## Saturday, April 23, 2016

### Lessons Learned I

I spent last week traveling and giving talks. I always learn a lot from this. One insight I got:  Real interest rates are really important in making sense of fiscal policy and inflation.

Harald Uhlig got me thinking again about fiscal policy and inflation, in his skeptical comments on the fiscal theory discussion, available here. At left, two of his graphs, asking pointedly one of the standard questions about the fiscal theory: Ok, then, what about Japan? (And Europe and the US, too, in similar situations. If you don't see the graphs or equations, come to the original.) This question came up several times and I had the benefit of several creative seminar participants views.

The fiscal theory says
$\frac{B_{t-1}}{P_t} = E_t \sum_{j=0}^{\infty} \frac{1}{R_{t,t+j}} s_{t+j}$
where $$B$$ is nominal debt, $$P$$ is the price level, $$R_{t,t+j}$$ is the discount rate or real return on government bonds between $$t$$ and $$t+j$$ and $$s$$ are real primary (excluding interest payments) government surpluses. Nominal debt $$B_{t-1}$$ is exploding. Surpluses $$s_{t+j}$$ are nonexistent -- all our governments are running eternal deficits, and forecasts for long-term fiscal policy are equally dire, with aging populations, slow growth, and exploding social welfare promises. So, asks Harald, where is the huge inflation?

I've sputtered on this one before. Of course the equation holds in any model; it's an identity with $$R$$ equal to the real return on government debt; fiscal theory is about the mechanism rather than the equation itself. Sure, markets seem to have faith that rather than a grand global sovereign default via inflation, bondholders seem to have faith that eventually governments will wake up and do the right thing about primary surpluses $$s$$. And so forth. But that's not very convincing.

This all leaves out the remaining letter: $$R$$. We live in a time of extraordinarily low real interest rates. Lower real rates raise the real value surpluses s. So in the fiscal theory, other things the same, lower real rates are a deflationary force.

The effect is quite powerful. For a simple back of the envelope approach, we can apply the Gordon growth formula to steady states. Surpluses $$s$$ grow at the rate $$g$$ of the overall economy. So, in steady state terms,
$\frac{B_{t-1}}{P_t s_t} = E_t \sum_{j=0}^{\infty} \frac{(1+g)^j}{(1+r)^j} \approx \frac{1}{ r - g}$
$\frac{P_t s_t}{B_{t-1}} \approx r - g \; \; (1)$
(and exact in continuous time). The left hand side is the steady state ratio of surpluses to debt. The right hand side is the difference between the real interest rate and the long-run growth rate.

So, with (say) a 2% growth rate g, and a 4% long-run interest rate r, surpluses need to be 2% of the real value of debt. But suppose interest rates decline to 3%. This change cuts in half the needed long-run surpluses! Or, holding surpluses constant, if long-run interest rates fall to 3%, the price level falls by half.

You can see the punchline coming. Long term real interest rates are really low right now. If anything, we're flirting with $$r \lt g$$, the magic point at which governments can borrow all they want and never repay the debt.

With this insight, Harald should have been asking of the fiscal theory, where is the huge deflation? And the answer is, well, we're sort of there. The puzzle of the moment is declining inflation and even slight deflation despite all our central bankers' best efforts.

Pursuing this idea, there is a larger novel story here about growth, interest rates, and inflation.

Obviously, there is an opposite prediction for what happens when real interest rates rise. Higher real rates, unless accompanied by higher surpluses, will drive inflation upwards.

In conventional terms, looking at flows rather than present values, suppose a government that is $20 Trillion in debt faces interest rates that rise from 2% to 5%. Well, then it has to increase surpluses by$600 billion per year; and if it cannot do so inflation will result.

A similar story makes sense for the cyclical falls in inflation. What happened to our equation in 2008?  Surpluses fell -- deficits exploded -- and future surpluses fell even more. Debt rose sharply. Why did we see deflation? Well, real interest rates on government debt fell to unprecedentedly low levels. This really isn't even economics, it's just accounting. The equation holds, ex-post, as an identity!

To think a bit more about real rates, growth, and inflation, remember the standard relation that the real interest rate equals the subjective discount rate (how much people prefer current to future consumption) plus a constant times the per capita growth rate
$r = \delta + \gamma (g-n)$
The constant $$\gamma$$ is usually thought to be a bit above one.

With $$\gamma=1$$ (log utility), then we have $$r-g = \delta-n$$. The magic land of unbounded government debt can occur because government surpluses can grow at the population growth rate, while interest rates are determined by the individual growth rate. But population growth is tapering off, and must eventually cease, and bondholders prefer their money now. With $$\gamma \gt 1$$ ,
$r-g = \delta - n + (\gamma-1)(g-n) \; \; (2)$
The new term is the per capita growth rate, which is positive, further distancing us from the land of magic.

More to the point, though, we now have before us the central determinant of long run real interest rates. Real interest rates are higher when economic growth is higher. And $$r-g$$ rises when economic growth $$g$$ rises.

So, going back to my equation (1), we actually had a puzzle before us. Higher real interest rates would mean lower values of the debt, and would thus be inflationary if not accompanied by austerity to pay more to bondholders. But higher real interest rates must come with higher economic growth, and higher economic growth would raise surpluses, helping the situation out. Which force wins? Well, equation (2) answers that question: With $$\gamma \gt 1$$, the usual case (a 1% rise in consumption growth comes with a more than 1% rise in real interest rates), higher growth g comes with higher still interest rates r, and thus remains an inflationary force, again holding surpluses constant.

All in all then, we have the hint of a fiscal theory Phillips curve: Inflation should be procyclical. In good times, interest rates rise and the real value of government debt falls, producing more inflation. In bad times, interest rates fall and the real value of government debt rises, producing less inflation.

Central banks have been absent in all this. The natural next question is, does this provide another reinforcing channel by which central banks might raise inflation if they raise interest rates? I don't think so, but one needs more equations to really answer the question.

What matters here are very long-term real interest rates, the kind that discount expectations of surpluses -- yes, we need some surpluses! -- 20 to 30 years from now to establish bondholder's willingness to hold debt today.

In no model I have played with can central banks affect real interest rates for that long. I think a quick look out the window convinces us that central banks cannot substantially raise interest rates in a slump, with supply of global savings so strong compared to demand for global investment. Long-term interest rates really must come from supply and demand, not monetary machination. Higher real interest rates require higher marginal products of capital, and thus higher economic growth, not louder promises, more speeches, or more energetic attempts to avoid the logic of a liquidity trap.

1. Thank you for your article Professor. Have a great weekend. MP

2. We are definitely in r<g territory and many economists are asking to monetize the debt to restore dynamic efficiency. But in your conclusion you rightly shift attention to the supply side and point out that we need to raise g to avoid getting in such messes.

3. Your reasonment is even more valid if precautionary savings go up in recession

4. Under the capital sigma is j=0, which holds the answer to the question IF we're looking at a corporation and use standard financial accounting (there is no difference between micro and macro on that one). At j=0 we look at unloading assets belonging to the bond issuing entity. Ricardian equivalence won't apply but it's neither necessary nor sufficient. That's the only way to explain the lack of inflation - plus the evident arbitrage among global goods and capital markets - assets belonging to the state itself are not being considered for sale, but they exist nonetheless, and furthermore they're assumed to be increasing in value.

5. "remember the standard relation that the real interest rate equals the subjective discount rate (how much people prefer current to future consumption) plus a constant times the per capita growth rate"

Out of interest could you please give a link to a document laying out a model which such a relationship arises?

Comments are welcome. Keep it short, polite, and on topic.

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