Start with basics. What are Neo-Fisherite ideas anyway? Narayana says
...in the absence of shocks, the equilibrium inflation rate should be constant if the nominal interest rate is pegged forever. The Fisher equation then implies that the inflation rate should move one for one with the nominal interest rate. This logic is sometimes referred to as “neo-Fisherian”.I would actually call these New Keynesian (NK) claims. For example, in "Interest and Prices," Mike Woodford takes pains to address the concern, which came out of the previous macro literature, that nominal interest rate pegs are unstable. Woodford's claim is that a Taylor rule that conforms to the Taylor principle (a greater than one-for-one increase in the nominal interest rate in response to an increase in inflation) will imply determinacy. That is, if there are no shocks, then the nominal interest rate is pegged at a constant forever, and the inflation rate is a constant - the inflation target. Further, in the basic NK model, if Woodford's claim is correct then, in the absence of shocks, if the central bank wants to increase its inflation target, then the nominal interest rate should increase one-for-one with the increase in the inflation target, and actual inflation will respond accordingly. Under basic NK logic, this behavior is supported by promises to increase the nominal interest rate in response to higher inflation - and this inflation never materializes in equilibrium.
But, whatever we think Neo-Fisherite or New Keynesian ideas are, Narayana is making a particular argument in his note, and we want to get to the bottom of it. I don't think the analogy part is particularly helpful though. There are two problems considered in Narayana's note. One is an asset pricing problem, and the other has to do with the properties of a particular NK model. As far as I can tell, the extent of the commonality is that solving each problem can involve geometric series. Otherwise, understanding one problem won't help you much with the other.
The asset pricing problem looks like a trick question you might give to unwitting PhD students on a prelim exam. The equilibrium one-period real interest rate is negative and constant forever, and we're asked to price an asset that pays out a constant real amount each period forever. Question: Solve for the steady state price of the asset. Answer: Dummy, there is no steady state price for the asset. Since a rational economic agent in this world values future payoffs more than current payoffs, if we compute the present value of the payoffs, it will be infinite.
Well, so what? On to the second problem. Narayana uses a version of the standard NK model. We're in a world with certainty - no shocks. I'll change the notation so I don't have to use Greek letters. From standard asset pricing, and assuming constant relative risk aversion utility, we can take logs and get
In general, we can solve to get the difference equation
So, suppose that the nominal interest rate is a constant R forever, and suppose that, in period T the inflation rate is i(T). Then, we can solve the difference equation (3) forward to get
What Narayana does is to take equation (5), and let T go to infinity, so he's only looking at the backward solution. As should be clear, I hope, that's not describing all the equilibria. By any conventional notion of what we mean by convergence and stability, the nominal interest rate peg is stable, and all the equilibria converge in the limit to R - r. The Fisher relation holds in the long run. As a practical implication of this, I've heard many people argue that, if the central bank holds its nominal interest rate at zero, then surely inflation will eventually rise to the 2% inflation target. Well, they can't be thinking about this model then. In any equilibrium with R = 0 forever and with inflation initially lower than some inflation target i*, inflation either falls to -r in the limit, or rises to -r in the limit. If -r < i*, the central bank will never achieve its target by staying at zero.
But, with a nominal interest rate pegged at some value forever, we have an indeterminacy problem - there exists a plethora of equilibria. This makes it hard to make statements about what happens when the interest rate goes up or down. For example, it's certainly correct that, if we set T=0 in equation (4), and think of time running from zero to infinity, solving the difference equation (3) forward, then given i(0), the inflation rate will be higher along the whole equilibrium path, if R rises. But i(0) is not predetermined - it's not an initial condition, it's endogenous and the first step in only one equilibrium path. Who is to say that economic agents don't treat R as a signal and jump to another equilibrium path? We might also be tempted to set i(0) = R*-r, then solve for the equilibrium path given R = R**, and think of that as describing the effects of an increase in the nominal interest rate from R* to R**, since an inflation rate of R* - r is the long run inflation rate when R = R*. Though that's suggestive, it's not precise, due to the indeterminacy problem.
So what to do about that? If we follow the usual NK approach, we would specify a Taylor rule
So, we might look for other policy rules that are better behaved. Here's one:
What Narayana seems to be getting at is that stickiness in expectations matters. In the example he gives in his note, fixed expectations in the infinite future can have very large effects today. You can see that in equation (5), for example, if we fix i(T) and solve backward. Indeed, it seems that conventional central banking wisdom comes from considering expectations as fixed, as is common practice in some undergraduate IS-LM/Phillips curve constructs. Take equation (1), fix all future variables, and an increase in the current nominal interest rate makes output and inflation go down. Indeed, sticky expectations is what George Evans and Bruce McGough have in mind. Here's their claim:
Following the Great Recession, many countries have experienced repeated periods with realized and expected inflation below target levels set by policymakers. Should policy respond to this by keeping interest rates near zero for a longer period or, in line with neo-Fisherian reasoning, by increasing the interest rate to the steady-state level corresponding to the target inflation rate? We have shown that neo-Fisherian policies, in which interest rates are set according to a peg, impart unavoidable instability. In contrast, a temporary peg at low interest rates, followed by later imposition of the Taylor rule around the target inflation rate, provides a natural return to normalcy, restoring inflation to its target and the economy to its steady state.We can actually check this out in Narayana's model. Following Evans-McGough (E-M), we'll assume a form of adaptive expectations. Let e(t+1) denote the expected rate of inflation in period t+1 possessed by economic agents in period t. Assume that
The question is, what happens for intermediate values of h? There are three cases: sticky expectations
Under no circumstances is the standard Taylor rule with d > 1 well-behaved. At best, if inflation is initially below target, the inflation target is only achieved in the limit, and at worst the central banker gets stuck at the zero lower bound forever. But, there are other rules. Here's one:
The critical value for inflation expectations is
In their paper, E-M say
As we have shown, the adaptive learning viewpoint argues forcefully against the neo-Fisherian view and in support of the standard view.As I hope I've made clear, that's overstated. I take the "standard view" to be (i) staying at the zero lower bound will eventually make inflation go up; (ii) a standard Taylor rule is the best the central bank can do. In Narayana's model, under adaptive learning, (i) is only correct under some parameter configurations - actual inflation and expectation inflation both have to be sufficiently sticky. Further, (ii) is never correct.