This post won't be as clear as I want it to be. I'm trying to get my head straight on something. Sorry.
Why are real interest rates positive? Turgot's answer was "Well, suppose they weren't, and never would be. Then the price of land would be infinite, because the present value of the rents would be infinite, so any landowner could sell off a tiny plot of land and use the proceeds to buy an infinite amount of consumption forever. And every landowner would want to do that, so land prices would fall, until they were finite, which means the interest rate would be positive." (OK, that's an extremely loose translation from the French. OK, I made it up.)
Could real interest rates ever be less than the growth rate, forever? Samuelson's answer was "Well, suppose they were. Then a totally useless asset, if it were in fixed supply, could become valuable, and its value would rise over time at the same rate as the growth rate of the economy, so the real interest rate would equal the growth rate." (Another very loose translation, from the math.)
Samuelson called that totally useless asset "money". People hold Samuelson's "money" only for the capital gains it provides. In Samuelson's model, people save enough "money" when young to live off when they are old and retired.
Stefan Homburg said that land is in fixed supply. And unlike Samuelson's "money", land isn't totally useless. Land pays rent. So land will always beat Samuelson's "money" as a form of savings. So Samuelson was wrong. People will always prefer to save by holding Turgot's land than by holding Samuelson's "money".
But in the limit, as the very long term rate of interest falls and approaches the very long term growth rate of the economy, because more people want to retire for longer and so want to save more, Turgot's land gets closer and closer to looking like Samuelson's "money". People hold it more and more for the capital gains, and less and less for the rents.
And if people ever lost confidence in land, from some irrational fear, they might switch from land to Samuelson's "money". It would be irrational because they would be losing confidence in a near bubble asset and switching instead to a pure bubble asset. But if they did that, the total value of "money" they hold would need to rise enough to replace the fall in the total value of land. Which is a very big increase in the real amount of "money". And if it didn't, so there weren't enough "money" to meet the increased demand, bad things would happen. Like a recession.
When people talk about rising house prices what they really mean is rising prices of the land the houses are built on. And farmland prices have been rising recently too.
Now for some math.
Here's an example where the math is simple.
Assume that land provides a service and people have Cobb-Douglas preferences for that service. That means that the price and income elasticities of demand for that service are both one. That means that people spend a constant fraction of their income on that service. And if the supply of land is fixed, that means that land rents will grow at the same rate as the growth rate of the economy. That means that the price of land, which equals the present value of the rents, will be determined by:
P(t) = R(t)/(r-g)
Where P(t) is the price of land at time t, R(t) the rent at time t, r the interest rate, and g the growth rate (both nominal or both real, it doesn't matter).
If r and g are constant over time, the price of land and land rents will both be rising at rate g.
The equation tells us that as r approaches g, the price/rent ratio rises towards infinity. It will look like a bubble, and will be close to being a bubble, but won't be a bubble. It is Samuelson's "money" that really is a bubble.
If I am right about this, the financial crisis was not caused by the bursting of a land bubble, because it wasn't a bubble. It was caused by the appearance of a "money" bubble. And the crisis will only end when Turgot's land once again replaces Samuelson's "money".
Stefan Homburg shows you don't need to make any special assumptions like Cobb-Douglas preferences and constant interest rates and growth rates to show that Turgot's land always beats Samuelson's "money". But his math is too hard for me. And the Cobb-Douglas case is easy to solve and understand. And I think it's roughly plausible, if we are talking about preferences to live in nice locations that are in fixed supply.
I thank commenter Herbert for tipping me off about Turgot and Homburg. This is helping me get my head straight on some ideas that have been floating around in my mind for some time. But I know I'm still not quite there yet.
Worthwhile Canadian Initiative 
Nick: ” But if they did that, the total value of “money” they hold would need to rise enough to replace the fall in the total value of land. Which is a very big increase in the real amount of “money”. And if it didn’t, so there weren’t enough “money” to meet the increased demand, bad things would happen. Like a recession.”
I’m not sure I understand it correctly but my interpretation is, should there be either deflation or growth of money supply big enough? In my mind there are two ways to increase total value of money. 1) raise the unit value of money holding money supply constant. Since the unit value of money is 1/P, deflation will raise the value of total money. 2) increase the quantity of money faster than increase in price level. Is that what you’re saying?
Chun: yes. Either the nominal quantity of “money” will have to rise a lot, or else the price level will have to fall a lot, to increase “M”/P by enough to equal the fall in the total value of land. And P didn’t fall at all (because prices are sticky), and “M” didn’t increase enough. (Note that Samuelson’s “money” is outside “money”, because it isn’t a liability of anyone.)
Nick,
Samuelson’s “bubble” only exists (as you know) in economies that exhibit a dynamic inefficiency (the equilibrium real rate of interest is less than the growth rate). This can be easily demonstrated in an OLG endowment model with zero assets. But once we add an asset (a Lucas tree, or income-generating land), the bubble equilibrium disappears. Homburg is correct: Turgot’s land always beats Samuelson’s money (assuming identical risk characteristics, of course). The way to think about is that land corresponds to money in the limit as the income from land approaches zero.
I have modeled the “bursting bubble” effect you are thinking about here: http://andolfatto.blogspot.com/2010/07/interpreting-recent-movements-in-money.html
In my model, a “bad news” shock lowers the expected return to capital spending, leading to a “flight to quality” (a substitution into money assets). There is a recession.
There is also this paper here by Narayana Kocherlakota that looks at the effects of a land price decline through the lens of a more traditional IS-curve analysis: http://www.sfu.ca/~dandolfa/kocherlakota%202013.pdf
The fundamental question you are asking, I think, is what causes an increase in the demand for money (more broadly, government securities)? Is it just psychology (irrational)? Or is it the byproduct of a rational pessimism? The answer to this question matters greatly, but as far as I can tell, no one really knows the answer.
David: thanks. Good comment.
“(assuming identical risk characteristics, of course)” Hmmm. I had forgotten about that. Assuming identical liquidity too.
I don’t know the answer either. But whatever it is, it seems to become more likely that something will do this, and have a bigger effect, as r-g approaches zero.
“P(t) = R(t)/(r-g)
Where P(t) is the price of land at time t, R(t) the rent at time t, r the interest rate, and g the growth rate (both nominal or both real, it doesn’t matter).”
I would not calculate the price of land using your method. First I look at the earning power of land and then capitalize that value. Then, I would look at the present value of land, which is present capitalized value (based on someone’s judgement). Thirdly, I would look at the expected inflation and capitalize that value. Then I would would look at taxes and make a judgement; how well does land fit with my future expectations of need for money or goal accomplishment?
I would consider the formula you suggested to be incredibly misleading, especially the negative value of g.
Roger: my formula does capitalise (calculates the present value of) the rents (earning power) of land.
Picking up a key part of Nick’s comments:
Since a lot hinges on that assumption, consider the data for the U.S., where the assumption would appear to be valid.
Nick: Your formula yields an infinite price of land at the r-g = 0 point. That discontinuity creates an invalid point.
Land can be assumed to provide an income stream that will increase with inflation. The present value of land is based on that notion; the present value of land (held in the hands of an informed owner) would already be inflated over land’s earning power from providing utility.
I would suggest the formula:
Present Value = capitalized-rental-value + present-value-of-inflation-adjusted-income-stream
with many ways of presenting the present-value-of-discounted-income-stream.
By separating the rental-value and inflation-based-capital-increase, the point of discontinuity is avoided.
Nick: Your formula yields a discontinuity at r-g = 0. It is incongruent to equate a currency-inflation-rate-equal-to-interest-rate situation to an infinite value of land.
If we assume that the present value of land (held in the hands of an informed owner) is the sum of two components, then the discontinuity can be avoided.
I would suggest the formula take the form of
Present value = utility-value + value-as-an-inflation-hedge
with ‘value-as-an-inflation-hedge’ calculated in many ways.
This avoids the discontinuity.
Nick,
Your formula is discontinuous at r-g = 0. It could not be programmed into a computer unless the r-g point was isolated by a trap.
I suggest reconstructing the present price by dividing it into a utility component and an inflation hedge component.
This is close to what happened in the most bubblicious areas where mortgage rates fell to local growth rates. What happens somewhere like China where eventually population will decline more than the productivity growths, rents and property values fall, (though it may induce more population growth)?
Ironman: very good find! thanks. (Ideally, we would want some sort of rental equivalent data, because inflation will distort the (nominal) interest payments on mortgages a bit, but it’s probably close enough to give the main story.)
Roger: but that’s the whole point of the post: as r-g approaches 0, price/rental ratios approach infinity.
Lord: I don’t know. I can’t do the math unless the growth rates and interest rates are (expected to be) constant over time. (I tried to figure out the formula where r-g was falling towards zero at a constant rate, but gave up.)
Well that certainly quantifies the thinking of many people in this country.
Nick,
I apologize for the triple post, each basically saying the same thing. My computer said the comment posted but when I rechecked your post, it was NOT there; so I tried again, thinking I had goofed some place. I managed to do that three times. Sorry.
Thanks for the thought-provoking post.
Roger: no worries about the triple post. It wasn’t your fault, it was our accursed spam filter. (I’m trying to “train” it, but it seems OK when you post under MechanicalMoney. It has renewed its hatred of me!)
This is extremely interesting, I’m very excited you’re writing about this. I have been thinking about these questions a lot lately. In my case the entry point is Cassel rather than Turgot. (I just read his Nature and Necessity of Interest for the first time.) But the logic is the same.
I think it is worth distinguishing the claim that i must be greater than zero, from the claim that i must be greater than g. The latter requires stronger assumptions than the former. The former also seems better supported empirically. History does not offer us many examples of sustained negative real interest rates without explosive growth of asset prices. It offers plenty of examples of sustained interest rates below growth rates, without any obvious instability of asset prices.
For the positive i result, all you need is a long-lived asset like land that is expected to provide some positive income in all future periods. But as long as the income from any asset that exists today will decline toward zero after some point in the future — which I think is perfectly reasonable; technology creates substitutes for everything, including land — there is no reason that the “natural” rate of interest cannot be below the growth rate.
(OK, I still need to read Samuelson 1958. Maybe there is a stronger argument that I am not seeing.)
The analysis here is about the demand side of the market for loans. We should also be thinking about the supply side. What I think is interesting there is that Cassel — and Bohm-Bawerk, Fisher, etc. — did not have the modern assumption of a bequest motive that makes consumption in the distant future interchangeable with consumption today. If people are not very interested in the income of their grandchildren, let alone remoter descendants, that also sets a floor on the rate of interest. Cassel has a very interesting argument that it is ultimately the length of a human lifetime that determines the minimum market interest rate, since people will never purchase an asset at a price that implies an appreciable fraction of the value comes from income flows more than 30 or 40 years in the future.
This kind of reasoning also strongly implies that the market rate of interest will always be above the socially optimal rate. It is very rational for me to discount goods many years in the future, since I may die before I am able to enjoy them. But for us collectively, the happiness of people in 2050 or 2100 should presumably get the same weight as the happiness of people alive today.
(And yes, I know that I might value an income stream in 2113 even though I won’t be around to receive it, because it can eventually be sold to someone who will be. That’s where liquidity has to come in to the story, I think.)
Leijonhufvud says that long-lived assets are normally undervalued, because “no one can hold land to maturity.” I think he is right.
I should add that without some account of the demand side like I suggest, your story work.
If an arbitrarily low interest rate corresponds to the discount rate used by landowners, they have no incentive to sell land now to finance expenditure today. It is true that as interest rates go to zero, the amount of present consumption that can be financed by the sale of a small quantity of land goes to infinity. But the present value of the future consumption foregone by the sale, ALSO goes to infinity. So for Turgot’s reasoning to go through, you need some independent reason for thinking people discount future consumption at a positive rate.
I meant, your story DOESN’T work.
A better approach IMO, which Cassel takes, is to focus not on land but on long-lived capital goods. Bridges, tunnels, harbors (his examples), railroads, many buildings, various kinds of IP, have very long useful lives. As interest rates fall the present value of these assets rises — or, equivalently, the cost of their services falls — very steeply. Long before long-term interest rates reached zero, the cost of this kind of investment would fall to a small fraction of its present value, and the resulting boom in investment demand would presumably halt the fall in interest rates.
So a condition of long rates reaching zero is absolute capital saturation, which is presumably a long ways off if it can be reached at all.
I think this argument is stronger than yours, because it does not depend on a divergence between the market interest rate and the discount rate used by market participants.
(OK I’ll shut up now.)
JW: It’s really good to hear you are thinking along similar lines. (Here is one of my very muddled earlier efforts!)
I can’t remember if I ever read Cassel.
“I think it is worth distinguishing the claim that i must be greater than zero, from the claim that i must be greater than g. The latter requires stronger assumptions than the former.”
That’s what I thought too, but my reading of Homburg is that he’s saying this is basically wrong; he’s saying that all it needs is infinitely-lived land. (That can’t be totally right, but I’m not up to figuring it out with full confidence that I understand his theorems!) We are talking about very long run interest rates and growth rates (in a sense of “very long run” that Homburg defines).
“If people are not very interested in the income of their grandchildren, let alone remoter descendants, that also sets a floor on the rate of interest.”
In Samuelson’s model there are no intergenerational bequests. If there are no assets at all, the natural rate of interest can be very negative. For example, if people can only work when young, and cannot work when old, they will want to save for their old age even if the interest rate is very negative. (There’s nobody they can lend to, except each other.) Introducing “money”, a useless asset with zero carrying costs but in fixed supply, pushes the interest rate up to the growth rate.
OK, I will read the Samuelson article.
Cassel frames the question in terms of a choice between an infinitely-lived asset and an annuity, for what it’s worth.
I wonder how reasonable is the assumption that there is such a thing as an asset in fixed supply.
JW: can we imagine a world in which there are two categories of goods. “A” goods can be traded for each other, at finite relative prices, and “B” goods can be traded for each other, at finite relative prices, but A goods are never traded for B goods, because nobody would ever sell an A good for B goods, even though everybody would like to buy A goods for B goods? (“A” goods are infinitely-lived assets like land, and B goods are everything else.)
I think in an OLG model, with finite lives and without bequests, Turgot’s argument works to rule out infinite land prices. If the price of land was infinite, you could sell a square inch of land and consume the whole of GDP for the rest of your finite life.
“A better approach IMO, which Cassel takes,…”
I remember my high school economics teacher saying that if r was zero forever it would be profitable to bulldoze the Rocky Mountains flat and convert them into productive farmland you could rent out!
You are correct just considering money and existing land stocks. There are more than these two assets to consider.
…stakeholder theory vs mere shareholder theory. In nations where the former is taught in biz schools and espoused by the MSM, bankers will act to create wealth rather than crash the economy.
Land is usually preferable vs money if those are the only two assets. But land isn’t very divisable; in the developing world, people often wind up selling land below some critical personal consumption farm size, and it is a terminal course towards manual labouring or death from that point. In this thought experiment, a bad enough harvest leads to the end of society without investment in grain storage.
…in your thought experiment, the citizens are dead in the long-term, just like the Nile Delta was doomed. It became a beauracratic mess of scribes copying past rhetoric, mostly because there was no where to hide from the church/rulers if you weren’t a snake or something.
In the real world, we are dead at our present trajectory, but money can be used in our fractional banking system for investment, hopefully in assets and teaching that lead to a better society. Land, is useless again without at least grain storage, as weather varies from year to year. Your people need metallurgy and rationality. The whole economy is basically a means to the end of creating a stable better future.
…finally figured out why the thought experiment is useless. It assumes your renters age from working age to being elderly, yet it assumes the market of potential renters stays perpetually the same. In addition to famines, there have been pandemcis and wars throughout history. During the Black Death, the earning potential of labourers doubled! The got some choice residences then, and became freeman moreso, for a few generations. You people must age, yet the # and market of renters is static: this is an illogical paradox. You can fix it, but any thought experiment without technological improvement or the fruits of technological improvement is useless and may distract students and dumb adults.
I remember my high school economics teacher saying that if r was zero forever it would be profitable to bulldoze the Rocky Mountains flat and convert them into productive farmland you could rent out!
Yes, I think this is the decisive argument against the possibility of long rates falling to zero.
(By the way, is it a usual thing to have a high school economics teacher? I went to a good public high school
In the US, and I am pretty sure there was no such thing as an economics class. The only economics I got in high school was Heilbroner’s Worldly Philosophers, assigned – for some reason – in a European history class… )
I wonder how important is the assumption that people save only to finance future consumption. If you allow agents whose objective is to maximize wealth, does that make it possible to have i less than g? If so, the existence of sustained periods of i less than g would be informative about the actual motives for saving. Zero i would still be out of the question in this case, because of the paving the Rockies thing.
I wonder whatever happened to rsj, who used to comment here. He was big on putting money in the utility function.
Nick,
In Samuelson’s model the bubble that is the value of fiat money in an OLG world is a stationary one. This is quite different from the sort of exploding bubble we saw/(see in Canada?) on housing. Makes a big difference.
I would like to add a few remarks on the interesting article and the discussion.
1. Regarding priority, one must keep in mind that Turgot is 18th century, whereas Cassel started writing in the 19th century, as did Bohm-Bawerk.
2. Turgot showed convicingly that i > 0 holds in a stationary state, where g = 0. In order to obtain the more general result i > g, Homburg needs one additional assumption, namely, that the land’s income share is bounded away from zero. If the land’s income share vanished in the limit, the economy would obviously behave like an economy without land.
3. What impresses me most about Turgot is that as late as 1935, Pigou (in his Economics of Stationary States) thought that i < 0 would well be possible. Other authors like Bohm-Bawerk, Cassel, or Fisher, attributed i > 0 to an alleged (if not somewhat ideological) premium for waiting, whereas Turgot had long shown that i > 0 holds independent of preferences, technologies, or bequest motives.
4. This reasoning has also an application to PAYG pension systems. These yield a return equal to g and must be enforced by law because i > g. And while Diamond 1965, in his famous national debt paper, thought that government debt could yield a Pareto improvement in case of i < g, the existence of land rules out that possibility.
5. For classroom purposes, the logic is as follows: Holding money in a stationary state keeps net worth constant. Holding land instead keeps net worth constant also, but yields an additional utility (if the land is used for a house) or additional profit (if the land is rented out). Hence no one would hold money in a perfect foresight equilibrium but manias and panics can disturb this mechanism in the short run.
6. Forecasters can take the nominal growth rate as a lower limit of the nominal market interest rate.
Nick: Your mathematical formula is perfectly correct.
Keystone: land is useful because you can grow stuff on it and live on it. Yes, there is always the risk that land may become useless in the future, but that is also true of Samuelson’s money.
JW: It wasn’t normal to have an economics teacher in high school in my day (late 60’s early 70’s). I went to a “public” (private) school in England. In those days, those very few who went on to grades 11 and 12 specialised in about 3 subjects in those last two years. IIRC our textbook was a slightly dumbed down version of a first year university text, maybe Lipsey. They would take bright young men (Mademoiselle was the only female in the whole school) straight out of their BA/BSc (no teacher training) and put them in front of the classroom, where most of them stayed for life. It was a very different world. I think the economics editor of the Guardian was in the same class. I would have learned a lot more if I had been paying more attention.
“I wonder whatever happened to rsj, who used to comment here.”
He’s busy commenting on my Naive vs Rational Expectations post!
Barkley: suppose you took Samuelson’s model, and made the length of the second period (retirement) slowly increase over time. I think you would get something like the model I have at the back of my mind here. The demand for the asset would slowly rise over time. In other words, I think Samuelson assumed a stationary model just to keep it simple.
Herbert: thanks for that very useful comment.
We should remember though that money (unlike Samuelson’s “money”) is more liquid than land, and so gets used as a medium of exchange, so there will be some demand for money even if land dominates it in rate of return. (And this also means that an excess demand for money, unlike an excess demand for land, can cause a recession in all trade.) But when people become satiated in liquidity services, money becomes like Samuelsonian “money”. (Not that you would probably disagree about any of this.)
@Nick (“We should remember though that money (unlike Samuelson’s “money”) is more liquid than land, and so gets used as a medium of exchange, so there will be some demand for money even if land dominates it in rate of return.”)
Agree. Money is useful as a means of payment, a idea perhaps represented best by Kimbrough 2006 who models money as an instrument for reducing transaction costs. My remarks pertained to the arguable function of money as a store of value. In this latter respect only, money is inferior to land as it yields neither direct utility nor rent payments.
Weird thought: would the policy recommendation be for the central bank to buy land? (To offset the land panic.)
You can also think about it as the arbitrage between owning and lending. If the rate is less than growth, then surplus accrues to owners, so lenders become owners, and vise versa.
Agree. Money is useful as a means of payment, a idea perhaps represented best by Kimbrough 2006 who models money as an instrument for reducing transaction costs. My remarks pertained to the arguable function of money as a store of value. In this latter respect only, money is inferior to land as it yields neither direct utility nor rent payments.
Can someone explain why land is assumed to dominate money in this stories where money has no convenience yield when money might still be safer? Why can’t money-as-government-debt be thought of as a senior interest in land?
cfaman: Good thought.
dlr: we could (I think) interpret government bonds as Samuelsonian “money”, provided those bonds violated the “no-ponzi” condition (they were pure bubble bonds). But then land would dominate those bonds, since land is productive.
Risk seems to me to be tricky. If “money” were in fixed supply, like land, they would seem to be equally risky in Homburg’s world. The fact that the government can print more money makes money riskier than land; but the fact of price stickiness might make money less risky than land, short-term.
Your take on this, Herbert?
Today’s story on farmland prices.
Is gold more like land or more like Samuelson’s “money”? It can be a useful asset, but most people who hold it do not use it. It’s not in exactly fixed supply, but fairly close to it.
http://research.stlouisfed.org/fredgraph.png?g=ol8
Also, at some point don’t we need to bring in Wicksell? In real economies, you typically finance an increase in current expenditure by issuing a new liability, not by selling an asset. I’m not sure how reliably intuitions from a world of fixed asset stocks carry over to a world with finance.
@Nick Rowe:
1. “Risk seems to me to be tricky. If “money” were in fixed supply, like land, they would seem to be equally risky in Homburg’s world. The fact that the government can print more money makes money riskier than land; but the fact of price stickiness might make money less risky than land, short-term. Your take on this, Herbert?”
2. “Is gold more like land or more like Samuelson’s “money”? It can be a useful asset, but most people who hold it do not use it. It’s not in exactly fixed supply, but fairly close to it.”
Both points have a common root. Turgot, Samuelson, Homburg have made contributions to pure theory, taking perfect foresight equilibria as a starting point. In reality there is no perfect foresight, which makes matters more interesting and also more challenging.
My best guess is that people have different priors regarding the risk of money, land, and gold. On the European continent, many – especially the elderly – buy land and gold because monetary investments have all too often ended in tears, Germany 1923 being the most cited but by far not the only example. Political risk is particularly difficult to model.
But these difficulties do not invalidate the theoretical core of the problem: Whenever the nominal growth rate exceeds the nominal interest rate for a while, investors are apt to try Ponzi games, which bring the interest rate up until intertemporal budget constraints are tight again.
@JW Mason: In postwar US history, the growth rate of nominal GDP and the nominal interest rate on Aaa bonds were both 6.0 percent on the average. Theory suggests that the nominal interest rate should exceed the nominal growth rate over an infinite horizon.
I’m playing the Catan Bug map, handicapped. There are 5 economic goods. In a thought experiment I could figure out what to do with all these sheep as the landlocked German best map doesn’t have ships…you probably only need a few more goods to get a useful model. But it doesn’t really apply to the 2008 crisis. The USA did alot of weird things during the Cold War. As mentioned, many USA bankers believe in shareholder only CSR. Some of these ones fooled some others. The others may not have been able to understand the derivative instruments. Here, it is the many different variables that were in part to blame, whereas a simplified two input model is easy to understand.
Maybe farm/houseland, renters/workers, technology, is enough to make your model work. The technology affects land, earnings power, productivity, and rate of tech growth. And your renters/workers have an oscillating population, and your farm yield fluctuate…technology ultimately gives you a reason for cash over land.
My future rant needs to incorporate mass murder as technologies, and a tyranny as a brake on technologies. I’m hoping people will keep track of what they read and learn, and neuroimaging; these can be modelled to select good future tyrants or to undo the sensor network as necessary. We didn’t need to publicly quantify human capital when the WMD industries emerged classified. The USA intelligence agencies suspect this all; are not as bullish as I am about the need for sensoring. Risk modelling is comparitively well handled by economics (it was finance that messed up the USA).
Nick: Peter Diamond (1965, National Debt in a Neo-classical Growth Model) generalized Samuelson 58 to a world with “capital,” which offers a gross real rate of return greater than 1. Just as in Samuelson, however, a bubble is welfare-improving when the growth rate exceeds the real interest rate. The bubble – whether it is fiat money or perpetually rolled-over government borrowing, – draws saving away from capital, allowing the real interest rate to increase.
Whoops: I read Herbert’s point 4 too late. So land invalidates the Diamond result as well? I’ll be damned!!
In postwar US history, the growth rate of nominal GDP and the nominal interest rate on Aaa bonds were both 6.0 percent on the average.
But the return on even Aaa corporate bonds incorporate some risk and liquidity premium. Surely if we are interested in the pure rate of interest, the Treasury rate is what we should be looking at.
Theory suggests that the nominal interest rate should exceed the nominal growth rate over an infinite horizon.
Indeed. So if that turns out not to be the case, we might need a different theory.
I’m suspicious of any result that relies on singularities to explain essential behavior. Yes, land finite supply but so is everything else. Much land sits empty because it is not economic to develop. Moreover land is substitutable with other factors of production — e.g. you can build higher buildings or you can build transit to areas where land is cheaper, bringing more land into production as the cost of land in the urban core rises. The house my parents bought costs the same, in real terms today as it did thirty years ago, even though they upgraded it by adding a swimming pool and refurbishing much of the interior. However the city they lived in is much bigger now and transportation costs have fallen.
Moreover, no asset is infinitely lived — the earth will be swallowed by the Sun at some point, and whatever contracts you sign now will be no longer in force long before then. You do not know what tax structure will be in place 10 years from now, nor do you know which government or unit of currency will be in use 100 years from now — which contracts will be honored by your grandchildren? Therefore we sharply discount assets whose tail carries most of the value as we know very little about those future payments.
That is enough reason to not bulldoze the rocky mountains to make farmland. You don’t know whether it will be economical to grow crops or use that land far into the future, you don’t know what the climate will be, what the transportation infrastructure will be, what markets for food there will be, etc. — regardless of what the long run interest rate today happens to be. And that assumes that you can find a lender who will lend to you for such a long term. I don’t think these types of considerations hinge on interest rates, and we see that rates of return on long run risky assets seem to be fairly stable, being dominated by the uncertainties of the long run.
I think it’s an interesting question to ask what then determines the long run discount rate when we know so little about what will happen in the future. Whatever the source of this discount is, it doesn’t have much to do with interest rates. Here is a graph about price to rent ratios and Baa yields. One of these has the familiar arc reflecting interest rate policies but the other does not.
FYI, anything land related, check out the lincoln institute, http://www.lincolninst.edu/.
Continuing with the pretense that Canada is also important (:P), I looked for a Canadian equivalent but found nothing. Perhaps someone here can point to a good data source.
And that assumes that you can find a lender who will lend to you for such a long term.
http://blogs.wsj.com/marketbeat/2011/05/11/mit-issues-rare-100-year-bond/
I think it’s an interesting question to ask what then determines the long run discount rate when we know so little about what will happen in the future. Whatever the source of this discount is, it doesn’t have much to do with interest rates.
And now we have arrived at Keynes…
Consider the more common way of expressing a simple no-arbitrage relationship between housing (land) prices and rents:
R(t)/P(t) = r(t) + m(t) – E [ i(t) ],
where R are rents, P is the price of housing, r is the real interest rate, m are costs due to maintenance and taxes, E is the expectation operator at time t, and i is the rate of increase in housing prices. The equation Nick presents is the special case in which we consider a steady state equilibrium with i(t) = g and m(t)=0 for all t.
As I understand it, Nick suggests that all the action is coming off changes in the real interest rate: if we let m=0 and macroeconomic forces push the real interest rate to the real growth rate, which under the conditions Nick gives is equal to the rate of housing price inflation in equilibrium, then the equilibrium level of P explodes, but it’s not a bubble.
The argument that P exploded because the real rate fell, but there is no bubble, has been battered around in the literature a lot. My non-specialist take is that the econometric evidence shows that prices are responsive to changes in the real rate, but not by as much as (simple) theories suggest, which makes bubbly explanations somewhat more plausible.
Also, the price level could both go up because the real interest rate fell AND there could be a bubble. Writing the equation as above makes it clear that expectations over price level changes matter, and that’s where a bubble (rational or otherwise) could form. The argument is essentially that decreases in the real rate generate increases in housing prices, and those increases in housing prices, through the mechanism embedded in the equation above, lead to further increases in prices, and in some places that became a bubble.
That figure is really intriguing. I want to say that the rent-price ratio is something like the natural rate if interest — or more broadly it is the rate of interest in models like the ones where discussing here. In which case, as rsj says, “interest rate” in the model and “interest rates” that we observe in the world are two different, unrelated phenomena.
… but then another part of me wonders, how much of the divergence is just inflation?
Another question: If the interest rate is — contra rsj — an intertemporal price, and if the interest rate is greater than the growth rate, that means the market price of total output at a later date is less than the market price of total output at an earlier date. Doesn’t r greater than g imply that the aggregate value of output is shrinking over time?
@JW Mason: You raise two interesting points, in different postings.
1. “But the return on even Aaa corporate bonds incorporate some risk and liquidity premium. Surely if we are interested in the pure rate of interest, the Treasury rate is what we should be looking at.”
Considering the economic argument behind the r>g result, I believe only the Aaa rate is relevant because investors have to pay this rate (or the comparable conventional mortgage rate) in order to make use of intertemporal arbitrage. The Aaa rate is also that rate which should equal the marginal productivity of capital in equilibrium. As an interesting corrollary, the bond market is not at a “zero lower bound”. The Aaa rate stands at 4.5 percent. It exceeds the nominal growth rate permanently since 2008.
2. “Another question: If the interest rate is — contra rsj — an intertemporal price, and if the interest rate is greater than the growth rate, that means the market price of total output at a later date is less than the market price of total output at an earlier date. Doesn’t r greater than g imply that the aggregate value of output is shrinking over time?”
Indeed r>g means that the present value of future output converges to zero. Moreover, it implies that the infinite series of all present values of future outputs remains finite. Under this premise “total intertemporal output” in a present value sense is a real number, and the proof of the first welfare theorem goes through. If r falls short of g, total intertemporal output is infinite and you can make some generations better off without making others worse off. This is the mathematical logic behind “dynamic inefficiency” and the results of Diamond (for national debt) or Aaron (for PAYG).