For wonks. And for Andy Harless, who encouraged me to do something like this.
Yep, there's still a burden on future generations, even with unemployment [update: if future taxes are increased as a result of current fiscal deficits].
Assume people live 2 periods. Lifetime utility is:
U = Log(consumption when young) + [(1/(1+n)]Log(consumption when old).
The personal Euler equation is:
(consumption when old/consumption when young)=(1+r)/(1+n). Where r is the real rate of interest.
Taking logs of both sides, we can re-write that as:
Utility when old - Utility when young = r-n. (IIRC, that is only approximately true, for small r and n, but it's good enough for me.)
Assume each person can produce 100 when young and 100 when old. Assume zero population growth, no government, no investment, closed economy. (I will relax some of these later).
Equilibrium 1: r=n; each person consumes 100 when young and 100 when old. Boring.
Now introduce a central bank that sets r. Assume that in period 1 the central bank screws things up and sets r above n. The result is a recession. (Like all New Keynesians, I am implicitly assuming that there is a tabu against consuming the goods you produce yourself, and a second tabu against barter exchange.) Assume that the recession lasts for one period only, because everyone knows the central bank will get it right in periods 2 and later. Assume that all of the costs of the recession are borne equally by the young in period 1, and that the old continue to produce and consume 100. (This is roughly realistic, in that old people keep their existing jobs and customers and it's new entrants that can't find jobs and customers).
Equilibrium 2: Consumption of cohort "A" when young in period 1 is 100(1+n)/(1+r). [Math error fixed, thanks to Lord.] (Which is less than 100). Everything else is 100.
A recession hurts the young of cohort A, and nobody else.
Now introduce a fiscal authority that makes bond-financed transfers to the young in period 1, sufficiently large to prevent the recession. (The central bank holds r constant in period 1 and does not respond to fiscal policy). Let the size of the transfer payments be F per young person.
Equilibrium 3: In period 1, the young in cohort A produce and consume 100 each, by assumption (because the fiscal policy is big enough to prevent the recession). (So do the old). In period 2, the old in cohort A produce 100 and consume 100 + F(1+r), because they sell their bonds to the young in cohort B. And the young in cohort B produce 100 and consume 100-F(1+r). In period 3, the old in cohort B consume 100+F(1+r)2 and the young in cohort C consume 100-F(1+r)2 . And so on. Note that r in periods 2 onwards will be above n, and (r-n) will be increasing over time.
Discussion. Fiscal policy makes cohort A (the ones who were young when the recession hit) doubly better off. They are fully employed and consume 100 when young (which is F more than they would consume if fiscal policy had not prevented the recession [update: I'm not 100% sure my math is right there, but I'm too wiped to check it]). But they gain an additional F(1+r) units of consumption when old. Their lifetime consumption has increased by F+F(1+r) relative to the recession, and by F(1+r) relative to what they would have consumed if the central bank had not screwed up. Cohort A is better off than it would have been if the central bank had not screwed up, provided the fiscal authority prevents a recession. All future cohorts B, C, etc. are slightly better off. They have higher lifetime consumption, but their consumption path is unsmoothed, which partly offsets that gain. But on net they must gain, because any individual in any future cohort can refuse to buy the bonds.
Now let's add taxes. The fiscal authority sees that the debt is growing at rate r. If r is positive, this is unsustainable (remember this economy has zero population growth and productivity growth), because eventually the young will be unable to buy the bonds from the old. Assume the government puts lump-sum taxes on the old in all cohorts from A onwards, just sufficient to pay the interest on the bonds.
Equilibrium 4: Cohort A consumes 100 when young and 100+F when old. Cohort B (and all future cohorts) consumes 100-F when young and 100+F when old. r stays above n by a constant amount, that is an increasing function of F.
Discussion. Fiscal policy makes cohort A much better off. Better off than if the central bank had not screwed up. All future cohorts have a lifetime consumption of 200, but their consumption is unsmoothed, so their lifetime utility will be lower. This loss in utility lasts forever. If the government ever paid off the debt, the cohorts that paid the extra taxes to repay the principal would have a lower lifetime consumption, as well as an unsmoothed consumption, and would be even worse off. Taxes make people worse off (duh). The value of the loss in utility of future gerations, measured in monetary terms, is F (duh, like equivalent variation, or is it compensating variation?). How much is the present value of what all future generations would be willing to pay to avoid the taxes needed to service the debt? It's the same as the present value of the taxes. Which is the same as the debt (if r exceeds the growth rate, because otherwise taxes aren't needed). The magnitude of the burden of the debt is the debt itself.
Now add private investment to the model. Assume that each cohort sells its capital and its bonds to the next cohort's young. If the investment opportunity is a "Crusonia plant", that grows at a fixed rate g, this won't make any difference to my results. See Steve Williamson for a proof. If instead we assume that the capital owned by the old increases the productivity of the young's labour, this will create an additional channel through which deficits will affect the utility of future cohorts. But I'm not quite sure how.
Now add government investment to the model. The government borrows to invest, and can give the returns of that investment to any future cohort it chooses. This will obviously affect my results a lot.
Someone else can do the math. Not my comparative advantage. And I gotta do other stuff now.