Paul Krugman asks a fair question. If anyone disagrees with what Mike Woodford says about the New Keynesian government expenditure multiplier, can they point to where exactly they disagree with Mike Woodford's model (pdf).
I am going to do just that.
I'm not 100% sure I'm right (we never should be). But I think Mike Woodford is making exactly the same mistake that Greg Mankiw made decades ago in an otherwise very lovely little paper which purported to show Keynesian results in a macroeconomic model with monopolistic competition.
In a nutshell: Mike Woodford is implicitly assuming that government expenditure is built with corvee labour. It doesn't buy goods from monopolistically competitive firms like everyone else. It just commandeers labour from everyone and makes us build goods for the government. That shifts the total labour supply curve right. And Woodford's workers are always on their labour supply curves. It's a supply-side multiplier!
I am and always have been cr*p at math. So let me give you the intuition of Woodford's model.
Assume that at some very very distant future date T government spending goes back to zero (or whatever). The equilibrium at that time pins down consumption at time T, C(T). Now assume the central bank holds the real rate of interest constant all the way from now to time T (Woodford makes that assumption). Given separability of the utility function in hours worked and consumption (which Woodford also assumes), that plus the consumption-Euler equation pins down the whole time-path of consumption all the way back to the present regardless of the level of government spending (which is what Woodford says too).
Now assume that the government requires everyone to work one hour per day producing goods for the government. Since consumption stays the same (and there is no investment in this model) hours worked to produce those consumption goods stays the same, and so everyone consumes less leisure because they are doing additional forced labour for the government. Y=C+G, and since C stays the same, Y must go up by whatever amount G went up. Et voila! Balanced budget multiplier of 1.00. BINGO!
(In a monopolistically competitive macro model, there's a gap between the real wage and the marginal product of labour, which helps make it all possible. And it can be welfare-improving because by assumption the government can get around buying goods produced by monopolistically competitive firms, so doesn't have to pay a markup.)
[Update: And the non-superneutralities of a Calvo Phillips curve lets him have the central bank hold the real interest rate fixed forever despite changes in G without the price level exploding or imploding. All the really nasty stuff is buried deep in footnote 8. Oh, God, no he really can't do this. If G changes, that changes the natural rate, so you can't hold the real rate constant forever and have G change any way you want it to change!!
"I have, however, made one crucial assumption about the supply side: I have
supposed that it is possible for monetary policy to maintain rt = ¹r at all times,
regardless of the chosen short-run path of government purchases. This assumption
is violated by the model with fully flexible wages and prices. However, under many
specifications of sticky prices or wages (or both), it is possible for monetary policy to affect real interest rates, and a path for monetary policy can be chosen under which rt = ¹r will hold, in the case of any path for government purchases satisfying certain bounds."]
Hey, if I believed Woodford's result, I would immediately insist on a massive increase in government spending from now until just before the end of time! But that result is nuts!
God I hate fancy math models! Yes, I know, sometimes rigour is needed; I just hate doing it myself. And they do hide a multitude of sins if the intuition isn't there along with the math. Oh God this takes me back decades to when I was first playing around with government expenditure multipliers and monopolistic macro in 1987. And lay awake all night trying to figure out where Greg Mankiw got his result.
But I still might be wrong. I never did understand math models. I don't think anyone really does. Tell me what you think about the model.
P.S. J.M. Keynes would reject Woodford's model. Because Woodford's model is on the labour supply curve the whole time (Keynes' Classical Postulate 2(?)). See Woodford's equation 1.4, which is what Keynes explicitly denied. Even worse, all the real action comes from the corvee labour shifting that labour supply curve to the right. That ain't Keynesian!
Worthwhile Canadian Initiative 
Which totally contradicts the results of Woodford’s model. He says the multiplier is one regardless of the duration of the increase in G, given his assumption of r constant. So I have taken him at his word, and said “OK, let’s make the duration permanent”. He says the time path of C will be unaffected, given r constant. It can’t be.
See where I’m coming from?
Now, I sort of see how he’s getting his results. Because he’s assuming there’s some future time T at which everything, including G and C, returns to normal. And he wants to assume r is held constant up until that future time T. And then he wants to take the limit as T goes off to infinity. But that result makes no sense at all. Because we know it can’t work at the limit, when the increase in G lasts forever.
I do see where you’re coming from, but I’m a little confused about the ultimate goal here. Is the idea that since Woodford’s model doesn’t make sense in a limiting case, there must be something wrong with it more generally?
In fact, I think the same model can give a multiplier of one for a permanent increase in G, although this depends on some quirks of the New Keynesian model and relies on a patently unrealistic specification of monetary policy.
The reason is that the log-linearized Keynesian “LRAS” curve, depicted in (y,pi) space, is not quite vertical, just extremely steep; if you take the NK Phillips curve pi_t = betapi_{t+1} + kappay_t and remove the time subscripts, you get pi = kappa/(1-beta)*y. For reasonable values of the discount parameter beta, 1/(1-beta) is really huge, so this implies that a permanent departure of the output gap** y from zero results in extremely high inflation. But it’s still possible in the model; nothing breaks down, and inflation is finite rather than infinite.
Now, in practice, do I care about or believe this mapping between permanent output gaps and inflation? No. There’s no way that a log-linearized Calvo pricesetting model is going to be accurate at such extremes, and the specter of “extremely high” inflation (as opposed to infinite inflation, or a breakdown of the model) is bad enough to make me want to avoid such policy anyway. But technically, within the context of the model, it works.
The other issue that arises when there is a “permanent” increase in G in this model is policy implementation. If there’s a date T “at which everything returns to normal”, then policy implementation is rather straightforward. From date T onward, the central bank is following some kind of standard policy (e.g. a Taylor rule) to make sure that everything is “normal” and consistent with its goals; before date T, the central bank doesn’t have to worry about unique implementation, since expectations for date T are pinned down, so it can literally just set the policy rate equal to a constant plus expected inflation to keep the real rate constant. If there is no terminal “T”, on the other hand, the central bank has to worry about uniqueness of equilibrium (which needs to be implemented with a full-fledged policy rule, not just an interest rate target) while also following the rather strange practice of keeping the equilibrium real rate constant in response to government spending shocks.
And that leads us to another quirk of the New Keynesian model: even though the distant long-run level of the output gap exerts enormous influence on the current output gap (one-for-one, if you hold real rates constant!), the monetary policy rule is the only thing pinning down the long-run output gap. In this case, we’re no longer engaging in a thought experiment where the central bank passively keeps the real rate constant in response to government spending shocks; rather, we’re talking about a central bank that is actively adjusting its long-run policy rule. And then the result becomes ambiguous: yes, the central bank could choose to increase the long-term output gap at precisely the rate consistent with a multiplier of one on government spending shocks, thereby incurring a very high rate of inflation. This is why there is formally no discontinuity. But it could also choose a long-term output gap of zero, and this would still be consistent with holding real rates constant. (After all, in this model permanent changes in government spending do not affect the natural real rate.) It could choose any other level too! That said, realistically speaking, if we suppose that the central bank is committed to the goal of keeping equilibrium real rates constant, but that it will choose the most efficient equilibrium among the multiplicity of equilibria consistent with that goal, then we’ll get the zero output gap, zero inflation equilibrium, and the “multiplier” will be the classical supply-side multiplier, lying somewhere in the open interval (0,1).
It’s very messy, which is why Woodford limited his discussion to a finitely lived change in the path of government spending. But I don’t think it’s the sort reductio ad absurdum that proves there is something fundamentally wrong with Woodford’s model. It just shows that the thought experiment of “holding real rates constant” isn’t very useful when evaluating permanent changes in government spending. And I think we can all agree with that.
**By the way, when I talk about the “output gap” here, I mean the modified output gap that appears in Woodford’s model, which doesn’t treat consumption and government spending symmetrically. If government spending increases, keeping this “output gap” at zero implies decreasing consumption spending less than one-for-one. The asymmetry arises because government spending is not substitutable with consumption spending in the household’s utility function, so that an increase in government spending strictly increases the neoclassical equilibrium level of total output. This is just a “wealth effect”, which you mention.
That reduction in the demand for leisure is a rightward shift in the labour supply curve, which shifts the LRAS curve right. If I rigged the parameter values just right, I could make that rightward shift in the LRAS curve big enough to make Y increase by delta G, so C stays the same. Which would make my results the same as Woodford’s results. Essentially, I would need to assume that any taxes are paid for by working harder, rather than by cutting consumption.
I don’t think it’s possible to “rig the parameter values just right” in this case so that you get this outcome from the supply side, unless you assume that consumption demand is perfectly inelastic or labor supply is perfectly elastic. The supply side is basically the condition u'(C) = v'(C+G). If as usual we assume u” < 0 and v” > 0, then an increase in G will always decrease the value of C that solves this equation.
On re-thinking, though, Mike Woodford’s math isn’t following my intuition on this, because he totally ignores the supply side in deriving his multiplier.
Essentially, the policy of keeping real interest rates constant, and having some future date T at which everything is normal, pins down the level of consumption. One can forget about the actual policy here and just think of this as vertical “aggregate demand” in (Y,pi) space. The supply side determines the amount of inflation that results from this policy, but it doesn’t determine output, because monetary policy is constructed in the precise way necessary to keep consumption invariant to supply considerations. That way, government spending increases total output one-for-one. (Of course, the supply side still matters for determining whether this form of policy is a good one, both by determining the inflation-output tradeoff and also, inflation aside, by determining the welfare loss from having a suboptimal output gap.)
I really like this as a benchmark result, because it provides a good intuition to compete with the “crowding out” intuition that’s so common. Ask the question: why should we expect a temporary change in government spending to cause a decrease in consumer spending? If there is an effect, it has to work through the optimization problem of the household. In the NK model with lump-sum taxes, there are basically three ways this can happen: (1) through the path of real interest rates, (2) through the long-run level of consumption, or (3) through an interaction between government spending and consumption in the utility function. Woodford’s point here is that if you don’t have (1)-(3), there’s simply no mechanism through which government spending can possibly affect consumer spending.
I think this kind of exercise—looking at the fundamentals of agents’ optimization problems, and spelling out what, exactly, is causing their behavior to change in response to some policy—is a very useful one. That’s why I like NK models, and I don’t like MV=PY. 🙂
This isn’t to say that there aren’t many other channels, lying outside the stripped-down NK model, through which temporary changes in government spending could potentially affect consumption spending. And it isn’t to deny that a more realistic specification of monetary policy would have different results. But…
Take the standard way we do multiplier analysis the old-fashioned way. ISLMADAS for example. We do it in three steps.
This brings me to the core of my argument. I think that Woodford’s approach in this paper is fundamentally an expansion of the traditional, ISLMADAS approach.
Why? Well, we all know that it’s impossible to say anything about the “multiplier” effects of fiscal policy without some specification of the monetary policy. But the true monetary reaction function is assuredly very complicated, and it’s not something that would be practical or desirable to include in every discussion of fiscal policy, or macro in general. (Imagine that whenever we had to talk about macro we had to start with some ugly empirical specification of monetary policy, with 10 variables and 5 lags, derived from some poorly identified VAR study. It would suck!)
Thus, in practice, we consider many models with idealized, unrealistic depictions of monetary policy—not in the hope that they will serve as literally accurate models, but in the hope that they will provide us some useful intuition, intuition that we can carry over into more complicated models and policy discussions.
In my view, this is the only cogent argument for using a framework like ISLMADAS. Implicit in that model is a deeply unrealistic specification of monetary policy. (To be honest, I’m not sure exactly what it is, but I think that it’s some kind of money or NGDP target.) There is no way that this specification comes close to describing monetary policy as it is actually practiced. I think most of us can agree on that—you and Scott Sumner might want policy to be governed by an NGDP target, but it isn’t yet! 🙂
Yet you still use this as an organizing framework for your thoughts. And that, I assume, is because it can provide some useful insights about macroeconomic policy, even if the monetary rule that closes the model has dubious empirical relevance. It’s a shortcut. And shortcuts can be very important tools in economics.
I think that’s the essence of Woodford’s “real interest rate constant” exercise. It’s not empirically accurate. As we’ve already discussed, it doesn’t make much sense as a long-term specification of policy. But it’s still a useful shortcut, a guide to intuition. In particular, it gives us a basis for thinking about real-world cases where monetary policy, in the short run, isn’t so far from a real interest rate target. This may be true (1) in the very short run in normal times, where inflation expectations are well-anchored and the central bank’s reaction to a temporary surge in government spending is delayed or nonexistent, and (2) at the zero lower bound, which is currently the main object of concern.
Now, one can argue that at the zero lower bound, monetary policy is more complicated than a simple “policy rate = 0” rule, since the central bank can make commitments about policy after the trap, and those commitments can change in response to fiscal policy. That’s where more elaborate exercises like Werning (2011) come in. But if you subscribe to the view that the zero lower bound is a true constraint, then this is a very useful guidepost indeed.
And Woodford has many other guideposts as well—the neoclassical guidepost, the strict inflation targeting guidepost, and so on. All empirically unrealistic depictions of monetary policy designed to convey a particular intuition. Just like ISLMADAS.
Matt: (Your comment got caught in our spam filter, which plays up from time to time.)
“I do see where you’re coming from, but I’m a little confused about the ultimate goal here. Is the idea that since Woodford’s model doesn’t make sense in a limiting case, there must be something wrong with it more generally?”
That’s partly it. Partly I’m also just trying to understand what is really going on in that model (and you are helping me a lot with that).
“The reason is that the log-linearized Keynesian “LRAS” curve, depicted in (y,pi) space, is not quite vertical,…”
That’s what I suspected. That’s what I was referring to by the “non-superneutralities of the Calvo pricing model”. Thanks for confirming that your intuition matches mine.
“Now, in practice, do I care about or believe this mapping between permanent output gaps and inflation? No.”
Me too. We are very much on the same page here.
“And that leads us to another quirk of the New Keynesian model: even though the distant long-run level of the output gap exerts enormous influence on the current output gap (one-for-one, if you hold real rates constant!),…”
Aha! Yes, that is the feature I’ve been trying to get my head around. Everything sort of “scales”, across time, in a way that old Keynesian IS curves don’t scale. Start with one equilibrium time-path for C, then double C at all periods in the time-path, holding the time-path of r constant, and you have a second equilibrium time-path, as far as the IS equation is concerned. That doesn’t happen with an Old Keynesian downward-sloping IS curve. It’s like having mpc=1 in an old Keynesian model. So you can’t really even talk about how much the New Keynesian IS curve shifts right for a permanent increase in G, because that “very long run” IS curve is horizontal. All you can say is that it doesn’t shift vertically up, if there’s a permanent increase in G. In that sense, a permanent increase in G doesn’t change the natural rate, and so it is like a bond-financed transfer payment under Ricardian Equivalence.
“(After all, in this model permanent changes in government spending do not affect the natural real rate.)”
Aha! You have just said the same thing! (As you can see, I’m taking your comment very slowly, one step at a time.)
I just want to complete your thought here, to check if I’ve understood you properly:
“That said, realistically speaking, if we suppose that the central bank is committed to the goal of keeping equilibrium real rates constant, but that it will choose the most efficient equilibrium among the multiplicity of equilibria consistent with that goal, then we’ll get the zero output gap, zero inflation equilibrium, and the “multiplier” will be the classical supply-side multiplier, lying somewhere in the open interval (0,1). ” And where exactly it lies between 0 and 1 depends on the elasticity of labour supply with respect to wealth (and stuff like that). Right?
“It just shows that the thought experiment of “holding real rates constant” isn’t very useful when evaluating permanent changes in government spending. And I think we can all agree with that.”
Basically agreed. Except I sort of find it useful in Old Keynesian models to break the multiplier down into 3 steps: how much does IS shift right?; how much does AD shift right?; how much does Y or P increase?
Now I have to change my first step to: how much does IS shift up?
This is very helpful Matt. I am really pleased to see our intuitions converging on this stuff. Thanks!
Now, what do you make of my monetarist derivation of the BBM? 😉
Matt: Now working through your 3.08 comment:
I’m with you until:
“In my view, this is the only cogent argument for using a framework like ISLMADAS. Implicit in that model is a deeply unrealistic specification of monetary policy. (To be honest, I’m not sure exactly what it is, but I think that it’s some kind of money or NGDP target.)”
In the simplest textbook model, Ms is constant, so it’s a money supply target.
“There is no way that this specification comes close to describing monetary policy as it is actually practiced. I think most of us can agree on that—you and Scott Sumner might want policy to be governed by an NGDP target, but it isn’t yet! 🙂”
I think of it as an inflation (forecast) target. Crudely, the Bank of Canada chooses: a horizontal LM curve for the very short run (6 weeks) because that’s how frequently it changes the nominal interest rate; a vertical LM curve in the short run (one year?) because it tries to keep Y close to what it thinks is potential Y; a horizontal AD curve at 2% inflation (you need to now put inflation on the axis to draw the LM curve) in the medium run (2 years).
Again, your comment is very helpful.
My main critique of Woodford’s model now shifts to my more recent post!