One last kick at this can.
"OK Nick. Your previous counterexamples have shown that debt can be a burden on future cohorts even if future national income is not affected. But that's not what we meant by 'future burden of the debt'. Those counterexamples don't show the country as a whole being worse off in future years. Obviously it isn't, because total consumption in any year is held constant in your examples"
I made up that quote, but I don't think it's a straw man.
Here's another counterexample, in which total consumption in any year is held constant, but there is still a burden on the country as a whole in those future years.
My counterexample is obvious really. What matters for economic welfare in any given year is not consumption in that year but the utility people get in that year from consumption in that year. If debt causes total utility to go down, even if total consumption stays the same, it seems reasonable to say there is a burden in that year. But why should debt cause total utility to go down, if it doesn't reduce total consumption? Because it may create or increase inequality of (marginal utility of) consumption between young and old. (Or it may reduce it, it depends).
Standard OverLapping Generations model. Lifetime utility function is:
U = Log(consumption when young) + Log(consumption when old)
Note that I have ignored time preference proper (discounting of future utility) to keep it simple.
No investment, no foreigners, no population growth, and no productivity growth. In the initial equilibrium, each person produces and consumes 100 apples when young and 100 apples when old. The rate of interest is 0%. There are one million people per cohort. All are identical.
Now suppose the government issues bonds and gives each person in the first cohort a bond worth B apples. In all subsequent periods the government taxes each person just enough to pay the interest on the debt, so the debt stays constant at B million.
Each young person therefore consumes 100-B when young and 100+B when old. (He saves to buy the bond when young, sells the bond when old, and collects interest, but also has to pay taxes equal to the interest costs per person.) The rate of interest will be determined by: (1+r) = (Marginal Utility of Consumption when young/Marginal Utility of Consumption when old) = (100+B)/(100-B). So the rate of interest will be positive, and taxes will be positive.
Total consumption in any period will be consumption of the young in one cohort plus consumption of the old in the previous cohort and will equal 200 million. The total utility from consumption in any given period will be the utility of the young in one cohort plus the utility of the old in the previous cohort.
So in any given period, total utility = Log(100-B) + Log(100+B) million utils. This is decreasing in B. The debt lowers total utility in any given period, even if it does not affect total consumption in any given period.
The important assumption in my example is that the rate of interest exceeds the growth rate (not originally, but it does when there is a debt). It would be possible to construct other examples in which the equilibrium rate of interest is less than the growth rate, and so debt increases total utility in any period.
The intuition is standard Utilitarianism. If debt creates or increases inequality of the marginal utility of consumption between young and old (which it does in my example) then debt reduces total utility. If debt reduces inequality of marginal utility of consumption between young and old (which it could if the rate of interest were less than the growth rate) then debt increases total utility.