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

Friday, November 6, 2015

Fisher without Euler

The Neo-Fisherian proposition is that raising the nominal interest rate (and keeping it elevated) will eventually cause inflation to rise (see Steve Williamson's explanation here.)

The basic idea revolves around the so-called Fisher equation:

R = r + E[p]

where R is the nominal interest rate, r is the real interest rate, and E[p] is the expected rate of inflation. If bond buyers expect inflation to increase then they'll ask for more compensation in the form of a higher nominal interest rate (a lower bond price).

The conventional idea is that monetary and fiscal policies (in particular, the expectation of how these policies will unfold over time) largely determined inflation expectations E[p]. In conventional (modern) macro economic theories, expectations are assumed to be formed "rationally" (i.e., in a manner that is consistent with the stochastic processes that actually govern the economy).

Neo-Fisherians reverse this conventional direction of causality. They argue that increasing R leads people to revise their inflation expectations upward. And because people have rational expectations, for these expectations to be consistent with reality, actual inflation will (somehow) have to increase.

As far as I can tell, this Neo-Fisherian proposition comes in two stripes. The first stripe is of the "cashless economy with Ricardian equivalence" variety--the models that Michael Woodford likes to use. In this class of models, "balance sheets don't matter." And because central bank money and government bonds are just ways of labeling the liabilities of the consolidated government sector, they don't matter for determining (among other things) the price-level. In this class of models, inflation expectations are somehow assumed to adjust to satisfy the Fisher equation. And then the price-setting behavior of firms (who set prices in an abstract unit of account but do not actually accept payment in any monetary object) adjusts in a manner that is consistent with higher expected inflations. Personally, I find this view implausible. Moreover, it's frustrating that no one promoting this view seems willing or able to explain how/why all this is supposed to happen (beyond repeating the phrase "the Fisher equation must hold" or "it's a rational expectations equilibrium").

The second stripe of this proposition, however, seems more plausible (at least, in principle) to me. In this world, balance sheets matter. The supply and composition of the government's assets and liabilities matter. And in particular, the time-path of the total nominal government debt (and its composition) matters for determining the price-level. The idea here is that when the central bank announces a higher R, there is a corresponding passive accommodation of central bank policy on the part of the fiscal policy to increase the rate of growth of total government debt (i.e., cut taxes, or engage in "helicopter drops"). If the fiscal authority behaves "passively" in this sense, then people will rationally expect higher inflation--and the higher inflation will actually transpire not because people expected it, but because the fiscal authority delivered it. I think this is an interpretation that even Nick Rowe agrees with (see here).

Both versions of the Neo-Fisherian proposition above seem to rely heavily on the notion of rational expectations. In my previous post, I speculated that the proposition might hold even if people had non-rational "adaptive" expectations. The idea I had there was that if a sudden increase in R caused to the price-level to jump up (instead of down, which is the usual presumption), then people with adaptive expectations will revise their inflation expectations upward (not downward).  An initial increase in the price-level might happen if, for example, the higher interest rate led to higher operating expenditures on the part of firms. Following this initial impulse, the actual path of inflation would be determined either by (stripe 1) the nature of learning dynamics or (stripe 2) the manner in which policy accommodates itself to the price shock (e.g., see Christiano and Gust, 1999).

In response to my post, Erzo Luttmer alerted me to his paper Fisher without Euler, in which he claims that the Neo-Fisherian proposition pops out of a model in which people are not forward-looking at all. The argument, as far as I can tell, relies heavily on how the government debt-service cost is financed. Let me try to explain (you can refer to Erzo's paper and short note to see whether I have it right).

Let's start with the government budget constraint,

G(t) - T(t) = q*B(t) - B(t-1)

where T(t) denotes tax revenue, G(t) government purchases, and B(t-1) denotes bonds maturing to cash at date t. Let 0 < q < 1 denote the price of a bond (1/q is the gross nominal interest rate, set by the Fed). For simplicity, I think we can set G(t) = 0 for all t, so that

T(t) = B(t-1) - q*B(t)

This makes it clear how a lower q (higher interest rate) means either  higher taxes and/or higher debt level. Now, let p(t) denote the price-level and define τ = T(t)/p(t). Assume that nominal debt grows at a constant rate, B(t) = μB(t-1). Now use this notation to rewrite the government budget constraint above as
τ = (1 - q*μ)*B(t-1)/p(t)

To close the model, we need a theory of the price-level. The simplest theory I can think of is the Quantity Theory: p(t) = B(t-1)/y(t), where y(t) is real income (and velocity is held constant), so that B(t-1)/p(t) = y(t). If we treat y(t) as exogenous, then it follows immediately that lowering the interest rate (increasing q) necessitates a decline in inflation (μ). So lowering the interest rate lowers the debt-service cost of debt which (for given real spending and taxation levels) means that the supply of nominal debt need not grow as quickly -- as the growth rate in the supply of "money" declines, so does inflation. The Neo-Fisherian result follows even without forward-looking behavior.

Erzo does not use the simple version of the Quantity Theory as I did here. Instead, he assumes that individuals adopt a simple behavioral rule (consumption function):

c(t) = α(y(t) - τ) + βB(t-1)/p(t)

where α is the propensity to consume out of disposable income and β is the propensity to consume out of wealth (here in the form of real bond holdings). If we let g(t) denote real government purchases, then goods-market-clearing requires:

c(t) = y(t) - g(t)

Erzo then combines these latter two equations to determine the price-level p(t), treating y(t) and g(t) as exogenous (as did I).

At the end of the day, it's a simple point. Still, I think it's an important one to keep in mind since I am reading in more than one place that the Neo-Fisherian proposition depends on rational expectations. Evidently, it does not.


  1. David: Good post. I think we are converging.

    Here is perhaps a simpler way to think about it.

    Start with an almost standard money demand function: M/P = L(Y, R-Rm), L1 > 0, L2 < 0, where Rm is the rate of interest paid on money, and R is the rate of interest paid on other assets, so R-Rm is the opportunity cost of holding money. (The only difference between this and the standard money demand function is that the standard money demand function assumes Rm=0).

    Now assume that interest on money is paid for by printing new money (so 5% interest on money is like an ongoing annual 1.05 for 1 stock split). This means that printing money to pay interest on money has no fiscal consequences.

    From then on, it's straight Quantity Theory. Increasing Rm *means* increasing the growth rate of M by exactly the same amount. So it's inflationary.

    I did a post on it once:
    (Maybe ignore the last line in that post?)

    1. Yes, this sounds exactly right. Alternatively, it works also if R = Rm, which is arguably closer to the present situation. The interesting thing here, I think, is that in this scenario, the inflationary consequences of increasing the interest rate follows even in the absence of rational expectations.

  2. David,

    I did an exercise earlier this year where I rewrote the Diamond growth model into an SFC format. ( and

    With suitable choice of parameters, this gives the same results as Diamond, but the different structure allows us to look at some questions slightly differently. The format requires a nominal interest rate, which can then be set as a policy tool. Not surprisingly, with rational expectations (which we need to mirror Diamond), the model is fully neo-Fisherian and the only effect of changing the nominal interest rate is a corresponding change in the inflation rate.

    However, setting out the model this way makes it easy to replace the RE assumption with adaptive expectations. This still gives the NF result, although it now takes a few periods to get there. (The rate of adaption can't be too quick or the model is not stable.)

    I think this is quite closely related to what you are saying here. The OLG structure obviously implies a consumption function of the form you describe.

    1. Nick, this looks very, very interesting. Can you please email me: Thanks.

  3. Thanks for this new, clarifying post.

    It seems clear that the first stripe of the Neo-Fisherian proposition is simply untenable: it is a case of 'immaculate' inflation, one where there is no transmission mechanism whatsoever that explains how higher nominal interest rate causes higher inflation (I discuss this issue in my recent comment (

    The second stripe, the one that you consider as more plausible, rests on active fiscal policy, without which there would neither be higher nominal interest rates nor higher inflation.

    But I would like briefly to consider the policy implications of this analysis. First, I would assume that we are in a ZLB world, where conventional monetary policies are ineffective and, thus, the central bank announces a higher nominal interest rate, in anticipation that the fiscal authority accommodates its policy and increases the rate of growth of total public debt: people will rationally expect higher inflation and higher inflation will follow (rightly as you note, not because people expect it, but because the fiscal authority delivers it).
    Two observations here:

    First, since we are at the ZLB, resource unemployment is is presumably large: why then should the fiscal impulse translate into higher inflation rather than larger output growth or a combination of both? If the fiscal impulse (partly) triggers output growth, then the nominal interest rate and inflation would not rise one-for-one. The Neo-Fisherian (strip two) result thus holds only at full employment, but why should the central bank want to adopt a Neo-Fisherian policy option and announce a higher nominal interest rate at full employment?!

    Second, still at ZLB, why should the central bank prefer to resort to the Neo-Fisherian policy option instead of agreeing with the fiscal authority on an overt monetary financing of the deficit (or helicopter money) program, whereby a given fiscal impulse would be fully financed with money creation? This way, the nominal interest rate would be held constant, there would be no new public debt creation, and the output multiplier effect would be larger.

    In the end, what is the advantage of the Neo-Fisherian policy option?

    1. Biago, thanks for your comments.

      First, this was not a post about any advantages of the Neo-Fisherian thought experiment of raising interest rates. It was just a post to see whether RE was necessary for the result.

      Having said this, I think I agree with most of your points about the desirability of raising rates in present circumstances. I talk a bit about this near the end of this paper:

  4. [Note: Biago, I accidentally deleted your comment, but recovered it from another source. Here it is.]

    Thank you for your reply and the link to the paper.
    Indeed, I fully appreciate your point and I didn't intend to attribute any advantages of the NF thought experiment to you. Indeed I think you brought clarifications to the issue. One thing I'd like to emphasize from my own comment is that, regardless of which expectations formation mechanism you assume, it may actually be the case that, at the ZLB, the NF policy option (central bank announcement on the interest rate + accommodating fiscal policy stance) might result in only modest inflation and a significant output effect: in fact, there would no NF result at all in this case. Besides, if the underlying transmission mechanism is correct, whereby the fiscal impulse leads to higher interest rate and a non-zero output effect, any expectations that would assume a one-for-one increase in nominal interest rate and inflation would not at all be rational, after all. Unless, of course, people believe that the economy is at a new (lower) potential output level but then, again, why would the central bank and the fiscal authority embark on a demand expansionary policy strategy, to start with...?! Thanks again

  5. I, as a fairly junior student of economics, wonder how the interpretation of the "expectations formation mechanism" of the NF model is affected were the constant velocity assumption implicit in this discussion, and also, from what I can tell, implicit in Erzo's behavioural rule, removed.

    I note that QTM on its own predicts a decrease in price levels with an increase in the money supply so long as V decreases more than M increases.

    If we assume V is dynamic, don't such changes occur due to changes in rational expectations? Can we then really say that the NF position does not require rational expectations?

    Admittedly, I feel like I am missing something fundamental in this discussion but I thought I would ask!

    I found these interesting:

    1. I think one could view V as being determined by forward-looking behavior, but that forward-looking is not equivalent to rational expectations.

      Good to see you reading FRB St. Louis research! :)