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.