House prices "bubbled" because Turgot's land beats Samuelson's "money"

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.

Comments

[Nick Rowe]

Jacques Rene: well, you could reasonably argue over whether we should or should not call it a "Ponzi scheme". It is like what Mr Ponzi did in some respects, because the proceeds of the loan are not invested in any real assets; Mr Ponzi just consumes them, and borrows more to pay interest and redemptions. But it is unlike what Mr Ponzi did in that it is possible for it to go on forever, without exploding, so rational people would lend even if they fully understood how the scheme works, and assumed everyone else is rational too.

Some economists call it a "rational bubble". I call it a "sustainable Ponzi".

[JW Mason]

Nick: if you never default,then it is not a Ponzi scheme

The definition of a Ponzi scheme here is an economic unit that repays its past borrowing only by issuing new debt. Such a unit may not ever default, even if it has no income.

[JW Mason]

Nick: I don't find your story quite convincing because you can also carry out your Ponzi scheme when r is greater than g. In that case, it's true, your debt-income ratio rises without limit, but so what? The basic setup does not have any limits on allowable debt-income ratios. And if we introduce a maximum debt-income ratio, that will exclude some non-Ponzi paths as well.

[Nick Rowe]

JW: if the flow of debt the old agents are selling exceeds the total income of the young agents, they won't be able to buy it. So if the debt/income ratio gets too high, the Ponzi scheme must collapse.

[rsj]

if the flow of debt the old agents are selling exceeds the total income of the young agents, they won't be able to buy it.

In that case, the debt will be repriced so that the younger generation *can* buy it. The repricing of debt (e.g. making it cheaper) is equivalent to an increase in interest rates. There is a finite amount of free lunch to be eaten, after which interest will go up if you try to eat more.

[Nick Rowe]

rsj: yes, and if the old agents who are now selling the debt understood that that will happen, they would not have bought the bonds in the first place, so the Ponzi scheme unravels back to the beginning.

[rsj]

Nick,

So you are saying that negative interest rates never occur?

Losing some asset value is the same as a (total) return that might be negative. If that is the equilibrium return, then that is what would clear the market.

I don't see how you can talk about the zero lower bound with a straight face while insisting that no one would ever tolerate a capital loss.

[Herbert]

@rsj: Nick Rowe's argument refer to steady states. The presence of land rules out a steady state where r < g. By the Turgot-Homburg argument, you necessarily have r > g.

But, answering your question, this does not imply that negative interest rates can never be observed outside steady states. In a period model, rt < gt can hold for some time, but not forever.

This is what Nick Rowe already indicated in the title of this thread: Land is a long-run stabilizer, as it rules out dynamic efficiency, but also a short-term troublemaker, as it induces people with adaptive expectations to try Ponzi games.