A trill perpetuity is a bond, which promises to pay the owner a one trillionth share of nominal GDP (that's the "trill" bit), every year forever (that's the "perpetuity" bit).
Trills, AFAIK, do not exist, though economists have thought about them. Perpetuities do exist (but are rare). Trill perpetuities do not exist. But they are a useful way of thinking about a problem that does exist.
That problem is dynamic inefficiency, and the possibility of sustainable Ponzi schemes, under uncertainty.
First lets think about dynamic inefficiency and sustainable Ponzi schemes in a world where it is known with certainty that the rate of interest and the growth rate of the economy will stay constant over time.
It is well understood that if the rate of interest is above the growth rate of GDP (both real, or both nominal, it doesn't matter) then we are living in a "normal" world where the long run government budget constraint is well-defined. The long run government budget constraint says that the present value of primary budget surpluses equals the value of the current stock of debt. (The "primary" budget surplus is defined as tax revenue minus government spending ignoring government spending to service the debt.) If the government runs a primary deficit today it must run a primary surplus some time in the future to pay at least the interest on the debt. If it doesn't do that, the debt will grow at the rate of interest, which is faster (by assumption) than the growth rate of GDP, so the debt/GDP ratio will grow without bound, and some day the next generation will be unwilling or unable to buy all the debt from the old generation because their income won't be big enough.
It is also well-understood (at least, by economists who have thought about this question) that things are very different if the interest rate is less than the growth rate. That's a weird world, where things are not normal. The government can borrow $100 today, spend it (or give it away), and then keep on borrowing to pay the interest, forever. The debt will grow at the rate of interest, but the economy (by assumption) grows faster than that, so the debt/GDP ratio will eventually shrink towards zero, and it's all perfectly sustainable. Somebody (whoever got the original $100) was made better off, and nobody was made worse off, because nobody had to pay higher taxes to pay for it. It's a free lunch, and any time there's a free lunch which isn't being eaten the economy is, by definition, not efficient. An economy in which the interest rate is below the growth rate is an economy that needs a (bigger) bubble/Ponzi scheme/chain letter swindle to operate efficiently.
In a weird world, but a world of certainty, we would know what to do. The government should borrow and spend, and not worry about paying even the interest on the debt. Borrowing and spending will eventually push up the rate of interest towards the growth rate. And it should keep on borrowing and spending until the world was just on the cusp of being normal. And then it should stop borrowing and spending, when the interest rate is just equal to the growth rate, so the debt/GDP ratio stays constant thereafter.
Trouble is, we don't live in a world of certainty, where nothing ever changes. Even if the world is weird right now, it might be normal again in the future. If the government borrows and spends when the rate of interest is below the growth rate, it might find in future that it's got a problem, if something changes and causes the rate of interest to rise or the growth rate to fall. Because the debt/GDP ratio will start to rise, unless it increases taxes or cuts spending. Maybe the government could cross its fingers and wait to see if the world turns weird again, and hope it turns weird again before the debt/GDP ratio gets too big?
Simon van Norden points me to a very good paper by Ball, Elmendorf and Mankiw (pdf JSTOR) that looks at this problem. They show the benefits of the borrow and spend strategy in a weird world. But they also note the risks of the world turning normal again, so that a Ponzi scheme becomes unsustainable. They try to quantify those risks by looking at past US experience of interest rates and growth rates.
I was reading that paper, and it was a hard paper, until my ADD kicked in, and I thought: "there must be an easier way to think of this". Then I thought of trill perpetuities.
The government never needs to rollover a perpetuity, so it never faces the risk that the interest rate will be higher when the bond matures, because a perpetuity is a bond that never does mature.
The annual coupon payment on a trill is indexed to nominal GDP. So the government never faces the risk that nominal GDP will fail to keep pace with the annual coupon payments on trills.
Put the two together and you get trill perpetuities.
In a normal world the value of a trill perpetuity is given by the present value of the coupon payments, which grow at rate g, discounted at the rate of interest r. The formula is P=1/(r-g) trillionths of current annual nominal GDP. So if r=5% and g=4% the market price of one trill perpetuity would be 100 trillionths of nominal GDP.
In a weird world the value of a trill perpetuity would be infinite. (I know a literal reading of the formula says it's negative, but trust me on this, because I don't want to have to resort to L'Hopital's rule, which I barely understand anyway).
The solution to dynamic inefficiency under uncertainty is for the government to sell one trill perpetuity. (Maybe broken up into a thousand or so pieces if that makes it smaller and more liquid).
If the world is weird, in the appropriate "expected present value to infinity" sense, the government gets enough revenue from selling a single trill perpetuity to pay off the national debt and go on a spending spree, and the sale of the single trill perpetuity is enough to stop the world being weird and dynamically inefficient. Which is a good thing.
If the world is normal, in the appropriate "expected present value to infinity" sense, it's no big deal, because it's just a single trill perpetuity. Use the revenue to pay down some of the normal debt.
[Update: In response to a comment from JKH, I clarified my explanation with this response:
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.
End Update.]
I think that's right. I don't know if anyone has thought of this before?
(This whole thing started out when I was thinking and worrying about Japan's debt/GDP ratio, which is rather large. Which is an excuse and a segue for:
Worthwhile Canadian Initiative 
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<r on equity.
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.