If we have:
80 c + 20 v + 20 s, this represents 20 workers, working a 10 hour day, being paid £1 a day. If wages rise from £1 to £1.25 then 20 v must now represent only 16 workers (16 x £1.25 = £20). But, 16 workers will produce only 75% of the new value as 20 workers. Twenty workers produced 20 v + 20 s= 40, so 16 will produce only 32. But, if they are paid £20 that leaves only £12 as surplus value, down from £20.
The rate of surplus value falls from 100% to 60%. For the rate of surplus value to remain constant, the working-day would have to rise, or increase in intensity, by 25%. But, in that case, wages have really stayed the same. The same applies in reverse if the 20 v were to represent 30 not 20 workers. For the rate of surplus value to remain the same, the working day would have to fall from 10 to 6 ⅔ hours.
“We have already in the main discussed to what extent c may in these divergent examples remain unchanged in terms of value expressed in money and yet represent different quantities of means of production changed in accordance with changing conditions. In its pure form this case would be possible only by way of an exception.
As for a change in the value of the elements of c which increases or decreases their mass but leaves the sum of the value of c unchanged, it does not affect either the rate of profit or the rate of surplus-value, so long as it does not lead to a change in the magnitude of v.” (p 62-3)
II) s' variable
“We obtain a general formula for the rates of profit with different rates of surplus-value, no matter whether v/C remains constant or not, by converting the equation:
p' = s' (v/C)
into
p'1 = s'1 (v1/C1) ,
in which p'1, s'1, v1 and C1 denote the changed values of p', s', v and C. Then we have:
p' : p'1 = s'1 (v/C) : s'1 (v1/C1) ,
and hence:
p'1 = (s'1/s1) × v1/v × C/C1 × p'.” (p 63)
In other words, p' is the original rate of profit. It can also be written as s' v/C, so if we know the rate of surplus value, s', and we know v, the variable capital, and we know C, the total value of the capital, we can calculate what the rate of profit is.
By the same token, some different rate of profit can be calculated in exactly the same way, if we know the values of s', v, and C of this other, or changed, capital that bring about this other rate of profit. We can distinguish between the two by labelling this other capital with a “1”, i.e. p1', s1', v1, and C1. But, then, because they are exactly the same things, it is obvious that p' stands in exactly the same proportion to p1' as s' v/C stands to s1' v1/C1.
The last formula, p1' = s1'/s x v1/v x C/C1 x p' is just a mathematical manipulation of the above to show how the changed rate of profit is related to the old rate of profit, and changes in accordance with the changes in the rate of surplus value, value of variable capital, and the total capital.
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