Part Five — The Production of Absolute and Relative Surplus-Value Chapters 16–18. Productive labour and subsumption, the changing magnitudes of the price of labour-power and surplus-value, and the formulae for the rate of surplus-value.

This short Part draws together the two forms of surplus-value analysed separately in Parts Three and Four and refines the concepts built along the way. Chapter 16 relates absolute and relative surplus-value, redefines productive labour under capital, and introduces the distinction between the formal and the real subsumption of labour under capital. Chapter 17 sets out the three variables that govern the relative magnitudes of the price of labour-power and surplus-value — the length of the day, the intensity of labour, and the productivity of labour — and works through their combinations. Chapter 18 reviews the formulae for the rate of surplus-value, exposing how the classical formulas understate exploitation, and states the conclusion the whole theory of surplus-value has been driving at: capital is command over unpaid labour.

Marx first revises the concept of productive labour. In the simple labour process, any labour that makes a product is productive; but with the collective worker, the individual need no longer touch the product to be productive — it is enough to be an organ of the combined labourer. Under capital the concept also narrows in a decisive way: the only productive labour is labour that produces surplus-value. Marx presses the point with a deliberately jarring example and a mordant conclusion.

The only worker who is productive is one who produces surplus-value for the capitalist, or in other words contributes towards the self-valorization of capital. If we may take an example from outside the sphere of material production, a schoolmaster is a productive worker when, in addition to belabouring the heads of his pupils, he works himself into the ground to enrich the owner of the school. … The concept of a productive worker therefore implies not merely a relation between the activity of work and its useful effect, between the worker and the product of his work, but also a specifically social relation of production, a relation with a historical origin which stamps the worker as capital's direct means of valorization. To be a productive worker is therefore not a piece of luck, but a misfortune.

Marx, Capital I, page 644. The ellipsis drops the sentence noting that whether the capital is laid out in a teaching factory or a sausage factory makes no difference to the relation.

Absolute surplus-value (lengthening the day) is the general foundation; relative surplus-value (revolutionizing the mode of production to shorten necessary labour) is built on it. This distinction lets Marx introduce a periodization that his later readers have made central. Capital can take over a labour process in its inherited technical shape and merely command it — the formal subsumption of labour, sufficient for absolute surplus-value — before it transforms that process from within into a specifically capitalist one, the real subsumption.

It therefore requires a specifically capitalist mode of production, a mode of production which, along with its methods, means and conditions, arises and develops spontaneously on the basis of the formal subsumption [Subsumtion] of labour under capital. This formal subsumption is then replaced by a real subsumption.

Marx, Capital I, page 645.

Against the temptation to naturalize surplus labour, Marx insists it rests on a natural basis only in the loosest sense, and that the productivity which makes it possible is historical, not given.

Besides, the capital-relation arises out of an economic soil that is the product of a long process of development. The existing productivity of labour, from which it proceeds as its basis, is a gift, not of nature, but of a history embracing thousands of centuries.

Marx, Capital I, page 647.

Holding to the assumption that labour-power is paid at its value, Marx isolates the three variables whose combinations determine the relative magnitudes of surplus-value and the price of labour-power.

On these assumptions, we have already found that the relative magnitudes of surplus-value and of price of labour-power are determined by three circumstances: (1) the length of the working day, or the extensive magnitude of labour, (2) the normal intensity of labour, or its intensive magnitude, whereby a given quantity of labour is expended in a given time and (3) the productivity of labour, whereby the same quantity of labour yields, in a given time, a greater or a smaller quantity of the product, depending on the degree of development attained by the conditions of production.

Marx, Capital I, page 655.

Marx works through the cases in turn — varying productivity and intensity with the working day constant, varying the day with productivity and intensity constant, and varying all three at once — to show how, on the assumption of payment at value, a rise in productivity or intensity redistributes the fixed value-product between wages and surplus-value in capital's favour, and how the day's length and its two components move against each other. The chapter is the systematic accounting behind the distinction Chapter 16 drew; its value for later writing is as the ledger of how the three levers of exploitation interact.

The final chapter of the Part compares the ways the rate of surplus-value can be written. Marx's own formulae express it as surplus-value over variable capital, or equivalently as surplus labour over necessary labour — a ratio of values or of the times in which those values are produced. Classical political economy, he shows, used a derivative and misleading form, measuring surplus labour against the whole working day (or surplus-value against the total value-product), which mathematically understates the rate of exploitation, because the denominator is larger than the necessary-labour portion alone. A third form expresses the rate as unpaid labour over paid labour, and Marx warns that "paid" and "unpaid" labour are only popular shorthand: the capitalist pays for labour-power, not for labour, and receives the right to dispose of the living power for the day.

From this the chapter, and with it the whole analysis of surplus-value, reaches its summary formulation. Whatever the surface form profit later takes, its substance is unpaid labour-time appropriated through the wage relation.

Capital, therefore, is not only the command over labour, as Adam Smith thought. 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-time. The secret of the self-valorization of capital resolves itself into the fact that it has at its disposal a definite quantity of the unpaid labour of other people.

Marx, Capital I, page 672.