Capital III, Chapter 3 - Part 3

Marx then sets out a series of examples of how variations of these factors influence the rate of profit. Marx says they can be considered as changes in one and the same capital, or more realistically comparisons between different capitals. In fact, most of this chapter was reworked by Engels with the help of Samuel Moore, to try to rationalise the various calculations. Engels admits it was a difficult task. In fact, it is impossible to keep to all of the constraints, set out earlier, and logically achieve the various changes described in the examples. On the one hand, the examples can be seen as a way of observing the various effects separate from their other implications, but, in reality, it just really obscures important points of analysis. The simple solution is what Marx says here, to consider the various examples as comparisons between completely separate capitals rather than changes within a particular capital.

I) s' constant, v/C variable 

This first example looks at two capitals C and C', where the rate of surplus value is the same, in each, but the amount of variable capital, relative to the total capital, v/C and v'/C' is different.

In turn, the proportional relation of capital C to C', C:C', can be represented as a fraction C'/C, which can be called E, whilst similarly, v'/v can be called e.

So, if C'/C = E, using the normal mathematical rule, that equality is maintained by doing the same thing to both sides, if we multiply both sides by C, then C'/C x C = C', and E x C = EC, so C' = EC.

By the same method, v' = ev.

Earlier, we saw that the rate of profit is also equal to the rate of surplus value, s', multiplied by v/C. Now, we can use these new variables to show the rate of profit, of C', is equal to the rate of surplus value multiplied by v'/C', which is also equal to ev/EC, meaning it is related to the proportion of v' to v, e, and C' to C, E.

Some numbers might make this easier to understand.

Firm 1 (C).

c 100 + v 100 + s 100; s' = 100%, p' = 50%.

Firm 2 (C')

c 150 + v 50 + s 50; s' = 100%, p' = 25%.

E = C'/C = 200/200 = 1.

e = v'/v = 50/100 = 0.5

p' (25%) = s' (100%) x v'/C' (50/200) = 100% x 1/4 = 25%.

which is also,

p' (25%) = s' (100%) x ev/EC.

e = 0.5 x v = 100 = 50

E = 1 x C = 200 = 200

So, p' (25%) = s' (100%) x 50/200 = p' (25%) = s' (100%) x 1/4 = 25%.

We can then relate the rate of profit of firm 1, p' to the rate of profit of firm 2, p1', as a proportion.

The rate of profit is equal to s' v/C. For the second firm it is s1' v'/C'.

So, p':p1' is the same thing as saying s' v/C:s' v'/C'.

But, we can then get rid of s' at it appears on both sides of the proportion. That leaves us with v/C:v'/C'.

So, all these proportions are the same.

p':p1' = s'v/C:s'v'/C' = v/C:v'/C'

Using the numbers from above.

p'(50%):p1'(25%) = 2:1

s'v/C = 100 x 100/200 = 50:s1'v'/C' = 100 x 50/200 = 25 = 2:1

v/C = 100/200 = 0.5:50/200 = 0.25 = 2:1

To put it simply into words, if the rate of surplus value remains constant, then the rate of profit will change proportionately to the proportion of variable capital to total capital (c+v). If the proportion of variable capital to total capital doubles, the rate of profit doubles, if it halves the rate of profit halves.

The importance of this will be seen later in considering the Law of the Tendency for the Rate of Profit to Fall. A warning should be made ahead of that, which is that we have assumed here that no other changes occur, for example, in productivity, or in the rate of surplus value, accompanying the variation of v to C. As we will see later, in fact, such a change of v to C is inextricably linked to such changes in productivity, and all of the implications which go with it.

In the above example, v was 100 and C 200. So, v/C = 1/2. Similarly, v' was 50 and C 200, giving v'/C' = 50/200 = 1/4. We can convert these to percentages by multiplying by 100. So, 1/2 x 100 = 50%, and 1/4 x 100 = 25%.

So, now we could simply replace v/C and v'/C' with these percentages so 50:25, in which case we would have p':p1' = v:v' = 2:1.

These two formulas p1' = s'ev/EC and p':p1'=v:v' cover all the possible variations of v/C.

Because the rate of profit and rate of surplus value are expressed as percentages, it is also easier to express c and v as percentages. In other words, to reduce the total capital (c+v) to 100, and then express c and v as a percentage of it.

“For the determination of the rate of profit, if not of the amount, it is immaterial whether we say that a capital of 15,000, of which 12,000 is constant and 3,000 is variable, produces a surplus-value of 3,000, or whether we reduce this capital to percentages:

15,000 C = 12,000c + 3,000v ( + 3,000s) 

100 C = 80c + 20v ( + 20s). 

In either case the rate of surplus-value s' = 100%, and the rate of profit = 20%.” (p 54) 

This is useful for comparing two different capitals, but not changes in the same capital as it tends to obscure the nature of the change. Sometimes, we need to know what has happened to the actual values of c,v and s to get a proper picture.

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