Now Marx makes his move. [This paragraph is actually quoted from page 127 of my book, UNDERSTANDING MARX. It seems to me somehow vaguely cheating to quote from a book rather than to write something new, but I did write it myself, after all.] The prices at which commodities exchange in the market are merely the surface of the capitalist market, the appearance. The underlying reality is the extraction of surplus value from the workers in the sphere of production. As Marx says in a letter to Engels in the year following the publication of volume one, "profit is for us first of all only another name or another category of surplus value ...[S]urplus value gets the form of profit, without any quantitative difference between the one and the other. This is only the illusory form in which surplus value appears."
The equality of profits with surplus value is an economy-wide fact, Marx claims. In any single industry, the particular capital intensity may cause prices to deviate from surplus value, so that the equality of the two, and the grounding of capitalism on exploitation, is hidden from view. But in the society as a whole, the total of profits, rent, interest and other unearned income must exactly equal the surplus labor value extracted from the workers. Marx adds the assertion of a second claim: That the total price of all the commodities sold in a time period must equal the total labor value of those commodities, even though the price of an individual commodity may lie above or below its labor value.
As it stands, Marx's solution to the problem posed by Ricardo is incoherent. The problem is one of units. Prices are measured in units of whatever is being used as money in the economy. Labor values are measured in units of labor time [hours, person-years, etc.] Strictly speaking, they cannot be equal unless one is deliberately stipulating the equality as a way of defining the money unit, in which case the equality is a tautology. But there is a simple way of capturing what Marx really intends to assert, namely in the following equality, which is definitely not a tautology:
(total profits/total surplus value) = (total prices/total labor values)
Since the numerator of each side of the equation is measured in money units, and the denominator is measured in labor time units, the units cancel out, leaving Marx asserting the equality of two pure fractions.
Marx also asserts one more equality, between the money profit rate and something he calls the value rate of profit. This latter quantity is simply the ratio between the surplus value extracted in the economy as a whole [ S] and the sum of the constant capital [C] and variable capital [V], measured in labor units. In other words:
The value rate of profit = S/(C+V)
So Marx claims that S/(C+V) = R
Is Marx right? Let us start by checking our little corn/iron/tools economy, the one that does not have equal organic composition of capital. We can do this, albeit a trifle tediously, by plugging into Marx's equation the values we got for prices, profits, labor values, and profit rate in that system. Here is what we get:
Total profits are simply the total money cost of all the capital inputs in the system, including labor [evaluated by means of the money wage], multiplied by the system wide profit rate, which is 1/3. When we carry out this calculation, we find that
Total profits ~ 110.28 money units [which, we will recall, was specified as units of corn, because we set Pc = 1.]
Total surplus value is calculated by subtracting the total amount of necessary labor in the economy [which is the labor value of all the real wages earned by the workers] from the total amount of labor employed in the economy. This is equal to 150(1 - Lw).
Total surplus value ~ 103.71 units of labor.
Total prices are arrived at by simply adding up the money price of the total output:
Total prices ~ 441.12 money units
Total values are calculated in analogous fashion by adding up the labor value of all the outputs.
Total values ~ 415.20 units of labor
Now we can check Marx's claim.
(total profits)/(total surplus value) ~ 1.063 money units/unit of labor
(total prices)/(total values) ~ 1.062 money units/unit of labor.
Marx is right! [Indeed, the minor difference is actually a result of rounding. The actual result is a strict equality.]
But this is just for our little corn/iron/tools model. What about the general case? Well, there is bad news, and there is good news. The bad news is that if you add a theology books sector, the equality break down. The good news is that the equality holds for EVERY POSSIBLE LINEAR REPRODUCTION MODEL IN WHICH THERE ARE NO LUXURY GOODS. What does all of this mean?
Well, when there are no luxury goods, the entire physical surplus is being used to expand the magnitude of production. In other words, an economy without any of its surplus diverted to luxury goods is engaged in the fastest possible rate of growth. "Accumulate, Accumulate, that is Moses and the Prophets to the capitalists," says Marx. Now, to the more alert among you, the following question may occur: As capitalists attempt to reinvest all of their profits in expanded production, how do we know that they will not be frustrated by bottlenecks and gluts -- too little of one input, too much of another? Can we be certain that balanced growth is in fact possible, whatever the technical specifications of production may be?
The answer, I am happy to say, is yes. There may indeed be mismatches between supply and demand at first, but it is a mathematical fact [proved in the Appendix of my book] that for any linear single-product system of production, there is some vector of activity of levels of the different industries, arrived at over time by the forces of competition, that supports a process of balanced growth, year after year. The growth rate, needless to say, is equal to the profit rate.
Interestingly enough, the great twentieth century mathematician John von Neumann proved a theorem in growth theory exactly along these lines. As a consequence, an economy embarked on maximum growth is said to be on a von Neumann balanced growth path.
[There is one really hinky other case in which Marx's equation is true, but as it has no apparent economic meaning, and involves some hairy propositions about maximal eigenvalues, I will leave it alone. You can find it discussed in Abraham-Frois and Berrebi's fascinating book, THEORY OF VALUE, PRICES, AND ACCUMULATION.]
This is the theoretical high point for Marx, the outer limits of the success of his version of the Labor Theory of Value. The key to the entire development in Marx is the distinction between labor and labor-power, which opens up the possibility of surplus value. Tomorrow, I will show you that Marx is all wrong, but that nevertheless his underlying intuition about capitalism is correct. I will also suggest an alternative theoretical formulation of this underlying intuition. Stay tuned.