Commenting 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.