Neo-Fisherism says, basically: "Excuse me, but I think you have the sign wrong." Conventional central banking wisdom says that increasing interest rates reduces inflation. Neo-Fisherites say that increasing interest rates increases inflation. Further, it's not like this is some radical, novel theory. Indeed, a cornerstone of Neo-Fisherism is:

*Neo-Fisherian Folk Theorem*: Every mainstream macroeconomic monetary model has neo-Fisherian properties.

Let me illustrate that. A nice, simple, version of the standard New Keynesian (NK) model is the one in Narayana Kocherlakota's slides from this conference put on by the Becker Friedman Institute. I'll use my own notation. NK's version of the NK model is a reduced form, with two equations. The first comes from a pricing equation for a nominal bond - what's often called the "NK IS curve," orHere,

*y*is the output gap, the difference between actual output and what is efficient, pi is the inflation rate,

*R*is the nominal interest rate,

*r*is the subjective rate of time preference, and

*a*is the coefficient of relative risk aversion. The second equation isThat's just a Phillips curve, with

*b >0*determined by the degree of price stickiness. In the underlying model, some fraction of firms is constrained to set prices to the average price from last period. Thus, there's no expectations term in the Phillips curve, as there's no forward-looking pricing. That makes the model easy to solve.

So, substitute for

*y*in equation (1) using the Phillips curve equation, to getSo, you can see why people think this type of model is a foundation for conventional central banking ideas. If inflation expectations are "anchored," which I guess means exogenous, on the right-hand side of the equation, then an increase in the current nominal interest rate would have to imply that the current inflation rate goes down. Indeed, if the central banker experiments, by choosing the nominal interest rate each period at random, then he or she will observe a negative correlation between inflation and nominal interest rates, which would tend to confirm conventional beliefs.

But consider the following. Suppose we look at the deterministic version of the model, and use (3) to solve for a first-order difference equation in the inflation rate:Then, an equilibrium is a sequence of inflation rates solving (4), and we can solve for output from (2). As is typical of monetary models, there's no initial condition to tie things down, so there are potentially many equilibria. We can say, however, that in a steady state, from (1),And then (2) givesSo, what "anchors" inflation and inflation expectations in the long run is the long run nominal interest rate. And then the Phillips curve determines output. That's the first Neo-Fisherian property of this standard model.

Next, from the difference equation, (4), if the nominal interest rate is a constant

*R*forever, then there is a continuum of equilibria, indexed by the initial inflation rate, and they all converge to a unique steady state, which is given by (6) and (7). To see this, start with any initial pi, and solve (4) forward. So, we know that the long run is Fisherian. But what about the short run?

We'll consider the transition to a higher nominal interest rate. In the figure, the nominal interest rate is constant until period T, and then it increases permanently, forever. In the figure, D1 is the difference equation (4) with a lower nominal interest rate; D2 is (4) with a higher nominal interest rate. We'll suppose that everyone perfectly anticipates the interest rate increase from the beginning of time. Again, there are many equilibria, and they all ultimately converge to point B, but every equilibrium has the property that, given the initial condition, inflation will be higher at every date than it otherwise would have been without the increase in the nominal interest rate. A straightforward case is the one where the equilibrium is at A until period T, in which case the inflation rate increases monotonically, as shown, to a higher steady state inflation rate. Inflation never goes down in response to a permanent increase in the nominal interest rate. That's consistent with what John Cochrane finds in a related model.

So, that's the second Neo-Fisherian property, embedded in this NK model. The NK model actually doesn't conform to conventional central banking beliefs about how monetary policy works. What's going on? From equation (1), an increase in the current nominal interest rate will increase the real interest rate, everything else held constant. This implies that future consumption (output) must be higher than current consumption, for consumers to be happy with their consumption profile given the higher nominal interest rate. But, it turns out that this is achieved not through a reduction in current output and consumption, but through an increase in future output and consumption. This serves, through the Phillips curve mechanism, to increase future inflation relative to current inflation. Then, along the path to the new steady state, output and inflation increase. But, if you read Narayana's Bloomberg post from five days ago, you would have noted that he thinks that lowering the nominal interest rate raises inflation and output:

Monetary policy makers should be seeking to ease, not tighten. Instead of satisfying a phantom need to “normalize” rates, the Fed should do what’s needed to get employment and inflation back to normal.Apparently he's thinking about some other model, as the one he constructed tells us the opposite.

For more depth on this, you should read this paper by Peter Rupert and Roman Sustek. Here's their abstract:

The monetary transmission mechanism in New-Keynesian models is put to scrutiny, focusing on the role of capital. We demonstrate that, contrary to a widely held view, the transmission mechanism does not operate through a real interest rate channel. Instead, as a first pass, inflation is determined by Fisherian principles, through current and expected future

monetary policy shocks, while output is then pinned down by the New-Keynesian Phillips curve. The real rate largely only reflects consumption smoothing. In fact, declines in output and inflation are consistent with a decline, increase, or no change in the ex-ante real rate.

Conventional central banking wisdom is embedded in Taylor rules. For simplicity, suppose the central banker just cares about inflation, and follows the ruleHere pi* is the central bank's inflation target. Under the Taylor principle,

*d > 1*, i.e. the central bank controls inflation by moving interest rates up when inflation goes up - and the nominal interest rate adjustment is more than one-for-one. It's well known from the work of Benhabib et el. that Taylor rules have "perils," and this model can illustrate that nicely. The difference equation determining the path for the inflation rate becomesIn the next figure, A is the intended steady state in which the central bank achieves its inflation target, and that is one equilibrium. But there are many equilibria for which the initial inflation rate is greater than -r and smaller than the inflation target, and all of these equilibria (like the one depicted) converge to the zero lower bound (ZLB), where the central banker gets stuck, with an inflation rate permanently lower than the target. Potentially, there could be equilibria with an initial inflation rate higher than the inflation target, which have the property that inflation increases forever. But in this model, that also implies that output increases without bound, which presumably is not feasible.

Rules with

*-1 < d < 1*all have the property that there are multiple equilibria, but these equilibria all converge to the inflation target - there's a unique steady state in those cases. Note that the Taylor rule central banker is Neo-Fisherian if

*d < 0,*and that this can be OK in some sense. But aggressive neo-Fisherism, i.e.

*d < -1 -2(a/b)*, is bad, as this implies that the inflation rate cycles forever without hitting the inflation target.

But if the central banker actually wants to consistently hit the inflation target, there are better things to do than (8). For example, consider this rule:Plug that into (4), and you'll getAnd so, (10) implies thatSo, under that forward-looking Taylor rule, the central bank always hits its target, and in equilibrium the central bank is purely Fisherian. If it wants to increase its inflation target - and actual inflation - it just increases the nominal interest rate one-for-one with the increase in the inflation target. So, I've lost count now, but I think that's Neo-Fisherian property 3 [see the addendum below. There's a glitch that needs to be fixed in the rule (10) to account for the ZLB.]

The rule (10) specifies out-of-equilibrium behavior that kills all of the equilibria except the desired steady state. Why does this work? If the central banker sees incipient inflation in the future, he or she knows that this will tend to increase current output, increase current inflation, and increase future output, which will also increase current inflation. To nullify these effects, the central banker commits to offset this completely, if it happens, with an increase in the nominal interest rate. In equilibrium the central banker never has to carry out the threat. Maybe you think that's not plausible, but that's the nature of the model. NK adherents typically emphasize forward guidance, and that's not going to work without commitment to future actions.

Some people (e.g. Garcia-Schmidt and Woodford) have argued that Neo-Fisherian results go out the window in NK models under learning rules. As was shown above, these models are always fundamentally Fisherian in that any monetary policy rule has to somehow adhere to Fisherian logic on average - basically the long-run nominal interest rate is the inflation anchor. But there can also be learning rules that give very Fisherian results. For example, suppose that the economic agents in this world anticipated that next period's inflation is what they are seeing this period, that isPlug that into equation (1), and we getSo, for this learning rule, inflation is determined period-by-period by the nominal interest rate - this is about as Fisherian as you can get.

Thus, if conventional central bankers are basing their ideas on some model, it can't be a mainstream NK model, since

*increasing the nominal interest rate makes inflation go up*in mainstream NK models. But don't get the idea that it's some other mainstream model they're thinking about. As the Neo-Fisherian Folk Theorem says, all the mainstream models have these properties, though some of the other implications of those models differ. For example, it's easy to show that one can get exactly the same dynamics from Alvarez, Lucas and Weber's segmented markets model. That's a model with limited participation in asset markets and a non-neutrality of money that comes from a distribution effect. Everyone in the model has fixed endowments forever, and they buy goods subject to cash-in-advance. The central bank intervenes through open market operations, but the people on the receiving end of the initial open market operation are only the financial market participants. The model was set up to deliver a liquidity effect, i.e. if money growth goes up, this increases the consumption of market participants (and decreases everyone elses's consumption), and this will reduce the real interest rate. Thus, you might think (like the NK model) that this produces the result that, if the central bank increases the nominal interest rate, then inflation will go down.

But, the inflation dynamics in the Alvarez et al. segmented markets model are identical to what we worked out above. In fact, the model yields a difference equation that is identical to equation (4), though the coefficients have a different interpretation. Basically, what matters is the degree of market participation, not the degree of price stickiness - it's just a different friction. And all the other results are exactly the same. But the mechanism at work is different. The quantity theory of money holds in the segmented markets model, so what happens when the nominal interest goes up is that the central bank has to choose a path for open market operations to support that. This has to be a path for which the inflation rate is increasing over time, but at a decreasing rate. This will imply that consumption grows over time at a decreasing rate, so that the liquidity effect (a negative real interest rate effect) declines over time, and the Fisher effect increases.

So, once you get it, you can form your own Neo-Fisherian support group. Moving from denial to advocacy is important.

**Addendum1**: Thanks to Narayana. This took some work, but this is a Taylor rule that assures that the central banker hits the inflation target period-by-period, implying that the nominal interest rate is constant in equilibrium, and will move one-for-one with the inflation target. If future inflation is anticipated to be sufficiently high, then the central banker follows the forward looking rule (10):This rule offsets incipient high inflation, and assures that the central bank hits the inflation target. But, low inflation is a problem for (16), as the ZLB gets in the way. So, if there is incipient low inflation, the central banker follows the rule: And the critical value for future inflation isHow does (17) work? Any equilibrium has to satisfy (4), but (4) and (17) implySo future inflation must be greater than the inflation target. But (17) says that the central banker chooses this rule only when future inflation is less than pi**, which is less than the inflation target. So this can't be an equilibrium. I like (17), as the central banker is Neo-Fisherian - he or she kills off low inflation with a high nominal interest rate.

**Addendum 2:**This is interesting too. Suppose the policy rule isThen there is a critical value for the initial inflation rate,such that, if the initial inflation rate is below this critical value, then the inflation rate goes to the inflation target in the next period and stays there. If the initial inflation rate is above the critical value, then the initial nominal interest rate is zero, and the inflation rate falls to the inflation target, and stays at the target forever. So, that's a Fisherian rule that has nice properties.

**Addendum 3:**Here's another one. Central bank follows rule (20) if current inflation is below the inflation target. Central bank follows rule (10) if current inflation is at or above the inflation target. With inflation below the target, this implies raising the nominal interest rate to get inflation to target. With inflation at or above the target, the central bank promises to raise the nominal interest rate in response to incipient inflation. At worst, this implies one period of inflation below target in equilibrium.