New-Keynesian explosions: The Cochrane interpretation and explosive solution

John Cochrane has some interesting comments on New Keynesian economics in his latest blog post on “New Keynesian Stimulus“. The interesting is not the part of the blog-literature to which it also contributes; the part about mudslinging in fiscal stimulus discussions, about which prominent economist got basic theory wrong, about who is acting most disrespectful and whatnot. I.e., the extremely counterproductive style of “debate” that was basically initiated by he-who-shall-go-unmentioned for once. I normally find that Cochrane behaves quite academic and adhere to scientific arguments (which is not entirely unfair given that he is a professor of economics), but even he has to defend himself every once in a while, and then the ball is rolling.

His post, however, does transcend the usual gibberish by advertising and describing his latest paper on New Keynesian theory. Here, “paper” means a peer-reviewed piece of academic research published in an international journal; not some opinionated self-published article. It is “Determinacy and Identification with Taylor Rules”, Journal of Political Economy  Vol. 119, No. 3, June 2011. (It, however, manages to slip in surprisingly many purely speculative statements, see below, but it definitely contains many interesting, provocative and analytically strong results.)

It is difficult in a blog post to do justice all of the contents of the long paper, but briefly, it has two main messages: 1) New-Keynesian models that includes a Taylor-type interest rule as a description of monetary policy do a poor job at providing a story for how aggregate prices are formed, or “determined.” 2) Those trying to infer anything about monetary policy conduct from econometric estimation of Taylor-type rules are in for an ugly surprise, as it is virtually impossible to identify their coefficients for a number of statistically based reasons (and even if one can overcome the identification problems, Cochrane does not find the estimated coefficients of interest due to message No. 1—indeed, he calls them “mongrels”).

Message 1) is interesting and arises from deep issues of equilibrium determination in economic theory. Basically, to pin down an equilibrium of a model, it is not just crucial to describe what happens in the model’s equilibrium, but also what happens “out-of” equilibrium. That is, a fully specified model must have a description of where the economy goes if it is not in equilibrium. One often says that it is the specification of “out-of-equilibrium” events that “support” the occurrence of the equilibrium of interest. If I have a model of peoples’ behavior when they embark on a trip onto a pedestrian bridge without fences across an abyss between two mountains, my prediction of an equilibrium where people walk straight and carefully on the bridge to get across, is supported by the out-of-equilibrium behavior where people walk carelessly and erratically, and occasionally falls into the abyss and die. If the model captures reality we should observe careful walkers, and never observe the out-of-equilibrium behavior. But it is nevertheless important for the prediction, since it is the embedded “threat” of falling into the abyss that keeps people walking carefully and straight.

Macroeconomics is full of such cases where one picks a particular equilibrium for scrutiny by ruling out others with more or less good arguments. In particular, many dynamic models involving expectations about the future have this feature. In the standard Ramsey growth model, for example, the determination of consumption is attained by ruling out “explosive” paths of consumption (as these will eventually not satisfy optimal consumer behavior). In asset-pricing models, a price based on fundamentals like dividends is attained by ruling out explosive paths for the asset price. I.e., by ruling out bubbles; both negative (which is easy) and positive (which is less so). In either case, equilibrium determination is attained by deeming the out-of-equilibrium events as unattractive for various reasons. And often the out-of-equilibrium paths for the variables under consideration are “explosive”, i.e., you fall into the abyss. Hence, the analyst rules them out and focuses on the often unique non-explosive equilibrium. Its relevance, of course, will depend crucially on the properties of the out-of-equilibrium, as those are what support the focus on the particular equilibrium.

In New Keynesian models, equilibrium determination is non-trivial. In fact, for a fixed nominal interest rate, the typical model will feature infinitely many non-explosive equilibria. Whatever is expected will happen. Such “indeterminacy” is of course not a desirable property, if the model to any extent should be used for normative guidance in real life (as that real life could be heavily disturbed if people just expects it—not nice). To obtain a unique non-explosive equilibrium in these models, one therefore has to specify particular policies, which will secure that any deviation from this unique equilibrium will lead to mayhem (make you fall of the bridge). One such policy is the famous Taylor rule for the nominal interest rate. It stipulates that the interest rate should respond to inflation sufficiently aggressive such that an increase in inflation leads to a more than one-for-one increase in the nominal interest rate thereby increasing the real interest rate. In the most common versions of the New Keynesian model, adherence to this “Taylor principle” in monetary policy leads to a unique equilibrium.

But given the story above, something bad out-of-equilibrium must be prescribed by the theory (even though we will never observe it, since it supports the choice of the particular equilibrium we will observe according to theory). Indeed, what secures uniqueness in the basic New Keynesian model is that a deviation from the equilibrium will make economic variables explode; e.g., inflation will explode over time (not just increase somewhat, but increase a lot and keep doing so). Cochrane is not convinced that such explosive out-of-equilibrium behavior is a viable support for the unique equilibrium. Therefore, he does not believe that Taylor rules are a way of determining prices and inflation. In his blog post he summarizes his position in plain language:

“For example, the common-sense story for inflation control via the Taylor rule is this:  Inflation rises 1%, the Fed raises rates 1.5% so real rates rise 0.5%, “demand” falls, and inflation subsides.  In a new-Keynesian model, by contrast, if inflation rises 1%, the Fed engineers a hyperinflation where inflation will rise more and more! Not liking this threat, the private sector jumps to an alternative equilibrium in which inflation doesn’t rise in the first place. New Keynesian models try to attain “determinacy” — choose one of many equilibria — by supposing that the Fed deliberately introduces “instability” (eigenvalues greater than one in system dynamics). Good luck explaining that honestly!”

I have no quarrels with the fact that it is instability that secures uniqueness—just like it is the missing fences that makes people walk carefully on the bridge. It is a mathematical result as he remarks (“eigenvalues greater than one”). In his words from the JPE paper, “to rule out equilibria, people must believe that the government will choose to blow up the economy” (p. 568). What I would like to quarrel with, is his particular interpretation of what happens out-of-equilibrium; that is, his colorful storytelling, which is constructed to sound sufficiently crazy to be impossible to “explain honestly.” Mostly, however, I will quarrel with his alternative solution to price determination (he mentions in the blog post that he “solves” the problem—I don’t think so), as well as the simple fact that the models he consider in detail with mathematical rigor are not New Keynesian at all (so there may be no problem to “solve”).

First, the storytelling. The intuition provided by Cochrane ignores output effects of monetary policy. But monetary-policy induced output effects are central in the basic New Keynesian model, and arise through the New Keynesian Phillips curve. In fact, they are the main reason that these models were dubbed something with “Keynes” in the first place: Demand plays a role for output determination. Moreover, an important determinant of inflation, apart from output, is expectations about future inflation. With this in mind, the explosive paths that are ruled out can instead be explained like this: “Assume the Fed follows a Taylor rule. If it responds sufficiently active toward inflation, this stabilizes inflation. For example, if inflation expectations go up for no underlying economic reason, this will increase current inflation. When the Fed then raises the nominal interest rate sufficiently, it depresses demand and output which reduces the initial impact on current inflation. The considered increase in expected inflation can therefore only be an equilibrium if inflation expectations keep on increasing, i.e., are on an explosive path. Hence, the Fed’s commitment to stabilize inflation implies that self-fulfilling equilibria can only be explosive”. This sounds perhaps less provocative, but is much more in accordance with the basics of the New Keynesian model. It is, however, no coincidence that Cochrane ignores output effects in his storytelling, as his formal model is one of a flex-price endowment economy. Yes, correctly understood: in endowment economies, there are by definition no output effects of anything—policy included.

Then, onto the proposed solution to price determination. Cochrane essentially adheres to what is known as the Fiscal Theory of the Price Level. This is a theory, to some controversial,which puts the interaction between monetary and fiscal policy at center stage and thus the government budget constraint. The theory secures that the government flow budget constraint is honored at all dates. This is not different from any other consistent model. Where it differs, is in terms of so-called “terminal conditions”, i.e., about what the model builder assumes about fiscal policy as time progresses far into the future. Many researches add a terminal condition to the government’s budget constraint that disallows explosive real debt. Essentially, for any path of prices (or other variables), this corresponds to an assumption that the government cannot run Ponzi schemes. With this assumption, one can solve the government’s flow constraint into a compact expression stating that current liabilities (like real debt) must match the present value of current and future net surpluses.

In the Fiscal Theory of the Price Level, the No-Ponzi-Game is not seen as a constraint on fiscal policymaking. Instead, it is viewed as an equilibrium condition that, by implication, only holds in equilibrium. Hence, the compact expression mentioned above becomes an equilibrium condition: The price level will adjust such that the real value of current liabilities exactly match the present value of current and future net surpluses. An example of such an equilibrium condition is given by equation (21) in Cochrane’s  JPE article. What does this imply for price determination? It implies that the government in theory can commit to an unsustainable path of net surpluses, i.e., a path that lead no explosive debt (and Ponzi schemes)—such policies are called “non-Ricardian”—for all price levels except the one that satisfies the equilibrium condition. Hence, the price level secures sustainability of fiscal policy even for the most bizarre fiscal paths. Put differently, it is a fiscal commitment to blow up the world that secures determinacy of the price level. In Cochrane’s words:

“If P is too low, then the real value of government debt explodes. In response to a shock, P jumps to the unique value that prevents such an explosion. (. . . ) If the price level is below the value specified by (21), nominal government bonds appear as net wealth to consumers. They will try to increase consumption. Collectively, they cannot do so; therefore, this increase in “aggregate demand” will push prices back to the equilibrium level. Supply equals demand and consumer optimization are satisfied only at the unique equilibrium.” – Cochrane (2011, JPE, p. 580)

Good luck explaining that honestly to a non-academic! Indeed, a more colorful storytelling of this mechanism could be: “Under the so-called fiscal theory of the price level determination, the government commits itself to engage in Ponzi schemes if the price level is slightly lower than desired. Not liking this threat of forever exploding debt, the private sector jumps to the price level where this threat will not be carried out by the government. So, determinacy is attained as the government deliberately introduces instability to the system“. To repeat: Good luck explaining that story “honestly”.

So, is Cochrane not just relocating the “blowing-up-the-world” story from the monetary authority to the fiscal authority? Yes, but he notes that the identity of the authority makes a fundamental difference in terms of whether one can support a unique equilibrium with the explosions they respectively make. In the case of the monetary explosions, the inflationary explosions are not seen as costly by Cochrane—they are merely nominal. There is nothing fundamental in the model that forbids them. We may not like them, we may not think they are realistic, but they are still valid equilibria. As for the fiscal explosions, these are real explosions, and as mentioned by Cochrane, ones that will be inconsistent with consumer optimization. So, within the model framework Cochrane formally presents, and if one accepts the Fiscal Theory of the Price Level, he has a valid point. Inflation is indeed irrelevant for individuals in the model economy. And therein lies the problem of his approach, as I see it. When inflation is irrelevant, it is a good indication that you do not have a model that is remotely New Keynesian. And indeed, as noted above, the model economy under scrutiny is a flex-price endowment economy. Obviously, prices are entirely irrelevant. Hence, Cochrane is basically using a straw-man argument against price determination by Taylor rules in New Keynesian theory. He presents verbally their flaws and the flaws of New Keynesian Theory, and then proves the flaws in a model that is classical. Endowment economies with flexible prices are by no standards New Keynesian.  I should think that this is not just my opinion, but an irrefutable fact.

Well, he is aware of this weakness: When he 13 pages later starts analyses of a truly New Keynesian model (to examine the empirical identification problems of Taylor rule coefficients), he immediately shows the simple linearized model, and do not make any proofs that Taylor-rule induced determinacy is based on explosive equilibria that are arbitrarily ruled out as in the flex-price model. Instead, the reader gets the rather disappointing message:

“One might complain that I have not shown the full, nonlinear model in this case, as I did for the frictionless model. This is a valid complaint, especially since output may also explode in the linearized nonlocal equilibria. I do not pursue this question here since I find no claim in any new-Keynesian writing that this route can rule out the nonlocal equilibria. Its determinacy literature is all carried out in simpler frameworks, as I have done. And there is no reason, really, to suspect that this route will work either. Sensible economic models work in hyperinflation or deflation. If they do not, it usually reveals something wrong with the model rather than the impossibility of inflation. In particular, while linearized Phillips curve models can give large output effects of high inflations, we know that some of their simple abstractions, such as fixed intervals between price changes, are useful approximations only for low inflation. The Calvo fairy seems to visit more often in Argentina.” – Cochrane (2011, JPE, p. 593)

I would say it is much more than a “valid complaint”. It is a reason to discard his criticism of price determination in New Keynesian models altogether. The paper raises, as said before, important issues, but its results are for models that are not New Keynesian: He shows that price level determination by active Taylor rules in flex-price models lacks solid foundation. Well, mostly would not really care that much, as Taylor rules in flex-price models involve counterfactual interest rate effects at business frequencies; cf. Jordi Galí’s textbook treatment “Monetary Policy, Inflation, and the Business Cycle” (in particular, Chapter 2). Actually, they perform so weird that contractionary interest-rate disturbances have expansive equilibrium effects on the interest rate (!).

More important, we should not as academics “suspect” things as Cochrane at this point apparently feels sufficient. We should prove them rigorously; I’m confident Cochrane would agree to that. Also, in the non-linearized New Keynesian model, inflation causes output dispersion which reduces output (one doesn’t see that in the standard linearized model when one examines small perturbations around zero inflation). So, explosive inflation would ultimately reduce output toward zero. That is a REAL explosion (or, in output terms, implosion). Would this be “allowed” as a reasonable equilibrium? I doubt it, but it obviously needs careful scrutiny.

[As an aside, most would never have been able to publish such speculation as the above, along with a more or less inside joke on “Calvo fairies” and Argentina in a Top-5 professional journal. I can, of course, not imagine that being located in Chicago makes it easier to slip these, at best vague, at worst silly, comments into the JPE because the journal is located in Chicago. I mean, it is a respected research outlet!]

So, as I see it, Cochrane raises numbers of interesting questions, but does not manage to provide a serious blow to the New-Keynesian literature as his abstract and first 20 pages seem to promise. In many respects his analyses are simply done in a framework too far from (if not orthogonal to) the New Keynesian framework.

As for his second main message, the paper is (also) an interesting read. He shows in many analytical examples and numerical simulations how estimations of Taylor rules are biased and/or meaningless. I like this a lot, but I am also biased in this respect, as I wrote similar stuff back in 2002 (albeit much less generally). Cochrane gracefully acknowledges this in the extensive online appendix to his paper: “Appendix B from John H. Cochrane, “Determinacy and Identification with Taylor Rules” (JPE, vol. 119, no. 3, p. 565)“. This appendix is a great paper in itself. My own humble output was eventually published as “Estimated Interest Rate Rules: Do they Determine Determinacy Properties? The B.E. Journal of Macroeconomics: Vol. 11: Iss. 1 (Contributions), Article 11 (2011).” The answer, by the way, is “no”.

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2 Responses to New-Keynesian explosions: The Cochrane interpretation and explosive solution

  1. Hi Henrik,

    I enjoyed very much reading your blog article. It is very insightful and provocative. However, it is worth thinking about something here. I believe that Cochrane’s frictionless two equation model in his JPE paper is an EXACT model. It is not loglinearized. The Fisher equation follows directly consumer’s first order condition which combined with a Taylor rule could give rise to explosive inflation in a flexibile price environment.

    However, the three equation NK model with Phillips curve is a LOGLINEARIZED model which is a good approximation around zero inflation (low inflation) steady state. It is pointless to argue about explosion (nominal or real) using such a model because the very premise of loglinearization makes any such explosion invalid.

    Thus Cochrane has a valid point (page 593 of his JPE 2009 article) saying that a sensible model should tell a good story about hyperinflation/deflation. I think the loglinearized three equation model precludes such story anyways.

    Thus by construction, this loglinearized NK model is orchestrated to pick only the stable solution (I think).

    Thus to make meaningful New Keynesian counterattack against Cochrane, one should develop an exact NK model which with Calvo fairy could potentially generate nominal or real explosion and then try to rule it out using equilibrium reasoning. Such a job is not easy.

    Parantap Basu

    • Thanks for the comment!

      @ “Thus to make meaningful New Keynesian counterattack against Cochrane, one should develop an exact NK model which with Calvo fairy could potentially generate nominal or real explosion and then try to rule it out using equilibrium reasoning. Such a job is not easy.”

      I am not per se interested in making a “new Keynesian counterattack” to Cochrane. I am merely pointing out that he make most of his criticisms of New-Keynesian models within classical models, which to me is weakening his arguments. He is purporting that the unique stationary equilibrium mostly analyzed in the New-Keynesian literature is supported by unreasonable out-of-equilibrium inflation “explosions”. I think the burden of proof lies with him, and it would therefore be interesting to see a formal global analysis of the issue. Suspicions are not just enough in my opinion.

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