Hilbert's Hotel and the National Debt when r < g

Hilbert's Hotel has infinitely many rooms. Even if every room is full, you can still make room for one more guest in room 1, by moving the guest currently in room 1 to room 2, moving the guest currently in room 2 to room 3, and so on forever.

When the rate of interest r is less than the growth rate of GDP g ("r < g"), the national debt in an infinitely-lived economy is a bit like Hilbert's Hotel. The government prints some bonds, which it gives (as a freebie) to generation 1 (this is like giving generation 1 a deficit-financed transfer payment, or tax cut). When they get old, generation 1 sells those bonds to the young in generation 2, in exchange for apples. So generation 1 eats more apples when old, and generation 2 eats fewer apples when young. When they get old, generation 2 sells those bonds to the young in generation 3. So generation 2 eats more apples when old, which more than compensates them for eating fewer apples when young (if it didn't more than compensate them, they wouldn't have been willing to buy the bonds at the prevailing market rate of interest). And so on forever.

If r < g, the government never needs to raise taxes to service the debt. It can just rollover the debt + interest, so the debt grows at rate r, while the economy grows at rate g, and if r < g the debt/GDP ratio is falling over time, so this is sustainable. Generation 1 is much better off (it gets free apples), and all subsequent generations are at least a little bit better off (they have a way of saving for their retirement that wasn't available before, and they chose it). A national debt makes all generations better off. There is no burden on any future generation.

But it won't work if the economy has a finite life; just like Hilbert's Hotel won't work if it has a finite number of rooms. The last generation is stuck with bonds it can't sell, and no compensation for consuming fewer apples when old young. So they wouldn't buy the bonds, and the whole thing unravels backwards. Unless the government raises taxes to retire the debt before the economy ends, but then those taxes would make the generations that pay them worse off.

Can we safely assume that the Canadian economy is infinitely-lived? No. But how much does this matter, if the Canadian economy will likely last for a very long time?

One way to tackle this question, and the one I'm going to use here (though there may be other better ways), is to start out with a finite-horizon economy, that survives for H periods, then ask what happens as H approaches infinity. Does the burden of the debt on the final generation approach zero as H approaches infinity? If so, this is not a problem we should worry about much.

I'm going to look at two examples. Both those examples will assume r=0 (to keep it simple for me, but not totally unrealistic at the moment, if we interpret r as the real (inflation-adjusted) interest rate, and g as the real growth rate).

In my first example, population is assumed constant, and all the GDP growth is coming from productivity growth, so GDP per capita grows at rate g. In this case I think the burden of the debt on future generations can be ignored. Even though the debt stays constant over time measured in apples (since r=0), future generations get richer and richer, so if we assume diminishing Marginal Utility of consumption, the burden of the debt on the final generation, measured in disutility, gets smaller and smaller, presumably approaching zero, as the horizon H approaches infinity. (Strictly, I need more than diminishing MU for this result; MU must approach zero in the limit as consumption approaches infinity, but I'm OK with that.) So Hilbert's Hotel still works in the limit (it's like the rooms get bigger and bigger as the numbers go up, so it's no big deal to squeeze two guests into the last room).

In my second example, per capita GDP is assumed constant, so all the GDP growth is coming from population growth. In this case I think the burden of the debt on the final generation cannot be ignored. Even though the per capita burden (whether measured in apples or disutility) shrinks as the population grows, we are multiplying that per capita burden by a growing population, so the total burden on the final generation does not approach zero as the horizon H approaches infinity.

Based on these two examples, my guess is that we should replace the "r < g" condition with an "r < per capita g" condition, if we are looking for the condition under which we can talk about sustainable debt-finance and reasonably ignore any burden on future generations. (We've also gotta be careful to distinguish between the average burden of the existing debt vs the marginal burden of extra debt, since extra debt could be a Bad Thing even if the existing level of debt is a Good Thing, though that is a topic for another post.)

[I thank someone who's name I have forgotten, but I think it was a commenter on an old post, for suggesting the relevance of Hilbert's Hotel, which I only DuckDuckGo'd last night. And I thank Ivan Werning and Foggy Anabasis for their replies on my initial Twitter musings on this topic. All without implication of course.

And yes, I am out of my math depth (or beyond my limits) on this question.

And though it sounds utterly arcane, it does seem to be policy-relevant, given current low interest rates.

And BTW, please leave a space either side of < and > in comments, because Typepad sometimes goes html nuts if you don't.]

Comments

[dlr]

i must be missing something obvious bc this seems wrong to me? in case 2, total debt stays flat while total gdp increases to infinity. or, debt per capita decreases toward zero while gdp per capita stays flat. both perspectives have the debt (total or per capita) approaching zero as H approaches infinity. it seems like you are trying to compare total debt (flat) with per capita gdp (flat), which not a ratio we care about.

start out with 2 people.
gdp per capita is 100.
debt is 100, or 50 pct of total gdp.
debt per capita is 50.
time passes.
population hits 2 million.
debt is still 100, but now .005 pct of gdp.
debt per capita is .00005
not approaching zero?

[Nick Rowe]

dlr: From a Utilitarian perspective, a loss of 1 util for 100 people is the same as a loss of 100 utils for 1 person.
Utils lost per person in the second example approach zero, but total utils lost don't approach zero.

[rsj]

Yes, in a finitely lived model, debt will be a transfer from the last generation to the first. However, the last generation will be richer if g > r, and so the burden will be smaller for them than if the first generation paid off the debt without accruing any interest.

Thus it's still a welfare maximizing shift. No, the the rich may not vote to accept the shift, but if we posit that only pareto optimizing shifts are allowed, then everything from unemployement insurance to a progressive income tax has to go.

My question is then why would one support a progressive income tax in a single period but not across multiple periods?

[Nick Rowe]

rsj: "My question is then why would one support a progressive income tax in a single period but not across multiple periods?"
Fair point.

[SV]

Isn’t this similar to the discussion on pay as you go vs fully funded social security?
http://www.econ.nyu.edu/user/violante/NYUTeaching/AMF/Spring13/lecture9_13.pdf

[Nick Rowe]

SV: yes. At the macro level (i.e. ignoring intragenerational individual differences) the national debt is equivalent to a PAYGO pension plan.

[Roger Sparks]

"The government prints some bonds, which it gives (as a freebie) to generation 1....."

If government does this once, why not again for generation 2, 3, etc?

Would it be more realistic to assume that, for each generation, bonds are given in exchange for resources and hours of labor?

[Hugo Andre]

Nice thought experiment! But although I agree most people would say that Canada will continue existing for a long time,
the same can't be said for r (less than) g. It could be that r (less than) g continues to hold for another 20 or 50 years but I think few people would confidently predict that relationship to hold for another 100 years or more.

Needless to say, if the relation reverses and r>g 50 years from now, the extra debt will be a burden for future generations. This is a reason to be cautious about taking on too much extra government debt, I think.

(for some reason, Typepad removes the last part of my sentences when I use the less than symbol.)

[Nick Rowe]

Roger: "Would it be more realistic to assume that, for each generation, bonds are given in exchange for resources and hours of labor?"

Maybe. But even here, an interesting counterfactual is that those same expenditures would have been tax-financed instead. And the difference between that counterfactual, and what the govt actually did, is a bond-financed tax cut, which is equivalent to a bond-financed transfer payment, which is equivalent to helicopter bonds.

"If government does this once, why not again for generation 2, 3, etc?"

If it does it too much, r rises above g, which makes things very different.

Hugo Andre: Thanks.

Yes, r < g might not last forever, even if Canada does. And that's possibly a more realistic scenario. But I think (I'm not sure about this) what I say above would still be (roughly) applicable.

Yep, Typepad thinks < and > are html instructions (or something), and has a funny turn. But if you leave a space either side, it seems to work OK.

[Roger Sparks]

Nick: "Maybe. But even here, an interesting counterfactual is that those same expenditures would have been tax-financed instead."

Would it be factual if we said it this way: When government expenditures are financed with both taxes and bonds, everyone getting funds directly from government can be considered as receiving more than they would get if government spending was limited to taxes collected?

If this is factual, we can find a hierarchy of beneficiaries: First are direct government payees. Next are those suppliers who benefit from government payee spending. Last would be savers and pension funds who delay re-spending until finding a need to spend.

This last group would have the opportunity to lend to government if government had a continuing need to borrow.

[Lord]

There would likely be an interaction between population growth and gdp per capita though, with population falling to maintain growth in gdp per capita.

[Lateral Mimas]

Some economy longevity detail: More technology suggests more shocks until implants and gene therapy are mature. Abstract or divergent thinking (neuro-imageable) helps predict simpler events. Lateral thinking (two or more brain memory areas used to solve one technology problem) is needed. An economy where the leaders/actuaries/NASA have only abstract thinking will not last as long as one where they have lateral thinking. For some peoples, just science is needed to learn pragmatism. Abstract thinking is good to learn them utilitarianism, but lateral thinking is a weapon they shouldn't have. Now, the risk of governments making robots armies is assessed, but not the risk of a single robot or a flukey nearer term AI reasoning train. We attack rent seekers and religious leaders as uncertain risks eventually. War time Bonds should be discussed here assuming even the near future.

[Mike Smitka]

Thinking of East Asia, where demographics imply a declining population (already occurring in Japan, starting soon in China and Korea), then you ought to (must!) use the working age population, which falls much faster given the dynamics generated by the sharp drop in fertility in the late 1960s and 1970s. You might also find this paper sobering: Hoshi, Takeo, and Takatoshi Ito. 2014. “Defying Gravity: Can Japanese Sovereign Debt Continue to Increase without a Crisis?” Economic Policy 29 (77): 5–44. Their denominator isn't GDP but total financial assets. Their simulations, given Japan's large debt stock, large deficit and declining population, are that debt will surpass total domestic financial assets. They have a hard time envisioning quite what might happen. But their crossing point, robust across parameter choices, is in the mid-2020s. Absent a large fiscal adjustment (my own long-ago calculation was 10% of GDP), they don't see how today's low interest rates can be adjusted. The political calculations over increasing taxes versus kicking the can down the road to yet-to-be-elected legislators continues to be "kick" – the bump in the consumption tax from 8% to 10% implemented this month is too small, and too many times delayed.

[Henry Rech]

"Even if every room is full, you can still make room for one more guest in room 1...."

Doesn't make sense - you have specified that every room is full???

[Ken Houghton]

Does assuming the bonds are either perpetuals or PIKs make a difference in either (especially the second) scenario?

Also, given you appear to be assuming n is growing in the second scenario, are shares of the bonds being reduced proportionately? (In simple terms, if Couple A raises two working children and Couple B raises four, do A1 and A2 have an IGT twice as large as each of B1-B4?)