# 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.