Dutch Capital Theory

Very few economists are aware of Dutch Capital Theory (DCT to us insiders). It was invented by my ancestor Nick van Rowe, over a century ago, but has been suppressed by a conspiracy of silence and ignorance ever since. Think of this as a teaching post.

Start with a very simple economy. Land is the only resource, all land is identical, and the quantity of land is fixed. Each acre of land produces 4 tons of wheat per year, without labour, fertiliser, seed, or anything else. Wheat cannot be stored from one harvest to the next. There are no other goods. There is a fixed population of identical people, who own the land. All markets are perfectly competitive. Nothing ever changes.

The annual rent on land would be 4 tons of wheat per acre per year, obviously.

What determines the price of land? What determines the rate of interest, if you borrowed wheat and promised to pay it back next year?

The answer is simple: preferences. More specifically: time-preference. If people are very patient the rate of interest will be low and the price of land will be very high. If people are very impatient the rate of interest will be very high and the price of land will be very low.

There is a very simple relationship between: the price of land P (measured in tons of wheat); the annual rent on an acre of land R (measured in tons of wheat per acre per year); and the rate of interest r (the extra tons of wheat per year you have to pay as interest to borrow one ton of wheat).

It's P=R/r. Or you can think of it as r=R/P. Arbitrage ensures that equation holds true. If r>R/P, the rate of return on owning land and renting it out would be less than the rate of interest, so everybody would want to sell land, driving P down. If r<R/P, the rate of return on owning land and renting it out would be greater than the rate of interest, so everybody would want to buy land, driving P up.

Or you can set the price of land equal to the Net Present Value of the rents, and get exactly the same answer, if you assume that rents are never expected to change and interest rates are never expected to change.

If you want to do some math, to show the exact relationship between preferences and the rate of interest, you can do so.

Assume V(t) = U(C(t)) + B.U(C(t+1)) + B^2.U(C(t+2)) + etc.

Where U(C(t)) is the Utility you get at time t from Consumption (of wheat) at time t. And B (which is a number between 0 and 1) is a subjective discount factor that measures how patient you are. We can rewrite B as 1/(1+i) where i measures your degree of impatience, or "rate of time preference proper". If i=0 then a person who is consuming the same amount of wheat every year is indifferent between consuming an extra ton this year and an extra ton next year.

The equilibrium condition is that 1/(1+r) = B.MU(C(t+1)/MU(C(t)), where MU(C(t) is Marginal Utility of Consumption at time t. Since consumption equals production in equilibrium, and production is constant every year, this reduces to 1/(1+r)=B. Or, more simply, r=i.

In this very simple model, where production and consumption never change over time, the rate of interest is determined by, and equal to, the rate of time preference proper. r=i.

The rate of interest cannot be determined by the marginal product of capital, because there isn't any capital in this model.

The rate of interest cannot be determined by the Marginal Physical Product of land. Because MPPL = 4 tons of wheat per acre per year in this model, and the rate of interest could be anything, depending on preferences. They don't even have the same units. Because the rate of interest will have the units 1/years, and MPPL will have the units tons/acre.year.

It is easy to complicate this simple model in lots of ways. Introduce lots of different goods in addition to wheat. Introduce lots of different types of land, some of which are better than others at growing different crops. Introduce labour into the production function. Introduce lots of different types of labour. Introduce monopoly, Etc. But, as my Dutch ancestor showed, none of these complications make any difference whatsoever to the basic insight of Dutch Capital Theory: that the rate of interest is equal to and determined by the rate of time preference proper, because 1/(1+r) = B.MU(C(t+1)/MU(C(t)) is still true. Provided consumption never changes over time.

It's when consumption starts changing over time that Dutch Capital Theory gets a little bit more complicated.

Lets look at some things that might cause consumption to change over time.

1. The sea level falls, so new land appears every year. Suppose the stock of land grows at 1% per year. Then production and consumption of wheat will grow at 1% per year. So the marginal utility of consumption will be falling over time. So the rate of interest will be greater than the rate of time preference proper. How much greater depends on how quickly Marginal Utility of Consumption diminishes as consumption increases.

2. Suppose there is a once in a lifetime flood, that reduces the stock of land temporarily for just one year. So production and consumption of wheat will be lower this year than next year. So the Marginal Utility of Consumption will be higher this year than next year. So the rate of interest on one-year loans of wheat will be higher this year than next year, and will be higher than the rate of time preference proper.

3. Once you have understood the effects of sea-level changes in 1 and 2, it's quite easy to figure out how more complicated patterns of actual and expected changes in the sea level will affect the rate of interest. Just plug them into the formula 1/(1+r) = B.MU(C(t+1)/MU(C(t)) to get the answer.

"Hang on Nick" I hear you saying. "What's any of this got to do with Capital Theory? You still don't have any capital in the model. You've only got land!"

And that would be the point at which my Dutch ancestor would deliver his punchline:

"Yes we do! "Capital" is just a strange name for land that we drained ourselves, instead of waiting for God to reduce the sea-level. Saving and investment is just like having a flood this year, and so consuming less wheat, barley, or whatever, this year, and a lower sea level and more land next year. It's just 1 and 2 together. Or maybe a more complicated pattern like 3, if you have to rebuild the dikes every few years. And growing new land is a business just like growing wheat or barley or any other business."

My Dutch ancestor was right, of course. But he did leave one thing out. The quantity of new land produced each year would be where the Price of land equals the Marginal Cost of producing new land. So the position of that Marginal Cost curve would also affect how much new land gets produced each year, which would also influence the current and future production and consumption of wheat, barley, and whatever. It's not just preferences that determine interest rates. Unless consumption never changes from year to year.

Comments

[Martin]

I was reading this paper (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2077604) yesterday; it reminds me a bit of your analysis here.

That is the dominant role played by the rate of time preference throughout the whole of the analysis.

I do have to say I prefer the DTC over the ATC, then again, I might be biased as a Dutchie.

[Simon van Norden]

Nick van Rowe, eh? Nice name! Has a ring to it.....

One thing I didn't quite follow in your entertaining post

"Saving and investment is just like having a flood this year, and so consuming less wheat, barley, or whatever, this year, and a lower sea level and more land next year."

I don't quite follow the investment bit. Why does adding more land mean foregoing some wheat (or corn, whatever) this year? It grows without labour, seed, etc., right?

[Yancey Ward]

Typo or not?

"If people are very patient the rate of interest will be low and the price of land will be very high. If people are very impatient the rate of interest will be very low and the price of land very high.

[Nick Rowe]

yancey. typo. fixed. thanks.

simon. if resources are producing investment goods they get taken away from producing consumption goods. they grow jute to make sandbags instead of wheat.

martin. should it be dct or dtc? (i just changed it).

that's 2 dutch guys!

[Greg Ransom]

What we are interested in is choice involving production goods with alternative productive uses of varying production times which are worn up with use -- THIS is the problem of choice across time involving tradeoffs between time discounting and output gains.

The Nick van Rowe model is no more than combining Frank Knight + Ludwig Mises on capital theory & interest and simplifying their assumptions to the bare essentials -- F. A. Hayek's work in this area is all about showing how this model does not engage the marginal valuation problem of multiple alternative production processes required more or less time with superior or inferior output, goods which are used up across time at varying rates.

Nick van Rowe writes,

"And growing new land is a business just like growing wheat or barley or any other business."

Now you are approaching the point where you have trade-offs between alternative production processes required more or less time with superior or inferior output -- now put REAL TIME into the model by including technological change, production mistakes, finite resource change, etc. -- ie production goods and production methods which will never be replicated, turning your toy (model) into a real boy in the real world, abandoning the scifi conformity world were all production is is always exactly the production, just varying the how much.

In other words, you are not there yet.

This result is already shown in Hayek's _The Pure Theory of Capital_ in a more sophisticated form. (It's also in Fisher):

"The quantity of new land produced each year would be where the Price of land equals the Marginal Cost of producing new land. So the position of that Marginal Cost curve would also affect how much new land gets produced each year, which would also influence the current and future production and consumption of wheat, barley, and whatever. It's not just preferences that determine interest rates."

But I don't get this:

"Unless consumption never changes from year to year."

[Greg Ransom]

Make that:

abandoning the scifi conformity world were all production is is always exactly the SAME production, just varying the how much.

[Greg Ransom]

Hayek goes through many of the pathologies with the unchanging world/scifi conformity strategy for doing capital theory/interest theory in his reply to Frank Knight -- F. A. Hayek, "The Mythology of Capital” http://mises.org/etexts/mythology.pdf

[Nick Rowe]

Greg: "What we are interested in is choice involving production goods with alternative productive uses of varying production times which are worn up with use -- THIS is the problem of choice across time involving tradeoffs between time discounting and output gains."

Yep. The DCT already has that. You have to repair the dikes occasionally. That's covered in example 3.

"...now put REAL TIME into the model by including technological change, production mistakes, finite resource change, etc..."

Yep. The DCT can handle that. Technological change is just like 1. A 1% annual improvement in productivity is just like the sea level slowly falling so you get 1% more land every year.

Production mistakes are like 3, where you don't know whether it will flood next year.

And finite resource change is like 1 or 2.

"This result is already shown in Hayek's _The Pure Theory of Capital_ in a more sophisticated form. (It's also in Fisher):"

Yep. Another case of simultaneous discovery, just like 1871. But Nick van Rowe did come up with the fundamental and specifically Dutch insight that capital is really land. Except the history books tell us that God made some land and our ancestors made some other land. But if we lost the history books, we would never be able to tell the difference. And only Ricardians would care. (And they spend most of their time in an unchanging steady state anyway.)

[Nick Rowe]

Greg: it occurs to me you might be missing the point of this post.

I don't think there's anything here an Austrian would object to or disagree with, other than: not enough uncertainty; not enough about coordination problems (BTW, did you read my very short run post, where I say something nice about Hayek?); it's not complicated enough; the Austrians are being very gently teased, almost in passing.

Others (not Austrians) might disagree *much* more strongly with what I say here.

[rsj]

Other than a great explanation as to where the phrase liquidating capital came from, I don't believe that there is much insight in this story for our world.

Rather than the constancy of consumption driving the constancy of rates, my gut says that consumption is itself driven by other factors. I.e. business profits decline, leading to a re-valuation of the capital stock, leading to a decline in wealth, forcing households to either re-adjust their consumption plans or to be forced to deviate from those plans via credit constraints. If you were to simplify this story, viewing business profits as earnings or interest, then the essential insight would be that rates drive consumption, not that consumption drives rates.

[Nick Rowe]

rsj: the Dutch were always very good at getting monetary policy exactly right. Or so my ancestor said. So the economy was always in equilibrium, because the central bank always allowed the rate of interest to adjust to equal the level predicted by Dutch capital Theory.

[Simon van Norden]

Ah! That explains why they chose a Dutchman as the first president of the ECB....

[rsj]

the Dutch were always very good at getting monetary policy exactly right

Hah! You are as dodgy as Dutch Finance.

And you've waved this magic wand before. In what sense did they "get things right"? That tells me that the real dynamics are different.

The laws of motion, if you are constrained to move on that slice, are different from the laws of motion in general.

Economics has to understand the general case, because otherwise the CB is assumed to have knowledge in excess of economics.

In physics we have "fictional forces". I.e. the centripetal acceleration experienced by a ball moving at the end of the string. In reality, someone must be pulling hard on that swing in order to keep the ball swinging. That person's reference frame does not have any fictional forces. But from the point of view of the ball, which doesn't see the person swinging it around, there is a mysterious force that constrains its movements to the circle. Physics, from the point of view of the ball, has 4 fundamental forces: gravity, strong, electro-weak, and centripetal. But we know that one of those is not fundamental, and should be fully explained by the other three. You can only describe that point by looking at what happens when the person lets go of the ball, or swings it differently. Then you get the real law of motion.

So I am not satisfied with the CB-as-ensurer-of-perfect prices story, anymore than I am satisfied with the Walrassian Auctioneer-as-ensurer-of-perfect prices story. I do not believe in either, and think that to get the capital story right, you need something more fundamental than what is presented here.

[Greg Ransom]

Actually, Hayek did the land is capital thing already -- he covers most of your Nick van Rowe stuff in The Pure Theory of Capital, addressing land being used up and repaired, etc.

"But Nick van Rowe did come up with the fundamental and specifically Dutch insight that capital is really land."

[Greg Ransom]

I'm not sure I miss the point, Nick. I like the point.

For my uses -- History of Econ thought / Phil of science -- I like how your model captures both Knight & Mises in the same model, and how that model fails to capture the valuation across time in production goods problem solved by Hayek and Fisher.

That was my point and interest, it's not a disagreement, it's an observation, and also an observation about what comes next in the analysis.

[Robert]

Consider an economy growing at a constant rate of growth with steady state prices. Two processes are operating, according to the above assumptions. In one, each acre of land is used to produce a yearly output of an unchanged acre of land and 4 tons of wheat. In the other a tons of wheat are used to produce an acre of land. Assuming a (competitive) uniform rate of profit, prices are described by two equations. The first is P(1 + r) = P + 4. The second is a(1 + r) = P. These are two equations in two unknowns. (As stated in the post, the first equation can be used to find a third unknown, the rent of land, in terms of the original unknowns.) There is only one economically meaningful solution. One need not talk about intertemporal preferences at all.

[Nick Rowe]

Greg: OK.

Robert: OK. And now assume that people get a little bit more or less impatient. The rate of interest rises or falls, and one or other of those two processes stops being used.

Everybody: I'm off canoeing for a few days. Somebody please defend the Dutch, if the English or Americans attack them. Good people the Dutch.

[Simon van Norden]

The English they can handle (as they did when they sailed their fleet up the Thames.)
Historically, it's the Spanish they worry about! (Okay....maybe Germany too.)

[Martin]

Nick,

when you get back and have time/or anyone else:

"But he did leave one thing out. The quantity of new land produced each year would be where the Price of land equals the Marginal Cost of producing new land. So the position of that Marginal Cost curve would also affect how much new land gets produced each year, which would also influence the current and future production and consumption of wheat, barley, and whatever. It's not just preferences that determine interest rates. Unless consumption never changes from year to year."

Bit of semantics, but is it really the interest rate that changes when consumption changes or is it something else and we just include in that what we call the interest rate?

For example: the forward price of an asset can be below the spot price of an asset, does this mean that the interest rate is negative? Usually it is said then that there is a cost of carry or that there are dividends.

Doesn't it make more economic sense to say that r=i, and we can calculate the rate of interest as 1/(1+r) = B.MU(C(t+1)/MU(C(t)), provided C(t+1) and C(t) are the same goods? C(t+1) and C(t), however are not the same goods, so that what we call the interest rate also includes the relative price of C(t+1) and C(t).

[Peter N]

If you look back few the last few months of postings, you'll find that a surprising (?) number of them concern the effects of aggregation. Aggregation is used every where in economics because of:

1) Historical reasons. Try to imagine the history of economics without capital, aggregate demand, aggregate labor, aggregate price levels etc.

2) Cultural reasons. This is how economics, particularly undergraduate economics is taught.

3) Practical reasons. Aggregation is necessary for most current theoretical economics

4) Political reason. Without aggregation micro-foundations are impractical (assuming you thought they otherwise were).

5) A need to get quantifiable results (even if they don't model the real economy). Economists who argued against the validity of aggregation are criticized for not being able to get the sort of results that users of aggregation can get. For example:

"the influences of Piero Sraffa and Joan Robinson, in particular, are of central importance. Even in that case, there is a flavour of necrophilia in the air. If one asks the question: what new idea has come out of Anglo-Italian thinking in the past 20 years?, one creates an embarrassing social situation. This is because it is not clear that anything new has come out of the old, bitter debates. Meanwhile mainstream theorizing has taken different directions. Interest has shifted from general equilibrium style (high-dimension) models to simple, mainly one-good models. Ramsey-style dynamic-optimization models have largely displaced the fixed-saving coefficient approach. The many consumers that Stiglitz implanted into neoclassical growth modelling did not flourish there. Instead the representative agent is usually now the model's driver.

Sraffa and Hayek may have won their arguments. Cambridge (US) conceded to Cambridge (UK). It doesn't seem to matter. You recently covered re-switching, but that's just the tip of the iceberg. The idea of a pool of undifferentiated capital is just silly.

I think we should follow what the Fed is doing and look at models based on finance. They're putting a fair amount of effort into this.

[Nick Rowe]

Simon: what's so interesting to me about the Dutch (from my very limited understanding of economic history) is how very close they came to beating the English to becoming the first modern country (Industrial Revolution and all that). As I understand it (probably not very well), the Dutch were ahead at one point, but fell behind. Probably because: there were more English than Dutch, so in a fight we would usually win (OK, except for that time they sailed up the Thames); the big debt the Dutch had from fighting the Spanish again and again.

Martin:

1. "Bit of semantics, but is it really the interest rate that changes when consumption changes or is it something else and we just include in that what we call the interest rate?"

I think it's mostly semantics. Dividends, expected capital gains/losses, carry cost, roll it all into one. I think Finance guys call it "yield" instead of interest?

2. Yep. With multiple consumption goods, C(t) is really a vector. And you get lots of equations like: [1/(1+r)].[P(t+1)/P(t)] = B.MU(Cb(t+1)/MU(Cb(t)) where r is the rate of interest on say wheat, and P(t) is the price of barley in terms of wheat, etc.

Or you can re-write those equations as 1/(1+rb) = B.MU(Cb(t+1)/MU(Cb(t))
where rb is the real interest rate on barley.

(I may have screwed up the maths in the above, since I usually do).

[Nick Rowe]

Peter N:

1. Is it any harder to aggregate capital than to aggregate labour or land? Is it because different machines (like different labour and different land) produce different goods, but also produce those different goods at different dates?

2. But there's something that puzzles me much more deeply about the whole Cambridge Capital Debates.

Re-read Robert's comment above, for example.

"Consider an economy growing at a constant rate of growth with steady state prices. Two processes are operating, according to the above assumptions. In one, each acre of land is used to produce a yearly output of an unchanged acre of land and 4 tons of wheat. In the other a tons of wheat are used to produce an acre of land. Assuming a (competitive) uniform rate of profit, prices are described by two equations. The first is P(1 + r) = P + 4. The second is a(1 + r) = P. These are two equations in two unknowns. (As stated in the post, the first equation can be used to find a third unknown, the rent of land, in terms of the original unknowns.) There is only one economically meaningful solution. One need not talk about intertemporal preferences at all."

I fired back an off-the-cuff response:
"Robert: OK. And now assume that people get a little bit more or less impatient. The rate of interest rises or falls, and one or other of those two processes stops being used."

I am a bit surprised that Robert didn't fire back with a quick counter: if the first process stops being used, there is zero consumption today, so MU(C(t)) probably becomes extremely high; and if the second process stops being used, growth will stop. But Robert could easily have handled that second case by simply assuming his second process was reversible, so growth could go negative if people suddenly get very impatient.

Let's put some numbers on it:

Assume 1 acre of land produces 4 tons of wheat per year. Assume 40 tons of wheat can be used to produce 1 acre of new land, and that this process is reversible. It follows immediately that the MC of land is 40 tons of wheat. The MC curve is flat, in both the positive and negative quadrants, so P=MC=40, regardless of preferences. And since the rent on land is 4, it follows immediately that the rate of interest (measured in wheat) is 10%, regardless of preferences.

In other words, if we do make a strong simplifying assumption about technology, we can determine the rate of interest from technology alone, regardless of preferences.

But, what Robert has done is reinvented "American (i.e. Solow-Samuelson, and certainly not Fisher) Capital Theory". Just redefine units so that we measure land in fortieths of an acre, and one unit of output is equivalent to one ton of wheat or one unit of land (i.e. capital). C+I=Y=4K where I=dK/dt. This is the "AK Growth Model", where "A" is 4 (Robert calls A "a").

With units defined that way, and with that technology, the rate of interest is equal to and determined by the marginal product of land (i.e. capital).

It was EXACTLY that same simplifying assumption that caused the whole Cambridge-Cambridge Capital controversy. It's a one-good model, where the capital good and the consumption good are exactly the same. (More generally, where the capital/labour ratio for the consumption good is always the same as the capital/labour ratio for the investment good, or in Marxian language, they have the same "organic composition of capital").

Cambridge UK economists wanted to explain prices from technology alone without talking about preferences. So they made a lot of simplifying assumptions that helped them do this. But they couldn't explain the rate of interest from technology alone.

Cambridge US economists made a simplifying assumption that let them explain the rate of interest from technology alone without talking about preferences. (Exactly the same assumption Robert makes above).

Instead of being very happy, Cambridge UK economists became very upset. They said that was a very special assumption about technology. They were right, of course. But why were they unhappy??

From my perspective, Cambridge UK and Cambridge UK were both on the same side. Both made a lot of extreme assumptions about technology (and only one type of labour etc.) to try to explain prices (and interest rates are just prices) without talking about preferences.

My hypothesis: Cambridge UK became upset because they didn't want to explain interest rates from either technology or preferences.

[Determinant]

Further, the Dutch were invaded by the French in 1795, had a puppet ruler under Napoleon (one of his brothers) and then were annexed outright to France in 1810. The Napoleonic Wars provided a kick to industrialization through the need to maintain the 800 ships of the Royal Navy. The Dutch got the political instability of the Napoleonic Wars.

[Peter N]

"My hypothesis: Cambridge UK became upset because they didn't want to explain interest rates from either technology or preferences."

It was more basic than that. Capital simply isn't well defined. It means different things and is valued different ways in different contexts. Now you can talk about book value, replacement cost, liquidation value, market value, sometimes excess earnings attributable. It depreciates, it can become impaired, in some cases it depends on whether it must be marked to market.

The rate at issue IIR was rate of return on capital (only in toy models is this the interest rate, and, of course, there's more than one interest rate. IBM pays 1.375% for 10 years and Spain pays over 7%). This obviously depends on how you value capital. You can't just aggregate the return in dollars, aggregate the monetary value of capital and from this calculate a rate. It won't result in anything useful.

Cambridge UK didn't believe that aggregating capital was valid. They had a mathematician's prejudice against drawing conclusions from incorrect mathematics.

"Cambridge UK economists wanted to explain prices from technology alone [to accomplish which they set out to prove this was impossible?!] without talking about preferences [which is impossible]. So they made a lot of simplifying assumptions that helped them do this."

Like what, exactly.

BTW Hayek observed that at 0% interest rates a permanent piece of return earning capital was infinitely valuable, since there is no NPV discount. Of course preferences, risk and uncertainty will save us from these ugly infinities. Maybe Hayek was part Dutch.

Value has been a problem for economics since at least the 17th century. Marx ended up using two different kinds - it didn't help. You can't combine intrinsic conserved value with market price value. Economics isn't thermodynamics. There's no equivalent to the conservation of energy. It seems unfair to be stuck with the aggravating second law without having the very useful first law, but that's (economic) life.

BTW, there's audio of some American Joan Robinson lectures on line, but it's unlistenable. Even an hour with Audacity couldn't fix it.

[Nick Rowe]

Peter N: "Like what, exactly."

Suppose we had a 2-good GE model. Apples and bananas. Put Qa on one axis and Qb on the other axis. Draw the PPF, and an Indifference curve that kisses the PPF, and a budget line that is tangent to both. The relative price of apples to bananas is the slope of the budget line. It is determined, in general by both the PPF and preferences. What assumption would we need to make to ensure we could ignore preferences, so that the relative price was determined by the PPF alone? We need to assume whatever is necessary to ensure that the PPF is a straight line. E.g. there is only one input (call it "labour") and all labour is identical. Technology is linear (no diminishing marginal products to labour). No joint production where there are synergies between producing wool and mutton. Probably others I don't know about or have forgotten.

They were willing to make all those assumptions about the intratemporal PPF so they could determine intratemporal prices without talking about preferences. But they were very unhappy when someone else made a similar assumption to make the intertemporal PPF a straight line so you could determine intertemporal prices (interest rates) without talking about preferences.

Put it another way; they were happy to add all the labour together, but unhappy to add all the machines together and add the production of new machines to the production of the consumption good.

"Value has been a problem for economics since at least the 17th century."

Well, if you try to do value theory without talking about people's evaluations of goods (preferences), you will run into problems.

"Maybe Hayek was part Dutch."

Very much so. If we lost the history books, we wouldn't be able to tell the difference between land and the pre-existing stock of capital. And it doesn't matter. And Hayek, like Jevons, would insist it doesn't matter. "Dutch" was just a rhetorical device that helped me make that point.

[Edeast]

Good work, on the communications ngdp stuff, but these recent posts reswitching, interest rates, and that massive one I couldn't understand are good too so dont think you can retire from blogging. Well you can if you want, just wanted to give some props. The only person waiting to hear from now is AdamP.

[Martin]

Nick,

"I think Finance guys call it "yield" instead of interest?"

Yup, I think so.

With regards to #2:

"Yep. With multiple consumption goods, C(t) is really a vector. And you get lots of equations like: [1/(1+r)].[P(t+1)/P(t)] = B.MU(Cb(t+1)/MU(Cb(t)) where r is the rate of interest on say wheat, and P(t) is the price of barley in terms of wheat, etc.

Or you can re-write those equations as 1/(1+rb) = B.MU(Cb(t+1)/MU(Cb(t))
where rb is the real interest rate on barley."

Yes. I was also more thinking along the lines as well of "C" changing over time whilst we still call it "C", bit like the iPhone 4 and iPhone 4S. I guess though that's the same as an increase in crop: i.e. the iPhone 4S is probably equivalent to a*iPhone 4 where a>1.

I never really gave the interest rate that much thought; I mean I know the math, the concepts, can draw the graphs etc, but didn't think much about what's behind it all. Suffice to say, I really appreciate the model, it made me read a bit into it more now.

[Greg Ransom]

The best study by far that I've found of the issues involved in the pure time preference theory of interest and its relation to intertemporal exchange involving production goods with alternative productivity & time dimensions is _The Austrian Subjective Theory of Interest_ by Ingo Pellengahr.

Highly recommended.

[Edeast]

If time preference is another word for risk tolerance, might make rsj happy. There's a good chance that the new firm, machine, dike might break. Now does the creation of capital, expand/change the shape of the ppf, hold it steady? Or do you introduce, technology as a separate term. I think it does change the ppf because it produces 4 per turn subject to floods.

[Edeast]

Nevermind.

[Nick Rowe]

Edeast: the two are related. The quicker Marginal Utility of consumption diminishes as consumption increases, the greater will be risk aversion, and (if consumption is growing over time) the greater will be the marginal rate of time preference.

[Edeast]

I worked through 1 and 2. With the equations as given. But what is the rational for the Lhs of the equillibrium condition. 1/(1+r)

[Edeast]

By rational i mean rationale. Looks like you assume what you are going to proove , just wonder if there is something obvious about it, that an outsider doesn't get. I get that you a discounting next periods marginal utility, is that in utils?, which cancel and then the r is wheat/time. Or something.

[Nick Rowe]

Edeast: It comes from the intertemporal budget constraint. If I hadn't been sloppy, I would have written:

Max V(t) = U(C(t)) + B.U(C(t+1)) + B^2.U(C(t+2)) + etc.

subject to C(t) + 1/(1+r).C(t+1) + 1/(1+r)^2.C(t+2) + etc = Y(t) + 1/(1+r).Y(t+1) + 1/(1+r)^2.Y(t+2) + etc

where Y(t) is non-interest income, and r is the rate of interest at which you can borrow or lend. What this says is that the Present Value of your consumption equals the Present Value of your income.

If you increase C(t) by 1 unit, by borrowing more or lending less, you must decrease C(t+1) by 1+r units, or decrease C(t+2) by (1+r)^2 units, etc. to repay the loan.

[Nick Rowe]

Intuition: If I increase C(t) by 1 unit of wheat, V increases by MU(C(t)) utils, but at the same time I decrease C(t+1) by 1+r units of wheat and V decreases by (1+r).B.MU(C(t+1)) utils. When I have chosen the V-maximising mix of C(t) and C(t+1), the two effects must net to zero. So

MU(C(t)) - (1+r).B.MU(C(t+1)) = 0

Rearrange to get: 1/(1+r) = B.MU(C(t+1)/MU(C(t))

[Edeast]

Id like to blame all of my spelling and grammar on the Ipad, but I've never been a literal liar.

[Edeast]

@9:02
Thats good, thanks.

[Edeast]

So the 1+r increases the MU(C) because it decreased the consumption, or quantity of C.

[Nick Rowe]

Edeast: you lost me there. Let me take a guess. If I increase C by 1 (very small) unit, my utility increases by MU(C) (by definition of MU(C)). If I increase C by 1+r units, my utility increases by (1+r).MU(C)

[Edeast]

Ya just trying to figure out why, (1+r).B.MU(C(t+1). I was sloppy in the comment. Just tracing why, 1/(1+r)C(t+1) flips up on top when we talk about the marginal utility.

[Edeast]

Forgot the negative sign. Ok. Actually forgot to read the whole comment.

[Edeast]

But shouldn't the 1+r go inside the C, and doesnt't marginal utility go down with each added quantity? I don't know why I 'm still commenting. I can see why total utility would go down next period due to less consumption, but i thought maginal utility would be higher.

[Martin]

Edeast,

the principles of inter-temporal choice are the same as those of ordinary choice; you maximize subject to a constraint and the result is that, graphically, the indifference curve is tangent to the slope of the budget-constraint. This point of tangency is expressed by the equation that Nick showed.

http://upload.wikimedia.org/wikipedia/commons/0/01/Intertemporal_Choice.jpg

This is basically the same picture as for a two-good model.

From the perspective of the individual, who can lend and borrow freely, consumption in the first period is exchanged against consumption in the second period until that point of tangency is reached.

From the perspective of the community, the interest rate has to be such that no one wants to exchange consumption from the first period against consumption in the second period.

The equation describes when both conditions are satisfied and the economy is in equilibrium.

Hope this clarifies?

[Edeast]

Thanks Martin, could you change the subject? I've got to catch a flight. Me at 11:18 still forgot the minus sign.

[Nick Rowe]

Edeast: the extra utility from eating 2 extra apples is twice the extra utility from eating one extra apple. As long as the apples are very very very small. That's the neat thing about calculus!

[Edeast]

Then why in example 1 does an increase in C lead to decrease in MU(c)

[Nick Rowe]

Because that wasn't a very very very small increase in C. It was a lot more wheat being produced and consumed. If it had been a very very very small drop in the sea level, it would have caused such a very very very small drop in MU and rise in r we would have ignored it.

[Andrew Lainton]

Fascinating - lets hope you really did have a Dutch Grandad but dont care if not it an interesting theory.,

I dont know how you can present this as a justification of a pure time preference approach as it is so similar to Von Thunen which is often (super-fiscally) presented as a pure productivity. In fact I would classify it as one of a range of models which unify productivity and time preference.

The 'pure' time approach to interest was criticised by McCulloch as time is

'A mere word a sound – is nothing – can do nothing '

To which Scrope - who first unified time preference and productivity theories of interest modified Seniors abstience approach

'No-one will sacrifice time by [merely] allowing it to operate on his property…that they do this acquire additional value due to natural laws –the sown wheat multiplying itself in its crop, the kept wine improving its flavour - is notorious.'

So time allows for interest by virtue of the opportunity cost of waiting during which the waiting provides the conditions for [natural or human induced] productivity to take place.

Scrope therefore set out a 'cost' based theory of interest. Bohm Berwerk essentially adopted the same structure but denied it was a cost. Instead, like you, he argued it was solely due to the foregone utility of discounting future abstained consumed.

This was thoroughly debunked by Thomas Carver Nixon and you should really read his definitive approach

'Interest does not correspond to any general discounting of future consumption of commodities, but only to the marginal discount or to the marginal sacrifice of saving [investing]. It must be sufficient to compensate the capitalist for saving [investing] the last increment of [finance]capital.'

Bohm Bawerck's, and (van) Rowe' approach is flawed because they are non-monetary and in a forthcoming article I show the flaws in Bohm Bawerck's examples.

Two points. Firstly marginal productivity. The example has no capital and so you are correct that it helps debunk the marginal productivity of capital approach, but what it does not debunk is that total factor returns are due to total factor productivity and the marginal return to finance capital (interest) is due to that marginal TFP. I show this graphically in my forthcoming article which is also a simple way of demonstrating wicksell effects.

Secondly dimensionality. If interest has units of money/years and we treat money as just another produced means of production then both interest and products have the same dimensionality - units/years - the issue then becomes one of numaraire.

1. Carver, T.N., Distribution of Wealth (1904),‘The Place of Abstinence in
the Theory of Interest,’ Quarterly Journal of Economics, October 1893.
2. Scrope, G.P., Principles of Political Economy. 1833: Longman, Rees, Orme, Brown, Green, & Longman.

[Nick Rowe]

Andrew: Thanks!

All my ancestors are English (at least, as far back as I know). My Dutch ancestor is literary licence.

"Bohm Bawerck's, and (van) Rowe' approach is flawed because they are non-monetary and in a forthcoming article I show the flaws in Bohm Bawerck's examples."

Agreed. I need to make it a monetary exchange economy. Must do that sometime.

"The example has no capital and so you are correct that it helps debunk the marginal productivity of capital approach, but what it does not debunk is that total factor returns are due to total factor productivity and the marginal return to finance capital (interest) is due to that marginal TFP."

You lost me a little there. TFP (in the simplest version of my model) is simply 4 tons of wheat per acre.

"Secondly dimensionality. If interest has units of money/years..."

That's where I disagree. The rate of interest (whether it's money or wheat or whatever) has the dimensions 1/years. Not $/years or tons/years.