Sign wars, and stability, with price level targeting

Suppose, just suppose, that everyone knows that the price level will be exactly 100 in 2084. (That's 70 years from now, to keep the math simple). Because in 2084 the central bank will redeem all the outstanding notes, in exchange for real goods, at a price of 100 notes per real good. And will then start afresh with a new money.

Also suppose, but just for simplicity, that the natural rate of interest is 3%, and everybody knows it will stay at exactly 3% until 2084.

Also suppose, initially, that prices are perfectly flexible.

If the central bank sets a nominal interest rate of 3%, and is expected to keep it at 3%, the equilibrium price level in 2014 will be 100, and there will be 0% inflation until 2084. The real interest rate is 3%, just like the natural rate.

1. What happens if the central bank instead suddenly raises the nominal interest rate to 4%, and is expected to keep it there? The answer is that the 2014 price level instantly halves to 50, and there is a steady 1% inflation rate until 2084. So the real interest rate stays at 3%.

2. And if the central bank instead suddenly cuts the nominal interest rate to 2%, and is expected to keep it there, the 2014 price level instantly doubles to 200, and there is steady 1% deflation until 2084. So the real interest rate stays at 3%.

Changes in nominal interest rates have the "right" sign in their effect on the price level, but the "wrong" sign in their effect on the inflation rate.

But we can easily eliminate that "right" sign on the price level, by changing the questions slightly.

1a. What happens if the central bank instead suddenly raises the nominal interest rate to 4%, and is expected to keep it there? And at the same time announces that the price level will be 200 in 2084, because it will redeem the outstanding notes at a price of 200? The answer is that the 2014 price level does not jump, and there is a steady 1% inflation rate until 2084. So the real interest rate stays at 3%.

2a. And if the central bank instead suddenly cuts the nominal interest rate to 2%, and is expected to keep it there, and at the same time halves the 2084 price level to 50, the 2014 price level does not jump, and there is steady 1% deflation until 2084. So the real interest rate stays at 3%.

It is very easy to get the "wrong" sign for the effect of nominal interest rates on inflation, without getting the "right" sign for the effect on the price level. But it is the change in the 2084 price level target, that accompanies that change in nominal interest rates, that is doing all the work.

(And the equilibrium path is a stable path, given standard assumptions. A lower than equilibrium price level today means higher expected inflation from now on, to get back to a price level of 100 in 2084. And higher expected inflation means a lower real interest rate for any given nominal interest rate. And a lower real interest rate means higher demand for goods. Which causes the price level to jump back up to the equilibrium path.)

If prices are sticky, it won't be quite as simple as in my answers above, because the price level won't jump up or down instantly when the central bank does something silly with the nominal interest rate. It will take time for the price level to adjust to the new equilibrium path. Raising the nominal interest rate, with no change in the 2084 price level, will cause inflation to decrease initially, and increase eventually. Cutting the nominal interest rate, with no change in the 2084 price level, will cause inflation to increase initially, then decrease eventually. You get the "right" sign initially, and the "wrong" sign eventually.

Pinning down the expected future price level provides a wonderful nominal anchor. It lets the central bank fall asleep at the nominal interest rate wheel, or do any stupid thing it wants with nominal interest rates, without causing the economy to explode into hyperinflation or implode into hyperdeflation. We don't need the central bank to follow the Howitt/Taylor Principle (raise nominal interest rates more than one-for-one with inflation) to do that job.

Price level path targeting, or NGDP level path targeting, amounts to much the same thing as pinning down the 2084 price level at 100. As long as everybody expects the central bank to wake up eventually, and do what it had promised to do in the past, we avoid nominal indeterminacy. We avoid the Wicksell Problem. The economy is self-stabilising.

Unfortunately, we don't live in a world like that. Nothing pins down the price level in 2084, or in any future year. There is no long run nominal anchor that could help the economy self-equilibrate despite central banks falling asleep at the nominal interest rate wheel, or doing silly things with it.

But we could live in a world like that, if we adopted price level path targeting, or NGDP level path targeting.

[For JP Koning. And for John Cochrane. And with thanks to Lorenzo from Oz, who told me the Bank of England once fell asleep at the nominal interest rate wheel for nearly a century. That was when the gold standard provided a long run nominal anchor.]

Comments

[Majromax]

Forgive me if this is a 101-level question, but *can* prices be so flexible as to make this situation true?

Imagine an apple-economy with overlapping generations and a real interest rate of 0%. Each worker picks two apples in a period, sells them on the market, and saves a fraction of the wages for their retirement. Each retiree spends their savings on the market for apple consumption. The price level is initially 100 with nominal rate 0%, and from the ancient past until now workers have sought to equalize consumption between working and retirement periods.

That means that at t=0, each retiree has 100; each worker expects to make 200, of which they plan to spend 100 on an apple and save 100.

Now, the CB commits to a future price level of 50. What happens in the economy?

In the market, 2 apples (per worker) are always produced. Each retiree bids 100, and each worker bids 2*p*c, where p is the final price level and c is the fraction of nominal income the worker wishes to consume this period. The ultimate price is p=(100 + 2*p*c)/2 (money bid divided by quantity produced). p=50 is a solution, but only if each worker decides c=0 -- they wish to defer all consumption to retirement. This is then a stable solution, as for the next period all now-retirees have saved 100 (which is also given by specification, since the stock of money is held by retirees). Unfortunately, this breaks money neutrality, since a nominal change has affected real preferences.

More problematically, the central bank cannot set a p=25 target consistently with this analysis, since this would require workers to have a negative propensity to consume. (p>100 is compatible with the analysis, however)

[Mike Sproul]

"the central bank sets a nominal interest rate"

Do you mean
1. the rate it pays on reserves?
2. the discount rate?
3. the rate earned by holders of currency?

Does the central bank set the rate for the whole world? Because if the CB raises its rate 1%, and the rest of the world is still at the old rate, then the CB would get no customers. Conversely, if the CB rate was 1% lower than the world rate, then the whole world would borrow from the CB, the CB would go broke, and there would be no redemption in 2084.

"Nothing pins down the price level in 2084, or in any future year. "

Unless the CB has a certain stock of gold now, and everyone thinks that they will still have that same gold in 2084, when they redeem.

[Nick Rowe]

Majromax: I haven't worked through your model exactly, but if money is not (super-)neutral in your model, then it does not have a natural rate, and so is different from my little model, which does have a monetary policy invariant natural rate of interest, by assumption.

Mike: The simplest version of my model is where the central bank issues notes, which pay no interest. And it borrows or lends notes freely at some announced nominal rate of interest (the discount rate). And so it earns monopoly profits from the spread between the nominal rate and the 0% paid on notes. And it hands those profits over to the government, every year, which spends them. But always keeps assets equal to its outstanding notes, so it can redeem them in 2084. Or, just consolidate the government and central bank.

[Majromax]

> I haven't worked through your model exactly, but if money is not (super-)neutral in your model, then it does not have a natural rate, and so is different from my little model, which does have a monetary policy invariant natural rate of interest, by assumption.

Thank you, that was it. My proposed model lacks land sales/ownership, which means no rent and therefore no natural interest rate. The remainder boiled down to accounting.

> But always keeps assets equal to its outstanding notes, so it can redeem them in 2084.

With that in mind, does this mean that in a land+apples economy, the central bank is redeeming notes for land rather than apples, so "price" means "price of land?"

If that's the case, then that's consistent with the idea of low interest rates leading to inflated asset prices, although I'm not sure if a superneutral price jump corresponds to a real-life rapid-but-finite-time price increase.

[JP Koning]

Thanks for the post, Nick.

"Price level path targeting, or NGDP level path targeting, amounts to much the same thing as pinning down the 2084 price level at 100... As long as everybody expects the central bank to wake up eventually, and do what it had promised to do in the past, we avoid nominal indeterminacy. We avoid the Wicksell Problem. The economy is self-stabilising. Unfortunately, we don't live in a world like that. Nothing pins down the price level in 2084, or in any future year. There is no long run nominal anchor that could help the economy self-equilibrate despite central banks falling asleep at the nominal interest rate wheel, or doing silly things with it. But we could live in a world like that, if we adopted price level path targeting, or NGDP level path targeting."

Because the Bank of Canada has a flexible inflation target, can't we assume a 2084 price level in the range of, say 90 to 110? And despite this being a range rather than one specific number, wouldn't this be sufficient to avoid nominal indeterminacy?

[Min]

"Suppose, just suppose, that everyone knows that the price level will be exactly 100 in 2084. (That's 70 years from now, to keep the math simple). Because in 2084 the central bank will redeem all the outstanding notes, in exchange for real goods, at a price of 100 notes per real good. And will then start afresh with a new money.

"Also suppose, but just for simplicity, that the natural rate of interest is 3%, and everybody knows it will stay at exactly 3% until 2084."

This real good, I suppose, is **not** an iPhone5. ;) What real good is associated with a natural rate of interest of 3%? Does that mean that, unlike the iPhone5, it increases at a rate of 3% per year per capita?

[JP Koning]

Btw, I think I agree with most of your post. See: http://jpkoning.blogspot.ca/2012/10/zero-percent-interest-rates-forever.html

[Market Fiscalist]

I'm a bit confused by this post.

You say:

"If prices are sticky, it won't be quite as simple as in my answers above, because the price level won't jump up or down instantly when the central bank does something silly with the nominal interest rate. It will take time for the price level to adjust to the new equilibrium path."

Doesn't this mean that a long-term price-level target isn't enough and a CB will need to take action to stabilize AD (or NGDP) even in the shorter term if recessions are to be avoided?

[Nick Rowe]

Majromax: It's low *real* (not nominal) interest rates that lead to high asset prices. And they don't really *lead* to high asset prices; they are the same thing.

"With that in mind, does this mean that in a land+apples economy, the central bank is redeeming notes for land rather than apples, so "price" means "price of land?""

It doesn't matter what it redeems notes in, as long as the value is the same.

JP: "Because the Bank of Canada has a flexible inflation target, can't we assume a 2084 price level in the range of, say 90 to 110? And despite this being a range rather than one specific number, wouldn't this be sufficient to avoid nominal indeterminacy?"

Not in this context, no. Because if the Bank of Canada falls asleep, and the 2014 price levels drifts away from 100, that causes an equal-sized drift in the 2084 price level target range. Inflation targeting allows "base drift".

Min: suppose the price of apples is declining at 1% per year relative to the price of bananas. Then if the banana natural interest rate was 3%, the apple natural interest rate would be 4%. You need to pick a price index to define a natural rate.

MF: "Doesn't this mean that a long-term price-level target isn't enough and a CB will need to take action to stabilize AD (or NGDP) even in the shorter term if recessions are to be avoided? "

Yes. If you want to avoid recessions, you need to stay awake. A 2084 price level target isn't enough. But the 2084 price level target will mitigate recessions, and also prevent explosions/implosions.

[The Market Fiscalist]

Thanks for the answer.

If I understand this correctly the future price level acts as an anchor because at any given moment people know if money is over or undervalued.

What if you relax the assumption about the natural rate being fixed and assume instead that it could vary exogenous between 1% and 5% and no-one knows what it is at any given moment. In this case it would not be so easy for people to evaluate the current value of money relative to its future value. How much would this weaken the effect of using expected future price level providing a nominal anchor?

[JP Koning]

Nick, this post has been very helpful. I just tried to read John Cochrane's paper, linked to from his most recent blog post, and got hopelessly lost. But this post helped clarify. It seems to me that Cochrane is arriving at the "wrong" sign via a different route than someone like Williamson?

[Nick Rowe]

JP: Thanks! (Actually, it helped me a lot too! Thinking through this with a determinate price level anchor in mind.)

The main difference between JC and SW, I think, is that JC actually thinks hard about stability/multiplicity of equilibria questions. SW can't see the problem. But I haven't summoned up the strength to try to read JC's latest paper, just his blog posts. But I think I can guess where he's going; he adopted the assumption of 4-period price setting to try to build some inflation inertia into his model. Maybe I should take a look.

TMF: My guess is that a random and uncertain natural rate wouldn't affect the conclusions much, except the price level would jump around more as information was revealed. But it would be a lot harder to model.

[Nick Rowe]

JP: OK, I have skimmed JC's paper.

Take my old post, where new money is paid as interest on old money:

http://worthwhile.typepad.com/worthwhile_canadian_initi/2014/08/if-new-money-is-always-paid-as-interest-on-old-money.html

So an increase in the interest rate paid on money increases the money growth rate, and raises the inflation rate. Now add some price stickiness, in section 3 of his paper. And (I think) you have his paper.