Rates, rents, shares, and capital theory

I think this is the technology that Paul Krugman has in mind:

1. C + Kdot = A.L^a.K^(1-a)

C is consumption, K is the capital stock, Kdot is investment, L is employment, A is a parameter that represents productivity, a is a parameter that (in competitive profit-maximising equilibrium) will equal labour's share of national income, so (1-a) is capital's share. (I've ignored physical depreciation of capital for simplicity).

Macroeconomists commonly assume this technology, mostly because it's easy to work with. But this technology creates a problem for Paul. That's because with this technology (in competitive profit-maximising equilibrium)  the (real) rate of interest equals the rental rate on capital goods (equals the Marginal Product of Kapital), and Paul wants to explain why capital income has gone up while the rate of interest has gone down. (So he has to assume that monopoly power has increased, and that what looks like an increase in capital income is really an increase in monopoly rents.)

Let me make one small change. Change 1 to:

2. C + Kdot/A = L^a.K^(1-a)

(I've divided both sides of 1 by 'A', then deleted the 'A' underneath 'C'.)

In the first, standard model, an improvement in productivity (an increase in A) affects the production of both consumption goods and capital goods equally. In the second model, an improvement in productivity affects the production of capital goods but not consumption goods.

That small change in the assumptions makes a big difference to the model.

In the first model, r = MPK

In the second model, r = MPK.A - Adot/A

In the first model, consumption and investment goods are perfect substitutes in production, with a marginal rate of transformation always equal to one, so that (under competition) the price of the capital good will always equal one (taking the consumption good as numeraire). This means that the rate of interest will always be equal to MPK. A decrease in 'a' will reduce labour's share, MPL and wages for labour, increase capital's share, MPK and rentals on capital, and raise the rate of interest. An increase in 'A' will raise both wages and capital rentals, and raise the rate of interest.

In the second model, consumption and investment goods are still perfect substitutes in production, but the marginal rate of transformation now equals 1/A, so that (under competition) the price of the capital good will always equal 1/A (taking the consumption good as numeraire). As A increases over time, capital goods become cheaper in terms of consumption goods. The rate of interest must equal the rate of return on owning capital goods, but that rate of return is lowered by the fact that the price of those capital goods is falling over time.

Here is an explanation of a higher capital share but lower real rate of interest: (1-a) has increased; but Adot/A has increased too.

Do I have any evidence to support my explanation? No. That's not what I am trying to do here.

What I am trying to do here is to show that the technology commonly assumed by many macroeconomists, mostly because it is easy to work with, is a very special technology. It distorts our thinking. It makes us think that an increase in capital rentals and an increase in the real interest rate are much the same thing. They aren't. My model is just an example that illustrates the possibility that one could go up while the other goes down. And my model is just the same as Paul's, except for one small change.

And people think that all the action in equation 1 comes from the right hand side, while the left hand side is just a boring national income accounting identity. They are wrong. I could forget the "Cobb-Douglas" bit, and replace the right hand side with "A.F(L,K)" and nothing important would change in this post. It's the left hand side that pins down the price of capital goods. And knowing MPK tells you nothing about r unless you also know the price of capital goods and how it is changing over time.

When all is said and done, Cambridge UK were wrong. Because "C+Kdot = F(L,K)" is not the same as "Neoclassical Capital Theory". But sometimes I think that they did have a point. Because "C+Kdot = F(L,K)" does play a bigger role in our thinking than it should.

[Addendum:  when national income accountants measure the stock of capital they measure it in value units rather than in physical units. An increase in A will generally cause an increase in the physical units of capital (because they are now cheaper to produce), but will also cause a lower price of those units, so the effect on the measured stock of capital will be ambiguous.

If you want to solve "my model", just add a saving function "r = r* + B.Cdot/C" and you should be good to go, provided your math is better than mine.]

Update: Hmmm. Maybe "my model" sorta does work empirically? Suppose somebody invents computers. Adot/A is a lot higher for computers than for other capital goods (or consumption goods). And so Adot/A slowly rises over time as more and more stuff gets computerised. Dunno.

Comments

[Greg Ransom]

Keep going .... and you'll make it all the way to where Hayek was in 1941, The Pure Theory of Capital.

[Greg Ransom]

If you figure out how to shorten the time length of a productive process in the manufacture of a capital good you have increased productivity, reducing the time cost of the process. The trade-off cost will be less to transfer resources forward across time to dedicate to the production of production goods. The producer will expend less in interest cost to produce the same in output.

Investors with the same time preference will then open up new long period production processes -- often turning new inpits into economic goods.

Economics is a science seeking an advancement in understanding of dynamic phenomena.

[J.V. Dubois]

Very nice article. I am glad that you responded to Krugman as I think it is one of the most interesting posts Krugman did during last few months (which makes me sad since this is how blog posts of top economics bloggers should look like now in a sane world where central bankers did their jobs)

Nothing important to add other than it is not as easy to say what "r" or "r*" is or if we can really substitute consumption and capital goods as easily given some structure of the economy.

Anyways this is great food for my mind to process, thanks for that (and keep that coming).

[W. Peden]

Capital theory and baiting Neo-Ricardians? You must be feeling energetic, Nick!

[Nick Rowe]

Greg: you are possibly right. But my brain and math aren't up to the job. But you've got to be careful when you talk about "shortening" and "lengthening" the time-structure of production. With multiple input and output times, the "length of the production process" is multi-dimensional. It works for whisky and trees, but not everywhere.

JV: thanks! Yep, my model 2 is still very restrictive, because the PPF between C and Kdot is still a straight line. All I have done here is make the slope of that PPF change over time, as A increases. Us NeoClassicals are really into curved PPFs, not straight lines. So God only knows how that straight line PPF came to be thought of as the epitomy of neoclassical capital theory.

W Peden: I'm actually trying to be nice to Cambridge UK in this post. Some of them don't need baiting anyway. They bait themselves. They were born angry. (But not all, I remember Luigi P as a very nice guy, for example.)

[Frank Restly]

Would a profit maximizing firm conciously raise its own real interest rate - ie would it produce more goods than are being demanded? A monopolistic firm may do just that (foregoing profitability for market share).

Are credit markets limited to funding the production of goods?

One final question - what are the units for productivity "A"? It seems to me that productivity in 1. cannot be the same as productivity in 2. because the units of measurement will not be the same.

[david]

Having argued with Greg before, he will not - he will never - write down a production function that would actually generate the claimed dynamics in his interpretation of Hayek. It would risk an re-interpretation that is less strong than he wishes. Hayekians stay literary so that they are not bound by petty consistency.

[david]

Anyway: equation 1 is just Y=C+I. Equation 2 would look like "Y=CA+I". That doesn't make any sense. The former exists as an accounting identity, conventionally speaking, with no economic meaning. As Restly points out, there is a units problem.

Now of course conventionally we smuggle in some economic meaning by presuming that there is no complicated substitution between C and I, whereas in fact we have plenty of empirical reasons to do so. We know that C varies in the real rate, for instance, even though Samuelson used to write Y=C(Y)+I(r), with a clean separation of variables.

What (I think) you're saying is: maybe the variables aren't separated, and a change in the real rate causes so much change in C that we don't see as much substitution from C to I as would be conventionally expected in Y=C(Y)+I(r) (or that a change in A causes any significant substitution between C and I). That is, it should be C(Y,r)+I(r) instead (or, as we do today, simply write planned expenditure as E(Y,r), and readily acknowledge that consumption is significantly affected by housing wealth, which is significantly affected by r) . That would make sense in terms of units. It would also generate your suggested dynamics, with an appropriate choice of C(*) (or E(*)).

But don't write Y=CA+I, that's just weird...

[W. Peden]

Nick Rowe,

To be fair to them, it would be frustrating if one got to the other end of "The Production of Commodities by Means of Commodities" and concluded anything other than that it's the best work ever and its lack of influence is a scandal. So people who have suffered through Sraffa are divided into the frustrated ("I wish I'd never read that!") and the enfuriated ("Why does almost no-one talk about this?!").

The frustration goes after a year or two, but I have no first-hand experience of the enfuriation. I suppose what I'm saying is that heterodox economists can be forgiven for being a bit trying at times, because they're just reacting rationally and with admirable social concern to the world's apparent ignorance of (or malice towards) the "truth".

[Nick Rowe]

Franks: C, K, and L all have different units. So strictly speaking there needs to be a parameter multiplying (or dividing) each one of them. But if those parameters don't ever change, we might as well ignore them, buy setting them equal to 1. They are just "nuisance parameters". It's when those parameters change that we need to include them. That's what I have done here.

david: "Anyway: equation 1 is just Y=C+I. Equation 2 would look like "Y=CA+I". That doesn't make any sense. The former exists as an accounting identity, conventionally speaking, with no economic meaning."

No. When we write C+Kdot=F(L,K) we are asserting something important about the technology. We are measuring C and K in physical units, and we are asserting that, for given K and L, if we produce one more unit of C we can produce one less unit of Kdot. The right hand side of that equation is an engineering relation, and the left hand side of that equation must be interpreted as an engineering relation too. Half of the Cambridge-Cambridge capital debate was about some people using value units when they should have been talking about physical units.

W Peden: OK, it's understandable that someone gets pissed off if they think they are saying something important but everyone else is ignoring them. But I have tried to explain my perspective to them a couple of times, and didn't seem to get anywhere either. And got very angry and obnoxious responses from one of them.

[Frank Restly]

Nick,

You specifically reference a changing A both in model 1. and model 2.

Model 1.: "An increase in 'A' will raise both wages and capital rentals, and raise the rate of interest."
Model 2.: "As A increases over time, capital goods become cheaper in terms of consumption goods."

[Nick Rowe]

Frank:
1. Stick in a "nuisance parameter" to get the units right.
2. Multiply that nuisance parameter by a second pure number parameter that changes.

[Nick Rowe]

Now I'm getting worried. I only built "my model" to illustrate a point, and chose it because it was very simple and a very small change from the standard model. Now I'm starting to believe it. It's computers that did it. They are an example of a capital-biased technical change, but they also have a very high Adot/A (Moore's Law and all that).

[Joseph]

Nick, I know this is not the point of the post, but in your model r = MPK.A - Adot/A. Presumably, over sufficient long time frames Adot would be monotonically increasing, just based off observation, leading A to grow over time. This would lead MPK.A to grow without bound and Adot/A to shrink to zero which lead to a secular increase in interest rates, which is not what has been observed. So this model could not be a long run model unless A grows in some weird manner.

[Nick Rowe]

Joseph: what I had in mind is that A would be growing by some percentage every year, like 1% per annum. So Adot/A=0.01

But with A growing over time, K would be growing over time too, so MPK would be falling over time. I think (I haven't checked) that for some sort of reasonable savings function there would be a quasi-steady state where MPK is falling and A is growing but MPK.A is constant over time.

I need to check. Currently thinking about a new post where I actually solve out this model. Not sure if my math is up to it. Pray for me.

[Jacques René Giguère]

Nick: In Canada, the correct formulation is "Pray for me Paul Henderson"...
http://www.imdb.com/title/tt0194274/

[Arin D]

Nick, do you know this paper by Karabarbounis and Neiman? They argue something similar.
http://www.nber.org/papers/w19136

[Robert]

Paul Krugman writes, "But something is clearly going on, and I don’t think simple capital bias in technology is enough."

So Krugman puts capital bias in changing technology aside in trying to outline what he thinks is going on. (I suspect I would think that Krugman's model, when fully expressed, relies too much on the fallacious marginal productivity theory of distribution and is confused, like so many academic models in economics, on dimensional analysis.) But anyways, I am not sure why Nick so laboriously tries to express capital bias technical change. He is certainly not elaborating Krugman's point. (I suppose that Rowe should adopt an aggregate production function of a form where Harrod-neutral and Hicks-neutral technical change can be distinguished - that is, specifically not a Cobb-Douglas function.)

Nick Rowe writes, "When all is said and done, Cambridge UK were wrong. Because 'C+Kdot = F(L,K)' is not the same as 'Neoclassical Capital Theory'." This is the logical fallacy known as a strawperson. For about a third of a century Sraffians have been explaining that their critique extends beyond aggregate neoclassical theory.

[J.V. Dubois]

david: It works mathematicaly if you divide F(L,K) to two functions: Fc(L,K) for production of consumption goods andd Fk(L,K) for production of capital goods. So you will have

C + Kdot = Fc(L,K)+ Fk(L,K)

you may also write it using Cobb-Douglas ending up with two different "productivities": Ac for consumption goods and Ak for capital goods. Then we assume that Ac = 1.

[Nick Rowe]

Robert: "But anyways, I am not sure why Nick so laboriously tries to express capital bias technical change."

What's so "laborious" about my equation 2? I made one small change to the left hand side, and get a very different result. And it's not obviously unreasonable to assume that technical change might happen more quickly for capital goods than for consumer goods. Computers would be an example.

We can have a nice fight about whether Neo-Ricardians or Neo-Classicals are wrong about everything some other time.

JV: Yep. And if we do it that way, we will also usually get a curved PPF between C and Kdot (unless both production functions always have the same K/L ratios for any given W/R ratio). In which case a change in saving preferences which causes a movement along that PPF will cause a change in the price of capital goods. But that complicates the model a bit.

[Nick Rowe]

Arin D: sorry that your comment got stuck in the spam filter. Stephen fished it out.

Good find. A much more complicated model than mine. They use a CES rather than Cobb-Douglas production function, and have imperfect comp too. I just wish they had solved out for the rate of interest (unless they did, and I missed it when I skimmed the paper), to see if they get the same result that a faster rate of technical change in producing capital goods reduces the rate of interest.

[Matthew]

"Suppose somebody invents computers. Adot/A is a lot higher for computers than for other capital goods (or consumption goods). And so Adot/A slowly rises over time as more and more stuff gets computerised."

This sounds to me like its on the right track. Most of the technological change--including Krugman's example of Apple--has been about inventing new goods, not inventing better ways to produce the same goods. I was working on a blog post about this, but it seems you beat me to it.

[Nick Rowe]

Matthew: I'm beginning to think that's maybe the right track too. I was trying to solve for the model's equilibrium time-path, but my math isn't up to the job. Looking forward to seeing your post.

[david]

No. When we write C+Kdot=F(L,K) we are asserting something important about the technology. We are measuring C and K in physical units, and we are asserting that, for given K and L, if we produce one more unit of C we can produce one less unit of Kdot. The right hand side of that equation is an engineering relation, and the left hand side of that equation must be interpreted as an engineering relation too. Half of the Cambridge-Cambridge capital debate was about some people using value units when they should have been talking about physical units.

... yes? There's only so much output Y. It's added up from C and I=Kdot. Hence, Y=F(L,K)=C+Kdot. All of these are measured in physical units, naturally, rather than value units (where you would write a quasi-budget-constraint instead). This is an accounting identity that says nothing about optimization behaviour. As time changes, the composition of Y may change arbitrarily.

How do you fit an A in there? If you want to model a decrease in the relative price of buying Kdot - a decrease in the price of capital goods - then you would need a price-driven optimization instead. You need a household-side consumption equation, not a Cobb-Douglas supply equation, to determine the intertemporal path of C versus Kdot.

[Nick Rowe]

david: With given existing L, K, and technology, there's a PPF between C and Kdot. (It may be curved, or it may be a straight PPF). If technology changes, that PPF will shift out. But it may not shift out parallel. In my model above, when A increases, the PPF swivels, rather than shifting out parallel, so there is no change in the maximum C that could be produced (if Kdot were 0) but there's a big increase in the maximum Kdot that could be produced (if C were 0).

Consumption/saving preferences help determine what point we choose on that PPF.

[david]

But consumption-saving preferences are not made against a C+Kdot=Y constraint, they are made against an intertemporal budget constraint, which is wholly different. C+Kdot=Y kinda looks like a budget constraint, but it isn't: Y is defined as C+Kdot in national accounting (without government or net exports). It isn't a variable that is independent from the perspective of the decision-making agent, in the way that p.x=M is.

[Determinant]

As a follow-up to W. Peden on Heterodox Economics,this quote from Joan Robinson sums your position up beautifully:

I very well remember Hayek's visit to Cambridge on his way to the London School. He expounded his theory and covered a black-board with his triangles. The whole argument, as we could see later, consisted in confusing the current rate of investment with the total stock of capital goods, but we could not make it out at the time.

It has also been pointed out that the Hayekian Triangle assumes equilibrium whereas Keynesianism assumes non-equilibrium.

[Nick Rowe]

david: in an extremely simple model, like Robinson Crusoe, consumption preferences are made against a C+Kdot/A=F(L,K) constraint.

In a very simple model, where people have a choice between buying consumption goods and buying capital goods from the firms that produce them, the constraint they face is C+Pk.Kdot=Y, where Pk is the price of capital goods (C is the numeraire).

In a more complex model, with monetary exchange, and borrowing and lending, it's the same C+Pk.Kdot=Y constraint, where Y+F(L,K), provided monetary policy is right (or we are in long run equilibrium), only now people buy IOUs from firms that use those IOU's to buy capital goods from other firms.

" Y is defined as C+Kdot in national accounting.."

No. Y is defined as C+Pk.Kdot in national income accounting (if nominal income is deflated by the CPI, rather than the GDP deflator).

Determinant: this is long run equilibrium analysis, not short run disequilibrium analysis. This is a long run growth model, not a short run business cycle model. It would fail miserably at the latter job. We are talking about the last few decades, not the last few years.

[Determinant]

I wasn't disparaging you, Nick. I was following up on W.Peden who was following up on Greg Ransom.

[Robert]

"I made one small change to the left hand side, and get a very different result."

Do you understand that you are elaborating an idea that Krugman says, in your linked post, is empirically unimportant? In other words, do you understand that you and Krugman disagree about what models are interesting for describing empirical trends in the USA?

"it's not obviously unreasonable to assume that technical change might happen more quickly for capital goods."

I, along with Dixit, find this assumption quite ad hoc. (See the post linked to by my name - I am extrapolating somewhat.)

[W. Peden]

Determinant,

I had no-one in mind in particular.

For a non-Apostle, Joan Robinson seems to have had a reasonable good Cambridge wit.

[Nick Rowe]

Determinant: Ah. OK.

Robert: As I understand him, Paul Krugman is looking for a model in which: labour's share falls; the real rate of interest falls. I think he thinks that both those two things are empirically important. His model is one way of fitting both those facts; mine is another way.

When I wrote this post, I too thought my assumption of quicker technical change for capital goods was quite ad hoc. That's why I said in the post "Do I have any evidence to support my explanation? No. That's not what I am trying to do here." But after writing this post, and thinking about Moore's Law, and about how quickly the quality-adjusted prices of computers have been falling, and about how computers are being more and more widely used, I'm beginning to think it might not be so ad hoc after all.

Which is more ad hoc: my assumption that technical change in producing capital goods has recently speeded up; or Paul Krugman's assumption that monopoly power has recently increased? I don't know.

[Alex1]

Ok, stupid question, but when you say that the interest rate is not the same as the capital rental rate, what exactly does "interest rate" mean? Is it the Wicksellian rate, ie the hypothetical rate that equates savings and investment, or is it the rate you would see on the capital markets whereas the capital rental rate is the wicksellian rate?

[Alex1]

Actually Im not even sure Im using the word Wicksellian correctly. There is just too many rates for my brain here...

[Alex1]

Ok, the more I think about this the more confused I get. I always seemed to think that there were only two rates that were interesting. The first was the hypothetical rate that equated AD and AS by equating saving and investment (and I always thought this was by definition equal to the rental rate of capital). The second was the financial interest rate which was set more or less by the CB and whose only significance lies in being (or not being) equal to the hypothetical "real" rate that equates everything.

But if I understand you correctly you are saying there is three rates. There is one saving and investment rate, one rate to buy capital goods (which can move in the opposite direction of the saving and investment rate?) and then there is a financial rate which, to equilibrate the economy must be set by the CB to hit the saving and investment rate regardless of the rental rate (or the other way round? Im completeley lost).

[Nick Rowe]

Alex1: let's suppose that the consumption good is wheat, and the capital good is computers. And let's measure all prices in tons of wheat.

The rental rate on capital R is how many tons of wheat you would pay to rent one computer for one year. It's like the wage on labour, or the rent on land, except you are renting a computer rather than a worker or a field.

The real rate of interest r is how many extra tons of wheat you would have to pay next year per ton of wheat borrowed this year. ("If you give me X tons of wheat now, I give you (1+r)X tons of wheat 1 year from now.")

The two are related, but not the same, because in equilibrium (assuming no risk) the rate of return on buying a computer, renting it out, and selling it next year, must equal the rate of interest. And if the price of a computer is Pk tons of wheat, that rate of return is r = R/Pk + Pkdot/Pk . The second term is the capital gain, (or loss if Pk is falling because computers are getting cheaper to produce). (I've ignored physical depreciation of computers in that example.)

Regarding Wicksell: well, this is a model without money and that's always in equilibrium. You might say that the rate of interest in this model is the same as the Wicksellian "natural rate of interest", but I'm not sure if it would be useful to say that, because it's only useful to talk about a "natural rate of interest" when you can contrast it with an actual rate of interest that is sometimes different. And in this model you can't do that.

[Alex1]

Ahh of course, that makes a lot more sense.
Thanks for explaining(I think its safe to say that I have learned more from this blog than quite a few courses I have had...)

[MiMo]

Meta: you should try to use MathJax - math would be much more readable

[J.V. Dubois]

Nick: "Which is more ad hoc: my assumption that technical change in producing capital goods has recently speeded up; or Paul Krugman's assumption that monopoly power has recently increased? I don't know."

I think it is both. And to add to that I am for some time thinking about "monopoly" in a different way that has something to do with branding. It is just a feeling but to me there seems to be a transformation, as people get richer and richer the actual quality and price of the product seems to be less important compared to emotional attachment that people have with the brand of the product. There are many real world examples of this:

1. There are numerous suppliers produce exactly the same goods (like food, clothing, TV display etc.) for different brands. Yet some people would never wear/eat/use the same product if it was branded by different company

2. There are many instances where you even have free alternative of very similar product and yet it people prefer paid product.

This is actually largely recognized by many industries and many times companies fear of something called "comodization of business" where people stop percieving difference in "quality" and start hunting prices. And branding which has a goal of "veblenization" of business is among the top weapons to use against this "dreaded" trend.

I cannot express it more clearly but to me having a company that has not an edge in production technology but in technology focused on abusing the way people create emotional attachments only to extract rents just does not feel right.

But maybe I am just overreacting. Maybe all this manipulation was with us all the time and it has no explanatory power for recent change in trends. Or maybe it is a a good thing. It is not a monopoly rent it is people now having something extra when they purchase a product. They pay market price for an emotion, for that extra utility when someone has this warm feeling that he has a phone from the best company in the world that he is in love with.

[Nick Rowe]

Alex1: and thanks for saying that.

One last thing I should have added: and in that very special case where the technology is such that the PPF between producing wheat and computers is a straight line of unchanging slope, we can always define units for "computers" so that the price of "one computer" is always and everywhere 1 ton of wheat, in which case r=R/Pk +Pkdot/Pk simplifies to r=R , because Pk=1 and Pkdot=0. (Or r=R-d if computers physically depreciate at rate d like a radioactive decay.) Which is where you get the simple standard model C+Kdot=F(L,K)

MiMo: you are probably right. But I am as scared of things like MathJax as I am of math!

JV: I think that Paul Krugman has something very similar in mind when he talks about increasing monopoly power. Yes, it's more like monopolistic competition.

It's not implausible to my eyes. But yes, to see if it would really fly empirically, we would need at least some sort of back of the envelope calculation to see if the effect was big enough to matter, and explain most of what was happening. So would I, of course.

Like you, I can't decide if it's a bad or good thing. Everybody else's attachment to brands is silly. But my MX6 is different. It's part of who I am. I am *not* one of those kids who drives a Civic! And when we have the basics like food and shelter, why shouldn't we take pleasure in spending our money on totally irrational things like supporting our favourite hockey team?

[edeast]

Mix in Dutch Capital Theory please.
Mix in all the models. I just want it to end.
Switching Debate. NPV, Time Preference. Expectations. Sticky Monopolistic Firms, as Capital Goods. Transaction costs everywhere. Edgeworthian Information Theory. Game Theory. Game Semantics. Tensorial Logic. Fuck.

[edeast]

Ignore, commenter, who is frustrated with brain. Can't seem to keep it all the concepts in it. Giving up my modelling quest, econ can stay outside of ecology. I'm done. I can't even build a model, let alone, jam entropy into it.