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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: https://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: https://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:

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.

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.

https://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, https://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.

https://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).

A slightly related question (or maybe unrelated)

Would a monetary policy pegging to NGDP have a smilar result as a monetary policy that pegs to nominal domestic land rentals?(when properly calculated)

JW: "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."

That is not obvious to be. Why not look at the rate on currency? Should we be subtracting something from the yield on illiquid assets, or adding something to the yield on liquid assets, if we want to find the "pure" interest rate?

rsj: is that rent/price ratio data for US farmland? Neat!

The rent/price ratio is a real yield, and the Baa rate is a nominal yield, so the change in expected inflation will explain part of the divergence and re-convergence between those two yields. The rent-price ratio is also a yield on a perpetuity, while the Baa yield is what, a 10 year rate??

Chris: by biggest empirical problem with the "bubble" explanation is that when bubbles are pricked they are supposed to burst, they are supposed to stay burst. Too many people, especially in the blogosphere, are too fixated on what happened to US house prices in 2007/8. There's something much bigger going on here. It's not just housing land it's farmland too. And it's not just the US. And even where they did seem to "burst" they are re-inflating again.

JW: "Doesn't r greater than g imply that the aggregate value of output is shrinking over time?"

It means the present value at time T, of GDP at time T+t, is shrinking as t increases. Yes, but I don't see where you are going with that.

Herbert: "The Aaa rate is also that rate which should equal the marginal productivity of capital in equilibrium."

Careful there. That only works if the price of the capital good is always equal to one consumption good. The rate of interest (measured in wheat) is the marginal physical product of land, divided by the price of land, plus the rate of appreciation of the price of land. For "land" substitute "tractors". (My old post.)

Prakash: I don't think so. rents/GDP ratio could change a lot.

JW: A second thought on whether we should add or subtract a liquidity premium: what we are really trying to do, in answering that question, is to merge a Samuelsonian analysis of dynamic efficiency with a Friedmanite analysis of the Optimum Quantity of Money. Which is too hard for my poor brain.

Start with a model with zero transactions costs. There's a "pure" rate of interest in that model. Now add transactions costs, and a liquidity spectrum of assets with different transactions costs. I think the most liquid asset (money) would now have a rate of interest below the pure rate, and the least liquid asset would now have a rate of interest above the pure rate. But if the holding period of the least liquid asset was very long, the effect of those transactions costs on its yield, relative to the pure rate, would be very small. (If it only changes hands every 50 years a 10% transactions cost would raise the equilibrium yield by 0.2%).

But then it seems to me that the thing we observe in the wild as the "interest rate" is the difference between the yields of the more and less liquid assets. It doesn't have any relationship with the interest rate in the model. No?

The interest rate in the model -- it seems to me -- is better captured by something like rsj's rent-price ratio. Not anything reported in the statistics as "interest."

JW: I think we should think of the "pure" interest rate from theory as being some sort of weighted average of the interest rates that we observe in the wild. And those exact weights should be......errr..... can I get back to you on that one?

The rent/price ratio on land is very definitely *an* interest rate. I think it would be a better approximation to the pure interest rate of a Samuelsonian type model than many other interest rates. Especially because it is a real interest rate, on a perpetuity. But the rent/price ratio will depend on the expected growth rates of real rents, which is one disadvantage of using it as a proxy for the interest rate of pure theory.

The rent price ratio is for residential property, with data from the lincoln institute. It is just as nominal as a Baa yield.

It is true that there are capital gains effects to take into account, but this is also true for bond yields. I.e. the total return is the current period interest plus the capital gain or loss. The lincoln institute does have data on land value changes as well, so I can post another graph taking that into account and compare it to stock markets, for example, and possibly bills as getting capital gains data for longer term bonds is harder to find.

rsj: if rents were permanently fixed in nominal terms, like the coupons on a bond, then the rent/price ratio would be a nominal yield. If we expect rents to rise with inflation, it's a real yield, like an indexed bond.

Nick, it is not a real yield because it is measured in nominal terms. Yes, rental yields change from time to time -- once the lease expires, but flexibility of changes in rental agreements does not make it a "real" yield.

I'm with Nick on this one. Buying a house entitles you to a future flow of housing services. To determine the ratio of the price of a housing services in the current period, to the price of a flow of housing services indefinitely far into the future, we do not need to know anything about the money price of housing services. That's why it's "real".

And the yields for bonds are not permanently fixed either. The price of the bond changes to take into account things like inflation and lower/higher yields offered by newer issues, and this causes a change in the "current" yield which is the current coupon divided by the current market price of the bond (as opposed to the nominal yield, which is the original price of the bond and the original coupon). Capital losses are a risk to bondholders and landholders alike. Both indices cited here measure the same thing, i.e. current yields -- current price of land and current rents versus current price of bond and current coupon.

J.W.,

Buying a house entitles you to a future flow of housing services.

Yes, and you can always buy these services with money (e.g. buy purchasing a bond and using the coupon to pay rent).

To determine the ratio of the price of a housing services in the current period, to the price of a flow of housing services indefinitely far into the future, we do not need to know anything about the money price of housing services.

If we don't want to pay more for the housing services than we need to, then money price of housing services needs to be taken into account.

This is a strange thing to be disagreeing about. It seems obvious to me that the rent-price ratio is a real rate. But you're a smart guy; I know you are not just confused. So evidently we are thinking of the real-nominal distinction differently, but I'm not sure how.

JW's argument is better than mine. If one acre of land rents for one ton of wheat, how many tons of wheat (how many "years' purchase" is how it used to be expressed) do I have to pay to buy one acre? We could talk about that ratio in a world without money.

...without technology, the rate of wheat spoilage is 25%, in the middle ages, for example. With 1st world technology, wheat cost 2%/yr to store. Your civilization needs galvanized steel and refrigeration equipment, else just like monkeys.

JW/Nick.

Suppose that rent was paid continuously and was pegged to inflation, so that you always received a payment in 1 fish per day per square foot of land rented out. In period 1, it costs 10 fish to buy one square foot of land, so we have a real interest rate.

How would you convert from a ratio of 1 fish per day per square foot of land to an equivalent consol yield (where the consol paid out dollars and was purchased with dollars) so that you receive the same real return? Suppose that in period 0, one fish costs $1, so that we know that a plot of land costs $10. And in dollar value, the plot of land pays out (in period n) out an equivalent of $1*(1+p)^n

So the plot of land is the same as a console paying out a rate of 10% + p. The yield is 10% + p and to get the "real" yield of the consol, you subtract p to get 10%.

To go from rates to real rates, you always subtract out inflation. It does not matter which type of asset generates the income stream (whether it is land or a consol), what matters is that you need to discount the stream of payments by inflation to get the real rate, or equivalently you need to add inflation to the real rate to get the nominal rate.

Just because we are talking about rental payments for housing versus dividend payments from shares or bond payments from consols doesn't matter. They are still nominal rates until you subtract out inflation.

@rsj: Regarding the nominal vs. real rate, it all depends on the specifications of the respective contract. Bond interest can be nominal or real, as with inflation indexed bonds. The same applies to rent contracts. So this is not a matter of principle.

Take a consol where the annual dollar coupon is not indexed to anything. Its yield is a nominal yield. Index that coupon to the price of the CPI basket of goods, it's a real yield. Index it to the GDP deflator, it's a real yield (but not exactly the same real yield as the consol indexed to the CPI, because we are using two different measures of inflation). Index it to the price of fish, it's also a real yield (again, with a different measure of inflation). Index it to the price of renting a house, it's again a real yield (again it's a different measure of inflation). But that last consol is equivalent to a house.

There are multiple real rates of interest, because there are multiple measures of inflation, because there are multiple real goods and baskets of real goods.

Fascinating discussion! I am currently teaching my students about rate of return equality in the world with money and capital and hence this post and discussions come at the right time. At the risk of redundancy let me add my 5 cents. In a world of perfect substitutability between capital and money, the returns would be equal. But along with uncertainly and risk aversion, you will have to pay someone to hold capital instead of money and that could explain the rate of return dominance puzzle. Another way of looking at it is suggested by Lagos and Rocheteau (2008). They start with an economy where money and capital are both valued. However, if for some reason "when the socially efficient stock of capital is too low to provide the liquidity agents need, they over accumulate productive assets to use as media of exchange". The situation could be easily reversed in inflationary times or if landlessness is rampant. This is easily imaginable in an agrarian setting like in India where cattle stock competes with money as a means to ensure future consumption as well as insurance. So cattle provides stream of consumption goods plus another avenue for ensuring consumption in uncertain times. In addition you can ensure that capital grows (except land perhaps) and hence have some control over capital accumulation. Given this and considering risk aversion, an asset portfolio of a typical agent will always feature a positive amount of both capital and money but capital will earn a positive and higher rate of return than money.

Nick,

yields measured in ratios of dollar amounts per time. The current yield is the current rent payment divided by the current price of the asset.

If inflation is 10% and the current yield is 20%, then the real yield is 10%. Do you agree?

If so, you agree that you must subtract inflation from the current yield to get the real yield. In the graph I gave, one yield is not "more real" than another -- both are current yields.

I think this is a bizarre exchange.

rsj-

Your reasoning would be correct if rent payments were fixed in perpetuity, like bond payments. But rents are renegotiated, usually every year. If the price of housing rises at the same rate as other stuff, then the yield on the apartment is already adjusted for inflation.

You don't think that the yield on an inflation adjusted bond is a nominal yield, that needs to be adjusted (again) for inflation?

Are you assuming that someone who purchases a house to rent out expects that the rental price, in dollars, will permanently remain at its current level? That's the only way I can make sense of what you are saying.

Part of the problem is the units. Normally, a nominal variable has $ in the units, and a real variable doesn't. But both real and nominal interest rates have the units 1/years.

Parag Waknis (who I am, oddly, also currently debating in the pages of Economic and Political Weekly) makes an interesting point. Part of the issue is whether liquidity preference dominates "solidity preference", as in modern economies, or whether solidity preference dominates, as in many premodern ones.

One way of thinking about this is to focus not on "the" interest rate but on the yield curve. (Tho in fact distinguishing these two things is not so straightforward, as Nick's most recent comments show.) In general, we see a positive yield curve. Interest rates are higher on long contracts than short ones. But why?

The conventional answer is that the lender assumes more risk with a long loan. But in fact, the risks are symmetrical. If inflation and/or interest rates fall during the life of the loan, the lender is happy and the borrower is sad; if they rise, the borrower is happy and the lender is sad. The long contract involves more risk, but the risk is borne equally by both parties. So why is it (normally) the borrower who pays the risk premium and the lender who receives it?

Leijonhufvud asks this question Keynesian Economics and the Economics of Keynes, and gives the following answer. on the one hand, people have finite consumption horizons. They don't care about the incomes of their distant descendants, and they don't even care that much about income in their own distant future, since too much can happen before they get to do anything with it. But on the other hand, the nature of modern production is that the most efficient processes involve very long-lasting capital goods. If you build a wooden shack, most of the value of it will come in the next year or two. But if you build a modern apartment building to rent out over the next 10 years, then you also now own a modern office building in 20 years or 50 years and in 100 years. There is no way to save money by only building the first 10 years of the building's life. Same goes for industrial investment, and of course same goes even more for investment in research and development, since knowledge does not decay at all. So in effect capital goods for use in the near future have a positive externality of capital goods available in the more distant future.

What this means is that people have claims on income farther into the future than they would have chosen, if it were possible to separate the sooner and later uses of capital goods. This "unwanted" supply of future income, relative to demand, is what creates an upward sloping yield curve.

In other words, efficient production produces more income stability than people need. Hence we speak of liquidity preference -- flexibility is what the financial system has to supply.

But in poor countries, production processes don't involve so much long-lived capital. The vast majority of the labor that goes into production takes place close in time to the final product. Output is less certain as well. (Also, people may have longer consumption horizons, if family lineages are more salient and if life is expected to continue more or less the same from generation to generation.) So there is no excess supply of future income, rather there is excess demand for it. So solidity preference will dominate liquidity preference and wealth will be held as land or cattle rather than money. (Which I think can be considered equivalent to an inverted yield curve.)

Part of the problem is the units. Normally, a nominal variable has $ in the units, and a real variable doesn't

Nope. A yield has units 1/time. A real yield has units 1/t. They have the same units.

You need to know whether or not inflation was subtracted from the nominal yield to get the real yield or not. That requires some knowledge of the methodology, but standard ratios such as rent/house price or coupon/bond price or earnings/stock price are not real yields.

Inflation only has to be subtracted from the yield if it is in there in the first place.

Again, rsj, would you say that inflation must be subtracted from the yield of an inflation-indexed bond to get its real yield?

Put another way, the current rent is NOT the yield of a plot of land in the same way that the coupon is the yield of a bond. The coupon is fixed over the life of the bond, so it incorporates today's expectations about what inflation will be over that time. The rent is not fixed, so it does not incorporate those expectations.

Again, rsj, would you say that inflation must be subtracted from the yield of an inflation-indexed bond to get its real yield?

Yes, of course. A bond may have a promise to pay out $100 per period if a given soccer team wins the world series, and only $50 per period if it does not. It may promise to pay out a higher yield if there is more inflation.

But when looking at the actual yield of this bond, we look at the actual coupon payment/bond price.

Put another way, the current rent is NOT the yield of a plot of land in the same way that the coupon is the yield of a bond.

Yes, it is. If you are to compare one to the another, you need to use the same definition for both. Then you can see which number is higher or lower and infer something about expected capital gains, or expected inflation, or measure some premium by measuring the difference in yields.

rsj: "Nope. A yield has units 1/time. A real yield has units 1/t. They have the same units."

Now read the rest of my comment, where I say precisely that. To quote myself: "But both real and nominal interest rates have the units 1/years."

I feel like we're just disagreeing on words. rsj's standard meaning of 'real' means relative to standard CPI inflation. Nick/JW's standard meaning of 'real' means relative to the asset's inflation, which in this case is housing.

Nick's explanation is very clear. But rsj's graph is still very striking to me -- I've been staring at it quite a bit.

Lot A has a price of $2 million, and can be rented for $100,000 this year, $100,000 next year, $100,000 the year after, $100,000 every year forever as far as you know. Lot B also sells for $1 million and rents for $100,000 this year, but next year you expect it to rent for $105,000, if the general inflation rate is 5 percent. And every year after that you expect the rent to increase at the inflation rate.

Both of these have a price-rent ratio of 0.05. Which is equivalent to a bond with a nominal yield of 5 percent? Which is equivalent to a bond with a real (inflation adjusted) yield of 5 percent?

Which of these scenarios do you think better describes the beliefs of people buying and selling property?

I came to a similar conclusion, and I think I convinced Scott Sumner that I was on to something. I agree that real rates and the inflation premium need to be separated. The real rate has a convex relationship to price, whereas the inflation premium has a more linear relationship to price because its effect is more through the ability of households to afford the monthly payment on a mortgage. The unprecedented combination of low real and nominal rates in the 2000's led to the price explosion.
Two strange facts that this explains:
1) In the late 1970's home prices were rising even as mortgage rates hit double digits (this is because real rates were low).
2) Home prices are currently rising again at relatively high price levels, even though mortgage credit markets are still very tight.

Here are two posts I did on it:
https://idiosyncraticwhisk.blogspot.com/2013/08/real-interest-rates-and-housing-boom.html
https://idiosyncraticwhisk.blogspot.com/2013/10/real-rates-vs-inflation-regarding.html

If nominal rates continue to be low over the next couple of decades, we need to get used to home prices moving more like bond prices, since they do not have constraints on demand that existed in past high rate environments.

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@Nick, rsw: You're both wrong.

Consider land vs. financial instruments from the perspective of an investor. If I have P_0 of surplus money right now and wish to consume that plus returns in one year, then I have two options:

*) First, I could purchase a one-year bond with yield y. After one year I would have my original capital stock back plus a return, giving me (1+y)*P_0 in nominal terms.

*) Second, I could purchase land, take rents (r) for a year, and then *sell the land*. I do *not* receive my original capital back, but instead I get (r+P_1), where P_1 is the sale price.

In the land case, the premium I realize is ((r+P_1)/P_0 - 1), which is then comparable to the bond yield y. The rent/price graph captures *only* the (r) portion of that equality and assumes no appreciation in value. For me to be indifferent to a bond versus temporary land ownership, y = (r+P_1)/P_0 must hold.

Since as per rsx's graph, the rent/price ratio is extremely stable over time and the interest rate is not, the price appreciation rate of land must in turn be quite volatile, which I think reflects history fairly well.

Using strictly rent/price to impute interest rates is unfortunately not going to work out, as price appreciation is not otherwise determined without further assumptions that may not hold.

Majromax,

rent/price ratio has been anything but stable for the past 15 years:
https://www.calculatedriskblog.com/2013/07/comment-on-house-prices-real-prices.html

rent/price has tracked pretty well with real rates over this time. It only lags now because of continued tight conditions in the mortgage credit market, leading to a relatively low number of non-cash home purchases.

Majormax,

No, you are wrong. Both bonds and land can experience capital gains. That is irrelevant to a comparison of yields.

"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.)"

I'm glad to hear that you made it up, because it is mathematical nonsense. Land prices would "fall", I suppose, but not to a finite value. They would remain forever infinite. Just as you can't get there from here, you can't get here from there. ;)

However, considering when he lived, it is almost certain that Turgot did not understand infinity very well. So he might have made such an argument, and I suspect that your outline is fairly accurate.

Even today, after the development of modern set theory, few people reason well about infinity, and even fewer reason well from infinity.

And in the human sciences, if you run into infinity in your reasoning, it is a good bet that you are overlooking something. It's a bad sign.

Small world indeed JW Mason! Interesting points though- will have to think more about them. Meanwhile, a couple of points that quickly come to mind:
1. Leijonhufvud's argument about risk being symmetrical- takes us into the world of modeling terms of trade between a lender and a borrower. I would rather argue that distribution of risk between borrower and lender depends on their relative bargaining power. In short you might have to carefully model the market structure of financial intermediation. But even if we assume that people have claims to future income and they don't care about them then presence of secondary markets should solve the problem.
2. Consumers have finite horizon or do not care about bequests is more of an assumption. Also, it might be more relevant to talk about degree of impatience rather than finiteness of the horizon. Given this, one would like to see what are the implications of households with different consumption horizons or different patience levels for asset prices. May be the consumption based CAPM could deliver some insights.

JW,

Both of these have a price-rent ratio of 0.05. Which is equivalent to a bond with a nominal yield of 5 percent?

Yes, that is a current yield of 5%. Of course, these bonds should have different yields in a competitive market because the inflation protections of one of the bonds should fetch a premium. But as you defined it, the bonds have the same current yield.

Just as a bond issued by a government *should* have a different yield than a bond issued by David Bowie. But you do not change the current yield when the bond behavior is different. I.e. you don't add what your subjective extra percent or so to a market-observed quantity and insist that your made-up yield is the yield of the bond, you look at the differences in market prices to gather information about how much the market values one offering over another. That valuation is subjective and cannot be quantified, whereas the ratio of price to rent is objective and can be observed.

The graph I showed was of current yields. Neither one nor the other is a real yield.

@ JW

Of course bonds can experience capital gains; that is why I chose the example of a one-year bond held to maturity. Land, of course, cannot be held to maturity so no direct equivalent is possible.

I think that capital gains on land are of a fundamentally different character than capital gains on financial instruments. For the latter, in an environment where interest rates remain fixed we would expect to see no capital gains. The prevailing rate of interest is reflected in a bond's yield precisely because its price adjusts such that no arbitrage is possible with newly-issued bonds.

Land doesn't behave that way, and especially not housing stock. Outside of local factors, houses are expected to appreciate in value over time, even with a constant interest rate. The mortgage bubble in the States relied on that phenomenon a bit too much, even. When appreciation is expected, then the rent-to-price ratio doesn't tell the full story on the effective yield of land holdings.

(of course, all this just begs the question of why appreciation is expected, if in theory fair value can be derived from a time-discounted stream of future exoected rents.)

Nick, why is land unique here, wouldn't the same apply to other fixed capital? Land has carry costs, but a very long depreciation schedule, it's not much different from say a computer that calculates bitcoin codes. Perhaps it comes the closest to the singularity but I think it gets resolved by treating money as scarce and not infinite. This is a physicist talking...

jesse: land is well, not unique, but,...they aren't making it any more, and it lasts forever. Physical capital depreciates, and if its price goes up more will be produced. Now physical capital will satisfy some of the desire for saving. But will it yield a return greater than the growth rate of the economy, if we produce a large enough amount of physical capital to satisfy all the desire to save? Maybe yes, and maybe no. If no, that's when land comes in.

But then old paintings etc. might also work, as well as land.

Nick, another thought occurred to me about real rates. We do have several examples of negative rates, including Japan pre 1990, China (today), and IIRC the US in the mid 1800s. All were investment driven growth models that used low real rates to increase national savings and investment, at the expense of consumption. This arguably led to asset price appreciation, but as is said, the books must balance. In other words the negative rates are indicative of transfers; they eventually lead to allocation decisions that are either productive or not productive. In the case of Japan, overinvestment was abetted by low (negative) cost of capital, but ultimately it fell apart as the investments could not return.

ie while you can look at the problem from the framework you suggest, it might be equally as instructive to look at it in the context of GDP components (saving, investment, consumption, etc.). Negative rates can and do exist, and there's agreat example in China right now, but in practice they self-correct after rebalancing takes place.

So I finally read the Samuelson article. It's really interesting!

I feel Nick misdescribed the argument a little bit. The optimal rate of interest in the model, in the absence of pure time preference, is equal to the rate of population growth. This is only the rate of output growth because the model excludes technical change; it's clear that if per capita income increased the optimal rate of interest would be less than the growth rate of aggregate income.

The main point of the paper, though, is that in a pure consumption-loan economy with no durable assets, the market rate of interest will not be the social optimum, it will be negative. People will end up with too much consumption when they are young and too little in retirement. So a mandatory pay-as-you-go social security system is welfare improving.

The possibility of solving the problem via money comes in almost as an afterthought, and it is clear that you can get a variety of interest rates depending on the properties of the durable asset. If the asset is liquid, then the interest rate will still be below the growth rate. If it is risky, the interest rate will be above the growth rate. And in any case the currently active population can always benefit from repudiating or diluting the claims of the currently retired asset owners, so the money/land solution does not avoid the political problems of the pay-go pension.

The article is pretty unambiguous in its conclusion that the problem of life-cycle saving is best solved through public provision. As for what it says about actual interest rates and asset prices -- I would say, nothing. Those are all abut the mix of risk, liquidity and the relative productivity of more roundabout methods of production -- all of which Samuelson abstracts away.


JW: Yep. It's a classic must-read paper (one of the few).

"it's clear that if per capita income increased the optimal rate of interest would be less than the growth rate of aggregate income."

I don't think that's right. Because if r were less than growth rate of income, you could run a stable ponzi and make all generations better off.

Well. Let's think about this. Here is my reasoning:

Let's follow Samuelson and assume no pure time preference.

If output per capita is rising then, in the absence of durable goods, consumption per capita must be rising at the same rate.

If r = g, then you can trade your whole consumption basket at one time for your whole basket at another time.

But growth in per capita output means that the later basket is larger than the earlier basket. And in the absence of pure time preference, that larger basket must provide more utility than the earlier one. (Though not proportionately, given convexity.) So the two baskets cannot trade at the same price, the later basket must be valued more than the earlier one. Which means r must be less than g.

Is there some mistake there?

Hm. I guess I see where I'm wrong. If we use Samuelson's U = log(C1) + log(C2) plus log(C3) then a given percentage change in consumption in any period produces the same change in utility. So some fraction of a consumption basket in one period, should trade for the same fraction in a different period. So ok, it seems we do want r = g.

JW: my intuition goes like this. If I could set up a ponzi scheme, where everyone voluntarily lends to me at rate r, and I never default, then I am better off, and nobody else is worse off, then the original allocation of resources cannot have been Pareto Optimal. And if r < g, I will never need to default.

Nick: if you never default,then it is not a Ponzi scheme. It's the normal working of an economy with money,savings, investing, forward-looking agents, peopled with individual with too much future income and so in need of liquidity...

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