Believe those who are seeking the truth. Doubt those who find it. Andre Gide

Friday, December 28, 2012

On the perils of Taylor rules

In the Seven Faces of "The Peril" (2010), St. Louis Fed president Jim Bullard speculated on the prospect of the U.S. falling into a Japanese-style deflationary outcome. His analysis was built on an insight of Benhabib, Schmitt-Grohe, and Uribe (2001) in The Perils of Taylor Rules.

These authors (BSU) showed that if monetary policy is conducted according to a Taylor rule, and if there is a zero lower bound (ZLB) on the nominal interest rate, then there are generally two steady-state equilibria. In one equilibrium--the "intended" outcome--the nominal interest rate and inflation rate are on target. In the other equilibrium--the "unintended" outcome--the nominal interest rate and inflation rate are below target--the economy is in a "liquidity trap."

As BSU stress, the multiplicity of outcomes occurs even in economies where prices are perfectly flexible. All that is required are three (non-controversial) ingredients: [1] a Fisher equation; [2] a Taylor rule; and [3] a ZLB.

Back in 2010, I didn't take this argument very seriously. In part it was because the so-called "unintended" outcome was more efficient than than the "intended" outcome (at least, in the version of the model with flexible prices). To put things another way, the Friedman rule turns out to be good policy in a wide class of models. But mostly, I figured that other factors were probably more important for explaining the events unfolding at that time.

Well, maybe I was a bit too hasty. Let me share with you my tinkering with a simple OLG model (similar to the one I developed here.) Unfortunately, what follows is a bit on the wonkish side. If you catch any errors, or otherwise have any comments to make, please let me know.


People live for two periods; they are "young" and then "old." Everyone only values consumption when old. Their objective is simply to maximize (expected) future consumption.

The young are endowed with some output y. The are also each endowed with an investment technology such that k units of output invested today yields f(k) units of output tomorrow. Assume that f(k) is increasing and strictly concave; i.e., f'' < 0 < f'.

The autarkic (also competitive) outcome is one where the young save their entire endowment and consume f(y) when old. The competitive equilibrium (gross) real rate of interest is equal to the marginal product of capital, r = f'(k). [Note that time-preference plays no role in determining the real rate of interest here.]

An economy with real debt

Assume that there is a government that issues one-period real debt b. Let r denote the gross real rate of interest paid on this debt. I assume that the government finances the carrying cost of its debt via a lump sum t tax applied to old agents. In a steady state,

[1] t = (r-1)b.

By construction, the savings decision is trivial: the young save all their income y. The interesting decision entails a portfolio allocation choice problem between capital and bonds, y = k + b. Conditional on a choice of b (hence, k), future consumption is given by:

[2] c = f(y-b) + rb - t 

Assuming an interior solution, (expected) rate of return equality implies:

[3] r = f'(y-b)

Technically, [3] determines bond demand. In equilibrium, the supply of bonds (determined by policy) must equal the demand for bonds. Hence, by choosing b in this model, the government can choose the prevailing real rate of interest. Lump-sum taxes are simply adjusted by way of [1] to finance the carrying cost of the debt. Equilibrium consumption is then given by [2]; i.e., c = f(y-b) + b.

The real GDP in this economy is given by Y = y + f(y-b). Notice that this model delivers a standard IS curve. That is, by increasing b, the government increases r, capital is crowded out, and output falls. Likewise, lowering the real interest rate stimulates (investment) demand, leading to an increase in output.

In what follows, I assume that a socially desirable outcome is associated with some 0 < b* < y. The "natural" rate of interest is defined as r* = f'(y-b*), and potential GDP is defined by Y* = y + f(y-b*). [Note that r* may be either greater or less than one. Some of you may argue that r* should equal 1 here. That's fine. The qualitative results below do not hinge on this issue.]

An economy with nominal debt

Let P denote the price of output denominated in some abstract unit of account. Let B = Pb the nominal debt. Let P+ denote "next period's" price level. Then a young person faces the following sequence of budget constraints:

Py = Pk + B 
P+C+ = P+f(k) + RB - T+

where R denotes the gross nominal interest rate, and T is the nominal lump-sum tax. Define Π+ = P+/P, the expected gross rate of inflation. Then using b =B/P and t = T/P, rewrite the budget constraints above as

y = k + b
c+ = f(k) + (R/ Π+)b - t+ 

Desired real bond holdings must now satisfy the condition

[4]  f'(y - b) R/ Π+

with the demand for nominal bond holdings given by B = Pb. If I define rR/ Π+ as the expected real rate of interest, then we see that [4] is equivalent to [3].

The government budget constraint is given by T+ = RB - B+. In real terms,

[5] t+(R/ Π+)b - b+ 

so in a steady state with b = b+, we have t = (r-1)b, which is equivalent to [1]. Consumption is given by [2].

The model to this point is riddled with indeterminacy, even restricting attention to steady states. What determines the nominal interest rate, the inflation rate, the price level, etc.? Note that this indeterminacy is not present in the model with real debt. In that world, I assumed that b was a policy instrument. This (along with the lump-sum tax instrument) pins down an equilibrium. In the world I am describing now, the government does not pick b. It need not even pick B if, in particular, it is willing to let demand determine quantity at a given rate of interest. How to proceed? As usual, in small steps.

A Monetarist regime

We can think of B as interest-bearing money. The nominal interest rate on money is commonly assumed to be zero, so R = 1. But there is nothing that requires this to be the case; we are free to pick any interest rate supportable by taxes here. The key assumption is that R is determined and that it is constant over time.

The monetarist views B (and the time path for B) as determined by policy. With the demand for real money balances determined by [4], market clearing requires B = Pb for all time. Since B is determined by policy, and b is determined by agents, the price level is determined by P = B/b. The inflation rate must therefore be determined by

[6]  Π+ = (B+/B)(b/b+)

Let B+ = μB. Now combine [6] with [4] to derive:

[7]  b+ = (μ/R) f'(y - b)b

which is a first-order difference equation in real money balances. The model has two steady states. In one, b = 0; in the other, b > 0 satisfies f'(y - b) = (R). It seems darn easy to construct the optimal policy here. Just set  (R) = r* and we're done.

Well, not so fast. As it turns out, even for the case of a fixed stock of money B, there generally exists a continuum of nonstationary equilibria indexed by an initial condition 0 < b0 < y, with the time path for b asymptotically approaching zero; see Figure 1 in Woodford (1984). Of course, since P = B/b with B fixed, this implies that the price level approaches infinity (in fact, these are hyperinflation dynamics). Isn't it interesting to note that Friedman's k percent rule is dynamically unstable here?

The undesirable hyperinflation dynamic here appears to an artifact of (among other things) the assumed passivity of policy (the nominal interest rate and money growth rate are held fixed forever). But evidently, there exists a simple "activist" policy rule that uniquely implements the desired outcome:

[8]  ln(R) = ln(R*) + α[ ln(Π+) - ln(Π*) ]

where Π* = μ (arbitrary), R* = r*Π*, and α = 1.  The policy rule [8] is a Taylor rule. The rule dictates that the policy rate be increased one-for-one with expected inflation. Such a policy keeps the expected real rate of interest pinned to its natural rate. As such, the economy is always at potential (this would not necessarily be the case if I was to introduce other shocks, of course). Note that the price level is now determined, P = B/b* with P+ = Π*P.

Would the ZLB restriction R ≥ 1 limit the ability of policy here? I do not think so. First, the problem in this model is a the possibility of a self-fulfilling hyperinflation -- deflationary equilibria do not exist. As such, policy only needs to threaten to raise, not lower, the nominal interest rate. Second, I believe that the optimal monetary policy may alternatively be expressed as a money growth rate that varies in proportion to the expected growth rate in real money demand. That is, targeting the inflation rate is feasible here (and contrary to Eagle (2006), an inflation target policy seems consistent with price level determinacy here).

A Wicksellian regime

Following the approach taken in the New Keynesian literature, we might instead assume that the quantity of nominal debt B (money) is entirely demand-determined. The only policy instrument is R (and, of course, the lump-sum tax). I assume that policy follows the Taylor rule [8] with α = 1.

As far as I can tell, all of the math developed in the previous section continues to hold. But giving up the quantity variable B as a policy instrument must have some implication. Indeed, it does. What we seem to lose is any fundamental economic force determining the price level and inflation rate. That is, the level of debt and its growth rate simply accommodate themselves to the prevailing price level and inflation rate, respectively. According to [8], exogenous movements in the expected rate of inflation (inflation shocks) are met one-for-one with movements in the nominal interest rate, leaving the real rate of interest pegged to its natural rate.

Note something interesting here: the inflation target Πis completely irrelevant. Inflation in this model can be whatever it "wants" to be. If the community expects an inflation rate Π+ < Π*, the inflation rate Π+ becomes a self-fulfilling expectation (and is hence a "rational expectation"). In this case, the monetary authority simply sets its policy rate R < R*. A situation like this can last indefinitely in this model.

Of course, everything works just fine here as long as the ZLB is not a constraint. Suppose, instead, that the policy rate is constrained by the ZLB, so that [8] becomes:

[9]  ln(R) = max{ 0, ln(R*) + α[ ln(Π+) - ln(Π*) ] }

Imagine that the economy is initially operating at potential with Π+ = Π* (without loss). Then, out of the blue, individuals suddenly believe that the inflation rate is going to be permanently lower Π+ = Π' < Π*. Moreover, suppose that this inflation shock is sufficiently large to make the ZLB bind. What happens?

What happens is that output drops permanently below potential (the economy continues to grow, however, at the rate implied by technological progress and population growth, both of which are normalized to zero here). Why does this happen?

It happens (here) because the real rate of interest rises above its natural rate, r' = 1/Π' > r*. The real interest rate is too high. The effect is to depress (investment) demand, r' = f'(k') implies k' < k*. The real GDP falls below potential, Y' = y + f(k') < Y*.

Because there is no nominal anchor for inflation in this economy, all sorts of bad things can happen at the lower bound. Contrary to the Friedman rule prescription, deflation is bad (generally, any inflation rate sufficiently low to make the ZLB bind). Not that the monetary authority could actually implement the Friedman rule if it wanted to. In this economy, the monetary authority has absolutely no control over the inflation rate!

A nominal anchor

An obvious way to provide a nominal anchor (in the model) is to adopt the monetarist approach and control the supply of the monetary aggregate. But perhaps this is something that is difficult to do in reality. What then?
Everything seems to hinge here on how individuals form inflation expectations. The theory here provides no guidance as to how these expectations should be formed. One can assert that individuals are likely to use the inflation target Π* as a nominal anchor. But this is just a bald-faced assertion. That is, if individuals do use Π* as a nominal anchor, then it will become a nominal anchor. The monetary authority, however, has no way enforcing the target Π* (unless it adopts a monetarist approach). 
Well then, let me assume a particular inflation expectation formation rule:

[10]  ln(Π+)  = (1 - ρ)ln(Π*) + ρln(Π) + δ[ ln(Y+) - ln(Y*) ] + ε

where 0 ≤ ρ ≤ 1, δ≥ 0, and where ε represents an inflation shock (say, i.i.d. and zero mean). Students may recognize [10] as a type of Phillips curve

Actually, now that I stare at [10], I see that the δ > 0 opens up another source of indeterminacy. It may be possible, for example, that if people suddenly expect a recession Y+ < Y*, that the downward revision in inflation forecasts implied by [10] could make the ZLB bind, generating a self-fulfilling prophecy.

Anyway, let's just set δ = 0 here. In the Wicksellian approach above, I adopted a special case of [10]; i.e., ρ = 1 and δ = 0;. But now, for 0 ≤ ρ < 1, any given inflation shock is mean-reverting (to the inflation target). The "lift off" date -- the date at which the monetary authority begins to raise its policy rate according to [9] depends on how quickly inflation expectations rise. The speed of adjustment here is governed by the parameter ρ--a lower ρ implies faster adjustment.

Is there anything the monetary authority can do here to "talk up inflation" (i.e., lower ρ)? We really can't say without a theory of expectation formation. But it seems to me that "promising to keep R = 1 for an extended period of time" may have the effect of increasing ρ, extending the period of adjustment. That is, by postponing the "lift off" date, agents may rationally expect inflation to remain below target for a longer period of time.

Concluding thoughts

Let me be clear that I do not think the 2008 drop in output below its previous trend was caused by a negative inflation shock. A negative inflation shock possibly played a role, but there had to be more to the story than this. In the analysis above, a negative inflation shock represents a movement along a stable IS curve; the real interest rate goes up, and output goes down. To make sense of recent events, we also have to consider shocks that shift the IS curve "leftward." (I describe just such a shock here.)

Nevertheless, I think it is interesting to explore what potential effects future downward revisions to inflation expectations may have on the economy at the ZLB. In the Wicksellian regime I study above, there appears to be no nominal anchor apart from what agents believe it to be. And if agents come to believe in a persistent deflation, it may come to pass, and the economy may be stuck below potential for a very long time. Convincing agents that the nominal interest rate is likely to remain at zero for a long time may be counterproductive, depending on how individuals interpret such policy announcements.

I want to stress, however, that while getting inflation and inflation expectations back to target (and firmly anchored to target) may be a solution to one problem, it is unlikely to be a solution to every problem currently facing the U.S. economy. To put it another way, suppose that the current real interest rate of -1% is too high relative to the current "natural" rate of -x%. Somehow driving the real return on bonds to -x% may then help things a bit, but it does nothing to address the more pressing question of why the "natural" rate is so low to begin with.


  1. David,

    This is a great piece of work. The simplicity of the model ought to make it the perfect reference framework for debate with some of the more prominent monetary factions in the blogosphere. I hope they read it carefully. And a pony.

    I do have one question (objection) though:

    “The monetary authority, however, has no way enforcing the target Π* (unless it adopts a monetarist approach).”

    What approach, *exactly* is this? OMOs? Unbacked helicopter drop? The former are clearly not going to work if the nominal natural rate is below zero. The latter, sure, that might cause inflation, unless they pre-empt a run on the Fed via financial repression. The trouble with monetarists is that when they bring out the Humean money doubling thought experiment they always forget about pesky things like debt and other nominal contracts which exist in quantities that are several orders of magnitude greater than the exchange medium. (Also, they forget that the quantity of the exchange medium *is* demand determined.)

    "it does nothing to address the more pressing question of why the "natural" rate is so low to begin with."

    We have a structural problem. The means of production are coordinated to produce goods for workers whose declining marginal product can no longer sustain the usual consumption gains without repeated rounds of leveraging. Production therefore has to be repurposed to make stuff for capitalists. Which is a painful process that takes time. I.e. it's a major drag on the growth outlook, which, all things being equal, will depress the natural rate. It's all very Marxian.

    1. Thanks, K.

      The "monetarist" approach in the model above refers to helicopter drops (and withdrawals) of the money supply via lump-sum injections (taxes). The assumption of lump-sum taxation is very powerful and unattractive (and yet, most of our monetary models, including NK models, assume the existence of such an instrument).

      By the way, the model here is one in which nominal debt exists. The old have non-indexed debt, and any surprise change in the price-level will have redistributive consequences. I talk about this here:

      Also, they forget that the quantity of the exchange medium is demand determined.

      By exchange medium, do you mean to include private money(demand deposit liabilities)? If so, I'm not sure I agree with your statement. Loan contracts typically specify quantity, interest rate, and collateral.

      With respect to your diagnosis of our structural problem--can you point me to a model that formalizes this idea?

  2. David,
    How can the ZLB be a floor on real rates when the forecast real rate is already negative? The implication is that the ZLB prevents the real rate from going from -2% to -5%. If so, then how was policy able to generate positive inflation expectations in the first place? Is the +2% inflation expectation occurring at the ZLB assumed to be "immaculately conceived" (independent of policy).

    It makes more sense that a ZLB constrains the real rate when there are outright deflation expectations (falling price level).

    1. Hi Diego,

      I'm not sure I fully understand your question, but let me try.

      How can the ZLB be a floor on real rates when the forecast real rate is already negative?

      I never said (I don't think) that the ZLB puts a floor on real rates (it is a floor on the nominal rate). If inflation expectations in the U.S. are roughly 2%, then the real interest rate is -2%. This is not a floor; they can obviously go lower if inflation expectations rise and the nominal interest rate remains at the ZLB.

      The implication is that the ZLB prevents the real rate from going from -2% to -5%.

      I don't see why.

      If so, then how was policy able to generate positive inflation expectations in the first place?

      I don't know. We would need a theory of expectation formation to answer this.

      It makes more sense that a ZLB constrains the real rate when there are outright deflation expectations (falling price level).

      I'm not sure I understand this. Just to be clear, the ZLB constrains the nominal rate, not the real rate, right? Maybe try rephrasing the question for me.

  3. I see - thanks for clarifying. I think I was getting at the question of whether initial conditions matter. Real rates during 1929-1933 were positive, as were they in Japan during the lost decade(s). Today they are abnormally negative. Are all three exhibit the same inability of RGDP to return to trend; but is the negative real rate (and on-trend inflation) one different in kind and not just degree? For instance, I would imagine that positive real rates spark adverse feedback loops in the credit channel, while negative ones do not.

    The better comparison is of 2012 with 1936. They share large and persistent output gaps, negative real rates, and strong recoveries in exports, consumer durables, and business investment (ex-structures). We happen to have a much larger service sector now, and it is this sector that has continued to lag its previous trend. So was the solution to the persistent 1936 output gap a "more negative" real interest rate? The answer for 1936 is much less clear than for 1932. Often it seems like economists miss this distinction.

    In short, maybe both 1936 and 2012 are periods of "rational pessimism" during which the IS curve has shifted leftward.

    1. If such a thing as an IS curve actually existed, you might be right. Unfortunately, IS curves are mythological, like unicorns and AD curves.

  4. Please address the more pressing question of why the "natural" rate is so low to begin with.

    Exceptionally informative

  5. I am not an economist. So I stumbled over the basics here: "People live for two periods; they are "young" and then "old." Everyone only values consumption when old. Their objective is simply to maximize (expected) future consumption."

    Seems to me the assumption that begins the equation does not reflect the reality of our economy. But again, I am not an economist. Just someone trying to fathom how our economy fell off a cliff in 2008.

    Where, in your equation, do you include innovative financial instruments like mezzanine ABS CDOs and the like? I believe tools like this contributed to the story of the collapse of our economy. When the federal government feels compelled to funnel billions of dollars to prop up the financial sector ($182 billion just to AIG, so kind of them to decline to sue us!), something other than inflation shock is at issue. Do you feel there are any regulatory remedies that could prevent such extreme failures in the future?

    The levels of debt we saw in both in company balance sheets and in personal accounts seem unprecedented to me. How does the model explain this?

    Again, not an economist. Just someone who remains shocked the catastrophic destruction of what once appeared to be a fairly solid economy.

    1. Main Street,

      This post was targeted mainly at academics and practitioners. I was interested in a narrow question related to the manner in which monetary policy is conducted. Certainly, there are no sophisticated financial instruments in the model. To answer the very good questions you raise, you may wish to consult Gary Gorton's fine work in the area:

  6. I have been reading and considering your post and Ballard's paper in light of current events.

    It seems to me that you have written one sentence that pretty profound and which condemns all the models and thinking of macro, being, "We really can't say without a theory of expectation formation." That is true across all models, save Keynes basic insights, such as his insight that fiscal policy can change the expectations of the average business man.

    For example, Ballard's paper thinks that what the Fed says is important and impacts expectations.

    Most certainly, if that is true, what the GOP said about the fiscal cliff is now saying about debt ceiling are important and have been important since 2011 and before.

    In fact, if he were as smart as he things he is, in the fall of 2010, President Bullard should not have been writing about the Fed and the Zero Bound. He should have been writing about the upcoming debt ceiling and how his models showed that would impact expectations, etc. etc.

    Yet, no macro theory deals with such because there is no theory of expectation formation.

    1. Well, I'm not sure whether Keynes actually had a *theory* of expectation formation. I'm sure that he, and many other people, remarked on how different factors could influence expectations.

      You may wish to consult the learning literature; see, for example,

  7. David,

    Long delay in replying. Sorry. Ragu Rajan agrees with my diagnosis of our structural problem. But I don't have a model. If I did, I think it would involve a (large) subset of households that are liquidity constrained (like Eggertsson Krugman 2010), but you'd also need two consumption goods, and labour retraining costs from producing the general consumption good to producing the luxury good. Wouldn't that work?

    "the model here is one in which nominal debts exist"

    Yes. Forget what I said about the internet monetarists. The real problem with them is that they confuse QE with a Humean money expansion. The latter is a helicopter drop (fiscal operation), while the former is basically *nothing* except to the extent that it effects the path of nominal short rate.

    By "exchange medium" I meant non-interest earning base (currency), the "hot potato" portion of the base, the quantity of which someone might (wrongly) consider to be under the the control of the Fed. The Fed can only effect the size of the whole base (through OMOs). Currency "refluxes" back to the banks in exchange for deposits at the option of households, and back to the Fed in exchange for reserves at the option of the banks. If reserves *don't* earn interest then a base expansion will result in the FF rate dropping to zero and there will be no hot potato anyways.

    The internet monetarists think the Fed can effect the exchange rate of nominal and real assets by exchanging interest bearing nominal assets with non-interest bearing nominal assets. I've been engaging in a (futile) debate with them (for way too long) that all OMOs can accomplish is to effect the exchange rate between interest bearing and non-interest bearing nominal assets, i.e. the nominal interest rate. Any *real* effects follow exclusively via the channel of the nominal interest change, and the quantity of the base has *no* independent real effects.

  8. Also, given the fact that labour markets are, in fact, utterly incomplete, and retraining from autoworker to jeweller might, in many cases, be impossible, I think we have the necessary conditions for a welfare loss. Maybe a luxury good tax and wealth transfer would be welfare improving.