Weird Fiscal multipliers in New Keynesian models under ZIRP

Paul Krugman has bravely and commendably been trying to understand intuitively the underlying reasons for the big differences between fiscal policy multipliers in a couple of New Keynesian models. Someone needs to do this job. (Take as read a general rant against economists who write down fancy mathematical models without understanding or explaining the results intuitively.)

A fiscal multiplier is the model's predicted change in output per dollar change in government spending (or taxes), other things equal. My guess is that the differences in multipliers has little to do with the models, and has a lot to do with the other things assumed equal. By messing with the other assumptions, it's easy to get fiscal multipliers ranging between plus and minus infinity, even in the same, standard model.

The model consists of three equations:

1. An Euler equation, which says that the ratio between consumption this period and expected consumption next period depends (negatively) on the expected real interest rate.

2. A Phillips curve in which inflation accelerates (deccelerates) if Consumption + Government spending is above (below) an exogenous full employment level.

3. A monetary policy reaction function, in which the central bank sets the nominal rate of interest to try to keep inflation on target and output at full employment. But the nominal interest rate cannot fall below zero.

Let's suppose we start out in period one at zero nominal interest rates with output (C+G) less than full employment.

First, let's assume, like Paul Krugman, that the economy will return to full employment in period 2, regardless of fiscal policy. So consumption in period 2 is pinned down by full employment output minus government spending in period 2. And assume the price level in period 2 is independent of fiscal policy, because the central bank adjusts the interest rate in periods 2 and later to make it independent.

Here are some multipliers given that assumption:

1. A temporary increase in G (period 1 only) has a multiplier of one.

2. A permanent increase in G (periods 1 and 2) has a multiplier of zero.

Those two multipliers are noted by Paul Krugman. A third one, he did not mention, is:

3. A future increase in G (period 2 only) has a multiplier of minus one. The reason is that future government spending crowds out future consumption one-for-one, given full employment; and a decrease in future consumptions causes an equal decrease in current consumption, via the Euler equation, given an unchanged real interest rate in period 1.

So, we can get a fiscal multiplier anywhere between plus one and minus one, depending on whether the bulk of the increased government spending happens before or after the return to full employment and nominal interest rates above zero.

Now, those first three multipliers all assumed that the date at which the economy returned to full employment was independent of fiscal policy. But then if fiscal policy does have an effect on current output, it will almost certainly also affect the time it takes for the economy to return to full employment.

Suppose fiscal policy has a positive multiplier, and speeds the return to full employment, so full employment returns in period 2 rather than period 3. If this implies increased consumption in period 2 (due to higher expected inflation and lower real interest rates), then the higher consumption in period 2 will cause higher consumption in period 1. The increased induced consumption will make the fiscal multiplier of a temporary increase in government spending greater than one.

How much greater than one? That depends. Suppose the economy is currently on the knife-edge between full employment and less than full employment at zero nominal interest rates. And suppose that if it slips below full employment, the economy will never recover, but will go into a permanent deflationary spiral. An infinitesimal increase in government spending could then make the difference between permanent full employment and permanent zero employment. The multiplier is infinite. Make less extreme assumptions, and the multiplier will be somewhere between one and infinity.

Now suppose fiscal policy has a negative multiplier, and slows the return to full employment. By a similar argument, the drop in induced consumption will make the multiplier bigger (in absolute value) than minus one. How much bigger? That depends, on how much it slows the return to full employment.

What's the fiscal multiplier in a New Keynesian model? To steal the old joke about the mathematician, the statistician, and the economist, on asked how much is 2+2: "what do you want it to be?".

It's not the model; it's the other assumptions that matter. And I haven't even started playing around with the assumption about monetary policy.

Comments

[Leigh Caldwell]

Maybe I'm oversimplifying, but to me the difference between the two papers was obvious: not because of any great cleverness on my part, but because Eichenbaum et al take pains to explain it in their first page. Cogan et al clearly indicate that the interest rate adjusts upwards as soon as the stimulus takes place, while Eichenbaum specifically holds it at zero for the first couple of years of their simulation.

This seems to be enough to explain the difference in multiplier; if so, the question then simplifies to: which of these conditions does the real world obey? Krugman and krew would argue that the zero interest rate will persist for at least the next year or two; Mankiw and his criw probably expect it to rise faster.

Both papers appear to use similar models (though Cogan's does not explain it in much detail, deferring to Smets-Wouters which I haven't read) so the key difference, as you suggest, seems to lie in the assumptions. But more specifically it seems to be the one specific assumption about the zero lower bound.

Am I missing some subtlety of the assumptions here? Of course there are other parameters in the models which could affect the results too, but I think this is the one that matters.

[brendon]

Combining the results of the papers seems to support the Keynesian idea that fiscal policy is effective in a liquidity trap but not a good idea in an expansion or a "normal" recession. Since we are in a liquidity trap, the Eichenbaum et al paper should be the most relevant. Why is context so often left out of these debates?

[Adam P]

Nick,

I think you've slipped a subtle misunderstanding of the model Krugman is using. You said "let's assume, like Paul Krugman, that the economy will return to full employment in period 2, regardless of fiscal policy". In Krugman's model this is a RESULT, not an assumption.

Krugman says "The simplest NK model is one in which... there’s a one-period “short run” in which prices are fixed, followed by an infinite-horizon long run of flexible prices".

Thus, Krugman is not just exogenously assuming the return to full employment, it is a consequence of full price adjustment (prices here would include wages).

Furthermore, this assumption is really just a way of defining what the periods are, period one is defined to have whatever length is long enough to allow full price adjustment to occur by period two. Thus, a period could be a couple of years.

[Adam P]

Leigh,

I think you're exactly right. It all depends on the fed accomodating a bit of inflation. In fact the cannonical NK phillips curve says that expected future inflation should show up in inflation today so, according to that model, without the fed accomodationg a bit of inflation neither fiscal nor monetary policy will produce a recovery.

Theory fleshed out here http://canucksanonymous.blogspot.com/2009/06/phillips-curve-in-liquidity-trap_01.html

[Adam P]

Nick,

when you say"read a general rant against economists who write down fancy mathematical models without understanding or explaining the results intuitively", I can't even begin to express how strongly I agree with you here.

In fact, I think you could say it better as simply: "read a general rant against economists who write down fancy mathematical models without understanding the results."

[Adam P]

Nick,

Just to flesh out what I'm saying in my first comment above, the point I was making applies when you say:

"Now, those first three multipliers all assumed that the date at which the economy returned to full employment was independent of fiscal policy. But then if fiscal policy does have an effect on current output, it will almost certainly also affect the time it takes for the economy to return to full employment."

This is not quite right. The economy having full employment in period 2 is a consequence of prices being fully flexible by the beginning of period 2, full employment in period 2 is in no sense a result of fiscal stimulus. The temporary first period stimulus simply increaes period one output.

Thus, all your examples in the second part of the post, after the statement I just quoted, don't apply to Krugman's model.

Moreover, this really goes to a distinction that people are often not careful about (I was trying to make this point in the discussion of your 'why fiscal poliy won't work competition'). We need to decide on what we mean for fiscal stimulu to "work". There are two distinct questions:

1) Can fiscal policy increase output in a liquidity trapped economy? Clearly yes.
2) Can fiscal policy end the recession, break the trap? Only by increasing expected inflation which could also be done by monetary policy (and doing it with monetary policy is clearly the better way).

[Nick Rowe]

Leigh: what you say sounds very plausible to me. The implications of "Holding monetary policy constant" depend very much on what precisely we mean by that.

As Brendon says, in normal times, when the Bank of Canada can and does adjust monetary policy to keep inflation on target, which means completely offsetting any effect of fiscal policy on AD, the fiscal multiplier is precisely zero. But at times like the present, it's more likely that the Bank will not offset any effect of fiscal policy, at least until AD is where the Bank wants it to be.

Yep, this illustrates precisely the general point. Forget about the precise details about the model. They barely matter, if at all. It's the assumptions about monetary policy, and what precisely is being held constant, that really do matter.

Adam: I interpret PK differently. But since neither of us can win that argument, let's argue instead about what he ought to have meant.

In normal times, outside a liquidity trap, and in a model where the Bank sets (say) M exogenously (it does not try to target inflation for example), it would make a lot of sense to assume that prices are fixed in period 1, and prices are flexible in period 2, and that price flexibility ensure we return to full employment in period 2. The length of the period is whatever time it takes for prices to adjust.

But in a liquidity trap, with i=0, it's not obvious that flexible prices in period 2 will get the economy back to full employment. Sure, a fall in P2, holding expected P3 constant, would reduce the real interest rate in period 2, and give you full employment in period 2. But then would P3 stay constant, and be expected to stay constant?

If there is inertia in inflation, and/or in expected inflation, it wouldn't work like that. For a given output gap, if prices fall from 1 to 2, they will fall even faster (or be expected to fall faster) from 2 to 3. So price flexibility in this case may make things worse, by increasing the real interest rate. (This is the guts of an old Brad DeLong(?) paper "Is price flexibility stabilising?")

In other words, if we think of monetary policy in Neo-Wicksellian terms (my maintained assumption throughout this post), where it's the nominal interest rate, it is decidedly unclear if the economy can ever escape from a liquidity trap by itself. It seems more likely the economy would go into a death-spiral of faster and faster deflation and expected deflation. (Strange it doesn't seem to happen, as in Japan, but that's another question).

Now we can always introduce some deus ex machina to get the economy back to full employment in period 2 (or sometime in the future). For example, people age, or cars wear out, so everybody starts buying stuff in period 2. This assumption makes logical sense, and allows us to solve the model. I thought this was what PK was doing.

Now, if PK is not doing that, and is instead assuming that fiscal policy is what gets us back to full employment in period 2, we have a problem with his multipliers. In a linear model, we calculate the multiplier by comparing output under two alternative fiscal policies: increase in government spending; no increase in government spending. But if the increase in government spending is what gets us back to full employment in period 2, then we can't hold Y2 constant when we compare Y1 under each of those two policies. And PK does hold Y2 constant when calculating the fiscal multiplier.

[Too Much Fed]

Do fiscal multipliers (more gov't debt/less gov't saving) depend on suckering the private sector into more debt?

[Nick Rowe]

There's a basic accounting identity that says:

(private Investment-private savings)+(Government spending-Taxes)+(exports-imports)= zero

(I-S)+(G-T)+(X-M)=0

Which we can think of as:

Net private borrowing + net government borrowing + net foreign borrowing (from us) = 0

In a closed economy (where X=M=0): an expansionary fiscal policy means government borrowing goes up and private borrowing goes down by the same amount.

In an open economy, it depends on the model, but the usual result is that net government borrowing goes up, and net private borrowing goes down, but by a smaller amount, so we borrow more from foreigners to fund part of the deficit.

[Nick Rowe]

Notice, by the way, that monetary policy (at least in a closed economy) has zero effect on net private borrowing. This must be true because if we hold fiscal policy constant, (G-T) does not change, so (I-S) cannot change either.

Magic, isn't it? How can that possibly be right? Because isn't the point of lowering interest rates to get people borrowing and spending? And yet the accounting identity says there cannot be any change in net private borrowing??

The key is that the private sector TRIES to increase net borrowing, but the only thing that happens is that as spending increases, income increases, so net borrowing stays the same.

[Too Much Fed]

Magic? Um no.

I think there are some points that need expanded upon.

Maybe start at the beginning?

[Nick Rowe]

Too much Fed: "Maybe start at the beginning?"

I'm not sure if it's what you wanted, but my latest post "Income = expenditure, and debt" is the result of my thinking about your question.