Using the variables defined at the end of the last Part, we can now set up a system of three simultaneous linear equations. Here they are:
100 + 2Lc + 16Li + 0Lb = 300Lc
90 + 9Lc + 12Li + 0Lb = 90Li
20 + 1Lc + 2Li + 2Lb = 40Lb
The first equation says that 100 units of labor directly applied to 2 units of corn, in which is embodied a quantity of labor two times the labor value of corn, and to 16 units of iron, in which is embodied sixteen times the labor value of iron, yields 300 units of corn, in which is embodied 300 times the labor value of corn. And so forth. Notice that the variable Lb does not appear in either of the first two equations. That means that an increase in the difficulty of producing theology books [problems with the proofs for the existence of God, perhaps] will have no effect whatsoever on the labor value of either corn or iron. It also means that we can treat the first two equations as a system of two linear equations in two unknowns, and after solving it we can simply plug the values of Lc and Lb into the third equation to find the value of Lb. [All of this is a mathematical representation of a very important set of facts about the economy, of course.]
So what is the solution to the first two equations? Well, if you will carry out the manipulations yourselves, you will find that Lc = 0.4 and Li = 1.2. Lb turns out to be equal to 0.6.
0.4 what? you may ask. 0.4 units of labor is the answer. That is how much direct and indirect labor ends up being embodied in one unit of corn. If the dimension of labor happens to be worker-years, and the dimension of corn happens to be metric tons, then the equations tell us that in this system it takes 0.4 worker-years of labor, directly and indirectly, to produce one metric ton of corn. [Real world factual plausibility is not an issue. We are doing economics here!]
We have now ascertained the labor values of the produced commodities in this system. Ricardo says that these labor values determine the natural or equilibrium prices at which these commodities sell in a laisser-faire marketplace. More precisely, he says that commodities exchange in proportion to their labor values. To find out whether he is right, we must still figure out what the equilibrium prices are, so that we can see whether they are proportional to the labor values.
In calculating labor values, we remained in the sphere of production, attending only to the quantities of inputs required for specified quantities of outputs, but now, as Marx would say, we move into the sphere of circulation. We must set up a new system of equations that is a bit more complicated to solve [second semester high school algebra], but before we can set up the equations, we must make a number of simplifying assumptions and behavioral assumptions about the capitalist economy in which all of this is taking place. Here are some of the things we must assume. [Marx, as we shall see eventually, has enormously insightful and important things to say about the historical, sociological, economic, and psychological conditions under which these assumptions are plausible, but I can only talk about one thing at a time, so they will have to wait for a while.]
First of all, we must assume that the goods being produced have become standardized, so that one unit of corn or iron or books is much like another. If the society is still in the stage of craft production, with hand crafted furniture and artisanal loaves of bread or wheels of cheese, it will be impossible to represent the production process by our simple equations. The workers too must be standardized, and stripped of their inherited skills, so that one unit of labor directly applied is much like another. In the sphere of circulation, we must assume that competition establishes a single price for each kind of good, a single wage rate for labor, and a single rate of profit on invested capital [rent is not yet an issue.] All of these assumptions are hidden behind the simple price equations we shall shortly be setting down. Marx was the first economist [indeed, the first thinker of any sort] to recognize the enormous significance of these assumptions. Much of the first several chapters of CAPITAL is devoted to spelling them out and analyzing them. Later on, I shall have a good deal to say about them.
To formulate our equations, we need some new symbols for the unknowns. [I am limited, unfortunately, by the fonts available to me, so this will be clumsier than I would like.] First of all, we need a variable for each of the prices of the goods produced: Pc for the price of corn, Pi for the price of iron, and Pb for the price of books. We shall use W for the money wage, and R for the rate of profit. [Ordinarily one would use the Greek letter pi for the profit rate, but such is life in cyberspace.] Here are the equations read off from the same data that yielded the labor value equations:
(100W + 2Pc + 16Pi + 0Pb) (1 + R) = 300Pc
( 90W + 9Pc + 12Pi + 0Pb) (1 + R) = 90Pi
( 20W + 1Pc + 2Pi + 2Pb) (1 + R) = 40Pb
Why (1 + R)? you may ask. Because the money that the capitalists get from selling their output [300Pc in the case of the corn sector] must be enough to cover the cost of production [the 1] plus enough to yield the going rate of profit on that cost [the R]. Hence (1 + R).
This is a system of three equations in five unknowns. Mathematicians call such a system "underdetermined" [no, that is not what Althusser means by "underdetermined," but that is another matter entirely.] What to do? Well, the first step is to eliminate one of the price variables. Remember that what we are interested in is relative prices, which is to say exchange ratios between different commodities. The classical practice, employed by Smith, Ricardo, Marx and all other classical Political Economists, is to select one commodity as the money in the system, give one unit of it the price 1, and then express all other prices as multiples of that unit commodity, or as it is usually called, numeraire. This can be an ounce of gold, a pound of silver [the British Pound Sterling] or, if one is a Masaai warrior, one cow. In this case, we shall choose corn as our numeraire, and set the price of one unit of corn equal to 1. When we plug that assumption into our equations, we get:
(100W + 2 + 16Pi + 0Pb) (1 + R) = 300
( 90W + 9 + 12Pi + 0Pb) (1 + R) = 90Pi
( 20W + 1 + 2Pi + 2Pb) (1 + R) = 40Pb
This system is still undetermined, by one degree [as we say], because there are now three equations and four unknowns. Notice that once again, the first two equations can be treated separately, because the price of books, Pb, does not appear in them. They constitute a system of two equations in three unknowns: Pi, W, and R.
There are two things we can do. The first is to reduce the equations to one by eliminating the price of iron, so that we get a single equation in W and R. If we do this, we find with some algebraic manipulations [which I must simply assume you folks can do on your own] that there is the following inverse relationship between W and R:
(1 + R) = 6/(2W + 1)
Inspection reveals that as the wage rises, the profit rate falls, and vice versa, which nicely demonstrates the fundamental Classical thesis that the interests of the working class and the capitalist class are diametrically opposed.
But we can also try just to solve the two equations for the price of iron. In general, when you have a system of two equations in three unknowns, you cannot do this, but if you go ahead and try, you will discover, to your amazement, that the Wage and the Profit Rate drop out, and the two equations yield the result Pi = 3. This is an extraordinary result. It seems that in this system, the price of iron [and also the price of books, which turns out to be 1.5] is totally independent of the wage rate and the profit rate. No matter how those fluctuate in inverse relation to one another, the prices remain the same.
Well, Ricardo said that the prices at which goods exchange -- their natural prices or values -- are proportional to the quantities of labor required directly and indirectly for their production -- their labor values. Is he right? Let us see.
The price of corn is 1 and the labor value is 0.4, so the ratio is 1/.4 = 2.5
The price of iron is 3 and the labor value is 1.2, so the ratio is 3/1.2 = 2.5
The price of books is 1.5 and the labor value is 0.6 so the ratio is 1.5/.6 = 2.5
HEY PRESTO! RICARDO IS RIGHT. TA DA!
Ah well, if life were only that easy. Stay tuned. Tomorrow we shall discover the secret of this remarkable result.