Low inflation in an atmosphere of low nominal interest rates brings up the issue of the Fisher Effect. Here is the long run Fisher equation for a “steady-state†nominal interest rate…
Inflation rate = steady-state nominal interest rate – natural real interest rate
The steady-state means that the nominal interest rate is projected to stay within a narrow range into the future. The natural real interest rate is the natural growth rate of an economy considering such factors as productivity, labor force growth and capital accumulation. The natural real rate is considered independent of monetary policy in the long run.
So how does inflation respond to the steady-state? Does it put up the white flag and surrender to the Fisher Effect or rebel against it?
Dynamic Equation for the Fisher Effect
If the Federal Reserve was to all of a sudden set the Fed rate at a steady-state rate and say that they would hold it there for a long time, inflation and the natural real rate would adjust over time according to the Fisher equation. This adjustment over time is described by a dynamic equation.
A dynamic equation shows changes to variables over time. Many dynamic equations lead asymptotically to a stable steady-state, and some lead to an unstable state. Here is the dynamic equation that I will use to look at the Fisher Effect.
 πt = Inflation rate at time period t.
Ï€0Â = Beginning inflation rate
Ï€*Â = Steady-state inflation rate if Fisher Effect is stable
α = autoregressive coefficient, which ultimately shows if the dynamic equation is stable or unstable. In the equation, it is raised to the time period t. When the autoregressive coefficient is between -1 and 1, the equation leads to a stable steady state. When α is greater than 1 or less the -1, the equation does not lead to a stable steady-state and is unstable.
The equation tracks changes in time by raising the autoregressive coefficient to t, the time periods. The equation comes from an ADL model (Autoregressive Distributed Lag). This particular equation assumes that the explanatory variable, in this case the nominal interest rate, stays constant at its long run mean. Since the Federal Reserve is projecting the Fed rate within a low narrow range for a couple years, I assume the Fed rate to be at a “long run steady-state†mean.
