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"I think we need to be picky on terminology here.

Marginal Product of Labour is the extra output per extra worker and equals the rental on a worker.

Marginal Product of Land is the extra output per acre of land and equals the rental on an acre of land.

Marginal Product of Capital is the extra output per extra machine and equals the rental on a machine.

What you are calling "mpc" you should be calling the Marginal Rate of Transformation between Present consumption and future consumption, and it equals 1 + the real interest rate."

I think of capital as more than just machines, and I think it captures the spirit of growth models better if you have a more abstract understanding of what k represents. To me capital is the fixed costs in micro models (most models don't have land in them), while labour (and commodities/resources) are the variable costs. The decision faced by the representative agent in a simple growth model is to invest or consume, investing increases production for the future, k represents the fixed cost stuff you can increase to increase future output, this includes machinery, but could really be anything.

I agree I think it is equivalent to the Marginal Rate of Transformation, in a model with no money. In a model that includes money, I'm not sure it is.

"Suppose e.g. the rate of interest fell because of a change in preferences, or demographics. Then (if the technology for transforming consumption into machines were non-linear) the price of machines would rise. So the Value of the total stock of machines would rise, even if the stock of machines didn't. So an econometrician who estimated a production function using the total *value* of the stock of machines would falsely think that the stock of machines had increased."

Is that it? Seems like a trivial issue.

"anon: it's actually easier to explain the point using land. Suppose an econometrician were estimating a production function Wheat = F(Land). And suppose he used the market value of all land as a proxy for land. Now suppose time preference halves, so the rate of interest halves, and the price of land doubles. The econometrician would falsely conclude that land productivity had halved."

Okay, but isn't that a problem of trying to measure capital rather than a problem with the implications of the model itself?

anon: "Okay, but isn't that a problem of trying to measure capital rather than a problem with the implications of the model itself?"

Yes. But it is important that we be at least conceptually clear on the difference.

It's not something I'm worried about because the only situation in which aggregate production functions actually need to be estimated is in estimating potential GDP (a statistic with limited uses), but I'm sure the CBO et al have a solution to this problem, or do they not?

By that I mean; I'm sure when the CBO estimates potential output using its parametrized model, it's not complete crap? They figured out a way to properly model the effect of a change in some measure of the capital stock?

Anon: If you write down say Y=F(N,K,L), then really, Y, N, K, and L are all vectors. Y is a vector of very different output goods, N a vector of very different workers, K a vector of very different capital goods, L a vector of very different natural resources. It might not be possible, even in principle, to construct scalars out of those vectors, so that the production function can be written as a function of those scalars. For example, suppose part of the capital stock consists of robots, that are identical to human workers. We might want to add the robots and human workers together, and keep the other parts of the capital stock separate.

I have no idea how big a problem this is empirically. Possibly some economists do.

Let's say you scrapped using scalars and instead used a vector of coefficients, would any profound underlying economic insight be found that couldn't be found using aggregated factors of production?

Anon: dunno about "profound", but consider my "robot" example. Some capital (robots) might be a substitute for labour (would reduce the MPL); other capital might be a complement to labour (would increase the MPL). Some capital might increase Y immediately; other capital might have a lagged effect on Y (wine ageing in barrels).

"Some capital might increase Y immediately; other capital might have a lagged effect on Y (wine ageing in barrels)."

Of course, my point is that I while this could be resolved with a more fluid and sophisticated production function, I still don't think this would actually yield useful economic insights about the real world that aren't already known or can't be shown from simple models.

There is a difference between whether the Fisher explanation of interest rates is correct (which it at least mostly used to be) and what economic and political conclusions should be drawn from. Economics really can't escape its history as a branch of moral philosophy. For instance you say

"Marginal Product of Capital is the extra output per extra machine and equals the rental on a machine."

First of all your capital may not be a machine and may not have a meaningful rental price. Fisher's definition of value in terms of rate and return is an accounting identity. Your modification is a non-trivial derivative of this and is only worth the trouble of making if you intend to draw some implication from it. And we've all been warned about drawing implications from accounting identities.

The implications that economists usually want to draw are explanations about the distribution of profits based on marginal product. But the motivation behind this is to argue that the marginal product represents an inherent productive virtue that is entitled to a just reward. This of course is exactly what Sraffa was attempting to refute and replace.

So the questions don't concern validity, but rather applicability, hidden agenda and equivocations on the term "value". I suspect Sraffa might have agreed.

"…in general the use of the term 'cost of production' has been avoided in this work, as well as the term 'capital' in its quantitative connotation, at the cost of tiresome circumlocution. This is because these terms have come to be inseparably linked with the supposition that they stand for quantities that can be measured independently of, and prior to, the determination of the prices of the products."

And Robert Murphy in his review of Production says from an Austrian perspective:

"Sraffa was perfectly correct to criticize the conventional, mainstream justification of the capitalists' income."

An economist with opposing views might call marginal product the explanation for the capitalists' income, but the meaning is the same.

Sraffa published Production in 1960. The US group didn't turn to the Fisher approach until Solow in 1963.

The Marshallian theory of interest Sraffa was opposing was logically incoherent, and that its concept of the value of CAPITAL was circular and arguably pernicious. Sraffa says:

'…in general the use of the term 'cost of production' has been avoided in this work, as well as the term 'capital' in its quantitative connotation, at the cost of tiresome circumlocution. This is because these terms have come to be inseparably linked with the supposition that they stand for quantities that can be measured independently of, and prior to, the determination of the prices of the products. (Witness the 'real costs' of Marshall and the 'quantity of capital' which is implied in the marginal productivity theory.) Since to achieve freedom from such presuppositions has been one of the aims of this work,
avoidance of the terms seemed the only way of not prejudicing the issue."

That should be clear enough. He saw marginal productivity valuation is the thin edge of the wedge of justification, which it had been historically and is in many quarters today. The Marshall approach isn't dead yet. It seems perfectly possible to use the Fisher method without drawing Marshallian conclusions, but that's not how it's usually done.

Many of the Cambridge UK contingent were socialists. Joan Robinson certainly was, and Kaldor was a Fabian. The question of whether the division of profit was the result of bargaining power between labor and capital or the natural result of the "value" of capital was rather important to them.

Peter N. If there were no time preference, in the sense that people didn't care when they consumed goods, the (real safe) rate of interest would be 0%, and there would be no interest income to explain or "justify". In such a world, owners of machines would still receive rents equal to (not "determined by") the marginal products of those machines, but the total rents over the life of the machine would exactly equal the marginal cost of producing the machine and the price they paid to buy the machine (I'm ignoring monopoly and uncertainty of course).

Just as any positive analysis of interest income needs to talk about time preference, I would think that any normative analysis about whether interest income is or is not justified would also have to talk about time preference. (Plus, you can have some people earning interest income even in a pure consumption economy where there is no capital so you can't even talk about the marginal product of capital). So if the CCC was deep down really a normative analysis (or a prelude to a normative analysis) of interest income, then I think it totally missed the point.

"In a world where people care about when they consume stuff, ought people consume more/less if they consume later/earlier?" That is the normative question about interest. (And any answer to that question should also apply to the case where real interest rates are negative, which is possible.)

Hi Nick:
In all fairness Sraffa never said that demand was irrelevant. What he disliked was subjective preferences associated to individual behavior, presumably because he thought, like the classical authors, that social preferences were based on fairly stable conventions. Think of the long time that it took for Tea to become the national beverage in England, going beyond being an exotic drink for the elites to becoming a habit of the lower classes. In that sense, classical authors didn't think that there was much to be said about those social preferences which are taken as given for the determination of prices (which can be determined purely by objective factors related to the cost of production). Mind you it is also far from clear that one learns a lot from the preference curves in conventional neoclassical economics, since the shape of the preferences are determined by the need of finding a solution (why should they be convex or homothetic? or whay aren't the individual preferences affected by the preferences of the group?). In fact, I highly recommed you check Ian Steedman, a Sraffian, that in his Schumpter Lectures (here http://www.amazon.com/Consumption-Takes-Schumpeter-Lectures-ebook/dp/B000OI0R3I) takes preferences very seriously, and it doesn't do any good for conventional neoclassical results.
Best,
Matias

Nick:

Like you I never really understood the Cambridge debates despite spending half my life trying. But I did get a couple of things:

(1) Joan and the UK Cambridge people believed capitalism is about profit and that Ricardo and Marx had an explanation for profit based on social class and an economy's ability to produce surplus. They could not accept a theory where there is no profit only interest rates determined by time preference and the intertemporal rate of transformation of currently produced goods into goods produced in the future(at least in GE/perfect competition/blah, blah, blah). The fact that Samuelson, Solow,et.al gave primacy to the rate of interest as the source of income for saving and insisted on calling the rate of profit the rate of interest(again in long run equilibrium)drove them nuts. They wanted a theory of capitalism and profit. Ricardo gave them that and Walras/Menger/Jevons did not in their opinion. Smith, Ricaro, Marx believed the rate of profit was critical in setting the maximum rate of interest rather than the other way round. For Joan, et. al. a theory of profit was critical. She did not think the Cambridge debates over capital (at least the aggregate capital vs heterogeneous capital debate)important but instead wanted a return to what she thought critical in a theory of capitalism: a theory of the rate of profit.

(2) they were not completely nuts in wanting a theory of profit given they and most uneducated in economic theory think the rate of profit should be at the center of any economic analysis of capitalism. Does the rate of profit show up in a first year course. No. We both wrote (or a least contributed to)a first year text where we duck and talk about a "normal rate of profit" which is never explained other than a crude reference to opportunity cost and the risk free rate of interest on short government bonds. In a second year course? No. In graduate school? No.

(3)Capital theory is important. Economists do not study it because it is so hard. I do not think there was a clear winner in the Cambridge Debates and it is sad that almost no economist knows there is still a problem here. Glad to see you are also uneasy about this.

In the end the philosophy of a scientist must be based on the fact that life is short and capital theory can occupy your life with little chance of resolving the debate so we (the royal we) are right in giving it a go at some point, then throwing up our hands in frustration, and then getting on with "normal science". But there are days when I am not sure my neoclassical/neokeynsian thinking is not completely wrong and that Smith/Ricardo/Marx/Joan Robinson were maybe right.

Hi Matias: On thinking more about Sraffa's quest, I can kinda sorta see why it might be an interesting question to ask. "What would an economy look like if it were replicating itself, and machine tools were used to make machine tools as well as consumer goods?" Just as I can see that it might be interesting to look at a Malthusian steady state, or a Solow Growth model steady state too (though Malthus and Solow do also talk about the process by which an economy would converge towards that steady state).

And I've been wondering too about how it relates to the Mengerian idea of capital goods of lower and higher orders (consumer goods are first order, goods that make consumer goods are second order, goods that make second order goods are third order, etc.). Presumably that Mengerian idea can be captured in the Sraffian matrix algebra.

But if technology changes more quickly than preferences, we would want to be looking at the path traced out by changing technology around stable preferences. And if technology changes more quickly than capital goods wear out, we never would see an economy replicating itself. The old machine tools are used to make their own newer and different replacements.

(As you might guess, these ideas are all now bubbling around in my brain, trying to form themselves into some sort of coherent order.)

Yep. preferences may not always be convex. And if they aren't we will probably get a corner solution. Which makes sense, because there are a lot of goods that many of us consume none of, and a lot of possible goods that don't get produced at all. Homotheticity isn't really needed, unless you want to talk about simple economies where everything scales up or down, or where you want to be able to talk about representative consumers.

The one thing I do really object to on the Cambridge UK side is this meme that's still floating around: "Neoclassical theory can't explain the rate of interest/prices of capital goods without circularity/logical incoherence, unless it assumes the one-good model." That is only true if you ignore time-preferences; but neoclassical theory does not ignore time-preferences. It's Neo-Ricardian theory which has to treat the rate of interest as exogenous, precisely because it ignores time-preference.

I have enjoyed reading Ian Steedman in the past (as well as talking with him when he visited Carleton ages ago). I will see if I can make any headway with his Schumpeter lectures.

Frank: sometimes I wonder, about the Cambridge debate, whether the whole thing isn't a whole lot clearer from a great distance. And whether those who invest their whole lives and identities and emotions in it sometimes cannot see the forest for the trees. And one very big picture that now stands out massively from a distance of 30 years is the role (or lack thereof) of time-preference. Whether you can or cannot aggregate capital goods is a side-issue. Even if you can aggregate all capital goods, if the capital good (like land) is different from the consumption good, technology does not pin down the price of the capital good, and you have to talk about time-preferences to determine the price of capital goods and the rate of interest. My Dutch Capital Theory post showed this quite simply.

On profit and interest: imagine a world where technology and prices never change, with perfect competition, in long run equilibrium, with no asymmetric information or principal-agent problems, where rates of profit are identical across all industries and across all time periods, etc., etc.. All assets (real and financial) would yield exactly the same rate of return. There would be no reason whatsoever to distinguish rates of profit from rates of interest. Yet in the real world we have the equity premium puzzle, which is presumably just a symptom of a much wider disparity between rates of return on safe financial assets and on those real assets you manage yourself. But if I wanted to explore this difference I am not sure I would turn to Cambridge UK for assistance. I might start looking at principal-agent problems, or maybe go Austrian and talk about entrepreneurial coordination processes in a world of ever-changing and uncertain opportunities. Sraffa's world would seem to be totally antithetical to any distinction between rates of profit and rates of interest.

"We both wrote (or a least contributed to)a first year text where we duck and talk about a "normal rate of profit" which is never explained other than a crude reference to opportunity cost and the risk free rate of interest on short government bonds."

Well, in the macro part, we do have a loanable funds model of the rate of interest. The investment curve is supposed to represent intertemporal transformation possibilites in production, and the savings curve is supposed to represent intertemporal preferences. Sometimes I have drawn an Irving Fisher diagram, but I'm not sure if they get the point.

For what it is worth, here is my exposition of a model of interest rates and profits:
http://robertvienneau.blogspot.com/2007/01/post-keynesian-model-of-growth-and_10.html
This model builds heavily on models presented by the Cambridge, UK, school concurrently with the CCC. (Matias will know of some more recent literature critical of these models that I have never really absorbed.) In this model, a very clear distinction exists between rates of interest and of profits.

Hi everyone, sorry I've been away so fell a bit behind. I'm writing a post so I will get into the more substantive issues there.

Martin,

I'm aware of Fisher (1930). But Fisher (1936) repudiated his own views on the basis that debt markets never clear and all debts are never paid, so I'm not sure why you'd appeal to him. There are also many alternative theories of the rate of interest - your (Fisher's) model seems to be based entirely on agents abstaining from consumption. But I'd agree with rsj that such decisions would be dwarfed by risk. I was also referring to the aggregation of consumer preferences rather than the interest rate.

Nick,

Indeed, there are problems with aggregating labour too - this is also a concern of mine! This paper is interesting:

http://www2.econ.iastate.edu/tesfatsi/aggregateprodfunctions.felipefisher.pdf

Robert is also correct that the conflation

(1) money and capital goods

(2) the rate of interest and the rate of profit

are problematic.

I don't agree that A-D is really just Sraffa in another form. But I will save that for later when I am not horribly jet lagged.

Here's what I find odd about this: economists love ceteris paribus. But if we are holding preferences constant, dear god! The model must be worthless.

Unlearning: (I hate jetlag. It always takes me over a week to recover when I go east across the Atlantic. My sympathies.)

Yep, preferences do seem to change over time. But technology definitely changes too. I wonder which one changes "more quickly"? (I'm not quite sure how we could even define "more quickly".) I was thinking of my MX6 (as usual). They made that generation of MX6 from 1993 to 1997 model years (and the similar Mazda 626 from 1993 to 2002), then they stopped production. The nearest replacement, the Mazda 6, has a totally different frame, body, and engine block. But most of those old MX6 and 626's were still on the road when they stopped production. They never did get replicated. We never got anywhere close to an equilibrium where new MX6's were produced to replace the old ones that were taken off the road. I wonder if this is true for capital goods more generally (at least, ever since the industrial revolution)?

Unlearningecon,

"I'm aware of Fisher (1930). But Fisher (1936) repudiated his own views on the basis that debt markets never clear and all debts are never paid, so I'm not sure why you'd appeal to him. There are also many alternative theories of the rate of interest - your (Fisher's) model seems to be based entirely on agents abstaining from consumption. But I'd agree with rsj that such decisions would be dwarfed by risk. I was also referring to the aggregation of consumer preferences rather than the interest rate."

I have numbered the points as this makes it easier for me to structure:

1. I am not aware of Fisher having repudiated his own views in 1936? Are you referring to his Debt-Deflation article?

2. Anyhow, I mentioned Fisher (1930) w.r.t. his theory of interest and his view of capital as this pretty much underlies the mainstream/neoclassical/textbook view. Fisher (1930) illustrates that the objections you raised are pretty much dealt with (in my opinion).

3. You'll find in Fisher (1930), as in the treatment in the textbook, that risk is also dealt with and can be incorporated into a theory of debt/capital. It's not an either or proposition.

For example, you could reformulate Nick's DCT model by using the exponential utility function:

u(t)=-Exp[-a*(c(t)+epsilon)]

where, a is the risk-preference (minus means there is aversion); if epsilon is normally distributed with epsilon ~ N[mu,sigma^2], you could write for the utility function:

u(t)=-Exp[-a*(c(t)+epsilon)]*-Exp[-a*mu+(a^2/2)*sigma^2]

insert the utility function into Nick's DCT, and you have theory of the interest rate with risk.

4. I don't get your point about aggregating consumer preferences. Why is that necessary/a problem?

It is somewhat surprising they did not discuss time preference. Perhaps they felt this was another variable rather than a constant and rather than permitting a dynamic solution, it resulted in an under determined set of equations. Certainly the 30s saw a significant sharp shift if not in time preference proper at least in perceived risk, so if you are contemplating the depression, perceived risk is somewhat circular as an explanation. One could say like Minsky the risk of lending is low when lending is low leading to more lending until lending is no longer low and only then does perceived risk jump. Does a change in debt level represent a change in expectations or a change in time preference or is there even a difference?

Heterogeneity gets into the model somehow, and with the DCT its in the consumer preferences, rather than the capital, or the commodities.

Nick,

You focus on 'time-preference' as an explanation for the interest rate, which Sraffa assumes is exogenous. But a vital part of endogenous monetary theory is, in fact, that the interest rate is exogenously set, so I don't see an issue for post-Keynesian theory as a whole (in fact my confirmation bias is feeling good as I've realised how compatible the two approaches are).

Time preference could also be modeled as a result of institutional factors, which actually seems more sensible to me. Lord made some good comments about this, and your comment on motor vehicles which I am not manly enough to understand actually appears to follow more of a social trend than an individual one.

Your formulation of A-D falls into a chicken and egg problem that is common in neoclassicism: where do prices come from? Sraffa examines production first (which, logically, must occur before consumption). I'm also not sure where the distribution between profits and wages comes in A-D (is it not assumed away?) Overall you seem to have to make some substantial 'tweaks' to get Sraffa.

Martin,

Indeed I am referring to debt deflation, though I wrote 1936 which is of course wrong (it was 1933). He abandoned the idea of debt markets in equilibrium and, as a result, that debts would not be paid was central to his hypothesis, and the problems that arose "[problems] would have far less serious results were they not conducted with borrowed money." This doesn't obliterate the idea of time preference altogether, but it does bring into question the idea that it can suffice to explain the interest rate.

As you say, it's entirely possible to incorporate risk, and I'm suggesting that it is probably more important than time preference. More importantly, both risk and time preference surely depend on institutional factors, no?

UnlearningEcon,

I have numbered the points again, as it makes it easier for me to keep order in things:

1. I looked up Fisher (1933): http://fraser.stlouisfed.org/docs/meltzer/fisdeb33.pdf, but I don't read it as abandoning equilibrium; he simply uses a different mental construct to discuss a different issue, which seems wholly appropriate.

Fisher (1933 p.337):
"2. Economic theory includes a study both of (a) such imaginary,
ideal equilibrium—which may be stable or unstable—and (b) disequilibrium. The former is economic statics; the latter, economic dynamics. So-called cycle theory is merely one part of the study of economic dis-equilibrium."

It's no different from when Nick discusses the interest rate in equilibrium, but the hot potato-effect in disequilibrium.

You should like Fisher (1933 p.341) though:

"While quite ready to change my opinion, I have, at present, a strong
conviction that these two economic maladies, the debt disease and the
price-level disease (or dollar disease), are, in the great booms and depressions, more important causes than all others put together, (...)

21. Disturbances in these two factors—debt and the purchasing power of the monetary unit—will set up serious disturbances in all, or nearly all, other economic variables. On the other hand, if debt and deflation are absent, other disturbances are powerless to bring on crises comparable in severity to those of 1837,1873, or 1929-33."

If you like Fisher (1933), you should compare this to Mishkin (2004 p.174-190?) I think, the textbook, with the chapter on financial crises, the explanation in Fisher (1933) and Mishkin (2004 and also later), is largely the same, though Mishkin gives more 'microfoundations' to the explanation in terms of adverse selection and moral hazard.

2. I think time-preference is a sufficient explanation for the interest rate, everything equal, time preference results in a positive rate of interest as we've seen in the DCT-model. The other factors you mentioned can also lead to a (positive) rate of interest, but time-preference as in "Beta" is sufficient.

3. Whether risk or time preference or whatever other factor is more important is an empirical question. I can't remember who said or wrote it, but someone said that the rate of interest fluctuated above some sort of floor. The floor is due to time preference, and the fluctuation is due to changes in risk and productivity according to that author.

Martin, not much to disagree with there.

If anybody wants to read my post on this issue, it is here:

http://unlearningeconomics.wordpress.com/2012/09/13/why-prefer-preferences/

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