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K.16
Formeln für die Rate des Mehrwerts
Chapter 15 varied the conditions determining surplus-value. Chapter 16 now fixes the formula for measuring its rate, separating exploitation from both a whole-day share and the profit-rate denominator.
Man hat gesehn, daß die Rate des Mehrwerts sich darstellt in den Formeln:
Formulae for the rate

We have seen that the rate of surplus-value is represented by the following formulae.

I.
Mehrwert
/
Variables Kapital
(
m
/
v
)
=
Mehrwert
/
Wert der Arbeitskraft
=
Mehrarbeit
/
Notwendige Arbeit
.
I.Surplus-value(s)=Surplus-value=Surplus-labour
Variable CapitalvValue of labour-powerNecessary labour
Die zwei ersten Formen stellen als Verhältnis von Werten dar, was die dritte als Verhältnis der Zeiten, worin diese Werte produziert werden. Diese einander ersetzenden Formeln sind begrifflich streng. Man findet sie daher wohl der Sache nach, aber nicht bewußt ausgearbeitet in der klassischen politischen Ökonomie. Hier begegnen wir dagegen den folgenden abgleiteten Formeln:
Formula I, strictly understood

The first two forms of Formula I express as value ratios what the third expresses as a ratio of the times in which those values are produced. The three interchangeable formulae are conceptually strict. Classical political economy therefore contains them in substance, though not as a consciously worked-out account; there we instead meet the derivative formulae that follow.

II.
Mehrarbeit
/
Arbeitstag
<In der autorisierten französischen Ausgabe setzt Marx diese erste Formel in Klammern, "weil sich der Begriff der Mehrarbeit in der bürgerlichen politischen Ökonomie nicht klar ausgedrückt findet".>
=
Mehrwert
/
Produktenwert
=
Mehrprodukt
/
Gesamtprodukt
.
II.Surplus-labour =Surplus-value=Surplus-product
Working-day Value of the ProductTotal Product
Eine und dieselbe Proportion ist hier abwechselnd ausgedrückt in der Form der Arbeitszeiten, der Werte, worin sie sich verkörpern, der Produkte, worin diese Werte existieren. Es wird natürlich unterstellt, daß unter Wert des Produkts nur das Wertprodukt des Arbeitstags zu verstehn, der konstante Teil des Produktenwerts aber ausgeschlossen ist.
Three forms, one proportion

In the authorized French edition, Marx brackets Formula II's first ratio because bourgeois political economy does not express the concept of surplus-labour clearly. One and the same proportion is expressed here in turn as labour-times, as the values embodied in those times, and as the products in which those values exist. “Value of the product” is understood to mean only the value-product newly created in the working day. The constant part of the total product-value is excluded.

In allen diesen Formeln ist der wirkliche Exploitationsgrad der Arbeit oder die Rate des Mehrwerts falsch ausgedrückt. Der Arbeitstag sei 12 Stunden. Mit den andren Annahmen unsres früheren Beispiels stellt sich in diesem Fall der wirkliche Exploitationsgrad der Arbeit dar in den Proportionen:
The false expression begins

In all Formula II’s forms, the actual degree of exploitation of labour — the rate of surplus-value — is falsely expressed. Let the working day be twelve hours. Under the assumptions of the earlier example, the real degree of exploitation is represented by the proportions that follow.

6 Stunden Mehrarbeit
/
6 Stunden notwendige Arbeit
=
Mehrwert von 3 sh.
/
Variables Kapital von 3 sh.
= 100%.
6 hours surplus-labour=Surplus-value of 3 sh.= 100%
6 hours necessary labourVariable Capital of 3 sh.
Nach den Formeln II erhalten wir dagegen:
Formula II’s different result

Formula II, however, gives a very different result.

6 Stunden Mehrarbeit
/
Arbeitstag von 12 Stunden
=
Mehrwert von 3 sh.
/
Wertprodukt von 6 sh.
= 50%.
6 hours surplus-labour=Surplus-value of 3 sh.= 50%
Working-day of 12 hoursValue created of 6 sh.
Diese abgeleiteten Formeln drücken in der Tat die Proportion aus, worin der Arbeitstag oder sein Wertprodukt sich zwischen Kapitalist und Arbeiter teilt. Gelten sie daher als unmittelbare Ausdrücke des Selbstverwertungsgrades des Kapitals, so gilt das falsche Gesetz: Die Mehrarbeit oder der Mehrwert kann nie 100% erreichen.17 Da die Mehrarbeit stets nur einen aliquoten Teil des Arbeitstags oder der Mehrwert stets nur einen aliquoten Teil des Wertprodukts bilden kann, ist die Mehrarbeit notwendigerweise stets kleiner als der Arbeitstag oder der Mehrwert stets kleiner als das Wertprodukt. Um sich zu verhalten wie 100 / 100 , müßten sie aber gleich sein. Damit die Mehrarbeit den ganzen Arbeitstag absorbiere (es handelt sich hier um den Durchschnittstag der Arbeitswoche, des Arbeitsjahrs usw.), müßte die notwendige Arbeit auf Null sinken. Verschwindet aber die notwendige Arbeit, so verschwindet auch die Mehrarbeit, da letztre nur eine Funktion der ersten. Die Proportion Mehrarbeit / Arbeitstag = Mehrwert / Wertprodukt kann also niemals die Grenze 100 / 100 erreichen und noch weniger auf 100+x / 100 steigen. Wohl aber die Rate des Mehrwerts oder der wirkliche Exploitationsgrad der Arbeit. Nimm z.B. die Schätzung des Herrn L. de Lavergne, wonach der englische Ackerbauarbeiter nur 1 / 4 , der Kapitalist (Pächter) dagegen 3 / 4 des Produkts18 oder seines Werts erhält, wie die Beute sich immer zwischen Kapitalist und Grundeigentümer usw. nachträglich weiter verteilt. Die Mehrarbeit des englischen Landarbeiters verhält sich danach zu seiner notwendigen Arbeit = 3 : 1, ein Prozentsatz der Exploitation von 300%.
A share is not exploitation

These derivative formulae do in fact express the proportion in which the working day, or its value-product, is divided between capitalist and worker. But if they are treated as direct expressions of capital’s degree of self-valorization, they yield a false law: surplus-labour or surplus-value could never reach 100%. Since surplus-labour is always an aliquot part of the working day, and surplus-value an aliquot part of the value-product, each is smaller than its whole. To stand at 100/100, numerator and denominator would have to be equal. For surplus-labour to absorb the whole day — the average day of the week or year is at issue — necessary labour would have to sink to zero. Yet if necessary labour disappears, surplus-labour disappears as well, because it is a function of it. Formula II can therefore never reach 100/100, much less exceed it. The attached note cites Rodbertus's third letter to Kirchmann: despite its false rent theory, it saw through the nature of capitalist production. Engels's addition to the third German edition qualifies Marx's praise after the later Rodbertus letters. There the capitalist's activity becomes a delegated economic and political function, profit a salary-form that may be regulated if it takes too much from wages, and Marx's book a polemic against the present form of capital supposedly confused with capital as such. Engels says Rodbertus's genuinely bold beginnings run aground in these ideological commonplaces. The rate of surplus-value, the actual degree of exploitation, can do both. Lavergne estimates that the English agricultural labourer receives one quarter and the farmer three quarters of the product or its value, whatever later division occurs among capitalist, landlord, and others. The calculation leaves out the part replacing constant capital. On this estimate surplus-labour stands to necessary labour as 3:1: exploitation is 300%. The attached note adds that Lavergne, a blind admirer of England, is likelier to have put the capitalist’s share too low than too high.

Die Schulmethode, den Arbeitstag als konstante Größe zu behandeln, wurde durch Anwendung der Formeln II befestigt, weil man hier die Mehrarbeit stets mit einem Arbeitstag von gegebner Größe vergleicht. Ebenso, wenn die Teilung des Wertprodukts ausschließlich ins Auge gefaßt wird. Der Arbeitstag, der sich bereits in einem Wertprodukt vergegenständlicht hat, ist stets ein Arbeitstag von gegebenen Grenzen.
The fixed-day habit

Formula II reinforced the school habit of treating the working day as a constant magnitude, because it always compares surplus-labour with a day of given size. The same is true when attention is confined to the division of the value-product: once the working day has been realized in a value-product, it necessarily appears as a day with fixed limits.

Die Darstellung von Mehrwert und Wert der Arbeitskraft als Bruchteilen des Wertprodukts - eine Darstellungsweise, die übrigens aus der kapitalistischen Produktionsweise selbst erwächst und deren Bedeutung sich später erschließen wird - versteckt den spezifischen Charakter des Kapitalverhältnisses, nämlich den Austausch des variablen Kapitals mit der lebendigen Arbeitskraft und den entsprechenden Ausschluß des Arbeiters vom Produkt. An die Stelle tritt der falsche Schein eines Assoziationsverhältnisses, worin Arbeiter und Kapitalist das Produkt nach dem Verhältnis seiner verschiednen Bildungsfaktoren teilen.19
The fraction conceals capital

Representing surplus-value and labour-power’s value as fractions of the value-product — a form arising from capitalist production itself, whose significance is disclosed later — conceals the specific capital relation. It hides the exchange of variable capital for living labour-power and the corresponding exclusion of the worker from the product. In its place comes the false appearance of an association in which worker and capitalist divide the product according to the different factors entering its formation. The attached note shows how easily developed capitalist cooperation can be stripped of its antagonistic character and retold as free association: Laborde does so explicitly, and Carey can perform the same trick even for slavery.

Übrigens sind die Formeln II stets in die Formeln I rückverwandelbar. Haben wir z.B.
Returning to Formula I

Formula II can always be reconverted into Formula I. For example, take the proportion that follows.

Mehrarbeit von 6 Stunden
/
Arbeitstag von 12 Stunden
, so ist die notwendige Arbeitszeit = Arbeitstag von zwölf Stunden minus Mehrarbeit von sechs Stunden, und so ergibt sich:
Mehrarbeit von 6 Stunden
/
Notwendige Arbeit von 6 Stunden
=
100
/
100
.
Surplus-labour of 6 hours
Working-day of 12 hours
then the necessary labour-time being 12 hours less the surplus-labour of
6 hours, we get the following result,
Surplus-labour of 6 hours=100
Necessary labour of 6 hours100
Eine dritte Formel, die ich gelegentlich schon antizipiert habe, ist:
A third formula

There is a third formula, already anticipated on occasion.

III.
Mehrwert
/
Wert der Arbeitskraft
=
Mehrarbeit
/
Notwendige Arbeit
=
Unbezahlte Arbeit
/
Bezahlte Arbeit
.
III.Surplus-value=Surplus-labour=Unpaid labour
Value of labour-powerNecessary labourPaid labour
Das Mißverständnis, wozu die Formel
The popular wording’s trap

The misunderstanding invited by the formula “unpaid labour / paid labour” — that the capitalist pays for labour rather than labour-power — is removed by the earlier argument. The formula is only a popular expression for “surplus-labour / necessary labour.”

Unbezahlte Arbeit
/
Bezahlte Arbeit
verleiten könnte, als zahle der Kapitalist die Arbeit und nicht die Arbeitskraft, fällt nach der früher gegebenen Entwicklung fort.
Unbezahlte Arbeit
/
Bezahlte Arbeit
ist nur populärer Ausdruck für
Mehrarbeit
/
Notwendige Arbeit
Unpaid labour,
Paid labour
into concluding, that the capitalist pays for labour and not for labour-power.
This formula is only a popular expression for
Surplus-labour,
Necessary labour
. Der Kapitalist zahlt den Wert, resp. davon abweichenden Preis der Arbeitskraft und erhält im Austausch die Verfügung über die lebendige Arbeitskraft selbst. Seine Nutznießung dieser Arbeitskraft zerfällt in zwei Perioden. Während der einen Periode produziert der Arbeiter nur einen Wert = Wert seiner Arbeitskraft, also nur ein Äquivalent. Für den vorgeschoßnen Preis der Arbeitskraft erhält so der Kapitalist ein Produkt vom selben Preis. Es ist, als ob er das Produkt fertig auf dem Markt gekauft hätte. In der Periode der Mehrarbeit dagegen bildet die Nutznießung der Arbeitskraft Wert für den Kapitalisten, ohne ihm einen Wertersatz zu kosten.20 Er hat diese Flüssigmachung der Arbeitskraft umsonst. In diesem Sinn kann die Mehrarbeit unbezahlte Arbeit heißen.
Purchased power, two periods

The capitalist pays the value of labour-power, or its price where price diverges from value, and receives in exchange disposal over living labour-power itself. Its use divides into two periods. During one, the worker produces only a value equal to labour-power’s value: an equivalent. In return for the price advanced for labour-power, the capitalist receives a product of the same price, as if it had been bought ready-made on the market. During surplus-labour, however, use of labour-power forms value for the capitalist without costing a further value-equivalent. This expenditure of labour-power comes to him gratis. In this sense, surplus-labour may be called unpaid labour. The attached note recalls the Physiocrats’ recognition that surplus-value was a disposable wealth the possessor had not bought and yet sold.

Das Kapital ist also nicht nur Kommando über Arbeit, wie A. Smith sagt. Es ist wesentlich Kommando über unbezahlte Arbeit. Aller Mehrwert, in welcher besondern Gestalt von Profit, Zins, Rente usw. er sich später kristallisiere, ist seiner Substanz nach Materiatur unbezahlter Arbeitszeit. Das Geheimnis von der Selbstverwertung des Kapitals löst sich auf in seine Verfügung über ein bestimmtes Quantum unbezahlter fremder Arbeit.
Command over unpaid labour

Capital is therefore not only command over labour, as Adam Smith says. It is essentially command over unpaid labour. All surplus-value, whatever particular form it later crystallizes into — profit, interest, rent, and so on — is in substance materialized unpaid labour-time. The secret of capital’s self-valorization resolves into its disposal over a definite quantity of other people’s unpaid labour.

K.16
Formulae for the Rate of Surplus-Value
Chapter 15 varied the conditions determining surplus-value. Chapter 16 now fixes the formula for measuring its rate, separating exploitation from both a whole-day share and the profit-rate denominator.
We have seen that the rate of surplus-value is represented by the following formulae:
Formulae for the rate

We have seen that the rate of surplus-value is represented by the following formulae.

I.Surplus-value(s)=Surplus-value=Surplus-labour
Variable CapitalvValue of labour-powerNecessary labour
I.Surplus-value(s)=Surplus-value=Surplus-labour
Variable CapitalvValue of labour-powerNecessary labour
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.
Formula I, strictly understood

The first two forms of Formula I express as value ratios what the third expresses as a ratio of the times in which those values are produced. The three interchangeable formulae are conceptually strict. Classical political economy therefore contains them in substance, though not as a consciously worked-out account; there we instead meet the derivative formulae that follow.

II.Surplus-labour =Surplus-value=Surplus-product
Working-day Value of the ProductTotal Product
II.Surplus-labour =Surplus-value=Surplus-product
Working-day Value of the ProductTotal Product
One and the same ratio is here expressed as a ratio of labour-times, of the values in which those labour-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.
Three forms, one proportion

In the authorized French edition, Marx brackets Formula II's first ratio because bourgeois political economy does not express the concept of surplus-labour clearly. One and the same proportion is expressed here in turn as labour-times, as the values embodied in those times, and as the products in which those values exist. “Value of the product” is understood to mean only the value-product newly created in the working day. The constant part of the total product-value is excluded.

In all of these formulae (II.), the actual degree of exploitation of labour, 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 labour will be represented in the following proportions.
The false expression begins

In all Formula II’s forms, the actual degree of exploitation of labour — the rate of surplus-value — is falsely expressed. Let the working day be twelve hours. Under the assumptions of the earlier example, the real degree of exploitation is represented by the proportions that follow.

6 hours surplus-labour=Surplus-value of 3 sh.= 100%
6 hours necessary labourVariable Capital of 3 sh.
6 hours surplus-labour=Surplus-value of 3 sh.= 100%
6 hours necessary labourVariable Capital of 3 sh.
From formulae II. we get very differently,
Formula II’s different result

Formula II, however, gives a very different result.

6 hours surplus-labour=Surplus-value of 3 sh.= 50%
Working-day of 12 hoursValue created of 6 sh.
6 hours surplus-labour=Surplus-value of 3 sh.= 50%
Working-day of 12 hoursValue 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 labourer. 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-labour or surplus-value can never reach 100%. 1 Since the surplus-labour is only an aliquot part of the working-day, or since surplus-value is only an aliquot part of the value created, the surplus-labour 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-labour may absorb the whole day (i.e., an average day of any week or year), the necessary labour must sink to zero. But if the necessary labour vanish, so too does the surplus-labour, since it is only a function of the former. The ratio
Surplus-labourorSurplus-value
Working-dayValue 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 labour. Take, e.g., the estimate of L. de Lavergne, according to which the English agricultural labourer gets only 1/4, the capitalist (farmer) on the other hand 3/4 of the product 2 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-labour of the English agricultural labourer is to his necessary labour as 3:1, which gives a rate of exploitation of 300%.
A share is not exploitation

These derivative formulae do in fact express the proportion in which the working day, or its value-product, is divided between capitalist and worker. But if they are treated as direct expressions of capital’s degree of self-valorization, they yield a false law: surplus-labour or surplus-value could never reach 100%. Since surplus-labour is always an aliquot part of the working day, and surplus-value an aliquot part of the value-product, each is smaller than its whole. To stand at 100/100, numerator and denominator would have to be equal. For surplus-labour to absorb the whole day — the average day of the week or year is at issue — necessary labour would have to sink to zero. Yet if necessary labour disappears, surplus-labour disappears as well, because it is a function of it. Formula II can therefore never reach 100/100, much less exceed it. The attached note cites Rodbertus's third letter to Kirchmann: despite its false rent theory, it saw through the nature of capitalist production. Engels's addition to the third German edition qualifies Marx's praise after the later Rodbertus letters. There the capitalist's activity becomes a delegated economic and political function, profit a salary-form that may be regulated if it takes too much from wages, and Marx's book a polemic against the present form of capital supposedly confused with capital as such. Engels says Rodbertus's genuinely bold beginnings run aground in these ideological commonplaces. The rate of surplus-value, the actual degree of exploitation, can do both. Lavergne estimates that the English agricultural labourer receives one quarter and the farmer three quarters of the product or its value, whatever later division occurs among capitalist, landlord, and others. The calculation leaves out the part replacing constant capital. On this estimate surplus-labour stands to necessary labour as 3:1: exploitation is 300%. The attached note adds that Lavergne, a blind admirer of England, is likelier to have put the capitalist’s share too low than too high.

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-labour is always compared with a working-day of given length. The same holds good when the repartition of the value produced is exclusively kept in sight. The working-day that has already been realized in given value, must necessarily be a day of given length.
The fixed-day habit

Formula II reinforced the school habit of treating the working day as a constant magnitude, because it always compares surplus-labour with a day of given size. The same is true when attention is confined to the division of the value-product: once the working day has been realized in a value-product, it necessarily appears as a day with fixed limits.

The habit of representing surplus-value and value of labour-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 labour-power, and the consequent exclusion of the labourer from the product. Instead of the real fact, we have false semblance of an association, in which labourer and capitalist divide the product in proportion to the different elements which they respectively contribute towards its formation. 3
The fraction conceals capital

Representing surplus-value and labour-power’s value as fractions of the value-product — a form arising from capitalist production itself, whose significance is disclosed later — conceals the specific capital relation. It hides the exchange of variable capital for living labour-power and the corresponding exclusion of the worker from the product. In its place comes the false appearance of an association in which worker and capitalist divide the product according to the different factors entering its formation. The attached note shows how easily developed capitalist cooperation can be stripped of its antagonistic character and retold as free association: Laborde does so explicitly, and Carey can perform the same trick even for slavery.

Moreover, the formulae II. can at any time be reconverted into formulae I. If, for instance, we have
Returning to Formula I

Formula II can always be reconverted into Formula I. For example, take the proportion that follows.

Surplus-labour of 6 hours
Working-day of 12 hours
then the necessary labour-time being 12 hours less the surplus-labour of
6 hours, we get the following result,
Surplus-labour of 6 hours=100
Necessary labour of 6 hours100
Surplus-labour of 6 hours
Working-day of 12 hours
then the necessary labour-time being 12 hours less the surplus-labour of
6 hours, we get the following result,
Surplus-labour of 6 hours=100
Necessary labour of 6 hours100
There is a third formula which I have occasionally already anticipated; it is
A third formula

There is a third formula, already anticipated on occasion.

III.Surplus-value=Surplus-labour=Unpaid labour
Value of labour-powerNecessary labourPaid labour
III.Surplus-value=Surplus-labour=Unpaid labour
Value of labour-powerNecessary labourPaid labour
After the investigations we have given above, it is no longer possible to be misled, by the formula
The popular wording’s trap

The misunderstanding invited by the formula “unpaid labour / paid labour” — that the capitalist pays for labour rather than labour-power — is removed by the earlier argument. The formula is only a popular expression for “surplus-labour / necessary labour.”

Unpaid labour,
Paid labour
into concluding, that the capitalist pays for labour and not for labour-power.
This formula is only a popular expression for
Surplus-labour,
Necessary labour
Unpaid labour,
Paid labour
into concluding, that the capitalist pays for labour and not for labour-power.
This formula is only a popular expression for
Surplus-labour,
Necessary labour
The capitalist pays the value, so far as price coincides with value, of the labour-power, and receives in exchange the disposal of the living labour-power itself. His usufruct is spread over two periods. During one the labourer produces a value that is only equal to the value of his labour-power; he produces its equivalent. Thus the capitalist receives in return for his advance of the price of the labour-power, a product ready made in the market. During the other period, the period of surplus-labour, the usufruct of the labour-power creates a value for the capitalist, that costs him no equivalent. 4 This expenditure of labour-power comes to him gratis. In this sense it is that surplus-labour can be called unpaid labour.
Purchased power, two periods

The capitalist pays the value of labour-power, or its price where price diverges from value, and receives in exchange disposal over living labour-power itself. Its use divides into two periods. During one, the worker produces only a value equal to labour-power’s value: an equivalent. In return for the price advanced for labour-power, the capitalist receives a product of the same price, as if it had been bought ready-made on the market. During surplus-labour, however, use of labour-power forms value for the capitalist without costing a further value-equivalent. This expenditure of labour-power comes to him gratis. In this sense, surplus-labour may be called unpaid labour. The attached note recalls the Physiocrats’ recognition that surplus-value was a disposable wealth the possessor had not bought and yet sold.

Capital, therefore, is not only, as Adam Smith says, the command over labour. It is essentially the command over unpaid labour. All surplus-value, whatever particular form (profit, interest, or rent), it may subsequently crystallize into, is in substance the materialization of unpaid labour. The secret of the self-expansion of capital resolves itself into having the disposal of a definite quantity of other people’s unpaid labour.
Command over unpaid labour

Capital is therefore not only command over labour, as Adam Smith says. It is essentially command over unpaid labour. All surplus-value, whatever particular form it later crystallizes into — profit, interest, rent, and so on — is in substance materialized unpaid labour-time. The secret of capital’s self-valorization resolves into its disposal over a definite quantity of other people’s unpaid labour.