The Relationship Between Nominal Interest Rates and Inflation: The Long Run

In the last post I discussed what I understand to be the conventional macroeconomic story about how interest rates and inflation rates behave relative to one another over short-to-intermediate time frames. While the story told doesn't capture all of the potential linkages (for example I have basically waived my hands and said the magic words with respect to role played by expectations, kinda like step 2 below),

it is, I believe, a fair representation of how most macro-economists view this relationship and it is consistent with both traditional Keynesian/Monetarist and New Keynesian theory.

Today I want to think about the long-run relationship between nominal interest rates and inflation. This topic is, I believe, less controversial since all internally consistent dynamic equilibrium models with money imply the same basic relationship between nominal interest rates and inflation. Equation (1) displays that long-run equilibrium commonly referred to as the Fisher equation. \begin{equation*} i_t = r_t + \pi^e\tag{1}\end{equation*} There are variants of (1) where we might include a risk premium, tax effects or allow the equilibrium real rate of interest to be time-varying but stationary, but all are consistent with the basic idea that in the long run, the equilibrium real interest rate should be independent of the inflation rate. This idea, independence of the real rate from inflation, does have some counter-arguments, like Mundell-Tobin effects and old-fashioned money illusion, but these effects should be short-lived so that in the long run the Fisher equation holds.

What determines trend inflation in the long run? Well there is not a whole lot of concensus on that point. Some believe that trend inflation is determined via the money supply growth rate, as in the standard quantity theory story \(MV = PY\)(Crowder, 1998). Others suggest it is a direct policy choice of the inflation target in a Taylor-type policy rule (Bernanke and Mishkin, 1997). And still others believe trend inflation is really determined by fiscal policy through the rate of deficit accumulation and the resulting seignorage requirements (Woodford, 1995). All of these "causes" of trend inflation imply that trend inflation leads or causes the trend in nominal interest rates. Causality in this sense is a little tricky since our conclusions about causality can depend on where we start the process.

Since we are interested in long-run trends and equilibria, let's start in a long-run equilibrium where output is equal to potential, unemployment is at the NAIRU and the equilibrium real rate (\(r_t\)) is constant. Note that in such a steady state the nominal interest rate is completely determined by equation (1), i.e. a constant real rate plus the rate of trend inflation. Changes that "originate" with the nominal rate will always end up back at the steady state. For example, assume the economy is in the steady state described. Now the central bank raises the nominal interest rate (whether this is through traditional open market operations or by raising the interest paid on reserves is not important) resulting in an (temporary) increase in the real rate of interest. This should lead to a fall in current spending (see previous post) and via some aggregate demand or Phillips curve relationship to lower inflation. Note that in the short-run, inflation moves in the opposite direction as the nominal interest rate.

But eventually the economy adjusts, either through intertemporal substitution effects, real wealth effects or just simple learning and modification of expectations, but eventually the central bank will reduce the nominal interest rate and inflation will return to target. The easiest way to see this is to use the simple 3-equation New Keynesian model that includes a Taylor rule \begin{equation*} i_t = r + \pi^* + \xi_y (Y_t - Y^*) + \xi_{\pi} (\pi_t - \pi^*)\tag{2}\end{equation*} an IS relationship \begin{equation*} Y_t - Y^* = -\sigma (r_t - r) + E_t(Y_{t+1} - Y^*) \tag{3}\end{equation*} and a New Keynesian Phillips curve \begin{equation*} \pi_t = \gamma E_t \pi_{t+1} + \kappa (Y_t - Y^*) \tag{4}\end{equation*} where \(i_t\) is the policy rate and \(Y^*\) and \(\pi^*\) are potential output and the inflation target.

Starting from an initial equilibrium where \(Y_t = Y^*\), \(r_t = r\) and \(\pi_t = \pi^*\), raising \(i_t\) leads to \(Y_t\) declining below potential (\(i_t \uparrow \rightarrow r_t \uparrow \rightarrow Y_t \downarrow\)) which then causes inflation to decline via (4). This situation is not consistent with long-run equilibrium. So this increase in the nominal rate is temporary or transitory. Eventually the nominal rate will fall and \(Y_t\) and \(\pi_t\) will rise to their equilibrium or targeted levels. So changes in the nominal interest rate that originate from the nominal interest rate are transitory, the effects will dissipate over time such that in the long run the nominal interest rate and inflation rate will return to their pre-shock levels.

So how could we get a permanent increase in the nominal rate? We would need a higher target inflation rate. If the central bank raises its inflation target, it will need to lower the policy rate initially, so that in (2) \(\pi^* = \pi_t\) initially but now \(\pi^* > \pi_t\). As before the lower nominal rate lowers the real rate temporarily which stimulates economic activity through the various effects already described. The Phillips relationship in (4) then tells us that as output rises above potential actual inflation begins to rise and will continue to do so until it equals the (now higher) inflation target. But as both output and inflation increase, the policy rate does also. The permanently higher inflation rate (and inflation target, remember trend inflation is a policy choice here) causes a permanently higher nominal interest rate. Trend inflation causes the nominal interest rate in the long run.

So where are we? In the short run nominal interest rates cause inflation. Raising (lowering) nominal interest rates leads to lower (higher) inflation temporarily. As the economy adjusts over time inflation will eventually return to trend and when inflation expectations are well anchored that will be the central banks target rate of inflation, i.e. the expected inflation in (1) will be the same as the target inflation in (2). But in order to change the nominal policy rate permanently there must first be a change in the trend (target) inflation rate, so that in the long run inflation causes the nominal interest rate.