New Keynesians just assume full employment without even realising it

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?

Here's Gali:

"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.

Comments

[Bill Woolsey]

Consumption is too low this period. The central bank lowers the interest rate to increase consumption this period. The equation used to represent the result of the lower interest rate has many solutions with two extremes. One is the desired effect where consumption rises today relative to an unchanged consumption in the future. The other is that consumption today stays the same, the lower interest rate doesn't have the desired effect at all, and instead it just lowers consumption in the future. You focus on that really bad possibility.

Why would a lower interest rate today lead to less consumption in the future? Only if people borrow more/save less today, and so have to pay debts or have less accumulated wealth for consumption tomorrow.

It seems to me, then, that the really bad result is impossible. The really good result seems to just ignore the adverse impact of the added debt/less wealth on future consumption.

And so, the mixed result are the more realistic ones. So, we consume more now and less in the future.

But in the future, if we consume below potential, the interest rate will be lower still. So, in the future, consumption won't be below potential.

So, all that can happen is that consumption today can rise. This isn't consumption rising above its future level but rising up to its future level.

I think this translates your problem of a low interest rate now and forever (right) just causing people to think that consumption will get lower and lower each period forever, into a situation where instead the interest rate gets lower and lower heading off to negative infinity. (I don't really know why it could need to get below -100%. Saving then is giving a gift and borrowing is accepting a gift. If haircuts are scarce, then there should be no problem with demand.)

With these representative agent models, net wealth, and so assets and debt have to be zero in equilibrium. This means that what debt and wealth mean to the people are only disequilibrium conjectures.

If the sunspot pushes down consumption this period, then the central bank sets an interest rate this period that is too low for that low level of consumption this period. Consumption then isn't too low. And everyone knows this, so the sunspot can't lower consumption. And even if the central bank is too slow to fix it this period, they will eventually.

[JKH]

“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… New Keynesians simply must put money back into the model."

I think I follow this because your explanation is so clear.

There’s nothing in your simplified model that would contradict your conclusion.

Whether its money that goes back in or not - something else EXPLICIT needs to go in in order to ensure a desired level effect beyond the rate of change effect.

[ivansml]

Not quite sure about your example (how can CB set real rate directly?), but in basic NK model as described in Gali, the logic is pretty much the same: any trajectories other than the equilibrium one would explode as t -> infinity, so they are ruled out. And even that requires monetary policy to respond strongly enough to inflation (even a hypothetical, off-equilibrium inflation), otherwise one could get indeterminacy and sunspots. I believe John Cochrane has a paper that criticizes NK models for similar reasons ("Determinacy and Identification with Taylor Rules", JPE 119(3)).

On the other hand, maybe we shouldn't interpret all the mathematical assumptions too literally, and just take NK model as a simplified representation of a particular economic story. In that case, do such mathematical issues really matter?

[Roger Gomis]

Nick, I wish that this topic you keep raising gets more attention because I find it extremely interesting. (In fact I think that a book reviewing all your quarrels with standard economics would be a great tool for graduate students.) I apologize for the unstructured comment I hope that most of it is useful for the topic you are considering.

I argued in your old post that with the threat of the CB implicit in a Taylor rule infinite consumption cannot be an equilibrium solution since the agent will anticipate physical limits (I tried to argue through the budget constraint but you proved me wrong). I am still trying to find a similar argument for avoiding consumption going to zero. But in this comment I want to argue that Galí is right.

I think that if one assumes that the effects of nominal rigidity vanish asymptotically is enough to get full employment, in the NK sense. I'm not sure why they would vanish asymptotically, but if in the long run we have flexible prices the output gap will go to zero, since under flexible prices the output equilibrium is the natural level of output. And its obtained by the intersection of labour supply and demand, the production technology and market clearing(assuming for simplicity Y(t)=N(t):

P(t)=MarkUp*W(t)

Using labour supply to find the real wage:

1=MarkUP*C(t)*f'(N(t))

Where f'(N(t)) is the marginal disutility from work, and using technology Y(t)=N(t) and market clearing Y(t)=C(t)

1=MarkUP*C(t)*f(N(t))

1=MarkUP*Y(t)*f'(Y(t))

And this gives the natural level of output, if the disutility from labour was simply disutility=N(t) (for simplicity), then the output is simply:

1/MarkUP=Y(t)

So if the effects of price rigidity vanish asymptotically we get the natural level of output and zero output gap.

Why would the effects vanish? I'm not sure, but I give a tentative explanation. All we need is that all firms in the limit set the price equal to the markup over marginal cost. If shocks vanish asymptotically (whatever kind of shock) then as time tends to infinity all firms will be able to set its price. Or maybe better explain, assuming vanishing shocks marginal cost in the limit is constant, and since a proportion theta of firms are able to update the price each period all firms (except theta^t proportion) will end up setting the price equivalent to the flexible price equilibrium, and thus the natural output will be recovered.

Of course this is not the case if the economy is regularly hit by new shocks. In that case I think I need the justification of why consumption cannot go to zero (decrease t+1 consumption always instead of increasing t) if the CB credibly promises to low the real rate every period.

I guess that using money in the utility function in a standard NK model, that is just used to find the money supply consistent with the appropriate nominal rate would not satisfy you.

Overall I find this topic really interesting and extremely disturbing at the same time and its related to a model I'm trying to construct. I have the feeling that under flexible prices labour supply and demand imposing market clearing and technology determines output, and the Euler equation just pins down the required real rate, and under sticky prices the reverse is true, but it's a bit confusing since in general equilibrium everything is simultaneous. In any case I'm considering cases in the zero lower bound and rigid prices and it gives this sensation.

[Benoit Essiambre]

It is clear in the mock economy you setup above that full employment would be reached quickly. I mean with no investment required (everybody already has a pair of scissors), people would simply meet in pair every week and barter a hair cut (you cut mine, I cut yours).

I guess a good model should take this activity into account and give us a prediction of how fast it would happen and the magnitude of the effect compared to other variables. The question becomes: Can we salvage the model? Can we modify it to add this aspect to it?

[primedprimate]

I am probably making some sort of fundamental error but I'd like to echo a commenter from your previous post and Roger Gomis: perhaps including disutility from labor (and its associated period by period, intra-temporal FOC) could resolve this issue.

In terms of Econ 101 type graphs, you have convinced me that AE coincides with the 45 degree line. However, Y (and therefore C) is pinned down (over the long run) by the interaction of labor demand and labor supply (production function and consumption-leisure decision). If you ignore AE in this scenario, you have the classical model and perhaps analogously, in the NK case, ignore nominal rigidities and you have RBC.

[JW Mason]

On the other hand, maybe we shouldn't interpret all the mathematical assumptions too literally, and just take NK model as a simplified representation of a particular economic story.

On one level this is right. These models were developed in order to retain something like an IS curve to justify countercyclical monetary policy, but within a framework of intertemporal optimization by rational agents. So in some sense the result of these models is all that matters and the particular way it's arrived at is not so important. New Keynesians already know that lower interest rates raise current output, they don't need Woodford or whoever to prove it.

But I don't think you can handwave away the logical problems with these models on those grounds. If all you want is Y_t = Y(r_t), Y' < 0, then just write that; getting there via an Euler equation is a complification, not a simplification.

[Kevin Donoghue]

Nick: "New Keynesians simply must put money back into the model."

I was with you until then. Honestly if you'd finished with "Delenda est Carthago" it would make as much sense to me. Why should the absence of money be picked on as the crucial flaw in the model? To me the real problem is that RE models in general, not just NK, are apt to have multiple equilibria. Time and again we see some implausible assumption being adopted for lack of a better way to make things determinate.

Is it so hard to admit that RE is a bit of a mess, whether it's Lucas or Woodford using it?

[JW Mason]

perhaps including disutility from labor (and its associated period by period, intra-temporal FOC) could resolve this issue.

I'd like to hear Nick's response to this but here's my sense of why it doesn't help. You are thinking that in the underemployment equilibrium, the marginal disutility of labor is lower than at the full employment equilibrium. So people should want to work more, i.e. there should be excess demand for haircuts and excess supply of labor. But that's only looking at one side of the market. Remember, the price of a haircut is just the disutility of the labor that produces it. So, yes, the marginal disutility of labor is lower in the underemployment equilibrium, but so is its marginal product.

Put it another way: Suppose I believe that my permanent income (in haircuts) is one haircut per week. Then I choose a consumption path of one haircut per week. And if everyone else does the same, then my lifetime income will in fact be one haircut per week. So there's no reason for me to change my behavior. (And there is no question of a price adjustment -- a haircut can't sell for a price other than one haircut.) It may very well be true that, given the utility of haircuts and the disutility of the labor to produce them, everyone would be better off giving and receiving two haircuts per week. But there is no way for the choices of individual rational agents to get us from the one-haircut to the two-haircut equilibrium.

[David Andolfatto]

Nick:

You are forgetting about the condition that determines employment in the model. What you say is true when the real interest rate and the real wage is set exogenously. But so what? These are not market-clearing prices. There are a lot of allocations that satisfy your Euler equation that do not satisfy market-clearing. Is this what you have claimed to discover?

David Andolfatto

[Nick Rowe]

Benoit has it exactly right: "It is clear in the mock economy you setup above that full employment would be reached quickly. I mean with no investment required (everybody already has a pair of scissors), people would simply meet in pair every week and barter a hair cut (you cut mine, I cut yours)."

If barter were possible, unemployment would be impossible, *even if the central bank set a really stupid r(t)*. The unemployed would just get together and barter their services. NK macroeconomists want to have unemployment when the CB sets r(t) wrong, and full employment when the CB sets r(t) right. NK models only make sense as a model of a monetary exchange economy. But they don't have money in the model. That's why, Kevin, I said they must put money back into the model. If there were a stock of money, then some sort of real balance effect would be included in individual agents' transversality condition, and unemployment and deflation would lead to agents holding too much real money, and they would plan to spend more than their expected incomes.

primed: if we allowed barter exchange, then we would have an RBC model. Equilibrium where the marginal disutility of labour = the marginal utility of a haircut. But in that model, even a stupid law that set r(t) too high ( a minimum interest rate law) couldn't prevent full employment.

Bill: "Why would a lower interest rate today lead to less consumption in the future? Only if people borrow more/save less today, and so have to pay debts or have less accumulated wealth for consumption tomorrow."

Remember, this is a model with identical agents. If they all have a declining path for C(t), they all have a declining path for Y(t) too. There is never any borrowing or lending along any equilibrium time-path.

Roger: "I think that if one assumes that the effects of nominal rigidity vanish asymptotically is enough to get full employment, in the NK sense. I'm not sure why they would vanish asymptotically, but if in the long run we have flexible prices the output gap will go to zero, since under flexible prices the output equilibrium is the natural level of output."

But if (say) you had a model where the AD curve were vertical, then no amount of price flexibility can get you to full employment. You need a downward-sloping AD curve (that cuts the LRAS curve) plus price flexibility, to get you to full employment. The standard NK model has a vertical AD curve, except it's a very thick AD curve. Contrast to the standard ISLM, where you get a downward-sloping AD curve in the usual case. Falling P means rising M/P with means a movement down along the AD curve to the right. But the NK model does not include M/P.

[Nick Rowe]

David: these are self-employed hairdressers. The production function is one haircut per hour. P and W are the very same thing. The supply of labour and the supply of haircuts are the same thing. The demand for labour and the demand for haircuts are the same thing. When there is unemployment, this means they cannot sell as many haircuts (= cannot sell as much labour) as they want to sell.

JW: my response is basically the same as yours. Except that the marginal product of labour is *always* one haircut per hour. But if you can't actually *sell* that extra haircut (for money) the marginal product is irrelevant. You can't cut hair if there isn't a customer sitting in the chair. And even if you could, you wouldn't gain by producing an extra haircut if you couldn't sell to anyone. In an underemployment equilibrium, every hairdresser would like to both sell and buy an extra haircut. But none will buy unless he can sell, and none can sell unless someone else buys.

[JW Mason]

Nick's last comment is very clarifying. (New) Keynesians believe both that the economy will not have full employment without appropriate monetary policy, and that it will reliably reach full employment with appropriate monetary policy. The challenge is to write down a model that has both those properties. (Or, really, to write a model that captures the most important facts about the world that give the economy both those properties.) The goal is a story in which monetary policy to be both necessary and sufficient. You don't solve the problem by telling a story in which resources are fully utilized without any need for a central bank.

Another, somewhat different way of looking at this same problem is, if markets in general set prices optimally, why does this one price, the interest rate, have to be set by a central planer?

One answer, which Nick obviously likes, is that money is special: It can only be produced by the government, and there is no substitutability between money and any privately produced goods. Of course that's not the only possible answer.

[Edited to fix typo NR]

[JW Mason]

Oops!

That sentence should be: "(New) Keynesians believe both that the economy will NOT have full employment without appropriate monetary policy..."

[Fixed. NR]

[Alex Godofsky]

JW:

I'd like to hear Nick's response to this but here's my sense of why it doesn't help. You are thinking that in the underemployment equilibrium, the marginal disutility of labor is lower than at the full employment equilibrium. So people should want to work more, i.e. there should be excess demand for haircuts and excess supply of labor. But that's only looking at one side of the market. Remember, the price of a haircut is just the disutility of the labor that produces it. So, yes, the marginal disutility of labor is lower in the underemployment equilibrium, but so is its marginal product.

(emphasis mine)

No, because per assumption the market for labor isn't clearing. If you reduce the number of haircuts done by half, it would be really really weird for the marginal product of an additional haircut to go down! In Nick's recessionary economy MP > MC and Qd < Qs.

[Nick Rowe]

JW: "(New) Keynesians believe both that the economy will not have full employment without appropriate monetary policy, and that it will reliably reach full employment with appropriate monetary policy. The challenge is to write down a model that has both those properties. (Or, really, to write a model that captures the most important facts about the world that give the economy both those properties.) The goal is a story in which [appropriate] monetary policy [is] both necessary and sufficient. You don't solve the problem by telling a story in which resources are fully utilized without any need for a [good] central bank [doing the right thing]."

Yes. Clearly stated.

[Kevin Donoghue]

Nick: "If there were a stock of money, then some sort of real balance effect would be included in individual agents' transversality condition...."

I'm out of my depth here but I think you've got too fond of this real-balance idea altogether. Unless I've got things wildly wrong (quite possible) simply adding a stock of money won't get you a real balance effect. You'll need a growing population, or OLG, or something like that. With a constant immortal population and separable utility there's no Pigou effect. At least that's what I gather from reading Benassy but I'm far too lazy to work through the proofs.

[Nick Rowe]

Kevin: the Pigou effect is a subset of the real balance effect. I haven't read Benassy (whose work I think highly of, as you know) on the Pigou effect, but I disagree with what you say he says on this.

[Frank Restly]

Nick,

Is the point you are making that New Keynesians have nominal interest rates in their models but no debt or debt level in them?

Curious, in your example, what action does a central bank perform to hit its desired "real" interest rate for haircuts?

Does it have a bunch of wigged mannequins to employ unemployed hair cutters - central bank creates demand for haircuts?

Does it have a robot that can give haircuts - central bank creates supply of haircutters?

I am having trouble following the metaphor all the way through.

[primedprimate]

Perhaps the argument being made by NK is that in the long run (so to speak) the economy resembles the barter economy assumed by RBC. In fact it could be argued (I am not saying NK argues this) that from the 'long arc of history' standpoint, even a monetary economy is a 'nominal rigidity'.

[Determinant]

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.

To be charitable, this is a modelling artifact. The modeller *should* have talked to the First Year Calculus Profs who would have clearly explained the difference between a derivative f'(x) and its underlying function F(x)+C. Derivation is not a linear operation, you lose information and the inverse operation, integration, produces a series of answers with C as a constant parameter.

What you are on to Nick is that C, the "anchor point" is completely absent in New Keynesian models whereas Old Keynesian models had them. You are trying to reinsert it through the use of money. Fair enough, it's a perfectly valid approach. But this whole argument also illustrates why mathematical rigour is good. Rigour ensures that we don't overlook anything or generate red herring answers that occur through imprecise model specifications.

[Nick Rowe]

primed: that would justify the assumption of full employment in the limit. Not sure though whether it makes even logical sense. If the economy reverts to barter at time T, how does the central bank set r(t) at time T-1? Dunno.

Frank: "Is the point you are making that New Keynesians have nominal interest rates in their models but no debt or debt level in them?"

No. It's that there is no stock of money.

"Curious, in your example, what action does a central bank perform to hit its desired "real" interest rate for haircuts?"

Ask the NKs that question. Implicitly, the CB borrows and lends money to any agent who wants to borrow or lend, at an interest rate r(t) (plus inflation). But there is no money ever borrowed or lent in equilibrium, because C(t)=Y(t).

[notsneaky]

I had another comment on this on the other post but I think it got lost in spam.

Here's a counter example, but not a very good one.

Suppose the CB's "monetary rule" is that each period they choose the interest rate to guarantee full employment. In other words they pick r(t)=((1+n)C(t+1)/100)-1. That's not "n" exactly... but it will be. That C(t+1) actually has an expectations operator on it. If the CB's policy is credible then the only rational expectation (or perfect foresight) is to believe that E(C(t+1))=100 as well which simplifies the rule to r(t)=n. In this case CB's policy acts as a coordinating mechanism for agents expectations.

So in this case you got an equilibrating mechanism. In fact you got no dynamics, it just jumps to full employment always (given absence of shocks etc). But it's not a very good counter example because the NK models don't usually have CBs in them which care only about full employment and don't give a fig about inflation. (Also, in this model why stop at 100? Since neither the CB nor the agents care about inflation, why not go for ... 101?)

Now consider the diametrically opposite case, where the CB only cares about inflation (inflation nutters). Consider... consider... consider... yup, there's absolutely no reason for why agents would expect C(t+1)=100 or any kind of other return to equilibrium. If you got a NK model with such a policy rule that returns to full equilibrium after a shock, then yeah there's probably some kind of cheating going on.

The intermediate case is where the CB's reaction is some kind of weighted average of inflation and employment. Remember that the Euler is just one equation in a 3 equation framework.

[Lord]

In the long run we reach full employment because our definition of full employment accommodates to our position and our experience becomes our reality.

[Sjysnyc]

I'm moving my (slightly modified) comment from the other thread over since the discussion has moved over here. Hopefully that's okay.

In that prior discussion I had an aside about the issue of labor markets in the NK model wasn't particularly useful -- indeed it was a distraction. I was expressing my skepticism about the usefulness of the NK/DSGE blend in general, and specifically about their labor market modeling assumptions as being the worst part.

But that's not particularly relevant to the discussion at hand. Which I take as to whether there is a meaningful trend/equilibrium/full employment that the model comes back to because of equilibriating forces within the model. In the terms you were asking: is consumption halving for everyone an equilibrium?

There I think the answer is no. There isn't just a optimization over consumption but over labor supply -- and as you note in your response to me the model assumes that the agent can work as much as he wants at the prevailing wage. And in the case where you halve consumption the agent would want to work substantially more. There is lurking somewhere in the depths of the model a first order condition with respect to labor supply as well. With a diminishing returns to labor production function around too. It's just when reducing it to the 3 variable system that gets swept under the table.

Or think about it in terms of the individual agents. Suppose one halves her consumption and expects everyone else to do the same. Then, at the prevailing wage she has an incentive to increase her hours of work. And by the assumptions of the model she can do so without affecting wages. She can consume more, but because she is also working more will not violate the transversality condition. But then, so does everyone else. And that gets you on the way back to an equilibrium with what they call full employment in the model.

With respect to your response to David above, the production function isn't one hair cut per hour. Instead it is F(n) = A_t * n^alpha (eq 5 in the Gali paper linked to earlier). There are diminishing returns to labor on the aggregate level and that matters.

[Nick Rowe]

Determinant: I think that's what I'm saying. (You use "C" to represent the constant of integration, presumably, rather than consumption.)

But normally economists are well aware of this problem. Somehow it slipped past them here, I think. Maybe due to confusing the individual agent with the economy as a whole, in a representative-agent model? Or more likely, from mistakenly applying an assumption which would make sense in the Old Keynesian ISLM model (where the effects of nominal rigidities really do disappear and ensure full employment in the long run, at least under certain assumptions) to a New Keynesian model which does not have a well-defined downward-sloping AD curve.

[Nick Rowe]

notsneaky: I had a look through the spam filter, and think I remember fishing out one of yours very early this morning. Nothing in there now.

I'm OK with assuming (for simplicity) the CB just targets full employment in this case. Given perfect information on shocks, that seems OK (let's ignore that, strictly speaking, that makes inflation indeterminate).

But I'm not sure I'm getting how the CB's setting r(t) in the way you suggest acts as a coordination mechanism. Wouldn't sacrificing a goat, or just using cheaptalk a la Schelling focal points, work even better than messing around with r(t)? Remember that C(t) can jump. And we normally assume the CB sets r(t) an instant before agents choose C(t), because agents choose C(t) after observing r(t).

Sjysync: no worries about moving your comments over here.

"With respect to your response to David above, the production function isn't one hair cut per hour. Instead it is F(n) = A_t * n^alpha (eq 5 in the Gali paper linked to earlier). There are diminishing returns to labor on the aggregate level and that matters."

It is in my model. I've simplified. But as long as we have non-decreasing marginal disutility of labour we still get a well-defined "full-employment" level of C(t). (And strictly, my Consumption-Euler equation assumes separability in C and L, I think).

" There isn't just a optimization over consumption but over labor supply -- and as you note in your response to me the model assumes that the agent can work as much as he wants at the prevailing wage. And in the case where you halve consumption the agent would want to work substantially more. There is lurking somewhere in the depths of the model a first order condition with respect to labor supply as well."

In my underemployment equilibrium, where C(t)=50, it is indeed true that the self-employed hairdressers want to sell more labour (i.e. sell more haircuts). But they can't, because nobody will buy any more than 50 haircuts. So yes, they are "off" their labour supply curves (i.e. "off" their output supply curves). That FOC is not satisfied. But the individual agent can't do anything about it. He is sales-constrained.

(In the background, of course, hairdressers are monopolistically competitive, and an individual hairdresser will cut his price (cut his wage) when the Calvo fairy touches him with her wand. But that makes no difference to my argument here. The speed or slowness of the Calvo fairy simply determines how quickly deflation will set in if the economy is in an underemployment equilibrium).

[Nick Rowe]

Damn it guys, everybody understood this stuff about the constrained labour demand curve (when firms were sales-constrained) back in the olden days of the 1970's, when Patinkin and Benassy and Clower ruled the roost. You young 'uns have so much to re-learn! When firms are sales-constrained, the labour demand curve is NOT the VMPL curve (or MRMPL curve under monopolistic competition). See Barro and Grossman 1971.

[Nick Rowe]

Lord: I can imagine a world in which that is true. But that's not the world of the NK model.

[Sjysnyc]

I think – though am not sure – that you’re right about the diminishing returns to labor not mattering. I’d have to crank through the algebra to be sure. But to be fair you’re simplifying away from their model.

But here is the rub: “In my underemployment equilibrium, where C(t)=50, it is indeed true that the self-employed hairdressers want to sell more labour (i.e. sell more haircuts). But they can't, because nobody will buy any more than 50 haircuts. So yes, they are "off" their labour supply curves (i.e. "off" their output supply curves). That FOC is not satisfied. But the individual agent can't do anything about it. He is sales-constrained.”

In the NK model that isn’t the way it works. There is a perfectly competitive market for labor among all the differentiated good producing firms and the workers. That market clears (by assumption) in the model and the wage is the same across all the differentiated product producers. In your example, though not in all NK models, wages are perfectly flexible every period. So any worker who drops there wage has infinite demand for labor. Each worker is infinitesimally small relative to the economy as a whole, so they have no effect on overall wages or labor supply.

You note “an individual hairdresser will cut his price (cut his wage) when the Calvo fairy touches him with her wand. But that makes no difference to my argument here.” But that isn’t the case. Price and wage are different in the model. Prices are what the individual firms charge while wages are what workers are paid.

In your scenario there is a _permanent_ difference between the marginal utility of consumption and the marginal disutility of labor. You’re implicitly not allowing the wage to adjust to clear that market. Each of the workers is happy to lower their wage and work more. Each of the firms would be happy to hire at a lower wage and increase their production (since production isn’t fixed, just the price of output). Instead you’re saying that doesn’t happen.

What you want is a different model. One where those adjustments cannot occur for some reason. Which is fine. But that doesn’t mean that under its own terms the NK model doesn’t have a equilibrating force. Half the consumption simply isn’t an equilibrium because every agent has an incentive to push away from it.

[notsneaky]

This conversation reminded me of Robert Hall's famous "consumption is a random walk" result. It's sort of similar isn't it? How does a CB stabilize that?

And speaking of Hall, this also sort of makes some sense of Paul Krugman's reporting that "Hall used to be famous at MIT for talks along the lines of “Not many people understand this, but the IS curve actually slopes up" " Hmmm. That seemed a bit cryptic but if you got that C(t+1) term in there, it does slope sort of slope upward, doesn't it?

Also, if we assume perfect foresight (except for a time zero shock which makes sure we don't start at full employment), can we just invert that Euler equation and write c(t+1)=c(t)+(r(t)-n) or is there something wrong with that?

[Frank Restly]

Nick,

"But there is no money ever borrowed or lent in equilibrium, because C(t)=Y(t)."

Incorrect. There is no net money borrowed or lent when C(t) = Y(t). Meaning there is no central bank that can push money into a system and pull money out of a system. You and I can lend to each other and still realize C(t) = Y(t). I borrow $10,000 from you at %5, you borrow $10,000 from me at 5%. Net debt between the two of us is $0 at 0% and so C(t) can equal Y(t) at every time t even though there is $20,000 of total debt outstanding.

[Andy Harless]

I think Siysnyc is right that typical NK models do have an auto-correct feature if you take them on their own terms. Nick's model simplifies away this feature by not separating labor and product markets. Of course it's noteworthy that the feature depends on an assumption (labor market clearing) that is extremely unrealistic in a way that undercuts our main motivation for doing macroeconomics in the first place. I mean, I mostly don't give two hoots (maybe I give one) that the economy isn't producing quite as much as it ideally could: if you care about output, then you want to do growth economics, not Keynesian macro. What mostly bothers me (and presumably most Keynesians and other demand-siders) about recessions is that a bunch of people are out of work. But in NK models this isn't even the case.

[dlr]

So Andy, does this auto-correct feature still exist in a NK macro model that does something a little weird (labor market power) to induce unemployment in a representative HH NK model? Like this from Gali, Smets, Wouters:

http://www.crei.cat/people/gali/gsw2011ma.pdf

[Nick Rowe]

Sjysnc and Andy: I strongly disagree. I will now modify my simple model to add a labour market, and show it changes nothing.

Assume each agent owns a salon, but is not allowed to work in his own salon. Each agent hires another agent to work in his salon, and works himself in another's salon.

In a barter economy , with W/P on the vertical axis, and L on the horizontal, the aggregate labour demand curve would be horizontal at 1 haircut per hour (because the MPL=1). With monopolistic competition, the aggregate labour demand curve would be horizontal at (1-1/e), where e is the elasticity of an individual salon's demand curve. And the aggregate labour supply curve would be upward-sloping, with a height equal to the (increasing) marginal disutility of labour. "Full employment" equilibrium L (full employment equilibrium C) is defined where those labour supply and demand curves cross. Suppose it's at 100 haircuts per year per salon (=100 haircuts per year per worker. And the full employment equilibrium W/P is (1-1/e).

In the underemployment equilibrium, the aggregate labour demand curve is horizontal at (1-1/e), ***until it hits 50 haircuts per year per worker***, and then it turns vertical and drops straight down. It's reverse-L-shaped. That's because the quantity of haircuts demanded is 50 haircuts per salon, so hiring 51 hours of labour cannot be profit-maximising, since that 51st hour of labour will be wasted, since there is no 51st customer. This is what Patinkin/Clower/Barro-Grossman called the "constrained labour demand curve", as opposed to the "notional labour demand curve" that we get if we ignore firms' sales constraint.

Assuming perfectly flexible nominal wages, (but sticky prices a la Calvo or whoever) the underemployment equilibrium real wage W/P is determined by the point where the labour supply curve crosses the vertical portion of the constrained labour demand curve. It will be strictly less than (1-1/e).

Yes, workers are "on" their labour supply curves. In that (rather stupid) sense, there is no "involuntary unemployment" in this model. But employment and real wages are lower than at full employment. And firm's are "on" their labour demand curves too. BUT THE CONSTRAINED LABOUR DEMAND CURVE IS VERY DIFFERENT FROM THE NOTIONAL LABOUR DEMAND CURVE.

Comparing this new version of my model to the original version: in underemployment equilibrium income is still 50 per agent. The income can now be decomposed into wage income plus profit income, but it still adds up to 50, and each agent is still working the same number of hours in both versions. In the new version, each individual agent *as worker* can work as many hours as he chooses. He chooses 50, because W/P is so low. But each agent as salon-owner would love to hire more labour and sell more haircuts, but cannot, because nobody wants to buy more.

Adding a labour market to the model, and assuming W is perfectly flexible, changes nothing at all. Barro Grossman showed this back in 1971.

[HJC]

Another excellent post Nick! This is something I've been wondering about as well. Do you know of any academic papers highlighting this? Also, how do you think it relates (if even slightly) to John Cochrane's criticisms of the NK models regarding their unrealistic assumption that they jump to a new equilibrium to ensure that inflation is determinate?

[Nick Rowe]

Sjysnc and Andy: BTW: the underemployment equilibrium I have described above is *exactly* what is going on in any NK model, if the CB sets r(t) too high (except I assumed constant MPL for simplicity and they normally assume diminishing MPL). The difference is that *I* am saying we can get that underemployment equilibrium *even if* the CB sets r(t) just right.

[Determinant]

Yep, Nick, "C" in my first post is a constant of integration, not Consumption. Us engineery types have our own jargon!

Anyway, the rest of this thread just seems to be talking in circles about why something isn't there that *should* be there, and everyone assumes is there. Blast it people, just change the model a bit so it has the parameter back in that you need!

Fudge factor, margin of safety, whatever. I don't care if you claim that all unemployed people wear blue hats, just make it explicit in the model, or else you will get lost!

C'mon, everyone back to First-Year Calc now.

[Nick Rowe]

notsneaky: it is sort of similar to hall's random walk consumption model. But only sort of. An individual agent who chooses C(t) where r(t)=n, but with random shocks to permanent income, will have C(t) follow a random walk. But permanent income is endogenous in this macro model.

(I have my own posts here on why the IS curve slopes upwards, BTW!)

"Also, if we assume perfect foresight (except for a time zero shock which makes sure we don't start at full employment), can we just invert that Euler equation and write c(t+1)=c(t)+(r(t)-n) or is there something wrong with that?"

Not sure. I think that means ruling out attacks of animal spirits, by assumption.

Frank: " You and I can lend to each other and still realize C(t) = Y(t)."

As I said explicitly in the post: All agents are identical. If I want to lend to you, you will want to lend to me, so you won't want to borrow from me. Therefore no loans in equilibrium.

HJC: thanks! I'm afraid I don't know of any papers on this subject.

"Also, how do you think it relates (if even slightly) to John Cochrane's criticisms of the NK models regarding their unrealistic assumption that they jump to a new equilibrium to ensure that inflation is determinate?"

I don't remember reading John Cochrane on that. It *might* be related. I don't know.

[HJC]

Nick: No problems, here is a link to a Cochrane paper if you are interested.

[notsneaky]

Not sure. I think that means ruling out attacks of animal spirits, by assumption.

That's why I make the exception for the zero-time time shock. If we're gonna analyze the model seriously and consider whether it's equilibratin' or not we have to start off away from full employment, inflation=target, and see whether it comes back or not. But we don't want to get bogged down in thinking about expectation formations and all that stuff. So let's assume that in time zero there's an unanticipated shock (either to output gap or inflation) but after that both consumers and the CB have perfect foresight, just for simplicity. Now, "assume perfect foresight" doesn't exactly jive with "there is an unanticipated shock" but for the purposes of taking the model apart, looking at all the axles, belts, rotors, and screws that make it up, I think it's a reasonable way of looking at it.

So. If I do get to invert the Euler equation we have (in logs, deviations from "full employment", all that)

c(t+1)=c(t)+(r(t)-n) ---- the upward sloping IS curve (except for initial period)

pi(t)=pi(t-1)+k*c(t) --- Philips curve

r(t)-n=f(whatever) --- monetary rule. Since CB has perfect foresight "whatever" means that you can pick any lags or leads you want to put in there.

This is actually the Ramsey model way of looking at it with the difference that r(t) is not determined by MPK but by a CB. And at that point it's simply a question of whether this system of difference equations has a stable steady state where c=0.

So here we are not assuming that everyone believes that the economy always returns to full employment, we're just asking what kind of f(whatever) function will get it back there and is consistent with such beliefs. Here's the weird thing. Because this system DOES have an upward sloping IS curve - the growth rate of consumption depends positively on r - that flips all the usual conclusions on their head. Iterate and work through the algebra. The CB should pick LOWER interest rates when inflation is high. It should pick LOWER interest rates when output is above equilibrium (to bring back down).

This is weirding me out somewhat.

[Frank Restly]

Nick,

"As I said explicitly in the post: All agents are identical. If I want to lend to you, you will want to lend to me, so you won't want to borrow from me. Therefore no loans in equilibrium."

Yes loans in equilibrium. You and I are both hair cutters and hair cut recipients. I want to ensure that I always have someone to cut my hair and you want to do the same. I do this by borrowing $10,000 from you and buying $10,000 in haircuts from you in advance. You do the same, borrowing $10,000 from me and buying $10,000 in hair cuts from me in advance.

We owe each other $10,000 and we owe each other $10,000 in future hair cuts.

You are making an assumption that borrowed money is always spent on goods delivered when the borrowed money is spent - the impatient borrower. Borrowed money can also secure goods that cannot be delivered until some time in the future.

[notsneaky]

I'm OK with assuming (for simplicity) the CB just targets full employment in this case. Given perfect information on shocks, that seems OK

Ok, but then I think that if you agree to this, then you're giving away the game. If there's one monetary policy which can lead back to full employment then there can be others. It's just a question of specifying an appropriate monetary rule policy (which is something like, choose appropriate initial r(t) then follow some monetary rule afterwards). Once you got that, and it's rational expectations (or perfect foresight, except for a time zero shock) all around, you got your equilibrating mechanism and a in-model justification for "consumers assume that output gets back to full-employment"

(let's ignore that, strictly speaking, that makes inflation indeterminate).

Yes, but that's just because the counter example was purposefully silly. Get a Philips curve and a different monetary rule then we can bring inflation back into it (I think).

But I'm not sure I'm getting how the CB's setting r(t) in the way you suggest acts as a coordination mechanism. Wouldn't sacrificing a goat, or just using cheaptalk a la Schelling focal points, work even better than messing around with r(t)?

It's the definition of equilibrium. Choices of CB and households have to be mutually consistent. Suppose folks insist on believing in 50 haircuts. Then the CB sets the interest rate at (1/2)*(1+n)-1 until the stupid consumers realize that expecting 50 haircuts for ever is just wrong. It's an-off equilibrium path so it never happens given that this model has (and has to have) rational expectations.

Maybe sacrificing goats would work too.

Remember that C(t) can jump. And we normally assume the CB sets r(t) an instant before agents choose C(t), because agents choose C(t) after observing r(t).

I'm not sure why this matters.

Let's try a phase diagram. You got your perfect foresight Euler (logs, deviations from full employment):

c(t+1)=c(t)+(r(t)-n))

That's just the Ramsey equation with r on the x-axis instead of capital. If r is less than n, c is falling. For dc/dt=0, r=n. Now we need a dr/dt=0 nulcline. Say the monetary rule is

r(t)=r(t-1)+b*c(t)

dr/dt=0 is the x-axis, full employment. There's at least one stable path.

If there's a period zero shock then the CB just needs to choose an initial r to get on that path and then follow the rule above. There is a weird implication here. If, say, the period zero shock is c(0)>0 (the shock just happened at the very end of the period and in this one instance only it was not perfectly foresighted), the CB needs to pick an initial r less than n in order to get on the declining c stable path. In other words, it needs to cut interest rates when output is above full employment just so it can raise them later. It's "discretion when shock happens, rule based monetary policy afterwards". That's the big "C", the constant of integration, you and Determinant are talking about, the initial condition which pins down the solution to the difference equations. Of course this is not the policy recommendation usually made or the implication that is drawn from these models.

[Nick Rowe]

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).

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."

Since there exists a physical upper limit on C, (and a limit less than that because even monopolistically competitive firms with sticky prices will ration haircuts if C(t) exceeds competitive equilibrium) this rules out any perfect foresight equilibrium path except C=100. Because r=2n forever means C(t) is growing forever, which can't happen.

(Actually, that's not perfectly true, since C(t)=0 for all t is also an equilibrium path for any r(t).)

Is my "proposed" monetary policy rule any different, (in the way it works) from yours?

[Nick Rowe]

Frank: People in this model would not want to buy haircut insurance from each other. They don't need it. People are identical, and always willing to sell more haircuts in equilibrium.

Stop throwing out red herrings. Stop trolling. This post is about NK models. If you don't have a clue about NK models, and what they assume and do not assume, then stop trying to change the subject.

[Frank Restly]

Nick,

"Then every agent has a bad case of animal spirits."

If it is possible that every agent can get a bad case of animal spirits and every agent knows that this is possible then every agent may want to buy haircut insurance.

[Sjysnyc]

You posit a hard cutoff of demand at 50 and no matter what you can’t get demand beyond that. And are stripping down the implicit model by only looking at one good. Again, this is a different model than the NK one. Whether it is a better model or not is a separate question. But it isn’t how the NK model works.

In the NK, there is the composite good made up of all the different monopolistically competitive firms goods. Each of those firms hires the non-differentiated labor in the market, and it’s all at the same wage. And if the prices of one of the intermediate goods is lower, then more of that is bought and goes into the composite good.

So suppose you have that initial drop in expectations down to producing 50. The Calvo pricing assumption in the model is that a fraction theta of the intermediate goods producers lower their prices substantially and increase their output. The model has smooth demand curves for all firms by the technology of the model—you can’t get the sharp drop off in demand you posit.

These intermediate goods producers who adjust then hire a large amount of labor in the spot market at the new lower wage but that still means that overall labor supply rises relative to the just 50 expectation. And all wages in the economy sink to that new lower level because the model has the labor market clearing. The marginal utility of consumption and marginal disutility of labor are equalized. Those firms that can’t adjust their prices see their demand cut back. Not sure of the sign of the change in profits for those firms since you have lower output, but costs of their only input (labor) has also declined, but those profits continue to be sent back to all the households in the economy since they all own equal shares of each company, so the households have income from that source as well.

But in the model, as stated and not saying the model is a good approximation of reality, there are forces that push back to what it calls the full employment equilibrium.

Now does this labor market make any sense? Nope, none at all. That was what I alluded to in my post on the other thread. There is no involuntary unemployment and attempted fixes such lotteries across agents are even worse. And as Andy notes above this makes the NK model not a great guide to think about what’s happening in the economy during a recession. And is one of the places where I have the major issues with the model. But the ability of firms to sell more output at a low enough price isn’t where I think the NK really falls down.

[Nick Rowe]

Sjysnyc: Ah, you may not realise this, but you are arguing with the guy who invented macro with monopolistically competitive firms, way back in 1987 before it was cool! I exaggerate, of course, but I did beat Blanchard and Kiyotaki to publication by a month or so, IIRC, (but nobody read my paper, possibly because theirs was better). I invented NK macro! (Well, not really.) But I did have monop comp in the back of my mind all the way through this post.

It doesn't make any difference (except the composition of aggregate demand between salons becomes indeterminate in the limit as we approach perfect competition).

Assume each salon has the the exclusive right to a particular style, and people have a taste for varity of styles, so each salon faces a downward-sloping demand curve, as a function of aggregate demand per salon, and the real price:

Ci = C.(Pi/P)^-e

Assume for simplicity the CB had targeted zero inflation since the beginning of time, so all salons have the same price when the animal spirits attack and C drops to 50. Each salon would like to cut its nominal price Pi, in order to cut its real price Pi/P, and move down along its demand curve to the point where MR=MC. A small fraction of the salons (those touched by the Calvo fairy) will do just that.

But if that fraction of the salons reduce their real price, all that means is that the remaining salons' real price is increased. The average real price across all salons is one, by definition. All this does is change the *distribution* of demand between firms. Price cuts are a "zero-sum" game.

Now, if we situated these salons in a normal macro model, with aggregate demand determined by (say) ISLM, or MV=PY, so you get a downward-sloping AD curve, the fall in the general price level P would also move the average salon down along that AD curve to the right, increasing C. So total consumption and employment (and W/P) would rise. But the NK model simply lacks that feature. P(t) does not appear in the model. Only P(t)/P(t+1) appears in the model, and it only influences demand via the gap between real and nominal interest rates.

(And if we take the limit as e approaches infinity, we get perfectly competitive salons.)

[Akshay]

Hello! (first comment from a long-time lurker)
Very insightful, as always. Now I do not have an expertise in NK models. If in your model, we add two (not unreasonable) assumptions:

1. Agents are not memoryless
2. At least one agent believes that the sunspot-shock is temporary

Then wouldn't the economy return to its equilibrium output without any need for change in nominal agregates (or even without money)? I'm not sure if NK-models state these assumptions explicitly (or if including them violates the canonical NK model).

[Sjysnyc]

I thought we weren’t supposed to use argument from authority anymore? Then wouldn’t we just say Mike Woodford is really smart and has lots of published papers that use NK models so they must be right.

The monopolistic competition does make a difference—or at least so it seems to me—as now you’re bringing in the very adjustment mechanisms that get you back to what counts as full employment in the NK model: allowing labor supply to increase and prices to adjust.

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.

At the same time, wages go down not only at those firms but at all firms. [In the background workers are getting not just wages but also the profits from the firms who have cut prices.] And for all the identical workers in the economy they are back on the equilibration of marginal utility of consumption and marginal disutility of labor. You’ve got the mechanism that allows labor supply to increase through the extra hiring—the other equilibrium condition I was talking about earlier. In contrast to your 9/11 648 post there _is_ an incentive to hire that extra labor now.

Or are you still saying that despite drop in price there can be no more than 50 total haircuts in the economy? Or that the extra haircuts just happen to be exactly canceled out by less at others? Despite the fact that nominal wages have now dropped substantially across the economy and you no longer have a difference between the marginal utility of consumption and the marginal disutility of labor?

And then if you think of dynamics that happens again next period as output goes up again as more firms have the ability to adjust their prices. So some further fraction now are producing more, and so it goes on each period with production slowly inching back up, and hence pushing employment back toward the NK model’s concept of full employment. [Well, kind of. This is all a little off as there isn’t any actual way for the consumption to fall from animal spirits in the model, but it does outline the equilibrium forces that push employment back to the model’s concept of full employment.]

[notsneaky]

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+a*pi(t), the PC is pi(t)=pi(t-1)+k*c(t). So r(t)-r(t-1)=(ak)*c(t)=b*c(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+v*c(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+b*c(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.

[BSF]

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

[Nick Rowe]

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).

[Ritwik]

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.

[Nick Rowe]

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.

[Frank Restly]

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).

[Karsten Howes]

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 href="http://faculty.chicagobooth.edu/john.cochrane/research/papers/cochrane_taylor_rule_JPE_660817.pdf">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.

[Nick Rowe]

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.

[Nick Rowe]

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.

[Kevin Donoghue]

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.

[Nick Rowe]

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.

[Nick Rowe]

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?

[Kevin Donoghue]

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?

[Tom Brown]

Well I just emailed Woodford to see what he thinks of all this claptrap. :D