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.
Worthwhile Canadian Initiative 
Semantics and the Debt Burden
Does government debt impose a burden on future generations? A relevant question given the high current government debt levels to which most people will answer with a clear “yes”: we are spending today and passing the bill to the next generation. But this answer is incorrect (or to be more precise it might be incorrect). The link between debt and burden on future generations is much more complex than what many think.
Recently, a debate has populated the economics blogosphere as some argue that that debt only imposes a burden when it is held externally, others coming up with counterexamples where this is not true (borrowing from Noah Smith a list of links to the debate: here, here, here, here, here or here.)
The debate becomes even more complex as the issue of desirability of another round of fiscal stimulus is mixed with the notion of intergeneration transfers associated to increasing government debt.
Unfortunately, economists tend to go in circles and debate the same subjects over and over again without reaching consensus, so when I went back a few months (January this year) I found a very similar debate with practically identical arguments being put forward by both sides.
The lack of consensus in this particular debate is much more about semantics that about disagreements on how the economy works. My reading of the debate is summarized well by Noah Smith long list of updates to his blog entry. In particular the following question: is government debt an indicator of the (fiscal) burden we are imposing on the next generations? And the answer is a clear no. Debt does not matter. What matters is taxes and spending, debt is just a vehicle to deal with imbalances between the two. Debt is not a burden per se but it can be the outcome of tax and spending decisions that lead to redistribution of resources.
We can construct examples where a government with high debt levels is not imposing any costs on future generations. We can also construct examples where a government with very little of no debt imposes large burden on a given generation (tax everyone under 50 and give the revenues as a transfer to everyone over 50).
And while seeing these debates come back without a resolution is frustrating, the advantage is that I can cut and paste below a longer and more detailed post that I wrote last time the debate happened. Just for those who still want to read more about it.
Debt does not matter. Spending and taxes do.
Monday January 2, 2012.
Paul Krugman makes the point that government debt matters less than most people think because in some cases we simply owe money to ourselves. He is right and what he has in mind is the notion that government debt is (in many countries) mostly held domestically. Japan is an extreme case where more than 90% of the government debt is held by its nationals but even in the US the majority of government debt is held by US citizens or institutions. For some it is debt but for others it is an asset, they cancel out from a national point of view.
We can think of an extreme case where government bonds are held by all taxpayers in proportion to their income – in a way that mimics tax rates. In that case, government debt is not imposing a future burden on anyone, it simply cancels out with the assets that all investors/taxpayers have.
How do future generations enter into this analysis? What if we try to pass the bill to future generations? Let’s start with the case of a closed economy/system. In a closed system (the world, no international trade or capital flows) the debt that the current generation has will end up in the hands of the future generation in one of two ways: either it gets simply passed to the next generation as a bequest or, alternatively, the current generation could try to sell their assets and spend all their wealth if they do not want to leave a bequest to their children. But the debt must be bought by someone. And given that this is a closed economy, it can only be bought by the future generations. In both cases the bond holders are also the taxpayers.
If we bring other countries into the picture then the analysis is different. The government debt that other countries hold is a claim on our current and future income and as such it is a financial burden that either the current generation or the future one will have to pay for. But Krugman’s point, which is correct, is that many make the mistake of assuming that government debt is equivalent to external debt and they overestimate the burden that it imposes on a country.
Let’s go back to the case of a closed economy: is it really true that debt does not matter? Not quite, because there are distributional issues of two types: first there is no perfect match between bond holders and taxpayers so it is not quite true that we owe money to ourselves. Some citizens owe money to others. The second distributional issue is about generations and here we need to go back to the example above to understand how difficult the analysis can get. The best way to understand the argument is to stop talking about debt and talk about spending and taxes, which is what really matters. A government spends some income today (builds a road, provides health services to the population). It decides not to tax anyone but instead it issues debt bought by the current generation. The government decides that it will only pay back the debt in the future when it raise taxes on the next generation, not the current one. Are we passing a burden to the next generation? It all depends on what the current generation does. If they decide to spend all their income and leave no bequests for their children then the answer is a clear yes. The current generation enjoyed services that they did not pay for themselves and did not compensate the next generation in any way for the future taxes they will have to pay. Just to be clear, the future generation will be holding the debt that the previous generation sold to them when they were spending their inheritance, but this is not a transfer of resources, the asset was sold at market price. So the fact that in the future bondholders are also the taxpayers does not mean that we are not passing a burden to the next generation.
There is a second scenario where there is no burden passed to the next generation. It can be that the current generation is responsible, understands that the government is asking future generations to pay for the goods and services that they enjoyed and they decide to leave a larger-than-planned bequest to their children so that they have resources to pay for all the taxes (you can think about the bequest being the government debt itself). In this case no burden is passed to the next generation.
This simple example () makes it clear that answering the question of what distributional impact government debt has across generations requires an understanding of the patterns of spending, taxes and saving of different generations. What matters is not debt but who enjoys the spending that the government does and who pays for it. Debt is just a vehicle that can be used to transfer resources across different individuals or generations. Debt is not a problem, the problem, from a generational point of view, is the potential mismatch between spending and taxes (even if future taxpayers are also the holders of government bonds when they are paid back).
() The example ignores many issues: the type of goods government buy, the possibility of default, the possibility of crowding out (government bonds displacing other forms of saving),…
Nick, in the context of this model I still don’t understand how the central bank can screw up and fix it without creating a burden somewhere. It looks like time is stopping. (I thought earlier you said there would be inflationary pressure in period two, so that r would need to be raised, wouldn’t that keep r higher than n in period two also?) In other words: since some time must elapse between the screw up and fix there must be someone who benefits before and after. That nothing would happen seems likes saying the screw up and fix happen simultaneously.
But have you have answered elsewhere why you do or don’t think monetary policy can create burdens more generally?
Antonio: OK. But that’s a bit like saying “Guns don’t kill, people and bullets do”!
anon: look, it’s a discrete time model. It can’t address the question of what happens if the central bank realises its mistake and fixes it halfway through the period. But simply imagine: the central bank is just about to screw up, by setting r too high, and then realises this would be a mistake just before the period begins, and does the right thing instead.
“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].”
What about spending cuts?