I do strange things in my sleep. I must.
I mean, I’m not sure what dreams really are, but I have the feeling that
they should be filled with emotional jabber, with things the mind enjoys or
fears or would like to do over but with superpowers. Dreams should be sexy or scary or strange and
slightly surreal. I woke up this morning
with a brain filled with algebra and graphs about how different people see the
relationship between deficit and debt. Well,
ok, that could be scary. But it’s
definitely not sexy, and doesn’t involve any superpowers. I really need some dream upgrades. I’ll have to look into that.
I’m usually a little reluctant to post a lot of algebra in
this blog. But last summer, while
sailing up the Potomac approaching the Wilson Bridge on our way back to the
Washington Sailing Marina, my friend Jeff told me he thought that would be
ok. He thought algebra would be
acceptable. And this bit of algebra ends
up with a picture that I think is interestingif you really trust me, you can
skip past the algebra and get to the pictures, but I’m leaving the algebra to
show how I got there. So, here it is.
Everyone is interested in debt as a share of GDP, and how
that changes over time. The algebra of
that is something like thisstart with plain nominal debt, not as a share of
anything:
I’m usingor at least approachingthe
symbols used in the
IMF’s chapter on debt overhang last fall (see page 104), but I
end up with a slightly different version because I’m looking forward and they
were looking back. That means I’m asking
what this year’s deficit will do to next year’s debt, rather than asking what
last year’s deficit did to this year’s debt.
I mention that chapter because it
keeps coming back to pester me. The
authors found only 26 episodes since 1875 in the entire IMF membership where
debt to GDP rose above 100%, and none where it rose above 300%. Only one, Israel in 1977, was above 250%. Why is
that? Why were there no instances of
debts of, say, 500%, or 1000%? Is 300%
some kind of limit, beyond which nations simply dissolve or crash something? Maybe.
But few of the countries in the study actually did crash. A few had high inflation rates in the 15
years after they reached a debt of 100%, but some also had deflation. It’s hard to know
what those extreme episodes mean, though, because they were usually during or
just after major wars.
In any case, the B in the equation above
is debt, i is the rate of interest paid on the debt, and
the D is deficit. The subscripts denote
time periodsprobably years. (I know
that the interest rate and the growth rate also change over time, and probably
the time subscript should be applied to them also, but for this part of the
algebra I only need the subscripts I used so why clutter everything up? This is informal algebra, comfortable
algebra, algebra en famille.) The equation says that next year’s debt will
be this year’s debt plus interest, plus whatever the primary (noninterest) deficit
adds to or subtracts from the debt.
I’m using capital letters so far, instead
of the lower case used in the IMF chapter, because in this version the symbols
are just talking about the raw numbers, nominal debt for example, rather than
debt as a share of GDP. But we want debt
as a share of GDP. So we need another
equation:
Where Y, of course, is GDP. I want
to look just at the numbers, at nominal Y, so is nominal growth rate. To get to something like the IMF version, we
would add that
Where is inflation and g is real growth rate. But I don’t need that decomposition to get
where I’m going, so I’ll stick with the nominal rate.
So the ratio of debt to GDP next year
will bejust substituting in from the equations above
Where the lower case letters are ratios of GDP, and the
upper case letters are nominal.
What we want to know, really, is whether debt as a share of
GDP is rising or falling. That is,
algebraically stated, we want to know whether is positive (debt/GDP is growing) or negative
(falling). I’ll call that difference just because I like using Greek letters. It looks sophisticated.
Since , the sign in that last version depends
completely on the sign of the phrase inside the last square brackets. Whether the debt as a share of GDP is
growing, stable, or shrinking is determined by whether
(I’m using the braces there to indicate
optionsgrowing, stable or shrinking if greater than, equal to or less than.)
But the strange thing, the thing that
crossed my eyes a little as I woke up this morning, was stating this condition
a different way. Let’s just look at the
case where debt as a share of GDP is shrinking, so I don’t have to write that
monster in braces all the time. So
shrinking debt/GDP implies that
Or
Or, assuming that debt is positive,
And this is where the weirdness comes
in. If this inequality holds, b, the
debt to GDP ratio, is declining. In this
form it looks as though the bigger b is, the more likely it is to decline. I mean for any given , the ratio gets smaller in absolute value as gets bigger.
That would make the inequality above easier to achieveright? So at high levels of debt to GDP, decreasing
that ratio would become easier, according to this. But can that be true? Is debt somehow selfdamping? That’s absolutely not what we keep hearing in
the political world. We keep hearing
that debt will become so huge that it will be “unsustainable”, that it will
accelerate out of control, that compounded interest will make it explode. But if nominal growth and interest rate were fixed, and we limited the primarydeficittoGDP
ratio d at some fixed positive value, then the weird result above would imply
that it is true, and that instead of
running away, debt as a share of GDP would have a tendency to slow down and
reverse itself, or at least slow to a stop, as it gets bigger. And if it were true, maybe that would explain
why nations never have debt to GDP ratios in the thousands of percent
range.
But of course it isn’t quite that easy. growth rate and interest rate are not fixed. In fact, they are a function of the deficit d, among many other things. Figuring out what the function might look like is complicated. I didn’t wake up with the algebra for that in my head and I don’t want to spend a lot of time trying to get at it. If anyone knows what that should look like, let me know. At a wild dreamstate guess thoughand for the moment it isn’t any more than thatwe could postulate that nominal growth and interest as a function of d have shapes that looks something like this:
Here’s the dreamy excuse I dreamed up for
these dreamy shapes, kind of nodding to a lot of different people’s beliefs
about the impact that deficits might have.
The nominal growth to the left of
the vertical dashed line is mostly real growth, and to the right it’s mostly
inflation. Interest rates are low until
we get out to the right, and then they rise because of inflation, and because
the high deficits are soaking up a lot of the available money. While interest rates are near zero the growth
curve is rising, because multipliers of the deficit are relatively large there,
so higher deficits result in higher growth; near the inflection point it’s flatter
because multipliers are smaller (because the central bank counteracts any
fiscal stimulus to avoid inflation). Out
to the right it’s big again because the deficits are so big the Fed has lost
control, and inflation is pushing nominal growth.
I’m not wedded to these shapes, they’re just the shapes I
woke up seeing. They might be
wrong. Still, it's not crazy to say that fiscal multipliers will be bigger when the economy is weak and we are far from full employment and smaller when we are close to full employment, either because it's hard to push the economy very far past that or because the central bank will push in the other direction to avoid inflation. And it's not crazy to say that interest rates will be low when the economy is weak, and rise as the economy gets stronger. So the critical part of the shape of these curves is at least not crazy: the slope of the difference between growth and interest rates will decline, or at least not rise, as deficits as a share of GDP increase.
If the shapes in the first graph are right then that difference, is the gap between the blue and red lines in the first picture, looks
something like this
I haven’t put a vertical axis on these graphs because it’s
not clear where it would go. It depends
on what the economy is doing. If we are
in a deep recession, the vertical axis will be farther to the left, farther
toward the flat low interest rate area; if we are in a boom it will be farther
to the right. But the interesting thing
happens when we put that axis inlet’s suppose we are in a weak economy, so
the axis is farther to the left. Then we
can complete the graph that was giving me a headache when I woke up like this:

The solid green line is the ratio of deficit to debt when the debt is just a bit over 100% of GDP. (It would be a 45 degree line if b, which is debt/GDP, were exactly 1.) If the deficit/GDP ratio is less than B and greater than A in this picture, then debt as a fraction of GDP is declining. Above B, of course, the size of the deficit just overwhelms any increase in GDP that may result, and we have a pretty icky outcome: growing debt and rising inflation. Below A, though, the economy is so weak that we are pushed into a slowly growing or even declining GDP, and the ratio begins to rise again: if the deficit is too low in this weakeconomy situation, then we also have an icky outcome, with weak or negative real growth and also rising debt as a fraction of GDP. That would not be true in a stronger economy; as the economy improves, the axis shifts to the right (and so does the green line, since it has to go through the origin).
But the curious thing is the dotted green line. This is a d/b line with higher b. And notice that the gap between A’ and B’ is
bigger than the gap between A and Bin this vision, a wider range of deficits will reduce the debt to GDP ratio if that ratio
is already high.
So maybe there is something in the idea that high debts
arewell, not selfcapping, but at least they push against an increasing
resistance as they get larger.
Does any of this make any sense at all? Or is this all just as surreal as dreams are
supposed to be? If it makes sense, then maybe debt as a fraction of GDP does naturally slow down as we get up past a couple hundred percentand there are several examples, such as Japan right now, where debts in that range have not created catastrophe. I'm not suggesting that debt really doesn't matter, or that we should ignore it; it does make a lot of things harder to do, and it is a nuisance. But maybe we also shouldn't panic about the specter of a runaway, exploding debt that seems, in this dream anyway, to be unlikely.
Of course it's just a simple model that came out of dream. It may disappear after enough coffee. And even if this does turn out to fantastic enough, surreal enough to be a proper dream, I still prefer dreams where I get to have superpowers.
Oh, and if you didn’t like the algebra, we can blame Jeff.
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