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CHAPTER III.: THE RELATION OF THE RATE OF PROFIT TO THE RATE OF SURPLUS-VALUE. - Karl Marx, Capital: A Critique of Political Economy. Volume III: The Process of Capitalist Production as a Whole 
Capital: A Critique of Political Economy. Volume III: The Process of Capitalist Production as a Whole, by Karl Marx. Ed. Federick Engels. Trans. from the 1st German edition by Ernest Untermann (Chicago: Charles H. Kerr and Co. Cooperative, 1909).
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THE RELATION OF THE RATE OF PROFIT TO THE RATE OF SURPLUS-VALUE.
WE have stated at the conclusion of the preceding chapter, and repeat it here, that we consider in this entire first part the amount of profit made by a certain capital to be equal to the full amount of surplus-value produced by means of this capital during a certain period of circulation. In other words, we leave aside for the present the fact that this surplus-value is split up into various secondary forms, such as interest on capital, ground-rent, taxes, etc., and that surplus-value is not identical, as a rule, with profit as appropriated on the basis of an average rate of profit, which will be discussed in part II.
So far as the quantity of profit is assumed to be equal to that of surplus-value, its magnitude, and that of the rate of profit, is determined by the relations of simple numerical magnitudes given or ascertainable in every individual case. The analysis, therefore, is first carried on purely on the field of mathematics.
We retain the terms used in volumes I and II. The total capital C consists of constant capital c and variable capital v, and produces a surplus-value s. The ratio of this surplus-value to the advanced variable capital, or s/v, is called the rate of surplus-value and designated by s'. Therefore s/v = s', and s = s'v. If this surplus-value is calculated on the total capital instead of the variable capital, it is called profit, p, and the ratio of the surplus-value s to the total capital C, or s/C, is called the rate of profit, p'. Accordingly, p' = s/C = s/(c+v). Now, substituting for s its equivalent s'v, we find p' = S'v/C = S'v/(c+v). And this equation may be expressed by the proportion p' : s' = v : C, or in words, the rate of profit is proportioned to the rate of surplus-value as the variable capital is to the total capital.
This proportion shows that the rate of profit, p', is always smaller than the rate of surplus-value, s', because the variable capital, v, is always smaller than the total capital, C, which is the sum of v + c, the variable plus the constant capital. The only exception to this rule is the practically impossible case, in which v = C, that is to say, in which no constant capital, no means of production, are advanced by the capitalist, but only wages.
However, our analysis must take into account a few other elements, which have a determining influence on the magnitude of c, v, and s. We shall mention them briefly.
There is, first, the value of money. We may assume this to be constant, throughout our analysis.
In the second place, there is the turn-over. We leave this element entirely out of consideration for the present, since its influence on the rate of profit will be treated later on in a special chapter. [We anticipate here only one point, namely that the formula p' = s' v/C is strictly correct only for one period of turn-over of the variable capital. But we may make it correct for an annual turn-over by substituting for s', the simple rate of surplus-value, the factor s'n, meaning the annual rate of surplus-value. The factor n in this term expresses the number of turn-overs of the variable capital during one year. (See chapter XVI, I, volume II.)—F. E.]
In the third place, the productivity of labor must be considered. Its influence on the rate of surplus-value has been thoroughly discussed in volume I, part V. The productivity of labor may also exert a direct influence on the rate of profit, at least of an individual capital. It has been demonstrated in volume I, chapter XII, that an individual capital may realize an extra profit, if it operates with a greater productivity than that of the social average and thereby produces its commodities at a lower value than the social average value of the same commodities. However, this case will not be considered for the present, since our premise in this part of the work is that the commodities are produced under normal social conditions and sold at their values. Hence we assume in each case that the productivity of labor remains constant. Under these circumstances the composition of the values of any capital invested in any line of industry, in other words, the proportion between the variable and constant capital, expresses a definite degree in the productivity of labor. As soon as this proportion is altered by other means than a mere change in the value of the material elements of the constant capital, or a change in the value of wages, it follows that the productivity of labor must likewise undergo a corresponding change. We shall see frequently, for this reason, that alterations affecting the factors c, v, and s imply also changes in the productivity of labor.
The same applies to the three remaining factors. namely the length of the working day, the intensity of labor, and the wages. Their influence on the mass and rate of surplus-value has been discussed in detail in volume I. It will be understood, therefore, that notwithstanding our assumption that these three factors remain constant there may be changes in v and s which may imply changes in the magnitude of these determining elements. In this respect we have but to remember that wages influence the quantity of surplus-value and the degree of the rate of surplus-value inversely from the length of the working day and the intensity of labor; that an increase of wages reduces the surplus-value, while a prolongation of the working day and an increase in the intensity of labor add to it.
Take it that a capital of 100 produces with 20 laborers by a working day of 10 hours and a total weekly wage of 20 a surplus-value of 20. Then we have 80 c + 20 v + 20 s, which implies that s' equal 100% and p' 20%.
Now let the working day be prolonged to 15 hours without an increase of wages. The total value produced by the 20 laborers is thereby increased from 40 to 60, since 10 : 15 = 40: 60. Seeing that v, the wages paid to the laborers, remains the same, the surplus-value rises from 20 to 40, and we have 80 c + 20 v + 40 s, implying that s' equals 200% and p' 40%. If, on the other hand, the working day remains unchanged at 10 hours, while wages fall from 20 to 12, the total value produced amounts to 40, but it is differently distributed. For v falls to 12, leaving a remainder of 28 for s. Then we have 80 c + 12 v + 28 s, whereby s' is raised to 233 1/3%, while the rate of profit, p', is as 28 to 92, or 30 10/23%.
We see, then, that both a prolongation of the working day (or a corresponding increase in the intensity of labor) and a fall in wages increase the mass, and thus the rate, of surplus-value. On the other hand, a rise in wages, other circumstances remaining the same, would lower the rate of surplus-value. Hence, if v rises through an increase of wages, it does not mean a greater, but only a dearer quantity of labor, and in that case s' and p' do not rise, but fall.
This indicates that a change in the working day, in the intensity of labor, and in wages cannot take place without at the same time altering v and s and their proportion, and therefore also p', which expresses the proportion of s to the total capital c + v. And it is also evident that a change in the proportion of s to v implies a corresponding change in at least one of the three determining elements of labor.
It is precisely this fact which reveals the specific organic relationship of variable capital to the movement of the total capital and its self-expansion, and also its difference from the constant capital. So far as it is a question of the generation of value, the constant capital is significant only for its value. It is immaterial for this question, whether a constant capital of, say, 1,500 p.st. represents 1,500 tons of iron at 1 p.st. each, or 500 tons of iron at 3 p.st. each. The quantity of the actual material, in which the value of the constant capital is incorporated, is immaterial for the question of the formation of value and the rate of profit. This rate varies inversely to the value of the constant capital, no matter what may be the proportion of the increase or decrease of the value of constant capital to the mass of its material elements.
It is different with the variable capital. Not its own value, not the labor incorporated in this capital, are of prime importance, but the fact that its own value implies the setting in motion of a grand total of labor whose quantity it does not express. This grand total of labor differs from the labor expressed in the value of the variable capital and paid by it in that it contains a certain amount of surplus-labor, which is so much greater, the smaller the value of the labor contained in the variable capital. Take it that a working day of 10 hours is equal to 10 shillings. If the necessary labor, which pays for the wages, or makes good the variable capital, is worth 5 shillings, then the surplus-labor amounts to 5 hours, or the surplus-value to 5 shillings. If the necessary labor amounts to 4 hours and is worth 4 shillings, then the surplus-labor is 6 hours and the surplus-value 6 shillings.
Hence, as soon as the value of the variable capital ceases to be an index of the amount of labor actually set in motion by it, as soon as the measure of this index is altered, the rate of surplus-value will vary inversely and at an inverse ratio.
Now let us pass on and apply the previously found equation of the rate of profit, p' = s' v/C, to the various cases possible. We shall change the value of the individual factors of s' v/C one after another and ascertain the effect of these changes on the rate of profit. In this way we obtain a number of different cases, which we may regard either as successively altered determinants of one and the same capital, or as different capitals existing side by side and compared with one another, no matter whether they exist in different lines of industry or different countries. In cases where the conception of some of our examples as successive conditions of the same capitals seems forced or impracticable, this objection is set aside by regarding them as illustrations of independent capitals.
We now separate the product s' v/C into its two factors s' and v/C. In the first place, we treat s' as a constant factor and analyze the effects of the possible variations of v/C. After that we treat the fraction v/C as constant and let s' go through its possible variations. Finally we treat all factors as variable magnitudes and thereby exhaust all cases from which rules concerning the rate of profit may be derived.
I. s' constant, v/C variable.
We make a general formula for this case, which comprises a number of sub-cases. Take two capitals C and C1, with their respective variable proportions v and v1, with equal rates of surplus-value s', and the rates of profit p' and p1'. Then p' = s' v/C and p1' = s' v1/C1.
Now let us make a proportion of C and C1, and v and v1, for instance let the value of the fraction C1/C = E, and that of v1/v = e. Then C1 = EC, and v1 = ev. Substituting in the above equation these values for p1', C1 and v1, we obtain P1' = s' ev/EC. Again, we may deduct a second formula from the above two equations, by transforming them into the equation p' : p1' = s' v/C: S' v1/C1 = v/C : v1/C1. Since the value of a fraction remains the same, if we multiply or divide its numerator or denominator by the same number, we may reduce v/C and v1/C1, to percentages, that is to say we may make both C and C1 equal to 100. Then we have v/C = v/100 and v1/C1 = v1/100. We may then drop the denominators in the above proportion and say that p' : p1' = v : v1. In other words, with any two capitals operating with the same rate of surplus-value the rates of profit are proportioned to one another as the variable capitals are to one another, calculated in percentages on their respective total capitals.
These two formulæ comprise all cases of variation of v/C.
Before we analyze these various cases, we make another remark. Since C is the sum of c plus v, of the constant and variable capital, and since the rates of surplus-value and of profit are generally expressed in percentages, it is convenient to assume that the sum of c plus v is also equal to 100, that is to say, to express c and v in percentages. It is immaterial for the determination, not of the mass, but of the rate of profit, whether we say that a capital of 15,000, composed of 12,000 of constant and 3,000 of variable capital, produces a surplus-value of 3,000, or whether we reduce this capital to percentages. So we may say that 15,000 C = 12,000 c + 3,000 v + (3,000 s), or that 100 C = 80 c + 20 v + (20 s). In either case the rate of surplus-value, s', equals 100% and the rate of profit, p', 20%.
The same is true in the comparison of two capitals. For instance, if we compare the foregoing capital with another, such as 12,000 C = 10,800 c + 1,200 v + (1,200 s), or 100 C = 90 c + 10 v + (10 s). In the last case, s' is 100% and p', 10%. And its comparison with the foregoing capital is easier by percentages.
On the other hand, if it is a question of changes taking place in the same capital, the expression by percentages is rarely convenient, because these peculiar alterations are almost always obliterated thereby. If a capital, expressed in percentages of 80 c + 20 v + 20 s assumes the percentages of 90 c + 10 v + 10 s, we cannot tell whether the change in the composition of percentages is due to an absolute decrease of v or an absolute increase of c, or to both. In order to ascertain this, we must have the absolute magnitudes in figures. But in the analysis of the following individual cases, everything depends on the question of the way in which the variations have been accomplished. Has 80 c + 20 v been changed into 90 c + 10 v by an increase of the constant capital without any change in the variable capital, for instance by changing 12,000 c + 3,000 v into 27,000 c + 3,000 v? Or has the same result been accomplished by leaving the constant capital untouched and reducing the variable capital, for instance by changing the above capital into 12,000 c + 1,333 1/3; v (corresponding to a percentage of 90 c + 10 v)? Or have both of the original capitals been changed into 13,500 c + 1,500 v (corresponding once more to percentages of 90 c + 10 v)? It is precisely these cases which we shall have to analyze, and in so doing we must dispense with percentages, or at least employ them only in a minor degree.
1. s' and C constant, v variable.
If v changes its magnitude, then C can remain unaltered only by a change in the opposite direction of c, the other component of C. If C consists originally of 80 c + 20 v, and if v is reduced to 10, then C can remain 100 only by an increase of c to 90; for 90 c + 10 v = 100. Generally speaking, if v is transformed into v ± d, into v increased or decreased by d, then c must be transformed into c + d, into c decreased or increased by the same amount, into c varying in the opposite direction from v, in order that the conditions of the present case be fulfilled.
Again, if the rate of surplus-value, s', remains the same, while the variable capital, v, changes, then the mass of surplus-value must change, since s = s'v, and since one of the factors of s'v, namely v, is invested with a different value.
The assumptions of the present case produce, aside from the original equation p' = s' v/C, still another equation by the variation of v, namely p1' = s' v1/C, in which v has become v1 and p1', the corresponding rate of profit, is to be sought.
It is found by the corresponding proportion:
p' : p1' = s' v/C : s' v1/C = v : v1.
That is to say, if the rate of surplus-value and the total capital remain the same, then the original rate of profit is proportioned to the new rate of profit produced by a change in the variable capital as the original variable capital is to the changed variable capital.
If the original capital was I) 15,000 C = 12,000 c + 3,000 v + (3,000 s), and if it is now II) 15,000 C = 13,000 c + 2,000 v + (2,000 s), then C is 15,000 and the rate of surplus-value 100% in either case, and the rate of profit of I), 20%, is proportioned to that of II), 13 1/3%, as the variable capital of I), 3,000, is to the variable capital of II), 2,000, that is to say 20% : 13 1/3% = 3,000 : 2,000.
Now, the variable capital may either increase or decrease. Take first an example in which it increases. Let a certain capital be constituted and operated as follows: I) 100 c + 20 v + 10 s. Then C equals 120, s' equals 50%, and p' equals 8 1/3%. Now let the variable capital increase to 30. In that case the constant capital must fall to 90, according to our assumption, which requires that the total should remain unchanged at 120. The amount of surplus-value produced will then rise from 10 to 15, the rate of surplus-value remaining constant at 50%. Our capital then is constituted as follows:
II) 90 c + 30 v + 15 s. C equals 120, s' equals 50%, and p', 12½%.
Now let us start out with the assumption that the wages remain unchanged. Then the other factors of the rate of surplus-value, namely the working day and the intensity of labor, must also be unchanged. Therefore the increase of v from 20 to 30 can signify only that more laborers are employed. In that case the total product in values also increases by one-half, from 30 to 45, and is distributed, the same as before, to 2/3 for wages and 1/3 for surplus-value. Simultaneously with the increase in the number of laborers the constant capital, the value of the means of production, has fallen from 100 to 90. We have before us, then, a case of decreasing productivity of labor combined with a simultaneous decrease of constant capital. Is such a case economically possible?
In agriculture and industries engaged in the extraction of substances, where a decrease in the productivity of labor and, therefore, an increase in the number of laborers are readily understood, this process is accompanied on the basis and within the scope of capitalist production, by an increase of constant capital, not by a decrease. Even if our assumed decrease of c were due merely to a fall in prices, an individual capital would be able to accomplish the transition from I) to II) only under very exceptional circumstances. But in the case of two independent capitals invested in different countries, or in different lines of agriculture or extractive industry, it would not be strange if more laborers (and therefore more variable capital) were employed on less valuable or fewer means of production in the case of one than in the other.
But let us have done with the assumption that the wages remain the same, and let us explain the rise of the variable capital from 20 to 30 by a rise of wages by one-half. Then we have another case. The same number of laborers continue to work with the same or slightly reduced means of production. If the working day remains unchanged, say at 10 hours, then the total product also remains unchanged. It was and remains 30. But this amount of 30 is now required to make good the consumed variable capital. The surplus-value would have disappeared. But we had assumed that the rate of surplus-value should remain constant at 50%, the same as in I). This is possible only if the working day is prolonged by one-half, increased to 15 hours. In that case 20 laborers produce in 15 hours a total value of 45, and all conditions would be fulfilled. We should have
II). 90 c + 30 v + 15 s. C would be 120, s', 50% and p', 12½%.
Under these circumstances the 20 laborers do not require any more instruments, tools, machines, etc., than in the case of I). Only the raw materials or auxiliary substances would have to be increased by one-half. If there were a fall in the prices of these materials, then the transition from I) to II) under the conditions of our assumed case might very well be accomplished even by an individual capital. And the capitalist would be somewhat compensated by increased profits for any loss incurred through the depreciation of his constant capital.
Now let us assume that the variable capital were to be reduced instead of increased. Then we have but to reverse our example. We have but to assume that II) is the original capital and to pass from II) to I). Then II), or 90 c + 30 v + 15 s changes into I), or 100 c + 20 v + 10 s, and it is evident that this transposition does not alter any of the conditions which regulate the respective rates of profit and their mutual relations.
If v falls from 30 to 20 because the number of laborers is reduced by one-third while the constant capital increases, then we have before us the normal case of modern industry, namely an increasing productivity of labor, an operation of a larger mass of means of production by fewer laborers. That this process is necessarily connected with a simultaneous fall of the rate of profit, will be demonstrated in the third part of this volume.
On the other hand, if v falls from 30 to 20 because the same number of laborers are employed at lower wages, while the working day remains the same, then the total product in values would remain 30 v + 15 s, or 45. Since wages have fallen to 20, the surplus-value would rise to 25, the rate of surplus-value from 50% to 125%, contrary to our assumption. In order to comply with the conditions of our case, the surplus-value, with its rate at 50%, must fall to 10. The total product must, therefore, fall from 45 to 30, and this is possible only by a reduction of the working day by one-third. Then we have, the same as before, 100 c + 20 v + 10 s. C equals 120, s', 50%, and p', 8 1/3%.
It need hardly be mentioned that this reduction of the working time with a fall in wages would not occur in practice. But this is immaterial. The rate of profit is a function of several variable magnitudes, and if we wish to know in what manner these variable magnitudes influence the rate of profit, we must analyze the individual effect of each seriatim, regardless of whether such an isolated effect is practicable with one and the same capital or not.
2) s' constant, v variable, C changed by the variation of v.
This case differs from the preceding one only in degree. Instead of c decreasing or increasing by as much as v increases or decreases, c remains constant. Under the modern conditions of great industry and agriculture the variable capital is but a relatively small part of the total capital. For this reason, the increase or decrease of the total capital, so far as either is due to variations of the variable capital, are likewise relatively small.
Let us start out again with a capital I) of 100 c + 20 v + 10 s. C equals 120, s' 50%, and p' 8 1/3%. This will then be transformed into II) 100 c + 30 v + 15 s, with C at 130, s' at 50%, and p' at 11 7/13%. The opposite case, in which the variable capital would decrease, would be symbolized by the transition from II) to I).
The economic conditions would be essentially the same as in the preceding case, and therefore require no reiteration. The transition from I) to II) implies a decrease in the productivity of labor by one-half. The assimilation of 100 c requires an increase of labor in II) by one-half over that of I). This case may occur in agriculture.9
While in the preceding case the total capital remained constant, owing to the conversion of constant capital into variable, or vice versa, there is in this case a tie-up of additional capital, if the variable capital is increased, and a release of previously employed capital, if the variable capital decreases.
3) s' and v constant, c and C variable.
In this case, the equation p' = s' v/C is changed into p1' = s' v/C1. After eliminating the same factors on both sides, we have p1': p' = C: C1. In other words, if the rates of surplus-value are the same and the variable capitals equal, the rates of profit are inversely proportioned to the total capitals.
Take it that we have three different capitals, or three different conditions of the same capital, for instance
I) 80 c + 20 v + 20 s; C = 100, s' = 100%, p' = 20%
Then we obtain the proportions:
20% : 16 2/3% = 120 : 100, and 20% : 25% = 80 : 100.
The general formula previously given for variations of v/C when s' remained constant was p1' = s' ev/EC. Now it becomes p' = s' v/EC. For since v remains unchanged, the factor e, or v1/v, becomes equal to 1.
Since s'v equals s, the mass of surplus-value, and since both s' and v remain constant, it follows that s is not affected by any variation of C. The mass of surplus-value is the same after the change that it was before.
If c were to fall to zero, p' would be equal to s', that is to say, the rate of profit equal to the rate of surplus-value.
The alteration of c may be due either to a mere change in the value of the material elements of constant capital, or to a change in the technical composition of the total capital, that is to say a change in the productivity of labor in that line of industry. In the last named case, the increase in the productivity of social labor due to the development of industry and agriculture on a large scale would bring about a transition, in the above illustration, from III to I and from I to II. A quantity of labor paid with 20 and producing a value of 40 would first work up means of production valued at 60. With a further increase in the productivity, and the same value, the means of production would be worked up to the amount of 80, and later on of 100. A reversion of this succession would imply a decrease in productivity. The same quantity of labor would work up a smaller quantity of means of production, the business would be cut down. This may occur in agriculture, mining, etc.
A saving in constant capital increases on the one hand the rate of profit, and on the other sets free some capital. It is, therefore, of great importance for the capitalist. We shall analyze this point later on, and likewise the influence of a change of prices of the elements of constant capital, particularly of raw materials.
We see once more, by this illustration, that a variation of the constant capital uniformly affects the rate of profit, no matter whether this variation is due to an increase or decrease of the material elements of c, or merely to a change in their value.
4) s' constant, v, c, and C variable.
In this case, the general formula indicated at the outset, namely p' = s' ev/EC, remains in force. It follows from this, assuming the rate of surplus-value to remain the same, that
a) the rate of profit falls, if E is greater than e, that is to say, if the constant capital increases to such an extent that the total capital grows at a faster rate than the variable capital. If a capital of 80 c + 20 v + 20 s is transformed so that it becomes 170 c + 30 v + 30 s, then s' remains at 100%, but v/C falls from 20/100 to 30/200, in spite of the fact that both v and C have augmented, and the rate of profit falls correspondingly from 20% to 15%.
b) The rate of profit remains unchanged only in the case that e equals E, that is to say, if the fraction v/C retain the same value even if the fraction is apparently changed, in other words, if its numerator and denominator are multiplied or divided by the same number. It is evident that the capital 80 c + 20 v + 20 s and the capital 160 c + 40 v + 40 s have the same rate of profit, namely 20%, because s' remains at 100% and v/C represents the same value, whether we write it 20/100 or 40/200.
c) The rate of profit arises, when e is greater than E, that is to say, when the variable capital grows at a faster rate than the total capital. If 80 c + 20 v + 20 s becomes 120 c + 40 v + 40 s, then the rate of profit rises from 20% to 25%, because s' has remained the same and v/C has risen from 20/100 to 40/160, or from 1/5; to ¼.
If the variation of v and C follows the same direction, we may look upon this change of magnitude up to a certain degree as though both of them varied in the same proportion, so that v/C would be regarded as unchanged to that extent. Beyond this point only one of them would then vary, and by this means we should reduce this complicated case to one of the preceding simpler ones.
For instance, if 80 c + 20 v + 20 s becomes 100 c + 30 v + 30 s, then the proportion of v to c, and also to C, remains the same up to the point of 100 c + 25 v + 25 s. Up to that point, the rate of profit remains likewise unchanged. We may then take our departure from 100 c + 25 v + 25 s. We find that later increased by 5 and became 30, so that C rose from 125 to 130. This is identical with the second case, that of the simple variation of v and the consequent variation of C. The rate of profit, which was originally 20%, rises by this addition of 5 v to 23 1/13, always assuming the rate of surplus-value to remain the same.
The same reduction to a simpler case can take place, whenever v and C change their magnitudes in opposite directions. For instance, let us start out once more from 80 c + 20 v + 20 s, and let this become 110 c + 10 v + 10 s. In that case, the rate of profit would have remained the same, if the variation had proceeded to the point of 40 c + 10 v + 10 s. It would still have been 20%. By adding 70 c to this intermediate form, the rate of profit is lowered to 8 1/3%. Thus we have reduced this case to a case of variation of one magnitude, namely of c.
Simultaneous variations of v, c, and C, do not, then, offer any new points of analysis. For they may be reduced in the last resort to cases in which only one factor is variable.
Even the only remaining case has actually been covered, namely that in which v and C are numerically unchanged, while their material elements experience a change of value, so that v stands for a changed quantity of assimilated labor and c for a changed quantity of assimilated means of production.
For instance, in the capital 80 c + 20 v + 20 s, let 20 v indicate originally the wages of 20 laborers working 10 hours daily. Then let the wages of each laborer increase from 1 to 1¼. In that case 20 v pay only 16 laborers instead of 20. Now, if 20 laborers produce in 200 working hours a value of 40, then 16 laborers will produce in 160 working hours a value of only 32. After deducting 20 v for wages, only 12 would remain for surplus-value. The rate of surplus-value would have fallen from 100% to 60%. But since our assumption is that the rate of surplus-value shall remain constant, the working day would have to be prolonged by one-quarter, from 10 hours to 12½ hours. If 20 laborers, working 10 hours daily, or 200 hours, produce a value of 40, then 16 laborers, working 12½ hours daily, or 200 hours, will produce the same value, and the capital of 80 c + 20 v produces the same surplus-value of 20.
Vice versa, if wages fall to such an extent that 20 v indicates the wages of 30 laborers, then s' can remain unchanged only in the case that the working day is reduced from 10 to 6 2/3 hours. For 20 × 10 = 30 × 6 2/3 = 200 working hours.
We have discussed previously in these diverging assumptions, to what extent c may express the same value in money, and yet represent different quantities of means of production corresponding to different conditions. In reality this case will very rarely be practicable in its purely theoretical form.
As for the change of value of the elements of c, by which their mass is increased or decreased, it touches neither the rate of surplus-value nor the rate of profit, so long as it does not imply a change of magnitude in v.
We have now exhausted all possible cases of variation of v, c, and C in our equation. We have seen that the rate of profit may fall, rise, or remain unchanged, while the rate of surplus-value remains the same, for the least variation in the proportion of v to c, or to C, is sufficient to change the rate of profit.
We have seen, furthermore, that there is everywhere a certain limit in the variation of v where the constancy of s' becomes economically impossible. Since every one-sided variation of c must also arrive at a certain limit where v can no longer remain unchanged, we find that every possible variation of v/C has certain limits, beyond which s' must likewise become variable. In the variations of s', which we shall now discuss, this interaction of the different variable magnitudes of our equation will become still plainer.
II. s' variable.
We obtain a general formula for the rates of profit with variable rates of surplus-value, no matter whether v/C remains constant or not, by converting the equation p' = s' v/C into p1' = s1' v1/C1. Here p1', s1', C1, and v1 indicate the changed values of p', s', C, and v. Then we have p': p1' = s'v/C: s1' v1/C1. This may be manipulated into
p1' = s1'/s' × v1/v × c/c1 × p'.
1) s' variable, v/C constant.
In this case we have the equations p' = s' v/C and p1' = S1' v/C. In both of them v/C is equal. Therefore p': p1' = s': s1. That is to say, the rates of profit of two capitals of the same composition are proportioned as the corresponding two rates of surplus-value. Since it is not a question, in the fraction v/C, of the absolute magnitude of v and C, but only of their proportion to one another, this applies to all capitals of equal composition, whatever may be their absolute magnitude.
80 c + 20 v + 20 s; C = 100, s' = 100% p' = 20%.
If the absolute magnitudes of v and C are the same in both cases, then the rates of profit are also proportioned to one another as the masses of surplus-value: p': p1' = s'v: s1'v = s: s1. For instance:
80 c + 20 v + 20 s; s' = 100%, p' = 20%.
Now, it is evident that with capitals of equal absolute composition, or equal percentages of composition, the rates of surplus-value can differ only when either the wages, or the length of the working day, or the intensity of labor are different. Take the following three cases:
I. 80 c + 20 v + 10 s; s' = 50%, p' = 10%.
In the case of I, the total product in values is 30, namely 20 v + 10 s, in II it is 40, in III it is 60. This may come about in three different ways.
First, if the wages are different, so that 20 v expresses in every individual case a different number of laborers. Take it that capital I employs 15 laborers for 10 hours per day at a wage of 1 1/3 p.st. and that these laborers produce a value of 30 p.st, of which 20 p.st. make good the wages and 10 p.st. are surplus-value. If wages fall to 1 p.st., then 20 laborers may be employed for 10 hours, and they will produce a value of 40 p.st., of which 20 p.st. make good wages and 20 p.st. are surplus-value. If wages fall still more, for instance to 2/3 p.st., then 30 laborers may be employed for 10 hours, and they will produce a value of 60 p.st., 40 p.st. of which will represent surplus-value after deducting 20 p.st. for wages.
This case, in which the percentages of composition of the capital, the working day, the intensity of labor, are constant, while the rate of surplus-value varies on account of the variation of wages, is the only one in which Ricardo's assumption is correct, to-wit, that "profits would be high or low, exactly in proportion as wages would be low or high." (Principles, chapter I, section III, page 18 of the "Works of D. Ricardo," edited by MacCulloch, 1852.)
Secondly, if the intensity of labor varies. In that case 20 laborers produce with the same means of production in 10 hours of daily labor 30 pieces of a certain commodity in I, 40 pieces in II, and 60 pieces in III. Every piece represents, aside from the value of the means of production incorporated in it, a new value of 1 p.st. Since every 20 pieces make good the wages of 20 p.st., there remain 10 pieces at 10 p.st. for surplus-value in I, 20 pieces at 20 p.st. in II, and 40 pieces at 40 p.st. in III.
Thirdly, the working day may vary in length. If 20 laborers work with the same intensity for 9 hours in I, 12 hours in II, and 18 hours in III, then their total products, 30:40: 60 vary in the proportions 9: 12: 18. And since wages are 20 in every case, the surplus-value is 10, or 20, or 40 respectively.
An increase or decrease in wages, then, influences the rate of surplus-value, and, since v/C was assumed as constant, also the rate of profit, inversely, while an increase or decrease in the intensity of labor, a lengthening or shortening of the working day, influence them in the same direction.
2) s' and v variable, C constant.
In this case the following proportion applies: p': p1' = s' v/C: s1' v1/C = s'v: s1'v1 = s: s1.
The rates of profit are proportioned to one another as the corresponding masses of surplus-value.
A variation of the rate of surplus-value, while the variable capital remains constant, signifies a change in the magnitude and distribution of the product in values. A simultaneous variation of v and s' also implies always a change in the distribution, but not always a change in the magnitude of the product in values. Three cases are possible.
a) The variation of v and s' takes place in opposite directions, but by the same amount, for instance:
80 c + 20 v + 10 s; s' = 50%, p' = 10%.
The product in values is equal in both cases, hence the quantity of labor performed likewise: 20 v + 10 s = 10 v + 20 s = 30. The difference is only that in the first case 20 are paid for wages and 10 remain for surplus-value, while in the second case wages are 10 and surplus-value 20. This is the only case in which the number of laborers, the intensity of labor, and the length of the working day remain unchanged, while v and s' vary.
b) The variation of s' and v takes place in opposite directions, but not by the same amount. In that case the variation of either v or s' is the greater.
I. 80 c + 20 v + 20 s; s' = 100%, p' = 20%.
Capital I pays for a product in values amounting to 40 with 20 v, II a value of 48 with 28, and III a value of 36 with 16. Both the product in values and the wages have changed. But a change in the product in values means a change in the amount of labor performed, and this implies a change either in the number of laborers, the hours of labor, or the intensity of labor, or in more than one of these.
c) The variation of s' and v takes place in the same direction. In that case it intensifies the effect of either.
90 c + 10 v + 10 s; s' = 100%, p' = 10%.
In these cases the three products in value are also different namely 20, 50, and 14. And this difference in the magnitude of the respective quantities of labor reduces itself once more to a difference in the number of laborers, the hours of labor, and the intensity of labor, or of several or all of these factors.
3) s', v and C variable.
This case offers no new points of view and is solved by the general formula given under II, in which s' is variable.
The effect of a change in the magnitude of the rate of surplus-value on the rate of profit is summed up, according to the foregoing, by the following cases:
1) p' increases or decreases in the same proportion as s', if v/C remains constant.
80 c + 20 v + 20 s; s' = 100%, p' = 20%.
2) p' rises or falls at a greater rate than s', if v/C moves in the same direction as s', that is to say, if v/C increases or decreases when s' increases or decreases.
80 c + 20 v + 10 s; s' = 50%, p' = 10%.
3) p' rises or falls at a smaller rate than s', if v/C changes in the opposite direction from s', but at a smaller rate.
80 c + 20 v + 10 s; s' = 50%, p' = 10%.
4) p' rises, while s' falls, or falls while s' rises, if changes in the opposite direction and at a greater rate than s'.
80 c + 20 v + 20 s; s' = 100%, p' = 20%.
s' has risen from 100% to 150%, p' has fallen from 20% to 15%.
5) Finally, p' remains constant, while s' rises or falls, if v/C changes in the opposite direction, but at exactly the same rate, as s'.
It is only this last case which requires some further explanation. We observed in the variations of v/C that the same rate of surplus-value may be an expression of different rates of profit. We see now that the same rate of profit may be based on different rates of surplus-value. So long as s' is constant, any change in the proportion of v to C is sufficient to call forth a difference in the rate of profit. But if s' varies in magnitude, it requires a corresponding inverse change of v/C in order that the rate of profit may remain the same. This happens but exceptionally in the case of one and the same capital, or of two capitals in one and the same country. Take it that we have a capital 80 c + 20 v + 20 s; C = 100, s' = 100%, p' = 20%. And let us assume that wages fall to such an extent that the same number of laborers may be bought for 16 v instead of 20 v. Then we have released 4 v, and other circumstances remaining the same, our capital will have the composition 80 c + 16 v + 24 s; C = 96, s' = 150%, p' = 25%. In order that p' may be 20%, as before, the total capital would have to increase to 120, the constant capital, therefore, to 104, thus, 104 c + 16 v + 24 s; C = 120, s' = 150%, p' = 20%.
This would be possible only if the fall in wages were accompanied by a change in the productivity of labor, which would require such a change in the composition of capital. Or, it might be that the money-value of the constant capital would increase from 80 to 104. In short, it would require an accidental coincidence of conditions such as occurs very rarely. In fact, a variation of s' which does not imply a simultaneous variation of v, and thus of v/C is practicable only under very definite conditions. It may happen in lines of industry in which only fixed capital and labor are employed, while the materials of labor are supplied by nature.
But this is not so in the comparison of the rates of profit of two different countries. For in that case the same rate of profit is based as a rule on different rates of surplus-value.
It follows from all of these five cases that a rising rate of profit may be the companion of a falling or rising rate of surplus-value; a falling rate of profit go hand in hand with a rising or falling rate of surplus-value; a constant rate of profit exist by the side of a rising or falling rate of surplus-value. And we have seen under No. I that a rising, falling, or constant rate of profit may be based on a constant rate of surplus-value.
The rate of profit, then, is determined by two main factors, namely the rate of surplus-value and the composition of the value of capital. The effects of these two factors may be briefly summed up in the manner stated hereafter. We may, in this summing up, express the composition of capital in percentages, for it is immaterial for this point which one of the two portions of capital is the cause of variation.
The rates of profits of two different capitals, or of one and the same capital in two different successive conditions, are equal
1) If the percentages of composition of capital are the same and the rates of surplus-value equal.
2) If the percentages of composition are not the same, and the rates of surplus-value unequal, provided that the products of the multiplication of the rates of surplus-value by the percentages of the variable portions of capital (s' and v) are the same, that is to say, the masses of surplus-value (s = s'v) calculated in percentages on the total capital; in other words, if the factors s' and v are inversely proportioned to one another in both cases.
They are unequal
1) If the percentages of composition are equal and the rates of surplus-value unequal, in which case the rates of profit are proportioned as the rates of surplus-value.
2) If the rates of profit are the same and the percentages of composition unequal, in which case the rates of profit are proportioned as the variable portions of capital.
3) If the rates of profit are unequal and the percentages of composition not the same, in which case the rates of profit are proportioned as the products s'v, that is to say, as the masses of surplus-value calculated in percentages on the total capital.10
[9.] The manuscript has the following note at this point: "Investigate later in what manner this case is connected with ground-rent."