## Various Formula for the Rate of Surplus-Value

Marx sets out the formulas he has previously derived, for measuring the Rate of Surplus Value and compares these with those used by the Classical Political Economists, which give a false calculation of the rate of exploitation.
Marx’s formula:

 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.” (p 497)

The formulas of Classical Political Economy.

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

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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.
On the basis of “II” the Rate of Surplus Value could never equal 100%, because that would equal the entire working day, leaving no time for necessary labour. But, if necessary labour sank to zero, so would surplus labour, because it is 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%.” (p 498-9)

This distinction is important, not just because it understates the degree to which the worker is exploited, but also because it acts to disguise the real social relation. Production under capitalism is necessarily co-operative. That is not just the case in relation to the fact of co-operative labour, but also the fact of a necessary co-operation between capital and labour – capital provides the means of production and labour provides the labour-power for their transformation. Bourgeois ideologists use this to present capitalist production as based on a free association between labour and capital for their mutual benefit. On this basis, the working day, the total product, and its value is then amicably divided between labour and capital as compensation for what both have contributed to the process.

Marx writes,

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’Esprit d’Association dans tous les intérêts de la communauté". Paris 1818. H. Carey, the Yankee, occasionally performs this conjuring trick with like success, even with the relations resulting from slavery.” (Note 2, p 499)

But, as Marx’s analysis has demonstrated, this is not at all an accurate picture of the real nature of capitalist production, or the relation of capital and labour. Labour does not enter a voluntary association with capital, but does so only because it has itself been stripped of the means of production. It has to sell its labour-power to capital as a commodity, in order to live. As capitalist production develops, even the specific nature of that labour-power is stripped away, as the artisan is first turned into the detailed worker, under the division of labour during manufacture, and then into the factory worker under modern machine industry.

The worker is first made formally subject to capital, but, at this stage, they could theoretically still sell the product of their labour, rather than their labour-power, if they could acquire means of production. Then, they are made really subject to capital, when their labour-power is reduced to that of factory labour, which can only ever be sold to the owners of factories!
There is nothing in reality free about this association, which is why the workers have to agree to hand over a part of the product of their labour to capital, without capital having paid anything for it. As Marx points out, in Value, Price & Profit

As to the limits of the value of labour, its actual settlement always depends upon supply and demand, I mean the demand for labour on the part of capital, and the supply of labour by the working men. In colonial countries the law of supply and demand favours the working man. Hence the relatively high standard of wages in the United States. Capital may there try its utmost. It cannot prevent the labour market from being continuously emptied by the continuous conversion of wages labourers into independent, self-sustaining peasants. The position of a wages labourer is for a very large part of the American people but a probational state, which they are sure to leave within a longer or shorter term. To mend this colonial state of things the paternal British Government accepted for some time what is called the modern colonization theory, which consists in putting an artificial high price upon colonial land, in order to prevent the too quick conversion of the wages labourer into the independent peasant.”

This relation is described in the formula:

 III. Surplus-value = Surplus-labour = Unpaid labour Value of labour-power Necessary labour Paid labour

But, as Marx points out, the danger of this formula is that it can lead to the false conclusion that Capital pays for labour not labour-power, and that it does not pay the full price for that labour-power.

As Marx has demonstrated, this is wrong. Capital purchases labour-power not labour. It pays the full price of that labour-power as a commodity like any other, and whose use value is its ability to perform useful labour. The worker then supplies their commodity as part of the exchange. They do so for a given number of hours determined by the normal working day. This working-day is divided into two parts. In one part, the new value created by the worker is equal to the value of the workers' labour power, and paid to them as wages. In the second part, the worker creates new value that is additional to what is required for their own reproduction, and which thereby constitutes a surplus value, which is appropriated by the capitalist.

This expenditure of labour-power comes to him gratis. In this sense it is that surplus-labour can be called unpaid labour.

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.” (p 500)

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