Marx does not present his explanation for the existence of profit in the form of equations. Instead, he poses a question that, he believes, has stumped his predecessors, and then presents his theoretical discovery as its solution. The problem is that even in the ideal case in which commodities of all sorts exchange in exact proportion to the quantities of labor directly and indirectly required for their production, which is to say exchange at their values, the capitalist somehow manages to exit from each cycle of production and exchange richer than he was before. Here is the way Marx poses the problem in the final paragraph of Chapter V, "Contradictions in the General Formula of Capital."
Our friend, Moneybags, who as yet is only an embryo capitalist, must buy his commodities at their value, must sell them at their value, and yet at the end of the process must withdraw more value from circulation than he threw into it at starting. His development into a full-grown capitalist must take place both within the sphere of circulation and without it. These are the conditions of the problem. Hic Rhodus, hic salta."
A few words about the ironic use of "Moneybags," which is an odd term to find an a work of theoretical economics. The German is geldbesitzer, whose literal translation would be "possessor of money." I use the old translation of Capital prepared from the 1887 Third Edition, after Marx's death, by Edward Aveling [the partner, but never actually the husband, of Marx's daughter] and Samuel Moore, and personally edited and overseen by Engels himself. To nineteen century English readers, "moneybags" would have conjured up the cartoons of political caricaturists like Thomas Nast -- a drawing of a fat little man in a top hat and cutaway coat with a dollar or pound sign on his vest. Those of you who have played the original version of the board game Monopoly will recall this character on the "Get out of jail free" card. The translation, which plays on the etymology of geldbesitzer as someone sitting on a bag of money, perfectly captures Marx's mocking tone.
The Latin tag at the end of the paragraph is yet another instance of Marx's rich, multi-lingual array of cultural references and invocations. It refers to a story about a man in ancient Rome who bragged of having made an extraordinary athletic broad jump when in Rhodes. His friends, unimpressed, challenged him to reproduce the feat, saying, "Rhodes is right here. Jump here!" Hic Rhodus, hic salta. Marx's habit of alluding in this way to the broad, deep range of literature in the Graeco-Roman Judeo-Christian tradition is no mere literary tic or bit of showing-off. It is his way of situating his economic theories in the context of Western Civilization, so that his readers will understand that he seeks to expose not merely the workings of the marketplace but the inner logic of the ideological rationalizations that have been elaborated to mystify and justify exploitation and oppression. But more of that later.
Immediately in the first paragraph of the next chapter, "The Buying and Selling of Labour-Power," Marx presents his solution to the puzzle, Whence profit? His solution, and the capstone to his entire critique of capitalism, is the distinction between labor and labor power -- between the work the laborer does for his wages, and the laborer's capacity for labor, which is what he rents out to the employer on an hourly, daily, or weekly contract.
Moneybags must be so lucky as to find, within the sphere of circulation, in the market, a commodity, whose use-value possesses the peculiar property of being a source of value, whose actual consumption, therefore, is itself an embodiment of labour, and, consequently, a creation of value. The possessor of money does find on the market such a special commodity in capacity for labour or labour-power.
In short, the capitalist pays for the labor of his worker a price equal to its cost of production, which is to say an amount in money just capable of paying for the food, clothing, and shelter that the worker must consume to produce his or her salable commodity, which is labor-power. Thus the worker is paid no more and no less than is paid to the coal seller, cloth seller, linen seller, and machine seller, each of whom sells his goods for a price exactly equal in value to the labor embodied in them.
Suppose, Marx says, by way of illustration, that the worker's necessaries require the expenditure of six hours of labor, directly or indirectly, somewhere in the economy, for their production. If the capitalist can get the worker to labor for more than six hours a day -- for twelve hours, let us say, which was in Marx's day the typical workday for a day laborer -- then the extra labor, the surplus labor, produced by the worker and embodied in the commodities she produces, will confer on those commodities a surplus value, which the capitalist will pocket when he exits the sphere of production and enters the sphere of circulation to sell his goods. This difference between the value embodied in labor and the cost of production of labor-power is, Marx says, the origin of profit. Notice, by the way, as the theologians, jurists, and philosophers supported by the largesse of the capitalist will be sure to explain, that the capitalist in thus enriching himself is not cheating either his workers or his customers. To his workers, the capitalist pays exactly the value of the labor-power he rents from them, namely an amount equal to the value embodied in that labor-power. And from his customers the capitalist demands only so much money as equals the value contained in what they are buying. It is for this reason that the capitalist can go to church each Sunday with a clear conscience, comfortable in the knowledge that he has fulfilled to the last jot and tittle the strictures of bourgeois justice. The little numerical example we carried through demonstrated with arithmetic rigor the correctness of Marx's claim.
If we now look a little more closely at this argument, we will discover that it does not actually prove what it is intended to prove. Marx seeks to demonstrate that profit is merely the monetary expression of the extra or surplus labor embodied by the workers in the produced output in the course of the production process. According to him, the value of the corn and iron, in our little model, is merely labor embodied in the corn and iron inputs and transferred from them to the output, without augmentation, during the production process. But the workers, when they labor in the factories or on the farms, create new value, for value is simply embodied labor. Now, we can distinguish between the worker's capacity to labor, or labor-power, and the actual labor performed by the worker. It is the worker's capacity for labor, or labor power, that the capitalist rents when he sets the worker to work in the factory or on the farm. And the difference between the value of that labor-power and the amount of new labor provided by the worker, that surplus labor, or surplus value [since value simply is embodied labor], is exactly, quantitatively, the capitalist's profit.
But look more closely at the labor value equations whose solution yielded the values, in our model, of λc, λi, and λb. Considered purely formally, without regard to the story Marx is telling us, what distinguishes labor from corn, iron, and books in those equations is the fact that the labor inputs are entered at full value -- 100 units of labor in the corn sector is entered as 100 units, where as all the other inputs -- of corn and iron -- are discounted by their labor values.
Suppose now we ask a question that neither Smith, Ricardo, nor Marx ever thought to ask: How much corn does it take, directly and indirectly, to produce one unit of iron, books, or labor? In short, what is the corn value of iron, books, or labor?
CORN VALUE?? What on earth am I talking about? There is no such thing as "corn value!" ["There is no crying in baseball."] Labor is the substance of value, not corn [or iron, for that matter.] But the claim that labor is the substance of value was not based on some sort of a priori intuition or philosophical argument. That claim was supposed by Smith and Ricardo and by Marx as well to be validated by the success that such a claim had in explaining the existence of natural prices [or, in modern jargon, equilibrium prices] in the marketplace.
Who would be crazy enough to ask such a question? Well, hem hem, the answer is, I would. I asked just this question in my 1981 article, "A Critique and Reinterpretation of Marx's Labor Theory of Value." How would we even go about calculating corn values? The answer is quite simple. Just set up equations on the basis of the Corn-Iron-Books model above, but introduce a new set of variables for corn values. [Greek does not have a letter "c," so we need to use the Greek letter k, or κ.] We use the same subscripts, but now κi will stand for "the quantity of corn directly and indirectly required for the production of one unit of iron." And we will, of course, have to introduce a variable κl to represent the quantity of corn directly and indirectly required for the production of one unit of labor. The variable κc in this set of equations will stand for the amount of corn directly and indirectly required to produce one unit of corn.
It should be intuitively obvious that every single conclusion we drew about labor values and surplus labor value and profits can be reproduced for corn values and surplus corn value and profits, without the need to introduce some sort of distinction between corn and corn power. If that is not intuitively obvious, then take it from me. I proved it [and I reproduced the proof in the Appendix to my book Understanding Marx.] The difference between the corn-value of a bushel of corn and a bushel of corn will be precisely the surplus corn value extracted from the corn inputs, and sure enough, the total surplus corn value extracted from the corn inputs will exactly equal the corn value of the physical surplus of labor, iron, and books.
How can we be sure that the corn value of corn will be less than 1, thus creating surplus corn value? Answer: So long as there is a physical surplus of some sort in the system, it is mathematically necessary that the quantity of any basic commodity required to produce one unit of that same commodity will be less than 1. [Basic commodities are commodities that are directly or indirectly required for the production of every commodity in the system.]
In short, by constructing our equations for the purpose, we can show that corn is exploited, or that iron is exploited, or that any other basic commodity is exploited. There is nothing special about labor. However, if we want corn values to be proportional to prices, we will have to construct a system that exhibits equal corn-organic composition of capital. Since the model we have been working with was carefully constructed by me to exhibit equal labor-organic composition of capital, not equal corn-organic composition of capital, it won't do.
So Marx was just plain wrong about the secret of the extraction of profit from the workers. BUT HE WAS NOT WRONG THAT PROFIT ORIGINATES IN THE EXPLOITATION OF THE WORKING CLASS. He just did not have the correct analysis. In fact, Marx knew perfectly well how and why it is that workers come to be exploited, and he told us at great length in Capital.
So what is the correct analysis? Funny you should ask. I will tell you tomorrow.