"Drop your shovels NOW!" On ending fiscal policy in NK models

I've been trying to get my head around this over the last few days. Still not sure I'm there yet.

But anyway:

It seems to me that the effect of fiscal policy at the Zero Lower Bound in New Keynesian models is extremely sensitive to timing the exit. This isn't about finding "shovel ready" projects that can be started relatively quickly; this is about finding projects that can be stopped instantly.

"Better late than never" is OK for starting fiscal policy; it is not OK for stopping fiscal policy. If we believe simple NK models.

Take an extremely simple NK model where we ignore private investment and foreigners, so C+G=Y.

If the economy is above the ZLB, the central bank offsets fiscal policy, to keep inflation on target and output at potential, so C+G=Y=Y*. Any increase in G causes 100% crowding out of C, because the central bank makes sure it does.

But if the ZLB is a binding constraint, C+G=Y < Y*. The central bank wants to cut interest rates to keep output at potential and inflation on target, but can't.

Consider fiscal policy for an economy that is temporarily at the ZLB.

1. Changes in government spending.

It is (reasonably) well-understood that a permanent increase in G has no effect on Y. Because future C falls by the same amount that G rises, So current C falls too by the same amount, via consumption-smoothing. So current G goes up and current C goes down but current Y stays the same.

It is also (reasonably) well-understood that a temporary increase in G causes Y to rise by the same amount, because C stays the same (again via consumption-smoothing).

Now suppose that the increase in G is temporary, but G does not return to normal until "one period" after the economy lifts off the ZLB. So in that period immediately after the ZLB ends, the higher G causes lower C, which via consumption-smoothing reduces current C during the ZLB periods. It's exactly like a permanent increase in G.

If the government workers drop their shovels "one period" too late (or are expected to do so), fiscal policy will not work at all.

How long is "one period"? It's as short as you want it to be.

2. Changes in marginal tax rates.

Lump sum taxes have no effect in a simple NK model, because of Ricardian Equivalence.

Changes in marginal tax rates do have an effect in NK models: first because of standard supply-side reasons (incentive effects on labour supply); second because at the ZLB those supply-side effects will affect inflation and (with nominal rates at 0%) hence real interest rates.

At the ZLB, if you are doing a temporary change in marginal tax rates you want to increase tax rates, to reduce current Y* and so reduce the output gap and so increase inflation (i.e. reduce the amount inflation falls below target) and so reduce real interest rates (i.e. reduce the amount they rise) to increase current C (i.e. prevent it falling as much). See Gauti Eggertsson (pdf).

At the ZLB, if you are doing a permanent change in marginal tax rates (bear with me please) you want to reduce tax rates, to increase future Y*, and so increase future C by the same amount, and so increase current C by the same amount (via consumption-smoothing). There is no effect on inflation, because current C and current Y* change by the same amount, leaving the output gap unchanged.

Now suppose you do a temporary change to marginal tax rates but don't return them to normal until "one period" after the economy lifts off the ZLB. It's exactly like a permanent change.

It's not just the magnitude, but the sign of the tax rate multiplier that changes, if there is any (expected) delay in returning tax rates to normal.

That's what I call extreme sensitivity to timing.

3. God only knows what to make of all this.

Is it an artefact of this extremely simple version of the New Keynesian model? Is it an artefact of my discrete-time setup.  [Update: maybe that's it. I'm thinking in discrete time rectangular steps, and maybe I should be thinking in continuous time triangular slopes. But all the same, it will still be very sensitive to timing the exit right.]

But if this simple version of the model is roughly right, the fiscal policy advice it gives cannot in practice be implemented. Governments can't usually stop doing things super-quickly.

Unless, maybe it requires some fancy footwork to coordinate monetary and fiscal policy?

But it's not enough to argue that the central bank can just delay raising interest rates for the same "one period". Because if you can do that, you don't need fiscal policy at all. Just tell the central bank to keep interest rates "too low for too long", to create the required amount of expected inflation.

Comments

[jonathan]

Good point! I don't think I've seen this idea explicitly brought out before.

I think the extreme sensitivity you observe is related to the "forward guidance puzzle" -- a small one-period change in future interest rates has a proportional effect on consumption in *every* period from its announcement to its implementation. Many people find this extreme sensitivity implausible -- hence the "puzzle".

Here's one way to think about your result: Suppose the government announced a one-period increase in government spending N periods in the future. Away from the ZLB, this would be no problem -- the Fed would lower interest rates in period N-1, and raise them in period N, so that consumption would fall just in period N, to leave the path of Y flat.

But now suppose the Fed couldn't lower interest rates in period N-1. Then if they still raise interest rates in period N, it's like negative forward guidance, and extremely powerful.

The fundamental "problem" is just the Euler equation.

[jonathan]

By the way, I think the optimal fiscal policy in a NK model at the ZLB is a decreasing path of G that approaches 0 as we approach the exit from the ZLB.

In this case, small changes in timing won't cancel out fiscal policy, because if the economy exits the liquidity trap one period sooner than anticipated, government spending will already be much lower than earlier in the liquidity trap, and so only a small fraction of the fiscal stimulus will be offset.

[Jussi]

I think Jonathan's point is good.

Moreoverly your model will break down if monetary effect is not instantaneous. It is super sensitive in that sense.

"It is also (reasonably) well-understood that a temporary increase in G causes Y to rise by the same amount, because C stays the same (again via consumption-smoothing)."

I appreciate that in the model labor market is efficient in a way that higher G will not mean higher Y even in the face of really high unemployement. That IMO seems highly implausible in reality. If higher G allows workers currently doing nothing (people without work) to do something even if only marginally productive I do not see how this could not be beneficial for the whole? As private side recovers it is certainly true that there is a trade-off and permanent G cannot have a positive effect (quite likely the other way around).

Plus I think Ricardian Equivalence doesn't hold all that well (isn't Krugman quite NK?): (http://krugman.blogs.nytimes.com/2011/12/26/a-note-on-the-ricardian-equivalence-argument-against-stimulus-slightly-wonkish/).

Relating: if we assume endogenous money creation (banking sector is alpha) would you Nick conclude that NGDP is determinated by G only?

[Roger Sparks]

OK. We have "Take an extremely simple NK model where we ignore private investment and foreigners, so C+G=Y".

We also assume that there are two principle economic managers that control the economy: The central bank (CB) and the government acting through fiscal policy.

The CB has a policy of C+G=Y=Y*. This is a steady state economy. Y1 will equal Y2.

Government has a goal of annual increases in Y. At the end of period two, government wants to see C+G=Y < Y*.

Now assume that interest rates are at ZLB.

Also assume that any increase in Y requires more money moving around (the velocity of money is assumed to be constant).

Where does the additional money come from?

I think the source of this new money is crucial to the discussion.

[Roger Sparks]

Correction: Government has a goal of annual increases in Y. At the end of period two, government wants to see C+G=Y less than Y*. Y1 is less than Y2.

(Your blog server does not like to see isolated "less than" symbols. Sorry about that.)

[Nick Rowe]

jonathan; thanks.

Good way of thinking about it (comparing it to the "forward guidance puzzle").

And I think your point about decreasing time-path of G is correct. And I think it's related to my update about triangles vs rectangles.

Very good comments.

Jussi: "Moreoverly your model will break down if monetary effect is not instantaneous. It is super sensitive in that sense."

Dunno. It's hard enough with just one lag. God knows what will happen with 2 or more lags. Ideally, we want a policy where the exact lags (or leads) don't matter too much. Which is why I like automatic stabilisers (like NGDP level-path targeting).

"If higher G allows workers currently doing nothing (people without work) to do something even if only marginally productive I do not see how this could not be beneficial for the whole?"

That's assuming the answer.

Roger: The standard NK model assumes that the central bank is in charge of AD, except at the ZLB. "The CB moves last."

You need to put a space either side of < . I will edit your comment so it reads right.

[Roger Sparks]

Nick: Thanks for the edit in my original. With a little practice using the preview option, I see what I did incorrectly.

The NK model assumption that the central bank is in charge of AD is slippery (in my opinion). Certainly the CB can raise interest rates or make money more difficult to obtain. Turning to economic reality, a change in difficulty is only one factor that influences decision makers as they decide whether to borrow or print money in their effort to accomplish something.

I think it is more fruitful to look at what is required to change Y from period to period. Y is a sum of money transactions measured in a time period. If Y changes from period to period, either the money used in the transactions flows more quickly (velocity increases) or more money is used (perhaps money from savings or perhaps money from monetary creation (someone mints a coin)).

Of course both velocity and the supply of money can change from period to period. The challenge is to separate the two possibilities into cause, effect, and relative magnitudes.

Now we can look at timing in a velocity-money supply framework. An early consideration should be the effect of taxes on the timing of decisions by both government and private spenders. Taxes may not be a factor in some of the spending decisions by government (government may borrow). We can be certain that a decision to stop spending will result in immediate freeze of an otherwise (probably) unstable Y.

[Jason Smith]

So wait: expectations of a delay in returning taxes to normal produce a change in expectations from an expected temporary change in tax rates to an expected permanent change in tax rates?

Can we add in rational expectations of an expected delay (obviously the government doesn't stop on a dime), so that people don't revise their expectations of a permanent/temporary tax change based on a delay in returning taxes to normal?

[Jussi]

"Dunno. It's hard enough with just one lag. God knows what will happen with 2 or more lags. Ideally, we want a policy where the exact lags (or leads) don't matter too much. Which is why I like automatic stabilisers (like NGDP level-path targeting)."

Hmm. I thought if the central bank doesn't immunize G after the lift-off G doesn't have the extreme sensitivity? Lets say the central bank will immunize at the step one after the lift-off but it takes one step to take effect (it has been said it takes 6 - 18 months a rate movement is absorbed into the economy). Now C is not immunized at the step one after the lift-off and thus smoothing doesn't take place at that step. Other than that I agree (NGDP part).

"That's assuming the answer."

Hah, agreed but IMO it is often assumed that higher G doesn't have any effect (you of course didn't), Ricardo and all. But why you did say

" ...future C falls by the same amount that G rises, So _current_ C falls too by the same amount, via consumption-smoothing. So current G goes up and current C goes down but current Y stays the same."

- then above is false if there is an output gap to fill with higher G because C is also higher because laid-offs are now producing something.

[Ralph Musgrave]

The answer is simple: don't combat recessions to any great extent with infrastructure projects. As Nick says, starting them quickly is often difficult, and stopping the construction of a road or bridge when they're half complete and just because a recession has ended is daft. There's plenty of other things to spend public money one, plus tax cuts are an option.

[Nick Rowe]

Roger: in the mid 1990's Canada tightened fiscal policy a lot, turning a big deficit into a big surplus. The Bank of Canada loosened monetary policy in response, letting the nominal interest rate and exchange rate fall, and there was no recession, and inflation stayed roughly on target.

Jason: I am assuming rational expectations.

Jussi: G can affect Y even if Ricardian Equivalence is true. Ricardian Equivalence is about taxes, not about government spending.

I didn't just *assume* changes in G would have no effect on Y (above the ZLB). I explained why. This is a consequence of the model. Read.This.Bit:

"If the economy is above the ZLB, the central bank offsets fiscal policy, to keep inflation on target and output at potential, so C+G=Y=Y*. Any increase in G causes 100% crowding out of C, because the central bank makes sure it does."

And if you don't understand what "consumption smoothing" means you will not understand this post.

[ThomasH]

I think Nick has a too narrow take on "fiscal policy" in a recession. If governments (and in the US context, that means state and local governments, too) adjust their taxing and spending according to borrowing costs and differences between market prices and marginal costs of expenditures (unemployment of factors of production = an output gap), G and deficits would rise and fall countercyclically in a standard "Keynesian" way. Probably very few expenditure categories would work off of extremely short term borrowing rates so few would be affected by the exact timing of when ST rates change from zero to non-zero. The proper fiscal rule (which the new Canadian Government sounds like it intends to follow) makes "fiscal policy" endogenous to monetary policy.

Whether a model in which G behaves in this fashion is NK or not I'll leave others to say.

[John Handley]

Nick,

"If the economy is above the ZLB, the central bank offsets fiscal policy, to keep inflation on target and output at potential, so C+G=Y=Y*. Any increase in G causes 100% crowding out of C, because the central bank makes sure it does."

That is unless the central bank thinks potential output is flexible price output. In that case, fiscal policy won't be offset fully. In practice, I think you're correct though. Real world central banks target output relative to a subjective idea of potential output because flexible price output is not observable. If, instead of making sure the output gap is zero at all times that the zero lower bound doesn't bind, the central bank uses a Taylor Rule, it is possible that the increase in output from the fiscal stimulus will cause the central bank to get off of the zero lower bound because the central banks policy rule has exceeded zero. I think this is the way that most NK economists think about it, but I could be wrong.

You could also think about this through a neo-fisherian and/or fiscal theory of the price level lens. Because monetary policy is stuck in a deflation trap (the second equilibrium of Taylor Rules), fiscal policy can change to being non-ricardian or 'active' and monetary policy can change to being 'passive' until the rate of inflation changes. This especially makes sense if you have a cash-in-advance model and the zero lower bound binds either because the central bank was stupid enough to promise a large deflation or the equilibrium real interest rate fell significantly. If either of these happen, the expected path of the price level is indeterminate and unrelated to the path of the money supply (that is unless the central bank shrinks the money supply until the constraint binds again). In this case, the fiscal authority can step in to ensure that there is an equilibrium for the price level and that that equilibrium is not consistent with the zero lower bound. At this point, the policy rules can return back to normal.

[Jussi]

Nick: Okay, I think I get it now. So even if "there can be a significant lag before interest rate
changes influence spending and saving decisions" (http://www.bankofcanada.ca/wp-content/uploads/2010/11/how_monetary_policy_works.pdf) that doesn't change anything in the model because of rational expectations. I apologize my ignorance. Rational expectations might be a standard NK assumption but it surely can give extremely sensitive results (and doesn't make much sense anyway).

[Jason Smith]

"Jason: I am assuming rational expectations."

I am not sure I understand the difference between:

a) adding a term in the model where model-consistent (i.e. rational) expectations don't suddenly change based on a 1-period delay

b) adding "rational expectations of an expected delay"

If rational expectations are model-consistent expectations, then adding rational expectations of X is the same thing as adding a term to the model that makes X happen. Or at least, that was my understanding.

Overall, however, I was being a bit tongue in cheek -- people revising their expectations to a permanent change in taxes from a temporary change in taxes based on an infinitesimal delay is a silly result. I'm not saying it's wrong mathematically, it's just silly. You should first question the scope of the model, not the model itself.

For example, how do you know you can add an infinitesimal delay to the end of fiscal policy in this model? It's not in the model originally. It produces the silly result. Maybe infinitesimal delays are out of scope of the model (i.e. there is an assumption that errors in the onset of fiscal policy dt << 1 so that t + 1 ≈ t + dt + 1)?

If you try to model atoms with classical electrodynamics, you get the ultraviolet catastrophe. The obvious conclusion was not that classical electrodynamics is bunk, but rather that maybe classical electrodynamics is bunk at short distances.

[Nick Rowe]

Jason: try re-reading it. The ***effect*** of a one-period delay is the same as a permanent change.

John: " If, instead of making sure the output gap is zero at all times that the zero lower bound doesn't bind, the central bank uses a Taylor Rule, it is possible that the increase in output from the fiscal stimulus will cause the central bank to get off of the zero lower bound because the central banks policy rule has exceeded zero. I think this is the way that most NK economists think about it, but I could be wrong."

And you could be right. Many NK models *do* assume the CB follows a Taylor Rule. But if the CB has knowledge about fiscal policy (which they do), it seems sensible to me that CBs will adjust that Taylor Rule so forecast inflation stays on target when fiscal policy changes.

[John Handley]

Nick,

"But if the CB has knowledge about fiscal policy (which they do), it seems sensible to me that CBs will adjust that Taylor Rule so forecast inflation stays on target when fiscal policy changes."

Except that fiscal policy changes need (do?) not have any impact on inflation expectations. As far as I know, there isn't really a demand-side effect of fiscal policy in New Keynesian models, just a real effect. I'm pretty sure any non-real effects stem from certain specifications of monetary policy. If the zero lower bound does not bind, a flexible-price-output-targeting New Keynesian central bank will do nothing in the event of a fiscal stimulus and, consequently, nothing will happen to inflation. I think this is part of the problem of monetary offset advocates. Yes, it is possible for the central bank to offset temporary changes in fiscal policy, but, if it's properly New Keynesian, it probably doesn't want to.

Of course, if the central bank were truly trying to ensure flexible price output and were truly "properly New Keynesian," it wouldn't be targeting a measure of output at all, but that's beside the point.

alternatively, again, you could start thinking in non-ricardian terms and see using fiscal policy to escape the zero lower bound in a different light. Namely, it's all about inflation and has relatively little to do with output.

[Nick Rowe]

John: "Except that fiscal policy changes need (do?) not have any impact on inflation expectations. As far as I know, there isn't really a demand-side effect of fiscal policy in New Keynesian models, just a real effect."

No. A temporary increase in G *will* have a demand side effect in a NK model, unless the CB offsets it by raising interest rates (which it will do when off the ZLB to keep inflation on target). That's what this post is about. (I am ignoring possible supply-side effects of increased G, for simplicity).

[nivedita]

"So in that period immediately after the ZLB ends, the higher G causes lower C, which via consumption-smoothing reduces current C during the ZLB periods"

I don't quite follow this -- the quantity that is smoothed is (C+G), not C alone, no? In the period after ZLB ends, C+G=Y*, so current (C+G)=Y* by smoothing? I'm assuming that by G, we mean consumption goods purchased by the government, which are redistributed somehow to individuals. I would want to smooth my total consumption, not just the portion that I financed out-of-pocket.

[Nick Rowe]

nive: No. C is smoothed, not C+G. But you are right that if G and C were perfect substitutes, we would have a very different model (in which even temporary changes in G would have no effect, because all it means is that the government is doing your shopping for you). The standard (unexamined) assumption in NK models is that G is neither substitute nor complement for C.

[Min]

"Consider fiscal policy for an economy that is temporarily at the ZLB.

1. Changes in government spending.

It is (reasonably) well-understood that a permanent increase in G has no effect on Y. Because future C falls by the same amount that G rises, So current C falls too by the same amount, via consumption-smoothing. So current G goes up and current C goes down but current Y stays the same."

OK. We can write that as

C - ΔC + G + ΔG = Y + ΔG - ΔC = Y

because

ΔC = ΔG

The text continues:

"It is also (reasonably) well-understood that a temporary increase in G causes Y to rise by the same amount, because C stays the same (again via consumption-smoothing)."

OK. We can write that as

C + G + ΔGt = Y + ΔGt

Where ΔGt represents a temporary change in gov't spending. C remains constant.

That also means that if at some time in the future, gov't spending returns to normal, we have

C + G + ΔGt - ΔGt = Y

"Now suppose that the increase in G is temporary, but G does not return to normal until "one period" after the economy lifts off the ZLB. So in that period immediately after the ZLB ends, the higher G causes lower C,"

No, it does not. As we have seen in the previous equation, C remains constant until G returns to normal, because the change in G is temporary. If the ZLB makes a difference, then it should appear in the previous equations.

[Min]

That said, it seems to me that your intuition is good. (Not that my opinion matters.)

Suppose, for instance, that the gov't decides to spend by building some pyramids or cathedrals. When the economy recovers, that spending will continue indefinitely, so that fewer consumer goods and services are produced, and consumption will be lower than if the pyramids or cathedrals were not being built. And we do not like that. It does seem important to have enough gov't spending that we can stop or reduce quickly as the economy recovers. One possibility might be unemployment benefits. :)

[Nick Rowe]

Min: implicit in consumption smoothing is C(t)=C(t+1)/(1+r(t))

Above the ZLB, the central bank chooses r. At the ZLB, it cannot choose r.

[Min]

@ Nick

Maybe so, but that has no relation to ΔG or ΔGt. And it violates the assumption that C remains constant if the change in gov't spending is temporary. The argument is incomplete, even with the additional equation. You have not made the connection between going off the ZLB and ΔGt.

Text: "How long is "one period"? It's as short as you want it to be."

As the length of the period goes to zero, so does r(t), which means that so does C(t+1) - C(t). What's the problem?

[BenjaminRizzo]

"But it's not enough to argue that the central bank can just delay raising interest rates for the same "one period". Because if you can do that, you don't need fiscal policy at all. Just tell the central bank to keep interest rates "too low for too long", to create the required amount of expected inflation."

I think though the point needs to be here that you can leave the leave the sooner than only monetary policy. I suppose it comes down to assuming that monetary policy is less effective at the lower bound.

[nivedita]

Nick, not following. In your model without investment, what can G be except consumption goods, and what can happen to these consumption goods other than individuals consuming them?

I'd have thought the idea is that at the ZLB, people are cutting their C, attempting to save money. In aggregate they cannot do so, which manifests itself by their output and income falling. The government steps in, issues debt to the people who want to save, and uses the money to buy consumption goods and redistributes them to the people. The full employment output gets produced and consumed, with the savers saving via government debt.

[Min]

@ Nick

OK, you have not filled out your argument, so let me give it a try. :)

First, let us consider

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

or

ΔC(t)/C(t) = r(t)

Since r(t) ≥ 0, consumption does not reduce over time, based upon changes in r. It may for other reasons, OC. Changes in r affect the rate of increase in consumption, but do not reduce consumption.

Since ΔGt is temporary, it does not affect ΔC. Away from the ZLB, even granting the CB omnipotence over r, it can only reduce its rate of increase, it cannot reduce it. Therefore the CB cannot make ΔC negative, as though ΔG were permanent.

Sorry, I cannot reproduce your argument.

[Min]

Hmmm. That is not all that clear.

Given the control of r by the CB away from the ZLB, it can only reduce the rate of increase in consumption, not consumption itself.

[Nick Rowe]

Smashes head on keyboard...

Nive: just assume G is building pyramids, that the government wants, but nobody else cares about. So it's neither C nor I. Yes, there are a whole lot of questions about the interaction between G and C and I and Y*, that I want to avoid in this post. Because they are ignored in the standard NK model. I have discussed them in other posts, and I can't remember where.

Min: are you some kind of engineer? r is the *real* interest rate. With a 2% inflation target, it is possible for r < 0 even if i cannot be < 0. Plus, stick a constant term in the equation, if you like (to represent time-preference). Plus, this whole equation is *expectational*. It is expected future C that is pinned down; current C can and will jump, on new information. You are thinking like an accursed engineer, where past C is pinned down by history. No. This ain't Newtonian mechanics.

[Min]

Nick Rowe: "You are thinking like an accursed engineer, where past C is pinned down by history."

No, I'm just going by what you said.

Nick, you still have not connected ΔC with ΔGt, to show that ΔC ≥ ΔGt, which is what you claim. (That should hold in both nominal and real terms.) You can assume that Chuck Norris runs the CB and that all the consumers are Jeanne Dixons. I am not going to argue with you about that over one blog post. But I still do not see an argument.

[Min]

Note:

When I said that ΔC ≥ ΔGt should hold in both nominal and real terms, I did not mean at the same time, but depending upon whether C and G in the equation C + G = Y were real or nominal. So that

C - ΔC + G + ΔGt ≤ Y

after adjusting for consumers' expectations. :)

[Min]

There is an interesting aspect to expectations in this kind of argument. A bare bones sketch of the argument, IIUC, goes like this.

Scenario 1, which is refuted.

C + G = Y < Y* , because of the ZLB. The gov't increases G temporarily in order to increase Y. C remains the same, because ΔGt is temporary, so now ΔY = ΔGt. After some time, the economy is no longer at the ZLB, but gov't spending continues at the higher level for one time period. In response to that, the CB lowers interest rates in order to reduce Y, which causes C to drop.

This scenario does not happen because when the gov't temporarily increases G consumers anticipate the later lowering of interest rates by the CB and adjust their consumption by an amount appropriate to the anticipated interest rates, which is enough to offset ΔGt with the equal and opposite ΔC, so that Y remains constant, just as though the increase in gov't spending were permanent.

That is the bare bones argument, I believe.

What I find of interest is that the second scenario does not happen, either.

Scenario 2.

C + G = Y < Y* , because of the ZLB. The gov't increases G temporarily to increase Y. C decreases by the same amount, so that Y does not increase. After some time, the economy is no longer at the ZLB, but gov't spending continues at the higher level for one time period. So does the offsetting decrease in C. The CB has no reason to reduce Y, and does not reduce interest rates. In the next time period gov't spending returns to normal, as does consumption.

This scenario does not happen, because when the gov't increases G temporarily, consumers anticipate the future inaction of the CB and do not alter C.

What is going on here? The consumers in both scenarios have incorrect expectations. Why? Because they are being illogical. They are not taking into account the fact that the CB's later action depends upon C. In scenario 1 the logical expectation is not that the CB will lower interest rates, but that the CB will lower interest rates IF C remains the same, that is, if ΔC = 0. Similarly, in scenario 2 the logical expectation is not that the CB will keep interest rates the same, but that it will do so IF ΔC offsets ΔGt. The correct expectations are not of what the CB will do, but of what it will do under different circumstances. The actions of the CB are not independent, but depend upon ΔC. Since the CB is not trying to affect previous consumption by what it does to interest rates after the economy is no longer at the ZLB, ΔC is an independent variable. (In real life the CB may try to affect expectations, but that is not part of any scenario.)