Transformation problem

Template:Marxist theory In 20th century discussions of Karl Marx's economics the transformation problem is the problem of finding a general rule to transform the "values" of commodities (based on labour according to his labour theory of value) into the "competitive prices" of the marketplace. This problem was first introduced by Marx himself in Chapter 9 of Capital's draft Volume III, where he also tried to solve it. The essential difficulty was this: given that he derived profit ("surplus value") from direct labour inputs and that the amount of direct labour input varied widely between commodities, how could he explain the tendency to an average rate of profit?

Overview

The production of any commodity generally requires both labour and some produced means of production (or capital goods), like tools and materials. The amount of labour so required is called the direct labour input into the commodity. But also the required capital goods have in their turn been produced (in the past) by labour and other capital goods; and so on for these other capital goods, and so on. The sum of all the amounts of labour, that were direct inputs into this backwards-stretching series of capital goods produced in the past, is called the indirect labour input into the commodity. Putting together the direct and indirect labour inputs, one finally gets the total labour input into the commodity, which may also be called the total amount of labour "embodied" in it, or its direct and indirect labour contents.

Now, according to Chapter I of Capital's Volume I, the Marxian "value" of a commodity is defined as its total amount of embodied labour. On the other hand, the (relative) "competitive price" of the same commodity is the ratio at which it is exchanged with other commodities in a competitive market system, and this ratio tends to be given by its relative cost of production. The transformation problem, studied by Marx in Chapter 9 of Capital's Volume III through a set of numerical examples, is hence the problem of finding a set of functions that transform the ratios between amounts of embodied labour into such relative prices.

Many mathematical economists assert that a similar set of functions does not generally exist, so that Chapter 9's transformation problem has formally no solution, outside two classes of very special cases. This was first pointed out long ago by, among others, Böhm-Bawerk (1896) and Bortkiewicz (1906). But it was only in the second half of the twentieth century that Leontief’s and Sraffa’s work on linear production models provided a framework within which to prove this result in a simple and general way.

Although he never actually mentioned the transformation problem, Sraffa’s (1960) Chapter VI on the "reduction" of prices to "dated" amounts of current and past embodied labour gave implicitly the first general proof, showing that the competitive price $P_{i}$ of the $i^{-th}$ produced good can be expressed as

P_{i}=\sum _{n=0}^{\infty }l_{in}w{(1+r)^{n}}

,

where $n$ is the time lag, $l_{in}$ is the lagged-labour input coefficient, $w$ is the wage and $r$ is the "profit" (or net return) rate. Since total embodied labour is defined as

E_{i}=\sum _{n=0}^{\infty }l_{in}

,

it follows from Sraffa’s result that there is generally no function from $E_{i}$ to $P_{i}$ , as was made explicit and elaborated upon by later writers, notably Ian Steedman in Marx after Sraffa.

A standard reference – with an extensive survey of the entire literature pre-1971 and a comprehensive bibliography – is Samuelson's (1971) "Understanding the Marxian Notion of Exploitation: A Summary of the So-Called Transformation Problem Between Marxian Values and Competitive Prices" Journal of Economic Literature 9 2 399–431.

Since the 1970s several major schools of Marxian economics have arisen in response to the transformation problem-related challenges of the neoclassical and Sraffian schools. Analytical Marxists accepted that the transformation problem disproved the Labour Theory of Value and based their Marxian social theory on a combination of the Fundamental Marxian theorem, game theory, and other neoclassical and mathematical tools. Empirical Marxists, including Anwar Shaikh, Moshe Machover, and Paul Cockshott, maintain that since empirical data bears out the correspondence of prices and labour values, the transformation problem is irrelevant. Followers of the Temporal Single System Interpretation and the New Interpretation argue that critics have misunderstood Marx's definition of value and that, correctly defined, there is no difference between value and price.

This article uses a very simplified linear production model to survey Marx's labour theory of value, starting from its precursors in British classical economics. It then offers a simple proof of the general lack of solution for the transformation problem, highlighting Marx's formal error in his attempt to find one. Finally, it summarizes some possible implications of this result, as Marxists and non-Marxists see them.

Implications

The lack of any function to transform Marx's "values" to competitive prices has important implications for Marx's theory of labour exploitation and economic dynamics, namely that there is no Tendency of the rate of profit to fall. This means that it is not pre-ordained that capitalists must exploit labour to compensate for a declining rate of profit. This implies that Marx's prophecy that worsening labour exploitation would result in an eventual revolution against the capitalist system and the establishment of communism is logically and mathematically false.

The mathematical proof that Marx's transformation problem has no general solution has been formally questioned [1] by proponents of the Temporal Single System Interpretation, who argue that the determination of prices by simultaneous linear equations (which assumes that prices are the same at the start and end of the production period) is logically inconsistent with the determination of value by labour time. Other Marxian economists accept the proof, but reject its relevance for some key elements of Marxian political economy. Still others reject Marxian economics outright, and emphasise the politics of the assumed relations of production instead. To this extent, the transformation problem—or rather its implications—is still today a controversial question.

British classical labour theory of value

Marx's value theory was developed from the labour theory of value discussed by Adam Smith and used by many British classical economists. It became central to his economics.

Simplest case: labour costs only

Consider the very simple example used by Adam Smith to introduce the subject. Assume a hunters’ economy with free land, no slavery and no significant current production of tools, where beavers $(B)$ and deer $(D)$ are hunted. In the language of modern linear production models, call the unit labour-input requirement for the production of each good $l_{i}$ , where $i$ may be $B$ or $D$ (i.e., $l_{i}$ is the number of hours of uniform labour normally required to catch either a beaver or a deer; notice that we need to assume labour as uniform in order to be able, later on, to use a uniform wage rate).

Then—Smith noticed—each hunter will be willing to exchange one deer (which costs him $l_{D}$ hours) for ${l_{D} \over l_{B}}$ beavers. The ratio ${l_{D} \over l_{B}}$ —i.e. the relative amount of labour embodied in (unit) deer production with respect to beaver—gives thus the exchange ratio between deer and beavers, the "relative price" of deer in units of beavers. Moreover, since the only costs are here labour costs, that ratio is also the "relative unit cost" of deer for any given competitive uniform wage rate $w$ . Hence the relative amount of labour embodied in deer production coincides with the competitive relative price of deer in units of beavers, which can be written as ${P_{D} \over P_{B}}$ (where the $P$ stands for absolute competitive prices in some arbitrary unit of account, and are defined as $P_{i}=wl_{i}$ ).

Capital costs

Things get less simple if production uses some scarce capital good as well. Suppose that hunting requires also some arrows $(A)$ , with input coefficients equal to $a_{i}$ , meaning that to catch for instance one beaver you need to use $a_{B}$ arrows, besides $l_{B}$ hours of labour. Now the unit total cost (or absolute competitive price) of beavers and deer becomes:

P_{i}=wl_{i}+k_{A}a_{i},(i=B,D)

where $k_{A}$ stands for the capital cost incurred in using each arrow.

Now, this capital cost is made up of two parts. First, there is the replacement cost of substituting the arrow when it is lost in production. This is $P_{A}$ , or the competitive price of the arrows, times the proportion $h\leq 1$ of arrows lost after each shot. Second, there is the net rental or return required by the arrows' owner (who might or might not be the same person as the hunter using it). This can be expressed as the product $rP_{A}$ , where $r$ is the (uniform) net rate of return of the system.

Summing up, and assuming a uniform replacement rate $h$ , the absolute competitive prices of beavers and deer may be written as:

P_{i}=wl_{i}+(h+r)P_{A}a_{i}

Yet, we still have to determine the arrows' competitive price $P_{A}$ . Assuming arrows are produced by labour only, with $l_{A}$ man-hours per arrow, we have:

P_{A}=wl_{A}

Assuming further for simplicity $h=1$ (all arrows are lost after just one shot, so that they are circulating capital), the absolute competitive prices of beavers and deer become:

P_{i}=wl_{i}+(1+r)wl_{A}a_{i}

Here $l_{i}$ is the amount of labour directly embodied in beaver and deer unit production, while $l_{A}a_{i}$ is the labour indirectly thus embodied, through previous arrow production. The sum of the two,

E_{i}=l_{i}+l_{A}a_{i}

gives the total amount of labour embodied.

It is now obvious that the relative competitive price of deer ${P_{D} \over P_{B}}$ can no longer be generally expressed as the ratio between total amounts of labour embodied. With $a_{i}>0$ the ratio ${E_{D} \over E_{B}}$ will correspond to ${P_{D} \over P_{B}}$ only in two very special cases: if either $r=0$ ; or, if ${l_{B} \over l_{D}}={a_{B} \over a_{D}}$ . In general the two ratios will not only differ: ${P_{D} \over P_{B}}$ may change for any given ${E_{D} \over E_{B}}$ , if the net rate of return or the wages vary.

As it will now be seen, this general lack of any functional relationship from ${E_{D} \over E_{B}}$ to ${P_{D} \over P_{B}}$ —of which Ricardo had been particularly well aware—is at the heart of Marx's transformation problem.

Marx’s labour theory of value

Labour as the "value-creating substance"

Marx defined the "value" of commodities as the total amount of socially necessary labour embodied in their production. He developed this special brand of the labour theory of value in the first Chapter of Capital's Volume I. Due to the influence of Marx's particular definition of value on the transformation problem, he is quoted at length where he argues as follows:

"Let us take two commodities, e.g., corn and iron. The proportions in which they are exchangeable, whatever those proportions may be, can always be represented by an equation in which a given quantity of corn is equated to some quantity of iron: e.g., 1 quarter corn = x cwt. iron. What does this equation tell us? It tells us that in two different things—in 1 quarter of corn and x cwt. of iron, there exists in equal quantities something common to both. The two things must therefore be equal to a third, which in itself is neither the one nor the other. Each of them, so far as it is exchange value, must therefore be reducible to this third." […]

"This common 'something' cannot be either a geometrical, a chemical, or any other natural property of commodities. Such properties claim our attention only in so far as they affect the utility of those commodities, make them use values. But the exchange of commodities is evidently an act characterised by a total abstraction from use value." […]

"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. […] 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." […]

"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."

Karl Marx, Capital Volume I, Chapter 1

However, contrary to popular belief, Marx did not deny the role of supply and demand influencing price:

It suffices to say that if supply and demand equilibrate each other, the market prices of commodities will correspond with their natural prices, that is to say, with their values as determined by the respective quantities of labor required for their production.

Value, Price and Profit Chapter 2

It may be incidentally noticed that the uniform-labour assumption of modern linear production models makes their labour inputs quantitatively equivalent to amounts of Marx's "human labour in the abstract". The quantitative aspects of Marx's value and price theories—which include the transformation problem—can thus be expressed in the mathematical language of such models: see Samuelson (1971).

Surplus value and exploitation

Within the quantitative relationships relevant to the transformation problem, labour plays a twofold role according to Marx. First, labour is itself a commodity which is produced, exchanged and used as a means of production—labour is thus a kind of capital in the sense that it is still an input. Marx refers to this as labour power. The "labour value" of each unit of labour is the amount of labour embodied in the goods that make up the real subsistence wages rate. This amount of labour will be denoted here as $l_{W}$ (with $0<l_{W}<1$ in any viable system). In our previous example, the Marxian value of the direct-labour input required by unit beaver and deer production is thus $l_{W}l_{i}$ . Like that of any other means of production (or capital), this value is entirely transmitted to the product.

However, this is not all. Being the "substance" of value, direct (or "living") current labour has for Marx the further property of creating and transmitting to the product a further amount of value, over and above its own. Formally, this property derives from the definition of value and the above assumption that $0<l_{W}<1$ : less than one unit of labour is embodied in the wage goods that pay for (or "produce", so to speak) one unit of labour. This extra value is called surplus value and denoted by $s$ . The amount of surplus value created in unit beaver and deer production will be denoted here as $s_{i}$ .

If the actual real wage is the subsistence wage used to calculate $l_{W}$ , all this surplus value created by labour will be received by the owners of the capital goods, called "capitalists". This is what Marx denoted as exploitation of labour.

Variable and constant capital

As labour produces in this sense more than its own value, the direct-labour input is called variable capital and denoted as $v$ . The amount of value which labour transmits to the deer in our previous example, varies according to the intensity of the exploitation. In our previous example one has $v_{i}=l_{W}l_{i}$ .

By contrast, the value of other inputs—in our example the indirect (or “dead”) past labour embodied in used up arrows—is transmitted to the product as it stands, without additions. It is hence called constant capital and denoted as c. The value transmitted by the arrow to the deer can never be greater than the value of the arrow itself. In our previous example one has $c_{i}=l_{A}a_{i}$ .

Value formulas

The total value of each produced good is obtained as the sum of the above three elements: constant capital plus variable capital plus surplus value. In our previous example:

p_{i}=c_{i}+v_{i}+s_{i}=l_{A}a_{i}+l_{W}l_{i}+s_{i}

Where $p_{i}$ stands for the (unit) Marxian value of beavers and deer.

However, from the definition of Marxian value as total labour embodied it must also be true that:

p_{i}=l_{A}a_{i}+l_{i}=E_{i}

Solving for $s_{i}$ the above two relationships one has:

{s_{i} \over v_{i}}={(1-l_{W}) \over l_{W}}=\sigma \forall i

This necessarily uniform ratio ${s_{i} \over v_{i}}=\sigma$ is called by Marx the rate of surplus value, and it allows to re-write Marx's value equations as:

p_{i}=c_{i}+v_{i}(1+\sigma )=l_{A}a_{i}+l_{W}l_{i}(1+\sigma )

Transformation of values into prices

Like Ricardo, Marx knew that relative labour values— ${p_{D} \over p_{B}}$ in the above example—do not generally tally with relative competitive prices— ${P_{D} \over P_{B}}$ in the same example. However, in the third volume of Capital he argued that competitive prices were obtained from his values through a transformation process, whereby capitalists redistributed among themselves the given aggregate surplus value of the system, in such a way as to bring about a uniform rate of return $r$ on the capital goods they owned in all production lines. This happened because of the capitalists' tendency to shift their capital towards the sectors where it earned higher returns. As competition become fierce in the sector, the rate of return comes down, while opposite will happen in the sector with low rate of return. Marx's attempt to give a detailed account of this process is found in chapter 9 of Volume III.

Marx's reasoning

The two following tables adapt the deer-beaver-arrow example already seen above (which, of course, is not found in Marx, and is only a useful simplification), to illustrate Marx's approach. In both cases it is assumed that the total quantities of beavers and deer captured are $Q_{B}$ and $Q_{D}$ respectively. It is also supposed that the subsistence real wage is one beaver per unit of labour, so that the amount of labour embodied in it is $l_{W}=E_{B}=l_{A}a_{B}+l_{B}<1$ .

**Table 1—Composition of Marxian values in the deer-beaver-arrow production model**
Sector	Total Constant Capital $Q_{i}c_{i}$	Total Variable Capital $Q_{i}v_{i}$	Total Surplus Value $\sigma Q_{i}v_{i}$	Unit Value $c_{i}+(1+\sigma )v_{i}$
Beavers	$Q_{B}l_{A}a_{B}$	$Q_{B}(l_{A}a_{B}+l_{B})l_{B}$	$\sigma Q_{B}(l_{A}a_{B}+l_{B})l_{B}$	$l_{A}a_{B}+(1+\sigma )(l_{A}a_{B}+l_{B})l_{B}$
Deer	$Q_{D}l_{A}a_{D}$	$Q_{D}(l_{A}a_{B}+l_{B})l_{D}$	$\sigma Q_{D}(l_{A}a_{B}+l_{B})l_{D}$	$l_{A}a_{D}+(1+\sigma )(l_{A}a_{B}+l_{B})l_{D}$
Total			$\sigma (l_{A}a_{B}+l_{B})(Q_{B}l_{B}+Q_{D}l_{D})$

Table 1 shows how the total amount of surplus value of the system—in the last row—is determined.

**Table 2—Marx's transformation formulas in the deer-beaver-arrow production model**
Sector	Total Constant Capital $Q_{i}c_{i}$	Total Variable Capital $Q_{i}v_{i}$	Redistributed Total Surplus Value $rQ_{i}c_{i}$	Resulting Competitive Price $v_{i}+(1+r)c_{i}$
Beavers	$Q_{B}l_{A}a_{B}$	$Q_{B}(l_{A}a_{B}+l_{B})l_{B}$	$rQ_{B}l_{A}a_{B}$	$(l_{A}a_{B}+l_{B})l_{B}+(1+r)l_{A}a_{B}$
Deer	$Q_{D}l_{A}a_{D}$	$Q_{D}(l_{A}a_{B}+l_{B})l_{D}$	$rQ_{D}l_{A}a_{D}$	$(l_{A}a_{B}+l_{B})l_{D}+(1+r)l_{A}a_{D}$
Total			$rl_{A}(Q_{B}A_{B}+Q_{D}a_{D})=\sigma (l_{A}a_{B}+l_{B})(Q_{B}l_{B}+Q_{D}l_{D})$

Table 2 then illustrates how Marx thought that this total would be redistributed between the two industries, as “profit” at a uniform return rate r over constant capital. First, the condition that total “profit” must equal total surplus value – in the last row of table 2 – is used to determine r. The result is then multiplied by the value of the constant capital of each industry, to get its “profit”. Finally, each (absolute) competitive price in labour units is obtained, as the sum of constant capital, variable capital and “profit” per unit of output, in the last column of Table 2.

Tables 1 and 2 parallel the tables in which Marx elaborated his numerical example in Chapter 9 of Capital's Volume III.

Marx's error and its correction

Later scholars argued that Marx's formulas for competitive prices were mistaken.

First, competitive equilibrium requires a uniform rate of return over constant capital valued at its price, not its Marxian value, contrary to what is done in Table 2 above. Secondly, competitive prices result from the sum of costs valued at the prices of things, not as amounts of embodied labour. Thus, both Marx's calculation of $r$ and the sums of his price formulas do not add up in all the normal cases, where—as in the above example—relative competitive prices differ from relative Marxian values. Marx noted this, but thought that it was not significant, stating in Chapter 9 of Volume 3 of Capital that "Our present analysis does not necessitate a closer examination of this point."

The simultaneous linear equations method of computing competitive (relative) prices in an equilibrium economy is today very well known. In the greatly simplified model of Tables 1 and 2, where by assumption the wages rate is given and equal to the price of beavers, the most convenient way is to express such prices in units of beavers, which means normalising $w=P_{B}=1$ . This immediately yields the (relative) price of arrows as

P_{A}=l_{A}

beavers.

Substituting this into the relative-price condition for beavers

1=l_{B}+(1+r)l_{A}a_{B}

gives the solution for the rate of return as

r={(1-l_{B}) \over (l_{A}a_{B})}-1

Finally, the price condition for deer can hence be written as

P_{D}=l_{D}+(1+r)l_{A}a_{D}=l_{D}+{a_{D}(1-l_{B}) \over a_{B}}

This latter result, which gives the correct competitive price of deer in units of beavers for the simple model used here, is generally inconsistent with Marx's price formulae of Table 2.

Ernest Mandel, defending Marx, explains this discrepancy in term of the time frame of production rather than as a logical error, i.e. in this simplified model, capital goods are purchased at a labour value price but final products are sold under prices which reflect redistributed surplus value. [2]

The non-transformation problem

Moreover, it is now easy to show that deer's relative embodied labour defined as $e_{D}={E_{D} \over E_{B}}$ can vary, while at the same time the correctly defined relative price $P_{D}$ remains constant. This follows from the fact that in our simple model $e_{D}$ varies with $l_{A}$ while—as long as the real wage is given— $P_{D}$ does not. Thus any change in the labour input required by arrow production will affect the Marxian value of deer relative to beavers, but will leave the economic value of deer (i.e., its exchange rate with beavers) totally unaffected.

The diagram on the right shows this through a numerical example. All the input coefficients, bar $l_{A}$ , have been given arbitrary numerical values. The two curves show the numerical corresponding values of $P_{D}$ and $e_{D}$ (on the vertical axis), calculated as $l_{A}$ varies along the horizontal axis. As one can see, $P_{D}$ is a horizontal line, showing constancy, while $e_{D}$ is a falling curve.

This appears to prove that the competitive exchange ratio between deer and beavers has actually nothing to do with relative total embodied labour, contrary to the above-quoted claim by Marx in Capital's Volume I.

Formal conclusions

Samuelson's eraser algorithm

As it has been shown in the literature, the above result holds true in general, including the more complicated models that Marx actually used. In Samuelson’s (1971) words, this means that the "transformation" of Marxian values into competitive prices must generally take the form of an eraser algorithm, described as follows:

"Contemplate the two mutually-exclusive alternatives of 'values' and 'prices'. Write down one. Now transform by taking an eraser and rubbing it out. Then fill in the other one. Voila! You have completed your transformation algorithm."

Special cases

This fails in just two special cases. The first and best known one is when no "transformation" is actually needed, because competitive prices and Marxian values happen to coincide to begin with. As already noticed, that is the case with either $r=0$ or (in the previous example) ${l_{B} \over a_{B}}={l_{D} \over a_{D}}$ . In Marxian terms, the latter condition is described as a uniform organic composition of capital in all production lines.

But there is also a second and less trivial case, unnoticed until relatively recent times. This is a development of Sraffa’s (1960) notion of "standard commodity", and has been called by Samuelson (1971) the case of "equal internal composition of (constant) capitals". It takes place when every production line happens to use all the various produced means of production (including the goods entering the real wage) in the same proportions among themselves. When this is so, the same proportions apply to both value and cost calculations. Marx's transformation procedure—based on value proportions—can then be rescued, as it produces the correct relative costs and competitive prices.

Yet, even when such very special conditions are met, prices can still be computed in the generally correct way, just based on information about input coefficients, with no need for any detour through Marxian values. Moreover, once prices have been thus directly determined, one can formally set up an inverse transformation process, whereby Marxian values are obtained from prices, rather than the other way round.

Implications and interpretations

The literature summarised above relates to the logical (mathematical) aspects of the transformation problem, as discussed by the numerical examples of Chapter 9 of Capital's Volume III. All the above formal conclusions have ceased to be controversial since more than a generation ago. To critics of Marx, their upshot appears to be that – since under competitive capitalism prices are generally unrelated to the direct and indirect labour contents of individual commodities – Marx's "value" of Capital's Volume I, and its attendant notions of surplus value, surplus labour and exploitation, are a purely fictional construct, of no theoretical or practical usefulness.

Yet, controversy still remains, as many Marxist theorists contend that there is no logical failure.

Mathematical vs historical transformation

Frederick Engels – the editor of Capital's Volume III – hinted since 1894 at a possibly alternative way to look at the whole matter. His view was that the pure Marxian "law of value" of Volume I and the "transformed" prices of Volume III applied to different periods of economic history. In particular, the "law of value" would have prevailed in the pre-capitalist exchange economies – from Babylon to the fifteenth century of the common era – while the "transformed" prices would have materialized under capitalism: see Engels' quotation by Morishima and Catephores (1975) p. 310.

Although Engels was reasoning on the mistaken presumption that Marx's transformation was formally correct, his basic idea does not depend on this, and was later taken up by Meek (1956) and Nell (1973). These authors argued that – whatever one must say of his interpretation of capitalism – Marx's "value" theory keeps its usefulness as a tool to interpret pre-capitalist societies, because – they maintained – in pre-capitalist exchange economies there were no "prices of production" with a uniform rate of return (or "profit") on capital. It hence follows that Marx's transformation must have had a historical dimension, given by the actual transition to capitalist production (an no more Marxian "values") at the beginning of the modern era. Then this true "historical transformation" could and should take the place of the failed mathematical transformation attempted by Marx in Chapter 9 of Volume III.

As Althusser and Balibar (1970) and others have pointed out, this proposal runs counter Marx's own ideas, as expressed in Marx (1859), where one reads:

"The point at issue is not the role that various economic relations have played in the succession of various social formations appearing in the course of history […], but their position within modern society." (En. trans. p.210, italics in the original.)

Moreover, the available quantitative research by economic historians has not generally endorsed Engels' view of antiquity and the Middle Ages as a "value epoch" in the Marxian sense: see for instance Hicks (1969) and Godelier (1973).

However, be that as it may, the proposed "historical" dimension of the transformation problem does not seem to affect in any way the (non-) applicability of Marx's notion of exploitation to contemporary capitalist societies.

Other Marxist views

Most interpretations of Marx's value theory consider it as an attempt to explain capitalist distribution as the result of labour exploitation, defined through the notion of surplus value. Yet, particularly if the wage is given, distribution is in itself determined by prices. Thus the transformation problem is central to the usefulness of Marx's economics in explaining exploitation, via a linkage from values (and hence surplus value) to prices. Indeed, no positive Marxian theory of labour exploitation through distribution could ever be built without some at least implicit reference to such linkage. Marx's reference was very explicit, and this is precisely why the transformation problem worried him so much.

According to the Temporal single-system interpretation of "Capital" advanced by Alan Freeman, Andrew Kliman and others, Marx's writings on the subject can be interpreted in such a way as to remove any supposed inconsistencies (Choonara, 2007).

However, there are also political-economic readings of Capital, such as Harry Cleaver's Reading Capital Politically, that re-define exploitation as a direct control of worked time, unrelated as such to distribution. These readings are usually associated with the autonomist strand of Marxism, which focuses on production as the key economic site within society. These readings of Capital are typically hostile to economics as such, and seek to side-step issues like the transformation problem by claiming that all social arrangements in capitalism (in particular, profit and distribution) are politically determined as contests between classes.

Notice should also be taken of the probabilistic interpretation of Marx advanced by Emmanuel Farjoun and Moshe Machover in Laws of Chaos (see references). They "dissolve" the transformation problem by reconceptualising the relevant quantities as random variables. In particular, they drop the assumption of strict equalisation of profit rates and instead consider an equilibrium distribution of firm profit rates. A heuristic analogy with the statistical mechanics of an ideal gas leads them to the hypothesis that this equilibrium distribution should be a gamma distribution.

Finally, there are Marxist scholars (e.g., Anwar Shaikh, Fred Moseley, Alan Freeman, Makoto Itoh, Gerard Dumenil & Dominique Levy, Duncan Foley) who accept that there exists no incontestable logical procedure by which to derive price magnitudes from value magnitudes, but still think that it has no lethal consequences on his system as a whole, for the following reasons.

As it has just been seen, in a few very special cases, Marx's idea of labour as the "substance" of (exchangeable) value would not be openly at odds with the facts of market competitive equilibrium. These authors have argued that such cases—though admittedly not generally observed—throw light on the "hidden" or "pure" nature of capitalist society. Thus Marx's related notions of surplus value and unpaid labour can still be treated as basically true, although the practical details of their workings are admittedly much more complicated than Marx thought.

In particular, some (e.g. Anwar Shaikh) have suggested, following the explicit remarks by Marx and Engels, that—since aggregate surplus value will generally differ from aggregate "profit"—the former should be in fact treated as a mere pre-condition for the latter, rather than a full explanation of it. Using input-output data and empirical proxies for labour-values, Anwar Shaikh & Ochoa have provided some statistical evidence to show that, although there may be no incontestable logical deduction possible of specific price magnitudes from specific value magnitudes even within a complex model (in contrast to a probabilistic prediction), even a "93% Ricardian theory" of labour-value appears to be a better empirical predictor of price than its rivals. Real capitalism just seems to function much more like ordinary people, businessmen and accountants understand it, and much less like neo-classical economists insist on portraying it, whatever the beavers and deer happen to be doing. That is probably one reason why, rightly or wrongly, Marx's theory retains its plausibility for many people.

Mainstream views

Mainstream scholars such as Samuelson question the assumption that the basic nature of capitalist production and distribution can be gleaned from unrealistic special cases. For example, in special cases where it apply, Marx's reasoning can be easily turned upside down, through an inverse transformation process, Samuelson argue that Marx's inference:

"Profit is therefore the [bourgeois] disguise of surplus value which must be removed before the real nature of surplus value can be discovered." (Capital Volume III, Chapter 2)

could with equal cogency be “transformed” into:

"Surplus value is therefore the [Marxist] disguise of profit which must be removed before the real nature of profit can be discovered." [Samuelson (1971) p.417]

To further clarify this point, it may be noticed that the special cases in question are also precisely those where J.B. Clark’s old model of aggregate marginal productivity holds strictly true, leading to equality between the equilibrium levels of the real wage rate and labour's aggregate marginal product: see the neo-Ricardian disputes of the Sixties and Seventies.^{[citation needed]} One would thus have a "pure" state of capitalist society where Marx's exploitation theory and its main supposed confutation were somehow both true.

This remarkable result, it is maintained, leads straight to the heart of the matter. Like Clark’s contention about the "fairness" of marginal-productivity wages, so Marx's basic argument—from the "substance" of value to the concept of exploitation—is claimed to be a set of non-analytical and non-empirical propositions. That is why, being non-falsifiable, both theories may be found to apply to the same formal and/or empirical object, though they are supposed to negate each other, as Karl Popper^{[citation needed]} and many others^[who?] had argued.

Marxian reply to mainstream views

The Marxian reply to this mainstream view is as follows:

it is firstly argued that it is never possible to provide any absolute scientific proof for the truth of any particular concept of economic value in economics, because the attribution of economic value itself always involves human and moral interpretations which go beyond facts and logic. By nature, the concept of economic value is not itself a scientifically provable concept but an assumption. Marx himself explicitly ridiculed the idea that he should be required to "prove his concept of value".

Secondly, however, the validity of any proposed theory of value depends on its explanatory, heuristic and predictive power, i.e. whether it makes possible a coherent interpretation of the known facts that can at least to some extent predict observable trends. In this sense, Marx evidently felt that he had "proved" the validity of his concept of value by the integrated theory of capitalist development which it made possible (see also law of value). What mattered was the application of the concept.

Thirdly, once a certain concept of economic value is assumed, certain predictions or explanations can be made on the basis of it, and those explanations or predictions can at least in principle be falsified by reference to logic and observables. And that concept of value can be compared with rival concepts and the rival theories they make possible, in order to establish which has more explanatory or predictive capacity.

Fourthly, modern philosophy of science rejects Popper's falsification theory as an adequate portrayal of science. Scientific statements are not necessarily falsifiable statements but instead fallible statements (they could be wrong) which, in principle, can be tested against observables, even if contingently we do not yet know technically how to do this. Scientists do not aim mainly to falsify theories, but confirm them to provide usable knowledge.

Finally, as Piero Sraffa showed clearly, the theory of the production and distribution of a surplus, however it might be devised, is logically independent of any particular theory of labour-exploitation. Labour-exploitation may occur and be conceptualised in various ways, regardless of which theory of value is held to be true. Consequently, if Marx's theory of labour-exploitation is false, this is a quite separate issue.

References

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Fred Moseley papers: [5]
Emmanuel Farjoun and Moshe Machover, Laws of Chaos; A Probabilistic Approach to Political Economy, London: Verso, 1983. [6]

Makoto Itoh, The Basic Theory of Capitalism.
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Hagendorf, Klaus: Labour Values and the Theory of the Firm. Part I: The Competitive Firm. Paris: EURODOS; 2009.