Long run, medium run, and short run Fisher curves

Here is a simple way to think about the correlations between nominal interest rates and inflation rates.

We need to distinguish between three "Fisher curves": the long run; the medium run; and the short run Fisher curve.

We need to distinguish between: the actual inflation rate; the central bank's inflation target; and what people expect the central bank's inflation target to be (the "expected inflation target").

Start with the (green) Long Run Fisher curve. It shows a positive correlation between nominal interest rate and inflation. This is what you observe when you vary the inflation target, and vary the expected inflation target, so that the two are always equal. It has a slope of one (assuming super-neutrality of money). If the inflation target were 0%, the nominal interest rate would have to equal the (real) natural rate of interest. Fluctuations in the natural rate would cause the LRFC to shift horizontally back and forth (not shown). If we look at long run or cross-country data, where inflation targets differ a lot, we should observe the LRFC.

The (blue) Medium Run Fisher curve shows what happens when we hold the inflation target fixed, and assume the expected inflation target equals the actual inflation target. We observe zero correlation between nominal interest rate and inflation. This is an immediate consequence of rational expectations on the part of the central bank, and the implied orthogonality of the central bank's inflation forecast errors with its information set (that information set presumably includes the nominal interest rate). Canada, over the last years of targeting 2% inflation, should look like the MRFC.

The (red) Short Run Fisher Curve shows what happens when we hold the expected inflation target fixed, and vary the actual inflation target. We observe a negative correlation between nominal interest rates and inflation. For example, if the central bank has a lower inflation target than people expect, it would need to set a higher nominal interest rate, to make real interest rates exceed the natural rate, to ensure that actual inflation is below what people expect. When we have a transition from one inflation target to the next, and if the expected inflation target does not immediately adjust, we would observe the SRFC.

For example: if the Bank of Canada cut the inflation target from 2% to 1% the blue MRFC would shift vertically down. If people immediately adjusted their expectations of the inflation target, the red SRFC would shift left, to intersect the blue MRFC on the green LRFC. Nominal interest rates would fall by 1%. But if the expected inflation target stayed constant in the short run, the red SRFC would not shift, and nominal interest rates would rise, to the point where the MRFC and SRFC intersect.

That's all.

Merry Christmas everyone!

Comments

[ReturnFreeRisk]

I think you might be on to something.

However, by this logic, FIVE YEARS INTO ZIRP, if the Fed wants inflation at 2% target, shouldn't the Fed funds be at 2% nominal?

[Nick Rowe]

RFR: that depends on what the natural rate is.

[ReturnFreeRisk]

For zero to be the new medium term, natural rate is negative. I am not a believer there. Eventually, the economy will be near its natural output level (zero output gap) and stable inflation. That means you are Japan. I guess my question is - can you draw the graph for Japan?

[JW Mason]

What's your definition of the natural rate?

[Nick Rowe]

JW: for this "model": that real interest rate compatible with actual inflation=inflation target=expected inflation target.

[Frank Restly]

JW Mason,

http://en.wikipedia.org/wiki/Knut_Wicksell

"Wicksell's most influential contribution was his theory of interest, published in his 1898 work, Interest and Prices. He made a key distinction between the natural rate of interest and the money rate of interest. The money rate of interest, to Wicksell, was merely the interest rate seen in the capital market; the natural rate of interest was the interest rate that was neutral to prices in the real market, or rather, the interest rate at which supply and demand in the real market was at equilibrium – as though there were no need for capital markets."

[Nick Rowe]

Which is not my definition here.

[Frank Restly]

The definition implied by the article on Wicksell above seems to treat two conditions as equal:

1. Supply and demand in the real market are at equilibrium
2. No need for capital markets

That seems at odds with the possibility that capital markets facilitate increases in productivity (same amount of goods produced in less time). Capital markets could still be required to achieve the desired level of productivity even if the supply and demand for goods is unchanged.

Nick, out of curiosity, does the definition you are invoking, also imply that capital markets are not required - hence the term "natural rate"? I would think no because the presence of a targeted positive inflation rate implies a constant disequilibrium between the supply and demand for goods, but I am not sure whether you want capital markets to try (and fail) to address the disequilibrium.

[Nick Rowe]

Frank: Wicksell's definition of the natural rate was problematic. The Wikipedia one is worse.

My definition is silent on "capital markets" (whatever they are). Except it does imply that interest rates and money exist, otherwise we could not talk about "interest rates" and "inflation". Positive inflation does not mean disequilibrium in goods markets (whatever that means).

[Frank Restly]

Nick,

Targeted positive inflation would produce a disequilibrium in goods markets (assuming the central bank has the all tools to meet that target). The same could be said for targeted negative inflation (again presuming the central bank has all the tools to hit that target).

[Ken Houghton]

Stipulating for model purposes that "capital markets" are frictionless intermediaries, I'm still stuck on (pi)^e=(pi). Are you stipulating also that d(pi)^e=d(pi), or vice versa?

If so, how would we ever move from the blue (medium-run) "curve"? Is (pi) in this presentation a function that depends in part on dY--or even expectations of dY?

Or am I missing something really basic?

[Nick Rowe]

Ken: If you are stuck on this, I think that means I have failed badly to explain it clearly enough! Oh well.

My previous post has the underlying model. (Though I think the picture here is more general than that particular model).

I'm not 100% sure what you are asking. Let me try a few things.

I'm not assuming frictionless capital markets. There can be many different interest rates. I am assuming superneutrality of money, so a permanent change in the inflation target leads to an equal change in nominal interest rates. If I relax that assumption, my LRFC won't have a slope of exactly one.

The Phillips Curve will be something like: actual inflation = expected inflation target + B.Y where Y is the output gap.

And the IS curve is something like: Y = -c(nominal rate - expected inflation target - natural rate).

And the central bank sets the nominal rate, conditional on the expected inflation target, and its expectation of the natural rate, to hit its inflation target.

Start with Canada. The inflation target is 2%, and everyone knows it's 2%. The Bank of Canada tries to set a nominal interest rate where the green LRFC crosses the blue MRFC. The green LRFC is shifting back and forth, so the BoC will need to keep changing the nominal interest rate accordingly. If the Bank of Canada gets it exactly right every period, we observe the blue MRFC. If the BoC makes random mistakes (it will) we will also observe points above and below the blue MRFC, but there will be zero correlation between inflation and nominal interest rates.

Now suppose the Bank of Canada cuts the inflation target to 1%, but doesn't tell anyone (or people don't believe it initially). Then we move down along the red SRFC.

[Nick Rowe]

Frank: since around 1970, was assume that actual inflation = expected inflation + "disequilibrium" in goods markets. An "expectations-augmented Phillips Curve".

[Frank Restly]

Nick,

Curiosity - should not the short run Fisher curve be a function of the interest rate differential between short and long term interest rates rather than a function of an interest rate itself. It would seem to me that you could generate a positive inflation rate by changing the short term rate ONLY when short term interest rates are lower than long term interest rates.

In other words, I don't think it would make a hill of beans of difference if long term interest rates were 5% and the central bank cut short terms rates from 12% to 11%.

[ReturnFreeRisk]

Interesting to me that there is no answer to my Japan question. Japan shoots all these monetary theories in the head.