As we said at the end of the last chapter, we're assuming here — and throughout this whole first Part — that the profit on a given capital equals the whole surplus-value that capital produces over a given period of circulation. For now we set aside two things: that this surplus-value actually splits into different sub-forms — interest, ground-rent, taxes, and so on — and that, in most cases, profit as it's actually appropriated under the prevailing rate of profit does not coincide with it at all. That's the subject of Part Two.
Once profit is counted as the same magnitude as surplus-value, its size — and the size of the rate of profit — is settled by arithmetic on figures that are either given or can be worked out in any individual case. So for now the inquiry moves on purely mathematical ground.
We keep the notation used in Volumes 1 and 2. The total capital C splits into constant capital c and variable capital v, and produces a surplus-value s. The ratio of this surplus-value to the variable capital advanced, that is s/v, we call the rate of surplus-value, and write it s'. So s/v = s', and therefore s = s'v. If instead of relating this surplus-value to the variable capital we relate it to the total capital, it's called profit (p), and the ratio of the surplus-value s to the total capital C, that is s/C, is called the rate of profit, p'. So we have:
If we substitute, for s, the value s'v we found above, we get:
an equation that can also be written as the proportion:
the rate of profit compares to the rate of surplus-value exactly as the variable capital compares to the total capital: whatever fraction of the whole capital the variable part is, the rate of profit is that same fraction of the rate of surplus-value.
It follows from this proportion that p', the rate of profit, is always smaller than s', the rate of surplus-value — because v, the variable capital, is always smaller than C, which is v plus c, variable and constant capital together. The one exception, impossible in practice, is the case where v = C, where the capitalist would advance no constant capital, no means of production at all, only wages.
Our inquiry now needs to take into account a further set of factors that have a determining effect on the size of c, v and s, and so need a brief mention.
First, the value of money. We can treat this as constant throughout.
Second, the turnover. We leave this factor entirely aside for now, since its effect on the rate of profit gets its own treatment in a later chapter. Engels adds here: the formula p' = s'v/C holds strictly only for a single turnover period of the variable capital, but we can make it hold for the annual turnover by putting s'n, the annual rate of surplus-value, in place of s', the simple rate of surplus-value — where n is the number of turnovers the variable capital makes within a year (see Volume 2, Chapter 16, I).
Third, we have to take into account the productivity of labour, whose effect on the rate of surplus-value was discussed at length in Volume 1.
Productivity can also have a direct effect on the rate of profit — at least on that of an individual capital — when, as shown in Volume 1, that capital works with a higher-than-average productivity, so that its products represent a lower value than the socially average value of the same commodity, letting it realize an extra profit. We leave this case out of account here too, though, since in this Part we're still working from the assumption that commodities are produced under socially normal conditions and sold at their values. So in every individual case we assume that the productivity of labour stays constant.
In fact, though, the value-composition of the capital invested in a branch of industry — that is, a given ratio of variable to constant capital — always expresses a certain degree of the productivity of labour. So as soon as this ratio changes for any reason other than a mere change in the value of the material elements of constant capital, or a change in wages, the productivity of labour must have changed too. And we'll often find, accordingly, that the changes going on in c, v and s also carry changes in the productivity of labour along with them.
The same holds for the three remaining factors: the length of the working day, the intensity of labour, and wages. Their effect on the mass and the rate of surplus-value was worked out at length in Volume 1. So it's understandable that, even though for simplicity we always assume these three factors stay constant, the changes going on in v and s can still carry changes in these underlying factors along with them.
Here it's worth a brief reminder: wages act on the size of surplus-value and the rate of surplus-value in the opposite direction to the length of the working day and the intensity of labour — a rise in wages reduces surplus-value, while lengthening the working day and raising the intensity of labour increase it.
Say, for example, a capital of 100 employs 20 workers on a 10-hour working day, at a total weekly wage of 20, and produces a surplus-value of 20. Then we have:
Now let the working day be lengthened, without any wage rise, to 15 hours. This raises the total value product of the 20 workers from 40 to 60 (10 : 15 = 40 : 60); since v, the wage actually paid, stays the same, the surplus-value rises from 20 to 40, and we have:
If, on the other hand, with a 10-hour working day, the wage falls from 20 to 12, we get the same total value product of 40 as before, but split differently: v drops to 12, leaving 28 for s. So we have:
So we see that both a longer working day (or, equally, greater intensity of labour) and a fall in wages raise the mass, and so the rate, of surplus-value; conversely, a rise in wages, other things being equal, would push the rate of surplus-value down. So when v grows because of a wage rise, it doesn't express a bigger quantity of labour, only a more expensively paid one — s' and p' don't rise, they fall.
This already shows that changes in the working day, the intensity of labour and wages can't happen without a simultaneous change in v and s and their ratio — and so also in p', the ratio of s to c + v, the total capital. And it's equally clear that changes in the ratio of s to v likewise involve a change in at least one of the three working conditions just mentioned.
This is exactly where the special organic relationship of variable capital to the movement of the total capital and its valorization — its making value grow — shows itself — and where its difference from constant capital comes out. As far as value-formation goes, constant capital matters only because of the value it has; it makes no difference to value-formation whether a constant capital of £1,500 represents, say, 1,500 tons of iron at £1 a ton, or 500 tons of iron at £3 a ton. The actual quantity of physical stuff that this value represents makes no difference at all to value-formation, or to the rate of profit, which varies in the opposite direction to this value — whatever the ratio between a rise or fall in the value of the constant capital and the mass of the physical use-values it represents.
It's quite different with variable capital. What matters about it, in the first place, isn't the value it has, the labour objectified in it, but rather this value taken as a mere index of the total labour it sets in motion, labour that is not expressed in it. This total labour differs from the labour expressed in it, and so paid for by it; and the smaller the labour contained in the variable capital itself, the bigger is the part of that total labour that forms surplus-value. Say a working day of 10 hours equals ten shillings. If the necessary labour — the labour that replaces the wage, that is the variable capital — is 5 hours = 5 shillings, then the surplus labour is 5 hours and the surplus-value is 5 shillings; if that necessary labour is 4 hours = 4 shillings, then the surplus labour is 6 hours and the surplus-value is 6 shillings.
So as soon as the value of the variable capital stops being an index of the mass of labour it sets in motion — as soon as the yardstick itself shifts, so that the same sum no longer stands for the same amount of labour — the rate of surplus-value changes too, in the opposite direction and in inverse proportion.
We turn now to applying the equation for the rate of profit above, p' = s'v/C, to the various possible cases. We'll let the individual factors of s'v/C change their value one at a time, and work out the effect of each change on the rate of profit. This gives us different series of cases, which we can view either as one and the same capital meeting changed circumstances one after another, or as different capitals existing side by side at the same time and brought in for comparison — say, in different branches of industry or different countries. So if treating some of our examples as successive states of one and the same capital over time seems forced or practically impossible, that objection falls away as soon as we take them instead as a comparison of independent capitals.
So we split the product s'v/C into its two factors, s' and v/C. First we treat s' as constant and examine the effect of the possible variations in v/C; then we treat the fraction v/C as constant and let s' run through its possible variations; and finally we let all the factors vary at once, exhausting in this way all the cases from which laws about the rate of profit can be derived.
This case covers several sub-cases, and we can write one general formula for all of them. Take two capitals, C and C1, with their variable parts v and v1. Both run at the same rate of surplus-value, s'. Call their rates of profit p' and p'1. Then:
Now put C and C1, and v and v1, into ratios with each other. Say the fraction C1/C equals E, and the fraction v1/v equals e. That means C1 = EC and v1 = ev. Put these values for C1 and v1 into the equation for p'1 above, and we get:
We can also draw a second formula out of the two equations above, by turning them into a proportion:
A fraction keeps its value if you multiply or divide the top and bottom by the same number. So we can turn v/C and v1/C1 into percentages, by setting both C and C1 equal to 100. Then v/C becomes v/100 and v1/C1 becomes v1/100, and we can drop the denominators from the proportion above, which gives us:
Take any two capitals that operate at the same rate of surplus-value. Their rates of profit stand to each other exactly as their variable parts do, once each variable part is worked out as a percentage of its own total capital.
These two forms cover every possible way v/C can vary.
Before we examine these cases one by one, one more remark. C is the sum of c and v, constant and variable capital, and since both the rate of surplus-value and the rate of profit are usually expressed in per cent, it is convenient to set c + v equal to 100 as well — that is, to express c and v as percentages. It makes no difference to the rate of profit — though it does make a difference to the amount of profit — whether we say that a capital of 15,000, made up of 12,000 constant and 3,000 variable capital, produces 3,000 of surplus-value, or whether we reduce this capital to percentages:
In both cases the rate of surplus-value is 100 per cent, and the rate of profit is 20 per cent.
The same goes if we compare two capitals with each other — say, the one above with another capital:
In both of these forms s' is 100 per cent and p' is 10 per cent, and setting the new capital next to the one before it, the percentage form makes the comparison far easier to take in.
But when it comes to changes taking place within one and the same capital, the percentage form can rarely be used, because it almost always blurs those changes. Suppose a capital moves from the percentage form:
into the percentage form:
then you cannot tell whether the changed percentage composition, 90c + 10v, came about through an absolute fall in v, an absolute rise in c, or both together. For that we need the absolute figures. And for examining the individual cases of variation that follow, everything turns on exactly how that change came about: whether 80c + 20v became 90c + 10v because, say, 12,000c + 3,000v turned into 27,000c + 3,000v by an increase in constant capital with variable capital unchanged (still 90c + 10v in percentage terms); or whether it took this form with constant capital unchanged, through a fall in variable capital, passing into 12,000c + 1,333⅓v (again 90c + 10v in percentage terms); or finally through a change in both amounts, say 13,500c + 1,500v (again 90c + 10v). It is exactly these cases we are going to examine one after another — which means giving up the convenience of the percentage form, or using it only as a secondary tool.
If v changes, C can stay the same only if the other part of C, constant capital c, changes by the same amount but in the opposite direction. Say C starts out as 80c + 20v = 100 and v then falls to 10: C can stay at 100 only if c rises to 90, since 90c + 10v = 100. Put generally: if v turns into v plus or minus d, then c must turn into c minus or plus d — it must move by that same amount d, but the opposite way, for this case's conditions to hold.
Likewise, with the rate of surplus-value s' held constant but variable capital v changing, the amount of surplus-value s must change too, since s = s'v, and in s'v one of the factors, v, is now taking a different value.
The assumptions of our case give us, alongside the original equation,
a second one, produced by varying v:
in which v has become v1, and p'1 is the changed rate of profit that follows from it, still to be found.
We find it through the corresponding proportion:
Or, put in words: with the rate of surplus-value and the total capital both unchanged, the original rate of profit stands to the new rate of profit produced by the change in variable capital, as the original variable capital stands to the changed variable capital.
Say the capital was originally as above:
and is now:
then C = 15,000 and s' = 100 per cent in both cases, and the rate of profit of case I, 20 per cent, stands to that of case II, 13⅓ per cent, as the variable capital of case I, 3,000, stands to that of case II, 2,000: that is, 20 per cent : 13⅓ per cent = 3,000 : 2,000.
Variable capital can either rise or fall. Let's first take an example where it rises. Say a capital is originally made up, and operates, as follows:
Now let variable capital rise to 30. By our assumption, constant capital must then fall from 100 to 90, so that the total capital stays unchanged at 120. And with the rate of surplus-value still at 50 per cent, the surplus-value produced must rise to 15. So we have:
We start from the assumption that wages are unchanged. If so, the other factors involved in the rate of surplus-value — the working day, and also the intensity of labour — cannot have shifted either. The rise in v, 20 becoming 30, can only mean one thing: half again as many workers are employed. The total value produced climbs by the same half, from 30 to 45, and is divided exactly as before — two-thirds to wages, one-third to surplus-value. But with more workers in the picture, constant capital, the value of the means of production, has dropped from 100 to 90. So we are looking at labour productivity falling while constant capital falls too. Can a case like that exist economically?
In agriculture and the extractive industries, where falling productivity — and so a growing number of workers employed — is easy enough to picture, this process — so long as we stay inside capitalist production and take it as given — goes together not with a decrease in constant capital but with an increase. Even if the fall in c above were brought about simply by a fall in prices, a single capital could make the move from case I to case II only under quite exceptional circumstances. But with two independent capitals, invested in different countries, or in different branches of agriculture or extractive industry, there would be nothing remarkable about one of them employing more workers — and so a larger variable capital — while working with less valuable or scarcer means of production than the other.
But let's drop the assumption that the wage stays the same, and explain the rise in variable capital from 20 to 30 by wages rising by half. Then we get a completely different case. The same number of workers — say 20 — go on working with the same means of production, or only insignificantly reduced ones. If the working day stays unchanged, at ten hours say, the total value product is also unchanged: it is still 30. But all 30 of that is now needed just to replace the variable capital advanced, 30; the surplus-value would have vanished. Yet we had assumed that the rate of surplus-value stayed constant, at 50 per cent as in case I. That is only possible if the working day is lengthened by half, to fifteen hours. The 20 workers would then produce a total value of 45 in fifteen hours, and every condition of the case would be satisfied:
In this case the 20 workers need no more means of labour — tools, machines, and so on — than in case I; only the raw or ancillary materials would have to increase by half. So if the price of these materials fell, the move from case I to case II would, on our assumptions, be economically workable for a single capital far more readily than before. And the capitalist would be compensated, at least to some extent, by a larger profit for whatever loss he suffered through the devaluation of his constant capital.
Now suppose variable capital falls instead of rising. Then all we need to do is reverse our example above: take case II as the original capital, and move from II to I.
Case II, 90c + 30v + 15s, then turns into case I, 100c + 20v + 10s, and it is obvious that this reversal changes nothing whatsoever in the conditions that govern the two rates of profit and the relation between them.
If v falls from 30 to 20 because a third fewer workers are employed while constant capital keeps growing, then we have in front of us the normal case of modern industry: the rising productive power of labour, with fewer workers handling larger masses of means of production. That this movement is necessarily bound up with a simultaneous fall in the rate of profit will emerge in Part Three of this book.
But if v falls from 30 to 20 because the same number of workers is employed at a lower wage, then with the working day unchanged the total value product would stay at 30v + 15s = 45 as before; and since v has fallen to 20, surplus-value would rise to 25, taking the rate of surplus-value from 50 per cent to 125 per cent — which would go against our assumption. To stay within the conditions of our case, surplus-value must instead fall to 10, so that the rate of surplus-value stays at 50 per cent, so the total value product falls from 45 to 30 — and that is only possible by cutting the working day by a third. Then we have, as above:
It hardly needs saying that this shortening of working time alongside a falling wage would not happen in practice. But that does not matter. The rate of profit is a function of several variables, and if we want to know how each of them affects the rate of profit, we have to examine the effect of each one on its own, one after another — regardless of whether that isolated effect would be economically possible for one and the same capital or not.
This case differs from the last one only in degree, not in kind. There, c fell exactly as much as v rose, or rose as much as v fell. Here, c stays fixed while v moves.
Under the conditions of large-scale industry and agriculture today, variable capital is only a relatively small part of the total capital. So when v changes, the resulting change in the total capital is also relatively small.
Let's start again from a capital:
That capital might then turn into:
The opposite case, where variable capital decreases, is pictured again simply by reversing the move — going from II back to I.
The economic conditions here are essentially the same as in the last case, so there's no need to go over them again. Going from I to II means a fall in the productivity of labour: handling the same 100 of constant capital now takes 50 per cent more labour in II than it did in I. This case can happen in agriculture.
But whereas in the last case the total capital stayed the same, because constant capital was converted into variable capital, or the reverse, here it's different: when the variable part grows, extra capital gets tied up; when it shrinks, capital that was previously in use gets set free.
In this case, the equation changes to:
and, cancelling the factors that appear on both sides, this gives the proportion:
when two capitals have the same rate of surplus-value and lay out the same absolute amount of variable capital, the larger the total capital, the lower its rate of profit.
Say we have three capitals, for example, or three different states of the same capital:
The resulting ratios are:
The general formula given earlier for variations in v/C with s' constant was:
since v doesn't change here, the factor e = v1/v comes out to 1.
Since s'v gives s, the amount of surplus-value, and since s' and v both stay fixed, a change in C doesn't touch s either — the amount of surplus-value stays exactly the same, whether C changes or not.
If c were to fall to zero, p' would equal s' — the rate of profit would equal the rate of surplus-value.
A change in c can come about either from a shift purely in the value of the material elements of constant capital, or from a change in the total capital's technical composition — that is, from a change in the productivity of labour in the branch of production concerned. In the second case, the rising productivity of social labour that comes with the growth of large-scale industry and agriculture would mean the shift happens in this order (in the example above): from III to I, then from I to II. A quantity of labour paid 20 and producing a value of 40 would at first work up a mass of means of labour worth 60; as productivity rises while the value of each of those means stays the same, the means of labour worked up would grow, first to 80, then to 100. The reverse order would mean falling productivity: the same quantity of labour could set fewer means of production in motion, and the scale of operation would be cut back — as can happen in agriculture, mining, and so on.
Saving on constant capital raises the rate of profit on the one hand, and sets capital free on the other — so it matters to the capitalist. We'll come back to this later, and to the effect of price changes in the elements of constant capital, raw materials especially.
Here too it turns out that a variation in constant capital affects the rate of profit in the same way, regardless of whether that variation comes from an increase or decrease in the physical, material elements of c, or simply from a change in their value.
In this case, the general formula given above for the changed rate of profit still holds:
It follows that, with the rate of surplus-value staying the same:
a) The rate of profit falls when E is bigger than e — that is, when constant capital grows so much that the total capital grows faster than the variable capital. Take a capital of 80c + 20v + 20s changing into 170c + 30v + 30s: s' stays at 100 per cent, but v/C falls from 20/100 to 30/200, even though both v and C have grown — and the rate of profit falls correspondingly, from 20 per cent to 15 per cent.
b) The rate of profit stays unchanged only when e equals E — that is, when the fraction v/C keeps the same value despite the apparent change, because the top and bottom are multiplied or divided by the same number. 80c + 20v + 20s and 160c + 40v + 40s obviously have the same rate of profit, 20 per cent, because s' stays at 100 per cent and v/C = 20/100 = 40/200 comes out the same in both examples.
c) The rate of profit rises when e is bigger than E — that is, when the variable capital grows faster than the total capital. If 80c + 20v + 20s becomes 120c + 40v + 40s, the rate of profit rises from 20 per cent to 25 per cent, because with s' unchanged, v/C = 20/100 has risen to 40/160 — from ⅕ to ¼.
When v and C change in the same direction, we can think of this change as happening in two stages: up to a certain point, both vary in the same proportion, so that v/C stays unchanged over that stretch. Beyond that point, only one of the two goes on changing — and that reduces this more complicated case to one of the simpler cases already covered.
Say, for example, 80c + 20v + 20s turns into 100c + 30v + 30s. The ratio of v to c, and so to C, stays unchanged over the stretch up to 100c + 25v + 25s — and up to that point, the rate of profit is untouched too.
So we can now take 100c + 25v + 25s as our new starting point. From there, v has risen by 5, to 30v, and C has risen accordingly from 125 to 130 — which is exactly the second case, a simple variation of v together with the resulting variation of C. The rate of profit, originally 20 per cent, rises through this addition of 5v, at the same rate of surplus-value, to 23 1/13 per cent.
The same reduction to a simpler case can happen even when v and C change in opposite directions. Take again 80c + 20v + 20s, and let it turn into 110c + 10v + 10s. If it had changed instead to 40c + 10v + 10s, the rate of profit would be the same as at the start, 20 per cent. Adding 70c to that intermediate form brings it down to 8⅓ per cent. So we've again reduced the case to one where a single variable changes — this time c.
So the simultaneous variation of v, c and C brings nothing new into view — it always comes back, in the end, to a case where only one factor is varying.
Even the one case still left over has, in fact, already been covered: the case where v and C stay numerically the same size, but their material elements undergo a change in value — so that v now stands for a changed quantity of labour set in motion, and c for a changed quantity of means of production set in motion.
In 80c + 20v + 20s, say 20v originally pays the wages of 20 workers, at 10 working hours a day. Now let each worker's wage rise from 1 to 1¼. Then 20v pays not 20 but only 16 workers. But if those 20 workers, in 200 hours of labour, produced a value of 40, then the 16 workers, at 10 hours a day — 160 hours of labour in all — will produce only a value of 32. Subtract 20v for wages, and only 12 is left of the 32 for surplus-value: the rate of surplus-value would have fallen from 100 per cent to 60 per cent.
But since, on our assumption, the rate of surplus-value has to stay constant, the working day would have to be lengthened by a quarter, from 10 hours to 12½. If 20 workers at 10 hours a day — 200 hours of labour — produce a value of 80, then 16 workers at 12½ hours a day — likewise 200 hours — produce the same value, and the capital of 80c + 20v still produces a surplus-value of 20, exactly as before.
The other way round: if wages fall so far that 20v now pays 30 workers, s' can only stay constant if the working day is cut from 10 hours to 6⅔ hours. 20 × 10 = 30 × 6⅔ = 200 hours of labour.
We have essentially explained already how c can stay the same in money terms under these opposite assumptions while still standing for the changed quantity of means of production that the changed conditions call for. In its pure form, this case would hardly ever arise.
As for a change in the value of c's elements that enlarges or shrinks their quantity but leaves the sum c unchanged: it touches neither the rate of profit nor the rate of surplus-value, so long as it doesn't bring about any change in the size of v.
With this, we've gone through every possible case of variation in v, c and C in our equation. We've seen that the rate of profit, with the rate of surplus-value held constant, can fall, stay the same, or rise — because the smallest change in the ratio of v to c, or in c itself, is enough to change the rate of profit too.
It has also become clear that, wherever v varies, a limit is always eventually reached where keeping s' constant becomes economically impossible. And since any one-sided variation of c must likewise reach a limit where v can no longer stay constant, it follows that limits are set on every possible variation of v/C — limits beyond which s' must become variable too. This interplay between the different variables in our equation will come out still more clearly when we turn now to variations in s' itself.
We can get a general formula for the rates of profit of two capitals whose rates of surplus-value differ — whether v/C stays the same or changes too. Start from the equation:
Let that equation turn into this other one:
Here p'1, s'1, v1 and C1 stand for the changed values of p', s', v and C.
That gives us:
And from that:
In this situation we have these equations:
Here v/C is the same in both cases. So it follows that:
The profit rates of two capitals with the same composition stand in the same ratio as their two rates of surplus-value. In the fraction v/C, what matters is not the absolute size of v and C but only the ratio between them — so this holds for every capital of that same composition, whatever its absolute size.
If v and C are the same absolute size in both cases, the profit rates also stand in the same ratio as the masses — the total amounts — of surplus-value:
For example:
It's now clear that for capitals with the same absolute or percentage composition, the rate of surplus-value can only differ if the wage, or the length of the working day, or the intensity of labour differs. Take the three cases:
a total value-product is produced: 30 in case I (20v + 10s), 40 in case II, and 60 in case III. This can come about in three different ways.
First, suppose the wages differ, so that the same 20v stands for a different number of workers each time. Say in case I, 15 workers work 10 hours at a wage of £1⅓ each and produce a value of £30, of which £20 replaces the wage and £10 is left as surplus-value. If the wage falls to £1, then 20 workers can be employed for 10 hours, producing a value of £40, of which £20 goes to wages and £20 to surplus-value. If the wage falls further to £⅔, then 30 workers are employed for 10 hours, producing a value of £60, of which, after deducting £20 for wages, £40 is left as surplus-value.
This case — constant percentage composition of capital, constant working day, constant intensity of labour, with the change in the rate of surplus-value caused only by a change in wages — is the only one where Ricardo's assumption holds:
Ricardo puts it this way: "Profits would be high or low, exactly in proportion as wages would be low or high." (Principles, ch. I, sect. III, p. 18 of Works of D. Ricardo, ed. MacCulloch, 1852.)
Or second, suppose the intensity of labour differs. Then, say, 20 workers with the same means of labour, working 10 hours a day, turn out 30 pieces of some commodity in case I, 40 in case II, and 60 in case III — where each piece, beyond the value of the means of production used up in it, represents a new value of £1. Since 20 pieces = £20 replace the wage each time, what is left for surplus-value is 10 pieces = £10 in case I, 20 pieces = £20 in case II, and 40 pieces = £40 in case III.
Or third, the working day is a different length. If, at the same intensity, 20 workers work 9 hours a day in case I, 12 in case II, and 18 in case III, their total product stands as 30 : 40 : 60, matching 9 : 12 : 18 — and since the wage is 20 each time, what is left for surplus-value is again 10, 20, and 40.
So wages move the rate of surplus-value in the opposite direction to themselves, while the intensity of labour and the length of the working day move it in the same direction — and when v/C stays constant, the rate of profit follows.
In this case, the following proportion holds:
The profit rates stand in the same ratio as the respective amounts of surplus-value.
A change in the rate of surplus-value with the variable capital staying the same meant a change in both the size and the distribution of the value-product. A simultaneous change in v and s' also involves a different distribution, but not always a change in the size of the value-product. Three cases are possible:
a) v and s' change in opposite directions, but by the same amount. For example:
The value-product is the same in both cases, and so is the quantity of labour performed: 20v + 10s = 10v + 20s = 30. The only difference is that in the first case, 20 is paid out as wages and 10 is left as surplus-value, while in the second case the wage is only 10, so the surplus-value is 20. This is the only case where, with v and s' changing at the same time, the number of workers, the intensity of labour, and the length of the working day all stay untouched.
b) s' and v also change in opposite directions, but not by the same amount in both. Then either the change in v or the change in s' dominates.
In case I, a value-product of 40 is paid with 20v; in case II, one of 48 with 28v; in case III, one of 36 with 16v. Both the value-product and the wage have changed — and a change in the value-product means a change in the quantity of labour performed, so either the number of workers, the length of the working day, or the intensity of labour, or several of these three together.
c) s' and v change in the same direction; then each reinforces the effect of the other.
Here too the three value-products differ — 20, 50, and 14 — and this difference in the amount of labour each time again comes down to a difference in the number of workers, the length of the working day, the intensity of labour, or several or all of these factors together.
This case raises no new points and is settled by the general formula already given above for s' variable.
So a change in the size of the rate of surplus-value works on the rate of profit in the following ways:
1. p' rises or falls in the same proportion as s', when v/C stays constant.
2. p' rises or falls by more than s', when v/C moves in the same direction as s' — that is, when v/C increases or decreases as s' increases or decreases.
3. p' rises or falls by less than s', when v/C changes in the opposite direction to s', but by a smaller proportion.
4. p' rises even though s' falls, or falls even though s' rises, when v/C changes in the opposite direction to s', and by a larger proportion than s' does.
s' has risen from 100 per cent to 150 per cent, while p' has fallen from 20 per cent to 15 per cent.
5. Finally: p' stays constant even though s' rises or falls, when v/C changes its size in the opposite direction, but by exactly the same proportion as s'.
It's only this last case that still needs some discussion. Just as we saw above, with variations in v/C, that one and the same rate of surplus-value can show up as very different rates of profit, so too we find that one and the same rate of profit can rest on very different rates of surplus-value. But while, with s' constant, any change at all in the ratio of v to C was enough to produce a difference in the rate of profit, when s' changes size, an exactly matching, opposite change in v/C has to occur for the rate of profit to stay the same. For one and the same capital, or for two capitals in the same country, this is only possible as a rare exception.
Let's take, for example, a capital:
And let's assume the wage falls so far that the same number of workers can now be had for 16v instead of 20v. Then, with everything else unchanged, and 4v freed up, we have:
For p' to still be 20 per cent, as before, the total capital would have to grow to 120 — that is, the constant capital to 104:
That growth would only be possible if, at the same time as the wage fell, a change in the productivity of labour occurred that required this altered composition of capital — or else if the money value of the constant capital rose from 80 to 104. In short, it would take an accidental coincidence of conditions that only occurs in exceptional cases. In fact, a change in s' that doesn't also bring a change in v, and so in v/C, is only conceivable under quite specific circumstances — namely in those branches of industry where only fixed capital and labour are employed, and the object of labour is supplied by nature.
But when comparing the profit rates of two countries, this is different. Here, one and the same rate of profit in fact usually expresses different rates of surplus-value.
From all five cases together, then, it follows that a rising rate of profit can correspond to a falling or a rising rate of surplus-value; a falling rate of profit to a rising or a falling one; and a constant rate of profit to a rising or a falling rate of surplus-value. That a rising, falling, or constant rate of profit can likewise correspond to a constant rate of surplus-value, we already saw under section I.
The rate of profit, then, is determined by two main factors: the rate of surplus-value and the value composition of capital. The effects of these two factors can be summed up briefly as follows — and here we can express the composition in percentages, since it makes no difference which of the two parts of the capital the change starts from:
The profit rates of two capitals, or of one and the same capital in two successive, different states, are equal:
1. when the capitals have the same percentage composition and the same rate of surplus-value.
2. when the composition is unequal and the rate of surplus-value is unequal, provided that in each case the rate of surplus-value multiplied by the percentage variable part of the capital — s' times v — comes out the same. That product is the amount of surplus-value reckoned as a percentage of the total capital (s = s'v). Put another way: s' and v stand in inverse proportion in the two cases.
The profit rates are unequal:
1. with the same percentage composition, when the rates of surplus-value are unequal — here they stand in the same ratio as the rates of surplus-value.
2. with the same rate of surplus-value and unequal percentage composition — here they stand in the same ratio as the percentage variable parts of the capital.
3. with unequal rate of surplus-value and unequal percentage composition — here they stand in the same ratio as the products s'v, that is, as the amounts of surplus-value calculated as a percentage of the total capital.
As we said at the end of the last chapter, we're assuming here — and throughout this whole first Part — that the profit on a given capital equals the whole surplus-value that capital produces over a given period of circulation. For now we set aside two things: that this surplus-value actually splits into different sub-forms — interest, ground-rent, taxes, and so on — and that, in most cases, profit as it's actually appropriated under the prevailing rate of profit does not coincide with it at all. That's the subject of Part Two.
Once profit is counted as the same magnitude as surplus-value, its size — and the size of the rate of profit — is settled by arithmetic on figures that are either given or can be worked out in any individual case. So for now the inquiry moves on purely mathematical ground.
We keep the notation used in Volumes 1 and 2. The total capital C splits into constant capital c and variable capital v, and produces a surplus-value s. The ratio of this surplus-value to the variable capital advanced, that is s/v, we call the rate of surplus-value, and write it s'. So s/v = s', and therefore s = s'v. If instead of relating this surplus-value to the variable capital we relate it to the total capital, it's called profit (p), and the ratio of the surplus-value s to the total capital C, that is s/C, is called the rate of profit, p'. So we have:
If we substitute, for s, the value s'v we found above, we get:
an equation that can also be written as the proportion:
the rate of profit compares to the rate of surplus-value exactly as the variable capital compares to the total capital: whatever fraction of the whole capital the variable part is, the rate of profit is that same fraction of the rate of surplus-value.
It follows from this proportion that p', the rate of profit, is always smaller than s', the rate of surplus-value — because v, the variable capital, is always smaller than C, which is v plus c, variable and constant capital together. The one exception, impossible in practice, is the case where v = C, where the capitalist would advance no constant capital, no means of production at all, only wages.
Our inquiry now needs to take into account a further set of factors that have a determining effect on the size of c, v and s, and so need a brief mention.
First, the value of money. We can treat this as constant throughout.
Second, the turnover. We leave this factor entirely aside for now, since its effect on the rate of profit gets its own treatment in a later chapter. Engels adds here: the formula p' = s'v/C holds strictly only for a single turnover period of the variable capital, but we can make it hold for the annual turnover by putting s'n, the annual rate of surplus-value, in place of s', the simple rate of surplus-value — where n is the number of turnovers the variable capital makes within a year (see Volume 2, Chapter 16, I).
Third, we have to take into account the productivity of labour, whose effect on the rate of surplus-value was discussed at length in Volume 1.
Productivity can also have a direct effect on the rate of profit — at least on that of an individual capital — when, as shown in Volume 1, that capital works with a higher-than-average productivity, so that its products represent a lower value than the socially average value of the same commodity, letting it realize an extra profit. We leave this case out of account here too, though, since in this Part we're still working from the assumption that commodities are produced under socially normal conditions and sold at their values. So in every individual case we assume that the productivity of labour stays constant.
In fact, though, the value-composition of the capital invested in a branch of industry — that is, a given ratio of variable to constant capital — always expresses a certain degree of the productivity of labour. So as soon as this ratio changes for any reason other than a mere change in the value of the material elements of constant capital, or a change in wages, the productivity of labour must have changed too. And we'll often find, accordingly, that the changes going on in c, v and s also carry changes in the productivity of labour along with them.
The same holds for the three remaining factors: the length of the working day, the intensity of labour, and wages. Their effect on the mass and the rate of surplus-value was worked out at length in Volume 1. So it's understandable that, even though for simplicity we always assume these three factors stay constant, the changes going on in v and s can still carry changes in these underlying factors along with them.
Here it's worth a brief reminder: wages act on the size of surplus-value and the rate of surplus-value in the opposite direction to the length of the working day and the intensity of labour — a rise in wages reduces surplus-value, while lengthening the working day and raising the intensity of labour increase it.
Say, for example, a capital of 100 employs 20 workers on a 10-hour working day, at a total weekly wage of 20, and produces a surplus-value of 20. Then we have:
Now let the working day be lengthened, without any wage rise, to 15 hours. This raises the total value product of the 20 workers from 40 to 60 (10 : 15 = 40 : 60); since v, the wage actually paid, stays the same, the surplus-value rises from 20 to 40, and we have:
If, on the other hand, with a 10-hour working day, the wage falls from 20 to 12, we get the same total value product of 40 as before, but split differently: v drops to 12, leaving 28 for s. So we have:
So we see that both a longer working day (or, equally, greater intensity of labour) and a fall in wages raise the mass, and so the rate, of surplus-value; conversely, a rise in wages, other things being equal, would push the rate of surplus-value down. So when v grows because of a wage rise, it doesn't express a bigger quantity of labour, only a more expensively paid one — s' and p' don't rise, they fall.
This already shows that changes in the working day, the intensity of labour and wages can't happen without a simultaneous change in v and s and their ratio — and so also in p', the ratio of s to c + v, the total capital. And it's equally clear that changes in the ratio of s to v likewise involve a change in at least one of the three working conditions just mentioned.
This is exactly where the special organic relationship of variable capital to the movement of the total capital and its valorization — its making value grow — shows itself — and where its difference from constant capital comes out. As far as value-formation goes, constant capital matters only because of the value it has; it makes no difference to value-formation whether a constant capital of £1,500 represents, say, 1,500 tons of iron at £1 a ton, or 500 tons of iron at £3 a ton. The actual quantity of physical stuff that this value represents makes no difference at all to value-formation, or to the rate of profit, which varies in the opposite direction to this value — whatever the ratio between a rise or fall in the value of the constant capital and the mass of the physical use-values it represents.
It's quite different with variable capital. What matters about it, in the first place, isn't the value it has, the labour objectified in it, but rather this value taken as a mere index of the total labour it sets in motion, labour that is not expressed in it. This total labour differs from the labour expressed in it, and so paid for by it; and the smaller the labour contained in the variable capital itself, the bigger is the part of that total labour that forms surplus-value. Say a working day of 10 hours equals ten shillings. If the necessary labour — the labour that replaces the wage, that is the variable capital — is 5 hours = 5 shillings, then the surplus labour is 5 hours and the surplus-value is 5 shillings; if that necessary labour is 4 hours = 4 shillings, then the surplus labour is 6 hours and the surplus-value is 6 shillings.
So as soon as the value of the variable capital stops being an index of the mass of labour it sets in motion — as soon as the yardstick itself shifts, so that the same sum no longer stands for the same amount of labour — the rate of surplus-value changes too, in the opposite direction and in inverse proportion.
We turn now to applying the equation for the rate of profit above, p' = s'v/C, to the various possible cases. We'll let the individual factors of s'v/C change their value one at a time, and work out the effect of each change on the rate of profit. This gives us different series of cases, which we can view either as one and the same capital meeting changed circumstances one after another, or as different capitals existing side by side at the same time and brought in for comparison — say, in different branches of industry or different countries. So if treating some of our examples as successive states of one and the same capital over time seems forced or practically impossible, that objection falls away as soon as we take them instead as a comparison of independent capitals.
So we split the product s'v/C into its two factors, s' and v/C. First we treat s' as constant and examine the effect of the possible variations in v/C; then we treat the fraction v/C as constant and let s' run through its possible variations; and finally we let all the factors vary at once, exhausting in this way all the cases from which laws about the rate of profit can be derived.
This case covers several sub-cases, and we can write one general formula for all of them. Take two capitals, C and C1, with their variable parts v and v1. Both run at the same rate of surplus-value, s'. Call their rates of profit p' and p'1. Then:
Now put C and C1, and v and v1, into ratios with each other. Say the fraction C1/C equals E, and the fraction v1/v equals e. That means C1 = EC and v1 = ev. Put these values for C1 and v1 into the equation for p'1 above, and we get:
We can also draw a second formula out of the two equations above, by turning them into a proportion:
A fraction keeps its value if you multiply or divide the top and bottom by the same number. So we can turn v/C and v1/C1 into percentages, by setting both C and C1 equal to 100. Then v/C becomes v/100 and v1/C1 becomes v1/100, and we can drop the denominators from the proportion above, which gives us:
Take any two capitals that operate at the same rate of surplus-value. Their rates of profit stand to each other exactly as their variable parts do, once each variable part is worked out as a percentage of its own total capital.
These two forms cover every possible way v/C can vary.
Before we examine these cases one by one, one more remark. C is the sum of c and v, constant and variable capital, and since both the rate of surplus-value and the rate of profit are usually expressed in per cent, it is convenient to set c + v equal to 100 as well — that is, to express c and v as percentages. It makes no difference to the rate of profit — though it does make a difference to the amount of profit — whether we say that a capital of 15,000, made up of 12,000 constant and 3,000 variable capital, produces 3,000 of surplus-value, or whether we reduce this capital to percentages:
In both cases the rate of surplus-value is 100 per cent, and the rate of profit is 20 per cent.
The same goes if we compare two capitals with each other — say, the one above with another capital:
In both of these forms s' is 100 per cent and p' is 10 per cent, and setting the new capital next to the one before it, the percentage form makes the comparison far easier to take in.
But when it comes to changes taking place within one and the same capital, the percentage form can rarely be used, because it almost always blurs those changes. Suppose a capital moves from the percentage form:
into the percentage form:
then you cannot tell whether the changed percentage composition, 90c + 10v, came about through an absolute fall in v, an absolute rise in c, or both together. For that we need the absolute figures. And for examining the individual cases of variation that follow, everything turns on exactly how that change came about: whether 80c + 20v became 90c + 10v because, say, 12,000c + 3,000v turned into 27,000c + 3,000v by an increase in constant capital with variable capital unchanged (still 90c + 10v in percentage terms); or whether it took this form with constant capital unchanged, through a fall in variable capital, passing into 12,000c + 1,333⅓v (again 90c + 10v in percentage terms); or finally through a change in both amounts, say 13,500c + 1,500v (again 90c + 10v). It is exactly these cases we are going to examine one after another — which means giving up the convenience of the percentage form, or using it only as a secondary tool.
If v changes, C can stay the same only if the other part of C, constant capital c, changes by the same amount but in the opposite direction. Say C starts out as 80c + 20v = 100 and v then falls to 10: C can stay at 100 only if c rises to 90, since 90c + 10v = 100. Put generally: if v turns into v plus or minus d, then c must turn into c minus or plus d — it must move by that same amount d, but the opposite way, for this case's conditions to hold.
Likewise, with the rate of surplus-value s' held constant but variable capital v changing, the amount of surplus-value s must change too, since s = s'v, and in s'v one of the factors, v, is now taking a different value.
The assumptions of our case give us, alongside the original equation,
a second one, produced by varying v:
in which v has become v1, and p'1 is the changed rate of profit that follows from it, still to be found.
We find it through the corresponding proportion:
Or, put in words: with the rate of surplus-value and the total capital both unchanged, the original rate of profit stands to the new rate of profit produced by the change in variable capital, as the original variable capital stands to the changed variable capital.
Say the capital was originally as above:
and is now:
then C = 15,000 and s' = 100 per cent in both cases, and the rate of profit of case I, 20 per cent, stands to that of case II, 13⅓ per cent, as the variable capital of case I, 3,000, stands to that of case II, 2,000: that is, 20 per cent : 13⅓ per cent = 3,000 : 2,000.
Variable capital can either rise or fall. Let's first take an example where it rises. Say a capital is originally made up, and operates, as follows:
Now let variable capital rise to 30. By our assumption, constant capital must then fall from 100 to 90, so that the total capital stays unchanged at 120. And with the rate of surplus-value still at 50 per cent, the surplus-value produced must rise to 15. So we have:
We start from the assumption that wages are unchanged. If so, the other factors involved in the rate of surplus-value — the working day, and also the intensity of labour — cannot have shifted either. The rise in v, 20 becoming 30, can only mean one thing: half again as many workers are employed. The total value produced climbs by the same half, from 30 to 45, and is divided exactly as before — two-thirds to wages, one-third to surplus-value. But with more workers in the picture, constant capital, the value of the means of production, has dropped from 100 to 90. So we are looking at labour productivity falling while constant capital falls too. Can a case like that exist economically?
In agriculture and the extractive industries, where falling productivity — and so a growing number of workers employed — is easy enough to picture, this process — so long as we stay inside capitalist production and take it as given — goes together not with a decrease in constant capital but with an increase. Even if the fall in c above were brought about simply by a fall in prices, a single capital could make the move from case I to case II only under quite exceptional circumstances. But with two independent capitals, invested in different countries, or in different branches of agriculture or extractive industry, there would be nothing remarkable about one of them employing more workers — and so a larger variable capital — while working with less valuable or scarcer means of production than the other.
But let's drop the assumption that the wage stays the same, and explain the rise in variable capital from 20 to 30 by wages rising by half. Then we get a completely different case. The same number of workers — say 20 — go on working with the same means of production, or only insignificantly reduced ones. If the working day stays unchanged, at ten hours say, the total value product is also unchanged: it is still 30. But all 30 of that is now needed just to replace the variable capital advanced, 30; the surplus-value would have vanished. Yet we had assumed that the rate of surplus-value stayed constant, at 50 per cent as in case I. That is only possible if the working day is lengthened by half, to fifteen hours. The 20 workers would then produce a total value of 45 in fifteen hours, and every condition of the case would be satisfied:
In this case the 20 workers need no more means of labour — tools, machines, and so on — than in case I; only the raw or ancillary materials would have to increase by half. So if the price of these materials fell, the move from case I to case II would, on our assumptions, be economically workable for a single capital far more readily than before. And the capitalist would be compensated, at least to some extent, by a larger profit for whatever loss he suffered through the devaluation of his constant capital.
Now suppose variable capital falls instead of rising. Then all we need to do is reverse our example above: take case II as the original capital, and move from II to I.
Case II, 90c + 30v + 15s, then turns into case I, 100c + 20v + 10s, and it is obvious that this reversal changes nothing whatsoever in the conditions that govern the two rates of profit and the relation between them.
If v falls from 30 to 20 because a third fewer workers are employed while constant capital keeps growing, then we have in front of us the normal case of modern industry: the rising productive power of labour, with fewer workers handling larger masses of means of production. That this movement is necessarily bound up with a simultaneous fall in the rate of profit will emerge in Part Three of this book.
But if v falls from 30 to 20 because the same number of workers is employed at a lower wage, then with the working day unchanged the total value product would stay at 30v + 15s = 45 as before; and since v has fallen to 20, surplus-value would rise to 25, taking the rate of surplus-value from 50 per cent to 125 per cent — which would go against our assumption. To stay within the conditions of our case, surplus-value must instead fall to 10, so that the rate of surplus-value stays at 50 per cent, so the total value product falls from 45 to 30 — and that is only possible by cutting the working day by a third. Then we have, as above:
It hardly needs saying that this shortening of working time alongside a falling wage would not happen in practice. But that does not matter. The rate of profit is a function of several variables, and if we want to know how each of them affects the rate of profit, we have to examine the effect of each one on its own, one after another — regardless of whether that isolated effect would be economically possible for one and the same capital or not.
This case differs from the last one only in degree, not in kind. There, c fell exactly as much as v rose, or rose as much as v fell. Here, c stays fixed while v moves.
Under the conditions of large-scale industry and agriculture today, variable capital is only a relatively small part of the total capital. So when v changes, the resulting change in the total capital is also relatively small.
Let's start again from a capital:
That capital might then turn into:
The opposite case, where variable capital decreases, is pictured again simply by reversing the move — going from II back to I.
The economic conditions here are essentially the same as in the last case, so there's no need to go over them again. Going from I to II means a fall in the productivity of labour: handling the same 100 of constant capital now takes 50 per cent more labour in II than it did in I. This case can happen in agriculture.
But whereas in the last case the total capital stayed the same, because constant capital was converted into variable capital, or the reverse, here it's different: when the variable part grows, extra capital gets tied up; when it shrinks, capital that was previously in use gets set free.
In this case, the equation changes to:
and, cancelling the factors that appear on both sides, this gives the proportion:
when two capitals have the same rate of surplus-value and lay out the same absolute amount of variable capital, the larger the total capital, the lower its rate of profit.
Say we have three capitals, for example, or three different states of the same capital:
The resulting ratios are:
The general formula given earlier for variations in v/C with s' constant was:
since v doesn't change here, the factor e = v1/v comes out to 1.
Since s'v gives s, the amount of surplus-value, and since s' and v both stay fixed, a change in C doesn't touch s either — the amount of surplus-value stays exactly the same, whether C changes or not.
If c were to fall to zero, p' would equal s' — the rate of profit would equal the rate of surplus-value.
A change in c can come about either from a shift purely in the value of the material elements of constant capital, or from a change in the total capital's technical composition — that is, from a change in the productivity of labour in the branch of production concerned. In the second case, the rising productivity of social labour that comes with the growth of large-scale industry and agriculture would mean the shift happens in this order (in the example above): from III to I, then from I to II. A quantity of labour paid 20 and producing a value of 40 would at first work up a mass of means of labour worth 60; as productivity rises while the value of each of those means stays the same, the means of labour worked up would grow, first to 80, then to 100. The reverse order would mean falling productivity: the same quantity of labour could set fewer means of production in motion, and the scale of operation would be cut back — as can happen in agriculture, mining, and so on.
Saving on constant capital raises the rate of profit on the one hand, and sets capital free on the other — so it matters to the capitalist. We'll come back to this later, and to the effect of price changes in the elements of constant capital, raw materials especially.
Here too it turns out that a variation in constant capital affects the rate of profit in the same way, regardless of whether that variation comes from an increase or decrease in the physical, material elements of c, or simply from a change in their value.
In this case, the general formula given above for the changed rate of profit still holds:
It follows that, with the rate of surplus-value staying the same:
a) The rate of profit falls when E is bigger than e — that is, when constant capital grows so much that the total capital grows faster than the variable capital. Take a capital of 80c + 20v + 20s changing into 170c + 30v + 30s: s' stays at 100 per cent, but v/C falls from 20/100 to 30/200, even though both v and C have grown — and the rate of profit falls correspondingly, from 20 per cent to 15 per cent.
b) The rate of profit stays unchanged only when e equals E — that is, when the fraction v/C keeps the same value despite the apparent change, because the top and bottom are multiplied or divided by the same number. 80c + 20v + 20s and 160c + 40v + 40s obviously have the same rate of profit, 20 per cent, because s' stays at 100 per cent and v/C = 20/100 = 40/200 comes out the same in both examples.
c) The rate of profit rises when e is bigger than E — that is, when the variable capital grows faster than the total capital. If 80c + 20v + 20s becomes 120c + 40v + 40s, the rate of profit rises from 20 per cent to 25 per cent, because with s' unchanged, v/C = 20/100 has risen to 40/160 — from ⅕ to ¼.
When v and C change in the same direction, we can think of this change as happening in two stages: up to a certain point, both vary in the same proportion, so that v/C stays unchanged over that stretch. Beyond that point, only one of the two goes on changing — and that reduces this more complicated case to one of the simpler cases already covered.
Say, for example, 80c + 20v + 20s turns into 100c + 30v + 30s. The ratio of v to c, and so to C, stays unchanged over the stretch up to 100c + 25v + 25s — and up to that point, the rate of profit is untouched too.
So we can now take 100c + 25v + 25s as our new starting point. From there, v has risen by 5, to 30v, and C has risen accordingly from 125 to 130 — which is exactly the second case, a simple variation of v together with the resulting variation of C. The rate of profit, originally 20 per cent, rises through this addition of 5v, at the same rate of surplus-value, to 23 1/13 per cent.
The same reduction to a simpler case can happen even when v and C change in opposite directions. Take again 80c + 20v + 20s, and let it turn into 110c + 10v + 10s. If it had changed instead to 40c + 10v + 10s, the rate of profit would be the same as at the start, 20 per cent. Adding 70c to that intermediate form brings it down to 8⅓ per cent. So we've again reduced the case to one where a single variable changes — this time c.
So the simultaneous variation of v, c and C brings nothing new into view — it always comes back, in the end, to a case where only one factor is varying.
Even the one case still left over has, in fact, already been covered: the case where v and C stay numerically the same size, but their material elements undergo a change in value — so that v now stands for a changed quantity of labour set in motion, and c for a changed quantity of means of production set in motion.
In 80c + 20v + 20s, say 20v originally pays the wages of 20 workers, at 10 working hours a day. Now let each worker's wage rise from 1 to 1¼. Then 20v pays not 20 but only 16 workers. But if those 20 workers, in 200 hours of labour, produced a value of 40, then the 16 workers, at 10 hours a day — 160 hours of labour in all — will produce only a value of 32. Subtract 20v for wages, and only 12 is left of the 32 for surplus-value: the rate of surplus-value would have fallen from 100 per cent to 60 per cent.
But since, on our assumption, the rate of surplus-value has to stay constant, the working day would have to be lengthened by a quarter, from 10 hours to 12½. If 20 workers at 10 hours a day — 200 hours of labour — produce a value of 80, then 16 workers at 12½ hours a day — likewise 200 hours — produce the same value, and the capital of 80c + 20v still produces a surplus-value of 20, exactly as before.
The other way round: if wages fall so far that 20v now pays 30 workers, s' can only stay constant if the working day is cut from 10 hours to 6⅔ hours. 20 × 10 = 30 × 6⅔ = 200 hours of labour.
We have essentially explained already how c can stay the same in money terms under these opposite assumptions while still standing for the changed quantity of means of production that the changed conditions call for. In its pure form, this case would hardly ever arise.
As for a change in the value of c's elements that enlarges or shrinks their quantity but leaves the sum c unchanged: it touches neither the rate of profit nor the rate of surplus-value, so long as it doesn't bring about any change in the size of v.
With this, we've gone through every possible case of variation in v, c and C in our equation. We've seen that the rate of profit, with the rate of surplus-value held constant, can fall, stay the same, or rise — because the smallest change in the ratio of v to c, or in c itself, is enough to change the rate of profit too.
It has also become clear that, wherever v varies, a limit is always eventually reached where keeping s' constant becomes economically impossible. And since any one-sided variation of c must likewise reach a limit where v can no longer stay constant, it follows that limits are set on every possible variation of v/C — limits beyond which s' must become variable too. This interplay between the different variables in our equation will come out still more clearly when we turn now to variations in s' itself.
We can get a general formula for the rates of profit of two capitals whose rates of surplus-value differ — whether v/C stays the same or changes too. Start from the equation:
Let that equation turn into this other one:
Here p'1, s'1, v1 and C1 stand for the changed values of p', s', v and C.
That gives us:
And from that:
In this situation we have these equations:
Here v/C is the same in both cases. So it follows that:
The profit rates of two capitals with the same composition stand in the same ratio as their two rates of surplus-value. In the fraction v/C, what matters is not the absolute size of v and C but only the ratio between them — so this holds for every capital of that same composition, whatever its absolute size.
If v and C are the same absolute size in both cases, the profit rates also stand in the same ratio as the masses — the total amounts — of surplus-value:
For example:
It's now clear that for capitals with the same absolute or percentage composition, the rate of surplus-value can only differ if the wage, or the length of the working day, or the intensity of labour differs. Take the three cases:
a total value-product is produced: 30 in case I (20v + 10s), 40 in case II, and 60 in case III. This can come about in three different ways.
First, suppose the wages differ, so that the same 20v stands for a different number of workers each time. Say in case I, 15 workers work 10 hours at a wage of £1⅓ each and produce a value of £30, of which £20 replaces the wage and £10 is left as surplus-value. If the wage falls to £1, then 20 workers can be employed for 10 hours, producing a value of £40, of which £20 goes to wages and £20 to surplus-value. If the wage falls further to £⅔, then 30 workers are employed for 10 hours, producing a value of £60, of which, after deducting £20 for wages, £40 is left as surplus-value.
This case — constant percentage composition of capital, constant working day, constant intensity of labour, with the change in the rate of surplus-value caused only by a change in wages — is the only one where Ricardo's assumption holds:
Ricardo puts it this way: "Profits would be high or low, exactly in proportion as wages would be low or high." (Principles, ch. I, sect. III, p. 18 of Works of D. Ricardo, ed. MacCulloch, 1852.)
Or second, suppose the intensity of labour differs. Then, say, 20 workers with the same means of labour, working 10 hours a day, turn out 30 pieces of some commodity in case I, 40 in case II, and 60 in case III — where each piece, beyond the value of the means of production used up in it, represents a new value of £1. Since 20 pieces = £20 replace the wage each time, what is left for surplus-value is 10 pieces = £10 in case I, 20 pieces = £20 in case II, and 40 pieces = £40 in case III.
Or third, the working day is a different length. If, at the same intensity, 20 workers work 9 hours a day in case I, 12 in case II, and 18 in case III, their total product stands as 30 : 40 : 60, matching 9 : 12 : 18 — and since the wage is 20 each time, what is left for surplus-value is again 10, 20, and 40.
So wages move the rate of surplus-value in the opposite direction to themselves, while the intensity of labour and the length of the working day move it in the same direction — and when v/C stays constant, the rate of profit follows.
In this case, the following proportion holds:
The profit rates stand in the same ratio as the respective amounts of surplus-value.
A change in the rate of surplus-value with the variable capital staying the same meant a change in both the size and the distribution of the value-product. A simultaneous change in v and s' also involves a different distribution, but not always a change in the size of the value-product. Three cases are possible:
a) v and s' change in opposite directions, but by the same amount. For example:
The value-product is the same in both cases, and so is the quantity of labour performed: 20v + 10s = 10v + 20s = 30. The only difference is that in the first case, 20 is paid out as wages and 10 is left as surplus-value, while in the second case the wage is only 10, so the surplus-value is 20. This is the only case where, with v and s' changing at the same time, the number of workers, the intensity of labour, and the length of the working day all stay untouched.
b) s' and v also change in opposite directions, but not by the same amount in both. Then either the change in v or the change in s' dominates.
In case I, a value-product of 40 is paid with 20v; in case II, one of 48 with 28v; in case III, one of 36 with 16v. Both the value-product and the wage have changed — and a change in the value-product means a change in the quantity of labour performed, so either the number of workers, the length of the working day, or the intensity of labour, or several of these three together.
c) s' and v change in the same direction; then each reinforces the effect of the other.
Here too the three value-products differ — 20, 50, and 14 — and this difference in the amount of labour each time again comes down to a difference in the number of workers, the length of the working day, the intensity of labour, or several or all of these factors together.
This case raises no new points and is settled by the general formula already given above for s' variable.
So a change in the size of the rate of surplus-value works on the rate of profit in the following ways:
1. p' rises or falls in the same proportion as s', when v/C stays constant.
2. p' rises or falls by more than s', when v/C moves in the same direction as s' — that is, when v/C increases or decreases as s' increases or decreases.
3. p' rises or falls by less than s', when v/C changes in the opposite direction to s', but by a smaller proportion.
4. p' rises even though s' falls, or falls even though s' rises, when v/C changes in the opposite direction to s', and by a larger proportion than s' does.
s' has risen from 100 per cent to 150 per cent, while p' has fallen from 20 per cent to 15 per cent.
5. Finally: p' stays constant even though s' rises or falls, when v/C changes its size in the opposite direction, but by exactly the same proportion as s'.
It's only this last case that still needs some discussion. Just as we saw above, with variations in v/C, that one and the same rate of surplus-value can show up as very different rates of profit, so too we find that one and the same rate of profit can rest on very different rates of surplus-value. But while, with s' constant, any change at all in the ratio of v to C was enough to produce a difference in the rate of profit, when s' changes size, an exactly matching, opposite change in v/C has to occur for the rate of profit to stay the same. For one and the same capital, or for two capitals in the same country, this is only possible as a rare exception.
Let's take, for example, a capital:
And let's assume the wage falls so far that the same number of workers can now be had for 16v instead of 20v. Then, with everything else unchanged, and 4v freed up, we have:
For p' to still be 20 per cent, as before, the total capital would have to grow to 120 — that is, the constant capital to 104:
That growth would only be possible if, at the same time as the wage fell, a change in the productivity of labour occurred that required this altered composition of capital — or else if the money value of the constant capital rose from 80 to 104. In short, it would take an accidental coincidence of conditions that only occurs in exceptional cases. In fact, a change in s' that doesn't also bring a change in v, and so in v/C, is only conceivable under quite specific circumstances — namely in those branches of industry where only fixed capital and labour are employed, and the object of labour is supplied by nature.
But when comparing the profit rates of two countries, this is different. Here, one and the same rate of profit in fact usually expresses different rates of surplus-value.
From all five cases together, then, it follows that a rising rate of profit can correspond to a falling or a rising rate of surplus-value; a falling rate of profit to a rising or a falling one; and a constant rate of profit to a rising or a falling rate of surplus-value. That a rising, falling, or constant rate of profit can likewise correspond to a constant rate of surplus-value, we already saw under section I.
The rate of profit, then, is determined by two main factors: the rate of surplus-value and the value composition of capital. The effects of these two factors can be summed up briefly as follows — and here we can express the composition in percentages, since it makes no difference which of the two parts of the capital the change starts from:
The profit rates of two capitals, or of one and the same capital in two successive, different states, are equal:
1. when the capitals have the same percentage composition and the same rate of surplus-value.
2. when the composition is unequal and the rate of surplus-value is unequal, provided that in each case the rate of surplus-value multiplied by the percentage variable part of the capital — s' times v — comes out the same. That product is the amount of surplus-value reckoned as a percentage of the total capital (s = s'v). Put another way: s' and v stand in inverse proportion in the two cases.
The profit rates are unequal:
1. with the same percentage composition, when the rates of surplus-value are unequal — here they stand in the same ratio as the rates of surplus-value.
2. with the same rate of surplus-value and unequal percentage composition — here they stand in the same ratio as the percentage variable parts of the capital.
3. with unequal rate of surplus-value and unequal percentage composition — here they stand in the same ratio as the products s'v, that is, as the amounts of surplus-value calculated as a percentage of the total capital.