Infinite Debt, Future Generations, and Hilbert’s Hotel

hilbert_hotelCommenting on my last post, Nick Rowe helpfully explained his argument that governments, by running deficits, can ‘steal from the future to give to the present’. I now see his point. Simon Wren-Lewis gives a similar explanation.

Wren-Lewis agrees that debt wouldn’t amount to ‘stealing from the future’ if the government never retired its debt. If it just kept rolling over the debt forever, there’d be no day of reckoning on which one generation had to pay for all the previous generations. But he thinks that government debt is always unsustainable in the long-run, since the average interest rate on it exceeds the average rate of GDP growth. This ignores the fact that the interest rate on Treasury debt is a policy variable, which I’ve talked about before; here I want to get onto something more mathematically interesting.

Rowe presents an interesting sort of Hilbert’s Hotel scenario to show that debt can be used to create intergenerational inequalities even when borrowing goes on forever. Imagine an infinite line of people, each holding one beer. Pick a person in the line (call her 1). Take a beer from the person behind her (call him 2) and give it to 1. Now take a beer from 3 and give it to 2, to replace what you took from him to give to 1. Take a beer from 4, give it to 3, and so on to infinity. You’ll end up with a situation in which 1 has two beers and everybody after 1 only has one beer. Unfair!

Well, replace ‘take’ with ‘borrow’, imagine that the line of people extends through time rather than space, and you’ve got a situation in which infinite borrowing creates intergenerational inequality.

This is perfectly right, but it doesn’t do the form of Hilbert’s Hotel justice. In the scenario Rowe describes, you could also make sure that everybody in the line had one more beer than she started off with. Take a beer from 2 and give it to 1: now 1 has two beers. Take a beer from 3, another beer from 4, and give both of these to 2. Now 2 has two beers. Take a beer from 5, another beer from 6, and give these to 3. Now 3 has two beers. Take a beer from 7, another beer from 8… in general, you can always give two beers to n by taking one from 2n-1 and another from 2n. Everybody ends up with two beers. So it seems that while infinite borrowing can create intergenerational inequality, it can also make everybody better off.

We can go further, and use infinite borrowing to give everybody an infinite number of beers. For each person n, take one beer from each of n^p, where p is any prime number larger than 2.

More generally, the number of proper subsets possessing the same cardinality as the set of all people in the line is the same as the number of people in the line! This is a surprising mathematical result, but it’s easy enough to prove. Thus it is possible to perform, for each member of the line, a borrowing operation that is equivalent to borrowing a beer from everybody else in the line and giving all these beers to that one member, without ever running out of people to borrow from.

So infinite borrowing can make every generation infinitely rich, without taking from anybody! Can it really be? Of course not. But we need to be clear on what is possible, and Rowe’s little Hilbertian exercise shows us this beautifully.

It shows us how, in the case where the government never pays back its debt, it can run a deficit of any size it likes – up to infinity. With beers, of course, you can’t do this, because you’d have to get the beers from future people before giving them to present people. But you could issue an infinite number of IOUs promising beers (‘to be funded by future revenues‘). The person who received those IOUs couldn’t use them to buy an infinite number of beers, obviously. But she could spend them buying, from the next person in the line, as many beers as he was capable of producing. The next person could then buy from the person after him as many beers as she was capable of producing. And so on. Everybody gets to have as many beers as the next person is capable of producing. That’s a lot of beer, and beer = happy.

Of course the exchange rate would be extreme: an infinite number of ‘beer credits’ would be paid in each case for a finite number of beers. So the purchasing power of each credit would be infinitesimal. Let’s stop reasoning about infinity; it almost always goes wrong. The government can achieve the beneficial result much more easily. Suspecting that person 2 in the queue can produce X more beers than person 1 is willing to buy, it issues person 1 with an IOU worth X beers to make up the gap. Person 1 is now capable of buying everything person 2 can produce while spending no more of her own resources than she is willing to spend. Now say the government suspects that person 2 is willing to buy more beers than person 3 is capable of producing. This could drive up the price of beer – the greatest of horrors. So the government should tax back some of person 2’s IOUs, to keep the price stable. Alternatively, if person 2 isn’t able to buy everything person 3 can produce, even with his IOU, then the government should issue another IOU to fill the gap.

This is just functional finance. The government lets its volume of outstanding IOUs grow and shrink as necessary to make sure that everything that can get produced does get produced (and sold) and the price-level doesn’t move. It maximises production subject to the inflationary constraint.

Functional finance depends on recognising that the government can issue as many or as few IOUs as it likes, so long as its debt is infinitely sustainable. That was my point, and I thank Nick Rowe for providing the Hilbertian example that helps to explain why it’s so. All it takes for the government’s debt to be sustainable is for the central bank to ensure that the rate on government debt stays less than or equal to the rate of growth.

‘But the central bank doesn’t make its rate adjustments for the government’s benefit! It raises rates when it wants to control inflationary pressures!’ Yeah. But if you had functional finance, it wouldn’t need to do that. And interest rates are turning out to be a pretty crook policy tool. And giving central banks power over the economy is a big hit to democracy. Functional finance is better on every measure. Again, though, that’s a separate debate.


19 thoughts on “Infinite Debt, Future Generations, and Hilbert’s Hotel

  1. NeilW

    It’s always important to remember that not everybody spends an IOU in a time period. I might want my beer on Friday, and it is very likely I’ll be able to get it because somebody else will want their beer on Tuesday when I don’t want it.

    This is the mathematical calculation that is used by electricians and telecoms engineers to size wires. Electricians call it a ‘diversity calculation’ and in telecoms it is a calculation involving an ‘Erlang number’.

    Both of which are mechanisms for working out the actual level of activity you can actually maintain on a physical structure. I’ve no doubt there are similar calculations in traffic management and Post Office queue design.

    What I’ve found interesting though is there is no such thing in economics.

    1. axdouglas Post author

      That’s very interesting. I guess that sort of analysis would be a bit like the microeconomic version of Godley’s dynamic path analysis.

      I can see why it wouldn’t exist in mainstream micro: when all you’re doing is solving for equilibrium (or stringing equilibriums together into a ersatz ‘dynamic’ model), then the calculation you describe is of no relevance.

  2. Nick Rowe

    Thanks for looking at this Alex.

    Is if “functional finance”? Or is it “monetary policy”? What is the bit of paper that gets passed back down the line in exchange for the beers that get passed up the line? Is it a bond, or is it money?

    If we define “money” as the “medium of exchange”, the one get that gets bought and sold when every other good gets bought and sold, then maybe the bits of paper in these models is money. (It depends on whether you interpret the “generations” literally or metaphorically: if it’s literally the paper is bonds, but if it’s metaphorically it’s money.

    Here’s another tiny toy model, with only one time period (so there are no bonds in the model, because bonds are IOUs for future periods), where a shortage of money causes recessions:

    1. axdouglas Post author

      It’s fiscal rather than monetary policy, because you’re giving people financial assets (or taking them) rather than swapping one asset for another. With monetary policy you swap white pieces of paper for green pieces of paper or vice-versa; you don’t give out new paper. Monetary policy doesn’t change anyone’s net worth. Thanks to @pio_trek_g on Twitter for reminding me of this.

      Notice also that if you define ‘money’ as ‘the medium of exchange’ then bonds aren’t excluded from the definition. You give me corn and I promise to give you sugar tomorrow. My promise – my bond – is the medium of exchange of sugar for corn. If you’re going to ask ‘are these money or bonds’, you need definitions of each that are mutually exclusive; otherwise the answer will just be ‘yes’.

  3. Scott Fullwiler

    Nice post, Alex!

    First, at least in the US, the interest rate on the national debt has been LESS than the growth of GDP since WWII. So, Wren-Lewis’s point–which is a standard argument in neoclassical macro–is wrong at least if the past 65+ years are a relevant time frame. Since that’s basically the time frame during which the US has had sovereign control over its own interest rates on the national debt (actually, a bit longer, into the 1930s), this would in fact seem to be the precisely correct time frame in the US case.

    Second, the issue discussed in the post is really about “taking” the real goods and services from one generation and giving it to another, not the “money.” So, one generation has to pay interest on the debt, or whatever other detrimental result of a deficit one wants to imagine, and thus the next generation can’t spend on real goods and services. But in the real world, the only applicable analogy is if one period’s deficit somehow–through debt service or whatever–forces a future generation to be poorer by having fewer real goods and services per capita. It’s quite obvious that this mostly means, for a sovereign currency issuer anyway, that a deficit was spent on “stupid stuff” or otherwise reduced the productive capacity of the macroeconomy in the future. It has little to do with the size of the deficit or debt, per se. In fact, it’s quite clear that the opposite has happened in Greece–the lack of ability to run deficits is ruining productive capacity and depreciating labor’s skills to the detriment of future generations.

    As a bit of an aside, I think the main applicability of the sort of “burden” on next generation stuff is to the “natural capital” of a nation–it quite obviously does not apply to the financial capital of a sovereign nation, and depending on the scenario, perhaps not even to the traditional fixed and labor capitals.

    1. axdouglas Post author

      Thanks, Scott.

      Good point about Wren-Lewis’ claim that ‘normally’ r>g. I just don’t know what ‘normally’ means.

      I argued with Nick quite a bit on my last post about whether it’s right to go from talking about taking stuff from one generation and giving it to another to talking about doing this with money. This was the swipe I made in my last post, which prompted Nick to reply. The idea, as he explained, is that since you can use money to redistribute real stuff among the people currently alive, if you repeatedly do this for the benefit of the old and at the expense of the young, and then suddenly stop, you could do something equivalent to ‘stealing from the future’. I still don’t quite agree, but it’s a tricky point, and I wanted to get on to talking about philosophy of maths. But this is one way it could work:

      Suppose we’re producing at capacity, and there are always only two overlapping generations. While A (old) and B (young) are alive, you issue money to A and somehow force B to save an equivalent amount – otherwise there’ll be inflation, since we’re at capacity. When B and C are alive, B spends the savings and you force C to save to avoid inflation. When C and D are alive, instead of letting C spend the savings, you tax them away, then D buys everything. So C has lost out and A has net gained.

      I observe that there are a lot of weird things about this. First, it’s not about debt at all. You could achieve the same effect by taxing B to pay A, then taxing C to pay B (which Wren-Lewis admits in one of those posts). It also doesn’t really amount to ‘stealing from the future’ in real terms, since somebody in the future must still get to consume everything that is produced. Finally, of course, how does the government, in real life, force savings? It can sell bonds, but there are plenty of liquidity options for a bond. Even if nobody wants to save, everybody can just pledge bonds as collateral to banks and get deposits.

      Of course we can and do depreciate natural capital and make future generations worse off (particularly if you think that a world at its current global mean temperature, with not too many extreme weather events, counts as natural capital). But to make the argument that that is ‘stealing from the future’ you need some normative measure of how many natural resources each future generation is entitled to use. That’s a different and tricky question over which philosophers have puzzled.

      1. NeilW

        “Finally, of course, how does the government, in real life, force savings? ”

        Forced saving is Tax. It’s just a hypothecated tax.

        Compulsory pension contributions are forced saving. Those have been introduced specifically so that those who have previously saved for a pension can cash out without a ‘loss’. Because pensions are always a current period issue and the saving system doesn’t work in the long run.

        All of these schemes boil down to paying a pension to people and taxing the current generation to cover that pension (again assuming other mechanisms are maintaining output at full) – which is simple redistribution.

        Essentially you tend not to be able to save real things over time, and the rate of interest paid to excess savers is a policy variable. Set it to zero, adjust taxes to make the auto-stabilisiers take up the slack, and policy limit the types of bank lending. Problem solved.

        All of the issues are down to believing monetary policy and fiscal policy are somehow separate and deal with different things. They are not and don’t. Time to get used to that idea. Occams Razor requires it.

      2. axdouglas Post author

        ‘Forced saving is Tax’ – yes, but then this ‘toy model’ isn’t about deficit spending at all, which is what it’s meant to be informing us about.

        So you start off assuming a balanced budget, then use the model to explain unbalanced budgets.

        A friend of mine said: ‘If economists did Hilbert’s Hotel, it would go like this: “Assume that the number of rooms is Aleph null. This is a reasonable assumption.”‘

      3. axdouglas Post author

        Alternatively, the argument is just invalid:

        1. Assume we’re at full capacity.
        2. So there is no deficit spending, since all spending is paid for by taxation or inflated away.
        3. Therefore deficit spending steals from the future.

        The conclusion doesn’t follow and is in fact inconsistent with the second premise.

      4. Scott Fullwiler

        Thanks Alex
        Nicks example then had nothing to do with deficits per see then as I notes. Redistribution of income or stuff within a period need have nothing to do with deficits. Neither does stopping the redistribution at some point. You dont need deficits to talk about any of that. And as Neil points out you can’t redistribute stuff across generations. As Randy Wray wrote years ago you cant bury Winnebagos. You can only leave future generations with more or less ability to create their own stuff.

      5. axdouglas Post author

        Yes, I agree. And as I said in my reply to Neil, a hidden premise in Nick’s argument is that there are no deficits: extra spending by the government is matched by forced saving of some kind, which is just tax.

        But my point went beyond this. I find it interesting that if you apply the Hilbert’s Hotel reasoning all the way to its logical conclusion, you get to functional finance: it is possible to use deficit spending to make sure that every generation is able to purchase everything that can be produced in its lifetime. This is the real-world approximation to exploiting the strange dynamics of Hilbert’s Hotel to give everybody an infinite number of beers. What HH tells us is that there can be the same number of As as Bs, even though every A is a B and not every B is an A. This is a surprising fact, and anybody using a HH construction ought to mention it. Of course it isn’t relevant to transferring beers through time, which you can’t do. But it is relevant to transferring financial assets through time, which you can.

        When you’re dealing with an infinite set of people (extended through space or time), what you can do is debit a certain number of people, credit the same number of people, and thereby put the whole set into net credit. How so? The number you’ve debited is the same as the number you’ve credited. But HH shows that you can do it so that while every person debited is also credited, not everybody credited is debited. Say you debit all the even numbers and credit all the natural numbers. The set as a whole is a net creditor, because everyone debited is credited the same amount, but not everyone credited is debited the same amount. And all without you being insolvent, since the number you’ve credited is the same number as you’ve debited! Weird, right? But that’s just a thing about infinite sets.

        Of course the government’s lifespan isn’t actually infinite. But if it is overthrown, debt repayment will be the least of its worries. So ‘infinite’ here can work as a proxy for ‘indefinite’ when working out the logic of the argument. It’s another way of looking at how the government doesn’t face a solvency constraint.

        This is mostly a nerdy logical concern rather than a concrete policy debate. But there are people like you for the concrete policy stuff. This was just a bit of fun. Fun in Hilbert’s Hotel, where three’s never a crowd and nor is any other countable set. Maybe I’m not using ‘fun’ properly.

  4. Min

    As I have argued in comments on “Worthwhile Canadian Initiative”, I do not think that it is scientifically useful to argue from infinity instead of from infinite limits of finite models. Let me make a small point that applies to both Hilbert’s Hotel and Rowe’s Bar. A guest comes to check in to Hilbert’s Hotel and, despite the fact that the hotel is full, a room is found for the new guest. Finding the room is not welfare enhancing. Unlike with finite hotels, a full hotel does not mean that there is no room at the inn. Infinity plus one equals infinity, it is not greater than infinity. Inequalities do not apply as they do in finite cases.

    In Rowe’s Bar a second beer is found for the first person in line, but that does not make that person better off than anybody else. As you show, an infinity of beers can be found for each person in line. Finite inequalities do not apply.

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