Das Kapital vol.3 part1, chapter 3: The Relation of the Rate of Profit to the Rate of Surplus Value

Das Kapital, Volume 3
part 1, chapter 3 The Relation of the Rate of Profit to the Rate of Surplus Value
(This post is part of an ongoing project: a close reading of volume 3 of Kapital, one post per chapter. I hope that others who are tackling this book for the first time might find my summaries and thoughts useful. I also hope that others might leave their own thoughts, criticisms, help, etc. here so that this blog might become a good resource for those brave souls who take on Vol. 3.)Photo 5This is not the most thrilling of chapters. There are more exciting ones to come. Bear with Marx. It’ll be worth it.

Before we can examine the formation of a general rate of profit and the way this causes prices to diverge from values we have more examining to do of this equation for the rate of profit: s/(v+c).

In order to do this we must make some simplifying assumptions. We assume that profit is not divided up into interest, rent, taxes, etc. This will be the subject of Part 4, 5 and 6. We also assume that there is not an average rate of profit. This will be the subject of part two. Marx is always making such simplifying assumptions. It is crucial to understand that he does so not to avoid analyzing exceptions to the law of value, but to make sure that each social relation is introduced into the equation in the right order. An element introduced into the picture too early could keep us from understanding the real nature of these social relations. We could end up with a fetishistic understanding of things. To take a popular example of such a mistake we might look at popular Austrian/conspiracy attacks on the Federal Reserve. They argue that value can be created by banks merely through debt and that this represents some sort of coercive distortion of the market. But such an analysis introduces the division of surplus value among different portions of the capitalist class before it examines the creation of surplus value in the first place. Of course banks are part of the capitalist class. Of course they extort surplus value by loaning capital. But these loans only create represent once they are repaid with actual value- value created by real people working as wage-laborers.

But back to chapter 3…
Much of chapters 3 through 6 is equations and mathematical examples. The reading is probably made more difficult from the fact that Marx continually introduces new mathematical examples rather than just varying the same one. He also seems determined to make every observation possible about the rate of profit, no matter how mundane. Let’s see if we can discern the reasons behind his tedious methods…

Marx designates the total capital with a capital C. Don’t confuse this with the lower case c used for constant capital. So instead of writing the equation for the rate of profit as s/(v+c) we could just write s/C, since C=v+c. Marx uses p’ to designate the rate of profit. He abbreviates the rate of surplus value (s/v) with s’.

(In my Moscow edition there are some mistakes in this chapter that make the math more confusing. For instance, on the first page of the chapter it should read s=s’v not s=sv. Later on in the first sub-heading it should read “s’ constant v/C variable” and not “c’ constant, v/C variable. I confirmed this by checking the version on marxists.org.)

After a series of fancy equations Marx concludes that the rate of profit is always less than the rate of surplus value because v is always less than C (except in the rare case of an firm where v=C, that is there is no expenditure on constant capital.)

Marx then lists 6 factors that can influence the rate of profit. These are 1. the value of money, 2. turnover time, 3. productivity of labor, 4. length of the working day, 5. intensity of labor, and 6. wages. For different reasons Marx will exclude these factors from the present analysis. Let us briefly review why.

1. The value of money only changes the quantitative expression of the rate of profit, not the actual amount of labor time appropriated by the capitalist. True, rapid changes in the value of money can impact long term investments in capital. But this requires an analysis of inflation and the monetary system and this is not the correct place in the argument to introduce these factors.

2. Turnover time will be discussed in chapter 4.

3. Changes in the productivity of labor are discussed extensively in Book 1. These only effect the rate of profit when an individual capitalist is producing above or below average productivity. This relative surplus value is temporary and thus not important to the present discussion, though of immense importance to us at other times in the argument.

4,5, and 6. Length of working day, intensity of labor and wages are all discussed extensively in Book 1. But Marx reviews their effect on the profit rate here for us anyway. Obviously, any increase in wages decreases the profit rate while an increase in the working day or the intensity of labor increases the profit rate. Marx gives us a couple short mathematical examples of this. Any change in the rate of surplus value implies some change in wages, intensity of labor or length of working day. The proportion between the money spent on wages (v) and the total amount of labor actually performed (v+s) is what is unique about variable capital. Constant capital does not have this property. The total value of constant capital is passed onto the product. It doesn’t matter whether this constant capital represents 100 tons of bricks or 200 yards of cloth. The specific use-value of constant capital is immaterial to the formation of value. But it is very important to know how much use-value a certain some of variable capital yields. If a capitalist can get more labor out of their variable capital for the same amount of money then their profit rate goes up.


Now begins “fun-time with the equation for the rate of profit” where Marx exhausts all possible variations in the equation p=s'(v/C). (I’m actually quite rusty at my algebra, so if anyone can write in and help me understand how Marx gets from p’=s/(v+c) to p=s'(v/C) that would be of great help. For readers who are equally as rusty as I am at algebra, you can get all the same results using the original equation for the rate of profit p=s/(v+c).)

Section 1: s’ constant; varying v/C
We start by holding the rate of profit constant while varying v/C. After some slightly numbing equations Marx concludes that since the rate of surplus value is held constant, changes in the rate of profit will come from the variation of v relative to the total capital, which is basically just a restatement of the equation. Marx then spends about a page apologizing for not being able to use simpler mathematical examples.

case 1: s’ and C constant, v variable
Since C stands for c+v the only way to increase v without changing C is to decrease c at the same rate at which we increase v. Let’s say I own a factory and I have the same amount of money to start the year off with as I did last year. But now they have these new machines that need less workers. So I spend more on machines and less on workers. My total expenditure (my cost price, c+v, C) stays the same but the proportion has changed. Now normally, in reality, I don’t have the exact same amount of money to invest in each production period. But holding this factor constant allows us to see more clearly the relation between v and the rate of profit.

Similarly, if we are holding s’ (which, remember stands for s/v) constant this means that any change in v must involve a simultaneous change in s for the proportion to remain constant. So if s was 5 and v was 5 we would have 5/5, or a 100% rate of exploitation. If v changes to 6, s must also change to 6 for there to remain a 100% rate of profit. Again, in real life there is rarely such a one-to-one change in both factors at once. But holding this ratio constant allows us to, again see what is unique about changes in v.

With these two constraints in place, a fixed rate of surplus value (s/v) and a fixed cost-price (c+v), then any change in v will have a corresponding change in the rate of profit. If the money laid out in wages go up then profit goes up, even though the rate of surplus value stays the same. If variable capital goes down then the rate of profit goes down. In these cases, an increase in v means an increase in the magnitude of surplus without a change in the rate of surplus value. Because c is shrinking with the increase in v, surplus goes up relative to cost. Of course, in real life an increase or decrease in v usually means changes in s and v. But we are holding these ratios constant now to make the point that more money spent on workers means more value created.

A change in v doesn’t just mean a change in wages. It could also be a result of a change in the intensity of labor or the length of the working day. If wages stay the same then an increase in v means an increase in the amount of laborers. Since we are holding C constant this means a decrease in the amount of constant capital. This seems counter to our understanding of the natural tendency of capitalism. We know that it is in the capitalist’s interest to decrease their reliance on wage labor by replacing them with machines, not the other way around. Capitalism tends toward an increase in the productivity of labor, not a decrease. But Marx reminds us that just such an inversion of the typical case can occur in agriculture and extractive industries like mining. When the land yields less and less product up, capital requires more and more workers to get the same output. This is a case of declining productivity of land. In our near future, as resources like oil become scarcer, or environmental degradation has an effect on agricultural productivity, we may see more such cases! But in such cases we usually have an increase in constant capital as well: more and more money poured into new technologies. So this case of rising v and decreasing c really is only found in comparisons between profit rates in different regions rather than changes in one industry over time. For instance, we might notice that rice production in rural china is more labor intensive and less capital intensive than in other parts of China where more productive technologies have replaced workers with modern machinery.

In the case of a rise in v due to an increase in wages, this would require an increase in the working day in order for s to remain the same. This certainly seems like a likely case. If the cost of living is going up and dragging wages up with it then capitalists will try to prolong the work day in order to compensate for this increased expenditure. On the other hand, rises in wages can also come about because of a strong union-movement. In this case it is much harder to increase the length of the working day. Instead, increasing the productivity of labor is the best option- but this involves a change in c/v, which leads us to our next point.

If v falls then the rate of profit falls (again, given constant s’ and constant C). Let’s first see what happens if the decrease in v comes from a decrease in workers employed. For C to remain constant this decrease in workers requires an increase in constant capital. This is a case of rising labor productivity, usually a result of machines displacing workers. But this rising productivity means a falling rate of profit. This case forms the cornerstone of Marx’s explanation for why capitalist economies go into crisis. He will expand more upon this idea in the 3rd part of the book: The Law of the Tendency of the Rate of Profit to Fall.

If the fall in v is due to a fall in wages, and not a decrease in workers, then we would also need to shrink the length of the working day in order to hold s’ constant. As Marx says, this case is highly unlikely. A capitalist would be mad to not jump at the chance to increase surplus value when wages fall.

case 2: s’ constant, v variable, C changes through the variation of v.

Here we set constant capital (little c) constant so that when we vary v then C must change also. (Because C=c+v any, if c is constant, a change in v changes C. In case 1, remember C was held constant which required a inverse change in c every time we changed v.) This means that we are examining changes in the famous “organic composition of capital”, the ratio of c to v (c/v). It will be Marx’s contention that a rise in the organic composition means a falling rate of profit. Much of the criticism of this theory relies on arguments that either there isn’t a tendency toward a rise in organic composition or arguments that a rise in organic composition does not lead to a fall in profits. The former objection lies within the bounds of Marx’s value theory, arguing that the cheapening of the means of production can stabilize profit rates. The latter objection, which goes by the name of the Okishio Theorum, is an assault upon the basic idea of Marx’s value theory, that only human labor can create value in a commodity economy. Okishio holds that an increase in physical productivity always means more value. Okishio’s “Theorum” has been roundly criticized by the Temporal Single System Interpretation of the LTV. But all this should wait for the part of the book on The Tendency Toward a Falling Rate of Profit”.


But I am getting ahead of myself again. Here in case 2 Marx points out that in modern industry the organic composition of capital is rather high already (high ratio machines to workers). This means that changes in v are rather small and don’t have a drastic effect on the profit rate. Perhaps this could be taken as a reason for a rising organic composition of capital. With the relatively small variability of v capitalists are forced to increase c in order to try to raise the productivity of labor.

In the example Marx gives a decrease in the productivity of labor causes a rise in profit. An increase in productivity causes a fall in profits. Timpani rattle. Spooky, orchestral strings tremolo on diminished chords.

case 3: s’ and v constant, c and therfore C variable

Here the capitalist increases or decreases expenditures on machines or raw materials while holding the rate of exploitation and variable capital constant. Thus the organic composition rises and falls freely. Immediately we see that the rate of profit is inversely proportional to the organic composition of capital. The more the organic composition rises, the more the profit falls.

This rise in c could come about through an increase in the actual number of machines or amount of raw material or through an increase in their price in the market. Rising c means a rising productivity of labor- that more stuff is being produced from the same amount of labor. But this “more stuff” doesn’t have more value. It does have higher cost. Thus profits fall.

4. s’ constant, v, c, and C all variable

Now Marx gets really wild and lets two variables vary. From this we learn that:
a. The rate of profit falls is c increases faster than v.
b. If they change in the same direction at the same rate the rate of profit stays the same.
c. The rate of profit rises if v rises faster than c.

None of this adds anything new to the analysis. We have learned all this from the earlier examples where one factor at a time was varied.

Along all of this analysis we have come upon limits where we can’t vary v/c any more without also varying s’. In examining the falling wage, for instance, we said that it is highly unlikely that this will happen without a corresponding increase in surplus value. Thus the next step is to examine changes in the rate of surplus value.

Section 2: s’ variable

This section opens with some equations that don’t particularly stir any great passions in me.

case 1: s’ variable, v/C constant

Obviously, in such a case rises in the rate of exploitation lead to a rise in the profit rate and vice versa.  Since we are holding v/C constant a change in v must have a corresponding change in c in order for the ratio to remain the same.

Marx takes a quick jab at Ricardo who only presents one factor for the variation in profit: changes in wages. It is a brief jab, and not knowing the wider context of the issue I’m not sure if I’m missing anything about Ricardo’s ideas. Marx argues that a change in v can also be the result of changes in the length of the working day or of the intensity of labor. All three factors have the same effect on the rate of profit.

case 2: s’ and v variable, C constant.
There are three possibilities here:
a. v and s’ vary in opposite directions by the same amount. This means that a decrease in wages means more surplus.
b. v and s’ vary in opposite directions but by different amounts. In this case the more rapidly changing variable is the determining element.
c. v and s’ vary in the same direction. In this case the two variables intensify each other. If v and s are shrinking profit falls. If they both rise profit rises.


case 3: s’, v and C variable
There is nothing new about this case that is not covered earlier. (Whew!)

Marx then makes some summary remarks about changes in the rate of surplus value:

1. If v/C is constant then the rate of profit changes in proportion to the rate of surplus value.
2. If v/C changes in the same direction and proportion as s’ then the rate or profit changes faster than the rate of exploitation.
3. If v/C changes inversely to s’ but at a slower rate p’ rises or falls at a slower rate.
4. if v/C changes at a faster rate than s’ then p’ rises while s’ falls, and inversely. This is crucial. It shows that the increase in organic composition can over-rule the increase in the rate of exploitation if it moves faster. A faster increase in the rate of exploitation could offset a falling rate of profit. So for us to establish that the Falling Rate of Profit is inevitable we not only have to show that the organic composition tends to rise, but that this rise outpaces any rise in the rate of exploitation.
5. p’ stays the same if an increase in v/C is proportionally offset by an increase in s’.

When the rate of profit in two different countries are compared it is usually differing rates of surplus value that determine the different profit rate.

From these five cases we notice that a falling or rising rate of profit can result from all sorts of different rates of surplus value. The direct link between surplus value and profit is severed. Thus, we can’t immediately see surplus value in the rate of profit. A firm could have a low rate of profit and a very high rate of surplus value. A firm could have a high rate of profit and a low rate of surplus value.

All in all the rate of profit depends on two factors: the rate of surplus value (s/v) and the value-composition (or organic composition) of capital (c/v). If the rate of surplus value rises, profit rises. If the organic composition rises then profit falls.

Next post

About kapitalism101

This entry was posted in Kapital Vol. 3 and tagged , , , , , , , , , , , , , , . Bookmark the permalink.

6 Responses to Das Kapital vol.3 part1, chapter 3: The Relation of the Rate of Profit to the Rate of Surplus Value

  1. Pingback: Das Kapital vol.3 Part 1 Chapter 2: The Rate of Profit « Kapitalism101

  2. To answer your request about Marx’s algebraic acrobatics:
    MArx tells us that the Rate of profit p’= s/c+v
    or, if you consider that C = c+v
    then we have p’=s/C
    fair enough

    s (surplus value itself) is of course the Rate of surplus value s’ (in other words s/v) multiplied by the outlay of variable capital v.
    so s= (s/v) v, in other words, s= s’v

    so: in the formula for the rate of profit
    Marx simply says that if you replace s (surplus value) in terms of how we arrive at surplus value (which is of course (s/v)v ,that is , the rate of surplus value s/v multiplied by the outlay of variable capital v ),
    THEN the rate of profit can also be expressed like this:

    p’= s/C = (s/v)v /C = s’v/C which is the same thing as =s'(v/c+v)

    And yes, always double check the frequent misprints of the moscow edition.In adddition to p. 49 as you pointed out, it happens again on page 50, second paragraph: p should read p’.

  3. Oliver C says:

    Realizing this is a bit late, and not a lot of posts have been made recently, can anyone explain to me the significance of the E and e in the p’1= s'(ev/EC) ? Because I don’t follow that equation.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s