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 
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:
http://idiosyncraticwhisk.blogspot.com/2013/08/real-interest-rates-and-housing-boom.html
http://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:
http://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…