It's the investment accelerator. Monetary tightening means lower expected NGDP; real interest rates can go either way. Think of this as a teaching post, to explain the intuition. Or look at Miles Kimball's great post.

Here's a thought-experiment. **Do not take this thought-experiment literally.** It's just my weird way of doing the math, where I start with the answer and work back to the question it answers. Sometimes it's easier this way. Like trying to figure out the relationship between policy X and outcome Y when you know that Y/2=X but you are really bad at algebra so you can't solve for Y=2X.

Imagine that expectations are totally non-rational. Expectations are just exogenous, and do not adjust to match what is actually happening. And one day everyone wakes up expecting a permanent recession. When they went to bed they expected that output would remain at 100 forever, and nothing would ever change. Now they expect that output will be 90 forever. A permanent 10% drop in output.

But the central banker is an New Keynesian macroeconomist who years ago made a bet with his grad skool classmates that expectations are rational. He has since learned that expectations are totally non-rational and exogenous, but he doesn't want to lose his bet. So he has to figure out a way to make the world adjust to match the expectations people hold, since the expectations people hold do not adjust to match the world.

**So what does the central banker need to do to the real interest rate to make those non-rational expectations of a permanent recession look like a rational expectations equilibrium?** (Remember, I'm just solving for the New Keynesian equilibrium in a weird way, to make the math easier.)

**Consumption only.** The central banker starts with a very simple New Keynesian model with no investment or government spending or net exports. So Y=C, and consumption is determined by infinitely-lived agents according to the permanent income hypothesis.

The central banker realises he should not change the real interest rate. Permanent income is 10% lower, so consumption will be 10% lower, so demand for output will fall by 10%, so output will fall by 10%, and stay there until expectations change again. He should do nothing to the real interest rate; just make sure the nominal rate falls over time as inflation falls over time. The pessimistic expectation validates itself.

**Consumption plus Investment.** The central banker now adds investment to his model. So Y=C+I. The central bankers realises that keeping the real interest rate constant won't work. If he keeps the real interest rate constant, then consumption falls by 10%, but investment may fall by more than 10%. Because if firms' desired Kapital/Output ratios stay the same, but output drops 10% while actual stock of Kapital stays constant, the actual K/Y ration will exceed the desired K/Y ratio, so investment will drop 100%, to zero. So that can't be an equilibrium in which expectations look rational, because Y will drop by more than 10%.

The central banker needs to make sure the desired K/Y ratio increases when Y falls by 10%, to ensure that desired investment falls by only 10%, and not to zero. How could that happen?

Hmmm. Maybe if nominal wages were much stickier than prices, so prices jump down relative to wages, so real wages jump up when the recession hits which might raise the desired K/L and K/Y ratio (firms want to replace labour with kapital), and it might raise it enough that investment falls only 10%?? No, that's daft. We don't see real wages suddenly jump up in a recession. Wages and prices seem to be roughly equally sticky.

**Nope, the central banker must cut the real interest rate.** That's the only way he can ensure that the expected permanent 10% recession becomes a rational expectation. If he cuts the real interest rate, consumption will fall by less than permanent income falls, and the desired K/Y ratio can rise by enough that investment falls, but not by 100%. So the demand for output falls by exactly 10%, and validates the expectation it would do so.

**Now let's change the thought-experiment, so the expected 10% recession is extremely short.** It's so short that expected permanent income barely changes at all.

In a model with only consumption, and no investment, the central banker would need to raise the real interest rate to make the expected short recession a rational expectation. Because consumption would barely fall at all if expected permanent income barely fell.

If we add investment to the model, it's a little more complicated.

If investment is nearly 10% of GDP, then a 100% drop in investment, plus the tiny drop in consumption, will add up to a 10% drop in output. Which is exactly the right size to make the expected short recession a rational expectation, if the central banker holds the real rate constant.

If investment is smaller than 10% of GDP, the central banker would need to raise the real interest rate. If investment is larger than 10% of GDP, the central banker would need to cut the real interest rate.

**Let's put it all together.**

The central banker would need to raise the real interest rate to make an expectation of recession a rational expectation if: people expect a short recession; investment as a share of GDP is small relative to the expected size of the recession.

**The central banker would need to lower the real interest rate to make an expectation of recession a rational expectation if: people expect a long recession; investment as a share of GDP is large relative to the expected size of the recession.**

As usual in economics, it depends. But if we can say what it depends on, that is useful.

**And no I am not saying that if a central bank cuts the real interest rate that will cause a recession.** What I am saying is that if the central bank fails to anchor expectations of NGDP growth, and this causes a recession, we could easily see that failure of monetary policy revealed as a fall in real interest rates.

**High or low real interest rates, even relative to an unchanged natural rate of interest, are a very bad measure of whether monetary policy is tight or loose. It's the NGDP expectations that matter. Monetary tightening means lower expected NGDP. Real interest rates could go either way.**Which is what Scott Sumner's been saying all along. I'm just saying it (here) in a New Keynesian way, because that's the language a lot of economists speak.

Nick,

I was with you up until you said "High or low real interest rates, even relative to an unchanged natural rate of interest, are a very bad measure of whether monetary policy is tight or loose. It's the NGDP expectations that matter." -- which seems to have nothing to do with what you talked about before. NGDP wasn't even mentioned in your model previously, and of course when we talk about monetary policy, we don't mean that the CB adjusts to make expectations rational, but that the CB adjusts to change expectations. For example, in a recession, there is panic and fear, and people are supposed to regain their animal spirits, or otherwise update their expectations to be more positive. That's at least what the CB has been doing all these years -- including it's lender of last resort role. So can you connect the dots a little more -- for example, add NGDP to your model to show that it's the thing that matters?

Posted by: rsj | November 30, 2017 at 04:47 AM

rsj: yep. I was afraid that bit of the post might not be as clear as it should be. But I'm still really pleased you were OK up to there!

Let's switch back to the old ISLM model for a bit, where the central bank sets the money supply, not the rate of interest. Start at full employment. Then suppose the central bank suddenly cuts the money supply, creating a recession. The standard IS curve slopes down, and says that the real interest rate will rise in this case. And I'm saying the IS curve might slope the wrong way, if people expect the recession to last a long time, or I/Y is big relative to the size of the recession.

Now lets change it to a New Keynesian model, where the CB sets a nominal interest rate. But it doesn't set that nominal interest rate in a vacuum. It looks at expected inflation, because it realises it's the real interest rate that counts. And it doesn't set that real interest rate in a vacuum either, because it's got some underlying target for prices and/or real output. And the simplest way to think about that underlying target (if prices are sticky in the short but not long run) is to simply multiply P and Y together to get NGDP. Now suppose something happens that causes people to expect the CB's target path for NGDP has dropped. Or that a shock hits, and people expect the CB will not respond to that shock to prevent NGDP falling. So people expect a recession. What happens to the real interest rate, relative to a world in which people were confident the CB would keep NGDP growing at trend?

Or you might prefer Miles Kimball's way of doing it, where the CB has a sort of Taylor Rule reaction function, and something causes it to shift left, a bit like the LM curve shifting left.

Posted by: Nick Rowe | November 30, 2017 at 06:47 AM

How much actual EVIDENCE is there that NGDP expectations are of significance? NGDP expectations are similar to Ricardian equivalence in that they both assume households are forward looking. There is very little evidence for RE far as I know.

Posted by: Ralph Musgrave | November 30, 2017 at 07:07 AM

Ralph: for this post, it wouldn't matter if half the agents were "Hand-To-Mouth", who consume all of their current income C=Y. (Which is my working assumption anyway.) All you need is that *some* are forward-looking.

Posted by: Nick Rowe | November 30, 2017 at 07:59 AM

I think to be abundantly clear, you might want to make explicit that you're talking about the real rate of debt. For a few paragraphs I was confused and thought you were thinking of the Wicksellian natural real rate. If monetary policy can affect this (and it might, if we include a continuum of investment opportunities in our model?), then that would be even more interesting.

Posted by: Majromax | November 30, 2017 at 02:29 PM