Fiscal policy in a simple New Keynesian OLG model with unemployment

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.

Comments

[Kevin Donoghue]

I'm clueless about this stuff, but my guru tells me Samuleson proposed a welfare criterion for models of this kind back in 1967 (a "turnpike refutation of the golden rule" or some such shite.)

[OGT]

r seems to be doing a great deal of work in this model. Why is r permanently above n, what restrains the CB from setting it lower than n in future periods?

Also, what restrains the government from taxing f? Making like Miles Kimball's lines of credit plan?

[Lord]

If r is set greater than n, how is (1+r)/(1+n) less than 1?

[Matias Vernengo]

Hi Nick:
As you noted in your debate with Noah, all the results here hinge on the relation of the rate of interest and growth. More importantly, if r is not an endogenous variable determined by productivity and intertemporal preferences, but a policy variable, then the scope for fiscal policy could be much bigger. Getting slightly off the model (and theory), one reading of the Bretton Woods era is that, by imposing capital controls and allowing low rates of interest at home (the reason why Keynes wanted capital controls in fact), it allowed for more expansionary fiscal policies.
Best,
Matías

[Nick Rowe]

Lord: Ooops! well-spotted. Edited and fixed.

Kevin: I read something about turnpike theorems once. But I don't remember much. I think in this model it would mean: if the debt is positive, it should be brought down to zero slowly. (But don't quote me on that).

OGT: in this model r = n if there is no debt, and the central bank gets monetary policy right. But r will be greater than n if there is debt, and so people consume less than 100 when young and they buy the bonds, and more than 100 when old and they sell the bonds. The personal Euler equation shows that. If the central bank sets r below n, there will be excess demand for goods and ever-accelerating inflation.

If the government taxes bonds 10%, rather than collecting lump sum taxes, all that happens is that r rises 10%.

[Nick Rowe]

Matias: Yep. If r is *permanently* less than the growth rate, there is no need ever to increase taxes. The government should just rollover the debt+interest forever. Then all cohorts will be better off. I've done a couple of posts on this. I like this one best, because it allows uncertainty about future r and growth rates.

The central bank is already setting the rate of interest where there is zero excess demand for goods. (Except in period 1, where it screws up and sets it too high.) If it sets r too low, the only result is excess demand for goods and eventual hyperinflation.

[rsj]

Nick,

Why can't r be lower than n in some cases (when there is a recession) and greater than n in other cases (when there is a boom)?

I.e. imagine that the CB "screws up" by not recognizing that r has changed. So the CB keeps it's rate constant.

This adds a twist because the government is able to borrow cheaply during recessions and more expensively during booms.

[Nick Rowe]

rsj: short answer: look at the consumption-Euler equation. If people expect C=100 next period, when old, then C can only be 100 this period if r=n. However, if people expect the central bank to screw up and create a recession next period, then the central bank would have to set r < n this period to prevent a recession this period.

[Nick Rowe]

rsj: "I.e. imagine that the CB "screws up" by not recognizing that r has changed..."

I think you meant to say "by not recognising that *n* has changed".

[david]

Most allusions to capital aggregation problems are much more lucid than that.

In any case Rowe is arguing with people who assume the existence of aggregatable K themselves.

[rsj]

yes, that n has changed.

[Госбанк]

The stochastic nature of the "sustainable" debt Ponzi game (when expected g > r), in some cases, implies that the probability of such a game failure, although low, is not zero. See Elmendorf and Mankiw or the following reference for details:

http://www.eea-esem.com/papers/eea-esem/2004/1819/ponzidebt.pdf

Besides, the "free lunch" debt rollover scenario implies a more or less hard limit on the principal ("debt ceiling").

[david]

@Госбанк

The linked paper doesn't seem to say what you cite. It says, on stochastic g: "If E[log(R1(t)/(1 + gt))] < 0 then a small enough Ponzi scheme based on one-period bonds fails only with arbitrarily small probability.".

[david]

@Sandwichman

Woe to the heterodox watching members of the orthodox have a go at each other. Nobody pays attention to you!

[Госбанк]

david:

"arbitrarily small" is a theoretical construct based on the ergodicity assumption for the growth rate process which is quite an assumption that is never true in real life, and "small enough" debt/GDP ratio. All we can say is that the probability of Ponzi game failure is "small" in practical terms.

Besides, "arbitrarily small" is not zero in mathematical sense either :)

[anon]

Doesn't this assume perfectly flexible prices? Otherwise wouldn't a recession and recovery be incoherent? How is this New Keynesian? And since some time, however small, *must* lapse between recession and recovery wouldn't *some* "generation", however small, be forced to consume less in period 1? Otherwise wouldn't this be overlapping time periods and not overlapping generations?

And even if we assume the young are the only ones unemployed by the recession, how would the old not be hurt unless the old can consume all of their own production? (Or is there perfect substitution, too.)

Also how realistic is it to rule out, if that's what you're doing, any harmful effects from period 1 unemployment? Otherwise, how could the central bank be sure to get things right in period 2? (In the do nothing scenario there may be deflationary pressure, in the "fix the recession" scenario there may be inflationary pressure.)

Finally, I'm still wondering why you don't think this applies to monetary policy used to fight recessions? It seems all economic transactions, possibly including barter, imply, winners and losers, and therefore burdens. (If I have to work slightly harder than a counterpart in a barter transaction he may be "burdening" me.)

[anon]

Edit, sorry, it seem like you have sticky prices with regard to interest rate changes, but flexible prices regarding fiscal policy, otherwise how could the old not consume less in period 1 after buying bonds? Or how could taxes not exist prior to introducing them? How can bonds be paid off without taxes?

[anon]

*Seems*. And never mind on the taxes. You're saying they'd keep borrowing until someone is forced to refuse to buy bonds.

[Ritwik]

Sandwichman has just proved that equity and debt markets don't exist.

Nick

Good model! But in an economy in equilibrium with 0 growth, r will be 0. r > n, so the trouble itself began when n turned -ve.

Thus, the consumption unsmoothing actually made every generation better off. (1/1+n) is greater than 1 when n is -ve, and hence more consumption when old leads to higher utility.

[Ritwik]

Sandwichman

Many of the earthworms and fungi that 'exist' were planted in the soil, ex-nihilo, by men and women who projected from past experience and pure fantasy. True story. Independent of these men & women, they don't exist, except in a trivial (or profound, your choice) Spinozan sense. Also true story.

And not that anyone will ever know, but if Keynes were alive in the 60s & 70s, I'd lay my bets on him pooh-poohing both Robinson & Solow, and aligning himself with, say Tobin & Hicks.

Point is, let's stay on Nick's model, shall we?

[Nick Rowe]

Госбанк: I had a read of that paper when someone linked to it on my trill perpetuity post. It's a good one. I'm not quite sure what to make of it.

anon; it's sticky prices, otherwise the central bank can't set a real interest rate, even temporarily.

I expect I could have had the unemployment in the recession shared equally by the old and the young. That would be what would happen if I had followed the standard NK approach, and introduced monopolistic competition formally. I don't think that would make much difference to the results though. Except that the old in period 1 would have shared some of the benefits of the fiscal transfer too. It wouldn't affect the results for cohorts B onwards.

"Also how realistic is it to rule out, if that's what you're doing, any harmful effects from period 1 unemployment?"

It isn't. But it keeps it simple and standard. And if fiscal policy prevents the recession, it doesn't matter anyway.

Ritwik: thanks!

The rate of interest exceeds the growth rate because people are impatient. n is a preference parameter, that measures time preference proper.

Sandwichman; you are off-topic and screwing up this comment thread.

[anon]

Okay, so you're saying the fiscal authority borrowed from and transferred to the young (cohort A) in period 1? If so, that's what confused me. I was thinking you borrowed from the old for some reason. Okay, that's making much more sense now.

But how is fiscal policy in this model actually working? Isn't there a contradiction between this and saying there won't be any inflationary pressure in period 2? Aren't you implicitly assuming prices can rise already? Similarly, isn't there something strange then about assuming that doing nothing won't go from bad to worse? (Or by "get it right in periods 2 and later" do you mean NGDP targeting?)

In light of Noah's post and that they'd eventually reject the bonds, I'm been thinking about this a little different: Everyone who buys bonds buys them voluntarily. What's more they apparently think whatever the price of the bonds it's an amount they can live without. Add a negative income tax to the model and no generation would ever have to consume less than what any one generation thought they could live with. Not only that but the entire burden can *always* fall *only* on someone who lent the government money of their own free will. The government doesn't even have to formally default as it can perform the whole operation through taxes.

Isn't risk priced into bonds? Does it really make sense to talk about a burden from a freely chosen risk? What are we saying: wealthy bondholders must be protected?

[Ritwik]

Nick

In general, I've now come to believe that a pure time preference in a world of certainty is not psychologically coherent (atleast for macro and capital theory), even though it is logically coherent. If the future is certain and known to be certain, why will anyone be impatient? Impatience is indistinguishable from certainty preference.

However, let's grant r>n>g, known and certain for all times in the future, for a moment. Fiscal policy is trivially useless in this model, as it's a -ve RoI proposition.

Until you let fiscal policy affect g or n, the welfare loss of period 1 can never be recouped, and can only be redistributed.

[Nick Rowe]

anon: "But how is fiscal policy in this model actually working? Isn't there a contradiction between this and saying there won't be any inflationary pressure in period 2?"

Intuitively, the way fiscal policy works is that by giving bonds to the young in cohort A, which they can sell to increase their consumption when old, they want to consume more when young, a la the Euler equation. The bonds are net wealth, so when they are given bonds, they consume more in both periods. There certainly would be inflationary pressure in period 2, but the central bank offsets that pressure by increasing r above n.

You lost me on the next bit. Being able to buy and sell bonds cannot make future cohorts worse off. Because they won't buy them if buying them made them worse off. It's the future taxes that make future cohorts worse off. Bonds create a burden if they imply future taxes (which they will if r > growth rate).

Ritwik: historically, real interest rates on safe loans on farmland in England were very high. Even in a stationary economy. Impatience is the only plausible reason.

[Ritwik]

Nick

Why would you assume that farmland loans were safe? What was the price volatility and transaction costs of selling farmlands?

[Nick Rowe]

BTW: the model I am using here is exactly the same as the model that is used by some who claim there is no burden on future generations, except: it's an OLG version of that model (which it must be to address the "future generations" question); I've simplified it.

[Nick Rowe]

Ritwik: I'm trying to recall the author and title of the book. Published about 6 years back. By an economic historian. About how the rich had more surviving kids than the poor. You bought the farmland by paying x years' rent in advance. The smaller is x, the higher the implied rate of interest. If you were a tenant farmer, who knows he will be farming the land anyway, the only risk is that rents might drop in the next x years.

[Ritwik]

Nick

It's not the tenant farmer's risk here, because he is the one 'borrowing'. If I'm the initial land owner, the reason that I am willing to 'sell' my land for, say, as less as 5 years rent is precisely because

1) I fear that rents are highly unstable.

2) I fear that tenant farmers are highly risky, and so unless someone pays me upfront, he must pay rent high enough so that he compensates for the fact that he may not be in a position to pay for the rent.

I get what you're saying about Baker/Krugman and OLG models, but I will be surprised if they agree with the 'let's assume fiscal policy does not impact g & n' conditions. Without that, fiscal policy is trivially an intergenerational redistribution, as your equilibrium 4 shows. Of course, your model has forced it out in the open as to what really is the key to decoding 'what does fiscal policy do' debates.

I find Tobin's model on government debt very useful, though it doesn't have OLG cohorts.

http://cowles.econ.yale.edu/P/cd/d05a/d0502.pdf

[anon]

It doesn't seem intuitive when they couldn't have saved their period 1 consumption anyway. By what mechanism does raising r cause a recession then? (It seems to easy to say they just felt wealthier so started spending more.)

Just to be clear your saying this model does contain inflation and deflation? If so you're saying the inflation will be tackled by the start of period 2, but r will be higher in period 2 than in period 1? Or will inflation "spill over" into period 2?

I'm agreeing with you that people won't buy bonds they *think* will harm themselves. I think this goes to what Ritwik is saying and what I tried to say to you before about the apple model: the government had no good reason to borrow to begin with. The only reason to do so would be to *corruptly* redistribute or transfer money as there was no possibility for a good outcome. In a reasonable model you'd assume there's the possibility for growth, only for whatever reason, it did not occur. Then people lending the government money would understand there is some risk involved. And since they'll be choosing to buy bonds, it seems wrong to say they could be burdened by someone who has not. And since we can always put the entire burden on the bondholder, imposing a burden is always a policy choice, not a result of the debt.

In other words: Apples really can't travel through time. QED.

[J.V. Dubois]

Nick: "The magnitude of the burden of the debt is the debt itself."

Yes and no. While I completely agree with you that unpaid debt is a burden on future generations, Keynesians believe in the Keynesian Multiplier. So stimulus is a way how we can solve a dynamic inefficiency, it is an inter-generation utility smoothing measure with positive social impact from the historical perspective.

And I think we should always engage in exactly such measures, because if you disagree, then we have to embark upon an exactly opposite path - refrain from consuming so that it can be consumed by future generation. At a very silly extreme it should be our moral obligation to go die alone in the woods on the "last hunt" when old and unproductive as in prehistoric times, so that we do not "burden" our children with our consumption.

[OGT]

Nick wrote: "Intuitively, the way fiscal policy works is that by giving bonds to the young in cohort A, which they can sell to increase their consumption when old, they want to consume more when young, a la the Euler equation. The bonds are net wealth, so when they are given bonds, they consume more in both periods. There certainly would be inflationary pressure in period 2, but the central bank offsets that pressure by increasing r above n."

OK, this is where you lose me (again). Why are they 'giving' them bonds, shouldn't the gov't be borrowing from somewhere? Or why not give them money? What's the point of the 'bond' if it's not borrowed? Wouldn't cohort A have to sell the bonds to increase their consumption in period 1? And then not get anything back from the oldies in an OLG?

Isn't the Kimball formulation give them lines of credit a more intuitive formulation for fiscal policy, especially in an OLG model. It seems like you're putting a round hole in a square peg and claiming to make an innovation.

[Wonks Anonymous]

Nick Rowe, the book you are thinking of is "A Farewell to Alms" by Greg Clark. And the argument wasn't just that the poor were dying out (he looked at their surnames, they tended to dissappear), the aristocracy kept killing each other off as well. It was relatively well-off farmers ("kulaks") that reproduced themselves, continually experiencing downward mobility until most of society acquired their patient traits. With a detectable effect on interest rates!

[rsj]

but their consumption is unsmoothed, so their lifetime utility will be lower. This loss in utility lasts forever.

Note that this has nothing to do with fiscal policy or monetary policy, but how the example was set up. Additional transfers that are budget neutral can undo this shift, which is an artifact of two facts:

1) that in the model, taxes are paid not as a percentage of income received but as lump sums levied irrespective of each cohort's income.
2) The economy has only one (and exactly one) recession. Not a series of recessions followed by booms, or a probability that each period will be in a recession, but exactly one recession.

[JF]

Prof. Rowe,

Interesting post. I might have missed something, but what is the source of the original transfer in your closed economy. From my understanding of the remainder of the post, it can not be the old. If not, what is the government transferring? Thanks

[Nick Rowe]

Wonks: Yes! That's the one. A great book.

Ritwik: They didn't borrow to buy the land. They bought the land in much the same way that people buy a bond. Land is like a perpetuity. The annual rent, divided by the price, tells you the yield on land. Sure, future dividends (rents) are uncertain, but if you are a tenant farmer, and always will be a farmer, you reduce your risk by buying the land you farm. Because you eliminate the uncertainty about future rents.

"Without that, fiscal policy is trivially an intergenerational redistribution, as your equilibrium 4 shows."

Well, they also keep saying that fiscal policy cures the recession, and my model captures that. I think, at the very least, I have put the onus back onto them, to build a model in which deficits (note I did not say balanced-budget increases in G) cure the recession and do not impose a burden on future cohorts. I'm sure that can be done, but again I repeat that my model is an absolutely standard NK model, except for OLG. I'm making all the same assumptions that e.g. Krugman and Eggertsson makes, except for OLG (and simplified).

[Nick Rowe]

JV: "While I completely agree with you that unpaid debt is a burden on future generations, Keynesians believe in the Keynesian Multiplier. So stimulus is a way how we can solve a dynamic inefficiency, it is an inter-generation utility smoothing measure with positive social impact from the historical perspective."

I think you are conflating two questions here (or maybe I misunderstand):

1. My model has a Keynesian multiplier (OK, it's less than one, I think, but it still works). Fiscal policy cures a recession due to deficient AD.

2. Dynamic inefficiency. Even at full employment, an increased debt can be a good thing if r is always less than the growth rate.

OGT. By definition, (with no government expenditure in the model) fiscal policy is bond-financed transfer payments (an increased transfer payment is just a tax cut). The only people the government can borrow from are the young agents, because the old know they are going to die so don't want to lend anything. And in my model I assumed that the young suffer the costs of the recession, so it seemed natural to give the transfer payment to the young. I could have assumed the government made the transfer payment to the old instead, which would have caused a bigger multiplier, but wouldn't have changed the rest of the results much, except F would be smaller for a given-sized screw-up by the central bank. A bond-financed transfer payment is exactly like giving people bonds (they can sell them if they wish).

rsj: sorry, but you lost me. And if I assumed taxes were a percentage of income, rather than lump-sum, this could only make the future burden worse, because there would be disincentive effects on production as well.

JF: the government borrows apples from the young (in exchange for a bond), then gives those same apples back to those same young. It is exactly as if the government gave the young bonds.

[JF]

Prof Rowe,

Hum... if the economy is already in recession, and the youg are producing less than 100, then the government can not borrow from them to bring back their consumption level to 100. Maybe the government gives them a bond that produces the incentive to increase production, but in this case this looks more like printing money, is it not?

[Nick Rowe]

JF: No. Here's the intuition: The government hands out bonds to the young. The young say: "Yipee, we are richer by F. Let's start spending half of that F now." So they spend part of it, and then they notice: "Wow! I've just noticed I've sold more apples, so my income is higher than it was before the government handed out those bonds, so I'm going to spend another half of that increase in income!". And so on, until full-employment is restored (provided the government got F to be exactly the right size).

[Wonks Anonymous]

Clark is in the news again, saying social mobility is fairly constant across time and space. The social status of your family surname centuries ago predicts where you will wind up:
http://www.npr.org/2012/10/16/162936707/movin-on-up-that-may-depend-on-your-last-name

[JF]

Thank you for your answer prof. Rowe,

I can accept your answer, but the same argument could be made by saying that the government printed money and gave it to the young, it seems. In that case, I don't really see why the government would like to use a transferable bond, even less attach an interest rate to it, with the risk of at some point having to punish later generations, which in your example it is forced to do. It seems to me your example is more saying that a poorly designed fiscal policy can hurt future generations, which I believe nobody would dispute, although in the greater debate that generated this post, as well as your previous posts on the subject, I guess you are saying that debt CAN be a burden on future generations, as opposed to debt IS ALWAYS a burden on future genetions.

[Stuart Ullman]

I apologize for being late to this discussion. I can't respond to these things while I'm at work, and when I got home, weary from a long day of labor, I was informed that I had cooking duties, and was given a glass of wine in compensation---the bottle was there beside the stove, so naturally I'm in exceptionally good condition to make a comment...

In general I'm a fan of your overlapping generations model, because it focuses on people as they travel through time rather than on a sterile abstraction like accounting periods. But I want to comment on this line from your original post, talking about equilibrium 4:

"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."

This last notion depends on your original equal distribution of income between young and old, which is unrealistic. If "young" and "old" mean 25 and 45, then this distributes far more than is realistic to the young, so the utility problem you point out in this quote is much worse than in your example. But if "young" means, say, 20 to 50, and "old" means 50 to 80, then your original example distributes way, way too much to the old. Income for those who are retired is a fraction of the income for those who are in their most productive years of life. And in this case, the permanent redistribution toward the old in equilibrium 4 may actually increase lifetime utility: it creates a smoother, rather than a lumpier, distribution between a cohort's young period and their old period. And in this case, it is an increased lifetime utility that is created, and that lasts forever. We have done all future cohorts to save while they are young so that they can eat when they are old.

I'm still not sure that the villain in your model is debt itself. The villain comes at the end, when this little country decides to redistribute income from the old to the young. In the example in your post they do this to pay off the debt, but it really doesn't matter what motivates them. If they end the long series of young-to-old redistribution at any time, for any reason, this problem is there: some cohort has to pay when they are young, but gets no return when they are old. The gist of your argument against debt, I think, is the assertion that compound interest will eventually *force* the nation to pay the debt off. If that happens then the nation is in trouble no matter what public purpose the debt was raised to pursue.

[Nick Rowe]

JF: I would re-interpret what you are saying as: they ought to use monetary policy (increase the money supply) to solve the recession, not use fiscal policy. Agreed.

Stuart: thanks!

1. Yes. Supposed I had made the young produce 150 and the old produce 50. Then (for smallish values of n) equilibrium r would be negative (and less than the growth rate). In which case, we are in the world of Samuelson 1958, and debt would be a good thing, and it would be bad policy to raise taxes at all.

Agreed. The villain of the model is not debt, but the taxes that the debt may (or may not, but it does in this model) entail. Agreed. But that's a bit like saying: "guns don't hurt people; bullets do." It depends whether the gun is firing blanks. If r > growth rate, guns/debt implies bullets/taxes.

[rsj]

rsj: sorry, but you lost me. And if I assumed taxes were a percentage of income, rather than lump-sum, this could only make the future burden worse, because there would be disincentive effects on production as well.

Heh. No, it can "only" make the burden better, because a progressive tax rate will increase labor supply due to backwards bending labor supply curves.

Really, this model has no relevance on the analysis of fiscal policy. To understand this, imagine if someone did the same analysis on the inter-generational transfers due to monetary policy. It would be strange right? Why? Because a generation supplies labor for about 40-50 years, but you do not have 40-50 year recessions. The frequency of recessions is much greater. So this model has nothing meaningful to say that isn't a 3rd or 4th order adjustment, IMO.

Deficit spending during downturns (which last about 1 year, not 40 years) is about intra-generational transfers. The same persoon who receives foodstamps and unemployment insurance will also pay more in taxes. Odds are that his children will not pay more in taxes due to deficit spending, because deficit spending has nothing to do with inter-generational transfers unless you are modelling it incorrectly.

Note that I am not saying that you cannot invent models that describe inter-generational transfers in terms of deficit spending, only that those models are incorrect characterizations of how we conduct deficit spending, and require making assumptions of 40 year recessions, or old people not being taxed, etc.

Deficit spending is first and foremost about income insurance. Think of auto insurance. One *could* argue that auto-accident insurance payouts are generation specific. If one did, they would be wrong. Odds are, the same person that receives the auto-accident payout also pays the premiums, just at different time periods in that person's working life, because the frequency of payouts is much greater than the working life of the person. So it is with recessions. If recessions occur every 10 years, and last 1 year, then each person will go through 4 recessions and 4 booms during their working life, and will have more than ample opportunity to repay benefits received during recessions during booms without passing high auto premiums onto their children. It is all about insurance.

Second, deficit spending is about risk. During recessions, the risk premium goes up and the divergence between risky and risk-free rates increases dramatically. It makes sense to supply more riskless debt in that case and to supply less, proportionally, during booms. If you are modelling this assuming that the risk-free rate alone is the only rate, you miss the bug story about recessions, IMO, and therefore about recession fighting. Recessions are about doubts that firms can earn a return in the first place, not about demands for higher returns.

Third, deficit spending is about allowing the CB to do its job. Risk free rates are negative. People are willing to take negative rates. But nominal rates cannot go negative, so the *assumption* is that in real terms not all the debt will be repaid due to inflation in the boom period. It then makes no sense to talk about the burden of repayment. There will be only partial repayment, because negative real rates are the equilibrating rates. It assumed that not all the debt will be repaid as the debt is nominal in nature and not real, and the CB is promising excess inflation during the subsequent boom.

Only to fourth order is deficit spending about inter-generational transfers. Because it is true that during the year or two of recession, there will be some retirees that exit the work-force. But those retirees themselves will have lived through multiple recessions and booms, so the odds of any particular retiree being a net beneficiary are small.

A fourth order effect. Not anything that you would use to judge fiscal policy, because the model is only approximate in the first place, and fourth order effects can be ignored.

[rsj]

Here is a graph of the spread of BAA-AAA yields. You see large spikes during recessions. So let's assume the risk-free rate is 3%. During the recession, people update their probability distributions and require higher returns of the firms that survive because they believe that more firms will fail to earn a return. It is a self-fulfilling prophecy. One *could* try to address this failure by attempting to lower the risk-free rate, but this would be wrong because the probability distribution estimate for risky debt is changing. So instead, the government steps in and buys from firms directly, ensuring that they have sales and in the process it increases the supply of risk-free debt. This is the most effective solution to the problem of increasing risk.

During the boom time, the government taxes some of that back, but of course not all of it because the risk-free rates are lower than the growth rate. Nevertheless, ignoring that, more taxes are paid during the booms and less benefit payments go out, so the budget position improves.

This is a far more effective stabilization technique to the problem of increasing doubts as to whether firms can sell their output.

It may not be a good stabilization technique for other problems, but the "can this firm find buyers" problem is the most common one.

[anon]

Nick,

How does monetary policy avoid the problem of burdens? If it causes inflation higher than the growth rate? (I keep asking this. Maybe it's a dumb question?)

(I'm now wondering if unorthodox monetary policy and, NGDP targeting (normal times), and fiscal policy don't favor different groups of people.)

Okay, I starting to get closer on this: 1. Interest rates rise. 2. Planned spending is cancelled. 3. Unemployment. 4. Demand falls. 5. Prices want to fall but can't. 6. Mass bankruptcies. 7. ???? 8. Another generation enters the work force to find everything back to normal? (It seems like there is no necessary bottom to a recession.)

I think I'm getting how the wealth effect is working, but am skeptical about how much people thinking they *might* be wealthier in the future will be effective at fighting the recession. Of course I don't know, but I'd guess wealth effects would be more likely in a boom. (Even if they did *think* they're wealthier, couldn't you have a double dip (or high inflation) in (or before) period 2, when people realize the economy hasn't grown?)

I still don't understand how inflation is working in this model. Any extra help here would be appreciated. (It seems strange that period 2 interest rates (if you are in fact saying that) would need to be raised but no one would be harmed.)

[Sandwichman]

Nick: "Sandwichman; you are off-topic and screwing up this comment thread."

Simply put, my point was that a formal model that ignores or abstracts from the ecological context is fundamentally flawed.

Stop right there.

[Nick Rowe]

Sandwichman: fair point. There are many ways we can benefit or burden future generations, and this model looks only at one way, and it's maybe not the most important one. And let's stop right there.

[Nick Rowe]

rsj: "No, it can "only" make the burden better, because a progressive tax rate will increase labor supply due to backwards bending labor supply curves."

No. That's wrong. It's doubly wrong.

1. The lump sum taxes on future cohorts will have an income (i.e. wealth) effect on the demand for leisure (supply of labour), and if leisure is a normal good (it is) that will increase employment and output. But that loss of leisure is a burden too, even though it is not counted in GDP. They suffer a loss in consumption. They share that loss in consumption between lower consumption of apples and lower consumption of leisure.

2. If we switch from lump-sum taxes to taxes that are an increasing function of employment (or consumption of apples), but consumption of leisure remains untaxed, that creates a substitution effect away from apples into leisure, (a movement along the upward-sloping income-compensated labour supply curve) so that there is a wedge between the MRS and MRT between apples and leisure, which creates an additional deadweight cost triangle burden.

Everything you read in the macro textbooks about relative price changes causing income effects and possible backward-bending labour supply curves is total cr*p. (That's probably too strong; there may be some I haven't seen that get this right).

[Nick Rowe]

rsj: "Deficit spending during downturns (which last about 1 year, not 40 years) is about intra-generational transfers."

True. Some of it will be. It depends on how quickly future taxes are increased. If I switchd my model to continuous time, then all those who got the transfer but who died (or stopped having taxable income) before taxes were increased would gain, and all others (including those born after or who didn't get the transfers) would lose. Gainers and losers would partly overlap. Part of the future burden would be borne by people getting the benefits. But if those people take that into account, we are back in the Ricardian world, for that sector of the population. And those who argue it's no burden because "we owe it to ourselves" would still be wrong.

[Nick Rowe]

anon: "How does monetary policy avoid the problem of burdens?"

Within the context of this simple model (which I don't really believe, but set up so I can communicate with my critics, who do believe it) monetary policy works by changing the real rate of interest. If, at the beginning of period 1, the central bank simply stopped screwing up, and dropped the real rate of interest so it equalled n, the recession would immediately end. The young in cohort A would immediately consume and produce 100 in period 1, instead of 100(1+n)/(1+r). Everything else would be 100 also.

[Antonio Fatas]

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),...

[anon]

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?

[Nick Rowe]

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.

[Too Much Fed]

"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?