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By this logic, the United States (assuming a succesful Sumner-esque 5% NGDP target) could stabilize debt/NGDP by running a deficit of, roughly, a quarter-trillion dollars.

And I agree with you! I'm just pointing out that this is a pretty stunning conclusion to the untrained eye

Squarely. Yep. Even to the trained eye, I always surprise myself when I do this little calculation!

Being an armchair economist, I really appreciate this quick exercise to ground my thoughts on the nation's debt. Thanks!

A relevant data series from the world bank:


The unexpected, fun and extremely useful conclusions you get at when you stop reasoning in terms of a coconut economy and think in money...

Primary deficit/ (g-r) = debt/GDP. Standard public finance steady state condition, no?

The big point about public debt is about the stability of r. Or, if in the quest to subdue the threat of a rising r, does the central bank engender high and accelerating inflation.

Thanks for the article, I will just add some numbers from the last census in 2011:

population - 33 476 688
population increase 2006-2011 > 5.29 per cent
The population of over-65ers has increased to almost five million over the last five years
The median age in Canada is today 40.6 years, up from 39.5 years five years ago and much higher than the 33.5 years of two decades ago
The average family in 1961 had 3.9 members, while today it has 2.9.

Nick: Thanks for a clear and useful post. As a useful encore, would you consider blogging about the difference between govt. consumption and govt. investment spending? And their implications?

OK Nick. Picked that up a while ago. It explains a large part of the Government of Canada's deficit/surplus position since WWII. From 1945 until 1970 Canada ran very small surpluses and deficits, so our national debt declined dramatically. The 1970's caused serious deficits to appear, we kept our heads above water in the 1980's and dealt with the unsustainable deficits in the 1990's.

I thought this was Applied Political Macro 1A03, no?


You forgot a couple factors - term structure and nature of the debt. Is it coupon type or accrual type and how quickly must the debt be rolled over?

Suppose the debt is all accrual type. Suppose bond holders require an extra 1.0% in interest payments for every 2.0% of extra duration. That gains you an extra 1% of allowable deficit.

One other thing,

Suppose a government sells securities that have a non-guaranteed potential rate of return and a success rate:

Potential Rate of Return (PRR) = (Real Potential GDP - Real GDP)/Real Potential GDP

Realized Rate of Return (RRR) = Success Rate (SR) * PRR

If the government sets the potential rate of return, the markets will try to maximize the realized rate of return (maximize success rate). If the market pushes the success rate above 1, then the government simply underestimated real potential GDP. If the markets subvert the success rate, then the realized rate of return will always be below potential.

In either case debt falls.

Your main point is obviously well-taken.

There is a minor nit to pick, though. When the growth rate of a quantity is a random variable rather than a fixed number, the expectation of the future value is in general less than the result of compounding the expectation of the rate. A common example: if increments in log GDP are normally distributed, then the expected future GDP is less than that implied by the expected growth rate by a rate of 0.5 x sigma^2, where sigma is the volatility of the growth rate.

For realistic values of GDP volatility, this doesn't make much difference in expectations. However, in addition to the adjustment to expectations, the future GDP level is also risky. For example the expected affordability rate of your assumptions is:

(1 + 2.5%)(1 + 2%) - 1 = 4.45% (approximately)

But I just ran 20 random samples assuming that inflation is fixed but GDP growth has 1.5% volatility, and happened to get a realized affordability rate of 3.92%. So a prudent government might not want to use all of that 27bn expected deficit room.

Phil: exp is convex so E[exp(X)] > exp(E[X])

Squarely Rooted,

US publicly held federal debt is $11.5 Tn. 5% of that is almost $600 Bn. If you count intra govt debt there is $16Tn so a deficit of $800Bn is sustainable.

Nick, everyone,

Assuming the short rate runs at or near NGDP growth (hasn't it almost always except during short periods of disequilibrium) it's really about the primary deficit. We can't expect to continue to be able to run massive primary deficits. Eventually g+pi and i will tend to converge.


You are making several assumptions:
1. The government must sell debt to fund deficits - they don't
2. The government must limit the duration of the liabilities that its sells to 30 years - they don't
3. The government must sell coupon securities to fund deficits - they don't

The real question is what happens to the US deficit when the economy exits the ZLB? Does it rise or fall? If the natural rate goes to 3%, interest on 16Tn rises to 800Bn so there is no longer any room for a primary deficit (at current debt levels). Maybe the sectoral balances types will claim that once out of the recession the primary deficit will return to zero as some kind of automatic consequence of inflation stabilization. (I can't claim to understand their logic well enough to reconstruct their arguments - empirically, though, there does seem to be some appeal to the hydraulic perspective).


I made none of those assumptions. The government may also issue currency in some assymptotically bounded ratio to the size of the economy.

But that doesn't change anything. The fact that the govt can use its seigniorage power to maintain some stable ratio of real value of currency to GDP helps its funding cost, but in no way modifies the fact that deficits are strictly bounded for non-accelerating rates of inflation.


"But that doesn't change anything. The fact that the govt can use its seigniorage power to maintain some stable ratio of real value of currency to GDP helps its funding cost, but in no way modifies the fact that deficits are strictly bounded for non-accelerating rates of inflation."

Um, yes it does.

Example 1: Picture a government that sells only debt that accrues interest (rather than debt that makes regular coupon interest payments). And so the interest payments that it makes in any year are a function of both how much debt matures in that year and what the average interest rate on that debt is. Now picture the same government extending the average duration of its debt 5% a year while the market demands only 2% more in interest for the added duration. Meaning that as long as the market is willing to make that tradeoff (5% more duration for 2% more interest) the government can run a 3% deficit indefinitely - inflation or no inflation.

Example 2: Picture a government that instead of selling debt (guaranteed claims on future tax revenue) to fund deficits, instead sells non-guaranteed equity to fund deficits. Suppose the government sets the rate of return on that equity to be equal to the real output gap and suppose that the only way for a government equity holder to realize that rate of return is through offsetting a tax liability. What happens? During recessions (large output gap) the rate of return rises but the success rate of previously purchased equity falls (fewer people have jobs and have a tax liability to offset). During growth periods the opposite happens - the rate of return falls, but the success rate of previously purchased equity rises.

Ritwik: minor typo I think. You forgot to divide the primary deficit by GDP.

It should be: (Primary deficit/GDP)/(g-r) = debt/GDP. But yep, what I am doing is just the same.

JB: Hmm. So population growth, and growth of working age population, is slowing. Maybe my 2.5% is a bit optimistic?

Simon: I see the importance of distinguishing government consumption from government investment. I wonder how easy it is to distinguish them in practice though? And if the I/C ratio were constant over time, would it matter? I'm still thinking about that.

Determinant: "I thought this was Applied Political Macro 1A03, no?"

Yep, but you wouldn't know it from reading the comments on regular news posts. Some of the better reporters note that the debt/GDP ratio fell. But we need some sort of longer run calculation, because GDP growth fluctuates year to year.

Phil: I'm thinking about that. If one took a geometric average of past growth rates as a base for your estimate of future growth rates, I *think* it would handle that problem?

K: "Eventually g+pi and i will tend to converge."

You can build simple OLG models where it doesn't. Samuelson 58. Unless the private sector can create it's own Ponzi scheme/bubble. Plus there's maybe a liquidity/safety premium on government debt.


I didn't mean literally converge. I just meant that in the long run they'll probably go back to within a percent or so of each other, where they've been historically when we weren't fighting runaway inflation or in a liquidity trap. I know there's no theoretical requirement that they be equal, even asymptotically. So *maybe* we can run a 1% or so primary deficit in perpetuity. But certainly not 5%-10% like the US has been doing for the last few years.

As to Phil's comment: the growth rate *is* the log of the geometric mean annual NGDP ratios (same as the mean of the log ratios). But that doesn't invalidate Phil's point (though he got it backwards) which is that the expected NGDP is greater than exp(expected NGDP growth) because of the convexity of the exp function and Jensen's inequality.

K: OK. I'm with you on convergence. Yep. Nominal interest rates and nominal GDP growth rates do tend to be correlated, and are normally closer together than they are now. I think I'm now starting to get Phil's point, but still mulling it over. My brain is a bit slow nowadays.


it is either the geometric average, or doing logs, but not both

ln (m1 x m2 x m3 ) ^1/3 = 1/3 [(ln (m1) + ln(m2) + ln(m3) ]

with m1 = 1 + rate_1, and I hope you get the natural extension to more components


you should for fairness include the coupon payments (r in %)in this.

This is a must in the moment you specify deficit as PRIMARY deficit (pd in % GDP). Then it is useful to also mention the tax t , the citizens fraction c, who have to pay on the interest r. Just to complete the nomenclature: g the real (as in: after inflation) total growth rate of a nation, and i the inflation

Delta (Debt / GDP) = pd + r * ( 1- c * t) - g -i

for more mature countries, who didnt lose a war, like Canada and the US

r * ( 1- c * t) - g -i

was pretty close to zero, in the long run ... : - )

[editorial remark: some other folks here, not Nick, have introduced r in a somewhat different way, which makes the tax / foreigner impact less obvious, but are, of course, first order correct, I would like to stay with my form, for a subsequent discussion, for folks interested beyond the Canadian case.]


The left-hand side of your expression is *exactly* what I said: the log of the geometric mean of the annual NGDP ratios. Your expression has *both* a logarithm *and* a geometric mean. The right-hand side is what I said in brackets.

"I hope you get the natural extension to more components"

Attitude, seriously? I'll assume this is just a foreign language/culture/communication problem.


On Phil's point, assume GDP after one year increases by a factor exp(g) where g is normally distributed with mean zero. Because exp is convex, a deviation in g on the upside changes GDP by more than the same deviation in g on the down side. Since those changes are equally likely in this example, the increased GDP cases will outweigh the decreased GDP changes and so expected GDP will be higher at the end of the year even though expected growth is zero.

In general E[f(X)] > f(E[X]) for any convex function f (Jensen's inequality).

Sorry, I meant three-quarters of a trillion dollars.

Was having a bad day.

K: Let's see.

One year of 0% growth and one year of 20% growth gives 20% total growth.

Two years of 10% growth gives 21% total growth.

I think I've got it!

Squarely: I was following the "rule of 10" (multiply Canadian numbers by 10 to get US numbers), and thought your quarter trillion seemed a little low. But three-quarter trillion (I believe you) is surprising!


On your equation:

Delta (Debt / GDP) = pd + r * ( 1- c * t) - g - i


dD/dt = pd * GDP + D * r
D = exp ( f(t) )
dD/dt = f'(t) exp ( f(t) ) = f'(t) * D

[ f'(t) - r ] * D = pd * GDP
D = [ pd * GDP ] / [ f'(t) - r ]
D/GDP = pd / [ f'(t) - r ]

For a stable debt to GDP ratio
pd = f'(t) - r

Meaning the primary deficit must equal dD/dt * 1/D minus the interest rate.

In the Clinton era a deficit of a "couple of hundred billiion dollars" (vagueness deliberate) was debt-to-GDP stable. It was Clinton's first proposal. He of course did much better.

While all this is useful to keep in mind surely (especially when provincial debt is included) we want to reduce debt substantially on the medium term? If there is another shock we will want to spend a dumptruck full of money in a hurry. I always get worried when Gaussian distributions are discussed in finance. The tails are much larger in reality than in a Gaussian model and the tails in reallity are not symmetric: if mean growth is 2.5%, a year with -2.5% growth is muck more likely than a year with 7.5% growth.


how about you think a little bit about the formula, and various simple cases, like several of the variables set to ZERO.

If you owe everything to your own people and tax every interest away, then the interest rate doesnt count.

You put all the interesting terms into your f(t), which then leads you to obviously ignore taxes, inflation, growth, foreign ownership.

Nick: I don't know how hard it is to distinguish between govt. investment spending and govt. consumption.
But I know that this is the way the US has reported its national accounts for some time. For example, see http://research.stlouisfed.org/fred2/categories/107

I should really know my own country's national accounts better....statcan splits government spending into consumption and investment as well (http://www5.statcan.gc.ca/cansim/a46?lang=eng&childId=3800064&CORId=3764&viewId=3)

Nick: I agree! Of course that assumes 5% NGDP growth at current levels of GDP, so if GDP plummets, say, plummets spectacularly that could cause a similarly spectacular spike in not just the gross nominal amount of debt but also the debt/NGDP ratio. Not that that could ever happen! Great moderation and all that.

Not to be too political in these comments, but in light of Speaker Boehner's comments:


to the effect that every increase in the debt limit must be accompanied by a corresponding reduction in projected deficits equal to the increase, it would be interesting to see a proposal that the US debt limit be adjusted from a nominal gross amount to a percentage of NGDP. Otherwise, under Boehner's proposal you could keep debt/NGDP perfectly stable or even decrease it, yet still hit the debt limit every year or two!

Also, this issue always gives me headaches - is it productive to apply this line of reasoning to current account balances?

Nick: I'm pretty new to thinking about govt. finance in terms of consumption and investment expenditures, but I think we pretty much agree that the distinction is important. To take just one example, the new PQ government here in Québec has announced that, in order to meet their deficit targets, they are going to delay spending on infrastructure investment. (I suspect much of this is repairing crumbling Montreal highways.) Assuming that these are positive NPV projects, I think delaying them makes us all (Québec taxpayers and all provincial bond investors) worse off.

I think the traditional macroeconomic analysis of the sustainability of government debt and deficits applies most closely to deficits on current expenditures. In contrast, investment spending on positive NPV projects might reasonably be expected to be self-financing (and if the projects are negative NPV, there is little point in pursuing them regardless of the state of government finances.)

Simon: I agree. We would certainly want to subtract self-financing government assets from the debt (and investment in such assets from the deficit) for public finance/"is this sustainable?" analysis. It gets a bit trickier for investments in bridges and schools, which might be self-financing indirectly, if they increase future potential GDP and future tax revenue.

Eyeballing those StatsCan numbers, I get the sense that government I doesn't move much. But it does look like some government I was preponed to 2010 from 2011 and 2012. Which makes sense from a policy perspective.

Squared: I think it makes sense. Canada has small current account deficits/surpluses, and net foreign asset positions, so it's not a big issue here. The US has a bigger CA deficit, but there's that whole "dark matter" debate, which means it is very hard to get a sense of what the US net foreign asset position truly is.

I try to keep this as short as possible:

g - -> 0


The common narrative is that mankind had made little progress on productivity until around 250 years ago, with growth rates well below 0.5%/year / detection limit. More than 50 % of the people were basically required to produce food.

Then the industrial revolution took off, first in England, Europe, the West …, multiplying GDP per capita by about a factor of 100, which can be translated into a yearly rate of gigantic ….. 1.8%

Add 0.7% population growth, and you come to the 2.5% total GDP growth, Nick Rowe and many others use as a rule of thumb, for the past.

So why not expect the same in the future?

1. There were 3 one off effects:
a) getting the people from the farm into the factory (let’s say factor 2),
b) getting the majority of women into the paid (and taxable) workforce, factor 1.6
c) educating people at all, from a very few years to 12 now in the most western countries (factor 2 – 3)

With those about half of the gains are explained by factors, we can not extend or repeat in the future.

Now, in a typical western country, about 1 % are working in agriculture, less than 20 % in manufacturing , with this magnificent economy of scale, and about 80 % in services, where there is little scaling. A haircut takes about the same time as 50 years ago, an hour of lawyer advice is still just that, to let him speak faster does not really help you : - ) the number of nurses and doctors per patient in a hospital is probably higher today than in 1950 (does anybody have numbers?)

As far as we understand this now, productivity per capita will from now on only increase with less than 1 %, maybe getting closer to 0.

2. Most western countries have fertility ratios substantially below 2.07, and that means a shrinking working population, masked in most places by some immigration and the time delays, the rest of the 3 one off factors, so far.
And that means that the total real GDP growth in many western countries will be pretty close to ZERO in the future, sometimes even a little negative.

Simply, you can’t “grow your way out of debt” anymore, as before.

The primary deficit has to get close to zero, or negative (means savings) in good years. Real rates will stay low, probably even a little below zero, IF, AND ONLY IF, the creditors believe, that you are really commited to pay in full and on time.
Otherwise, much faster than in the past, the spiral starts, and you become a client of the IMF, not designed to be a pleasant experience.

Overall, Canada (current account, budget balance) is in an acceptable shape, a little worse than the Euro area.

One last thing, if you get into competition with new players, like China, or for several south west European countries, with highly educated, eager eastern Europeans, the price of your warez drop, and you have negative productivity rates. Those folks wanna have their part of the cake too.


I want to point to an assumption that underlies this part of your comment:

2. Most western countries have fertility ratios substantially below 2.07, and that means a shrinking working population, masked in most places by some immigration and the time delays, the rest of the 3 one off factors, so far. And that means that the total real GDP growth in many western countries will be pretty close to ZERO in the future, sometimes even a little negative.

Simply, you can’t “grow your way out of debt” anymore, as before.

A heuristic I always keep around is that GDP growth is actually a function of 2-3 different things. One way to conceptualize it is (all in "real" terms):

GDP growth = population growth + (productivity growth + mysterious other forces)

The truly tautological way to express it is:

GDP growth = (GDP growth per capita)*(capita)

The assumption underlying your comment is that we won't see GDP growth per capita increase at a rate that offsets the decline in population (or at least GDP-producing population).

That may, or even probably, is true. But it isn't necessarily true.

Perhaps thanks to Moore's law we'll see vast productivity increases in previously labor-intensive sectors. If we saw that in health care all bets are basically off.

Nick: Thank you - good point about "dark matter," too easy to assume that "the data" is all equal (especially when it all looks so equally shiny and authoritative on FRED).

Few short points between copies...
Dark matter: good point. The U.S. borrows governementally at low rates and lend-DFI privately and lives on the spread. Plus seignoriage. What counts is not tha balance sheet but the income statement. The Italian government essentially did the same in the national scheme until the ECB told them it was bad.
Investment vs consumption: as long as we haven't found a way of integrating human capital in the accounts, it is idle talk. Why would the pay of a mason putting a brick wall around my class consudered an investment while me putting something in heads that will last for decades ( and is self-financing) is consumption, is beyond me.
PQ postponing physical investment? Waiting till the mess is cleared will earn a 30% rate of return...


"How about you think a little bit about the formula, and various simple cases, like several of the variables set to ZERO. If you owe everything to your own people and tax every interest away, then the interest rate doesnt count. You put all the interesting terms into your f(t), which then leads you to obviously ignore taxes, inflation, growth, foreign ownership."

No I didn't put all of the interesting terms into f(t).

GDP = Real GDP * (1 + Inflation Rate) - Inflation Rate and Growth Included
pd = ( Taxes - Expenditures ) / GDP - Taxes Included

Yes there are tax implications for who owns the debt, though it would be a little silly for a government to pay interest on its debt only to tax it all right back. Also, a lot of the debt is also owned on a tax deferred basis (mutual funds) and through intra-governmental holdings (Social Security trust) and so identifying each and every owner's tax implications is tedious.

The important part that you missed (and what f(t) really represents) is the choice between spending the interest "r" or re-investing it. If the interest income is spent in the same period that it is received then the interest payments themselves are part of GDP. Meaning your term r * (1 - c*t) is 0 even if the interest rate is non-zero.

What f(t) really represents is the save versus spend preference or liquidity preference if you prefer.

@ Squarely and Frank

1. Nomenclature / formulaes
These nomenclature thing travel very badly in blogs. On the other hand , especially here, this creates a lot of misunderstandings, so I give it a try.

The very most of these relations, like the GDP / capita equation of Squarly and the averageing over multiple years equation form Frank above are products and very often of the type (1 +m1) * (1+m2), and m1, m2 small, percentage numbers.
It is very convenient to shorten the multiple multiplications and divisions in this way:
Ln ( 1+ m1) = m1 + small error, [ x –x^2/2+x^3/3 …..]
usually below the accuracy we know m1 anyways, as long as m1 is small
example: m1 = 10 % -> ln (1+m1) = .0953
then it is easy to add /subtract all these small terms, simply multiply with the number of years.

You just shouldn’t forget to do the exponential at the end of it for large numbers.
Like my 1.8% over 250 years, above -> exp (.018 *250) = 90

This is very practical in long term calculations, since doing 50 multiplications, or 10 division …
Doing an arithmetic average of 10 yearly (logarithmic) inflation rates is a lot easier, and less error prone, then doing a 10th root of a number, especially in your head.

2. Accuracy of data
For agriculture productivity is relatively easy. A bushel of wheat is that, and I can count how many the farmer turns out. Same for manufacturing, a car is a car … Yeah , these are also 2 simplifications.
How do you measure now the productivity of a haircutter, a lawyer, professor, nurse ….?

De facto: salary paid / number of hours accounted.

In this way the OECD finds, that Germany lacks horribly, by 30 % or so, over the last 20 years, in service productivity relative to its neighbors. Urgent action needed, SUBITO ! And the recipes are on hand, no apprenticeship, sent all nurses etc 12 years to school, and then at least 3 years university , … as in most other countries.
And with 75% of Germans working services this should be the dominant factor.

Well we do export our discounters (Aldi, Lidl) and their model to other countries, people there are happy that it reduces their consumer prices. We do export our apprenticeship model to Italy, because they like it, just the opposite what the OECD recommends on their unquestionable facts …. : - )

The truth of course is, that the accounting of the OECD (And IMF and Worldbank are no better) is sheer nonsense.
But when I don’t believe their stories about my country, why should I believe their stories about Greece or Portugal?

I ll gave 3 examples, why I do not see any significant scaling in services.
Moores law only describes semiconductors, and is not replicated anywhere else. This is a historically absolutely unique event for one specific technology, I am part of, and it is petering out in usefulness.
Miracles can happen, but I would not base conservative government policy on the expectation of them. All what is required is that people do not overstep the income by 1 or 2 %, and adjust to the real numbers, if divergences occur. The longer you wait , the harder it gets.
Improving on this productivity is the holy grail, I have here a book in front of me, printed 1979, where people discuss the various long term government programs to improve things on IT, bio science, cancer, fusion, etc. Every idea where there could be big boosts are welcome

The term r * (1 – c* t) gets paid out of the government purse independent of what the owners of the debt are doing with it, you are mixing up debt and GDP.

because people got very creative with the accounting of "investments", not only in "human capital", we purged that out of an earlier version of our financial stability rules (§115 GG)


"The term r * (1 – c* t) gets paid out of the government purse independent of what the owners of the debt are doing with it, you are mixing up debt and GDP."

Yes the term r * (1 - c*t) gets paid out of the government purse indepedent of what the owners are doing with it, BUT the term - ( g + i ) is not independent of r * ( 1 - c*t ).

If the interest payments are all spent in the same time frame that they are received then g+i will be larger than if the interest payments are re-invested or stuffed in a matress.

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