**Summary:**

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

**Topics:**

Nick Rowe considers the following as important: finance, Fiscal Policy, Macro, Nick Rowe

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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