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Great post.

Nick, As you know I don't have a good grasp for these sorts of models. Is the price level pinned down in Williamson's model, or just the inflation rate (as in many NK models?) If not, could there be some sort of overshooting implied? Say a increase in the liquidity premium on government bonds caused the steady state inflation rate to rise from 1% to 1.5%, but also caused a 40% fall in the equilibrium price level right now. So that even the long run expected price level is lower. This is just a stab in the dark, but I thought I would throw it out there.

Scott: Thanks! Brad DeLong, Noah Smith, and Steve Williamson have been arguing about the exact same thing, in comments on Noah's blog. I think Steve is saying no. But I would need to check more carefully to be sure. It all depends on exactly what is being held constant in fiscal policy, which is hard to see from the equations (what's exogenous and what's endogenous) because you might get some version of the fiscal theory of the price level, though Steve said not when I asked him. I'm more concerned about the methodological point.

"Update: as I was driving down Autoroute 5, pressing the gas pedal, watching the speedo, I suddenly remembered Andy Harless had made a similar point a couple of years back about umbrellas not causing rain. Sorry Andy. (But cars are cooler than umbrellas."

Interestingly, for various reasons, I happened to be rereading that very post last night, and guess who turns up in the comment thread to argue that umbrellas cause rain?

"If the central bank targets the nominal interest rate at a low enough rate forever, you have to get deflation. By arguing against this you're making yourself look silly."

[Link here NR]


P.S. I wonder, has Williamson ever read Friedman's "The Role of Monetary Policy"?


And if so, did Williamson think it made Friedman look silly?

Another really good post Nick, and yes, the responses of both Andolfatto and Williamson are simply missing the point, still.

However, the equilibrium equations that SW writes down say even less about causality than you imply. It's not that you can't reverse the order of causation, it's that sets of equilibrium conditions alone don't specify any causual ordering at all.

Asking what determines what else amongst endongenous variables is like asking what caused heads to be facing up after you flip a coin. Is heads facing up because tails is facing down? Or is tails facing down because heads is facing up?

Let me give another example:

Did Japan have deflation because the yen was too high or was the yen too high because Japan had deflation?

Answer 1) Deflation in Japan lowers the price of Japanese goods making them cheaper to foreigners, the increased demand for Japanese goods from foreigners bids up the Yen as they buy yen to buy the goods. This makes it sound like deflation makes the yen appreciate.

Answer 2) Yen appreciation raises the cost of Japanese goods to foreigners, this lowers overal demand for Japanese goods and thus lowers their prices. This makes it sound like yen appreciation causes the deflation.

The correct answer of course is neither is the cause, both the Japanese inflation rate and the value of the yen are endogenous so nether cause the other (just like the coin flip example).

Causality can only come from exogenous factors.

A sufficiently expressive formal language would enable economists to build into their models whatever ideas they have about the relationships between economic variables. Part of the problem, then, may be that the formal vocabularies of contemporary macro economists are insufficiently expressive. Using Pearl's calculus of counterfactuals, for example, would clarify the desired causal interpretation of Williamson's model in such a way as to permit others to draw out the implausible implications you believe his interpretation to have formally (which, ideally, would convince Williamsln that his model is not sensible even if it is coherent). This would also allow econometricians to evaluate the model against the data more completely by embedding statistical techniques of causal inference (instrumental variables, propensity score-matching, etc.) within a fuller characterization of the system under study. A fuller grasp of the theory of nonlinear dynamical systems might also enable macroeconomists to more completely specify the disequilibrium properties of their models in a way that would clarify the stability issues that have been raised.

I value informal models as much as you do, Nick, and I think Williamson's demand that this argument be carried out on an entirely formal plane is probably excessively pedantic. Nevertheless, sometimes you have to fight fire with fire, and pointing out the artificial way in which Williamson and other economists are restricting what goes into their models by building alternative models that are not so restricted might be the only way to decisively win this argument.

Mark and Andy: good comments.

Ram: it's funny you mention counterfactual conditionals. I think those are very relevant to this question. I actually started to write a post on that subject a couple of days back, but I scrapped it because I couldn't do it right. I must check up on Pearl. Last I read about counterfactual conditionals, which was a long time ago, it was all about second-best most plausible possible worlds, subject to the constraint of the counterfactual.

"Is the price level pinned down in Williamson's model, or just the inflation rate (as in many NK models?) If not, could there be some sort of overshooting implied?"

In Williamson's model the initial price level is determined by fiscal policy but monetary policy determines the rate of inflation. In my opinion, the way the model is constructed, sudden shifts in the price level cannot happen.

Interestingly, both Brad DeLong and Noah Smith seemed confused about this fact, indicating to me they haven't carefully read Williamson's paper. In fact I'm beginning to suspect that 99% of the people commenting on this haven't actually read his paper.

"If you increase the growth rate of the money supply, that will cause the inflation rate to rise, and that in turn will cause the nominal interest rate to rise."

In the long run, maybe. But is it obvious that the immediate response of the nominal rate to an unexpected sudden increase in the monetary growth rate would be a rise rather than a fall?

> But an inverted pendulum is unstable, and you move the bottom end south initially, then north, if you want the top end to move north.

This is an even more apt metaphor than it appears at first, as with the proper (oscillatory but specified in advance) forcing at the base, an inverted pendulum can be made stable.

I'm suspicious of any economic model where an agent (naive or representative) can earn an economic profit by performing actions that drive the system away from equilibrium. That's the case in a "low interest rates invariably mean deflation" model, since if we start with positive inflation expectations an agent would find it profitable to purchase real goods on margin (even at zero real but positive expected nominal return), increasing AD and thus (by a Philips curve) increase inflation.

Nick, I'd love to read that post if you still have it. There are other formal frameworks for causal inference, but Judea Pearl's (2000) is probably the most comprehensive and connected with the questions macroeconomists are interested in. It's always fascinated me how sophisticated economists are when jt comes to conventional mathematical models and frequentist statistics, but how rarely they draw upon more recent developments in the theory of formal modeling. Of course, as a statistician who studied applied math and the philosophy of science, thinking about models as such is kind of my day job, so maybe I'm expecting too much.

"I must check up on Pearl."

Then you want his book "Causality": http://www.amazon.ca/Causality-Judea-Pearl/dp/052189560X. But he also has a blog you can skim: http://www.mii.ucla.edu/causality/.

Ram, econometricians are actually keenly interested in the sorts of models and issues Pearl investigates, and Pearl himself acknowledges econometrics "gets it right" and attributes the genesis of formal causal modeling to Haavelmo's work in the 1940s. See the discussions between Pearl and Heckman over the last decade or so, for example.

Which brings me back to Nick's excellent post: one way of interpreting Nick's point is that this symbol: "=", is much trickier than it looks. If I write,

y = a + bx + e

as an econometrician I typically mean, "y is totally caused by x and by e." But the symbol "=" dividing that expression does not clearly indicate that that's what I mean, as Nick's example shows.

I recall reading, but am too lazy to look up a cite, that the Cowles commission econometricians argued over whether some symbol other than "=" ought to be used in structural econometric equations, where this other symbol would mean "the object on the left of this symbol is totally caused by the object on the right." It's probably a shame that they didn't run with that.

We can nonetheless restate Nick's model in such a way as to more or less avoid the problem. Let D be the deviation between the position of the speedometer and the speed of the car. Then (setting a bunch of coefs to 1 for simplicity)-

N(t) = S(t) + D(t)
S(t) = G(t) + dS(t-1)

and we see that if I reach into the system and exogenously manipulate D, that changes the position of the needle but not the speed of the car, assuming D does not cause G (there is no wire connecting the needle to the gas pedal).

Thomas Tooke observed way back in the 1840's that changes in the price level PRECEDE changes in the money supply. Thomas Sargent observed the same thing about the European hyperinflations of the 1920's. Here's a good explanation: The government has issued $100, backed by 100 oz of silver, so $1=1 oz. Then the government loses 10 oz, so the $100 is now backed by only 90 oz, and $1=.9 oz. Real balances in the hands of the public drop from 100 oz to 90 oz, so the public (desperate to restore their real balances to 100 oz.) sells $11 of bonds to the government in exchange for $11 of new currency. This restores the public's real balances to 100 oz.

Inflation came first, followed by money growth.

"Simply read the same equation from right to left."


If its meaning changes depending on which direction you read it from, it's NOT an equation.

OK. To be fair, there is a difference between the direction of reading that is pragmatic. We typically treat the side of the equation that is read first as **given** and the side that is read second as **deduced**. Now, it is true that temporally, the given comes before the deduced, and in efficient causality, the cause comes before the effect. But it is often true that we deduce the cause from the effect.

You have stated that you have observed that aN(t) = bS(t), approximately, where a and b have the same sign. That means that aN(0) = bS(0) and aN(1) = bS(1). Which in turn means that you have observed a(N(1) - N(0)) = b(S(1) - S(0)), or adN(0) = bdS(0). But you have also observed that dS(0)/dN(0) < 0, which is a contradiction, when the CB has changed N(t). That is, there are certain exceptions to the general rule, under certain specified conditions.

Fine. Humans can reason with exceptions, called defeasible reasoning. But as scientists we do not simply want to say that there are exceptions, but to understand the exceptional cases and the conditions under which they apply. IOW, the model is underspecified. It is not enough to say, OMG! They are turning the needle the wrong way!

"But as scientists we do not simply want to say that there are exceptions, but to understand the exceptional cases and the conditions under which they apply."

Yes, that is my take on it.

Nick's point about automotive engineers is apt - of course any such individual would understand that the vehicle must necessarily find itself working against gravity and friction more often than it finds itself working with them and would set their expectations and make their design decisions accordingly. The car designer knows that the bank of Canada can't do anything except attempt to reflect the forward speed of the vehicle as a welfare increasing public information service.

Modelling hills as shocks is of little use without knowing at least the tendency towards encountering each kind of shock.

If shocks are held to come from the 'real world' then they are necessarily not random, and necessarily follow their own irreversible dynamics which must impose real constraints on the observed behaviour and hence evolution of the subsystem (here, the monetary apparatus) under study.

The system as a whole is thus evolved primarily to deal with externally imposed irreversibility and should not be expected to seek to maintain a reversible path through its own internal state space over anything but the shortest timescales** under these circumstances since there would be no possibility of succeeding.

** some level of reversibility over some timescale which is small relative to the timescale of external shocks must of course be possible in order to effect a control loop. But the whole must exhibit irreversible dynamics regardless.

Nick, you are on fire lately. This is awesome stuff.

But I can't help complaining: Isn't your criticism of Williamson at least in the same genus as your criticism of the New Keynesians (based on Cochrane's handling of multiple equilibria etc. and how the Euler equation didn't actually recommend an increase in G to fix Aggregate Demand)? But back when I thought you were hitting it out of the park on that issue, I was frustrated that DeLong/Krugman weren't explaining exactly why you were wrong.

Now, when your awesome insights happen to be attacking someone who was skeptical of QE, all of a sudden you're the coolest guy on the Internet.

I realize I am biased, so I hope you will settle this one way or the other.

Chris, I agree. My points apply more to macroeconomic theorists than to econometricians. And to be sure, some macroeconometricians understand the work of Pearl. Still, consider what Williamson is doing here. He obviously has some ideas about causation. He may even have some ideas about disequilibrium dynamics. They're not in the model, though. They're in the English that is presented alongside the model. In some cases, they're not even presented at all. Why not? Presumably, the benefits of using a formal model extend to everything in the model, and not just the parts that only require some algebra. And this is a perfect opportunity to formalize those things, since the whole issue being debated is what his model implies causally, or about stability, etc. But this is pretty common in macroeconomic theory, it seems to me. Macroeconometrics may be better on this score, but if theorists outsource the development of their theory whenever the math gets hard, or uses recent techniques, this seems like a recipe for generating poorly considered models.

Silly boy, that's why we put covers over the speedometers, to discourage you from experimenting. If people start to discover that moving the needle does in fact change the velocity of the car they will discover that the world is utterly unlike the illusion we've been successfully promoting for all these years, and there goes our main source of income.

Sorry all for not responding to these very good comments. You will probably see why when you see my latest post. Words really are much clearer than math, in this case. But I don't know what to say any more. What the hell happened? How did it happen?

Nick, I said the same thing in comments at Noah's blog when this thing first hit. I used the analagy of weight and blood pressure:

"And as Andolfatto points out, there are macro models out there that are very similar to New Keynesian models, but have one small twist that makes the "QE-causes-deflation" equilibrium the stable one!"

Krugman was not just arguing that Williamson's equilibrium was unstable; he was also arguing that raising interest rates would not get us to that equilibrium. As I put it, he was arguing against the concept of immaculate equilibrium. That if you find an equilibrium, we will get there, even if it takes a miracle; or that just achieving one factor in the equilibrium will automatically make us achieve all of the others.

An analogy is, suppose it's an equilibrium for a man to be at 240 lbs. and have blood pressure of 160/100. And it's also an equilibrium for him to be 170 lbs. and 130/80. That doesn't mean that if he takes powerful blood pressure medication to bring him to 130/80, that will also make his weight drop to 170. Nor will it give him many of the other benefits of eating healthy and dropping to 170.

Williamson seems to say that there's an equilibrium (in a model that really excludes a lot that matters) where interest rates are a lot higher than they are now and inflation is a lot higher than it is now. So, if we raise interest rates we will get to that equilibrium. He seems to be saying that raising interest rates will also raise inflation, just like taking blood pressure medicine to lower our blood pressure, will also lower our weight. But the causation is not there, at least for the blood pressure medication example. That will not get us to that equilibrium. Other measures must be taken, including measures outside of the model, like eating healthy foods and exercising.

At: http://noahpinionblog.blogspot.com/2013/12/does-qe-cause-deflation.html?showComment=1385968110631#c7859598410803540360

BUT Nick, here's the thing, and it's so aggravating when you deal with Stephen, and his group of often assinine assault anonymi. He can always say, you just don't know what's going on because you are not fluent in the super-mathematical models in my area (And here he seems to imply that his equilibrium is not unstable, and there's a proven path for how we would get there. It's: obscure math, obscure math, math, math.) And, of course, it's gigantic hours to try to discern the meaning and intuition behind all that math. Few people in the world will do this besides the 700 people out of 7 billion who are paid handsomely to do this as a career.

In every other area of econ I've ever looked at you can understand the econ quite well without having to pull teeth to get it out of the equations. There are lots of books and articles and posts that tell you what's going on at least in a way that's clear for economists not in that specialty, and usually for people with just bachelors in econ, if not just well educated laypeople.

You know that the papers have been done with all of the math to prove internal consistency, but you don't have to go to them to learn the lessons, and intuition. There are good books, articles, and posts for that.

But Stephen's econ is a terrible exception. It's pull it out of the raw math or nothing. And I know Stephen sometimes tries to put it into clear words for people not in his area. But I don't think he usually succeeds, and I don't necessarily fault him for that. It may be especially hard with his econ. Still, contrast his posts on this recent brouhaha with Krugman's first, which was not that hard to understand, even for someone who just had some intermediate econ in college, if not just micro and macro I.

This aggravated me so much that one summer I actually did pull teeth, equation by equation, trench warfare, with Wallace's seminal, AER '81 "Wallace Neutrality" paper, to finally see if Stephen could actually be right about the amazing things he was saying at the time:

"No, in a liquidity trap, if the Fed purchases gold, it does not change the price of gold, just as it will not change the prices of Treasury bonds if it purchases them." – Stephen Williamson

"The Fed can buy all the government debt it wants right now, and that will be irrelevant, for inflation or anything else." – Stephen Williamson

– Links at: http://richardhserlin.blogspot.fr/2012/09/want-to-understand-intuition-for.html

So, I put in about 80 hours on this paper. I'm still paying the price for that with my wife! And I found out there were some real factors there, but in the real world they we're far from 100% in their strength, so the Williamson quotes were, in fact, very untrue. I enjoyed it, and it went well, but I'll rarely be able to do that, and that was just the beginning. By the way, two posts came out of it that were quoted favorably by Bruegal and Miles Kimball:

"Richard Serlin (HT Mark Thoma) gives the bottom line intuition of Wallace neutrality." – Bruegal, at: http://www.bruegel.org/nc/blog/detail/article/885-blogs-review-wallace-neutrality-and-balance-sheet-monetary-policy/

"Richard has kindly given me permission to quote at some length from his nice explanation of the logic behind Wallace Neutrality and why it might not hold in the real world" – Miles Kimball, at: http://blog.supplysideliberal.com/post/63442789541/wallace-neutrality-roundup-qe-may-work-in-practice

So, it is doable; I'm not just some disgruntled idiot, but the time cost to pull it out of the pure math is enormous, and this area looks unique in that you just can't find good verbal and/or graphic and/or light and clear math, explanations of the implications and intuition.

So, what's Stephen's reply? It seems to be No, my equilibrium is stable, Yes, there is a logic outlined for why we would move to it. And, it's all in the math. And very few people will know that math or have time to fight equation by equation to decipher it.

Of course, my aggravation level is nearing the point where I may do another Wallace on Stephen's recent AER. I may be sleeping on the couch for a while.

Yep, I had the same reaction to the portion of the claim that says "therefore, inflation must fall." Very well-expressed.

Nick Edmonds: "In the long run, maybe. But is it obvious that the immediate response of the nominal rate to an unexpected sudden increase in the monetary growth rate would be a rise rather than a fall?"

It's not obvious. It could go either way. It depends on how quickly expectations and the inflation rate adjust. If they adjust slowly, the nominal rate will fall temporarily.

Ram, and Phil: I checked, and found I had deleted my draft post. Which is probably just as well. But I may return to the topic. I read a little of Judea Pearl, who looks really good, but mostly over my head. It turns out my own understanding of counterfactual conditionals is way out of date. I had a vague idea that was some sort of mix of Lewis possible worlds and that Bayesian theory.

Chris and Min: but I had thought that anything with "=" in it *was* an equation, and you could read it either way around! Because algebra says you can do that, and maths must be true, right? ;-) Dunno. That's why I resort to words.

Mike: news about a future increase in the money supply will cause the price level (and the inflation rate) to rise now, according to standard quantity-theoretic reasoning too.

Bob: Here's my ranking:

1. The fiscal policy stuff is small beer. People not reading their models' results right.

2. The indeterminacy problem is potentially strong ale, that should affect how we model.

3. This problem here is 100 proof spirits (liquor). Or, someone's been drinking the KoolAid.

Richard: that's a good analogy! But I like mine better. I can talk about cars, and my knowledge of how they work!

I agree with your second comment. And those stupid full-of-themselves idiots EJMR commenters who hang around. That is why, despite all else, I must really give Steve credit for his latest post, which is all in words, and is very clear about how he reasons.

"I always thought that anything with "=" was an equation and you could read it either way.
An equation merely tell you that the values on both sides are equal. It doesn't tell you how you got at these values. Measured? Computed? Just given to you?
It never tells you anything about the process. Never.
A stoechiometric equation tells you the starting and end products. It doesn't tell you how to synthetize the product. (As Svante Arrhenius said at Balaclava:"It's a nice stoechiometric equation. It's not industrial chemistry.")

Jacques, yes that's very interesting. I had a similar reaction when I began programming. In some languages, X = Y is not the same as Y = X. In the first case it's put the current value of Y into X. In the second it's put the current value of X into Y. So, in the first case Y is causal; in the second X is. They end up equal in the end, but equal to different numbers.

Richard. Yep. I am old enough to have learned FORTRAN as my first programming language. ( I even learnrd APL. Nothing ever beat that one. How our generation was dismayed when the first micro came in early '80's and we hasd to learn what? BASIC and Pascal?)
In FORTRAN "=" means "assign the value Y to variable X". So you write X=Y.Very different indeed from X=Y. Not easy to switch your reasoning from algebra to code, especially if you are coding algebra...
I have to remind students that when they encounter the following at the bottom of a statistical table 2001=100. it doesn't even have the meaning of FORTRAN. It means "assign the value 100 as the value of the variable in the year 2001 ,variable whose name is defined somewhere else.".
The same glyph having three different,if related meanings. And then you wonder why your students are confused.

With the car example, there's no causality in any direction. Going up a steep hill, pressing the gas pedal may not be enough to even maintain speed or speedometer needle position. Going downhill, the speed may increase and needle may move without pressing the gas pedal.

If the car is up on blocks, pressing the gas pedal may move the needle but won't make the car go faster. Similarly, if the car is free-falling, it will go faster without the needle moving.


"news about a future increase in the money supply will cause the price level (and the inflation rate) to rise now, according to standard quantity-theoretic reasoning too."

But which is more likely: that people forecast future changes in the money supply and adjust prices in advance? or that people react to tight money conditions by withdrawing more money from banks?

Mike: those are the same thing, with opposite signs. Tight money means people expect prices to fall, which increases the demand for money.

Wouldn't a simpler (at least to me) way to put it is this:

Forget about stability, Franklin Fisher or telling stories. That has nothing to do with it.

The problem is that in Williamson's model inflation is a (possibly time varying) *parameter*. It's consumption, of good 1, which adjusts to inflation not vice versa. Then the usual MRS condition pins down consumption of good 2 as a function of the interest rate. The CB sets the interest rate. That's it. Adding in the government debt constraint makes the model over determined, where we've "solved" for a parameter.

(alternatively the government picks its debt level. Adding in a central bank makes the model over determined)

notsneaky: I'm afraid you lost me.


"Tight money means people expect prices to fall, which increases the demand for money."

We mean different things by 'tight money'. I had in mind an initial situation where the bank issued $100, backed by 100 oz., so $1=1 oz. Then the bank lost 10 oz, so the $100 are now backed by 90 oz, $1=.9 oz, and real balances have fallen from 100 oz worth of dollars to only 90 oz worth. The tightness of the money supply (i.e., the 10 oz shortfall) does not cause prices to rise. (i.e., $1 is still worth 0.9 oz.)

People react to the 10 oz drop in real balances by asking the bank to issue another $11, in exchange for 10 oz worth of bonds or other securities.

Nick Rowe: "Chris and Min: but I had thought that anything with "=" in it *was* an equation, and you could read it either way around! Because algebra says you can do that, and maths must be true, right? ;-) Dunno. That's why I resort to words."

Nick, I apologize for my over-reaction. :(

Nick, I missed your response: "Chris and Min: but I had thought that anything with "=" in it *was* an equation, and you could read it either way around! Because algebra says you can do that, and maths must be true, right? ;-) Dunno. That's why I resort to words."

If the equation is part of a structural model, it's actually not an equation! Sort of. Consider the "equation":

y = bx + u.

What does the parameter b mean? If this is part of a structural econometric model, then b means "the effect of a one-unit change in x on the mean of y, holding u constant."

But if I treat the symbol "=" in that "equation" as implying I can move things around at will, I can rewrite that equation:

x = dy + e,

where d=(1/b) and e=(1/b)u, which implies, if again we read that as structural, that a one-unit change in y causes a d unit change in the mean of x. But if the complete structural model is, say,

y = bx + u
x = cz + v.

so y does not appear in the structural equation for x, then the model says that a one-unit change in y causes no change rather than a d unit change in the mean of x!

So the symbol "=" does not mean "is equivalent to" in structural models, and these "equations" cannot be reversed without changing their meaning. The symbol "=" means something akin to the assignment operator in a computing language.

See Pearl (2001) chapter 5 for more.

The general answer to this problem is that the original equality is incorrect. Proof: if you can move the speedometer needle without changing the car's speed,
then it isn't generally true that the needle reading equals the speed.
Likewise, if you can change the interest rate without changing inflation, then that equality also isn't generally correct.
Or, economics isn't physics!

(A variant on a comment others have made.)

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