Das Kapital/Chapter 18
Chapter 18: Various Formula for the Rate of Surplus-Value
We have seen that the rate of surplus-value is represented by the following formulae:
Surplus-value (s) Surplus-value Surplus-labor I. == == - Variable Capital (v) Value of labor-power Necessary labor
The two first of these formulae represent, as a ratio of values, that which, in the third, is represented as a ratio of the times during which those values are produced. These formulae, supplementary the one to the other, are rigorously definite and correct. We therefore find them substantially, but not consciously, worked out in classical Political Economy. There we meet with the following derivative formulae.
Surplus-labor Surplus-value Surplus-product II. - == == - Working-day Value of the Product Total Product
One and the same ratio is here expressed as a ratio of labor-times, of the values in which those labor-times are embodied, and of the products in which those values exist. It is of course understood that, by "Value of the Product", is meant only the value newly created in a working-day, the constant part of the value of the product being excluded.
In all of these formulae (II.), the actual degree of exploitation of labor, or the rate of surplus-value, is falsely expressed. Let the working-day be 12 hours. Then, making the same assumptions as in former instances, the real degree of exploitation of labor will be represented in the following proportions.
6 hours surplus-labor Surplus-value of 3 sh. - == - == 100%. 6 hours necessary labor Variable Capital of 3 sh.
From formulae II. we get very differently,
6 hours surplus-labor Surplus-value of 3 sh. - == - == 50%. Working-day of 12 hours Value created of 6 sh.
These derivative formulae express, in reality, only the proportion in which the working-day, or the value produced by it, is divided between capitalist and laborer. If they are to be treated as direct expressions of the degree of self-expansion of capital, the following erroneous law would hold good: Surplus-labor or surplus-value can never reach 100%.  Since the surplus-labor is only an aliquot part of the working-day, or since surplus-value is only an aliquot part of the value created, the surplus-labor must necessarily be always less than the working-day, or the surplus-value always less than the total value created. In order, however, to attain the ratio of 100:100 they must be equal. In order that the surplus-labor may absorb the whole day (i.e., an average dy of any week or year), the necessary labor must sink to zero. But if the necessary labor vanish, so too does the surplus-labor, since it is only a function of the former. The ratio
Surplus-labor Surplus-value - or - Working-day Value created
can therefore never reach the limit 100/100, still less rise to 100+x/100. But not so the rate of surplus-value, the real degree of exploitation of labor. Take, e.g., the estimate of L. de Lavergne, according to which the English agricultural laborer gets only 1/4, the capitalist (farmer) on the other hand 3/4 of the product  or its value, apart from the question of how the booty is subsequently divided between the capitalist, the landlord, and others. According to this, this surplus-labor of the English agricultural laborer is to his necessary labor as 3:1, which gives a rate of exploitation of 300%.
The favorite method of treating the working-day as constant in magnitude became, through the use of formulae II., a fixed usage, because in them surplus-labor is always compared with a working-day of given length. The same holds good when the repartition of the value produced is exclusively kept insight. The working-day that has already been realized in given value, must necessarily be a day of given length.
The habit of representing surplus-value and value of labor-power as fractions of the value created a habit that originates in the capitalist mode of production itself, and whose import will hereafter be disclosed conceals the very transaction that characterizes capital, namely the exchange of variable capital for living labor-power, and the consequent exclusion of the laborer from the product. Instead of the real fact, we have false semblance of an association, in which laborer and capitalist divide the product in proportion to the different elements which they respectively contribute towards its formation. 
Moreover, the formulae II. can at any time be reconverted into formulae I. If, for instance, we have
Surplus-labor of 6 hours , Working-day of 12 hours
then the necessary labor-time being 12 hours less the surplus-labor of 6 hours, we get the following result,
Surplus-labor of 6 hours 100 == -. Necessary labor of 6 hours 100
There is a third formula which I have occassionally already anticipated; it is
Surplus-value Surplus-labor Unpaid labor III. == - == . Value of labor-power Necessary labor Paid labor
After the investigations we have given above, it is no longer possible to be misled, by the formula
Unpaid labor , Paid labor
into concluding, that the capitalist pays for labor and not for labor-power. This formula is only a popular expression for
Surplus-labor -. Necessary labor
The capitalist pays the value, so far as price coincides with value, of the labor-power, and receives in exchange the disposal of the living labor-power itself. His usufruct is spread over two periods. During one the laborer produces a value that is only equal to the value of his labor-power; he produces its equivalent. This the capitalist receives in return for his advance of the price of the labor-power, a product ready made in the market. During the other period, the period of surplus-labor, the usufruct of the labor-power creates a value for the capitalist, that costs him no equivalent.  This expenditure of labor-power comes to him gratis. In this sense it is that surplus-labor can be called unpaid labor.
Capital, therefore, it not only, as Adam Smith says, the command over labor. It is essentially the command over unpaid labor. All surplus-value, whatever particular form (profit, interest, or rent), it may subsequently crystallize into, is in substance the materialization of unpaid labor. The secret of the self-expansion of capital resolves itself into having the disposal of a definite quantity of other people's unpaid labor.
^ Thus, e.g., in "Dritter Brief an v. Kirchmann von Rodbertus. Widerlegung der Ricardo'schen Lehre von der Grundrente und Begrundung einer neuen Rententheorie". Berlin, 1851. I shall return to this letter later on; in spite of its erroneous theory of rent, it sees through the nature of capitalist production.
NOTE ADDED IN THE 3RD GERMAN EDITION: It may be seen from this how favorably Marx judged his predecessors, whenever he found in them real progress, or new and sound ideas. The subsequent publications of Robertus' letters to Rud. Meyer has shown that the above acknowledgement by Marx wants restricting to some extent. In those letters this passage occurs:
"Capital must be rescued not only from labor, but from itself, and that will be best effected, by treating the acts of the industrial capitalist as economic and political functions, that have been delegated to him with his capital, and by treating his profit as a form of salary, because we still know no other social organization. But salaries may be regulated, and may also be reduced if they take too much from wages. The irruption of Marx into Society, as I may call his book, must be warded off.... Altogether, Marx's book is not so much an investigation into capital, as a polemic against the present form of capital, a form which he confounds with the concept itself of capital."
("Briefe, &c., von Dr. Robertus-Jagetzow, herausgg. von Dr. Rud. Meyer", Berlin, 1881, I, Bd. P.111, 46. Breif von Rodbertus.) To such ideological commonplaces did the bold attack by Robertus in his "social letters" finally dwindle down. F.E.
^ That part of the product which merely replaces the constant capital advanced is of course left out in this calculation. Mr. L. de Lavergne, a blind admirer of England, is inclined to estimate the share of the capitalist too low, rather than too high.
^ All well-developed forms of capitalist production being forms of cooperation, nothing is, of course, easier, than to make abstraction from their antagonistic character, and to transform them by a word into some form of free association, as is done by A. de Laborde in "De l'Espirit d'Association dans tout les interets de la communaute". Paris 1818. H. Carey, the Yankee, occassionally performs this conjuring trick with like success, even with the relations resulting from slavery.
^ Although the Physiocrats could not penetrate the mystery of surplus-value, yet this much was clear to them, viz., that it is "une richesse independante et disponible qu'il (the possessor) n'a point achetee et qu'il vend." (Turgot: "Reflexions sur la Formation et la Distribution des Richesses", p.11.)
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