Optimal fiscal and monetary policy in a hybrid Old/New Keynesian model with and without the ZLB

Update: see update at very end.

Some people aren't as rational and future-oriented as agents in New Keynesian models. Some people aren't as present-oriented as agents in Old Keynesian models. A hybrid Old/New Keynesian model, with both types of agents, looks attractive.

But other people's hybrid models looked too complicated, so I decided to build my own. All you really need is one equation. (OK, plus a couple more little ones to help explain it.)

I discuss the first-best solution. Then the second-best solution, if the Zero Lower Bound on nominal interest rates is a binding constraint.

The results are not what you might expect. If the ZLB is a binding constraint, the government should reduce the growth rate of government spending, and/or increase the growth rate of taxes, to ensure the economy remains at potential output.

(Paul Krugman again misses this point. "If some of them are instead liquidity-constrained, the increase in income from the rise in G will lead to some increase in C as well, and we have a story that is even closer to the old Keynesian version." It is not the rise in G; it is the fall in the growth rate of G that does the work, which is far away from the Old Keynesian version.)

I explain this weird result, and reconcile it with the Old Keynesian intuition.

The Old Keynesian agents are straight out of the textbook. C(t)=A(t)+B(Y(t)-T(t)), where A is autonomous consumption, B is the marginal propensity to consume, and (Y-T) is current disposable income. I've written it A(t) to allow for shocks to autonomous consumption.

The New Keynesian agents are also straight out of the (different) textbook. If the real rate of interest r(t) exceeds their rate of time preference n(t), their current consumption will be less than their permanent income, and will be growing over time. Assume it's Cdot(t)=R(r(t)-n(t)), where R(.) is some function representing the Euler equation with the properties R(0)=0 and R' > 0, where Cdot(t) is consumption growth. I've written it n(t) to allow for shocks to the rate of time preference.

What we are looking for is a fiscal and monetary policy (I look at both together) that will keep output at potential at all times. Ignoring investment and net exports to keep it simple, that means we want C(t)+G(t)=Y*(t) for all t, where Y*(t) is potential output. Or, taking the derivative with respect to time, we want Cdot(t)+Gdot(t)=Y*dot(t) for all t. Consumption growth plus government expenditure growth needs to equal potential output growth.

Assuming some fraction k of agents are Old Keynesian, and (1-k) are New Keynesian, we want a fiscal/monetary policy that ensures:

k[Adot(t)+B(Y*dot(t)-Tdot(t))] + (1-k)[R(r(t)-n(t))] + Gdot(t) = Y*dot(t)

That's the model.

If k=1 and you integrate with respect to time (delete the "dot"s) you get the first-year Keynesian Cross model's solution for fiscal policy G(t) and T(t) to keep output at potential.

If k=0 you get the New Keynesian model, where Tdot(t) disappears because Ricardian Equivalence holds.

(I've cheated a little because strictly speaking k should represent expenditure shares rather than population shares, but I don't think this matters much.)

The government/central bank has three instruments: G(t), T(t) and r(t), and it only needs one instrument to keep the economy at potential. But if the government wants to maximise agents' utility, it will need to use all three instruments.

Philosophically this is a bit tricky, as Frances Woolley explains, because the government is acting in loco parentis for the non-rational Old Keynesian agents. If A(t) increases, for example, does this represent a change in their underlying preferences that should be accommodated, or is it just an irrational whim that should be suppressed?

The simplest assumption (though one that is certainly contestible) is that the true preferences of the Old Keynesian agents are exactly like the preferences of the New Keynesian agents; they just don't act on them. Given this assumption, we can solve for optimal fiscal and monetary policy.

The full solution is left as an exercise for the reader. Here are my answers:

T(t) should alone be used to offset shocks to A(t). If the Old Keynesian agents go on an irrational consumption binge, raise taxes enough to stop them, without changing G(t) or r(t).

Set G(t) so that the marginal utility of G(t) is equal to the marginal utility of C(t). (Taxes are assumed non-distorting.) This means that if Y*(t) increases, both C(t) and G(t) should increase.

Set r(t) to ensure that output is at potential, given G(t) and T(t). This means that r(t) must respond positively to n(t) and to Y*dot(t), and negatively to Gdot(t) and positively to Tdot(t). (Look at the equation!)

That is the first best solution for fiscal and monetary policy.

Now suppose that monetary policy is constrained by the Zero Lower Bound. The central bank cannot cut r(t) to implement the first-best solution. What are the second best solutions for G(t) and T(t)?

Inspect the equation above. To offset an r(t) that is too high (because of the ZLB), the government needs to reduce Gdot(t) and/or increase Tdot(t), if it wants to keep output growing at potential.

That result is very different from the Old Keynesian model, which says you should increase the level of G(t) and/or reduce the level of T(t), if you want to avoid a recession.

Here's the intuition:

Let's start in a stationary equilibrium, to keep it simple. All the dot variables are zero, and nothing else is changing. Then the rate of time preference n(t) falls. If G(t), T(t) and r(t) were unchanged, the drop in n(t) would cause the New Keynesian agents to want to cut C(t) and increase Cdot(t). (They want to save now, so they can grow their consumption over time.) But this would cause a recession. The first best optimal policy is to keep G(t) and T(t) constant, and reduce r(t), so that Cdot(t) stays at zero and C(t) stays constant. But if the ZLB prevents r(t) from being cut, the second best policy will be to reduce Gdot(t) and increase Tdot(t). Both these policies allow the New Keynesian agents to have a positive Cdot(t) without cutting C(t). By reducing the growth of government spending, and reducing the growth of consumption by the Old Keynesian agents (by growing taxes), the permanent income of the New Keynesian agents is now higher than their current income, and this offsets their reduced rate of time preference, and prevents them wanting to save part of their current income.

Why is this result counter-intuitive, if you approach this with an Old Keynesian intuition? It's because if an Old Keynesian economist hears that agents have an increased desire to save, he interprets that as a reduction in A(t), to which the first best optimal response is a cut in T(t), not a cut in r(t). A reduction in A(t) never causes the ZLB to be a binding constraint on the first-best solution in the first place, if we assume it's an irrational whim that should be suppressed.

Update: I have to confess: these results still creep me out. So I'm not surprised if you have the same reaction. Are they "right"? Well, if by "right" you mean "do they follow from the model?", then yes, I think they do. But would I implement them? No, I wouldn't. Despite my giving the intuition, there is still something weird going on here, and I don't know what it is. My hunch is that the indeterminacy problem of the New Keynesian model is at the root of all this. I'm beginning to wonder if the indeterminacy of equilibrium output (under interest rate control) mightn't be a feature, rather than a bug, of the New Keynesian/Neo-Wicksellian model. And policy recommendations that are based on solutions that assume away the indeterminacy problem (which is what NK macroeconomists do) are deeply flawed. The recession was just the indeterminacy problem telling us it refuses to be assumed away.

It's sort of weird. Macroeconomists like Roger Farmer are trying to build indeterminacy into their models to help them explain the world; while New Keynesians, who already have indeterminacy in their models to begin with, are trying to get rid of it by assuming it away.

Comments

[Andy Harless]

Bring it the time consistency issue. The first-best second-best way to reduce Gdot (second best for the zero bound but first best for the path of G conditional on that) is both to raise current G and to cut future G. But when you get to the future and the ZLB is no longer binding, your first best policy will be to go back to equalizing marginal utilities of G and C, which means it will be suboptimal to keep your promise to cut G relative to that path. It appears to me that the time consistent "second best second best" solution is to make all the burden of cutting Gdot fall on current G, which is to say, raise current G just as the Old Keynesians would tell you. Of course if we don't have to worry about time consistency, then the zero bound is no longer a problem, because you can commit to a temporarily higher inflation rate and don't have to tinker with G at all.

[Nick Rowe]

Andy: I think those are good points. I have my doubts though about the practicalities of raising current G and reducing Gdot immediately. In practice, G isn't a jump variable, because it takes time for governments to slowly increase the level of spending. So an increase in G in practice means an increase in Gdot for a year or so.

[Alex Godofsky]

Yeah, but Nick, your solution requires a permanent reduction in G below the optimal rate. Integrated across all of time that seems like a big deal.

[Nick Rowe]

Alex: Fair critique, but only if the ZLB binds permanently. Once n(t) rises again, you can slowly bring G(t) back up to the first-best solution. Or, maybe raise the inflation target, as Andy says, if you think this is going to be a recurrent problem.

(BTW, I'm ignoring the fact I'm supposed to be a Market Monetarist in all these posts. I'm taking the Keynesian models and arguments about the ZLB at face value.)

[Lord]

Are those in gdp terms or absolute terms? Since gdp is plunging it would be almost impossible for G to drop faster in gdp terms.

[Lord]

Hoover the NK?

[Min]

"To offset an r(t) that is too high (because of the ZLB), the government needs to reduce Gdot(t) and/or increase Tdot(t), if it wants to keep output growing at potential."

"Too high" is ambiguous. If interest rates "want" to be negative, but cannot be because of the Zero Lower Bound, they are in some sense too high. However, that does not necessarily mean that they are too high for the desired growth in GDP. In fact, if GDP growth is too low, that is prima facie evidence that the interest rate is not too high in that regard.

[Nick Rowe]

Lord: I think everything is in absolute terms.

Min: I'm not sure I follow you there. When I say "too high" I mean "higher than first best". My little model assumes GDP is always growing at some exogenous potential, because fiscal and monetary policy keep it there.

[Frank Restly]

k[Adot(t)+B(Y*dot(t)-Tdot(t))] + (1-k)[R(r(t)-n(t))] + Gdot(t) = Y*dot(t)

Nick here r(t) is the real interest rate but monetary policy sets a nominal interest rate.

"T(t) should alone be used to offset shocks to A(t)."

Wouldn't you be better served by using T(t) to offset increases in r(t)?

[Nick Rowe]

Frank: look at the equation. It tells us that Tdot(t) can offset shocks to Adot(t) (and therefore T(t) can offset shocks to A(t)). But it tells us that it is Tdot(t) and not T(t) that can offset shocks to n(t) or r(t).

[Frank Restly]

Nick,

Okay. Would a first best solution use Tdot(t) to offset shocks to r(t)?

[Nick Rowe]

Frank: No. That question doesn't even make sense. r(t) is a control variable, so there are no shocks to r(t), unless you hit the ZLB. And if you hit the ZLB you are no longer doing first best. Stop.

[ignoramus]

Something doesn't make sense to me: would cause the New Keynesian agents to want to cut C(t) and increase Cdot(t).

I can understand this in the "story-telling" sense, but not at all in the mathematical sense.
If they're going to cut C(t) suddenly, you can't derivate to Cdot(t) (or else, Cdot(t) will go negative before it can go back to positive).

[Nick Rowe]

ignoramus: Assume Y*(t) does not jump (to keep this simple).

In an Old Keynesian model, if the government wants to keep C(t)+G(t) at Y*(t), that means it cannot let C(t) jump down unless it makes G(t) jump up at the same time. But in this model, if G(t) jumps up, that will cause the C(t) of New Keynesian agents to jump down by an equal amount, so that doesn't help. So this means the government cannot let C(t) jump. So if the government sees that C(t) of the New Keynesian agents is about to jump down (because n(t) jumps down), it must jump r(t) down to prevent it happening. And if it cannot lower r(t), because of the ZLB, the only thing it can do to prevent C(t) of the NK agents jumping down is cut Gdot(t). (Or increase Tdot(t)).

[Kevin Donoghue]

From Nick's update: "My hunch is that the indeterminacy problem of the New Keynesian model is at the root of all this."

Well if we assume an exogenous money supply and a simple additive demand-for-money function the particular indeterminacy problem that bothers you goes away. But does it change your argument at all?

[Nick Rowe]

Kevin: good question. Dunno. My head's not clear.

The basic problem is that the IS curve (in levels) is horizontal. And it's upward-sloping in rates of change. So changing the level of G does nothing, and changing Gdot has the "wrong" sign, because increasing Gdot shifts the IS right but that lowers the natural rate of interest.

If you want to restore the standard OK result, you need to make the IS curve, in levels, slope down. But I don't think it does. Introducing an investment accelerator into the model would make it slope up too.

Now my head is blanking again.

[Nick Rowe]

Kevin: yes, I think it would.

An increase in M/PY would reduce desired saving, and would increase the r at which desired Cdot=0, and S=I. Therefore an increase in Y would reduce r along the IS curve, which makes the LRIS curve slope down.

At its simplest, outside money changes the New Keynesian agent's long run budget constraint. They can plan to have permanent consumption greater or less than permanent income, which Woodfordian agents can't. And (Ms-Md)/P tells us how much their desired permanent consumption will exceed their permanent income.

[Andy]

I'm confused, how can the result be unaffected by the value of k?

[Nick Rowe]

Andy: k affects the magnitudes, but not the signs of the results (unless k=0 or k=1).

[Andy Harless]

"In practice, G isn't a jump variable, because it takes time for governments to slowly increase the level of spending"

OK, but, as I understand it, your method is to assume that gov't has already solved the policy problem and then ask what that solution must be. In practice, when a recession hits, the government has ipso facto failed to solve the policy problem, so your method doesn't apply. The typical situation wherein fiscal policy becomes an issue is one where we start out away from even the second-best second-best outcome. If you start out in an unintentional recession, I don't think temporarily pushing Gdot in the "wrong" direction will have the same effect as it would when you start out at equilibrium. In particular, Y will no longer be constrained by supply at time zero, so the inverse relationship between C and G, and therefore between Cdot and Gdot, will not exist initially.

I won't try to do the out-of-equilibrium math, but I can reason verbally from the PIH. Suppose you get hit with an unexpected recession that takes you to the ZLB, and you expect to remain there for 5 years. The typical Keynesian proposal, subject to what is practically possible and then stylized to make it easier to think about, is to keep G where it is for a year, then raise it for 4 years, then lower it again. Under the PIH, the effect of raising G for 4 years is a small permanent tax increase, so consumption will decline a little in the first year. But not a lot, because at any reasonable discount rate, 4 years is a small fraction of forever. So yes, in the purely NK case -- which is to say, without the liquidity constraints that almost all actual New Keynesians believe in -- Y goes down slightly, and r* gets slightly more negative, during the period of shovel-unreadiness. But because the change is small, you only need to throw a small wrinkle into the model to reverse it. I think a small accelerator effect will do the trick, for example. (In practice, part of the reason you can't raise G immediately is that the private sector has to get ready the increase in G: government suppliers need to hire more staff etc.) Anyhow, in practice most NK's will say that (1) any realistic model would include liquidity constraints and (2) a fiscal stimulus in the absence of shovel-ready projects should include a temporary tax cut in the first year. Those changes will surely be enough to offset the out-of-equilibrium effect of a positive Gdot.

I don't think there is anything very weird going on here. Was it Milton Friedman who said, "To spend is to tax"? That's an issue that policymakers need to take into account. It makes things look a little weird when you use the "assume the problem is already solved" method and work in continuous time with the simplest possible model, but I don't think it has a huge effect on the optimal policy recommendation for most real-life situations.

[Min]

Nick Rowe: "My little model assumes GDP is always growing at some exogenous potential, because fiscal and monetary policy keep it there."

That's sort of what I had in mind when I said that "too high" is ambiguous. I thought that rates were considered too high at the Zero Lower Bound because proper monetary policy would have them be negative. But you seem to mean something else.

Nick Rowe: "When I say "too high" I mean "higher than first best".

Check. :)

k[Adot(t)+B(Y*dot(t)-Tdot(t))] + (1-k)[R(r(t)-n(t))] + Gdot(t) = Y*dot(t)

At the ZLB r(t) = 0 and that becomes

k[Adot(t)+B(Y*dot(t)-Tdot(t))] + (1-k)[R(-n(t))] + Gdot(t) = Y*dot(t)

IIUC, you say that it would be better if r(t) were negative, and, holding everything else on the left side constant, if r(t) went negative, then Y*dot(t) would go down, and that is desirable. Similarly, if we held everything on the left side constant but Gdot(t), then reducing Gdot(t) would also reduce Y*dot(t), which is what we want.

Or do I misunderstand your argument?

[Min]

k[Adot(t)+B(Y*dot(t)-Tdot(t))] + (1-k)[R(-n(t))] + Gdot(t) = Y*dot(t)

Actually, that should be rewritten as

{k[Adot(t) - (B)Tdot(t)] + (1-k)[R(-n(t))] + Gdot(t)}/(1-(k)B) = Y*dot(t)

unless (k)B = 1.

[Nick Rowe]

Andy: "OK, but, as I understand it, your method is to assume that gov't has already solved the policy problem and then ask what that solution must be."

Correct.

" Under the PIH, the effect of raising G for 4 years is a small permanent tax increase, so consumption will decline a little in the first year."

That would be the *direct* effect on C(1) from the increase in permanent taxes. But if C(1) changes, Y(1) will change by an equal amount, which causes a further change in C(1) and C(2), and C(3) etc, and these cause further changes in Y(1), Y(2), etc, which cause....and the whole indeterminacy problem rears its ugly head, because the mpc out of permanent income is 1. Notice that I have ducked the indeterminacy question by assuming that my equation, which is only a necessary condition for Y(t)=Y*(t) for all t, is in fact a sufficient condition.

We have to *just assume* the economy returns to Y*(t) after the ZLB is past, and solve backwards from there.

Min: "... if r(t) went negative, then Y*dot(t) would go down, ..."

Y*(t) and Y*dot(t) are exogenous. Y* is potential output, not actual output. I am using the "Type 2" method from my previous post.

[Min]

Min: "... if r(t) went negative, then Y*dot(t) would go down, ..."

Y*(t) and Y*dot(t) are exogenous. Y* is potential output, not actual output. I am using the "Type 2" method from my previous post.

OK. If Y*dot(t) went down, then r(t) would go negative, but it can't so Gdot(t) goes down. Right?

[Nick Rowe]

Min: yep. But that's not a great thought-experiment, since we would want Gdot to go negative if Y*dot went negative, even in the first best. But if r can't go down, Gdot would go down by more than in the first best.

It's better to think of n going down.

[Market Fiscalist]

'the second best policy will be to reduce Gdot(t) and increase Tdot(t). Both these policies allow the New Keynesian agents to have a positive Cdot(t) without cutting C(t). "

Are you assuming that in the future y will always be at y* ? I can see that Cdot(t) for NK-agents = (Ydot(t) - (G + (Cdot(t) for OK-agents))). But won't the reduced G and (Cdot(t) for OK-agents) likely reduce y rather than just giving a bigger share to NK-agents ?

Am I missing something ?

[Market Fiscalist]

I meant Gdot(t)and ydot(t) rather than G and y in last comment.

[Frank Restly]

Nick, r(t) is not a control variable available to monetary policy makers. A central bank sets a nominal rate of interest not a real rate.

[Nick Rowe]

MF: "Are you assuming that in the future y will always be at y* ?"

Yes. See my post here.

Frank: NKs assume that by controlling i the CB controls r. Because expected inflation adjusts slowly.

[Min]

Nick, I am obviously not getting your argument. If you are trying to change actual GDP, it does not appear in your equation, so it is unclear what relevance the equation has.

Nick Rowe: "We have to *just assume* the economy returns to Y*(t) after the ZLB is past, and solve backwards from there."

Working backwards from the desired solution is a good heuristic. But it is not clear to me that you have worked backwards to where you are starting from.

Let me offer an parody to suggest what in general I am not understanding. I am driving along the freeway, wondering what is the best way to get home. I realize that when I get home I will park the car. So I put on the brakes.

[Roger Sparks]

"The Old Keynesian agents are straight out of the textbook. C(t)=A(t)+B(Y(t)-T(t)), where A is autonomous consumption, B is the marginal propensity to consume, and (Y-T) is current disposable income. I've written it A(t) to allow for shocks to autonomous consumption."

The heterodox opinion is that the text books may be misleading. Here is why:

Consumption (C) IS consumption but tax (T) is INCOME (to the government). As a result, here we are combining apples-and-oranges in the same equation; private consumption is being added to government income; Y - T is private income, not private consumption.

If we identify government expenditures as government consumption (G), then, making sure that all the terms are describing consumption, we could correctly write

C(t)=A(t)+B(Y(t)-G(t)).

The difference between the two equations is G - T = government deficit. Private income would equal private consumption if government deficit was equal to zero.

We also have a second problem: The equation is strictly backward looking. This is because none of the terms carry information from past to future. When we take the derivative of any of these terms, we get a constant. The resulting constant is strictly dependent upon past measurements.

This is unlike many equations in nature where the base equation contains a squared term. The derivative of a squared term is a variable to the first power; but the important thing here is that the variable is continuous throughout the series, and thus carries information from past to future. This continuum of information does not occur in a derivative of these economic equations.

You can see why this heterodox commentator is quickly stopped by this logic.

[Market Fiscalist]

In which case I'm not sure I get the intuition behind the model.

NK-agents are supposed to be super-rational and consumption-smooth based on expectations of future income. The reason they expect higher future income in your model is that they expect increased future taxes to increase (unmatched by increases in G) which will cause OK-agents to consume less and leave them to consume the balance of Y*. However if they are super-rational they would also expect that future falling OK-consumption will cause y to fall. If y < y* their expectations of future income will be indeterminate (they get a bigger share of a unknown sized pie). This means the impact on current NK-agent's C is also indeterminate.

[Nick Rowe]

Min: did you read my post where I thought I had explained what I am doing here? Or wasn't it as clear as I thought it was?

Roger: "The heterodox opinion is that the text books may be misleading. Here is why:

Consumption (C) IS consumption but tax (T) is INCOME (to the government). As a result, here we are combining apples-and-oranges in the same equation; private consumption is being added to government income; Y - T is private income, not private consumption."

Of course Y-T is not the same as private consumption. I didn't say it was.

[Peter N]

What is the integral of R(r(t)-n(t)) and why is R a function only of r(t) and n(t)?

[Nick Rowe]

Peter N:
For NK agents, C(t) is determined by Maximising the present value of U(C), where future utility is discounted at rate n, subject to the present value of consumption equals the present value of disposable income, discounted at rate r.

You are asking about the New Keynesian model's indeterminacy (so is Market Fiscalist). I have assumed this problem away, just like the New Keynesian economists do. Read this post where I explain it.

[Peter N]

Suppose k=1. Then Y* is completely unaffected by r(t). This means that r(t) influences C of the OK agents solely through its effects on the NK agents. This doesn't feel right.

Also with just

[A(t)+B(Y*(t))-T(t))] + G(t) = Y(t)

an increase in Gdot(t) increases Y*dot(t) and BY*dot(t).

let T be 0 B be, 5. A 25 and G 25 then BY*= Y* -A-G so Y*=100 C= 75
increase G to 55 and Y* = 160 C = 105

IF
"This means that r(t) must respond positively to n(t) and to Y*dot(t), and negatively to Gdot(t) and positively to Tdot(t)"

for this to be true, G and Y* must move in opposite directions, and this has to be caused by the behavior of the NK agents dominating that of the OK agents.

But this can't be true for all values of k, since k can be arbitrarily close to 1.

I'm confused.

[Nick Rowe]

Peter: You need to distinguish Y and Y*. Y* is exogenous.

Also, by assumption the OK agents don't respond to r. They only respond to (Y-T). They don't think about the future, so don't think about the relative price of current to future goods.

But despite that, this is confusing. My head is not as clear on it as I would like it to be.

[Nick Rowe]

Look guys. Let me explain what's going on in this post:

Normally, what economists do is this: they solve for Y as a function of fiscal and monetary policy. Then they ask: if Y is less than Y* (there is a recession), what would we need to do to fiscal and/or monetary policy to increase Y?

That is not what I am doing here. Because the math is too hard.

Instead I am assuming Y=Y* always. Yes, I am assuming full employment always. Then I am solving for fiscal and monetary policy that is compatible with full employment. Because the math is a lot easier.

[Min]

Nick Rowe: "if Y is less than Y* (there is a recession), what would we need to do to fiscal and/or monetary policy to increase Y?

"That is not what I am doing here. Because the math is too hard.

"Instead I am assuming Y=Y* always. Yes, I am assuming full employment always. Then I am solving for fiscal and monetary policy that is compatible with full employment."

In my parody, Y = Y* is getting home, and my point was that what you do at home is different from what you do to get home.

But, of course, when you have a homeostatic system, "home" is not a static condition, and what you do to maintain homeostasis may be similar to what you do to return to it in the first place. For instance, I may exercise to get fit, but if I stop exercising when fit, I can become unfit.

Now, paradoxical interventions are not uncommon in dysfunctional family systems. So it may be that paradoxical interventions could work with dysfunctional economies. But in family systems paradoxical interventions work because they are resisted. And they are paradoxical because there is more to the dynamics than meets the eye.

In this case I suspect that the paradox arises from the assumption that Y = Y*. Under such a condition, do we even confront a Zero Lower Bound? Does "Rates are too high" mean the same thing as when Y <> Y*?

[Peter N]

So the equation says for OK

A(t) + G(t) = constant

perfect crowding out at equilibrium full employment.

And if y And if we want an effect from r we add I(r(t))

However if G crowds out I rather than C it gets a bit more complicated.

If you unsimplify your model, will you get the same trend in answers?

[Nick Rowe]

Peter N: "So the equation says for OK

A(t) + G(t) = constant"

NO! You just failed first year macro!

A + B(Y-T)+G=Y

Y= [1/(1-B)][A+G-BT] is the solution to the normal question. (You can add dots if you like).

and:

Gdot-BTdot = -Adot + (1-B)Y*dot is the solution to my question. (You can delete the dots if you like)

Min: If k=0, so we have a pure NK economy, we have a system that borders on what you call "paradoxical". My equation is then a necessary condition for Y=Y* for all t, but it is not a sufficient condition. That's because of the indeterminacy problem. There are multiple equilibria. And I worry about this. And I wonder, like you, if it does affect my results, if the economy ever gets away from Y=Y*

BUT:

1. The NK economists *just assume* this indeterminacy problem away, so I am doing the same here. They can hardly complain about me doing it when they all do it themselves without even realising they are doing it!

2. If k > 0 this indeterminacy problem *might* go away even without just assuming it away. (In other words, having a few OK agents around *might* resolve the indeterminacy problem in NK models). I think it might, but I haven't checked the math. And I would need to fix my little cheat of assuming k means population shares rather than expenditure shares.

[Roger Sparks]

Try writing

(1) C(t)+G(t)+C*dot(t)+G*dot(t)=Y*(t)

C(t)+G(t) are locates that tell us where we were the last time we measured. All of the * terms are predictions of future measurements.

Now bring in Old Keynesian (OK) and New Keynesian (NK) theory.

OK theory has all income spent on consumption. This implies that: if it was measured, it must have been spent. In this theory, investment is just another form of consumption. (Yes, investment is for future use, but, investment also involves people working and consuming resources with the goal of building an improved engine-of-production.)

NK theory adds thought that the rate of interest influences consumption decisions. To the extent that interest rates influence spending decisions and psychology, this addition to theory should be a good addition. (One limitation is that interest rates only are likely to influence a definable portion of economic decision makers. This limitation may be very important for macroeconomic planning.)

Returning to equation 1, terms C(t)+G(t) are history. If we desire to influence
Y*(t), we must influence C*dot(t)+G*dot(t). Now we are back to applying conventional tools, including Keynesian tools, to shift spending and working patterns.

[Market Fiscalist]

"Instead I am assuming Y=Y* always. Yes, I am assuming full employment always. Then I am solving for fiscal and monetary policy that is compatible with full employment"

I think I have it. This post is a Nick Rowe parody of NK economics - aimed at demonstrating the absurdity of assuming y = y* (which seems to be a feature of NK model.

Am I correct ?

[ignoramus]

Nick: And if it cannot lower r(t), because of the ZLB, the only thing it can do to prevent C(t) of the NK agents jumping down is cut Gdot(t). (Or increase Tdot(t))

.

Still confused. There seems to be a lack of a mechanism here. At ZLB the Nk Cdot would decrease as r (t) - n (t) is negative. Lowering Gdot doesn't "make room" for increased consumption. Y*dot is already greater than the lhs at the ZLB. If r should decrease as Gdot increases then Gdot should increase if we want to decrease r but can't. Decreasing Gdot only makes the lhs smaller I don't see s mechanism whereby that forces a jump in C or a change in n

[Nick Rowe]

Roger: "OK theory has all income spent on consumption."

No it does not. Read Keynes General Theory. He explicitly says that the marginal propensity to consume is between 0 and 1. Which is what I have assumed here. My B is his mpc.

"In this theory, investment is just another form of consumption."

No it is not. Again, read Keynes. He says that Investment and Consumption are both types of demand, but they have very different determinants.

And if you did say that "OK theory has all income spent on consumption (plus investment plus government spending)." You would be asserting Say's Law, which Keynes again explicitly rejected. (Or merely stating an accounting identity which all economists accept for a closed economy, but which has no theoretical content.)

"Try writing

(1) C(t)+G(t)+C*dot(t)+G*dot(t)=Y*(t)"

No, because it does not make mathematical sense. You could restate it in discrete time so it did make mathematical sense. But it would be wrong, unless you added Y*dot(t) to the right hand side. And if you did say that C(t) and G(t) are predetermined, I would simply reply: "what you call t I call t-1", which gets us nowhere.

What does calling yourself a "heterodox economist" mean? Does it have any content at all, except by negation?

[Nick Rowe]

MF: "Am I correct ?"

No. You are totally wrong. AAAAAAAAAAAAAAAARRRRRRRRRRRRGGGGGGGHHHHHHHHHHHHHHHHH!!!!!!!!!!

You are still not getting it. Please read my post, the other one I linked to twice in comments above, asking you to read.

Or try this. Suppose G is the position of the gas pedal and Y is the speed of the car. Suppose they are related via this equation: Y=sG. Suppose I want to drive at the speed limit, which is Y*=100. Where should I put the gas pedal? Answer: assume Y=Y*, and solve for G. G=(1/s).100

Do you get it now? Because it's just like that, except we have gas and brakes and tranny, and the relation is a complicated dynamic one, where both speed and acceleration are affected, and there are hills and headwinds on the road, and the speed limit varies. And the car has a mind of its own.

[Nick Rowe]

ignoramus: "There seems to be a lack of a mechanism here."

See Min's comment, and my reply, immediately above. About indeterminacy, and sufficient vs necessary conditions.

[Roger Sparks]

"Try writing

(1) C(t)+G(t)+C*dot(t)+G*dot(t)=Y*(t)"

Would you prefer to write

(2) C(t)+G(t)+C*dot(t)+G*dot(t)= Y(t)+Y*dot(t)

where Y*(t)=Y(t)+Y*dot(t)?

I think both equation 1 and 2 are correctly written.

It is also true that Y(t)=Y*(t) if Y*dot(t)=C*dot(t)+G*dot(t)=zero.

I think our goal is to predict and influence the future. The future is an extension of the past.

Our predictions must accommodate introductions of new capital (money) into our equations. This introduction is like the size of the earth increasing each year by an amount to be decided by the earth expansion committee. Predict the future surface area of the earth.

Now complicate the earth expansion by increasing only part of the surface and leaving the remainder unchanged. Now predict which parts of the earth will be expanded by the earth expansion committee and by how much each will expanded.

There is no intent to be negative, but grasping the problem from different perspectives increases the probability of finding a model we can grasp.

[Market Fiscalist]

Ooops. I apologize Nick. I did not mean to elicit that kind of response :(

I think I see what you are doing. If you could get all the agents in the model to expect that y=y* then what values would you have to set the various variables at t=0 to be be consistent with meeting these expectations? (I think I can also see how that would useful for an NGDPT approach).

My concern is this: If everyone always expected y=y* then you can probably set the variables to what you want and things will work out well (all the dot variables will be adjusted in the future to compensate for any that set "wrong" in the present. But probably if r, g and t were all set to crazy values then people would stop expecting y=y* and y would stop being = y*!

So the only measure of what the "right" values for these variables is whether they lead to the agents in the model expecting y=y* in the future. Rather than developing a theory to calculate these values you would be better of "crowd-sourcing" them and setting them to whatever values happen to work.

[Peter N]

k[Adot(t)+B(Y*dot(t)-Tdot(t))] + (1-k)[R(r(t)-n(t))] + Gdot(t) = Y*dot(t)

assume k=1

Adot(t)+B(Y*dot(t)-Tdot(t)) + Gdot(t) = Y*dot(t)

Adot(t)-BTdot(t) + Gdot(t) = Y*dot(t)- BY*dot(t)

integrate

A(t)-BT(t) + G(t) = Y*(t)- BY*(t)

A(t) + (GT(t)-BT(t)) = (1-B)Y*(t)

A(t) + N(t) = (1-B)Y*(t)

where N(t) depends on fiscal policy.

Y* is exogenous, so it doesn't depend on A(t), G(t) or T(t)

which is what I meant constant. I said assume T was 0, which removes the BT term. You're not the only one who can simplify a model. Increase N and you decrease A.

Y obviously isn't a constant if Y can be unequal to Y*.

Then

A(t) + (GT(t)-BT(t)) = (1-B)Y(t) or as you wrote

Y= [1/(1-B)][A+G-BT]

So it seems we are and were in agreement.

Now, however, a change in G can just increase Y. Of course, this doesn't actually happen.

Now supposedly G competes with I, so I'm putting it back in

A(t) + (GT(t)-BT(t)) + I(t,r) = (1-B)Y(t)

I was asking what effect the addition of I would have both when K = 1 and when it doesn't.

Without I and as long as Y is close to Y*, everything in the model will work as you say AFAICS, BUT if Y* - Y is too large or the addition of I changes the behavior, it seems things might work differently. It looks like you're saying if monetary policy keeps Y close to Y*, G negatively affects C.

[Majromax]

Is it fair to consider an investment-free model here? Or, for that matter, one without money?

If there is no investment, then what precisely are consumers doing with their non-consumed surplus? For that matter, how are G(t) and T(t) not identical? Provided the central bank does not monetize the deficit (generally frowned upon), without the capacity for investment (or disinvestment) then the resulting bonds issued to finance government spending must exactly match the un-spent consumer surplus.

Permitting investment allows the accumulation or drawdown of some un-consumed but stored surplus, such as a granary full of wheat.

My intuition suggests that perhaps the differences are reconcilable if we allow for actual return on investment, much as in traditional "stimulus" programs.

Also, speaking as a mathie, I think you can get some very qualitatively different results if 0 < k < 1. Then instead of an algebraic, steady-state relationship (either after integrating once or directly), you end up with a bona-fide differential relationship that can admit a solution path.

[Nick Rowe]

MF: no worries. I was losing it a bit!

"I think I see what you are doing. If you could get all the agents in the model to expect that y=y* then what values would you have to set the various variables at t=0 [and for all times t] to be be consistent with meeting these expectations [and actions]?"

You've got it! [with my little bit added in brackets, because they actually have to act so as to fulfill their expectation of full-employment, and they won't do that if policy is wrong, even if they do expect full employment, and you have to do it not just when t=0 but for all t.]

But tell me, was it my gas pedal/speed limit analogy that helped you get it, and you found my other post didn't explain it clearly? (As a teacher, I need to know things like that.)

[Nick Rowe]

Roger: "I think both equation 1 and 2 are correctly written."

no. They make even less sense now.

"Our predictions must accommodate introductions of new capital (money) into our equations."

Money and capital are not the same thing.

This is not getting us anywhere. Let's stop.

[Nick Rowe]

Peter N:

"Increase N and you decrease A."

Let's say it the other way around: if A decreases the government needs to increase N.

[Nick Rowe]

Majromax: Is it fair to consider an investment-free model here?"

New Keynesians often do this, for simplicity, so if it's fair for them it's fair for me, since this post is me speaking to New Keynesians.

"Or, for that matter, one without money?"

Same answer. Money is not explicitly in the NK model, but it has to be there implicitly. (I think this is a big problem, but this is not something NK's can complain about.)

The rest of your questions are intelligent questions but mostly first year questions. Sorry, but I'm not going to explain the answers, except to say "bonds and money". This old post may help

"Also, speaking as a mathie, I think you can get some very qualitatively different results if 0 < k < 1. Then instead of an algebraic, steady-state relationship (either after integrating once or directly), you end up with a bona-fide differential relationship that can admit a solution path."

I'm not sure I understand you. (Probably because I'm bad at math.)

[Market Fiscalist]

I think it was the "AAAAAAAAAAAAAAAARRRRRRRRRRRRGGGGGGGHHHHHHHHHHHHHHHHH!!!!!!!!!!" that shocked me into reading your earlier post more carefully, after which it made sense.

(Also I had read some of your earlier posts on the hidden assumptions in the NK model about y=y*, and somehow had formed the opinion that this was a bad thing for a model to include)

[Nick Rowe]

MF: Aha! So I have now learned a new pedagogical technique! Saying "AAAAAAAAAAAAAAAARRRRRRRRRRRRGGGGGGGHHHHHHHHHHHHHHHHH!!!!!!!!!!"!

(Yep, my earlier stuff on the NK's just assuming *expected future* Y=Y* would have confused it. Because I'm doing that too here.)

[Majromax]

> I'm not sure I understand you. (Probably because I'm bad at math.)

I'll retract this point, at least for now. Going back to try to clarify my point, I think I was partially mis-reading the sample model, and in so doing mentally dropping the derivatives from the terms. Clearly, I'm not caffeinated enough today.

[Majromax]

... but I do have one further question that I hope is slightly above 101-level. How do New Keynsian agents in this model work at the zero lower bound? In the derivation, you say:

> The New Keynesian agents are also straight out of the (different) textbook. If the real rate of interest r(t) exceeds their rate of time preference n(t), their current consumption will be less than their permanent income, and will be growing over time. Assume it's Cdot(t)=R(r(t)-n(t)),

... which of course makes sense. But at the ZLB, the real interest rate is somewhere between zero and negative something, so it cannot exceed the agents' time preference, provided they do not actively value spending later more than spending now.

If credit doesn't exist, than agents cannot spend more than their income and Cdot(t) = 0 at the ZLB, and the New Keynsian agents don't enter into the analysis anymore. If credit does exist, then New Keynsian agents can spend more now and have Cdot(t) < 0, which would suggest that Gdot(t) > 0 to compensate?

I've probably made another first-week error here.

[Min]

Nick Rowe: "My equation is then a necessary condition for Y=Y* for all t, but it is not a sufficient condition. That's because of the indeterminacy problem. There are multiple equilibria. And I worry about this."

Don't worry, be happy! :)

Multiple equilibria are a feature, not a bug. IMO, all human systems are chaotic (in the technical sense, so they can appear to be stable for a long time) or on the edge of chaos, typically with multiple (quasi-) equilibria. Dysfunctional systems are typically in a suboptimal equilibrium, which can last a long time. IMO the so-called New Normal is such a suboptimal equilibrium, and needs to be recognized as such, or it will become a self-fulfilling prophecy. One feature of a dysfunctional system is doubling down by policy makers. X isn't working, so we need more X. See Europe and the US. {weak grin}

[Nick Rowe]

Min: "Multiple equilibria are a feature, not a bug."

I think I said the same thing (or that they *might* be a feature not a bug) in a comment just recently, but now I can't find it. (spam filter strikes again?) And recessions might just be the indeterminacy problem telling us we can't just assume it away.

And I said how ironic it is that economists like Roger Farmer are trying to put indeterminacy *into* their models while New Keynesians are trying to just assume it out of their models!

[ignoramus]

After much thinking, I can better explain it to myself.

I'm going to rearrange to C=Y-G, so:

k[Adot(t)+B(Y*dot(t)-Tdot(t))] + (1-k)[R(r(t)-n(t))] = Y*dot(t) - Gdot(t)

And now I can understand this, as at the ZLB by decreasing Gdot(t) we increase the r.h.s. allowing to increase consumption... but...

for the OK if we take C(t) = A(t) + B(Y*(t) - T(t)), we really should be substituting back
So we'd get:

Cdot(t) = k[Adot(t)+B(Cdot(t) + Gdot(t)-Tdot(t))] + (1-k)[R(r(t)-n(t))]

Cdot(t) * [1 - kB] = k[Adot(t)+B(Gdot(t)-Tdot(t))] + (1-k)[R(r(t)-n(t))]

Now I'll put back the r.h.s.

Cdot(t) * [1 - kB] = k[Adot(t)+B(Gdot(t)-Tdot(t))] + (1-k)[R(r(t)-n(t))] = [Ydot(t) - Gdot(t)] * [1 - kB]

IF we want to preserve NK consumption, then I do think that you're right about the indeterminacy. Since we've only specified a condition on the derivative, it is possible to integrate and to get a different constant (i.e., it is underspecified and shocks to absolute consumption are certainly possible). And I think I understand better now when you say:

Both these policies allow the New Keynesian agents to have a positive Cdot(t) without cutting C(t). By reducing the growth of government spending, and reducing the growth of consumption by the Old Keynesian agents (by growing taxes), the permanent income of the New Keynesian agents is now higher than their current income, and this offsets their reduced rate of time preference, and prevents them wanting to save part of their current income.

I think the tie-in to reduction of the current income (or at the least in making its rate decrease), isn't well represented in the equations presented. It should be possible to capture that in some term.

Thanks for taking the time to respond to all our comments!

[Peter N]

Seen this one?

An Analytical Critique of ‘New Keynesian’ Dynamic Model

http://ns.fujimori.cache.waseda.ac.jp/PKR/2013/asada.pdf

"The most notorious problem of the prototype NK dynamic model is the phenomenon that is called the sign reversal in New Keynesian Phillips curve, pointed out by Mankiw (2001) clearly."

and

"Lemma 3.1(1) means that the prototype NK model is accompanied by equilibrium that is unstable in the mathematically orthodox sense.
The equilibrium of NK dynamic model will never be reached. This undermines the basis of the NK theory, however. Hence, the NK literature adopts the trick that makes the unstable system try to mimic a ‘stable’ system by using the so-called jump variable technique."

They draw on this paper by Mankiw:

http://scholar.harvard.edu/mankiw/files/royalpap.pdf

[Nick Rowe]

Peter N: The Mankiw paper (I had seen before) is about an empirical problem with the Phillips Curve assumed in NK models. Mankiw is right, IMO, but that problem has nothing important to do with the rest of the NK model. Plus, Mankiw and others have been working on fixing it.

The Asada paper looks (I think) closer to the indeterminacy problem I am talking about. But he lost me when he said "Jacobian", because I have forgotten what that means, and why it matters. Then it wandered off into Lemmas. It's probably a good paper, but not good for me, unfortunately.

I think his Old Keynesian model is just a tarted up early 1970's ISLM with adaptive expectations, but I didn't look too closely.

[Peter N]

"2. Formulation of the Prototype NK Dynamic Model
The simplest version of the prototype NK dynamic model may be formulated as follows."

Philips curve derived from the optimizing behavior of the imperfectly competitive firms with costly price change

π(t) = E(t)[π(t+1)] + α(y(t) - y*) + ε(t)

IS curve derived from the Euler equation of consumption

y(t) = E(t)[y(t+1)] -β(r(t) - E(t)[π(t+1)] - ρ0) + ξ(t)

Taylor rule

r(t) = ρ0 +π* +γ1(π(t) -π*) + γ2(y(t) - y*)

Simplifying and assuming rational expectations, you have 2 equations in two variables Δπ and Δy. The jacobian is the matrix of the coefficients. From this you calculate the characteristic equation. Its roots determine the stability. If the absolute value of 1 root > 1 the equations are unstable giving a saddle point equilibrium. If 2 have absolute value > 1 the equations are totally unstable, then all paths diverge either monotonically or cyclically. In this case, at least 1 root is > 1.

...

"The equilibrium of NK dynamic model will never be reached. This undermines the basis of the NK theory, however. Hence, the NK literature adopts the trick that makes the unstable system try to mimic a ‘stable’ system by using the so-called jump variable technique. In the NK dynamic model, both of two endogenous variables π(t) and y(t) are considered to be jump variables or not-predetermined variables , the initial values of which are freely chosen by the economic agents. only the initial conditions of the endogenous variables are chosen so as to ensure the convergence to the equilibrium point."

[Nick Rowe]

Peter N: OK. I'm not worried about Y(t) being a jump variable. If new information arrives in a jump, you jump your spending up or down. And if you don't, it means you have habit persistence, or adjustment costs, which need to be built into the model anyway. I'm more worried about inflation being a jump variable, but then I think the Calvo Phillips Curve is wrong, empirically.

[Min]

Well, I have slept on this and done some more thinking about this. Back to basics.

Suppose a simple linear model that holds when y(t) = z(t). That is,

1) Ax(t) + B = y(t)

2) Ax(t) + B = z(t)

Now, starting when y = z, let the system be perturbed so that y > z, but the difference is small. Now we have

3) Ax(t) + B ≃ y(t)

4) Ax(t) + B ≃ z(t)

Ax(t) + B does not tell us how to return to the condition where y(t) = z(t).

But let's see what we can get out of this. When y(t) = z(t) we have

A xdot(t) = ydot(t) = zdot(t).

If we hold y(t) constant, we can use A xdot(t) = zdot(t) to increase z(t) so that it equals y(t). With a small perturbation, that might work. Or we can hold z(t) constant and use A xdot(t) to decrease y(t) so that it equals z(t). Even assuming that one or the other works, we do not know which to try. And that is true even if one of y(t) or z(t) is exogenous, and we cannot raise or lower it. Exogeneity or endogeneity is not part of the equation. A kind of relativity holds, so that increasing z(t) to equal y(t) and decreasing y(t) to equal z(t) are equivalent operations. We simply cannot tell from the equation whether to increase x(t) or decrease x(t), even when doing one or the other will work.

OC, in real life, we can often try one and reverse course if it is not working. We can also use what we have learned from previous experience. :)

[Nick Rowe]

Min: I'm not 100% sure I understand you correctly, but assuming I do:

For the OK model, I can solve for Y as a function of G and shocks, or G as a function of Y and shocks, and I know what's going on. I know if I increase G then Y increases. So if Y < Y*, I know I need to increase G.

For the NK model, it's not that simple, but only because of the indeterminacy problem. There is only one path of r compatible with a given path of Y and shocks, but there are many paths of Y compatible with a given path of r and shocks. NK keep *just assuming*, that if Y < Y*, it means r > r*, so you need to cut r (which is true in an OK model like ISLM, but is not true in a NK model.

[Min]

Nick Rowe: "For the OK model, I can solve for Y as a function of G and shocks, or G as a function of Y and shocks, and I know what's going on. I know if I increase G then Y increases. So if Y < Y*, I know I need to increase G."

Right. You have another equation that tells you about Y, whether Y = Y* or not. So you can use that information.

Nick Rowe: "For the NK model, it's not that simple, but only because of the indeterminacy problem. There is only one path of r compatible with a given path of Y and shocks, but there are many paths of Y compatible with a given path of r and shocks. NK keep *just assuming*, that if Y < Y*, it means r > r*, so you need to cut r (which is true in an OK model like ISLM, but is not true in a NK model."

Many thanks, Nick. :)

Then, if indeterminacy is a feature, the thing to do is to go empirical, to experiment, as Roosevelt did during the Great Depression. And as the US is not doing now.

[Min]

Also, OC, if there are multiple equilibria, you can change the payoffs. But that means a level of gov't intervention that is not now popular.

[Nick Rowe]

"OC" = ?

Or, recognise that restoring determinacy means putting M back into the model, and into the policy rule in the world, and not just r.

[Nick Rowe]

"You have another equation that tells you about Y, whether Y = Y* or not."

It's really the same equation, except you read it from left to right rather than right to left.

NK only has an r=F(Ydot) equation, not an r=F(Y) equation

[Min]

Moi: ""You have another equation that tells you about Y, whether Y = Y* or not."

Nick Rowe: "It's really the same equation, except you read it from left to right rather than right to left."

What I am trying to say is something like this.

1) Ax + B = y

is not the same as

2) Ax + B = y , given that y = y*.

The first equation gives you more information than the second, information that you can use.

[Min]

Information about the relationship between x and y, I mean. :)

[Min]

Moi: " OC, if there are multiple equilibria, you can change the payoffs."

Nick Rowe: "OC" = ?"

Changing the payoffs alters the game. I expect that you will still get chaos, but chaotic systems can remain relatively stable for a long time. Like the solar system. :)