My Interpretation of the Thought of Karl Marx
Part Six: Enter the Mathematics
(Several of the comments call for a response but I will postpone that in order to get on with my exposition.)
There is, of course, vastly more to be said about Marx’s views in volume 1, but I wish at this point to turn to the modern mathematical interpretation of classical Political Economy which I mentioned at the very beginning of this exposition. Unlike their modern descendants, the classical economists made almost no use at all of formal mathematics. Marx had a go at working out some numerical examples but mostly he botched it and it added very little to his explanation of his theories. When the modern mathematical re-interpreters of Ricardo and Marx undertook to translate their theories into equations, they had to formalize a decision that Ricardo and the others had made more or less without explicitly stating it.
In order to reduce the complexity of real economic activity to equations, economists must in effect choose between supposing that there is one dominant technique for the production of each distinct commodity and supposing that there are an infinite number of techniques for the production of each distinct commodity. To put this point as simply and formulaically as I can, they have to decide whether they are going to use linear algebra or calculus. The neoclassical assumption of an infinity of alternative ways of combining inputs to produce an output lends itself to analysis using calculus and the classical assumption of a single dominant technique of production for each commodity finds its most natural expression in systems of linear equations.
Linear algebra makes it possible to handle formally any finite number of commodities, each one represented by a single vector of inputs per unit of output. One can then manipulate what is called the unit input matrix to derive a variety of powerful conclusions. Since I may have lost many of you at this point, let me give a very simple example which will serve quite adequately to illustrate what I want to say. All this is laid out precisely and at length in my book, Understanding Marx.
Suppose we are talking about an elementary economy in which there are only two commodities, corn and iron. (What we call corn does not grow in England, of course, but the term “corn” was used by the English to mean “the dominant grain of a region,” hence the great debate in the early 19th century in parliament as well as in the writings of the political economists over the so-called “corn laws” regulating the importation of grain from abroad.) I have invented the following little corn/iron economy to illustrate what I want to say. Since I am now doing economics, there is no need for me to worry about the real-world relevance of what I am saying.
Suppose it takes 100 units of labor, 2 units of seed corn, and 10 units of iron to produce 300 units of corn. If we use the Greek letter l for labor value or quantity of embodied labor with subscripts indicating whether we are talking about the labor value of corn or the labor value of iron, and if we recall, what is essential, that direct labor must be entered at par since it is labor directly, not indirectly, contributed to the production of the output, then we can write the labor value equation for the corn sector in the following way:
100 + 2lc+ 10li = 300lc
In words, this equation says that when producing 300 units of corn, the 100 units of labor directly applied in production and thus embodied in the corn output, added to the amount of labor embodied in the two units of corn used in production and the amount of labor embodied in the 10 units of iron used in production taken altogether equally amount of labor embodied in the 300 units of corn that are the output of the production process.
Using the letter p with appropriate subscripts to stand for the prices of corn and iron, the letter w to stand for the wage paid for the labor, and the Greek letter π to stand for the rate of profit, we can write the corresponding price equations for this little corn/iron model. The price equation in the corn industry looks like this:
(100w + 2pc+ 10pi)(1 + π) = 300pc
If I choose the appropriate input quantities for the iron sector and carry out a series of mathematical manipulations with which I shall not trouble you, I can demonstrate that the prices of corn and iron are proportional to their labor values, as Ricardo claimed, and also, what is really quite interesting, that these prices and labor values are independent of the wage and the profit rate, which vary inversely to one another, thereby also demonstrating the class conflict between labor and capital. All very impressive.
I spent a very great deal of time plowing through 10 or more thick difficult mathematical economics texts in each of which the theories of Ricardo and Marx were explored in excruciating detail. The big take away from all these books was that both Ricardo and Marx had been, contrary to the conventional wisdom, brilliant intuitive mathematical economists, much of whose theoretical work was sustained by this sophisticated 20th century analysis.
But as I toiled away at my studies I noticed something curious and eventually troubling. There were symbols in the equations in these books for just about everything that Ricardo and Marx had talked about – symbols for quantities of inputs, symbols for quantities of outputs, symbols for prices of commodities, symbols for quantities of labor, and also the wages of labor, symbols for profit rates earned by capitalists. But nowhere in the equations could I find a symbol for Marx’s signature concept, labor power. Since in Capital Marx had made the distinction between labor and labor power the key to his solution to the central problem of the origin of profit, it seemed to me that any modern mathematical rendering of his theories should have somewhere a symbol for that central concept, but it was nowhere to be found. In effect, the mathematical economists I was reading, all of whom were extremely sympathetic to Marx’s theories, seemed to be saying that his story about the distinction between labor and labor power was simply, in Pooh Bah’s immortal phrase from the Mikado, merely “corroborative detail designed to lend an air of verisimilitude to an otherwise bald and unconvincing narrative.”
At this point a thought occurred to me. (Now, this is a sad tale but I feel compelled to tell it nonetheless.) Since there is no symbol for labor power, the distinction between labor and labor power plays no role at all in the formal analysis of Marx’s theories. To be sure, in the labor value equations the labor inputs are valued at par and in the price equations the price of labor has a letter, w, all its own, but the first is an assumption, not a conclusion, and the second is simply a notational convention. Could one write a set of equations that permitted us to calculate the iron value of labor and corn, rather than the labor value of corn and iron?
Iron value!? What on earth would an iron value be? Nobody ever talks about iron values or corn values but only about labor values. There might in fact be a distinction between labor power and labor and no corresponding distinction between corn power and corn or iron power and iron but if the distinction between labor power and labor did not enter into the equations then that would make no difference.
So I had a go at setting up some corn value and iron value equations and seeing what I would get. The first question – quite important – was whether one could always be sure in calculating iron values or corn values that when the equations were solved those values would be positive. After all, it would not make much sense to say that the amount of corn directly or indirectly required for the production of a unit of iron was negative! Well, a little mathematical manipulation (with some help from friends in the UMass economics department) revealed that so long as there was any surplus of any commodity anywhere in the system, all the corn values, or iron values, or labor values, or x-values in the system would necessarily be positive.
One of Marx’s most striking claims, demonstrated to be correct by the modern mathematical reinterpretations, was that the labor value of the physical surplus is exactly equal to the surplus labor extracted in the production process from the workers – a lovely mathematical demonstration of the fact of exploitation. But a little more manipulation with the equations demonstrated that this was also true for corn values or iron values. The corn value of the physical surplus was exactly equal to the surplus corn value extracted from the corn inputs in the system, and so forth.
In fact every single theoretical claim made by Marx in Volume 1 of Capital (all the claims in volumes two and three, for that matter) could be replicated using these nutty notions of corn value and iron value.
I was remarkably pleased with myself when I had reached these conclusions for I thought that I was the first person in the entire history of the discussion and commentary on the theories of Karl Marx to have even thought of this, let alone to have demonstrated it mathematically. I told this story on my blog 10 years ago. Let me close this episode in my exposition by reproducing what I wrote there:
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.
Sic transit gloria mundi