# Chapter 18 - Various Formula for the Rate of Surplus-Value

Economic Manuscripts: Capital Vol. I - Chapter Eighteen

Karl Marx. Capital Volume One

Chapter Eighteen: Various Formula for the Rate of Surplus-Value

 I. Surplus-value ( s ) = Surplus-value = Surplus-labour Variable Capital v Value of labour-power Necessary labour

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

 I. Surplus-value ( s ) = Surplus-value = Surplus-labour Variable Capital v Value of labour-power Necessary 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.

 II. Surplus-labour = Surplus-value = Surplus-product Working-day Value of the Product Total 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.

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.

 6 hours surplus-labour = Surplus-value of 3 sh. = 100% 6 hours necessary labour Variable Capital of 3 sh.

From formulae II. we get very differently,

 6 hours surplus-labour = 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 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%.  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-labour or Surplus-value 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 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  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%.

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 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. 

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

</table> 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 Working-day of 12 hours
 Surplus-labour of 6 hours = 100 Necessary labour of 6 hours 100
There is a third formula which I have occasionally already anticipated; it is
 III. Surplus-value = Surplus-labour = Unpaid labour Value of labour-power Necessary labour Paid labour
After the investigations we have given above, it is no longer possible to be misled, by the formula </table> into concluding, that the capitalist pays for labour and not for labour-power. This formula is only a popular expression for
 Unpaid labour, Paid labour