It is easy to identify interest rate neutrality in equations but difficult to digest intuitively.

The equation says that interest rate = real rate plus expected inflation, \[i_t = r + E_t\pi_{t+1.}\]This is easy one way: if people expect too much inflation, they demand higher nominal interest rates to offset the falling dollar. This leaves the real interest rate unchanged.

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

But in our economy, the Fed sets the nominal interest rate and the rest must be adjusted. In the short run with constant prices and other frictions, the real price may change, but in the end the real price is determined by real things and the expected inflation must rise. We can study that in the long run by excluding constant prices and other frictions, and then the expected inflation rises immediately.* Rises.* Higher interest rates *upload* inflation. How does this really work? What is economic strength?

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

(Fed Chair Powell explains the standard view well while sparring with Senator Warren here. The clip is great in many dimensions. No, the Fed can’t increase the supply. No, nothing of what Senator Warren is talking about will affect the show either. The elephant is in The room, the massive financial stimulus, is not mentioned by either party. Just why each party is silent about this is an interesting question.)

This is a beautiful case *An individual *Causality goes in the opposite direction to *Equilibrium* causation. This happens a lot in the macro economy and can cause a lot of confusion. It is also an interesting case of confusion between expected inflation and unexpected inflation. Besides confusing the relative prices of inflation, this is common and easy to do. Hence the post.

Start with the first-order consumer case, or the “IS curve” of the new Keynesian models,\[ x_t = E_t x_{t+1} – \sigma (i_t – E_t \pi_{t+1}-r) \] With \(x = \) the post-linear consumption or output or output gap, \(i = \) the nominal interest rate, \(\pi = \) inflation and \(r\) equal to the discount rate or the long-term real interest rate. For the individual, the expected rate of interest and inflation—the price levels p_t and p_{t+1} and thus \pi_{t+1}=p_{t+1}-p_t—are given, external. (Minus not divided by , all in records.) The consumer chooses consumption \(x\) subject to budget constraints. If the Federal Reserve raises interest rates and rates are not yet adjusted, the consumer would like to lower consumption today (x_t) and raise consumption tomorrow (x_{t + 1}). This is the standard and correct intuition.

Now, wanting to reduce consumption today pushes the price level down today, and consuming more tomorrow raises the price level tomorrow. More deeply, let’s pair this first condition with equilibrium in the endowment economy, with constant \(x_t = x\). In english, fix the supply – there is only so much output \(x\) to contend with so prices need to be adjusted so that people are satisfied with buying what’s on the shelves, nothing more, nothing less. (We can also associate it with a Phillips curve, then determine elastic prices.) The current price level p_t falls relative to the expected future price level p_{t+1} until consumer demand equals supply, so E_t \pi_{t + 1} = i_t\). *Expected inflation will rise to meet the interest rate. *As promised, and exactly through the traditional mechanism.

This logic tells us that a higher interest rate results in higher inflation in the future, from this year to next. Now, you can get higher inflation at a lower initial price \(p_t\) or from a higher subsequent price \(p_{t + 1}\). The graph below shows the two possibilities, and (green) the median probability.

So the original intuition could be correct: higher interest rates may lower current demand and lower \(p_t\). (blue line) which produces less *old post *Inflation \ (\pi_t = p_t-p_{t-1}\) and higher *is expected* Inflated \ (E_t \ pi_{t + 1} = E_t (p_{t + 1} -p_t)\). Intervention can “lower inflation” in this sense. This is how standard (neo-Keynesian) models work.

If we stop here, the confusion is just semantics. As is so often the case in life, you can resolve a lot of seemingly intractable arguments just by defining the terms more carefully. Higher interest rates can reduce current inflation. Fixed prices and other frictions can lead to this regression period. In terms of price recovery, inflation going up in the future, well, we often see that — inflation returns as it did in the 1970s — or maybe the Fed doesn’t leave interest rates alone long enough to see it. The long run is a long time.

But there is another possibility. The expected higher inflation may all come from a higher price level in the future, rather than from the current lower price level; The red line is not the blue line. Which is higher \(p_{t + 1}\) or lower \(p_t\)? This first condition is not sufficient to answer this question. You need either New Keynesian equilibrium selection policies or financial theory to determine which one. Either way, it’s about fiscal policy. In order for an unexpected drop in inflation to occur, Congress must raise tax revenue or reduce spending to pay bondholders with more valuable cash. If Congress rejects, we’ll get the top line, more inflation in the future, no inflation cuts today. If Congress continues, we can have the bottom line. Fiscal and monetary policies always work hand in hand.

But this post is all about the narrow question: Why are high interest rates soaring *Expected future* inflation? If it occurs by lowering the current price level, which results in an unexpected deflation, then that is consistent with the question. So part of the axiomatic problem was understanding the question, and in the verbal discussion one side (traditional, implicit) was talking about current unexpected inflation, while the other side (Fischer) was talking about expected future inflation. Both could be right!

For an individual, the price levels and expected inflation are exogenous and the consumption decision \(x\) is endogenous. In equilibrium, the stop is \(x\) exogenous, followed by expected price and inflation levels. This is the same clever reversal of Lucas’ famous asset pricing model. An individual chooses depreciation while seeing asset prices. In equilibrium, changes in endowment lead to changes in asset prices.

(Thanks to the colleagues who pressured me into finding a good intuition for this result.)