MARX BOOK DAY FOURTEEN

As we have seen, Smith, Ricardo, Mill, and Marx and their many lesser compatriots wrote their works of political economy with little or no mathematical elaboration. The sophisticated mathematical economists who returned to that tradition in the 1960s, 1970s, and 1980s brought to bear on it the powerful tools of linear algebra and associated branches of mathematics. For a while in the late 1970s and early 1980s I was content simply to read book after book, following the arguments given by the authors and deepening my own understanding of the classical tradition of political economy. But after I had plowed through two or three thousand pages of mathematical economics, several things began to dawn on me.

 

First of all, as I have struggled very hard to make clear, the central theme of Marx’s analysis of capitalism is that it is thoroughgoingly mystified. It portrays the marketplace as a realm, in Marx’s wonderful phrase, of Freedom, Equality, Property, and Bentham, whereas the truth is that is a thoroughly mystified misrepresentation of the reality of capitalism, which is a realm of exploitation and domination. But in the formal reconstructions of the arguments of the classical political economists, none of this mystification appears. The equations are precise, transparent, and – as one author after another demonstrates – essentially correct. The labor value of the physical surplus in each system is in fact equal precisely, mathematically, to the surplus labor extracted from the labor inputs. What is more – and this will be gone into in considerable detail later on – it is even possible to demonstrate that in an extremely important subset of all the possible cases, the money profit appropriated by all of the capitalists in the system can be derived from that surplus labor value.

 

In my attempt to make the modern formal reconstructions of classical political economy available to an audience not familiar with linear algebra, I constructed little corn – iron models like System C with variables for the amount of corn, the amount of iron, the labor value of corn, the labor value of iron, the price of corn, the price of iron, the wage rate and the profit rate. But in the books I was reading, the authors did not bother with such elementary examples. It was page after page of matrices with variables like a₁, a₂, … a_n and such.

 

One day, as I was working my way through a page of mathematics, a thought occurred to me. “I have been asking how much labor is required, directly or indirectly, to produce a unit of iron. That is the labor value of a unit of iron. I wonder how much corn is required, directly or indirectly, to produce a unit of iron.  What is the corn value of a unit of iron?”

 

“THE CORN VALUE OF A UNIT OF IRON!  What on earth are you talking about! There is no such thing as the corn value of a unit of iron. This is the LABOR theory of value, for God sake!”  But then I thought to myself, “Why can’t I calculate the corn value of a unit of iron?”

 

“Well,” I asked myself, “is that question even meaningful? How do I know that the amount of corn required directly or indirectly to produce a unit of iron is always a positive number? And since there is nothing special about corn in relation to iron, the question I am really asking is whether any input into production can serve as the basis for such a question."

 

Nobody in all of the books I had read had ever even hinted at such a question, but by this point I had acquired sufficient familiarity with the mathematics so that, with very little difficulty, I was able to prove the following proposition: So long as there is a physical surplus anywhere in the system of any of the inputs into the system (restricting myself, of course, to inputs that are required directly or indirectly in all lines of production, which excludes luxury products like theology books), all of the a_i values in the system will be positive quantities of the input a_i.

 

Fairly quickly, with very little difficulty, I was able to demonstrate that all the propositions Marx and his predecessors had asserted regarding labor values could be demonstrated equally well for corn values, iron values, or a–values, indeed for any input a required directly or indirectly in all lines of production.

 

But this, I reflected, striking though it might be, was irrelevant to Marx’s argument, because that argument dependent on the distinction between labor-power and labor, and there was certainly no distinction to be drawn between corn–power and corn, or iron–power and iron, or, in the general case, a_i–power and a_i.

 

And then I realized something enormously important, something that, so far as I could tell, no one else had ever thought before me. The distinction between labor–power and labor seemed to be an essential part of Marx’s argument in Capital but there was no symbol in the formal mathematical reconstructions of Marx’s argument in Capital that corresponded to the concept of labor–power!

 

Let me say that again. In the formal mathematical reconstructions of Marx’s argument, there were symbols for corn, iron, and all the other inputs into production, there was a symbol for labor with units of hours, weeks, years, or whatever, there were symbols for the prices of the inputs into production, for the labor values of the inputs into production, there were symbols for the wage and for the profit-rate.  But there was no symbol for labor–power. That term played a central role in Marx’s argument in Capital but played no role at all in the formal reconstruction of his argument.

 

My first thought was, So, Marx was wrong.  But then I thought, He wasn’t wrong.  He understood, in a way that nobody before him really had, that capitalism depended on the existence of a large mass of propertyless men and women who were compelled, if they wanted to live, to sell their labor as though it were a commodity.  As Marx said in chapter 6, “For the conversion of his money into capital, therefore, the owner of money must meet in the market with the free labourer, free in the double sense, that as a free man he can dispose of his labour-power as his own commodity, and that on the other hand he has no other commodity for sale, is short of everything necessary for the realisation of his labour-power.”

 

The question I faced was precisely this: How could I represent formally in the equations both the reality of exploitation and the appearance of free equality? How could I introduce the mystification into the equations, as it were?  The labor/labor–power distinction was Marx’s unsuccessful effort to solve this problem, but his fundamental insight into what was happening in capitalism was, I was convinced, absolutely correct.

 

Tomorrow I will explain the solution I came up with, a solution which I published forty-two years ago.