Here is a response emailed to me from Nick Rowe:
I’m crap at math, and don’t really understand transversality conditions either. So take this with a grain of salt.
L’Hopital’s Theorem is useful here. You are trying to find what happens to a ratio, in the limit, when both top and bottom approach infinity. L’Hopital says take the derivative of both top and bottom wrt time, and the limit of the new ratio is the same as the limit of the original ratio. If bottom grows faster than top, the limit is zero, and infinite if vice versa. (Very loosely speaking).
Start with a very different simple model. An overlapping generations model where people live 2 periods, and the old sell their government bonds to the young. That model has a natural limit to the debt/GDP ratio. The debt cannot be bigger than the youngs’ income, otherwise they will be unable to buy the debt.
Can the government in that model rollover the debt forever, borrowing to pay the interest, so the debt grows at rate r? That depends:
If r > g (the growth rate of GDP) then no. Because the young would eventually be unable to buy it. So the top of that second term must grow more slowly than the bottom in the limit, so that second term becomes zero, so the debt must equal the Present Value of future budget surpluses.
If r < g then yes. It’s sustainable forever. And money is like that, because currency (for example) pays minus 2% real interest (if the inflation target is 2%). A government that has currency as its only liability can run deficits forever (printing more currency to finance spending) because currency is so liquid people are prepared to hold it (in limited amounts) even at a negative real interest rate.
Now change the model, to one with infinitely-lived agents. It’s not obvious whether there is any upper limit on the debt/GDP ratio. Under Ricardian Equivalence, there needn’t be. And with a single agent (like in your example) there won’t be, because that agent as taxpayer can easily owe himself as bondholder a zillion dollars, and it’s a wash.
Coupla other points:
1. None of this is about *subjective* (time-preference) interest rates. We are talking market interest rates. Preferences matter only insofar as they affect what those market interest rates will be, at which people are willing to hold the stock of debt.
2. In a finite horizon model, where people know in advance that the world will end at time T, it’s different. The government must pay off the debt at time T-1, or nobody will buy it at T-2, etc., and the whole thing unravels backwards. So that second term must be zero.
3. A model where the world has a d% probability of ending each period is more like the infinite world. The only difference is that the market interest rate will now be higher (because individuals now subjectively discount future consumption for 2 reasons).
Dunno if this helps.