diverging inflation. There may be solid economic reasons to reject such paths (see below), but Woodford suggests that the mere fact that inflation is not observed to be diverging exponentially ought to be enough grounds to reject those paths.
Maybe the lack of determinacy is a feature, rather than a bug. As Woodford (2000) showed, there are paths of fiscal policy for which running a Taylor rule cannot guarantee the stabilization of inflation. Particularly extreme such fiscal policies, are consistent with expectations of asymptotic runaway inflation, Taylor principle notwithstanding. So the fact that the basic NK model is consistent with both convergent and divergent paths just means that it is consistent with both (unspecified) Ricardian and runaway fiscally dominant regimes.
John Cochrane takes up the case of possibly non-Ricardian fiscal policy, and finds that quantity of government debt along with the fiscal theory of the price level can, in fact, provide the nominal anchor required for determinacy in the NK model (much like the real balance does for monetarists). If Cochrane is correct, then I think it follows that agent belief in sufficiently bounded primary surpluses can provide the determinacy that’s needed for eliminating diverging paths (or to put us in the divergent path in the case of insufficiently bounded deficits/surpluses).
A couple of other interesting takes I found:
McCallum 2011 agrees with Cochrane that the divergent paths cannot be ruled out a priori, but he invokes a kind of ratex version of the anthropic principle: Since it is known that the divergent paths of the NK model are not learnable, and since the agents in fact have model consistent expectations that they must have learned, they must be in a world with convergent paths. I.e. it’s not consistent to assume both ratex and divergent paths.
Minford and Srinivasan (2012) don’t buy it: if the agents really learned the model they would also know about the divergent paths, and there is nothing to prevent jumping into a different path. They “fix” the problem by adding a fixed inflation target rule that kicks in contingent on being on a divergent path. Since that makes those paths non-divergent, the model is now well determined in the unique convergent equilibrium.
My feeling is that any issue that can be remedied by adding a rule, which by virtue of having been added never has to be invoked, must be a pretty minor issue. It’s a bit like eliminating infinite Ponzi game paths (infinite profit, over infinite time, with infinitely small probability), to make models arbitrage free when the support of the probability space is not finite. It’s a reasonable axiom that makes expectations well defined, but if you really don’t like it you can use a finite horizon model, at the cost of having to add more complicated boundary conditions. Then, if you want, you can take the limit of that model as T->infinity and then the resulting model won’t have the bad paths.
“That’s why, Kevin, I said they must put money back into the model”
Money or FTPL or a contingent different target. But none of those things necessarily change the dynamics of the converging paths. Eggertsson and Woodford (2003) show that under very general conditions, money in the utility function has no impact at all on the NK model dynamics. But I’m guessing you could use it to rule out the diverging paths, just like the FTPL.
Karsten: thanks for your comment. I appreciate it. You are much more up on this literature than I am. I haven’t fully digested it, but a couple of immediate thoughts:
1. I didn’t specify the Phillips curve equation in my post. In principle, it could be anything from totally fixed prices to an equation with inflation inertia, or whatever. I think I’m making a point here, about the level of employment being indeterminate, that is independent of (though parallel to) the question of whether inflation is determinate.
2. Suppose the supply of land is perfectly inelastic. Suppose I compare two models of the demand for land. In model A, it’s a negative function of the price of land, and a positive function of the expected rate of increase in that price. In model B, it’s a function of the expected rate of increase only; the price of land does not appear in the model. Model A has divergent paths, but in some sense those divergent paths are pathological. It determines a price of land. Model B does not determine the price of land. The NK model is like model B — except it’s not just P that is indeterminate, it’s the level of output that’s indeterminate too. It only determines expected Ydot. Standard macro models (like ISLM for example) are like model A.
Not sure if that’s clear.
Shorter version: here’s the NK model:
1. expectedYdot = F(r)
2. Pdot – expectedPdot = G(Y)
You can ignore 2, and Pdot, and still see that Y is indeterminate.
Karsten Howes: “Eggertsson and Woodford (2003) show that under very general conditions, money in the utility function has no impact at all on the NK model dynamics.”
Gali shows that this is true in his model provided the utility function is separable. The thing is, he does have money in his model; just not in the way Nick wants.
Kevin: that’s sort of right. But even if you just threw money into the U function, separable U or not, that still ought to give you a Pigou effect, and that would pin down Y to full employment in the long run. Because in the C=50 case, we know that P would eventually fall towards 0, so M/P would rise to infinity, so an individual would want to consume more than his income from cutting hair.
Maybe: put money in anyhow, like in U, so Y is determinate, then take the limit of this model as the U of money disappears? Y is only indeterminate at the limit, not in the limit?? Is that what Woodford is sorta doing (implicitly or explicitly?
Nick, I’m not sure that long-run Y really is indeterminate in Gali’s model. I suspect he just throws in the “assumption that the effects of nominal rigidities [on Y] vanish asymptotically” because it’s just too difficult to actually prove that they do. (He reminds me of Bertrand Russell’s wisecrack about axioms having the advantages of theft over honest toil.) But the fact that it’s hard to prove just makes it an open question. It’s also very hard to see how an output gap (say for simplicity a positive one) can be sustained forever. With adaptive expectations it’s easy; just inflate faster and faster. Obviously that won’t work with RE. Still, maybe there’s a way if the central bank is really wild?
Well I just emailed Woodford to see what he thinks of all this claptrap. 😀