In the first case, where surplus labour-time is increased by a lengthening of the working day, this gives a much more extended working-day, because the rate of surplus value was already taken as high. A much lower rate of surplus value would have given a different result. For example,
v 8 + s 2 x 10 workers = 80 v + 20 s = 100 hours.
To double s,
v 8 + s 4 x 5 workers = 40 v + 20 s = 60 hours.
The total social working day falls much more here, because the working day only needs to be extended by a small amount, to double the surplus labour, and compensate for the reduced workforce.
So, Marx is wrong on several counts. He is wrong that the total quantity of products would remain unchanged, if the total surplus value remains the same. He is wrong that the total expended on wages falls in half, if surplus value rises, because of a rise in relative surplus value. Wages may fall much more than that, i.e. by 80%, in the example above (Part 52). He is also wrong that the quantity of raw material processed remains the same, because in both cases, the length of the social working day is reduced, so that less is produced and less material is thereby needed.
In order to obtain this final conclusion, Marx assumes that the length of working day remains constant, but half the workers produce the same level of output as before, as a result of the introduction of additional fixed capital. On this basis, total output remains constant, and workers are paid the same wage, but with half the workers, this amounts to half the total wage bill. But, unless we take it that this is just one firm taking advantage of introducing this fixed capital, its not at all clear that this would result in the additional surplus value per worker that Marx seeks, for the reasons he has set out elsewhere. Moreover, the additional fixed capital employed so as to obtain this result, itself requires an increase in the value of constant capital.
The fact that 5 workers working a 10 hour day now produce the same 1000 units of output that previously 10 workers, working a 10 hour day, produced, previously, means that the previous value of these 1000 units (100 hours) has fallen to 50 hours, as a result of the rise in productivity. Suppose that the workers are producing linen. They work a 10 hour day, of which 4 hours is necessary labour and six hours surplus labour. Say the product of an hour's labour is equal to £1, so the workers are paid £4 in wages and produce £6 of surplus value. If there are 10 workers that is a total product value of £100, which we will say is represented by 1000 metres of linen.
In that case, 400 metres of linen are required to cover wages and 600 metres constitute surplus value. If then the number of workers falls to 5, but they continue to produce the same quantity of linen, its value must fall, because it now represents only 50 hours of labour, whereas previously it represented 100 hours of labour. Each worker still requires £4 as wages, but with only 5 workers to pay rather than 10, wages fall to £20. But, £20 now represents more linen.
Previously, 1000 metres of linen represented 100 hours, equals £100, but now represents only £50. To cover the £20 of wages, for the 5 workers, now requires 400 metres of linen to be sold, leaving a surplus product of 600 metres, with a value of £30. In fact, therefore, the rate of surplus value here remains unchanged at 150%, because s' remains constant, whilst v is halved, therefore, s falls from £60 to £30. Marx seems to have fallen into the trap he had initially criticised, of confusing the value of labour-power with the value of the product of that labour-power.
Marx's conclusion from his example, that the rate of profit would rise, is also false, because his assumption that the surplus value remains constant is itself false. In fact, because the surplus value would halve, in line with v, the rate of profit would fall, because c would remain constant so s/c+v would fall.
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