I am going to adopt certain notational conventions that have become more or less standard in the literature that has grown up to discuss Marx's theories. Let us use the Greek lower case letter λ to stand for the quantity of labor directly and indirectly required to produce a unit of some commodity. Subscripts will indicate which commodity is being referred to. Thus λc will stand for the quantity of labor directly and indirectly required to produce one unit of corn. In the language that Marx used, following Smith and Ricardo, λc will be the Labor Value of Corn. In like manner, λi and λb will be the Labor Values of Iron and Books respectively.
If we now look back at the input data I presented two posts ago in the input/output table labeled the Corn, Iron, Theology Books System, will see that this table does not yet present us with equations. It simply tells us how much of each input is required for a certain amount of output. But we can convert that information into a system of equations in the following way.
Look first at the line labeled "Corn Sector." This line tells us that to produce 300 units of corn, we need 100 units of labor. We also need as much labor as it took in past cycles of production to produce 2 units of corn. This is labor "indirectly required," in the sense that it is not required directly in this cycle of production, but was required earlier to produce those 2 units of corn, and is now, as Ricardo says, "embodied" in the 2 units of corn needed in this cycle in the Corn Sector. We don't yet know how much labor that is, so we must list it simply as 2 times the amount of labor required directly or indirectly to produce 1 unit of corn, which is to say 2λc units of labor. By the same reasoning, we will need 16λi units of labor. And all of this -- 100 units of labor plus 2λc units of labor plus 16λi units of labor, will have to be equal to 300 units of corn times the amount of labor directly or indirectly required for the production of a single unit of corn, or 300λc units of labor. In mathematical symbols, this means:
100 + 2λc + 16λi + 0λb = 300λc
Exactly the same process of reasoning allows us to convert the information about the other two sectors into Labor Value equations, thus:
90 + 9λc + 12λi + 0λb = 90λi20 + 1λc + 2λi + 2λb = 40λb
We now have a little system of three linear equations in three unknowns, λc, λi, and λb. Some of you will wonder why I have devoted so many words to explaining something that is self-evidently obvious to you. Quite simply, long experience has taught me that getting the elementary steps in an argument clear is the secret to making the entire subject clear. If you are irritated by my verbosity, you can console yourself with the thought that when I get to talking about the reasons for Marx's strange choice of literary style, I will go into just as much detail about the nuances of ironic discourse, which you may find quite useful, even though it will bore your more humanistically inclined fellow readers.
Those of you who remember your high school algebra will know that this little system of three linear equations in three unknowns is quite easily solved [since the equations are mathematically independent of one another.] I shan't try your patience further. The solution is:
λc = 0.4 units of laborλi = 1.2 units of labor
λb = 0.6 units of labor.
Now let us return to the question, How do the surplus getters get the surplus? We know that the physical surplus in our little system is 246 units of corn, 49 units of iron, and 38 units of books. What is the Labor Value of this surplus? Easily enough calculated, now that we know the labor values of corn, iron, and theology books. It is:
246 (0.4) + 49 (1.2) + 38 (0.6) units of labor, or 168 units of labor, all of which, of course, is appropriated by the entrepreneurs who own and run the companies that produce the corn, iron, and books.
Now take a look at the information concerning the "production" of labor, which thus far we have not paid any attention to. The workers must consume 42 units of corn and 21 units of iron each year to enable them to work. The Labor Value of their consumption basket is 42 units of labor. But the workers each year produce 210 units of labor directly in the three sectors combined. So they are contributing more labor to the production process than they are consuming, and all of that extra labor is bestowed upon, or embodied in, the output of the system, which is then appropriated by the entrepreneurs and sold in the marketplace. How much more labor are the workers embodying in the product than they are consuming? Well, the answer is obviously
210 - 42 = 168 units of labor.
And that is precisely the Labor Value of the physical surplus appropriated by the capitalists. The labor embodied in the physical surplus is the surplus labor extracted from the workers in the production process. That physical surplus is converted by the capitalists in the market into money, yielding them a Surplus Value, over and above what they paid for their inputs, or, as we are accustomed to call it, Profit. The capitalists find themselves richer after each cycle of production by exactly the amount of Surplus Value yielded up by the workers. And what is the technical term for extracting more value from an input than one paid for it?
To summarize: Capitalism rests on the exploitation of the working class. Sound familiar?
Before bringing this Part to a close, let me add one little complication with the most profound implications for an understanding of the political struggles of the working class. In the table referred to above, I specified the consumption of the workers. This allowed me to calculate the precise quantity of surplus labor extracted from them. But of course the consumption of the workers is variable, a result of the relative bargaining power of the workers and the capitalists. The capitalists try to drive down wages to bare subsistence, drawing on what Marx called "the reserve army of the unemployed" to defeat efforts by the workers to raise their wages. The workers counter by organizing and striking. Even what is considered "bare subsistence" is itself historically and politically determined, as Ricardo, to his great credit, clearly recognized and said. The entire history of the labor movement can be written as the story of this struggle.
Now, if we leave the exact quantities of worker consumption unspecified, we can then manipulate the labor value equations [and the price equations, which I will introduce tomorrow] to demonstrate that the wage and the profit rate [or worker consumption and the size of the surplus] are directly contrary to one another. They vary inversely. As one grows larger, the other must grow smaller. Thus, contrary to all the ideological blather of the capitalists, class warfare is built structurally and unavoidably into the nature of capitalism as such. This may strike the readers of this blog as obvious, but try telling that to Thomas Friedman!