Mirror : Equalisation of rate of profit or equalisation of rate of surplus value

Warning – statistical reasoning required

This post presents arguments about the rate of profit using forms of statistical reasoning that will be unfamiliar to many readers. But I am asking you to try and perservere with the argument. Some properties of economic relations can only be understood by an in-depth examination of statistical distributions.

In the analysis of capitalism presented in Vol 1 of Capital Marx has the money value of commodities shadowing their labour value.

This matches how Ricardo initially presents value at the start of the first chapter of his Principles.

Both Ricardo and Marx later go on to modify their presentation to take into account an assumed tendency for profit rates to equalise. Ricardo presents this modification almost immediately towards the end of his first chapter. Marx’s modifications were published posthumously by Engels in what appeared as Vol 3 of Capital.

The underlying assumption in both the Principles and Capital 3 is that movement of capital out of low profitable branches and into highly profitable ones will tend to equalise profit rates.

It is interesting that  Marx and the classicals were willing to allow for some things to be tendential: movement of profit rates over time, tendencies of prices to oscillate around labour values or natural prices. When it comes to profit rates across industries, instead of a tendential treatment the examples given in the Principles and Capital 3 assume a single uniform profit rate.

This assumption carried over first to the critics of Marx like Boehm Bawerk and then to Sraffian school. The whole debate from Ricardo to Steadman was conditioned on the assumption of profit rate equalisation.

One may ask why, if the classicals were willing to allow a stochastic relationship between market price and natural price, did they resort to the assumption of strict equalisation when it came to the rate of profit?

Why was it that it took until the publication of the Laws of Chaos for this assumption to be questioned?

It is certainly much easier to think of economic data in terms of averages than in terms of complete statistical distributions. Easier to think of just the average profit rate and assume this applies generally than to recognise that there is always a spread around the average. If you are making up simple examples, as those from Ricardo to Steadman did, then you really have little choice. To give examples involving a spread of profit rates you would need to include dozens of different industries with randomly distributed profit rates. That would be messy. It would not tell a clear story. It would be hard to draw conclusions.

But if we accept that profit rates will actually be dispersed, and that at most there is a tendency for profit rates to equalise, how would we expect this to be manifest?

Let us focus on the key distinction between price of production theory and value theory. In the classical presentation by Marx the first step is to assume uniform ratios of s/v and then show that in the presence of different c/v ratios this leads to a dispersal of profit rates s/(c+v). This dispersal is then taken to contradict an assumption of profit rate equalisation so a second scheme is constructed in which profit rates equalise, but in consequence the ratio s/v becomes dispersed.

So equalisation of one ratio implies the dispersal of the other.

Two class standpoints

There is an implicit acceptance that one dispersal is ok ( dispersal of s/v ) whereas the other dispersal (s/(c+v) ) is not. But behind this acceptance there is an implicit assumption – that capital has more agency than labour. It is taken as given that owners of capital will act to maximise their profit, shifting from low profitable branches to high ones. But there is no analogous assumption made about labour.

Why should workers accept being in an industry in which the profit to wage ratio is much higher than in others?

They have two options after all. They could simply change jobs to move to the industries in which the wage share was higher. Alternatively they could strike for higher wages, knowing that if their employers were making abnormally high profits from each worker then they would have a good chance of winning.

The profit rate equalisation theory only looks at struggle between the owners of capital to maximise their return. It ignores the struggle between capital and labour. And I mean that literally. The books and articles about the transformation problem prior to the publication of Laws of Chaos, just never discuss the class struggle when looking at profit rate equalisation.

Paradoxically, insofar as discussion of class struggle in this context occurs at all, it is thanks to Ricardo not Marx. Ricardo was interested in differential capital compositions because he wanted to know how wage rises would have a different effects on prices of various commodities. His conclusion was that wage rises would lead to a relative fall in the price of capital intensive commodities. But neither he, nor the Neo-Ricardians concerned themselves with the ability of different capitalists to resist wage demands.

In Capital the assumed relationship is:


What happens if we assume that there does indeed exist a tendency towards equalisation of profit rates but that it is only enough to control the dispersion of the rate of profit. What does this weaker hypothesis imply for the relationship between the dispersion of s’= s/v and r= s/(c+v). Intuitively it is fairly clear that if we assume a constraint on the dispersion of r then the dispersal of s’ will necessarily be greater, since, in price of production theory the s/v ratio has to accomodate the random variation in c/v on top of any independent variation in profit between industries.

One can test this with a simple spreadsheet linked here .  I suggest you dowload it an examine it in Excel or Liber Office. This has about 40 industries all have a constant capital of 100, but have variable capitals normally distributed with a mean of 50 and a standard deviation of 18. We can the set the rate of profit to also be normally distributed with a controlled standard deviation. If you download the spreadsheet we can see that the default rate of profit is 20% with a standard deviation of the profit rate of 12% – actually quite a widely dispersed profit.

How do we measure dispersion

The way to measure dispersion in absolute terms is the standard deviation. Basically this is a technique for averaging the  spread about the mean. Since some instances in the spreadsheet have profit rates above 20% and some below, ranging from around 6% to around 40%, one wants to add these discrepancies from 20% up in a way that ignores the sign. What statisticians normally do is square the disparities, take the mean of the squared disparities and then take the square root of that. The squaring is to map both positive and negative disparities to positive squared values, you then take the mean of these to get the average. But that is still too big because of the squaring step, so you take the square root to cancel it out.

This gives you an absolute positive number, but it does not tell you how wide the dispersion of profit rates is in relative terms. If the average profit rate was 20% then a standard deviation of 2% is not much – only a 1/10th of the total level of profit. If the rate of profit fell to 4% and the standard deviation was still 2%, then in relative terms it would be a big spread – a spread by half.

So to measure the spread of the rate of profit you have to divide the standard deviation in profit by the average rate of profit. This measure standard deviation/ average is called the coefficient of variation of the rate of profit.

One can apply the same technique to the rate of surplus value and get the coefficient of variation of s/v by dividing the standard deviation of s/v by the average rate of surplus value.

In the example spreadsheet the coefficient of variation of the rate of profit is set by the random number generator to be 0.6. The spreadsheet also calculates the coefficient of variation of the rate of surplus value. Since the spreadsheet uses random number generation recalculates whenever you load it, or whenever a cell is altered the coefficient of variation you get will vary. But if you repeatedly recalculate, you find that the coefficient of variation of the rate of surplus value tends to be greater than that of the rate of profit.

By altering the cell labeled rstd ( cell h6) you can tune the dispersion of the rate of profit and see that there is a strong tendency for the dispersion of the rate of surplus value to be greater. Columns L and M shows the result a large number of runs giving dispersions of r and s/v. Column N shows the ratio of the two dispersions for each.

X axis coefficient of variation of rate of profit, y axis coefficient of variation of the rate of surplus value.


  • On average  the dispersion of s/v is 2.9 times that of r
  • The graph shows the relationship between them
  • Note that as the dispersion of r increases so does that of s/v, but to a greater extent.

Why does this happen? It is because the rate of profit is given a mean and standard deviation that are independent of the organic composition. The profit to wage ratio is then subject to the sum of two random disturbances

  1. Due to random variations in profitability independent of organic composition
  2. Due to the effect of organic composition – since the assumption of independence of the rate of profit on organic composition means that the profit to wage ratio is the only ‘route out’ for this random noise.

So what can we conclude?

If we assume a weakened price of production theory in which the dispersion of the rate of profit is not zero, but is constrained to be within a specific coefficient of variation, then we should expect that the spread of the rate of surplus value will be greater than that of the rate of profit.

Way back in the mid 1990s Allin Cottrell and I did an empirical study of the rates of profit and rates of surplus value in the sectors of the UK economy described in the national input output table. We got the following measures for the empirical dispersions of the rate of profit and rate of surplus value:

Emprirical data UK Cottrell and Cockshott 1995
Indicator Coefficient of variation
Rate of profit 0.608
Rate of surplus value 0.423


The actual value of the surplus value dispersion was lower than that for the rate of profit.

We concluded in our original paper that this was pretty conclusive evidence that the tendency of the rate of profit to equalise either did not operate or, at the very least, was significantly weaker than a tendency towards equalisation of the profit/wage ratio.

At the time we had not run the sort of simulation performed by the just released spreadsheet. The spreadsheet simulation reinforces our conclusion from 1995.

The dispersion of the rate of profit was the maximum that I tried on my spreadsheet ( which is why I set that as the profit dispersion in the uploaded spreadsheet ). On the basis of running multiple simulations of the spreadsheet the expected dispersion of the rate of surplus value would have been at least 1.

Consider the implications of a coefficient of variation of the rate of surplus value greater than 1. It implies that the standard deviation in the rate of surplus value would be greater than the rate of surplus value itself. In consequence a substantial number of industries would be showing negative rates of surplus value – they would be running at a loss.  You can verify this by repeatedly rerunning the simulation ( press F9 button for libre office ).

Let me go over this again.

  • If you assume that there is even an imperfect tendency of the rate of profit to equalise,
  • If you then assume that it results in the empirically observed dispersion of profit rates
  • The consequence would be that the dispersion of s/v would be so high that a significant fraction of whole industries would have negative surplus value.

To those of my readers familiar with the work of Farjoun and Machover will remember that in the Laws of Chaos, they predicted that to avoid a prohibitive number of firms running at a loss the dispersion of the rate of surplus value require around 3 standard deviations between the rate of surplus value and the loss making point – that is to say they predicted a coeffiecient of variation of s’ of around 0.33. Well the actual dispersion of the UK rate shown by the 1984 IO table was slightly higher. But only slightly. There are about 2.5 standard deviations separating the mean rate of surplus value from the loss making point.

I said above that the prediction is that the spread of s/v must be greater than the spread of r if r is independent of c/v. Since in fact r has the greater spread, it must be the case that r is not independent of c/v, and this is indeed what we found.


The narrow constraint on the rate of surplus value forces the profit rate of industries with high organic composition to be low.

In our 1995 paper we concluded that the empirical evidence supported the predictions of Farjoun and Machover.

The spreadsheet demonstrates the principles of the Farjoun and Machover argument in a way that allows you to play around with the dispersion of the rate of profit and see what the implications are for the spread of rates of surplus value.

It reinforces the theoretical incoherence of price of production theory.

What is the alternative

But what do you have to assume in order to get a model with the observed situation where the dispersion of the rate of surplus value is lower than that for the rate of profit?

Well if you assume that the basic law is one of a tendency for the rate of surplus value to equalise, then you can set up an analogous spreadsheet like the one here. In this case the rate of surplus value is controled by a Gaussian random number generator with a specified mean and standard deviation. You then compute what the consequent dispersion of the rate of profit is as shown below for a similar series of runs.


In these runs the dispersion of the rate of profit tends to be greater than that of the rate of surplus value, though the effect is less marked at high surplus value dispersion levels.

This is consistent with the observed data for the UK.

So we conclude that a theory of the tendency of the rate of surplus value to equalise is supported by the empirical data for the UK.

Abstract and Concrete Labour

Paul Cockshott 15/11/19

The issue of abstract and concrete labour is relatively minor. The two concepts dealt with in a few very clear paragraphs at the beginning of Capital. But unfortunately it has, over the last couple of decades, been mystified by some Marxist. Misleading claims have been put about to the effect that :

  • Abstract labour only exists under capitalism
  • In socialist economies there is only concrete labour
  • That there is no division of labour in non capitalist economies

Marx uses the concept on the second page of Capital where he writes:

If then we leave out of consideration the use value of commodities, they have only one common property left, that of being products of labour. But even the product of labour itself has undergone a change in our hands. If we make abstraction from its use value, we make abstraction at the same time from the material elements and shapes that make the product a use value; we see in it no longer a table, a house, yarn, or any other useful thing. Its existence as a material thing is put out of sight. Neither can it any longer be regarded as the product of the labour of the joiner, the mason, the spinner, or of any other definite kind of productive labour. Along with the useful qualities of the products themselves, we put out of sight both the useful character of the various kinds of labour embodied in them, and the concrete forms of that labour; there is nothing left but what is common to them all; all are reduced to one and the same sort of labour, human labour in the abstract. (Capital 1 Chap 1, page 28 )

So he is saying that each type of commodity has its own special physical qualities, and that these qualities are given to it by the special actions of the different types of labour : spinning, joinery, masonry. But if we consider commodities in general, bearing in mind that they exchange with one another, it can not be the specific character of labour the labour that made them that is important. It is the fact that they are all made by human labour. He says human labour in the abstract to emphasise that it is people doing it, whatever these people were doing.

It is a unique ability of our species, shared by no other species currently alive on this planet, to be able to apply ourselves readily to a vast variety of tasks. The combination of hands with large brains gives us this adaptability. It was only human labour that Marx considered as important in commodities.

At the time Marx was writing, we were not the only species working. There was a lot of horse labour going on in the English economy. In terms of physical effort probably more was done by horses than by people. I have seen figures for the horse population of late Victorian England as being 3.3 million, the human population was 21 million. A human male is hard put to sustain 75 watts of output, women and children considerably less. A horse power is 735 watts, so we can estimate that horses were delivering around twice as much work as people in Marx’s England.

But this work by our equine sisters was all traction. Horses were not spinning, engaging in cabinet making or bricklaying. There was no horse labour in the abstract; all they did was pull or carry riders.

The important thing to understand about commodities is that they were all produced by people working, irrespective of what kind of work people were doing. And because we are not concerned with exactly what they were doing, we can measure labour in units of time:

A use value, or useful article, therefore, has value only because human labour in the abstract has been embodied or materialised in it. How, then, is the magnitude of this value to be measured? Plainly, by the quantity of the value-creating substance, the labour, contained in the article. The quantity of labour, however, is measured by its duration, and labour time in its turn finds its standard in weeks, days, and hours. (Capital 1 Chap 1, page 29 )

Clearly if we are measuring labour in units of time, we are ignoring what the person was doing, and only taking into account that they were working at something.

The argument in Capital is in the context of commodities whose value, he says, comes from the division of labour. If we think about the division of labour, it is obviously human labour in general that is being divided between specific trades or professions. But the claim of some Marxists that abstract labour and the division of labour are something specific to capitalism does not follow. Just because Marx is writing about the division of labour under capitalism here, does not imply that capitalism is necessary for a division of labour. Indeed he explicitly makes this point:

To all the different varieties of values in use there correspond as many different kinds of useful labour, classified according to the order, genus, species, and variety to which they belong in the social division of labour. This division of labour is a necessary condition for the production of commodities, but it does not follow, conversely, that the production of commodities is a necessary condition for the division of labour. In the primitive Indian community there is social division of labour, without production of commodities. Or, to take an example nearer home, in every factory the labour is divided according to a system, but this division is not brought about by the operatives mutually exchanging their individual products. Only such products can become commodities with regard to each other, as result from different kinds of labour, each kind being carried on independently and for the account of private individuals. (Capital 1 Chap 1, page 30 )

So the argument goes that the division of human labour between different activities gives rise to the exchange value of commodities, but this only occurs if the division of labour occurs in a society of private individuals producing independently. In other social organisations, there can be a division of labour without commodities.

So there is every reason to suppose that a division of labour and therefore human work in the abstract will also exist in communist societies – even if there was no commodity production in them. We may hope that communist societies will tend to free people from a narrow subordination to this division of labour, so that people may vary their tasks either during the week or from year to year. This is what Marx was getting at when, many years earlier, he wrote

For as soon as the distribution of labour comes into being, each man has a particular, exclusive sphere of activity, which is forced upon him and from which he cannot escape. He is a hunter, a fisherman, a herdsman, or a critical critic, and must remain so if he does not want to lose his means of livelihood; while in communist society, where nobody has one exclusive sphere of activity but each can become accomplished in any branch he wishes, society regulates the general production and thus makes it possible for me to do one thing today and another tomorrow, to hunt in the morning, fish in the afternoon, rear cattle in the evening, criticise after dinner, just as I have a mind, without ever becoming hunter, fisherman, herdsman or critic.(German Ideology, Ch 1)

The fact that one person may do different concrete tasks at different times abolishes neither the division of labour nor abstract labour as Marx points out in Capital :

There are, however, states of society in which one and the same man does tailoring and weaving alternately, in which case these two forms of labour are mere modifications of the labour of the same individual, and not special and fixed functions of different persons, just as the coat which our tailor makes one day, and the trousers which he makes another day, imply only a variation in the labour of one and the same individual. Moreover, we see at a glance that, in our capitalist society, a given portion of human labour is, in accordance with the varying demand, at one time supplied in the form of tailoring, at another in the form of weaving. This change may possibly not take place without friction, but take place it must.(Capital 1, page 31)

I am not quite sure where this prejudice about the abstract labour only existing under capitalism, and ceasing to exist in a future economy comes from. It clearly does not come from a straighforward reading of Capital. I suspect there exists a substantial number of Marxists who put off reading Capital for a while. In that period they ‘prepared themselves’ by reading commentators on Capital. Perhaps they read Heinrich, Rubin etc. But this means that by the time they read Marx, they already have certain ideas about what they should expect to find there. They read Marx through lenses they have borrowed. And these introduce a certain confirmation bias.

I always encourage people to read great thinkers in their own words first. Whether it is Einstein, Darwin or Marx you want to study, read them in the original first. Only read commentators afterwards. If you have read the original you are in a position to critically assess the commentators. If not, you may give excessive weight to the commentary.

Response to Horowitz

Every now and then I get requests to respond to specific criticisms of our1 work. I usually just make a brief response in an email when asked a question about what a critic has said. But like buses, you wait 20 years then three come at once. In the last week three people from three countries have asked me to respond to a 1996 article by Horowitz Money, money prices, and the socialist calculation debate2.

Here is a brief response. It is brief both because his own criticisms of us are short, and because we have replied at length in the past to the same or very similar points.

Horowitz accuses us of only focusing on von Mises critique of socialist calculation. He acknowledges that we have demonstrated that computationally the problem of socialist calculation is tractable, but that we have ignored the arguments of Hayek. Hayek, Horowitz wrote, made a distinct critique of socialism, not that millions of equations were too complex to solve, but that on epistemological grounds socialist calculation was impossible.

Horowitz is correct that the Hayek critique of socialism is not identical to that of Mises, but by the time Horowitz article was published in 1996 we had already produced a report3 dealing with Hayek’s arguments. As a technical report this had perhaps limited circulation but a journal article4 covering the same ground has also been published. Our response to Hayek was extended ten years later in more detail5.

Horowitz’s article touches only on one of all the points we raise against Hayek, that of so called ’tacit knowledge’. Hayek argued that a function of market incentives was to draw out from economic actors what he called tacit knowledge. By this he means knowledge that they did not know that they even had until the right monetary incentives drew it out from them.

We are very skeptical of this notion. Observe that the existence of tacit knowledge is so defined as to be unmeasurable and undetectable. Once any knowledge is writen down or communicated, then it no longer meets the criterion of being tacit. So how do we know that this tacit knowledge exists in the first place?

The hypothesis that explicit knowledge has tacit knowledge as its precondition is untestable by definition. Tacit knowledge like the existence of the soul, is an unfalsifiable proposition. The sorts of examples that Hayek gave – the particular and specialised knowledge that a shipping clerk has – are singularly unconvincing. As we wrote in 1994:

Further, even the sort of ‘particular’ knowledge which Hayek thought too localized to be susceptible to centralization is now routinely centralized. Take his example of the information possessed by shippers. In the 1970s American Airlines achieved the position of the world’s largest airline, to a great extent on the strength of their development of the SABRE system of computerized booking of flights (Gibbs, 1994). Since then we have come to take it for granted that our local travel agent will be able to tap into a computer network to determine where and when there are flights available from just about any A to any B across the world. Hayek’s appeal to localized knowledge in this sort of context may have been appropriate at the time of writing, but it is now clearly outdated.

Horowitz argues that production functions are not knowable because they are continuously in the process of being created by entrepreneurial activity. This he says, precludes planning in the absense of monetary economy.

The first point I should make here is that Marxist economists do not think that there is such a thing as a production function. It is a concept of bourgeois economics. We did say that the Austrian school is distinguished from other bourgeois economics schools by thinking that the production function is constantly mutable and subject to change. But Marxists reject the whole notion of a production function6. Instead socialist planning theory analyses production in terms of sets of techniques7.

These are two very different concepts.

The production function in neoclassical theory typically takes the form


where P is production, L is labour used, K is capital used. Clearly this is a non linear function, and a highly abstract one in that K and L are just sums of money. K is not broken down in to any concrete list of goods used. This abstraction was the focus of the Cambridge capital theory debates of the 1960s where Sraffian economists showed that even if one assumes perfect competition and uniform profit rates, you can not define the value of capital independently of both the concrete lists of components used in each technique or the distribution of national income between the working and capitalist classes.

The model used by Sraffa to critique the neoclassical model was a matrix specifying how much of each input is used by each production process. Sraffa uses only one production technique per process. This is itself a simplified version of what the mathematical planning theory of Kantorovich used, since Kantorovich allows multiple possible production techniques to make each product.

Now how does this collection of technique fit in with what Horowitz says?

How about his claim that the knowledge of how to produce things is only created or elicited by entrepreneurial activity?

Well he is certainly right in that the knowledge of the techniques needed to make things is itself the result of activity. But it is not the activity of the mythical entrepreneur. It is, in advanced industrial society, the collective effort of design teams and bureaus. And the process of knowledge production is very objective. It can not function without extensive documentation. As we wrote in 2007:

One component of a cybernetic control system has to be distributed. Clearly it is the Airbus factories that have the information about what parts are used to make an A340, the car plants have the information about what parts are used to make a Mondeo. This information approximates to what Hayek and the Austrian school of economics call contextual or tacit knowledge – but it is of course no longer human knowledge. Literally nobody knows what parts go into an A340. The information, too vast for a human to handle, is stored in a relational database. At an earlier stage of industrial development it would have been dealt with by a complex system of paper records. Again the knowledge would have been objective – residing in objects rather than in human brains. The very possibility of large scale, co-ordinated industrial activity rests upon the existence of such objectivised information. The information to construct the parts explosion is generated by a computerised design process within the collaborating factories of Airbus Industrie. In a cybernetically controlled socialist economy, the parts explosion data for the A340, along with the parts explosion data for other products would have to be computationally combined to arrive at a balanced production plan.

I suspect that the Austrian school writers have never actually worked in a modern design bureau. They certainly have a rather abstract idea about how the knowledge our industrial society depends on is actually produced.

When Horowitz says that without money prices experts would have no way of determining what was technically feasible but economically irrational, he is falling back from Hayek to Mises. This argument that only money provides a way of aggregating different inputs in order to chose the cheapest technique was the core of Mises argument. But at least Mises recognised that if one had access to labour values, these would serve just as well. Horowitz forgets this concession by Mises. He also seems unaware of Kantorovich’s Objective Valuation, arising out of linear programs, so this particular argument by Horowitz was already long obsolete in 1996.

1Cottrell, Cockshott and Michaleson’s

2Money, money prices, and the socialist calculation debate Steven Horwitz Advances in Austrian Economics,ISBN: 978-0-76230-055-6, eISBN: 978-1-84950-019-7, ISSN: 1529-2134, Publication date: 31 May 1996

3Cottrell, A., & Cockshott, W. P. (1994). Information and Economics: A Critique of Hayek. Department of Computer Science, University of Strathclyde.

4Cockshott, W. Paul, and Allin F. Cottrell. “Information and economics: a critique of Hayek.” Research in Political Economy 16 (1997): 177-202.

5Cottrell, Allin, and W. Paul Cockshott. “Against hayek.” (2007).

6The classic Marxist deconstruction of it is Shaikh, Anwar. “Humbug production function.” Capital Theory. Palgrave Macmillan, London, 1990. 191-194.

7Kantorovich, Leonid V. “Mathematical methods of organizing and planning production.” Management science 6.4 (1960): 366-422.

New website

 I have created a new, paid for, website to bring together material that I previously had spread over several other sites. I am in the process of trying to collect as many of my publications and draft publications together here as I  can.

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