Value, prices and probabilities

What is the connection between value and price? Moshé Machover concludes his discussion of the labour theory of value

This is an edited version of the second half of a talk given on January 21 at a weekend school on the ‘Fundamentals of political economy’ sponsored by the CPGB. The first part of this article, based on the first half of my talk, was published in the last issue of the Weekly Worker.[1] We looked briefly at the basics of the labour theory of value (LTV), as Marx presents it in the first volume of Capital, clarifying the notion of the (exchange) value of the commodity and the vital distinction between labour and labour-power.

There is a lot more that can be said about these basic matters, and Marx devoted to them other writings, ranging from the popular Value, price and profit (a talk delivered in 1865 at a meeting of the general council of the First International) to the massive collection of critical research notes published long after his death as Theories of surplus value. But I shall leave it there. Instead, I would like to turn to some difficulties - problems arising in connection with Marx’s LTV; some of which he deals with, others which he does not.

There are two kinds of problem. First, those regarding the definition and measurement of the quantity of value itself. I will mention some which are, in my opinion, relatively slight difficulties that can be fairly easily resolved - one of them in at least two different ways. The second kind of problem is the exact connection between the value of a commodity and its price. These are the most serious difficulties.

Problems of definition

It must be stressed at the outset that value is a theoretical quantity that cannot be directly observed. When you look at a transaction in which some good (or service) is sold and bought, you can observe the price being paid for it. But its value can, at best, only be estimated using rather complicated calculations and assumptions. Contrary to the claims made by some authors, this does not disqualify value as a scientific concept. In fact, many quantities used in the most exact sciences are not directly observable. Take, for example, the apparently simple physical quantity, temperature. It seems straightforward enough: you place a thermometer in some substance and read off its temperature. But what is the relationship between the reading of the thermometer and the temperature of that piece of substance as defined in modern theoretical physics? This turns out to be a surprisingly tangled tale. A few years ago a philosopher of science won a prize for a 300-page treatise dealing with this ‘simple’ issue.[2]

So let me now mention some of the problems regarding the definition of value. First, there are commodities that apparently do not require any work for their production. If you buy a copy of some software then, yes, there was labour involved in producing the original software; but the actual production of this extra copy involves virtually zero labour: you just click and download it. Does it have no value?

There are at least two ways of resolving this problem, which is why I do not think it is a serious one. Briefly, one answer is that the commodity produced is the original software written by the programmer; and when you buy a copy you are not really buying a part of that commodity, but being charged rent for using it. The owner of the original software has copyright, an information monopoly on it, just as a landlord has monopoly on a piece of real estate.

Another way of looking at this problem is by noting that it is not essentially different from what happens when you buy a copy of a newspaper. The labour involved in producing one extra copy is negligible. In effect, the value of the entire edition of the paper is divided by the number of copies actually sold, so each such copy carries an equal share of that total. But the copies that remain unsold have zero value, because only a commodity that gets sold has value. The same applies to a service commodity such as a train journey (note, by the way, that a commodity need not be a physical object).

Then there is a problem with unique one-off artefacts, such as a work of art. Intuitively, some works of art have value way beyond the amount of labour embodied in them. At any rate, their price can be huge, and some of them are priceless. Well, we can put these aside: the LTV is applicable to commodities that are, in principle, reproducible (a so-called reproduction of a painting does not really reproduce it …).

A more serious difficulty is one that Marx himself raises in volume 1 of Capital, which is the distinction between several kinds of labour, skilled and unskilled. Apparently skilled labour contributes more value per unit of time to the product. Clearly, skilled labour-power may have greater value than unskilled, because it takes more work to (re)produce it: this is what training is all about. But does this mean that it creates greater value per unit of time? Marx says it does: “Skilled labour counts only as simple labour intensified, or rather, as multiplied simple labour, a given quantity of skilled being considered equal to a greater quantity of simple labour.”[3] But by how much should unskilled labour be “multiplied” to yield a given sort of skilled labour? Marx tries to resolve this, but in my view what he says is circular, or at best vague. He does not actually tell you how to quantify skilled labour and “reduce” it to a multiple of unskilled labour. And then he simply puts the problem aside.[4] This is a remaining difficulty; but if you follow the logic of what I will later enlarge upon, I think it more or less dissolves.

Then there is the problem of joint production. Suppose several commodities are produced together, in one process. The stock example is the work of a shepherd. The sheep yield wool, milk and meat. So certain parts of the labour are specific: the shearing labour you can allocate to the production of wool (although you have to shear the sheep anyway, otherwise they will die of heat); the labour of milking is for the milk and the labour of slaughtering is for the meat. But what about the overall work of the shepherd in tending the flock, etc? How are you going to allocate this labour to the milk, meat and wool? This is a serious problem that has exercised people. I will not go into it here. It is discussed from a Marxian viewpoint in the Langston memorial volume, a collection of essays devoted to the value controversy.[5]

Value and price

But the difficulty I want to concentrate on is the connection between value and price. The original idea behind the LTV was that value determines relative price: the price of a commodity is proportional to its value. (Note: proportional rather than equal, because they are measured in different units. Price is measured in pounds, dollars, euros, etc, and value is measured in worker-hours, or workers-years and so on.) In the first part of this article I raised the question of whether it was ever like this, under simple commodity exchange, in pre-capitalist commodity exchange. Was price proportional to value? Adam Smith believed it had been so in older societies. I suppose there must have been a strong correlation between the amount of labour needed to produce something and the price it would fetch in market exchange, otherwise people would not have come up with this idea. How strong that correlation was is a serious question for economic historians.

But in the modern capitalist system, in which the capitalist mode of production predominates, this (like everything else …) becomes more complicated. Strict value-price proportionality contradicts another law, which arises from the competitive nature of a ‘free’ market economy.

Marx recognises this difficulty. The first approximation, which is used in the first volume of Capital, is value-price proportionality. But then Marx recognises that in the capitalist mode of production this proportionality cannot prevail. Marx believed - as did Adam Smith, David Ricardo, as well as many economists after Marx - that the rate of profit across the economy tends to equalise. This is due to competition. If the rate of profit in a given sector of production is higher than average, then capital investment will flow into it, lead to increased production, and competition will reduce the rate of profit down to the average.[6] There is an idea, as it were, of an equilibrium situation (Marx does not use the word ‘equilibrium’, but this is what later economists have called it); a state of ideal equilibrium, which is never actually reached, but works as a tendency pushing the rate of profit across the economy towards uniformity.

But then prices cannot be proportional to values. This is because in an industry, or a firm, in which there is a high capital intensity (high organic composition) - that is, the invested (‘fixed’) capital per worker (or per worker-hour) is greater than average - the amount of surplus value extracted per unit of invested capital is smaller than in an industry where the organic composition of capital is low. If profit comes from the exploitation of labour in the form of surplus value, from the difference between the value of labour-power and the actual labour that is done in the process of production, then value-price proportionality would imply that in industries with higher capital intensity - where there is a greater investment of capital per worker - the rate of profit would be lower than in other industries. This contradicts the idea that there is a tendency for the rate of profit to become uniform, to equalise.

Marx tries to deal with this in the third volume of Capital, especially in chapters 9 and 10. He introduces a sort of link between values on the one hand and market prices on the other. This is because market prices are concrete and directly observable; they are what you actually pay the shop, or the supplier. They are subject to all sorts of contingent and incidental influences, such as fluctuations in supply and demand, ‘special offers’, etc. Values, however, represent theoretical quantities that would be extraordinarily complicated to calculate exactly for every minute input. Marx introduces in between these two another theoretical concept: the price of production. The price of production is not the actual price that you pay in the shop, nor is it observable; it is a theoretical quantity that, according to Marx, would do two things.

1. It would equalise the rate of profit in all sectors of production. This is a theoretical situation that does not exist in reality, but it is supposed to be a limit position of equilibrium.

2. In a hypothetical, purely theoretical situation in which all commodities would be sold at their prices of production, this uniform rate of profit in money terms would be equal to the rate of profit calculated globally, across the whole economy, in terms of value.

This global (or average) rate of profit in terms of value is defined as follows. You take the surplus value S produced in the whole economy over a unit of time - say, a year - and you divide it by the value K of the capital invested - the fixed capital, not the constant capital[7] - in the whole economy. So the rate of profit calculated in terms of value is r = S/K. For example, if S is a tenth of K, then r = 1/10 (or 10%) per annum. And this, Marx said, is going to be the rate of profit according to which the prices of production are determined. Once you have determined the price of production, the actual market price is this plus ‘noise’ (Marx does not put it like this; but this is how it would be expressed today). The price of production of a given commodity is the ‘centre of gravity’ around which its market price is supposed to fluctuate.

Now, in Capital Marx actually tries to work out the prices of production and to show how they are determined. Here he introduces a very important mechanism, which does not work for him, but is important nevertheless, and that is schemes of reproduction. These are the schemes where the same commodities enter both as inputs and as outputs. (The germ of this idea was the tableau économique introduced by François Quesnay in 1759 and used by 18th century French economists, known as the ‘physiocrats’.)

Marx (as edited by Engels) deals with the problem in a very simplified form. He assumes an economy (in effect, what would now be called an economic model) with just three or four types of commodities as output, and just one type of non-labour input. He also assumes, for simplicity, that the whole of the fixed capital is constant capital (in other words, that all the invested capital is used up in one year). But the exercise does not work out. This was suspected a long time ago, but becomes clearer when these schemes, now known as price-profit equations, are written out in modern mathematical notation in a much less simplified form.[8]

The problem is that if you assume that each commodity has a unique price of production and that when all commodities are sold and bought at these prices the rate of profit is uniform across the whole economy, then this rate of profit (in money terms) turns out in general to be different from r (the global rate of profit in value terms). Alternatively, you can ‘force’ the uniform rate of profit in the equations to be equal to r, but then the price-profit equations do not balance: you get one ‘price of production’ for a given type of commodity when it is bought as input, and a different ‘price of production’ for the very same type of commodity when it is sold as output. In my opinion, this makes the notion of price of production quite arbitrary and devoid of explanatory power. This was discovered by the students of Sraffa, the so-called neo-Ricardians, in the 1950s or 1960s and gave rise to a controversy between them and the Marxists.

The value controversy

Well, no wonder Marx could not see this problem with the idea of price of production. He was a moderately good mathematician, but by no means an expert. But even had he been one, no mathematician at that time knew how to handle such equations in proper generality. The precise algebraic theory that deals with this kind of situation depends on a theorem finalised by two mathematicians, Oskar Perron and Georg Frobenius, in the early 20th century.

The idea of schemes of reproduction is extremely useful - in economic planning, for example, calculating quantities and values. Someone who did make use of it and got a Nobel prize for his efforts was Wassily Leontief. He was born in 1905 in Berlin of a Russian family and graduated at a very early age (he was a mathematical and economic prodigy) and at age 19 began working in the Soviet Union for the economic planning committee, Gosplan, using a variant of Marx’s schemes of reproduction as a planning device - which is actually a very sound idea. He left the Soviet Union very early and ended up in America, where he became a very famous economist and developed the so-called Leontief input-output analysis that won him the Nobel prize in 1973. Not many people realise that an idea based on Marx’s third volume of Capital has been acclaimed by mainstream economics in this way!

The controversy between the neo-Ricardians, led by the economist, Ian Steedman,[9] and the Marxists was raging through the 1970s. Both sides assumed that there is a theoretical state of equilibrium in which the rate of profit becomes uniform. No-one claimed that this actually happens in reality; but it was assumed to be the limiting ideal situation towards which the economy tends. The neo-Ricardians concluded that Marx’s LTV is, so to speak, without any real value, as it does not explain prices. The Marxists for the most part tried to patch up the idea of prices of production as a bridge between values and market prices. In my opinion, the main motive for these orthodox attempts was to acquit Marx of error or inconsistency rather than to provide a realistic explanation of the connection between values and prices.[10]

Deceptively attractive

Then, in about 1980, Emmanuel Farjoun came up with a radical, unorthodox idea, which we later developed together in a jointly authored book.[11] What he said was: ‘Wait a moment: this is all wrong. A situation in which the rate of profit is uniform across the economy is not a state of equilibrium, even as a theoretical limiting state. The argument for it is deceptively attractive, but it is fallacious.’ And the reason for this, the explanation, was suggested by analogy with a branch of physics called statistical mechanics.

In the 19th century it became established that heat is actually the kinetic energy of the molecules in any piece of matter - say, a volume of gas. Put simply, heat is the movement of molecules. In a famous series of experiments done by the Lancashire brewer, James Prescott Joule, who was an amateur scientist, he showed the rate at which mechanical energy is converted into heat. And he has a unit of energy named after him: the joule. (These experiments, whose “result would have delighted old Hegel”, performed by “an Englishman whose name I can’t recall”, are mentioned enthusiastically in a letter by Engels to Marx, dated July 14 1858.[12])

The idea was, originally, that if you take a volume of gas at a given constant temperature, the molecules are rushing all over the place and they collide with one another. Now, the fast molecules will collide with slower molecules and slow down, and the slower molecules will get hit by the faster ones and speed up; and so at equilibrium the speed of all the molecules will equalise. The higher the temperature, the greater this uniform speed. And Joule actually made a calculation of what would be the speed of the molecules of a given mass of gas at a given temperature.

But then statistical mechanics was initiated by two famous scientists, James Clerk Maxwell and Ludwig Eduard Boltzmann, about the time Capital was being written. And they said, no, this is a deceptive argument. Actually at equilibrium there is no uniformity of speed: this is impossible; even if at one moment the molecules were to travel at a uniform speed, then that uniformity would be scrambled in an instant. What really happens is that at equilibrium there is a certain statistical distribution (known as the Maxwell-Boltzmann distribution). And they worked out what this statistical law was. This at first sight seems counter-intuitive, but it is correct and the whole later theory of heat, thermodynamics, is based on it.

But the same logic also applies to the economic argument. At equilibrium, the rates of profit of the multitude of firms in an economy are distributed according to a definite statistical law. A situation in which the rate of profit equalises across the economy is not even a possible theoretical state of equilibrium. So it is not only that the whole notion of prices of production does not work for explaining market prices in terms of values: it is pointless, because it is based on wrong assumptions.

In a hypothetical situation in which each commodity is sold and bought at its ‘price of production’ so as to yield a uniform rate of profit in money terms, this rate of profit would in general not be the same as r (the global rate of profit in value terms). But nor would such a situation be a state of equilibrium. So Farjoun and I proposed to excise the whole notion of prices of production; it does not do what it is supposed to do, nor make any sense for the reasons given.

LTV without the bridge

Then what is the connection between value and price? And how much of the labour theory of value remains without prices of production? What we argue is that a capitalist economy is normally at or near a dynamic state of equilibrium.

Please do not misinterpret this as a hunky-dory stasis, in which each individual firm is in a stable state. On the contrary, the rates of profit of individual firms can change or fluctuate rather rapidly; but their statistical distribution ­- that is, the proportion of the total fixed capital that yields a given rate of profit - is normally stable or changes fairly slowly. Only in times of major crisis is there a rapid shift in the distribution. A similar statistical logic applies also to market prices: at equilibrium, not only does the rate of profit have a statistical distribution, but also the price of each type of commodity. There is no such thing as the price of a given commodity - say, a kilo of sugar - even on a given day. There is a distribution of prices. Do a survey of the prices charged for a kilo of sugar in the various shops and supermarkets in London on a given day, and you will see (this is what shopping around is all about).

It turns out that, although there is no theoretical way of connecting the individual prices of individual commodities to their respective individual values, there is a statistical connection that can be established without the bridging concept of prices of production. Take two big ‘baskets’ - two large random samples of commodities of diverse types. Then the ratios between their respective total values and total prices will, with extremely high probability, be very close to equality. So there is a macro relationship between prices and values, but the relationship is statistical rather than individual. Even if you gave me the value of every commodity at a given moment, if such a thing were possible, I would not be able to calculate the price of any individual commodity. But for a whole basket the relationship is very close to proportionality. In other words, if you are talking about big macro baskets of commodities it does not matter whether you reason in terms of values or in terms or price, as they are virtually identical (or, strictly speaking, proportional).

What about the rate of profit? Take the global rate of profit calculated across the whole economy in terms of price, which econometricians can actually calculate: you take the total price of the annual surplus and divide it by the total price of the capital invested. It turns out, as a corollary of what I have just said, that with a high probability this global rate of profit will be almost exactly the same as the rate of profit calculated in terms of value. I think this resolves the issue in a positive way, because it saves the core of the labour theory of value. What Marx wanted to show with his reproduction schemes was not that he could calculate the price of each commodity. What he was trying to show was that the global rate of profit over the whole economy is equal to what it would be if you calculated it in terms of values: and this turns out to be correct.

There are other benefits of focusing not on the supposedly uniform numerical rates of profit, but on their statistical distribution. An example is what happens at a time when the average rate of profit moves up or down. If the average rate of profit represents all rates of profit and it declines, say, from 10% to 7% per annum, then it seems no big deal: 7% per annum is still quite handsome, thank you very much. But if you focus on distribution, then it gets interesting. Remember that the average rate of profit is exactly that: an average along the overall distribution of the different rates of profit across the different firms of the whole economy. Firms with a profit rate of 3% or less become losing firms. The average has not changed very much, but a lot of firms will go bust. On the other hand, even when the average rate of profit plummets, there are still many firms making large profits.

We lose the whole idea of the prices of production, but there are gains. The whole notion of the productivity of labour makes very good sense in statistical terms. If you take a commodity over a long period of time - say, a bushel of corn - its value in terms of labour will tend to decline: it takes less labour to produce it. You can actually show that. And the only way you can show it is with a statistical argument. A firm producing this commodity will certainly want to reduce its costs of production. This may be done by introducing labour-saving devices, which will tend to reduce the direct labour per unit produced. But it could also be done through capital savings, so that less money is spent on fixed capital. But less money spent on fixed capital does not mean that the value of fixed capital is going down, because there is no one-to-one relation between value and price, which is also what Marx says. Perhaps the price of the fixed capital goes down, but its value does not. However, the statistical argument shows that with very high probability the value, the total labour content, of the commodity tends to decline over time; and thus that the productivity of labour tends to rise.

In conclusion I would like to say that Marxist theory grows more vigorously if you prune it judiciously.


1. ‘The centrality of labour-power’, March 15.

2. Hasok Chang Inventing temperature: measurement and scientific progress Oxford 2004.

3. K Marx Capital Vol 1, chapter 1.

4. “For simplicity’s sake we shall henceforth account every kind of labour to be unskilled, simple labour; by this we do no more than save ourselves the trouble of making the reduction” (ibid).

5. See, in particular, E Farjoun, ‘The production of commodities by means of what?’ in R Langston, E Mandel and A Freeman Ricardo, Marx, Sraffa: the Langston memorial volume London 1984.

6. For statements to this effect quoted from several authors, see F Farjoun and M Machover Laws of chaos: a probabilistic approach to political economy London 1983.

7. This is a vital distinction: in Marx’s terminology, constant capital consists of the non-labour inputs used up (consumed) in production during the given period. Fixed capital is the invested capital used, but not necessarily used up, in the process. The rate of profit (in money or value terms) is calculated relative to fixed capital.

8. For the precise mathematical form of the price-profit equations, see http://eprints.lse.ac.uk/36428.

9. I Steedman Marx after Sraffa London 1977.

10. For a later attempt in this vein, see A Kliman Reclaiming Marx’s ‘Capital’: a refutation of the myth of inconsistency Lanham 2006.

11. F Farjoun and M Machover Laws of chaos: a probabilistic approach to political economy London 1983. See also Farjoun and Machover, ‘Probability, economics and the labour theory of value’ New Left Review No152, pp95-108, 1985.

12. www.marxists.org/archive/marx/works/1858/letters/58_07_14.htm.