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Nick,

Don’t you want to stipulate that the central bank (or some bank) must buy the bonds?

Otherwise, you need to demonstrate where the infinite cash came from to buy them.

And otherwise, you’ve designed a security that’s too expensive to sell for its value.

And otherwise, you can only sell the security for less than its value, which craters the solution to the problem.

But it should be a piece a cake with the help of some bank.

Nice article and I think in some basic level you are right. I will only argue with two “details”:

1. I do not think that the weird world is dynamically inefficient. Not really in a way what “dynamically inefficient” means. The point is that since the debt is just nominal, not acruing it today but taking it tomorrow is not inefficient. Think about is as in this example: Imagine that we have exactly 14 billion units of some rare natural resource in our universe. This resource is naturally replenished at the rate of one unit a year, it cannot be artificially created and it never depreciates. So is it "dynamically inefficient" for first generation to discover this resource not to use all 14 billion units at once? The resource is a "free lunch" laying there in the universe waiting to be used. I would say no, because this pool could be used by the next generation or the one after – into infinity. If we knew when the universe will die, we could make the resource utilization more efficient. If we knew how the technology of utilization of the resource will evolve in time so people may take more utility for one piece of resource, then we could make yet some different reallocation. If we knew that there is a small chance of some disaster to occur that could wipe out humanity and that the chance of our survival would depend on the amount of resource mobilized to solve it, the allocation would yet again change dramatically.

I think debt behaves in the same way.

2. The trill idea is very neat but I do not think that its price should be infinite. If I remember correctly, the present value of perpetuities is calculated as PV = C/i (see here: http://en.wikipedia.org/wiki/Present_value) so if we say that the C is one trillionth of the NGDP and nominal interest (calculated as expected inflation + real GDP growth) is for instance 5%, then PV of such perpetuiti is exactly 20 trillionths of current NGDP.

Maybe I misunderstood your point in the article when writing 2: What you probably had in mind is, that if government would be more efficient reallocating resources then private sector (interest rate permanently higher then bond rates), all the world's investment would end up in the hand of the government. It is as if government machine is more efficient than private machine at multiplying wealth no matter how much wealth you throw at any of it.

But I still do not think that the price would be "infinite" as people also need to buy something else beside government versis private bonds/stocks. But I now probably get what you were trying to say.

Nice post.

The infinite rather than negative value of a trill perpetuity in a weird world is simple - it doe snot need L'Hospital's. The standard formula for a perpetuity's value no longer applies because the series diverges. The value is infinite.

JKH: do you remember that old hypothetical story about what would have happened if someone had saved \$1 back in the year dot, and had lent it out at interest, and had re-lent the interest, and so on, down to the present day. And how many trillions of dollars it would be worth now? And bigger than the whole world's wealth?

Well, that story is wrong. Because if someone had done that (and it hadn't been ripped off or defaulted on, etc.) the rate of interest would have been lower than it actually was, because saving would have been higher. Or the world's wealth would have grown faster than it actually did, because investment would have been higher.

Back to my story. Let me tell it slightly differently. Instead of selling the one trill perpetuity, the government gives it away (maybe split up into 33 million bits, so every Canadian gets a share). What would be the competitive equilibrium market price of that trill perpetuity?

If the world were weird (i.e. if r were less than g), then the value and market price of that trill perpetuity would be infinite (see Ritwik's explanation). But if the value were infinite, aggregate demand would be infinite too (even if everybody wanted to spend only a tiny fraction of their infinite wealth, which they would, because they would still have an infinite amount of wealth left). But then AD would exceed AS (which is finite). So the rate of interest would have to rise (or the central bank would have to raise the rate of interest, if you prefer), until AD=AS, but that can only happen if AD is only finite, which can only happen if the value of the trill perpetuity is finite, which can only happen if r is not less than g.

In other words, if the government were to give away one trill perpetuity that would make it impossible for the world to stay weird, so the value of the trill would in fact be finite, immediately people realised that the government was giving away one trill perpetuity. Imagining the possibility of a trill perpetuity with infinite value is like a proof by contradiction. It's like a good that has infinite value only if none exist. As soon as the supply increases above zero, the price becomes finite.

Think of a bond like TIPS but linked to GDP rather than inflation, and perpetual. If the yield is <0%(+GDP) then it's safe to sell them.

Scratch the perpetual part...that wouldn't work. But you could make the maturity really long.

JV: I'm not sure, but I think you might be misunderstanding.

It's easiest to think about this stuff in a world with no investment (though it does generalise to a world with investment). Imagine Samuelson's Exact Consumption Loan model. The young want to save so they can eat when they are old and can't work. But there is no savings vehicle. They all try to lend to each other, but nobody wants to borrow, because they all face the same problem. So the rate of interest is very low (maybe negative). By issuing bonds, the young now have something they can buy and sell again when they are old to the next cohort. All cohorts are better off.

Now suppose that goods can be stored costlessly until your old age. The equilibrium (real) rate of interest is 0%, and the old no longer starve. But if the population is growing (or if productivity is growing), the government can still do better still, by issuing bonds (starting a PAYGO pension plan) that pays a rate of return equal to the growth rate. Sure, the first cohort, which gets the pension but doesn't pay into the plan, benefits a lot. But all cohorts benefit.

The value of the trill perpetuity must be infinite if r less than g. (BTW, NOBODY USE THE "LESS THAN" SIGN, because typepad thinks it's a hypersomething.) PV=C/i is the correct formula for a perpetuity if the coupon stays the same over time. But if the coupon is growing at rate g (which it will with trills), it's PV=(one trillionth today's NGDP)/(i-g).

Ritwik: "The standard formula for a perpetuity's value no longer applies because the series diverges."

OK. That's a good way to think of it intuitively. But don't we have to use L'Hopital for a formal proof? Because you have got this term with (1+g)^t in the numerator and (1+r)^t in the denominator, both of which become infinite as t goes to infinity, so we need L'Hopital to prove the term goes to zero if r greater than g and infinity if r less than g?

Nick

You could use L'Hospital's, but I think it's simpler. Replace (1+g)/(1+r) as x. Now this simply reduces to x+x^2+x^3+...., where x>1 and is finite. This series diverges as x^n --> infinity as n --> infinity for all x>1, and x/x-1 is finite.

Is that very different from corporate finance? It looks like this perpetual trill is not different from a regular share in a company.

The only shocking thing I read was that Nick does not understand L'Hopital's Rule.

This is first-year calculus. Just take the derivative of both the numerator and denominator until you get to a finite expression. If your first round does not succeed, wash, rinse, repeat.

If the expression diverges then L'Hopital's Ryle will never generate a finite expression.

Perpetuities exist. The UK Government still has perpetual Consolidated Loans ("Consols") dating from the 1750's and War Loan perpetuities outstanding from the Crimean War. They have no redemption date.

Similar to JKH's comment, we're just talking about public debt. Because of random GDP expectations, or "bubbles" etc. private banking would create money to purchase the trills. The disappearance of instability of the public debt then reappears in the private sector?

Nick: Ok, I revoke my claim I was wrong. It is indeed dynamically efficient from the point of view of lifetime utility of consumption. So in that sense it indeed is a "free lunch" that it is a pareto improving measure. What I got confused about is something I would call a "virtual burden" to further generation. For instance imagine that some generation would in some kind of moral fit tear up half of the bonds. That would create more space for the future generation to consume more and at the same time to sell create some new bonds that they in turn may sell the yet unborn young.

Also on trills you are right, it was very neat. So by all means ignore my comments and keep posting these posts that give me headaches :D

Nick Rowe | July 12, 2012 at 08:03 AM

Nick, that's excellent

Just wondering - is there anyway you can transfer some part of that intuition to an explanation of why interest rates eventually had to come down in the Volcker period in order to ensure "fiscal sustainability"

As Zorblog notes, a trill is simply an equity share of 'the economy'. The simple act of the issuance of a trill, hence, will not lead to r>g. Trills will trade at a discount (usually) to almost any form of government debt. A Trill is a beta one asset, not a zero beta asset, and this 'risk' will be seen and priced.

Ok, that was unclear. What I meant to say was r(trill) will not collapse to r( risk free). r(trill) > g. r (risk free)< g. The issuance of a trill will push up its own interest rate, but not the interest rate on any other form of government debt.

Ritwik: "Replace (1+g)/(1+r) as x"

Aha! Yep, that's much simpler.

Determinant: "This is first-year calculus. Just take the derivative of both the numerator and denominator until you get to a finite expression. If your first round does not succeed, wash, rinse, repeat."

That's what I thought it was, but I don't understand why it works. Plus (and this will shock you) I can't remember how to take the derivative of y = (1+g)^t wrt t. (I used to know once, but I forgot).

Zorblog: "It looks like this perpetual trill is not different from a regular share in a company."

Plus Ritwik: "The issuance of a trill will push up its own interest rate, but not the interest rate on any other form of government debt."

Yep. Good point. (And the yield on a risky share in a regular company may be higher than that on a safe government bond). But my interpretation of that point is that we cannot tell whether we are living in a normal or weird economy just by looking at the interest rate on existing nominal bonds. We need to know the interest rate on trill perpetuities. And we can only guess what that would be, unless we issue them. And if we issue them, that will make the economy normal.

But both your questions lead me to ask myself: could some other good be playing the role of trills already? Something in fixed supply, that grows in value in proportion to NGDP? Maybe beachfront land? I tried to get my head around this is an old post, but failed.

jt: see my reply above to JKH.

JV: You have a point there. All cohorts gain, but the first cohort gains most. And every future cohort has an incentive to burst the old bubble and start a new one. I don't know how to handle that. Maybe they daren't for fear that if they did it, the next cohort would expect some future cohort would do it to them, and wouldn't buy the new bonds. (I vaguely remember discussing that question with JKH in an old post.)

JKH: Thanks! (Actually, I surprised myself by explaining it more clearly than I thought I would be able to!)

"Just wondering - is there anyway you can transfer some part of that intuition to an explanation of why interest rates eventually had to come down in the Volcker period in order to ensure "fiscal sustainability""

Hmmm. Probably. But nothing comes to my head right now.

Nick, I used to think Japan is weird because its NGDP keeps falling. Now I see it would be weird if it's NGDP didn't fall!

Seriously, are there two risk issues here? Ritwik mentions unstable NGDP growth. But even if NGDP and RGDP grew at precisely constant rates, wouldn't trills have more default risk? They are real bonds and hence it's not clear the government can meet the future interest obligations---unlike with nominal perpetuities.

And finally---first step toward a NGDP futures market?

Nick, I do not understand your reply to Ritwik. Why do we need to know the trill perpetuity rate and how would it help if we did? Ritwik's point is that we know in advance that the trill rate will be greater than g even as the nominal bond rate remains less than g. So when we actually issue a trill zero information is provided by observing the rate; we could still be in either a weird or normal world.

Parenthetically, I note that Ritwik has contrived to render the "less than" symbol without bringing on the apocalypse. By using "<" instead?

Umm ... I meant by using ampersand l t.

Ritwik: "Replace (1+g)/(1+r) as x"

Aha! Yep, that's much simpler.

Determinant: "This is first-year calculus. Just take the derivative of both the numerator and denominator until you get to a finite expression. If your first round does not succeed, wash, rinse, repeat."

That's what I thought it was, but I don't understand why it works. Plus (and this will shock you) I can't remember how to take the derivative of . (I used to know once, but I forgot).

It works because the expression y(x) = f(x)/g(x) is a compound form of two functions. Consider the two functions f(x) and g(x) separately. One goes off away from the x-axis into infinity, the other converges to zero. This is a contest of strength, and the one that is stronger gets further quicker. So it depends on the rate of curve of the functions f(x) and g(x), which is the derivative. Eventually the expression will either be an infinite limit (f(x)/0 is not infinity, it approaches infinity, but if you see the expression it means f(x) won) or a constant, in which case g(x) won.

You take the derivative of y = (1+g)^t wrt t by taking the logarithm of both sides.
log(y) =log (1+g)^t) Invoke another math rule about logs to powers.
log y = t*log(l+g)

Now differentiate.

1/y*dy = dt*log(1+g)

Now y and dt trade places.

dy/dt = y*log(1+g). This is a function of the form dy/dt = Ky where K =log(1+g).

This is contained in a first-year calc book, which may be found either in the library, in the math department or in the backpacks of the first years. I you can either visit the math department and buy the profs a coffee or buy a frosh a coffee and borrow their calc book.

Scott: I think trills are like shares in a country. The default risk would be if the government was either unable to collect the same share of GDP in taxes, or if it were unwilling to. Sure, a government that can print money is never unable to pay the obligations on a nominal bond. But it could "default" via unexpected inflation. Trills are going to have a lower default risk than inflation indexed bonds.

"And finally---first step toward a NGDP futures market?"

yep. ;-)

Two friends out hunting. One accidentally shoots the other. He dials 911.
"I've just accidentally shot my friend. He's dead."
Operator replies "Can you first make absolutely sure he's dead?"
BANG!

Before we issue trills, we can't know for sure if the demand price for trills at a quantity of zero is infinite. (Does the demand curve asymptote to the P axis, or does it hit the P axis at a finite price?)

But we know for sure the demand curve won't have an infinite price at any quantity strictly greater than zero. (BANG! Yep, the price is finite.)

Sorry. That's still not as clear as I want it. Did you see my response to JKH?

Determinant

Sorry to nit-pick, but these appear as two functions simply because we express them so algebraically. Economically, all that we are calculating is the PV of each cash flow. And then figuring out where the summation of all PVs converges to a finite value or diverges to infinity. So (1+g)/(1+r) is as economically meaningful a term as either 1+g or 1+r. The series itself is a summing up of the powers of the composite function and I'm not sure why we'd wish to engage individually with the (1+g)^t or (1+r)^t forms.

In any case, L'Hospital's is best used for ratios when the two series both diverge, but their ratios converge, and we don't quite know what that ratio is and so find it out using L'Hospital's. If it is trivial to show that both series may diverge, but the ratio either diverges to infinity or converges to 0, then I wouldn't use L'Hospital's and stick with the algebra.

Scott

If NGDP grows at a constant rate, then for ceteris paribus of the fiscal stance (tax rates/ spending decisions), a Trill is as good as a nominal risk-less payment. In this scenario, in equilibrium, the discount rate on trills reflects not default risk, but investors' expectations of the timing of apocalypse, which can be inferred from the duration of the trill (1+r)/r. Obviously, if investors are relatively sanguine about the world (as well as the monetary policy governing the world), trills will trade at NGDP growth rate + epsilon, epsilon tends to 0.

Ritwik: "If NGDP grows at a constant rate, then for ceteris paribus of the fiscal stance (tax rates/ spending decisions), a Trill is as good as a nominal risk-less payment."

Yep. I would say: a trill is as good as a nominal bond, if you are sure the central bank will stick to NGDPLPT.

I think this is an invented problem. Nominal Risk free borrowing rates have always been well below the growth rate of the economy. If the government incurs nominal debts and can issue currency, then its borrowing rates will be risk free. At the same time, the rental rate of capital has been well above the growth rate of the economy, e.g. http://www.economics.harvard.edu/files/faculty/40_Assessing_Dynamic_Efficiency.pdf

So there is no dynamic inefficiency and yet the government can continue to borrow whatever it wants and not worry. Ricardian equivalence cannot hold, as the government is the one with the consol, in the sense that its taxing authority is an asset that yields a fixed percentage of NGDP, but its liabilities are short dated risk free nominal debts.

I don't see any problem or weirdness. Rather, the weirdness is just an artifact of a model that assumes the rental rate on capital is the same as the risk-free rate.

Issuing an NGDP consol would be pure evil. The poor, who are credit constrained, would sell their consols to the rich at distress prices when a crisis hits them and they need to eat or buy shelter. A small class of "consol" gentry would emerge. There is only one actor in the economy that has a right to an NGDP consol, and that is the government, due to its taxing power. We should not allow anyone else to collect a perpetual tax on NGDP.

I also want to argue, if it's not already obvious, that there is an enormous demand for risk-free assets. One of my drinking buddies is a retiree, and he was complaining that before the crisis, his CDs were earning \$1000/month, but now he gets only \$40 a month in interest. I asked him why he doesn't buy bonds. He never touches bonds or stocks. It is all a casino to him. He keeps his money in a bank account and that's it. There are many people like that, and of course the banks are happy to oblige.

In addition to that, pension funds and other institutional investors have an huge demand for risk-free debt. Finally, there is a financial engineering demand for risk-free nominal bonds.

Going back to household demand for safety and liquidity, if the government were to squeeze out the banks and not allow them to create money, then the entire Federal debt could monetized over night just to meet the deposit demand of households. That is seignorage income lost to the government and handed over to the private sector banking system.

Economists massively under-estimate the demand for a guaranteed return, and equally underestimate the costs and difficulty of arbitrage between risky and riskless rates. Merely the costs of determining whether a company is under-priced or over-priced are so high that no one bothers and the firm can only borrow against collateral (land or risk-free bonds) whose future value is cost-effective to price.

Ritwik:

Nick wanted to take the derivative of y = (1+g)^t wrt t and said he couldn't remember how, so I did it for him.

If he wishes to change the constant, you get dy/dt = y*((1+g)/(1+r)). You could even express this as an exponential, but that's overkill.

What Nick wants to do with that is his call, I'm not an economist.

Showing is not proving, so it depends on how much rigour you want.

I think that it is difficult to buy bonds if you can not connect the dots. This is why I understand your drinking buddy.
Of course that banks are making money on people like him, but on the other hand, he still have not lost anything. This generation will not be that secure.

I find this article very interesting, but it is just a theory. In a more complicated world, government can not follow this simple assumption of surplus into debt´s interests. It would be perfect of course. If it would be so, there would not be anything that insane like the bubble in home prices in Vancouver. The only trill perpetuity I see is a big and real debt.

\$2.50/hr today. Really thought I'd push it above \$3/hr; would love Chretein's Trills. The enlightened Saudis, future investors in solar, give their citizens a six figure sum at adulthood.
Have to assume no tinker with GDP formula, no massive privatizations and stuff. Agree with JV Dubios. I use 20x payment/yr as the value of an annuity, unless itself is big enough to affect economic growth. Still having trouble with transfinite math. Your Trill is infinite in stated duration, but it isn't infinite in value. For instance, the value of 2x infinity (only even #s) is less than the value of infinity. Trills will always have a finite value when it comes to savings. So you need to prevent neutrons from splitting into protons and electrons in 4T yrs before you can ignore that Canada won't be around forever.
This parralels what I'm thinking about QALY. QALY accounts for quality-of-life in addition to longevity. I think this is a good way to judge politicians and efficiency of nations. A car gives you some mobility. An electric car (Zenn is dead CPC) does the same for our future trade partners. Transit-taxi saves you income and lesser mobility...telecommunication shares ideas. Being able to work where you want from where you want...QALY captures what GDP should turn into. It gets tricky when considering longer-term stuff or negative Q-of-L dystopias...I can't imagine why AB wouldn't want to give out Trills to cdns. I really need to get laid after \$2.50/hr wage after 4 cities, 3 of them in Western Canada, wouldn't employ me. I don't think I will; I hate RW too much.
There is an objective optimal tax/trill rate. Can be put as math. You have to determine the QALY of your output, as well as worker productivity levels. If your output is humane and your productivity high, you can optimally neocon your income. If your output is GHGs and your productivity lagging (ie. no R+D) you should be taxed more probably to pay out Trills and new industries. A key is how much will we have in our human capital. If the stuff hits the fan and we panic, should invest more in redundancy and capital-intensive emergency infrastructures.

Keystone: I have unpublished two of your comments that had wandered totally off-topic. Your comment above is barely coherent and barely relevant to the topic.

rsj: "So there is no dynamic inefficiency and yet the government can continue to borrow whatever it wants and not worry."

To my way of thinking, that sentence sounds self-contradictory. If the government can borrow and spend, without ever raising taxes, that means there's a free lunch that's not being eaten. And I would define an economy with uneaten free lunches as inefficient.

But on the other hand, IIRC, I have seem some economists define "dynamic inefficiency" in terms of the rate of return on *capital* (as opposed to govt bonds) being below the growth rate. Which I think is how you are defining it here. I would say that's sufficient, but not necessary.

I'm probably going to be away from the internet for a few days, so won't be responding to comments.

But people have decaying discount rates (citations...), and in businesses this can be reflected by things like monthly/quarterly reporting requirements driving demand for current cash to be safe from fluctuations. What would the median and marginal auction buyer pay for interest rates if there are different buyers managing different cash flow and planning constraints? No one can spend an infinite (or even relatively high) price for a bond like this because we need money now, whether as individuals, businesses or as government. So ... maybe the contradiction associated with the trill bond is not quite completely valid, so maybe r < g is possible? Probably not much lower though. I'm sure there are some creative financiers who have some interesting solutions for related problems, for services to very long lived Japanese, or perhaps institutional investors like pension funds, or those who figure it will simply take Japan an eternity to pay off its debt.

Excuse me, Nick, I see that I was not clear. Yes, I read and understood your reply to JKH. But what I was driving at was similar rsj's remark: "the weirdness is just an artifact of a model that assumes the rental rate on capital is the same as the risk-free rate."

Our chain of reasoning here has been: 1) if r < g, then the value of a trill would be infinite; 2) no asset can have infinite value; 3) therefore when we issue a single trill, circumstances must change so that the value of a trill is finite; 4) the circumstance that will change is r such that r > g. But this reasoning rests on a conflation between r_trill and r_nominal.

All of which reminds me of a bad old joke of my own:

1. A ham sandwich is better than nothing.
2. Nothing is better than anything.
3. Therefore, a ham sandwich is better than anything.

Another version: there is a close analogy between the trill argument and the St. Petersburg paradox. The concept of utility, though it may be flawed as the psychologists tell us, is nevertheless a useful one. But Bernoulli was wrong to think that the contradiction of the paradox offered support for utility because the paradox requires other assumptions which are known with certainty to be false. For example, that the game is played infinitely fast (or else that players are infinitely long-lived and value a future dollar at least as much as today's dollar.) Or again, as in the trill case, the St. Petersburg paradox requires the "house" to be a riskless credit in arbitrarily large amounts, which is impossible.

Nick: "In a weird world the value of a trill perpetuity would be infinite."

The value of a trill perpetuity would be infinite only if weird world continues forever. As Ritwik@09:02AM showed, the value is x+x^2+x^3+...., so if x becomes less than 1, i.e., r becomes greater than g at some point in the future, the value is finite. Therefore, only people who believe they live in a world of certainty which will never become normal again would perceive the value of a trill perpetuity as infinite. I suspect that few would believe such thing, so the impact of actual issuance of trill perpetuity would be minimal.

BTW, x+x^2+x^3+....+x^n = x(1-x^n)/(1-x). So if x is less than 1, it converges to 1/(1-x) as n goes to infinity. If x is greater than 1, (1-x^n) goes to minus infinity, so x(1-x^n)/(1-x) goes to infinity.

"I don't know if anyone has thought of this before?"

I now remember similar argument took place a while ago (in comment section) in the context of PV of seigniorage. As for money, r=0, so g would be greater than r only if it remains positive. So the government+CB obtain infinite seigniorage if they make money grow at GDP growth rate and keep GDP growth rate positive. This should be a lot easier than keeping GDP growth rate greater than trill interest rate, but that's what the Japanese government+BOJ have been failing...

To my way of thinking, that sentence sounds self-contradictory. If the government can borrow and spend, without ever raising taxes, that means there's a free lunch that's not being eaten.

No, it does not mean that. There could be several options here:

Government borrows and spends, but the economy is at full output and prices go up. As NGDP goes up, tax receipts go up (taxes are levied as a percentage of nominal income). As inflation goes up, the debt burden (which is nominal) naturally falls. Crowding out of consumption occurs, because the government purchased more consumption output reducing that available to the private sector. Consumption prices rose.

Government borrows and spends, but the economy is at full output, and the CB steps in and hikes interest rates so that prices do not go up. In that case, crowding out occurred on the savings front, and the price of borrowing rose -- but because the CB did it, not because Treasury did it.

There is an output gap, in which case there *is* a free lunch. But we know that output gaps are periods when there are free lunches.

All I'm saying is that the government can always "afford" to spend as much as it wants. But there are consequences to that, vis-a-vis crowding out of consumption or savings, that may prevent a free lunch from occuring. You only get the free lunch if output is below potential, and the size of the lunch is the size of the gap.

But I think that output is almost always somewhat below potential, so there is probably always a free snack, if not a free lunch.

And the main issue here is that (some) want to impose budget constraints on the government that are not operative constraints at all, but are always tautologically met. Government will never not be able to service its debt, or worry about being able to service its debt. For a government that is just a non-issue. The only thing that matters is the effect on the real economy of the fiscal policy. So you cannot say "This would be a great thing to do if only we could afford it". If it's a great thing to do, then do it, and you will always be able to afford it. If the loss in welfare due to crowding out is worse than the gain in welfare from the project, then the project is not a great thing to do and we shouldn't do it. But not because we can't afford to do it, or because we will be "burdening future generations". There have been many future generations, none of which have been burdened with the need for governments to run surpluses, but many of which have been burdened with poor infrastructure and in insufficient quantity of public goods.

Trills can't pay a 1 trillionth NGDP forever. They can only pay NGDP weighted for counter-party risk. Increasing the weight can never compensate for this.

So Phil Koop is right, that your analysis requires that "requires the "house" to be a riskless credit in arbitrarily large amounts, which is impossible." is correct.

Other factors are also at work, like "In the long term, we're all dead". What would you be willing to pay me for a bond that payed \$1 billion (real \$) on 7-13-3012?

What would you pay for a Roman bond from 7-14-0012 for 1 billion real sesterces (@ around \$6) payable tomorrow from the Roman Republic?

Investment in fat-tailed risk can only be profitable to a gambler with a finite lifetime. The gamble is that the probability of a catastrophic event is small in the period. And if it is large, it won't be made survivable by not gambling. If you live in a Las Vegas destroyed in a nuclear holocaust, not gambling at the Mirage won't save you.

RSJ is also correct that it is a bad idea to auction off the government's power to tax. It caused problems when the French and the Romans tried it. There's only 100% of fixed percentages available, and you would soon be faced with Henry VIII's problem:

"religious houses in the 16th century controlled appointment to about a two fifths of all parish benefices in England,[1] disposed of about half of all ecclesiastical income,[2] and owned around a quarter of the nation's landed wealth."

You may recall how Henry solved it.

The long term risk of governments has historically been very very close to 100%. Autonomous religious organizations seem to last the longest (a few like Ma Mundeshwari Temple [108 AD!], Saint Anthony's and Saint Catherines [early Byzantine] are doing OK, and Mt. Athos [reign of Justinian] is thriving). Organizations with historical physical gaps, like the Vatican (1870-1929 being the latest), don't count, since they would have defaulted on any creditors.

The "free lunch" arises when you *want* to crowd out investment because excessive investment is a drag on growth. You get higher consumption today and tomorrow by substituting public debt for investment.

The "free lunch" arises when you *want* to crowd out investment because excessive investment is a drag on growth.

Only if the rental rate on capital is less than the growth rate, which isn't the case for observable economies. But the interest rate on government debt is less than the growth rate for observable economies.

Nick's centerpiece of g > r implies something nonsensical in a standard economy. First, a few assumptions: (1) Y = C; (2) dY/Y = gdt + sdB; (3) Unique stochastic discount factor M; (4) CRRA representative utility with risk aversion b and discounting rate p. Second, a quick elaboration: (1) This assumption is to Nick's advantage; (2) Nick said the economy grows at rate g in our uncertain economy characterized by my choice of dB; (3) Complete markets and no arbitrage; (4) The usual suspect.

Ahem.

g > r implies E(dY/Y) = E(dC/C) > -E(dM/M). So let's look at the SDF: E(dM/M)/dt = -p - bg + b(b+1)s^2/2. Plugging in the negative of the last expression with our drift for Y (and identically C) we get: g > p + bg - b(b+1)s^2/2. For clarity, let's look at log utility: g > p + g - s^2 or s^2 > p. This last expression (s^2 > p) is the point of contention. Nick's statement requires a huge variability in output's (consumption's) dynamics. With a discounting rate of 0.05, the required variability is over 0.22--10 times reality's estimate.

Adding in a growing, deterministic inflation process would change the log equation to pi + s^2 > p, which could take a bite out of the claim, but for reasonable numbers (pi = 0.02, although now I'm not so sure...) it still requires something unrealistically volatile.

"It is well understood that if the rate of interest [on gov't bonds] is above the growth rate of GDP (both real, or both nominal, it doesn't matter) then we are living in a "normal" world "

Could you please explain what is normal about it? It seems crazy to me. (OC, we could both be right. ;))

Thanks. :)

Wouldn't this already imply that the current value of NGDP is infinite? If so, there is something wrong somewhere.

Bonds would be price on expected future interest rates in a normal world, adjusted for probability of getting to normal, so there would be an anchor around par.

Wouldn't this already imply that the current value of NGDP is infinite? If so, there is something wrong somewhere.

Why? Take the future NGDP of the whole world. That includes population growth, invention, innovation, etc. Perhaps in the future we will be colonizing other planets, or even creating our own universes in basements. Who is to say that the entire future of the human race needs to be finite in NPV terms? We are going to be imposing these arbitrary constraints on the future of humanity just to make garbage model not produce garbage results?

Something can grow forever, but because there is a discount factor, the current value is not infinite.

Ok. So there is a zero chance that current value of NGDP is infinite, because that would imply zero value of money ( infinite inflation) which is not possible because money would be abandoned before that happens. That allows us to price trills on a probability of "return to normal," which is how the market would price them, I am sure.

Infinite NPV of "the economy" does not mean infinite prices, as you cannot buy "the economy". You can only buy shares in firms that presently exist, but, in NPV terms 100% of the future value of the economy is in firms that do not yet exist selling products that have not yet been invented.

rsj: "Issuing an NGDP consol would be pure evil. The poor, who are credit constrained, would sell their consols to the rich at distress prices when a crisis hits them and they need to eat or buy shelter. A small class of "consol" gentry would emerge."

It is totally wrong, because:

1. If we suppose that the improvident poor did behave this way, they would behave in the exact same way whether or not consols exist. The rich can buy (say) 10 year bonds from the poor, and just roll then over every 10 years.

2. The policy I am advocating is for the government to issue ONE SINGLE trill perpetuity (maybe split into 33 million pieces). Lets assume rsj's nightmare scenario comes true. A small class (say 10 people) of Consol gentry buys up all 33 million pieces of that ONE trill perpetuity, and uses the income generated to support itself in idle luxury, forever. Lets see. One trillionth of Canadian NGDP,....should be enough to buy them....one medium double-double at Tim Hortons, once a year, with 10 straws.

Colin: "Nick's centerpiece of g > r implies something nonsensical in a standard economy."

Is your "standard economy" a model with infinitely-lived agents? (I think it is.) Most models where g>r are Overlapping Generations models (Samuelson's 1958 OLG model is where this all started.) Government bonds (or an unfunded govt pension plan, or "Money" in Samuelson) are supposed to provide an asset so the young can save and dissave when they retire.

Colin: to explain further, your assumption that Y=C is what's at issue. It's not the fact that it ignores I and G and NX. It's that what happens to the C and Y of a representative agent and what happens to aggregate C and Y are quite different. In an OLG model without some asset as a savings vehicle, the C and Y of an individual agent falls as he ages, even though aggregate C and Y may be increasing over time.

rsj: there are lots of different types of inefficiences (and associate uneaten free lunches) in economics. Roughly speaking, we can divide them into 3 groups:

1. Standard Micro stuff. Externalities etc.

2. Standard Macro business cycle stuff, due to bad monetary/fiscal/AD policy in the face of shocks with sticky prices or some other sort of nominal rigidity.

3. The sort of dynamic inefficiency stable Ponzi game chain-letter stuff I am talking about here. This inefficiency may exist even if all micro markets function perfectly, and there are perfectly flexible prices and no shocks, and even in a barter model.

It's best not to conflate 2 and 3. It just muddles things.

Phil and himaginary and Peter N and Robert:

(I like the St Petersburg Paradox relation).

The original models showing dynamic inefficiency assumed no shocks, no uncertainty, and an infinitely-lived government and economy. And they only had one interest rate.

My guess is that those models would generalise quite easily to a world where there were some fixed probability that the world would end with a bang.

My guess is that they would not generalise to a world where people knew in advance the world would end at time T, because the last generation of young wouldn't buy the bonds, so the whole RE equilibrium would unravel backwards.

My guess is that those models would also generalise quite easily to a world with two interest rates, but where the interest rate on government bonds was safe and the interest rate on equity was risky. If g>r on government bonds, we have a free lunch, even if g

What about a world where there is uncertainty about future growth and future r's and lots of different r's on different government bonds? My way of thinking is that issuing a single trill perpetuity provides a very cheap insurance (one trillionth of GDP per year) against dynamic inefficiency. The benefit/cost ratio is not infinite (as it would be in a true Ponzi scheme, which has zero costs), but it is very very large, because the costs are very very small.

I do not know (nor does anyone, because we can't see the future), whether there exists a true free lunch associated with dynamic inefficiency. But if anyone thinks there might be a problem, then this is a good solution. Because this lunch, though not strictly free, is very very cheap, and might be a cure for an ongoing problem of a shortage of savings vehicles.

Max: "The "free lunch" arises when you *want* to crowd out investment because excessive investment is a drag on growth. You get higher consumption today and tomorrow by substituting public debt for investment."

That is true. But I'm not sure if it is the best way of understanding it. Because we can get the free lunch even where investment is impossible.

Min: "Could you please explain what is normal about [a "normal" world]?"

We think it normal that if you borrow you have to repay. Any entity with a finite life has a well-defined intertemporal budget constraint, where the present value of debt is zero. That's the "normal" state of affairs for normal (i.e. finite-lived) agents. Does that generalise to infinitely-lived agents? In a world where r>g, the answer is "yes", the present value of debt is still zero. So I call that a "normal" world.

himaginary: yep, currency pays zero nominal interest, (and pays negative 2% real in Canada, given the 2% inflation target). So currency metts the definition of a Ponzi game. And Milton Friedman's Optimal Quantity of Money argument said the government should let us eat this free lunch, by either paying interest on currency, or else creating deflation equal to the real interest rate. But currency is special, because it is also the medium of account, and of exchange. We might not want 3% deflation, in a world where prices are sticky.

Very general questions (ideas for a future post!) ...
(1) what are the pros and cons between issuing 1 vs. 1000 trills?
(2) related question: is there a generally-accepted model for the optimal term structure of gov obligations (i.e. why \$XXXB in 3 year, \$XXB in 30 -year, \$XXB in TIPs etc.). You often hear answers like: "trying to match the private sector's demand for XXyr assets","acts as a price discovery mechanism, e.g. TIPs vs. normal bonds"
(3) if the purpose of trills is to help price discovery, how useful is this? I.e. is 1 trill enough market information to help the central bank do NGDPLT; similarly, has TIPs implied inflation really helped CBs target inflation?

Nick: "If the world were weird (i.e. if r were less than g), then the value and market price of that trill perpetuity would be infinite"

I don't see that. Why would you discount it at the risk free rate? You are assuming people are indifferent wrt NGDP risk, which seems very unlikely. I'd assume people are extremely averse to low states of NGDP. If the rental rate of equity capital is 10%, or whatever, in nominal terms, then I'd expect the required return of NGDP shares to be high too. Both are definitely high beta assets, and totally different from fixed rate perpetual bonds in an inflation targeted economy.

Sorry rsj, I am not trying to be thick, but doesn't infinite NPV indeed require an infinite price level? The real value of the current GDP is fixed by technology, labour, capital and is finite. The price level therefore must be infinite in order to generate an infinite NPV. Since an infinite price level means zero value for money, it will not happen because money would have been discarded....

Nick, the relevant "r" for dynamic efficiency is the r of capital, not the r of government debt, right?

No government budget constraint (r_money < g) doesn't imply dynamic inefficiency. Reducing investment reduces growth. This is a normal world.

Sorry rsj, I am not trying to be thick, but doesn't infinite NPV indeed require an infinite price level?

No, infinite NPV of something that is not traded does not require an infinite price level.

Think of it this way, low rate basically correspond to longer time horizons. It means money (with the shortest maturity) is more valuable and not less valuable. As the maturity stretches out, NPV of "the economy" begins to rely more and more on firms that do not yet exist today, and so cannot be traded. With a very low discount rate, you care very much about what the economy will be like in, say, 500 years, or 10,000 years. But you cannot trade those firms (if we even have firms then) and other effects come to dominate. That long horizon does not mean money is worthless.

jt:

1. Hmmm. Dunno. The answer probably depends on the risk-aversion of bondholders and taxpayers, plus what sort of risk they face.

2. Hmmm. Dunno either. That's a question I have sometimes asked myself. Again, the answer probably depends on risk aversion, time-horizons, and the risks faced, by bondholders and taxpayers. There presumably (I hope, given the policy relevance) is some sort of literature on this, but it's not one I'm familiar with.

3. I don't see trill prepetuities (in this post) as designed for price discovery. I see them as changing the equilibrium. I *think* TIPs have given inflation targeting central banks a little bit more information that has helped them target inflation better. But I don't know for sure. I do know (at least in principle) how to find out, but it would require running a couple of regressions. Here's my old post saying how.

K: "I don't see that. Why would you discount it at the risk free rate?"

I don't think I would want to discount it at the risk free rate. I would want to discount it at the rate on trills. That is what "r" is supposed to mean in this context (yes, I wasn't clear on this, sorry.) What we are looking for is a stable ponzi that is risk free for the government.

Max: "Nick, the relevant "r" for dynamic efficiency is the r of capital, not the r of government debt, right?"

Well, I read some economists saying that, but it doesn't seem right to me. Because, for example, we can imagine a world without any capital at all, and there could still be dynamic inefficiency. The original Samuelson 1958 model was like that. To my way of thinking, if the government can borrow, and keep rolling over the loan + interest forever, and rational people knowing that are still willing to lend, there's a free lunch.

(And remember what Milton Friedman said about the inefficiency of money paying less interest than govt bonds. Optimum Quantity of Money argument.)

Robert: rsj is right. But let me try to explain it my way.

Example 1: Suppose NGDP=\$100, and it stays constant at \$100 forever. If the rate of interest were 0%, the NPV of NGDP, from now until infinity, would be NPV=\$100+\$100+\$100 etc. for ever, which is infinite.

Example 2: Now suppose r=5%. Then NPV=\$100 + \$100/1.05 + \$100/(1.05)^2 etc. which is finite.

Example 3. Now suppose r=5% but the growth rate of NGDP is 6%. Then NPV = \$100 + \$100(1.06)/1.05 + etc. which is infinite.

Well, I read some economists saying that, but it doesn't seem right to me. Because, for example, we can imagine a world without any capital at all, and there could still be dynamic inefficiency. The original Samuelson 1958 model was like that. To my way of thinking, if the government can borrow, and keep rolling over the loan + interest forever, and rational people knowing that are still willing to lend, there's a free lunch.

Is seignorage a "free lunch", or is an exchange that benefits both parties? How do you define free lunch?

This is no different than seignorage. Think of money as a 0 maturity liability of the government. People are willing to keep holding a stock of money without ever being "repaid". There is a continuuum from the zero maturity asset to the one day maturity asset up until the consol. Think in terms of this smooth continuum.

Now, for a given yield curve, the household sector as a whole has a certain asset demand for risk-free assets. It may be that they demand \$100 of 0-maturity money, \$200 of T-bills at 3% and \$400 of consols at 5%. Perhaps the economy grows at 6%, and the rental yield on capital is 8%.

If we fast forward the economy (assumed to be in equilibrium) by a 100 years, then there will still be a demand for \$100 of money, \$200 of bills, and \$400 of consols. Those liabilities will be continuously rolled over by the government and will never be "paid back".

I think this is an important difference between micro and macro. A representative agent is not a person that borrows and repays, an RA is really the household *sector*. This sector maintains a certain level of mortgage debt, for example, and holds a certain level of government debt. The macro equivalent of repaying is something like debt to income levels returning to some long run ratio. But then you need a theory that determines what the ratio should be, and what accounts for its movements.

There is no reason to believe, IMO, that just because the household sector wants to hold \$100 of T-Bills at a rate less than the growth rate of the economy, that there is a free lunch economically speaking. They are obtaining value in exchange for holding those t-bills.

If r=5% and NGDP=\$100, then NPV (We do mean net present value, right?) collapses to 100/.05, which is \$2,000.

My argument is that, sure, if the growth rate is higher than the discount rate, that would imply an infinite present value in theoretical terms. But, practically, that cannot be right. Nothing can be of infinite monetary value today UNLESS MONEY IS OF NO VALUE TODAY, i.e. x/0 is infinite.

At this point, money will not be used, thus it is impossible to measure anything with it.

The price of a trill will never be infinite because investors will never accept a growth rate higher than the discount rate in perpetuity. When valuing stocks that have grown 30% annually over the last five years, for example, they will routinely assume a future growth rate of, say 3%, in order to get a tractable solution. Or they will assume that at some point the equity reaches par (or book value) at some point in the future when cost of capital equals return on capital.

I have read wha you have to say but I am not convinced!
Trills would be calued in the same way.

Robert, how would the market "allow" or "not allow" an interest rate for something that is not traded? But if you know of a way to purchase a share of NGDP growth, let me know! That would indeed be magical.

I was interested and trying to understand the exchange between Colin and Nick, and found this paper. Now I realized that what Colin was talking corresponds to "the risk-free rate puzzle" referred in this paper. And, yes, an attempt to solve this puzzle uses OLG model, as Nick noted.

Ah, I see the problem and why we are talking past each other.

1) I assume that the instrument IS being traded.
2) When I say the price cannot be infinite, I am simply saying that implies an infinite supply of the unit of account.

If I am wrong that this is the difference between us, then I suspect it may be due to an inability on my part to understand.

Oh, and as for a share of NGDP growth, no, I know not of perfect a way - until trills are invented. But a broad stockmarket ETF is perhaps closest.

hgimaginary,

That is a *great* paper. And it makes a lot of sense, too. I wonder why it has not received more attention.

Robert,

A stock market ETF will not give you a share to a perpetual slice of NGDP growth.

The point being that when some synthetic asset has a theoretical price of infinity according to some valuation assumptions, then

1) those assumptions are wrong, or
2) the synthetic asset cannot be traded, or
3) the synthetic asset cannot be traded for its theoretical value (it will be sold only under circumstances of distress for less than the theoretical value)

Note that I omitted your preferred option, 4) Money becomes worthless. The reason why is that large nominal savings demands that are so large as to create the theoretical infinite price for the stream of cash-flows are the exact opposite of what makes money worthless.

Rsj,

I think we are finally agreeing!

ETFs - yes, indeed. I agree. I don't believe they give you NGDP growth.
1) yes
2) yes
3) yes

4) no. (so close) large savings demands can indeed push interest rates to zero because of a huge demand for money but they do so temporarily. (Perhaps for a month, a year, or a hundred years....) But they CANNOT push rates to zero or indeed below growth rates IN PERPETUITY.

Well, Robert, that is just an assertion. The historically observed risk-free rates have indeed been below the growth rate of the economy, at least for intermediate maturities. And yet money has value. You need to reconcile that empirical observation with your beliefs.

..one way to reconcile it is by looking at quantities, rather than the micro assumption that assets are available in infinite quantities. We know in the U.S. that the households sector wishes to hold deposits that pay almost no interest in quantities roughly equal to the half of GDP or so. And they want to hold government bonds roughly equal to half of GDP that pay about half the growth rate of the economy. So as long as the quantity of risk-free bonds supplied is less than this, barring changes in aggregate preferences, risk-free rates can continue to be below the growth rate of the economy. So think in terms of a demand curve, pick a quantity of debt, and you get an equilibrium rate for that quantity. Nothing requires the quantity of debt to be such that the equilibrium rate needs to be higher than the growth rate.

Ok. I agree it is an assertion. Your latest comments are going to require a bit of pondering on my part. As for empirical stuff. Yes, can be below growth rate for a time, but not on average.

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