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Wednesday, February 9, 2011


I write these next words with tears in my eyes -- not because it turns out that Marx is wrong. No, I am a grown-up, I have lived through twenty years of Reagan, Bush, and Bush without breaking. I have shown that I can take it. The tears are for my one claim to economic fame. In 1981, I published an essay entitled "A Critique and Reinterpretation of Marx's Labor Theory of Value," in a journal called PHILOSOPHY AND PUBLIC AFFAIRS. [I believe it is available on-line.] In that essay, I proved an extremely important theorem that shows that Marx was wrong to impute the exploitative capacity of capitalism to the labor/labor power distinction. I was, I firmly believed, the first person ever to realize the underlying logical flaw in Marx's argument, and to demonstrate it mathematically. The proof was not much from a mathematical point of view. Indeed, when I had first actually proved the theorem several years earlier, I was ignorant of linear algebra, and had used nothing but elementary algebra and some ingenious labeling moves. After the essay appeared [since I made the mistake of publishing it in a philosophy journal, almost no one read it who was capable of appreciating it], the brilliant, mathematically extremely sophisticated Marxist John Roemer published a reply and criticism in the journal in which, in passing, he pointed out that the same theorem had been published two years earlier by Josep M. Vegara in a monograph entitled ECONOMIA POLITICA Y MODELOS MULTISECTORIALES. [For those who are interested, the proof appears in section 3.5 of that book.] As I am linguistically challenged, and do not even read Spanish, let alone Hungarian or Japanese, I had of course not noticed Vegara's book. In the essay, I go on to develop an entirely original alternative analysis of exploitation, which in due course I shall repeat here [Roemer criticized that too]. But since this was my one shot at a genuine formal proof to which my name could be attached, I still weep, all these years later, for the lost opportunity. I am not the only person to whom this has happened, needless to say. At some point in the 70's, Samuel Bowles presented to his class at UMass an elegant formal refutation of Marx's famous claim about the falling rate of profit. When he finished, one of the graduate students said, "Sam, Okishio proved that in 1960." Bowles was reduced to publishing a note in 1981 in the CAMBRIDGE JOURNAL OF ECONOMICS with the rather modest title "Technical Change and the Profit Rate: A Simple Proof of the Okishio Theorem." I think I will stick to philosophy. In that field, if someone publishes an idea before you do, all you need to do is put brackets around your essay and a negation sign in front of the brackets, and then publish. You will get just as much respect.

So, herewith an informal, very simple, and elegant exposition of Vegara's Theorem. [God, how I hate writing those words.] The purpose of formal representations of arguments is to exhibit certain very general structural features of the arguments without the distraction of the content. Suppose I present you with the following argument: All Martians are vegans. All vegans are politically correct pompous asses. Therefore, all Martians are politically correct pompous asses. If you have never had an elementary course in logic, you may protest that there are no Martians, and that you are offended by the statement about vegans. These natural reactions are likely to blind you to the fact that the argument is a perfectly valid instance of what, in the old days, was called a syllogism in Barbara, namely All A are B, All B are C, therefore All A are C. [As a mnemonic device to help them remember which syllogisms are valid, the medievals gave all the various possibilities names in which in which a sequence of three vowels stood for the major and minor premises and the conclusion. Since "All A are B" was labeled a proposition in the "a" form, the name Barbara, with syllables aaa, stands for a syllogism in which each of the three propositions is of the form "All x are y."]

Marx, as we have seen, makes a number of formal claims about the relationship between the physical surplus, the quantity of labor directly indirectly required to produce a commodity, the prices at which goods exchange in a market ruled by competition, and the profit rate. Some of these claims are true, others are true under certain interesting circumstances, and still others are false. Marx also makes the very strong claim that all of these theses rest on the distinction, unique to the labor input in production, between labor and labor power. Taken all together, he claims to have shown rigorously that capitalism and its profits rest on the exploitation of the working class. What is more, he argues, on the basis of all of these claims, that exploitation is completely compatible with bourgeois morality, whose fundamental command is to give equals for equals.

We have translated many of Marx's claims into formal mathematical assertions, and have been able by this device to establish exactly when, and under what circumstances, they are true. Suppose now that we look back at the equations we formulated for the purpose of calculating labor values and prices and see whether, in those equations, there is some formal representation of Marx's signature innovation, the distinction between labor and labor power.

On first inspection, the answer would seem to be yes. Here, for example, is the equation we formulated, in the corn/iron/books model, for the labor values in the corn sector:

100 + 2Lc + 16Li + 0Lb = 300Lc

The labor inputs are distinguished by the fact that in the process of production, each unit contributes one full unit of labor value to the corn output, whereas the corn and iron inputs contribute only an amount equal to their labor value. The amount of labor required to produce a unit of labor is, we saw, necessarily LESS than one unit of labor [this follows necessarily from the fact that the system as a whole produces a physical surplus], and yet it contributes as full unit of labor to the output. The difference between those two quantities -- the amount of labor required to produce a unit of labor and a unit of labor -- is precisely the surplus labor that is transformed, in the sphere of circulation, into profit.

But suppose now we ask a question that it never occurred to Smith or Ricardo or Marx to ask, a question [I cannot let this go] which, for a brief moment, I thought I was the first person ever to ask: How much corn, directly and indirectly, does it take to produce one unit of iron, one unit of books, one unit of corn itself, AND ONE UNIT OF LABOR? Why on earth ask that question? There is no plausible distinction between corn and corn power, after all. But that is just the point. If we look at the equations, the only thing that tells us which input is labor, which is, corn, which is iron is the label. Suppose we keep the same physical characteristics of the model, but rewrite the equations so that they are set up to discover how much corn it takes, directly and indirectly, to produce one unit of each input. Following our labeling practice, let is call Ci the amount of corn, directly and indirectly required to produce one unit of iron, Cl the amount of corn directly and indirectly required to produce one unit of labor, Cb the "corn value" of books, and Cc the "corn value" of corn itself.

We can certainly do this. Nothing stops us from formulating such equations. But will they have any economic meaning? Do we have any assurance that we won't get absurd results, such as a negative value for the "corn value" of books? And, most important of all, will it turn out that it takes less than one unit of corn to produce a unit of corn, so that there is some "surplus corn value" generated each time we use corn as an input into production?

The answer to all of these questions, and any other similar questions we might ask, is a simple YES. EVERY SINGLE RESULT WE PROVED IN OUR VARIOUS SYSTEMS WITH REGARD TO LABOR, LABOR VALUES, AND SURPLUS LABOR VALUE, CAN BE PROVED IN EXACTLY A PARALLEL MANNER FOR CORN VALUES, OR FOR IRON VALUES OR TOOL VALUES. The only thing we cannot do is replicate these results for "book values," because books are not required as directly or indirectly as an input into every line of production. They are a luxury commodity, as I called them.

It is necessarily true that all of the corn values or iron values or tools values] will be positive, that the corn [or iron or tools] value of the physical surplus will exactly equal the surplus corn [or iron or tool] value extracted from the corn inputs, that in an economy embarked on a von Neumann balanced growth path:

(total money profits)/(total surplus corn value) = (total money prices)/(total corn values)

and so on and on.

Clearly, Marx and Smith and Ricardo are right that there is something special about labor that sets it apart from all the other inputs into production, but the distinction between labor and labor power is not it, because the revised equations we wrote [well, we did not actually rewrite all of them, but you get the idea] appear to demonstrate that in our system corn is exploited! What on earth is going on?

I have an answer [and this one IS in some sense original with me, I think], but I must prepare now to lecture this afternoon about Michael Oakeshott, so my answer will have to wait until tomorrow.


Chris said...

So Capitalism will still rest on exploitation, albeit Marx was wrong about his theories regarding surplus value?

Robert Paul Wolff said...

Yes. I will provide what I believe is a betyter analysis of the root of that exploitation, which is actually more in tune with Marx's own theories. Tomorrow. :)

hari said...
This comment has been removed by the author.
hari said...

This may seem tangential or irrelevant just as the un- pc nature of the claim about Martians.

But I still don't understand this as a description of a capitalist economy.

In a capitalist economy there is cut-throat competition. This means that capitalists are not selling outputs over time at the prices that they bought them.

Because Marx assumes a constant value of money and continuous rising productvity there should be a downward trend in prices.

But that then means that the prices of outputs can't be solved with the prices of the inputs in simultaneous equations. If you proceed as you do, prices can't change interperiodically. That means that you are not formalizing Marx's theory of how prices are determined.

As Alan Freeman has long argued, Marx's dynamic theory can't be modeled via the use of simultaneous equations.

Now once we drop the assumption of price fixity for inputs and outputs, will your challenge to Marx still stand?

Maybe it will, so this challenge to your simultaneous equation framework does not undermine your result.

But I am not sure.

Ian J. Seda Irizarry said...

I have to say I'm deeply disappointed with the claims made in this posting, as if nobody has ever dealt with Okishio's claims. To make matters worst, Okishio himself published (i think in 2001 in CJE) an article where he basically accpets the irrelevance of his position regaarding his "refutation" of MArx's claims. But apart from that, I think Anwar Shaikh's answer to Okishio still holds the test of time. Basically he points out how Okishio rules out fixed capital in his modelling of constant capital, thereby ruling out the component that reflects the competition (not perfect competition as the neoricardians you celebrate that supposedly describes what Marx means with competition))where unit costs are going down for the firm, thereby attracting surplus created in other spheres, which then gives the impression (one of the surface vs appearance cases in Marx) as if capital were productive- Marx then combines this with his discussion of how the rate of profit is understood in equation form that also seems to reflect the productive capital. Again, verytime I hear about these "refutations" I just cross my arms in disbelief as if the argument has been settled. And well, recently Andrew Kliman wrote another book which approaches the issue via another way which I honestly cant recall.

Robert Paul Wolff said...

Why don't you write a guest post on the subject in which you take the time to explain it clearly and at length? It is anciallary to the line of my argument, but it is certainly interestinbg, and worth talking about.

Ian J. Seda Irizarry said...

Thanks for the opportunity. I'll try to, although right now I'm trying to finish my dissertation for your colleagues Resnick and Wolff and if they find out I'm using my fellowship for this, I don't think the outcome will be overdetermined :-)
Still, for those interested, here is one of Shaikh's writings on the topic:,%20notes%20on%20Dobb's%20theory%20of%20.pdf

Apart from that I want to apologize for my style. After re-reading what I initially wrote I noticed that maybe I was still fuming after watchingn last night the movie "Biutiful" with Javier Bardem. I highly recommend it, especially to those that think that "fixing" things by regulating the financial sector will actually solve any fundamental problems. Unfortunately many inspired by the Marxian tradition have fallen for this keynesian perspective...

Robert Paul Wolff said...

Absolute rule: Dissertation comes first. This space will always be open to you when you are ready to try your hand at a guest post.

Give my regards to Steve and Rick.

Noumena said...

Among mathematicians, it's not unusual to name a theorem after several independent discoverers. I've no problem calling it the Vegara-Wolff Theorem.

Robert Paul Wolff said...

I can live with that!