Interest rate neutrality is easy to express in equations, but is difficult to understand intuitively.

The equation states that interest rate = real interest rate plus expected inflation, \[i_t = r + E_t\pi_{t+1.}\]In one sense, it’s easy: if people expect a lot of inflation, then they charge higher nominal \(i_t\) interest rates to offset the falling value of the dollar. Thus the real \(r\) interest rate remains unchanged.

(Note: This post uses Mathjax equations. If you can’t see them, come to the original.)

But in our economy, the Fed sets the nominal interest rate and the rest has to adjust. In the short term, the real interest rate can change with stalled prices and other frictions, but ultimately the real interest rate is determined by real things and the expected inflation must increase. We can look at this over the long term, ignoring the stalled prices and other frictions, and then the expected inflation rises immediately.* Rises.* Higher interest rates *to lift* Inflation. How does this really work? What is the economic power?

Standard intuition overwhelmingly says that higher interest rates cause people to spend less, which lowers inflation. The equations seem to hide a kind of subtlety.

(Fed Chairman Powell does a good job of explaining the default view while sparring with Senator Warren here. The clip is great on several dimensions. No, the Fed can’t increase supply. No, none of what Senator Warren is talking about will either Impairing supply. Elephant in the room, massive fiscal stimulus, not mentioned by either side. Why both are silent on this is an interesting question.)

That’s a nice case *Individually *Causality goes in the opposite direction of *balance* Causality. This often happens in macroeconomics and can cause a lot of confusion. It’s also an interesting case of confusing expected inflation with unexpected inflation. Coupled with confusing relative prices for inflation, this is common and easy to do. Hence this post.

Start with the consumer’s first-order condition, or the “IS curve” of New Keynesian models, \[ x_t = E_t x_{t+1} – \sigma (i_t – E_t \pi_{t+1}-r) \] where \(x = \) consumption, output, or output gap after linearization, \(i = \) nominal interest rate, \(\pi = \) inflation, and \(r\) equal to the discount rate or real long-term interest rates. For the individual, the interest rate and expected inflation – the price levels \(p_t\) and \(p_{t+1}\) and hence \(\pi_{t+1} = p_{t+1} – p_t\ ) – are given, exogenously. (Minus not divided by, these are all in logs.) The consumer chooses the consumption \(x\) considering a budget constraint. If the Fed is raising interest rates and prices aren’t adjusting yet, the consumer wants to decrease consumption today \(x_t\) and increase consumption tomorrow \(x_{t + 1}\). This is standard intuition and correct.

Now the desire to reduce consumption today pushes the price level down today, and the desire to consume more tomorrow increases the price level tomorrow. Let’s pair this first-order condition more precisely with equilibrium in a capital wealth economy, with \(x_t = x\) constant. In English, Fix Supply – there is only so much output \(x\) that prices have to be adjusted until people are happy to buy what’s on the shelves, no more no less. (We can also couple it to the Phillips curve and then specify flexible prices.) The current price level \(p_t\) will fall relative to the expected future price level \(p_{t+1}\) until consumer demand is equal supply, i.e. \ (E_t \ pi_ {t + 1} = i_t + r \). *The expected inflation increases to meet the interest rate. *As promised and with exactly the traditional mechanism.

This logic tells us that the higher interest rate creates higher future inflation from this year to next year. Now you can get higher inflation by lower initial price \(p_t\) or higher later price \(p_{t+1}\). The graphic below shows the two options and (green) an intermediate option.

So the original intuition may be correct: higher interest rates could depress current demand and lower \(p_t\). (Blue line) This generates less *retroactively *Inflation \(\pi_t=p_t-p_{t-1}\) and higher *expected* Inflation \(E_t\pi_{t+1}=E_t(p_{t+1}-p_t)\). In this sense, intervention can certainly “lower inflation”. This is how standard (New-Keynesian) models work.

If we stop here, the confusion is only semantic. As is often the case in life, you can resolve many seemingly stubborn arguments by being more careful in defining terms. Higher interest rates can lower current inflation. Firm prices and other frictions can prolong this phase of the decline. As for price recovery and higher future inflation, we see that a lot – inflation coming back like it did in the 1970s – or maybe the Fed isn’t leaving interest rates on long enough to see it. Long term is a long time.

But there is another possibility. Perhaps the higher expected inflation comes from a higher future price level, not from a lower current price level; the red line not the blue line. What is it – higher \(p_{t+1}\) or lower \(p_t\)? This first-order condition is not sufficient to answer this question. You need either a New Keynesian equilibrium selection policy or a fiscal theory to determine which it is. In both cases, fiscal policy matters. To achieve an unexpected drop in inflation, Congress must increase tax revenues or cut spending to pay off bondholders with more valuable cash. If Congress refuses, we get the top line, more future inflation, no inflation reduction today. If Congress cooperates, we can determine the bottom line. Fiscal and monetary policy always work together.

But this post addresses the narrow question: why are interest rates rising? *expected future* Inflation? If this is done by lowering the current price level and creating unexpected deflation, then that agrees with the question. So part of the intuitive problem was understanding the question, and in the verbal debate one side talked (conventionally, implicitly) about unexpected current inflation, while the other (Fisherian) side talked about expected future inflation. Both can be right!

For the individual, the price level and the expected inflation are exogenous and the consumption decision \(x\) is endogenous. In equilibrium, the endowment \(x\) is exogenous, and the price level and expected inflation follow. This is the same clever inversion of Lucas’ famous asset price model. Individuals choose consumption by seeing asset prices. In equilibrium, changes in endowment lead to changes in asset prices.

(Thanks to the colleagues who pushed me to find good intuition for this result.)