Interest, capital, MRScc=(1+r)=1+(MPK/MRTci)+(dMRTci/dt)/MRTci

That's the equilibrium condition for the real rate of interest in a competitive economy. I will explain what it means a little later.

This is intended as a simple "teaching" post, and because I have a strange feeling that the theory of interest and capital is becoming topical again in the blogosphere, and that a post like this might be helpful.

Suppose you had a competitive economy that produced and consumed both apples and bananas. How would you explain what determines the equilibrium relative price of apples and bananas? This is how I would do it.

I would draw a simple diagram like this:

The red Production Possibilities Frontier shows all the combinations of apples and bananas that can be produced given resources and technology. The slope of the PPF represents the Marginal Rate of Transformation of apples into bananas: how many extra bananas can be produced if you produce one less apple.

The blue Indifference Curve shows all the combinations of apples and bananas that give the consumer the same level of utility. (There is a different indifference curve for each level of utility, but I have only drawn one.) The slope of the indifference curve is the Marginal Rate of Substitution of apples for bananas: how many extra bananas would you need to consume to give the same level of utility if you consumed one less apple.

The black budget line shows all the combinations of apples and bananas that can be bought or sold for the same level of income. The slope of the budget line shows the relative price of apples in terms of bananas: how many extra bananas you can buy or sell with the same income if you buy or sell one less apple.

In competitive equilibrium, where producers are maximising profits and consumers are maximising utility, and demand equals supply, all three curves have the same slope. The equilibrium condition is:

MRSab = Pa/Pb = MRTab

The relative price (and the quantity of each good) is determined both by the PPF and by the indifference curve(s). You need both to explain prices.

Only in special cases do you not need both. Like if the PPF was a straight line you don't need the indifference curve to explain relative prices. Like if the Indifference curve was a straight line, you don't need the PPF to explain prices. (But even in these cases you need to look at the other curve to check you aren't at a corner solution where only one good gets produced and consumed). Or if one of the two curves is kinked, so its slope is undefined at that kink, so only the slope of the other curve determines the relative price. (But even in these cases you still need the kinked curve to tell you where you are on the other curve.)

OK. That's how I would explain what determines the relative price of apples and bananas. (If you want to add carrots to the model, just add a third dimension to the diagram.)

Irving Fisher used basically the same diagram to explain what determines the competitive equilibrium real rate of interest.

The only thing different between the two diagrams is what's on the vertical axis. Instead of a trade-off between apples and bananas, we have a trade-off between apples today and apples "tomorrow" (or next year). The slope of the budget line is (1+r), where r is the real rate of interest (the nominal rate of interest, minus the inflation rate on apples). If r=5% per year, if you consume 100 less apples today you consume 105 more apples next year.

We now interpret the slope of the indifference curve as the Marginal Rate of intertemporal Substitution, between consumption this year and consumption next year. Call it MRScc.

And we now interpret the slope of the PPF as the Marginal Rate of intertemporal Transformation, between consumption this year and consumption next year. Call it MRTcc. The equilibrium condition is:

MRScc = (1+r) = MRTcc.

If you want to add a third time period, just add a third dimension to the diagram.

Here is an example of a special case of the Irving Fisher diagram, where the PPF is kinked.

The economy can produce 100 apples each year. The green PPF assumes you cannot store apples from one year to the next. The alternative blue PPF assumes you can store apples from one year to the next. (If 10% of the stored apples rotted, you would need to replace my "200" with "190", and if 100% of the stored apples rotted you get the green PPF). In this example, people choose not to store apples. But the equilibrium rate of interest is still determined in this example. It is determined by the the slope of the indifference curve at the kink. The slope of the PPF is undefined at the kink. If we wanted to, we could complicate this simple model by having different people with different preferences. More patient people would lend apples to less patient people, at a positive rate of interest, but no apples get stored.

We don't need "capital", or its marginal product, to determine the rate of interest.

But let's go back to the more general case of the Irving Fisher diagram, where the PPF is curved. Where is "capital" in this model? And where is the Marginal Product of Kapital? Does MPK determine r? Does MPK=r? "No", is the answer to both those questions.

The equilibrium condition is MRScc=(1+r)=MRTcc. MPK is one of the things, but not the only thing, that affects MRTcc. And MRTcc is equal to (1+r), but it does not determine (1+r).

Suppose that the economy can produce capital goods ("investment" is the quantity of capital good produced each year) and consumption goods. There is a PPF between consumption and investment, just like the PPF between apples and bananas in my first diagram. The slope of that PPF is the Marginal Rate of Transformation between consumption and investment. Let's call it MRTci. In equilibrium MRTci=Pk, where Pk is the price of capital goods in terms of consumption goods. If the PPF is curved, like in my diagram, Pk will need to increase if we want producers to produce more investment and less consumption goods. And so we may not assume that Pk will be the same next year as this year, unless the PPF is a straight line, whose slope never changes from one year to the next.

And if you want two different capital goods, just draw a 3-dimensional PPF.

MPK is defined as the extra apples produced per extra existing machine, holding technology and other resources constant, and holding the production of new machines constant. If we move along the PPF between consumption and investment this year, we will have a bigger stock of capital goods next year, which will shift out next year's PPF. MPK tells us how much it shifts out, per extra machine.

What is the relationship between: MRTcc (the slope of the intertemporal PPF); MRTci (the slope of the consumption-investment PPF); and MPK?

This year is year 0. Next year is year 1. The year after next is year 2. Suppose we consume one less apple this year. That lets us produce 1/MRTci(0) extra machines this year. That lets us produce MPK(1)/MRTci(0) extra apples next year. But it also lets us produce 1/MRTci(0) fewer machines next year, and go back to the same number of machines we would have had the year after next. And producing 1/MRTci(0) fewer machines next year lets us produce MRTci(1)/MRTci(0) extra apples next year.

So if we consume one less apple this year, we can consume MPK(1)/MRTci(0) + MRTci(1)/MRTci(0) extra apples next year. That is the slope of the intertemporal PPF.

So the equilibrium condition becomes:

MRScc = (1+r) = MRTcc = MPK(1)/MRTci(0) + MRTci(1)/MRTci(0)

Or, we could rewrite it as (where dMRTci/dt is the rate at which MRTci is changing over time):

MRScc = (1+r) = 1+(MPK/MRTci)+(dMRTci/dt)/MRTci

Or, since MRTci=Pk, we could write it as:

MRScc = (1+r) = 1+(MPK/Pk)+(dPk/dt)/Pk

If capital exists, the real rate of interest is equal to, but not determined by, the Marginal Product of Kapital divided by the real price of the machine, plus the capital gains from appreciation of the real price of machines. (The latter, of course, is only an expectation, which matters a lot in the real world. And all these models also leave out money, which also matters a lot in the real world, because bad monetary policy causes recessions, and recessions push the economy to a point inside the PPF.)

Rather than saying "MPK determines r", it would be more true to say "MRScc determines r, which determines the prices of capital goods". And the only thing wrong with saying that is that is that MRScc is not a fixed number, but depends on the expected growth rate of consumption, which in turn depends on our ability to divert resources to producing extra capital goods instead of consumption goods, and the productivity of those extra capital goods.

Or, we could even say that the central bank determines r, which in turn determines both the expected growth rate of consumption and the prices of capital goods. And the only thing wrong with saying that is that a central bank which sets r ignoring the equilibrium condition will really mess up the economy badly.

(Introducing physical depreciation of machines, so that new and old machines have different prices, is left as an exercise for the reader, because I will mess up the arithmetic.)

This post is also for Bob Murphy. Bob is really good on the theory of capital and interest, and already knows all this stuff and more, but he can't draw diagrams.

Comments

[Bob Murphy]

Great post Nick! And my mind sees the economy as a n-good system in continuous time. Diagrams cannot capture my imagination.

[W. Peden]

"Marginal Product of Kapital"

The Maoist phase continues!

"We don't need "capital", or its marginal product, to determine the rate of interest."

Is this a controversial claim?

[Nick Rowe]

Thanks Bob!

W Peden: "Is this a controversial claim?"

Hmm. I don't know. It certainly shouldn't be. But I suspect some might be a bit lost if they were asked what determined the rate of interest in a world where there were no additional investment opportunities, so that MRTcc is indeterminate, and they can't use MPK=r. For example, where existing land and labour has a positive VMP, but there is nothing we can do to create new land or improve the future productivity of existing land or labour.

[Nick Rowe]

W Peden: put it this way: Austrians should have no difficulty with that claim. Irving Fisher would have had no difficulty with that claim 100 years ago. Cambridge UK types should have no difficulty with that claim (Luigi Pasinetti for example wrote a paper around 1980 of a model with no capital and a positive rate of interest). But maybe some never got the memo, and think there is no standard micro theory of r without capital and its marginal product.

[Mike Sproul]

I suppose this might be gilding the lily, but you should have been drawing a COMMUNITY PPF for two people. The diagram you showed was showing curves for only one person, which isn't really right, since the person is part of a market economy.

[Link here NR]

Then you can inscribe an Edgeworth box inside the community PPF. General equilibrium can then be pictured by drawing a budget line that is tangent to both PPF's, and to the indifference curves of both traders. At that point, the slopes of all the relevant curves are equal to each other.

Once a pair of indifference curves are drawn tangent to each other at the point of equilibrium, you can slide them along each other to form the community indifference curve, which would be tangent to the community PPF.

[Nick Rowe]

Mike: you are totally right, of course. But the thought of trying to draw an Edgeworth box inside the Irving Fisher diagram, and trying to explain it all, filled me with dread. So I resorted to that terrible macro cheat, and assumed a representative agent!

If anyone else questions me on that, I will ask you to handle it for me!

[Bob Murphy]

W. Peden, both Nick and I were aghast at this passage from Piketty's book:

Technology naturally plays a key role. If capital is of no use as a factor of production, then by definition its marginal productivity is zero. In the abstract, one can easily imagine a society in which capital is of no use in the production process: no investment can increase the productivity of farmland, no tool or machine can increase output, and having a roof over one’s head adds nothing to well-being compared with sleeping outdoors. Yet capital might still play an important role in such a society as a pure store of value: for example, people might choose to accumulate piles of food (assuming that conditions allow for such storage) in anticipation of a possible future famine or perhaps for purely aesthetic reasons (adding piles of jewels and other ornaments to the food piles, perhaps). In the abstract, nothing prevents us from imagining a society in which the capital/income ratio β is quite high but the return on capital r is strictly zero. In that case, the share of capital in national income, α = rXβ, would also be zero. In such a society, all of national income and output would go to labor.

That sure looks like he is saying that if MPK = 0, then we can conclude that the real rate of interest is zero and thus the interest income earned by capitalists is zero.

[W. Peden]

Nick Rowe,

This subject makes some strange allies!

"What major points did Joan Robinson, Irving Fisher and Friedrich Hayek all agree on?" would be quite an interesting history of economic thought essay question!

[Kevin Donoghue]

Bob Murphy: "That sure looks like [Piketty] is saying that if MPK = 0, then we can conclude that the real rate of interest is zero and thus the interest income earned by capitalists is zero."

No, he's saying that if a variable on one side of an equation is set to zero then both sides of the equation are zero. Note that r is not the real rate of interest. It's the return on capital.

[Nick Rowe]

Bob: that paragraph is very worrying. It sure sounds like a pure productivity theory of interest. But does he talk about time preference elsewhere, and redeem himself?

W.P. Good question! My crappy answer:

All three agreed that "r=MPK and MPK determines r" is wrong (outside of a simple model where the capital good and consumption good are the same).

Hayek would I think agree with Fisher if you drugged Hayek into accepting a perfect foresight model that was always in equilibrium, but I think JR would disagree, and still say it's all fundamentally flawed, because it's circular, so r is indeterminate. Dunno, that's just my guess.

[Nick Rowe]

Kevin: look at my third picture, with the green PPF. Suppose we can convert 1 apple into 1 machine, but that machine is totally useless. The MPK=0 for that machine, and it is impossible to convert that machine back into 1 apple. Then we get the square green PPF. The rate of return on investment in that machine is 0, but it is not true that the rate of interest must equal 0. MRTcc is undefined at that point.

[Edward Lambert]

Above MRTci=Pk in equilibrium,
Yet, MRSab = Pa/Pb = MRTab,
then terms follow MRTci=Pc/Pk.
Is Pk=Pc/MRTci?
Also "Pk is the price of capital goods in terms of consumption goods" and "Pk will need to increase if we want producers to produce more investment and less consumption goods." So as Pk increases relative to Pc, the slope of equilibrium MRTci decreases showing more investment on the PPFci?
Then when MRTci=Pc/Pk =(1+r) through association... If central bank policy maintains r below equilibrium from year to year suppressing an equilibrium right-shift along the PPFci to consuming the extra production, Pk is being artificially raised relative to Pc so that, investment in capital is encouraged for job creation. However, if Pk itself is suppressed in real terms say from cheaper capital goods being purchased from foreign producers as is happening ... Pc (inflation?) would be even more suppressed than Pk showing low inflation tendencies.
I realize lower inflation directly raises r in real time, but the equilibrium equation is opposite, Pc/Pk = (1+r) implies change in Pc = change in (1+r) holding Pk constant. This looks like the long run Fisher Effect to me.
In other words, moving toward equilibrium and raising nominal rates would raise r, which would increase the slope of MRTci, and would imply Pc rising relative to Pk, and maybe a turn-around of persistent low inflation.
Shoot me down...

[Kevin Donoghue]

Nick, I've no complaints about your post. What I don't get is why, given that Piketty warns against conflating the return on capital with the rate of interest, you insist on reading "rate of interest" when he wrote "return on capital". I guess your faith in Bob has led you astray.

[Nick Rowe]

Edward:

1. I have assumed Pc=1 (we measure all prices in apples).

2. I always get muddled about whether "substituting apples for bananas" means more apples and fewer bananas or vice versa. Even after singing "Substitute" by The Who to myself. So I always get confused between MRTab and MRTba. But if you look at the pictures you can work out what it should be.

3. If the central bank sets r above the equilibrium, there will be insufficient demand for goods, so we are inside the PPF's, because somebody will be unable to sell what they can produce, so some resources are unemployed, and this model does not tell us what happens then. If the central bank sets r above the equilibrium, there will be excess demand for goods, so somebody will be rationed, so won't be able to buy what they want to buy, and this model does not tell us what happens then.

[Nick Rowe]

Kevin: OK, but if his "r" does not mean the rate of interest, so the rate of interest can be positive, why would all of national income go to labour (even ignoring land rents)?

The whole thing is muddled by his using "capital" to mean "wealth" as well as "produced means of production".

Let's just say it is a very unclear paragraph. (OK, we all have our bad days.) And I fear some would simply interpret it as saying: "If MPK=0 then the rate of interest must be zero too", and think that is right.

[HJC]

Nick & W. Peden: I don't think Nick's Fisherian capital theory is too controversial with regard to the capital debates. He's abandoned the aggregate production function that Joan Robinson found circular (except in the one-good model). But it seems to me that it also does away with the concept of *the* real rate of interest in the Wicksellian sense, which I think will have other implications. The quote from Piketty looks like he's using the aggregate function to *explain* capital/wage shares via their marginal product. This is controversial with regard to the Cambridge debates.

[Kevin Donoghue]

The whole thing is muddled by [Piketty] using "capital" to mean "wealth" as well as "produced means of production".

He makes it clear at the outset that he always means wealth:

To summarize, I define “national wealth” or “national capital” as the total market value of everything owned by the residents and government of a given country at a given point in time, provided that it can be traded on some market. It consists of the sum total of nonfinancial assets (land, dwellings, commercial inventory, other buildings, machinery, infrastructure, patents, and other directly owned professional assets) and financial assets (bank accounts, mutual funds, bonds, stocks, financial investments of all kinds, insurance policies, pension funds, etc.), less the total amount of financial liabilities (debt). If we look only at the assets and liabilities of private individuals, the result is private wealth or private capital. If we consider assets and liabilities held by the government and other governmental entities (such as towns, social insurance agencies, etc.), the result is public wealth or public capital.

He makes it very clear that he rejects any notion of restricting the definition of capital to produced means of production.

[Nick Rowe]

HJC: "But it seems to me that it also does away with the concept of *the* real rate of interest in the Wicksellian sense, which I think will have other implications."

Let's introduce a second consumption good, bananas. Taking Pa as numeraire, the model will determine Pb, and how Pb changes over time. r is the real apple interest rate, and if rb is the real banana interest rate, we get rb = r - (dPb/dt)/Pb. If the central bank sets an interest rate so the real interest rate on apples equals the apple natural rate, the real interest rate on bananas will also automatically equal the banana natural rate.

I don't think there's a problem here. (But Bob does, I think, and Bob has thought about this question more than me. David Laidler also thinks there's a problem here, and I think David Glasner does too. So there's another internecine war among monetarists, plus weird bedfellows, on this question. Austrians plus Cambridge UK plus some but not all monetarists.)

My own problem with the "natural rate" is that, if the central bank misses (and it almost certainly will, in practice), the fact that it missed will have real effects and will almost certainly cause a change in the whole time-path of the natural rate.

[Nick Rowe]

Let me modify what I just said, on the multiple natural rates question:

If someone (an Austrian???) says: "The central bank should set the nominal rate equal to the real natural rate", then there is a problem. Which natural rate? The apple or the banana natural rate?

But if someone (a New Keynesian) says: "The central bank should target 2% inflation rate on a basket of one apple plus one banana, and set the nominal rate so that the real rate for that basket equals the natural rate for that basket", there is no problem. If it hits one natural rate it hits them all.

That probably wasn't clear.

[Nick Rowe]

Kevin: OK, but if by "capital" he always means "wealth", then when he says "the return on capital r is strictly zero" he means the rate of interest is strictly zero.

[HJC]

Nick: I think what you are describing was also called the 'neutral' or 'optimum' rate by Keynes in the GT (p 243), although his might have been explicitly nominal. Basil Moore also believes that there is an interest rate associated with full employment. But they both rejected loanable funds.

[Nick Rowe]

HJC: Let me put it a third way: if someone said "the central bank should target 0% inflation", but didn't specify whether he was talking about the inflation rate on apples or bananas, or a basket, that would be a problem. And it would be the same problem as someone saying "the central bank should set the nominal rate equal to the natural real rate.", without specifying which natural real rate.

[HJC]

Nick: On second thoughts perhaps I am off track with the neutral rate, you are more thinking about a path of money idea.

[HJC]

Nick: Have you got any links to Laidler etc on this?

[Nick Rowe]

HJC: David Laidler on Natural Hazards What you get when you cross a monetarist with Cambridge UK!

[Nick Rowe]

Actually, the published version of David's paper is different from the draft version I remember. So what I said above isn't quite right.

[srin]

You are of course aware that the SMD results imply that at the aggregate level the relationship between marginal rate of intertemporal substitution and interest rates is not quite well-behaved as in the micro level! So, particular interest rate may well be consistent with different consumption vectors. Even though MSRCC would technically "determine" the interest rate, it really tells you nothing at all. The empirical problems with PIH and lack of sensitivity of consumption to interest rates should tell you that this is not a mere theoretical issue.

[HJC]

Nick: Many thanks, even before reading it, I see that it's in terms of the 'neutral' interest rate. It would be nice if all these terms had accepted meanings!

[Nick Rowe]

srin: there are lots of things to worry about, in this model like any other. I worry most about monetary policy and sticky prices, and whether expectations will adjust to bring the economy towards this equilibrium, or whether the equilibrium moves further away from us if we are off it. My guess is that these are bigger issues than distribution effects. But I could be wrong.

HJC: The Bank of Canada uses the term "neutral rate" as, I think, a euphemism for "natural rate + 2%"!

[HJC]

Nick: I've just noticed that there's a handy glossary in the Laidler paper.

[HJC]

Nick: I've read the paper, thanks, it seems to confirm a lot that I was thinking: Wicksellian rate only valid in the one-good (& one-agent!) model; Joan Robinson's circularity argument; difficulties with constructing an appropriate index of heterogeneous goods; etc.

This quote is interesting (p. 9) 'There will of course, still exist at any moment some level of the real (inflation-adjusted [i.e. not pure real]) market rate of interest that would induce just the right overall amount of current saving to provide for the economy's overall demand for investment, but there is no reason to think that this rate will remain constant, let alone be equal to some [pure] real natural rate of interest uniquely dependent upon unchanging elements in the economy's structure.' Although to be post-Keynesian (in the sense of Moore) I think the words 'saving' and 'investment' need to be swapped. He highlights the ambiguity of the two forms of real rate at the end of page 2.

There's not too much that I would disagree with, but not much that seems monetarist either. I must have missed something!

[Nick Rowe]

HJC: at the end of page 2, David is simply making the distinction between the actual real rate and the natural real rate. That distinction is very standard in New Keynesian/Neo Wicksellian macro, and central to their analysis. They say that if the bank sets the nominal rate such that the actual real rate is above the natural rate, inflation will fall, and if it sets it below, inflation will rise.

Monetarists are generally unhappy with central banks using interest rate control. One of the reasons is that we think the natural rate will fluctuate over time, so it is hard to get it right. And setting an actual real rate above or below the natural rate will itself cause fluctuations in the natural rate, which makes it even harder, because it's not just hitting a moving target, but a target that moves away from your shots. But once you have a model like I've sketched here, with just one capital good and one consumption good, I really don't see what qualitative difference adding even more capital and consumption goods makes. 2 goods (one capital and once consumption) is different from 1 good, because Pk is no longer bolted down by technology, but 2, 3, 4, 5, are all the same.

[Kevin Donoghue]

Piketty's terminology is a bit unsatisfactory; he says MPK when he should say MEC; see link.

Having said that, I don't believe he is at all confused. All in all, it's a very clearly-written book.

http://anyoldbullshit.blogspot.ie/2014/05/piketty-and-marginal-product-of-capital.html

[HJC]

Nick: Surely the actual real rate is ex post? The ex ante version is the Fisher definition. Don't you really mean the market real rate! Yes, I can see the 'moving target' references in the paper, didn't realise that that was monetarist though. So, the effect of adding more goods is where you and Laidler disagree.

[Nick Rowe]

Kevin: I just read your post, and I agree with it. (I prefer calling MEC "internal rate of return on investment", just to avoid that ambiguity.) I have no idea whether he is generally confused, or whether there's just a few unfortunate passages. I would be more worried about the distinction between capital and land.

I doubt I will read his book. 700 pages means 700 blog posts I won't read. Which investment would have the higher, er, marginal product? marginal efficiency? And my bloggers ADD is getting so bad I can't even read a novel nowadays.

[Nick Rowe]

HJC: Yep. I wasn't clear. I meant ex ante. But "actual" can mean "as opposed to natural", as well as "as opposed to expected".

I don't think the "moving target" thing is *exclusively* monetarist. Funny thing is, a year or two back we had a big argument on the CD Howe MPC about the Bank of Canada saying it thought the neutral rate had changed. Some didn't get it: "But if inflation gets back to target, and the unemployment rate goes back to normal, shouldn't the interest rate go back to normal too?" And throughout the 2000's, I was telling them: "4% is the new 5%", then "3% is the new 4%", etc.

I *think* that's where me and David disagree, but I confess I'm not sure I understand what he's saying there, at times.

[Kevin Donoghue]

Nick, Piketty and novels are complements, not substitutes. Somebody (AFAICR it was Tyler Cowen) complained that Piketty didn't mention Trollope. But actually, Barchester Towers, Dr Thorne and Phineas Finn are all pretty good accounts of Pikettyland, with lots of men pursuing wealthy women simply because work doesn't pay nearly as well as marrying money.

But you will read the book. Just think how many of those 700 blog posts will be about Piketty, not to mention the fun you'll have yourself, blogging about his thesis.

[Steve]

Nick: I always seem to read your posts when the comments section is already very long. Quite an interesting exchange. Seems to me that there are (at least) three aggregation problems here. (1) Measuring aggregate capital (Cambridge controversy). (2) Measuring current and future consumption (there are many short-term and permanent relative price changes within any consumption basket). (3) Preferences: I can understand that you did not want to incorporate two different consumers using an Edgeworth box. If there are many consumers with differing rates of time preference, standard neoclassical theory says that it's the most patient individual that eventually winds up owning everything as long as he/she is immortal. Just takes a minor modification of neoclassical theory to get a non-degenerate distribution of wealth, though. Time preference could depend endogenously on wealth. We could also tell the story of the patient founder of the company, his dull son, and his profligate grandson (sorry for the sexism if you're reading this Frances). What bother me about Piketty, though, is that there are MANY standard neoclassical models that (1) lead to balanced growth, (2) satisfy r > g along the balanced growth path and (3) have a well-determined share of capital income (which would equate to wealth if there is no land, human capital, etc.). This certainly holds in most models where there is one good that is both capital and consumption (with or without endogenous growth, i.e. constant returns to scale in reproducible factors), and probably extends to many models with heterogeneity across inputs and outputs. In a model with only labour and constant returns to scale, all income will go to labour, and if there's exogenous technological change we would still get r > g, where r is the real interest rate and (obviously) not the rate of return to capital. The diagrammatic illustration of the relationship between growth and the interest rate is useful.

[HJC]

Nick: Got it, thanks. That paper was good, I would read more if you care to recommend any.

[Robert]

Tangent warning:

Alll this Piketty stuff seems to miss another agglomeration of power problem, admittedly more US-centric: a member of either the Bush or Clinton families has been at the upper reaches of political power almost constantly for the last 34 years. Longer , if you count BUSH I's time heading the CIA.

Just wait till they start intermarrying.....

[Nick Rowe]

Kevin: yep, but those marriageable prospects owned *land*! (I did read that dreadful "It is a truth universally..." book.)

Steve: "Just takes a minor modification of neoclassical theory to get a non-degenerate distribution of wealth, though."

That to me seems to be the big question. Assortative mating, number of kids, regression to the mean, risk of blowing the lot through cards, drugs, horses, religious cults, etc.

The one book I should read is "The son also rises". I read his last one.

Robert: yep. Were the events of 1776 really worth it? George 3 and his descendants would have been fine as hereditary leaders, with a little less power!

HJC: "Although to be post-Keynesian (in the sense of Moore) I think the words 'saving' and 'investment' need to be swapped."

Why? If we vary r, and move along the intertemporal indifference curve, we trace out the consumption/saving choice of households, and if we move along the intertemporal PPF we trace out the consumption/investment choice of firms. Put S and I together, and we get the dreaded "Loanable Funds" model of r, to which many PKs seem to have such an aversion!

[HJC]

Nick: You're going to hate this, but since you asked... For Moore there is 'no volitional "supply of savings" function' and 'saving is the accounting record of investment.' It's based the view that 'economies are complex adaptive systems, where change is continual, a position of general equilibrium in all markets is never observed.' See here for a concise version of his views. Other post-Keynesians will have other versions but mostly they are based on the concept of effective demand and monetary economies. In general, I think that the post-Keynesian view is that, in a monetary economy, investment causes saving, as opposed to the saving causes investment view of the classical one-good economy. (Leave aside the neo-Walrasian view that neither causes either.)

[Jason]

Some pretty 3D pictures of the two-good market:

http://informationtransfereconomics.blogspot.com/2014/05/equilibrium-in-two-good-market.html

[ZDENPR]

Nice post, but the whole discussion rests on extremely unrealistic assumptions.

Let me point out some of them:

1) "Suppose that the economy can produce capital goods ("investment" is the quantity of capital good produced each year) and consumption goods. There is a PPF between consumption and investment, just like the PPF between apples and bananas in my first diagram."

- Only if you assume FULL utilisation of resources and FULL employment (something not seen in capitalist economies since World War II).

2) "If we move along the PPF between consumption and investment this year, we will have a bigger stock of capital goods next year, which will shift out next year's PPF. MPK tells us how much it shifts out, per extra machine."

- Not necessarily true if you assume feedback from demand for consumption goods to investment in fixed assets (picking up on your apple example: why the hell would someone in a capitalist economy plant MORE apple trees if the demand for apples is DECREASING?).

With less demand for consumption goods, there will be LESS, not more, demand for investment goods, and as a result, the stock of capital will DECREASE, not increase.

(Again, in order to circumvent this objection, you'd need to assume very specific circumstances where the investment is unaffected by the lack of demand for consumer goods - I think USSR industrialization in the 1930s could be regarded as one such model case, but again, hardly relevant for the world in which we live now).

By the same token,

to assume that "the expected growth rate of consumption (...) depends on our ability to divert resources to producing extra capital goods instead of consumption goods, and the productivity of those extra capital goods" seems to me to be extremely stretched.

- There's simply nothing that is telling the capitalist entrepreneur that there will be future demand for the extra consumption goods that she would produce with the new capital equipment; quite the contrary: the capitalist entrepreneur is just seeing a DROP in the demand for her consumer products, and she would have to be a fool (or have rational expectations quivalent to some sort of omniscience) to assume that this means there will be more demand for her consumer products in the future.

Again firms in a free market economy usually don't behave that way.

3) Any effects of interest rate change are going to be too small to offset the effective demand effect discussed under my point 2. The central bank can do much less than you think it can. Entrepreneurs invest only when they can reasonably expect their investments to be profitable (i.e. when they expect demand for the consumer goods produced with the capital equipment to be acquired), not because they find it easier to finance the investments.

[Almar]

At first I thought I understood the initial post, but following Nick's reference back to his third diagram I looked at it again. If there is no storage (so that we have the green ppf), a single representative consumer and no monetary assets, what meaning can be given to the budget line? There is a point on the indifference curve,and an associated MRS,but it seems it cannot be used to trade-off consumption between periods. One could define the MRS here as a rate of interest, but I don't see that it has any connection to what is usually meant by a rate of interest.

[Curmudgeon_Killjoy]

Suppose that the market value of everything owned is the appropriate definition of national wealth or national capital.

Even so, the sum of real assets plus financial assets less financial liabilities (marked-to-market) double counts real assets and financial assets. Financial assets represent legal claims to real (non-financial) assets and the income streams generated by real assets.

[Declan]

Nick,

Piketty seems very sceptical of "pure" land values (both agricultural and urban), arguing that they are impossible to separate from the value of improvements and mostly due to them (see p47). Seemed a little questionable to me too.

Kevin,

"He makes it very clear that he rejects any notion of restricting the definition of capital to produced means of production."

He seems to regard marketable wealth as a decent proxy for produced means of production (although with lots of SR fluctuations due to asset bubbles, wars etc). There is a Solow-Swan model in the background most of the time when he discusses K/Y. Also see above comment re. land.

[Declan]

More generally, the book's not that hard to read. OK it's nearly 700 pages, but he says economists can skip the first hundred. The prose is simple and repetitive, and there are lots of graphs, so it is easy to skim.

Part 2 on the K/Y ratio is a bit over a hundred pages (short version, K/Y rises as growth rates fall, hence capital incomes and inheritance rise in importance).

Part 3 on individual income distribution is about 250 pages. The remainder on policy isn't particularly original.

Or just skim the pics at http://piketty.pse.ens.fr/en/capital21c2

[Bill Woolsey]

Nick:

Wouldn't capital reswitching involve the "third" axis?

The simple examples of reswitching I have read involve a single interest rate.

Do the more sophisticated approaches use a yield curve?

The simple examples of reswitching that I have seen involve two techniques. Is it usual to explicitly consider allocating some resources to both techniques?

[Paragwaknis]

"Rather than saying "MPK determines r", it would be more true to say "MRScc determines r, which determines the prices of capital goods". And the only thing wrong with saying that is that is that MRScc is not a fixed number, but depends on the expected growth rate of consumption, which in turn depends on our ability to divert resources to producing extra capital goods instead of consumption goods, and the productivity of those extra capital goods."

You just captured the essence of consumption based asset pricing theory! Also, of interest is idea of stochastic discount factors: John Cochranne has an interesting Presidential address that focuses on it. You can find it here- http://www.afajof.org/details/video/2870771/2011-Presidential-Address.html

[JKH]

Nick,

“MPK is defined as the extra apples produced per extra existing machine, holding technology and other resources constant, and holding the production of new machines constant. If we move along the PPF between consumption and investment this year, we will have a bigger stock of capital goods next year, which will shift out next year's PPF. MPK tells us how much it shifts out, per extra machine.”

Question: why am I not seeing an MPK function somewhere in here that is defined as the alternative of producing extra machines per extra existing machine? Isn’t that type of function also inherent in the transformation function? Or is it redundant to the required math?

[Tom Brown]

O/T, Jason has an answer for your unresolved question Nick (the one for which you're still in the "I don't know" camp):
http://informationtransfereconomics.blogspot.com/2014/05/do-monetary-aggregates-measure-money.html?showComment=1399925805457#c4001174889208369862

[Jon]

Fascinating situation in California right now because of the drought. Cali produces 80% of the worlds tradable almond market. Almond trees require water (a decent amount four acre-feet). Other fruit trees are the same. The fascinating part though is that farmers are cutting back on the less water intensive crops in favor of the more intensive fruit trees which take years to mature but can be killed in a single year of not providing water. So water is this years capital investment. But new fruit trees would not be planted this year because they take many more years to bear fruit.

Lots of path dependence in the growth of the capital stock...

[Bob Murphy]

Kevin,

Not sure if you're still checking this, but in regards to your blog post: You're acting like it's an unfortunate choice of terminology on Piketty's part, that he uses MPK when he should be using MEC. But no, what he did makes perfect sense, *if* it's legitimate to switch back and forth between "capital" meaning physical machinery etc. versus "capital" meaning a sum of money.

Because if we're measuring capital as money, then the MPK = MEC. Do you agree?

[Bob Murphy]

(E.g. with physical machinery MPK might be 10 apples per additional hour of machine-time. But if we measure output and machinery in dollar terms, them MPK might be $10 / $200 = 5% = MEC.)

[Kevin Donoghue]

Bob,
I don't think that works. The MPK is meaningful even in a one-period model. It's just a partial derivative. But just to define the MEC we have to assume there's more than one period involved.

Piketty writes very clearly for the most part, setting out his definitions early on and sticking with them. But that paragraph is a mess. Maybe there's a way it could be rescued, but as it stands I think it should have a here-be-dragons warning beside it.

[Nick Rowe]

My mind is still on the farm in England!

JKH: "Question: why am I not seeing an MPK function somewhere in here that is defined as the alternative of producing extra machines per extra existing machine? Isn’t that type of function also inherent in the transformation function? Or is it redundant to the required math?"

It's implicit/redundant. Let Kc and Lc be the amounts of machines and labour used to produce consumption goods, and Ki and Li be the mounts of machines and labour used to produce new machines (investment). let there be production functions C=F(Kc,Lc) and I=H(Ki,Li). If we assume Kc+Ki=K, and Lc+Li=L, (existing stocks of machines and labour are fully employed), we can derive a PPF between I and C just like we can derive a PPF between apples and bananas. (And unless F and H are identical functions, that PPF will generally be curved out rather than a straight line).

At any point on that PPF, there will be an MPK for C (=dC/dKc) and an MPK for I (=dI/dKi). And if competitive firms are maximising profits, the two will be related, like this: Pk.MPKi = Pc.MKPc = R where R is the rental per machine. And the slope of the PPF will be -MPKc/MPKi.

(In the above, I am assuming all functions are smooth, to keep it simple.)

[Bob Murphy]

Kevin, I thought that too at first, and it made me wonder if I (and Nick?) were being sloppy when saying r=MPK only under certain circumstances. Because if the dimensions aren't even the same, then obviously r can't equal MPK, period.

But then I thought about it more, and I think it works out; the sloppiness is in thinking that MPK is denominated just in units of output.

Look, we all agree that w=MPL right? Well the wage is a *rate*, or if we're thinking in just output units, then we're implicitly having a rate in mind.

And it's also not right (as some people, not you, have suggested) that the interest rate is a pure number, because we usually mean "per year" at the end and just take it for granted.

Anyway Kevin there are all sorts of passages that have dragons flying around. In several spots Piketty is clearly discussing r as being due to the technological characteristics of adding machinery, land, etc. to production process. So it's clearly MPK.

But then he talks about r influencing how fast financial capital grows, which is clearly an interest rate.

[Major_Freedom]

Nick:

"because bad monetary policy causes recessions"

Does bad monetary policy ever cause booms?

[Nick Rowe]

Bob: if we are talking about the use of capital goods to produce those very same capital goods, then MPK has the units 1/years. And if that MPK is independent of the age of the machine, so that employing one extra machine produces an extra flow of MPK extra machines per year, then the real interest rate on machines (nominal rate minus inflation rate on machines) will equal MPK. Probably the easiest way to think about it.

MF: Yes, probably. Good monetary policy will cause neither. Bad monetary policy will cause both. But booms and recessions don't seem to be symmetrical. Milton Friedman's plucking model seems to match the data closer than a symmetric model.

[JKH]

thanks Nick