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Can't comment on Woodford's model, but I do find interesting is comparing your views and Krugman's on the roles of intuition and formal mathematics. Krugman likes to emphasize formal math--often in the form of deliberately simplified and unrealistic toy models--as a check on our pre-mathematical intuitions. Even very simplified, toy versions of the world are full of feedbacks and nonlinearities and other things that make it difficult to reason verbally about how they'll behave. Math forces you to make your assumptions explicit and precise, and forces you to logically derive all of their implications. This isn't to say that intuition is irrelevant; one reason we do the math is to teach ourselves new, better intuitions. That certainly the way I often use math in my own work.

So I'm interested to find you suggesting here that math can be a way to paper over lack of intuition, or justify bad intuition, rather than a way to better train our intuition. I'm intrigued and will need to think more about it. But my first reaction is to side with Krugman. I'm way out of my depth here, but Krugman seems to cite Woodford's model because it undermines certain intuitions about the effects of accounting identities and anticipation of future taxes. If Woodford's model does that only by assuming that the government can commandeer labor, well, the math has still undermined those intuitions, hasn't it? Or at least clarified the precise contexts in which those intuitions hold?

I mean, every simple model (and Woodford's model, for all its fancy math, certainly makes some radical simplifying assumptions) is going to make some highly unrealistic assumptions--that's what makes it simple. And for that reason, every simple model is going to behave oddly in some ways. Which to me just means that we have to be precise about the particular intuitions about which a particular model has something useful to say.

My suspicion, thought, is that you're not really so far from Krugman on the roles of intuition and math. I've always had the sense that your wonderful thought experiments aren't really so very different in "style" from Krugman's preferred toy models like IS-LM. I mean, yes, your thought experiments involve money, or are designed to reveal the importance of money, and Krugman's aren't. But your thought experiments are still descriptions of highly simplified worlds, and further they could easily be expressed mathematically (right?) So are your differences with Krugman not really about preferring intuition over math, and more about the specific mathematical models that you find most useful as a shaper of, and check on, your intuitions?

Jeremy: "My suspicion, thought, is that you're not really so far from Krugman on the roles of intuition and math."

Your suspicions are correct. And if I had to pick on some economist for relying on math and forgetting intuition, it certainly wouldn't be Paul Krugman.

I'm just a bit peeved at having to face the stress of working through Woodford's model and being stunned to see what's really driving his results. Which is so very very different from what Keynesians normally think of as the mechanism through which government expenditure works. Woodford's effects are basically coming from the supply-side!!) Forcing those workers (who are already working as many hours as they choose at the market wage) to work extra hours for the government!

Jeremy: "So are your differences with Krugman not really about preferring intuition over math, and more about the specific mathematical models that you find most useful as a shaper of, and check on, your intuitions?"

I'm not sure about that. I can't put my finger on it. But in this particular case, I don't think the model is saying the same thing as his intuition is saying.

I think the thing we have lost sight of is that we are trying to prove the integrity of the following two statements:

1) Spending supported by taxes pretty obviously won’t work: If the government taxes A by \$1 and gives the money to B, B can spend \$1 more. But A spends \$1 less and we are not collectively any better off.

and...

2) You apply a multiplier to the bridge builders, then you've got to apply the same multiplier with a minus sign to the people you taxed to build the bridge.

Which are both saying the multiplier is zero right?

Is it plausible that the effect of the gap between real wages and marginal product of labor would be to drive the multiplier from one to zero?

If so do those conditions even remotely resemble the situation in North America from 2007-2012?

Mike Woodford is implicitly assuming that government expenditure is built with corvee labour.

I'm certainly not going to say that this is false, but I'd like a pointer to whatever in the paper supports it. AFAICT the idea is that the government buys output on the same terms as everyone else. Are you basing this on earlier Woodford papers?

Michael: "2) You apply a multiplier to the bridge builders, then you've got to apply the same multiplier with a minus sign to the people you taxed to build the bridge."

Yes, but you've also got the bridge (if the bridge builders weren't doing anything before).

"Yes, but you've also got the bridge (if the bridge builders weren't doing anything before)."

This goes to points 5 and 6 in your previous post, I see.

But what is it about the gap between real wages and marginal product of labor we are supposed to take away here?

Kevin: look at 1.2, 1.8, and 1.11. Only the consumption good has monopolistic competition. Now sure, I have simplified a little, because the government buys the raw intermediate good in a perfectly competitive market, rather than labour. But we can replace Woodford's production function 1.2 with Y=H with no loss of generality.

Actually, the government doesn't really *buy* the raw intermediate good. It just takes it. There is no monetary exchange explicit in this model. Which is a whole other question I didn't want to get into.

And there's at least one other nasty buried deep in this model too, with that Calvo Phillips Curve and its non-superneutrality, which is the only thing that allows Woodford to keep the real rate constant while varying G without violating the markup over costs implied by the fixed elasticity of substitution between varieties and real wages varying with G.

Michael: "But what is it about the gap between real wages and marginal product of labor we are supposed to take away here?"

Not easy to explain clearly and quickly. In a perfectly competitive model there is never any gap. In a monopolistic model with neutrality and superneutrality of money, and with fixed elasticity of varieties in the Dixit Stiglitz preferences, that gap would be fixed, and anticipated changes in G would cause real interest rates to change. Woodford can only avoid that and keep real interest rates constant while G fluctuates with his Calvo Phillips Curve.

If that's "simple analytics," what are complex analytics? :)

A bit more rigor will cut through the confusion, i.e. we need to formalize the hidden assumption shared by all of these Keynesian models and put all of this in rigorous mathematical equations.

So, here we go, cutting through the verbal confusion with some perhaps difficult but strictly rigorous math:

Let "J" be short for Jesus, and let "J" = "and then a miracle occurs". Whenever your model specifies that, e.g. Y increases via G increasing, cut through the verbal confusion, and introduce rigor into your economics, using the formula GJ + I + C = Y, where J giver gives clarity to the notion the any change in G means more vale, wealth, goods in terms of dollars, etc. as a function of J.

Are neutrality and non-superneutrality synonymous terms?

Mike: money is neutral if the level of money supply and prices has no real effect. Money is superneutral if the growth rate of money and rate of inflation has no real effect.

Nick: So non-superneutrality implies growth in the rate of money has an effect but does not imply anything with respect to the effect of the level of money, right?

Edit: "growth in the rate of" immediately above should read "the growth rate of"...

Michael: correct.

Michael: "2) You apply a multiplier to the bridge builders, then you've got to apply the same multiplier with a minus sign to the people you taxed to build the bridge."
Nick: "Yes, but you've also got the bridge (if the bridge builders weren't doing anything before)."

Well, no. It depends on who you taxed.

For example, if you were taxing a class of people who were saving money and sending it to relatives in a different country then the multiplier would be different (higher) than if they weren't.

More generally, if the gov't taxes A and gives it to B then the multiplier depends on both who A and B is and what their respective marginal propensity to spend rates are.

If B is very poor and is behind on rent and food and if A is a super saver who wasn't going to spend the money immediately but in, say, 5 years time then the multiplier is going to be high. If you also say the time effects cancel out then, again, it depends on who A is. For example, if A is a profit sharing owner of a business where B has spent the money (say the owner of the grocery store) then B's income goes up as well in the short term as net profits increase as revenues goes up. So net deferred saving is less than the amount actually taxed.

A converse reverse Robin Hood example would have a similar but negative effect on the multiplier.

Sorry typo: in the above "B's income goes up" should be "A's income goes up".

What I'm wondering about is why people would even care about this murky controersy. To me, all it shows is that the income-expenditure approach to macro is deeply misguided. A quasi-monetarist has no trouble with positing that some budget-balanced policy choices (possibly including tax-funded government spending) might cause an expansionary effect by decreasing the demand for money balances and increasing velocity. Sure, one can try and make the same arguments in income-expenditure terms, but this introduces a lot of additional compexity; I don't really see the point.

in the case of any path for government purchases satisfying certain bounds

Surely this caveat means that his result would hold only for a small delta in G, which would counter your:

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!

Since it wouldn't be a small change.

To be honest I find this entire arguement a bit silly. I think Krugman is right to the extent that you cannot logically prove that the multiplier is zero (which is what he stated in the last post). That doesn't mean it is different from zero though...

As you said in point 7 of your previous post it all depends on the substitue-complement relationship of leisure-private good-public goods... One could settle the score with experiments, but those are more or less impossible in economics... I think sometimes we need to accept that we cannot know...

You can make the assumption that monetary policy can offset any fiscal policy action, which gives zero multiplier again (given a certain monetary regime). It's not a bad assumption, it sounds right, but can you prove it? I think not... Arguing about which assumption is right without the benefit of experimenting to determine who is really right is a bit pointless...

For example, I think it's standard practice to assume that GDP will grow after a natural disaster... Why does that induce people to work more when we generally think of taxation as inducing people to work less? The effect on the individual is somewhat similar...

Nick,

I haven't read the entire Woodford paper but I looked at the first few sections and it's enough to see that this:

"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"

is wrong. Woodford's economy moves along the labour supply curve, it does not shift it's labour supply curve.

Even in section 2 Woodford preview's this point:

"In the next section, I present the equations of a particular
familiar model of price adjustment (the model with °exible wages and Calvo-style
staggered adjustment of prices), and show how it is possible to determine the mon-
etary policy required to keep the real interest rate constant in that model.

...

It may seem surprising that the multiplier in this baseline case is independent of
the degree of °exibility of prices and wages; there thus appears to be a discontinuity
in the case of complete °exibility (and full information), where the multiplier is given
by (1.7). The explanation is that the derivation of (2.3) requires that it be possible
for monetary policy to maintain a constant real interest rate despite an increase in
government purchases; and while such a policy is technically possible, according to
the model of price adjustment presented in section 3.1, for any positive degree of
price stickiness, as the degree of price stickiness becomes small, the required degree
of in°ation becomes extreme."

Basically, the increase in government purchases leads to an increase in inflation and thus the CB lowers the nominal rate to keep the real reate fixed. The increase in labour supply is accomplished by an increase in the the real wage. Those firms that can adjust their output price do so and maintain their margin, those that can't compress their margins to meet the higher demand which they are willing to do because initially p > mc.

I think my comment was eaten...

I'm lost right from the start. Probably a stupid question because this is way over my head, but nonetheless...

If the model is driven by the representative agent maximizing the (discounted) difference between the utility of consumption and the disutility of working (for wages to consume), then why would they ever agree to supply labour for zero wages? Seems to me it would never be utility maximizing to do that. But I'm probably wrong.

Adam: The production function is Y=F(H) where H is labour. Invert it to get H=G(Y). The utility function, IIRC, is U=u(C) - v(H). Substitute the inverse production function into the utility function to get: U=u(C) - v(G(Y)). Rewrite the utility function as U=u(C)-z(Y). Note that z(Y) has all the normal properties. Now substitute in Y=C+G and we get the utility function as U=u(C)-z(C+G).

It is exactly as if the government took away G of your hours, and let you choose how many extra hours C you would sell to firms to produce the Dixit-Stiglitz good. That's like a wealth effect of a lump sum tax on your leisure.

I'm re-thinking my critique though. There are two things going on in this paper: the supply side effects of the tax (plus the fact that the government doesn't buy the Dixit-Stiglitz good but buys Y instead, and so pays no markup); the demand-side consumption smoothing effects. As far as I can see, the exact numerical value of 1.00 for the multiplier is coming purely from the demand side. The supply side multiplier is there, but plays no role in the exact numerical calculation.

Patrick: with a tax you are always forced to give up something for free. Normally, you give up some of your money income for free, and the government uses it to buy stuff that you willingly sell. But the government could also just take some of your labour, or take some of the goods you produce.

Adam: The way I originally thought of it, 25 years ago, is that if you are doing macro with monopolistic competition, then if the private sector has Dixit Stiglitz preferences then the government ought to have D-S preferences too. And a benchmark assumption would have the same elasticity of substitution in each, so the markup of price over marginal costs is the same for private and government purchases.

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!

Nope, he assumes that government expenditure is purchased from monopolistically competitive firms---though this actually doesn't make much difference in his argument, except for one minor point I'll mention later.

This is apparent, for instance, from the discussion on page 5 (when he's introducing the monopolistically competitive framework, and before he introduces sticky prices), where he says that "H_t is the common quantity of labor hired by each firm" and mentions that relation (1.2), which was previously the representative firm's production function Y = f(H), must now be satisfied as an aggregate production function for monopolistic producers. Y includes G, as is apparent in equation (1.11) and elsewhere, and since Y is a function of the quantity of labor hired by each firm, it's clear that G must be produced by firms as well.

This is also clear from the nature of his results in the "zero lower bound" part of the paper. Equation (4.7) gives the multiplier from an increase in government spending during the liquidity trap. This multiplier depends on the supply side of the economy only through the parameter kappa, which in turn depends only on the nu_u and nu_v, which are the elasticities of u' (the marginal utility function) and v' (the marginal disutility of supplying output). As defined on page 3, the function v combines workers' utility from leisure and firms' production function f. If the government was commandeering labor from workers directly, only the former would be relevant---but since they're both bundled up in a single parameter (the marginal disutility of supplying output), the government must be obtaining output through the normal production process.

But even if the government was obtaining goods by directly commandeering labor, it wouldn't matter much for the "multiplier" analysis. Roughly speaking, the multiplier at the zero lower bound is composed of two parts: (1) a multiplier of 1, from the benchmark case where the real interest rate is constant, and (2) an additional multiplier, resulting from changes in the real interest rate arising from increases in expected inflation, which in turn arises from increases in government spending. Government spending matters for expected inflation because it increases the real marginal costs of firms. If the government is purchasing goods from these firms, this increase in real marginal costs is a combination of (A) the concavity of the production function and (B) the convexity of workers' marginal disutility of labor. If the government is commandeering labor, then we just have (B). In practice, that's a (probably small) quantitative difference, not a qualitative one.

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!!

Narrow answer: that's why he adds the qualification "for government purchases satisfying certain bounds". If it's impossible to keep the real interest rate at a given level for an arbitrary path of government purchases, that's because the anticipated deviation between the natural rate and the target rate results in infinite (or impractically high) inflation. It would be a distraction to spell out the exact relationship between arbitrary paths of government purchases and anticipated inflation, so he skims over it and mentions "certain bounds".

Longer answer: the "holding the real interest rate constant" exercise is meant as a pedagogical device, a useful benchmark, not as a realistic model. You have to understand the context here. There were two widely cited papers written about the multiplier near the zero lower bound (Chistiano, Eichenbaum, Rebelo and Cogan, Cwik, Taylor, Wieland), both of which used extremely opaque and difficult-to-parse computational models, with the bells and whistles typically added in the computational NK literature. The two papers found different results (the multiplier was higher in CER than CCTW), which was ultimately traceable to the simple fact that more spending came online during the zero lower bound in CER's model than CCTW's---but since there was no simple, theoretical treatment of the issue, many people didn't readily understand this point, and there was a lot of totally unnecessary confusion.

Woodford's paper was designed to cut through this muddle and explore the issue at a more intuitive level. It starts with several intentionally unrealistic thought experiments (perfect competition, monopolistic competition with flexible prices, stable real interest rates, strict inflation targeting, etc.) to clarify the conceptual terrain and provide useful benchmarks. It proceeds to consider the case of the zero lower bound, which is the application everyone really has in mind. This is the meat of the paper.

In this case, real-world departures from the "stable real interest rate" benchmark are actually a feature of government stimulus, not a bug. At the zero lower bound, nominal interest rates are (temporarily) fixed; given this, higher anticipated government spending during the trap will result in higher expected inflation and therefore lower real rates, giving an added kick to the stimulus.

In practice, I wouldn't be surprised if this "added kick" is very small, meaning that the multiplier in a Ricardian model like Woodford's is indeed quite close to one. (This is why the "multiplier = 1" benchmark is so useful conceptually!) Of course, there are many other considerations---the non-Ricardian effects that formed the basis for Old Keynesian multipliers, the welfare effects of stimulus under various labor market regimes, the possibility of monetary policy commitment undoing the zero lower bound by itself, etc. But Woodford's paper is still a very valuable addition to the literature.

Sorry, an important addendum to the above comment, without which it's hopelessly muddled: when I say 'v' I mean 'tilde v'. Woodford is using 'v' as the disutility of labor, and 'tilde v' as the disutility of output. Once he's substituted to make 'tilde v', he uses that almost exclusively.

Rewrite the utility function as U=u(C)-z(Y). Note that z(Y) has all the normal properties. Now substitute in Y=C+G and we get the utility function as U=u(C)-z(C+G).

It is exactly as if the government took away G of your hours, and let you choose how many extra hours C you would sell to firms to produce the Dixit-Stiglitz good. That's like a wealth effect of a lump sum tax on your leisure.

I didn't see this when I wrote my comment. First, note that your 'z' is Woodford's 'tilde v'.

Second, 'tilde v' and 'z' are compositions of the worker's disutility from labor and the firms' inverse production functions. Expressions like z(C+G) implicitly reflect the fact that stimulus spending is being done through firms, in a way that's exactly symmetrical to consumption spending.

Nevertheless, I think you're driving at a very important point here. In casual conversation about the multiplier, Keynesians often talk about multipliers of 1 or greater as if they reflect "free" output. This is not true in Woodford's model.

Let
MUC = marginal utility from consumption
MUG = marginal utility from government spending
MDP = marginal disutility of production

In normal times, if the government is optimizing correctly, we have MUC = MUG = MDP. But in a liquidity trap, a gap opens up between MUC and MDP. In the absence of additional government spending, we have MDP < MUG < MUC. The "multiplier = 1" justification for additional government spending is to close the gap between MUG and MDP. If the multiplier is greater than 1, then the "multiplier - 1" portion also increases MUC. Unless the multiplier is much greater than 1, however, the optimal fiscal policy will not close the output gap completely. And unless MDP is literally zero, spending is never "free", even if the multiplier is 1.

I think a lot of Keynesians implicitly feel that MDP is very low in a recession, even close to zero---which may well be right, though it depends on a lot of specific detail about industries and labor markets. Or maybe they feel that we started out with MUG > MUC, or that MUG is basically flat (that there are so many great, readily available spending opportunities out there that we can spend hundreds of billions and not even come close to exhausting them). But it's important to make this case explicitly---multipliers alone don't tell you very much about the welfare effects of government spending.

One more stupid question, then I'll shut-up: would it be right to say that Gov't imposes a lump sum tax on wages and buys output from firms with it?

Patrick, a lump-sum tax is unaffected by the amount of labour supplied or anything else. So "a lump sum tax on wages" looks like a contradiction in terms to me. The simplest way to think about the tax in Woodford's model, AFAICT, is that the government demands a fixed amount of goods from each agent. You can't avoid the tax by spending less or working less.

Kevin Donoghue: "the government demands a fixed amount of goods from each agent"

Or equivalently, borrows it permanently, from whomever. If the government can engineer a permanent increase in debt/GDP (if the short rate is asymptotically less than nominal growth), it can pull a fast one on RE and the auctioneer at the end of time. I think this is the Keynesian "savings rat-hole" that Scott Sumner hates so much. It's an extra degree of freedom, the excess demand for which, like money, can cause recessions. (That was for you, Nick).

Kevin - I was just thinking that they would have to earn the money to pay the tax bill.

Matt: this is very interesting and helpful. I've got a busy day teaching today. Will collect my thoughts and come back later.

Patrick, sure they must work to pay the tax; but since the tax is lump-sum it doesn't affect the leisure-income tradeoff. If they work an extra hour the additional income is untaxed.

Yup. Thanks. I got that. Really just trying to understand how Nick got to corvee labour from lump sum taxes paid out of wages earned.

Nick, there is no money in Woodford's model...the usual problem...

Lars Christensen: "there is no money in Woodford's model...the usual problem..."

Why is that a problem, if it works fine without money?

K,

"Why is that a problem, if it works fine without money?"

Very Friedmanesque (Friedman the methodologist, not Friedman the economist).

W Peden: Indeed, Friedman, the economist, saw a central role for the money demand. But as Matt Rognlie pointed out in one of his last posts (hopefully not the last forever) there just isn't enough zero-interest money in the economy for seignorage to have a material role in macro control. Then there's this paper by Woodford which discusses how technology is eliminating interest-free money (which nobody wants) and why it makes no difference whatsoever in the CB's ability to control the yield curve - or, in fact, makes it easier.

Matt: Again, your comments are very useful. Let me make a couple of points:

1. Suppose, as you say, the government is buying final goods from the monopolistically competitive firms, just like the households do. Then where is the government's D-S utility function? Does the government have a taste for variety in the same way that households do? Does the government have the same elasticity of substitution between varieties as households? If not, then do firms price discriminate and charge the government a higher or lower price than households? And if the firms cannot price discriminate, and the government has a different elasticity of substitution, then when the government starts buying this would change the elasticity of an individual firm's demand curve and so this would change the equilibrium markup and the gap between the marginal disutility of output and the marginal utility of consumption.

I don't even see any assumption along those lines, which is why I think he's got the government taking raw intermediate output (which is equivalent to taking labour).

2. Take the standard way we do multiplier analysis the old-fashioned way. ISLMADAS for example. We do it in three steps:

We first ask how much the IS shifts right (which is like holding r constant).

Next we ask about the LM curve, and figure out how much the rightward shift in the IS moves the ISLM intersection to the right, which is also the amount the AD curve shifts right (which is like holding P constant).

Finally we ask about the AS curve, and figure out how much the AD/AS intersection moves right.

Woodford is rolling these three steps into one.

Suppose within that modelling framework someone asks me "what is the effect of a *permanent* increase in G, (by delta G), assuming Ricardian Equivalence?" My answer would be immediate. If we are talking permanent, then we are on the LRAS curve, which is vertical. So Long Run Y is pinned down by the LRAS curve. *If* the LRAS curve does not shift (I'm coming back to this later), then permanent disposable income falls by delta G. Assuming (for simplicity, and because I think this is consistent with Woodford's model) an mpc of one out of permanent disposable income, C will fall immediately by delta G. Therefore the IS curve does not shift right . Therefore the AD curve does not shift right. Therefore neither real income, nor the real interest rate, nor the price level, will change. The only change is a change in the composition of demand with G rising permanently by delta G and C falling permanently by delta G. (It looks exactly like a bond-financed transfer payment under Ricardian Equivalence, even though it isn't.)

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.

Now let me back up and reconsider whether the LRAS curve shifts. (Note that Mike Woodford's calculation of the multiplier (in the case where he is assuming r constant) doesn't refer to the Phillips Curve/supply side at all. It's all coming from the IS equation, plus the Y=C+G identity.) The tax (though it's implicit in his model, and regardless whether or not you think it's equivalent to corvee labour) will have a wealth effect on the demand for leisure, which falls as a result of the increase in permanent taxes needed to finance delta G. 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. But that's a supply-side multiplier. "Let's tax those lazy workers to make them build pyramids so they all work harder and increase Y!"

That's where I'm coming from.

A *temporary* increase in G (assuming it is like pyramids and doesn't affect the MU of C) would have an effect on AD, and that would increase Y under sticky prices. Because the increase in permanent taxes would be less than delta G if delta G is temporary. But a permanent increase in G? No way.

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. And what he is saying makes a lot more sense for a temporary increase in G during a time when inflation has fallen below target and for some reason the monetary authorities are unable/unwilling to do anything.

The intuition behind the balanced budget multiplier result is in fact very simple. Take a representative agent model where everybody is working 4 hours a day when they want to work 8 hours a day, but if they all worked 8 hours a day and produced double the output there would be an excess demand for money. So they spend 4 hours a day sitting idle. So the government comes along and tells them they all have to spend 4 hours a day building pyramids instead of sitting idle. Which they do. The monetary exchange economy continues on as before, with them all working 4 hours a day to produce consumption goods which they buy and sell for money.

That, at root, is what is happening in Woodford's model.

Lars: Over the years, I have done a number of posts, and had some very fruitful arguments with Adam P., on whether or not there is money in money in NK models' like Woodford's. My view is that those models make no sense except as models of monetary exchange economies, even though money is not explicit. The easiest way to think of them is that everyone has a chequeing account at the central bank, with unlimited overdraft facilities, and the central bank sets the rate of interest it pays or collects on balances in that chequeing account.

Nick, if the government is taking raw intermediate output, why is there no mention at all of intermediate goods in the paper? There's umpteen references to government purchases. My reading is that the government is indifferent as to variety and so ends up purchasing the same bundle as consumers do.

Kevin: that's just terminology. You can think of New Keynesian models as having a competitive market in a raw intermediate good Y, and monopolistically competitive firms have a monopoly on converting that intermediate good into a specific variety of finished consumer goods, with one variety per firm, and put a markup on the cost, because consumers have a taste for variety. Labour produces ice cream. Firms buy ice cream and each adds it's own unique patented flavour. Adding the flavour costs nothing.

If the government were indifferent to variety firms would price discriminate, since the government has a perfectly elastic demand curve for any given variety. And if price discrimination were banned, government purchases would make each firm's demand curve more elastic, which would reduce the profit-maximising markup.

Nick, I've seen both kinds of NK models and maybe they are equivalent in some sense. But if so, why not let Woodford do it his way? I can't see how firm j can discriminate given that the jth firm only produces an infinitesimal fraction of total output and government can always buy from firm k instead. It seems to me that symmetry ensures the government will buy an equal share from all equal segments of the continuum of firms.

Kevin: what I meant was that if the government doesn't care which variety it eats, it will have a perfectly elastic demand curve for any particular variety, while households care about variety and have a downward-sloping demand curve for any particular variety. So if firms can price discriminate, they will charge a lower price to the government (they will sell ice-cream to the government at cost).

Nick, by the same token the households don't care what kind of ice-cream they give to the government as taxes. So I'm not sure where that leads. Obviously Woodford is assuming that readers are very familiar with the literature and can fill in these gaps. Sadly, I'm not, so I'd best bow out and leave it to you and Adam P to debate it. Thanks.

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 = beta*pi_{t+1} + kappa*y_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).

My main critique of Woodford's model now shifts to my more recent post!

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