Inflation ≠ Expected Inflation

Neo-Fisherian arguments seem to rest on the idea that expected inflation is somehow related to current inflation - almost to a point of equivalence. A typical argument would be: look at the fisher relation $i_t = r + E_t \pi_{t+1}$. Notice that the nominal interest rate, $i_t$, and the expected inflation rate, $E_t \pi_{t+1}$, are related. Increasing the nominal interest rate must therefore cause the rate of inflation to increase. Before you accuse me of debating a straw man, read this from John Cochrane's recent post:

If you parachute down from Mars and all you remember from economics is the Fisher equation, this looks utterly sensible. Expected inflation = nominal interest rate - real interest rate. So, if you peg the nominal interest rate, inflation shocks will slowly melt away. Most inflation shocks are individual prices that go up or down, and then it takes some time for the overall price level to work itself out.

The problem with this argument is that the current rate of inflation is never modeled; the central bank can choose expected inflation, but there is no reason that the actual rate of inflation must change in response to higher expected inflation. There are a couple of ways around this problem. In the interest of keeping the model as simple as possible, you could assume that the central bank sets the nominal interest rate in response to the current inflation (i.e. a Taylor Rule) or, in the interest of coming of with a more structural model, you could try and come up with a variable that actually does cause current inflation (e.g. the money supply).

The Taylor Rule approach is the way that most economists have gone in the last twenty years or so. Positive deviations of the nominal interest rate from the level implied by the Taylor Rule result in lower rates of inflation. This is itself enough to prove that, as long as a central bank follows a Taylor Rule, a Neo-Fisherian analysis is wrong. There are still some valid contentions that a Neo-Fisherian might make though: a.) central banks set interest rates by discretion, not by adherence to a Taylor Rule b.) Taylor Rules don't actually produce a unique equilibrium value for the initial rate of inflation or the initial price level. In order to deal with contention a, it is clear that a more structural model of inflation is necessary since interest rates clearly do not cause inflation. Contention b is a bit more complicated. In order to make sense of it, it is helpful to look at the coefficient on inflation in the Taylor Rule. If that coefficient is less than one, then any initial rate of inflation will converge to the central bank's inflation target; there are multiple equilibria. Alternatively, the coefficient can be greater than one which will cause the rate of inflation in the future to explode unless the initial rate of inflation is equal to the target rate. The only reason that this calibration works is because economists have chosen to rule out explosive solutions which may make sense for real variables, but does not make sense for nominal variables like inflation.

Contentions a and b leave two options for revision to the conventional approach: come up with a more structural model of inflation or come up with a model that determines a unique equilibrium for "passive" Taylor Rules (i.e. Taylor Rules where the coefficient on inflation is less than one). For some reason, the price determination literature failed to go down the first route and instead chose to come up with 'the fiscal theory of the price level'. Basically, the fiscal authority can threaten to disobey its budget constraint unless the initial inflation rate does not jump to the correct level. I don't really understand how this is all that much better than the trick with active Taylor Rules though. After all, both involve threats to either cause a hyperinflation or not pay off debt at some point that force the initial inflation rate to be on target. 

Because of this, it seems obvious to me that the structural path should be taken. There should be a way to determine a unique equilibrium rate of inflation without requiring that fiscal or monetary policy be intentionally unstable. Of course, a cursory analysis using something like the money supply is easy. Current inflation is caused by current money growth and expected inflation is caused by expected money growth, so interest rates and inflation will go up in response to an increase in the growth rate of the money supply that is expected to be persistent. The debate should end there.

P.S. I don't necessarily mean to say that applies to the current situation; the zero lower bound is special both in theory and in practice.

P.P.S. For the more visually oriented, here's a graph that illustrates the explosive behavior of inflation under a Taylor Rule:

Two Views of Price Determinacy

Intro:

All the following views of price determination revolve around the consolidated government budget constraint that combines the actions of the central bank and the fiscal authority into one equation. The price level is determined by a combination of fiscal and monetary policy which can result in unconventional effects.

The Model:

The fiscal and monetary authority collectively obey a budget constraint:

$$ (1) \: B_t + P_t s_t = R_{t-1}B_{t-1} $$

where $ B_t $ is the total stock of government liabilities (including money and government bonds), $ s_t $ is the lump sum tax levied by the government, $ P_t $ is the price level, and $ R_t $ is the gross nominal interest rate on government debt.

The central bank targets the price level and sets $ R_t $ according to a policy rule:

$$ (2) \: R_t = \frac {P_t^\phi}{\beta} $$

$ \phi $ measures is the central banks' reaction to an off-target price level and $ \beta $ is the factor at which future actions are discounted by agents.

The fiscal authority targets a level of nominal debt by setting $ s_t $ according to a policy rule:

$$ (3) \: s_t = \left(\frac {1}{\beta} - 1\right) \bar B + \phi^f (B_t - \bar B) $$

where $ \bar B $ is the target level of debt and $ \phi^f $ measures the fiscal authority's reaction to an off-target debt level.

The nominal interest rate is related to $ \beta $ by the Fisher Equation (which is technically a bond pricing equation drawn from the first order conditions of the consumers maximization problem):

$$ (4) \: R_t \beta = \frac {E_t P_{t+1}}{P_t} $$

Monetary and fiscal policy can either be active or passive. Monetary policy is active when $ \phi > 1 $ and passive when $ \phi < 1 $. Fiscal policy is active when $ \phi^f $ is positive and passive when $ phi^f  \leq 0 $. In order for the price level to be determined, one or both of the government policy instruments must be active.

View One: Neo-Fisherian:

Neo-Fisherians contend that the fiscal authority responds only passively to government debt, so when the nominal interest rate is increased above the rate suggested by the policy rule, the amount of government debt will increase with no fiscal response. This forces the inflation rate to increase. The mechanism by which this occurs is best illustrated by iterating the budget constraint forward in time and dividing by the price level:

$$ (5) \: \frac {B_{t-1}}{P_t} = E_t \sum_{j=0}^{\infty} \beta^{j} s_{t+j} $$

The increase in government liabilities, without any fiscal policy response, causes the price level to rise.

View Two: Everyone Else:

For the conventional wisdom ($ \uparrow R_t $ causes $\downarrow P_t $) to be the case, the fiscal authority needs to commit to commit to have active fiscal policy ($ \phi^f > 0 $). When the central bank raises the nominal interest rate and the amount of government debt in (5) goes up, the right side of the equation goes too, and the price level falls.

Alternatively, the total stock of government liabilities can be set to zero forever, as in New Keynesian models, changes in the nominal interest rate have no fiscal effects, so the price level is determined by only two equations:

$$(1a)\: R_t \beta = \frac {E_t P_{t+1}}{P_t} $$

$$(2a)\: R_t = \frac {P_t^\phi}{\beta} $$

Conclusion:

Different views on how the price level is determined depend on how monetary and fiscal policy are seen to be interacting. On one hand, if monetary and fiscal policy are active, then monetary policy effectively has fiscal backing. On the other hand, if fiscal policy is passive, then the Neo-Fisherian hypothesis is correct. A third possible equilibrium is one with passive monetary policy and active fiscal policy. This is a fiscal theory of the price level regime that may have occurred between 1950 and 1970 in the US.

The monetary-fiscal regime that a country is in at a given moment can have an important impact on the price level. For example, the US is likely once again in an active fiscal/passive monetary regime, so fiscal policy plays an important role in the determination of the price level.