Some simple arithmetic of debt/GDP and population growth

I had always known that a given debt/GDP ratio would be more worrying for a country with declining population than for a country with a growing population. There will be fewer future people to carry the same future burden. But I had never sat down to do the arithmetic, until just now. What surprised me was just how big this effect was. So if you have never done the arithmetic yourself, or seen anyone else do it, read on.

Suppose the real interest rate is r, the growth rate of per capita GDP is y, and the growth rate of population is g. Define the "debt burden" as the percentage of GDP needed to service the debt to prevent the debt/GDP ratio rising over time.

Then: debt burden = (r-y-g)debt/GDP

In a stationary economy (y=0, g=0) the government needs to set taxes higher than program expenditure so that the resulting "primary surplus" is just big enough to pay the interest on the debt, and prevent the debt rising over time. (Both the interest rate, and the debt, are measured in "real" terms, i.e. adjusted for inflation.)

To my mind, the "primary surplus" needed to prevent the debt/GDP ratio rising over time is a good measure of the burden of the debt. It represents taxes we pay that cannot be spent on the things we want to spend it on.

In a growing economy (y+g>0) the debt burden is smaller (for given real interest rate and debt/GDP ratio). That's because the debt can be allowed to grow at rate y+g while keeping the debt/GDP ratio constant.

Let's plug in some numbers. Suppose r-y=2%. That seems roughly plausible for advanced countries, with real interest rates on government debt around 3%, and real income per person growing around 1%. Now compare two countries: the first has 1% population growth; and the second 0%.

The country that has a growing population has half the debt burden of the country with constant population, if both have the same debt/GDP ratio. And the country with a growing population can handle twice the debt/GDP ratio with the same debt burden.

It's that second way of thinking about the arithmetic that surprised me.

Presumably there is some maximum level of debt burden that a population is willing to accept, because they are paying taxes, but getting nothing in return. (Well, they did get something in return, but that was yesterday's deficits, paying for yesterday's generation's benefits). Past a certain point, if the debt burden gets too high the population may refuse to pay, or may emigrate to avoid the burden, which just raises the burden on those who remain.

Maximum debt/GDP = maximum debt burden/(r-y-g)

All other things equal, if we just looked at projections for population growth rates, my guess is that Canada can handle nearly double the maximum debt/GDP ratio that (say) Greece can handle, since our population will grow at nearly 1 percentage point higher.

Of course, other things matter too, but population growth matters a lot. I hadn't realised the effects of population growth were so big.

Comments

[Andrew F]

Is that one reason to be less worried about the USA, because they have such a high rate of population growth (rather astonishing, really, for such a large country).

[Ironman]

At least you're starting on the right track. ;-)

One wrinkle you might want to consider in your analysis is the relationship that might exist between Canada's debt burden per capita and its maximum income tax rates, should one exist, which might help you determine where the relevant tipping points might be.

[Nick Rowe]

Andrew: Yes, the US has a higher population growth rate than (say) most of Europe, both due to higher birth rate and higher immigration. That makes its debt/GDP ratio less worrying.

Ironman: interesting posts. But I notice you define "debt burden" differently than I do here. You define it as debt/GDP ratio (a stock/flow). I define it as debt service costs/GDP required to hold debt/GDP constant (a flow/flow). My guess is that people's willingness to pay taxes depends on what services they are getting in return.

[Kosta]

I like the approach Nick and I agree that a growing population reduces the debt burden, but I don't think that assumption that r, the per capita growth rate, is independent of y, the population growth rate, is true. I think the equation (Max Debt)/GDP = (Max Debt)/(r-y-g) needs to be tweaked.

I believe that countries with a higher growth rate (be it from higher birth rates or immigration) have more low-paying jobs.

You can see this by comparing the hours worked and productivity of population growth countries like France to the U.S. (a high population growth country). France has a productivity as high, if not higher than the U.S. in spite of its citizens working less hours than in the U.S. (and not being paid as well). The higher productivity stems in part from the lack of low paying jobs such as service in France. The lack of low paying jobs is in turn a response to the lack of people willing to take those jobs, which arises from the low birth rate and lack of immigration.

I haven't looked at the numbers to be sure (or worked out the math), but if you have a country with limited population growth, you would think that country would focus its energies on keeping and improving the most productive jobs, which should lead to a higher per capita growth rate even if the overall growth rate is lower. I think this is partially borne out by the tendency of the work force in low population growth countries to be well educated and well trained (and I'm thinking of France, Germany and Japan here). Conversely, if you have a country with a higher population growth rate, its more important to find employment for the extra population, rather than having them educated and/or trained to fill the most productive jobs. Hence the lower education levels in the U.S. and the greater number of low-skilled jobs, be they construction or fruit-picking.

The net effect should be that the better trained workforce (i.e. the workforce of the country with the lower population growth), should have a higher per capita economic growth rate.

Also, if you consider that the capital per economy is constrained, then the country with the lower growth rate will be better able to put that capital to productive uses, which again should increase the per capita growth rate.

p.s. Being of Greek heritage and having lived in Greece, I'm not sure that the above arguments hold for that economy.

[Mark Dowling]

One of the reasons for the catastrophic change in fortunes of the Irish economy was the decision of many of the European immigrant workers who had caused an upward spike in population numbers to leave. This combined with a massive borrowing increase to deal with a yawning deficit due to the disappearance of the taxes the property sector was spewing into the Government finances has turned a country with a low debt/GDP ratio into a basket case in three years.

Because much of Canada's immigration and therefore population growth is based on family ties it is unlikely to be halted quite as abruptly.

[Jon]

Which governments in Canada do the borrowing?

Here in the US, we have Federal Debt, State Debt, County Debt, Municipal Debt, School Board Debt. These are not insignificant sums. For instance, our local school board in a town of 10000 has close to 50M in debt. Sure 50M is a small number but 10000 is small relative to 300M.

[Luis Enrique]

this is the book I've always meant to read along these lines, but keep putting it off because it looks too depressing.

[Ironman]

Very good point, Nick, although I would suggest that given the dynamics for when debt as a stock is involved, people's willingness to pay taxes often depends more on what services they are okay with having already received (such as paying down debts following major wars, etc.) rather than the services they are presently receiving.

As for the taxes themselves, if history is any guide, tax rates are frequently determined not by what it directly costs to get services in return, but rather by what tax revenues are required to be allowed to continue increasing the stock of debt with a nation's given rates of GDP growth and population growth, which is where the flow/flow relationship you've developed might offer some important insights.

[Arturo]

Nick, interesting, but how would you handle a situation where real int rates are less than y + g? That's arguably where the US stands, and the model result is that increasing debt to GDP ratios create less and less of a burden. Do you impose a minima, rework the model, or could there be a valid macro argument in there along the lines of Lerner's functional finance?

P.S. Re Canada vs Greece, demographic ratios appear to matter too, perhaps more than overall growth: http://investments.miraeasset.us/en/ourMarkets/outlookView.do?board_id=1125&group_id=1&pageNo=1
If so, watch out, Kosta's countrymen might be buying Alberta real estate in ten years. ;)

[RSJ]

How are you getting r-y = 2% as a "plausible" scenario? I'm not aware of this happening to any industrialized nation.

Generally y > r.

Perhaps if you are particularly inept and decide to solely sell long bonds would you get r equal to y, and it may even exceed it on occasion, but generally not over the long run. (I am ignoring the sad situation of countries stuck on a currency peg or who are otherwise paying "non-risk free" rates)

For the U.S. we have had, as a post WW2 average:

r = 1.7%
y = 2%

[Scott Sumner]

Nice post, a few comments:

It might be simpler to subtract NGDP growth (per capita) from nominal interest rates. Then you avoid having to calculate real interest rates. My hunch is that r-y is a bit less than 2% for most developed countries, which makes your point even stronger. If it were 1.5%, then the countries with 1% pop. growth would have only one third the debt burden of the ZPG countries.

And also note that Japan's population is now falling.

Right now the fiscal situation of Canada, Australia and East Asia (ex-Japan) looks way way better than the US, Japan and Europe.

[RSJ]

Ok, let me try again.

Government debt is sold at different maturities, and governments have a "debt management strategy", one of whose core missions is to manage the duration of that debt as well as the weighted average time to maturity.

Generally the maturity of government debt is about 5 years. You can look this data up in OECD.stat or directly from the debt issuing branch of government. Therefore the interest rate paid is not going to be the long bond rate, or even the 10 year rate, but close to the 5 year rate.

For the U.S., the actual rate has averaged 1.7% in real terms. For Canada, over the last decade, that real rate was 1.6% (both average and median). In both cases, this was less than the per capital real GDP growth rate -- i.e. y > r.

All of this is fairly easy to check via pubic records.

[Lord]

UK/US demonstrates lower population growth can go with higher income growth. Whether it can do so after world population growth peaks later this century is more doubtful. Japan's debt is internal (yen) so it shouldn't be too difficult for them. As g falls, r will have to as well, as it has done there.

[Alex Plante]

A few weeks ago I went to the Statistics Canada website to look up the statistics for government debt. They have a table that neetly consolidates all govertment debt and financial assets, however I was unable to find information on government-owned real assets. It seems to me that it would be interesting to know the net worth of the public sector. Some of the debt service is part of the cost of maintaining a public infrastructure.

[RSJ]

"As g falls, r will have to as well, as it has done there."

Exactly!

If D = Debt, G = GDP, N = deficit, k = NGDP growth rate, and r = rate of interest on debt.

Then
dD/dt = N + rD

So applying the chain rule to d(D/G)/dt and substituting, we get

d(D/G)/dt <= 0 iff

N/G < (k-r)(D/G)

If you really believe that r > k -- i.e. that the *risk-free* rate of government debt is greater than the GDP growth rate, then the only way you can reduce the debt/GDP ratio is to run surpluses.

But generally countries do not reduce the debt/GDP by running surpluses, they reduce the Debt/GDP ratio by stabilizing the Deficit/GDP ratio and letting time take its course, and this is enough to prove that r < k.

Typically r = 60% or so of k, depending on the maturity of the debt. This *must* be the case, for risk-free debt, as no one will pay a long term risk-free rate higher than the long term equity returns -- everyone would be buying bonds in that case. And unless you believe that P/E multiples quickly shoot off to infinity, then the equity returns cannot be higher than k either. So the long bond will not be greater than k over long time periods, and the 5-6 year bond will be about 50-60% of the long bond. Only in situations of very slow NGDP growth and/or very fast pop. growth, will the per-capita gdp growth rate be less than 60% of the real GDP growth rate.

[Too Much Fed]

"real income per person growing around 1%"

I think that is a bad assumption for at least 50% of the population in the high wage countries.

What happens if real income per person is at least 4% for the few and 0% or "negative %" for the many?

[Nick Rowe]

Finally I have the time to respond!

Kosta and Mark: yes, I have assumed that y and g are exogenous, and they probably aren't. Diminishing returns, for example, could make growth in real income per capita depend negatively on population growth. Mark's point worries me the most; if the debt burden increased, it may cause more people may emigrate, so the debt burden increases on the remaining population. If this effect is strong enough, the debt burden per person may spiral out of control.

25(?) years ago, before Ireland became a Celtic Tiger, I confess I did think that Ireland might be doomed by its debt. It's easy for young Irish workers to emigrate. I was wrong then, because I did not foresee the economic reforms, and how successful they would be. But maybe it's time to worry about that again? (Or, maybe all the other places they could emigrate to are in a similar situation?)

Jon: as in the US, all Canadian levels of government borrow. I think the provincial debts are about as big as the Federal. My own municipality has debt equal to its annual tax revenue, but does have offsetting assets.

Luis: yes, I think we macroeconomists should pay more attention to demographics. Stephen did a post on the aging Canadian population about a month ago.

Ironman: I ignored dynamics, assuming everything is constant as a fraction of gdp per person. But yes, they are probably important.

Arturo and RSJ: I deliberately ignored what would happen if r is less than y, or y+g. Weird things happen in that case. Ponzi finance becomes sustainable indefinitely. Present Values become infinite. Some asset prices can become infinite. I did a post on infinite asset prices about a month ago. But many commenters assured me that r could never be less than y, so this couldn't happen!

Empirically it's a trickier matter. For some historical periods, r has certainly been less than y. But was that ex post, or ex ante? were the same measures of inflation used for real interest rates and for real GDP growth? (That might be a good reason to follow Scott Sumner's suggestion and compare nominal interest rates to nominal GDP growth instead). Even if it's true exante for some periods, could it continue indefinitely? Remember that in the past few decades the participation rate has increased as more women entered the labour force, and the baby boomers also caused the employment/population ratio to increase. Both these effects would increase the growth rate of GDP per total population. But they can't continue indefinitely into the future.

Does my assumption that r-y=2% looking forward make sense? Dunno. Maybe it's too high. But if it were smaller than 2%, then, as Scott says, this would only strengthen the effects of population growth. (And 2% keeps the arithmetic simple!)

Lord: even if all the debt is internal, the young and future generations will have to pay the taxes to the old generation that owns the debt. Will they continue to do so? I know there's the argument that "we owe it to ourselves". But if the owners of the government debt consider it an asset, and not something they must bequeath to the next generations, then the next generations must consider it a liability.

Alex: My guess is that it's difficult to value some of governments' assets. Provincial parks, for example. And what do you include or exclude? Plus, accountants have an awful tendency to value such assets at historical cost.

Too much Fed: It depends who pays the taxes to pay the burden of the debt. And who gets the benefits of government program spending. If both are proportional to income, then I don't think it affects my analysis much. If a greater percentage of the net burden falls on the rich (either because of progressive taxes, or because benefits don't increase proportionately to income), then the rich pay a higher proportionate burden of the debt, and the worry is that the rich will emigrate.

[RSJ]

Nick,

It is not "tricky" -- precision is tricky, but this is not a question of precision. And ex-post = ex-ante over the long run: from 1960-1980, we consistently underestimated future inflation, and long term rates were too low, but from 1980-2000 the reverse was true. Over the long run, these expectation errors cancel, and over that time period, the average/median return of the long bond was the same as the NGDP growth rate.

Here is long run historical data from measuringworth.com (a great resource for economic history).

The U.K. has good time series on long term government securities. From 1830-2009, the delta between (annual) long term rates and 10 year trailing NGDP rates was about zero (average = -0.23%, median = +0.14%). If you compare current long term rates to 10 year future NGDP growth, the result is also about zero (average = -.31%, median = 0.25%).

For the U.S., the time series uses New England municipal bonds and corporates, but this is good enough -- from 1800-2009, the average spread between the annual nominal LT rate and 10 year trailing NGDP CAGR was +0.05%, and the median spread was 0.26%. The spread between the LT rate and the 10 year leading NGDP was -.06% (average) and +.06% (nominal) -- also about zero, or within measurement error of the original indices.

http://2.bp.blogspot.com/_fevQMK7kLEI/S5h1c57wjfI/AAAAAAAAAJ8/mMIFy1OBXfA/s1600-h/UK_LT_NGDP.png

http://3.bp.blogspot.com/_fevQMK7kLEI/S5h1cfsftPI/AAAAAAAAAJ0/QUDJ0u0Z7cQ/s1600-h/US_LTR_NGDP.png

200 years of data should be enough convince you that this is not a "special case", but the expected return over long time periods. And where is the data of those who say that risk free rates will consistently be in excess of the economic growth rate? That is not an obscure or "tricky" prediction -- it should easily be verified or disproved.

"Weird things happen in that case. Ponzi finance becomes sustainable indefinitely. Present Values become infinite. Some asset prices can become infinite. "

Asset prices shooting to infinity would require earning yields of *zero*. As long as there is positive long run growth, then setting the earnings yields to converge to this growth does not result in infinite asset prices. But if there is no positive future growth, then there is no long-term return, and no reason to invest. Only in the zero growth case do wierd things happen.

And because the tax revenue is tied to NGDP, government is *not* a ponzi scheme and is able to offset interest payments with tax revenue, as the latter grows at a faster rate. What you are citing as the typical case -- *that* would be the ponzi scheme case, in which debt service exceeds income.

Paul Krugman's recently showed a nice time series for UK debt/GDP ratios, and how they were brought down several times -- primarily without running surpluses, and this is why it is possible. As soon as you cap the Deficit/GDP ratio, the Debt/GDP ratio starts coming down as well.

[RSJ]

and re: the rich emigrating, that is not a worry, as they will need to sell the currency of their native country, exchanging it for the new currency. Their currency will stay in the country, even they themselves do not, and it will be taxed at whatever rate is appropriate for the buyer of that currency.

We do not live in a gold-standard world where people can move money around across borders without crossing forex markets first.

[Nick Rowe]

RSJ: suppose you ran a chain letter in which every year you give x% of per capita income to the person before you in the chain, and sent out 100+g letters for every 100 you received. The rate of return on the chain letter would be y+g, which would exceed r. Yet the chain letter is stable, and can continue indefinitely. Ponzi schemes work.

Suppose you had an asset in fixed supply (oceanfront land) that earned income proportional to GDP. The dividends per share (per acre) would grow at y+g. Infinite present value.

On emigration: yes for immobile assets. But human capital is the problem. That's why they built the Berlin Wall.

[RSJ]

Nick,

Before engaging further -- let me first find out. Are you:

1) Agreeing that historical long term interest rates are about the GDP growth rate, but you believe this causes some theoretical problems and so you want to hash those out (e.g. "Theory says X, and data says Y, therefore let's fix X")

or

2) Trying to raise theoretical issues in an attempt to avoid confronting the data (e.g. "Theory says X therefore Y must not be Y")

Honest question.

[Nick Rowe]

RSJ: I was asking myself the exact same question, driving into work this morning!

A bit of both.

1. I wanted to do a simple post on population and debt, avoiding tricky theoretical issues.

2. I've been "burned" on a very closely-related question, on my "infinite asset prices" post. I couldn't convince people that it was theoretically possible.

3. But ultimately I want to fix the theory. (And get my head around the whole question).

[RSJ]

OK, I just want to distinguish the issue of what real government interest costs are -- which to me is purely an econometric question -- with the model-theoretic issues you raise. There are real public policy affects of *assuming* that government interest costs will be greater than the long run growth rate of government tax revenue, and seeing how there is wealth of data on this point, these debates should not be determined by coastal real-estate property valuations.

[Nick Rowe]

RSJ: "There are real public policy affects of *assuming* that government interest costs will be greater than the long run growth rate of government tax revenue,..."

Yes. I'm gathering up courage to try to think about tackling them. But one of the side-effects is that an economy with r less than the growth rate "needs" ponzi-finance/bubbles, to function efficiently.

If you have any more thoughts on this issue, I would be interested to hear them.

[Mickey D]

I would like to see some data on total debt of various countries, taking into account federal, state (prov.) and local debt levels. Any idea about a source for this?

[RSJ]

Hi Nick,

I will make some more comments about this later, but here are my four cents.

1. "But one of the side-effects is that an economy with r less than the growth rate "needs" ponzi-finance/bubbles, to function efficiently." ---> No, an economy with incomes that grow at r, "needs" the long run rate of return to also be r. If it were less, then there would be free money in the form of borrowing to buy capital. If it were greater, then there would be free money in the form of shorting capital (e.g. borrowing to buy capital-making capital). So as soon as capital is endogenous, then arbitrage opportunities drive the long run rate of profit to the long run growth rate of revenue. If you are in a pre-industrial economy in which capital is not endogenous, or there are other frictions that prevent arbitrage, then it's a different situation.

2. re: beach-front land.

The prices would not be infinite if the income growth rate = discount rate. There is always a trade-off between location and structure, and as long as MRS(L,S) is finite, then land is not infinitely priced. To see what is going on here, look at a simple utility function for the potential renter.

u(L,S) = L^(1-a)*S^a.

This is a model of monopolistic competition in which landlords, each with an endowment of L compete for tenants by adding structure, increasing S until MR = MC. They will not be able to raise the rent without making improvements -- how many improvements? If Incomes grow at a rate r, u(L,S) will grow at a rate r, which means S^a will grow at a rate r, which means the structure will grow at a rate r/a > r.

Then profits = incomes - improvements will grow at a slower rate than r, so the discounted sum of cash-flows will be finite.

In general, over the long run, you have revenues growing at r, but those revenues require re-investment (even net of depreciation), and the rate of that re-investment is increasing due to diminishing returns. As a result, it is never the case that any asset can be infinitely priced, as the cash-flows of each particular investment ( = revenues - investments) will not grow at the same rate as revenue over the long run. By the way, this is why taxation conquers all. It is impossible to purchase any asset whose return will be greater, over the long run, than the right to tax all incomes at a fixed rate. Therefore it is impossible for governments to be unable to pay their debts simply by capping deficits.

3. Re: Gordon model --> this model is applicable only in the case that cash-flows grow at a fixed rate, in which case:

Dividend/(discount rate - growth rate of Dividend) = NPV.

What this is telling you is that if the discount rate is 3%, then you are agnostic between receiving $.03 forever, or $.02 now that grows at 1% or $.01 now that grows at 2%.

They all have a NPV of $1.

In the limiting case that growth rate --> discount rate, the Dividend --> 0. This does not mean that NPV --> infinity, but that it is of the indeterminate form 0*infinity. The source of the indeterminacy is that you are trying to model an asset without any cash-flows. E.g. a painting or a statue, or collectible -- these will grow with GDP, but pay no cash-flows. As soon as you start earning a cash-flow, then you are using capital with diminishing returns and these force your NPV to be finite. The value of land is the residual of revenue -- construction costs, and measures the monopoly markup of the location. In this way, the price of land is determined by the rate of return of the productive economy (ignoring bank lending bubbles).

4. Re: chain-mail ponzi scheme -- I'm not sure what you are trying to show here. Is it

1) That chain-mail companies obtain credit at the risk-free rate?
2) If 1) holds, and if also everyone in the economy acts irrationally, then ponzi schemes can keep going?

Madoff kept his ponzi scheme going for several decades, but just because something is mathematically possible does not mean that it is mathematically possible over the long run in a competitive model.