## The Relation of the Rate of Profit to the Rate of Surplus-Value

At this stage of the analysis, Marx makes clear that the simplifying assumption is that the profit of an individual capital is equal to its surplus value, during a given period of circulation. The later analysis will deal with the reality that the surplus value is shared with money-capitalists, merchant-capitalists, landlords, and the state; that in the formation of a general rate of profit, competition will tend to result in the total social surplus being divided according to the size of capital; and that the actual rate of profit is modified by the rate of turnover of capital. Marx also assumes that the value of money remains constant.

The relations between the rate of surplus value and rate of profit, are reducible to several simple ratios, and so can be analysed via a number of formulas.

The following designations are used.

C = c+v

c = constant capital advanced for one turnover period

v = variable capital advanced for one turnover period

s = surplus value produced in one turnover period

s' = s/v = rate of surplus value

n = the number of turnovers of the variable capital in the year, as defined in Volume II

s'n = the annual surplus value

s'v = s

p = profit

p' = s/C = s/c+v = rate of profit for one turnover period.

“Now, substituting for s its equivalent s'v, we find

p' = s' (v/C) = s' v/(c + v)

which equation may also be expressed by the proportion

p' : s' = v : C ;

the rate of profit is related to the rate of surplus-value as the variable capital is to the total capital.” (p 50)

So, unless there is no constant capital at all, which never happens, the rate of profit, s/c+v, must always be smaller than the rate of surplus value, because v is always less than C (c+v); the rate of surplus value is a function of v; and the rate of profit a function of v + c = C; the ratio of the rate of surplus value to the rate of profit will be proportional to the relation of v to C.

For example,

(1) c 100 + v 100 + s 100.

C = 200, v/C = 50%, s' = s/v = 100/100 = 100%, p' = s/c+v = 100/200 = 50%

(2) c 150 = v 50 + s 50

C = 200, v/C = 25%, s' = s/v = 50/50 = 100%, p' = s/c+v = 50/200 = 25%.

So,

(1) v/C = 50%, p' = 50%, s' = 100%,

(2) v/C = 25%, p' = 25%, s' = 100%

or in both cases p':s' = v:C

=

(1) p' (50):s' (100) = v(100):C(200)

(2) p' (25):s' (100) = v(50):C(200)

=

(1) p'(1):s'(2) = v(1):C(2)

(2) p'(1):s'(4) = v(1):C(4)

Although the effect of the rate of turnover of capital is dealt with separately, later, Marx notes that the rate of profit covering several circulation periods, can be calculated by using s'n, the annual rate of surplus value, rather than s'.

As well as the assumption that the value of money remains constant, and that only one turnover is being considered, a third consideration is the consequences of productivity. An individual capital, that enjoys higher productivity, for whatever reason, will produce commodities that have a lower individual value than their social value, i.e. the labour-time that is actually required for their production is less than the average socially necessary labour-time, required for these commodities production. As a result, this firm will make higher profits, on its sale of these commodities, than other producers of the same commodity, producing at or below the average level of productivity.

As another simplifying assumption, it is taken that all commodities are produced under the average conditions.

“In effect, the value-composition of a capital invested in a branch of industry, that is, a certain proportion between the variable and constant capital, always expresses a definite degree of labour productivity. As soon, therefore, as this proportion is altered by means other than a mere change in the value of the material elements of the constant capital, or a change in wages, the productivity of labour must likewise undergo a corresponding change, and we shall often enough see, for this reason, that changes in the factors c, v, and s also imply changes in the productivity of labour.” (p 51)

For similar reasons, its assumed that the length and intensity of the working day and wages remain constant. However, changes in v and s required to analyse the relation between the rate of surplus value and rate of profit, may imply changes in these factors, because they are their determining elements.