You will notice that I have constructed the sector devoted to the production of devotional literature so that iron, corn, and theology books are required as input. You may wonder in what sense theology books can be required as input into the production of theology books. Let us just suppose that the writers of the new books in each cycle of production read and re-read the existing books so assiduously that they wear them out. Analytically speaking, what matters here is that none of the output of the books sector is required as input into any other sector but itself. [For those of a more formal turn of mind, the square matrix of unit input coefficients is semi-positive and partially decomposable, with one of the submatrices representing the sector of basic commodities and another representing the sector of luxury commodities, but we need not go into that.]
Now, let us get to it. According to Ricardo, the natural price, or value, of a commodity [which is to say, something reproducible, like corn or iron, not something non-reproducible, like land or an Old Master] is determined by the quantity of labor that is required, directly or indirectly, to produce it. By labor indirectly required Ricardo means the labor that was required to produce the non-labor inputs into its production. If we look at the Corn Sector in the little model above, we will see that 100 units of labor are required to produce 300 units of corn, so each unit of corn requires one-third of a unit of labor directly. But each unit of corn also requires 1/150 of a unit of corn, as seed presumably [I get this number by dividing the left hand side of the Corn Sector line through by 300, to find out what are called the "unit input requirements" of the Corn Sector.] Now, we have just seen that each unit of corn requires 1/3 of a unit of labor, so the corn that is used up in the production process of the Corn Sector must have required 1/450 of a unit of labor in the previous cycle of production. This labor is described by Ricardo as being "embodied" in the seed corn, and as being transferred to the output when the seed corn is used in the production process. The same sort of calculation shows that a unit of corn requires 4/75 of a unit of iron. Now, looking at the Iron Sector, we find that each unit of iron requires one unit of labor to be produced, so that means that the iron that goes into one unit of corn production must itself have required 4/75 of a unit of labor in the previous cycle of production. So, thus far, we see that a unit of corn requires 1/3 of a unit of labor directly, and (1/450 + 4/75) or 1/18 of a unit of labor indirectly -- 1/18 of a unit of embodied labor.
But wait, you will protest [and you will be right], in that previous cycle of production to which you have been alluding, corn and iron, as well as labor, were required to produce the corn and iron that are being used in this cycle, so there are some extra little bits of embodied labor carried over from an even earlier cycle, that must be added in. It doesn't take much mathematical imagination to see that we have here the makings of an infinite sum of bits of embodied labor. Two questions immediately present themselves: First, does this infinite sum converge on some finite quantity? and Second, if it does, what does it converge on? How much total labor, directly and indirectly, does it take to produce one unit of corn? Or, as we have learned to say, What is the Labor Value of Corn?
Well, it turns out that we have here a problem whose answer can be arrived at by setting up and solving a little system of simultaneous linear equations. The Table we have been working with gives us a great deal of information about the production of corn, iron, and theology books, but there are three quantities that it does not directly tell us about. namely, the Labor Values of corn, iron, and theology books. These are, as they say in elementary algebra classes, the unknowns [or, as Donald Rumsfeld would say, the known unknowns.] So, I am going to choose three symbols to stand for these unknowns. Let us let Lc stand for the labor value of corn, which is to say the total amount of labor directly and indirectly required for the production of one unit of corn. Li will stand for the labor value of iron, and Lb for the labor value of books. Three unknowns, three equations pretty easily arrived at. We can solve that baby! Tomorrow, I will show you the equations [back to the scanner -- ugh], tell you what the solution is [you are going to have to work out that bit of elementary math yourselves], and move on to the next step in the process of checking to see whether Ricardo's Labor Theory of Value is true.